Handbook of X-Ray Spectrometry Revised and Expanded (Practical Spectroscopy, V. 29)

ISBN: 0-8247-0600-5 First edition was published as Handbook of X-Ray Spectrometry: Methods and Techniques This book is ...

Author: by Rene Van Grieken (Editor) | A. Markowicz (Editor)

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ISBN: 0-8247-0600-5 First edition was published as Handbook of X-Ray Spectrometry: Methods and Techniques This book is printed on acid-free paper. Headquarters Marcel Dekker, Inc. 270 Madison Avenue, New York, NY 10016 tel: 212-696-9000; fax: 212-685-4540 Eastern Hemisphere Distribution Marcel Dekker AG Hutgasse 4, Postfach 812, CH-4001 Basel, Switzerland tel: 41-61-261-8482; fax: 41-61-261-8896 World Wide Web http:==www.dekker.com The publisher oﬀers discounts on this book when ordered in bulk quantities. For more information, write to Special Sales=Professional Marketing at the headquarters address above. Copyright # 2002 by Marcel Dekker, Inc. All Rights Reserved. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microﬁlming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 PRINTED IN THE UNITED STATES OF AMERICA

Preface to the Second Edition

The positive response to the ﬁrst edition of Handbook of X-Ray Spectrometry: Methods and Techniques and its commercial success have shown that in the early 1990s there was a clear demand for an exhaustive book covering most of the specialized approaches in this ﬁeld. Therefore, some ﬁve years after the ﬁrst edition appeared, the idea of publishing a second edition emerged. In the meantime, remarkable and steady progress has been made in both instrumental and methodological aspects of x-ray spectrometry. This progress includes considerable improvements in the design and production technology of detectors and in capillary optics applied for focusing the primary photon beam. The advances in instrumentation, spectrum evaluation, and quantiﬁcation have resulted in improved analytical performance and in further extensions of the applicability range of xray spectrometry. Consequently, most of the authors who contributed to the ﬁrst edition of this book enthusiastically accepted the invitation to update their chapters. The progress made during the last decade is reﬂected well in the chapters of the second edition, which were all considerably revised, updated, and expanded. A completely new chapter on microbeam x-ray ﬂuorescence analysis has also been included. Chapter 1 reviews the basic physics behind x-ray emission techniques, and refers to extensive appendices for all the basic and generally applicable x-ray physics constants. New analytical expressions have been introduced for the calculation of fundamental parameters such as the ﬂuorescence yield, incoherent scattering function, atomic form factor, and total mass attenuation coeﬃcient. Chapter 2 outlines established and new instrumentation and discusses the performances of wavelength-dispersive x-ray ﬂuorescence (XRF) analysis, which, with probably 15,000 units in operation worldwide today, is still the workhorse of x-ray analysis. Its applications include process control, materials analysis, metallurgy, mining, and almost every other major branch of science. The additional material in this edition covers new sources of excitation and comprehensive comparisons of the technical parameters of newly produced wavelength-dispersive spectrometers. Chapter 3 has been completely reconsidered, modiﬁed, and rewritten by a new author. The basic principles, background, and recent advances are described for the tubeexcited energy-dispersive mode, which is invoked so frequently in research on environmental and biological samples. This chapter is based on a fresh look and follows a completely diﬀerent approach. iii

iv

Preface to the Second Edition

Chapter 4 reviews in depth the available alternatives for spectrum evaluation and qualitative analysis. Techniques for deconvolution of spectra have enormously increased the utility of energy-dispersive x-ray analysis, but deconvolution is still its most critical step. The second edition includes discussions of partial least-squares regression and modiﬁed Gaussian shape proﬁles. Chapter 5 reviews quantiﬁcation in XRF analysis of the classical and typical ‘‘inﬁnitely thick’’ samples. In addition to being updated, the sections on calibration, quality control, and mathematical correction methods have been expanded. Chapter 6, on quantiﬁcation for ‘‘intermediate-thickness’’ samples, now also includes the presentation of a modiﬁed version of the emission-transmission method and a discussion of both the accuracy and limitations of such methods. Chapter 7 is a completely original treatment by a new author of radioisotope-induced and portable XRF. It discusses semiconductor detectors, including the latest types, analyzes in detail the uncertainty sources, and reviews the recent and increasingly important applications. Since the appearance of the ﬁrst edition, synchrotron-induced x-ray emission analysis has increased in importance. Chapter 8 was updated and modiﬁed by including a comprehensive review of the major synchrotron facilities. Although its principles have been known for some time, it is only since the advent of powerful commercial units and the combination with synchrotron sources that total reﬂection XRF has rapidly grown, mostly now for characterization of surfaces and of liquid samples. This is the subject of the substantially modiﬁed and expanded Chapter 9. The new authors have taken a radically diﬀerent approach to the subject. Polarized-beam XRF and its new commercial instruments are treated in detail in a substantially revised and expanded Chapter 10. Capillary optics combined with conventional ﬁne-focus x-ray tubes have enabled the development of tabletop micro-XRF instruments. The principles of the strongly growing microbeam XRF and its applications are now covered thoroughly in an additional chapter, Chapter 11. Particle-induced x-ray emission analysis has grown recently in its application types and particularly in its microversion. Chapter 12 discusses the physical backgrounds, instrumentation, performance, and applications of this technique. The sections dealing with the applications were substantially expanded. Although the practical approaches to electron-induced x-ray emission analysis— a standard technique with wide applications in all branches of science and technology— are often quite diﬀerent from those in other x-ray analysis techniques, a treatment of its potential for quantitative and spatially resolved analysis is given in Chapter 13. The new and expanded sections deal with recent absorption correction procedures and with the quantitative analysis of samples with nonstandard geometries. Finally, the completely updated and revised Chapter 14 reviews the sample preparation techniques that are invoked most frequently in XRF analysis. The second edition of this book is again a multiauthored eﬀort. We believe that having scientists who are actively engaged in a particular technique covering those areas in which they are particularly qualiﬁed outweighs any advantages of uniformity and homogeneity that characterize a single-authored book. The editors (and one coworker) again wrote three of the chapters in the new edition. For all the other chapters, we were fortunate to have the cooperation of truly eminent specialists, some of whom are new contributors (see Chapters 3, 7, 9, 10 and 11). We wish to thank all the contributors for their considerable and (in most cases) timely eﬀorts.

Preface to the Second Edition

v

We hope that novices in x-ray emission analysis will ﬁnd this revised and expanded handbook useful and instructive, and that our more experienced colleagues will beneﬁt from the large amount of readily accessible information available in this compact form, some of it for the ﬁrst time. An eﬀort has been made to emphasize the ﬁelds and developments that have come into prominence lately and have not been covered in other general books on x-ray spectrometry. We also hope this book will help analytical chemists and other users of x-ray spectrometry to fully exploit the capabilities of this powerful analytical tool and to further expand its applications in material and environmental sciences, medicine, toxicology, forensics, archeometry, and many other ﬁelds. Rene´ E. Van Grieken Andrzej A. Markowicz

Preface to the First Edition

Scientists in recent years have been somewhat ambivalent regarding the role of x-ray emission spectrometry in analytical chemistry. Whereas no radically new and stunning developments have been seen, there has been remarkably steady progress, both instrumental and methodological, in the more conventional realms of x-ray ﬂuorescence. For the more specialized approaches—for example, x-ray emission induced by synchrotron radiation, radioisotopes and polarized x-ray beams, and total-reﬂection x-ray ﬂuorescence— and for advanced spectrum analysis methods, exponential growth and=or increasing acceptance has occurred. Contrary to previous books on x-ray emission analysis, these latter approaches make up a large portion of the present Handbook of X-Ray Spectrometry. The major milestone developments that shaped the ﬁeld of x-ray spectrometry and now have widespread applications all took place more than twenty years ago. After wavelength-dispersive x-ray spectrometry had been demonstrated and a high-vacuum x-ray tube had been introduced by Coolidge in 1913, the prototype of the ﬁrst modern commercial x-ray spectrometer with a sealed x-ray tube was built by Friedmann and Birks in 1948. The ﬁrst electron microprobe was successfully developed in 1951 by Castaing, who also outlined the fundamental concepts of quantitative analysis with it. The semiconductor or Si(Li) detector, which heralded the advent of energy-dispersive x-ray ﬂuorescence, was developed around 1965 at Lawrence Berkeley Laboratory. Acceleratorbased particle-induced x-ray emission analysis was developed just before 1970, mostly at the University of Lund. The various popular matrix correction methods by Lucas-Tooth, Traill and Lachance, Claisse and Quintin, Tertian, and several others, were all proposed in the 1960s. One may thus wonder whether the more conventional types of x-ray ﬂuorescence analysis have reached a state of saturation and consolidation, typical for a mature and routinely applied analysis technique. Reviewing the state of the art and describing recent progress for wavelength- and energy-dispersive x-ray ﬂuorescence, electron and heavy charged-particle-induced x-ray emission, quantiﬁcation, and sample preparation methods is the purpose of the remaining part of this book. Chapter 1 reviews the basic physics behind the x-ray emission techniques, and refers to the appendixes for all the basic and generally applicable x-ray physics constants. Chapter 2 outlines established and new instrumentation and discusses the performances of wavelength-dispersive x-ray ﬂuorescence analysis, which, with probably 14,000 units in operation worldwide today, is still the workhorse of x-ray analysis with applications in a wide range of disciplines including process control, materials analysis, metallurgy, vii

viii

Preface to the First Edition

mining, and almost every other major branch of science. Chapter 3 discusses the basic principles, background, and recent advances in the tube-excited energy-dispersive mode, which, after hectic growth in the 1970s, has now apparently leveled oﬀ to make up approximately 20% of the x-ray ﬂuorescence market; it is invoked frequently in research on environmental and biological samples. Chapter 4 reviews in depth the available alternatives for spectrum evaluation and qualitative analysis; techniques for deconvolution of spectra have enormously increased the utility of energy-dispersive x-ray analysis, but deconvolution is still its most critical step. Chapters 5 and 6 review the quantiﬁcation problems in the analysis of samples that are inﬁnitely thick and of intermediate thickness, respectively. Chapter 7 is a very practical treatment of radioisotope-induced x-ray analysis, which is now rapidly acquiring wide acceptance for dedicated instruments and ﬁeld applications. Chapter 8 reviews synchrotron-induced x-ray emission analysis, the youngest branch, with limited accessibility but an exponentially growing literature due to its extreme sensitivity and microanalysis potential. Although its principles have been known for some time, it is only since the advent of powerful commercial units that total reﬂection x-ray ﬂuorescence has been rapidly introduced, mostly for liquid samples and surface layer characterization; this is the subject of Chapter 9. Polarized beam x-ray ﬂuorescence is outlined in Chapter 10. Particle-induced x-ray emission analysis is available at many accelerator centers worldwide; the number of annual articles on it is growing and it undergoes a revival in its microversion; Chapter 11 treats the physical backgrounds, instrumentation, performance, and applications of this technique. Although the practical approaches to electron-induced x-ray emission analysis, now a standard technique with wide applications in all branches of science and technology, are often quite diﬀerent from those in other x-ray analysis techniques, a separate treatment of its potential for quantitative and spatially resolved analysis is given in Chapter 12. Finally, Chapter 13 brieﬂy reviews the sample preparation techniques that are invoked most frequently in combination with x-ray ﬂuorescence analysis. This book is a multi-authored eﬀort. We believe that having scientists who are actively engaged in a particular technique covering those areas for which they are particularly qualiﬁed and presenting their own points of view and general approaches outweighs any advantages of uniformity and homogeneity that characterize a single-author book. Three chapters were written by the editors and a coworker. For all the other chapters, we were fortunate enough to have the cooperation of eminent specialists. The editors wish to thank all the contributors for their eﬀorts. We hope that novices in x-ray emission analysis will ﬁnd this book useful and instructive, and that our more experienced colleagues will beneﬁt from the large amount of readily accessible information available in this compact form, some of it for the ﬁrst time. This book is not intended to replace earlier works, some of which were truly excellent, but to supplement them. Some overlap is inevitable, but an eﬀort has been made to emphasize the ﬁelds and developments that have come into prominence lately and have not been treated in a handbook before. Rene´ E. Van Grieken Andrzej A. Markowicz

Contents

Preface to the Second Edition Preface to the First Edition Contributors 1 X-ray Physics Andrzej A. Markowicz I. Introduction II. History III. General Features IV. Emission of Continuous Radiation V. Emission of Characteristic X-rays VI. Interaction of Photons with Matter VII. Intensity of Characteristic X-rays VIII. IUPAC Notation for X-ray Spectroscopy Appendixes I. Critical Absorption Wavelengths and Critical Absorption Energies II. Characteristic X-ray Wavelengths (A˚) and Energies (keV) III. Radiative Transition Probabilities IV. Natural Widths of K and L Levels and Ka X-ray Lines (FWHM), in eV V. Wavelengths of K Satellite Lines (A˚) VI. Fluorescence Yields and Coster–Kronig Transition Probabilities VII. Coefﬁcients for Calculating the Photoelectric Absorption Cross Sections t (Barns=Atom) Via ln–ln Representation VIII. Coefﬁcients for Calculating the Incoherent Collision Cross Sections sc (Barns=Atom) Via the ln–ln Representation IX. Coefﬁcients for Calculating the Coherent Scattering Cross Sections sR (Barns=Atom) Via the ln–ln Representation X. Parameters for Calculating the Total Mass Attenuation Coefﬁcients in the Energy Range 0.1–1000 keV [Via Eq. (78)] XI. Total Mass Attenuation Coefﬁcients for Low-Energy Ka Lines XII. Correspondence Between Old Siegbahn and New IUPAC Notation X-ray Diagram Lines References

iii vii xv 1 1 1 2 3 7 17 31 34 36 40 49 53 56 58 68 74 76 78 87 91 92 ix

x

Contents

2 Wavelength-Dispersive X-ray Fluorescence Jozef A. Helsen and Andrzej Kuczumow I. II. III. IV. V. VI. VII.

Introduction Fundamentals of Wavelength Dispersion Layout of a Spectrometer Qualitative and Quantitative Analysis Chemical Shift and Speciation Instrumentation Future Prospects References

3 Energy-Dispersive X-ray Fluorescence Analysis Using X-ray Tube Excitation Andrew T. Ellis I. II. III. IV. V.

Introduction X-ray Tube Excitation Systems Semiconductor Detectors Semiconductor Detector Electronics Summary References

4 Spectrum Evaluation Piet Van Espen I. II. III. IV. V. VI. VII. VIII. IX. X.

Introduction Fundamental Aspects Spectrum Processing Methods Continuum Estimation Methods Simple Net Peak Area Determination Least-Squares Fitting Using Reference Spectra Least-Squares Fitting Using Analytical Functions Methods Based on the Monte Carlo Technique The Least-Squares-Fitting Method Computer Implementation of Various Algorithms References

5 Quantiﬁcation of Inﬁnitely Thick Specimens by XRF Analysis Johan L. de Vries and Bruno A. R. Vrebos I. II. III. IV. V. VI.

Introduction Correlation Between Count Rate and Specimen Composition Factors Inﬂuencing the Accuracy of the Intensity Measurement Calibration and Standard Specimens Converting Intensities to Concentration Conclusion References

95 95 100 104 150 169 173 189 191 199 199 200 214 230 236 236 239 239 240 245 260 264 268 278 300 306 315 336 341 341 343 350 359 362 402 403

Contents

xi

6 Quantiﬁcation in XRF Analysis of Intermediate-Thickness Samples Andrzej A. Markowicz and Rene´ E. Van Grieken I. II. III. IV.

Introduction Emission-Transmission Method Absorption Correction Methods Via Scattered Primary Radiation Quantitation for Intermediate-Thickness Granular Specimens References

7 Radioisotope-Excited X-ray Analysis Stanislaw Piorek I. II. III. IV. V. VI. VII. VIII.

Introduction Basic Equations Radioisotope X-ray Sources and Detectors X-ray and g-ray Techniques Factors Affecting the Overall Accuracy of XRF Analysis Applications Future of Radioisotope-Excited XRF Analysis Conclusions Appendix: List of Companies that Manufacture Radioisotope-Based X-ray Analyzers and Systems References

8 Synchrotron Radiation-Induced X-ray Emission Keith W. Jones I. II. III. IV. V. VI. VII. VIII. IX. X. XI. XII.

Introduction Properties of Synchrotron Radiation Description of Synchrotron Facilities Apparatus for X-ray Microscopy Continuum and Monochromatic Excitation Quantitation Sensitivities and Minimum Detection Limits Beam-Induced Damage Applications of SRIXE Tomography EXAFS and XANES Future Directions References

9 Total Reﬂection X-ray Fluorescence Peter Kregsamer, Christina Streli, and Peter Wobrauschek I. II. III. IV. V. VI. VII.

Introduction Physical Principles Instrumentation Chemical Analysis Surface Analysis Thin Films and Depth Proﬁles Synchrotron Radiation Excitation

407 407 408 415 423 430 433 433 435 442 456 469 474 495 496 497 498 501 501 503 506 507 523 524 525 530 532 542 545 551 555 559 559 560 567 574 583 588 590

xii

Contents

VIII. IX.

Light Elements Related Techniques References

10 Polarized Beam X-ray Fluorescence Analysis Joachim Heckel and Richard W. Ryon I. II. III. IV. V. VI. VII.

Introduction Theory Barkla Systems Bragg Systems Barkla-Bragg Combination Systems Secondary Targets Conclusion References

11 Microbeam XRF Anders Rindby and Koen H. A. Janssens I. II. III. IV. V.

Introduction and Historical Perspective Theoretical Background Instrumentation for Microbeam XRF Collection and Processing of m-XRF Data Applications References

12 Particle-Induced X-ray Emission Analysis Willy Maenhaut and Klas G. Malmqvist I. II.

III. IV. V. VI. VII. VIII.

Introduction Interactions of Charged Particles with Matter, Characteristic X-ray Production, and Continuous Photon Background Production Instrumentation Quantitation, Detection Limits, Accuracy, and Precision Sample Collection and Sample and Specimen Preparation for PIXE Analysis Applications Complementary Ion-Beam-Analysis Techniques Conclusions References

13 Electron-Induced X-ray Emission John A. Small, Dale E. Newbury, and John T. Armstrong I. II. III. IV. V.

Introduction Quantitative Analysis Microanalysis at Low Electron Beam Energy Analysis of Samples with Nonstandard Geometries Spatially Resolved X-ray Analysis References

595 597 599 603 603 605 610 618 627 627 628 629 631 631 637 646 667 696 712 719 719

720 727 739 748 750 783 792 797 811 811 816 857 876 909 926

Contents

xiii

14 Sample Preparation for X-ray Fluorescence Martina Schmeling and Rene´ E. Van Grieken

933

I. II. III. IV. V. VI. VII. Index

Introduction Solid Samples Fused Specimen Liquid Specimen Biological Samples Atmospheric Particles Sample Support Materials References

933 934 944 948 958 965 968 970 977

Contributors

John T. Armstrong, Ph.D. Gaithersburg, Maryland

National Institute of Standards and Technology,

Eindhoven, The Netherlands

Johan L. de Vries, Ph.D.*

Oxford Instruments Analytical Ltd., High Wycombe,

Andrew T. Ellis, Ph.D. Buckinghamshire, England Joachim Heckel, Ph.D. many

Spectro Analytical Instruments, GmbH & Co. KG, Kleve, Ger-

Jozef A. Helsen, Ph.D.

Catholic University of Leuven, Leuven, Belgium University of Antwerp, Antwerp, Belgium

Koen H. A. Janssens, Ph.D. Keith W. Jones, Ph.D.

Brookhaven National Laboratory, Upton, New York

Peter Kregsamer, Dr. techn., Dipl. Ing.

Lublin Catholic University, Lublin, Poland

Andrzej Kuczumow, Ph.D. Willy Maenhaut, Ph.D.

Ghent University, Ghent, Belgium

Klas G. Malmqvist, Ph.D. Sweden

Lund University and Lund Institute of Technology, Lund,

Andrzej A. Markowicz, Ph.D. Dale E. Newbury, Ph.D. Maryland

Atominstitut, Vienna, Austria

Vienna, Austria

National Institute of Standards and Technology, Gaithersburg,

* Retired. xv

xvi

Contributors

Stanislaw Piorek, Ph.D.{ Anders Rindby, Ph.D. Go¨tebo¨rg, Sweden Richard W. Ryon, B.A. nia

Niton Corporation, Billerica, Massachusetts

Chalmers University of Technology and University of Go¨tebo¨rg,

Lawrence Livermore National Laboratory, Livermore, Califor-

Martina Schmeling, Ph.D.

Loyola University Chicago, Chicago, Illinois

John A. Small, Ph.D. Maryland

National Institute of Standards and Technology, Gaithersburg,

Christina Streli, Ph.D.

Atominstitut, Vienna, Austria

Piet Van Espen, Ph.D.

University of Antwerp, Antwerp, Belgium

Rene´ E. Van Grieken, Ph.D.

University of Antwerp, Antwerp, Belgium

Bruno A. R. Vrebos, Dr. Ir.

Philips Analytical, Almelo, The Netherlands

Peter Wobrauschek, Ph.D.

{

Atominstitut, Vienna, Austria

Previous aﬃliation: Metorex Inc., Princeton, New Jersey.

1 X-ray Physics Andrzej A. Markowicz Vienna, Austria

I.

INTRODUCTION

In this introductory chapter, the basic concepts and processes of x-ray physics that relate to x-ray spectrometry are presented. Special emphasis is on the emission of the continuum and characteristic x-rays as well as on the interactions of photons with matter. In the latter, only major processes of the interactions are covered in detail, and the cross sections for diﬀerent types of interactions and the fundamental parameters for other processes involved in the emission of the characteristic x-rays are given by the analytical expressions and=or in a tabulated form. Basic equations for the intensity of the characteristic x-rays for the diﬀerent modes of x-ray spectrometry are also presented (without derivation). Detailed expressions relating the emitted intensity of the characteristic x-rays to the concentration of the element in the specimen are discussed in the subsequent chapters of this handbook dedicated to speciﬁc modes of x-ray spectrometry.

II.

HISTORY

X-rays were discovered in 1895 by Wilhelm Conrad Ro¨ntgen at the University of Wu¨rzburg, Bavaria. He noticed that some crystals of barium platinocyanide, near a discharge tube completely enclosed in black paper, became luminescent when the discharge occurred. By examining the shadows cast by the rays. Ro¨ntgen traced the origin of the rays to the walls of the discharge tube. In 1896, Campbell-Swinton introduced a deﬁnite target (platinum) for the cathode rays to hit; this target was called the anticathode. For his work x-rays, Ro¨ntgen received the ﬁrst Nobel Prize in physics, in 1901. It was the ﬁrst of six to be awarded in the ﬁeld of x-rays by 1927. The obvious similarities with light led to the crucial tests of established wave optics: polarization, diﬀraction, reﬂection, and refraction. With limited experimental facilities, Ro¨ntgen and his contemporaries could ﬁnd no evidence of any of these; hence, the designation ‘‘x’’ (unknown) of the rays, generated by the stoppage at anode targets of the cathode rays, identiﬁed by Thomson in 1897 as electrons. The nature of x-rays was the subject of much controversy. In 1906, Barkla found evidence in scattering experiments that x-rays could be polarized and must therefore by waves, but W. H. Bragg’s studies of the produced ionization indicated that they were 1

2

Markowicz

corpuscular. The essential wave nature of x-rays was established in 1912 by Laue, Friedrich, and Knipping, who showed that x-rays could be diﬀracted by a crystal (copper sulfate pentahydrate) that acted as a three-dimensional diﬀraction grating. W. H. Bragg and W. L. Bragg (father and son) found the law for the selective reﬂection of x-rays. In 1908, Barkla and Sadler deduced, by scattering experiments, that x-rays contained components characteristic of the material of the target; they called these components K and L radiations. That these radiations had sharply deﬁned wavelengths was shown by the diﬀraction experiments of W. H. Bragg in 1913. These experiments demonstrated clearly the existence of a line spectrum superimposed upon a continuous (‘‘White’’) spectrum. In 1913, Moseley showed that the wavelengths of the lines were characteristic of the element of the which the target was made and, further, showed that they had the same sequence as the atomic numbers, thus enabling atomic numbers to be determined unambiguously for the ﬁrst time. The characteristic K absorption was ﬁrst observed by de Broglie and interpreted by W. L. Bragg and Siegbahn. The eﬀect on x-ray absorption spectra of the chemical state of the absorber was observed by Bergengren in 1920. The inﬂuence of the chemical state of the emitter on x-ray emission spectra was observed by Lindh and Lundquist in 1924. The theory of x-ray spectra was worked out by Sommerfeld and others. In 1919, Stenstro¨m found the deviations from Bragg’s law and interpreted them as the eﬀect of refraction. The anomalous dispersion of x-ray was discovered by Larsson in 1929, and the extended ﬁne structure of x-ray absorption spectra was qualitatively interpreted by Kronig in 1932. Soon after the ﬁrst primary spectra excited by electron beams in an x-ray tube were observed, it was found that secondary ﬂuorescent x-rays were excited in any material irradiated with beams of primary x-rays and that the spectra of these ﬂuorescent x-rays were identical in wavelengths and relative intensities with those excited when the specimen was bombarded with electrons. Beginning in 1932, Hevesy, Coster, and others investigated in detail the possibilities of ﬂuorescent x-ray spectroscopy as a means of qualitative and quantitative elemental analysis. III.

GENERAL FEATURES

X-rays, or Ro¨ntgen rays, are electromagnetic radiations having wavelengths roughly within the range from 0.005 to 10 nm. At the short-wavelength end, they overlap with g-rays, and at the long-wavelength end, they approach ultraviolet radiation. The properties of x-rays, some of which are discussed in detail in this chapter, are summarized as follows: Invisible Propagated in straight lines with a velocity of 36108 m=s, as is light Unaﬀected by electrical and magnetic ﬁelds Diﬀerentially absorbed while passing through matter of varying composition, density, or thickness Reﬂected, diﬀracted, refracted, and polarized Capable of ionizing gases Capable of aﬀecting electrical properties of liquids and solids Capable of blackening a photographic plate Able to liberate photoelectrons and recoil electrons Capable of producing biological reactions (e.g., to damage or kill living cells and to produce genetic mutations)

X-ray Physics

3

Emitted in a continuous spectrum whose short-wavelength limit is determined only by the voltage on the tube Emitted also with a line spectrum characteristic of the chemical elements Found to have absorption spectra characteristic of the chemical elements

IV.

EMISSION OF CONTINUOUS RADIATION

Continuous x-rays are produced when electrons, or other high-energy charged particles, such as protons or a-particles, lose energy in passing through the Coulomb ﬁeld of a nucleus. In this interaction, the radiant energy (photons) lost by the electron is called bremsstrahlung (from the German bremsen, to brake, and Strahlung, radiation; this term sometimes designates the interaction itself). The emission of continuous x-rays ﬁnds a simple explanation in terms of classic electromagnetic theory, because, according to this theory, the acceleration of charged particles should be accompanied by the emission of radiation. In the case of high-energy electrons striking a target, they must be rapidly decelerated as they penetrate the material of the target, and such a high negative acceleration should produce a pulse of radiation. The continuous x-ray spectrum generated by electrons in an x-ray tube is characterized by a short-wavelength limit lmin , corresponding to the maximum energy of the exciting electrons: lmin ¼

hc eV0

ð1Þ

where h is Planck’s constant, c is the velocity of light, e is the electron charge, and V0 is the potential diﬀerence applied to the tube. This relation of the short-wavelength limit to the applied potential is called the Duane–Hunt law. The probability of radiative energy loss (bremsstrahlung) is roughly proportional to q2 Z2 T=M20 , where q is the particle charge in units of the electron charge e, Z is the atomic number of the target material, T is the particle kinetic energy, and M0 is the rest mass of the particle. Because protons and heavier particles have large masses compared to the electron mass, they radiate relatively little; for example, the intensity of continuous x-rays generated by protons is about four orders of magnitude lower than that generated by electrons. The ratio of energy lost by bremsstrahlung to that lost by ionization can be approximated by 2 m0 ZT ð2Þ M0 1600m0 c2 where m0 the rest mass of the electron.

A.

Spectral Distribution

The continuous x-ray spectrum generated by electrons in an x-ray tube (thick target) is characterized by the following features: 1. Short-wavelength limit, lmin [Eq. (1)]; below this wavelength, no radiation is observed.

4

Markowicz

2.

3.

Wavelength of maximum intensity lmax , approximately 1.5 times lmin ; however, the relationship between lmax and lmin depends to some extent on voltage, voltage waveform, and atomic number. Total intensity nearly proportional to the square of the voltage and the ﬁrst power of the atomic number of the target material.

The most complete empirical work on the overall shape of the energy distribution curve for a thick target has been of Kulenkampﬀ (1922, 1933), who found the following formula for the energy distribution; ð3Þ IðvÞ dv ¼ i aZðv0 vÞ þ bZ2 dv where IðnÞ dn is the intensity of the continuous x-rays within a frequency range ðn; n þ dvÞ; i is the electron current striking the target, Z is the atomic number of the target material, n0 is the cutoﬀ frequency ð¼ c=lmin Þ above which the intensity is zero, and a and b are constants independent of atomic number, voltage, and cutoﬀ wavelength. The second term in Eq. (3) is usually small compared to the ﬁrst and is often neglected. The total integrated intensity at all frequencies is I ¼ iða0 ZV20 þ b0 Z2 V0 Þ

ð4Þ

in which a0 ¼ aðe2 =h2 Þ=2 and b0 ¼ bðe=hÞ. An approximate value for b0 =a0 is 16.3 V; thus, I ¼ a0 iZV0 ðV0 þ 16:3ZÞ

ð5Þ

The eﬃciency Eﬀ of conversion of electric power input to x-rays of all frequencies is given by Eff ¼

I ¼ a0 ZðV0 þ 16:3ZÞ V0 i

ð6Þ

where V0 is in volts. Experiments give a0 ¼ ð1:2 0:1Þ 109 (Condon, 1958). The most complete and successful eﬀorts to apply quantum theory to explain all features of the continuous x-ray spectrum are those of Kramers (1923) and Wentzel (1924). By using the correspondence principle, Kramers found the following formulas for the energy distribution of the continuous x-rays generated in a thin target: pﬃﬃ AZ e dv; IðvÞ dv ¼ 316p 3m0 V0 c3 IðvÞ dv ¼ 0; 2

2 5

v < v0 v > v0

ð7Þ

where A is the atomic mass of the target material. When the decrease in velocity of the electrons in a thick target was taken into account by applying the Thomson–Whiddington law (Dyson, 1973), Kramers found, for a thick target, 8pe2 h Zðv0 vÞ dv IðvÞ dv ¼ pﬃﬃﬃ 3 3lm0 c3

ð8Þ

where l is approximately 6. The eﬃciency of production of the x-rays calculated via Kramers’ law is given by Eff ¼ 9:2 1010 ZV0

ð9Þ

which is in qualitative agreement with the experiments of Kulenkampﬀ (Stephenson, 1957); for example,

X-ray Physics

Eff ¼ 15 1010 ZV0

5

ð10Þ

It is worth mentioning that the real continuous x-ray distribution is described only approximately by Kramers’ equation. This is related, inter alia, to the fact that the derivation ignores the self-absorption of x-rays and electron backscattering eﬀects. Wentzel (1924) used a diﬀerent type of correspondence principle than Kramers, and he explained the spatial distribution asymmetry of the continuous x-rays from thin targets. An accurate description of continuous x-rays is crucial in all x-ray spectrometry (XRS). The spectral intensity distributions from x-ray tubes are of great importance for applying fundamental mathematical matrix correction procedures in quantitative x-ray ﬂuorescence (XRF) analysis. A simple equation for the accurate description of the actual continuum distributions from x-ray tubes was proposed by Tertian and Broll (1984). It is based on a modiﬁed Kramers’ law and a reﬁned x-ray absorption correction. Also, a strong need to model the spectral Bremsstrahlung background exists in electron-probe x-ray microanalysis (EPXMA). First, ﬁtting a function through the background portion, on which the characteristic x-rays are superimposed in an EPXMA spectrum, is not easy; several experimental ﬁtting routines and mathematical approaches, such as the Simplex method, have been proposed in this context. Second, for bulk multielement specimens, the theoretical prediction of the continuum Bremsstrahlung is not trivial; indeed, it has been known for several years that P the commonly used Kramers’ formula with Z directly substituted by the average Z ¼ i Wi Zi (Wi and Zi are the weight fraction and atomic number of the ith element, respectively) can lead to signiﬁcant errors. In this context, some improvements are oﬀered by several modiﬁed versions of Kramers’ formula developed for a multielement bulk specimen (Statham, 1976; Lifshin, 1976; Sherry and Vander Sande, 1977; Smith and Reed, 1981). Also, a new expression for the continuous x-rays emitted by thick composite specimens was proposed (Markowicz and Van Grieken, 1984; Markowicz et al., 1986); it was derived by introducing the compositional dependence of the continuum x-rays already in the elementary equations. The new expression has been combined with known equations for the self-absorption of x-rays (Ware and Reed, 1973) and electron backscattering (Statham, 1979) to obtain an accurate description of the detected continuum radiation. A third problem is connected with the description of the x-ray continuum generated by electrons in specimens of thickness smaller than the continuum x-ray generation range. This problem arises in the analysis of both thin ﬁlms and particles by EPXMA. A theoretical model for the shape of the continuous x-rays generated in multielement specimens of ﬁnite thickness was developed (Markowicz et al., 1985); both composition and thickness dependence have been considered. Further reﬁnements of the theoretical approach are hampered by the lack of knowledge concerning the shape of the electron interaction volume, the distribution of the electron within the interaction volume, and the anisotropy of continuous radiation for diﬀerent x-ray energies and for diﬀerent ﬁlm thickness. B.

Spatial Distribution and Polarization

The spatial distribution of the continuous x-rays emitted by thin targets has been investigated by Kulenkampﬀ (1928). The author made an extensive survey of the intensity at angles between 22 and 150 to the electron beam in terms of dependence on wavelength and voltage. The target was a 0.6-mm-thick Al foil. Figure 1 shows the continuous x-ray intensity observed at diﬀerent angles for voltages of 37.8, 31.0, 24.0, and 16.4 kV ﬁltered by 10, 8, 4, and 1.33 mm of Al, respectively (Stephenson, 1957). Curve (a) is repeated as a dotted line near each of the other curves. The angle of the maximum intensity varied from

6

Markowicz

Figure1 Intensity of continuous x-rays as a function of direction for different voltages. (Curve (a) is repeated as dotted line.) (From Stephenson, 1957.)

50 for 37.8 kV to 65 for 16.4 kV. Figure 2 illustrates the intensity of the continuous x-rays observed in the Al foil for diﬀerent thicknesses as a function of the angle for a voltage of 30 kV (Stephenson, 1957). The theoretical curve is from the theory of Scherzer (1932). The continuous x-ray intensity drops to zero at 180 , and although it is not zero at 0 as the theory of Scherzer predicts, it can be seen from Figure 2 that for a thinner foil, a lower intensity at 0 is obtained. Summarizing, it appears that the intensity of the continuous x-rays emitted by thin foils has a maximum at about 55 relative to the incident electron beam and becomes zero at 180 . The continuous radiation from thick targets is characterized by a much smaller anisotropy than that from thin targets. This is because in thick targets the electrons are rarely stopped in one collision and usually their directions have considerable variation. The use of electromagnetic theory predicts a maximum energy at right angles to the incident electron beam at low voltages, with the maximum moving slightly away from perpendicularity toward the direction of the elctron beam as the voltage is increased. In general, an increase in the anisotropy of the continuous x-rays from thick targets is observed at the short-wavelength limit and for low-Z targets (Dyson, 1973).

X-ray Physics

7

Figure 2 Intensity of continuous x-rays as a function of direction for different thicknesses of the A1 target together with theoretical prediction. (From Stephenson, 1957.)

Continuous x-ray beams are partially polarized only from extremely thin targets; the angular region of polarization is sharply peaked about the photon emission angle y ¼ m0 c2 =E0 , where E0 is the energy of the primary electron beam. Electron scattering in the target broadens the peak and shifts the maximum to larger angles. Polarization is deﬁned by (Kenney, 1966) Pðy; E0 ; En Þ ¼

ds?ðy; E0 ; En Þ dskðy; E0 ; En Þ ds?ðy; E0 ; En Þ þ dskðy; E0 ; En Þ

ð11Þ

where an electron of energy E0 radiates a photon of energy En at angle y; ds?ðy; E0 ; En Þ and dskðy; E0 ; En Þ are the cross sections for generation of the continuous radiation with the electric vector perpendicular (?) and parallel (k) to the plane deﬁned by the incident electron and the radiated photon, respectively. Polarization is diﬃcult to observe, and only thin, low-yield radiators give evidence for this eﬀect. When the electron is relativistic before and after the radiation, the electrical vector is most probably in the ? direction. Practical thick-target Bremsstrahlung shows no polarization eﬀects whatever (Dyson, 1973; Stephenson, 1957; Kenney, 1966).

V.

EMISSION OF CHARACTERISTIC X-RAYS

The production of characteristic x-rays involves transitions of the orbital electrons of atoms in the target material between allowed orbits, or energy states, associated with ionization of the inner atomic shells. When an electron is ejected from the K shell by electron bombardment or by the absorption of a photon, the atom becomes ionized and the ion is left in a high-energy state. The excess energy the ion has over the normal state of the atom is equal to the energy (the binding energy) required to remove the K electron to a state of rest outside the atom. If this electron vacancy is ﬁlled by an electron coming from an L level, the transition is accompanied by the emission of an x-ray line known as the Ka line. This process leaves a vacancy in the L shell. On the other hand, if the atom contains suﬃcient electrons, the K shell vacancy might be ﬁlled by an electron coming from an M level that is accompanied by the emission of the Kb line. The L or M state ions that remain may also give rise to emission if the electron vacancies are ﬁlled by electrons falling from further orbits.

8

A.

Markowicz

Inner Atomic Shell Ionization

As already mentioned, the emission of characteristic x-ray is preceded by ionization of inner atomic shells, which can be accomplished either by charged particles (e.g., electrons, protons, and a-particles) or by photons of suﬃcient energy. The cross section for ionization of an inner atomic shell of element i by electrons is given by (Bethe, 1930; Green and Cosslett, 1961; Wernisch, 1985) Qi ¼ pe4 ns bs

ln U UE2c;i

ð12Þ

where U ¼ E=Ec;i is the overvoltage, deﬁned as the ratio of the instantaneous energy of the electron at each point of the trajectory to that required to ionize an atom of element i, Ec;i is the critical excitation energy, and ns and bs are constants for a particular shell: s ¼ K: s ¼ L:

ns ¼ 2; bs ¼ 0:35 ns ¼ 8; bs ¼ 0:25

The cross section for ionization Qi is a strong function of the overvoltage, which shows a maximum at U ﬃ 3–4 (Heinrich, 1981; Goldstein et al., 1981). The probability (or cross section) of ionization of an inner atomic shell by a charged particle is given by (Merzbacher and Lewis, 1958) ss ¼

8pr20 q2 fs Z 4 Zs

ð13Þ

where r0 is the classic radius of the electron equal to 2.81861015 m, q is the particle charge, Z is the atomic number of the target material, fs is a factor depending on the wave functions of the electrons for a particular shell, and Zs is a function of the energy of the incident particles. In the case of electromagnetic radiation (x or g), the ionization of an inner atomic shell is a result of the photoelectric eﬀect. This eﬀect involves the disappearance of a radiation photon and the photoelectric ejection of one electron from the absorbing atom, leaving the atom in an excited level. The kinetic energy of the ejected photoelectron is given by the diﬀerence between the photon energy hn and the atomic binding energy of the electron Ec (critical excitation energy). Critical absorption wavelengths (Clark, 1963) related to the critical absorption energies (Burr, 1974) via the equation l(nm) ¼ 1.24=E(ke V) are presented in Appendix I. The wavelenghts of K, L, M, and N absorption edges can also be calculated by using simple empirical equations (Norrish and Tao, 1993). For energies far from the absorption edge and in the nonrelativistic range, the cross section tK for the ejection of an electron from the K shell is given by (Heitler, 1954) pﬃﬃﬃ 7=2 32 2 2 Z5 m 0 c2 pr0 tK ¼ 3 ð137Þ4 hv

ð14Þ

Equation (14) is not fully adequate in the neighborhood of an absorption edge; in this case, Eq. (14) should be multiplied by a correction factor f(X ) (Stobbe, 1930): 1=2 4X arccot X D e fðXÞ ¼ 2p hv 1 e2pX where

ð15Þ

X-ray Physics

9

X¼

D hv D

1=2 ð15aÞ

with 1 m 0 c2 D ﬃ ðZ 0:3Þ2 2 ð137Þ2

ð15bÞ

When the energy of the incident photon is of the order m0 c2 or greater, relativistic cross sections for the photoelectric eﬀect must be used (Sauter, 1931).

B.

Spectral Series in X-rays

The energy of an emission line can be calculated as the diﬀerence between two terms, each term corresponding to a deﬁnite state of the atom. If E1 and E2 are the term values representing the energies of the corresponding levels, the frequency of an x-ray line is given by the relation v¼

E1 E2 h

ð16Þ

Using the common notations, one can represent the energies of the levels E by means of the atomic number and the quantum numbers n, l, s, and j (Sandstro¨m, 1957): ! 2 E ðZ Sn;l Þ2 1 3 2 ðZ dn;l; j Þ ¼ þa Rh n2 n3 l þ 12 4n a2

ðZ dn;l; j Þ4 jð j þ 1Þ lðl þ 1Þ sðs þ 1Þ n3 2lðl þ 12Þðl þ 1Þ

ð17Þ

where Sn;l and dn;l; j are screening constants that must be introduced to correct for the eﬀect of the electrons on the ﬁeld in the atom, R is the universal Rydberg constant valid for all elements with Z > 5 or throughout nearly the whole x-ray region, and a is the ﬁnestructure constant given by a¼

2pe2 hc

ð17aÞ

The theory of x-ray spectra reveals the existence of a limited number of allowed transitions; the rest are ‘‘forbidden.’’ The most intense lines create the electric dipole radiation. The transitions are governed by the selection rules for the change of quantum numbers: Dl ¼ 1;

Dj ¼ 0 or 1

ð18Þ

The j transition 0 ? 0 is forbidden. According to Dirac’s theory of radiation (Dirac, 1947), transitions that are forbidden as dipole radiation can appear as multipole radiation (e.g., as electric quadrupole and magnetic dipole transitions). The selection rules for the former are Dl ¼ 0 or 2;

Dj ¼ 0; 1; or 2

The j transitions 0 ? 0, and 0 $ 1 are forbidden. The selection rules for magnetic dipole transitions are 1 1 2? 2,

ð19Þ

10

Markowicz

Dl ¼ 0;

Dj ¼ 0 or 1

ð20Þ

The j transition 0 ? 0 is forbidden. The commonly used terminology of energy levels and x-ray lines is shown in Figure 3. A general expression relating the wavelength of an x-ray characteristic line with the atomic number of the corresponding element is given by Moseley’s law (Moseley, 1914): 1 ¼ kðZ sÞ2 l

ð21Þ

where k is a constant for a particular spectral series and s is a screening constant for the repulsion correction due to other electrons in the atom. Moseley’s law plays an important role in the systematizing of x-ray spectra. Appendix II tabulates the energies and wavelengths of the principal x-ray emission lines for the K, L, and M series with their approximate relative intensities, which can be deﬁned either by means of spectral line peak intensities or by area below their intensity distribution curve. In practice, the relative

Figure 3

Commonly used terminology of energy levels and x-ray lines. (From Sandstro¨m, 1957.)

X-ray Physics

11

intensities of spectral lines are not constant because they depend not only on the electron transition probability but also on the specimen composition. Considering the K series, the Ka fraction of the total K spectrum is deﬁned by the transition probability pKa, which is given by (Schreiber and Wims, 1982) pKa ¼

IðKa1 þ Ka2 Þ IðKa1 þ Ka2 Þ þ IðKb1 þ Kb2 Þ

ð22Þ

Wernisch (1985) proposed a useful expression for the calculation of the transition probability pKa for diﬀerent elements: 8 11 Zi 19 < 1:052 4:39 104 Z2i ; ð23Þ pKa;i ¼ 0:896 6:575 104 Zi ; 20 Zi 29 : 1:0366 6:82 103 Zi þ 4:815 105 Z2i ; 30 Z2i 60 For the L level, split into three subshells, several electron transitions exist. The transition probability pLa, deﬁned as the fraction of the transitions resulting in La1 and La2 radiation from the total of possible transitions into the L3 subshell, can be calculated by the expression (Wernisch, 1985) 8 39 Zi 44 < 0:944; ð24Þ pLa;i ¼ 4:461 101 þ 5:493 102 Zi : 7:717 104 Z2i þ 3:525 106 Z3i ; 45 Zi 82 Radiative transition probabilities for various K and L x-ray lines (West, 1982–83) are presented in detail in Appendix III. The experimental results, together with the estimated 95% conﬁdence limits, for the relative intensity ratios for K and L x-ray lines for selected elements with atomic number from Z ¼ 14 to 92 have been reported by Stoev and Dlouhy (1994). The values are in a good agreement with other published experimental data. Because the electron vacancy created in an atom by charged particles or electromagnetic radiation has a certain lifetime, the atomic levels E1 and E2 [Eq. (16)] and the characteristic x-ray lines are characterized by the so-called natural widths (Krause and Oliver, 1979). The natural x-ray linewidths are the sums of the widths of the atomic levels involved in the transitions. Semiempirical values of the natural widths of K, L1 , L2 and L3 levels, Ka1 and Ka2 x-ray lines for the elements 10 Z 110 are presented in Appendix IV. Uncertainties in the level width values are from 3% to 10% for the K shell and from 8% to 30% for the L subshell. Uncertainties in the natural x-ray linewidth values are from 3% to 10% for Ka1;2 . In both cases, the largest uncertainties are for low-Z elements (Krause and Oliver, 1979).

C.

X-ray Satellites

A large number of x-ray lines have been reported that do not ﬁt into the normal energylevel diagram (Clark, 1955; Kawai and Gohshi, 1986). Most of the x-ray lines, called satellites or nondiagram lines, are very weak and are of rather little consequence in analytical x-ray spectrometry. By analogy to the satellites in optical spectra, it was supposed that the origin of the nondiagram x-ray lines is double or manyfold ionization of an atom through electron impact. Following the ionization, a multiple electron transition results in emission of a single photon of energy higher than that of the characteristic x-rays. The majority of nondiagram lines originate from the dipole-allowed deexcitation of multiply

12

Markowicz

ionized or excited states and are called multiple-ionization satellites. A line where the initial state has two vacancies in the same shell, notably the K shell, is called a hypersatellite. In practice, the most important nondiagram x-ray lines occur in the Ka series; they are denoted as the Ka3 ;a4 doublet, and their origin is a double electron transition. The probability of a multiple-electron transition resulting in the emission of satellite x-ray lines is considerably higher for low-Z elements than for heavy and medium elements. For instance, the intensity of the AlKa3 satellite line is roughly 10% of that of the AlKa1 ; a2 characteristic x-rays. Appendix V tabulates wavelengths of the K satellite lines. A new class of satellites that are inside the natural width of the parent lines was observed by Kawai and Gohshi (1986). The origin of these satellites, called parasites or hidden satellites, is multiple ionization in nonadjacent shells. D.

Soft X-ray Emission-Band Spectra

In the soft x-ray region, the characteristic emission spectra of solid elements include continuous bands of width varying from 1 to 10 electron volts (eV); the same element in vapor form produces only the usual sharp spectral lines. The bands occur only when an electron falls from the outermost or valency shell of the atom, the levels of which are broadened into a wide band when the atoms are packed in a crystal lattice. Investigation of the emission-band spectra is of great signiﬁcance in understanding the electronic structure of solid metals, alloys, and complex coordination compounds. E.

Auger Effect

It has already been stated that the excess of energy an atom possesses after removing one electron from an inner shell by whatever means may be emitted as characteristic radiation. Alternatively, however, an excited atom may return to a state of lower energy by ejecting one of its own electrons from a less tightly bound state. The radiationless transition is called the Auger eﬀect, and the ejected electrons are called Auger electrons (Auger, 1925; Burhop, 1952). Generally, the probability of the Auger eﬀect increases with a decrease in the diﬀerence of the corresponding energy states, and it is the highest for the low-Z elements. Because an excited atom already has one electron missing (e.g., in the K shell) and another electron is ejected in an Auger process (e.g., from the L shell), the atom is left in a doubly-ionized state in which two electrons are missing. This atom may return to its normal state by single- or double-electron jumps with the emission of diagram or satellite lines, respectively. Alternatively, another Auger process may occur in which a third electron is ejected from the M shell. The Auger eﬀect also occurs after capture of a negative meson by an atom. As the meson changes energy levels in approaching the nucleus, the energy released may be either emitted as a photon or transferred directly to an electron that is emitted as a high-energy Auger electron (in the keV range for hydrogen and the MeV range for heavy elements). Measurements of the energy and intensity of the Auger electrons are applied extensively in surface physics studies (Auger electron spectroscopy). F.

FluorescenceYield

An important consequence of the Auger eﬀect is that the actual number of x-ray photons produced from an atom is less than expected, because a vacancy in a given shell might be

X-ray Physics

13

ﬁlled through a nonradiative transition. The probability that a vacancy in an atomic shell or subshell is ﬁlled through a radiative transition is called the ﬂuorescence yield. The application of this deﬁnition to the K shell of an atom is straightforward, and the ﬂuorescence yield of the K shell is oK ¼

IK nK

ð25Þ

where IK is the total number of characteristic K x-ray photons emitted from a sample and nK is the number of primary K shell vacancies. The deﬁnition of the ﬂuorescence yield of higher atomic shells is more complicated, for the following two reasons: 1. Shells above the K shell consist of more than one subshell; the average ﬂuorescence yield depends on how the shells are ionized. 2. Coster–Kronig transitions occur, which are nonradiative transitions between the subshells of an atomic shell having the same principal quantum number (Fink, 1974; Bambynek et al., 1972). In case Coster–Kronig transitions are absent, the ﬂuorescence yield of the ith subshell of a shell, whose principal quantum number is indicated by XðX ¼ L; M; . . .Þ, is given as oX i ¼

IX i nX i

ð26Þ

X for the shell X is deﬁned as An average or mean ﬂuorescence yield o X ¼ o

k X

ð27Þ

X NX i oi

i¼1

where NX i is the relative number of primary vacancies in the subshell i of shell X: nX i NX i ¼ Pk

X i¼1 ni

;

k X

NX i ¼1

ð28Þ

i¼1

The summations in Eqs. (27) and (28) extend over all k subshells of shell X. For the deﬁnition of the average ﬂuorescence yield, the primary vacancy distribution must be ﬁxed; X generally is not that is, Coster–Kronig transitions must be absent. It is noteworthy that o a fundamental property of the atom, but depends both on the atomic subshell ﬂuorescence X yields oX i and on the relative number of primary vacancies Ni characteristic of the method used to ionize the atoms. In the presence of Coster–Kronig transitions, which modify the primary vacancy distribution by the transfer of ionization from one subshell with a given energy to a subshell with less energy, the average ﬂuorescence yields can be calculated by using two X is regarded as a alternative approaches. In the ﬁrst, the average ﬂuorescence yield o with a vacancy distribution linear combination of the subshell ﬂuorescence yields oX i modiﬁed by Coster–Kronig transitions: X ¼ o

k X i¼1

X VX i oi ;

k X i¼1

VX i >1

ð29Þ

14

Markowicz

where VX i is the relative number of vacancies in the subshell i of shell X, including vacancies shifted to each subshell by Coster–Kronig transitions. The VX i values can be expressed in terms of the relative numbers NX i of primary vacancies and the Coster–Kronig transition probability for shifting a vacancy from a subshell Xi to a higher subshell Xj , denoted as f X ij (Bambynek et al., 1972): X VX 1 ¼ N1

ð30Þ

X X X VX 2 ¼ N2 þ f 12 N1

VX 3

¼

NX 3

þ

X fX 23 N2

þ

ðfX 13

þ

fX 12

X fX 23 ÞN1

X is a linear combination of In an alternative approach, the mean ﬂuorescence yield o : the relative numbers of primary vacancies NX i X ¼ o

k X

X NX i ni

ð31Þ

i¼1

where nX i represents the total number of characteristic x-rays that result per primary vacancy in the Xi subshell. The transformation relations between the coeﬃcients nX i and the subshell ﬂuorescence yields oX i follow from Eqs. (29) through (31) and are given in Fink (1974) and Bambynek et al. (1972). X Among the ﬂuorescence yield oX i , the Auger yield ai , and the Coster–Kronig X transition probabilities fij , the following relationship must hold (Krause, 1979): X oX i þ ai þ

k X

fX ij ¼ 1

ð32Þ

i¼1

The mean Auger yield aX is given by ax ¼

k X

X VX i ai

ð33Þ

i¼1

The values of the K, L, and M shell ﬂuorescence yields, the Coster–Kronig transition probabilities, as well as the Auger yields are given in Appendix VI. Although, in principle, the K shell ﬂuorescence yield ok can be calculated theoretically, experimental data are applied in practice. The following semiempirical equation, due to Burhop (1952), gives values correct to a few percent between Z ¼ 23 and Z ¼ 57 and less accurate values outside these limits: 1=4 oK ¼ 6:4 102 þ 3:40 102 Z 1:03 106 Z3 ð34Þ 1 oK The ﬂuorescence yield for the K series can also be calculated from a diﬀerent equation: 1=4 oK ¼ 0:217 þ 0:03318Z 1:14 106 Z3 ð35Þ 1 oK which gives quite good agreement with the experimental values for almost all elements. Based on a critical review of the available experimental and theoretical data, a recom L ; and o M values together with the following analytical expressions mended set of oK ; o were given by Hubbell (1989) and Hubbell et al. (1994):

X-ray Physics

15

" oK ð1 Z 100Þ ¼

3 X

Ci Zi

i¼0

#4 8 < :

" 1þ

3 X i¼0

Ci Zi

#4 91 = ;

ð36Þ

with C0 C1 C2 C3

¼ 0:0370 0:0052 ¼ 0:03112 0:00044 ¼ ð5:44 0:11Þ 105 ¼ ð1:25 0:07Þ 106

L ð3 Z 36Þ ¼ 1:939 108 Z3:8874 o " #4 8 " #4 91 < = 3 3 X X L ð37 Z 100Þ ¼ Ci Zi 1þ Ci Zi o : ; i¼0 i¼0

ð37Þ

with C0 C1 C2 C3

¼ 0:17765 ¼ 0:00298937 ¼ 8:91297 105 ¼ 2:67184 107

M ð13 Z 100Þ ¼ 1:29 109 ðZ 13Þ4 o

ð38Þ

Other useful expressions for the calculation of the ﬂuorescence yields oK ð12 Z 42Þ and oL3 ð38 Z 79Þ have been proposed by Hanke et al. (1985), based on literature and experimental data: oK ¼ 3:3704 101 6:0047 102 Z þ 3:3133 103 Z2 3:9215 105 Z3 oL3 ¼ 4:41 102 4:7559 103 Z þ 1:1494 104 Z2 1:8594 107 Z3

ð39Þ ð40Þ

N data is a theoretical work of McGuire (1974) For the N shell, the best source of o which provides oN1 ; oN2 , and oN3 values for 25 elements over the range 38 Z 103 and oN4 ; oN5 , and oN6;7 values for 20 elements over the range 50 Z 103. N can be calculated from (Hubbell, 1989) The average ﬂuorescence yield o N ¼ o

7 X 1 NN oN 32 i i i¼1

ð41Þ

where NNi are the numbers of electrons in each Ni subshell. A comparison of the total x-ray yields for bulk samples (including both the probability of ionization and the ﬂuorescence yield) in terms of photons per steradian per incident quantum for electrons, protons, and x-ray photons is shown in Figure 4.

16

Markowicz

Figure 4 Total x-ray yields for excitation by electrons, protons, and primary x-ray photons as a function of energy of the exciting quantum. (From Birks, 1971a.)

G.

Fine Features of X-ray Emission Spectra (Valence or Chemical Effects)

Because characteristic x-ray emission is a process in which the innermost electrons in the atom are concerned, it is reasonable to suppose that the external, or valence, electrons have little or no eﬀect on the x-ray emission lines. However, this is not fully true for K lines of low-Z elements and L or M lines of higher-Z elements, where the physical state and chemical combination of the elements aﬀect the characteristic x-rays (Clark, 1955). The changes in ﬁne features of x-ray emission spectra with chemical combination can be classiﬁed into three groups: (1) shifting in wavelength (Kallithrakas-Kontos, 1996), (2) distortion of line shape, and (3) intensity changes (Kawai et al., 1993; Rebohle et al., 1996). Wavelength shifts to both longer and shorter wavelengths result from energy-level changes due to electrical shielding or screening of the electrons when the valence electrons are drawn into a bond. Generally, the so-called last or highest-energy member of a given series is most aﬀected by chemical combination; maximum energy shifts are of the order of a few electron volts. Distortion of an x-ray emission line shape gives some indication of the energy distribution of the electrons occupying positions in or near the valence shell. The changes in the characteristic x-ray intensity are a result of alterations in excitation

X-ray Physics

17

probabilities of the electrons undergoing transitions. Certain x-ray lines or bands appear or disappear with chemical combinations. In the case of the K series, the most noticeable chemical eﬀects on x-ray emission are seen in spectra from low-Z elements (4 Z 17). The L series shows as large or even larger changes with chemical combination of the elements than K series. The valence eﬀects in L spectra have been observed for elements of the ﬁrst transition series and others nearby in the periodic table. Because the ﬁne features of x-ray emission spectra may be applied to determine how each element is chemically combined in the sample (speciation), the valence eﬀects found numerous applications in such ﬁelds as physics of solids and surface or near-surface characterization. VI.

INTERACTION OF PHOTONS WITH MATTER

Interactions of photons with matter, by which individual photons are removed or deﬂected from a primary beam of x or g radiation, may be classiﬁed according to the following: The kind of target, such as electrons, atoms or nuclei, with which the photon interacts The type of event, such as absorption, scattering, or pair production, that takes place These interactions are thought to be independent of the origin of the photon (nuclear transition for g-rays versus electronic transition for x-rays); hence, we use the term ‘‘photon’’ to refer to both g- and x-rays here. Possible interactions are summarized in Table 1 (Hubbell, 1969), where t is the total photoelectric absorption cross section per atom (t ¼ tK þ tL þ ) and sR and sC are Rayleigh and Compton collision cross sections, respectively. The probability of each of these many competing independent processes can be expressed as a collision cross section per atom, per electron, or per nucleus in the absorber. The sum of all these cross sections, normalized to a per atom basis, is then the probability stot that the incident photon will have an interaction of some kind while passing through a very thin absorber that contains one atom per square centimeter of area normal to the path of the incident photon: stot ¼ t þ sR þ sC þ

ð42Þ

The total collision cross section per atom stot , when multiplied by the number of atoms per cubic centimeter of absorber, is then the linear attenuation coeﬃcient m per centimeter of travel in the absorber: 1 cm2 g N0 atoms m ¼ stot ð43Þ r cm cm3 g atom A where r is the density of the medium and N0 is Avogadro’s number (6.0225261023 atoms=g atom). The mass attenuation coeﬃcient m (cm2=g) is the ratio of the linear attenuation coeﬃcient and the density of the material. It is worth mentioning that the absorption coeﬃcient is a much more restricted concept than the attenuation coeﬃcient. Attenuation includes the purely elastic process in which the photon is merely deﬂected and does not give up any of its initial energy to the absorber; in this process, only a scattering coeﬃcient is involved. In a photoelectric interaction, the entire energy of the incident photon is absorbed by an atom of the medium.

Nuclear Compton scattering (g, g0 ) Z

Nuclear coherent scattering (g, g) Z2

Coherent resonant scattering (g, g)

Delbru¨ck scattering Z4

Compton scatteringa sC Z

Rayleigh scatteringa sR Z2

Photoelectric effecta Z4 low energy t Z5 high energy Nuclear photoelectric effect: reactions (g, n) (g, p), photofission Z (E 10 MeV) 1. Electron–positron pair production in field of nucleous, Z2 (E 1.02 MeV) 2. Electron–positron pair production in electron field Z (E 2.04 MeV) 3. Nucleon–antinucleon pair production (E 3 GeV) Photomeson production (E 150 MeV)

Major effects of photon attenuation in matter, which are of great importance in practical x-ray spectrometry. Source: From Hubbell, 1969.

a

Interactions with mesons

Interaction with electrical field surrounding charged particles

Interaction with nucleus or bound nucleons

Interaction with atomic electrons

Inelastic (incoherent)

Elastic (coherent)

Scattering Absorption

Classiﬁcation of Photon Interactions

Type of interaction

Table 1

Two-photon Compton scattering Z

Multiphoton eﬀects

18 Markowicz

X-ray Physics

19

In the Compton eﬀect, some energy is absorbed and appears in the medium as the kinetic energy of a Compton recoil electron; the balance of the incident energy is not absorbed and is present as a Compton-scattered photon. Absorption, then, involves the conversion of incident photon energy into the kinetic energy of a charged particle (usually an electron), and scattering involves the deﬂection of incident photon energy. For narrow, parallel, and monochromatic beams, the attenuation of photons in homogeneous matter is described by the exponential law: I ¼ I0 em

t

ð44Þ

where I is the transmitted intensity, I0 is the incident intensity, and t is the absorber thickness in centimeters. If the absorber is a chemical compound or a mixture, its mass attenuation coeﬃcient can be approximately evaluated from the coeﬃcients mi for the constituent elements according to the weighted average: m¼

n X

Wi mi

ð45Þ

i¼1

where Wi is the weight fraction of the ith element and n is the total number of the elements in the absorber. The ‘‘mixture rule’’ [Eq. (45)] ignores changes in the atomic wave function resulting from changes in the molecular, chemical, or crystalline environment of an atom. Above 10 keV, errors from this approximation are expected to be less than a few percent (except in the regions just above absorption edges), but at very low energies (10–100 eV), errors of a factor of 2 can occur (Deslattes, 1969). For situations more complicated than the narrow-beam geometry, the attenuation is still basically exponential, but it is modiﬁed by two additional factors. The ﬁrst of these, sometimes called a geometry factor, depends on the source absorber geometry. The other factor, often called the buildup factor, takes into account secondary photons produced in the absorber, mainly as the result of one or more Compton scatters, which ﬁnally reach the detector. The determination of the buildup factor, deﬁned as the ratio of the observed eﬀect to the eﬀect produced only by the primary radiation, constitutes a large part of g-ray transport theory (Evans, 1963). In subsequent sections, only major eﬀects of photon attenuation are discussed in detail.

A.

Photoelectric Absorption

In the photoelectric absorption described partially in Sec. V.A, a photon disappears and an electron is ejected from an atom. The K shell electrons, which are the most tightly bound, are the most important for this eﬀect in the energy region considered in XRS. If the photon energy drops below the binding energy of a given shell, however, an electron from that shell cannot be ejected. Hence, a plot of t versus photon energy exhibits the characteristic ‘‘absorption edges.’’ The mass photoelectric absorption coeﬃcient tN0 =A at the incident energy E (keV) can approximately be calculated based on Walter’s equations (Compton and Allison, 1935):

20

Markowicz

8 30:3Z3:94 > < AE3

tN0 4:30 ¼ 0:978Z AE3 > A : 0:78Z3:94 AE3

for E > EK for EL1 < E < EK for EM1 < E
ð46Þ

Based on available experimental and theoretical information for approximately 10,000 combinations of Z and E covering 87 elements and the energy range 1 keV to 1 MeV, the following ln–ln polynomials for the photoeﬀect cross section tj have been ﬁtted in incident photon energy between each absorption-edge region (Hubbell et al., 1974): ln tj ¼

1;2;or P3

Aij fln½E ðkeVÞgi

ð47Þ

i¼0

In this polynomial, the total photoeﬀect cross section tj represents one of the following sums: t1 ¼ tM þ tN þ tO þ

or

tN þ tO þ ;

EM1 < E < EL3 or E < EM5 t2 ¼ tL þ tM þ tN þ tO þ ; EL1 < E < EK t3 ¼ tK þ tL þ tM þ tN þ tO þ ;

ð48Þ

E > EK

The values of the ﬁtted coeﬃcients Aij for the ln–ln representation are given in Appendix VII (McMaster et al., 1969). In multiple-edge regions (e.g., between L1 and L3 edge energies), the photoelectric absorption cross sections are also obtained via Eq. (47) by using the following constant ‘‘jump ratios’’ j (t just above an absorption edge divided by t just below that absorption edge): j ¼ 1:16 j ¼ 1:64 ¼ 1:16 1:41 j ¼ 1:1 j ¼ 1:21 ¼ 1:1 1:1 j ¼ 1:45 ¼ 1:1 1:1 1:2 j ¼ 2:18 ¼ 1:1 1:1 1:2 1:5

for for for for for for

EL2 < E < EL1 EL3 < E < EL2 EM2 < E < EM1 EM3 < E < EM2 EM4 < E < EM3 EM5 < E < EM4

ð49Þ

Simple expressions for calculating the values of the energies of all photoabsorption edges are given in Sec. VI.D. The experimental ratio of the total photoelectric absorption cross section t to the K shell component tK can be ﬁtted with an accuracy of 2–3% by the equation (Hubbell, 1969) t ﬃ 1 þ 0:01481 ln2 Z 0:000788 ln3 Z tK

ð50Þ

Based on the tables of McMaster et al. (1969), Poehn et al. (1985) found a useful approximation for the calculation of the jump ratios (called also jump factors) for the K shell ( jK ) and L3 subshell ( jL3 ): jK ¼ 1:754 10 6:608 101 Z þ 1:427 102 Z2 1:1 104 Z3

for 11 Z 50

jL3 ¼ 2:003 10 7:732 101 Z þ 1:159 102 Z2 5:835 105 Z3

for 30 Z 83

ð51Þ

X-ray Physics

21

As already mentioned [Eq. (49)], the values of the jump factors at the L2 and L1 absorption edges are constant for all elements and equal to 1.41 and 1.16, respectively. Tabulated values for the photoelectric absorption cross sections for the elements 1 Z 100 in the energy range of 1 keV to 100 MeV are also available in the work of Storm and Israel (1970), which provides the photon cross sections for all major interaction processes as well as the atomic energy levels, K and L x-ray line energies, weighted average energies for the K and L x-ray series, and relative intensities for K and L x-ray lines. When the apparently sharp x-ray absorption discontinuities are examined at high resolution, they are found to contain a ﬁne structure that extends in some cases to about a few hundred electron volts above the absorption edge. The ﬁne structure very close to an absorption edge (less than or equal to 50 eV above the edge) is generally referred to as the Kossel structure and is designated as XANES (x-ray absorption near-edge structure). Peaks and trenches in this region, which can diﬀer by a factor of 2 or more from the smoothly extrapolated data, can be described in terms of transitions of the (very low energy) ejected electrons to unﬁlled discrete energy states of the atom (or molecule), rather than to the continuum of states beyond a characteristic energy (Sandstro¨m, 1957; Koningsberger and Prins, 1988; Behrens, 1992b). Superimposed on the Kossel structure is the so-called Kronig structure [extended x-ray absorption ﬁne structure (EXAFS)], which usually extends to about 300 eV above the absorption edge (occasionally to nearly 1 keV above an edge). The Kronig structure can be described in terms of interference eﬀects on the de Broglie waves of the ejected electrons by the molecular or crystalline spatial ordering of neighboring atoms (Hasnain, 1991; Behrens, l992a). The oscillations of the absorption coeﬃcient are of the order of 50% in the energy region 50–60 eV above an absorption edge and of the order of 15% in the region beyond 200 eV above the edge. Modulations of the absorption coeﬃcient in the energy region above an absorption edge can be described theoretically in terms of the electronic parameters (Lee and Pendry, 1975). Through a Fourier transform relationship, the modulations are closely related to the radial distribution function around the element of interest (Sayers et al., 1970). Because both the Kossel and the Kronig ﬁne structures can vary in magnitude and in energy displacement of the features, depending on the molecular, crystalline, or thermal environment of the atom, they can be applied for local structural analysis of various materials, including powders, disordered solids, and liquid and amorphous substances (Lagarde, 1983, Behrens, 1992a, 1992b; Koningsberger and Prins, 1988).

B.

Compton Scattering

Compton scattering (Compton, 1923a, 1923b) is the interaction of a photon with a free electron that is considered to be at rest. The weak binding of electrons to atoms may be neglected, provided the momentum transferred to the electron greatly exceeds the momentum of the electron in the bound state. Considering the conservation of momentum and energy leads to the following equations: hn ¼

hn0 1 þ gð1 cos yÞ

gð1 cos yÞ T ¼ hn0 hn ¼ hn0 1 þ gð1 cos yÞ 1 y tan j ¼ cot 1þg 2

ð52Þ ð53Þ ð54Þ

22

Markowicz

with g¼

hn0 m 0 c2

where hn0 and hn are the energies of the incident and scattered photon, respectively, y is the angle between the photon directions of travel before and following a scattering interaction, and T and f are the kinetic energy and scattering angle of the Compton recoil electron, respectively. For f ¼ 180 , Eqs. (52) and (53) reduce to ðhnÞmin ¼ hn0

1 1 þ 2g

ð55Þ

and Tmax ¼ hn0

2g 1 þ 2g

ð56Þ

The diﬀerential Klein–Nishina collision cross section dsKN =dO (deﬁned as the ratio of the number of photons scattered in a particular direction to the number of incident photons) for unpolarized photons striking unbound, randomly oriented electrons is given by (Klein and Nishina, 1929) dsKN r20 hn 2 hn0 hn ¼ þ sin2 y dO 2 hn0 hn hn0

cm2 electron sr

ð57Þ

where sr is an abbreviation for steradian. Substitution of Eq. (52) for Eq. (57) gives the diﬀerential cross section as a function of the scattering angle y: dsKN r20 1 þ cos2 y ¼ dO 2 ½1 þ gð1 cos yÞ2 ( ) g2 ð1 cos yÞ2 1þ ð1 þ cos2 yÞ½1 þ gð1 cos yÞ

cm2 electron sr

ð58Þ

For very small energies hn0 m0 c2 , the expression reduces to the classic Thompson scattering cross section for electromagnetic radiation on an electron: dsTh r20 ¼ ð1 þ cos2 yÞ dO 2

cm2 electron sr

ð59Þ

For low energies of incident photons (approximately less than a few tens of a kiloelectron volt, the angular distribution of Compton-scattered photons is symmetrical about y ¼ 90 ; at higher incident photons energies, the Compton scattering becomes predominantly forward. The diﬀerential Klein–Nishina scattering cross section dsSKN =dO for unpolarized radiation, deﬁned as the ratio of the amount of energy scattered in a particular direction to the energy of incident photons, is given by dssKN hn dsKN ¼ hn0 dO dO

cm2 electron sr

ð60Þ

X-ray Physics

23

The average (or total) collision cross section sKN gives the probability of any Compton interaction by one photon while passing normally through a material containing one electron per square centimeter: Zp

dsKN 2p sin y dy dO 0 ( ) lnð1 þ 2gÞ lnð1 þ 2gÞ 1 þ 3g 2 1 þ g 2ðg þ 1Þ ¼ 2pr0 þ g2 1 þ 2g g 2g ð1 þ 2gÞ2

sKN ¼

cm2 electron ð61Þ

Again, at the low-energy limit, this cross section reduces to the classic Thomson cross section: 8 sTh ¼ pr20 ¼ 0:6652 1024 3

cm2 electron

ð62Þ

At extremely high energies hn0 m0 c2 , Eq. (61) reduces to 1 1 cm2 ln 2g þ sKN ¼ pr20 g 2 electron

ð63Þ

The average (or total) scattering cross section, deﬁned as the total scattered energy in photons of various energies hn, is given by Zp

ssKN

dssKN 2p sin y dy dO 0 " # 2ð1 þ gÞð2g2 2g 1Þ 8g2 2 lnð1 þ 2gÞ ¼ pr0 þ þ g3 3ð1 þ 2gÞ3 g2 ð1 þ 2gÞ2

¼

cm2 electron

ð64Þ

The usual Klein–Nishina theory that assumes that the target electron is free and at rest cannot be directly applicable in some cases. Departures from it occur at low energies because of electron-binding eﬀects and, at high energies, because of the possibility of emission of an additional photon (double Compton eﬀect) and radiative corrections associated with emission and reabsorption of virtual photons; these corrections are discussed in the work of Hubbell (1969). The total incoherent (Compton) collision cross section per atom sC, involving the binding corrections by applying the so-called incoherent scattering function S(x, Z ), can be calculated according to Z1 ( 1 2 ½1 þ gð1 cos yÞ2 sC ¼ r0 2 1 " # ) g2 ð1 cos yÞ2 cm2 2 1 þ cos y þ ð65Þ ZSðx; ZÞ 2pdðcos yÞ 1 þ gð1 cos yÞ atom where x ¼ sinðy=2Þ=l is the momentum transfer parameter and l is the photon wavelength (in angstroms).

24

Markowicz

The values of the incoherent scattering function S(x, Z ) and the incoherent collision cross section sC are given by Hubbell et al. (1975). A useful combination of analytical functions for calculating S(x, Z ) has recently been proposed by Szalo´ki (1996): 8 3 X > < s1 ðx; ZÞ ¼ di ½expðxÞ 1i ; 0 x xi Sðx; ZÞ ¼ i¼1 > : s2 ðx; ZÞ ¼ ½Z s1 ðx; ZÞ t2 g1 ðxÞ þ t2 g2 ðxÞ þ s1 ðx1 ; ZÞ x1 < x ð66Þ where g1 ðxÞ ¼ 1 exp½t1 ðx1 xÞ and g2 ðxÞ ¼ 1 exp½t3 ðx1 xÞ The parameters for the calculation of the S(x, Z ), including the critical value of x1, are given by Szalo´ki (1996) for all elements (Z ¼ 1–100) and any values of x. The average deviation between the calculated [Eq. (66)] and tabulated data (Hubbell et al., 1975) is slightly above 1%. The incoherent collision cross sections sC can also be calculated by using ln–ln polynomials already deﬁned by Eq. (47) (by simply substituting tj with sC and taking i ¼ 3). The values of the ﬁtted coeﬃcients for the ln–ln representation for sC valid in the photon energy range 1 keV to 1 MeV are given in Appendix VIII. To complete this subsection, it is worth mentioning the Compton eﬀect for polarized radiation. The diﬀerential collision cross section ðdsKN =dOÞpp for the plane-polarized radiation scattered by unoriented electrons has also been derived by Klein and Nishina. It represents the probability that a photon, passing through a target containing one electron per square centimeter, will be scattered at an angle y into a solid angle dO in a plane making an angle b with respect to the plane containing the electrical vector of the incident wave: dsKN r20 hn 2 hn0 hn 2 2 þ ¼ 2 sin y cos b dO pp 2 hn0 hn hn0

cm2 electron sr

ð67Þ

The cross section has its maximum value for b ¼ 90 , indicating that the photon and electron tend to be scattered at right angles to the electrical vector of the incident radiation. The scattering of circularly polarized (cp) photons by electrons with spins aligned in the direction of the incident photon is described by 2 dsKN 2 hn ¼ r0 hn0 dO cp hn0 hn hn0 hn cm2 sin2 y þ ð68Þ cos y hn hn0 hn hn0 electron sr The ﬁrst term is the usual Klein–Nishina formula for unpolarized radiation. The þ sign for the additional term applies to right circularly polarized photons.

X-ray Physics

C.

25

Rayleigh Scattering

Rayleigh scattering is a process by which photons are scattered by bound atomic electrons and in which the atom is neither ionized nor excited. The incident photons are scattered with unchanged frequency and with a deﬁnite phase relation between the incoming and scattered waves. The intensity of the radiation scattered by an atom is determined by summing the amplitudes of the radiation coherently scattered by each of the electrons bound in the atom. It should be emphasized that, in Rayleigh scattering, the coherence extends only over the Z electrons of individual atoms. The interference is always constructive, provided the phase change over the diameter of the atom is less than one-half a wavelength; that is, whenever 4p y ra sin <1 l 2

ð69Þ

where ra is the eﬀective radius of the atom. Rayleigh scattering occurs mostly at the low energies and for high-Z materials, in the same region where electron binding eﬀects inﬂuence the Compton scattering cross section. The diﬀerential Rayleigh scattering cross section for unpolarized photons is given by (Pirenne, 1946) dsR 1 2 ¼ r0 ð1 þ cos2 yÞ j Fðx; ZÞj2 2 dO

cm2 atom sr

ð70Þ

where Fðx; ZÞ is the ‘‘atomic form factor,’’ Z1 Fðx; ZÞ ¼

rðrÞ4pr

sin½ð2p=lÞrs dr ð2p=lÞrs

ð71Þ

0

where rðrÞ is the total density, r is the distance from the nucleus, and s ¼ 2 sinðy=2Þ. The atomic form factor has been calculated for Z < 26 using the Hartree electronic distribution (Pirenne, 1946) and for Z > 26 using the Fermi–Thomas distribution (Compton and Allison, 1935). At high photon energies, Rayleigh scattering is conﬁned to small angles; at low energies, particularly for high-Z materials, the angular distribution of the Rayleigh-scattered radiation is much broader. A useful simple criterion for judging the angular spread of Rayleigh scattering is given by (Evans, 1958).

0:0133Z1=3 yR ¼ 2 arcsin E ðMevÞ

ð72Þ

where yR is the opening half-angle of a cone containing at least 75% of the Rayleighscattered photons. In the forward direction, jFðx; ZÞj2 ¼ Z2 , so that Rayleigh scattering becomes appreciable in magnitude and must be accounted for in any g- or x-ray scattering experiments. The total coherent (Rayleigh) scattering cross section per atom sR can be calculated from

26

Markowicz

1 sR ¼ r20 2

Z1 ð1 þ cos2 yÞjFðx; ZÞj2 2p dðcos yÞ 1

3 ¼ sTh 8

ð73Þ

Z1 ð1 þ cos2 yÞjFðx; ZÞj2 dðcos yÞ 1

2

cm atom

The values of the atomic form factor F(x, Z ) and the coherent scattering cross section sR are given in the work of (Hubbell et al., 1975). Recently, Szalo´ki (1996) proposed a useful combination of analytical functions to calculate F(x, Z): 8 0 x x1 f11 ðx; ZÞ ¼ a expðb1 xÞ þ ðZ aÞ expðcxÞ; > > > f ðx; ZÞ ¼ f ðx ; ZÞ expðb ðx xÞ; < x 1 x x2 12 11 1 2 1 ð74Þ F1 ðx; ZÞ ¼ f13 ðx; ZÞ ¼ f12 ðx2 ; ZÞ exp½b3 ðx2 xÞ; x 2 x x3 > h ib4 > > : f14 ðx; ZÞ ¼ f13 ðx3 ; ZÞ x ; x 3 x x4 x3 where 1 Z 7 8 < f21 ðx; ZÞ ¼ a expðb1 xÞ þ ðZ aÞ expðcxÞ; 0 x x1 F2 ðx; ZÞ ¼ f22 ðx; ZÞ ¼ f21 ðx1 ; ZÞ exp½b2 ðx1 xÞ; x 1 x x2 : f23 ðx; ZÞ ¼ f22 ðx2 ; ZÞ exp½b3 ðx2 xÞ; x 2 x x3

ð75Þ

where 8 Z 100 The parameters for the calculation of F(x, Z ), including the critical values of x1, x2, and x3, are given by Szalo´ki (1996) for all elements (Z ¼ 1–100) and the momentum transfer x from 0 to 15 A˚71. The average deviation between the calculated [Eqs. (74) and (75)] and tabulated data (Hubbell et al., 1975) is less than 2%. The simplest method for calculating the coherent scattering cross section sR consists in applying the ln–ln representation [see Eq. (47) with sR instead of tj and i ¼ 3]. The values of the ﬁtted coeﬃcients for ln–ln polynomials for calculating sR in the photon energy range 1 keV to 1 MeV are given in Appendix IX.

D. Total Mass Attenuation Coefficient An extensive review of current tabulations of x-ray attenuation coeﬃcients has been given by Hubbell (1984). Diﬀerences between various compilations of total mass attenuation coeﬃcients result from uncertainties in our knowledge of partial cross sections for the interaction of photons with matter as a function of elemental atomic number Z and photon energy E. Present discrepancies are disturbing, to say the least, frequently amounting to 5–10% in the photon energy region below 10 keV and rising to as much as 30% near an absorption edge. Hubbell (1982) has tabulated mass attenuation coeﬃcients and mass energy absorption coeﬃcients for photon energies from 1 keV to 20 MeV for 40 elements ranging from hydrogen (Z ¼ 1) to uranium (Z ¼ 92) and for 45 mixtures and compounds of dosimetric interest. The uncertainty ranges for the total mass attenuation coeﬃcient values in the tabulation of McMaster et al. (1969) have been estimated by Hubbell et al. (1974). These ranges of uncertainties fall into four categories. Category I (uncertainty below 2%) applies over the energy region 6–40 keV (except near absorption edges) for the following elements: C,

X-ray Physics

27

Mg, Al, Ti, Fe, Ni, Cu, Zn, Zr, Mo, Pd, Ag, Cd, Sn, La, Gd, Ta, W, Pt, Au, Pb, Th, and U. In this category, the photon energy region above 100 keV is also included for all elements in which incoherent scattering comprises more than 90% of the total cross section. Category II (uncertainty of 2–5%) applies to the energy region 2–6 keV for all elements, 6–40 keV for all elements not speciﬁed in category I, and above 40 keV except for the scattering-dominated region speciﬁed in category I. In category III (uncertainty of 5–15%), the authors (Hubbell et al., 1974) included (1) the elements hydrogen, helium, and lithium, (2) the energy region 1–2 keV for elements, and (3) the regions containing K, L, M, and N absorption edges, and the ﬁne-structure regions extending from 200 eV to 1 keV above each of these regions. The experimental uncertainties in cases 1 and 2 greatly exceed 15%. Category IV (uncertainty above 15%) applies to the photon energy region about 200 eV above an absorption edge (Kossel and Kronig ﬁne-structure regions) for all elements. Based on the tables published by McMaster et al. (1969), Wernisch et al. (1984) developed an algorithm for the calculation of the total mass attenuation coeﬃcient valid for the photon energy range from 1 to 50 keV and for 73 elements (11 Z 83). The authors have applied the simple expression m

cm2 ¼ Hedþk g

ð76Þ

ln E

Values of H, d, and k have been obtained from least-squares ﬁts applied to the data published by McMaster et al. (1969); they are given in Figure 5. The values of the edge energies EK, EL1, EL2, EL3, EM1, EM2, EM3, EM4, and EM5 can simply be calculated from (Wernisch et al., 1984) Ei ¼ ri þ si Z þ ti Z2 þ ni Z3

keV

ð77Þ

the parameters ri, si, ti, and ni for various absorption edges i are given in Table 2. Another ﬂexible semiempirical scheme to calculate the total mass attenuation coefﬁcient m for a very wide photon energy range (0.1–1000 keV) has been proposed by Orlic et al. (1993) m ¼ exp½ p1 þ p2 ðln lÞ þ p3 ðln lÞ2 þ p4 ðln lÞ3 þ sKN ZN0 A1

ð78Þ

where sKN is the average incoherent collision cross section gives by Eq. (61). The values of the ﬁtting parameters p1, p2, p3, and p4 are constant for each element and within the energy regions deﬁned by two adjacent absorption edges or for energies beyond the K absorption edge; the parameters are given in Appendix X. The experimental and theoretical values of total interaction cross sections [stot, Eq. (42)] and the mass attenuation coeﬃcients m for the elements (1 Z 98) in the energy range 4.9–24.9 keV have been collected and compared by Arndt et al. (1992). Most of the available tabulations of x-ray attenuation coeﬃcients do not include the photon energy region below 1 keV that corresponds to the energies of characteristic K x-rays of light elements (Z < 11). Experimental data in this energy region are incomplete and it should not be assumed that the accuracy of the available tabulated values is better than 15% (Veigele, 1974; Appendix XI). E.

Diffraction, Refraction, and Dispersion

When a beam of monochromatic x-rays falls onto a crystal lattice, a regular periodic arrangement of atoms, a diﬀracted beam only results in deﬁnite directions. The phenomenon of x-ray diﬀraction at an ordered array of atoms (or molecules) can also be

28

Markowicz

Figure 5 Definition of the energy ranges (a) and values of the parameters H, d, and k (b) applied for the calculation of the total mass attenuation coefficients according to Eq. (76). (From Wernisch et al., 1984.)

X-ray Physics

29

Table 2 Values of the Parameters ri,si,ti, and ni Applied for Calculating the Energies of Absorption Edges via Eq. (77) Applicability range i K L1 L2 L3 M1 M2 M3 M4 M5

ri

si 1

1.304610 4.5066101 6.0186101 3.3906101 8.645 7.499 6.280 4.778 2.421

ti 3

2.633610 1.5666102 1.9646102 4.9316102 3.9776101 3.4596101 2.8316101 2.1846101 1.1726101

ni 3

9.718610 7.5996104 5.9356104 2.3366103 5.9636103 5.2506103 4.1176103 3.3036103 1.8456103

5

4.144610 1.7926105 1.8436105 1.8366106 3.6246105 3.2636105 2.5056105 2.1156105 1.3976105

%a

Zmin

Zmax

3.5 2.2 2.3 1.9 0.4 0.4 0.4 0.4 0.4

11 28 30 30 52 55 55 60 61

63 83 83 83 83 83 83 83 83

a Standard deviation of calculated energies [Eq. (77)] relative to the energies from the tables of McMaster et al. (1969). Source: From Wernisch et al., 1984.

interpreted as a reﬂection of an incident x-ray beam by the interior planes of a crystal (Bragg reﬂection). By elementary calculation of the diﬀerence in path between two coherent rays, W. L. Bragg found the reinforcement condition for reﬂection (known as Bragg’s equation or law): yn nl ¼ 2d sin ð79Þ 2 where n is the order of reﬂection, d is the interplanar spacing, and yn=2 is the angle of reﬂection (or Bragg angle) deﬁned as the angle between the reﬂecting plane of the crystal and the incident or reﬂected beam. The ﬁrst-order reﬂection (n ¼ 1) is normally strongest, and the reﬂected intensity decreases as n increases. Bragg’s law as given in Eq. (79) is only a ﬁrst approximation, as the refraction in the crystal interferes with the angle of reﬂection. Because the refractive index of x-rays is slightly less than unity, the deviations from Bragg’s law, Eq. (79), were not observed in the early years until methods were found for precise measurements of x-ray wavelengths. The refraction is accounted for by ascribing a slightly diﬀerent value dn of the lattice constant to each order of reﬂection; the simple Bragg’s law [Eq. (79)] can thus be written as (Sandstro¨m, 1957) 4d2 d yn nl ¼ 2d 1 2 2 sin ð80Þ n l 2 where d ¼ 1 n0 for small photon absorption (l < 1 A˚) and n0 is the refractive index for x-rays. d is a small positive number of the order 105 for heavy elements and 106 for light elements at l ¼ 1 A˚ and is proportional to l2. Because the value of d is positive, total reﬂection occurs back into air when an x-ray beam meets a surface at a large enough angle of incidence. Provided no absorption occurs, the critical glancing angle ytr (tr, total reﬂection) is deﬁned by (Sandstro¨m, 1957) pﬃﬃﬃﬃﬃ sin ytr ¼ 2d ð81Þ

30

Markowicz

For l ¼ 0.1 nm, the value of ytr is of the order 103 for light elements and 56103 for heavy elements, increasing in proportion to l The refractive index n0 for a medium containing one type of atom can be calculated from (Hirsch, 1962) n0 ¼ 1

Nl2 e2 Fð0Þ 2p m0 c2

ð82Þ

where N is the number of atoms per unit volume and Fð0Þ is the atomic scattering factor at zero scattering angle. Equation (82) shows that n0 depends on the wavelength; this phenomenon is called dispersion. The anomalous dispersion causes the quantity d=l2 , Eq. (80), to vary slightly with the wavelength. The variations become important only in the neighborhood of the absorption edges of the constituents of the crystal. Dividing the quantity d=l2 into one normal part ½ðd=l2 Þn and one anomalous part ½ðd=l2 Þa , the theory of anomalous dispersion leads to an expression of Bragg’s law that can be written as (Sandstro¨m, 1957) 4d2 d 4d2 d yn nl ¼ 2d 1 2 sin ð83Þ 2 2 2 n n 2 l n l a Combined with Bragg’s law in its uncorrected form, Eq. (79), this expression becomes (Sandstro¨m, 1957) 4d2 d yn =2 4d2 d l ¼ 2d 1 2 l ð84Þ sin n n n2 l 2 a l2 n The values of ðd=l2 Þa giving a correction for anomalous dispersion can be determined experimentally. The theory of anomalous dispersion has been applied by Sparks (1974, 1975) to explain the inelastic angular-independent scattering from elements having an absorptionedge energy just above the energy of the incident x-rays. The observed intensity of the inelastically scattered radiation was found to be dependent on the nearness of the energy of an absorption edge to the energy of the incident x-rays. The energy of the inelastic peaks is shifted from the incident energy by the binding energy of the most tightly bound shell from which electrons could be photoejected by the incident radiation. F.

X-ray Raman Scattering

Immediately after the discovery of Raman scattering in the visible-wavelength region, a similar eﬀect concerning x-ray radiation was experimentally examined (Davis and Mitchell, 1928; Krishnan, 1928). X-ray Raman scattering appears as a band spectrum having a short-wavelength edge corresponding to a deﬁnite energy loss equal to the K electron-binding energy EK of the element. This inelastic eﬀect was observed, for example, when CrKa and CuKa radiation was scattered by solids of light elements, such as lithium, beryllium , boron, and by graphite (Suzuki, 1966; Suzuki et al., 1970; Suzuki and Nagasawa, 1975). The shape of the Raman band is similar to that of the soft x-ray K absorption spectrum of the solids. Mizuno and Ohmura (1967) have found the following two conditions for x-ray Raman scattering: 4p a <1 l0

ð85Þ

X-ray Physics

31

and hv0 EK

ð86Þ

where a is the mean radius of charge distribution of the K electrons and l0 is the wavelength of the incident x-rays. The intensity of the x-ray Raman scattering I(y, l) is given by (Suzuki and Nagasawa, 1975) ( 2 2 ) 4p a y 4p a y 2 ð87Þ sin þT2 ðlÞ sin Iðy; lÞ ﬃ ð1 þ cos yÞ T1 ðlÞ l0 2 l0 2 where T1(l) and T2(l) factors are related to the dipole and multipole transitions, respectively. Although according to this equation a slight displacement of the peak position of the Raman band is expected with the scattering angle, the peak position does not shift in most experiments. In general, x-ray Raman scattering gives information about the unoccupied states above the Fermi level of the relevant solids. Moreover, this kind of inelastic scattering by electrons in solids should sometimes be taken into account in x-ray spectrum evaluation.

VII.

INTENSITY OF CHARACTERISTIC X-RAYS

This section provides some of the necessary background information for subsequent chapters dealing with quantitative x-ray analysis. Derivation of any relationship between excitation source intensity and measured characteristic x-rays is sometimes complex and is presented in detail in many relevant books, such as those by Jenkins et al. (1981) and Tertian and Claisse (1982) on XRF analysis and by Goldstein et al. (1981) and Heinrich (1981) on EPXMA. A.

Photon Excitation

When continuous (polychromatic) radiation is used to excite the characteristic x-rays of element i in a completely homogeneous sample of thickness T (cm) and when enhancement eﬀects are neglected, the intensity of the ﬂuorescent radiation Ii (Ei) is described by Ii ðEi Þ dO1 dO2 ¼

dO1 dO2 eðEi Þ 4p sin C1 E Zmax 1 exp½rTðmðE0 Þ csc C1 þ mðEi Þ csc C2 Þ ai ðE0 Þ I0 ðE0 Þ dE0 mðE0 Þ csc C1 þ mðEi Þ csc C2

ð88Þ

Ec;i

with ai ðE0 Þ ¼

Wi t0i ðE0 Þoi pi

1 1 ji

ð89Þ

where dO1 and dO2 are the diﬀerential solid angles for the incident (primary) and emerging (characteristic) radiation, respectively; eðEi Þ is the intrinsic detector eﬃciency for recording a photon of energy Ei ; Ec;i and Emax are the critical absorption energy of element i and the maximum energy in the excitation spectrum; r is the density of the specimen (in g=cm3); C1 and C2 are the eﬀective incidence and takeoﬀ angles, respectively; mðE0 Þ and mðEi Þ are the total mass attenuation coeﬃcients (in cm2=g) for the whole specimen [Eq. (45)] at

32

Markowicz

energies E0 and Ei , respectively; I0 ðE0 Þ dE0 is the number of incident photons per second per steradian in the energy interval E0 to E0 þ dE0 ; Wi is the weight fraction of the ith element; and t0i ðE0 Þ is the total photoelectric mass absorption coeﬃcient for the ith element at the energy E0 (in cm2=g). Because large solid angles O1 and O2 are used for the excitation and characteristic x-rays in practical spectrometers, Eq. (88) should also be integrated over these ﬁnite solid angles. Such calculations can often be omitted; however, and for a given measurement geometry, an experimentally determined geometry factor G can be applied. As seen from Eq. (88), the intensity of characteristic x-rays is modiﬁed by the eﬀects of primary [mðE0 Þ] and secondary [mðEi Þ] absorption in the specimen; this is a major source of the so-called matrix eﬀects in XRF analysis. If the excitation source is monochromatic (emits only one energy), Eq. (88) simpliﬁes to eðEi Þai ðE0 ÞI0 ðE0 Þ sin C1 1 exp½rTðmðE0 Þ csc C1 þ mðEi Þ csc C2 Þ mðE0 Þ csc C1 þ mðEi Þ csc C2

Ii ðEi Þ ¼ G

ð90Þ

The enhancement eﬀect, consisting of an extra excitation of the element of interest by the characteristic radiation of some matrix elements, modiﬁes the equations for the intensity Ii ðEi Þ. In the case of monochromatic photon excitation, a factor, 1 þ Hi , should be included in Eq. (90), where Hi is the enhancement term deﬁned as Hi ¼

m X 1 1 Wk o k 1 m ðEk Þmk ðE0 Þ 2mi ðE0 Þ k¼1 jk i lnð1 þ mðE0 Þ=½mðEk Þ sin C1 Þ lnð1 þ mðEi Þ=½mðEk Þ sin C2 Þ þ mðE0 Þ= sin C1 mðEi Þ= sin C2

ð91Þ

where mi ðE0 Þ and mi ðEk Þ are the total mass attenuation coeﬃcients for the ith element at the energy of incident radiation ðE0 Þ and characteristic radiation of the element k ðEk Þ, respectively, mk ðE0 Þ is the total mass attenuation coeﬃcient for the element k at the energy E0, and mðEk Þ is the total mass attenuation coeﬃcient for the whole specimen at the energy Ek . 1. Thin-SampleTechnique If the total mass per unit area of a given sample m ¼ rT is small, Eq. (90) simpliﬁes to Iithin ðEi Þ ¼

G eðEi Þai ðE0 ÞI0 ðE0 Þm sin C1

ð92Þ

The relative error resulting from applying Eq. (92) instead of Eq. (90) does not exceed 5% when the total mass per unit area satisﬁes the condition m thin

0:1 mðE0 Þ csc C1 þ mðEi Þ csc C2

ð93Þ

A major feature of the thin-sample technique is that the intensity of the characteristic x-rays, Iithin , depends linearly on the concentration of the ith element (or on its mass per unit area); it is equivalent to the fact that, in the thin-sample technique, matrix eﬀects can safely be neglected.

X-ray Physics

33

2. Thick-SampleTechnique By the term ‘‘thick sample,’’ we mean a sample whose mass per unit area (thickness) is greater than the so-called saturation mass (thickness). The saturation thickness is deﬁned as a limiting value above which practically no further increase in the intensity of the characteristic radiation is observed as the sample thickness is increased. If the total mass per unit area of a given sample is suﬃciently large, Eq. (90) simpliﬁes to Iithick ðEi Þ ¼

GeðEi Þai ðE0 ÞI0 ðE0 Þ mðE0 Þ þ sin C1 =ðsin C2 ÞmðEi Þ

ð94Þ

The relative error resulting from applying Eq. (94) instead of Eq. (90) does not exceed 1% when the total mass per unit area satisﬁes the condition m thick

B.

4:61 mðE0 Þ csc C1 þ mðEi Þ csc C2

ð95Þ

Electron Excitation

For a thin foil (deﬁned as a foil of thickness such that the beam electron undergoes only one scattering act), the intensity of the characteristic x-rays Iitf (in photons per second) is given by Iitf ¼ Ki I0 Qi oi N0

1 rTWi Ai

ð96Þ

where Ki is a factor depending on the measurement geometry and detection eﬃciency and the other symbols have the following dimensions: I0 ¼ electrons=s, Qi ¼ ionizations per electron for 1 atom=cm2, oi ¼ photons=ionization. N0 ¼ atoms=mol, 1=Ai ¼ mol=g, r ¼ g=cm3, and T ¼ cm. For a bulk target, Eq. (96) must be integrated over the electron path, taking into account the loss of energy by the electron beam:

Iibulk

Wi ¼ I0 oi Ki N0 Ai

ZEc;i

Qi ðEÞ dE dE=dðrxÞ

ð97Þ

E0

where dE=dðrxÞ is the electron energy loss per unit of distance traveled in a material, given by the Bethe equation (Heinrich, 1981; Bethe and Ashkin, 1953): n dE 1 X Wj Z j E ¼ 78; 500 ln 1:166 dðrxÞ E j¼1 Aj Jj

keV=g cm2

ð98Þ

eVÞ where Jj is the mean ionization potential of element jðJj ¼ 9:76Zj þ 58:5Z0:19 j (Goldstein et al., 1981). Strictly speaking, the Bethe equation is only valid for the electron energy E > 6:3Jj . Below this limit, the energy loss of the electrons should be described either by the modiﬁcation of Rao-Sahib and Wittry (1972) [Eq. (99)] or by the modiﬁcation of Love et al. (l978) [Eq. (l00)]:

34

Markowicz n dE 1 X Wj Z j pﬃﬃﬃﬃ keV=g cm2 ¼ 0:6236 105 pﬃﬃﬃﬃ dðrxÞ A E j¼1 j Jj n X dE Wj Z j 1 ¼ 1=2 5 dðrxÞ A J j j 1:18 10 ðE=Jj Þ þ 1:47 106 E=Jj j¼1

ð99Þ keV=g cm2

ð100Þ

However, to describe the intensity of characteristic x-rays emitted, three eﬀects should be considered additionally (Heinrich, 1981; Goldstein et al., 1981; Love and Scott, 1981): 1. 2. 3.

C.

Absorption of characteristic x-rays within the specimen Electron backscattering Secondary ﬂuorescence by characteristic x-rays and=or bremsstrahlung continuum produced by an electron beam

Particle Excitation

For a thin, uniform, homogeneous target, the intensity of characteristic x-rays Itti is given by the simple formula I tti ¼ Ki I0 N0 ss;i ðE0 Þoi

1 mi Ai

ð101Þ

where mi is the areal density of the element with atomic number Zi and atomic mass Ai . For a thick, homogeneous target, the intensity of characteristic x-rays, Iithick , from the element i of concentration Wi can be calculated by (Campbell and Cookson, 1984) Iithick ¼

0 X Ki oi N0 ss;i ðEÞTi ðEÞ dE I 0 Wi SðEÞ Ai E

ð102Þ

0

where S(E) is the stopping power and Ti ðEÞ is the photon attenuation factor. The latter is given by 0 1 ZE sin C1 dE C B Ti ðEÞ ¼ exp@mðEi Þ ð103Þ A SðEÞ sin C2 E0

More details on various x-ray analytical techniques are provided in subsequent chapters.

VIII.

IUPAC NOTATION FOR X-RAY SPECTROSCOPY

The nomenclature commonly used in XRS to describe x-ray emission spectra was introduced by M. Sieghahn in the l920s and is based on the relative intensity of lines from diﬀerent series. A new and more systematic notation for x-ray emission lines and absorption edges, based on the energy-level designation, was developed by the International Union of Pure and Applied Chemistry (Jenkins et al. 1991). Because the new notation, called the IUPAC notation, replaces the Siegbahn’s notation, some characteristic features of the new nomenclature must be mentioned. The IUPAC notation prescribes Arabic numerals for subscripts; the original notation uses Roman numerals (e.g., L2 and L3 ,

X-ray Physics

35

instead of LII and LIII ). In the IUPAC notation, states with double or multiple vacancies should be denoted by, for example, K2 ; KL1 , and Ln2;3 , which correspond to the electron conﬁgurations 1s2 ; 1s1 2s1 , and 2pn , respectively. X-ray transitions and x-ray emission diagram lines are denoted by the initial (placed ﬁrst) and ﬁnal x-ray levels separated by a hyphen. To conform with the IUPAC notation of x-ray spectra, the hyphen separating the initial and ﬁnal state levels should also be introduced into the notation for Auger electron emission process. The IUPAC notation is compared with the Sieghahn notation in Appendix XII.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu

Atomic number Element

918 504 226.953 106.9 64.6 43.767 31.052 23.367 18.05 14.19 11.48 9.512 7.951 6.745 5.787 5.018 4.397 3.871 3.437 3.070 2.757 2.497 2.269 2.070 1.896 1.743 1.608 1.488 1.380

A˚

K edge

0.014 0.025 0.055 0.116 0.192 0.283 0.399 0.531 0.687 0.874 1.08 1.303 1.559 1.837 2.142 2.470 2.819 3.202 3.606 4.037 4.495 4.963 5.462 5.987 6.535 7.109 7.707 8.329 8.978

keV

258 225 197 143 105 81.0 64.2 52.1 43.2 36.4 30.7 26.8 23.4 20.5 18.3 16.3 14.6 13.3 12.22 11.27

A˚ keV

0.048 0.055 0.063 0.087 0.118 0.153 0.193 0.238 0.287 0.341 0.399 0.462 0.530 0.604 0.679 0.762 0.849 0.929 1.015 1.100

L1 edge

564 365 248 170 125 96.1 75.6 61.1 50.2 41.8 35.2 30.2 27.0 23.9 21.3 19.1 17.2 15.6 14.2 13.0

A˚ keV

0.022 0.034 0.050 0.073 0.099 0.129 0.164 0.203 0.247 0.297 0.352 0.411 0.460 0.519 0.583 0.650 0.721 0.794 0.871 0.953

L2 edge

564 365 253 172 127 96.9 76.1 61.4 50.6 42.2 35.5 30.8 27.3 24.2 21.6 19.4 17.5 15.9 14.5 13.3

A˚ keV

0.022 0.034 0.049 0.072 0.098 0.128 0.163 0.202 0.245 0.294 0.349 0.402 0.454 0.512 0.574 0.639 0.708 0.779 0.853 0.933

L3 edge A˚ keV

M4 edge

APPENDIX I: CRITICAL ABSORPTION WAVELENGTHS AND CRITICAL ABSORPTION ENERGIES

A˚

keV

M5 edge

36 Markowicz

30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63

Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs Ba La Ce Pr Nd Pm Sm Eu

1.283 1.196 1.117 1.045 0.980 0.920 0.866 0.816 0.770 0.728 0.689 0.653 0.620 0.589 0.561 0.534 0.509 0.486 0.464 0.444 0.425 0.407 0.390 0.374 0.359 0.345 0.331 0.318 0.307 0.295 0.285 0.274 0.265 0.256

9.657 10.365 11.100 11.860 12.649 13.471 14.319 15.197 16.101 17.032 17.993 18.981 19.996 21.045 22.112 23.217 24.341 25.509 26.704 27.920 29.182 30.477 31.800 33.155 34.570 35.949 37.399 38.920 40.438 41.986 43.559 45.207 46.833 48.501

10.33 9.54 8.73 8.107 7.506 6.97 6.46 5.998 5.583 5.232 4.867 4.581 4.298 4.060 3.83 3.626 3.428 3.254 3.085 2.926 2.777 2.639 2.511 2.389 2.274 2.167 2.068 1.973 1.889 1.811 1.735 1.665 1.599 1.536

1.200 1.30 1.42 1.529 1.651 1.78 1.92 2.066 2.220 2.369 2.546 2.705 2.883 3.054 3.24 3.418 3.616 3.809 4.018 4.236 4.463 4.695 4.937 5.188 5.451 5.719 5.994 6.282 6.559 6.844 7.142 7.448 7.752 8.066

11.87 10.93 9.94 9.124 8.416 7.80 7.21 6.643 6.172 5.755 5.378 5.026 4.718 4.436 4.180 3.942 3.724 3.514 3.326 3.147 2.982 2.830 2.687 2.553 2.429 2.314 2.204 2.103 2.011 1.924 1.843 1.767 1.703 1.626

1.045 1.134 1.248 1.358 1.473 1.59 1.72 1.865 2.008 2.153 2.304 2.467 2.627 2.795 2.965 3.144 3.328 3.527 3.726 3.938 4.156 4.380 4.611 4.855 5.102 5.356 5.622 5.893 6.163 6.441 6.725 7.018 7.279 7.621

12.13 11.10 10.19 9.39 8.67 8.00 7.43 6.89 6.387 5.962 5.583 5.223 4.913 4.632 4.369 4.130 3.908 3.698 3.504 3.324 3.156 3.000 2.855 2.719 2.592 2.474 2.363 2.258 2.164 2.077 1.995 1.918 1.845 1.775

1.022 1.117 1.217 1.32 1.43 1.55 1.67 1.80 1.940 2.079 2.220 2.373 2.523 2.677 2.837 3.001 3.171 3.351 3.537 3.728 3.927 4.131 4.340 4.557 4.780 5.010 5.245 5.488 5.727 5.967 6.213 6.466 6.719 6.981 0.7967

0.9448 0.9951 1.0983 1.1575

15.56

13.122 12.459 11.288 10.711

11.552 11.013

13.394 23.737

15.89

1.0732 1.1258

0.9257 0.9734

0.7801

X-ray Physics 37

64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92

Atomic number

Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Fr Ra Ac Th Pa U

106.759 109.741 112.581 115.610

0.116 0.113 0.110 0.108

keV

50.215 51.984 53.773 55.599 57.465 59.319 61.282 63.281 65.292 67.379 69.479 71.590 73.856 76.096 78.352 80.768 83.046 85.646 88.037 90.420 93.112 95.740 98.418 101.147

A˚

K edge

0.247 0.238 0.231 0.223 0.216 0.209 0.202 0.196 0.190 0.184 0.178 0.173 0.168 0.163 0.158 0.153 0.149 0.415 0.141 0.137 0.133 0.130 0.126 0.123

Continued

Element

Appendix I

1.477 1.421 1.365 1.317 1.268 1.222 1.182 1.140 1.100 1.061 1.025 0.990 0.956 0.923 0.893 0.863 0.835 0.808 0.782 0.757 0.732 0.709 0.687 0.665 0.645 0.625 0.606 0.588 0.56

A˚ 8.391 8.722 9.081 9.408 9.773 10.141 10.487 10.870 11.271 11.681 12.097 12.524 12.968 13.427 13.875 14.354 14.837 15.338 15.858 16.376 16.935 17.490 18.058 18.638 19.229 19.842 20.458 21.102 21.764

keV

L1 edge

1.561 1.501 1.438 1.390 1.338 1.288 1.243 1.199 1.155 1.114 1.075 1.037 1.001 0.967 0.934 0.903 0.872 0.843 0.815 0.789 0.763 0.739 0.715 0.693 0.671 0.650 0.630 0.611 0.592

A˚ 7.938 8.256 8.619 8.918 9.260 9.626 9.972 10.341 10.732 11.128 11.533 11.953 12.380 12.817 13.266 13.731 14.210 14.695 15.205 15.713 16.244 16.784 17.337 17.904 18.478 19.078 19.677 20.311 20.938

keV

L2 edge

1.710 1.649 1.579 1.535 1.482 1.433 1.386 1.341 1.297 1.255 1.216 1.177 1.140 1.106 1.072 1.040 1.008 0.979 0.950 0.923 0.897 0.872 0.848 0.825 0.803 0.782 0.761 0.741 0.722

A˚ 7.250 7.517 7.848 8.072 8.361 8.650 8.941 9.239 9.554 9.874 10.196 10.529 10.867 11.209 11.556 11.917 12.3 12.655 13.041 13.422 13.817 14.215 14.618 15.028 15.439 15.865 16.293 16.731 17.160

keV

L3 edge

3.485 3.608 3.720

1.804 1.880 1.958 2.042 2.126 2.217 2.307 2.404 2.504 2.606 2.711

6.87 6.59 6.33 6.073 5.83 5.59 5.374 5.157 4.952 4.757 4.572

3.557 3.436 3.333

1.4415

keV

8.601

A˚

M4 edge

3.729 3.618 3.497

7.11 6.83 6.560 6.30 6.05 5.81 5.584 5.36 5.153 4.955 4.764

8.847 8.487

A˚

3.325 3.436 3.545

1.743 1.814 1.890 1.967 2.048 2.133 2.220 2.313 2.406 2.502 2.603

1.4013 1.4609

keV

M5 edge

38 Markowicz

Np Pu Am Cm Bk Cf Es Fm

0.105 0.102 0.099 0.097 0.094 0.092 0.090 0.088

118.619 121.720 124.816 128.088 131.357 134.683 138.067 141.510

Source: From Clark, 1963 and Burr, 1974.

93 94 95 96 97 98 99 100

0.553 0.537 0.521 0.506 0.491 0.477 0.464 0.451

22.417 23.097 23.793 24.503 25.230 25.971 26.729 27.503

0.574 0.557 0.540 0.525 0.509 0.494 0.480 0.466

21.596 22.262 22.944 23.640 24.352 25.080 25.824 26.584

0.704 0.686 0.669 0.653 0.637 0.622 0.607 0.593

17.614 18.066 18.525 18.990 19.461 19.938 20.422 20.912

X-ray Physics 39

40

Markowicz

APPENDIX II: CHARACTERISTIC X-RAY WAVELENGTHS (A) AND ENERGIES (keV) Table 1

K Series Diagram Lines (A˚)a

Line

a1,2

a1

a2

Approximate intensity

150

100

50

Li B B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

230 113 67 44 31.603 23.707 18.307 14.615 11.909 9.889 8.339 7.126 6.155 5.373 4.729 4.192 3.744 3.360 3.032 2.750 2.505 2.291 2.103 1.937 1.791 1.659 1.542 1.437 1.341 1.256 1.177 1.106 1.041 0.981 0.927 0.877 0.831 0.788 0.748 0.710 0.676 0.644 0.614 0.587

8.338 7.125 6.154 5.372 4.728 4.191 3.742 3.359 3.031 2.749 2.503 2.290 2.102 1.936 1.789 1.658 1.540 1.435 1.340 1.255 1.175 1.105 1.040 0.980 0.926 0.875 0.829 0.786 0.747 0.709 0.675 0.643 0.613 0.585

8.341 7.127 6.157 5.375 4.731 4.194 3.745 3.362 3.034 2.753 2.507 2.294 2.105 1.940 1.793 1.661 1.544 1.439 1.344 1.258 1.179 1.109 1.044 0.984 0.930 0.880 0.833 0.791 0.751 0.713 0.679 0.647 0.617 0.590

b1

b3 15

14.460 11.574 9.559 7.960

b2

b4

5

<1

11.726 9.667 8.059 6.778

5.804 5.032 4.403 3.886 3.454 3.089 2.780 2.514 2.285 2.085 1.910 1.757 1.621 1.500 1.392 1.296 1.207 1.129 1.057 0.992 0.933 0.879 0.829 0.783 0.740 0.701 0.665 0.632

1.393 1.208 1.129 1.058 0.993 0.933 0.879 0.830 0.784 0.741 0.702 0.666 0.633 0.601

0.572 0.546 0.521

0.573 0.546 0.521

1.489 1.381 1.284 1.196 1.117 1.045 0.980 0.921 0.866 0.817 0.771 0.728 0.690 0.654 0.621 0.590 0.562 0.535 0.510

0.866 0.815 0.770 0.727 0.689 0.653 0.620 0.561 0.534

X-ray Physics Table 1

41

Continued

Line

a1,2

a1

a2

Approximate intensity

150

100

50

0.559 0.535 0.512 0.491 0.470 0.451 0.433 0.416 0.401 0.385 0.371 0.357 0.344 0.332 0.321 0.309 0.299 0.289 0.279 0.270 0.261 0.253 0.244 0.236 0.229 0.222 0.215 0.209 0.202 0.196 0.191 0.158 0.180 0.175 0.170 0.165 0.161 0.156 0.152 0.148 0.144 0.144 0.140 0.133 0.131 0.126

0.564 0.539 0.517 0.495 0.475 0.456 0.438 0.421 0.405 0.390 0.376 0.362 0.349 0.337 0.325 0.314 0.304 0.294 0.284 0.275 0.266 0.258 0.250 0.241 0.234 0.227 0.220 0.213 0.207 0.201 0.196 0.190 0.185 0.180 0.175 0.170 0.165 0.161 0.157 0.153 0.149 0.149 0.145 0.138 0.136 0.131

Ag Cs In Sn Sb Te I Xe Cs Ba La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Fr Ra Ac Th Pa U

47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92

0.561 0.536 0.514 0.492 0.472 0.453 0.435 0.418 0.402 0.387 0.373 0.359 0.346 0.334 0.322 0.311 0.301 0.291 0.281 0.272 0.263 0.255 0.246 0.238 0.231 0.224 0.217 0.211 0.204 0.198 0.193 0.187 0.182 0.177 0.172 0.167 0.162 0.185

0.135 0.128

a Conversion equation: E (keV) ¼ 12.4=l (A˚). Source: From Clark, 1963.

b1

b3

b2

b4

5

<1

0.498 0.476 0.455 0.436 0.418 0.401 0.385 0.360 0.355 0.342 0.329 0.317 0.305 0.294

0.487 0.465 0.445 0.426 0.408 0.391 0.376

0.486

0.274 0.265 0.256 0.246 0.238 0.231 0.223 0.216 0.208 0.203 0.196 0.191 0.185 0.179 0.174 0.169 0.164 0.160 0.155 0.151 0.147 0.143

0.267 0.258 0.249 0.239 0.231

15 0.497 0.475 0.455 0.435 0.417 0.400 0.384 0.369 0.355 0.341 0.328 0.316 0.305 0.294

0.346 0.333 0.320 0.309 0.297 0.287

0.444 0.425 0.407

0.319 0.307

0.283 0.274 0.264 0.255 0.246 0.237 0.230 0.222 0.215 0.208 0.202 0.195 0.190 0.184 0.179 0.173 0.168 0.163 0.159 0.154 0.150 0.146 0.142 0.138 0.134 0.131 0.127 0.127 0.124 0.117 0.115 0.111

0.135 0.132 0.128 0.128 0.125 0.118 0.116 0.112

0.217 0.203 0.197 0.190 0.185 0.179 0.174 0.169 0.164 0.159 0.155 0.150 0.146 0.147 0.138 0.133

0.184 0.179 0.174 0.168 0.163 0.159 0.154 0.150 0.146 0.141 0.138

0.114

0.114

0.108

0.108

42 Table 2

Markowicz L Series Diagram Lines (A˚)

Line Approximate intensity Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

a

a1

a2

b1

b2

b3

b4

b5

b6

b7

b9

110

100

10

50

20

6

4

1

<1

<1

<1

36.393 31.393 27.445 24.309 21.713 19.489 17.602 16.000 14.595 13.357 12.282 11.313 10.456 9.671 8.990 8.375

36.022 31.072 27.074 23.898 21.323 19.158 17.290 15.698 14.308 13.079 12.009 11.045 10.194 9.414 8.735 8.126

7.318 6.863 6.449 6.070 5.725 5.406

7.325 6.870 6.456 6.077 5.732 5.414

7.075 6.623 6.211 5.836 5.492 5.176

4.846 4.597 4.368 4.154 3.956 3.752 3.600 3.439 3.290 3.148

4.854 4.605 4.376 4.162 3.965 3.781 3.609 3.448 3.299 3.157

2.892

2.902

19.429 17.757 15.742 14.269 13.167 12.115 11.225

8.930

5.586 5.238 4.923

6.788 6.367 5.983 5.632 5.310 5.013

6.821 6.403 6.018 5.668 5.346 5.048

6.984 6.519 6.094 5.710 5.361 5.048

4.620 4.374 4.146 3.935 3.739 3.555 3.385 3.226 3.077 2.937

4.372 4.130 3.909 3.703 3.514 3.339 3.175 3.023 2.882 2.751

4.487 4.253 4.034 3.834 3.644 3.470 3.306 3.152 3.009 2.874

4.532 4.289 4.071 3.870 3.681 3.507 3.344 3.190 3.046 2.912

4.487 4.242 4.016 3.808 3.614 3.436 3.270 3.115 2.971 2.837

3.155 3.005 2.863 2.730

3.792 3.605 3.430 3.268 3.115 2.973 2.839 2.713

2.683

2.511

2.628

2.666

2.593

2.485

2.478

X-ray Physics

43

b10

b15

b17

g1

g2

g3

g4

g5

g6

g8

‘

Z

s

t

<1

<1

<1

10

1

2

<1

<1

<1

<1

3

1

<1

<1

410 260 180

5.384 5.036 4.726

3.799 3.611 3.437 3.274 3.121 2.979 2.847 2.720

4.182 3.944 3.725 3.523 3.336 3.162 3.001 2.852 2.712 2.582

2.492

2.348

67.84 56.212 47.835 41.042 36.671 31.423 27.826 24.840 22.315 20.201 18.358 16.693 15.297 14.081 12.976 11.944 11.069 10.293 9.583

67.25 56.813 47.325 40.542 35.200 30.942 27.375 24.339 21.864 19.73 17.86 16.304 14.940 13.719 12.620 11.608 10.732 9.959 9.253

6.754 6.297 5.875 5.497 5.151 4.387

8.363 7.836 7.356 6.918 6.517 6.150

8.042 7.517 7.040 6.606 6.210 5.847

2.026 2.778 2.639 2.511 2.391

4.288 4.045 3.822 3.616 3.426 3.249 3.085 2.932 2.790 2.657

5.503 5.217 4.952 4.707 4.480 4.269 4.071 3.888 3.716 3.557

5.204 4.922 4.660 4.418 4.193 3.983 3.789 3.607 3.438 3.280

2.174

2.417

3.267

2.994

6.045 5.644 5.283 4.953 4.654 4.380 3.897 3.685 3.489 3.307 3.137 2.980 2.835 2.695 2.567 2.447 2.237

2.233

44 Table 2

Markowicz Continued

Line Approximate intensity Ba La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Fr Ra Ac Th Pa U Np Pu Am

56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95

a

a1

a2

b1

b2

b3

b4

b5

b6

b7

b9

110

100

10

50

20

6

4

1

<1

<1

<1

2.776 2.665 2.561 2.463 2.370 2.283 2.199 2.120 2.046 1.976 1.909 1.845 1.785 1.726 1.672 1.619 1.569 1.522 1.476 1.433 1.391 1.352 1.313 1.277 1.242 1.207 1.175 1.144 1.114

2.785 2.674 2.570 2.473 2.382

2.404 2.303 2.208 2.119 2.035

2.516 2.410 2.311 2.216 2.126

2.555 2.449 2.349 2.255 2.166

2.482 2.379 2.282 2.190 2.103

2.382 2.275 2.180 2.091 2.009

2.376 2.282 2.188 2.100 2.016

2.210 2.131 2.057 1.986 1.920 1.856 1.796 1.738 1.682 1.630 1.580 1.533 1.487 1.444 1.402 1.363 1.325 1.288 1.253 1.218 1.186 1.155 1.126

2.567 2.458 2.356 2.259 2.166 2.081 1.998 1.920 1.847 1.777 1.710 1.647 1.587 1.530 1.476 1.424 1.374 1.327 1.282 1.238 1.197 1.158 1.120 1.083 1.049 1.015 0.982 0.952 0.921

1.882 1.812 1.746 1.682 1.623 1.567 1.514 1.463 1.416 1.370 1.327 1.285 1.245 1.206 1.169 1.135 1.102 1.070 1.040 1.010 0.983 0.955 0.929

1.962 1.887 1.815 1.747 1.681 1.619 1.561 1.505 1.452 1.402 1.353 1.307 1.263 1.220 1.179 1.141 1.104 1.068 1.034 1.001 0.969 0.939 0.908

2.000 1.926 1.853 1.785 1.720 1.658 1.601 1.544 1.491 1.441 1.392 1.346 1.302 1.260 1.218 1.179 1.142 1.106 1.072 1.039 1.007 0.977 0.948

1.779

1.856 1.788 1.723 1.659 1.599

1.862 1.792

1.494

1.485

1.387 1.342 1.298 1.256 1.215 1.177 1.140 1.106 1.072 1.040 1.010 0.981 0.953 0.926 0.900

1.946 1.875 1.807 1.742 1.681 1.622 1.567 1.515 1.466 1.419 1.374 1.331 1.290 1.252 1.213 1.179 1.143 1.111 1.080 1.050 1.021 0.993 0.967

1.395 1.350 1.306 1.264 1.224 1.186 1.149 1.115 1.082 1.050 1.019 0.990 0.962 0.935

1.384 1.336 1.291 1.247 1.204 1.165 1.126 1.090 1.054 1.021 0.986 0.957 0.927 0.898

1.030 1.005

1.017

0.840 0.814

0.858 0.836

0.803

0.841

0.807

0.871

0.817

0.769

0.956 0.933 0.911 0.890 0.868 0.849

0.968 0.945 0.923 0.901 0.880 0.860

0.766 0.742 0.720 0.698 0.678 0.658

0.794 0.774 0.755 0.735 0.719 0.701

0.755 0.732 0.710

0.793 0.770 0.748

0.765 0.746 0.726

0.828 0.803 0.789

0.775 0.755 0.736

0.723 0.701 0.681

0.669

0.707

0.691

Source: From Clark, 1963.

1.577

X-ray Physics

45

b10

b15

b17

g1

g2

g3

g4

g5

g6

g8

‘

Z

s

t

<1

<1

<1

10

1

2

<1

<1

<1

<1

3

1

<1

<1

2.387 2.290 2.195 2.107 2.023

2.442 2.141 2.048 1.961 1.878

2.138 2.046 1.960 1.879 1.801

2.134 2.041 1.955 1.874 1.797

2.075 1.983 1.899 1.819 1.745

1.870 1.800 1.731 1.667

1.726 1.657 1.592 1.530 1.473 1.417 1.364 1.316 1.268 1.222 1.179 1.138 1.098 1.061 1.025 0.991 0.958 0.927 0.897 0.868 0.840 0.814 0.786

1.659 1.597 1.534 1.477 1.423 1.371 1.321 1.274 1.228 1.185 1.144 1.105 1.068 1.032 0.998 0.966 0.934 0.905 0.876 0.848 0.822 0.796 0.771

1.655 1.591 1.529 1.471 1.417 1.364 1.315 1.268 1.222 1.179 1.138 1.099 1.062 1.026 0.992 0.959 0.928 0.898 0.869 0.842 0.815 0.790 0.764

1.606 1.544 1.485 1.427 1.374 1.323 1.276

1.494 1.392 1.343 1.299 1.254 1.212 1.172 1.133 1.097 1.062 1.028 0.996 0.964 0.934 0.905

1.372 1.328 1.437 1.287 1.247 1.339 1.208 1.293 1.171 1.137 1.166 1.072 1.128 1.041 1.090 1.012 1.056 0.984 1.022 0.957 0.989 0.931

1.185 1.143 1.103 1.065 1.028 0.993 0.959 0.928 0.897 0.867 0.839 0.812 0.761

2.309 2.222 3.135 2.862 2.205 3.006 2.740 2.110 2.023 2.892 2.620 2.020 1.936 2.784 2.512 1.935 1.855 2.675 2.409

1.708

1.518 1.462 1.406 1.355 1.307 1.260 1.215 1.173 1.132 1.094 1.057 1.022 0.988 0.956 0.925 0.895 0.867 0.840

1.243 1.198 1.155 1.114 1.074 1.037 1.001 0.967 0.934 0.903 0.873 0.845 0.817 0.791 0.765

2.482 1.632 2.395 2.312 2.234 2.158 2.086 2.019 1.955 1.250 1.894 1.204 1.836 1.161 1.782 1.120 1.728 1.081 1.678 1.044 1.630 1.008 1.585 0.974 1.541 0.941 1.499 0.010 1.460 0.880 1.422 0.852 1.385 0.824 1.350 0.799 1.317 1.283

2.218 2.049 1.898 1.826 1.757 1.695 1.635 1.478 1.523 1.471 1.421 1.374 1.328 1.285 1.243 1.202 1.164 1.127 1.092 1.058

1.831 1.776 1.663 1.723 1.612 1.672

1.352 1.414 1.279 1.342 1.244 1.308 1.210

0.716 0.776 0.838 0.844 0.694 0.682 0.675 0.649 0.717 0.673 0.680 1.167 0.908 0.730 0.708 0.687

0.653 0.642 0.635 0.611 0.675 0.632 0.640 1.115 0.855 1.011 1.080 0.634 0.624 0.617 0.594 0.655 0.613 1.091 0.830 0.615 0.605 0.598 0.577 0.635 0.595 0.601 1.067 0.806 0.904 1.035 0.597 0.579 0.562

46 Table 3

Markowicz M Series Diagram Lines (A˚)

Line K Cu Ru Rh Pd Ag Cd In Sn Sb Te Ba La Ce Pr Nd Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Th Pa U

a1 19 29 44 45 46 47 48 49 50 51 52 56 57 58 59 60 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 90 91 92

a2

b

g

l 680 170

26.85 25.00 21.80 20.46

14.88 14.06

14.51 13.78

12.675 Band Band Band Band Band Band Band 8.460

Band Band 10.744 10.253 9.792 9.364 8.965 8.593 8.246 7.909 7.600 7.304 7.022 6.756 6.504 6.267 6.037 5.828 5.623 5.452 5.250 5.075 4.909 3.942 3.827 3.715

8.139

8.155 7.840

7.539 7.251 6.983 6.528

7.546 7.258 6.990 6.490

6.261 6.046 5.840

6.275 6.057 5.854 5.666

5.461 5.285 5.118 4.138 4.022 3.910

Source: From Clark, 1963.

5.472 5.299 5.129 4.151 4.035 3.924

17.94 16.92 15.93 12.700 12.064 11.534 10.997 10.504 9.599 9.211 8.844 8.485 8.144 7.865 7.545

18.38

14.22 13.57 12.98 12.43 11.86 11.37

7.023 6.761 6.543 6.312 6.088 5.887 5.681 5.501 5.320 5.145

10.48 10.07 9.69 9.32 8.96 8.63

4.825 4.674 4.531 3.679 3.577 3.480

6.97 6.74 6.52 5.24 5.08 4.95

8.02 7.74 7.47

X-ray Physics Table 4 Atomic number 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

47

Energies of Principal K and L X-ray Emission Lines (keV) Element Li Be B C N C F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn

Kb2

8.328 8.976 9.657 10.365 11.100 11.863 12.651 13.465 14.313 15.184 16.083 17.011 17.969 18.951 19.964 21.012 22.072 23.169 24.297 25.454 26.641 27.859 29.106

Kb1

1.067 1.297 1.553 1.832 2.136 2.464 2.815 3.192 3.589 4.012 4.460 4.931 5.427 5.946 6.490 7.057 7.647 8.264 8.904 9.571 10.263 10.981 11.725 12.495 13.290 14.112 14.960 15.834 16.736 17.666 18.621 19.607 20.585 21.655 22.721 23.816 24.942 26.093 27.274 28.483

Ka1

Ka2

0.052 0.110 0.185 0.282 0.392 0.523 0.677 0.851 1.041 1.254 1.487 1.486 1.40 1.739 2.015 2.014 2.309 2.306 2.622 2.621 2.957 2.955 3.313 3.310 3.691 3.688 4.090 4.085 4.510 4.504 4.952 4.944 5.414 5.405 5.898 5.887 6.403 6.390 6.930 6.915 7.477 7.460 8.047 8.027 8.638 8.615 9.251 9.234 9.885 9.854 10.543 10.507 11.221 11.181 11.923 11.877 12.648 12.597 13.394 13.335 14.164 14.097 14.957 14.882 15.774 15.690 16.614 16.520 17.478 17.373 18.410 18.328 19.278 19.149 20.214 20.072 21.175 21.018 22.162 21.988 23.172 22.982 24.207 24.000 25.270 25.042

Lg1

2.302 2.462 2.623 2.792 2.964 3.144 3.328 3.519 3.716 3.920 4.131

Lb2

Lb1

2.219 2.367 2.518 2.674 2.836 3.001 3.172 3.348 3.528 3.713 3.904

0.344 0.399 0.458 0.519 0.581 0.647 0.717 0.790 0.866 0.943 1.032 1.122 1.216 1.517 1.419 1.526 1.638 1.752 1.872 1.996 2.124 2.257 2.395 2.538 2.683 2.834 2.990 3.151 3.316 3.487 3.662

La1

La2

0.341 0.395 0.492 0.510 0.571 0.636 0.704 0.775 0.849 0.928 1.009 1.096 1.166 1.282 1.379 1.480 1.587 1.694 1.692 1.806 1.805 1.922 1.920 2.042 2.040 2.166 2.163 2.293 2.290 2.424 2.420 2.558 2.554 2.696 2.692 2.838 2.833 2.994 2.978 3.133 3.127 3.287 3.279 3.444 3.436

48

Markowicz

Table 4

Continued

Atomic number

Element

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

Sb To I Xe Cs Ba La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Fr Ra Ac Th Pa U Np Pu Am Cm Bk Cf Es Fm

Kb2

Kb1

Ka1

Ka2

Lg1

Lb2

Lb1

La1

La2

30.387 31.698 33.016 34.446 35.819 37.255 38.728 40.231 41.772 43.298 44.955 46.553 48.241 49.961 51.737 53.491 55.292 57.088 58.969 60.959 62.946 64.936 66.999 69.090 71.220 73.393 75.605 77.866 80.165 82.526 84.904 87.343 89.833 92.386 94.976 97.616 100.305 103.048 105.838 108.671 111.575 114.549 117.533 120.592 123.706 126.875 130.101 133.383 136.724 140.122

29.723 30.993 32.292 33.644 34.984 35.376 37.799 39.255 40.746 42.269 43.945 45.400 47.027 48.718 50.391 52.178 53.934 55.690 57.576 59.352 61.282 63.209 65.210 67.233 69.298 71.404 73.549 75.736 77.968 80.258 82.558 94.922 87.335 89.809 92.319 94.877 97.483 100.136 102.846 105.592 108.408 111.289 114.181 117.146 120.163 123.235 126.362 129.544 132.781 136.075

26.357 27.471 28.610 29.802 30.970 32.191 33.440 34.717 36.023 37.359 38.649 40.124 41.529 42.983 44.470 45.985 47.528 49.099 50.730 52.360 54.063 55.757 57.524 59.310 61.131 62.991 64.886 66.820 68.794 70.821 72.860 74.957 77.097 79.296 81.525 83.800 86.119 88.485 90.894 93.334 95.851 98.428 101.005 103.653 106.351 109.098 111.896 114.745 118.646 120.598

26.109 27.200 28.315 29.485 30.623 31.815 33.033 34.276 35.548 36.845 38.160 39.523 40.877 42.280 43.737 45.193 46.686 48.205 49.762 51.326 52.959 54.579 56.270 57.973 59.707 61.477 63.278 65.111 66.980 68.894 70.320 72.794 74.805 76.868 78.956 81.080 83.243 85.446 87.681 89.942 92.271 94.648 97.023 99.457 101.932 104.448 107.023 109.603 112.244 114.926

4.347 4.570 4.800 5.036 5.280 5.531 5.789 6.052 6.322 6.602 6.891 7.180 7.478 7.788 8.104 8.418 8.748 9.089 9.424 9.779 10.142 10.514 10.892 11.283 11.684 12.094 12.509 12.939 13.379 13.828 14.288 14.762 15.244 15.740 16.248 16.768 17.301 17.845 l8.405 18.977 19.559 20.163 20.774 21.401 22.042 22.699 23.370 24.056 24.758 25.475

4.100 4.301 4.507 4.720 4.936 5.156 5.384 5.613 5.850 6.090 6.336 6.587 6.842 7.102 7.368 7.638 7.912 8.188 8.472 8.758 9.048 9.346 9.649 9.959 10.273 10.596 10.918 11.249 11.582 11.923 12.268 12.620 12.977 13.338 13.705 14.077 14.459 14.839 l5.227 15.620 16.022 16.425 16.837 17.254 17.667 18.106 18.540 18.980 19.426 19.879

3.543 4.029 4.220 4.422 4.620 4.828 5.043 5.262 5.489 5.722 5.956 6.206 6.456 6.714 6.979 7.249 7.528 7.810 8.103 8.401 8.709 9.021 9.341 9.670 10.008 10.354 10.706 11.069 11.439 11.823 12.210 12.611 13.021 13.441 13.873 14.316 14.770 15.233 15.712 16.200 16.700 17.218 17.740 18.278 18.829 19.393 19.971 20.562 21.l66 21.785

3.605 3.769 3.937 4.111 4.286 4.467 4.651 4.840 5.034 5.230 5.431 5.636 5.846 6.039 6.275 6.495 6.720 6.948 7.181 7.414 7.654 7.898 8.145 8.396 8.651 8.910 9.173 9.441 9.711 9.987 10.266 10.549 10.836 11.128 11.424 11.724 12.029 12.338 12.650 12.966 13.291 13.613 13.945 14.279 14.618 14.961 15.309 15.661 16.018 16.379

3.595 3.758 3.926 4.098 4.272 4.451 4.635 4.823 5.014 5.208 5.408 5.609 5.816 6.027 6.241 6.457 6.680 6.904 7.135 7.367 7.604 7.843 8.087 8.333 8.584 8.840 9.098 9.360 9.625 9.896 10.170 10.448 10.729 11.014 11.304 11.597 11.894 12.194 12.499 12.808 13.120 13.438 12.758 14.082 14.411 14.743 15.079 15.420 15.764 16.113

X-ray Physics

49

APPENDIX III: RADIATIVE TRANSITION PROBABILITIES Radiative Transition Probabilities for K X-ray Lines

20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100

Ca Ti Cr Fe Ni Zn Ge Sc Kr Sr Zr Mo Ru Pd Cd Sn Te Xe Ba Ce Nd Sm Gd Dy Er Yb Hf W Os Pt Hg Pb Po Rn Ra Th U Pu Cm Cf Em

Ka2 =Ka1 0.505 0.505 0.506 0.506 0.507 0.509 0.511 0.513 0.515 0.518 0.520 0.523 0.526 0.528 0.531 0.533 0.536 0.537 0.542 0.545 0.549 0.551 0.556 0.560 0.565 0.568 0.572 0.576 0.580 0.585 0.590 0.595 0.600 0.605 0.612 0.619 0.624 0.631 0.638 0.646 0.652

e e

Kb1 ¼ KM2 þ KM3 þ KM4;5 : Kb2 ¼ KN2;3 þ KO2;3 : c Kb ¼ Kb1 þ Kb2 : d Ka ¼ Ka1 þ Ka2 : Source: From West, 1982–83. e

b

e

a

e

Atomic number Element

Kb3 =Kb1

ðKb1 þ Kb3 Þ=Ka1

Kb1 a =Ka1

0.516 0.517 0.518 0.519 0.521 0.522 0.523 0.525 0.526 0.527 0.529 0.531 0.532 0.534 0.535 0.537 0.539 0.541 0.542 0.544 0.546 0.548 0.550 0.552 0.554 0.556

0.116 0.137 0.155 0.172 0.189 0.202 0.215 0.225 0.235 0.244 0.251 0.258 0.264 0.270 0.275 0.280 0.285 0.290 0.294 0.298 0.303 0.307 0.310 0.314 0.317 0.320 0.324 0.326 0.330 0.333 0.336 0.339 0.342 0.345 0.348 0.351 0.354 0.356 0.359 0.362 0.364

0.116 0.137 0.156 0.171 0.187 0.202 0.215 0.225 0.235 0.244 0.252 0.259 0.265 0.271 0.277 0.282 0.287 0.292 0.297 0.301 0.306 0.311 0.314 0.318 0.322 0.325 0.329 0.332 0.336 0.339 0.343 0.346 0.350 0.353 0.356 0.360 0.363 0.366 0.370 0.374 0.377

e

Table 1

Kb2 b =Ka1

Kbc =Kad

0.006 0.013 0.022 0.034 0.043 0.048 0.051 0.054 0.056 0.060 0.064 0.670 0.076 0.082 0.085 0.088 0.089 0.090 0.090 0.091 0.092 0.094 0.097 0.100 0.103 0.106 0.110 0.113 0.118 0.123 0.125 0.130 0.134 0.138

0.069 0.095 0.114 0.128 0.133 0.137 0.142 0.153 0.164 0.175 0.185 0.193 0.201 0.209 0.216 0.222 0.226 0.232 0.240 0.244 0.247 0.250 0.253 0.256 0.259 0.261 0.263 0.267 0.270 0.274 0.277 0.282 0.285 0.288 0.291 0.295 0.299 0.301 0.305 0.309 0.312

50 Table 2 Atomic number 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94

Markowicz Radiative Transition Probabilities for L1 X-ray Lines Normalized to Lb3 ¼ 100 Element

Lb3

Lb4

Lg3

Lg2

Kr Sr Zr Mo Ru Pd Cd Sn Te Xe Ba Ce Nd Sm Gd Dy Er Yb Hf W Os Pt Hg Pb Po Rn Ra Th U Pu

100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100

— — — 71.0 64.9 61.9 61.0 61.4 62.6 64.1 65.5 66.7 67.5 69.7 68.0 67.8 67.6 67.5 67.6 68.3 69.6 71.7 74.8 78.8 83.9 89.8 95.0 102.0 110.0 120.0

10.8 14.7 18.1 21.0 23.4 25.5 27.2 28.5 29.5 30.3 30.8 31.2 31.4 31.5 31.5 31.4 31.4 31.4 31.4 31.6 31.9 32.3 33.0 34.0 35.2 36.8 38.7 41.0 43.8 47.5

— — — — — — — — — — — — — — — — — 18.5 20.0 21.5 23.2 25.0 26.9 28.8 30.9 33.1 35.3 37.7 40.2 44.0

X-ray Physics Table 3 Atomic number 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94

51

L2 X-ray Lines Normalized to Lb1 ¼ 100 Element

Lb1

LZ

Lg1

Lg6

Zn Ge Se Kr Sr Zr Mo Ru Pd Cd Sn Te Xe Ba Ce Nd Sn Gd Dy Er Yb Hf W Os Pt Hg Pb Po Rn Ra Th U Pu

100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100

12.3 10.4 8.75 7.40 6.25 5.25 4.35 3.63 3.0 2.6 2.35 2.25 2.2 2.16 2.12 2.10 2.10 2.10 2.10 2.10 2.12 2.13 2.16 2.20 2.23 2.28 2.33 2.40 2.45 2.50 2.60 2.80 2.30

— — — — — 0.91 6.71 10.6 13.1 14.5 15.3 15.4 15.6 15.6 15.7 15.8 15.9 16.1 16.5 17.0 17.6 18.4 19.2 20.1 20.9 21.7 22.3 22.8 23.2 23.4 23.7 24.0 24.2

— — — — — — — — — — — — — — — — — — — — — — 0.375 1.73 2.42 2.98 3.45 3.88 4.29 4.74 5.25 5.88 6.65

52 Table 4

Markowicz L3 X-ray Lines Normalized to La1 ¼ 100

Atomic number 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94

Element Ti Cr Fe Ni Zn Ge Se Kr Sr Zr Mo Ru Pd Cd Sn Te Xe Ba Ce Nd Sm Gd Dy Er Yb Hf W Os Pt Hg Pb Po Rn Ra Th U Pu

Source: From West, 1982–83.

La1

Lb2,15

La2

Lb5

Lb6

Ll

100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100

— — — — — — — — — 2.43 6.40 9.55 12.1 13.9 15.4 16.4 17.2 17.8 18.2 18.5 18.8 19.2 19.6 20.0 20.5 21.2 21.9 22.7 23.5 24.4 25.3 26.2 26.8 27.3 27.5 27.5 27.0

— — — — — — — — — — 12.5 12.2 12.1 11.9 11.7 11.5 11.3 11.2 11.1 11.1 11.0 11.1 11.1 11.2 11.2 11.3 11.3 11.4 11.4 11.5 11.5 11.4 11.4 11.3 11.1 11.0 10.5

— — — — — — — — — — — — — — — — — — — — — — — — — — 0.242 0.873 1.74 2.62 3.24 3.85 4.28 4.69 4.94 5.20 5.40

— — — — — — — — — — — — — — — — — — — — — — — — — 1.15 1.28 1.38 1.46 1.55 1.59 1.65 1.70 1.75 1.80 1.85 1.89

40.37 26.13 15.35 10.29 7.56 5.96 4.98 4.36 3.98 3.75 3.65 3.58 3.55 3.56 5.59 3.62 3.67 3.73 3.79 3.86 3.92 3.99 4.07 4.15 4.23 4.32 4.42 4.53 4.65 4.78 4.93 5.09 5.27 5.46 5.69 5.93 6.18

X-ray Physics

53

APPENDIX IV: NATURAL WIDTHS OF K AND L LEVELS AND Ka X-RAY LINES (FWHM), IN eV Level Element

X-ray line

K

L1

L2

L3

Ka1

Ka2

(0.000) (0.000) 0.00l 0.004 0.015 0.032 0.054 0.083 0.126 0.152

(0.000) (0.000) 0.001 0.004 0.014 0.033 0.054 0.087 0.128 0.156

0.24 0.30 0.36 0.43 0.49 0.57 0.65 0.72 0.81 0.89

0.24 0.30 0.36 0.43 0.49 0.56 0.64 0.72 0.80 0.89

10 11 12 13 14 15 16 17 18 19

Ne Na Mg Al Si P S Cl Ar K

0.24 0.30 0.36 0.42 0.48 0.53 0.59 0.64 0.68 0.74

( < 0.1) (0.2) 0.41 0.73 1.03 1.26 1.49 1.58 1.63 1.92

20 21 22 23 24 25 26 27 28 29

Ca Sc Ti V Cr Mn Fe Co Ni Cu

0.81 0.86 0.94 1.01 1.08 1.16 1.25 1.33 1.44 1.55

2.07 2.21 2.34 2.41 2.54 2.62 2.76 2.79 2.89 3.06

0.17 0.19 0.24 0.26 0.29 0.34 0.37 0.43 0.52 0.62

0.17 0.19 0.22 0.24 0.27 0.32 0.36 0.43 0.48 0.56

0.98 1.05 1.16 1.26 1.35 1.48 1.61 1.76 1.94 2.11

0.98 1.06 1.18 1.28 1.37 1.50 1.62 1.76 1.96 2.17

30 31 32 33 34 35 36 37 38 39

Zn Ga Ge As Se Br Kr Rb Sr Y

1.67 1.82 1.96 2.14 2.33 2.52 2.75 2.99 3.25 3.52

3.28 3.38 3.53 3.79 3.94 4.11 4.28 4.44 4.67 4.71

0.72 0.83 0.95 1.03 1.13 1.21 1.31 1.43 1.54 1.65

0.65 0.76 0.82 0.94 1.00 1.08 1.17 1.27 1.39 1.50

2.32 2.59 2.78 3.08 3.33 3.60 3.92 4.26 4.63 5.02

2.39 2.66 2.92 3.17 3.46 3.73 4.06 4.92 4.79 5.18

40 41 42 43 44 45 46 47 48 49

Zr Nb Mo Tc Ru Rh Pd Ag Cd In

3.84 4.14 4.52 4.91 5.33 5.77 6.24 6.75 7.28 7.91

4.78 3.94 4.25 4.36 4.58 4.73 4.93 4.88 4.87 5.00

1.78 1.87 1.97 2.08 2.23 2.35 2.43 2.57 2.62 2.72

1.57 1.66 1.78 1.91 2.00 2.13 2.25 2.40 2.50 2.65

5.40 5.80 6.31 6.82 7.33 7.90 8.49 9.16 9.79 10.56

5.62 6.01 6.49 6.99 7.56 8.12 8.67 9.32 9.91 10.63

50 51 52 53

Sn Sb Te I

8.49 9.16 9.89 10.6

2.97 3.13 3.32 3.46

2.84 3.00 3.12 3.25

2.75 2.87 2.95 3.08

11.2 12.0 12.8 13.7

11.3 12.2 13.0 13.8

54 Appendix IV

Markowicz Continued Level

Element

X-ray line

K

L1

L2

L3

Ka1

Ka2

54 55 56 57 58 59

Xe Cs Ba La Ce Pr

11.4 12.3 13.2 14.1 15.1 16.2

3.64 3.78 3.92 4.06 4.21 4.34

3.40 3.51 3.57 3.68 3.80 3.89

3.13 3.25 3.32 3.41 3.48 3.60

14.6 15.6 16.5 17.6 I8.6 19.8

14.8 15.8 16.8 17.8 18.9 20.1

60 61 62 63 64 65 66 67 68 69

Nd Pm Sm Eu Gd Tb Dy Ho Er Tm

17.3 18.5 19.7 21.0 22.3 23.8 25.2 26.8 28.4 30.1

4.52 4.67 4.80 4.91 5.05 5.19 5.25 5.33 5.43 5.47

3.97 4.06 4.15 4.23 4.32 4.43 4.55 4.66 4.73 4.79

3.65 3.75 3.86 3.91 4.01 4.12 4.17 4.26 4.35 4.48

20.9 22.2 23.6 24.9 26.4 27.9 29.4 31.1 32.7 34.6

21.3 22.5 23.8 25.2 26.7 28.2 29.8 31.5 33.1 34.9

70 71 72 73 74 75 76 77 78 79

Yb Lu Hf Ta W Re Os Ir Pt Au

31.9 33.7 35.7 37.7 39.9 42.1 44.4 46.8 49.3 52.0

5.53 5.54 5.63 5.58 5.61 6.18 7.25 8.30 9.39 10.5

4.82 4.92 5.02 5.15 5.33 5.48 5.59 5.69 5.86 6.00

4.60 4.68 4.80 4.88 4.98 5.04 5.16 5.25 5.31 5.41

36.5 38.4 40.5 42.6 44.9 47.2 49.6 52.1 54.6 57.4

36.7 38.7 40.7 42.9 45.2 47.6 50.0 52.5 55.2 58.0

80 81 82 83 84 85 86 87 88 89

Hg Tl Pb Bi Po At Rn Fr Ra Ac

54.6 57.4 60.4 63.4 66.6 69.8 73.3 76.8 80.4 84.1

11.3 12.0 12.2 12.4 12.6 12.8 13.1 13.3 13.4 13.6

6.17 6.32 6.48 6.67 6.83 7.01 7.20 7.47 7.68 7.95

5.50 5.65 5.81 5.98 6.13 6.29 6.41 6.65 6.82 6.98

60.1 63.1 66.2 69.4 72.7 76.1 79.7 83.4 87.2 91.1

60.8 63.8 66.8 70.1 73.4 76.8 80.5 84.2 88.1 92.0

90 91 92 93 94 95 96 97 98 99

Th Pa U Np Pu Am Cm Bk Cf Es

88.0 91.9 96.1 100 105 109 114 119 124 129

13.7 14.3 14.0 14.0 13.5 13.3 13.6 13.8 14.0 14.3

8.18 8.75 9.32 9.91 10.5 10.9 11.4 11.8 12.2 12.7

7.13 7.33 7.43 7.59 7.82 8.04 8.26 8.55 8.75 9.04

95.2 99.3 103.5 108 113 117 122 127 132 138

96.2 100.7 105.4 110 115 120 125 130 136 141

X-ray Physics APPENDIX IV

55 Continued Level

Element 100 101 102 103 104 105 106 107 108 109 110

Fm Md No Lr

X-ray line

K

L1

L2

L3

Ka1

Ka2

135 140 145 150 156 162 168 174 181 187

14.4 14.8 15.1 15.9 16.2 16.5 16.8 17.0 17.2 17.6

13.1 13.6 14.0 14.4 14.9 15.5 15.9 16.4 17.0 17.5

9.33 9.61 9.90 10.1 10.5 10.8 11.2 11.6 12.0 12.3

144 150 155 161 167 173 179 185 193 200

148 153 159 165 171 177 184 190 198 205

193

18.1

18.1

12.7

206

211

Source: From Krause and Oliver, 1979.

56

Markowicz

APPENDIX V: WAVELENGTHS OF K SATELLITE LINES (A) Atomic number 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

Element Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag

aII

aI

a3

11.860 11.830 9.845 9.820 8.302 8.283 7.100 7.095 7.082 6.140 6.131 6.117 5.370 5.354 5.341 4.725 3.739 3.356 3.029 2.747 2.502

Source: From Clark, 1963.

3.728 3.347 3.021 2.740 2.496 2.282 2.095 1.930 1.784 1.653 1.536 1.431 1.138 1.250 1.173

a3I

a3II

a4

11.810

5.339

9.804 8.267 7.073 6.109 5.344 5.334

a4I

a5

a6

a7

b0

11.742 11.711 9.749 9.724 8.226 8.206 7.069 7.035 7.017 7.026 6.075 6.063

bI

6.816 6.838 4.415

3.721 3.340 3.015 2.733 2.491 2.279 2.093 1.928 1.782 1.651 1.535 1.430 1.335 1.250 1.172 1.101 1.036

3.718 3.338 3.013 2.731 2.489 2.277 2.091 1.926 1.780 1.650 1.533 1.428 1.334 1.248 1.170 1.100 1.035

3.724 3.343 3.018 2.737 2.493 2.280 2.094 1.929 1.783 1.652

3.717 3.337 3.013 2.731 2.490 2.278 2.091 1.927 1.781 1.650 1.534 1.429 1.335 1.249 1.171 1.101 1.036

0.8727

0.8646

0.8721

0.7836

0.7828 0.7432 0.7065

0.7832 0.7436 0.7070

3.496 3.133 2.819 2.551 2.320 2.118 1.783 1.645 1.522

1.061 0.9958 0.9367 0.8832 0.8835 0.7881 0.7457 0.7056 0.6706 0.6369 0.5766 0.5493

0.5833

0.5837 0.5007

3.101 2.789 2.522 2.291 2.090 1.914 1.760 1.623 1.502 1.394 1.296 1.209

X-ray Physics

b1x

bx

5.800 5.028

6.753 5.792 5.023

57

bV

bII

b IIIII

5.712 4.400 3.882 3.449 3.087

bIII

bIV

b5

b6

b7

b8

b9

b10

1.116 1.044 0.9786 0.9194

1.064 0.9996 0.9403

1.050 0.9854 0.9629

1.054 0.9889 0.9297

1.047 0.9228

0.8155 0.7699 0.7218 0.6890 0.6531

0.8365 0.7911 0.7492 0.7105 0.6741

0.8234 0.7778 0.7362 0.6973 0.6619 0.6288

0.8259 0.7802 0.7382 0.6993 0.6636 0.6303

0.8193 0.7736 0.7315 0.6924 0.6568 0.6235

0.6256

0.6519

0.5693 0.5427

0.5708 0.5439 0.5189 0.4954

0.5639 0.5374

0.5399

0.5607 0.5452

0.4902

0.4920

5.691

4.395 3.441 3.082 2.772 2.506 2.277 2.078

3.412 3.054 2.749 2.489 2.262 2.066 1.895 1.742 1.607 1.487 1.380 1.282

1.749 1.614 1.494 1.391 1.295 1.207 1.128 1.042 0.9770 0.9177 1.8745 0.8135 0.7681 0.7261 0.6870 0.6510

0.5589

3.404 3.048 2.744

0.5808

0.4943

58

Markowicz

APPENDIX VI: FLUORESCENCE YIELDS AND COSTER^KRONIG TRANSITION PROBABILITIES Table 1

K Shell Fluorescence Yield oK

Atomic number 6 7 8 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

Element C N O Ne Na Mg Al Si P S Cl Ar K Sc Ca Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru

oK

Atomic number

Element

oK

0.0009 0.0015 0.0022 0.0100 0.020 0.030 0.040 0.055 0.070 0.090 0.105 0.125 0.140 0.165 0.190 0.220 0.240 0.26 0.285 0.32 0.345 0.375 0.41 0.435 0.47 0.50 0.53 0.565 0.60 0.635 0.665 0.685 0.71 0.72 0.755 0.77 0.785 0.80

45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 82 92

Rh Pd Ag Cd In Sn Sb Te I Xe Cs Ba La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb In Hf Ta W Re Os Ir Pd Au Hg Pb U

0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.875 0.88 0.89 0.895 0.90 0.905 0.91 0.915 0.92 0.925 0.93 0.93 0.935 0.94 0.94 0.945 0.945 0.95 0.95 0.95 0.955 0.955 0.96 0.96 0.96 0.96 0.965 0.965 0.965 0.97 0.97

Source: From Birks, 1971b and Fink, 1974.

X-ray Physics Table 2

59

Experimental L Subshell Fluorescence Yields oi

Atomic number

Element

o1

o2

54 56 65 67

Xe Ba Tb Ho

68

Er

70

Yb

71

Lu

72

Hf

73

Ta

74

W

75 76 77

Re Os Ir

78

Pt

79

Au

80

Hg

81

Tl

0.07 0.02

0.319 0.010 0.373 0.025

82

Pb

0.07 0.02 0.09 0.02

0.363 0.015

83

Bi

0.12 0.01 0.095 0.005

0.32 0.04 0.38 0.02

0.06 0.18

0.165 0.018 0.170 0.055 0.185 0.060 0.188 0.011

0.25 0.02 0.257 0.013

0.331 0.021

0.39 0.03 0.319 0.010

o3 0.10 0.01 0.05 0.01 0.188 0.016 0.22 0.03 0.169 0.030 0.21 0.03 0.172 0.032 0.20 0.02 0.183 0.011 0.22 0.03 0.251 0.035 0.22 0.03 0.228 0.025 0.27 0.01 0.25 0.03 0.191 0.228 0.013 0.254 0.025 0.207 0.272 0.037 0.284 0.043 0.290 0.030 0.244 0.262 0.036 0.262 0.31 0.04 0.317 0.029 0.291 0.018 0.276 0.31 0.04 0.317 0.025 0.40 0.02 0.32 0.05 0.367 0.050 0.300 0.010 0.37 0.07 0.386 0.053 0.306 0.010 0.330 0.021 0.337 0.315 0.013 0.32 0.35 0.05 0.354 0.028 0.367 0.36 0.37 0.05 0.362 0.029 0.40 0.05

60 Table 2

Markowicz Continued

Atomic number

Element

90

Th

91 92

Pa U

96

Cm

o1

0.28 0.06

o2

o3

0.552 0.032 0.55 0.02

0.340 0.018 0.42 0.517 0.042 0.46 0.05 0.44 0.500 0.040 0.515 0.034 0.63 0.02

f13

f23

Source: From Bambynek et al., 1972.

Table 3

Measured L Shell Coster–Kronig Yields

Atomic number

Element

f12 0.66 0.07 0.41 0.36

56 65 67 68 70 73

Ba Tb Ho Er Yb Ta

74 75 77 78 79

W Re Ir Pt Au

80

Hg

81

Tl

0.17 0.05 0.14 0.03

82

Pb

83

Bi

0.15 0.04 0.17 0.05 0.19 0.05

0.76 0.10 0.57 0.10 0.56 0.07 0.56 0.05 0.57 0.03 0.61 0.08 0.58 0.05

0.18 0.02

0.58 0.02

92 93

U Np

0.10 0.04

0.55 0.09

94

Pu

96

Cm

0.038 0.022

0.68 0.04

Source: From Bambynek et al., 1972.

< 0.14

0.25 0.13

0.43 0.28

0.19 < 0.36 0.27 0.03 0.30 0.04 0.46 0.06 0.50 0.05 0.51 0.13 0.61 0.07

0.74 0.04

0.066 0.014 0.205 0.034 0.255 0.025 0.142 0.009 0.148 0.010 0.20 0.04

0.22 0.22 0.04 0.08 0.02 0.188 0.010 0.25 0.13 0.169 0.010 0.159 0.013 0.164 0.016 0.156 0.010 þ0.14 0.06 7 0.06 0.164 0.23 0.12 þ0.05 0.02 70.02 0.22 0.08 0.24 0.08 0.188 0.019

X-ray Physics Table 4

61

Theoretical L Subshell Fluorescence Yields oi and Coster–Kronig Yields fij a

Atomic number

Element

o1

13 14 15 16 17 18 19 20 22 24 26

Al Si P S Cl Ar K Ca Ti Cr Fe

3.05-6 9.77-6 2.12-5 3.63-5 5.60-5 8.58-5 1.15-4 1.56-4 2.80-4 2.97-4 3.84-4

28 29 30 32 33 34 35 36

Ni Cu Zn Ge As Se Br Kr

4.63-4

37 38 40

Rb Sr Zr

42

Mo

44 47

Ru Ag

50

Sn

51 54 56 60

Sb Xe Ba Nd

65 67

Tb Ho

70 74

Yb W

79 80 83 85 90 93

Au Hg Bi At Th Np

a

5.23-4 7.70-4 1.40-3 1.30-3 1.85-3 2.19-3 1.32-2 3.00-3 3.97-3 3.96-3 5.75-3 6.34-3 7.74-3 1.02-2 1.01-2 1.30-2 1.30-2 3.11-2 5.84-2 4.46-2 7.46-2 6.00-2 0.112 0.094 0.112 0.115 0.138 0.105 0.098 0.120 0.129 0.197

o2

o3 2.40-3 1.08-3 4.1-4 2.9-4 2.3-4 1.9-4 2.1-4 2.1-4 1.18-3 3.29-3 5.59-3 1.49-3 8.02-3 3.83-3 1.08-2 1.44-2 9.74-3 1.78-2

f12

f13

0.629 0.636 0.652

0.325

0.622

0.322 0.266 0.282 0.302

0.624 0.671 0.547 0.616

2.36-2 1.23-2

0.230 0.225

0.686 0.585

2.24-2 2.94-2 1.89-2 3.50-2 2.45-2 4.18-2 5.47-2 4.30-2 6.56-2 5.67-2 6.16-2 9.12-2 9.07-2 0.133 0.120 0.166 0.203

2.43-2 2.95-2 2.01-2 3.73-2 2.59-2 4.50-2 6.02-2 4.49-2 7.37-2

0.249 0.236 0.271 0.166 0.048 0.057 0.052 0.064 0.052 0.072 0.164 0.179 0.168 0.207 0.165

0.646 0.648 0.522 0.689 0.692 0.779 0.786 0.695 0.784 0.693 0.316 0.274 0.336 0.303 0.332

0.287 0.271 0.357 0.352 0.417 0.422 0.529 0.460

0.268 0.253 0.327 0.321 0.389 0.380 0.461 0.472

0.202 0.178 0.180 0.195 0.160 0.083 0.101 0.069 0.082 0.069

0.309 0.317 0.316 0.332 0.324 0.644 0.618 0.656 0.612 0.575

2.69-3 3.57-3 7.72-3 8.85-3 9.94-3 1.09-2 2.20-2 1.19-2

6.33-2 9.70-2 8.99-2 0.135 0.120 0.160 0.201

f23

0.982 0.975 0.971 0.968 0.964 0.965 0.962 0.955 0.313 0.317 0.302

1.43-3

f12 þ f13

Figures following a sign indicate powers of 10. For example, 3.05-6 means 3.0561076. Source: From Bambynek et al., 1972.

7.24-2 9.97-2 0.109 2.49-2 4.13-2 5.95-2 7.64-2 8.97-2 9.22-2 0.107 0.115 0.118 0.123 0.124 0.126 0.136 0.152 0.130 0.162 0.136 0.138 0.173 0.151 0.141 0.142 0.131 0.138

0.123 0.117 0.132 0.108 0.101 0.100 0.102 0.209

Cm

96

0.06

0.037 0.007 0.035 0.002

0.029 0.002

0.023 0.001

M o

0.030 0.006

0.026 0.005

b

0.037 0.005

0.032 0.006

0.030 0.006

0.016 0.003

0.013 0.0024 0.024 0.005

oLMb

oLMa

Corrected for a 20% contribution from double M shell vacancies. Uncorrected values. Source: From Bambynek et al., 1972.

a

Os Au Au Pb Pb Bi Bi Bi U Np

Element

o1 þ f12o2

þ 0.0089 0.0075 70.0075

þ 0.003 0.002 70.002

Measured M Shell Fluorescence Yields and Coster–Kronig Probabilities

76 79 79 82 82 83 83 83 92 93

Atomic number

Table 5

o5 ¼ 0.06 0.012

oi

þ 0.0051 n1 ¼ 0.068 0.023 o2 ¼ 0.0046 n2 ¼ 0.062 0.019 70.0046 n3 ¼ 0.080 0.006 n4,5 ¼ 0.075 0.012 o5 ¼ 0.075 0.012

n2 ¼ 0.0080 0.029 n3 ¼ 0.062 0.005 n4 ¼ 0.065 0.012 n4 ¼ 0.065 0.012 n4,5 ¼ 0.081 0.016

n1 ¼ 0.065 0.014

ni

62 Markowicz

X-ray Physics Table 6

Adopted Values of Fluorescence and Coster-Kronig Yieldsa ok

Z 5 6 7 8 9

63

o1

o2

o3

f1

f12

f13

f23

B C N O F

1.7E-03 2.8E-03 5.2E-03 8.3E-03 0.013

10 11 12 13 14

Ne Na Mg Al Si

0.018 0.023 0.030 0.039 0.050

2.9E-05 2.6E-05 3.0E-05

1.2E-03 7.5E-04 3.7E-04

1.2E-03 7.5E-04 3.8E-04

0.962 0.965 0.959

0.32 0.32 0.32

0.64 0.64 0.64

15 16 17 18 19

P S Cl Ar K

0.063 0.078 0.097 0.118 0.140

3.9E-05 7.4E-05 1.2E-04 1.8E-04 2.4E-04

3.1E-04 2.6E-04 2.4E-04 2.2E-04 2.7E-04

3.1E-04 2.6E-04 2.4E-04 2.2E-04 2.7E-04

0.951 0.944 0.939 0.934 0.929

0.32 0.32 0.32 0.31 0.31

0.63 0.62 0.62 0.62 0.62

20 21 22 23 24

Ca Sc Ti V Cr

0.163 0.188 0.214 0.243 0.275

3.1E-04 3.9E-04 4.7E-04 5.8E-04 7.1E-04

3.3E-04 8.4E-04 1.5E-03 2.6E-03 3.7E-03

3.3E-04 8.4E-04 1.5E-03 2.6E-03 3.7E-03

0.920 0.912 0.902 0.894 0.885

0.31 0.31 0.31 0.31 0.31

0.61 0.60 0.59 0.58 0.57

25 26 27 28 29

Mn Fe Co Ni Cu

0.308 0.340 0.373 0.406 0.440

8.4E-04 1.0E-03 1.2E-03 1.4E-03 1.6E-03

5.0E-03 6.3E-03 7.7E-03 8.6E-03 0.010

5.0E-03 6.3E-03 7.7E-03 9.3E-03 0.011

0.877 0.868 0.856 0.847 0.839

0.30 0.30 0.30 0.30 0.30

0.58 0.57 0.56 0.55 0.54

0.028 0.028

30 31 32 33 34

Zn Ga Ge As Se

0.474 0.507 0.535 0.562 0.589

1.8E-03 2.1E-03 2.4E-03 2.8E-03 3.2E-03

0.011 0.012 0.013 0.014 0.016

0.012 0.013 0.015 0.016 0.018

0.831 0.822 0.815 0.809 0.804

0.29 0.29 0.28 0.28 0.28

0.54 0.53 0.53 0.53 0.52

0.026 0.032 0.050 0.063 0.076

35 36 37 38 39

Br Kr Rb Sb Y

0.618 0.643 0.667 0.690 0.710

3.6E-03 4.1E-03 4.6E-03 5.1E-03 5.9E-03

0.018 0.020 0.022 0.024 0.026

0.020 0.022 0.024 0.026 0.028

0.800 0.797 0.794 0.790 0.785

0.28 0.27 0.27 0.27 0.26

0.52 0.52 0.52 0.52 0.52

0.088 0.100 0.109 0.117 0.126

40 41 42 43 44

Zr Nb Mo Tc Ru

0.730 0.747 0.765 0.780 0.794

6.8E-03 9.4E-03 0.010 0.011 0.012

0.028 0.031 0.034 0.037 0.040

0.031 0.034 0.037 0.040 0.043

0.779 0.713 0.712 0.711 0.709

0.26 0.10 0.10 0.10 0.10

0.52 0.61 0.61 0.61 0.61

0.132 0.137 0.141 0.144 0.148

45 46 47 48 49

Rh Pd Ag Cd In

0.808 0.820 0.831 0.843 0.853

0.013 0.014 0.016 0.018 0.020

0.043 0.047 0.051 0.056 0.061

0.046 0.049 0.052 0.056 0.060

0.705 0.700 0.694 0.688 0.681

0.10 0.10 0.10 0.10 0.10

0.60 0.60 0.59 0.59 0.59

0.150 0.151 0.153 0.155 0.157

64 Table 6

Markowicz Continued

Z

ok

o1

o2

o3

f1

f12

f13

f23

50 51 52 53 54

Sn Sb Te I Xe

0.862 0.870 0.877 0.884 0.891

0.037 0.039 0.041 0.044 0.046

0.065 0.069 0.074 0.079 0.083

0.064 0.069 0.074 0.079 0.085

0.439 0.448 0.495 0.461 0.466

0.17 0.17 0.18 0.18 0.19

0.27 0.28 0.28 0.28 0.28

0.157 0.156 0.155 0.154 0.154

55 56 57 58 59

Cs Ba La Ce Pr

0.897 0.902 0.907 0.912 0.917

0.049 0.052 0.055 0.058 0.061

0.090 0.096 0.103 0.110 0.117

0.091 0.097 0.104 0.111 0.118

0.470 0.474 0.478 0.482 0.485

0.19 0.19 0.19 0.19 0.19

0.28 0.28 0.29 0.29 0.29

0.154 0.153 0.153 0.153 0.153

60 61 62 63 64

Nd Pm Sm Eu Gd

0.921 0.925 0.929 0.932 0.935

0.064 0.066 0.071 0.075 0.079

0.124 0.132 0.140 0.149 0.158

0.125 0.132 0.139 0.147 0.155

0.488 0.490 0.492 0.493 0.493

0.19 0.19 0.19 0.19 0.19

0.30 0.30 0.30 0.30 0.30

0.152 0.151 0.150 0.149 0.147

65 66 67 68 69

Tb Dy Ho Er Tm

0.938 0.941 0.944 0.947 0.949

0.083 0.089 0.044 0.100 0.106

0.167 0.178 0.189 0.200 0.211

0.164 0.174 0.182 0.192 0.201

0.493 0.492 0.490 0.487 0.483

0.19 0.19 0.19 0.19 0.19

0.30 0.30 0.30 0.30 0.29

0.145 0.143 0.142 0.140 0.139

70 71 72 73 74

Yb Lu Hf Ta W

0.951 0.953 0.955 0.957 0.958

0.112 0.120 0.128 0.137 0.147

0.222 0.234 0.246 0.258 0.270

0.210 0.220 0.231 0.243 0.255

0.478 0.472 0.465 0.457 0.447

0.19 0.19 0.18 0.18 0.17

0.29 0.28 0.28 0.28 0.28

0.138 0.136 0.135 0.134 0.133

75 76 77 78 79

Re Os Ir Pt Au

0.959 0.961 0.962 0.963 0.964

0.144 0.130 0.120 0.114 0.107

0.283 0.295 0.308 0.321 0.334

0.268 0.281 0.294 0.306 0.320

0.485 0.552 0.603 0.640 0.672

0.16 0.16 0.15 0.14 0.14

0.33 0.39 0.45 0.50 0.53

0.130 0.128 0.126 0.124 0.122

80 81 82 83 84

Hg Tl Pb Bi Po

0.965 0.966 0.967 0.968 0.968

0.107 0.107 0.112 0.117 0.122

0.347 0.360 0.373 0.387 0.401

0.333 0.347 0.360 0.373 0.386

0.690 0.696 0.696 0.694 0.689

0.13 0.13 0.12 0.11 0.11

0.56 0.57 0.58 0.58 0.58

0.120 0.118 0.116 0.113 0.111

85 86 87 88 89

At Rn Fr Ra Ac

0.969 0.969 0.970 0.970 0.971

0.128 0.134 0.139 0.146 0.153

0.415 0.429 0.443 0.456 0.468

0.399 0.411 0.424 0.437 0.450

0.685 0.682 0.677 0.672 0.664

0.10 0.10 0.10 0.09 0.09

0.59 0.58 0.58 0.58 0.58

0.111 0.110 0.109 0.108 0.108

90 91 92 93 94

Th Pa U Np Pu

0.971 0.972 0.972 0.973 0.973

0.161 0.162 0.176 0.187 0.205

0.479 0.472 0.467 0.466 0.464

0.463 0.476 0.489 0.502 0.514

0.660 0.664 0.652 0.642 0.605

0.09 0.08 0.08 0.07 0.05

0.57 0.58 0.57 0.57 0.56

0.108 0.139 0.167 0.192 0.198

X-ray Physics Table 6

65

Continued

Z

ok

o1

o2

o3

f1

f12

f13

f23

95 96 97 98 99

Am Cm Bk Cf Es

0.974 0.974 0.975 0.975 0.975

0.218 0.228 0.236 0.244 0.253

0.471 0.479 0.485 0.490 0.497

0.526 0.539 0.550 0.560 0.570

0.595 0.587 0.580 0.573 0.565

0.05 0.04 0.04 0.03 0.03

0.55 0.55 0.54 0.54 0.54

0.203 0.200 0.198 0.197 0.196

100 101 102 103 104

Fm Md No Lr

0.976 0.976 0.976 0.977 0.977

0.263 0.272 0.280 0.282 0.291

0.506 0.515 0.524 0.533 0.544

0.579 0.588 0.596 0.604 0.611

0.556 0.548 0.540 0.538 0.531

0.03 0.02 0.02 0.01 0.01

0.53 0.53 0.52 0.53 0.52

0.194 0.191 0.189 0.185 0.181

105 106 107 108 109

0.977 0.978 0.978 0.978 0.978

0.300 0.310 0.320 0.331 0.343

0.553 0.562 0.573 0.584 0.590

0.618 0.624 0.630 0.635 0.640

0.522 0.513 0.505 0.497 0.488

0.01

0.51 0.51 0.50 0.50 0.49

0.178 0.174 0.171 0.165 0.163

110

0.979

0.354

0.598

0.644

0.477

0.48

0.158

a Note: 1.7E-03 ¼ 1.761073; designation for L shell is omitted, read, for example, o1 ¼ oL1 ; f1 ¼ f12 þ f13 . Source: From Krause, 1979.

66

Markowicz

Table 7 Z

Auger Yields for K, L1, L2 and L3 Levelsa aK

a1

a2

a3

Z

aK

a1

a2

a3

B C N O F

0.998 0.997 0.995 0.992 0.987

50 51 52 53 54

Sn Sb Te I Xe

0.138 0.130 0.123 0.116 0.109

0.524 0.513 0.504 0.495 0.488

0.778 0.775 0.771 0.767 0.763

0.936 0.931 0.926 0.921 0.915

10 11 12 13 14

Ne Na Mg Al Si

0.982 0.977 0.970 0.961 0.950

0.038 0.035 0.041

0.999 0.999 1.000

0.999 0.999 1.000

55 56 57 58 59

Cs Ba La Ce Pr

0.103 0.098 0.093 0.088 0.083

0.481 0.474 0.467 0.460 0.454

0.756 0.751 0.744 0.737 0.730

0.909 0.903 0.896 0.889 0.882

15 16 17 18 19

P S Cl Ag K

0.937 0.922 0.903 0.882 0.860

0.049 0.056 0.061 0.066 0.071

1.000 1.000 1.000 1.000 1.000

1.000 1.000 1.000 1.000 1.000

60 61 62 63 64

Nd Pm Sm Eu Gd

0.079 0.075 0.071 0.068 0.065

0.448 0.444 0.437 0.432 0.428

0.724 0.717 0.710 0.702 0.695

0.875 0.868 0.861 0.853 0.845

20 21 22 23 24

Ca Sc Ti V Cr

0.837 0.812 0.786 0.757 0.725

0.080 0.088 0.098 0.105 0.114

1.000 0.999 0.999 0.997 0.996

1.000 0.999 0.999 0.997 0.996

65 66 67 68 69

Tb Dy Ho Er Tm

0.062 0.059 0.056 0.053 0.051

0.424 0.419 0.416 0.413 0.411

0.688 0.679 0.669 0.660 0.650

0.836 0.826 0.818 0.808 0.799

25 26 27 28 29

Mn Fe Co Ni Cu

0.692 0.660 0.627 0.394 0.560

0.122 0.131 0.143 0.152 0.159

0.995 0.994 0.992 0.963 0.962

0.995 0.994 0.992 0.991 0.989

70 71 72 73 74

Yb Lu Hf Ta W

0.049 0.047 0.045 0.043 0.042

0.410 0.408 0.407 0.406 0.406

0.640 0.630 0.619 0.608 0.597

0.790 0.780 0.769 0.757 0.745

30 31 32 33 34

Zn Ga Ge As Se

0.526 0.493 0.465 0.438 0.411

0.167 0.176 0.183 0.188 0.193

0.963 0.956 0.937 0.923 0.908

0.988 0.987 0.985 0.984 0.982

75 76 77 78 79

Re Os Ir Pt Au

0.041 0.039 0.038 0.037 0.036

0.371 0.318 0.277 0.246 0.221

0.587 0.577 0.566 0.555 0.344

0.732 0.719 0.706 0.694 0.680

35 36 37 38 39

Br Kr Rb Sr Y

0.382 0.357 0.333 0.310 0.290

0.196 0.199 0.201 0.205 0.209

0.894 0.880 0.869 0.859 0.848

0.980 0.978 0.976 0.974 0.972

80 81 82 83 84

Hg Tl Pb Bi Po

0.035 0.034 0.033 0.032 0.032

0.203 0.197 0.192 0.189 0.189

0.533 0.522 0.511 0.500 0.488

0.667 0.653 0.640 0.627 0.614

40 41 42 43 44

Zr Nb Mo Tc Ru

0.270 0.253 0.235 0.220 0.206

0.214 0.278 0.278 0.278 0.279

0.840 0.832 0.825 0.819 0.812

0.969 0.966 0.963 0.960 0.957

85 86 87 88 89

At Rn Fr Ra Ac

0.031 0.031 0.030 0.030 0.029

0.187 0.184 0.184 0.182 0.183

0.474 0.461 0.448 0.436 0.424

0.601 0.589 0.576 0.563 0.590

45 46 47 48 49

Rh Pd Ag Cd In

0.192 0.180 0.169 0.157 0.147

0.282 0.286 0.290 0.294 0.299

0.807 0.802 0.796 0.789 0.782

0.954 0.951 0.948 0.944 0.940

90 91 92 93 94

Th Pa U Np Pu

0.029 0.028 0.028 0.027 0.027

0.179 0.174 0.172 0.171 0.190

0.413 0.389 0.366 0.342 0.338

0.537 0.524 0.511 0.498 0.486

5 6 7 8 9

X-ray Physics Table 7 Z

67

Continued aK

a1

a2

a3

Z

aK

a1

a2

a3

95 96 97 98 99

Am Cm Bk Cf Es

0.026 0.026 0.025 0.025 0.025

0.167 0.185 0.184 0.183 0.182

0.326 0.321 0.317 0.313 0.307

0.474 0.461 0.450 0.440 0.430

105 106 107 108 109

0.023 0.022 0.022 0.022 0.022

0.178 0.177 0.175 0.172 0.169

0.269 0.264 0.256 0.251 0.247

0.382 0.376 0.370 0.365 0.360

100 101 102 103 104

Fm Md No Lr

0.024 0.024 0.024 0.023 0.023

0.181 0.180 0.180 0.180 0.178

0.300 0.294 0.287 0.282 0.275

0.421 0.412 0.404 0.396 0.389

110

0.021

0.169

0.244

0.356

a Designation for L shell is omitted; read, for example, a1 aL1. Source: From Krause, 1979.

68

Markowicz

APPENDIX VII: COEFFICIENTS FOR CALCULATING THE PHOTOELECTRIC ABSORPTION CROSS SECTIONS (BARNS=ATOM) VIA ln^ln REPRESENTATION Table 1

E > EK a

Atomic number

Element

Atomic weight

H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh

1.008 4.003 6.940 9.012 10.811 12.010 14.008 16.000 19.000 20.183 22.997 24.320 26.970 28.086 30.975 32.066 35.457 39.944 39.102 40.080 44.960 47.900 50.942 51.996 54.940 55.850 58.933 58.690 63.540 65.380 69.720 72.590 74.920 78.960 79.920 83.800 85.480 87.620 88.905 91.220 92.906 95.950 99.000 101.070 102.910

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

A0 2.44964 6.06488 7.75370 9.04511 9.95057 1.06879 þ 1 1.12765 þ 1 1.17130 þ 1 1.20963 þ 1 1.24485 þ 1 1.26777 þ 1 1.28793 þ 1 1.31738 þ 1 1.32682 þ 1 1.33735 þ 1 1.37394 þ 1 1.36188 þ 1 1.39491 þ 1 1.37976 þ 1 1.42950 þ 1 1.39664 þ 1 1.43506 þ 1 1.47601 þ 1 1.48019 þ 1 1.48965 þ 1 1.43456 þ 1 1.47047 þ 1 1.42388 þ 1 1.45808 þ 1 1.44118 þ 1 1.36182 þ 1 1.39288 þ 1 1.34722 þ 1 1.30756 þ 1 1.32273 þ 1 1.35927 þ 1 1.30204 þ 1 1.35888 þ 1 1.34674 þ 1 1.27538 þ 1 1.33843 þ 1 1.39853 þ 1 1.28214 þ 1 1.26658 þ 1 1.21760 þ 1

A1

A2

A3

73.34953 73.29055 72.81801 72.83487 72.74173 72.71400 72.65400 72.57229 72.44148 72.45819 72.24521 72.12574 72.18203 71.98174 71.86342 72.04786 71.71937 71.82276 71.54015 71.88644 71.40872 71.66322 71.88867 71.82430 71.79872 71.23491 71.38933 79.67736-1 71.18375 79.33083-1 73.18459-1 74.79613-1 77.73513-2 1.83235-1 1.37130-1 73.05214-2 3.82736-1 2.20194-3 1.91023-1 6.97409-1 2.81028-1 71.17426-1 7.51993-1 8.85020-1 1.19682

74.71370-2 71.07256-1 72.41738-1 72.10021-1 72.15138-1 72.00530-1 72.00445-1 72.05893-1 72.34461-1 72.12591-1 72.74873-1 72.99392-1 72.58960-1 73.16950-1 73.39440-1 72.73259-1 73.54154-1 73.28827-1 73.94528-1 72.83647-1 74.14365-1 73.31539-1 72.71861-1 72.79116-1 72.83664-1 74.18785-1 73.86631-1 74.78070-1 74.13850-1 74.77357-1 76.11348-1 75.72897-1 76.60456-1 76.94264-1 76.83203-1 76.51340-1 77.32427-1 76.38940-1 76.86616-1 77.89307-1 76.86607-1 75.91094-1 77.87006-1 78.11144-1 78.66697-1

7.09962-3 1.44465-2 2.62542-2 2.29526-2 2.27845-2 2.07248-2 2.00765-2 1.99244-2 2.19537-2 1.96489-2 2.50270-2 2.67643-2 2.22840-2 2.73928-2 2.88858-2 2.29976-2 2.90841-2 2.74382-2 3.23561-2 2.26263-2 3.34355-2 2.62065-2 2.15792-2 2.17324-2 2.22095-2 3.21662-2 3.03286-2 3.66138-2 3.12088-2 3.62829-2 4.58138-2 4.31277-2 4.92177-2 5.02280-2 4.95424-2 4.77616-2 5.29874-2 4.60070-2 4.97356-2 5.64531-2 4.86607-2 4.17843-2 5.58668-2 5.73759-2 6.06931-2

X-ray Physics

69

Table 1

Continued

Atomic number

Element

Atomic weight

Pd Ag Cd In Sn Sb Te I Xe Cs Ba La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Rn Th U Pu

106.400 107.880 112.410 114.820 118.690 121.760 127.600 126.910 131.300 132.910 137.360 138.920 140.130 140.920 144.270 147.000 150.350 152.000 157.260 158.930 162.510 164.940 167.270 168.940 173.040 174.990 178.500 180.950 183.920 186.200 190.200 192.200 195.090 197.200 200.610 204.390 207.210 209.000 222.000 232.000 238.070 239.100

46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 86 90 92 94 a

A0 1.39389 þ 1 1.33926 þ 1 1.15254 þ 1 1.18198 þ 1 1.30323 þ 1 9.06999 1.16656 þ 1 1.21075 þ 1 1.10857 þ 1 1.13757 þ 1 1.02250 þ 1 1.09780 þ 1 1.02725 þ 1 1.10156 þ 1 1.17632 þ 1 1.13864 þ 1 1.19223 þ 1 1.16168 þ 1 9.91968 1.13818 þ 1 1.14845 þ 1 8.75203 1.20195 þ 1 1.25613 þ 1 7.42791 1.26387 þ 1 7.58160 8.65271 7.57541 1.36944 1.37534 þ 1 1.25506 þ 1 1.27882 þ 1 4.96352 1.97594 þ 1 1.52879 þ 1 8.63374 9.44293 1.51782 þ 1 1.34336 þ 1 1.37951 þ 1 1.82787 þ 1

Notation abbreviated as in Appendix VI, Table 4.

A1

A2

A3

1.64528-1 4.41380-1 1.07714 1.45768 7.90788-1 3.28791 1.71052 1.43635 2.08356 1.94161 2.67835 2.23814 2.74562 2.22056 1.79481 2.05593 1.79546 1.97533 3.03111 2.14447 2.10451 3.71822 1.84815 1.57523 4.28955 1.55476 4.47037 3.73117 4.28874 7.79444 1.02122 1.63090 1.63605 5.79212 71.97990 2.73664-1 3.69400 3.44965 3.49021-1 1.34805 1.23983 71.17371

76.62170-1 76.93711-1 78.31424-1 78.88529-1 77.62349-1 71.26203 79.48281-1 78.82038-1 71.01209 79.83232-1 71.12648 71.03549 71.14174 71.02216 79.36661-1 79.88180-1 79.42902-1 79.70901-1 71.17520 79.99222-1 79.89870-1 71.29273 79.39582-1 78.90467-1 71.35167 78.81094-1 71.42808 71.26359 71.34998 71.99822 77.77126-1 78.75676-1 78.98523-1 71.61842 72.76981-1 76.38890-1 71.21312 71.19886 76.37638-1 78.13282-1 78.01545-1 73.68344-1

4.76289-2 4.82085-2 5.79120-2 6.05982-2 5.27872-2 8.53470-2 6.53213-2 6.03575-2 6.90310-2 6.7l986-2 7.62669-2 7.02339-2 7.74162-2 6.90465-2 6.35332-2 6.69106-2 6.44202-2 6.58459-2 7.86751-2 6.75569-2 6.69382-2 8.55026-2 6.38106-2 6.09779-2 8.66136-2 6.02036-2 9.39044-2 8.23539-2 8.65200-2 1.26225-1 5.38811-2 5.92011-2 6.18550-2 1.02911-1 2.68856-2 4.57495-2 7.74601-2 7.83484-2 4.51377-2 5.55664-2 5.53596-2 2.98738-2

70

Markowicz

Table 2

EL1 < E < EKa

Atomic number

Element

Atomic weight

Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs Ba La Ce

22.997 24.310 26.970 28.086 30.975 32.066 35.457 39.944 39.102 40.080 44.960 47.900 50.942 51.996 54.040 55.850 58.933 58.690 63.540 65.380 69.720 72.590 74.920 78.960 79.920 83.800 85.480 87.620 88.905 91.220 92.906 95.950 99.000 101.070 102.910 106.400 107.880 112.410 114.820 118.690 121.760 127.600 126.910 131.300 132.910 137.360 138.920 140.130

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58

A0 1.02355 þ 1 1.05973 þ 1 1.08711 þ 1 1.12237 þ 1 1.15508 þ 1 1.18181 þ 1 1.20031 þ 1 1.22960 þ 1 1.24878 þ 1 1.27044 þ 1 1.28949 þ 1 1.31075 þ 1 1.32514 þ 1 1.34236 þ 1 1.35761 þ 1 1.36696 þ 1 1.38699 þ 1 1.39848 þ 1 1.42439 þ 1 1.43221 þ 1 1.44795 þ 1 1.46813 þ 1 1.46431 þ 1 1.47048 þ 1 1.48136 þ 1 1.49190 þ 1 1.49985 þ 1 1.50114 þ 1 1.51822 þ 1 1.52906 þ 1 1.52088 þ 1 1.53494 þ 1 1.55086 þ 1 1.54734 þ 1 1.55757 þ 1 1.55649 þ 1 1.56869 þ 1 1.59668 þ 1 1.62101 þ 1 1.58638 þ 1 1.57557 þ 1 1.61087 þ 1 1.64086 þ 1 1.63098 þ 1 1.65418 þ 1 1.66217 þ 1 1.63134 þ 1 1.65862 þ 1

A1

A2

72.55905 72.89818 72.77860 72.73694 72.92200 72.64618 72.41694 72.63279 72.53656 72.55011 72.40609 72.53576 72.49765 72.51532 72.49761 72.39195 72.50669 72.48080 72.58677 72.62384 72.54469 72.69285 72.48397 72.38853 72.42347 72.42418 72.39108 72.28169 72.38946 72.38703 72.20278 72.26646 72.33733 72.23080 72.24976 72.17229 72.22636 72.38363 72.51838 72.19010 72.04460 72.27876 72.48214 72.31679 72.46363 72.48972 72.20156 72.36288

71.19524-1 2.34506-1 1.75853-1 1.27557-1 2.54262-1 79.68049-2 72.40897-1 77.36600-2 71.04892-1 79.43195-2 71.77791-1 79.57177-2 7 1.06383-1 71.01999-1 71.05943-1 71.37648-1 78.69945-2 78.88115-2 76.67398-2 72.64926-2 77.57204-2 72.08355-2 77.96180-2 71.05877-1 79.14590-2 78.76447-2 79.59473-2 71.26485-1 78.81174-2 79.12292-2 71.36759-1 71.16881-1 79.87857-1 71.19454-1 71.13377-1 71.27652-1 71.12223-1 78.01104-2 75.49951-2 71.13539-1 71.40745-1 79.29405-2 75.07179-2 78.54498-2 75.42849-2 74.49623-2 79.80569-2 76.54708-2

X-ray Physics Table 2 Atomic number 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 86 90 92 94 a

71

Continued Element

Atomic weight

Pr Nd Pm Sm Eu Cd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Rn Th U Pu

140.920 144.270 147.000 150.350 152.000 157.260 158.930 162.510 164.940 167.270 168.940 173.040 174.990 178.500 180.950 183.920 186.200 190.200 192.200 195.090 197.200 200.610 204.390 207.210 209.000 222.000 232.000 238.070 239.100

Notation abbreviated as in Appendix VI, Table 4.

A0 1.67179 þ 1 1.65964 þ 1 1.68368 þ 1 1.68725 þ 1 1.70692 þ 1 1.71159 þ 1 1.71499 þ 1 1.73446 þ 1 1.76583 þ 1 1.77988 þ 1 1.74250 þ 1 1.69795 þ 1 1.72638 þ 1 1.64329 þ 1 1.72410 þ 1 1.72533 þ 1 1.78750 þ 1 1.73525 þ 1 1.65270 þ 1 1.73636 þ 1 1.74240 þ 1 1.71857 þ 1 1.77379 þ 1 1.77963 þ 1 1.75348 þ 1 1.75028 þ 1 1.85481 þ 1 1.75258 þ 1 1.75519 þ 1

A1

A2

72.40326 72.26073 72.38881 72.39051 72.48046 72.47838 72.45507 72.54821 72.72523 72.74671 72.51103 72.22577 72.37189 71.82851 72.30313 72.23874 72.61051 72.28550 71.76315 72.21112 72.23911 72.08470 72.37745 72.37691 72.23353 72.13876 72.61281 72.07237 72.02162

76.12619-2 78.72426-2 76.45041-2 76.01080-2 74.47055-2 74.37107-2 74.71370-2 73.17606-2 78.19409-4 72.87580-3 73.29454-2 77.32557-2 74.95994-2 71.32268-1 75.91006-2 77.27338-2 71.36093-2 75.88047-2 71.35232-1 77.30934-2 76.63720-2 78.53294-2 74.33223-2 74.55883-2 75.96161-2 77.24638-2 77.90574-3 77.23932-2 78.22940-2

72 Table 3

Markowicz EM1 < E < EL3 a

Atomic number 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79

Element Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs Ba La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au

Atomic weight 69.720 72.590 74.920 78.960 79.920 83.800 85.480 87.620 88.905 91.220 92.906 95.950 99.000 101.070 102.910 106.400 107.880 112.410 114.820 118.690 121.760 127.600 126.910 131.300 132.910 137.360 138.920 140.130 140.920 144.270 147.000 150.350 152.000 157.260 158.930 162.510 164.940 167.270 168.940 173.040 174.990 178.500 180.950 183.920 186.200 190.200 192.200 195.090 197.200

A0

A1

1.22646 þ 1 1.24133 þ 1 1.25392 þ 1 1.26773 þ 1 1.27612 þ 1 1.28898 þ 1 1.30286 þ 1 1.31565 þ 1 1.32775 þ 1 1.34508 þ 1 1.35434 þ 1 1.36568 þ 1 1.37498 þ 1 1.38782 þ 1 1.40312 þ 1 1.41392 þ 1 1.41673 þ 1 1.43497 þ 1 1.44115 þ 1 1.45572 þ 1 1.46268 þ 1 1.47125 þ 1 1.47496 þ 1 1.47603 þ 1 1.49713 þ 1 1.50844 þ 1 1.51863 þ 1 1.52693 þ 1 1.53379 þ 1 1.54353 þ 1 1.55131 þ 1 1.56006 þ 1 1.57063 þ 1 1.57159 þ 1 1.58415 þ 1 1.59225 þ 1 1.60140 þ 1 1.60672 þ 1 1.61269 þ 1 1.61794 þ 1 1.62289 þ 1 1.62758 þ 1 1.63068 þ 1 1.62613 þ 1 1.63564 þ 1 1.64233 þ 1 1.65144 þ 1 1.67024 þ 1 1.64734 þ 1

72.68965 72.53085 72.41380 72.39750 72.37730 72.26021 72.38693 72.36655 72.43174 72.50201 72.50135 72.48982 72.44737 72.48066 72.61303 72.57206 72.48078 72.52756 72.49401 72.56792 72.55562 72.54324 72.48179 72.45068 72.53145 72.56341 72.58287 72.58174 72.57086 72.59006 72.59623 72.61328 72.63481 72.60843 72.64040 72.65289 72.67903 72.67587 72.67886 72.67715 72.67128 72.66622 72.66148 72.60672 72.62453 72.63163 72.64832 72.71631 72.57834

X-ray Physics Table 3

Continued

Atomic number 80 81 82 83 86 90 92 94 a

Element

Atomic weight

Hg Tl Pb Bi Rn Th U Pu

200.610 204.390 207.210 209.000 222.000 232.000 238.070 239.100

A0

A1

1.65903 þ 1 1.66564 þ 1 1.67131 þ 1 1.67078 þ 1 1.69000 þ 1 1.70483 þ 1 1.70353 þ 1 1.72953 þ 1

72.60670 72.61593 72.61538 72.58648 72.60945 72.58569 72.56903 72.62164

Notation abbreviated in Appendix VI, Table 4.

Table 4

E < EM5 a

Atomic number 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 86 90 92 94 a

73

Element Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Rn Th U Pu

Atomic weight 147.000 150.350 152.000 157.260 158.930 162.510 164.940 167.270 168.940 173.040 174.990 178.500 180.950 183.920 186.200 190.200 192.200 195.090 197.200 200.610 204.390 207.210 209.000 222.000 232.000 238.070 239.100

Notation abbreviated as in Appendix VI, Table 4. Source: From McMaster et al., 1969.

A0 1.55131 þ 1 1.56006 þ 1 1.57063 þ 1 1.57159 þ 1 1.58415 þ 1 1.59225 þ 1 1.60140 þ 1 1.60672 þ 1 1.61269 þ 1 1.39111 þ 1 1.39813 þ 1 1.40548 þ 1 1.41313 þ 1 1.42536 þ 1 1.42392 þ 1 1.42795 þ 1 1.43422 þ 1 1.43785 þ 1 1.44398 þ 1 1.45195 þ 1 1.45473 þ 1 1.45771 þ 1 1.46832 þ 1 1.47243 þ 1 1.47730 þ 1 1.49036 þ 1 1.48535 þ 1

A1 72.59623 72.61328 72.63481 72.60843 72.64040 72.65289 72.67903 72.67587 72.67886 72.40380 72.40841 72.42829 72.47214 72.32582 72.35326 72.21971 72.40183 72.34834 72.32838 72.33016 72.26773 72.25279 72.30940 72.12905 71.91192 72.12148 71.87733

74

Markowicz

APPENDIX VIII: COEFFICIENTS FOR CALCULATING THE INCOHERENT COLLISION CROSS SECTIONS C (BARNS=ATOM) VIA THE ln^ln REPRESENTATION Atomic Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

Element H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd

A0

A1

A2

A3

72.l5772 72.56357 71.08740 76.90079-1 77.91177-1 79.87878-1 71.23693 71.73679 71.87570 71.75510 79.67717-1 75.71611-1 74.39322-1 74.14971-1 74.76903-1 76.56419-1 77.18627-1 76.82105-1 73.44007-1 79.82420-2 71.59831-1 72.30573-1 73.08103-1 73.87641-1 72.47059-1 73.42379-1 74.28804-1 75.04360-1 75.70210-1 74.20535-1 73.58218-1 73.34383-1 73.39189-1 74.32927-1 74.48001-1 73.91810-1 71.28039-1 7.99161-2 6.29057-2 3.66697-2 2.02289-4 75.62860-2 7.57616-2 74.24981-2 71.60399-1 72.67564-1

1.32685 2.02536 1.03368 9.46448-1 1.21611 1.46693 1.74510 2.17686 2.32016 2.24226 1.61794 1.35498 1.30867 1.34868 1.46032 1.65408 1.74294 1.74279 1.49236 1.32829 1.39055 1.45848 1.52879 1.59727 1.49722 1.57245 1.64129 1.70040 1.75042 1.63400 1.60050 1.60237 1.62535 1.72833 1.76082 1.73010 1.53044 1.38397 1.41577 1.45207 1.49347 1.55778 1.44950 1.54639 1.64861 1.73740

73.05620-1a 74.48710-1 71.90377-1 71.71142-1 72.39087-1 72.93743-1 73.54660-1 74.49050-1 74.75412-1 74.47640-1 72.87191-1 72.22491-1 72.11648-1 72.22315-1 72.51331-1 72.98623-1 73.19429-1 73.17646-1 72.54135-1 72.13747-1 72.25849-1 72.39160-1 72.52768-1 72.66240-1 72.38781-1 72.53198-1 72.66013-1 72.76443-1 72.84555-1 72.53646-1 72.44908-1 72.45555-1 72.50783-1 72.77138-1 72.85099-1 72.76824-1 2.27403-1 71.92225-1 71.99713-1 72.08122-1 72.17419-1 72.33341-1 72.04890-1 72.26470-1 72.50238-1 72.69883-1

1.85025-2 2.79691-2 7.79955-3 6.51413-3 1.17686-2 1.56005-2 1.98705-2 2.64733-2 2.80680-2 2.55801-2 1.31526-2 8.30141-3 7.54210-3 8.41959-3 1.07202-2 1.42979-2 1.58429-2 1.56467-2 1.07684-2 7.73065-3 8.51954-3 9.38528-3 1.02571-2 1.11523-2 8.93208-3 9.85822-3 1.06512-2 1.12628-2 1.16930-2 9.27233-3 8.61898-3 8.71239-3 9.09103-3 1.11735-2 1.17865-2 1.11280-2 7.39033-3 4.78611-3 5.33312-3 5.95139-3 6.62245-3 7.85506-3 5.64745-3 7.18375-3 8.93818-3 1.03248-2

X-ray Physics

Appendix VIII Atomic Number 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 86 90 92 94

75

Continued Element Ag Ce In Sn Sb Te I Xe Cs Ba La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Rn Th U Pu

a Notation as in Appendix VI, Table 4. Source: From McMaster et al., 1969.

A0

A1

A2

A3

71.66475-1 75.16701-2 78.17283-2 1.42151-2 1.56362-2 74.07579-2 74.04420-2 72.82407-3 1.84861-1 3.44376-1 4.09104-1 4.39881-1 4.49124-1 4.37283-1 4.05823-1 3.55383-1 2.80316-1 2.73133-1 2.57539-1 2.42685-1 2.28493-1 2.15233-1 2.02656-1 2.02248-1 1.97176-1 1.99469-1 1.96871-1 1.91015-1 1.89644-1 1.16448-1 7.19908-2 4.20186-2 1.56916-2 1.14587-1 1.47052-1 1.82167-1 1.89860-1 1.96619-1 1.70890-1 1.08277-1 3.88791-2

1.65794 1.57426 1.55865 1.55754 1.57175 1.64267 1.65596 1.64039 1.50030 1.38742 1.33075 1.30925 1.30351 1.31370 1.33837 1.37733 1.44016 1.43842 1.45064 1.46266 1.47438 1.48545 1.49625 1.48804 1.50264 1.50233 1.50623 1.51240 1.50867 1.57615 1.61204 1.63611 1.65406 1.58076 1.56695 1.54661 1.56125 1.60080 1.65561 1.74158 1.82229

72.48740-1 72.27646-1 72.24492-1 72.24736-1 72.28753-1 72.47897-1 72.51067-1 72.47642-1 72.13333-1 71.86356-1 71.70883-1 71.64548-1 71.61841-1 71.62866-1 71.67229-1 71.74941-1 71.88641-1 71.86137-1 71.87591-1 71.89102-1 71.90559-1 71.91908-1 71.93234-1 71.89143-1 71.92474-1 71.91385-1 71.91396-1 71.91922-1 71.89570-1 72.05532-1 72.13186-1 72.17964-1 72.20982-1 72.02968-1 72.00347-1 71.95793-1 72.00932-1 72.13800-1 72.29702-1 72.54104-1 72.76009-1

8.66218-3 7.05650-3 6.85776-3 6.91395-3 7.26386-3 8.80567-3 9.04874-3 8.82144-3 6.24264-3 4.24917-3 3.04111-3 2.52641-3 2.27394-3 2.29377-3 2.55570-3 3.06213-3 4.01226-3 3.75240-3 3.79932-3 3.85628-3 3.90903-3 3.95645-3 4.00233-3 3.62264-3 3.85751-3 3.74011-1 3.70889-3 3.71450-3 3.49584-3 4.66731-3 5.20497-3 5.52670-3 5.70751-3 4.35692-3 4.20901-3 3.90772-3 4.36768-3 5.51717-3 6.92516-3 8.95056-3 1.07392-2

76

Markowicz

APPENDIX IX: COEFFICIENTS FOR CALCULATING THE COHERENT SCATTERING CROSS SECTIONS R (BARNS/ATOM) VIATHE ln^ln REPRESENTATION Atomic number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

Element H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag

A0 71.19075-1 1.04768 1.34366 2.00860 2.61862 3.10861 3.47760 3.77239 4.00716 4.20151 4.26374 4.39404 4.51995 4.64678 4.76525 4.92707 5.07222 5.21079 5.25587 5.32375 5.43942 5.55039 5.65514 5.77399 5.84604 5.93292 6.01478 6.09204 6.17739 6.23402 6.28298 6.33896 6.39750 6.45637 6.51444 6.57129 6.59750 6.62203 6.67096 6.72275 6.79013 6.84600 6.87599 6.93136 6.97547 7.03216 7.06446

a

A1

A2

A3

79.37086-1 78.51805-2 1.81557-1 74.61920-2 72.07916-1 72.60580-1 72.15762-1 71.48539-1 75.60908-2 4.16247-2 1.34662-1 1.37858-1 1.40549-1 1.62780-1 1.68708-1 1.65746-1 1.49127-1 1.35618-1 1.88040-1 2.06685-1 2.00174-1 1.97697-1 1.99533-1 2.03858-1 2.13814-1 2.25048-1 2.37959-1 2.52277-1 2.73123-1 2.84312-1 2.91334-1 2.91512-1 2.88866-1 2.86737-1 2.86324-1 2.87711-1 3.02389-1 3.24559-1 3.25075-1 3.23964-1 3.11282-1 3.02797-1 3.26165-1 3.34794-1 3.46394-1 3.49838-1 3.63456-1

72.00538-1 74.03527-1 74.23981-1 73.37018-1 72.86283-1 72.71974-1 72.88874-1 73.07124-1 73.32017-1 73.56754-1 73.70080-1 73.59540-1 73.52441-1 73.58563-1 73.60383-1 73.59424-1 73.52858-1 73.47214-1 73.59623-1 73.61664-1 73.59064-1 73.57694-1 73.57487-1 73.59699-1 73.59718-1 73.61748-1 73.64056-1 73.66568-1 73.72360-1 73.72143-1 73.69391-1 73.65643-1 73.61747-1 73.58794-1 73.57027-1 73.56311-1 73.56755-1 73.61651-1 73.60613-1 73.59463-1 73.55233-1 73.51131-1 73.58969-1 73.63497-1 73.67794-1 73.70099-1 73.73597-1

1.06587-2 2.69398-2 2.66190-2 1.86939-2 1.44966-2 1.35181-2 1.51312-2 1.67303-2 1.87934-2 2.07585-2 2.14467-2 2.02380-2 1.93692-2 1.96926-2 1.97155-2 1.95505-2 1.89439-2 1.84333-2 1.93085-2 1.93328-2 1.91027-2 1.89866-2 1.89691-2 1.92225-2 1.91459-2 1.93024-2 1.94754-2 1.96586-2 2.01638-2 2.00525-2 1.97029-2 1.92895-2 1.88788-2 1.85618-2 1.83557-2 1.82470-2 1.81706-2 1.84800-2 1.83325-2 1.81890-2 1.78231-2 1.74403-2 1.80482-2 1.84429-2 1.87885-2 1.89983-2 1.92478-2

X-ray Physics

Appendix IX Atomic number 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 86 90 92 94 a

77

Continued Element

A0

A1

A2

A3

Cd In Sn Sb Te I Xe Cs Ba La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Rn Th U Pu

7.09856 7.12708 7.16085 7.19665 7.23464 7.27415 7.31469 7.33490 7.35812 7.39532 7.44255 7.48347 7.52334 7.56222 7.60020 7.63711 7.66938 7.70798 7.74188 7.77470 7.80643 7.83711 7.86662 7.89137 7.91803 7.94534 7.97266 7.99940 8.02574 8.05150 8.08084 8.10524 8.12542 8.14399 8.15996 8.17489 8.22553 8.27843 8.33010 8.38174

3.72199-1 3.82082-1 3.85512-1 3.85543-1 3.82493-1 3.77223-1 3.70315-1 3.76825-1 3.79361-1 3.69895-1 3.71328-1 3.68431-1 3.66462-1 3.65055-1 3.64134-1 3.63957-1 3.59752-1 3.65345-1 3.67107-1 3.69722-1 3.73226-1 3.77547-1 3.82933-1 3.86034-1 3.87021-1 3.87299-1 3.87704-1 3.88739-1 3.90458-1 3.93143-1 3.95790-1 4.00576-1 4.05858-1 4.08692-1 4.18031-1 4.27916-1 4.51478-1 4.79056-1 4.78314-1 4.77085-1

73.75345-1 73.76855-1 73.76481-1 73.75054-1 73.72715-1 73.69728-1 73.66280-1 73.65713-1 73.64099-1 73.59376-1 73.59642-1 73.57689-1 73.56048-1 73.54511-1 73.53086-1 73.51909-1 73.48899-1 73.50031-1 73.49433-1 73.49132-1 73.49147-1 73.49441-1 73.50126-1 73.49756-1 73.48881-1 73.47926-1 73.47155-1 73.46726-1 73.46658-1 73.47052-1 73.48032-1 73.49340-1 73.50329-1 73.49802-1 73.52330-1 73.55068-1 73.62056-1 73.67657-1 73.67250-1 73.66556-1

1.93481-2 1.94151-2 1.93305-2 1.91608-2 1.89194-2 1.86280-2 1.83025-2 1.81843-2 1.79817-2 1.75406-2 1.75852-2 1.74099-2 1.72620-2 1.71214-2 1.69894-2 1.68783-2 1.65890-2 1.66927-2 1.66273-2 1.65862-2 1.65710-2 1.65780-2 1.66173-2 1.65480-2 1.64406-2 1.63299-2 1.62372-2 1.61751-2 1.61455-2 1.61573-2 1.62345-2 1.63264-2 1.63772-2 1.62888-2 1.64660-2 1.66601-1 1.71556-2 1.74621-2 1.74129-2 1.73422-2

Notation as in Appendix VI, Table 4. Source: From McMaster et al., 1969.

Sy

H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn

Z

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0.2292 0.2702 0.3200 0.3771 0.4378 0.5004 0.5637 0.6282 0.6946 0.7690

EL1

77.00266 72.38938 72.50341 72.03564 72.91417 0.15632 2.10634 3.34768 3.81364 3.05305 2.76125 3.00797 2.62786 0.68453 2.44853 2.03061 2.33976 2.38673 2.53782 2.65431

l1

5.19086 3.24818 3.43696 3.43789 4.13834 2.00149 0.28244 70.83777 71.04042 70.24003 0.29160 0.19834 0.68939 3.86200 1.46227 2.27680 1.88017 1.99445 1.85115 1.83552

l2

l3

70.34413 70.11121 70.12806 70.13520 70.22931 0.32080 0.92539 1.34549 1.42378 1.25060 1.09998 1.11670 1.01331 70.43637 0.69759 0.31534 0.55084 0.50415 0.62868 0.65972

EL1 < E < EK

70.04820 70.11707 70.16880 70.17980 70.17486 70.16451 70.16546 70.16625 0.04226 70.12878 70.07434 70.11829 70.11481 70.14231 70.15525

l4

0.2838 0.4016 0.5320 0.6854 0.8669 1.0721 1.3050 1.5596 1.8389 2.1455 2.4720 2.8224 3.2029 3.6074 4.0381 4.4928 4.9664 5.4651 5.9892 6.5390

EK 76.48071 73.51566 72.36523 71.38914 70.58244 0.11476 0.63927 1.08730 1.42132 1.81194 2.07992 2.39196 2.62043 2.88077 3.06672 3.28801 3.42667 3.53219 3.75863 3.93329 4.00179 4.11127 4.21463 4.35405 4.44148

k1 3.27538 2.60589 2.88866 3.05349 3.07947 3.09245 3.08731 3.07389 3.06182 3.04034 3.04059 3.00737 2.99387 2.96887 2.95078 2.92821 2.90289 2.88277 2.85955 2.83775 2.81618 2.79692 2.77190 2.75878 2.72589

k2

k3 0.04385 0.26361 0.15389 0.07919 0.05871 0.03213 0.01587 0.00035 70.00946 70.02320 70.02978 70.04492 70.05681 70.06174 70.07943 70.08514 70.09061 70.10191 70.09992 70.11226 70.11678 70.11885 70.13140 70.13335 70.14641

E > EK

70.00866 70.03787 70.02935 70.02484 70.02647 70.02494 70.02538 70.02438 70.02708 70.02568 70.03575 70.02734 70.03063 70.02724 70.03227 70.03094 70.02881 70.03142 70.02689 70.02972 70.02898 70.02784 70.03049 70.03142 70.03205

k4

Table1 Parameters for Calculating the Total Mass Attenuation Coefﬁcients for Photon Energies Above the K Absorption Edge (k1, k2, k3, k4) and for Energies Between L1 and EK Absorption Edge (l1, l2, l3, l4); Z ¼ Atomic Number, Sy ¼ Symbol of Corresponding Element

APPENDIX X: PARAMETERS FOR CALCULATING THE TOTAL MASS ATTENUATION COEFFICIENTS IN THE ENERGY RANGE 0.1^1000 keV [VIA EQ. (78)]

78 Markowicz

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Te Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs Ba La Ce Pr Nd

0.8461 0.9256 1.0081 1.0961 1.1936 1.2977 1.4143 1.5265 1.6539 1.7820 1.9210 2.0651 2.2163 2.3725 2.5316 2.6977 2.8655 3.0425 3.2240 3.4119 3.6043 3.8058 3.0180 4.2375 4.4647 4.6983 4.9392 5.1881 5.4528 5.7143 5.9888 6.2663 6.5488 6.8348 7.1260

2.41689 2.60591 2.69756 2.75840 2.87225 2.93846 3.02673 3.12353 3.19106 3.30117 3.36926 3.46256 3.54696 3.63965 3.71677 3.79880 3.86387 3.93865 3.99914 4.07189 4.12607 4.19974 4.24309 4.30447 4.35252 4.40598 4.43690 4.51763 4.53737 4.61780 4.65770 4.71490 4.77277 4.83442 4.87359

2.80062 2.55718 2.72893 2.72647 2.72029 2.73406 2.74409 2.74414 2.76121 2.73599 2.74007 2.74320 2.73914 2.73646 2.73020 2.72885 2.72657 2.72262 2.70967 2.71531 2.71194 2.70350 2.70139 2.69926 2.69160 2.68835 2.68323 2.67963 2.66911 2.66299 2.65946 2.65505 2.64710 2.64088 2.63355 70.00696 0.18669 0.04244 0.04391 0.04441 0.03264 0.01747 0.01165 70.01683 0.01398 0.00224 70.10161 70.00741 70.01290 70.01273 70.01869 70.02278 70.02487 70.02706 70.02854 70.03442 70.03661 70.04152 70.04319 70.04781 70.04963 70.05424 70.05663 70.05780 70.06277 70.07172 70.07121 70.07113 70.07717 70.07273

70.01904 70.06696 70.03220 70.03423 70.03527 70.03438 70.03117 70.03096 70.02249 70.03428 70.03125 70.02723 70.03104 70.03036 70.03038 70.02896 70.02881 70.03004 70.03l74 70.03173 70.02941 70.02694 70.02776 70.03320 70.02689 70.03001 70.03161 70.03296 70.02658 70.02884 70.03577 70.03442 70.03077 70.03325 70.02815

7.1120 7.7089 8.3328 8.9789 9.6586 10.3671 11.1031 11.8667 12.6578 13.4737 14.3256 15.1997 16.1046 17.0384 17.9976 18.9856 19.9995 21.0440 22.1172 23.2199 24.3503 25.5140 26.7112 27.9399 29.2001 30.4912 31.8138 33.1694 34.5614 35.9846 37.4406 38.9246 40.4430 41.9906 43.5689

4.56600 4.64390 4.77308 4.81074 4.89850 4.93990 5.00743 5.07635 5.11759 5.19474 5.23284 5.29354 5.34695 5.41166 5.45290 5.50180 6.47503 5.57355 5.60095 5.63478 5.65532 5.68392 5.68137 5.72030 5.76598 5.74957 3.74700 5.79401 5.80690 5.82191 5.84551 5.85321 5.87476 5.90293 5.90905

2.70395 2.67587 2.65503 2.62267 2.60534 2.57045 2.55814 2.53376 2.50569 2.48145 2.44844 2.41378 2.38309 2.37364 2.33638 2.30251 3.03067 2.24202 2.21314 2.17349 2.14514 2.05134 2.09840 2.04363 2.05890 1.98639 1.96282 1.93222 1.91476 1.87715 1.87466 1.82364 1.79700 1.77311 1.74652 70.15515 70.16255 70.16936 70.18383 70.18393 71.19961 70.19972 70.20666 70.21581 70.22156 70.23505 70.24899 70.25992 70.25521 70.26879 70.28165 70.04965 70.29909 70.30745 70.32262 70.32915 70.34706 70.36341 70.36014 70.34949 70.37446 70.37979 70.38924 70.39242 70.040213 70.39966 70.41682 70.42193 70.42727 70.43341

70.03352 70.03403 70.03476 70.03729 70.03594 70.03856 70.03827 70.03872 70.03969 70.04024 70.04200 70.04399 70.04534 70.04393 70.04558 70.04737 70.03050 70.04910 70.04983 70.05195 70.05230 70.05473 70.05665 70.05573 70.05406 70.05693 70.05738 70.05845 70.05865 70.05948 70.05904 70.06098 70.06126 70.06168 70.06218

X-ray Physics 79

Sy

Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Fr Ra

61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88

7.4279 7.7368 8.0520 8.3756 8.7080 9.0458 9.3942 9.7513 10.1157 10.4864 10.8704 11.2707 11.6815 12.0998 12.5267 12.9680 13.4185 13.8799 14.3528 14.8393 15.3467 15.8608 16.3875 16.9393 17.4930 18.0490 18.6390 19.2367

EL1

Continued

Z

Table 1

4.93318 4.95712 5.00931 5.03443 5.08055 5.11495 5.15537 5.19648 5.23938 5.26855 5.31163 5.34305 5.37806 5.41320 5.44913 5.47715 5.50865 5.54158 5.57564 5.60041 5.62415 5.65349 5.69400 5.73466 5.76863 5.74682 5.78319 5.80588

l1 2.62474 2.61647 2.61226 2.60481 2.60115 2.59238 2.58071 2.57148 2.56466 2.55541 2.55490 2.54268 2.53595 2.53128 2.52042 2.52666 2.50428 2.50551 2.48980 2.47759 2.46173 2.45575 2.47086 2.46840 2.45600 2.43686 2.43220 2.41932

l2 70.08002 70.07625 70.08680 70.08912 70.07597 70.08230 70.08885 70.09459 70.09047 70.10174 70.09208 70.10077 70.09548 70.10036 70.10577 70.09007 70.10492 70.09847 70.10793 70.11241 70.12251 70.12044 70.10349 70.09608 70.10153 70.11031 70.10993 70.11330

l3

EL1 < E < EK

70.02998 70.02487 70.03205 70.03162 70.02286 70.02541 70.02644 70.02802 70.02454 70.03012 70.02484 70.02734 70.02359 70.02631 70.02724 70.02154 70.02495 70.02303 70.02561 70.02570 70.02850 70.02698 70.02277 70.01973 70.02099 70.02272 70.02259 70.02255

l4 45.1840 46.8342 48.5190 50.2391 51.9957 53.7885 55.6177 57.4855 59.3896 61.3323 63.3138 65.3508 67.4164 69.5250 71.6764 73.8708 76.1110 78.3948 80.7249 83.1023 85.5304 88.0045 90.5259 93.1050 95.7299 98.4040 101.1370 103.9219

EK 5.92214 5.92934 5.94975 5.92827 5.95498 5.95991 5.94804 5.96408 5.88779 6.04546 6.07395 6.00771 6.05159 6.00250 6.03030 6.11400 6.09889 6.15378 6.15303 6.19862 6.17641 6.17648 6.30427 6.21512 6.18945 6.24492 6.31471 6.26760

k1 1.70715 1.69904 1.67827 1.64215 1.63471 1.61100 1.55826 1.54243 1.43854 1.57974 1.57891 1.48367 1.50856 1.43225 1.43738 1.50446 1.45990 1.53187 1.47494 1.50676 1.46912 1.45115 1.55736 1.43087 1.38290 1.47699 1.51865 1.45136

k2

k3 70.44407 70.44407 70.44795 70.45629 70.45467 70.46029 70.47717 70.47949 70.50808 70.45878 70.45631 70.48598 70.47273 70.49552 70.48952 70.46496 70.47717 70.44176 70.46441 70.45079 70.46009 70.46277 70.42520 70.46426 70.47598 70.43939 70.42408 70.44373

E > EK

70.06326 70.06312 70.06335 70.06402 70.06359 70.06408 70.06595 70.06606 70.06870 70.06334 70.06301 70.06608 70.06439 70.06672 70.06580 70.06309 70.06429 70.05974 70.06253 70.06097 70.06183 70.06196 70.05792 70.06195 70.06300 70.05879 70.05721 70.05917

k4

80 Markowicz

Ac Th Pa U

19.8400 20.4721 21.1046 21.7574

5.83228 5.85156 5.87741 5.88224

2.39643 2.40038 2.35466 2.34307 70.12322 70.11549 70.14418 70.15054

70.02407 70.02204 70.02817 70.03027

106.7553 109.6509 112.6014 115.6061

6.32123 6.39823 6.28773 6.30701

1.48583 1.55418 1.41526 1.44776 70.42883 70.40447 70.44597 70.43028

70.05744 70.05479 70.05907 70.05716

Source: From Orlic et al., 1993. Reprinted with kind permission from Elsevier Science – NL, Sara Burgehartstraat 25, 1055 KV Amsterdam, The Netherlands.

89 90 91 92

X-ray Physics 81

S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1085 0.1487 0.2122 0.2118 0.2370 0.2631

Sy E0 =Eedge

Z

3.98381 72.71337 2.07772 8.53825 5.10006 7.28382 6.39592 2.83998 77.38556 2.13747 2.21508 4.76476 4.28916 2.32551 4.23694 2.60598 4.27624 5.14580 6.74588 7.26941 11.95287 15.89070 11.09104 5.18978 6.15438 5.83085 6.35098 5.57003

m1

71.96207 2.78307 70.84611 74.62277 71.85097 73.26298 72.45282 0.44991 8.17416 1.11467 1.10224 71.05687 70.66344 1.22642 70.64439 0.84383 70.65284 71.51272 73.03410 73.69837 78.69443 712.8788 78.15432 71.99566 72.90381 72.56434 73.21302 72.03397

m2 1.29140 0.23822 1.21097 1.97295 1.29907 1.60507 1.38399 0.61932 71.29292 0.46032 0.47160 1.09423 1.02071 0.45413 1.08840 0.68376 1.15806 1.47391 1.98313 2.28437 4.04104 5.51561 4.08776 2.09278 2.42622 2.36756 2.67445 2.13536

m3

E0<E<EL3

70.13371 70.05767 70.14523 70.19719 70.14560 70.16673 70.14698 70.08033 0.07794 70.06687 70.06697 70.12565 70.12207 70.06650 70.13570 70.10262 70.15244 70.19017 70.24674 70.28824 70.48832 70.65735 70.52451 70.32184 70.36813 70.37176 70.41914 70.33831

m4

1.0197 1.1154 1.2167 1.3231 1.4358 1.5499 1.6749 1.8044 1.9396 2.0800 2.2223 2.3705 2.5202 2.6769

EL3

77.206 47.827 20.461 6.9330 0.0065 74.6062 75.6370 3.2959 2.0366 1.9840 2.2290 2.724 2.813 3.044

l31

727.639 716.433 75.2040 0.6640 3.8833 6.2403 6.9490 2.4609 3.1926 3.2996 3.2290 2.999 2.999 2.907

l32

EL3 < E < EL2

1.0428 1.1423 1.2478 1.3586 1.4762 1.5960 1.7272 1.8639 2.0068 2.1555 2.3067 2.4647 2.6251 2.7932

EL2

9.4520 6.2330 5.4560 5.1580 3.9491 3.8764 3.4100 4.0979 3.9355 3.9621 3.9270 3.981 3.996 4.051

l21

70.2320 1.1180 1.4580 1.6080 2.1968 2.2734 2.5440 2.2097 2.3328 2.3596 2.4170 2.425 2.447 2.452

l22

EL2 < E < EL1

0.2292 0.2702 0.3200 0.3771 0.4378 0.5004 0.5637 0.6282 0.6946 0.7690 0.8461 0.9256 1.0081 1.0961 1.1936 1.2977 1.4143 1.5265 1.6539 1.7820 1.9210 2.0651 2.2163 2.3725 2.5316 2.6977 2.8655 3.0425

EL1

Table 2 Parameters for Calculating the Total Mass Attentuation Coefﬁcients for Photon Energy Between L1, L2, and L3 Absorption Edges and for Energies Between L3 and Lower Energy Limit E0 or M Absorption Edge

82 Markowicz

44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78

Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs Ba La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt

0.2901 0.3180 0.3425 0.3766 0.4129 0.4530 0.4946 0.5379 0.5827 0.6293 0.6775 1.2171 1.2928 1.3613 1.4346 1.5110 1.5753 1.6540 1.7228 1.8000 1.8808 1.9675 2.0468 2.1283 2.2065 2.3068 2.3981 2.4912 2.6009 2.7080 2.8196 2.9317 3.0485 3.1737 3.2960

9.99349 6.55219 6.32821 5.65378 3.12801 2.78289 2.57173 2.52338 4.98562 3.00387 3.00203 3.16785 3.19080 3.23265 3.29411 3.35891 3.41765 3.45084 3.49847 3.54762 3.58135 3.61719 3.67219 3.72181 3.77845 3.82800 3.86458 3.91793 3.94527 3.99165 4.02988 4.07043 4.10244 4.14636 4.18356

78.27969 73.56318 73.29827 72.26740 1.54731 2.29146 2.70219 2.80299 71.29951 2.36374 2.56894 2.42799 2.48392 2.53710 2.53524 2.54117 2.51947 2.59000 2.55440 2.56798 2.57591 2.61430 2.57359 2.55757 2.52004 2.53261 2.50726 2.48402 2.52598 2.50855 2.51689 2.52499 2.53312 2.52893 2.53390

5.01647 2.98931 2.95809 2.49736 0.70511 0.30064 0.11208 0.07373 2.27938 0.26067 0.10775 0.12755 0.08776 0.05145 0.05465 0.04839 0.05777 0.00563 0.02671 0.01585 0.00022 70.03278 70.00002 0.010760 0.05588 0.03019 0.06374 0.08777 0.03404 0.05734 0.04574 0.03041 0.01448 0.01458 0.00520 70.76636 70.48554 70.49645 70.43066 70.16236 70.09544 70.07092 70.06562 70.44589 70.09483 70.06179 70.05218 70.04429 70.03744 70.03926 70.03842 70.04029 0.02894 70.03367 70.03181 70.02718 70.01918 70.02988 70.03537 70.04817 70.03886 70.05192 70.06146 70.04239 70.05289 70.05071 70.04512 70.03948 70.04009 70.03783

2.8379 3.0038 3.1733 3.3511 3.5375 3.7301 3.9288 4.1322 4.3414 4.5571 4.7822 5.0119 5.2470 5.4827 5.7234 5.9642 6.2079 6.4593 6.7162 6.9769 7.2428 7.5140 7.7901 8.0711 8.3579 8.6480 8.9436 9.2441 9.5607 9.8811 10.2068 10.5353 10.8709 11.2152 11.5637

3.320 3.653 4.146 4.111 3.971 4.293 4.834 4.637 4.618 4.577 4.572 4.215 4.363 4.401 4.449 4.469 4.463 4.500 4.536 4.553 4.583 4.633 4.655 4.695 4.727 4.767 4.802 4.847 4.866 4.911 4.940 4.979 5.006 5.050 5.084

2.770 2.586 2.250 2.321 2.454 2.217 1.753 1.940 1.955 2.048 2.073 2.528 2.384 2.397 2.402 2.438 2.500 2.528 2.502 2.556 2.543 2.537 2.572 2.558 2.581 2.592 2.569 2.560 2.629 2.584 2.624 2.629 2.661 2.673 2.666

2.9669 3.1469 3.3303 3.5237 3.7270 3.9380 4.1561 4.3804 4.6120 4.8521 5.1037 5.3594 5.6236 5.8906 6.1642 6.4404 6.7215 7.0128 7.3118 7.6171 7.9303 5.2516 8.5806 8.9178 9.2643 9.6169 9.9782 10.3386 10.7394 11.1361 11.5440 11.9687 12.3850 12.8241 13.2726

4.303 4.091 4.344 4.217 4.328 4.620 4.592 4.584 4.554 4.625 4.599 4.563 4.601 4.698 4.702 4.757 4.778 4.854 4.861 4.924 4.927 4.985 5.014 5.047 5.086 5.128 5.082 5.181 5.214 5.237 5.276 5.302 5.323 5.355 5.375

2.302 2.502 2.337 2.487 2.420 2.196 2.238 2.279 2.323 2.318 2.375 2.482 2.470 2.404 2.470 2.467 2.490 2.449 2.469 2.428 2.467 2.423 2.420 2.426 2.414 2.398 2.815 2.431 2.377 2.468 2.318 2.373 2.314 2.361 2.278

3.2240 3.4119 3.6043 3.8058 3.0180 4.2375 4.4647 4.6983 4.9392 5.1881 5.4528 5.7143 5.9888 6.2663 6.5488 6.8348 7.1260 7.4279 7.7368 8.0520 8.3756 8.7080 9.0458 9.3942 9.7513 10.1157 10.4864 10.8704 11.2707 11.6815 12.0998 12.5267 12.9680 13.4185 13.8799

X-ray Physics 83

Au Hg Tl Pb Bi Po At Rn Fr Ra Ac Th Pa U

79 80 81 82 83 84 85 86 87 88 89 90 91 92

3.4249 3.5616 3.7041 3.8507 3.9991 4.1494 4.3170 4.4820 4.6520 4.8220 5.0020 5.1823 5.3669 5.5480

E0 =Eedge

Continued

4.22770 4.26093 4.29302 4.32934 4.36991 4.41866 4.46126 4.45433 4.49562 4.52666 4.56932 4.59153 4.63772 4.65216

m1

2.52409 2.51954 2.51974 2.51778 2.52184 2.52158 2.51394 2.51411 2.51108 2.50733 2.50607 2.50190 2.49776 2.49170

m2 0.01426 0.02092 0.01369 0.00460 70.01213 70.00971 0.00640 70.00666 70.01358 0.00692 70.01644 70.00815 70.00936 70.02703

m3

E0<E<EL3

70.04262 70.04665 70.04376 70.03614 70.02654 70.03298 70.04593 70.03871 70.02823 70.05016 70.03124 70.03948 70.03490 70.01144

m4 11.9187 12.2839 12.6575 13.0352 13.4186 13.8138 14.2135 14.6194 15.0312 15.4444 15.8710 16.3003 16.7331 17.1663

EL3 5.125 5.156 5.184 5.220 5.257 5.301 5.347 5.334 5.376 5.410 5.454 5.490 5.525 5.541

l31 2.690 2.643 2.630 2.644 2.642 2.625 2.642 2.637 2.647 2.653 2.662 2.696 2.666 2.672

l32

EL3 < E < EL2

13.7336 14.2087 14.6979 15.2000 15.7111 16.2443 16.7847 17.3371 17.9065 18.4843 19.0832 19.6932 20.3137 20.9476

EL2 5.406 5.430 5.451 5.512 5.536 5.555 5.574 5.535 5.650 5.644 5.761 5.883 5.798 5.829

l21 2.303 2.337 2.323 2.502 2.453 2.387 2.335 2.281 2.494 2.411 2.594 2.800 2.534 2.571

l22

EL2 < E < EL1

Source: From Orlic et al., 1993. Reprinted with kind permission from Elsevier Science – NL, Sara Burgehartstraat 25, 1055 KV Amsterdam, The Netherlands.

Sy

Z

Table 2

14.3528 14.8393 15.3467 15.8608 16.3875 16.9393 17.4930 18.0490 18.6390 19.2367 19.8400 20.4721 21.1046 21.7574

EL1

84 Markowicz

E0

0.1085 0.1085 0.1085 0.1085 0.1085 0.1328 0.1328 0.1487 0.1717 0.1717 0.1511 0.1511 0.1511 0.1833 0.2122 0.2770 0.1926 0.2122 0.3117 0.3117 0.2770 0.2770 0.2770 0.2770 0.1717 0.2122 0.2770 0.2122

Z

41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68

n2

n3

n4

EM5

m42

m41

EM5 < E < EM4 EM4

m32

m31

EM4 < E < EM3 EM3

m22

2.621 2.098 2.132 2.162 1.551 2.193 2.234 2.255 2.278 2.308 2.296 2.342 2.363 2.368

m21

EM3 < E < EM2

115.013 769.4208 14.9364 71.0543 106.662 767.3126 15.2602 71.1408 3.5107 2.3889 70.4178 0.0324 89.5750 757.9740 13.8424 71.1025 80.5898 757.4238 15.0813 71.3052 76.3873 1.0408 1.9749 70.3214 29.0389 718.5282 5.5664 70.5470 11.0289 76.3556 2.6658 70.3046 174.946 7138.963 38.3880 73.5077 42.9485 731.1679 9.1280 70.8669 39.8331 730.6663 9.5340 70.9502 37.0196 728.3109 8.8955 70.8939 14.2756 79.7352 3.9179 70.4554 23.1086 718.4405 6.7995 70.7732 32.5527 727.5567 9.7373 71.0874 0.7330 2.364 31.6390 728.1233 10.4351 71.2218 0.7905 0.7961 3.768 2.088 1.0622 3.897 28.6047 724.1290 8.8027 71.0079 0.8473 0.8485 3.554 2.190 1.1234 3.870 3.9181 70.9835 1.6398 70.2735 0.8861 0.9013 3.194 2.376 1.1854 3.852 4.6993 0.0130 0.6969 70.0975 0.9349 0.9511 3.317 2.355 1.2422 5.314 20.5246 716.5997 6.5042 70.7721 0.9843 0.9999 5.484 1.419 1.2974 3.879 16.9684 713.1825 5.4255 70.6587 1.0269 96.373 735.542 1.0515 6.609 0.921 1.3569 3.845 11.8798 77.7898 3.5371 70.4381 1.0802 89.021 733.267 1.1060 6.352 1.023 1.4198 3.825 5.6437 71.4828 1.4474 70.2096 1.1309 83.564 731.660 1.1606 6.316 1.039 1.4806 3.825 6.6183 72.4260 1.7586 70.2443 1.1852 76.154 729.235 1.2172 8.597 0.048 1.5440 3.792 75.6112 10.4330 72.6765 0.2602 1.2412 73.753 728.683 1.2750 6.289 1.062 1.6113 3.863 1.4535 3.1211 70.1907 70.0138 1.2949 69.682 727.456 1.3325 6.096 1.157 1.6756 3.809 4.077 0.1743 0.9232 70.1528 1.3514 65.046 725.898 1.3915 5.446 1.471 1.7412 3.811 1.6315 2.7196 0.0767 70.0607 1.4093 60.848 724.477 1.4533 5.353 1.523 1.8118 3.847

n1

E0 < E < Eedge

1.0650 1.1367 1.2044 1.2728 1.3774 1.4028 1.4714 1.5407 1.6139 1.6883 1.7677 1.8418 1.9228 2.0058

EM2

3.983 3.957 3.970 4.007 2.489 3.993 4.039 3.986 4.005 4.032 4.027 4.060 4.092 4.087

m12

2.072 2.097 2.114 2.120 2.863 2.167 2.170 2.206 2.220 2.217 2.242 2.243 2.245 2.269

m11

EM2 < E < EM1

1.2171 1.2928 1.3613 1.4346 1.5110 1.5753 1.6540 1.7228 1.8000 1.8808 1.9675 2.0468 2.1283 2.2065

EM1

Table 3 Parameters for Calculating the Total Mass Attenuation Coefﬁcients for Photon Energies Between M1, M2, . . ., M5 Absorption Edges and for Energies Between M5 and Lower Energy Limit E0

0.1926 0.1085 0.1085 0.1085 0.1085 1.1085 0.1085 0.1085 0.1085 0.1328 0.1511 0.2122 0.1926 0.1717 0.1926 0.2122 0.3117 0.3924 0.3924 0.3924 0.3924 0.2770 0.2770 0.2770

69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92

75.0355 71.8380 70.3276 70.1868 71.9405 71.1407 70.3842 71.8806 72.9368 1.5113 3.5698 3.6427 3.6296 4.6393 5.7697 5.6131 5.1757 4.2011 4.5198 5.5170 6.3317 4.1761 4.3146 3.9954

n1

Continued

9.4382 6.2849 4.8522 4.8402 6.5977 5.9151 5.1929 6.8129 7.9106 3.4177 1.1260 1.0962 1.1014 70.1055 71.3933 71.3279 70.6452 0.6074 0.1763 71.1906 72.2638 0.8164 0.8409 1.2801

n2

72.1146 71.0736 70.6175 70.6304 71.1795 70.9671 70.7225 71.2684 71.6078 70.1106 0.7499 0.7581 0.7912 1.2868 1.7810 1.8314 1.5552 1.0490 1.2693 1.9034 2.3875 1.0031 0.9488 0.7662

n3

E0 < E < Eedge

0.1726 0.0593 0.0113 0.0127 0.0672 0.0439 0.0155 0.0733 0.1054 70.0581 70.1652 70.1655 70.1770 70.2451 70.3078 70.3245 70.2925 70.2281 70.2654 70.3623 70.4354 70.2383 70.2273 70.2038

n4 1.4677 1.5278 1.5885 1.6517 1.7351 1.8092 1.8829 1.9601 2.0404 2.1216 2.2057 2.2949 2.3893 2.4840 2.5796 2.6839 2.7867 2.8924 2.9999 3.1049 3.2190 3.3320 3.4418 3.5517

EM5 63.572 54.494 57.769 55.928 53.589 50.662 30.913 54.605 48.393 45.233 43.502 40.458 31.751 20.254 9.171 70.200 74.893 77.059 72.615 0.627 2.581 3.793 3.639 3.662

m42 726.240 722.411 724.531 724.196 723.567 722.588 712.446 725.802 722.958 721.695 721.199 719.895 715.013 78.092 71.075 5.161 8.501 10.172 7.198 4.924 3.529 2.621 2.790 2.788

m41

EM5 < E < EM4

1.5146 1.5763 1.6394 1.7164 1.7932 1.8716 1.9489 2.0308 2.1161 2.2019 2.2911 2.3849 2.4851 2.5856 2.6876 2.7980 2.9087 3.0215 3.1362 3.2484 3.3702 3.4908 3.6112 3.7276

EM4 5.841 5.900 6.893 7.170 6.974 6.894 6.854 6.900 7.249 7.209 7.102 6.847 6.055 5.339 4.475 3.586 3.141 3.222 3.649 3.982 4.255 4.355 4.369 4.409

m32 1.257 1.222 0.681 0.527 0.627 0.671 0.666 0.581 0.343 0.344 0.393 0.540 1.064 1.574 2.212 2.895 3.276 3.251 2.961 2.716 2.525 2.455 2.484 2.460

m31

EM4 < E < EM3

1.8845 1.9498 2.0236 2.1076 2.1940 2.2810 2.3673 2.4572 2.5507 2.6454 2.7430 2.8471 2.9566 3.0664 3.1769 3.3019 3.4260 3.5280 3.6638 3.7918 3.9098 4.0461 4.1738 4.3034

EM3 3.889 3.916 3.876 3.916 3.957 4.019 4.063 4.032 4.126 4.166 4.199 4.187 4.191 4.259 4.277 4.327 4.372 4.382 4.405 4.391 4.460 4.495 4.526 4.527

m22 2.370 2.373 2.422 2.420 2.420 2.408 2.405 2.445 2.410 2.408 2.414 2.445 2.467 2.443 2.459 2.459 2.457 2.442 2.458 2.501 2.475 2.462 2.477 2.492

m21

EM3 < E < EM2

2.0998 2.1730 2.2635 2.3654 2.4687 2.5749 2.6816 2.7922 2.9087 3.0265 3.1478 3.2785 3.4157 3.5542 3.6963 3.8541 4.0080 4.1590 4.3270 4.4895 4.6560 4.8304 5.0009 5.1822

EM2 4.114 4.125 4.129 4.158 4.146 4.178 4.214 4.228 4.307 4.323 4.317 4.382 4.314 4.453 4.383 4.460 4.494 4.500 4.523 4.511 4.599 4.576 4.603 4.614

m12

2.278 2.287 2.309 2.311 2.343 2.344 2.346 2.356 2.327 2.338 2.370 2.344 2.419 2.332 2.420 2.395 2.402 2.386 2.398 2.442 2.387 2.437 2.457 2.458

m11

EM2 < E < EM1

Source: From Orlic et al., 1993. Reprinted with kind permission from Elsevier Science – NL, Sara Burgehartstraat 25, 1055 KV Amsterdam, The Netherlands.

E0

Z

Table 3

2.3068 2.3981 2.4912 2.6009 2.7080 2.8196 2.9317 3.0485 3.1737 3.2960 3.4249 3.5616 3.7041 3.8507 3.9991 4.1494 4.3170 4.4820 4.6520 4.8220 5.0020 5.1823 5.3669 5.5480

EM1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

Atomic number

H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co

Element

Absorber C 4.48 þ 1 2.77 7 1 4.89 þ 2 4.06 þ 3 1.08 þ 4 2.29 þ 4 3.75 þ 4 2.75 þ 3 4.71 þ 3 6.90 þ 3 9.73 þ 3 1.36 þ 4 1.80 þ 4 2.44 þ 4 2.92 þ 4 3.47 þ 4 4.19 þ 4 4.95 þ 4 5.17 þ 4 5.24 þ 4 5.92 þ 3 7.15 þ 3 7.80 þ 3 8.45 þ 3 9.22 þ 3 1.11 þ 4 1.20 þ 4 1.40 þ 4 1.54 þ 4

B 6.67 þ 1a 1.83 7 1

1.81 þ 3 1.20 þ 4 2.98 þ 4 5.82 þ 4 3.93 þ 3 7.32 þ 3 1.19 þ 4 1.74 þ 4 2.45 þ 4 3.72 þ 4 4.25 þ 4 5.62 þ 4 6.44 þ 4 7.48 þ 4 7.49 þ 4 8.18 þ 4 7.44 þ 3 9.51 þ 3 1.14 þ 4 1.31 þ 4 1.43 þ 4 1.54 þ 4 1.68 þ 4 2.09 þ 4 2.23 þ 4 2.61 þ 4 2.86 þ 4

1.51 þ 2 1.25 þ 3 3.67 þ 3 8.40 þ 3 1.53 þ 4 2.44 þ 4 1.83 þ 3 2.68 þ 3 3.73 þ 3 5.70 þ 3 7.26 þ 3 1.00 þ 4 1.23 þ 4 1.58 þ 4 1.93 þ 4 2.38 þ 4 2.67 þ 4 3.00 þ 4 3.51 þ 4 3.49 þ 4 3.97 þ 3 4.35 þ 3 4.77 þ 3 5.61 þ 3 6.13 þ 3 7.10 þ 3 7.85 þ 3

N 3.16 þ 1 3.92 7 1 5.85 þ 1 5.07 þ 2 1.60 þ 3 3.78 þ 3 7.22 þ 3 1.22 þ 4 1.73 þ 4 1.27 þ 3 1.79 þ 3 2.71 þ 3 3.52 þ 3 4.89 þ 3 6.10 þ 3 7.93 þ 3 9.74 þ 3 1.21 þ 4 1.39 þ 4 1.59 þ 4 1.94 þ 4 2.21 þ 4 2.34 þ 4 2.20 þ 4 1.66 þ 4 3.13 þ 3 3.46 þ 3 4.00 þ 3 4.41 þ 3

O 2.36 þ 1 5.25 7 1 2.62 þ 1 2.23 þ 2 7.57 þ 2 1.86 þ 3 3.63 þ 3 6.29 þ 3 9.23 þ 3 1.21 þ 4 9.12 þ 2 1.39 þ 3 1.83 þ 3 2.55 þ 3 3.22 þ 3 4.22 þ 3 5.19 þ 3 6.52 þ 3 7.50 þ 3 8.59 þ 3 1.07 þ 4 1.24 þ 4 1.34 þ 4 1.45 þ 4 1.57 þ 4 1.58 þ 4 1.68 þ 4 2.31 þ 3 2.58 þ 3

F 1.83 þ 1 6.77 7 1 1.22 þ 1 1.06 þ 2 3.84 þ 2 9.59 þ 2 1.93 þ 3 3.43 þ 3 5.14 þ 3 6.94 þ 3 8.54 þ 3 7.54 þ 2 1.01 þ 3 1.42 þ 3 1.80 þ 3 2.37 þ 3 2.92 þ 3 3.69 þ 3 4.27 þ 3 4.89 þ 3 6.13 þ 3 7.19 þ 3 7.79 þ 3 8.59 þ 3 9.40 þ 3 1.09 þ 4 1.16 þ 4 1.31 þ 4 1.22 þ 4

Ne 1.46 þ 1 8.49 7 1

Emitter wavelength (A˚) energy (keV)

APPENDIX XI: TOTAL MASS ATTENUATION COEFFICIENTS FOR LOW-ENERGY K LINES

6.32 þ 0 5.58 þ 1 2.09 þ 2 5.44 þ 2 1.10 þ 3 1.98 þ 3 3.02 þ 3 4.17 þ 3 5.23 þ 3 7.11 þ 3 5.93 þ 2 8.37 þ 2 1.07 þ 3 1.41 þ 3 1.74 þ 3 2.21 þ 3 2.57 þ 3 2.93 þ 3 3.71 þ 3 4.38 þ 3 4.74 þ 3 5.27 þ 3 5.81 þ 3 6.75 þ 3 7.27 þ 3 8.22 þ 3 8.84 þ 3

Na 1.19 þ 1 1.04 þ 0 3.76 þ 0 3.15 þ 1 1.20 þ 2 3.26 þ 2 6.73 þ 2 1.21 þ 3 1.90 þ 3 2.67 þ 3 3.44 þ 3 4.55 þ 3 5.65 þ 3 5.06 þ 2 6.63 þ 2 8.93 þ 2 1.09 þ 3 1.39 þ 3 1.63 þ 3 1.85 þ 3 2.36 þ 3 2.75 þ 3 2.99 þ 3 3.34 þ 3 3.70 þ 3 4.30 þ 3 4.67 þ 3 5.31 þ 3 5.73 þ 3

Mg 9.89 þ 0 1.25 þ 0

2.23 þ 0 1.77 þ 1 6.82 þ 1 1.89 þ 2 3.94 þ 2 7.22 þ 2 1.14 þ 3 1.63 þ 3 2.11 þ 3 2.84 þ 3 3.53 þ 3 4.39 þ 3 4.07 þ 2 5.52 þ 2 6.78 þ 2 8.64 þ 2 1.01 þ 3 1.15 þ 3 1.47 þ 3 1.74 þ 3 1.90 þ 3 2.13 þ 3 2.36 þ 3 2.74 þ 3 3.00 þ 3 3.42 þ 3 3.71 þ 3

Al 8.34 þ 0 1.49 þ 0

X-ray Physics 87

28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

Atomic number

Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs

C 4.48 þ 1 2.77 7 1 1.81 þ 4 1.96 þ 4 2.26 þ 4 2.42 þ 4 2.64 þ 4 2.86 þ 4 3.03 þ 4 3.22 þ 4 3.28 þ 4 3.56 þ 4 3.47 þ 4 3.10 þ 4 3.06 þ 4 2.96 þ 4 2.23 þ 4 1.41 þ 4 4.18 þ 3 4.64 þ 3 5.22 þ 3 5.48 þ 3 5.75 þ 3 6.06 þ 3 6.27 þ 3 6.50 þ 3 6.58 þ 3 6.99 þ 3 7.27 þ 3 7.52 þ 3

B 6.67 þ 1a 1.83 7 1

3.34 þ 4 3.65 þ 4 4.03 þ 4 4.18 þ 4 4.41 þ 4 4.41 þ 4 4.49 þ 4 4.19 þ 4 3.94 þ 4 3.52 þ 4 2.83 þ 4 2.04 þ 4 6.41 þ 3 4.28 þ 3 4.66 þ 3 4.99 þ 3 5.49 þ 3 5.87 þ 3 6.09 þ 3 6.34 þ 3 6.45 þ 3 6.61 þ 3 6.64 þ 3 6.71 þ 3 6.65 þ 3 5.77 þ 3 5.79 þ 3 5.91 þ 3

Continued

Element

Absorber

Appendix XI

9.19 þ 3 9.96 þ 3 1.17 þ 4 1.26 þ 4 1.40 þ 4 1.55 þ 4 1.69 þ 4 1.86 þ 4 2.05 þ 4 2.17 þ 4 2.29 þ 4 2.34 þ 4 2.46 þ 4 2.67 þ 4 2.29 þ 4 2.29 þ 4 2.47 þ 4 2.48 þ 4 2.21 þ 4 1.10 þ 4 4.10 þ 3 4.42 þ 3 4.70 þ 3 4.94 þ 3 5.09 þ 3 5.52 þ 3 5.88 þ 3 6.13 þ 3

N 3.16 þ 1 3.92 7 1 5.25 þ 3 5.94 þ 3 6.56 þ 3 7.10 þ 3 7.89 þ 3 8.80 þ 3 9.64 þ 3 1.07 þ 4 1.19 þ 4 1.28 þ 4 1.37 þ 4 1.51 þ 4 1.61 þ 4 1.78 þ 4 1.85 þ 4 1.78 þ 4 1.95 þ 4 1.94 þ 4 1.80 þ 4 1.91 þ 4 2.01 þ 4 2.23 þ 4 9.12 þ 3 3.36 þ 3 3.53 þ 3 3.88 þ 3 4.17 þ 3 4.40 þ 3

O 2.36 þ 1 5.25 7 1 3.01 þ 3 3.20 þ 3 3.73 þ 3 4.08 þ 3 4.56 þ 3 5.10 þ 3 5.61 þ 3 6.28 þ 3 6.98 þ 3 7.57 þ 3 8.21 þ 3 9.06 þ 3 9.77 þ 3 1.08 þ 4 1.14 þ 4 1.19 þ 4 1.30 þ 4 1.38 þ 4 1.46 þ 4 1.47 þ 4 1.47 þ 4 1.36 þ 4 1.38 þ 4 1.48 þ 4 1.75 þ 4 1.18 þ 4 2.84 þ 3 3.02 þ 3

F 1.83 þ 1 6.77 7 1 1.80 þ 3 2.01 þ 3 2.25 þ 3 2.47 þ 3 2.75 þ 3 3.08 þ 3 3.38 þ 3 3.97 þ 3 4.23 þ 3 4.63 þ 3 5.04 þ 3 5.58 þ 3 6.03 þ 3 6.69 þ 3 7.09 þ 3 7.45 þ 3 8.16 þ 3 8.79 þ 3 9.42 þ 3 1.00 þ 4 1.04 þ 4 1.10 þ 4 1.08 þ 4 1.13 þ 4 1.13 þ 4 1.02 þ 4 1.08 þ 4 1.15 þ 4

Ne 1.46 þ 1 8.49 7 1

Emitter wavelength (A˚) energy (keV)

1.02 þ 4 9.47 þ 3 5.26 þ 3 1.52 þ 3 1.70 þ 3 1.95 þ 3 2.17 þ 3 2.44 þ 3 2.66 þ 3 2.92 þ 3 3.20 þ 3 3.55 þ 3 3.86 þ 3 4.27 þ 3 4.55 þ 3 4.90 þ 3 5.37 þ 3 5.80 þ 3 6.23 þ 3 6.66 þ 3 7.09 þ 3 7.51 þ 3 7.88 þ 3 8.26 þ 3 8.35 þ 3 8.24 þ 3 8.72 þ 3 9.79 þ 3

Na 1.19 þ 1 1.04 þ 0 6.58 þ 3 7.04 þ 3 7.51 þ 3 6.92 þ 3 4.72 þ 3 1.30 þ 3 1.53 þ 3 1.68 þ 3 1.72 þ 3 1.89 þ 3 2.08 þ 3 2.32 þ 3 2.53 þ 3 2.80 þ 3 2.99 þ 3 3.57 þ 3 3.92 þ 3 4.23 þ 3 4.54 þ 3 4.86 þ 3 5.84 þ 3 6.20 þ 3 6.50 þ 3 6.82 þ 3 6.90 þ 3 6.93 þ 3 6.75 þ 3 6.67 þ 3

Mg 9.89 þ 0 1.25 þ 0

4.26 þ 3 4.52 þ 3 4.96 þ 3 5.23 þ 3 5.63 þ 3 5.32 þ 3 4.86 þ 3 1.01 þ 3 1.14 þ 3 1.25 þ 3 1.38 þ 3 1.52 þ 3 1.66 þ 3 1.84 þ 3 1.97 þ 3 2.12 þ 3 2.32 þ 3 2.51 þ 3 2.69 þ 3 2.90 þ 3 4.48 þ 3 4.75 þ 3 4.99 þ 3 5.25 þ 3 5.32 þ 3 5.39 þ 3 5.30 þ 3 5.29 þ 3

Al 8.34 þ 0 1.49 þ 0

88 Markowicz

56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90

Ba La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Fr Ra Ac Th

4.09 þ 3 3.76 þ 3 6.79 þ 3 8.74 þ 3 1.05 þ 4 1.26 þ 4 1.48 þ 4 1.58 þ 4 1.49 þ 4 1.87 þ 4 2.04 þ 4 2.16 þ 4 2.32 þ 4 2.43 þ 4 2.28 þ 4 2.15 þ 4 2.13 þ 4 2.04 þ 4 1.93 þ 4 1.76 þ 4 1.58 þ 4 1.38 þ 4 1.13 þ 4 9.09 þ 3 6.62 þ 3 5.42 þ 3 4.19 þ 3 3.17 þ 3 2.54 þ 3 2.28 þ 3 2.52 þ 3 2.69 þ 3 2.66 þ 3 2.49 þ 3 1.99 þ 3

7.64 þ 3 7.74 þ 3 7.98 þ 3 8.77 þ 3 9.43 þ 3 1.04 þ 4 1.13 þ 4 1.20 þ 4 1.12 þ 4 1.24 þ 4 1.37 þ 4 1.47 þ 4 1.60 þ 4 1.72 þ 4 1.79 þ 4 1.75 þ 4 1.80 þ 4 1.82 þ 4 1.85 þ 4 1.84 þ 4 1.67 þ 4 1.61 þ 4 1.57 þ 4 1.49 þ 4 1.47 þ 4 1.29 þ 4 1.14 þ 4 1.01 þ 4 8.62 þ 3 6.67 þ 3 4.13 þ 3 3.02 þ 3 2.42 þ 3 2.59 þ 3 2.12 þ 3

6.21 þ 3 6.32 þ 3 7.14 þ 3 7.60 þ 3 7.92 þ 3 8.40 þ 3 8.95 þ 3 9.42 þ 3 9.39 þ 3 9.93 þ 3 1.06 þ 4 1.13 þ 4 1.21 þ 4 1.29 þ 4 1.35 þ 4 1.36 þ 4 1.27 þ 4 1.32 þ 4 1.37 þ 4 1.39 þ 4 1.39 þ 4 1.44 þ 4 1.50 þ 4 1.51 þ 4 1.50 þ 4 1.40 þ 4 1.38 þ 4 1.35 þ 4 1.34 þ 4 1.31 þ 4 1.21 þ 4 1.10 þ 4 9.31 þ 3 8.17 þ 3 6.44 þ 3

4.52 þ 3 4.66 þ 3 5.16 þ 3 5.45 þ 3 5.64 þ 3 5.92 þ 3 6.26 þ 3 6.58 þ 3 6.60 þ 3 7.38 þ 3 7.83 þ 3 8.29 þ 3 8.86 þ 3 9.49 þ 3 9.94 þ 3 1.01 þ 4 9.96 þ 3 1.04 þ 4 1.09 þ 4 1.12 þ 4 1.12 þ 4 1.15 þ 4 1.12 þ 4 1.16 þ 4 1.17 þ 4 1.21 þ 4 1.23 þ 4 1.26 þ 4 1.33 þ 4 1.23 þ 4 1.16 þ 4 1.17 þ 4 1.17 þ 4 1.15 þ 4 1.11 þ 4

3.13 þ 3 3.26 þ 3 3.57 þ 3 3.77 þ 3 3.89 þ 3 4.07 þ 3 4.27 þ 3 4.49 þ 3 4.53 þ 3 4.99 þ 3 5.26 þ 3 5.56 þ 3 5.94 þ 3 6.36 þ 3 6.67 þ 3 6.86 þ 3 7.20 þ 3 7.54 þ 3 7.90 þ 3 8.18 þ 3 8.36 þ 3 8.36 þ 3 8.83 þ 3 9.23 þ 3 9.30 þ 3 9.55 þ 3 8.93 þ 3 9.23 þ 3 9.68 þ 3 1.02 þ 4 1.02 þ 4 1.05 þ 4 1.04 þ 4 1.01 þ 4 8.83 þ 3

1.16 þ 4 4.21 þ 3 2.45 þ 3 2.59 þ 3 2.68 þ 3 2.79 þ 3 2.93 þ 3 3.08 þ 3 3.12 þ 3 3.39 þ 3 3.56 þ 3 3.76 þ 3 4.00 þ 3 4.27 þ 3 4.47 þ 3 4.62 þ 3 4.86 þ 3 5.11 þ 3 5.36 þ 3 5.57 þ 3 5.74 þ 3 6.09 þ 3 6.43 þ 3 6.74 þ 3 6.95 þ 3 7.24 þ 3 7.10 þ 3 7.35 þ 3 7.72 þ 3 8.14 þ 3 8.26 þ 3 7.52 þ 3 7.75 þ 3 7.88 þ 3 7.85 þ 3

7.51 þ 3 7.89 þ 3 8.81 þ 3 9.49 þ 3 8.29 þ 3 2.51 þ 3 2.01 þ 3 2.12 þ 3 2.17 þ 3 2.34 þ 3 2.47 þ 3 2.61 þ 3 2.77 þ 3 2.96 þ 3 3.10 þ 3 3.22 þ 3 3.38 þ 3 3.53 þ 3 3.70 þ 3 3.87 þ 3 4.02 þ 3 4.27 þ 3 4.51 þ 3 4.68 þ 3 4.77 þ 3 5.13 þ 3 5.34 þ 3 5.55 þ 3 5.82 þ 3 5.70 þ 3 5.72 þ 3 5.99 þ 3 6.22 þ 3 6.35 þ 3 6.22 þ 3

6.36 þ 3 6.56 þ 3 7.12 þ 3 6.10 þ 3 6.19 þ 3 6.59 þ 3 6.94 þ 3 7.32 þ 3 4.57 þ 3 1.61 þ 3 1.73 þ 3 1.87 þ 3 2.01 þ 3 2.14 þ 3 2.32 þ 3 2.42 þ 3 2.54 þ 3 2.45 þ 3 2.57 þ 3 2.69 þ 3 2.92 þ 3 3.11 þ 3 3.28 þ 3 3.28 þ 3 3.31 þ 3 3.88 þ 3 4.04 þ 3 4.20 þ 3 4.42 þ 3 4.57 þ 3 4.46 þ 3 4.57 þ 3 4.68 þ 3 4.49 þ 3 4.64 þ 3

5.21 þ 3 5.22 þ 3 5.43 þ 3 5.26 þ 3 5.47 þ 3 5.73 þ 3 5.84 þ 3 4.88 þ 3 5.01 þ 3 5.49 þ 3 5.68 þ 3 5.70 þ 3 2.84 þ 3 1.56 þ 3 1.47 þ 3 1.54 þ 3 1.62 þ 3 1.77 þ 3 1.86 þ 3 1.95 þ 3 1.99 þ 3 2.12 þ 3 2.24 þ 3 2.29 þ 3 2.36 þ 3 2.50 þ 3 2.61 þ 3 2.72 þ 3 2.88 þ 3 3.04 þ 3 3.04 þ 3 3.18 þ 3 3.30 þ 3 3.40 þ 3 3.48 þ 3 X-ray Physics 89

Pa U Np Pu

Element

C 4.48 þ 1 2.77 7 1 2.38 þ 3 2.30 þ 3 2.41 þ 3 2.39 þ 3

B 6.67 þ 1a 1.83 7 1

2.34 þ 3 2.57 þ 3 2.93 þ 3 3.73 þ 3

Continued

Notation as in Appendix VI, Table 4. Source: From Veigele, 1974.

a

91 92 93 94

Atomic number

Absorber

Appendix XI

3.44 þ 3 2.70 þ 3 2.62 þ 3 2.72 þ 3

N 3.16 þ 1 3.92 7 1 1.13 þ 4 1.09 þ 4 9.65 þ 3 6.59 þ 3

O 2.36 þ 1 5.25 7 1 9.45 þ 3 9.39 þ 3 9.74 þ 3 1.01 þ 4

F 1.83 þ 1 6.77 7 1 8.35 þ 3 8.36 þ 3 8.59 þ 3 8.24 þ 3

Ne 1.46 þ 1 8.49 7 1

Emitter wavelength (A˚) energy (keV)

5.96 þ 3 6.03 þ 3 6.34 þ 3 6.66 þ 3

Na 1.19 þ 1 1.04 þ 0 4.97 þ 3 4.96 þ 3 5.21 þ 3 5.40 þ 3

Mg 9.89 þ 0 1.25 þ 0

3.71 þ 3 3.76 þ 3 3.74 þ 3 3.89 þ 3

Al 8.34 þ 0 1.49 þ 0

90 Markowicz

X-ray Physics

91

APPENDIX XII: CORRESPONDENCE BETWEEN OLD SIEGBAHN AND NEW IUPAC NOTATION X-RAY DIAGRAM LINES Siegbahn

IUPAC

Siegbahn

IUPAC

Ka1 Ka2 Kb1 KbI2

K-L3 K-L2 K-M3 K-N3

La1 La2 Lb1 Lb2

L3-M5 L3-M4 L2-M4 L3-N5

Lg1 Lg2 Lg3 Lg4

L2-N4 L1-N2 L1-N3 L1-O3

KbII2 Kb3 KbI4 KbII4 Kb4x KbI5 KbII5

K-N2 K-M2 K-N5 K-N4 K-N4 K-M5 K-M4

Lb3 Lb4 Lb5 Lb6 Lb7 Lb07 Lb9 Lb10 Lb15 Lb17

L1-M3 L1-M2 L3-O4,5 L3-N1 L3-O1 L3-N6,7 L1-M5 L1-M4 L3-N4 L2-M3

Lg04 Lg5 Lg6 Lg8 Lg08 LZ Ll Ls Lt Lu Lv Ma1 Ma2 Mb Mg Mz

L1-O2 L2-N1 L2-O4 L2-O1 L2-N6(7) L2-M1 L3-M1 L3-M3 L3-M2 L3-N6,7 L2-N6(7) M5-N7 M5-N6 M4-N6 M3-N5 M4,5-N2,3

Source: From Jenkins et al., 1991.

Siegbahn

IUPAC

92

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REFERENCES Arndt UW, Brown PJ, Colliex C, Cowley JM, Creagh DC, Dolling G, Fink M, Freund AK, Hilderbrandt R, Howie A, Hubbell JH, Gjønnes, Lander G, Lynch DF, McAuley WJ, Pynn R, Ross AW, Rowe JM, Sears VF, Spence JCH, Steeds JW, Valvoda V, Willis BTM, Wilson AJC, Zvyagin BB. In: Wilson AJC, ed. International Tables for Crystallography. Vol. C. Mathematical, Physical and Chemical Tables. Dordrecht: Kluwer Academic Publishers, 1992, p 162. Auger P. J Phys 6:205, 1925. Bambynek W, Crasemann B, Fink RW, Freund HU, Mark H, Swift CD, Price RE, Venugopala Rao P. Rev Mod Phys 44:716, 1972; erratum: 46:853, 1974. Behrens P. Trends Anal Chem 11:218, 1992a. Behrens P. Trends Anal Chem 11:237, 1992b. Bethe HA. In: Annalen der Physik, Band 5. Leipzig: Verlag von Johann Ambrosius Barth, 1930, p 325. Bethe HA, Ashkin J. Experimental Nuclear Physics, Vol. 1. New York: Wiley, 1953, p 252. Birks LS. Electron Probe Microanalysis. New York: Wiley–Interscience, 1971a, p 51. Birks LS. Electron Probe Microanalysis. New York: Wiley–Interscience, 1971b, p 172. Burhop EHS. The Auger Eﬀect and Other Radiationless Transitions, New York: Cambridge University Press, 1952. Burr A. In: Robinson JW, ed. Handbook of Spectroscopy, Vol. I. Cleveland, OH: CRC Press, 1974, p 24. Campbell JI, Cookson JA. Nucl Instrum Methods Phys Res B3:185, 1984. Clark GL. Applied X-Rays, New York: McGraw-Hill, 1955, p 144. Clark GL. In: Clark GL, ed. Encyclopedia of X-Rays and Gamma Rays. New York: Reinhold Publishing, 1963, p 1124. Compton AH. Phys Rev 21:15, 1923a. Compton AH. Phys Rev 22:409, 1923b. Compton AH, Allison SK. X-Rays in Theory and Experiment. New York: Van Nostrand, 1935, p 537. Condon EU. In: Condon EU, Odishaw H, eds. Handbook of Physics, Part 7. New York: McGrawHill, 1958, p 7. Davis B, Mitchell DP. Phys Rev 32:331, 1928. Deslattes RD. Acta Crystallogr A25:89, 1969. Dirac PAM. The Principles of Quantum Mechanics. 4th ed. Oxford: Clarendon Press=Oxford University Press, 1958, p 136. Dyson NA. X-rays in Atomic and Nuclear Physics. London: Longman Group, 1973, p 42. Evans RD. In: Flu¨gge S, ed. Encyclopedia of Physics, Vol. XXXIV. Berlin: Springer-Verlag, 1958, p 218. Evans RD. In: Gray DE, ed. American Institute of Physics Handbook. New York: McGraw-Hill, 1963, p 8. Fink RW. In: Robinson, JW, ed. Handbook of Spectroscopy, Vol. I. Cleveland, OH: CRC Press, 1974, p 219. Goldstein JI, Newbury DE, Echlin P, Joy DC, Fiori C, Lifshin E. Scanning Electron Microscopy and X-ray Microanalysis. New York: Plenum Press, 1981, p 105. Green M, Cosslett VE. Proc Phys Soc 78:1206, 1961. Hanke W, Wernisch J, Po¨hn C. X-ray Spectrom 14(1):43, 1985. Hasnain SS, Ed. X-ray Absorption Fine Structure, Proceedings of the Conference Held in York, August 1990. Chichester: Ellis Horwood, 1991. Heinrich KFJ. Electron Beam X-Ray Microanalysis. New York: Van Nostrand–Reinhold, l98l, p 234. Heitler W. The Quantum Theory of Radiation. 3rd ed. London: Oxford University Press, 1954, p 207.

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Hirsch PB. In: Thewlis J, ed. Encyclopaedic Dictionary of Physics. Oxford: Pergamon Press, 1962, p 800. Hubbell JH. Photon Cross Sections, Attenuation Coeﬃcients, and Energy Absorption Coeﬃcients from 10 keV to 100 GeV. Washington, DC: National Bureau of Standards, l969. Hubbell JH. Int J Appl Rad Isot 33:1269, 1982. Hubbell JH. Proceedings of the Workshop on New Directions in Soft X-Ray Photoabsorption, Asilomar Conference Center, Paciﬁc Grove, CA, 1984. Hubbell JH. Report NISTIR-89-4144, U.S. Department of Commerce, National Institute of Standards and Technology, August 1989. Hubbell JH, McMaster WH, Kerr Del Grande N, Mallett JH. In: Ibers JA, Hamilton WC, eds. International Tables for X-Ray Crystallography, Vol. 4. Birmingham, UK: Kynoch Press, 1974, p 47. Hubbell JH, Veigele VJ, Briggs EA, Brown RT, Cromer DT, Howerton RJ. J Phys Chem Ref Data 4(3):471, 1975. Hubbell JH, Trehan PN, Singh N, Chand B, Mehta D, Garg ML, Garg RR, Singh S, Puri S. J Phys Chem Ref Data 23(2):339, 1994. Jenkins R, Gould RW, Gedcke D. Quantitative X-Ray Spectrometry. New York: Marcel Dekker, 1981. Jenkins R, Manne R, Robin J, Senemaud C. Pure Appl Chem 63(5):735, 1991. Kallithrakas-Kontos N. Spectrochim Acta Part B51:1655, 1996. Kawai J, Gohshi Y. Spectrochim Acta 41B:265, 1986. Kawai J, Nakajima K, Gohshi Y. Spectrochim Acta 48B:1281, 1993. Kenney RW. In: Besanc¸on RM, ed. Encyclopedia of Physics. New York: Reinhold Publishing, 1966, p 86. Klein O, Nishina Y. Z Phys 52:853. 1929. Koningsberger DC, Prins R, eds. X-ray Absorption. Chemical Analysis Vol. 92, New York: Wiley, 1988. Kramers HA. Philos Mag 46:836, 1923. Krause MO. J Phys Chem Ref Data 8(2):307, 1979. Krause MO, Oliver JH. J Phys Chem Ref Data 8(2):329, 1979. Krishnan KS. Nature 122:961, 1928. Kulenkampﬀ H. Ann Phys 69:548, 1922. Kulenkampﬀ H. Ann Phys 87:597, 1928. Kulenkampﬀ H. Handbuch der Physik, Bd. 23. Berlin: Geiger-Scheel 1933, p 433. Lagarde P. Nucl Instrum Methods 208:621, 1983. Lee PA, Pendry JB. Phys Rev B11:2795, 1975. Lifshin L. Adv X-ray Anal 19:113, 1976. Love G, Scott VD, Scanning 4:111, 1981. Love G, Cox MG, Scott VD. J Phys D 11:7, 1978. Markowicz AA, Van Grieken RE. Anal Chem 56:2049, 1984. Markowicz AA, Storms HM, Van Grieken RE. Anal Chem 57:2885, 1985. Markowicz AA, Storms H, Van Grieken R. X-ray Spectrom 15:131, 1986. McGuire EJ. Phys Rev A9: 1840, 1974. McMaster WH, Kerr Del Grande N, Mallett JH, Hubbell JH. Compilation of X-Ray Cross Sections. Lawrence Radiation Laboratory (Livermore) Report UCRL-50174, Sec. II, Rev. 1, University of California, 1969. Merzbacher E, Lewis HW. In: Flu¨gge S, ed. Encyclopedia of Physics, Vol. 34. Berlin: SpringerVerlag, 1958, p 166. Mizuno Y, Ohmura Y. J Phys Soc Japan 22:445, 1967. Moseley HGJ. Philos Mag 27:703, 1914. Norrish K, Tao GY. X-ray Spectrom 22:410, 1993. Orlic I, Loh KK, Sow CH, Tang SM, Thong P. Nucl Instrum Methods Phys Res B74:352, 1993.

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Pirenne MN. The Diﬀraction of X-Rays and Electrons by Free Molecules. New York: Cambridge University Press, 1946. Poehn C, Wernisch J, Hanke W. X-ray Spectrom 14(3):120, 1985. Rao-Sahib TS, Wittry DB. Proceedings of the 6th International Conference on X-Ray Optics and Microanalysis. Tokyo: University of Tokyo Press, 1972, p 131. Rebohle L, Lehnert U, Zschornack G. X-ray Spectrom 25:295, 1996. Sandstro¨m AE. In: Flu¨gge S, ed. Encyclopedia of Physics, Vol. 30. Berlin: Springer-Verlag, 1957, p 78. Sauter F. In: Annalen der Physik, Band 9. Leipzig: Verlag von Johann Ambrosius Barth, 1931, p 217. Sayers DE, Lytle FW, Stern EA. Adv X-ray Anal 13:248, 1970. Scherzer O. Ann Phys 13:137. 1932. Schreiber TP, Wims AM. X-ray Spectrom 11:42, 1982. Sherry WM, Vander Sande JB. X-ray Spectrom 6:154, 1977. Smith DGW, Reed SJB. X-ray Spectrom 10:198, 1981. Sparks CJ Jr. Phys Rev Lett 33:262, 1974. Sparks CJ Jr. In: Ramaseshan S, Abrahams SC, Hodgkin D, eds. Proc. Inter-Congress Conference on Anomalous Scattering held in Madrid, Spain, 22–26 April l974. Copenhagen: Munksgaard, 1975, p 175. Statham PJ. X-ray Spectrom 5:154, 1976. Statham PJ. Proc Annu Conf Microbeam Anal Soc 14:247, 1979. Stephenson ST. In: Flu¨gge S, ed. Encyclopedia of Physics, Vol. 30. Berlin, Springer-Verlag, 1957, p 337. Stobbe M. In: Annalen der Physik, Band 7. Leipzig: Verlag von Johann Ambrosius Barth, 1930, p 661. Stoev KN, Dlouhy JF. X-ray Anal 37:697, 1994. Storm E, Israel HI. Nucl Data Tables A7:565, 1970. Suzuki T. J Phys Soc Japan 21:2087, 1966. Suzuki T, Nagasawa H. J Phys Soc Japan 39:1579, 1975. Suzuki T, Kishimoto T, Kaji T, Suzuki T. J Phys Soc Japan 29:730, 1970. Szalo´ki I. X-ray Spectrom 25:21, 1996. Tertian R, Claisse F. Principles of Quantitative X-Ray Fluorescence Analysis. London: Heyden, 1982. Tertian R, Broll N. X-ray Spectrom 13:134, 1984. Veigele WJ. In: Robinson JW, ed. Handbook of Spectroscopy, Vol. I. Cleveland, OH: CRC Press, 1974, p 155. Ware NG, Reed SJB. J Phys E 6:286, 1973. Wentzel G. Z Phys 27:257, 1924. Wernisch J. X-ray Spectrom 14:109, 1985. Wernisch J, Po¨hn C, Hanke W, Ebel H. X-ray Spectrom 13:180, 1984. West RC, ed. Handbook of Chemistry and Physics. 63rd ed. Cleveland, OH: Chemical Rubber Co, 1982–83, p E-l88.

2 Wavelength-Dispersive X-ray Fluorescence Jozef A. Helsen Catholic University of Leuven, Leuven, Belgium

Andrzej Kuczumow Lublin Catholic University, Lublin, Poland

I.

INTRODUCTION

Moseley’s law, formulated soon after the discovery of x-rays by Ro¨ntgen (Ro¨ntgen, 1896), represented the direct and deﬁnite onset of the use of x-ray spectrometry in chemistry. The ﬁrst successful period was completed in the beginning of the 1920s by creating order in the periodic table of the elements and by the discovery of the missing elements (Weebs and Leicester, 1967). X-ray ﬂuorescence (XRF) spectrometry has some unique features so that it became an irreplaceable tool for the analyst. Not may techniques had such a brilliant start to their analytical careers! gThe ﬁrst commercial instrument became available around 1940 and was derived from the x-ray goniometer of a diﬀraction instrument. A reasonable estimation of the number of wavelength-dispersive (WDXRF) instruments working today is 15000, of which some 20% are multichannel spectrometers. In addition, about 3000 instruments working in this mode are attached to the electron microprobes, mainly in Japan. Some 3000 energy-dispersive (EDXRF) instruments are working independently, and some 1500–2000 others are attached to electron and other [e.g., proton (Uda et al., 1987, Folkmann and Frederiksen, 1990)] microprobes; a few hundred instruments are connected to synchrotrons as excitation sources. About 300 instruments are working now in total reﬂection (TRXRF) and several in grazing emission modes (Klockenka¨mper, 1997). The average price of a new WDXPF instrument in 1998 is of the order of magnitude of $250,000, $100,000 for EDXRF, and $200,000 for TRXRF (Claes et al., 1997) and this is not expected to rise much in the coming years. The calculation of the total money involved in the x-ray instrument market made hereafter was based on the 50% amortization estimate per instrument. We consider in our account only these parts of Particle-induced x-ray emission (PIXE), electron microscope, and synchrotron arrangements which can be treated as x-ray attachments. An auxiliary market was created also around the main equipment business, covering the spare parts, recent supplements, and service connected with the maintenance. Then, the commercial value of the instruments running today may be estimated at some US$4 billion (Markowicz and Van Grieken, 1986; Van Grieken et al., 1986). An important body of scientiﬁc literature relates to x-ray spectrometry 95

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(Markowicz and Van Grieken, 1986; van Grieken et al., 1986; Jenkins, 1984; de Vries, 1986; Klockenka¨mper, 1987; Kikkert, 1983; Potts and Webb, 1992; To¨ro¨k et al., 1996, To¨ro¨k et al., 1998; Szalo´ki et al., 2000; Brill, 1996; Ellis et al., 1997 and Quiseﬁt, 1996). Literature about essentially diﬀraction and scattering problems can also be very useful for the researchers working in the ﬁeld of XRF (Fewster, 1996). About 10% of the WDXRF instruments are applied in geological research and prospecting (sequential and simultaneous spectrometers). Most of the simultaneous instruments are installed in industry, and the sequential instruments are distributed over industry, analytical laboratories, and research institutes at universities and public services. X-ray ﬂuorescence spectrometry is unique in element analysis of inanimate matter: In qualitative analysis, it has unrivaled selectivity for all elements between beryllium and uranium and also for transuranium elements and an extremely wide dynamic range in quantitative analysis (ppm to 100%). Although the plasma emission spectrometry [Direct Current Plasma (DCP) or Inductively Coupled Plasma (ICP)], atomic absorption spectrometry (AAS), neutron activation analysis (NAA), and diﬀerent kinds of mass spectrometry (MS) have lower or much lower detection limits, the wide dynamic range remains a unique feature of XRF (over ﬁve decades). Moreover, it possesses the scarce virtue among spectral methods—the upper useful limit reaches the concentration of several tens of percent of the total concentration for the main component. The detection limits at the ppm level without preconcentration and the limits of precision and accuracy that can be obtained (connected, in turn, to sample homogeneity and counting statistics) determine the boundaries of its application. Earlier books describing the techniques are mentioned in the reference list (Compton and Allison, 1935; Jenkins and De Vries, 1967; Birks, 1969; Dyson, 1973; Jenkins, 1974; Bertin, 1975; Tertian and Claisse, 1982; Lachance and Claisse, 1995, Jenkins et al., 1995; Buhrke et al., 1998—the latter is a kind of a guide for sample preparation in XRF and XRD, but is cited here, because the problem is very important and nontrivial). The purpose here is to describe the basic principles that allow wavelength dispersion of x-rays and to describe all the components of a spectrometer. The phenomenon of x-ray ﬂuorescence is only a small part of a much wider problem of the interaction of charged particles or photons with matter (Feldman and Mayer, 1986). Diﬀerent kinds of secondary particles are then emitted (Fig. 1a). Various combinations of exciting=emitted particles are possible, giving birth to diﬀerent analytical techniques, with the combination ‘‘photons as exciting particles=emitted photons’’ as a base for the x-ray ﬂuorescence. The angles the trajectories of the exciting or emitted particles create with the surface of the sample deﬁne normal XRF and PIXE methods and total reﬂection versions: TXRF (Chapter 9) and TRPIXE (Vis and van Langevelde, 1991; van Kan and Vis, 1996a, 1996b, 1997). In addition, for the x-ray ﬂuorescence, a grazingemission method can be mentioned (GEXRF) (Sasaki, 1990; Becker et al., 1983; de Bokx and Urbach, 1995; Urbach and de Bokx, 1996; Tsuji et al., 1995; Pe´rez and Sa´nchez, 1997) (Fig. 1b). Insofar as the fundamental principles are concerned, we may recall the law of Moseley (Moseley, 1913, 1914), which is the basis of the qualitative application of XRF: pﬃﬃﬃ n¼

rﬃﬃﬃ c ¼ kðZ sÞ l

ð1Þ

where n is the frequency (in Hz), c is the speed of light (in m=s), l is the wavelength (in m), k is the constant for a given series, and s is the screening constant. The relationship

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97

Figure 1 Phenomena accompanying the interaction of x-rays with matter: (a) different applications for analytical purposes (abbreviations of methods coupled with the use of particular phenomena are mentioned; (b) different versions according to the incidence and emergence angles (I—total reflection XRF or PIXE; II—grazing emission XRF; III—grazing incidence–grazing emission XRF).

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between the wavelength of the lines belonging to the same spectral series and the atomic number is illustrated by Figure 2. For high-Z elements, the XRF spectra become more complex and full exploitation for qualitative analysis of mixtures is not always simple. Practical unraveling of spectra is dealt with in Chapter 4. The intensity of any spectral line is proportional to the number of atoms emitting photons of energies attributed to this line. However, a simple linear proportionality is not the rule. From the concepts of mass attenuation coeﬃcients introduced in Chapter 1, it should be clear that the intensity of an analyte line of one species is attenuated by atoms of the same species and by any other atoms present in the matrix. Matrix refers to all elements present in a sample except the analyte element. Attenuation by absorption represents a ﬁrst complication compromising the proportionality. If the matrix contains elements with absorption edges of slightly lower energy than the energy of the characteristic line of the analyte, strong absorption can occur. On the one hand, this results in further attenuation of the analyte line and, on the other hand, in enhancement of the spectral line related to the said absorption edge of the other matrix element. This phenomenon is known as secondary ﬂuorescence. The process may be repeated with respect to all matrix elements with an absorption edge with a lower energy than the energy of a ﬂuorescence line emitted by another matrix element. Secondary and higher-order ﬂuorescence is the major complication deteriorating the simple proportionality between intensity and concentration. In Figure 3, three typical kinds of relationship between the relative intensity (measured intensity divided by the intensity obtained for the pure element) and concentration (expressed as weight fraction) are represented. Curve I is obtained when the analyte line undergoes absorption only. If enhancement occurs, the analyte line intensity is higher than expected from primary excitation only and the curve is situated above the diagonal. A similar curve, however, can be obtained when the mass attenuation coeﬃcient of the matrix is lower than the related coeﬃcient of the element for its own radiation, as shown in the case for Fe in FeO (curve II). In a special case, the mutual relation between attenuation

Figure 2

Representation of Moseley’s law of K and L spectral lines.

Wavelength-Dispersive X-ray Fluorescence

99

Figure 3 Absorption phenomena in a heavy matrix (curve I), in a light matrix (curve II), and in a neural matrix (curve III).

and enhancement coeﬃcients may be balanced such that both eﬀects cancel precisely and then the diagonal is approximated (curve III). The eﬀects just discussed are all energy (wavelength) dependent. This means that any calculations converting intensities into concentration should include the attenuation and higher-order ﬂuorescence eﬀects and also integrate over all energies present in the exciting primary beam above the value of the absorption edge and over all ﬂuorescence lines. This is an ab initio approach for conversion of intensities into concentration implemented by several authors. We mention here the articles by Gillam and Heal (1952), Sherman (1954, 1955), Shiraiwa and Fujino (1966, 1967), who in the 1950s and the 1960s proposed suitable equations. Sparks (1976) and Li-Xing (1984) implemented small corrections and reﬁnements concerning the details of these expressions, and de Boer (1990) and de Boer and Brouwer (1990) gave solutions to some diﬃcult exponential integrals related to the enhancement. Important contributions to the numerical side of quantitative XRF are due to Fernandez (1989), Fernandez and Molinari (1990), Rousseau et al. (1996), and Mantler (1984). The ﬁrst two authors prepared commercial, customer-oriented versions of the numerical programs XRFPc and CiROU, CiLAC, CiLT, and CiREG, respectively, and Mantler paved the way for the analyses of multilayers. It should be emphasized that Fernandez solved the transport equations for x-ray beams, the very general and powerful but diﬃcult tool for the description of particle beams. Likewise, Gardner and Hawthorne (1975) obtained similar results by Monte Carlo simulation of x-ray excitation. A speciﬁc approach was presented by Dane et al. (1996). These authors solved the problem of quantiﬁcation in situations where either an incomplete data set is available or where a reasonable assumption about composition and=or depth proﬁle cannot be done. Then, the idea of processing whole sets of solutions (populations or strings or generations) has been introduced, instead of processing the particular concentration values in consecutive iterations. In the fundamental parameter methods, the incidence and takeoﬀ angles are the important parameters. As a rule, they are kept constant during the traditional run of the analysis. However, there have been signiﬁcant eﬀorts to use the angle-resolved version

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of x-ray XRF spectrometers for depth proﬁling of the samples (Gries, 1992; Schmitt et al., 1994). For a complete treatment of equations deriving the line intensities in x-ray ﬂuorescence spectrum from the elemental sample composition, the reader referred is to Chapter 5.

II.

FUNDAMENTALS OF WAVELENGTH DISPERSION

The crystal monochromator is the heart of a wavelength-dispersive spectrometer. Wavelength dispersion of electromagnetic radiation in the x-ray region cannot be performed, as a rule, by normal gratings but only by diﬀraction on crystals or, for the long-wavelength regions, on multilayers. We brieﬂy explain the principles because the construction features of the monochromator are directly derived from them. Consider a monochromatic beam of x-rays of wavelength l with their electrical vectors of equal amplitude in phase along any point of the direction of propagation. Assume further that the beam is parallel and is incident on a crystal at an angle W between a given crystal plane (and all the planes parallel to this ﬁrst) and the incident beam direction. The beam is scattered and diﬀracted rays of equal l result but interfering constructively only in those directions for which the phase relationship is conserved. This happens at an angle W for scattered rays 1 and 2 (Fig. 4), for which the path diﬀerence ABC of ray 2 with ray 1 is equal to an integral number n of wavelengths. From Figure 4, it is clear that AB þ BC ¼ d sin W þ d sin W ¼ nl or, according to Bragg and Bragg (1913), who ﬁrst formulated this relation, written as nl ¼ 2d sin W

ð2Þ

where n denotes the number of wavelength diﬀerences between the rays scattered by the adjacent planes. Following the recommendation of the IUPAC (Freiser and Nancollas, 1987), n is the order of diﬀraction. If n ¼ 1, the diﬀerence is one wavelength and the diﬀraction is said to be of ﬁrst order. If n ¼ 2, the diﬀerence is two wavelengths and the diﬀraction is second order, and so on. All x-rays emitted at angles diﬀerent than W cancel because they are out of phase and destructive interference occurs. From Figure 4, one can note that in the Bragg scattering,

Figure 4

Derivation of the Braggs’ law. ABC is the path difference.

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101

the incident and exit angles are equal and, in this sense, there is some analogy to mirror reﬂection in the classical optics. However, diﬀraction is by no means equal to reﬂection in classical optics, because the diﬀraction is a volume not a surface process. It is less eﬃcient (with a great loss of intensity) and is performed only under particular angular conditions; that is, according to Bragg’s law. Full analogy between the reﬂection of x-rays and the reﬂection of optical rays happens only at grazing incident angles (i.e., smaller than the critical Snell’s angle). This is intrinsically a consequence of the refraction index, which is slightly smaller than 1 for x-rays. There are continuous eﬀorts to construct a real mirror on the basis of multilayers, which would eﬃciently reﬂect x-rays under large incident angles, at least for soft x-rays (Kearney et al., 1991). Thus, the static condition for obtaining diﬀraction of a monochromatic x-ray beam in some direction in the volume surrounding the analyzing crystal is given. What happens in the case of a polychromatic beam of x-rays? For a crystal, one set of planes is selected (for diﬀerent reasons) and d is constant. If only ﬁrst-order diﬀraction is considered and constructive interference must be realized for all l present in the incident beam, then W is the only variable: l ¼ ðconstantÞðsin WÞ

ð2aÞ

The signal arriving from the diﬀraction angle W is detected by a detector placed on a goniometer arm. The detector rotates around an axis through the macroscopic plane of the analyzing crystal. For a source at a ﬁxed position, the detector rotates over an angle 2W. The wavelength is calculated from constant ¼ 2d and sin W. Note that this holds only for the ﬁrst-order diﬀraction. If the second-order diﬀraction is used, the wavelength is equal to half of that value. The maximum wavelength lmax that can be diﬀracted in the ﬁrst-order diﬀraction by a crystal is equal to 2d because sin W 1. Because wavelength and energy are related, one more equations must be given, allowing us to convert wavelengths into energy units: E¼

12398:5 l

ð3Þ

where the energy E is in electron volts (eV) and l is in angstroms (A˚). It is a common practice that the units used by x-ray spectroscopists are still electron volts or kiloelectron volts for energy and angstroms for wavelength. When joules and nanometers are used, as required by the international rules, the numerical value of the conversion constant becomes 1.9864561016 (nm=J). The presence of diﬀerent wavelengths of diﬀerent order on the same goniometer position has a particular consequence for wavelength-dispersive spectrometry. For a given position of the goniometer (and detector), one may have a ﬁrst-order wavelength, say 0.6 A˚, second-order diﬀraction of l ¼ 0.3 A˚, third-order diﬀraction of l ¼ 0.2 A˚, or values of energy 20.66, 41.32, or 61.98 keV, respectively. Figures 5 and 6, reproduced from the work of Arai (1987) demonstrate a practical situation and Table 1 reprints a fragment from x-ray tables by Cauchois and Senemaud (1978). Similar compendia of spectral lines and attenuation coeﬃcients are easily available (Birks, 1974), as a rule covering the range of elements between lithium (Z ¼ 3) to plutonium (Z ¼ 94). Much more controversial are the values of the mass attenuation coeﬃcients for low energies and for low-Z elements at the same time (Henke et al. 1982, 1988). If necessary, one can have access to the calculation for even heavier elements (Soﬀ et al., 1977). It is easy to understand the practical value of such calculations for the analysis. However, it is much more diﬃcult to estimate the accuracy of numerical calculations for transuranium elements. To sort out x-rays with

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Figure 5 Higher-order overlapping of L series x-rays of Cr, Fe, and Ni to Ka of carbon. (From Arai, 1987.)

Figure 6

Overlapping of CKa by higher-order x-rays of Ni, Fe, and Cr. (From Arai, 1987.)

wavelengths belonging to diﬀerent spectral orders, a pulse-amplitude selector or detector is needed that can make the distinction between energy levels. A detector is a device of which the principle can be most easily explained by assuming the particle character of electromagnetic radiation. This is one reason that the characteristic of the impacting photons is often expressed in terms of energy, not wavelength.

Wavelength-Dispersive X-ray Fluorescence Table 1

103

Fragment of Detailed Tables of X-ray Lines

l (uX) 1251.56 1251.64(a) 1252.192 1252.58 1252.6 1252.82 1255.17 1255.43 1255.50 1256.60 1256.71 1257.1(a) 1257.12 1257.66 1258.74 1259.210 1259.8 1260.30 1260.6 1260.620(a) 1261.15(a) 1261.23 1261.956 1262.68 1263.086 1263.428(a) 1264.6 1264.99(a) 1265.0 1265.24(a) 1268.520(s) 1268.66 1268.68(a) 1270.2 1270.35(a)

l (mA˚) 1254.19 1254.27 1254.824 1255.21 1255.2 1255.45 1257.81 1258.07 1258.14 1259.24 1259.35 1259.7 1259.76 1260.30 1261.39 1261.856 1262.4 1262.95 1263.2 1263.269 1263.80 1263.88 1264.608 1265.33 1265.741 1266.083 1267.3 1267.65 1267.7 1267.90 1271.186 1271.33 1271.35 1272.9 1273.02

Element 42 58 62 73 90 73 73 32 69 75 73 71 90 73 54 68 90 74 90 58 64 73 42 90 42 58 74 70 91 69 92 90 68 74 64

Mo Ce Sm Ta Th Ta Ta Ge Tm Re Ta Lu Th Ta Xe Er Th W Th Ce Gd Ta Mo Th Mo Ce W Yb Pa Tm U Th Er W Gd

Transition

Notation usuelle

Ordre

KMIV KMIV KLII abs. LIII LINIV LIIIOIV,V LIIINVI,VII KLII KLI LIMII LIIIOIII LIINI abs. LII LIIIOII KLII KLIII LIIPII,III LIMIII LIIPI KMIII KMIV,V LIIIOI KMIII LIIOIV KMII KMII LIIINIII LIINIV LIINIV LINIII LIINI LINIII LIOIV,V LIIMV KMIII

KbII5 KbII5 Ka2 — — Lb5 Lb7 Ka2 — Lb4 — Lg4 — — Ka2 Ka1 — Lb3 — Kb1 Kb5 Lb7 Kb1 Lg6 Kb3 Kb3 — Lg1 Lg1 Lg3 Lg5 Lg3 — — Kb1

2 4 4 1 2 1 1 1 5 1 1 1 2 1 3 5 2 1 2 4 5 1 2 2 2 4 1 1 2 1 2 2 1 1 5

E (keV)

n=R

ðn=RÞ1=2

9.8776

725.99

26.9441

9.8758 9.8573 9.8552

725.85 724.49 724.34

26.9416 26.9163 26.9135

9.8460 9.8452 9.842

723.66 723.60 723.4

26.9010 26.8998 26.8956

9.8377

723.05

26.8897

9.8171

721.54

26.8615

9.8099

721.01

26.8516

9.784 9.7807

719.1 718.86

26.8158 26.8116

9.7788

718.72

26.8090

9.7523 9.741

716.77 715.9

26.7726 26.7566

Source: From Cauchois and Senemaud, 1978.

The diﬀerences between wavelength-dispersive (WD) and energy-dispersive (ED) XRF are due not only to the diﬀerent detectors but also to other factors: 1. The brightness of a WD spectrometer is very low, the attenuation in a crystal being responsible for an important part of losses. This problem may be overcome by the use of radiation sources of signiﬁcant intensity, the synchrotron being the best known but bound to the availability of this large facility. 2. The crystal is only the dispersive device, not the detecting device. The situation is diﬀerent in EDXRF, for which detectors play a double role: as the dispersive device and as the detector at the same time.

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3.

4.

Because Bragg’s law is geometric in character, the angular conditions for the collimation of primary and secondary beams are very severe for WDXRF (8– 25 msr), not the same as for EDXRF (150 msr) (Wollman et al., 1997). This restriction on the geometrical eﬃciency still makes the overall photon collection eﬃciency of a wavelength-dispersive spectrometer worse. An attractive aspect of EDXRF is the simultaneous collection of the whole spectrum, whereas a typical WDXRF device is exclusively sequential. However, the maximum count rate for an EDXRF instrument is 30 kcps for the whole spectrum, which severely limits the total number of accumulated counts and, consequently, limits the precision (counting statistics). Similarly, the limitation on the total count rate leaves a small margin for trace element analysis. From the assumption, the trace element, if present in a sample, can emit only a very small part of the total radiation emitted by a whole sample (and there is also the background, participating in the total amount of counts).

Simultaneous WD instruments are composed of a series of individual crystal spectrometer (channels) operating simultaneously, but the number of channels is limited. A comparison of wavelength- and energy-dispersive versions of x-ray spectrometers is performed in a recent publication by Brill (1996).

III.

LAYOUT OF A SPECTROMETER

The main parts of a spectrometer can be represented in the simplest form by Figure 7. The analyzing crystal is the central point of the wavelength-dispersive instrument. On the lefthand side of the crystal, we ﬁnd (1) the source of excitation, (2) the ﬁlters and devices for shaping the exciting radiation (collimators and masks), and (3) the sample. On the righthand side, we ﬁnd (4) devices for shaping the diﬀracted beam (collimators) and (5) the detector. Signals from the detector are fed into the electronic circuitry where they are shaped to be processed by the computer software for data analysis. This arrangement can be reduced or made more complicated according to the demand. However, all possible reductions lead to less ﬂexible devices, and complication does not necessarily enhance the quality of the instrument.

Figure 7 A wavelength-dispersive spectrometer: FC ¼ proportional flow counter; SD ¼ scintillation detector. (Reprinted with permission of Siemens AG.)

Wavelength-Dispersive X-ray Fluorescence

A.

105

Sources

A variety of radiation sources, emitting either charged particles or g- or x-rays of suﬃcient energy, are used for excitation of some or all elements of the periodic table and some or most of the spectral lines of analytical interest. Other chapters deal with excitation by protons [proton-induced or sometimes particle-induced x-ray emission (PIXE) (Chapter 12)], by electrons [electron microprobe (Chapter 13)], or by x-rays emitted from secondary targets or from a synchrotron (Chapter 8). Excitation by x-rays or soft g-rays from radioisotopes and x-rays from low-power tubes is mainly restricted to energy-dispersive spectrometers (Chapter 3). The ideal excitation source would be a tunable x-ray laser (monochromatic and intense, allowing the best choice of exciting wavelength and often selective excitation), but this cannot be expected in the near future. Nevertheless, it is worth reading some treatises about recent and future progress in the ﬁeld (Nagel, 1982; Jamelot, 1995; Fill, 1995; Crasemann, 1994; London, 1993). There are serious reasons for the slow progress in x-ray laser construction. The primary reason is that the population inversion of electrons means a much larger deviation from the energetical equilibrium if done between the levels, allowing the emission of x-rays. Consequently, it demands an excessive pumping power, turning the pumped matter into a plasma. One must be aware that the x-ray laser would be even much more useful in the ﬁelds of x-ray microscopy, holography, litography, or for the research of time-resolved phenomena than as a source for XRF analysis on a routine basis. The synchrotron is another modern source of x-ray radiation, but it cannot be considered as a tabletop instrument and a device for easy, inexpensive, and routine-based applications. From a practical point of view, vacuum x-ray tubes are the overwhelming choice among other potential excitation sources. High-power tubes are the only ones dealt with in detail in this chapter; low-power tubes are discussed in Chapter 3. All modern tubes owe their existence to Coolidge’s hot-cathode x-ray tube as presented in Physical Review some 85 years ago (Coolidge, 1913). They essentially consist of a sealed glass tube containing a hot tungsten ﬁlament for the production of electrons, a cooled anode, and a beryllium window. From a variety of modiﬁcations proposed over more than three-quarters of the century, two geometries have emerged as the most suitable for all practical purposes [the end-window tube (EWT) and the side-window tube (SWT)], but now the preponderance of the EWT pushes the SWT out of the market. Perhaps, a future deeper orientation of the WDXRF toward the microprobe applications will renew the interest in SWT, which is much better in deriving parallel x-ray beams. The general requirements of the x-ray tube as a source are as follows: 1. Suﬃcient photon ﬂux over a wide spectral range, with increasing emphasis on the intensity of the long-wavelength tail of the white spectrum. The actual vivid interest in low-Z element analysis will certainly activate research in this direction. 2. Good stability of the photon ﬂux (< 0.1% at least). Long-term stability reduces the frequency of recalibration in routine analyses; short-term stability is an absolute requirement for obtaining an acceptable precision. 3. Switchable tube potential (10–100 kV), allowing the creation of the most eﬀective excitation conditions for each element. Still, in more high-power constructions available on the market, putting a ﬁlter in the beam path is an easier solution than switching the voltage (Shimadzu XRF-1700). For sure, the selection of the spectral region for the targeted excitation by the use of a single x-ray tube will never be as good as in dedicated synchrotron beam lines with

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4.

monochromators of the adequate quality, but the low cost of tube operation is an obvious advantage. The intensity of the analyte lines varies considerably with excitation conditions. An extreme example is given by Vrebos and Helsen (1985a, 1985b) for simulated Al–Mo alloys. Freedom from too many interfering lines from the characteristic spectrum of the tube anode.

Freedom from interfering lines is important. The scattered characteristic lines of the anode may spectroscopically interfere with analyte lines, disturbing the qualitative recognition, the peak intensity estimation for the line of interest, and the accuracy of subsequent conversion of intensity to concentration. This is found in the results obtained after correction by some algorithms as well as by fundamental parameter calculations. Although there is some kind of a remedy by the use of more eﬃcient spectral deconvolution methods (Remond et al., 1993; 1996), it occurs at the expense of a greatly increased computational eﬀort and depends heavily on the assumptions made a priori. An x-ray tube is characterized by its anode element (a single element or two elements as in dual-anode tubes), its input power [expressed in watts (W) or kilowatts (kW), typically between 0.2 and 5 kW for high-power tubes], the voltage range between anode and cathode (10–100 kV for a SWT; limited to 60 kV for an EWT for the majority of commercial instruments), tube current [milliampere (mA), typically up to 60 mA or sometimes to 150 mA for high-power tubes], and an open (SWT) or closed (EWT) anode cooling circuit. The photon output of the tube or, more importantly, the photon ﬂux hitting the sample (expressed in counts per second per watt of input power) is determined by a, the incidence angle of an electron beam on the anode, the takeoﬀ angle b (for SWT), the distance t to the beryllium window, the thickness d of the beryllium window, and the distance between window and sample, t0 (Fig. 8). The radiation output power is rather poor with respect to the input power and is of the order of 1%, making the device a very ineﬃcient transformer of electron current to electromagnetic radiation. For an energy balance, see Bertin (1975) and some remarks by Wollman et al. (1997). The impact of the electrons creates an excitation volume from which the white and the characteristic radiation of the target escape. This phenomenon represents the ﬁrst distinct diﬀerence between a SWT and an EWT. As shown in Figure 9a, the ‘‘escape’’ path of the photons in SWT geometry is, on average, less than in the EWT geometry. An immediate consequence of this characteristic is that dual anodes are used only in a SWT, where a light element (e.g., scandium) covers a heavier element (e.g., molybdenum). By switching the excitation voltage, x-rays are produced either in the upper (lighter) element layer or in the substrate (higher-Z element), resulting in two distinct tube spectra with diﬀerent yields in the low- and high-wavelength regions (Fig. 9b). The popular dual-target tube arrangements are now Rh=Cr and Rh=W. The smaller the distance t, the higher the output. A decrease of t to half of the original value increases the intensity roughly by a factor of 4. However, the reduction of t in the classical setup is limited. The smaller the value of t, the more intense the bombardment by the electrons and subsequent heating of the window by the scattered electrons (Fig. 8). In a SWT, this bombardment is rather intensive because both the anode and the window are in this geometry at ground potential. In a EWT, to the contrary, the ﬁlament and the window are both at ground potential, and heating of the window is negligible. For an EWT, however, t can be reduced to a certain extent only because it faces the anode at high potential (Fig. 10).

Wavelength-Dispersive X-ray Fluorescence

Figure 8

107

Details of the geometry of a side-window x-ray tube and sample.

1. Alternative Con¢gurations To overcome the absorption of the low-wavelength tail of the continuum, a windowless conﬁguration was considered. This is an obvious solution, but it requires that the whole spectrometer with sample, collimators, crystals, and ﬂow counter be evacuated to the low pressure suitable for securing an acceptably long life for the tube ﬁlament and conducting analyses in the required range of the soft x-rays. Nordfors (1956) advocated a dual-anode tube with two anodes physically separated. They were excited by deviating the electron beam to either of the two anodes (Fig. 11). Lack of stability prevented this solution from gaining commercial interest. Probably, a modiﬁed version of this idea was introduced in the series 2000 x-ray spectrometers of Diano (dual target þ dual ﬁlament; see Section VI). Tubes with exchangeable anodes are another possibility for solving the problem of comfortable switching between diﬀerent anodes. To the best of our knowledge, this solution is not commercially available because of the tedious exchange operation and diﬃculties in rigorous repeating of previous conditions of the tube action. Using the sample as an anode is another solution, but then the application is bound to conducting materials. In a modiﬁed form, this solution is implemented in scanning electron microprobes. Because this ‘‘anode’’ cannot be adequately cooled, the input power must be restricted to very low values, entailing low count rates and, consequently, reducing the precision that can be obtained within reasonable counting times. Another possibility is to apply an existing 18-kW rotating anode x-ray tube generator. Suﬃcient intensity is obtained. Part of this gain can be sacriﬁced and Boehme (1987) and Nichols et al. (1987) introduced the microdiﬀraction beam collimator into the beam

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Figure 9 (a) Principle of excitation in a side-window tube. (b) Principle of excitation in a dualanode x-ray tube (side window); Mo, W, or Au (for A) and Sc, Rh, or Cr (for B) are commercially available anode components. The spectral feature of the tube spectrum depends on the energy of the incident electrons.

derived from such a high-power tube. This collimates the beam to about 30 mm and allows microﬂuorescence. Squeezing the beam to more or less the same diameter is possible when applying the polycapillary lens, even more so with the single capillary. These solutions are very restrictive for the beam intensity; thus, a reasonable device conﬁguration includes an energy-dispersive detector instead.

2. Optimization Because the takeoﬀ angle is a very important parameter, the surface of the anode can be formed in steps in such a way that the yield of low-wavelength radiation is optimized. As is clear from Figure 12, the step-shaped surface decreases the escape depths. In a conventional EWT, the reduction of the anode to window distance t is limited (Fig. 10). According to an old idea of Thordarson (1939), Botden et al. (1952) developed an x-ray tube with a gold anode evaporated directly on the beryllium window. The tube was used for surface radiotherapy. After many years, Phillips returned to Botden’s construction and again reduced t to zero by sputtering the anode target element directly on the beryllium window (Fig. 13). In a SWT, the anode is at ground potential, which

Wavelength-Dispersive X-ray Fluorescence

Figure 10

109

Principle of end-window x-ray tubes.

Figure 11 X-ray tube with two separate anodes: (1) anodes: (2) filaments; (3) cooling; (4) separating brass wall; (5) insulator. (After Norfords, 1956.)

Figure 12

X-ray tube with stepwise configuration of the anode.

considerably facilitates and simpliﬁes the safety requirements. This geometry enhances the intensity by an order of magnitude, allowing a much lower input power: 200 W for an intensity equivalent to the earlier 3-kW EWT. As a result of the low input power, the heat produced in the target by the electron beam is easily dissipated and no supplementary water cooling is required. The tube construction is reduced and only the most basic components are left. The commercial name for this development is T3 or TTT (target transmission tube; Fig. 14c). It is used exclusively in simultaneous WDXRF instruments, especially in the new version of the PW1660 (Philips Technical Materials Simultaneous X-Ray Spectrometer System PW 1660) (see Section VI).

110

Figure 13

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Anode elements deposited on the beryllium window.

3. End-WindowTube The essential parts of an EWT are displayed in Figure 14a. The ring-shaped cathode and the cooling circuit of the anode are typical features of EWT construction. The latter consists of a closed circuit ﬁlled with deionized water. Deionized water is necessary to reduce the conductivity and enhance the safety because the circuit is at anode potential! Electrons are diverted to the anode surface by electron-optics elements. Because the ﬁlament and window are at equal potential, no electrons hit the window. Heating is mainly by radiation from the anode, and an external cooling circuit is provided, which may be connected to normal water supplies because it is not in contact with a voltage source. Although the construction just described is in a common use now, it cannot be considered as an ideal source of x-rays. Recently, van Sprang and Bekkers (1998) and Kuczumow et al. (2000) have discussed the deviations from the homogeneity in the endwindow x-ray tube output. The geometry of the excitation system, as described by an angle supported on the tube axis – the sample point – the goniometer entrance axis, deviates from the formal geometry (about 83 ). The ‘‘real,’’ not formal, geometry of the device can be read out from the shift in the Compton-scatter energy. It creates the special demands for all the constructors of microprobes and milliprobes. They must avoid using the conventional end-window tube. There are also the consequences for the normal analysis: The uniformity of the samples and spinning operation are essential for getting reproducible results. Otherwise, particular spots on a surface of the sample can be irradiated in a diﬀerent manner. 4. Side-WindowTube An expanded view of a SWT is shown in Figure 14b. The essential diﬀerences from an EWT are the distance between cathode and anode, an earthed anode, and a single cooling circuit for the anode only. This circuit may be directly connected to the normal water supply because it is at ground potential. The geometric arrangement of the cathode supports higher potentials of the cathode with respect to the anode. This is the basic reason that tube potentials of 100 kV are allowed. Side-window tubes with ﬁne-focus anodes are used in many microprobe constructions. They are appreciated for their photon output, easy to transform into a parallel, collimated beam. A consequence of the regulated anode potentials is the possibility of use of dual anodes. The available dual-anode tubes are summarized later in Table 10. 5. Tubes with Rotating Anode The electrical input in an x-ray generator is high; however, a ﬁnal output of about 1% of primary power is not very impressive. This must be due to the low eﬃciency of conversion of the electrical energy into the energy of x-ray photons. One should remember that photon losses are even much greater in further parts of the spectrometer (especially

Wavelength-Dispersive X-ray Fluorescence

111

Figure 14 Different types of construction of commercial x-ray tubes: (a) end-window tube (EWT) (courtesy of Siemens AG); (b) side-window tube (SWT) (courtesy of Philips Analytical); (c) target transmission tube (TTT) (courtesy of Philips Analytical); (d) tube with a rotating anode (Vekemans 2000 courtesy of B. Vekemans).

112

Figure 14

Helsen and Kuczumow

Continued

in collimators and crystals). The desparate need for more eﬃcient excitation sources emerges with many more subtle applications of x-rays: the microprobe constructions, (micro)diﬀraction experiments with organic or biochemical materials, research on the ﬁne structure of the absorption edge, focusing of x-rays, or the application of the total reﬂection, for example. If one does not have access to the synchrotron, no patience to wait for the beam time or no money for paying great sums for simple analyses, then the tube with a rotating anode seems to be the best solution.

Wavelength-Dispersive X-ray Fluorescence

Figure 14

113

Continued

According to our knowledge, three companies, Rigaku, Siemens, and Enraf-Nonius, constructed high-power tubes with a rotating anode. The anode, in the shape of a wide cylinder placed on a narrower cylindrical rotary shaft, can rotate with a speed of 100 rps. Mo, Cu, and Ag are the materials most frequently applied for the construction of the anodes. The sides of the anode are under intensive electron bombardment, but the high heat load is dissipated by the intensive rotation coupled with cooling by water, ﬂowing inside the anode with a rate of 8–15 L=min. The system is evacuated by a prevacuum oilrotary pump followed by a turbo-molecular pump. It enables one to work with a voltage of 60 kV and a current of 300 mA, sometimes even greater. It gives a power of the order of 20–30 kW. Recently, the most powerful generator available is the Ru-1500 by Rigaku with a 90-kW load. The application of power of the order of 20–30 kW means roughly a 10-fold increase in the intensity as compared to the conventional x-ray tubes. When coupled with a Kumakhov lens, rotating anode tubes can be used as a powerful microfocusing source. 6. Compact Flash X-Ray Sources An x-ray tube is a very conventional x-ray source, dating back to the times of Ro¨ntgen and Coolidge. The eﬀorts described earlier aimed at a better and more eﬃcient construction of the x-ray tube. Diﬀerent versions of the tube were by no means the only solution for constructing a better and more convenient source. The most pronounced progress of the last decades is the synchrotron, but it is still far from an ideal device in terms of price and size. Much more promising for most XRF laboratories seems to be a compact ﬂash x-ray source (Germer, 1979). Such sources are now being developed intensively, giving, for example, a ﬂux of photons with energies between a few kiloelectron volts to 200 keV, in pulses lasting between 10 and 50 ns, with a repetition frequency up to 50 Hz. Doses are very intensive, up to several roentgens per shot (Pouvesle et al., 1996). Primary x-rays from

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such sources can be used for the excitation of x-ray ﬂuorescence, being especially useful for real-time monitoring of kinetics of fast processes in ballistics and in the simulation of accidents. It seems that the polycapillary x-ray focusing waveguide would be an ideal supplement to this kind of x-ray source, giving the impetus for the development of a new generation of pulse x-ray spectrometer. This device would combine the time resolution of the source with the spatial resolution of the focusing polycapillary semilens. 7. High-Voltage Generator A high-voltage generator supplying up to 60 kV for an EWT or up to 100 kV for a SWT and an output power of 3–4 kW (or much lower, 0.2 kW for TTT tubes) is required. A special generator is needed if one wants the high-power rotating anode tube. The conventional power supplies were very cumbersome because of the size of the transformer. The new generation belongs to the so-called switched power supplies. Such a power supply is basically a dc=dc converter in which a dc voltage is electronically switched at high frequencies (several kilohertz) and fed into an inductance–capacitance network. The output power is not continuously regulated but is controlled through pulse-width modulation. One possible scheme of a switched power supply is given in Figure 15. The mains are rectiﬁed (a), high-frequency switched (b), transformed (c), and rectiﬁed and smoothed (d). The output voltage is sensed (e) and compared to a reference voltage (f), and this signal monitors the pulse-width modulator (g), which, in turn, commands the switching circuit (b). In the high-frequency transformer, a ferrite core is used; this is a lightweight component. The whole setup allows the size of the generator to be reduced to a fraction of the volume of classical generators. The whole system is generally monitored by microprocessor. Power switching of the tube is done preferentially along the isowatt curve (constant input power), all under microprocessor control. The system has a number of safety switches on ﬂow and temperature controllers of cooling water with microswitches on all panels, which when

Figure 15

Principle of switched power supply: dc=dc convertor.

Wavelength-Dispersive X-ray Fluorescence

115

removed could expose the customer either to the electrical circuitry (high voltage of the tube supply or the mains voltage) or to unacceptable radiation levels. B.

X-ray Tube Spectrum

The ﬁrst attempts to propose a quantitative description of the tube intensity as a function of wavelength were made by Kulenkampﬀ (1922) and Kramers (1923). The quantitative treatment of intensities of ﬂuorescent x-rays (obtained by polychromatic excitation) by fundamental parameter programs requires knowledge of the intensity function of the exciting beam. This explains the continuous interest in the rigorous functional or numerical description of the tube spectrum. The description of the tube spectrum is divided into two parts, the ﬁrst concerning the continuum spectrum [bremsstrahlung; see Seltzer and Berger (1985)] and the second concerning the characteristic lines of the anode material superimposed on the continuum. The relative importance of these two contributions depends on the target element: For a tungsten target, the characteristic L lines constitute the minority of the emitted radiation, about 24%, and the K lines from the copper tube represent about 60% of the emitted radiation at 45 kV. The most popular Rh x-ray tube derives about 4% of the emitted x-rays as RhK and 20% as RhL photons; the rest is emitted as bremsstrahlung (45 kV). The positive side of the latter tube is its relatively good output of the low-energy photons, with a possibility of the excitation of the lines of heavier elements by RhK lines. In the continuum research, the main eﬀort was focused on the accuracy of Kramers’ law. Some deviations of this law from experimental results were corrected by the introduction of a nonintegral exponent m in the wavelength-dependent term in Kramers’ law (Brunetto and Riveros, 1984): Zr l Nl / 2 1 Kramers law l l0 m Zr l 1 Brunetto’s result Nl / 2 l l0

ð4aÞ ð4bÞ

where l0 is the wavelength at the short-wavelength (high energy) end of the continuum and r and m are constants related to the power-dependent terms: atomic number Z and energy function, respectively. The constants were determined by ﬁtting the theoretical curves to the experimental results. Brunetto and Riveros’ result was conﬁrmed later by Tertian and Broll (Tertian and Broll, 1984). By no means does it exhaust the solutions; Many other descriptions of bremsstrahlung spectra have also been published; to mention only the best known, there are those by Rao-Sahib and Wittry (1972), Reed and Ware (1973), Fiori et al. (1976), and Statham (1976a, 1976b). Finally, the reader can ﬁnd the interesting articles by Small et al. (1987) and Trincavelli et al. (1998) summarizing the state of the art in this area. Other corrections must be added to Eqs. (4): the absorption correction [often that of Philibert; see Pella et al. (1985) and Markowicz and Van Grieken (1984)] and the exponential correction term including the absorption of the beryllium window. Furthermore, in the article by Pella et al. (1986), an interesting algorithm was proposed for taking the characteristic line intensities into account. For this purpose, these authors made use of the equation proposed by Green and Cosslett (1961). The ratio of peak to background intensity was found to be a function of overvoltage U0 and atomic number Z:

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" 2 # Nchar U0 1 a U0 ln U0 1 ¼ exp 0:5 þ d 1:17U0 þ 3:2 b þ Z4 Ncon U0 1

ð5Þ

Overvoltage U0 is the ratio of the initial electron energy E0 to the excitation energy of a given shell Eex . The symbols a, b, and d are experimentally determined constants. The intrinsic merit of this formula is its dependence on the physical parameter U0 and on Z [see also the work by Murata and Shibahara (1981) with Monte Carlo estimations of the penetration depth of electrons]. Recently, a series of papers on the numerical description of the tube output has been published by Ebel et al. (1989) and Schoßmann et al. (1995, 1997). The solutions were derived from the diﬀerent depth distribution functions approximating the electron penetration in anode materials. Apart from all eﬀorts concentrating on the numerical estimation of x-ray tube emission, experimentally recorded spectra may always be used [collected in Gilfrich and Birks (1968), Gilfrich et al. (1971), Gilfrich (1974), Brown et al. (1975), and Loomis and Keith (1976)]. The same kind of calculation as for the spectral x-ray tube output is very helpful for the description of the background in wavelength-dispersive XRF analysis. The background is mainly determined by the scattered bremsstrahlung radiation from the tube. The theoretical estimation of the continuous radiation should always be helpful for solving background problems (Arai and Omote, 1988; Omote and Arai, 1989; Arai and Shoji, 1990; Arai, 1991). Sometimes, the knowledge of background can be applied for analytical aims (Kuczumow et al., 1992). In this case, a selected bremsstrahlung channel provides quantitative (but not qualitative!) information about the sample contents. A similar approach has been applied in an article by Kuczumow et al. (1995) for the extraction of the information included in the coherently scattered signal in the energydispersive mode on the capillary version of x-ray spectrometry. A similar extraction of the information from the Rayleigh signal in wavelength-dispersive XRF and from the selected bremsstrahlung channel in the electron microprobe is possible as well (Kuczumow et al., 1999). It concerns special samples (e.g., the biological ones) and the information obtained is connected with the description of the matrix. It brings additional knowledge about the density of matrix, which is sometimes of essential signiﬁcance in data evaluation. The articles just mentioned described the analysis of tree rings and other similar periodic structures, where the variability of the density of the matrix was, by far, more important than the variability in the chemical composition.

C.

Collimators and Masks

The wavelength-dispersive mode of operation depends strongly on rigorous geometric constraints on goniometer construction and analyzing crystal. Ideally, only a parallel beam of ﬂuorescence radiation should be diﬀracted and stray radiation should be absent. Stray radiation enhances the background. Diverging beams result in worsening of the spectral resolution. Converging beams act in the same direction, but the latter are often necessary for making microanalysis. If applied in such conditions, they demand a special numerical treatment to obtain analytical results with a given level of conﬁdence. Useful versions of fundamental parameter calculations for converging beams are now available (Chang and Wittry, 1994).

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Masks, which reduce the size of the ﬂuorescent beam in front of the collimator, cut oﬀ x-rays emitted or scattered by an area larger than the sample area. Three positions for collimators in the optical path of the spectrometer are possible: (1) between the sample and the crystal, (2) between the crystal and the detector, and sometimes (3) an auxiliary position between two detectors working in tandem. Collimators are not the ideal devices because they deliver a divergent photon beam with the angle of divergence a given by a a ¼ arctan ð6Þ l where a is the spacing between the blades and l is the length of the blades. In addition to divergence, there is also a substantial loss of counts while passing the collimator; for example, for a radiation of about 10 keV passing by a typical 10060.25 mm Soller collimator, the transmission is close to 105. On the bonus side of the use of collimators, it must be said that the resolution is improved very substantially (see Fig. 16). Thus, one needs to ﬁnd a compromise between better resolution and higher brightness. In the determination of light elements, for which the intensity is relatively low (because poor excitation and eﬃcient attenuation) but relatively well-spaced spectral lines exist, the demand for intensity prevails over resolution and, whenever possible, a coarse collimator (or none at all) is usually suﬃcient. Further considerations on this topic are in Section III.D.2. related to multilayers. In all focusing spectrometers (with curved crystals), the use of collimators is superﬂuous and the role of collimators is taken over by pinholes or slits. D.

Dispersive Elements

Both the primary photon beam derived from the x-ray tube (or another source of radiation) and the secondary or ﬂuorescence beam derived from the sample can be monochromatized. The monochromatization of the primary beam is not essential but can be made for more eﬃcient excitation of the selected elements by photons of chosen wavelengths or for radical simpliﬁcation of calculation procedures (the fundamental parameter calculation is then transformed to simple Lachance–Traill–Tertian correction). Monochromatization may be compared to the action of a narrow bandpass ﬁlter by which only a small band of the whole spectrum is transmitted. The width of the bandpass is related to the spectral resolution. A number of parameters characterizes this process.

Figure 16 Two spectra from the same sample demonstrate how an auxilliary collimator reduces line overlap when measuring heavy elements (a) without and (b) with auxillary collimator. (Courtesy of Phillips Analytical. From Phillips Materials.)

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The ratio DE=E, where DE denotes the energy width of transmitted radiation band and E the energy of photons to be transmitted (the mean energy in case of a wide band), is called the relative spectral resolution of the monochromator. DE is generally expressed as full width at half-maximum (FWHM) of the transmitted band (a peak in intensity–energy representation). The reciprocal of spectral resolution, E=DE, is called the resolving power. The same can be expressed in wavelength units as Dl=l by transformation through Eq. (3). Both expressions are numerically equivalent but of opposite sign. For the wavelengthdispersive spectrometer, the angular dispersion (Bragg’s dispersion) is important and is obtained by diﬀerentiation of Bragg’s law: dW n ¼ dl 2d cos W

ð7Þ

The monochromatization of the primary beam, if required, is made by the use of ﬁlters or secondary targets. The absorption edges of the elements present in the ﬁlter should be at lower wavelengths than the wavelength domain to be transmitted. Secondary targets emit their own characteristic radiation without a white background (except the scattered part of the primary beam) and with a population density of peaks depending on the atomic number of the secondary target. In both of these cases, the monochromatization is far from being perfect. A relatively simple application of double total reﬂection from the quartz plates was proposed with success for monochromatization in TXRF (Schwenke and Knoth, 1982). The criteria for the monochromatization of the primary beam diﬀer substantially, depending on the application. They become very sharp in some methods of research [photoelectron spectroscopy, extended x-ray absorption ﬁne structure (EXAFS)]. Then, the methods of monochromatization of the beam are diﬀerent also, depending very much on new discoveries in x-ray and related optics. In general, the strong monochromatization of the beam is associated with a great intensity loss and can be made for the very eﬃcient sources of x-rays as synchrotrons. The eﬀorts for the virtual numerical monochromatization of the real exciting beam by the introduction of the so-called ‘‘equivalent wavelength’’ concept failed. A new equivalent wavelength is needed for each analytical situation (e.g., change in composition). The gain from the simpliﬁcation of the spectrum is doubtful as compared with the great increase in the numerical eﬀort. The monochromatization of the secondary beam is the key problem. Dispersion of radiation is needed for the speciﬁc detection of characteristic lines. There are, however, a number of analytical problems in which the requisite dispersion is low. In limited cases, it may be suﬃcient to subdivide the spectrum into two spectral windows, from which one is to be detected. In other cases, the cutting oﬀ of a relatively wide spectral window may be suﬃcient, as is done in the Philips MiniMate device, where only a sealed gas proportional counter is applied for the photon energy discrimination. In practice, this selection of wide spectral domains is easily performed by energy-dispersive systems and pulse-height selection, by ﬁlters or by a set of balanced ﬁlters. A spectrometer for general use over the widest range of elements, however, should be equipped with monochromators of suﬃcient overall resolution, as is obtained by crystal diﬀraction. 1. Crystals According to the orientation to the possible crystallographic planes of the crystal, diﬀerent values for d are preferentially diﬀracting and, coupled with it, a diﬀerent resolution results

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119

for a given wavelength. A few values has been calculated for LiF in diﬀerent orientations and these are listed in Table 2. From the dispersion, it can be expected that the Ka line of manganese can be separated from the chromium Kb by LiF (220), not by LiF (200). The values in Table 2 are calculated for ideal crystals and a parallel beam. For real crystals, the dispersion and resolving power are less favorable; some real values for a complete spectrometer are provided later. In Table 3, a list is presented for crystals currently available for spectrometers: lithium ﬂuoride (LiF), silicon (Si), germanium (Ge), pyrolytic graphite (PG), indium antimonide (InSb), pentaerythritol (PE or PET), ethylenediamine-d-tartrate (EDDT), ammonium dihydrogen phosphate (ADP), thallium hydrogen phosphate (TlAP), and multilayers with their commercial designations. Important characteristics other than the interplanar distances codetermine the ultimate usefulness of an analyzing crystal: spectral resolution, mosaicity, reﬂectivity, stability, the thermal expansion coeﬃcient, and the spectral range. a. Resolution The spectral resolution DE=E of the crystals oscillates about the value 102, with few exceptions. The data for the resolution of diﬀerent crystals are included later in Figure 26a, a summarizing ﬁgure on resolution [see also Kuczumow and Helsen (1989)]. The spectral resolution for the crystals is, generally speaking, better than that of other dispersive devices, such as detectors or ﬁlters, but the diﬀerence with respect to the energy-dispersive spectrometers becomes less favorable for the short wavelengths. The progress in superconducting tunnel junctions detectors (Finkbeiner et al., 1991; De Korte, 1992; Le Grand et al., 1993) and microcalorimeter detectors (Lessyna et al., 1993; McCammon et al., 1993; Silver et al., 1996) during the last decade shows that these devices would have even better spectral resolution in the medium energy range (i.e., above 4–5 keV) than the crystals (see Fig. 26b). However, it is a question of compromise among price, necessity to work at liquid-helium temperatures, the very value of spectral resolution necessary for a given analysis, and the loss of intensity in crystals on one hand and the poor count rates allowed in the newest detectors on the other hand, which decides what kind of resolving device will be the future choice for commercial applications. b. Mosaicity and Reﬂectivity Real crystals have all kinds of imperfections, and mosaicity is one of them. Mosaicity refers to the existence of ‘‘blocks’’ within the crystal with sizes of the order of magnitude of 100 A˚, which have slightly diﬀerent orientations and lead to widening the diﬀracted peaks. This happens to widely diﬀering degress: topaz, EDDT, ADP, and gypsum exhibit low mosaicity, quartz and LiF a little higher mosaicity, and PET a signiﬁcantly higher

Table 2 LiF: d Spacings for Different Orientations and Angular Dispersion Crystal

2d (A˚)

Dispersion

LiF LiF LiF LiF

1.652 1.800 2.848 4.027

— — 0.5154 0.2902

(422) (420) (220) (200)

Note: Calculated for W angles corresponding to MnKa (2.102 A˚) and n ¼ 1.

120 Table 3

Helsen and Kuczumow Currently Available Crystals

Crystal

2d (nm)

Element range

LiF(420)

0.180

Ni–U

LiF(220)a LiF(200)a

0.2848 0.4027

V–U K–U

Si(111) Ge(111)a,b

0.6271 0.6532

P, S, Cl P, S, Cl

PG(002)a InSb(111)a,b PET(002)b

0.6708 0.784 0.8742

P, S, Cl Si Al–Cl

EDDT(020)

0.8808

Al–Cl

ADP(101) TIAP(100) PbSt LTC

1.064 2.575 10.0 15.6

Mg O–Mg F, C Up to Be

PX-1a PX-3a PX-4a OVO 55 OVO 100 OVO 160 OVO H300 PX6

4.93 19.5 12.2 5.5 10.0 16.0 30.0

O–Mg B C Mg, Na, F C, O B, C Be, B

a

Remarks High resolution, special applications for heavy-element K lines High resolution, good intensity General purpose, wide range, best diffracted intensity Supresses even orders Supresses even orders, good for intermediate- and low-Z elements Good intensity, poor resolution Good intensity, low spectral contamination, soft Lower intensity than PET, problems with stability Still lower intensity Especially F, Na, Mg, poisonous Good resolution, low intensity Longest wavelengths known for semicrystals’ structures Low resolution, good intensity Low resolution, good intensity Low resolution, good intensity Low resolution, good intensity Low resolution, good intensity Low resolution, good intensity One of the largest 2d available up to now

Also available as curved in simultaneous instruments. Also available as transversely curved for sequential instruments.

b

mosaicity. A result is a slight angular widening of the diﬀracted peak on one hand, but on the other hand, the intensity of the peak is generally enhanced. Widening is smaller than widening due to other geometric factors of the whole spectrometer and the loss can be tolerated while compensated by the increase in intensity. Some mechanical treatments enhance the mosaicity, and LiF crystals are easily improved in that way. The angular intensity distribution of a diﬀracted peak, the rocking curve, can be determined by doublecrystal spectrometer. In Figure 17, the peak height of the represented curve obtained by a high-resolution spectrometer is determined by atomic scatter factors and the space group of the analyzing crystal, whereas the width at half-maximum is determined by the mosaicity of the analyzing crystal and the divergence of the incident and diﬀracted beams. The surface under the curve is the integral reﬂection or, when the curve is normalized to the intensity of the incident beam, the integral reﬂection coeﬃcient. This coeﬃcient is very much dependent on mosaicity. Abraded LiF has a 10-fold increase in the reﬂection coeﬃcient compared to freshly cleaved LiF (36105 to 46104 rad), measured by using CuKa radiation (Birks, 1969). LiF crystals have also the great advantage of low absorption by their constituting atoms. Topaz and quartz, otherwise good crystals, have

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Figure 17 Rocking curve that would be obtained from a real crystal-diffracting parallel monochromatic radiation (rocking curves are actually measured on double-crystal spectrometers): P ¼ the peak diffraction coefficient; R ¼ the integral reflection coefficient. (From Birks, 1969.)

poor reﬂectivity properties, whereas PET is very good in this respect. Reﬂection constants as a function of wavelength are given in Figure 18. c.

Stability and Temperature

The mechanical stability of most crystals is satisfactory, but there are exceptions. Gypsum can eﬄoresce (especially in a high vacuum); PET has a tendency to change phases on aging

Figure18 Single-crystal integral reflection coefficients of graphite, LiF (200), LiF (220), and KAP. [From Gilfrich et al., 1971). Reprinted by permission of Analytical Chemistry. Copyright # American Chemical Society.]

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and it is soft so that it is easily damaged when manipulated. The temperature inside the measuring chamber of a spectrometer is kept as constant as necessary (< 1 or < 0.01 C if chemical peak shifts have to be measured). In these circumstances, damage of the organic crystals by high temperature is not likely to occur. When very precise measurements of peak position are to be made for speciation purposes, the thermal-expansion coeﬃcient is important. PET is very disadvantageous in this respect; topaz is in the opposite situation. It should be noted that the characteristic spectral lines of the crystal can be superimposed on the spectrum, making it obscure or even unﬁt to use, especially when it suggests the presence of some elements in a false way. Crystals of excellent resolving features, Be and Ge (111), have their own lines in the soft x-ray region. d. Spectral Range The useful spectral range covered by one crystal is limited. The maximum wavelength is of trigonometric origin and is imposed by Bragg’s law, namely by the condition that sin W < 1 and lmax < 2d. In practice, the goniometer is scanned for values of 2W varying from a few degrees to about 150 . However, for higher values of 2W, an angular dispersion widens the peak proﬁles. On the contrary, in the low-2W range, only a small fraction of x-rays emitted by the sample is intercepted unless the analyzing crystal is very long. The surface of the crystal projected on a plane perpendicular to the x-ray beam may be smaller than the width of the beam emerging from the collimator. Thus, 2W values as well as the macroscopic dimension determine the useful spectral range of a crystal. For x-rays of low energies, the interplanar distances of real crystals become too small. Some substances, namely salts of heavy metals with organic acids with long chains (otherwise the soaps of heavy metals), can take over the role of dispersive structures if their organic chains are regularly arranged. There are special techniques for making such quasicrystals. The individual layers are called Langmuir–Blodgett ﬁlms. The ends of hydrophobic chains of adjacent layers join each other, and the heavy metal ends are connected to the next heavy-atom ends from the next molecules. Many features of such ﬁlms depend on the length of the chain. A great number of similar structures with a wide variety of interplanar distances 2d (up to 156 A˚ in the case of lead melissate) have been synthesized. Langmuir–Blodgett structures have diﬀerent negative properties: for example, they are soft and not very stable, strongly hydrophobic, and of low reﬂectivity. Some of them, like thallium adipate, are highly poisonous substances. e. Curved Crystals and Other Focusing Systems Not only are ﬂat crystals used in the goniometer. Some focusing systems, such as the Johann, Johansson, and Cauchois arrangements, may be used (Fig. 19). Then, the crystal must be curved, with the curvature radius equal to the diameter of the Rowland circle and, at the same time, to the Bragg curvature, being half that of the Rowland circle. Threedimensional crystals can sometimes meet both of these conditions. The focusing arrangements are extremely useful in all cases in which the sample area is small, losses of intensity are prohibitive, and scanning over large angles is not necessary. In these arrangements, x-rays are collected in one point by focusing on the detector. An obvious virtue of such an arrangement is found in the analysis of small sample areas and by a using focused exciting beam, as in electron or proton microprobes. In such situations, the curved crystal geometry allows the collection of x-rays from a relatively large solid angle of diverging x-rays to a small spot (Fig. 20). A consequence is that the collimators normally placed between the sample and the crystal, and the crystal and the detector must be replaced by slits and pinholes. The focusing geometry is a very economical arrangement for saving intensity (the brightness of such a device is estimated to be up to two orders of magnitude greater than

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123

Figure19 Curved crystal x-ray arrangements in (a) Johann, (b) Cauchois (in transmission), and (c) Johansson version. S ¼ entrance slit of radiation; D ¼ exit slit of diffracted radiation; d ¼ crystal lattice spacing.

Figure 20

Focusing of x-rays by a logarithmically curved crystal. (Courtesy of Siemens AG.)

that of conventional ﬂat geometry). The curved crystal geometry is applied in simultaneous instruments in which scanning is not used and an optimized arrangement is chosen for each wavelength (element) implemented. Then, logarithmically curved crystals are used. In a new version of the sequential instrument (Venus), Philips used the reverted channels taken from the simultaneous instrument. Crystals applied in that construction are curved as in a simultaneous device and their position is additionally adjusted by a screw to trap as much intensity as possible. In general, during the last two decades, great progress took place in the planning and construction of the optical elements based on curved reﬂecting or diffracting surfaces of crystals and multilayers; most of the developments were in the synchrotron version of x-ray spectroscopy (Ice and Sparks, 1984). The curved crystal version of the x-ray microprobe is not the only known solution. The progress in x-ray optics during last decades enabled the easy concentration of x-rays while being transmitted through a single capillary (Rindby, 1986; Carpenter 1989; Thiel et al., 1989) or even through a semilens based on the bundle of capillaries (Kumakhov and Komarov, 1990; Yiming et al., 1994; Dagabov et al., 1995; Xunliang et al., 1995; X-Ray Optical Systems Inc., 1997, 1998). This ﬁrst possibility, due to the great loss of intensity resulting from a poor interception of the primary beam by the narrow inlet of a single capillary, is limited to the energy-dispersive version of x-ray spectrometry (Attaelmanan et al., 1994). The second version is more interesting from the point of view of WDXRF. A properly curved capillary bundle can serve as a kind of semifocusing lens with a well-deﬁned

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Figure 21 Cross section of a typical multicapillary ‘‘lens’’: (a) focusing configuration, from wide spot (divergent source) to another spot (semifocus); (b) configuration transforming the divergent into the parallel beam. (Courtesy of X-Ray Optical Systems Inc., Albany, NY. and personally from Dr. J. P. Bly.)

curvature and moderately-deﬁned focus (Fig. 21). The intensity of the beam leaving the ‘‘lens’’ is, of course, lower than the intensity derived from the x-ray tube, but still suﬃcient for many applications. Moreover, the primary collimator is removed in this system and the gain from this removal is partially balancing the loss resulting from the bundle construction and operation. One can estimate the focal distance f of such a lens as df ¼ d0 þ 2 f ycr

ð8Þ

where d is the spot size, d0 is the diameter of the capillary end, and ycr is the critical angle of the total reﬂection. As a rule, the polycapillary array is introduced to concentrate the primary beam, but it can also be used for the collection of secondary x-rays outgoing from the sample, before reaching the curved crystals in nonconventional arrangements. 2. Multilayers The limitation of XRF analysis in the region of soft x-rays is one of the most serious limitations of crystal monochromators, although none of the ‘‘ﬁrst principles’’ constituting the background of XRF method puts this limitation explicitly. Moseley’s law determines the rules for qualitative analysis of any element except hydrogen and helium, and the Shiraiwa and Fujino equations govern quantitative analysis. Irresolvable problems occurred until the late eighties because of troublesome absorption, lack of an adequate dispersive device, and bad detection while passing to longer wavelengths. The concept of multilayers or, strictly speaking, layered synthetic microstructures (LSMs) or multilayer interference mirrors (MIMs) solved at least part of the problems. Two approaches to the idea of a suitable analyzing device in the soft x-ray domain have been applied from the very beginning (Underwood and Barbee, 1981). The ﬁrst resulted from consideration of Bragg’s equation [Eq. (2)]. This indicates that if any crystal of a longer interplanar distance is found, it will be possible to disperse longer wavelengths in the soft x-ray range. The potential candidates for long-period crystals or quasicrystals, such as clay minerals of the chlorite (14 A˚) or illite–montmorillonite (25–30 A˚) type, are not suﬃciently ordered in this respect (Weaver and Pollard, 1973). Unfortunately, natural crystals, irrespective of their diﬀerent crystallographic planes, oﬀer only limited possibilities in this spectral range; speciﬁcally, their reﬂectivity is very low. The class of structures called Langmuir–Blodgett ﬁlms was more promising, but the reﬂectivity and thermal, mechanical, and chemical stabilities were rather poor.

Wavelength-Dispersive X-ray Fluorescence

125

A correct solution resulted from generalization of Bragg’s law, namely that diffraction also occurs in media consisting of layers of diﬀerent refraction coeﬃcients—in other words, in sites of diﬀerent periodically changing electron densities. Such structures may be formed in an artiﬁcial way. They may be treated as crystallike substances ordered along the c axis (i.e., spacing d or 2d), but with uncontrolled arrangement within the ab plane. The structure in the ab plane can be amorphous, provided that the optical and mechanical properties are uniform in these directions. The ﬁrst eﬀort of synthesis was made in 1940, but the Au–Cu multilayer of Du Mond and Youtz (1940) did not survive more than a few days because of deterioration by interdiﬀusion. The problems of thermal, radiative, diﬀusional, and chemical instabilities of multilayers have recently been solved. Signiﬁcant progress in this ﬁeld has taken place in the last three decades, hence some years before the application of multilayers in WDXRF. The second approach was the result of an increasing interest in x-ray optics. As a branch of physics, x-ray optics is nearly as old as the discovery by Ro¨ntgen, but because of the speciﬁc properties of x-rays, its progress has been slow. All existing substances exhibit complex refractive indices: m ¼ 1 d ib

ð9Þ

where m is the refraction index, d is the so-called unit decreament, a real number, and the imaginary term b is related to absorption. The refraction indices in the x-ray region are all below but very close to 1. This makes all eﬀorts of x-ray focusing by lenses in a traditional sense an impractical task. Franks (1977) gives an example of a lens with a curvature radius of 10 mm built of a material with d ¼ 5 105 . The focal length of this lens is 100 m. In such a situation, only glancing, grazing, and diﬀracting features of materials have been of practical interest for a long time. Otherwise, the grazing optics conditions for x-rays are extremely severe. However, the need for mirrors, monochromators, and focusing devices for the x-ray region is increasing rapidly in the ﬁelds of synchrotron radiation, plasma research and diagnostics, x-ray microscopy, x-ray ﬂuorescence analysis, diﬀraction, thermonuclear fusion, x-ray laser, x-ray astronomy, and even x-ray waveguides. The invention of multilayers deﬁned this time as a complex set of particular layers consisting of materials of periodically diﬀerent refractive indices gave great impetus to progress in this branch of optics. Some important properties of the materials used to produce the multilayers are discussed in the next section. a. Nature of Materials In principle, three kinds of material are necessary for the synthesis of multilayers. First, there must be a support (sometimes called a substrate), such as a piece of ﬂat or curved silica layer, with an ideally polished surface. This condition is very important because the roughness of the surface is translated to the deposited layers, correctable almost exclusively when one of the layers consists of amorphous materials such as carbon. After years of progress in the ﬁeld, when the roughness of the interfaces have probably reached the limit of about 2 A˚, the most important contribution in it is from the deviations of the substrate from ﬂatness (Kortright, 1996). Two sources of relatively inexpensive and precisely polished supports were found: silicon wafers as used in microelectronics (curved) and glass or fused-silica optical elements for low-scatter mirrors (ﬂat or curved). The additional polishing is the last step in the preparation of the substrate for further manufacturing of the multilayer. An even more complicated role for the substrate is described by Nicolosi et al. (1986) in whole coal analysis. The authors determined the contents of

126

Helsen and Kuczumow

C, N, O, and F using multilayered microstructure PX2 and the contents of P, S, K, and Ca by taking advantage of diﬀracting properties of the supporting wafer of silicon (111) on which the multilayer was settled. Heavy metals (Hf, Pb, W, Mo, Pt, Ir, Ru, Rh, Os, Ni, Co, Fe, and Cr) and light elements or compounds [Al, B, B4C (Ovonic Synthetic Materials Co., 1987), BN, Be, C, Si, SiO2, SiC, Ti and Sc—the last with the high reﬂection] may be used as materials for alternating high and low optical densities, at the same time being strong and weak scatterers, respectively. Sometimes, materials from the ﬁrst group are called absorbers, and those belonging to the second group are called spacers. The reactive materials and substances with a low melting point are excluded from the list. Both components of the multilayer should not share a common crystalographic structure, to avoid the epitaxial growth. The selection rule for the materials can easily be understood from Figure 22 for boron. The mass attenuation coeﬃcients for the boron K line, as presented versus the atomic number of the absorber, have several minima. One falls in the region of boron (weak self-absorption) and carbon, and the next very deep minimum is in the region of niobium=molybdenum. It puts the constraints on the materials, which can be used for the determination of boron. The ﬁrst, a low-density material, can be carbon or even better B4C; the second, a high-density material, should be niobium or molybdenum. Indeed, Philips made the structure PX3, composed of Mo=B4C, with a period of 195 A˚, adviced for the determination of boron. However, when a new multilayer should be synthesized

Figure 22 Mass attenuation coefficient for the BKa line drawn against the possible absorbers in the periodic table. Arrows show the deep minima in coefficient value. (Courtesy of Bruno Vrebos’ inspiration.)

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with high reﬂectivity in mind, then the attention is put, instead, on a pair of materials with a small value of the scattering Debye–Wallner factor [see Eq. (10)]. Here, the W=B4C tandem is a very good one. As a rule, the low-electron-density compounds are selected from the elements with K absorption edges above the energy of spectral lines to be measured. In that sense, boron and carbon are selected for multilayers of spacing above 50 A˚ and beryllium and carbon for spacings above 120 A˚. b. Smoothness This parameter is of great importance on each level, starting from the supporting substrate to the last upper layer. All rough places, uneven layers, disturbances in interfaces, interdiﬀusion, and chemical interaction regions or defects, act in a manner analogous to the thermal motions in classical diﬀraction analysis. During synthesis, the extent of roughness can vary in a diﬀerent manner: In some cases, it tends to smoothness, in others to further growth. The use of amorphous materials to form some layers favors smoothing. The greater interplanar distances are, the more pronounced the roughness is. On the other hand, the smoothness probably has its limits, even in the case of multilayers with relatively small interplanar distances, with the Debye– Wallner parameter s tending to reach a value of about 2 A˚. It probably results from the smoothness level obtained during the substrate production. Finally, it puts some domination of the Debye–Wallner parameter over other possible factors inﬂuencing the quality of multilayers in the region where the interplanar distance is not very long (say, a few tens of angstroms). c. Reﬂectivity and Thickness of a Multilayer Classical dispersive crystals exhibit low reﬂectivity in the soft x-ray range. The situation was greatly improved in multilayer reﬂectors in which a single layer at normal incidence angles exhibits a reﬂection coeﬃcient of the order of 104 rad. This coeﬃcient is, however, proportional to the square of the electrical ﬁeld amplitude (i.e., to a value of the order of 102 for the amplitude reﬂection coeﬃcient). Thus, a multilayer consisting of 100 single layers may provide almost the maximum reﬂectivity if the absorption can be neglected (Underwood and Atwood, 1984; Underwood, 1986, http:==www-cxro.lbl.gov==multilayer=survey.html, 1998). For long wavelengths, the results are not so impressive; for example, in the very important region of the so-called water window (284–530 eV), the reﬂection coeﬃcient was lower than 3% (Kozhevnikov et al., 1994). The region is of essential signiﬁcance for the construction of an x-ray microscope based on zone plates for biological research (Kirz et al., 1995). These results are in accordance with the calculations by Rosenbluth concerning numbers of layers necessary for ‘‘saturation’’ of reﬂectivity [data cited by Barbee (1986)]. Peak reﬂectivities of some commercially available multilayers [e.g., Philips materials PX1, PX2 and PX3 (Nicolosi et al., 1986; Van Eenbergen and Volbert, 1987) and the OVONYX line of multilayers [(Ovonic Synthetic Materials Co., 1987)] are often compared to classical pseudocrystal dispersive structures, like lead octodecanate and thallium acid phthalate. The predominance of the multilayer structures over classical structures is striking, and spectral resolution is the only parameter in which the classic crystals are better (Van Eenbergen and Volbert, 1987; Henke et al., 1986). Still, when using multilayers, one must remember the weak points inherently coupled with those devices: great refraction eﬀects (Martins and Urch, 1992; Luck et al., 1992), deviating the eﬀective 2d value of the structure from the real value by as much as 5%, and poorly shaped background with ‘‘ghost’’ lines in some cases.

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Quality of Layers and Boundaries

The structure of the multilayer ought to exhibit geometric features. This condition is very strict. Thus, ﬂat layers or ﬁgured structures must be uniform, the latter with controlled curvature, without defects but with clearly deﬁned optical parameters. All deviations from an ideal model, easy to estimate by the reﬂectivity characteristics, are, as a rule, coupled by the use of the Debye–Waller factor into the eﬀective roughness parameter seﬀ, including both thickness errors and imperfect boundaries (Spiller and Rosenbluth, 1986). The Debye–Waller factor DW is the Gaussian-type term describing the reduction of reﬂected amplitude at a boundary: " # 2ps cos a 2 DW ¼ exp 2 ð10Þ l where s is the width of a smooth transition layer and a is the propagation angle of electromagnetic radiation, measured from the normal to the boundary between a medium with one refractive index to another. The question arises, however, whether the errors result from a rapid or from a mild change of the optical features on the boundaries. Any roughness of a given layer decreases its reﬂectivity (Payne and Clemens, 1993), but a random distribution of roughness may bear higher reﬂectivities of the higher-order peaks. Moreover, many other physical factors can inﬂuence the uniformity and expected layer characteristics: Both the support and the particular monolayers can have diﬀerent slopes, densities are subject to ﬂuctuations, the elements can diﬀuse, or some chemical elements can mutually react (e.g., W with C or Mo with Si) (Rosen et al., 1993). It should be emphasized that despite of so many obstacles, the magnitude of the roughness in such a totally artiﬁcial structure can be kept in the range of an atom radius (Barbee, 1986). The numerical description of multilayers is based on the matrix method (a matrix expresses the features of a layer or interlayer region, and for the characteristics of a multilayer, the product of matrices is written) or on the recursive use of layer–reﬂection– transmission characteristics. Deviations of empirical results (mainly reﬂectivity measurements) from the model predictions are associated with roughness. In this case, seﬀ, the roughness parameter, results from the best ﬁt of empirical data to the model. The selection of an adequate method of synthesis and subsequent control of production and quality is a key question. Many methods were checked, and the aim of all these processes was the deposition in which one atom follows another. The physical vapor condensation, chemical vapor deposition, electrochemical covering, and sequential adsorption of ﬁlms are the most promising methods. However, for the time being, the ﬁrst method prevails. Two basic versions of physical vapor deposition are used—with thermal and sputter sources. In the thermal source technique, the covering materials are evaporated by electron beam or laser evaporation. The vapor source and samples are in a ﬁxed position. The time of evaporation is a parameter controlled by shutters or by pulsed laser irradiation. An in situ monitoring system was used as a feedback system for controlling the shutter. Spiller (1981) invented a system for thermal deposition. He used a soft x-ray reﬂectometer (a source plus a detector), placed in an evaporation chamber to register the reﬂectivity of successive layers. The open loop is set according to the indications of maximum reﬂectivity for a given layer. It is possible by this method to obtain a multilayer with an area up to 25 cm2. The rates of deposition vary in the range of 2–100 A˚=s and must be carefully controlled because the successive layers must be commensurate. Sometimes, especially

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when multilayers of large surface are produced, a special motion of the substrate in the evaporation chamber may be necessary to guarantee the same evaporation rate for all points of the surface. Other diﬃculties arise from this type of deposition: the high vacuum required in the evaporation chamber (< 102 Pa), radiation heating of the deposited ﬁlm, and the problems with thermal sources (alloys may segregate during melting). The sputter source systems (Barbee, 1981) belong to another processing method. The glow discharge, magnetron, electron cyclotron, ion beam, and triode devices may serve as sputter sources. The plasma discharge is always the ﬁrst step of the process. The plasma ﬂux is directed by the electrical potential diﬀerence to the cathode surface, where the ions and atoms dislodge the maternal atoms. The secondary atoms and ions establish the proper ﬂux (Sigmund, 1981), which passes to the substrate to deposit a layer on it. Samples are placed on a special rotatory table, appearing at ﬁxed intervals under the secondary ﬂux courses. The rotation speed, precisely controlled, is the regulating factor for a designated thickness of the layer. The purity of secondary ﬂuxes and their energies also inﬂuence the quality of the multilayer. The main errors in this method result from the ion and atom inclusion into layer (there are high-energy tails in the ﬂuxes) and from heating caused by ﬂux energy deposition in thin layers. However, substrate rotation may overcome many of these defects. Sputter rates range from 1 to 20 A˚=s; the covered surfaces may be quite large. Today, the magnetron sputtering is the most common method, with the best results obtained as far as the reﬂectivity of the products is concerned. After presenting two basic methods of layer formation, some attention must be paid to methods of control. As mentioned earlier, reﬂectivity estimations are of fundamental importance in multilayer quality determination, both experimentally and in theory (Spiller, 1981). A comparison of reﬂectivity predicted and obtained in an experiment is instructive, allowing, at least, understanding the source of the roughness (Henke et al., 1986; Luck et al., 1992). Diﬀerent peak, absolute, and integrated reﬂectivities were measured with the use of both laboratory x-ray sources and synchrotron facilities. These results conﬁrmed the layered structures of the synthesized materials; as a rule, reﬂectivity was smaller than predicted, but the results show that the optical parameters estimated until now (Henke et al., 1982, 1988) are reasonably correct, with the shorter the wavelength concerned, the better. The methods just discussed only allow the deduction of an indirect image of the multilayer structure. Other methods give direct insight into the structure. Thus, the diffraction data may be very useful, but they require large sample volumes. Progress in capillary version of the diﬀractometer (Bilderback et al., 1994), allowing the observation of the samples as small as 56103 mm3, may bring a solution to the structural problems. Very small samples are needed for obtaining the structural information by electron microscopy, especially in combination with a special technique of sample preparation, socalled microcleavage (Lepeˆtre et al., 1986). Wedge-shaped strips of materials are placed in the electron beam in such a manner that the planes of particular layers are parallel to the beam direction. Both the transmission image and the diﬀraction pattern can be obtained. The in-depth and lateral variations, the roughness, and the visual image of the structure are in the ﬁeld of observation. High-resolution electron microscopy can give even better results, and electron energy loss spectroscopy (EELS) may provide the complementary chemical information on the subsequent layers (Lepeˆtre et al., 1986). Requirements for the application of multilayers in XRF may be summarized as follows: They should satisfy Bragg’s law for the assumed wavelength domain within permissible goniometer angles, have a high reﬂectivity and a good spectral resolution,

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suppress diﬀraction peaks of higher order, and do not absorb soft radiation too intensively. Flat or curved multilayers can be used. Dedicated designs are described in the literature for the application of curved multilayer optics (Van Eenbergen and Volbert, 1987; Gilfrich et al., 1982). The geometric requirements imposed on the curvature of multilayers for XRF are modest in comparison with their applications in other ﬁelds of x-ray optics. The increasing spectral distance between K lines of the neighboring elements when going to lower-Z elements may require a dedicated multilayer for each element. Sometimes, great diﬀerences in the mass attenuation coeﬃcients for the lines of neighbouring elements of interest exist for a given material—attenuator. Then, one can construct a multilayer eﬃciently diﬀracting the radiation of one element while deeply suppressing it for other elements. The dedicated multilayers are the preferable choice for the simultaneous WDXRF instrument, with an optimized channel for each element. Multilayers can be also applied in a special version of WDXRF analysis, called grazing emission x-ray analysis (de Bokx and Urbach, 1995). Curved multilayers are also used in electron microprobes (Ovonic Synthetic Materials Co., 1987). From what has just been described, it may be concluded that in order to analyze a useful range of soft x-rays, at least three diﬀerent multilayers have to be selected. Such combinations are currently available commercially (e.g., for example from Osmic, Philips, Siemens, Rigaku, and Shimadzu) (Ovonic Synthetic Materials Co., 1987; Van Eenbergen and Volbert, 1987; D. K. G. de Boer, personal communication, 1998): PX1 PX3 PX4 PX5 PX6

2d ¼ 49:3 A˚ 2d ¼ 195 A˚ 2d ¼ 122 A˚ 2d ¼ 112 A˚ 2d ¼ 300 A˚

oxygen to magnesium boron and possibly for beryllium (Nicolosi et al., 1987) carbon nitrogen beryllium

and from the Ovonic Synthetic Materials Company (1987): OV-040A

2d ¼ 40 A˚

fluorine to silicon

with multilayers changing 2d spacing every 20 A˚ up to OV-140B

2d ¼ 140 A˚ carbon to oxygen

and also the MoB4C structure: OV-H series

2d ¼ 244 A˚ beryllium to boron

up to OV-300H

2d ¼ 300 A˚

Probably, materials with even greater interplanar distances can also be supplied. However, recent scientiﬁc (except the XRF ﬁeld) interest in multilayers is centered, instead, more on multilayers with periods of 10–20 A˚ for their excellent reﬂectivity or for their polarizing abilities (Scha¨fers et al., 1998). These materials are described in the current literature as completely thermally stable; thus, they do not need temperature stabilization when used in x-ray spectrometers (Nicolosi et al., 1986; Barbee, 1986; Van Eenbergen and Volbert, 1987). Some sensitivity to the damaging inﬂuence of intense radiation [above 0.15 J=cm2 (Kohler et al., 1985)], conﬁrmed later in an article by Kortright et al. (1991) and by a comprehensive report by MacGowan et al. (1993), may limit the application of the multilayer for monochromatization of dense ﬂux of x-rays, which usually happens in synchrotron, plasma,

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or laser research. Perhaps, the more important conclusion from the latter study is that, after the impact of high power on multilayers, the amorphous carbon layer transforms partially into a graphite form. It improves the reﬂectivity but greatly deteriorates the stability and changes the interplanar distances. The phenomenon of ultrafast graphitization ( 40 ps) can, in the future, be applied for the construction of x-ray switches. These data can be compared with the results cited in a report by Barbee (1986). He claims that during the sputtering process of multilayer synthesis, very high temporary doses of energy can be deposited in the layers (power density ﬃ 105 W=cm3=s1), which involves a signiﬁcant increase in local temperatures (50–180 C). The relative spectral resolution of dispersive devices is often expressed in the form of the important parameter DE=E or Dl=l, which is the basis for their choice in XRF applications. Unfortunately, this is the weakest point of the multilayer. In a commercially available specimen, the relative spectral resolution changes from about 0.025 to over 0.1 while passing from structures with 2d ¼ 40 A˚ to those with 2d ¼ 244 A˚, respectively, for diﬀerent measured lines (Ovonic Synthetic Materials Co., 1987), and for the structure with 2d ¼ 75 A˚, it changes from 0.025 to 0.04 in the wavelength range 7–75 A˚ (Henke et al., 1988), which is about three times worse (Ovonic Synthetic Materials Co., 1987) or even more (Henke et al., 1986) than for a good classical pseudocrystal. The comparison of spectra of the same sample by using multilayers with diﬀerent spacings, as in Figure 23, is very instructive (Van Eerbergen and Volbert, 1987). Relatively broad internal bandpasses of multilayers diminish the resolution of the whole spectrographic device, seriously increasing the detection limits. This is of special importance to the analysis of the L or M series, and avoids the eﬀorts to search the chemical shifts in x-ray spectra [although see Habulibaz et al. (1996), where chemical shift is investigated for the SiL spectrum, diffracted on a multilayer with an interplanar distance 300 A˚]. The relatively wide bandpass of the multilayer can be used in a constructive way. As proven by Vrebos (personal communication, 1997; see Fig. 24), the imminent width of the BKa peak is so great that the contribution from the secondary beam collimation to the total width of the signal is negligible. In that situation, the analyst can remove the ﬁne or medium collimation or

Figure 23 Comparison of sensitivity and resolution for layered synthetic microstructures with 2d values of 5, 12, and 16 nm from bottom to top, respectively. Increased sensitivity together with decreased resolution on the drop in 2d is evident. (From Van Eenbergen and Volbert, 1987.)

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Figure 24 Comparison of the spectra of the boron K line, obtained by a multilayer PX-3 with a fine, medium, and coarse collimator. (Courtesy of Bruno Vrebos and Philips Analytical.)

even the collimation completely. The gain is double. It results from the greater intensity given by the multilayer as such, as well as from the lack of the collimator, which is always one of the most intensity-restrictive parts of the WD spectrometer. This kind of work with a limited collimation is possible only in the region of soft x-rays, where the population of lines is rare and in the absence of the very soft L and M lines of heavier elements. The suppression of higher-order diﬀraction peaks gives better spectral purity and may at least partially compensate for the poorer resolution of multilayers. The isolation of the weak MgKa line from the dominant CaKa line is possible by the use of PX1 structure for the analysis of cement (Fig. 25). This is an extreme example of the superiority of multilayers over a classical crystal in particular cases (Nicolosi et al., 1986). Another great advantage of multilayers is their great reﬂectivity. It should be added that the peak reﬂectivity is 3–10 times greater than that of corresponding classical crystals. Multilayers were frequently used in the routine determination of light elements. Many examples can be cited, including carbon (Nicolosi et al., 1986, 1987; Van Eenbergen and Volbert, 1987; Philips Application Note No. 745) and boron determination (Van Eenbergen and Volbert, 1987; Nicolosi et al., 1987; Adamson et al., 1991; Philips Application Note No. 737 and No. 803; Uhlig and Mu¨ller, 1991; van Sprang and Bekkers, 1998), and beryllium detection (Nicolosi et al., 1987; Siemens Lab Report X-ray Analysis, 1994; Anonymous article, The Rigaku J., 1997). In the analysis, advocated by Rigaku, the application of the high-power 4-kW generator was the additional factor, enhancing the detection power for beryllium determination. The detection limit for that element was found to be in the region of a few tenths of a percent. Determinations of light elements were made using both ﬂat optics sequential spectrometers and curved optics in simultaneous instruments (Van Eenbergen and Volbert, 1987). After overcoming the technical diﬃculties, the results obtained from such analyses may seem quite trivial (Nicolosi et al., 1987; Bonvin et al., 1995). However, not always are things going in that usual way. Van Sprang and Bekkers, (1998) emphasized the number of problems, coupled with light-ele-

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Figure 25

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2W scans for Mg in cement using TlAP and PX1. (From Nicolosi et al., 1986.)

ment determination. For the analysis of boron in glass, only the combination of methods gave good results in a whole range of the detectable concentrations: The fundamental parameter method using the ﬂuorescent signals for the low B concentrations and the use of the incoherently scattered radiation as a measure of the light-element fraction above 4%. One should note some special features of x-ray spectrometry in the soft x-ray range: The type of analysis changes from bulk to surface measurements; the inﬂuence of the chemical state and the subsequent wavelength shifts on the analysis can be signiﬁcant; the wavelength of soft x-rays is comparable to the surface roughness; and the problem of the surface quality becomes extremely important. Considering the so-called total information depth of x-rays [see Sec.IV.G.1, Eq. (19)], it can be calculated that in the case of iron determination in a steel sample, the eﬀective energy of exciting photons being 20 keV, the secondary x-rays arriving at the detector emerge from depths up to 62 mm. A carbon signal, emitted by a coal sample excited in the same manner, emerges from depths of only 8.3 mm. The problem looks even more dramatic if carbon is determined in a steel sample: In this case, signals emerge from depths as small as 0.45 mm! This means that the analytical volume for iron is about 140 times greater than that for carbon and the information on both constituents really refers to totally diﬀering volumes of the steel sample. Moreover, the method of excitation is also critical in this case: The excitation of light elements is most eﬃcient if made with relatively soft x-rays, the energy of which does not substantially exceed the value of the absorption edge. Excitation by use of hard radiation is not proﬁtable in this case, as photons are mainly stopped inside the sample. The most restricting stage is the very short escape path of the characteristic photons for low-Z elements. The eﬀect of the chemical state can be observed during the analysis of some elements using soft x-rays. The wavelengths of La1,2 lines of chromium and manganese are 0.573 and 0.637 A˚, respectively, and should be analyzed in the region otherwise reserved for the

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K series of oxygen and ﬂuorine, hence by the use of the PX-1, PX-4 or OV-120B pseudocrystals. The question is whether the limited spectral resolution of multilayers is suﬃcient to show such subtle eﬀects as chemical shifts and to separate the mentioned lines of metals from the disturbing lines of oxygen or ﬂuorine if present in the sample. The wavelengths of characteristic x-rays in the K series are longer than atomic radii starting from Ga (GaKa ¼ 1.337 A˚; atomic radius of Ga ¼ 1.26 A˚) and passing to the lighter elements. Radiation with a wavelength comparable to the dimensions of an obstacle always result in surface problems, with scattering and diﬀraction at the head. One should take into account that even the most perfectly polished sample has a surface roughness of the order of at least one atomic radius and that the best layers in multilayer structures have a roughness, expressed as the eﬀective roughness seﬀ, on the order of one to several atomic radii (Barbee, 1986; Kortright, 1996). These latter remarks are not intended to weaken interest in multilayer optics and its application in XRF, but only to evoke a special awareness of potential problems. 3. Spectral Resolution Figure 26a shows the features of commonly used dispersive devise (Heinrich, 1981; Caciuﬀo et al., 1987; Plotnikow and Pszenicznyj, 1973; Fitzgerald and Gantzel, 1971; Burkhalter and Campbell, 1967; Gohshi et al., 1982a, 1982b; Potts et al., 1985; Sparks, 1980; Salem and Lee, 1976; Bent, 1970; Bandas et al., 1978; Gilfrich, 1987). The width of the bandpass DE (expressed as FWHM whenever possible) is plotted versus the energy of radiation E; the parameter is relative to the spectral resolution DE=E. A line of given value of DE=E is called the isoresolving line. These lines divide the whole ﬁgure into two zones of diﬀerent resolutions (or bandpasses). Note that both the typical dispersing devices and typical detectors are collected in Figure 26a. Only a few detecting devices with no resolving power (the reciprocal of the relative spectral resolution) at all exist (e.g., Geiger-Mu¨ller counter), and no detector exists with perfect, inﬁnite resolving power. Real dispersive devices exhibit very contrasting features, with the parameter DE=E varying from almost 1 (scintillator counter or semiconductor photocathode) to systems with DE=E as small as 104 [Ge (111) crystals and two- or three-crystal spectrometers]. Even better dispersive devices are needed, for example, for x-ray inelastic scattering spectroscopy in which relative energy resolution near 107 is wanted. Indeed, we can cite here some exemplary articles announcing the achievement of similar values: 1:14 107 at a level of 14:4 keV (Chumakov et al., 1996) or 2 108 for 25:7 keV (Verbeni et al., 1996). XRF spectrometry has rather modest demands for resolution. We see that for introducing the characteristics of the devices with spectral resolution on a level of 108 in our diagram, the scale of Figure 26a should be prolonged by approximately four orders of magnitude at the bottom! Another interesting conclusion can be drawn from Figure 26a. Nearly all dispersive structures exhibit variable relative spectral resolution, but only KAP, topaz, and Ge (111) crystals and, to some extent, also multilayer structures can be considered as isoresolving systems. The microcalorimeters show rather unusual behavior, compared with other detectors. They have more or less the same absolute spectral resolution in the whole useful range (2–20 keV). As we can see from the shape of the DK and DL curves, nature does not demand isoresolving behavior from the dispersive structures. Rather, these structures, with resolution curves parallel to the curves mentioned, would be optimal for spectrometric aims: The Si–Li detector seems to have the better features. Generally speaking, a bandpass range of 101–103 is considered analytically useful.

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Figure 26 (a) Spectral resolution of different devices: DZ ¼ spectral difference between analogous lines of adjacent elements; DW ¼ width of spectral line; Ka1 7 Ka2 ¼ difference between Ka1 and Ka2 lines; (b) the imminent features of x-ray spectra extracted from (a) for better characterization of the analytical demands; (c) characteristics of new microcalorimeter detectors as compared with typical Si–Li and LiF (200) dispersive structures.

Systems with bandpasses below 103 are used only in spectrophysical work. For making a proper choice of the dispersive structure for study of chemical shifts, one has to take into account the curves DW and Ka1 7 Ka2 (or even La1 7 La2) from Figure 26b. Figure 26b shows the spectral characteristics, extracted from Figure 26a, as a better demonstration of the immanent features of x-ray spectra. Depending on what matters in analysis, the adequate choice of the resolving system should be made. The regions occupied in the diagram by the most common [SiLi and crystal (e.g., LiF)] or most promising (microcalorimeter) detectors are shown in Figure 26c for easier comparison with data from Figure 26b. Figure 27 represents some example of the parallel analysis of the same sample, based on spectra from the electron microprobe acquired by the WDXRF system (TLAP), EDXRF system with Si(Li), and EDXRF system with the

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Figure 26

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Continued

microcalorimeter. For the time being, some preponderance of the LiF detector is observed for light element determinations (below 4 keV) (Wollman et al., 1997), but even this was endangered when Wollman announced the energy resolution of 2.0þ=0.1 eV at 1.5 keV level (Wollman et al., 2000). Spectral resolution is only one parameter determining the selection of a spectrometer. Other parameters, however, such as speed of measurement, count rate capability, geometrical collection eﬃciency, and ease of operation may aﬀect the choice of the most suitable instrument. E. The Goniometer The goniometer is basically a very simple construction and is the analog of a diffractometer as used in x-ray diﬀraction (the crystallographic research preceded the x-ray ﬂuorescence method). The essential features are shown in Figure 7. The sample is positioned in front of the tube window. To account for possible heterogeneities, the sample is

Wavelength-Dispersive X-ray Fluorescence

Figure 26

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Continued

spun about its axis at 30 rotations per minute. The analyzer crystal is positioned in the center of the diﬀractometer circle, rotates around an axis perpendicular to the plane of the drawing, and passes through the macroscopic plane of the crystal. The detector moves along the diﬀractometer circle supported by the goniometer arm. As imposed by the Bragg condition, the crystal rotates over an angle 2W, realizing equal angles of incidence and diﬀraction with respect to the crystal plane. The crystal is diﬀracting the incident ﬂuorescent beam of the opening angle determined by the slit width and length of the collimator. Rotation of crystal and detector are either mechanically coupled through gears with a ratio of 1 : 2 or mechanically decoupled but moved by separate computer-controlled stepping motors, but both arrangements are still implemented (cf. Tables 8–10 in Sec. VI). The idea of mechanical decoupling and optical position control was ﬁrst introduced by Applied Research Laboratories (see Sec. VI). This technique oﬀers a number of advantages: Alignment of crystals is no longer necessary. They can be positioned on a known spectral line and the oﬀsets are memorized by the computer.

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Figure 27 (a) NIST microcalorimeter spectrum of AlGaAs (solid line) compared to parallelly determined EDS spectrum [with commercial Si(Li) detector]; (b) WDS spectrum of the same sample (with TLAP applied as a crystal), spectrum transformed from wavelength into energy representation. Observe still a little better performance of the WDS spectrometer for the considered energy region 1–1.6 keV. (From Wollman et al., 1997.)

The detectors can be juxtaposed on the goniometer arm and brought into position as required by the program. Tandem arrangement of ﬂow and scintillation is no longer necessary. Slewing and scanning speed range is virtually unlimited. Optical position control (Moire´ fringes or optical encoder) allows very high angular precision and accuracy. Low-energy radiation is strongly attenuated by air. The Ka radiation of a pure copper sample (sample 2’’, measured on a Philips 1410 sequential spectrometer, SWT with chromium anode, 45 kV and 60 mA) produces 67800, 33200, and 2660 cps, respectively, in vacuum and 48000, 16800, and 890 cps, respectively, in air. The results mentioned are not easily repeated on recent computer-monitored instruments (e.g., Philips 2400) which do not routinely allow working under air at normal pressure. To decrease the absorption of photons, all wide-range spectrometers operate in a controlled atmosphere (vacuum or helium). The vacuum cabinet contains the entire spectrometer, except for some instruments such as the scintillator detector. Attenuation of the high-energy radiation is low, and to reduce the size of the vacuum chamber, the scintillation detector was and still is placed by some producers outside the chamber. A long rectangular window of a mechanically strong low-density polymer (Mylar) along the diﬀractometer circle allows the high-energy photons to reach the detector. High angular precision and accuracy is required for positioning the goniometer arm; therefore, severe mechanical constraints are imposed on its construction. Typical is an

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angular reproducibility of 0.001 or less and a mechanical resolution of a few thousands of a degree or less (cf. Sec. VI). In most modern designs, 2W and W axes are decoupled. Scanning (step size can be software selected) or measuring at preprogrammed positions (e.g., element line, background left and right of the line for many elements) is possible. Typical angular scan speeds (2W) are in the range from 0.1 to 180 =min and slewing speeds can be up to 4800 =min. The slewing speed is the maximum rotation speed of the goniometer arm for passing from one position to another without recording. The 2W range covered is from 4 to 152 , with slight diﬀerences between manufacturers. Because of the high slewing speeds and full automation, the analysis time is severely reduced compared to the older intsruments and, when not too many elements have to measured, is often not prohibitively long. Jenkins et al. (1984) and Croke and Nicolosi (1987) announced a dual-channel sequential spectrometer. A twin primary collimator was installed consisting of an upper, ﬁne collimator and a lower, coarse collimator. The beam is ‘‘divided’’ into two parts, each striking its own crystal. Two crystals are mounted close together but with a 15 inclination diﬀerence. The diﬀracted beams are detected by two detectors 30 apart. This ﬁxed angle between both detectors, however, limits the beneﬁts of a dual-channel instrument. Two alternative constructions should be discussed here. Simultaneous or multichannel instruments are equipped with monochromators set by the manufacturer (or by the customer) at a ﬁxed position for the detection of a single element. Usually, curved crystal optics are used, and the whole setup is optimized for that speciﬁc element with an appropriate curved crystal, entrance and exit slits, and detector. The crystals are logarithmically or logarithmically and cylindrically curved as explained in Sec. III.D.1.e, see also Figure 20. To give some more ﬂexibility to multichannel instruments, one or two channels can be equipped with a scanner. The optics are either focusing (e.g., Bruker) or nonfocusing (e.g., Philips). The ﬂat crystal arrangement is principally a reduced size of the normal monochromator. The focusing arrangement is complicated by the fact that crystal and detector must be moved to focus the diﬀracted beam on the detector at any position. The scan range of 2W for the Bruker scanner is 30 –120 and 10 –100 for the Philips scanner. Full details of typical construction diﬀerences are given in Sec. VI on instrumentation. Two hybrid, diﬃcult-to-classify spectrometers are discussed in this section: MDX1000 (Oxford) and the Venus 100 (Philips). The so-called ‘‘ﬂexi-channel’’ of MDX1000 is an energy-dispersive spectrometer. The Venus 100 has only ﬁxed channels, but they are sequentially presented to the ﬂuorescent beam. Moreover, manufacturers are focusing more and more on dedicated hyphenated instruments combining x-ray ﬂuorescence with diﬀraction measurements, such is an ARL 8600S and Philips CubiX. F.

Detectors

The objective of a detector is the transformation of photon energy into an electrical pulse. Pulses are counted over a period of time, and the count rate, expressed in counts per second or any other unit of time, is a measure of the intensity of the detected x-ray beam. From the theoretical point of view, the interaction of photons with matter is explained by the particle character of electromagnetic radiation, not by its wave nature. Four main classes of detectors may be distinguished for use in x-ray spectrometry according to the medium responsible for the energy transformation: gas detectors, scintillation detectors, semiconductor detectors, and recently introduced detectors supported on superconductive elements (tunnel junctions and microcalorimeters). They diﬀer in the eﬃciency of detecting photons of a given energy, determining the range of energy or wavelength for which they

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are suited. The width of distribution of pulse amplitudes as a result of impact of photons of the same energy determines the spectral resolution. Until recently, the best performance in energy discrimination has been obtained with semiconductor detectors and the full width at half-maximum (FWHM) is for the typical XRF domain in the range of 130 eV. This allows electronic pulse-height selection discriminating the energy of the impacting photons (at the expense of count rate, a maximum of 30 kcps). In WD spectrometers, pulse-height discrimination is required for removal of noise of low amplitude and of higher-order peaks of other elements interfering with the analytical line. It is a disadvantage typical for diﬀraction spectrometers. High-resolution detectors are not required in WD spectrometers because the interfering photons diﬀer so substantially in energy that moderate discrimination is suﬃcient (a few hundred electron volts). High-resolution detectors constitute the heart of energy-dispersive devices and are discussed extensively in Chapter 3. Now, it is obvious that new types of device, superconductive junction detectors and microcalorimeters, exhibit energy discrimination an order of magnitude better. Of course, in the future they might be the obvious choice in applications demanding better spectral resolution, but the availability of the superconductive detectors is still a problem, as well as the maximum count rates they can manage and their routine maintenance and price. Only gas-ﬁlled and scintillation detectors will be discussed here as the devices supplementing crystal dispersive operation. Some attention will be paid also to the microcalorimeter counters, because the stormy period in their development and because of the prospects for the construction of the future generation of XRF (see Sec. VII). 1. Gas-Filled Detectors In this type of detector, the energy-exchange process occurs between photons and gas atoms or molecules in a strong electric ﬁeld. When a photon strikes an atom, there is a given probability that the quantum of energy hn of the photon is imparted completely to an orbital electron of the atom. As a result of this gain in energy, the electron emerges from the atom with a kinetic energy of Ekin ¼ hn W (Einstein’s photoelectric equation). W is the work function of the electron for leaving its orbital and is a speciﬁc constant of the atom. In the case of argon, hn

Ar ! Arþ þ e The electron imparts its kinetic energy to other atoms, creating a series of electron–cation pairs on its path through the gas in a number equal to Ekin divided by the energy needed to expel an electron. Values for diﬀerent gaseous and solid detector media are included in

Table 4

Effective Ionization Potentials (eV)

He Ne Ar Kr Xe NaI (Tl) Si (Li) Ge

27.8 27.4 26.4 22.8 20.8 50 3.8 3

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Table 4. Subsequently, the cations and electrons are accelerated toward the cathode and anode of the detector by the gradient of the electrical ﬁeld between both electrodes, forming the avalanches of new ion pairs on their way to the electrodes. The discharge at the electrodes gives rise to a current pulse in the external electrical circuit. For a range of electrode potentials, the current pulse is proportional to the energy of the photon, and in spectrometers, the gas-ﬁlled detectors are exclusively used in this range. This is why these detectors are called proportional counters. Above this potential, in the Geiger–Mu¨ller plateau (Fig. 28), any photon entering the detector gives rise to a pulse of maximum amplitude without relation to the initial photon energy. The ratio of the number of ion pairs discharged at the electrodes to the ion pairs originally formed is called the ampliﬁcation factor, generally denoted A. It is a complex exponential function of ionization potential of the gas, anode potential, radii of detector chamber and anode wire, and the mean free path of the electron. It implies that stability of applied potential and gas pressure is very important for reducing statistical errors on pulse amplitude. The basic parts of a gas-ﬁlled detector are a metal rectangular housing as cathode, a thin wire passing through the central axis as an anode isolated from the cathode, a ﬁlling gas, and a window either on one side or on two opposite sides of the housing. This gas ﬂows through or is enclosed in the housing (Fig. 29). The diameter of the anode wire should be small. Realistic diameters are in the range of 40–80 mm. a. Flow-Proportional Counters These detectors cover the widest wavelength range and are used in sequential spectrometers exclusively for long wavelength ( > 2 A˚) and also in simultaneous instruments. The windows are made of thin foils of Mylar or polyester of 0.6, 1.5, 2 or 6 mm or polypropylene 1, 2, or 6 mm thick. On the inner side, windows are coated with gold or aluminum to create a homogeneous electrical ﬁeld inside the detector. Because of these very thin windows, there is a leak of the ﬁlling gas that is compensated by a

Figure 28

Region of proportionality for gas detectors.

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Figure 29

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Construction of proportional counters: (a) flow counter; (b) sealed counter.

constant ﬂow of gas through the counter (Fig. 29a). This is called a ﬂow-proportional counter. The gas is Ar with CH4 at a concentration of 10%, and the ﬂow is about 0.5 dm3=h. For detectors in simultaneous instruments, helium is also used, or 88% He þ 12% CO2. For very speciﬁc applications, other combinations of gases may be useful. The gas is stabilized for pressure, ﬂow, and temperature. This is of signiﬁcance because a 1% change in gas pressure or temperature introduces a much more serious 6–7% change in pulse amplitude, which can discredit the results of the analysis (Short, 1991). The ampliﬁcation factor in the proportional range is between 102 and 106. The current pulse is conducted over a resistor by which it is converted into a voltage pulse that is ampliﬁed, shaped, possibly discriminated in height, and counted. The count rate is a maximum of 26106 cps. The operating voltages range from 1000 to 3000 V. b. Sealed Proportional Detector Basically, the construction is the same as in the ﬂow-proportional detector but the windows are thicker and no constant ﬂow compensation for a leak is needed (Fig. 29b). The gases used are Ne, Kr, and Xe with Al, Be, mica, Mylar, or polypropylene windows. According to the element for which the radiation is to be detected, the most suitable combination of crystal, gas, and windows is chosen; for example, a PET crystal with a Ne detector with Al (for the determination of magnesium!) or a LiF (100) crystal with an Ar or Kr detector and a Be window for the detection many of medium-Z elements. If the ﬁlament is 80 mm thick, it is stiﬀ and easy to install, has a long lifetime, and is easily exchanged. Because the proportional counter is often used in tandem with a scintillation counter, two windows are present on both sides of the main body. The absorption

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probability of photons decreases with photon energy; the high-energy photons pass to the scintillation counter through the proportional counter without being essentially absorbed. The sealed detectors are applied in simultaneous instruments on their own and are not used in tandem with a scintillation counter. There is no need for a tandem arrangement because each channel is optimized for one element with the best-suited combination of detector and crystal. c. Pulse-Height Distribution In the detector, a photon generates a number of ion pairs proportional to the pulse amplitude. The formation of ion pairs is subject to statistical ﬂuctuations, which means that the number of pairs formed oscillates around the most probable value. A record of pulses and their amplitudes as they emerge from the detector as a function of time is represented in Figure 30. When these pulses are collected according to amplitude, a distribution emerges as shown on the right side of Figure 30. The width of the distribution is measured at half-maximum and is designated FWHM. This determines the energy resolution of the detector. It is small for semiconductor detectors, such as Si(Li) (about 130–150 eV for the 5.9-keV MnKa lines from a standard 55Fe source; see Chapter 3), about 900 eV for gas proportional counters, and 3500 eV for a scintillation counter, all measured for the same photon energy. Note that this distribution diﬀers from a distribution in intensity for a normal spectral line (intensity versus wavelength) because of natural and instrumental line broadening. d. Escape Peak As mentioned in the beginning of this section, a very probable method of energy exchange between a photon and an atom of the counter gas is the removal of an outer-shell electron. The probability of removal of inner-shell electrons is not zero, however, and this phenomenon happens in a number of interactions, in which case the speciﬁc K or L radiation is emitted. Because the wavelength of this radiation is on the long-wavelength side of the absorption edge, the mass attenuation coeﬃcient is low and this characteristic photon has a good chance of escaping from the detector. The remainder of the photon energy is transferred to the electron of which the kinetic energy is generating a number of ion pairs but necessarily in a smaller number as an electron having received the whole photon energy. Thus, a number of pulses is created, distributed around an amplitude proportional to the energy of this electron. The absorption edge for ArK is 3203 eV; the energy of the FeKa line is 6403 eV. Pulses centered around 6403 7 3203 ¼ 3200 eV are generated, and

Figure 30 Amplitude–time record of impulses from the detector (left) and transformation of it into a pulse-height distribution (right).

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about 3200=26 ¼ 123 ion pairs are formed, 26 eV being the energy needed per creation of electron–ion pair. These pulses are present in the pulse-amplitude distribution as a second maximum on the low-energy side of the iron peak position. For Ne, Kr, or Xe, photon energies greater than 0.87, 14.32, or 35.58 keV, respectively, are needed to excite the K radiation. For lower energies, only L radiation is excited, resulting in an escape peak closer to the main pulse distribution. Due to the constant distance of escape peaks in relation to their ‘‘parent’’ ﬂuorescence peaks and high intensity in some cases, they can be used sometimes instead of the originals. e.

Pulse-Height Discrimination

The energy resolution of a proportional counter, a few hundred electron volts at least, is not suﬃcient for use in energy-dispersive spectrometers (but see the simpliﬁed MiniMate construction by Philips). The spectral line separation in wavelength-dispersive spectrometers is done by the crystal monochromator. As is obvious from Bragg’s law, however, higher-order lines of other elements and the continuum radiation of the tube may be present in the same spectral window (angular position) of the crystal monochromator. In spectrometry in the ultraviolet or visible region, the higher-order diﬀractions of the gratings are removed by ﬁlters. This is not easily done with x-rays. For the same angle 2W, a second-order line has a substantially diﬀerent wavelength or energy and is far enough apart to give two distinct pulse-amplitude distributions at the exit of the counter. For x-rays, electronic pulse-height selection or discrimination fulﬁlls the role of ﬁlters in the optical region. The pulse-height analysis constitutes a necessary attribute to the wavelength-dispersive systems. Within an interval of 0.1 around 2W ¼ 37 for LiF (200) as analyzer crystal, we ﬁnd AuLa1 in ﬁrst order, E ¼ 1:2764 keV, ThLg8 in second order E ¼ 0:6390 keV, SnKb2 in third order, E ¼ 0:4259 keV, and LaKb2 in fourth order, E ¼ 0:3201 keV. The energy diﬀerence with the second-order line is 0.6374 keV and the energy resolution of a proportional counter is about 0.3 keV. The electronic circuits of all commercial instruments are able to discriminate between the pulses of both lines and, consequently, also the other higher-order lines. 2. Scintillation Detectors The energy exchange occurs in this type of detector in a medium of higher density and with high-Z elements in the matrix, namely a thallium-doped sodium iodide [NaI(Tl)], and as can be expected, this detector is only eﬃcient for high-energy photons (< 2 A˚ or > 6 keV). The outer orbital electron for an iodide ion requires about 30 eV to be knocked out. Hence, the originally ejected electron is imparted with almost all the initial energy of the photon. The ejected electron dissipates its energy by promoting valence-band electrons to an excited state 3 eV above ground level, an energy emitted on deexcitation as a photon of 3 eV or l ¼ 410 nm. The intensity of the emitted light pulse is proportional to the number of electrons excited by the x-ray photon (ﬃ hv=3 eV). The pulses are detected by a photomultiplier in which the light pulse produces a few photoelectrons from the cathode material (e.g., indium antimonide). These electrons are accelerated inside the vacuum tube between the cathode and the ﬁrst intermediary anode, called a dynode (Fig. 31). They gain kinetic energy and generate, in turn, a higher number of electrons from the dynode. This process is most often repeated 10 times, resulting in a substantial multiplication of the original number of electrons. The multiplication depends on the potential diﬀerences between the successive dynodes (100–150 V), and the order of magnitude of this multiplication is 106, which is called the ampliﬁcation factor A. In general, the photomultiplier is a front-end

Wavelength-Dispersive X-ray Fluorescence

Figure 31

145

Layout of a scintillation counter.

tube, and the iodide crystal is in close contact with the tube window. The crystal itself has on its outer faces a reﬂective coating (but transparent for the photons above 6 keV) and a coating against moisture. In the external electronic circuit, the electrons provoke a current pulse of which the amplitude is ultimately proportional to the energy of the photon captured in the crystal. The conversion of the x-ray photons into a current pulse occurs over diﬀerent steps: the initial formation of a photoelectron (I ! I0 þ e ), the quantiﬁed dissipation of the kinetic energy of this electron in 3-eV steps, emission of 410-nm light photons, the production of photoelectrons from the photocathode of the photomultiplier, and the multiplication of electrons. Although the production of the number of light photons in the crystal (hn=3 eV) is higher than the number of ion pairs created in a ﬂow counter (hv=30 eV), the subsequent conversion into electrons is ineﬃcient and roughly 1 electron is produced for every 10 light photons! The statistical errors of each process, particularly in the photomultiplier tube, propagate into the error of the output current pulse, thus substantially widening the distribution of pulse amplitudes and, consequently, reducing the spectral resolution. The DE is approximately two to three times worse than for a ﬂow-proportional counter (see Fig. 26a), making a scintillation counter unsuitable for ED spectrometry. However, the resolution is still suﬃcient for a discrimination of higher-order spectra. Hardware for pulse amplifying, shaping, and counting are much the same as for the proportional counters. Also, an escape peak is present, with its energy equal to the initial photon energy minus the energy at the iodine K absorption edge. As already discussed, in many sequential spectrometers, ﬂow-proportional and scintillation detectors are used in tandem, coupled in diﬀerent ways. In another arrangement, both detectors are shifted over 30 . A secondary collimator is placed in front of the scintillation detector but inside the vacuum chamber; the scintillation counter is outside the chamber, but close to the exit window. According to the manufacturer (Rigaku), this allows a gain in count rate of 15%. 3. Alternative Con¢gurations Some interesting modiﬁcations of the detector have been proposed in the article by Ebel et al. (1983). These authors considered the possibility of reducing count losses between

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analyzer crystal and detector (see Fig. 32) by removing the secondary collimator and allowing the detector to collect a divergent beam and, at the same time, by using a positionsensitive wire detector (with an energy resolution of 100 eV for detected photons with energies of about 10 keV). Position-sensitive constructions are popular in many physical applications, unfortunately not in the XRF ﬁeld. Potentially, it may lead to new solutions for WDXRF. Another interesting eﬀort is the use of a scintillation gas instead of the scintillation crystal (Kikkert, 1983; Palicarpo, 1978). This is the so-called gas-scintillation proportional counter (GPSC). As is clear from Figure 26a, the resolution is substantially better than that of its ‘‘parents.’’ The production of avalanches of ion pairs by the expelled electrons is the same as described previously. The discharge currents at the electrodes are not measured, however, but, rather, the scintillation pulses in the ultraviolet part of the optical spectrum are detected by a photomultiplier as in other scintillation counters. An essential diﬀerence with conventional gas proportional counters is that the space charge during the passage of a photon is avoided. The freedom-of-charge-related phenomena enable using higher counting rates (up to 2.56104 cps; Dos Santos et al., 1993). The lifetime of a GPSC can be quite long, up to several years, if there is no gas leakage in the meantime. The ability to use counters with large windows positively diﬀerentiates this kind of construction from room-temperature solid-state detectors, such as HgI2. Their pulse amplitude is also proportional to the energy of the incident photon. The counter can be supplied with a digital rise-time discrimination analyzer to suppress the pulses

Figure 32 Geometry with position-sensitive detector: T ¼ x-ray tube; specimen with point P irradiated by primary x-rays; crystal; mp ¼ mirror plane; M ¼ mirror point of P; PSD ¼ positionsensitive detector; l ¼ distance between mirror point and detector wire. (From Ebel et al., 1983.)

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forming the background (Simo˜es et al., 1997). This results in a much better performance of the counter. 4. Calorimetric X-ray Detectors Calorimetric detectors operate in extreme conditions—at temperature of liquid helium. The essential part of the device is a low-temperature, high-Z superconducting material with a minimal heat capacity. X-rays, even single photons, are absorbed and their energy is transferred to phonons. The detector is thermally coupled to a cold bath. The increase in temperature, which results from the absorption, is measured and is proportional to the energy of the photon. The temperature is measured by a doped semiconductor thermistor, as small as possible to minimize its heat capacity. It converts the experienced temperature change into a change in resistance. The thermalization time of the detector is in the range of 1–50 ms, and the time necessary to equalize the temperature of the detector to that of the cooling device is 100–500 ms. There is an analogy to the dead time of gas-ﬁlled and gasscintillation detectors. Still, there is a possibility for that type of device to allow count rates as high as 104 cps. Now, in more modern constructions (Irvin et al., 1996), the semiconductor thermistor is substituted by a superconducting transition-edge sensor (TES) thermometer. It is a strip of superconductor material biased within the transition range from the normal to superconductive state. The greater thermal sensitivity of the device allows the use of normal metal absorbers as a ﬁrst stage of the detector. Unlike semiconductors and superconductors, metals can dissipate the absorbed energy into freeelectron excitations, not phonon ones. The free-electron deexcitation process is quick and eﬃcient. The use of TES leads to a change in the ampliﬁer: A less noisy superconducting quantum-interference-device current ampliﬁer is introduced. As a whole, this new combinations exhibits a surprising feature: The energy resolution of the detector, as measured for the MnKa line, is better than the thermodynamic energy ﬂuctuations in the microcalorimeter (7.2 eV in comparison with 10.4 eV). The total spectrometer eﬃciency obtained up to now is reaching 80% in some low-energy regions below 5 keV and can still be improved. Among all the detectors mentioned in the present chapter, this one does not make use of the ionization phenomenon. Strictly speaking, this type of detectors can measure energy transfers even in these processes, in which no ionization takes place. It has been recently demonstrated for, e.g., particle detection and impact energy measurements in time-of-ﬂight mass spectrometry of single biomolecules (Hilton et al., 1998). The quantum eﬃciency of such an x-ray detector can be very good ( 95%) and the absolute spectral resolution is 7 eV in the range 0.2–20 keV; see Figure 26b (Silver et al., 1997). The expected absolute spectral resolution can potentially be even better than 1 eV in the mentioned range (Moseley, 1984). Of course, we are fully aware that this type of detector is not intended to compete with wavelength-dispersive systems, but we must emphasize here that the time of unbeaten behavior of crystals over all other spectral-resolving devices has gone. It seems to be a good place to present the comparison between wavelength and most modern energy-dispersive detectors at their present status. 5. Dead Time and Shift of Maximum Pulse Amplitude At high count rates (from a few thousand counts per second on!), the recorded count rate deviates proportionally from the real count rate. The reason for this discrepancy is the slow decay of the anode charge. Cation sheaths screen the cathode. This reduces the

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eﬀective voltage between the electrodes, resulting in a lower acceleration of the electrons and, consequently, in a smaller number of ion pairs formed. Fouling of the anode wire by impurities in the counter gas has the same eﬀect. This phenomenon shifts the maximum of the pulse-amplitude distribution to pulses from apparently lower-energy photons. Moreover, when a photon is detected, the detector is, for a short time, unable to detect the next incident photon. This period of time is called the dead time. The value of the dead time is about 1–2 ms for scintillation counters, about 0.2 ms for proportional counters, and 200 ms for a Geiger–Mu¨ller detector. To overcome this problem, one can either choose another detector (but the choice is very limited), lower the intensity of the exciting beam by masks or absorbers, change the geometry and the optical arrangement (as is done in the case of synchrotron radiation), or use electronic anticoincidence circuits or appropriate mathematical corrections. Now, after the introduction of the high-power 4-kW option to the modern spectrometers, masks can be inserted for the analysis of the major elements (e.g., in Shimadzu instruments). The mask avoids problems with too great a count rate and its introduction is probably easier than tube power change during the analysis. The relation between the real and recorded count rate is Nr ¼

Nm 1 N m td

ð11Þ

where Nr is the real count rate (count=s or cps), Nm is the measured count rate (cps), and td is the dead time (s). If we allow a given level of discrepancy, say a%, then the following equation is valid: a ¼ N r td 100 a

ð12Þ

The real count rate and dead time are thus connected by a hyperbolic relation. If the allowed value of discrepancy must be kept constant, the left-hand side of the hyperbolic equation is constant. However, this is only part of the problem. As already stated, a photon, arriving just at the end of the dead time, is registered with reduced amplitude. It is only after the socalled recovery time that a new photon is registered with normal amplitude. An improved method for calculation of the dead time was proposed by Bonetto and Riveros (1984). These authors used the well-known fact that the second-order peaks are signiﬁcantly less intensive than their ﬁrst-order analogs, but the ratio of the intensities should be constant. If it is not constant, the deviation results fully from dead-time losses in the ﬁrst-order signal. They derived the relation N m1 ¼ B AN m1 N m2

ð13Þ

where B ¼ Nr1 =Nr2 and A ¼ Btd . The indices 1 and 2 refer to the order of the spectrum. A summary of the constituting parts of spectrometers and the wavelength ranges in which they are used is given in Figure 33. Note that diﬀerent excitation sources are used in diﬀerent branches of XRF, namely radioisotopes for EDXRF and tubes for WDXRF; there are also examples of synchrotron radiation. The x-ray laser is mentioned, but its appropriate version for XRF applications does not yet exist.

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Figure 33 Review of different elements serving the construction of a spectrometer: optimal range (crossed area), feasible range (hatched area), and possible range (open area). (Adapted from Bertin, 1975.)

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QUALITATIVE AND QUANTITATIVE ANALYSIS

In this section, we review a number of items important to both qualitative and quantitative analyses. The qualitative application of XRF spectrometry is obvious. In simple cases, quite unequivocal assignments of spectral lines can be made based on a library of Ka or La lines or, in case of doubt, when an overlap exists, Kb or other L or M lines are checked to conﬁrm or reject a line assignment. A few more complex situations are discussed. Quantitative analysis may also be simple. In the low-concentration range, quasilinear relationships are found between intensity and concentration. The same holds for determinations of higher concentrations but in a very narrow range for samples with quite comparable matrices. Interpolation on standard curves, obtained on suitable standard samples, allows a quick and suﬃciently accurate quantitative determination. Another intensity–concentration relation, valid for binary samples, is based on the relative intensities of Ka and Kb lines in the case that the absorption edge of another element is located in between these two lines. In these boundary cases, matrix eﬀects are negligible or interbalanced but always exist. In normal practice, they compromise the conversion of intensity into concentration in such a way that more or less complicated mathematical methods must be used for obtaining any accurate quantitative determination. The approach by fundamental parameter programs or correction algorithms is dealt with in Chapter 5. Between simple qualitative work and rigorous quantitative work, however, an area of applications is situated in which knowledge of the exact concentration or the complete qualitative composition is less important than the overall aspect of a spectrum for recognizing classes of samples (i.e., pattern recognition). An introductory discussion of this aspect is given in this section. Another point of interest in qualitative as well as to quantitative work is the measurement of the background. A good estimation of the background is necessary for obtaining net count rates, but decreasing the background allows lowering the detection limits. The origin and magnitude of the background is bound to the physical processes involved in the method of excitation and, thus, indirectly to the type of spectrometer. This aspect is also treated in this section. Finally, the said disturbing phenomena associated with the excitation of ﬂuorescence radiation, as bremsstrahlung and the scattered radiation, can be used to increase the total analytical information and they also have some interesting applications. A.

Background

The background in WDXRF has four main sources: (1) the coherent and incoherent scattering of source radiation, (2) the presence of characteristic radiation of other than sample origin (materials of sample cup masks, sample cups, collimators, spectrometer housing, for example), (3) the detector, and (4) environmental noise (i.e., resulting from the threshold natural radiation of materials and even from a cosmic origin). The background from characteristic radiation from construction materials of the spectrometer can be minimized by careful construction and geometric arrangement of the device. Scattering from sample cup masks can eﬃciently be reduced by masks in front of the primary collimator. This contribution is especially great when small samples are analyzed. The beam mask reduces the irradiated area and may totally eliminate scatter from sample cup masks. Higher-order scatter increases with increasing wavelength. Pulse-height selection, as explained earlier, supresses counts from higher-order diﬀraction, but this technique is most eﬃcient in the long-wavelength region and is decreasingly eﬃcient toward the short-wavelength end. The short-wavelengths contribution to the background can be reduced by

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installing adequate ﬁlters. A 100-mm titanium foil removes the characteristic lines of chromium from a chromium anode tube, signiﬁcantly improving the conditions for the detection of Cr, Mn, Fe, Co, and Cu (Tertian and Claisse, 1982). For Cr tube radiation and a LiF (200) analyzer crystal, removal of scattered chromium radiation may be necessary when detecting the neighboring ﬁrst-order lines. As Table 5 shows, not very many common elements are interfering. For steel analysis, only manganese really represents a problem. Because chromium is present in prevailing concentrations in the environment (tube anode!), manganese determination is also diﬃcult with an aluminum ﬁlter. The simplest solution in such a case is the use of a LiF (220) crystal with better resolution in this region (but also with lower intensity). If tubes with other targets are used, tables are similar to Table 5 but with even more complex interference if targets like Mo and Rh are applied. The other elements listed in Table 5 refer mostly to less common elements and less intense lines. Other higher-order lines may interfere as well if the concentration of elements, from which the lines are derived, is high. For special cases, a convenient combination of tube targets, analyzer crystals, and ﬁlters can almost always be selected. Let us look at an example for the rhodium target tube. In the determination of palladium, PdKa1 overlaps with the Compton peak of RhKb (Fig. 34). A molybdenum ﬁlter in the primary beam reduces this peak to a very small level, allowing a perfect estimation of the location of the maximum and the determination of the intensity of the palladium line. Another eﬃcient solution for background suppression is the use of linearly polarized x-rays. An excellent example was given by Kno¨chel et al. (1983) and it is reproduced in Figure 35. The analysis of nitrogen with a small concentration of xenon was made by a synchrotron excited XRF. Some special geometries allowing the use of polarized x-rays have been proposed by Wobrauschek and Aiginger (1980, 1983, 1985, 1986) for the more conventional XRF spectrometers. For EDXRF systems, a special commercial polarizing design is available (SPECTRO A.I. Gmbh). The subject is of great signiﬁcance to synchrotron-based XRF and discussed as such in Chapter 8. A careful estimation of the background is apparently the simplest way to take the background into account! A ﬁrst approach, but with limited application, is measurement on blanks—simple at ﬁrst glance, but the adequate blank is often a rarity. Moreover, the blank ought to have the same scattering properties. Intensive eﬀorts for mathematical modeling of background were performed but frequently for other purposes. The most common method of background evaluation is the linear (or even nonlinear) interpolation from background countings at a suitable angle above and below the peak position (suitable means not too far away for an easy linear interpolation, outside the tailings of the peak, and not at the position of an adjacent peak). Consequently, a good

Table 5 CrKa Pm La2 La Lb2 Ba Lg5 La Lb7 Ce Lb6 Pm La1 V Kb

Interfering Lines for CrKa and CrKb with LiF 200 crystal DE (eV)

D ( 2W)

CrKb

10 30 40 30 20 20 10

0.05 0.31 0.60 0.23 0.31 0.31 0.56

Ho Pr Mn Pm Ba

Ll Lb7 Ka Lb1 Lg4

DE (eV)

D ( 2W)

< 10 20 50 10 30

0.04 0.24 0.61 0.17 0.30

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Figure 34 Spectra of Pd excited by a Rh tube and recorded with and without a primary beam molybdenum filter (0.1 mm Mo). The peaks are labeled PdKa, PdKb, RhKa, and RhKb with their respective order indices. (Courtesy of ARL.)

spectral resolution of the system is essential because only in this case are peaks very narrow and do not overlap, and the background level is easily reached in the intervals between peaks. Finally, we refer to a method based on the relationship of ﬂuorescent and scattered radiation intensities versus the value of attenuation coeﬃcients. No measurements of background for the estimation of net line intensities is required in this case (Bougault et al., 1977). If we cannot overcome the background problem, it would be advantageous to use the background or scattered tube lines for increasing the information about the sample. There are very interesting examples of such work and they will be shown in Sec. IV.C.

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Figure 35 Synchrotron-excited spectra of gaseous nitrogen with a small amount of xenon taken in the direction of polarization and perpendicular to the orbit plane. Counts normalized to stored electron current are integrated over the lifetime of the detection system. DORIS was operated at 3.3 GeV and about 55 mA. (From Kno¨chel et al., 1983. Reprinted by permission of the author and Elsevier Science Publishers.)

B.

Qualitative Spectrometry

Although the number of spectral lines is very limited compared to atomic emission spectra in the ultraviolet or visible region, overlapping of lines in x-ray spectra does exist. A classic example can be found in the works of Gentry et al. (1976), Sparks et al. (1978), and Sparks (1980), and an example is reproduced in Figure 36. At energies at which K lines of elements between palladium and cesium (about 22 and up to 30 keV) can be present, additionally the tails of L lines of uranium and thorium are found, and also the theoretically calculated M lines of the superheavy elements of Z ¼ 110 [in the time which passed since 1976 this element has been discovered (Hofmann et al., 1994)] and 126 should be expected. The erroneous attribution of some lines to these hypothetical elements was the subject of extended research and long discussions (Sparks et al., 1978; Sparks, 1980). In the software of modern spectrometers, special searching blocks exist for the assignment of lines: The detected line is compared with data compiled in the program. After the initial assignment, other lines of the element are identiﬁed according to data from the library. The initial assignment is eventually rejected if not conﬁrmed by other lines (Fig. 37). Multichannel wavelength-dispersive instruments are not suited for this type of work, except for elements implemented on the spectrometer (maximum 28 elements). Sequential instruments can cover all elements (from beryllium on). The spectra can be displayed on the monitor screen of all modern spectrometers, documented, memorized as ﬁles, and printed. Particular parts of the spectrum can be scanned occasionally with higher resolution and sensitivity (step scanning) according to the time available for the analysis and the information required. Parts of particular interest can be displayed separately with a linear intensity axis or rescaled to square root or logarithmic scales. The peaks can be identiﬁed on the display by the angle 2W, by element name, either by wavelength or energy.

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Figure 36 Fluorescence spectrum from a monazite particle excited with 37-keV synchrotron radiation (ORNL DATA) shows an improved signal-to-background ratio over that excited with 5.7-MeV protons (FSU DATA). Data points every 20 eV in ORNL spectrum and every 61.8 eV in FSU spectrum. (From Sparks, 1980.)

Figure 37 Example of qualitative analysis by rejection. The suspected MoLa peak in the lower spectrum was rejected after comparison with the standard containing molybdenum (upper spectrum). (Courtesy of Phillips Analytical.)

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The spectrum can be scanned linearly with respect to 2W or in a more optimal way. If Bragg’s and Moseley’s laws are combined, the following relationship among Z, the atomic number, and W is found: rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ m þs ð14Þ Z¼ k2d sin W Z is proportional to the inverse square root of sin W. Jenkins et al. (1979) proposed computer monitoring of the scan speed according to this equation. It optimizes the scan speed, which represents a high gain in time especially in the low-Z range where the angular distances between the lines are long. The newest applications of WDXRF softwares try to avoid the old-fashioned treatment of data. One such traditional manual estimation was that concerning the possible spectral interferences. Normally, the analyst has to decide which place in a spectrum is relatively empty of disturbing peaks and compromising background contributions. This job was rather time-consuming, with the wide use of spectral tables and other datasets concerning, for example, intensity ratios. The results were not always encouraging. Now, virtual synthesis of wavelength-dispersive spectra is in progress (Reed and Buckley, 1996), in the Internet demonstration version also (Buckley, 1998). The expected pure-element spectra are loaded from the memory and installed in a region of interest of the anticipated spectrum. Those partial spectra are apparently corrected for the expected concentrations of the sample constituents. From the result shown on a screen of the computer, one can easily estimate whether the region of interest is suitable for a given analysis or should be excluded. Such a procedure makes the selection of lines for the qualitative analysis much easier, as well as the estimation of the background and also the assessment of the detection limits. C.

Special Quantitative Applications

Although the main quantitative treatment of XRF has been left to Chapter 5, still there are some special and nonconventional applications which proved to have potential practical meaning. Their signiﬁcance becomes obvious when, considering the excitation process, one wonders how great a part of the radiation is transformed into the scattered form and how it can supplement the essential quantiﬁcation based on characteristic signals. As stated earlier, monochromatic excitation simpliﬁes the quantitative treatment of spectral data considerably, but this case is only seldom met in WDXRF. The most common excitation source is the x-ray tube, in which a set of characteristic lines superimposed on a white spectrum is generated. For this reason, only the fundamental coeﬃcient method can be considered as rigorous and somewhat simpliﬁed in comparison with a complete fundamental parameter method. How the correction terms can be derived in a rigorous manner for the case of monochromatic excitation was shown in an article by Kuczumow (1982). In its original formulation, the Lachance–Traill equation is (Lachance and Traill, 1966) ! X Wi ¼ R i 1 þ ð15Þ aij Wj j6¼i

with aij deﬁned as a constant written as aij ¼

mjC1 ðEÞ þ mjC2 ðEi Þ miC1 ðEÞ miC2 ðEi Þ miC1 ðEÞ þ miC2 ðEi Þ

ð15aÞ

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where m denotes the mass absorption coeﬃcients for primary or secondary photons of energy E or Ei, respectively, corrected for the incident C1 and takeoﬀ C2 angles. It has been proved (Claisse and Quintin, 1967) that this expression is valid only for boundary conditions: monochromatic excitation and absence of enhancement. In all other conditions, a’s are not constant, as experimentally shown (Tertian and Claisse, 1982; Lachance, 1981). It is impossible to derive a more ﬂexible algorithm with constant coeﬃcients directly from Shiraiwa and Fujino’s equation (Kuczumow, 1982; Rousseau and Claisse, 1974). Other algorithms similar to that of Tertian (Rousseau, 1974) and Lachance COLA (Lachance, 1981) are not recognized as fundamental. An algorithm that can be considered partially fundamental is that of Claisse and Quintin (1967), but it suﬀers from inaccuracies in the approximations introduced. Later, Rousseau (1984, 1987) and Kuczumow and Holland (1989) gave examples of algorithms derived from fundamental assumptions. Kuczumow and Holland describe the change in a when enhancement is included (monochromatic excitation) as follows: ! X aij Wij þ Daik Wk Wi ¼ R i 1 þ ð16Þ j6¼i;k

where all a’s are deﬁned as previously but Da is deﬁned as Daik ¼

m0kC2 ðEi Þ mkC2 ðEi Þ miC1 ðEÞ miC2 ðEi Þ

ð16aÞ

The new term m0kC2 ðEi Þ was introduced in Eq. (16). Although dependent on the excitation condition E and composition, this new term has the great advantage of being dependent on the latter in a simple way. The possibility of deriving the Lachance–Traill equation directly from the Shiraiwa and Fujino equation under the conditions applicable to Eq. (16) is also well established (Kuczumow and Holland, 1989). The next interesting case concerns scattered radiation as a source of analytical algorithms. Similar to the derivation made by Kuczumow (1982), an expression on the basis of coherently scattered radiation is obtained (Kuczumow, 1988): X coh coh coh þ R a b ð17Þ Wi ¼ Rcoh Wj i i ij ij j6¼i

where acoh ij ¼ bcoh ij ¼

mjC1 ðE0 Þ þ mjC2 ðE0 Þ miC1 ðE0 Þ miC2 ðE0 Þ miC1 ðE0 Þ þ miC2 ðE0 Þ coh sj ðE0 Þ scoh i ðE0 Þ

ð17aÞ ð17bÞ

and Ricoh results from the comparison of a coherently scattered line from the sample and from the pure element i and scoh is the mass coherent scatter coeﬃcient (for the set of incident and takeoﬀ angles, as always in the conditions of the analysis). For Comptonscattered radiation, analogous expressions are obtained: X Wi ¼ Rcom þ Rcom acom bcom ð18Þ Wj ij i i ij j6¼i

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with mjC1 ðE0 Þ þ mjC2 ðEcom Þ miC1 ðE0 Þ miC2 ðEcom Þ miC1 ðE0 Þ þ miC2 ðEcom Þ scom ðE Þ 0 j ¼ com si ðE0 Þ

acom ¼ ij

ð18aÞ

bcom ij

ð18bÞ

is the ratio of the Compton peak intensity of the sample to that of the pure eleRcom i ment. Equations (17) and (18) may be called Lachance–Traill equations for scattered radiation by their general appearance as well as by the way they have been derived. These equations are less sensitive to the type of components and changes in compositions than their analogs for ﬂuorescent radiation, which otherwise is an advantage in analyses of samples widely varying in composition. From assumption, they include only the terms for strictly deﬁned energy values, even if the analysis is carried out with a polychromatic source! In Table 6, the coeﬃcients a and b from Eqs. (15)–(18) are collected, calculated for the K lines of La and Sm, the main components of samples consisting of La2O3 þ Sm2O3 excited by 60-keV photons at the incidence and emergence angles equal to 80 and 70 , respectively (the example is taken from EDXRF, but it is also instructive here). When applied to nonhomogeneous samples, also the a’s are proved to be dependent not only on the composition and the energy but also on particle size for dispersions or layer thickness for piles of layers of diﬀerent composition (Helsen and Vrebos, 1984a, 1984b). Kuczumow et al. (1992) have proved that parallely with the conventional analysis of binary Au–Cu alloys by ﬂuorescent lines (Fig. 38), it is possible to get similar information using coherently or incoherently scattered tube radiation. Sometimes, the results are even better than for the direct ﬂuorescence analysis. Still other possibilities of using the scattered lines exist. Kuczumow et al. (1995) have proved, by a proper mathematical transformation of the scattered Rayleigh signal, that for light, chemically homogeneous matrices (as biological materials, e.g., wood), this new function of scattered radiation, called the corrected scattered intensity, strictly follows the density changes in the analyzed material (Fig. 39). The use of Rayleigh signals eliminates the necessity of transmission measurements and brings important auxiliary information (e.g., about the density). Although the method just described has been applied up to now for x-ray capillary-microprobe-type measurements only, it may be used generally (Luggar and Gilboy, 1994).

Table 6

a and b Coefﬁcients for Fluorescent and Scattered Signals La2O3

af Daf acoh bcoh acom bcom

(La2O3 Sm2O3)

0.292 0.895 0.285 1.153 0.281 1.000

Sm2O3 (Sm2O3 La2O3)

1.009 — 0.222 0.867 0.219 1.000

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Figure 38 Analytical curves in the system R=W-1 versus w w0 for the CuKa line in Au–Cu alloys. The solid line with open circles denotes the authors’ fluorescence measurements; the solid line with crosses denotes Rayleigh measurements, and the solid line with solid square denotes Compton measurements. R is the intensity relative to the intensity measured for the pure element and W is the composition expressed as weight ratio; w represents the ratio of the mass attenuation coefficient of the sample to the sine of the takeoff angle C2 . The subscript O refers to the values for the pure element o. (Modiﬁed from Kuczumow et al., 1992.)

D.

Pattern Recognition

The spectrum of a sample of complex composition is characterized by a set of lines, documented by their position and intensity. If we imagine an n-dimensional space with the intensities on the coordinate axes, the set of intensities determines a point in this space. Such an intensity space is convertible into an n-dimensional space of concentrations, and the point in the intensity representation has its equivalent in the concentration space (Klimasara and Berry, 1987). The aim of pattern recognition is to attribute a sample to a given class. A class is in a discrete way a set of points of similar features in the intensity or concentration space or, in a continuous way, a subspace inside a complete n-dimensional space. This subspace is restrained by a closed hyperplane, which can be approximated by a number of hyperplanes of smaller dimensionality. Inherent to all experimental determinations, the determination of intensity is also bound to a certain degree of uncertainty. This implies that a point, as such, cannot be determined, only a small subspace of limited size, limited by the assumed conﬁdence levels. Equally ill-deﬁned for the same reasons is the boundary hyperplane surrounding the subspace occupied by a class of samples. The ‘‘thickness’’ of this ‘‘space ring’’ surrounding the subspace is determined in relation to each

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Figure 39 Line scans from x-ray microprobe measurements that show the correspondence between transmission (dotted) and corrected scattered (solid) data, and the sequence of tree rings of Silesian Pinus silvestris. The linear scan was made using a 89-mm capillary and a step size of 89 mm. (Modified from Kuczumow et al., 1995.)

coordinate axis by the appropriate multiplicity of the projection of standard deviation on this axis. 1. Delimitation of Classes If we consider classes as separate subspaces in n-dimensional conﬁguration space, it is obvious that each of these classes has its own boundary hyperplane or closed space ring. If these hyperplanes can be determined in an analytical way, we are able to know the boundaries of classes. In simple cases, a class can be delimited by construction of a lowdimensional hyperplane that closely approximates the real boundary. The simplest example is the trivial division of the whole conﬁguration space on two categories; occupied space–empty space (zero–one decision). It can be done by assuming the limiting surface (‘‘wall’’) inbetween the mentioned subspaces. If classes are separated by empty spaces, the diﬀerentiation of classes is even easier. 2. Method of Nearest Neighbours Here, the classes are considered as small subsets of points. The classiﬁcation of each new point relies on a comparison with its nearest neighbors. Knowledge of the distance between the sample and the nearest neighbors is essential. The accuracy of the method depends on the way the distance is determined and the degree of complexity of the boundaries between classes, although it can be simpliﬁed by imposing proper statistical weights on learning samples. The method of the nearest neighbors has further implications.

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Once the sample is placed in relation to the nearest neighbors, which are nothing but the standards, a separate program can draw the average from the composition of standards, taking the distances from the neighbors as the statistical weights for the estimation of the inﬂuence of each standard on the ﬁnal result (Klimasara and Berry, 1987). However, this way of using pattern-recognition methods is the same as very conventional method of comparison with standards! Of course, standards are no longer considered in a conventional sense as some numerical values in relation to some elements, but rather as the full patterns [compare also the genetic algorithm by Dane et al. (1996)]. 3. Reduction in Number of Dimensions All points and classes are included in the multidimensional space generated by the number of elements present in the sample; however, some variables may be less dependent on the related coordinates, and then the number of variables necessary to diﬀerentiate the sample may be reduced. The number of dimensions may also be reduced if some variables are correlated; the most obvious way of coupling the elements is by joining them in a chemical compound. 4. Pattern Grammar Approach Another version of ‘‘pattern recognition’’ is also possible. The full spectra for diﬀerent standards and=or samples can be collected and treated as some analytical claims (sentences), with peaks considered to be analogs of single words, and the whole set of spectra as a constituent of a ‘‘spectral language.’’ At the same time, we have the complete chemical knowledge of the same standards, with standard composition treated as the sentences, their particular components as the single words, and such second system is the part of another, ‘‘chemical language.’’ The task of the chemist is to translate sentences from spectral to chemical language while working with the real samples. Even more, the translation from one language to another can be treated as mapping (rather linear than nonlinear!). 5. Examples Samples may be classiﬁed into groups according to their origin. A primitive version of such x-ray methods was demonstrated by Lagarde et al. (1984). They observed the characteristic features of spectra of ceramics samples from western Africa and then put the samples in order ‘‘by the naked eye’’ according to their origin. A classic reference here is the work of Kowalski et al. (1972), which was followed by many other articles on archeology. Kowalski recognized the origin of obsidian artifacts by studying a set of intensities of 10 trace elements and projected the image of this set from 10-dimensional conﬁguration space into a 2-dimensional space (nonlinear mapping). Similar applications in geology are obvious. Lo I Yin et al. (1989a, 1989b) proposed the application of the pattern-recognition-supported XRF to the initial selection and analysis of Mars Rover geological samples. They discovered on this occasion that it is not necessary to use sophisticated detection systems. On the contrary, they applied probably the most primitive version of an x-ray spectrometer with a proportional counter as detector. The resolving power of such a detecting system belongs to the worst among the existing ones (compare Fig. 26a), but even in these conditions, the results of the sample-selection procedure are good. In such classiﬁcation procedures, putting emphasis on advanced data analysis can be a substitute for using a sophisticated high-resolution spectrometer for special analyses in diﬃcult conditions.

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If only a limited amount of data is available, samples can only be subdivided in groups. Discrimination of the alloys 663, B-1900, and 1455 can be done on the basis of their zirconium content (Kevex Material Analyzer Mode 6500 brochure). Nickel–cobalt alloys were subdivided on the basis of a logical analysis of successive spectral responses: Is the ratio ‘‘contents of elements A=contents of element B’’ greater than some assumed level X: A=B > X (Fig. 40)? Complete analysis of the alloy was not necessary. The same is often done when samples can be compared to standards [e.g., the ‘‘framing action’’ by software (description of Siemens software)]. An important application of pattern recognition may be found in ‘‘two-stage analysis.’’ For complex samples with nonlinear dependence of intensity on concentration, the analysis can be broken down in steps with a smaller number of dimensions (lower degree of complexity) or the samples may ﬁrst be subdivided in classes before applying correction algorithms. The appropriate sets of diﬀerential equations exist: they establish the relation between the value of relative intensities [Claisse–Thinh systems (Claisse and Thinh, 1979)] or between increments of intensities [Kloyber et al. (1980) and Kuczumow (1984) systems] and increments of concentrations. These increments are calculated with respect to a central standard that represents the origin of a local system of coordinates. In

Figure 40 Selective procedure of the KEVEX Material Analyzer Model 6500 based on successive analysis of the element ratios. (Courtesy of KEVEX Corp.)

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this local system, a certain set of diﬀerential equations is valid and is of linear character (if not, it is not proﬁtable to use it in such a restricted space). After attributing the samples to given classes, the right set of diﬀerential equations may be used for accurate determination of composition. The origin of the local system of coordinates may, at the same time, be the central point of a given class. When the delimitation of classes becomes problematical, a point in the conﬁguration space may be designated and a set of diﬀerential equations, valid about this point, may be determined. From this, the boundaries of classes can be found and determined by some leading vectors introduced in the central points of the classes. Yap (1987) and Yap and Tang (1984, 1985) posed the reciprocal question in relation to ‘‘pattern recognition’’: how to extract some number of diﬀerentiating features from the known classes, diﬀering them by time and place of origin. This was used to analyze pottery samples. Another name for this method is the ‘‘training phase’’ of the analysis. E.

Analysis on the Basis of Neural Networks

The pattern-recognition method leads us to quite diﬀerent levels of reasoning. The most popular recent way of working with a spectrum is to transform it numerically and show it in two-dimensional intensity–wavelength or equivalent intensity–2W or intensity–energy diagrams. Some logical artifacts result from the numerical transformation of the original sets of signals. The interesting comparison between functioning of the electronic chip in the binary system of information and the biological cell in the complex system of information was presented by Cholewa during 15 ICXOM, Antwerp (Cholewa, 1998) (see Fig. 41). At the same time, without such a numerical presentation, a common naked-eye observation allows one to compare a fresh spectrum to other similar pictures compiled in our memory. Our way of reasoning is not numerical at all, but resembles some procedures of pattern recognition. It is simply a process of attributing a sample to some class. In a happy case, one can recall the same pattern in memory and then classify the sample as a part of existing and well-described objects. One of the possible ways of reasoning similar to the ways of functioning of the human brain is by applying so-called ‘‘fuzzy logic.’’ In that type of logic, the object can belong to the proper set with some probability, not necessary equal to 1 (belongs) or 0 (does not belong). The spectrum can be recognized in this manner, with some artiﬁcial probabilistic rules set in the very beginning of the process. An instructive example is taken from an article by Otto et al. (1992). For the qualitative estimation of an x-ray spectrum, the following rule is very useful: If the Ka line is supposed to be in a spectrum, then the Kb line should be associated with it, being ‘‘about 5’’-fold weaker. Neural networks open another approach to the problem. They imitate the action of neurons joined by synapses in the human brain. The artiﬁcial network has a layer of input cells and a layer of output cells. The cells from layers just mentioned are connected, some of them directly and the majority indirectly, by the additional cells, grouped in one or more additional, hidden layers. Each middle cell collects one or more signals from previous cells, gives those signals the proper statistical weight (e.g., according to previously imposed rules), and sums them up. Next, the modiﬁed signal is sent from the middle cell into one or many cells in the next hidden layer or, ﬁnally, to the output layer. The initial information (e.g., spectrum) given to the input layer is transformed into a useful set of information (e.g., about the chemical composition) in the output layer (Otto et al., 1992; Larsen et al., 1992; Bishop, 1992; Bos and Weber, 1991; Luo et al., 1997). Otherwise,

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Figure 41 Drawing showing the different degrees of complicacy involved in the action of (left) the electronic chip and (right) the single biological cell, also from the informatic point of view. The electronic chip includes only one bit of the binary information; the cell is an incomparably more complicated system. It is a basis for the differences in computing and brain thinking. (Courtesy of Dr. Marian Cholewa.)

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it resembles the ‘‘pattern grammar’’ method previously described. The method can be applied for the classiﬁcation of very complex spectra, for clustering the datasets, for coupling the sets of spectra to the groups of compounds, and for nonlinear data transformation between sets with diﬀerent dimensionalities.

F.

Detection Limits

The importance of the correct determination of detection limits has been acknowledged by many chemists, especially those involved in environmental regulation. The accepted deﬁnition is as follows: ‘‘The limit of detection is the lowest concentration level that can be determined to be statistically signiﬁcant from an analytical blank’’ (Nomenclature, 1978). The limit of detection is expressed as a concentration cL or an amount qL and is derived from the smallest measure IL, I being the instrument reading. How the data are treated in the statistics is described in Chapter 5. For a thorough discussion we may refer to the article by Long and Winefordner (1983). This article also contains interesting references to articles of Kaiser, Boumans, and others, which are therefore not repeated here. For the derivation of some simple statistical dependencies, the small, almost forgotten book of Beers (1958) is instructive. Finally, we can recommend the excellent discussion of the physical origin of background sources and detection limits in PIXE and synchrotron-excited XRF given by Sparks (1980), which is also valid for WDXRF. Here, we only invoke that according to the International Union of Pure and Applied Chemistry (IUPAC) recommendation, the characteristic signal is statistically attributed to the element when it is elevated not less than 3s(s ¼ standard deviation) above the background level. It makes the conﬁdence level equal to 99.86%. The detection limit is then calculated for the standard deviation multiplicity k ¼ 3 and is expressed as a function of intensity in a unit of time (R ¼ N=t) with t as the counting time and with the sensitivity m expressed in cps=% or cps=ppm. In Table 7 a series of values for cL are given for randomly chosen elements collected from manufactures’ documents. No general list can be included because detection limits as well as sensitivities depend heavily on matrix composition and on the experimental conditions. Instead, this is intended as an indication of order of magnitude. Each discussion of detection limits provides methods for deciding whether a signal value IU for a unknown was greater than background reading IB with a given level of conﬁdence. Another important approach is advocated and applied by Clayton et al. (1987). They state that ‘‘traditional techniques for determining detection limits have been concerned only with providing protection against type I errors, or false positive conclusions (i.e., reporting an analyte as present when it is not). They have not considered type II errors, or false negative assertions (i.e., reporting an analyte as not present when it is).’’ In their article, they discuss a test of the null hypothesis c ¼ 0 versus alternatives that c > 0, where c denotes the true but unknown concentration of an analyte in a medium and for two models of calibration curves with known or with unknown parameters. The test is applied to chromatographic data. Their theory can certainly be applied to XRF data, but the attempt has not yet been reported.

G.

Limits of XRF

Compared to ultraviolet (UV)–visible emission spectra, x-ray spectra are ‘‘oligoline’’ ones and very convenient with respect to the resolution of the spectrometers. The UV-visible

Wavelength-Dispersive X-ray Fluorescence Table 7

165

Detection Limitsa

Matrix

Element

Terephthalic acid Low-alloy steel

Al–Mg alloy

Aluminum Glass powder Phosphosilicate glass Cement

Copper alloys

Fe Co C Al Si Cr Mg Si Ti Mn Cu Mg P B B2O3 Na2O MgO Al2O3 SiO2 SO3 P2O5 Be

cL 0.15 0.18 80, 240 4, 4.5 2.1, 4.1 2, 1.7 7 5 3 2 1 10.5 1.3 1% 0.4% 36 27 22 50 24 32 0.2%

Data source Philips Philips, Siemens

Philips

Siemens Broll, 1985 Philips Siemensb

Philips

Source: Data from Philips Analytical ‘‘X-Ray Spectrometry Application Notes nos. 504, 622, 623, 737, 808, 913, 916; Siemens SRS Awendung 84=8, 86=1, 86=5, 86=6; and Broll, 1985. a ppm (unless otherwise speciﬁed) for 100 s counting time and k ¼ 3. b With use of multilayer OVO 55.

emission spectrum of iron contains some 2000 lines between 200 and 800 nm; its complete x-ray counterpart has a dozen important lines between 1.757 A˚ (Kb1) and 0.615 A˚ (Lg)! For everyday work, however, only two lines, 1.936 and 1.940 A˚ (Ka1,2) and 1.757 A˚ (Kb), play a role. The densely populated L series spectrum such as that of gold has a set of about 20 lines between 0.9 and 1.5 A˚, a spectrum that can be managed by a spectrometer with a relative spectral resolution of 102. The prospects for the x-ray spectrometric range should be bright, but what are the limits to this method? 1. Application Limits All elements from beryllium on can be detected and=or quantitatively determined. On the high-Z element or low-l side, the spectral resolution may be a problem for pure spectroscopic studies, but seldom or never for qualitative or quantitative applications. Because the emitted energy of photons is high and the attenuation coeﬃcients are low, the critical depth is great and, consequently, the analyzed volume is very representative of the composition of the bulk. The critical depth is the thickness dcrit of a layer parallel to the surface, from which 99% of the intensity sensed by the detector is recruited.

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It is a function of the takeoﬀ angle C2 , the linear absorption coeﬃcient mr of the matrix, and the density: ln 1 II1t ln102 4:61 dcrit ¼ lim ! ð19Þ r=sinC2 mr=sinC2 mr=sinC2 m where It is the intensity emitted by a thin foil and I 1 is the intensity emitted by an inﬁnitely thick sample. For long wavelengths, mr becomes high while dcrit and analyzed volume become low. If the sampled volume is low, the analysis may be called a near-surface determination, not representative for the bulk (see discussion in Sec. III.D.2.d). In that sense, the results of the determination of heavy and light elements in the sample including diﬀerent kinds of elements can be nonequivalent. The heavier elements are then determined in the bulk, whereas the light ones are determined on the surface only. This can lead to surprising results if the process of surface segregation is occurring. Moreover, during recent density determinations (Tsuji and Wagatsuma, 1996) of coating layers by a similar method, Grazing emission x-ray ﬂuorescence, striking diﬀerences have been revealed between surface and bulk densities; for example, the density of a 27-nm-thick coating lead layer was equal to hardly 35% of the normal, the bulk value of the lead density (Tsuji et al., 1997). It is a true limit by the force of nature, although it may be exploited for near-surface determination of light elements. From the concept of critical depth, it is obvious that coatings can be analyzed up to certain thicknesses depending on the absorption (and enhancement!) characteristics of the coating and substrate; the overall thickness range is from 0.01 mm to a few hundred micrometers for the normal angle versions of XRF. Apart from the analysis of macrosamples, WD spectrometers may also be ﬁt to electron microprobes, where they supplement the ED spectrometer for better spectral resolution of light elements. Curved crystal optics is generally used because of optimum eﬃciency (compare the quantum eﬃciency of a classical WD spectrometer ﬃ108, curved crystal WD ﬃ 106, and ED spectrometer ﬃ104). WD spectrometers are also needed for heavy-particle-induced emission; because of the presence of many satellite lines, good resolution is required (Watson et al., 1977; Chapter 12). The dynamic range of analysis covers about ﬁve orders of magnitude, from 103 to 100%. Below 103%, preconcentration procedures must be used. The intensity and uniformity of exciting beams delivered by conventional end-window tubes were not suﬃcient to allow spot analysis. However, Boehme (1987) and Nichols et al. (1987) published the results of two-dimensional mapping executed with small beams from high-power sources (rotating anode tubes), allowing lateral resolution of about 30 mm. This oﬀsets the boundaries to a new domain: high-spatial-resolution x-ray spectrometry. Still, the possibilities of the tube-excited, slit-based WDXRF microprobe are relatively poor in comparison with an x-ray capillary-based EDXRF system or especially proton and electron microprobes (Chapters 11–13). However, for many applications, the lateral resolution mentioned earlier is quite suﬃcient and this technique is time-saving. Moreover, the tube can be coupled with the polycapillary device. Note, that, for example, the diameter of the spot obtained from the polycapillary lens is in a range 50–500 mm, but the careful work with such a beam can give a somewhat better spatial resolution. In Sec. VI.B.3, we will discuss the milliprobes, a new solution which seems to be the most suitable for the wavelength-dispersive mode of operation.

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2. Limits of Precision and Accuracy An analyst is always threatened by errors in the analysis outliers, random, and systematic errors. When a given operation or measurement provokes an oﬀset between the true and measured values, a systematic error is said to be introduced. This oﬀset may be due to a wrong calibration of any kind or to a wrong standard. Such errors can only be detected by analyzing a sample from the same batch with other techniques. Random errors and outliers can be dealt with by statistical methods. One example of such a method is described by Plesch (1981) and implemented in the software package Spectra 310 (Siemens). A calibration with n standards is assumed to give a standard deviation. The standard deviation ss is calculated according to sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ X ðC0 Ci Þ2 i ss ¼ ð20Þ np where Ci is the true weight fraction and C0i the result obtained; p is the number of parameters used in the ﬁt: p ¼ 2 for a linear and p ¼ 3 for a parabolic ﬁt. If the concentration Ca of an element in one specimen diﬀers considerably from C0a , the presence of an outlier is suspected. The suspected value is eliminated, and a new standard deviation is calculated. Ca is an outlier with a probability of P% according to the inequality (F-test) (S1=S2)2 > Fp. The threshold values for Fp are listed in the current statistical literature. Application of the F-test in fact checks whether a statistically signiﬁcant diﬀerence exists between the standard deviation calculated with and without the suspected result. Possible causes of outliers are numerous, including all kinds of human mistake, temporary instrument failure, and heterogeneity of specimens. Recently, the numerical programs helping to segregate the data on the ‘‘true’’ ones and outliers are available. They are based on the Grubbs test (Grubbs, 1969; Rousseeuw and Leroy, 1987). A measurement of ﬂuorescent radiation is, as in any other experimental determination, subject to minimal error. The standard deviation of a radiation measurement is equal to the square rootpof ﬃﬃﬃﬃ the number of accumulated counts N. The real standard deviation is greater than N because of errors of instrumental origin. Another source of error is the conversion of intensity into concentration and, last but not least, inherent to the sample, microheterogeneity, which is diﬃcult to foresee and to quantify and is present even when all normal precautions for careful sample preparation are taken. Heterogeneities caused by sample preparation are dealt with elsewhere. These artifacts should be absent here. Segregation, however, which is the origin of what we call microheterogeneity, is a bulk (or sometimes surface) phenomenon of crystallographic origin: solid solutions of compounds or elements that are not perfectly soluble in each other, segregating into multiphase components. The ‘‘degree’’ of heterogeneity in the composition of the segregated phase as well as in the size of the phases depends on the history of the solution (thermal, chemical, or mechanical treatment). It is not easily mastered and is diﬃcult to quantify. Its importance for XRF analysis has been acknowledged by several authors; Examples are tin and lead in solders (Glade and Post, 1970), silicates in fused beads (Novosel-Radovic et al., 1984), and silicon in Si–Al alloys (Michaelis and Kilday, 1962). The ﬁrst report we are aware of is an article by Claisse (1957) on the determination of FeS in a sulfur matrix. A systematic study has been devoted to this phenomenon by Helsen and Vrebos (1984a, 1984b, 1986a, 1986b) and Vrebos and Helsen (1983, 1985a, 1985b, 1986), both by Monte Carlo simulation on hypothetical mixtures and by measurements on samples of known segregation. It was found by simulation on a (hypothetical) dispersion of spherical particles of iron in a chromium matrix

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that even for a particle diameter of 1 mm, an error is introduced in the relative intensity of FeKa of 6% (on the level of cFe ¼ 0.5) with respect to the perfectly homogeneous solution. The eﬀect was proved experimentally on alloys of Al–Si, Al–Mo, and Al–Zn. The extent of the eﬀect changes with time, as in fused beads. This was demonstrated by the determinations of Novosel-Radovic et al. (1984) on ﬂuxed silicates and also by Monte Carlo simulations by Vrebos and Helsen (1985a, 1985b). Microsegregation imposes a true physical limit to the accuracy obtainable by XRF spectrometry on quite a substantial number of samples, including minerals and alloys. 3. High-Z Limit We already insisted on the low-Z limit in Secs. III.D.1 and III.D.2, but what about the high-energy end of the spectrum? Commercially available generators and side-window tubes allow working at high voltages up to 100 kV, bringing even the K line of bismuth into the potential analytical spectrometric range. If a tungsten target is used, however, characteristic lines represent only a minority of the integrated emitted intensity, thus, excitation is bound mostly to continuous radiation, the maximum of which lies at 50 keV. Thus, only elements with K absorption edges below this value can be eﬃciently excited (Eu, Z ¼ 63, and all elements below). The above remark is even more valid for the spectrometers with end-window tubes, most commonly used in present-day instruments, of which the high voltage is limited to 60 kV. Possible excitation of K lines for all the elements from the periodic table by the tubes operating in voltages up to 160 kV is bound to metal–ceramic low-power tubes, cooperating with EDXRF systems and is outside the scope of this chapter. Thus, the available voltage limitation is the ﬁrst one. If we return to Figure 26b and compare the spectral resolution of the LiF spectrometer to spectral distances between analogous lines in the K series, it is obvious that spectral resolution is worse above 35 keV (for K lines of Pr, Z ¼ 59). This is a second limitation. Most of goniometers have their optimal 2W range between 10 –15 and 70 : At the higher 2W end, the angular dispersion becomes insuﬃcient; at the lower 2W end, the intensity becomes very low. If we choose LiF (420) with the shortest interplanar distance, 2d ¼ 1.802 A˚, useful measurements can be made at 10.0 corresponding to photons of 39.6 keV (Sm, Z ¼ 62 in the ﬁrst-order spectrum). This is a third limitation. Studying Figure 26 further, we note that the spectral resolution of the LiF spectrometer is below the spectral resolution of the semiconductor Si–Li detector for photons of energy exceeding 17 keV, which corresponds to MoKa, Z ¼ 42. This is the fourth and most serious limitation in the K series. For analysis in the L series, there is no such limitation because the whole energy range lies within the capabilities of wavelength-dispersive devices. However, using the new microcalorimetric detectors with extraordinary and relatively constant spectral resolution, one can anticipate the preponderance of the spectral resolution factor for wavelength-dispersive crystals only in a region below 4–5 keV, which means the Ka line of V, Z ¼ 23 (see Fig. 27). Then, the ability to make the eﬃcient trace analysis (because of higher count rates) and the availability of the device are still the advantageous sides of application of the crystal dispersion in the future. The wide voltage range of generators should not be used without some awareness of inherent drawbacks. For example, the K absorption edge of Mo is 20 keV. Thus, each x-ray tube with characteristic lines above 20 keV and working above twice this 20 kV for the maximum of bremsstrahlung generation is quite good for the excitation of Mo. The potential, however, should not exceed this value substantially much because the maximum

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of the white radiation would shift more and more away from the molybdenum absorption edge and even the spectral characteristics do not improve. If we divide the mass coeﬃcient of photoelectron excitation by the sum of mass coeﬃcients for Compton and coherent scatters (this is a kind of measure of peak-to-background ratio), we obtain the values 76 and 18 for tube-excitation potentials of 40 and 100 kV, respectively! This is a substantial loss in peak-to-background ratio while passing to higher voltages. The discussion in this section shows that the advantage of high-energy photons (generated by very high voltages in SWT) does not compensate for the drawbacks and this probably explains why most manufacturers ﬁt EWT tubes to the instruments. During the last years, manufacturers rather resigned from high-voltage generators and tubes, even in high-power 4-kW constructions. The use of a higher current is now a preferred option. Thus, we expect that only spectrometers with a high voltage up to 60 kV will dominate the market in the future. The eﬀective spectral range above 30–40 keV should probably be left for typical energy-dispersive detectors. In the preceding paragraphs, we tried to show that although the spectrometers are almost perfect, there are limits to precision and accuracy due to the nature of the sample and to the mathematical conversion of intensity to concentration. These uncertainties are generally greater than those introduced by instrumental parameters.

V.

CHEMICAL SHIFT AND SPECIATION

The photoelectric excitation of atoms and subsequent speciﬁc deexcitation by emission of x-rays is the basis of all qualitative and quantitative uses of XRF. The deexcitation process was considered to take place between unique excited and ground quantum states. If these states were always stable, x-ray emission lines were to be found at strictly determined wavelengths or energies. However, electron density and quantum states are inﬂuenced by the environment of the atom with consequent wavelength shifts, or so-called chemical shifts, of the emitted lines. Such techniques as nuclear magnetic resonance or Mo¨ssbauer spectroscopy collect their spectroscopic information essentially from measurement of chemical shifts. As a matter of fact, all kinds of spectral lines of electromagnetic radiation are subject to chemical shifts. All these shifts are small and detectable only if the spectral linewidths are suﬃciently small and the instrument has the required spectral resolution. For x-rays, the line shift is at most a few electron volts. An angular error DW ¼ 0.02 on the determination of a peak position at l ¼ 1 A˚ corresponds to 5 eV. Moreover, small temperature changes of the analyzing crystal already shifts the position over similar values. The sensitivity of peak position to the chemical environment of the atom was observed at the very beginning of x-ray spectrometry by Lindh and Lundquist (1924). The following characteristics of lines are sensitive to changes: line position (chemical shift) (Sumbaev et al., 1968; Sumbaev, 1970; 1976), line shape (Urch and Wood, 1978), and mutual ratios of line intensities (Asada et al., 1975). A detailed summary of the eﬀect can be found in the book by Agarwal (1979), for example. In spectrometers with limited resolution, such as diﬀerent energy-dispersive systems, most spectral shifts are hardly detectable. Chemical shift research has been done on high-resolution spectrometers. Gohshi et al. (1982a, 1982b) constructed an excellent two-ﬂat-crystal instrument for studying chemical shifts. Asada et al. (1975) and Haycock and Urch (1978) improved the use of commercially available instruments for this purpose: by Fourier transform and iterative procedures for peak position detection or by using ﬁner collimators. When analyzing

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crystals with large thermal expansion coeﬃcients, such as PET, are used, a temperature variation of 1 C gives rise to peak shifts greater than the chemical shifts. Helsen and Wijnhoven (1972) solved this problem by inserting standards at regular intervals and by using statistical techniques for the determination of the peak position. Sumbaev and Coworkers (1968, 1970, 1976) made experiments with an instrument in a Cauchois arrangement. In an article by Kataria et al. (1986), it was proved that even using a common solid-state Si–Li detector with moderate energy resolution, it was possible to detect chemical eﬀects. Similarly, Habulibaz et al. (1996) performed experiments with a multilayer with an exact 2d spacing of 300 A˚, which means with poor spectral resolution. They managed to register SiL spectra for diﬀerent compounds, clearly showing the inﬂuence of the chemical environment. Theoretically, chemical shifts can be calculated by the self-consistent ﬁeld approach for the four quantum states involved (Manne, 1981). Unfortunately, the calculations are largely dependent on initial assumptions, the choice of which is diﬃcult and somewhat random and the results are far from unequivocal. The experimental results of Sumbaev seem to be more spectacular. He connected the chemical shift DEKL to Pauling ionicity, i, valency m, and the parameter p, which was determined by lifetime ratios of electrons on the levels K or L: DEKL ¼ im½ pK DEK;ðZ:Z1Þ pL DEL;ðZ:Z1Þ

ð21Þ

He achieved an even more surprising result by presenting chemical shifts per unit ionicity versus valence (Fig. 42). This dependence is linear and may serve as proof of both the importance of chemical shift and the accuracy of Pauling’s scale. The shift can be related to the character of the bond in the molecule or the crystal. During the formation of a chemical bond, a rearrangement of electrons occurs, thus the formation of ions or bonds of varying degree of ionicity, the inﬂuence of which is reﬂected on all electron energy levels, sometimes more on the K level than on others levels. An example may be the shift between the Ka position of metallic aluminum and aluminum in diﬀerent coordination states (IV and VI) (Helsen and Wijnhoven, 1972). Asada and Co-workers (1973, 1975, 1976) also made measurements in which the resulting chemical shift was attributed to the ionic bond character. They investigated the

Figure 42 Chemical shift as a function of valence. The energy shift divided by the ionicity is plotted (solid line). The dotted curve illustrates the uncorrected values of DE. (From Sumbaev, 1970. Reprinted with permission of J. B. Adashko.)

Wavelength-Dispersive X-ray Fluorescence

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changes in the ratio of some spectral lines, such as those for the Kb5 and Kb1 satellite line, and they connected them to the oxidation number of the analyte element. The results are again surprisingly elegant (Fig. 43). Clear proof for the dependence between the shape of lines and the oxidation number was given for manganese (Urch and Wood, 1978) and also between the chemical shift and the coordination number for diﬀerent elements (Gohshi, 1981). The shift of AlKa for coordination states 0, IV, and VI was given by Helsen and Wijnhoven (1972). The application of the determination of chemical shift to chemical speciation is obvious. In geological samples, sulfur can be distinguished according to its diﬀerent oxidation states (sulfur, sulﬁde, sulﬁte, sulfate, and hyposulfate). Birks and Gilfrich (1978) made measurements for sulfur using a portable low-power WDXRF instrument. In environmental and biological samples, the oxidation state of chromium is important: Cr(III) does not pass the cell wall, but Cr(VI) does and is suspected of being carcinogenic. Iwatsuki and Fukasawa (1987) reported the determination of chemical states of arsenic, selenium, and bromine. The determination of the valence of iron in welding fumes by measuring the energy shift and the intensity ratio of Kb0 to Kb1,3 has yielded very ﬁne results (Fig. 44) (Tanninen et al., 1985). Kataria et al. (1986) managed to measure the varying ratio between the total line intensities of Kb and Ka as due to

Figure 43 1975.)

Satellite peak intensity ratio Kb5 =Kb1 versus oxidation number. (From Asada et al.,

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Figure 44 Relative chemical shift DE and the average valence of iron in standard samples and four welding fumes. The squares indicate the statistical uncertainties: SS ¼ stainless steel; MS ¼ mild steel; MMA ¼ manual metal arc technique; MIG ¼ metal inert gas technique. (From Tanninen et al., 1985.)

diﬀerent oxidation states of manganese using an instrument of very moderate resolving power. This discussion of other than elemental or quantitative information from x-ray spectra has been intentionally restricted to x-ray spectrometry. A counterpart exists in the spectrometry of the ejected electrons, but that discussion belongs to the domain of spectrometry of Auger electrons or primary ejected electrons, for which higher resolutions are available or better speciation is possible. Another and sometimes severe drawback of these techniques is the required high vacuum and surface charging for nonconducting specimens. It should also be mentioned that Raman spectrometry is possible in the x-ray domain, but this ﬁeld of research is only really accessible when synchrotron radiation is available (Sparks, 1974; Eisenberger et al., 1976a, 1976b), although performance on conventional instruments is feasible (Suzuki et al., 1970). In analytical determinations by synchrotron radiation, the Raman signals are often viewed as an additional disturbance (Jaklevic et al., 1988). Not many researchers are working in the ﬁeld of speciation by WDXRF; nevertheless, it is a ﬁeld of increasing importance, with theoretical as well as practical interest. It gives more than simple elemental information. Moreover, the shift of lines aﬀects the intensity measurements for quantitative analysis, and stresses the use of the right standards for speciﬁc cases. In extreme cases, it may even be compulsory for quantitative determinations to scan over the peak maximum for each determination and locate in each case the intensity at the maximum for good precision and accuracy. If ultimate accuracy should be required, correction for peak heights may be involved. The few examples given point out that for studying the chemical shift, shape, and intensity measurements of spectral lines, the use of dedicated instruments is indicated, although a certain amount of useful work can be performed on commercially available instruments. A very interesting discussion of the problems immanently connected with the analytical application of soft x-rays can be found in the work by van Sprang and Bekkers (1998). These remarks have the obvious implications for the speciation analysis.

Wavelength-Dispersive X-ray Fluorescence

VI.

173

INSTRUMENTATION

The physical principles of a WD spectrometer do not allow dramatic diﬀerences in the general layout of the spectrometer. A practical instrument unites a number of construction compromises for speed, versatility, resolution, intensity, cost, user friendliness, and other factors, and all combinations have their own merits and their own drawbacks. Buying an instrument is a decision that in itself is a compromise because all beneﬁts wanted by a customer are seldom or never united within a single instrument. In this section, the main features of the most important commercially available instruments are juxtaposed without formulating a global or ﬁnal judgment.

A.

Electrical and Electronic Features

In modern instruments, all or almost all functions are controlled by a computer. As the electronic conﬁguration is subject to continuous change, no details will be given here. Mentioned here are a few aspects which were diﬃcult to automate or, when feasible, were hardly automated in older instruments: High voltage generators: The costly electromechanical control of conventional generators is replaced by computer-controlled solid-state generators, switched or high frequency. An example of exploitation of this ﬂexible control is the switching of the tube along the isowatt curve, modifying the high voltage but keeping the power constant, an interesting feature if diﬀerent intensities of the primary x-ray beam are required for diﬀerent elements in the analysis of a complex sample (major, minor elements) by a sequential spectrometer. A consequence of the mechanical decoupling of W and 2W axes lifts the necessity of manual alignment of the crystals by giving (and memorizing) a correcting oﬀset angle to the W axis for each crystal. A tandem disposition for gas and scintillation counters is no longer a necessity. The one or the other is brought into the right position under software control. The sin W potentiometer for continuous adjustment of the manual setting of the lower level and window of the pulse-height discriminator is replaced by the multichannel analyzer and electronic and electro-optical detection of the position of the goniometer. Automatic digital gain control corrects for pulse shrinking. Dead time is automatically corrected. This feature ensures linear response in most instruments to 2 Mcps for a gas-ﬂow-proportional counter and to 1 or more Mcps for a scintillation counter. The main remaining electromechanical parts are the motor(s) of the goniometer in the sequential spectrometer (dc or stepper) or alignment motors for the crystals in the simultaneous spectrometers, the vacuum pumps, the electrovalves for gas ﬂow (He, N2, counter gas), cooling water and compressed air, temperature control and fans, safety switches, sample lock and changer (internal and external), crystal turret, collimators, diaphragms, attenuator, and primary beam ﬁlters. An aspect totally absent in the older instruments was remote servicing. The full computer control allows remote servicing over a modem which substantially reduced the mean service time, especially in situations, where the service engineer is not close to the user’s facility.

174

B.

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Spectrometer Layout

Commercially available spectrometers are classiﬁed according to the x-ray tube used, the position of the sample, sequential or simultaneous acquisition of the ﬂuorescent radiation, hybrid forms, and some special dedicated designs. They will be discussed in the following order: Sequential instruments with EWT or SWT, sample down or sample up Simultaneous instruments with sample surface up or down Hybrid or dedicated designs 1. Sequential Spectrometers with EWTor SWT A typical example of geometry with sample surface down (i.e., the x-ray tube positioned under the sample) is represented in Figure 45 (layout of the Siemens-Bruker SRS-3400). The merit of the sample-surface-down geometry is in the analysis of solutions, not easily feasible with a sample-surface-up geometry. The positions of primary beam ﬁlters, sample=spectrometer seal, collimator masks or diaphragms, and spectrometer optics are clearly shown. The front end of the x-ray tube and all other parts except the scintillation counter are enclosed in a vacuum chamber. The sample=spectrometer seal is inserted in order to isolate the sample lock from the goniometer during sample loading. The sample lock and goniometer are evacuated by their own vacuum pump. The aim of this particularity is the maintenance of a monochromator atmosphere of constant composition,

Figure 45 A spectrometer layout with EWT and sample-surface-down geometry. (Siemens SRS3400. Courtesy of Bruker.)

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which may contribute to the reproducibility of the determination of light elements and prevent the contamination of the monochromator, particularly the collimator, from contamination by powdery samples. A drawback is the x-ray absorption when light elements are to be determined. Philips recently introduced a spectrometer with the same geometry (PW 2400 or 2404; Fig. 46) with an end-window x-ray tube, called the Super Sharp Tube (SST). The suﬃx ‘‘super sharp’’ points to the angle of conical nose of the tube, allowing a decrease of the sample distance to the window. The duplex detector (replacing the classical tandem fpc=sc construction) is a combination of a ﬂow counter and a sealed xenon detector (improved performance for Ka for Ti to Cu and for La to W). The scintillation counter is giving an oﬀset angle with respect to the gas counters. By this, the scintillation counter can be enclosed in the vacuum chamber without increasing the volume of the chamber. The selection of the detector is software controlled and became possible by the decoupling of the W and 2W axes. Shown in Figure 47 is the method by which one or two additional ﬁxed channels can be added. Fixed channel(s) reduce(s) the analysis time for routine analysis of samples with one or two elements constantly present or for samples with an internal standard. ARL

Figure 46 Spectrometer with EWT and sample-surface-down position but with the addition of a fixed channel. This channel is the similar to the ones used in simultaneous spectrometers. (Philips PW2404) (Philips Technical Materials. Sequential X-Ray Spectrometer System PW 2404. Courtesy of Philips.)

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Figure 47 Heart of the ARL 9400 spectrometer (EWT, sample surface down). Two such monochromators, operating simultaneously, are integrated in one spectrometer, model, ARL 8420S. (Courtesy of ARL.)

realizes the simultaneous determination of more than one element by enclosing within the vacuum chamber a complete second monochromator of their unique extremely compact design as represented in Figure 47. It is characterized by the complete absence of gears and a positioning system by counting Moire´fringes resulting from interference of two grating systems. A development of the last few years is the substitution of diaphragms by a smalldiameter aperture allowing analysis of an area as small as approximately 1 mm2 (see Sec. VI.A.3 about milliprobes). An appropriate mechanism is moving the sample, which allows element mapping. This feature is realized in the XRF-1700 or 1500 series of Shimadzu (Fig. 48). It is an instrument with sample-surface-up geometry and ﬁtted with an attenuator, reducing the intensity by a factor of 10 (for a major element determination); the merit of this is that no tube power switching is required for major and minor element determination.

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Figure 48 Sample-surface-up spectrometer, fitted with small-aperture collimator diaphragms and an attenuator; all parts are enclosed in the vacuum chamber (Shimadzu XRF-1700). (Courtesy of Shimadzu.)

Only one manufacturer still produces an instrument with SWT. A number of instruments of a former generation are still in use and the tubes are still available. Diano Corporation continues the production of a spectrometer with sample-surface-up geometry and SWT (cf. Table 10). The merits and disadvantages of EWT or SWT were previously discussed. However, if a good homogeneity of the beam is required, a ﬁne-focus SWT is by far a better solution. Sample holders of slightly diﬀerent design for solids and liquids, for sample surface up or down, are oﬀered by the manufacturers. The holders for instruments with samplesurface-up geometry are ﬁtted with a spring mechanism to press the sample against the sample mask, both for sequential and simultaneous instruments. The holders adapting insert for liquids have a shape that can be recognized automatically and will not be accepted if the spectrometer is not under a He atmosphere (or eventually switches automatically from vacuum to He). 2. Simultaneous Instruments All simultaneous instruments are equipped with end-window tubes with very similar construction features. Up to 30 monochromators may be arranged radially around the tube head, with alternating high and low takeoﬀ angles. One or more monochromators may be replaced by programmable goniometers of the ﬂat (a normal ﬂat crystal goniometer of reduced size) or curved crystal type. The latter is small but limited in angular range; the former has a large angular range but is large and occupies at least two positions otherwise taken by ﬁxed monochromators. As already mentioned, Philips oﬀers a small, low-power target transmission tube (TTT tube) used in the spectrometer PW 1660 (Philips Technical

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Materials. Simultaneous X-ray Spectrometer System PW 1660) or X0 CEM capable of simultaneous determination of only 14 ﬁxed elements or 12 elements when equipped with a programmable goniometer. In most instruments the samples are placed in the surface-up position. Some instruments (e.g., Oxford and PW2600 from Philips) irradiate in the sampledown position. The monochromators can be ﬁtted to the outside of the vacuum cabinet, in which case they need not be vacuum tight. Figure 49 shows a generally used geometry. In the ARL sequential instruments, ﬁxed channels (up to eight) or a complete second monochromator can be added. In Philips’ sequential spectrometer (PW 2400), an optional two ﬁxed channels may be installed, making this instrument partly simultaneous. This is made feasible by the use of the special x-ray front end tube, the model Super Sharp Tube. 3. Milliprobes In an obvious eﬀort toward the new ﬁelds of application, two companies introduced options, which seem to be restricted up to now to the very special, laboratory-made instruments. Rigaku and Shimadzu independently constructed the analytical systems with special masks: nonmovable (Rigaku) and scanning type (Shimadzu). In both versions, the diameter of the opening is 1 mm. Coupled with a 4-kW generator and high-power tube, it can carry out the analysis of the small pieces of a sample. In the Shimadzu version,

Figure 49 Layout of a simultaneous spectrometer with sample-surface-up geometry, typical for Bruker MRS4000 or Philips 1606 and with the arrangement at two levels with different takeoff angles. (Courtesy of Bruker.)

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the opening is scanned before a front of the sample, thus enabling the elemental mapping of the sample. The suggested application would be for the spot composition and density tracing of the sample, as it has been checked elsewhere (Kuczumow et al., 1995; Luggar and Gilboy, 1994). The construction of the milliprobe requires special attention concerning the x-ray tube used. The conventional end-window tube, due to its output inhomogeneouity, might be used only in the Rigaku arrangement, with a constant mask and the sample moving in front of the tube. Otherwise, the relatively small output of such tubes prevents their use as a radiation source for the milliprobes. C.

Intercomparison of Instruments

Only a limited number of manufacturers produce x-ray spectrometers. Hereafter, we will discuss the instruments of seven manufacturers subdivided in three classes: sequential and simultaneous instruments and a third class that we called hybrid spectrometers, because they have characteristics belonging to both former classes. The series numbers together with the names of the manufacturer are given in Table 8. In Tables 9–11, the numbers in the top row of the third, ﬁfth, and last column refer to the manufacturer and series number of Table 8. The reader will ﬁnd the actual addresses of the manufacturer’s headquarters at the end of this subsection. 1. Sequential Instruments In Table 9, a summary is given of the main characteristics of the current instruments as a guide to the market for the interested reader and for those who are considering buying a new instrument or replacing an older instrument. a.

General Remarks and Items Common to All Instruments A question mark in the table means that the data were not available in the brochures. When nothing is marked, it means that the given item is not applicable.

Table 8

Available Wavelength-Dispersive Spectrometers Sequential

Manufacturer ARL Diano Oxford Philips

Rigaku Siemens Shimadzu

Series

Simultaneous

Table 9

Series

Hybrid

Table 10

9400 2000

1 2

8600S

1

PW2400 PW2404

3 30

PW1606 PW2600 X-CEM

2 3 4

RIX2100 RIX3100 SRS3400 XRF1700 XRF1500

4 40 5 6 60

MRS 4000

5

Series

Table 11

MDX1000 Venus 100 PW4100=20 PW4120=20 PW4110=10

1 2 20 3

Primary beam filters number Elements Sample Sample surface Sample turret: number of positions Sample=spectrometer seal (yes, no) Optics Diaphragms: number of positions Collimators: selectable (yes, no), number

Cr, EA-75, 250 W,Pt,W=Cr, Pt=Cr,Mo=Cr None — Up 4 n

— 1

Rh, 3GN, 75

W, Mo, 125

Cu, Al, Fe

Down 2

y

3

y, 4

3

SWT

EWT, 3

4, 40

EWT, 3 or 4

100 or 150=? ? ? 0.0005

3 or 4 60=?

y, 3

4

n

Down 2

Al, Cu

y, 2 or 3

6

n

Up 6

Zr, Ti, Al, Cu

Others on request Rh=Cr,Rh,=C, Rh=W 5 1 or 5

Rh, SS, 75 or 125 Rh

EWT, 3

10–125=1 < 2,5 Isowatt curve 0.0005

0–80=0.1 ? ? 0.0005

?–100 ? ? 0.0001

3, 30 3, 4 20–60=1

2

Manufacturer

3 3 ?–60=? (opt.: ?–70=?) 15–60=1

1

Presentation of Sequential Instruments

Generator Power (kW) Voltage range=step (kV=step) Current range=step (mA) Switch time (s) Switch mode Stability D%/% mains Source X-ray tube: geometry, load (kW) Anode, type, Be window thickness (mm) Option other anodes

Table 9

100 or 140=? — — 0.0002

3 or 4 60=?

6, 60

y, 4

4

y

Down 2

Cu, Al

10

3

5

n

Up 8 or 2

Al, Ti, Ni, Cr

4

Rh, AGRh66G, Rh 75 or 125 Mo, W, Au, Cr Rh=Cr, Rh=W

EWT, 3 or 4

5–150=1 Fast — 0.0001

3 or 4 10–60=1

5

180 Helsen and Kuczumow

Dead-time correction Computer instrument monitoring peripherals Remote servicing (yes, no) Sample changer: type (x, y), positions

Goniometer angular range ( 2y) fpc angular range ( 2y) sc and 2 axes (coupled=decoupled) Positioning technology Continuous scanning ( 2y=min) Step scanning ( 2y=min) Slewing speed ( 2y=min) Angular accuracy ( 2y, y) Angular reproducibility ( 2y, y) Temperature stabilization ( C) Additional fixed channels (yes, no, or number) Parts included in vaccum Counting electronics Count rate (kcps): fpc sc Sealed Duplex Pulse-shift correction

Attenuator (yes, no) Crystal changer: number of positions Detectors: extra (yes, no) 4 or 6 Sealed Xe

9 n

None All ? ?

y

All

2000 1500

All

2

Automatic

y xy, 6, 24–168

? x(?), 50, 100

Pulse-height discrim. Automatic Full

2000 1000

All except sc

None

?

< 0.05

n x, 4 or 10

None

0.1

0.005–0.1 1000 ? 0.0005

? 0.1–240

8–148 5–118 ?

10 n

y

0.0001–2 2400 0.0025 0.0001

y xy, 12–98

0.01–? 600 0.001 ?

0.001–1.00 4800 0.001 0.0002

Optical sector 0.006–120

Automatic Full

? 0.1–10

Moire´ fringe 0.25–128

8 Sealed Xe, Ne, Kr 13–148 8–104 dec

n

2000 1500 1500 3500 Graphical display Automatic, dynamic Graphical display Automatic Partly Full

c

17–152 0–115 dec

0–146

n

n

y xy, 8, 58, 100, 110, 138

Automatic Full

Automatic

2000 2000 4000

All except sc

None

0.01

— 1000 0.00001 0.001

Stepper motor 1–200

4–150 4–112 dec

8 n

n

n ?

Automatic Full

Automatic

2000 1000

All

None

0.3

0.002–1 1200 ? 0.0003

Stepper motor 0.01–180

7–148 0–118 dec

10 n

y Wavelength-Dispersive X-ray Fluorescence 181

Continued

Installation Instrument dimensions: H*W*D (mm) Weight (kg) Electricity supply (V, kVA) Cooling water: bar, max. temp. ( C), flow (L=min) Compressed air: bar, flow (L=min)

Table 9

1400*860*1346 590 208–480, 10 > 1, 29, 19 —

450 220, 6 > 2, 20, 0.5–5

—

2

920*1080*760

1

4–5, ?

5

None

6–10, 20

700 115–400, 8 4–6, —, 1.5

1400*1420*1054 1070*1890*880

4, 40

580 750 200–240, 8, 5 200–220, 10 > 3.5–8, 20, 6–11 ?, < 30, 3.5–11

1092*1410*920

3, 30

Manufacturer

—

780 200–220, 8 or 9 > 1.5–3, 30, 3–10

1350*1770*1080

6, 60

182 Helsen and Kuczumow

Wavelength-Dispersive X-ray Fluorescence

183

The analyzing range of all instruments goes from Be to U and the transuranians, except for Diano 2000, for which the range is limited to B–U. The mains power requires a stability from 10% to þ10%, exceptionally from 15% to þ10%. A single phase is required, except for Rigaku, which requires three phases. Most commonly oﬀered by all manufacturers are rhodium anodes for x-ray tubes. Primary beam ﬁlters are used to remove interfering lines from the tube target element and=or to improve signal-to-background ratio: Al: improves peak-to-background ratio for Pb and As in light matrices; removes RhKa line and the CrK lines Cu: determination of Ru, Rh, Pd, Ag, and Cd; removal of interfering lines from Rh Fe: removes the Ka line of Rh Ti: determination of Cr, Co, Fe, Zn; improves peak-to-background ratio Zr: measurement of CdKa, removes the Ka line of Rh Most manufacturers supply other ﬁlters on request. Sample dimensions: Most instrument allow a sample size about 52 mm in diameter (Diano: 63 mm) and about 30 mm in height (Bruker: 57 mm). To cope with small heterogeneities, the samples are spun at 30 rpm (or 60 rpm for Shimadzu) on most instruments (for Diano, the spinner is only available on the sample changer with 10 positions). The collimators are quoted in most brochures as extra-coarse, coarse, medium, ﬁne (high intensity to high resolution). Only Bruker mentions the divergence angles: 0.077 , 0.15 , 0.46 , 0.93 , 1.0 , 1.90 , and 1.54 . Crystals: All sequential spectrometers make use of ﬂat crystals. Counters: A gas-ﬂow-proportional counter (fpc) and a scintillation counter (sc) are standard on all instruments, except Diano, in which the sealed Xe detector is standard and a scintillation counter is an option. The gas ﬂow (Ar þ 10% CH4) is 1–2 L=min. The electronic counters are all based on a multichannel analyzer (Shimadzu and Rigaku probably still use a pulse-height discriminator) with an automatic pulse shift and dead-time correction. Available on all spectrometers is the choice of vacuum or He as the spectrometer atmosphere, with nitrogen as option for many. All companies supply the control unit by personal computer in more or less sophisticated versions with software for monitoring the instrument, for qualitative analysis, and for a number of possibilities for conversion of intensity to concentration. All of these items are subject to change and not discussed in detail here. The conversion algorithms are discussed in detail in Chapter 5. The software systems supplied by Bruker, ARL, Rigaku, and Shimadzu are all very user friendly. Installation data are added to give the potential buyer an idea of the required space and infrastructure. The dimensions and weight of the instruments are given without an external sample changed ﬁt. b.

Remarks About the Individual Instruments (in Alphabetical Order as in Table 8) ARL 9400: As far as we are aware, ARL was the ﬁrst manufacturer (and still the only one) to use counting Moire´ fringes for its goniometer positioning. The goniometer is a complete gearless construction and it has the highest slew rates

Element range Goniometer

Be windows (mm) Primary beam filters positions Elements Primary beam attenuators (Yes, No) Samples Sample surface Size: diameter, height (mm) Sample changer (positions) Monochromator (fixed channels) Takeoff angle ( ) Maximum number Crystals Detectors Be–U

29, 44 28 (a) x

? 30, 20, 11 (c) Curved Gas counter, sc

Be–U

Up 51, 40 6, 72, 300

Up 75, 50 11, 32, or 8 (b)

Cu, Al, Fe

125, 75 10

3 Rh, Cr, Mo

3 Rh, W

125, 75 4 (a)

3 ?–60=? ?–75=? 0.0005

2

3 ?–60=? ?–100=? 0.0001

1

Presentation of Simultaneous Instruments

Generator Power (kW) Voltage (kV), step (kV) Current (mA), step (mA) Stability D%/% mains Source X-ray tube: Load (kW) Anodes

Table 10

29, 44 28 Log. curved

Down (b) 51, 40

y, 14

3 Rh, other on request 125, 75 3

3 5–60=1 (a) 5–125=1 0.0005

3

Manufacturer

fc, sc

Gas (Ar flow, Ne, Kr sealed)

Be–U

26, 24 (a) 28

52, 57 8, 58, 100, 138

y

Rh 66 G Rh, Mo, W, Au, Cr 125, 75

3 20–60=1 5–150=1 0.0001

5

25, 39, 5 14

40, ?

TTT* Cr

0.2 50 4 0.0005

4

184 Helsen and Kuczumow

?

305–8, < 20, 6–10 4–5, ?

580

6–10, 20

900 3–8, 20, 3

900

1000 > 3, < 18, 5

F–U

1800*1810*950

1190*815*1230

Vac, air

Up to 4

2 (a)

1804*1232*877

Vac, other on request

fpc

8

1350*1400*950

Vac, other on request

sc, Kr-sealed

2 (d) Slits 4 (PE, InSb, Ge, LiF 200, 220) Cyl. curved 0.06–76 0.01 480 ? ? < 0.005 85–47

n

B–U Vac, He, air

B–U Vac

2 (c) 1 1 (LiF 200, 220, Ge) Flat 0.06–76 0.01 480 ? ? < 0.01 10–100

y

5–120

0–115

XRD channel (yes, no) Installation Dimensions: H*W*D (mm) Weight (kg) Water supply(bar, max. C, L=min) Compressed air (bar, L=min)

0.06–76 0.01 1000 ? ? 0.003 5–120

Flat 0.25–128 0.001–1 4800 0.015 0.001 0.0002 17–155

Type Scan speed ( 2y=min) Step scan ( 2y=step) Slewing speed ( y=min) Angular accuracy ( 2y) (LiF) Angular resolution ( 2y) Angular reproducibility ( ) Angular range gas counter ( 2y) sc ( 2y) Detector Element range Spectrometer environment

(a)

1 (d) 3 9

Maximum number Collimators Crystals number

Wavelength-Dispersive X-ray Fluorescence 185

186

Helsen and Kuczumow

Table 11

Hybrid Instruments Manufacturer 2, 20

1 Generator Voltage (kV)=current (mA)=power (W) Stability D%=%mains Max. mains variation (V) X-ray source Tube Anode External Cooling Sample Sample surface up, down Spinning (rpm) Dimension (diameter, height in mm) Sample changer Spectrometer chamber Monochromator (fixed) Element covered

Crystals Additional spectrometers Dimensions (H*W*D in mm) Weight (kg)

3

40–50=4=200

50=4=200

id.

0.007 210–250

0.0005 180–264

id. id.

Similar (a) Rh None

Target transmission Sc None

id. Cr id.

Down

Down

48, 40

26–47, 40

id. 30 id.

? Vac, He optional

Single, 10 Vac, He

Single Vac

F–U

S, S-bg., Zr, Ni, Ni-bg., Mn, Mn-bg., V, V-bg Curved None

Al, Si, Na, Mg, Ti, Mn, K, Ca, P, S, Fe id. id.

id.

1200*750*960

id.

180

Flat, curved optional Flexi-channel (EDXRF) 1090*800*800 160

of all spectrometers. In the series 9420S, two goniometers or additional ﬁxed channels can be ﬁtted, adaptable to a great variety of analytical situations, excellent for rapid nonroutine industrial control, reducing the analysis time by a factor of 2 with respect to a conventional sequential spectrometer. Separate vacuum pumps support the vacuum for the sample lock and the monochromator. Diano 2000: This oﬀers dual-target tubes. The targets are push-button selected and no moving parts are present (dual ﬁlament). Two collimators are oﬀered: a 0.032-in. spacing ‘‘beam tunnel’’ or a 0.010-in.-spacing ‘‘diﬀracted beam soller slit.’’ The instrument is not temperature stabilized because the 45-kg bronze bull gear goniometer has a suﬃciently high thermal capacity, avoiding rapid temperature changes. It is, however, not suﬃcient when one wants to measure peak shifts. The crystal in the turret must be individually tuned. The W and 2W axes are coupled but with a bisector arrangement and moved by a dc motor. The standard detectors are a ﬂow-proportional counter and sealed Xe, the scintillation counter is an option. The counting is also by a pulse-height selector. Diano oﬀers a ﬁxed-sample and ﬁxed-element mode for analysis.

Wavelength-Dispersive X-ray Fluorescence

187

Philips PW 2400, 2404: Both types diﬀer only in the power output of the generator. The tube power is switched along the isowatt curve (i.e., by keeping the power constant when either the voltage or current is changed). The ﬁxed channels that can be added have as the detector either a ﬂow-proportional counter (3000 kcps), Xe-sealed (1000 kcps), Kr-sealed (2000 kcps), Ne-sealed (2000 kcps) proportional counters, or a scintillation counter (1500 kcps). The positioning of the goniometer is done by optical disk, a sophisticated highperformance system, giving the highest angular accuracy and reproducibility. The goniometer is driven by dc motors. The external sample changer accepts samples of nonuniform size. Rigaku 2100, 3100: Both types diﬀer only in power output of 3 and 4 kW. The collimators are either standard=high resolution or standard high resolution=ultralight elements. The attenuator in front of the monochromator crystal reduces the intensity by a factor 10. The advantage should be that it avoids power switching of the x-ray tube. The sample-surface-up geometry makes it diﬃcult, if not impossible, to analyze solutions. A diaphragm of 1 mm diameter allows small-spot analysis. A built-in cleaning system removes the impurities of the wire in seconds in this instrument because the ﬂow-counter eﬃciency deteriorates by contamination of the wire electrode. Siemens (Bruker) SRS 3400: The instrument is rather conventional and easy to service. The positioning of the decoupled arms of the goniometer is done by stepper motors. The positioning of the sample holders is very reproducible. The external sample changer accepts nonuniform samples also. An excellent software SPECTRA PLUS is supplied for qualitative, semiquantitative, and quantitative analyses. The instrument was formerly sold under the name of Siemens but is taken over by Bruker Analytical X-Ray Systems (Bruker AXS). Shimadzu XRF 1700, 1500: The diﬀerence between both instruments is that the selection of ﬁlters, diaphragms, attenuator, and crystals is automated in the 1700 model. A special feature of this instrument is the presence of diaphragms of diameter 1, 3, 10, 20, and 30 mm. The 1-mm diaphragm together with an appropriate sample positioning system allows element mapping. 2. Simultaneous Instruments with Sample Surface Up or Sample Surface Down A summary of the characteristics of the instruments capable to determine simultaneously a number of the elements is given in Table 10. The instruments are composed by a few till maximum 30 small-size high-eﬃciency spectrometers as discussed earlier, optimized for one particular element. The set of spectrometers is dedicated to a particular series of samples as used in, for example, the cement or steel industry. To attribute some ﬂexibility to these instruments, one or more scanning goniometers are added. a.

Remarks and Features Common to All Instruments All manufacturers use x-ray tubes with either a 75- or 125-mm-thick Be window. The detectors oﬀered are ﬂow-proportional gas counter (Ar), sealed gas counters ﬁlled with Kr, Ne, or Xe, and a scintillation counter. The linearity of the photon count rate is equal or similar to the one given for sequential instruments. Temperature stabilization is either individual for each spectrometer or for the ensemble.

188

Helsen and Kuczumow

Electricity power supply is single phase, 10–15 kVA. ARL oﬀers the possibility of adding an additional x-ray diﬀraction device within the ﬂuorescence spectrometer. Other manufacturers oﬀer integrated sample transport from the spectrometer to an x-ray diﬀraction instrument. The CubiX XRF from Philips is a simultaneous spectrometer for up to 14 elements, similar to the X’Cem. It is integrated in the same module with an x-ray diﬀraction instrument; both can be juxtaposed, allowing a rapid sample transport from one instrument to another. Both devices are operated by a single personal computer. Cubix represents a modular concept for integrated analysis, including elemental and crystallographic determination. b.

Remarks About the Individual Instruments (in Alphabetical Order as in Table 8) APR 8600S 1. 2. 3. 4.

The primary beam ﬁlter takes the place of three monochromators and cannot be ﬁt alongside an XRD channel. Eight positions for cassettes for large samples (Ø75 mm and height of 50 mm); 11 and 32 positions for small samples (Ø52 mm and height of 30 mm). Thirty monochromators can be ﬁtted or 20 plus 1 goniometer, or 11 plus an XRD channel. The goniometer is the same as the module of the sequential instrument.

Philips PW1600: Maximum two ﬁxed channels can be substituted by a programmable goniometer (with ﬂat crystals and parallel beam collimation). One programmable channel occupies space as for one ﬁxed channel. Philips PW2600 1. 2. 3.

Switching time of the generator is 2 s and the tube is switched along the isowatt curve. Particular feature of this instrument is that it allows liquid samples to be handled also. Two diﬀerent goniometers can be ﬁtted: (1) one with a ﬂat crystal and parallel beam collimation and it takes the space of one ﬁxed channel; (2) one with a cylindrically curved crystal (logarithmic spiral optics) and two slits and it takes the space of three ﬁxed channels. Optional beam attenuators can be ﬁtted, enabling all channels to handle concentrations from ppm levels up to 100%.

Philips X9 Cem: The x-ray tube is a low-power Target Transmission Tube (TTT), (Sec. IIIA). Because of the low power (200W), this instrument does not need external water cooling. Siemens MRS 4000:

High power (4 kW) and high speed instrument.

3. Hybrid Instruments The instrument from Oxford MDX1000 is classiﬁed as hybrid because it combines a WD ﬁxed channel with what is called a Flexi channel, an EDXRF unit. The x-ray tube is similar to the target transmission tube, produced by Philips, but no other details are given in their brochures. The instrument series Venus 100 from Philips are called hybrid because Philips uses ﬁxed channels, which are, however, switched sequentially in front of the sample. The instruments are delivered for speciﬁc purposes: The PW4110=10 is provided for the analysis of industrial minerals, particularly for the ceramic industry, and the PW4100=20 and

Wavelength-Dispersive X-ray Fluorescence

189

PW4120=20 for the determination of, for example, nickel, sulfur, and vanadium in petroleum products. The element range can be adapted to speciﬁc purposes. Both instruments from Oxford and Philips are simple, low-cost, but high-performance devices which can be used in harsh industrial environments; they are simple to install and are characterized by small dimensions, light weight, no water cooling, no compressed air, and no severe requirements for the stability of the mains. Hybrid instruments are presented in Table 11. 4. Instruments Dedicated to Special Applications and Supplements Two manufacturers, Siemens and Philips, supply for the semiconductor industry transformed versions of their sequential spectrometers SRS 3300 (power up to 4 kW, mapping possible) and PW 2800, respectively. The main feature is the sample area which must accept silicon disks as large as 10 in. in diameter. The sample introduction system is automated and meets the standards for working in a cleanroom environment. The manufacturers also provides as many auxiliary devices as possible. Advanced ‘‘analysis lines’’ are available from Rigaku, Philips, and Siemens, which oﬀer a complete set of the mills, ﬂuxers, and polishing devices for the XRF and XRD laboratory. Worth noting is that, on the other hand, during the last years, some new companies appeared on the market, oﬀering not whole instruments as such but only some specialized parts of instrumentation. Among the most important participants in this kind of activity, the following should be mentioned: Advance Hivolt, oﬀering x-ray generators, especially with 4 kW power X-Ray Optical Systems Inc., supplying x-ray focusing and collimating polycapillary optical devices plus designing the optical systems with x-ray semilens Osmic Inc., the main producer of multilayers Corporation Scientiﬁque Claisse Inc., producing ﬂuxers for sample preparation Contact with these producers can be very meaningful alternative in case of constructing new, home-made, and dedicated version of an instrument. We conclude this subsection on the intercomparison of instruments with a list of the addresses of the headquarters of the manufacturers: ARL Applied Research Laboratories S.A., En Vallaire Ouest C, case postale, CH1024 Ecublens, Switzerland Diano Corporation, 271 Salem Street, Woburn, MA 01801, USA Oxford Instruments, 19=20 Nuﬃeld Way, Abingdon, Oxon OX14 1 TX, England Philips Analytical X-Ray BV, Lelyweg 1, 7602 EA Almelo, The Netherlands Rigaku International Corporation, 3-9-12, Matsubara-cho, Akishima-shi, Tokyo 196, Japan Shimadzu Corporation, International Marketing Division, 3, Kanda-Nishikicho 1-chome, Chiyoda-ku, Tokyo 101, Japan Siemens, A.G., now: Bruker AXS GMBH, Oestliche Rheinbrueckenstr. 50, D-76187, Karlsruhe, Germany

VII.

FUTURE PROSPECTS

The time passed since the ﬁrst edition of this handbook has been one of the most stormy periods in research and applications of x-rays since their discovery! It is well known how

190

Helsen and Kuczumow

much Ro¨ntgen was confused, although totally convinced, by the impossibility to demonstrate the electromagnetic nature of x-rays through reﬂection, diﬀraction, or interference experiments! For many years, it remained impossible and further progress was very slow and tedious. Recent years have brought the real breakthrough. We know of three recent realizations of x-ray lenses: Two diﬀractive: 1.

2.

Zone plates (in other words, Fresnel diﬀracting structures), earlier restricted to the very soft x-rays. They are very important for the construction of the x-ray microscopes and for the observation of living biological objects in the so-called ‘‘water window.’’ But now, by combining the Fresnel diﬀraction patterns with Bragg’s phenomenon, the special Bragg–Fresnel lens was constructed, well suited for focusing hard synchrotron radiation (Aristov et al., 1986; Snigirev, 1995; Snigirev et al., 1995; Chevallier and Dhez, 1997). Snigirev refractive lens (Snigirev et al., 1996; 1998; Elleaume, 1998; Hastings, 1996; Eskildsen, et al., 1998), consisting of one or two mutually perpendicular arrays of holes drilled in low-Z dense material (Be, Al). Depending on the number and shape of the holes, they can focus hard x-ray beams,

One version using total reﬂection: 1.

Kumakhov polycapillary lenslike device, which might be somewhat poor in focusing capability (spot is 50–500 mm wide) but very eﬃcient with respect to the fraction of the primary input power turned into a small spot on the analyzed surface. Interesting products were announced by X-Ray Optical Systems Inc., Advertising Materials.

In parallel, great progress was realized in the construction of energy-dispersive detectors with increasingly better spectral resolution. The most promising device seems to be the microcalorimeter detector, a device with very good spectral resolution parameters. Moreover, its spectral resolution is more or less constant over the whole spectral range covered (1–20 keV). With such a device, the spectrum deconvolution problems, so characteristic for devices with poor spectral resolution, should vanish. Potentially, it lowers the detection limits, by narrowing the energy region in which the very sharp peak is placed. Only a very restricted number of background counts is detected in that region, thus improving the peak-to-background count rate ratio. However, the detection limit depends on the maximum count rate allowed for these detectors, which is, at the moment, very restricted. Recent discussions show (Ladbury, 1998) that intensive work is going on to overcome this shortcoming in the nearest future. The aim is to get the device with an area of 40 mm2, with count rates up to 105 cps. Another progress took place in the construction of time-resolved excitation sources. The classic radiation generator for the purpose is synchrotron. Probably much more convenient, the ‘‘portable’’ and switchable version of a time-resolved x-ray source will be a compact-ﬂash source. Furthermore, the construction of the x-ray laser—an old dream!—is also progressing, maybe slowly, but in a promising direction. Collected together, all available information about this progress justiﬁes hope on the construction of new versions of existing instruments as well as the advent of new ones. It is worth noting that in the present chapter we considered an XRF spectrometer mainly as a device used for the determination of elemental composition. What is now discussed testiﬁes that future instruments will give both compositional and spatial resolution (microprobe) and time resolution (kinetic probe). New kinds of lenses and normal incidence

Wavelength-Dispersive X-ray Fluorescence

191

mirrors give an impetus to the construction of x-ray microscopes also in the hard part of the x-ray spectrum. One can easily imagine a microscope (but one easily construct!) working in the hard x-rays regime supported by two diﬀerent x-ray lenses—one acting as an ocular lens and one as an objective lens. The new version of the microprobe could then eﬃciently concentrate x-rays (e.g., by Kumakhov polycapillary) on the conﬁned areas and then record the ﬂuorescent radiation with the excellent spectral resolution of a microcalorimeter as the energy-dispersive detector with the whole spectrum completed in one operation. Such a device is now under construction in the National Institute of Standards and Technology (Wollman et al., 1997a, 1997b; Anonymous, 1997). The microcalorimeter detector, attached to a scanning electron microscope, has, for the time being, a spectral resolution of 7.2 eV 0.4 eV, calculated in the standard way for the MnKa line, or 2.0 eV for 1.5 keV (Wollman et al., 2000). Count rates are below 150 cps, but it is, by far, not the last word in this matter. A newcomer in the generation of possible applications of the classical WDXRF, rather unexpectedly brought to the awareness of scientists, was that the extended ﬁne structure of ﬂuorescent spectra emitted by solid samples can be observed (Hayashi et al., 1997; Kawai et al., 1998). Up to now, this ﬁeld was reserved for the absorption mode, mostly in combination with synchrotron radiation. The discovery of the extended ﬁne structure in experiments performed under normal laboratory conditions opens a new ﬁeld in the study of local structure in condensed matter and makes this method available to lower-budget laboratories.

ACKNOWLEDGMENTS All manufacturers of instruments are gratefully acknowledged for supplying documents about their instruments. Mr. P. Van der Aa, Dr. B. Volbert, and Dr. ir. B. A. R. Vrebos from Philips Analytical and Mr. S. Ornigg, Dr. K. E. Mauser, and Dr. N. Broll from Siemens (Bruker) AG deserve our gratitude for invaluable advice and critical reading of the manuscript. Dr. D. de Boer is acknowledged for supplying us with his knowledge on the most recent progress in multilayer research. A. Kuczumow thanks the Catholic University of Leuven for a fellowship that enabled him to participate in the ﬁrst edition of this work and the Flemish Government for a fellowship helpful in the present revised version.

REFERENCES Adamson BW, Juchli K, Matula G. Adv X-Ray Anal 34:131, 1991. Agarwal BK. X-Ray Spectroscopy. Berlin: Springer-Verlag, 1979. Anonymous. J Res NIST 102:392, 1997. Anonymous. Rigaku J 14:37, 1997. Arai T. Adv X-Ray Anal 30:213, 1987. Arai T. X-Ray Spectrom 20:9, 1991. Arai T, Omote K. Adv X-Ray Anal 31:507, 1988. Arai T, Shoji T. Adv X-Ray Anal 33:521, 1990. Aristov VV, Gaponov SV, Erko AI, Salaschenko NN. Optics News 12:128, 1986. Asada E. Jpn J Appl Phys 12:1946, 1973. Asada E. Jpn J Appl Phys 15:1417, 1976. Asada E, Takiguchi T, Suzuki Y. X-Ray Spectrom 4:186, 1975. Attaelmanan A, Larsson S, Rindby A, Voglis P, Kuczumow A. Rev Sci Instrum 65:7, 1994.

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3 Energy-Dispersive X-ray Fluorescence Analysis Using X-ray Tube Excitation Andrew T. Ellis Oxford Instruments Analytical Ltd., High Wycombe, Buckinghamshire, England

I.

INTRODUCTION

The scope of this chapter is laboratory and industrial x-ray ﬂuorescence (XRF) analysis systems in which x-ray tubes are used for excitation and energy-dispersive (ED) semiconductor detectors, as opposed to Bragg-diﬀraction (wavelength) dispersion devices, are used. Such ED detectors directly measure the energy of the x-rays by collecting the ionization produced in a suitable semiconductor material. The coverage of this chapter speciﬁcally excludes the following topics, which are covered in other chapters: on-line EDXRF systems (see Chapter 7), total-reﬂection XRF (TXRF) (see Chapter 9), polarized beam XRF (see Chapter 10), and x-ray microﬂuorescence (XRMF) (see Chapter 11). Early approaches to EDXRF used gas proportional counters or scintillation detectors to determine directly the energy of the x-rays. Such systems were limited in their application because of the inherently poor energy resolution, which precluded the separation of characteristic x-rays of adjacent elements in the periodic table. The limited actual energy resolution of such detectors led them to be known as nondispersive, but this drawback has been eﬀectively countered in some benchtop instruments through the use of novel arrangements of primary and secondary beam ﬁlters (Ross, 1928; Kirkpatrick, 1939; Kirkpatrick, 1944; Field, 1993). The real breakthrough in EDXRF came in the late 1960s (Bertolini et al., 1965; Bowman et al., 1966; Elad and Nakamura, 1966; Aitken, 1968) with the arrival of solid-state semiconductor diode detectors and their associated pulse processing circuits. These detection systems were developed through the early 1970s, often in electron microscopes, to the point at which practical x-ray spectrometry with an energy resolution of 200 eV or less became possible (Frankel and Aitken, 1970; Landis et al., 1971; Heath, 1972). Although the energy-resolution capability of semiconductor detectors remained greatly inferior to that achieved by wavelength-dispersive (WD) XRF systems, the increased eﬃciency inherent in the energy-dispersive method compensated in many analytical applications and permitted the use of a multiplicity of experimental geometries not practical with WDXRF. A wide variety of EDXRF analytical systems based on radioisotopoe sources, x-ray tubes, charged-particle accelerators, microprobe electron beams, and synchrotron light sources have been developed in recent years.

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Within the scope of this chapter, only EDXRF systems in which x-ray tubes are used for excitation will be considered further. The material in this chapter discusses the instrumental aspects of the energy-dispersive method, with particular emphasis on the use of low-power, compact x-ray tubes in combination with semiconductor detectors. The basic components of an EDXRF instrument within the scope of this chapter are depicted schematically in Figure 1. Each of the main subsystems of such an instrument is covered in detail in the following sections.

II.

X-RAY TUBE EXCITATION SYSTEMS

In EDXRF spectrometry, there is no physical discrimination of the secondary radiation that leaves the sample and enters the detector. This means that all photons of all energies in this secondary beam interact with the detector. The detector and its associated signal processing chain have a limited capacity to process these events and this is typically in the range 150 kcps. As a direct consequence, EDXRF has a limited total counting capacity

Figure 1

Basic components of a tube-excited EDXRF instrument.

EDXRF Analysis Using X-ray Tube Excitation

201

for all element x-ray emissions and it is, therefore, essential that the content of the incident spectrum be optimized for its useful information. This may require that more than one acquisition be made under diﬀerent excitation conditions in order to control the range of energies needing to be collected in the detection system. Alternatively, the excitation system may be optimized to increase the intrinsic peak-to-background ratio through the use of alternative geometries such as TXRF or polarization EDXRF. In both these cases, however, higher-power x-ray tubes are typically required and there is still the need to apply some type of selectivity to the excitation system. This selectivity can be achieved in TXRF using more than one tube target material—typically Mo and W. In the case of polarization EDXRF, more than one tube high voltage or polarizing element is likely to be applied. This selectivity of excitation employed in EDXRF is completely the reverse of conventional WDXRF practice. In WDXRF, the high discrimination power of the dispersion and detection process and its associated losses coupled with very simple detection and counting systems allow very high count rates to be processed for each narrow wavelength band. Thus, the detection process is highly selective and the count rates in an individual element channel with a modern WDXRF system can be well in excess of 106 CPS. Typically, the simple rule of thumb in WDXRF would be to use twice to three times the kilovoltage (kV) of the absorption edge of the highest element of interest and then to apply the maximum allowed current at that setting. In summary, we may regard the essential operational diﬀerence between EDXRF and WDXRF spectrometry as follows: WDXRF uses broad-band excitation and selective detection. EDXRF uses selective excitation and broad-band detection. There are various ways in which selective excitation is achieved in EDXRF spectrometry. The use of TXRF and polarization is covered elsewhere in this volume and will not be considered further here. The most important modes in which selective excitation is realized in EDXRF are as follows: Selection of tube anode material Variation of tube voltage (kV) Use of primary beam ﬁlters Use of secondary targets (and associated ﬁlters) The selection of tube anode material is typically a once-oﬀ decision made at the time of purchase in view of the intended application. The high purchase cost of x-ray tubes generally precludes operational changes and low-power dual or multiple (Kis-Varga, 1988) anode x-ray tubes are not commercially available. The types of x-ray excitation arrangements are shown schematically in Figure 2. The following subsections describe typical arrangements for excitation and describe how optimum excitation conditions can be determined. Earlier work on this subject may be found in the XRF literature (Sandborg and Shen, 1984; Vane and Stewart, 1980; Gedcke et al., 1977). A.

Direct and Filtered Direct Excitation

In Figure 2, the beam (1) represents the conﬁguration used for direct excitation of the sample by the x-rays emitted from the anode. A primary x-ray beam ﬁlter can be used to modify the spectrum from the x-ray tube that is ﬁnally used to excite the elements in the sample. The optimum selection of kV and primary beam ﬁlter are critically important for obtaining the best data from an EDXRF system. As with all spectrometry methods, the

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Figure 2 An x-ray tube and typical excitation geometries. The high-energy electron beam strikes the anode to produce x-rays. (1) Direct or direct filtered: the flux (1) is used directly to strike sample (1) or is first passed through a primary beam x-ray filter. (2) Secondary target: the primary flux (1) strikes a secondary target 2, which generates a flux of x-rays (2), which strikes the sample at 2. (3) Transmission geometry: the beam (3) is directed through the anode to the sample.

principal driver for best precision and detection limits is peak-to-background (P/B) ratio. However, as mentioned earlier, the limited total counting capacity of the EDXRF system and the analysis of multielement samples are added complications in deriving optimum excitation conditions. In general, the kV governs the sensitivity and the primary beam ﬁlter governs the background. The energy distribution of the spectrum arriving at the sample governs the eﬀectiveness of excitation for element XRF lines. In order to excite x-ray ﬂuorescence, it is necessary to have incident x-ray energies above the absorption edge for the element’s line series to be excited. In order to have adequate excitation, there needs to be a high intensity of x-rays with energies higher than the edge and conventional wisdom has suggested that the tube kV should be 1.52 times the absorption edge of interest. This overvoltage ensures that there is a substantial proportion of the x-ray tube output spectrum available for excitation of the element line(s) of interest. Figure 3 shows the unﬁltered x-ray tube spectrum scattered from a thin-polymer-ﬁlm sample when the x-ray tube is operated at 5, 10, and 15 kV. The observed scatter spectrum is a reasonable representation of the spectrum striking the sample and clearly shows the following:

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203

Figure 3 Scattered, unfiltered excitation spectrum provided by a silver target x-ray tube operated at (1) 5, (2) 10, and (3) 15 kV.

The high-energy cutoﬀ at the operating kV. This is also known as the DuaneHunt limit. The bremsstrahlung ‘‘hump’’. The intense characteristic AgL lines (around 3 keV) from the tube anode. The intensity of the AgL lines increases with the kV (i.e., as the overpotential increases). The tube L lines of Ag and Rh are particularly eﬀective for exciting the light elements in the range 12.5 keV. Once higher-energy lines (e.g., FeK at 6.4 keV) are sought, the L lines provide no excitation, as they are lower in energy than the absorption edge (FeK at 7.11 keV) and the bremsstrahlung hump provides the excitation. Despite the highest intensity of the bremsstrahlung being lower than AgL lines, it is the integrated excitation spectrum, higher in energy than the absorption edge, which is important. When the x-ray tube kV exceeds the energy of the K edge of the tube anode material, the characteristic K lines will begin to dominate the excitation spectrum. This is shown in Figure 6, which depicts the scattered excitation spectrum with the tube operated at 35 kV, at which the AgK characteristic lines are strongly excited. The basic sensitivity for analysis is governed by the kV applied to the x-ray tube and an example of this is seen in Figure 4. The sample was a pressed pellet and the kV was set in the range 616 kV using a vacuum path in the spectrometer and no primary beam ﬁlter. At each kV, the rhodium-target x-ray tube current was adjusted to give 50% deadtime. The K-series line intensities were used and the sensitivity for each element has been normalized to the value at 10 kV. From the plots, the following are clear: The Si intensity falls oﬀ as the kV increases. This is due to the increasing distance of the tube main spectrum from the SiK absorption edge and to the increased relative contribution of the other elements to the total counts in the spectrum. This latter

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Figure 4 details).

Effect of kV on sensitivity for the major elements in a geological sample (see text for

factor is important and derives from the limited total count rate for the whole spectrum. The FeK lines are not excited until at least 8 kV is applied. The FeK absorption edge is at 7.111 keV and the maximum sensitivity for Fe is at approximately twice this value. The general rule of thumb in which the kV is set to twice the highest absorption-edge energy is a reasonable place to start. However, as more elements are sought across a wider energy range, there usually comes a point at which the ‘‘balance’’ of the spectrum is unfavorable for the analytical problem being solved. When this happens, split the spectrum into two or more areas of interest and apply a kV to best suit each of these element ranges. In any event, as the kV and the number of elements increase, the importance of using a primary beam ﬁlter and splitting the spectrum into optimized energy regions increases. B.

Primary Beam Filter Selection

A primary beam ﬁlter acts as an x-ray absorber and it is placed between the x-ray tube and the sample to modify the x-ray tube output spectrum to which the sample is exposed. In general, the kV is selected ﬁrst to ensure high excitation sensitivity and then an appropriate ﬁlter is chosen. The ﬁlter acts to reduce scattered background in the region of interest and to reduce the excited intensity of lower-energy peaks. The characteristics of the ﬁlter are deﬁned by its x-ray absorption curve, which is controlled by selecting the material and its thickness. Filters are typical thin, pure metal foils of thickness in the range 10500 mm. Some typical ﬁlters, the range of kV used with each, and the applicable optimum element range (K lines) are shown in Table 1.

EDXRF Analysis Using X-ray Tube Excitation

205

The x-ray tube spectrum provided by unﬁltered radiation can be seen in Figure 3. The eﬀect on the x-ray tube spectrum of applying a thin and thick aluminum ﬁlter is seen in Figure 5, where a tube voltage of 15 kV is used. The aluminum acts as a simple absorption ﬁlter, having a single absorption edge at 1.56 keV. From the plots, it is clear that the aluminum ﬁlters both completely absorb the AgL lines as a consequence of the high-mass-absorption coeﬃcient of around 700 cm2=g. The hump at lower energy provided by the spectrum ﬁltered though thin Al yields good P=B ratios for elements in the S to V range (2.35 keV). The absence of the AgL (or RhL, MoL) lines from the tube removes their spectral interference and avoids the need for these x-rays to consume counting capacity. The thick aluminum ﬁlter ‘‘pinches’’ the low-energy side of the bremsstrahlung hump, providing high P=B ratios for the elements with lines in the range 38 keV. The combination of kV and an absorption ﬁlter provides an energy region in which the P=B ratio is optimal. On the low-energy side of this region, the excitation is suppressed, which allows more of the system’s counting capacity to be used in the region itself. On the high-energy side of the region, the excitation from the tube is tuned to provide a high integrated intensity above the energy of the absorption edges of the element lines of interest. The use of a thick copper ﬁlter with maximum kV is another example of this type of ﬁlter. The copper ﬁlter used is generally thick enough to completely absorb the x-ray tube anode K lines, which are of high energy (AgKa is 22.1 keV). Such an arrangement provides for the eﬀective determination of Ag and Cd using a silver anode x-ray tube. At the other end of the energy scale, a cellulose ﬁlter is a weakly absorbing ﬁlter which suppresses the AgL lines. The so-called Regenerative Monochromatizing Filter (RMF) is a particularly important ﬁlter in tube-excited EDXRF spectrometry. The ﬁlter acts on the x-ray tube anode K lines so the kV must be suﬃciently high above the K absorption edge of the anode material to generate intense K characteristic emission. The same element as the anode material is used to preferentially transmit the characteristic K x-rays generated in the anode. The ﬁlter takes its beneﬁt from the fact that an element has a low-mass-

Table 1

Typical Primary Beam Filters and Their Range of Use

Filter

Thickness

kV range

Elements

450

All, NaCa

No filter

N=A

Cellulose Thin aluminum Thick aluminum

Single sheet 2575 mm 75200 mm

510 812 1020

SiTi SV CaCu

Thin anode element

2575 mm

2540

CaMo

Thick anode element

100150 mm

4050

CuMo

Copper

200500 mm

50

>Fe

a

Comments Optimum for light elements with 48 kV excitationa Suppresses tube L linesa Removes tube L linesa Good for transition element metal alloys RMFb, wide range and good trace analysis RMFb, good for trace analysis of heavy-element L lines. Suppresses tube K lines

Needs to be used in conjunction with vacuum or He path to remove the air in the spectrometer that would severely attenuate the low-energy lines and result in ArK lines in the detected spectrum. b Regenerative monochromatizing ﬁlter (RMF). The same element as the anode material is used as the ﬁlter material to preferentially transmit the characteristic K x-rays generated in the anode.

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Figure 5 Scattered excitation spectrum provided by a silver target x-ray tube operated at 15 kV. The plots show the effect of two thicknesses of an Al primary beam filter: (1) unfiltered; (2) thin Al filter; (3) thick Al filter.

absorption coeﬃcient for its own characteristic lines, as these lie immediately below the absorption-edge jump. In the case of Ag, the AgKa1,2 lines are at 22.1 keV and the absorption coeﬃcient of Ag for this energy is only 14 cm2=g. The eﬀect of two silver primary ﬁlters, of diﬀerent thickness, on the spectrum from a silver anode x-ray tube operated at 35 kV is shown in Figure 6. The unﬁltered 35-kV excitation (curve 1) provides equally intense AgL and AgK characteristic lines and a broad bremsstrahlung hump centered on 1215 keV. Any element in the 515 keV range will have the beneﬁt of excitation by both the AgK characteristic lines and the bremsstrahlung. However, there exists the disadvantage of sitting on scattered background derived from the radiation in the bremsstrahlung hump. With the thin Ag ﬁlter in place (curve 2), the AgL characteristic lines are completely absorbed and the bremsstrahlung hump is reduced to a low-energy tail on the AgK characteristic lines. The resulting excitation yields high P=B ratios for lines in the energy range 412 keV. Use of the thicker Ag ﬁlter (curve 3), reduces even further the residual bremsstrahlung below the AgK lines and provides pseudomonochromatic radiation, which yields high P=B ratios for lines in the energy range 515 keV. The strongly ﬁltered AgK line excitation provides excellent sensitivity and background reduction and a clear, intense Compton-scatter peak, which is good for trace analysis where a matrix correction based on Compton scatter is used. The same can be said for anode and ﬁlter materials of Mo, Rh, and Pd, which also provide characteristic K lines in this important region of the spectrum. A speciﬁc and well-documented example of optimum selection of x-ray tube anode, excitation kV, and primary beam ﬁlter can be seen in work by Potts et al. (1986). In that work, a Co anode x-ray tube and an iron primary beam ﬁlter were used to remove the

EDXRF Analysis Using X-ray Tube Excitation

207

Figure 6 Scattered excitation spectrum provided by a silver target x-ray tube operated at 35 kV. The plots show the effect of two thicknesses of Ag primary beam filter: (1) unfiltered; (2) thin Ag filter; (3) thick Ag filter.

CoKb line which would otherwise strongly excite the Fe that typically dominates this part of the spectrum in geological analysis. Most modern EDXRF systems provide the ability to use more than one set of conditions per analysis. Depending on the analytical needs, the kV and ﬁlter are chosen to optimize the important parts of the spectrum and the data from spectra at each of these conditions are consolidated to provide the best P=B values. In order to establish the optimum conditions and make the best use of analysis time, the best strategy is to select a typical sample and then to make measurements with the diﬀerent conditions available. Use criteria such as detection limits to decide upon the optimum conditions for the required analysis problem. Often, the instrument manufacturer’s recommendations can conveniently be taken as the starting point for any investigation. C.

Secondary Target Excitation

The basic geometry of secondary target excitation is shown in Figure 2. Use of this mode of excitation for EDXRF spectrometry was ﬁrst described by Jaklevic et al. (1972), who incorporated the secondary target inside a small, low-power x-ray tube. The high P=B ratios that they reported were encouraging and Porter (1973) subsequently described the ﬁrst work in which the secondary target was external to the x-ray tube. The primary radiation from the x-ray tube strikes the secondary target, which then emits characteristic radiation and generates some scattered bremsstrahlung radiation. The emitted radiation is pseudomonochromatic and the degree of monochromaticity is further enhanced, as in the case of the RMF in direct excitation, with a ﬁlter between the secondary target and the sample. Filters are of the same material as the target and are

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typically only used with the higher-Z targets (e.g., Mo, Ag, and Gd). With increasing distance below the exciting lines, the absence of bremsstrahlung lower in energy than the exciting lines results in a very rapid reduction in sensitivity. The provision of a number of secondary targets with associated ﬁlters, which can usually be selected automatically, provides a range of discrete and narrow excitation bands. This arrangement can be conveniently thought of as providing the equivalent of a number of transmission target x-ray tubes, each with diﬀerent anode materials. In order to generate intense characteristic radiation from the secondary target, the tube kV should be two to three times the K absorption-edge energy of the element from which the secondary target is made. The K lines are used, as they oﬀer the highest ﬂuorescence yield, which means that there are lower losses than with other line series. Typical secondary targets are, in decreasing energy order of K line emission: Gd, Sn, Ag, Mo, Ge, Cu, Fe, Ti, and Al. The optimization of x-ray tube kV, secondary target, and ﬁlter was described by Spatz and Lieser (1979). The ﬂuorescence yield decreases as the target atomic number decreases and the efﬁciency of x-ray production falls oﬀ rapidly below Ti. The result of this is that the lighter elements Si to K are not well excited by secondary targets and there are few suitable elements available to provide targets with K lines in this important region. One approach is to use a reduced kV and a scattering target which scatters the x-ray tube L lines and bremsstrahlung, which, as was seen for direct excitation, is eﬀective in this energy region. A stable polymer block provides a suitable target for this arrangement, although the overall eﬃciency is much lower than with direct excitation. There are two other arrangements that eliminate this limitation. The ﬁrst is to use a second x-ray tube whose geometry provides for direct, and perhaps direct ﬁltered excitation. Such an arrangement adds considerable cost and complexity but does oﬀer optimum performance. It is possible for both tubes to be run from the same high-voltage power supply. The second approach is to provide mechanical repositioning of the x-ray tube such that it can be operated either in secondary target or in direct and direct ﬁltered mode. The positional repeatability of the mechanical arrangement needs to be excellent if this approach is adopted. The mechanical complexity, which inevitably adds cost and some possibility for unreliability, has not stopped this approach from being used in the past with some success in one commercially available instrument. One disadvantage of this approach is the time that is needed for the mechanical switching, which can be a severe penalty if a number of acquisitions are required for a complete analysis—which is usually the case. Very recently, a so-called Wide Angle Geometry was described by Yokhin (2000), in which a single, ﬁxed x-ray tube was used to deliver direct unﬁltered or ﬁltered and secondary target excitation. This arrangement was made possible using the wide cone x-ray beam from the special x-ray tube and a proprietary x-ray beam distributor. The reduction in overall excitation eﬃciency when using secondary targets is usually oﬀset by increasing the x-ray tube power available. A tube providing close geometric coupling and power in the range 100400 W is typically used to oﬀset the low overall eﬃciency of excitation. In order to eﬃciently excite the GdK lines, it is preferable to use an x-ray tube capable of being operated at 60 kV. A high-Z anode such as tungsten or gold provides the highest bremsstrahlung intensity and, therefore, the best excitation eﬃciency for the higher-Z secondary targets. The high degree of monochromatization in ﬁltered secondary target systems provides for very high P=B ratios for elements with absorption edges immediately below the target x-ray emission lines. The background in secondary target geometry can be further reduced using 3-axis or Cartesian geometry as described by Standzenieks and Selin (1979), Christensen et al. (1980), and Bisgard et al. (1981).

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209

Despite secondary target excitation gaining wide acceptance through the early 1980s, the use of direct and direct ﬁltered excitation has remained as the most common excitation arrangement. This is, perhaps, partly due to the restricted energy range oﬀered by each secondary target+ﬁlter combination and the general need for unﬁltered direct excitation to obtain the highest sensitivity for the light elements. In the 1990s, the use of polarized EDXRF (see Chapter 10) tends to have supplanted secondary target EDXRF for the provision of high P=B ratios in some applications. In addition, improved detection systems have continued to improve the performance of instruments using direct ﬁltered excitation. An application in which secondary target systems have a valuable advantage is when the secondary target lines are selected to be just below the absorption edge of a major element, but just above the absorption edges of trace elements. One example of this is the use of a copper secondary target for the determination of low levels of Mn and Fe in Cu alloys. In this example, the CuK lines from the target do not excite the Cu but are highly eﬀective for exciting Fe. The detection and counting system is not dominated by CuK lines, neither is the FeKa peak interfered with by the Si escape peak from CuKa. Such applications are few in number and, in the case of the metals industry, analysts may often prefer to use WDXRF spectrometers. D.

X-ray Tubes

X-rays were ﬁrst discovered over a century ago (see Michette and Pfauntsch, 1996) by Ro¨ntgen (1896a, 1896b), and since that time, they have become used in a great variety of applications. These applications encompass medical and dental x-ray radiology and imaging, industrial and baggage inspection, industrial nondestructive testing (NDT), industrial thickness gauging, and x-ray spectrometry in all its guises. In most, if not all, of these ﬁelds, the convenience of the high-voltage x-ray tube has substantial beneﬁts in terms of convenience, safety, and availability of appropriate spectral and intensity output. The modern x-ray tube used for EDXRF analysis is based on a design introduced by Coolidge as long ago as 1913 (Coolidge, 1913). The basic design of x-ray tubes typically used in EDXRF instrumentation consists of the following: A radiation-shielded glass envelope An independent tungsten ﬁlament with current control unit A high-melting-point metallic anode A high-voltage connection to the anode A beryllium foil exit window Some means of dissipating the heat generated In order to operate, the x-ray tube requires a controllable, stable high-voltage power supply capable of providing typically 560 kV. In addition, there is a low-voltage power supply for providing current to heat the ﬁlament, thereby controlling the electron beam current and output intensity. X-ray tubes used for EDXRF analysis are typically operated with a power in the range of 1400 W. For direct excitation, the power output is typically, no greater than 50 W. Reproducibility of analytical results is achieved for x-ray tube excitation by operating for a measured amount of time at a ﬁxed emission current (dc operation). This approach depends on the reasonable assumption that the total output x-ray ﬂux is proportional to the number of electrons striking the anode. In some types of x-ray tube, a control grid can be used to modulate or pulse the emission current. This can be used to modulate the electron beam current to ensure constant output of radiation from the tube in dc operation or it can be used to rapidly

210

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switch the electron beam on and oﬀ for pulsed operation. If the pulsing is synchronized with x-ray detection, increased output countrate can be achieved by reducing pulse pileup (see Sec. IV.C). The key features of the x-ray tube used in EDXRF instruments are as follows: Easily shielded and safe to operate Compact and able to be placed close to the sample Stable output over an extended period, typically months Well-deﬁned spectral output High-voltage controllable in small increments, typically 1 kV, throughout the operating range (typically 550 kV) Beam current controllable in small increments, typically 10 mA, throughout the operating range (typically 101000 mA). The geometry and designs typically used and the characteristics of each are described in the following sections. Further reading on the subject can be found in Skillicorn (1982, 1983). 1. Side-Window Geometry Side-window geometry is the oldest and, perhaps, the most common x-ray tube geometry used in EDXRF instruments and it is suitable for both high-kV and high-power operation. Figure 7 shows the essential details of this design. The geometric coupling of tube anode spot to sample is limited ﬁrst by the distance of the spot itself from the exit window and then, potentially, by the tube encapsulation itself. Typically, these tubes are operated in the range 550 kV. They can dissipate 50 W with the need only for forced-air cooling in order to keep the tube external body below a temperature of, say, 55 C. It is important not to exceed the maximum temperature stated by the x-ray tube manufacturer. Higher powers (up to 100 W or 120 W) normally need some form of liquid cooling, which is typically achieved through the use of a closed-cycle liquid heat exchanger. The energy spectrum that is output from a side-window x-ray tube is governed by the tube anode material, the angle of the anode, the kV applied to it, and the exit window’s material and thickness. Tube anode materials are usually Rh, Ag, or W, although Ti, Cr, Co, Cu, Mo, Pd, or Au is also available. The selection of anode material depends on the intended application, but Rh, Ag, and Pd are generally the most popular, as they are highly compatible with tube manufacture and operation. In addition, their K and L characteristic emission lines are in regions of the energy spectrum where they oﬀer high sensitivity with minimum spectral interference. Dual-target x-ray tubes have been used in attempts to maximize ﬂexibility and performance. These eﬀorts have met with limited success in EDXRF, in part due to the spectral cross-contamination that is typically experienced in the designs used to date. There has been, however, considerable success in the use of dual-target x-ray tubes for WDXRF, where the problems of spectral contamination have much less impact. Typically, the anode is cut at an angle of 20 35 from normal to the incident electron beam and the exit window is usually made from beryllium foil of thickness in the range 50250 mm. The tube output spectrum can be measured or estimated and such information is important in analytical correction procedures, particularly those using fundamental parameter calculations for which accurate and computationally convenient tube output spectrum models are required (see Chapter 2, Sec. I.B for further details.)

211

Figure 7

Side-window x-ray tube.

EDXRF Analysis Using X-ray Tube Excitation

212

Ellis

It should be remembered that the tube output spectrum, especially with higherpower tubes, may change with time. One way in which this can occur is through deposition of a thin tungsten ﬁlm on the inside of the tube output window through evaporation of tungsten from the hot ﬁlament. Such deposition is unlikely to aﬀect the low-power tubes typically used in EDXRF systems and, in any event, is usually taken care of through the normal restandardization process. 2. End-Window Geometry The design features of such an x-ray tube are shown in Figure 8. The same factors of operation and tube spectrum apply to this type of tube as was applied to side-window tubes. The tube spectrum is altered by the fact that the takeoﬀ angle is normal to the anode surface and so there is minimal self-absorption by the target material. This high takeoﬀ angle makes such a design particularly good for low-energy excitation. One of the important beneﬁts of the end-window design is the very tight coupling of anode spot to sample that can be achieved with a ‘‘sharp-nose’’ design. The spot-to window coupling can be close with suitable electron optics (e.g., 12 mm in the Varian EG50 design). The use of a high positive voltage on the target minimizes the eﬀect which repelled electrons would have of causing an unacceptable amount of exit-window heating. This reduced heating allows thinner beryllium windows to be used, which further improves the low-energy x-ray output. 3. TransmissionTarget Geometry The design features of such an x-ray tube are shown in Figure 9. In this type of x-ray tube, the geometry is basically end window, but in this embodiment, there is a metal foil target parallel to and inside the beryllium exit window. The incident high-voltage electron beam

Figure 8

End-window x-ray tube.

EDXRF Analysis Using X-ray Tube Excitation

Figure 9

213

Schematic of a transmission target x-ray tube.

strikes and enters one side of the thin-ﬁlm target and the resulting radiation is heavily ﬁltered as it passes through the foil and exits the far side of the target. This novel design was initially developed (Jaklevic et al., 1972) in order to reduce the bremsstrahlung from the x-ray tube, which would then reduce the scattered background in the spectrum and thereby improve detection limits. The initial design used a molybdenum transmission target of 0.12 mm thickness and, when operated at 42 kV and up to 25 W, was shown to give an output spectrum consisting principally of characteristic MoK radiation. The output spectrum was comparable to an 125I radioisotope source coupled to a molybdenum-foil secondary target. This type of x-ray tube was made commercially available by WatkinsJohnson (Hershyn, 1975a, 1975b). The spectrum output from this type of tube was investigated by Zulliger and Stewart (1975), who calculated and measured spectra from Cr, Mo, and W transmission target x-ray tubes. The targets were either thin (50 mm) metal foils or thin ﬁlms (510 mm) on a beryllium substrate of 0.250.5 mm thickness. The thin targets mean that only low-power tubes (< 100 W typically) can be made successfully. These authors found, as predicted, a substantial reduction in bremsstrahlung compared to the characteristic lines and quoted detection limits for a number of applications. The Cr target was found, not unsurprisingly, to be best for the light elements S and Ca. Although the energy range covered was claimed to be wide, the intrinsic ﬁxed ﬁltration of this design makes it a less valuable general excitation source compared to a side- or end-window tube with an external, selectable ﬁlter. A novel, multiple-target transmission tube has even been described (Kis-Varga, 1988) in which the ﬁve anode materials (Fe, Cu, Ge, Mo, and Ag) were changed by a magnetic mechanism. The intrinsic ﬁltration of the low-energy bremsstrahlung makes this type of tube much less suitable for excitation of low energy (i.e., < 2-keV) x-ray lines. Consequently, their usage is limited to speciﬁc applications where the intrinsic ﬁltration and potential for close geometric coupling of the tube anode spot to the sample are major beneﬁts. In fact, in one design, the WatkinsJohnson tubes did not have a target closely coupled to the exit window, which, when combined with the intrinsic lossess in the target, led to low output ﬂux from the tubes and, consequently, reduced sensitivity. Finally, these tubes are not useful when operated at low voltages (Skillicorn, 1982) due to strong absorption in the target. This makes them much less sensitive than solid-target, side- or end-window tubes for selectively exciting the light elements in the presence of large amounts of heavier elements.

214

III.

Ellis

SEMICONDUCTOR DETECTORS

In this type of detector, the total ionization produced by each x-ray photon striking the detector is converted to a voltage signal with amplitude proportional to the incident energy. Specially designed preampliﬁcation and processing electronics are employed to maintain the linearity of the voltage signal with respect to the original charge pulse. A multichannel analyzer accumulates an energy spectrum of the sequential events in a histogram memory. Because the energy analysis does not depend in any way on the diffraction or focusing of the x-rays by the detector, the geometry of the system is relatively insensitive to the placement of the detector with respect to the sample. An exception to this is in polarized excitation XRF (Chapter 10), where the orientation angle can be used to suppress scattered background. This insensitivity to geometry and ability to place the detector element close to the sample provides for a large solid angle and high geometric eﬃciency. In addition, the mechanism by which the ionization signal is measured is not restricted to a narrow energy region, thus allowing the simultaneous detection of x-rays over a wide dynamic range of the emission spectrum. The principal advantages of EDXRF derive from its capability for simultaneous detection, with high geometric eﬃciency, of characteristic x-rays from multiple elements. A.

Fabrication and Operating Principle

The energy-dispersive detecting element is based on the simple, semiconductor diode structure shown in Figure 10. The example shown is a structure typical for lithium-drifted silicon, Si(Li), detectors, although the basic elements are similar for high-purity germanium (HPGe) and other semiconductor detectors. The device is fabricated on a cylindrical wafer of high-quality semiconductor material with rectifying p or n contacts on opposing end surfaces. The bulk of the material is characterized by a very low concentration of free-charge carriers. This reduced free-carrier concentration is achieved either with the use of extremely pure material, in the case of HPGe detectors, or through the charge compensation of p-type silicon with lithium donor atoms in the case of Si(Li) detectors. A typical dimension for such a detector crystal is 1080 mm2 active area with a thickness of 35 mm. In the geometry shown in Figure 10, the lithium-diﬀused region acts as an n contact and the metal surface barrier (typically an evaporated Au layer) serves as the p rectifying contact. When the diode is reverse-biased, any remaining free carriers are swept out of the bulk by the applied ﬁeld and an active depletion region is created. In this condition, the only current that ﬂows between electrodes on the respective contacts at each end of the diode cylinder is due to thermally generated charge carriers, which are excited above the narrow band gap of the semiconducting material. In order to minimize the noise from these thermally generated charges, the detector crystal is operated at reduced temperature, which is typically achieved using cryogenic cooling by means of a Dewar vacuum ﬂask ﬁlled with liquid nitrogen. Other cooling strategies can be adopted (see Sec. III.C). When an x-ray photon is incident upon the active volume (i.e., depletion region) of the diode, it normally interacts by photoelectric absorption to create an inner-shell vacancy in the semiconductor material together with an energetic photoelectron. This photoelectron interacts with the atoms in the semiconductor crystal lattice to produce multiple low-energy ionization events. This process continues until the electron comes to rest at the end of a total range, which is short compared with the dimensions of the crystal. The energy associated with the inner-shell vacancy is also absorbed in the crystal, in most

EDXRF Analysis Using X-ray Tube Excitation

Figure 10

215

Cross section of a typical Si(Li) detector showing electrode contacts and active volume.

cases following the emission of Auger electrons or multiple low-energy x-rays and subsequent re-absorption. The result of these multiple ionization process is the essentially instantaneous production of a large number of free electronhole pairs in the sensitive volume of the detector structure. These free-charge carriers form as a cloud from which they are separated by the ﬁeld gradient. The front detector electrical contact, being biased at 500 V, for example, attracts the hole and repels the electrons, the latter negative charge cloud is swept to the rear contact and converted, by means of the FET, to an ampliﬁed voltage pulse. The number of charge carriers produced is directly proportional to the energy of the x-ray photon incident on the detector. Thus, the number of charges collected results in a voltage pulse whose magnitude is, in turn, directly proportional to the energy of the detected photon. B.

Crystal Materials

The characteristics of energy-dispersive semiconductor x-ray spectrometers derive primarily from the properties of the materials used to make the device itself. Although the most common material for semiconductor detectors in EDXRF is silicon, devices based on the use of germanium or compound semiconductors such as mercury(II) iodide (HgI2), CdTe, CdZnTe, or GaAs have also been used to varying degrees. In particular, mercury (II) iodide detectors (Swierkowski et al., 1974; Slapa et al., 1976; Ponpon and Sieskind, 1996) have been used successfully in both research and commercial products for many years. Early devices with adequate performance at room temperature were ﬂown on space missions. More recently, devices have been used in hand-held EDXRF spectrometers where the beneﬁt of ambient temperature operation, with or without thermoelectric (Peltier) cooling of the ﬁeld-eﬀect transistor (FET), have been successfully exploited. Iwanczyk et al. (1996) compared the performance of HgI2, Si PIN photoiodide, and CdTe detectors for this latter, important application. Large arrays of up to 100 HgI2 detectors

216

Ellis

have also been fabricated (Iwanczyk et al., 1995; Patt et al., 1995). Dabrowski et al. (1981) have been particularly successful in designing, fabricating, and characterizing HgI2 detectors throughout the past two decades and their work should be consulted for further details of this important type of EDXRF detector. Although high-purity germanium (HPGe) is widely used for detectors in electron microscopes as a result of its slightly better resolution than Si(Li) detectors, the use of HPGe detectors is less popular in photonexcited EDXRF spectrometry. One reason for this lack of acceptance may lie in the high escape peak intensities (see Sec. III.E.1) which can strongly interfere in trace analysis. Table 2 lists properties relevant to x-ray detection for Si, Ge, and HgI2, which are the materials most commonly employed in commercial EDXRF spectrometers. A variety of other compound semiconductor materials, such as GaAs, CdTe, and CdZnTe, have been used for applications in which speciﬁc properties are particularly beneﬁcial. These properties include the following: Higher atomic number for increased photoelectric cross section Larger band gap for lower thermal leakage at room temperature A lower band gap corresponding to a smaller average energy for creation of holeelectron pairs (Knoll, 1979), which oﬀers improved energy resolution. Recalling the direct relationship described earlier between photon energy and number of charge carriers and, therefore, the detectors signal resulting from detection of a particular x-ray photon, one can use the data from Table 2 in the following example for detection of CaKa1 (KL3), which has an energy of 3.691 keV: No. electronhole pairs produced ¼ 3691=3.86 ¼ 956 Charge on electron ¼ 1.661019 C Therefore, Charge from CaKa1 interaction ¼ 1.5361016 C Assuming a feedback capacitance in the ﬁrst stage ampliﬁcation of 0.1 pF, the resulting output voltage pulse is 1.53 mV. From this, we can see the very small magnitude of the signals involved. This has led to very great eﬀorts having been made in the design of detector and signal processing systems to minimize all sources of electronic noise which would otherwise interfere very strongly with the signal being sought. In addition to the conventional Si(Li) detector, Si PIN (Haselberger et al., 1996; Cesareo et al., 1996) and Si drift detectors (Lechner et al., 1996; Bertuccio et al., 1996) have also become available in recent years. The active area of these devices is generally

Table 2

Semiconductor Detector Material Properties

Detector properties Atomic number Atomic=molecular weight Density (300 K) g=cm3 Band gap (300 K) (eV) Average energy E per electronhole pair (77 kV) (eV) Fano factor (77 K) a

At 300 K.

Si

Ge

HgI2

14 28.09 2.33 1.115 3.86

32 72.60 5.33 0.665 2.96

80.53 454.45 6.3 2.13 4.15a

0.12

0.08

EDXRF Analysis Using X-ray Tube Excitation

217

310 mm2 and the depletion thickness is of the order of 300 mm, which reduces detection eﬃciency (see Sec. III.F) compared to Si(Li) detectors. In the following section, the details of detector operation are discussed, with particular emphasis on those aspects that impact EDXRF spectrometry most directly. These include the detector energy resolution, detection eﬃciency, spectral response, and counting system throughput.

C.

Cooling Systems

The need to minimize the electronic noise of the detection system is paramount for obtaining the best energy resolution. The main source of noise in the detector itself is leakage current, which derives from the generation of charge carriers in the absence of x-rays through, for example, thermal vibrations of the detector crystal lattice. Whatever the cooling system used, the intention of it is to minimize leakage, and the lower the temperature, the less leakage will occur. The boiling point of liquid nitrogen (77 K) is a typical temperature used for the detector crystal, although the FET may require a diﬀerent and often higher temperature to minimize its noise contribution and, therefore, obtain optimum detector resolution. The need for reduced operating temperature results in the need for careful design in the insulation from ambient temperatures of the detector crystal. Eﬀective insulation is typically achieved by enclosing the detector head (crystal and FET) assembly in a evacuated cryostat. In order for the low-energy x-rays (< 2 keV) to reach the detector crystal, the vacuum is usually retained by a thin (typically 550 mm) beryllium entrance window (Fig. 11). In recent years, the thinnest windows (5, 8, and 12 mm) have become available with coatings which ensure that there is no leakage of He through the Be window. Any helium getting through the window would degrade the high vacuum within the cryostat and lead to increased temperature, leakage current, and deteriorating performance. The original and perhaps simplest way of cooling a detector unit is to attach the measurement head via a cold-ﬁnger assembly to a dewar vessel containing liquid nitrogen (LN). The dewar itself is typically a complex technological component if it is to provide high performance and long operational lifetime. The modern LN cryostat delivers optimum resolution and the ability to be cycled through multiple warm-up=cooldown cycles without degradation in performance. The inevitable and natural boiling-oﬀ from the dewar of LN over a period (1 L per day is typical) means that the dewar needs regular replenishment. In order to minimize the frequency of replenishing LN, a large dewar is preferable, although this desire for a long LN holding time may lead to a rather bulky system. LN volumes of 715 L are typical, although some large systems may use volumes up to 25 L and some small, hand-held units may use 1 L or less. In any event, the following are important: Use only liquid nitrogen: Liquid air, although delivering the required temperature range, will fractionate, yielding O2, which is an explosion hazard in the chemical laboratory. Ensure that there is no ice or dirt in the LN used. The presence of ice or dirt in the dewar can cause microphony, which can degrade energy resolution. Ice and dirt can be eliminated by pouring the LN through a funnel equipped with a wire screen (e.g., a motor fuel metal ﬁlter funnel). Observe safe LN handling procedures. Avoid overﬁlling and spillages as liquid nitrogen can cause embrittlement of polymers and cable insulation or result in

218

Ellis

Figure 11

Cross section of a groove-type Si(Li) detector mounted in its cryostat end cap.

condensation from the atmosphere being deposited on components that are sensitive to moisture. It is essential that a Si(Li) detector be kept cold throughout the time that the bias is applied. If this is not done, then there is the possibility of damage to the detector assembly by, for example, migration of the drifted Li ions from the compensated region of the crystal. Should this happen, the detector will fail and, for this reason, it is usual for an automatic bias shutoﬀ system to be implemented. Such systems typically use a temperature sensor or a LN level sensor to switch oﬀ the detector bias in case of the dewar warming up. Other detectors, such as HPGe or high-purity Si, for which drifting is not required, do not exhibit this particular problem but may still suﬀer some problems if thermally cycled repeatedly, unless the cryostat assembly is speciﬁcally designed to do this. The simplicity of the LN-ﬁlled dewar has led to its widespread use in EDXRF systems. However, the diﬃculty in obtaining LN in remote areas, or the need for a no-LN solution, has led to the introduction in the past decade or so of thermoelectric (Peltier eﬀect) coolers or mechanical refrigerating units. The mechanical units may employ the JouleThompson (JT) eﬀect or some other cycle such as the Stirling cycle to obtain essentially LN temperatures at the detector head. The major beneﬁt of this approach is that there is no diﬀerence in performance in terms of detector energy resolution from the conventional dewar LN systems, but the coolers come with penalties of increased complexity, cost, and bulk. HPGe detectors, which typically need liquid-nitrogen temperature to operate, may use these mechanical coolers, but other detectors which will function at higher temperatures, albeit with degraded energy resolution, are suitable for use with thermoelectric (Peltier eﬀect) coolers.

EDXRF Analysis Using X-ray Tube Excitation

219

Peltier coolers can oﬀer a very compact cryostat=detector assembly that requires only electrical power to provide its cooling. Multistage (three to six) coolers may be used to provide a DT cooling, from ambient, of 50 C to a maximum of perhaps 120 C. In an ambient room temperature of 20 C, such coolers deliver working detector temperatures typically in the range 30 C to 90 C, which are suitable for Si PIN, HgI2, and extremely low-leakage Si(Li) detectors. Energy resolution in the range 200350 eV (see Sec. III.D) is typically available with these systems and both Si PIN and HgI2 detectors are commercially available with Peltier cooling. More recently, Peltier-cooled Si drift detectors, which oﬀer energy resolution better than 150 eV, have been developed (Lechner et al., 1996; Bertuccio et al., 1996) and are now commercially available.

D.

Energy Resolution

The energy resolution of the semiconductor spectrometer system determines the ability of a given system to resolve characteristic x-rays from multielement samples. It is normally deﬁned as the full width at half-maximum (FWHM) of the pulseheight distribution measured for a monoenergetic x-ray at a speciﬁed energy. A convenient choice of x-ray energy at which resolution is quoted is the weighted mean MnKa line at 5.895 keV, as this emission line is readily available from 55Fe radioisotope sources. The MnK x-rays are of suﬃciently low energy that the contribution to the FWHM of the unresolved Ka1 þ Ka2 doublet (5.898 and 5.887 keV, respectively) and the intrinsic width of the emission lines can be neglected. This is not true for higher-energy characteristic x-rays (e.g., MoK lines) for which the separation of the Ka1 þ Ka2 doublet and the intrinsic width of the emission line must be considered. Figure 12 shows a typical pulseheight spectrum of MnK x-rays taken from an 55Fe source. Typical state-of-the-art 30-mm2 Si(Li) detectors achieve a FWHM better than 140 eV at 5.9 keV, although values as low as 130 eV are achievable. However, this number is but one indicator of the quality of an EDXRF detector system, and other factors such as maximum count rate or the presence of background and artifacts may be more important in many analytical applications. Peltier-cooled Si PIN detectors oﬀer FWHM values typically in the range 165220 eV, whereas Si drift detectors claim < 150 eV. The interdependence of energy resolution and count rate is discussed below, although it is worth noting at this point that the user can exert a large degree of control over this by making an appropriate selection of pulse processor settings. If one neglects the natural linewidth of the x-ray lines, the instrumental energy resolution of a semiconductor detector x-ray spectrometer is a function of two independent factors. One of these is determined by the properties of the detector itself; the other is dependent on the nature of the electronic pulse processing employed. In some systems, a pulser or strobe signal may be injected into the measurement system in order to monitor the resolution of the electronic system independent of any peak broadening due to the detector itself. The measured FWHM of the x-ray line (DETotal) is the sum, in quadrature, of the contribution due solely to detector processes (DEDet) and that associated with the electronic pulse processing system (DEElec): qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ DETotal ¼ DE2Det þ DE2Elec ð1Þ The DEDet component is determined by the statistics of the free-charge production process occurring in the depleted volume of the diode. The average number of electronhole pairs

220

Ellis

Figure 12

Pulseheight spectrum of MnK x-rays taken with a 30-mm2 Si(Li) detector.

produced by an incident photon can be calculated as the total photon energy divided by the mean energy required to produce a single electronhole pair (see Sec. III.B). If the ﬂuctuation in this average were governed by Poisson statistics, the standard deviation would be rﬃﬃﬃﬃ pﬃﬃﬃ E s¼ n¼ ð2Þ e In semiconductor devices, the details of the energy-loss process are such that the individual events are not strictly independent and a departure from Poisson behavior is observed. This departure is taken into account by the inclusion of the Fano factor in the expression for the detector contribution to the FWHM: rﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ FE s ¼ Fn ¼ ð3Þ e Taking sðEÞ pﬃﬃﬃﬃﬃﬃ ¼ Fn E

ð4Þ

and, rearranging,

pﬃﬃﬃﬃﬃﬃﬃﬃ DEDet ¼ 2:35 FeE

ð5Þ

where e is the average energy required to produce a free electronhole pair, E is the energy of the photon, F is the Fano factor (see Table 2), and the factor 2.35 converts the standard deviation to FWHM for a Poisson distribution. A typical value for the detector contribution to resolution, also known as the dispersion, is 120. Examination of the values of e and F listed in Table 2 shows that for an

EDXRF Analysis Using X-ray Tube Excitation

221

equivalent energy, the detector contribution to the resolution is 28% less for the case of Ge compared to Si. This results in better energy resolution being obtained for HPGe detectors and could be an important consideration in the choice of detector for certain applications. However, this theoretical advantage in energy resolution of Ge over Si(Li) is mitigated in many systems because of the dominance of electronic noise (e.g., from the FET). In addition, the very high GeK escape peak probability (see Sec. III.E.1) is likely to be a serious limitation to the use of HPGe detectors in most EDXRF work. HPGe detectors are widely used in electron microscopes, where energies < 10 keV are likely to be of most interest. E.

Spectrum Artifacts

The simplicity of the element line spectra encountered in XRF spectrometry is one of its major beneﬁts when compared with other atomic spectroscopy methods. However, a few peaks can arise in the EDXRF spectrum from sources other than the elements in the sample and it is wise to be aware of their nature. Unlike basic detector eﬃciency and spectrum background, which are deﬁned by the selection of detector window and the detector manufacturing process, the end user can exert some control over the eﬀects of spectrum artifacts. From the spectra presented so far, it is clear that the bremsstrahlung continuum and the Compton (and Rayleigh)-scatter peaks arise from the interaction of the exciting radiation with the sample. These features are not regarded as artifacts. 1. Escape Peaks For an incident x-ray with energy higher than the SiK absorption edge, the detection process will involve the generation, through x-ray ﬂuorescence of the detector material, of SiK x-rays. The vast majority of these will themselves immediately be absorbed in the detector volume and contribute to the overall charge collected for the original incident x-ray photon. This is the normal mechanism of x-ray detection, which was described earlier. There is, however, a ﬁnite probability that the SiKa x-rays produced will escape from the detector volume and not contribute to the charge collected for the original photon that was detected. The resulting energy detected will be reduced by 1.740 keV, which is the energy of the SiKa x-ray that escaped. The greatest probability of escape arises close to the front of the detector crystal from which SiK x-rays can more easily escape. X-rays detected far within the volume will still generate SiK x-rays, but they will all be absorbed before they can reach the outer surface of the crystal and escape. The probability of generating the SiKb x-ray and of it escaping is extremely low and this eﬀect can safely be ignored. The result of the SiK escape process is a peak in the collected spectrum, which is 1.74 keV lower in energy than the parent peak. The Si escape peaks for TiKa and TiKb are clearly seen, 1.74 keV below the TiK lines, in Figure 13 (indicated by 1 and 2, respectively) (see also Table 3). In many detectors, the incomplete charge-collection tail on the lowenergy side of the parent peak will obscure the Si escape peak from TiKb. The escape probability is highest for lines closest to the SiK absorption edge at 1.838 keV and may be calculated as described by Statham (1976) and Dyson (1974). The intensity falls rapidly from around 3% of the parent peak at 23 keV to around 0.1% of the parent peak at 10 keV. At energies above 10 keV, the eﬀect is negligible, but below this energy, it is important that spectrum-processing software takes account of the peak. Typically, the processing software will make a correction in which the Si escape peak is removed and the most comprehensive packages will reinstate the lost intensity to the parent peak. If such packages are available, then make use of them before carrying out

222

Ellis

Figure13

Peak artifacts from a Ti spectrum collected by a Si(Li) detector (see Table 3 for details).

qualitative analysis in order to avoid classic errors such as the Si escape peaks from ZnK being identiﬁed as Co! In the case of HPGe, the escape probability for GeL lines is negligible in XRF and can be ignored. The escape of the GeK lines has, however, a very high probability, and for Se, the escape peak intensity is around 16% of the parent peak. This is due to the combination of the high detector eﬃciency for GeK x-rays and the much greater depth within the crystal from which the GeK lines can escape. There is further complication in that there are many GeK lines and each of these lines will result in an escape peak. The complexity and intensity of Ge escape peaks can easily be seen in the spectrum from a molybdenum oxide sample, shown in Figure 14. The principal GeK lines and their characteristic energies are as follows: GeKa: 9.876 keV GeKb1: 10.981 keV GeKb2: 11.100 keV The assignments of the numbered GeK escape peaks arising from the detection of the MoK lines are listed in Table 4.

Table 3 Peak No. 1 2 3 4 5

Artifact Peak Energy and Assignments for Figure 13 Energy (keV)

Assignment

2.8 3.2 9.0 9.4 9.9

TiKaSiK escape TiKbSiK escape TiKa þ TiKa sum peak TiKa þ TiKb sum peak TiKb þ TiKb sum peak

EDXRF Analysis Using X-ray Tube Excitation

Figure 14 details).

223

GeK escape peaks from Mo spectrum collected by HPGe detector (see Table 4 for

The energy and intensity of these GeK escape peaks interferes very strongly over a wide and important region of the spectrum. The interference is considerable from the GeK escape peaks of Nb and Mo in stainless steels and corrections for these intense escape peaks are not suﬃciently accurate to retain the intrinsic performance of EDXRF for what would be a reliable and straightforward analysis using a Si(Li) detector. The severe limitation posed by the Ge escape peaks in EDXRF is also demonstrated in Figure 15 (see also Table 5) for a typical determination of minor and trace heavy metals in a low-Z matrix. In this case, the complex GeK escape peaks from the Rh backscatter peaks completely dominate the important part of the spectrum for trace analysis. In conclusion, the Si escape peak is small and easily corrected, but the number and high intensity of the GeK escape peaks severely limit the value of the HPGe detector for tube-excited EDXRF. In the case of energy-dispersive (ED) microanalysis, however,

Table 4 Peak No. 1 2 3 4 5

GeK Escape Peak Assignments and Relative Intensity for Figure 14 Energy (keV)

Assignment

% of parent

6.4 7.57 8.6 9.74 10.12

MoKaGeKb1 MoKaGeKa MoKbGeKb MoKb1GeKa MoKb2GeKa

1.5 9.6 1.0 8.8 2.5

224

Ellis

Figure 15 Comparison of Si(Li) and HPGe detectors for tube-excited EDXRF trace analysis of low-Z sample matrix (see Table 5 for details).

the HPGe detector has beneﬁts in that the intensity of the GeL escape peaks is negligible and the energy resolution is better than that of Si(Li) detectors. 2. Sum Peaks Sum peaks arise from a speciﬁc form of peak pileup (see Sec. IV.C) where two events from high-intensity peaks arrive in the pulse processing electronics so close together in time that the pileup inspector cannot recognize them as two events. The eﬀect of this is for the signals to be seen as one and for them to be registered at the energy that is the sum of the two. The sum peaks of Ti are seen in Figure 13, in which the most intense peak (3) is from the sum of two TiKa events. The smallest peak (5), from the sum of two TiKb events is only visible when the parent peak is very intense. At ﬁrst glance, peaks 3 and 4 may appear like a typical Ka, Kb

Table 5 Peak 1 2 3 4

GeK Escape Peak Assignments for Figure 15 Assignment RhKa ComptonGeKa RhKa RayleighGeKa RhKb ComptonGeKa RhKb RayleighGeKa

EDXRF Analysis Using X-ray Tube Excitation

225

doublet, but the peak separation is too small for K lines in the region of the spectrum in which they occur. In addition, the relative line ratio is not the same as for a characteristic K line series at this energy. Consequently, sum peaks are unlikely to be incorrectly identiﬁed as elements, but they may interfere with important lines in a particular analysis. This is particularly the case in some environmental analyses, (e.g., where high concentrations of Fe lead to intense FeK lines). This will result in FeK sum peaks in the region of the K lines of Se and Br and the L lines of toxic heavy elements such as Hg, Tl, and Pb. Some spectrum-processing software packages are able to correct for sum peaks and should be used to minimize potential errors in analysis (See Chapter 4). If it is found that such features are not available, then reducing the count rate will substantially reduce the eﬀect of the sum peaks. In order not to lose analytical performance, however, selection of alternate excitation conditions (e.g., using a more absorbing beam ﬁlter) will ‘‘rebalance’’ the spectrum in favor of the peaks at higher energy and minimize the interference. 3. Diffraction Peaks Whenever a crystalline sample is measured in an EDXRF spectrometer, there is the possibility that the conditions for Bragg diﬀraction will be met. This is exacerbated by the divergent geometry of the incident and detected beam and the variety of excitation conditions available. The worst cases are where unﬁltered primary radiation is used, as there are then many energies and angles which increase the probability of meeting the Bragg condition for a crystalline component in the sample. The more monochromatic the exciting beam is, the less likely it is that the Bragg condition will be met. Figure 16 shows the diﬀraction peak obtained from a boric acid pellet which was excited with a silver anode x-ray tube operated at 5 kV with no primary beam ﬁlter and in an approximately 90 geometry. The spectrum is overlaid with one from a sulfur pellet measured under exactly the same conditions. It can be seen that the diﬀraction peak is broader than a K series line at that energy and the low-energy tailing is more pronounced. There is some possibility for misidentiﬁcation, but the most likely problem arises in any spectrum processing not setup to take account of this peak. In cases such as silicon wafers, where the crystal planes are highly oriented, the sample can be tilted or rotated to minimize the often sharp diﬀraction peaks that can be observed. Alternatively, eﬀective use can be made of additional collimation to constrain more closely the angles that can satisfy the Bragg condition or of alternate excitation conditions, which change or reduce the incident energies available for diﬀraction. Figure 17 shows a comparison of the spectra taken from the same two samples under diﬀerent excitation conditions. The increased kV and use of an absorption ﬁlter completely change the energy distribution of the excitation spectrum and eliminate the diﬀraction peak. This second set of conditions is particularly eﬀective for lines in this part of the spectrum and eﬀectively removes problems from diﬀraction peaks. Once a diﬀraction peak is identiﬁed as causing a problem, a change of excitation conditions is often the best measure to adopt. The use of monochromatic or polarized excitation or of secondary targets in Cartesian geometry will also eliminate diﬀraction peaks. 4. System Contamination Peaks Careful screening of all the components in an EDXRF spectrometer is required in order to eliminate spurious element peaks in measured spectra. Each spectrometer designer will adopt diﬀerent approaches and materials for this purpose and, thus, there is a variety of

226

Ellis

Figure 16 Spectrum from boric acid, overlaid with one from a sulfur pellet, showing the diffraction peak. Conditions: Ag target x-ray tube, 5 kV, no primary beam filter.

Figure 17 Spectrum from boric acid, overlaid with one from a sulfur pellet, showing removal of the diffraction peak seen in Figure 16. Conditions: Ag target x-ray tube, 15 kV, thin Al primary beam filter.

EDXRF Analysis Using X-ray Tube Excitation

227

potential spectrum contaminant peaks. Careful design will eliminate them, although, in some cases, the use of a simple blank correction may be required in some trace analysis. Detector components can be a cause of some system peaks. Beryllium entrance windows usually contain some trace elements that, in extreme cases such as TXRF, can be seen in blank spectra. The use of thick beryllium windows exacerbates this potential problem, but window thicknesses commonly used (850 mm) should pose no practical problem. The Si internal ﬂuorescence peak and absorption edge caused by the dead layer are eﬀects not seen in practice and need very careful characterization with specialized conditions if they are to be seen at all. The most likely source of spurious spectrum peaks is sample handling and presentation, both of which are largely in the control of the user. F.

Detection Efficiency and Entrance Windows

One of the more important advantages of semiconductor spectrometers is the absolute eﬃciency with which ﬂuorescence x-rays are detected and their energies measured. This is the result of the intrinsically high photoelectric absorption eﬃciencies for semiconductor materials in the x-ray energy range and from the large solid angles achieved in typical EDXRF geometries. The solid angle is determined by the area of the detector and the sampledetector distance and varies with the design of the system. Typical areas are 1080 mm2 for Si(Li) detectors. Although the additional solid angle is advantageous for many applications, one must realize that the added capacitance associated with increased active area results in an increased contribution in the electronic noise of the system. This, in turn, results in an increased energy resolution of the detection system. The intrinsic eﬃciency of the semiconductor device can be approximated by a simple model in which the probability of detecting an x-ray incident on the detector is assumed to be the probability of photoelectric absorption within the sensitive volume. This can be expressed as eðEÞ ¼ emt ð1 esd Þ where e(E) is the energy-dependent intrinsic eﬃciency of the detector, t is the thickness of any absorptive layer between the sample and the detector, m ¼ m(E ) is the mass absorption coeﬃcient of the absorptive layer, d is the detector thickness, and s ¼ s(E ) is the photoelectric mass-absorption coeﬃcient for the detector material. Figure 18 shows plots of the above equation for the case of a 3-mm-thick Si(Li) and a 5-mm-thick HPGe detector. The poor eﬃciency at low energies is assumed to be determined by the combined absorption of the 25-mm beryllium entrance window and a nominal 2-cm air pair between sample and detector. The energies of the characteristic K emission lines for several elements are also shown. The plots show the near-unity intrinsic eﬃciencies for both detectors over a wide range of useful x-ray energies. The Ge detector is eﬃcient at much higher energies than Si because of the higher atomic number and subsequent larger photoelectric cross section. The simple model gives a semi-quantitative picture of the eﬃciency behavior of semiconductor spectrometers. However, several other factors must be considered for quantitative calibration of a ﬂuorescence spectrometer. The concept of a thin entrance window that either absorbs or transmits an incident x-ray does not describe cases in which the secondary electrons, either photoelectric or Auger, are emitted into the active volume from the window layer. Similarly, detailed studies of the low-energy eﬃciency have indicated the presence of an absorbing layer on the surface of both Si and Ge devices

228

Ellis

Figure 18 Calculated intrinsic photopeak efficiency for Si and Ge detectors as a function of incident photon energy. The high-energy limits are established by the photoelectric cross section of the detector material and the diode thickness. The low-energy cutoff is determined either by absorption in the thin Be window or in a nominal 2-cm air path from sample to detector or fluorescence source.

associated with the surface layer of the semiconductor material itself. This layer is comprised of an evaporated metal contact used to form the rectifying Schottky barrier and a thin surface layer (the dead layer) of inactive semiconductor material from which charge cannot be collected. The thickness of evaporated metal, normally Au, can be determined by direct measurement during the manufacturing process and is typically in the range of 10 nm. The eﬀective ‘‘dead layer’’ of semiconductor material is a more complicated parameter to determine. Empirical studies have attempted to measure the absorption inherent in the semiconductor surface layer. Such studies established that the eﬀective thickness is determined largely by the absorption length of the low-energy photons and the chargetransport characteristics of the associated ionization products. In the simplest model, there is competition from the rate of diﬀusion of the electronhole distribution, against the gradient of the applied ﬁeld. For distances near the entry contact, a part of the charge can diﬀuse into the contact and be lost to the signal before the electrical ﬁeld can sweep it to the opposing electrode. This loss of charge can be interpreted in terms of an eﬀective ‘‘window’’ thickness of typically 0.2 mm silicon equivalent. In addition, thin evaporated contacts are deposited on the entry surface that can absorb incident low-energy photons. Detailed experiments have indicated that the charge-collection eﬃciency for events near the surface depends in a complex way on low-energy x-ray and charge-transport properties (Llacer et al., 1977; Goulding, 1977). As a consequence of these entry-windowrelated eﬀects, the eﬃciency for low-energy photons can be reduced. The events lost from the photopeak can then appear in a continuum background below the full-energy peak, where they reduce detectability and interfere with spectral analysis. Similarly, secondary electrons or photons originating from the photoelectric absorption events that occur initially in the active volume can escape. This results in a collected charge pulse of reduced

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229

amplitude. An easily observed manifestation involves the observation of discrete peaks in the spectrum. Continuous-loss processes involving electron escape can also occur, although the probability is small. These and related mechanisms that reduce the amplitude for a given event can lower the photopeak eﬃciency relative to the simple model described. There have been studies in which the eﬃciencies of Si(Li) and Ge have been carefully measured using calibrated sources of x-rays spanning the energy region of interest (Cohen, 1980; Szoghy et al., 1981; Campbell and McGhee, 1986). These indicate that the maximum intrinsic photopeak eﬃciency is reduced by a few percent relative to the curves shown in Figure 18 and is slightly higher at the upper energy cut oﬀ than calculated. These results substantiate the overall validity of the simple photoelectric absorption model but demonstrate the limitations if precise results are required. For laboratory applications of EDXRF, it is not necessary to explicitly determine the photoelectric eﬃciency function because it is included in the overall calibration factor of the instrument. In any event, the internal eﬃciency of the detector itself is deﬁned by the design and manufacture of the detector and this is out of the control of most users. The eﬃciency curve shown in Figure 18 illustrates that for conventional EDXRF measurements, the absorption of ﬂuorescence x-rays in air and in the Be entry window limits the accessible energy range to photons greater than approximately 2 keV. The absorption losses in the air path can be signiﬁcantly reduced by the common practice of employing a vacuum or He atmosphere. A vacuum path is preferable due to higher x-ray transmission, and if helium is used, the beryllium entrance window must be able to eliminate leakage into the cryostat with subsequent loss of its vacuum. Coated beryllium windows are commonly used nowadays to minimize helium leakage into the cryostat. The necessity of a beryllium window to maintain vacuum integrity between the cryostat enclosure and atmospheric pressure does not constitute a serious limitation for most analyses. However, there are applications in which the detection of x-rays with energies below 1 keV (e.g., for F) becomes necessary. The relatively high detection limits achieved by EDXRF systems because of the poor energy resolution (compared with WDXRF) combined with absorption by the beryllium window generally make EDXRF the second choice for these very low-energy x-ray lines. A beryllium window thickness of 8 mm is typically used for very light-element analysis, although windows as thin as 5 mm have been used. The use of extremely thin windows may lead to increasing concerns for helium integrity and equipment cost. High-strength thin windows made from low-atomic-number elements that are capable of withstanding a full 1-atm-pressure diﬀerential have become commercially available in the last decade. These include self-supporting 0.5-mm diamond polycrystalline ﬁlms and 0.25-mm windows composed of a vapor-deposited amorphous material consisting of 90% boron by weight with nitrogen and oxygen for the remainder. These windows exhibit signiﬁcant x-ray transmission for photons well below 1 keV and have found widespread use in x-ray microanalysis. G.

Detector Background

Eﬀects associated with the partial collection of the photoelectric signal by the detector have a small eﬀect on its eﬃciency. However, these incomplete charge-collection processes can have consequences that are far more serious for analytical performance through their eﬀect on spectral background. The eﬀect of incomplete charge collection of these events is to produce tailing on the low-energy side of a peak. In addition, it produces a continuum of events that appear as a shelf on the low-energy side of major peaks in the spectrum. This

230

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background tailing and shelf interfere with the measurement of lower-energy ﬂuorescence x-rays and is a signiﬁcant factor in obtaining the lowest detection limits for the determination of trace quantities on the low-energy side of major peaks. Studies designed to reduce this background have indicated that in addition to fundamental x-ray and electron-energy-loss processes, a more signiﬁcant background resulting from incomplete charge collection from the detector active volume is normally the dominant contribution (Goulding et al., 1972; Jaklevic and Goulding, 1972). This process is an artifact of the detector operation in which the collection of the free charge from the depleted volume is inhibited as a result of nonuniformities in the applied electrical ﬁeld. These nonuniformities are typically associated with edge eﬀects at the periphery of the cylindrical detector. The incomplete charge-collection background can be reduced either by external collimators that prevent the incident radiation from interacting in the periphery of the detector or by internal electronic collimation brought about by the use of a guard-ring structure (Goulding et al., 1972). The tailing can sometimes be improved by increasing the detector bias, but this is not an option available to the majority of users. There has been much work on characterizing the detector line shape and the eﬀects on background and tailing of incomplete charge collection (Campbell and Wang, 1991; Campbell et al., 1997; Heckel and Scholz, 1987; Lepy et al., 1997). Such studies have led to a greater understanding of the processes involved and to peak-ﬁtting procedures that can take account of them. However, improvements in this form of background can only be achieved through improvements in detector design and manufacturing processes. The P=B ratio of a detector is a standard measurement of detector quality. The measurement is made using a 55Fe radioisotope source to ensure that there is no other contribution to background. The measurement cannot be made using x-ray tube excitation, as it is extremely diﬃcult to eliminate the residual bremsstrahlung continuum. The peak is taken from the most intense channel in the MnKa peak and the background is taken as the average intensity in the channels from 0.9 to 1.1 keV. A high-quality Si(Li) detector will deliver a P=B ratio in excess of 10,000:1 and have minimal low-energy tailing.

IV.

SEMICONDUCTOR DETECTOR ELECTRONICS

A.

Sources of Electronic Noise

The contribution to resolution associated with electronic noise (DEElec) is the result of random ﬂuctuations in thermally generated leakage currents within the detector itself and in the early stages of the ampliﬁer components. Although these processes are intrinsic to the overall measurement process, there are methods for limiting the impact on the ﬁnal system resolution. Figure 19 is a schematic diagram of a typical pulse processing system employed in a semiconductor detector x-ray spectrometer. The pulse processing can be divided between the charge integration, which takes place in the preampliﬁer, and the voltage ampliﬁcation and pulse shaping, which occur in the main ampliﬁer (pulse processor). The function of the charge-sensitive preampliﬁer and subsequent ampliﬁcation stages is to convert the integrated charge pulse, produced by collection of the photoelectrically induced ionization, to a voltage pulse that can be measured and stored in the multichannel pulseheight analyzer (MCA). The ﬁrst stage of the process occurs in the FET, which is attached to the rear of the detector crystal. The method of charge restoration used for the FET has a fundamental eﬀect on the electronic noise of the overall detection system and is

EDXRF Analysis Using X-ray Tube Excitation

Figure 19

231

Typical pulse processing system used for energy-dispersive detectors.

particularly important. Early detectors used pulsed optical charge restoration, but newer systems make use of integrated solid-state devices incorporating low-noise-junction FETs to minimize electronic noise (Statham and Nashashibi, 1988; Lund et al., 1995, Lund et al., 1996). However, the pulse processing must also achieve a very important goal of amplifying the weak charge signal to a measurable level while suppressing random ﬂuctuations in the signal amplitude produced by thermal noise. This is achieved by generating a carefully deﬁned pulse shape in the main ampliﬁer, which restricts the Fourier frequency components in the ﬁnal signal in such a way that the signal contributions are emphasized relative to the noise ﬂuctuations. The most common pulse shapers employed in modern semiconductor spectrometers generate output pulses that are Gaussian, triangular, or cusp shaped (Fairstein and Hahn, 1965; Kandiah et al., 1972; Landis et al., 1982). Each is capable of achieving adequate energy resolution for EDXRF analysis; the diﬀerences are due mainly to the eﬀective time interval required for processing a pulse. Because the relative amplitude of the noise contribution is a strong function of the characteristic time constant associated with the pulse shaper, the diﬀerence between pulse shapes becomes important for applications in which high counting rates are important. A detailed analysis of the eﬀects of various pulse-shaping options on spectrometer performance is presented by Goulding and Landis (1982). More recently, suﬃciently fast, sampling analog-to-digital converters (ADCs) have become available and these have resulted in the availability of digital pulse processors (Warburton et al., 1988). These devices can implement the optimum noisereduction ﬁlters in conjunction with much higher count rates than conventional analog pulse processors. As pointed out below in the following subsection, however, it is not always desirable to operate at the minimum-noise shaping time because an unacceptably large dead time may result. Although EDXRF instruments are designed with a suitable compromise between energy resolution and count rate capabilities, an understanding of these trade-oﬀs on the part of the user is important for optimum usage of the instrument.

232

B.

Ellis

Resolution and Count Rate

There exists a great deal of literature on the nature of the detection and pulse processing systems and the great strides which have been made over the past 20 years to improve resolution and increase count rate. For many years, the complete integration of the detection and electronics systems within the instrumentation has meant that the individual choice of and matching by the EDXRF user of detector, main ampliﬁer, ADC unit, and MCA is no longer required. Once the application is determined, the choice of system will typically revolve around achievable analytical performance. The user will typically be left with a relatively straightforward selection of the single signal processing setting providing the required resolution and count rate (see Fig. 20). There is only a single setting and, unfortunately, resolution and rate go in opposite directions when this setting is changed. In one direction, the ampliﬁer time constant will increase, which gives better resolution (lower FWHM) at the expense of a lower count rate into the MCA memory. In the opposite direction, resolution degrades but count rate increases. Once the EDXRF system is chosen, this control is really the only way in which the user can inﬂuence resolution and count rate. It is a little like choosing a car where you select the type of vehicle most suited to your use. That vehicle then has a gearing selection system that allows you some degree of trade-oﬀ in control. Low gears give highest accuracy in maneuverability (i.e., resolution) but lowest speed (i.e., count rate). Conversely, the highest gears give a high speed (rate) at the expense of accurate maneuverability! At one end of the range, best resolution is obtained with lowest total count rate, whereas at the other, highest count rate is delivered with the worst energy resolution. When severe peak overlap is the dominant limitation to analytical performance, the best resolution should be selected. In this case, which is typical for the light elements, the

Figure 20

Plots of energy resolution (at 5.9 keV) and count rate for a Si(Li) detector.

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233

excitation conditions will need adjusting to make best use of the limited count rate available. Where peak separation is good, such as above 15 keV, or where there are no severe overlaps, select the fastest count rate. In both cases, the current is used to adjust for optimum dead time (50% typically) and the measurement time is used to control the precision. C.

Pulse Pileup

The count-rate limitations associated with a semiconductor spectrometer are an inherent property associated with the ﬁnite pulse processing time required by the electronic shaping network. When a random sequence of pulses is incident on the detection system, some of the events cannot be processed without ambiguity. To appreciate the fundamental nature of this limitation and its relationship to system performance, some elementary concepts of electronic pulse processing must be considered. Figure 21 illustrates the time sequence of pulses that occur at various stages in the pulse processing chain. Trace A shows the output of the charge-integrating preampliﬁer. The steps at times 1, 2, and 3 represent the charge integrals of discrete events. Traces B and C are the outputs of the fast (short shaping time) ampliﬁer and an associated discriminator that serves as a timing marker for the individual events. The main shaping ampliﬁer output is shown as the Gaussian pulse shapes (in trace D). For each event, a total pulse processing time (neglecting analog-to-digital conversion in the pulseheight analyzer) of td is required after the arrival of the pulse and before the system is ready to accept the next event. Although the average counting rate detected by the system can be well below the frequency deﬁned by the reciprocal of the pulse width td, that the events are statistically uncorrelated implies that the events are not uniformly distributed in time. There then exists a probability that two pulses will occur within the same processing time interval. This is illustrated by the overlap of pulses 2 and 3 in the trace. At a low average counting rate, this pulse overlap is not a limiting factor. As the average counting rate increases, however, a point is reached at which there is a signiﬁcant probability that a second event will occur before the ﬁrst event has been fully analyzed. If the two events occur within a time less than the shaping time of the ampliﬁer, the charge signals are indistinguishable and an erroneous ‘‘pileup’’ energy signal results. Modern spectrometers employ some form of rejection circuitry to eliminate these pileup events from the pulseheight spectrum. Typical systems rely on the inspection of the fast discriminator output to determine if two pulses have occurred in rapid succession. Appropriate logic is then employed to gate the output of the processor to eliminate the resultant ambiguous energy signals. This is shown in traces E and F of Figure 21, in which the logic signal causes the output to be inhibited when the pulses overlapped to produce an ambiguous pileup energy output. Because the fast shaper that generates the discriminator output has inherently poorer energy resolution relative to the slow channel, the pileup rejection circuit is limited in the minimum energy of pulses it can detect. However, in typical tube-excited EDXRF, the major fraction of pulses occur from scattered high-energy radiation and a pileup rejector system is an important feature in the system design. The pileup probability is obviously a function of the characteristic shaping time tp, which, in turn, establishes the eﬀective dead time, td, of the system. This probability is independent of the details of the speciﬁc type of pileup rejection circuit used to eliminate the ambiguous events. The number of events that experience pileup and are consequently eliminated from the spectrum can be estimated for the case of a totally random arrival

234

Ellis

Figure 21 Time sequence of pulses during the processing of a series of x-ray events; (A) the output of the preamplifier is represented as a series of voltage steps reflecting the integrated charge from the individual events; (B) the output from a fast shaping amplifier that operates with a shorter time constant than the main shaping system; (C) a fast discriminator timing pulse derived from B; (D) the output of the main shaper, with the pileup of pulses 2 and 3 indicated; (E) the dead-time gates; and (F) the final output with the ambiguous 2 and 3 pileup pulses rejected.

distribution. For a series of events randomly distributed in time with an average frequency N0, the probability P that no pileup events occur within a characteristic time td after a given event can be expressed as P ¼ N0 eN0 td From this expression, we can calculate the fraction of events transmitted through the system. Figure 22 is a plot of nonpileup output rate versus input rate expressed in terms of a characteristic pulse processing time. The input rate for which the output rate is a maximum is seen to be the reciprocal of the shaping time. The output rate at this point is reduced by a factor of 1=e. It should be emphasized that this behavior is a fundamental consequence of random arrival statistics and a ﬁnite measurement time. However, one should be aware that it is always possible to reduce the characteristic processing time to achieve an increase in counting rate at some sacriﬁce in energy resolution. It is also worth noting that, by working at too high a dead

EDXRF Analysis Using X-ray Tube Excitation

235

Figure 22 Nonpileup output rate (N0) as a function of input rate (N1) scaled as a function of a characteristic shaping time. Maximum output is 0.37 of the input rate that is equal to the reciprocal of the time constant.

time, the throughput curve inverts and the output rate begins to fall. Further increase of the dead time will eventually lead to paralysis of the system and negligible output countrate. Although diﬀerent algorithms can be used to correct for events lost to pileup, there is no way to eliminate the eﬀect through passive pulse processing (Gedcke, 1972; Hayes et al., 1978; Statham, 1977). Because diﬀerent manufacturers of x-ray equipment vary in their approaches to pileup rejection, the only way to evaluate the throughput of the system is using a variable intensity source of radiation. The input rate can be determined by scaling the fast discriminator output, which is sometimes available from the instrument data system. The output rate can be simultaneously measured in the pulseheight analyser. A plot of nonpileup output rate versus input rate can then be generated and compared to the ideal case shown in Figure 22. When the excitation source can be switched oﬀ in a time interval that is short compared to the characteristic pulse processing time, it is possible to increase the average output counting rate by eliminating the eﬀects of pileup. Such pulsed excitation systems rely on the ability to detect an event in the fast channel and to shut oﬀ the excitation before a second pileup event can occur. In this mode of operation, the output rate can equal the input rate up to the point at which the system is continuously busy. This method of pileup control has been implemented using pulsed x-ray tubes (Jaklevic et al., 1976; Stewart et al., 1976), although the increased complexity of the electron optics tends to lead to increased cost. D.

Dead-Time Correction

The presence of pulse pileup must be considered in any system designed for quantitative measurements because it causes the eﬃciency with which pulses are processed to be rate dependent. EDXRF analytical systems are designed to correct for this discrepancy through a variety of approaches. The most straightforward involve the use of a live-time

236

Ellis

clock consisting of a gated oscillator and scaler. The oscillator clock is turned oﬀ when the system is busy processing pulses. In this way, the duration of the measurement in terms of a live-time interval is extended to correct for those periods when the system is incapable of processing a pulse. Alternative methods rely on the direct measurement of the fast discriminator pulses to keep track of those events that were missed. An additional correction is added to compensate for those intervals in which pileup occurs (Bloomﬁeld et al., 1983; Hayes et al., 1978). A simple empirical method used to correct for dead-time losses is a measurement of the ratio of input to output counts as a function of input rate over the range of values normally encountered. Subsequent analyses require that the input rate be measured for each experiment. The live-time correction can then be applied using the previously determined response function. Because all live-time correction methods have some limitation on the range of counting rates over which they can be used, it is important that methods be devised to evaluate the precision with which such corrections are generated. A carefully prepared series of standards of varying concentrations represents a direct approach. A potentially more precise method involves the use of a single thin-ﬁlm standard. Variable-mass targets are placed behind the standard to vary the total counting rate over the range of interest. If the variable mass target is chosen so that the variable intensity of scattered or ﬂuorescence radiation does not induce ﬂuorescence in the thin-ﬁlm standard, the measured intensity of ﬂuorescence from the standard should be independent of total counting rate. The ability of the dead-time correction system to compensate for pileup can then be empirically evaluated. All modern EDXRF systems incorporate eﬀective dead-time correction circuitry, which is likely to be at least as accurate as experiments designed to test this standard feature.

V.

SUMMARY

The technology that makes chemical analysis with EDXRF practical is based on the use of semiconductor x-ray detectors and associated pulse processing and data acquisition systems. The present chapter attempted to explain to the analyst the basic concepts behind the operation of these components and the manner in which they inﬂuence overall system performance. The trade-oﬀs one must make between such parameters as excitation conditions, detector resolution, count rate, and other design variables determine how eﬀectively one can tailor a given instrument or experimental apparatus to a speciﬁc application. Furthermore, a thorough understanding of the factors that limit performance should enable one to implement experimental tests to determine the eﬀectiveness of a particular approach and evaluate various commercial options.

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4 Spectrum Evaluation Piet Van Espen University of Antwerp, Antwerp, Belgium

I.

INTRODUCTION

This chapter deals with (mathematical) procedures to extract relevant information from acquired x-ray spectra. Smoothing of the spectrum results in a graph that can be easier interpreted by the human observer. To determine which elements are present in a specimen, peak search methods are used. To obtain the analytically important net peak areas of the ﬂuorescence lines, a variety of methods, ranging from simple summation to sophisticated least-squares-ﬁtting procedures, are to the disposal of the spectroscopist. Spectrum evaluation is a crucial step in x-ray spectrometry, as much as sample preparation and quantiﬁcation. As with any analytical technique, the performance of x-ray ﬂuorescence analysis is determined by the weakest step in the process. Spectrum evaluation in energy-dispersive x-ray ﬂuorescence analysis (EDXRF) is certainly more critical than in wavelength-dispersive spectrometry (WDXRF) because of the relatively low resolution of the solid-state detectors employed. The often-quoted inferior accuracy of EDXRF can, to a large part, be attributed to errors associated with the evaluation of these spectra. As a consequence of this, most of the published work in this ﬁeld deals with ED spectrometry. Although rate meters and=or strip-chart recorders have been employed in WD spectrometry for a long time, the processing of ED spectra by means of computers has always been more evident because of their inherent digital nature. Some of the techniques to be discussed have their roots in g-ray spectrometry developed mainly in the sixties; for others (notably the spectrum-ﬁtting procedures), EDXRF has developed its own specialized data processing methodology. The availability of inexpensive and fast personal computers together with the implementation of mature spectrum evaluation packages on these machines has brought sophisticated spectrum evaluation within the reach of each x-ray spectrometry laboratory. In this chapter, various methods for spectrum evaluation are discussed, with emphasis on energy-dispersive x-ray spectra. Most of the methods are relevant for x-ray ﬂuorescence, particle-induced x-ray emission (PIXE), and electron beam x-ray analysis [electron probe xray microanalysis (EPXMA), scanning electron microscopy – energy dispersive x-ray analysis (SEM–EDX), and analytical electron microscopy (AEM)]. The principles of the methods and their practical use are discussed. Least-squares ﬁtting, which is of importance not only for spectrum evaluation but also for qualiﬁcation procedures, is discussed in detail in Sec. IX. Section X presents computer implementations of the main algorithms. 239

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FUNDAMENTAL ASPECTS

The aims of spectrum evaluation is to extract analytically relevant information from experimental spectra. Obtaining this information is not straightforward because the spectral data are always corrupted with measurement noise. A.

Amplitude and Energy Noise

In x-ray spectra, we can distinguish between amplitude and energy noise. Amplitude noise is due to the statistical nature of the counting process, in which random events (the arrival of x-ray photons at the detector) are observed during a ﬁnite time interval. For such a process, the probability of observing N counts when the ‘‘true’’ number of counts is N0 is given by the Poisson distribution (Bevington and Robinson, 1992): PðN; N0 Þ ¼

NN 0 N0 e N!

ð1Þ

The number of counts in each channel of an x-ray spectrum as well as the sum over a number of channels obey this Poisson distribution. For a Poisson random variable, the population standard deviation is equal to the square root of the true number of counts: pﬃﬃﬃﬃﬃﬃ sN0 ¼ N0 ð2Þ The sample standard deviation, which is an estimate of the true standard deviation, therefore can be calculated as the square root of the observed number of counts: pﬃﬃﬃﬃ ð3Þ sN ¼ N sN0 The statistical nature of the counting process (Poisson statistics or counting statistics) causes the typical channel to channel ﬂuctuations observed in x-ray spectra. That the uncertainty of the data can be calculated from the data itself [Eq. (3)] is of great importance for the spectrum evaluation methods. Energy noise, on the other hand, causes the characteristic x-ray lines of ED spectra to appear much wider than their natural linewidth of about 510 eV. Part of this line broadening is due to the nature of photon-to-charge conversion process in the detector, and part of it is associated with the electronic noise in the pulse ampliﬁcation and processing circuit, as discussed in Chapter 3. As a result, x-ray photons with energy E, which on average correspond to a pulse height stored in channel i, from time to time gives rise to slightly higher or lower pulses, causing the x-ray events to be stored in channels above and below i, respectively. Accordingly, characteristic x-ray lines appear as relatively broad (140250 eV), nearly Gaussian-shaped peaks in the spectrum. The peaks observed in wavelength-dispersive spectra are also wider than the natural linewidth because of imperfections in the diﬀraction crystal and the ﬁnite size of the collimators. B.

Information Content of a Spectrum

In the absence of these two noise contributions, spectrum evaluation would be trivial. A spectrum would consist of a well-deﬁned continuum on which sharp characteristic lines were superimposed. The intensity of the continuum and the net x-ray line intensities could be determined without error. Any remaining peak overlap (e.g., between AsKa and PbLa, where the separation of 8 eV is less than the natural width of the AsKa line) could be dealt with in an exact manner.

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Unfortunately, we cannot eliminate the noise in the measurements completely. It is possible, however, to reduce the noise in various ways. Amplitude noise (i.e., counting statistics) can be reduced by acquiring the spectrum for a longer period of time or by using a more intense primary beam. The eﬀect of energy noise can be lowered by using a detector and associated electronics of good quality and by shielding the system from external sources of electronic noise. Although these suggestions may sound straightforward and not appropriate in the context of spectrum evaluation, it is important to realize that once a spectrum has been acquired, the information content remains constant. No spectrum processing procedure, no matter how sophisticated, can produce more information than present originally. It is therefore much more eﬃcient to employ optimal experimental conditions when acquiring the data rather than to rely on mathematical techniques in an attempt to obtain information which is not present in the ﬁrst place (Statham and Nashashibi, 1988). From this point of view, spectrum processing can be seen as any (mathematical) procedure that transforms the information content of a measured spectrum into a form that is more useful for our purposes (i.e., more accessible). As is indicated in Figure 1, most of the procedures that calculate this ‘‘useful’’ information require some form of additional input. Sometimes, this extra information is intuitive and not clearly deﬁned; in other cases, additional information is used in the form of mathematical model. In this respect, not the complexity of model but rather the ability to accurately describe the physical reality is of relevance. When we use a procedure to estimate the net peak area of a characteristic line by summing the appropriate channels and interpolating the continuum left and right of the peak, explicitly some ‘‘additional information’’ is given to this ‘‘spectrum processing’’ method in the form of peak and continuum boundaries. In addition (implicitly), a certain mathematical model is assumed regarding the shape of the peak and the continuum. Provided this model is correct, this relatively simple procedure returns a correct estimate of the net peak area as good as one could obtain by a complicated ﬁtting procedure. The important distinction between simple and more sophisticated spectrum evaluation procedures lies in the incorporated ﬂexibility and applicability. A simple area estimation based on the integration of peak and continuum regions is not generally applicable in real-world situations in which peak overlap, curvature in the continuum, and peak tailing occur. Procedures employing more complex models are adaptable to each speciﬁc situation, yielding reliable peak estimates. Spectrum evaluation procedures should

Figure 1 Spectrum evaluation seen as an information process: spectrum processing requires additional information to extract useful information from measured spectral data.

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therefore be compared on the basis of the explicit and implicit assumptions that are made in the model(s) they employ. C.

Components of an X-ray Spectrum

To evaluate an x-ray spectrum correctly, it is necessary to understand all the phenomena that contribute to the ﬁnal appearance of the spectrum. This includes the two main features, characteristic lines and continuum, and also a number of spectral artifacts, which become important especially in trace analysis work (Van Espen et al., 1980). 1. Characteristic Lines The characteristic radiation of a particular x-ray line has a Lorentz distribution. Peak proﬁles observed with a semiconductor detector are the convolution of this Lorentz distribution with nearly Gaussian detector response function, giving rise to what is known as Voigt proﬁle (Wilkinson, 1971). Because the Lorentz width is of the order of only 10 eV for elements with atomic number below 50, whereas the width of the detector response function is of the order of 160 eV, a Gauss function is an adequate approximation of the line proﬁle. Only for K lines of elements such as U and Th does the Lorentz contribution become signiﬁcant and need to be taken into account (Gunnink, 1977). A more close inspection of the peak shape reveals some distinct tailing at the low energy side of the peak and a shelf extending to zero energy. This is mainly due to incomplete charge collection caused by detector imperfections (dead layer and regions of low electric ﬁeld) as discussed in Chapter 3. The eﬀect is most pronounced for low energy x-rays. For photons above 15 keV Compton scatter in the detector also contributes to deviation from the Gaussian shape. The distortion caused by incomplete charge collection has been described theoretically (Joy, 1985; Heckel and Scholz, 1987). Various functions have been proposed to model the real peak shape more accurately (Campbell et al., 1987). The observed emission spectrum of an element is the result of many transitions, as explained in Chapter 1. The resulting x-ray lines, including possible satellite lines, need to be considered in the evaluation of an x-ray spectrum. A more detailed discussion on the representation of the K and L spectra and the peak shape is given in Sec. VII. 2. Continuum The continuum observed in x-ray spectra results from a variety of processes. The continuum in electron-induced x-ray spectra is almost completely due to the retardation of the primary electrons (bremsstrahlung). The intensity distribution of the continuum radiation emitted by the sample is in ﬁrst approximation given by Kramer’s formula (Chapter 1). Absorption in the detector windows and in the sample causes this continuous decreasing function to fall oﬀ at low energies, giving rise to the typical continuum shape observed. The attenuation of this bremsstrahlung by major elements in the sample also causes absorption edges to appear in these spectra. Continuum modeling for electron-induced x-ray spectra has been studied in detail by a number of authors (Statham, 1976a, 1976b, Smith et al., 1975; Sherry and Vander Sande, 1977; Bloomﬁeld and Love, 1985). For particle-induced x-ray emission, a similar continuum is observed also mainly due to secondary electron bremsstrahlung. Other (nuclear) processes contribute here, making it virtually impossible to derive a real physical model for the continuum. Special absorbers placed between the sample and detector further alter the shape of the continuum.

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In x-ray ﬂuorescence, the main source of the continuum is the coherent and incoherent scattering of the excitation radiation by the sample. The shape can therefore become very complex and depends both on the initial shape of the excitation spectrum and on the sample composition. When white radiation is used for the excitation, the continuum is mainly radiative and absorption edges can also be observed. With quasimonoenergetic excitation (secondary target, radioisotope), the incomplete charge collection of the intense coherently and incoherently scattered peaks is responsible for most of the continuum (see Chapter 3). Also here, realistic physical models for the description of the continuum are not used. The incomplete charge collection of intense ﬂuorescence lines in the spectrum further complicates the continuum. The cumulative eﬀect of the incomplete charge collection of all lines causes the apparent continuum at lower energies to be signiﬁcantly higher that expected on the basis of the primary continuum generating processes. 3. Escape Peaks Escape peaks result from the escape of SiK or GeK photons from the detector after photoelectric absorption of the impinging x-ray photon near the edge regions of the detector. The energy deposited in the detector by the incoming x-ray is diminished with the energy of the escaping SiK or GeK photon. Typical examples of the interference due to Si escape peaks are the interference of TiKa (4.51 keV) by the FeKa escape at 4.65 keV and the interference of FeKa by the CuKa escape. For a Si(Li) detector, the escape peaks is expected 1.742 keV (SiKa) below the parent peak. Experimentally, it is observed that the energy diﬀerence is slightly but signiﬁcantly higher, 1.750 keV (Van Espen et al., 1980). Ge escape peaks are observed 9.876 (GeKa) and 10.984 keV (GeKb) below the parent peak. The width of the escape peaks is smaller than the width of the parent peak and corresponds to the spectrometer resolution at the energy of the escape peak. The escape fraction f is deﬁned as the number of counts in the escape peak Ne divided by the number of detected counts (escape þ parent). Assuming normal incidence to the detector and escape only from the front surface, the following formula can be derived for the escape fraction (Reed and Ware, 1972): Ne 1 1 m m 1 K ln 1 þ I ¼ oK 1 f¼ ð4Þ r Np þ Ne 2 mI mK where mI and mK are the mass-attenuation coeﬃcient of silicon for the impinging and the SiK radiation, respectively, oK is the K ﬂuorescence yield, and r is the K jump ratio of silicon. Using 0.047 for the ﬂuorescence yield and 10.8 for the jump ratio, the calculated escape fraction is in very good agreement with the experimentally determined values for impinging photons up to 15 keV (Van Espen et al., 1980). Equation (4) is also applicable for estimating the escape fraction in Ge detectors, provided that the parameters related to Si are substituted with these of Ge. Knowing the energy, width, and intensity of the escape peak, corrections for its presence can be made in a straightforward manner. 4. Pileup and Sum Peaks With modern pulse processing electronics, the pileup eﬀects are suppressed to a large extend. With a pulse-pair resolution time of a few 100 ns or less, only true sum peaks are observed. These sum peaks are, within a few electron volts, located at their expected position, and they are slightly wider (5%) than normal peaks located at the same energy in the spectrum (Van Espen et al., 1980). The count rate of a sum peak is given by

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N_ 11 ¼ tN_ 1 N_ 2

ð5Þ

N_ 12 ¼ 2tN_ 1 N_ 1

ð6Þ

and

with N_ 11 the count rate (counts=s) in a sum peak due to the coincidence of two x-rays with the same energy, N_ 12 the count rate of a sum peak resulting from two x-rays with diﬀerent energy, and t the pulse-pair resolution time in seconds. Sum peaks are often found when a few large peaks at lower energy dominate the spectrum. Typical examples are PIXE spectra of biological and geological samples. The high count rate of the K and CaK lines produces sum peaks that are easily observed in the high-energy region of the spectrum where the continuum is low. It is important to note that the intensity of sum peaks is count-rate dependent, they can be reduced and virtually eliminated by performing the measurement with a lower primary beam intensity. A method for correcting for the contribution of sum peaks in least-squares ﬁtting has been proposed by Johansson (1982) and is discussed further in Sec. VII. 5. Discrete Character of a PulseHeight Spectrum Another aspect of spectral data that should be mentioned is that a pulseheight spectrum is a discrete histogram representing a continuous function. Digitization of this continuous function, especially the Gaussian peaks, into too few channels causes considerable systematic errors. If the peak contains less than 2.5 channels at the FWHM, the peak area estimate, for example, obtained by summing the channel contents is largely overestimated. This lower limit of 2.5 channels at FWHM corresponds to a spectrometer gain of 60 eV=channel for a peak width of 150 eV. In practice, 40 eV=channel or lower is recommended, otherwise peak position and width determinations and the results of spectrum ﬁtting become unreliable. 6. Other Artifacts A number of other features might appear in an x-ray spectrum and can cause problems during the spectrum evaluation. In the K x-ray spectra of elements with atomic number between 20 and 40, one can detect a peaklike structure with a rather poorly deﬁned maximum and a slowly declining tail (Van Espen et al., 1979a). This structure is due to the KLL radiative Auger transition, which is an alternative decay mode of the K vacancy. The maximum is observed at the energy of the KLL Auger electron transition energy. The intensity of the radiative Auger structure varies from approximately 1% of the Ka line for elements below Ca to 0.1% for elements above Zn. For chlorine and lower atomic number elements, the radiative Auger band overlaps with the Ka peak. In most analytical applications, this eﬀect will not cause serious problems. The structure can be considered as part of non-Gaussian peak tail. The scattering of the excitation radiation in x-ray ﬂuorescence is responsible for most of the continuum observed in the spectrum. When characteristic lines are present in the excitation spectrum, two peaks might be observed. The Rayleigh (coherently)-scattered peak has a position and width as expected for a normal ﬂuorescence line. The Compton (incoherently scattered) peak is shifted to lower energies according to the well-known Compton formula and is much broader than a normal characteristic line at that energy. This broader structure, resulting from scattering over a range of angles and Doppler eﬀect,

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is diﬃcult to model analytically (Van Dyck and Van Grieken, 1983). The structure is further complicated by multiple scattering phenomena in the sample (Vincze et al., 1999). Apart from these commonly encountered scattering processes, it is possible to detect xray Raman scattering (Van Espen et al., 1979b). Again, a bandlike structure is observed with an energy maximum given by the incident photon energy minus the electron-binding energy. The Raman eﬀect is most prominently present when exciting elements with atomic number Z 7 2 to Z þ 7 with the K radiation of element Z. In this case, Raman scattering occurs on L electrons. For x-ray excitation energies between 15 and 25 keV, the Raman scattering on the K electrons of Al to Cl can also be observed. Because of its high-energy edge, the eﬀect may appear as a peak in the spectrum with possible erroneous identiﬁcation as a ﬂuorescence line. The intensity of the Raman band increases as the incident photon energy comes closer to the binding energy of the electron. The observed intensity can amount to as much as 10% of the L ﬂuorescence intensity for the elements Rh to Cs when excitation with MoK x-rays is used. When the excitation source is highly collimated diﬀraction peaks can be observed in the x-ray ﬂuorescence spectrum of crystalline materials. It is often diﬃcult to deal with these diﬀraction patterns. They can interfere with the ﬂuorescence lines or even be misinterpreted as being ﬂuorescence lines, giving rise to false identiﬁcation of elements.

III.

SPECTRUM PROCESSING METHODS

Spectrum processing refers to mathematical techniques that alter the outlook of the spectral data. This is often done, using some digital ﬁlter, to reduce the noise, to locate peaks, or to suppress the continuum. In this section, various methods of ﬁltering are discussed. Because of its relation to the frequency domain, the concept of Fourier transformation is introduced ﬁrst. A.

Fourier Transformation, Convolution, and Deconvolution

One can think of an x-ray spectrum as consisting of a number of components with different frequencies. In the spectrum shown in Figure 2, one recognizes a nearly constant component (the continuum) as well as a component that ﬂuctuates from channel to channel ( ¼ fast). The latter is obviously the noise due to counting statistics. Peaks, then, must have frequency components intermediate between these two. The frequency characteristics of a spectrum can be studied in the Fourier space. For any discrete function f(x), x ¼ 0, . . . , n 7 1 (e.g., a pulseheight spectrum), the discrete Fourier transform is deﬁned as n1 1 X j2pux fðxÞ exp FðuÞ ¼ n x¼0 n n1 1 X ux ux fðxÞ cos 2p j sin 2p ð7Þ n x¼0 n n pﬃﬃﬃﬃﬃﬃﬃ with j ¼ 1 and u ¼ 0, . . . , n 7 1. F(u) is a complex number. The real part, R(u), and the imaginary part, I(u), represent respectively the amplitude of the cosine and the sine functions that are necessary to describe the original data. The square of F(u) is called the power spectrum,

¼

jFðuÞj2 ¼ R2 ðuÞ þ I2 ðuÞ

ð8Þ

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Figure 2 A 256-channel pulseheight spectrum (single Gaussian on a constant continuum) and the Fourier-filtered spectrum.

and gives an idea about the dominant frequencies in the spectrum. Because there are n diﬀerent nonzero real and imaginary coeﬃcients, no information is lost by the Fourier transform and the inverse transformation is always possible: n1 X j2pux fðxÞ ¼ FðuÞ exp ð9Þ n u¼0 Figure 3 shows that power spectrum of the pulseheight distribution in Figure 2 (a single Gaussian on a constant continuum). The frequency (inverse channel number) is deﬁned as u=n, with n ¼ 256 and u ¼ 0, . . . , n=2. The amplitude of the zero frequency jFð0Þj2 , which is equal to the average of the spectrum, is not shown. The dominating low frequencies originate from the continuum and from the Gaussian peak, whereas the higher frequencies are caused mainly by the counting statistics. It is clear that if we eliminate those high frequencies, we are reducing this noise. This can be done by multiplying the Fourier transform with a suitable function: GðuÞ ¼ FðuÞHðuÞ An example of such a function is a high-frequency cutoﬀ ﬁlter:

1; u ucrit HðuÞ ¼ 0; u > ucrit

ð10Þ

ð11Þ

which sets the real and imaginary coeﬃcients above a frequency ucrit to zero. If we apply this ﬁlter to the Fourier transform of Figure 3 using ucrit ¼ 0.05 and then apply the inverse Fourier transformation [Eq. (9)], the result as shown by the solid line in Figure 2 is obtained. The peak shape is preserved, but most of the statistical ﬂuctuations are eliminated.

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Figure 3

247

Fourier power spectrum of the pulseheight distribution shown in Figure 2.

If we would cut oﬀ at even lower frequencies, peak distortions at the top and at the base of the peak would become more pronounced. This Fourier ﬁltering can also be done directly in the original data space. Indeed, the convolution theorem says that multiplication in the Fourier space is equivalent to convolution in the original space: GðuÞ ¼ FðuÞHðuÞ , fðxÞ hðxÞ ¼ gðxÞ

ð12Þ

The convolute at data point x is deﬁned as the sum of the products of the original data and the ﬁlter centered around point x: X fðx x0 Þhðx0 Þ ð13Þ gðxÞ ¼ fðxÞ hðxÞ ¼ x0

h(x) is called a digital ﬁlter and is the inverse Fourier transformation of H(u). In general, this convolution or ﬁltering of a spectrum yi with some weighing function is expressed as yi ¼

j¼m 1 X hj yiþj N j¼m

ð14Þ

where hj are the convolution integers and N is a suitable normalization factor. The ﬁlter width is given by 2m þ 1. Fourier ﬁltering with the purpose to reduce or eliminate some (high or low) frequency components in the spectrum can thus be implemented as a convolution of the original data with a digital ﬁlter. This convolution also alters the variance of the original data. Applying the concept of error propagation, one ﬁnds that the variance of the convoluted data is given by

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s2y ¼ i

m 1 X h2 yiþj N2 j¼m i

ð15Þ

when the original data follows a Poisson distribution (s2y ¼ y). Because the measured spectrum y(x) is itself a convolution of the original (x-ray emission) signal f(x), with the instrument (or detector) response function h(x), it is, in principle, possible to restore the measured signal if this response function is know. This can be accomplished by dividing the Fourier transform (FT) of the measured spectrum by Fourier transform of the (nearly Gaussian) response function, followed by the inverse Fourier transform (IFT) of the resulting quotient: ) FT IFT yðxÞ!YðuÞ YðuÞ ¼ FðuÞ ! fðxÞ ð16Þ FT HðuÞ hðxÞ!HðuÞ The fact that the detector response function changes with energy (becomes broader) and, more importantly, the presence of noise prohibits the straightforward application of this Fourier deconvolution technique. Indeed, in the presence of noise, the measured signal must be presented by yðxÞ ¼ fðxÞ hðxÞ þ nðxÞ

ð17Þ

and its Fourier transform YðuÞ ¼ FðuÞHðuÞ þ NðuÞ

ð18Þ

YðuÞ NðuÞ ¼ FðuÞ þ HðuÞ HðuÞ

ð19Þ

or

At high frequencies, the response, H(u), goes to zero while N(u) is still signiﬁcant, so that the noise is emphasized in the inverse transformation. This clearly shows that the noise (counting statistics) is the ultimate limitation for any spectrum processing and analysis method. A clear introduction to Fourier transformations related to signal processing can be found in the work of Massart et al. (1998). Algorithms for Fourier transformation and related topics are given in the work of Press et al. (1998). Detailed discussions on Fourier deconvolution can be found in many textbooks (Jansson, 1984; Brook and Wynne, 1988). Fourier deconvolution in x-ray spectrometry based on maximum a posteriori or maximum entropy principles is discussed by several authors (Schwalbe and Trussell, 1981; Nunez et al., 1988; Gertner et al., 1989). Gertner implemented this method for the analysis of real x-ray spectra and compared the results with those obtained by simple peak ﬁtting. The problem that the deconvolution algorithms are limited to systems exhibiting translational invariance was overcome by a transformation of the spectrum, so that the resolution becomes independent of the energy. B.

Smoothing

pﬃﬃﬃ Because of the uncertainty y on each channel content yi , ﬁctitious maxima can occur both on the continuum and on the slope of the characteristic peaks. Removal or suppression of these ﬂuctuations can be useful during the visual inspection of spectra (e.g., for locating small peaks on a noisy continuum) and is used in most automatic peak search and

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continuum estimation procedures. Although smoothing can be useful in qualitative analysis, its use is not recommended as part of any quantitative spectrum evaluation. Smoothing, although reducing the uncertainty in the data locally, redistributes the original channel content over the neighboring channel, thus introducing distortion in the spectrum. Accordingly, smoothing can provide a (small) improvement in the statistical precision obtainable with simple peak integration but is of no advantage when used with leastsquares-ﬁtting procedures in which assumptions about the peak shapes are made. 1. Moving Average The most straightforward way of smoothing (any) ﬂuctuating signal is to employ the ‘‘box-car’’ or moving-average technique. Starting from a measured spectrum y, a smoothed spectrum y* is obtained by calculating the mean channel content around each channel i: yi ¼ yi ¼

þm X 1 yiþj 2m þ 1 j¼m

ð20Þ

This can be seen as a convolution [Eq. (14)] with all coeﬃcients hj ¼ 1. The smoothing eﬀect obviously depends on the width of the ﬁlter, 2m þ 1. The operation being a simple pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ averaging, the standard deviation of the smoothed data is reduced by a factor 2m þ 1 in regions where yi is nearly constant. On the other hand, such a ﬁlter introduces a considerable amount of peak distortion. This distortion depends on the ﬁlter width-to-peak width ratio. Figure 4 shows the peak distortion eﬀects when a moving-average ﬁlter of widths 9, 17, and 25 is applied to a peak with P full width at half-maximum (FWHM) equal to nine channels. Being a unit area ﬁlter hj =N ¼ 1 with N ¼ 2m þ 1, the total counts in the peak is not aﬀected in an appreciable way other than by rounding errors.

Figure 4 Distortion introduced by smoothing of a peak with a moving-average filter 9, 17, and 25 points wide. The FWHM of the original peak is nine channels.

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Figure 5 shows the eﬀect on the peak height and width when applying this type of ﬁlter with diﬀerent sizes. The peak distortion is caused by the fact that in the calculation of yi , the content of all neighboring channels is used with equal weight. Consequently, by employing a nonuniform ﬁlter h, which places more weight on the central channels and less on the channels near the edge of the ﬁlter, smoothing can be achieved with less broadening eﬀects. 2. Savitsky and Golay Polynomial Filters Another way of dealing with statistical ﬂuctuations in experimental data is by drawing a best-ﬁtting curve through the data points. This idea resulted in the development by Savitsky and Golay (1964) of a general type of smoothing ﬁlters with very interesting features. The method is based on the fact that nearly all experimental data can be modeled by a polynomial of order r, a0 þ a1 x þ a2 x2 þ þ ar xr , when the data are conﬁned to a suﬃciently small interval. If we consider a number of data points around a central channel io , such as yi0 2 ; yi0 1 ; yi0 ; yi0 þ1 ; yi0 þ2 , a least-squares ﬁt with the function. yðiÞ ¼ a0 þ a1 ði i0 Þ þ a2 ði i0 Þ2

ð21Þ

can be made. Once we have determined the coeﬃcients aj , the value of the polynomial at the central channel i0 can be used as the smoothed value: yi ¼ yði0 Þ ¼ a0

ð22Þ

This concept is schematically illustrated in Figure 6. By moving the central channel to the right (from i0 to i0 þ 1), the next smoothed channel content can be calculated by repeating the entire procedure.

Figure 5 Percent change in peak height and width introduced by filtering with a moving-average and a Savitsky and Golay polynomial filter as a function of the filter width-to-peak FWHM ratio, (2m þ 1)/FWHM.

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Figure 6 Concept of polynomial smoothing. A parabola is fitted trough the points i0 3 to i0 þ3. The value of the parabola at i0 is the smoothed value ( ).

At ﬁrst sight, this smoothing method would require a least-squares ﬁt for each channel in the spectrum. However, the fact that the x values are equidistant allows us to formulate the problem in such a way that the polynomial coeﬃcients can be expressed as simple linear combinations involving only yi values: j¼m 1 X ak ¼ Ck; j yiþk ð23Þ Nk j¼m This means that it is possible to implement the least-squares-ﬁtting procedure more eﬃciently as a convolution of the spectrum with a ﬁlter having appropriate weights. For this second-order polynomial, the coeﬃcients are given by C0; j 3ð3m2 þ 3m 1 5j2 Þ ¼ ð2m 1Þð2m þ 1Þð2m þ 3Þ N0

ð24Þ

Smoothing with a ﬁve-point second-degree polynomial ð2m þ 1 ¼ 5Þ thus becomes 1 yi ¼ a0 ¼ ð3yi2 þ 12yi1 þ 17yi þ 12yiþ1 3yiþ2 Þ ð25Þ 35 In general, for a polynomial of degree r ﬁtted to 2m þ 1 points, this can be written as yi ¼

j¼m 1 X Crm; j yiþj Nrm j¼m

ð26Þ

where the convolution integers Crm;j and the normalization factors Nrm do not depend on the data to be smoothed but are a function only of the polynomial degree r and on the ﬁlter

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half-width m. Table 1 lists the coeﬃcients of polynomial smoothing ﬁlters with widths between 5 and 25 points. The coeﬃcients for a second- and a third-degree polynomial are identical. In comparison with the moving-average ﬁlter of the same width, polynomial smoothing ﬁlters are less eﬀective in removing noise but have the advantage of causing less peak distortion. The distortion eﬀect as a function of the ﬁlter width-to-peak width ratio is given in Figure 5. When the ﬁlter becomes much wider than the peak, the smoothed spectrum features oscillations near the peak boundaries, as illustrated in Figure 7. An interesting feature of this type of ﬁlters is that they can produce not only a smoothed spectrum but also a smoothed ﬁrst and second derivative of the spectrum. If we diﬀerentiate Eq. (21) and take the value at the center position, dyðiÞ 0 y ¼ ¼ a1 ð27Þ di i¼i0 d 2 yðiÞ ¼ 2a2 ð28Þ y00 ¼ di2 i¼i0 it follows from Eq. (23) that the smoothed ﬁrst and second derivative of the spectrum can also be calculated by means of suitable convolution coeﬃcients. For instance, for the ﬁrst derivative of a second-order polynomial, using ﬁve data points this becomes y0 i ¼

1 ð2yi2 yi1 þ yiþ1 þ 2yiþ2 Þ 10

ð29Þ

The corresponding convolution integers for the calculation of the smoothed ﬁrst and second derivative are listed in Tables 24. The use of the derivative spectra is illustrated in next section dealing with peak search methods. The FORTRAN implementation of the Savitsky and Golay ﬁlters is given in Sec. X. Variations on these smoothing strategies, such as the use of variable-width ﬁlters are reviewed by Yule (1967). The eﬀect of repeated smoothing on the accuracy and precision of peak area determination is discussed by Nielson (1978). A more comprehensive treatment on polynomial smoothing can be found in Enke and Nieman (1976) and its references. Table 1 Savitsky and Golay Coefﬁcients for Second- and Third-Degree (r ¼ 2 and r ¼ 3) Pj¼m Crm; j yiþj , Filter Width ¼ 2m þ 1 Polynomial Smoothing Filter; yi ¼ ð1=Nrm Þ j¼m j ðCrm; j ¼ Crm;j Þ m

Nrm

0

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6 7 8 9 10 11 12

35 21 231 429 143 1105 323 2261 3059 805 5175

17 7 59 89 25 167 43 269 329 79 467

12 6 54 84 24 162 42 264 324 78 462

3 3 39 69 21 147 39 249 309 75 447

2 14 44 16 122 34 224 284 70 422

21 9 9 87 27 189 249 63 387

36 0 42 18 144 204 54 342

11 13 7 89 149 43 287

78 6 24 84 30 222

21 51 9 15 147

136 76 2 62

171 21 33

42 138

253

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Figure 7 Effect of the smoothing of a peak with a Savitsky and Golay polynomial filter. The FWHM of the original peak is nine channels.

3. Low Statistics Digital Filter Most smoothing techniques originate from signal processing and were initially introduced in the ﬁeld of g-ray spectroscopy. Also in the PIXE community, considerable attention has been devoted to a number of aspects of spectrum processing and evaluation. Within the framework of continuum estimation (see later), a smoothing algorithm was developed that removes noise from a spectrum on a selective basis (Ryan et al., 1988). The method provides an n-point means smoothing in regions of low statistics (few counts) while avoiding spreading of the base of peaks and degradation of valleys between peaks. Table 2 Savitsky and P Golay Coefﬁcients for Second-Degree (r ¼ 2) Polynomial, First-Derivative Filter; yi ¼ ð1=Nrm Þ j¼m j¼m Crm; j yiþj , Filter Width ¼ 2m þ 1 j ðCrm; j ¼ Crm;j Þ m

Nrm

0

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6 7 8 9 10 11 12

10 28 60 110 182 280 408 570 770 1012 1300

0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1

2 2 2 2 2 2 2 2 2 2 2

3 3 3 3 3 3 3 3 3 3

4 4 4 4 4 4 4 4 4

5 5 5 5 5 5 5 5

6 6 6 6 6 6 6

7 7 7 7 7 7

8 8 8 8 8

9 9 9 9

10 10 10

11 11

12

Nrm

12 252 1,188 5,148 24,024 334,152 23,256 255,816 3,634,092 197,340 1,776,060

m

2 3 4 5 6 7 8 9 10 11 12

0 0 0 0 0 0 0 0 0 0 0

0

8 58 126 296 832 7,506 358 2816 29,592 1,222 8,558

1

3 22 142 532 1,796 17,842 902 7,372 79,564 3,350 23,806

2

1 67 193 503 1,489 13,843 673 5,363 56,881 2,365 16,649 86 294 1,578 18,334 1,002 8,574 95,338 4,098 29,562

4

300 660 14,150 930 8,700 101,900 4,530 33,450

5

1,133 4,121 643 7,481 96,947 4,567 35,003

6

12,922 98 4,648 78,176 4,130 33,754

7

j ðCrm; j ¼ Crm;j Þ

748 68 43,284 3,140 29,236

8

6,936 10,032 1,518 20,982

9

3,938 8,602

30,866

12

Crm; j yiþj , Filter

11

j¼m

Pj¼m

84,075 815 8,525

10

Table 3 Savitsky and Golay Coefﬁcients for Third-Degree (r ¼ 3) Polynomial, First-Derivative Filter; yi ¼ ð1=Nrm Þ Width ¼ 2m þ 1

254 Van Espen

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255

Table 4 Savitsky and Golay Coefﬁcients for P Second- and Third-Degree (r ¼ 2 and r ¼ 3) j¼m Polynomial Second-Derivative Filter; yi ¼ ð1=Nrm Þ j¼m Crm; j yiþj , Filter Width ¼ 2m þ 1 j ðCrm; j ¼ Crm;j Þ m

Nrm

0

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6 7 8 9 10 11 12

7 42 462 429 1,001 6,188 3,876 6,783 33,649 17,710 26,910

2 4 20 10 14 56 24 30 110 44 52

1 3 17 9 13 53 23 29 107 43 51

2 0 8 6 10 44 20 26 98 40 48

5 7 1 5 29 15 21 83 35 43

28 6 2 8 8 14 62 28 36

15 11 19 1 5 35 19 27

22 52 12 6 2 8 16

91 25 19 37 5 3

40 34 82 20 12

51 133 37 29

190 56 48

77 69

92

The rather heuristic algorithm is discussed in Section X, together with the computer implementation. The eﬀect of the ﬁlter is illustrated in Figure 8, where it is compared with the other smoothing methods discussed.

C.

Peak Search Methods

Several methods have been developed for the automatic localization of peaks in a spectrum. Nearly all methods follow a strategy where the original spectrum is transformed into a form that emphasizes the peaklike structures and reduces the continuum, followed by a decision whether these peaklike structures are statistically signiﬁcant. The latter involves some adjustable parameter(s) controlling the sensitivity of the peak search. Although visual inspection of the spectrum still appears to be the best method, peak search algorithms, which are heavily used in g-ray spectrometry, may have some value in x-ray spectrometry. Their use in energy-dispersive x-ray analysis as part of an automated qualitative analysis procedure is hampered by the extreme peak overlap in these spectra. More elaborated procedures involving artiﬁcial intelligence techniques have been considered for this (Janssens and Van Espen, 1986; Janssens et al., 1988). Peak search procedures usually involve three steps: (1) transformation of the original spectrum so that continuum contribution is eliminated, peaks are readily locatable, and overlapping peaks are (partially) resolved; (2) signiﬁcance test and approximate location of the peak maximum; and (3) more accurate peak position estimate in the original spectrum. The various peak search algorithms mainly diﬀer in the choice of the transformation. Some methods use the ﬁrst and second smoothed derivative of the spectrum. This method is illustrated in Figure 9. The sign change (crossing of the x axis) of the ﬁrst derivative and the minimum of the second derivative are quire suitable to detect the peaks in the original spectrum. Other methods employ some form of correlation technique, which is basically the convolution of the original spectrum with a ﬁlter that approximates the shape of the peak and, therefore, emphasizes the peak. If a zero-area correlator (ﬁlter) is used, the continuum is at the same time eﬀectively suppressed. The simplest and most eﬀective correlators belong to the group of zero-area rectangular ﬁlters. These ﬁlters have a central window with

256

Van Espen

Figure 8 Effect of various smoothing methods applied to part of a x-ray spectrum: (A) original spectrum, (B) nine-point Savitsky and Golay filter, (C) nine-point moving average, (D) low-noise statistical filter.

constant and positive coeﬃcients and two side lobes with constant and negative coeﬃcients. Convoluting an x-ray spectrum with this kind of ﬁlter yields spectra in which the continuum is removed and peaks are easily locatable. They are similar to inverted second-derivative spectra. An important representative of this group of ﬁlters is the ‘‘top-hat’’ ﬁlter, which has a central window with an odd number of channels w and two side windows each n channels wide. The value of the ﬁlter coeﬃcients follows from the zero-area constraint: 8 1 < 2n ; n w2 k < w2 1 ð30Þ hk ¼ ; w2 k þ w2 : w1 2n ; þ w2 < k w2 þ n The ﬁltered spectrum is obtained by the convolution of the original spectrum with this ﬁlter: yi ¼

k¼nþw=2 X

hk yiþk

ð31Þ

k¼nw=2

The eﬀect of this ﬁlter on a typical spectrum is shown in Figure 10. The variance of the ﬁltered spectrum is obtained by simple error propagation: s2y ¼

k¼nþw=2 X

i

k¼nw=2

h2k yiþk

ð32Þ

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Figure 9 Peak search using first and second derivatives: (A) doublet, peak width (s) four channels, separation eight channels, on a constant continuum. (B) First and (C) second smoothed derivative using a five-point Savitsky and Golay filter.

If yi is signiﬁcantly diﬀerent from zero, a peak structure is found and the top of the peak can approximately be located by searching for the maximum. Thus, i corresponds to the position of a peak maximum in the original spectrum if yi > rsyi

ð33Þ

258

Van Espen

Figure 10 Result of applying a top-hat filter. Dotted line: typical X-ray spectrum; solid line: filtered spectrum.

and yi1 yi > yiþ1

ð34Þ

In Figure 11, the positive part of the ﬁltered spectrum (w ¼ 9 and n ¼ 5) and the decision line ðrsy Þ for r ¼ 1 and 4 are displayed. If required, other peak features can be obtained from the ﬁltered spectrum: The distance between the two local minima is a measure of the width of the peak and the height at the maximum is related to the net peak area. Because the width and heights of the peaks in the ﬁltered spectrum strongly depend on the dimensions of the ﬁlter, it is important that its dimensions are matched to the width of the peaks in the original spectrum. From considerations of peak detectability (signal-tonoise ratio) and resolution (peak broadening), it follows that the optimum width of the positive window w is equal to the FWHM of the peaks (Robertson et al., 1972). The width of the negative side windows should be chosen as large as the curvature of the continuum allows. A reasonable compromise between sensitivity to peak shapes and rejection of continuum is reached when n equals FWHM=2 to FWHM=3. Typical values for the sensitivity factor r are between 2 and 4. Higher values result in the loss of small peaks; lower values will cause continuum ﬂuctuations to be interpreted as peaks. Other zero-area rectangular ﬁlters, variations to the top-hat ﬁlter, are also in use, such as the square-wave ﬁlter with typical coeﬃcient sequence 1, 1, 2, 2, 1, 1 (Philips and Marlow 1976; McCullagh, 1982) and the symmetric square-wave ﬁlter with coeﬃcients 1, 1, 1, 1 (Op De Beeck and Hoste 1975). A detailed account of the performance of this ﬁlter is given by Op De Beeck and Hoste (1975). A method using a Gaussian correlator function is discussed by Black (1969).

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Figure 11 Peak search using the positive part of the top-hat filtered spectrum and the decision level for one and four times the standard deviation. The original spectrum is shown at the bottom.

Once the peak top is approximately located in the ﬁltered spectrum, a more precise maximum can be found by ﬁtting a parabola over a few channels around the peak. For a well-deﬁned peak on a low continuum (or after continuum subtraction), the channel content near the top of the peak can be approximated by a Gaussian: "

ðx mÞ2 yi h exp 2s2

# ð35Þ

The logarithm of the data then is a simple polynomial ln yi ¼

m2 ln h 2 2s

þ

m m2 2 x x s2 2s2

ð36Þ

If we ﬁt ln yi with a polynomial a0 þ a1 x þ a2 x2 , where x represents the channel number, the position of the peak m is obtained from m¼

a1 2a2

ð37Þ

260

Van Espen

with an accuracy of 0.1 channel or better if the continuum contribution is small and if the peak is interference free [i.e., if Eq. (35) accurately describes the data]. An estimate of the peak’s width and height is obtained at the same time: rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 FWHM ¼ 2 2 ln 2s ¼ 2:3548 ð38Þ 2a2 a2 h ¼ exp a0 þ 1 ð39Þ 4a2 To obtain a reliable estimate of the parameters, only the channels in the FWHM or, at most, within the FWTM region of the peak must be included in the ﬁt. As a somewhat simpler and faster alternative, one can ﬁnd an estimate of the peak maximum by ﬁtting the parabola over the three top channels of the peak. If i is the peak maximum found in the ﬁltered spectrum, a better estimate of the maximum in the original spectrum is found by m¼iþ

1 yi1 yiþ1 2 yi1 þ yiþ1 2yi

ð40Þ

This method might be preferred for small peaks when the continuum cannot be disregarded. A FORTRAN implementation of a peak search algorithm is given in Section X.

IV.

CONTINUUM ESTIMATION METHODS

Except for some special quantiﬁcation procedures (e.g., the peak-to-background method in electron microscopy), the relevant analytical information is found in the net peak areas and continuum is considered a nuisance. There are, in principle, three ways to deal with the continuum: (1) the continuum can be suppressed or eliminated by a suitable ﬁlter; (2) the continuum can be estimated and subtracted from the spectrum prior to the estimation of the net peak areas; and (3) the continuum can be estimated simultaneously with the other features in the spectrum. The ﬁrst approach is discussed in Section VI, where the continuum is removed from spectra by applying a top-hat ﬁlter followed by linear leastsquares ﬁt of the spectrum with a number of (also ﬁltered) reference spectra. Least-squares ﬁt (linear or nonlinear) with analytical functions (Sec. VII) allows the simultaneous estimation of continuum and peaks, providing a suitable mathematical function can be found for the continuum. In this section, we discuss a number of procedures that aim to estimate the continuum independently of the other features in the spectrum. Once estimated, this continuum can be subtracted from the original spectrum and all methods for further processing, ranging from simple peak integration to least-squares ﬁtting can be applied. Any continuum estimation procedure must fulﬁll two important requirements. First, the method must be able to reliably estimate the continuum in all kinds of situations (e.g., small isolated peaks on a high continuum as well as in the proximity of a matrix line). Second, to permit processing of a large number of spectra, the method needs to be nearly free of user-adjustable parameters. Although a number of useful continuum estimation procedures has been developed, it must be realized that their accuracy in estimating the continuum is not optimal. In one way or another, they rely on the diﬀerence in frequency response of the continuum compared to other structures such as peaks, the former mainly consisting of low

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261

frequencies (slowly varying). Because the peaks also exhibit low frequencies at the peak boundaries, it is diﬃcult to control the method in such a way that it correctly discriminates between peaks and continuum. This results in either a small underestimation or overestimation of the continuum, introducing potentially large relative errors for small peaks. In this respect, the ﬁtting of the continuum with analytical functions may provide more optimal results (Vekemans et al., 1994). A considerable advantage of the methods discussed here is that they do not assume an explicit mathematical model of the continuum. Constructing a detailed and accurate analytical model for the continuum based in physical theory is nearly impossible except for some simple geometry and particular excitation conditions. Most often, some polynomial type of function must be chosen when ﬁtting a portion of the spectrum with analytical functions. A.

Peak Stripping

These methods are based on the removal of rapidly varying structures in a spectrum by comparing the channel content yi with the channel content of its neighbors. Clayton et al. (1987) proposed a method which compares the content of channel i with the mean of its two direct neighbors: mi ¼

yi1 þ yiþ1 2

ð41Þ

If yi is smaller than mi, the content of channel i is replaced by the mean mi. If this transformation is executed once of all channels, one can observe a slight reduction in the height of the peaks while the rest of the spectrum remains virtually unchanged. By repeating this procedure, the peaks are gradually ‘‘stripped’’ from the spectrum. Because the method tends to connect local minima, it is very sensitive to local ﬂuctuations in the continuum due to counting statistics. This makes smoothing of the spectrum, as discussed in the previous section, prior to the stripping process mandatory. Depending on the width of the peaks after typically 1000 cycles, the stripping converges and a more or less smooth continuum remains. To reduce the number of iterations, it might be advantageous to perform a log or pﬃﬃﬃﬃ square root transformation to the data prior to the stripping: y0i ¼ logðyi þ 1Þ or y0i ¼ yi . After the stripping, the continuum shape is obtained by applying the inverse transformation. A major disadvantage of this method is that after a number of cycles, the bases of partially overlapping peaks are transformed into broad ‘‘humps,’’ which take much longer to remove than isolated peaks. The method was originally applied to PIXE spectra but proves to be generally applicable for pulseheight spectra. In Figure 12, this method is applied to estimate the continuum of an x-ray spectrum in the region between 1.6 and 13.0 keV. The spectrum results from a 200-mg=cm2 pellet of NIST SRM Bovine Liver sample excited with the white spectrum of an Rh-anode x-ray tube ﬁltered through a thin Rh ﬁlter (Tracor Spectrace 5000). Because of the white tube spectrum, a considerable continuum intensity was observed, increasing quite steeply in the region above 10 keV. To calculate the continuum, the following algorithm was used: (1) the square root of the original spectrum was taken; (2) these data were smoothed with a 10-point Savitsky and Golay ﬁlter; (3) a number of iterations were performed applying Eq. (41) over the region of interest; (4) the square of each data point was taken (back transformation) to obtain the ﬁnal continuum shape. In Figure 12, the continuum after 10, 100, and, ﬁnally, 500 iterations is shown. As a generalization of the above-discussed method, the average of two channels a distance w away from i can be used:

262

Van Espen

Figure 12 Continuum estimate after 10, 100, and 500 iterations obtained with simple iterative peak stripping.

mi ¼

yiw þ yiþw 2

ð42Þ

Ryan et al. (1988) proposed using twice the FWHM of the spectrometer at channel i as the value for w. They reported that only 24 passes were required to produce acceptable continuum shapes pﬃﬃﬃ in PIXE spectra. During the last eight cycles, w is progressively reduced by the factor 2 to obtain a smooth continuum. To compress the dynamic range of the spectrum, a double-log transformation of the spectrum, log[log( yi+1)+1], before the actual stripping was proposed. In combination with the low statistics digital ﬁlter, this procedure is called the SNIP algorithm (Statistical Nonlinear Iterative Peak clipping). A variant of this procedure is implemented in the procedure ‘‘SNIPBG’’ given in Sec. X. Instead of the double logarithmic, we employed a square root transformation, and a Savitsky and Golay smoothing is performed on the square root data. The width w is kept constant over the entire spectrum. The value of w is also used as the width of the smoothing ﬁlter. Using this implementation, the continuum of the above-discussed spectrum is calculated and represented in Figure 13. The width was set 11 to channels approximately corresponding to the FWHM of the peaks in the center of the spectrum and 24 iterations were done. Apart from delivering a smoother continuum with smaller ‘‘humps,’’ this method executes much faster than the original method proposed by Clayton. B.

Continuum Estimation Using Orthogonal Polynomials

Another interesting continuum estimation procedure was introduced by Steenstrup (1981), who applied the method to energy-dispersive x-ray diﬀraction spectra. The spectrum is ﬁtted using orthogonal polynomials, and the weights of the least-squares ﬁt are iteratively adjusted so that only channels belonging to the continuum are included in the ﬁt.

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Figure13 Continuum estimate using statistical nonlinear iterative peak clipping (SNIP algorithm) with 24 iterations.

The method is generally applicable to pulseheight spectra and can be implemented as an algorithm that needs little or no control parameters. The continuum is described by a set of polynomials up to degree m: m X yðiÞ ¼ cj Pj ðxi Þ ð43Þ j¼0

where Pj(xi) is an orthogonal polynomial of degree j. As an example of m ¼ 3, the function becomes yðiÞ ¼ c0 þ c1 ðxi a0 Þ þ c2 ððx1 a1 Þðxi a0 Þ b1 Þ

ð44Þ

The least-squares estimates of the parameters cj are given by cj ¼

n X wi yi Pj ðxi Þ gj i¼1

ð45Þ

where wi are the weights of the ﬁt. A detailed discussion on orthogonal polynomial ﬁtting can be found in Sec. IX. Because the polynomial terms Pj ðxi Þ are orthogonal, no matrix inversion is required to obtain the results. Orthogonal polynomials of a much higher degree can be ﬁtted to the experimental data without running into problems with illconditioned normal equations and oscillating terms. The goal of this continuum estimation method is to ﬁt the continuum with the above-described orthogonal polynomial of degree m and to interpolate under the peaks. This can be achieved by careful manual selection of only those data pairs ðxi ; yi Þ that belong to the continuum. The more elegant approach proposed by Steenstrup consisted of using all channels and automatically adjusting the weights in such a way that the continuum contributions are emphasized. If yðiÞ is a polynomial approximation of the continuum of degree m, then yi yðiÞ if i is a continuum channel, otherwise yi > yðiÞ.

264

Van Espen

A better approximation to the continuum can then be found by choosing small weights for the data points where yi > yðiÞ and repeating the ﬁt. The following weighting scheme is proposed by Steenstrup: pﬃﬃﬃﬃﬃﬃﬃﬃ 1 if yi yðiÞ þ r yðiÞ wi ¼ ð46Þ yðiÞ pﬃﬃﬃﬃﬃﬃﬃﬃ 1 wi ¼ if y > yðiÞ þ r yðiÞ ð47Þ i ½ yðiÞ yi 2 where r is an adjustable parameter. Typically, r ¼ 2 is used. If yi p is ﬃﬃﬃﬃﬃﬃﬃ normally distributed ﬃ (which is approximately the case for yi > 30 counts), yi yðiÞ þ 2 yðiÞ holds for 97.7% of the channels containing only continuum. A too high value of r will cause the inclusion of the tails of the peaks into the ﬁt. With the new weights, a new polynomial ﬁt of degree m is performed. The process is stopped when the new polynomial coeﬃcients cj are within one standard deviation from the previous ones. We have also obtained good results by setting the weights eﬀectively to zero when the channel content is statistically above the ﬁtted continuum [Eq. (47)]. The method can be made even more unsupervised by including a procedure to automatically select the best degree of the polynomial. This can be done by ﬁtting (as described earlier) successive polynomials with higher degrees and testing the signiﬁcance of each additional polynomial coeﬃcient. If the coeﬃcient Cmþ1 is statistically not signiﬁcant diﬀerent from zero, jCmþ1 j < 2sCmþ1

ð48Þ

a polynomial of degree m is retained. Figure 14 shows the spectrum of bovine liver sample (same as Fig. 12) and the ﬁtted continuum using a fourth-, ﬁfth-, and sixth-degree orthogonal polynomial. A value of 1.5 was chosen for r [Eqs. (46) and (47)] and the weights were set to zero for the channels not belonging to the continuum; wi ¼ 0 in Eq. (47). Table 5 gives the data on the ﬁts. The ﬁnal number of continuum channels retained, the number of weight adjustments performed, and the value and standard deviation of the highest polynomial coeﬃcient are shown. The total number of channels in the ﬁtting region was 571. Because the value of the seventh degree polynomial coeﬃcient is insigniﬁcant [Eq. (48)], it is concluded that a sixth-degree orthogonal polynomial is needed to model the continuum. Section X lists the FORTRAN code that implements the complete procedure to estimate the continuum of a given spectrum by an mth-degree orthogonal polynomial. V.

SIMPLE NET PEAK AREA DETERMINATION

In both in WDXRF and EDXRF, the concentration of the analyte is proportional to the number of counts under the characteristic x-ray peak corrected for the continuum. At constant resolution, this proportionality also exits for the net peak height. In EDXRF, preference is given to the peak area. In WDXRF, the acquisition of the entire peak proﬁle is very time-consuming and the count rate is usually measured only at the peak maximum. A.

Peak-Area Determination in EDXRF

The most straightforward method to obtain the net area of an isolated peak in a spectrum consists of interpolating the continuum under the peak and summing the continuumcorrected channel contents in a window over the peak, as shown in Figure 15.

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Figure 14 Continuum estimate obtained by fitting a fourth-, fifth-, and sixth-degree orthogonal polynomial.

The net peak area, Np , of an isolated peak on a continuum is given by NP ¼

iP1 X

½ yi yB ðiÞ ¼

iP2

X

yi

i

X

yB ðiÞ ¼ NT NB

ð49Þ

i

where NT and NB are the total number of counts of the spectrum and the continuum in the integration window iP1 iP2 . The uncertainty of the estimated net peak area due to counting statistics is pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sNP ¼ NT þ NB ð50Þ The continuum yB ðiÞ is calculated by interpolation, assuming a straight line (see Fig. 15): yB ðiÞ ¼ YBL þ ðYBR YBL Þ

i iBL iBR iBL

ð51Þ

Table 5 Data on the Fit of the Continuum of the Spectrum Shown in Figure 14 Using Orthogonal Polynomials Degree of polynomial 4 5 6 7

No. of channels used

No. of weight adjustments

Highest degree coeﬃcient and standard deviation

126 199 267 266

7 16 6 6

(3.98±0.20)106 (1.88±0.09)107 (7.66±0.28)1010 (3.66±2.03)1013

266

Van Espen

Figure 15

Determination of the net peak area by interpolating the continuum.

YBL and YBR are the values of the continuum at the channels iBL and iBR , left and right from the peaks, respectively. These values are best estimated by averaging over a number of channels: YBL ¼ YBR ¼

iBL1 1 X NBL yi ¼ nBL i¼iBL2 nBL

1

iBR1 X

nBR

i¼iBR2

yi ¼

NBR nBR

ð52Þ ð53Þ

The number of channels in the continuum windows are nBL ¼ iBL2 iBL1 þ 1 and nBR ¼ iBR2 iBR1 þ 1. The center position of the continuum windows (not necessarily an integer number!) used in Eq. (51) are iBL ¼ ðiBL1 þ iBL2 Þ=2 and iBR ¼ ðiBR1 þ iBR2 Þ=2. If both continuum windows have equal width, nBL ¼ nBR ¼ nB =2, and are positioned symmetrically with respect to the peak window ðiP iBL ¼ iBR iP Þ a much simpler expression is obtained for the net peak area: nP ð54Þ NP ¼ NT ðNBL þ NBR Þ nB where NBL and NBR are the total counts in the left and right continuum windows, respectively, each nB =2 channels wide, and nP equals the number of channels in the peak window. Applying the principle of error propagation, the uncertainty in the net peak area is then given by sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 2 nP ðNBL þ NBR Þ ð55Þ sNP ¼ N T þ nB

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From this equation, it can be seen that, in principle, the continuum should be estimated using as many channels as possible ðnB Þ to minimize random errors due to counting statistics. In practice, the width will be limited by curvature of the continuum and by the presence of other peaks. Most often, the total width of the continuum ðnB ¼ nBL þ nBR Þ is taken equal to or slightly larger than the width of the peak window. The optimum width of the peak window to minimize counting statistics depends on the peak-to-continuum ratio (Jenkins et al., 1981). For low peak-to-continuum ratios, the peak height being only a fraction of the continuum height, the optimum width of the integration window is 1.17 times the FWHM of the peak. If the ratio of the peak height to the continuum height is larger than 1, a slightly wider window is optimal, although, in practice, the improvement in precision is negligible. The method does not deliver the total net peak area, only a fraction of it. Integrating over 1.176FWHM (from 1:378s to þ1:378s) covers only 83% of the peak. To cover 99% of the peak, the window should be 2.196FWHM (±2.579s). If position or width changes are observed from one spectrum to the other, care should be taken to deﬁne the windows in such a way that cover the same part of the spectrum. The wider the peak window, the less sensitive the method for peak shift. Although such a peak-area-determination method seems naively simple, if correctly used it provides results that are as accurate and precise as the most sophisticated procedures. The premises for use are that the peak window should be known to be free from interferences, that there should be no peaks in the continuum windows, and that the continuum should be linear over the extent of the windows. A peak search procedure can, in principle, be used to setup the windows automatically. However, its practical use is limited by complexity of the EDXRF spectra. Moreover, the use of such automated procedures is hazardous because no measure can be given for the presence or absence of systematic errors. Because of this restrictions, a simple peak integration method cannot be used as a general tool for spectrum processing. In a limited number of applications, good results can be obtained. An evaluation of various peak integration methods is given by Hertogen et al. (1973). B.

Net Count Rate Determination in WDXRF

In WDXRF, the count rate at the 2Y angle of the peak maximum corrected for the continuum is used as analytical signal. The continuum is estimated at a 2Y position on the left or right side of the peak. Continuum interpolation as described in the previous section is also possible. If NP is the number of counts accumulated during a time interval tP at the top of the peak and NB is the number of the continuum counts accumulated during time tB , then the net count rate, I, is given by I ¼ IP IB ¼

NP NB tP tB

and the uncertainty in the net count rate due to counting statistics is given by sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ NP NB sI ¼ þ 2 t2P tB

ð56Þ

ð57Þ

Various counting strategies can be considered and the eﬀect on the precision can be estimated using Eq. (57) (Bertin, 1970). In a ‘‘optimum ﬁxed time’’ strategy, the minimum

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uncertainty is obtained when, for a total measurement time t ¼ tP þ tB ; tP and tB are chosen in such a way that their ratio is equal to the square root of the peak-to-continuum ratio: rﬃﬃﬃﬃﬃ tP IP ð58Þ ¼ tB IB Under, these conditions, the uncertainty in the net intensity is given by pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ IP þ IB sI ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ tP þ tB

VI.

ð59Þ

LEAST-SQUARES FITTING USING REFERENCE SPECTRA

In this section, two techniques based on linear least squares are discussed. The ﬁlter-ﬁt methods makes use of library spetra, measured or calculated spectra of pure compounds, that are used to describe the spectra of complex samples. The other method is based on partial least-squares (PLS) regression, a multivariate calibration technique. In this case, no spectrum evaluation in the strict sense is performed, but, rather, relations between the concentrations of the compounds in the samples and the entire spectrum are established. In this way, quantitative analysis is possible without obtaining net peak areas of the characteristic lines. A.

Filter-Fit Method

1. Theory If a measured spectrum of an unknown sample can be described as a linear combination of spectra of pure elements constituting the sample, then the following mathematical model can be written: ymod ¼ i

m X

aj xji

ð60Þ

j¼1

the content of channel i in the model spectrum and xji the content of channel i in with ymod i the jth reference spectrum. The coeﬃcients aj are a measure of the contribution of pure reference spectra to the unknown spectrum and can be used for quantitative analysis. The values of the coeﬃcients aj are obtained via multiple linear least-squares ﬁtting, minimizing the sum of the weighted squared diﬀerences between the measured spectrum and the model: " #2 n2 n2 m X X X 1 1 2 2 w ¼ ½ yi yðiÞ ¼ yi aj xji ð61Þ s2 s2 i¼n1 i i¼n1 i j¼1 where yi and si are the channel content and the uncertainty of the measured spectrum, respectively, and n1 and n2 are the limits of the ﬁtting region. A detailed discussion of the least-squares-ﬁtting method is given in Sec. IX. The assumption of linear additivity [Eq. (60)] holds normally reasonable well for the characteristic lines in the spectrum, but not for the continuum. To apply this technique, the continuum can be removed from the unknown spectrum and from the reference spectra using one of the procedures described in Sec. IV before the actual least-squares ﬁt.

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269

Another, frequently used approach is to apply a digital ﬁlter to both unknown and reference spectra. This variant is known as the ﬁlter-ﬁt method (Schamber, 1977; Statham, 1978; McCarthy and Schamber, 1981) and is discussed in some detail below. By the discrete convolution of a spectrum with top-hat ﬁlter [Eqs. (30) and (31)], the low-frequency component (i.e. the slowly varying continuum) is eﬀectively suppressed as discussed in Sec. III. Apart from removing the slowly varying continuum, a rather severe distortion of the peaks is also introduced. If we apply this ﬁlter to both the unknown spectrum and the reference spectra, the nonadditive continuum is removed and the same type of peak distortion will be introduced in all spectra, allowing us to apply the method of multiple linear least-squares ﬁtting to the ﬁltered spectra. Equation (61) then becomes " #2 n2 m X X 1 y0 aj x0ji w2 ¼ s0 2 i j¼1 i¼n1 i

ð62Þ

Where y0i and x0ji are the ﬁltered unknown and reference spectra, respectively, s0i 2 is the variance of y0i given by X s02 h2k yiþk ð63Þ i ¼ k

The least-squares estimates of the contribution of each reference spectrum is then given by (see Sec. IX) aj ¼

m X

a1 jk bk ;

j ¼ 1; . . . ; m

ð64Þ

k¼1

with n2 X 1 0 0 yx 02 i ji s i¼n1 i n2 X 1 0 0 ajk ¼ x x s0 2 ki ji i¼n1 i

bj ¼

ð65Þ ð66Þ

The uncertainty in each coeﬃcient aj is directly estimated from the error matrix: s2aj ¼ a1 jj

ð67Þ

Schamber (1977) suggested the following equation for the uncertainties, taking into account the eﬀect of the ﬁlter: s2aj ¼

nw 1 a n þ w jj

ð68Þ

Where w is the width of the central positive part of the ﬁlter and n is the width of the negative wings. A measure of the goodness of ﬁt is available through the reduced w2 value: w2n ¼

1 w2 ðn2 n1 þ 1Þ m

ð69Þ

which is the w2 value of Eq. (62) divided by the number of points in the ﬁt minus the number of reference spectra. A value close to 1 means a good ﬁt, indicating that the reference spectra are capable of adequately describing the unknown spectrum.

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Most of the merits and the disadvantages of the ﬁlter-ﬁt method can be deduced directly from the mathematical derivation given in the preceding paragraphs. The most interesting aspect of the ﬁlter-ﬁt method is that it does not require any mathematical model for the continuum and that, at least in principle, the shape of the peaks in the unknown spectrum are exactly represented by the reference spectra. Reference spectra should be acquired with good counting statistics, at least better than the unknown spectrum, because the least-squares method assumes that there is no error in the independent variables x0ji . Reference spectra can be obtained from single-element standards. Only the portion of the spectrum that contains peaks needs to be retained as reference in the ﬁt. Multielement standards can be used if the peaks of each element are well separated. The reference spectra must provide an accurate model of the peak structure present in the unknown spectrum. This requires that reference and unknown spectra are acquired under strictly identical spectrometer conditions. Changes in resolution and, especially, energy shifts can cause large systematic errors. The magnitude of this error depends on the degree of peak overlap. Peak shifts of more than a few electron volts should be avoided, which is readily possible with modern detector electronics. If shifts are observed over long periods of operations of the spectrometer, careful recalibration of the spectrometer is required or, better, the reference spectra should be acquired again. Also, peak shift and peak broadening due to diﬀerences in count rate between standards and unknown must be avoided. Diﬀerential absorption is another problem that might inﬂuence the accuracy of the model. Because of the diﬀerence in x-ray attenuation in the reference and the unknown, the Kb to Ka ratios might be diﬀerent in the two spectra. This becomes especially problematic if the Kb line is above and the Ka line below an absorption edge of a major element of the unknown sample. The magnitude of the error depends on the peak overlap. Careful selection of the samples to produce the reference spectra is therefore required. The procedure requires that a reference spectrum be included for each element present in the unknown. The method allows no mechanism to deal with sum peaks. Apart from removing the continuum, the ﬁlter also has some smoothing eﬀect on the spectrum and causes the peak structure to be spread out over more channels. This is equivalent to ﬁtting a spectrum with a somewhat lower resolution than originally acquired. Therefore, the precision and detection limits attainable with the ﬁlter-ﬁt method are slightly worse than optimal. The width of the ﬁlter is important in this respect. Schamber (1977) suggests taking the width of the top of the ﬁlter equal to the FWHM resolution of the spectrometer, u ¼ FWHM. The width of the wings can be taken as v ¼ u=2. 2. Application The calculation procedure is quite simple and requires the following steps. The top-hat ﬁlter is applied to the unknown spectrum and the m reference spectra [Eqs. (30) and (31)], and the modiﬁed uncertainties are calculated using Eq. (63). Next, the vector b of length m and the m m square matrix a are formed using Eqs. (65) and (66), summing over the part of the spectrum one wants to analyze (n1 n2 ). Only the relevant part, such as the Ka or the Ka plus the Kb, of the reference spectra needs to be retained; the rest of the ﬁltered spectrum can be set to zero. After calculating the inverse matrix a1 , the contribution of each reference to the unknown and its uncertainty are calculated using Eqs. (64) and (67) or (68). A computer implementation of the ﬁlter-ﬁt method is given in Sec. X. The method was used to analyze part of an x-ray spectrum of a polished NIST SRM 1103 brass sample (Fig. 16A). The measurements were carried out using a Mo x-ray tube and a Zr secondary target and ﬁlter assembly. Spectra of pure metals (Fe, Ni, Cu, Zn, and Pb) were acquired

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271

Figure 16 (A) Part of the x-ray spectrum of NIST SRM 1103 brass sample. (B) Top-hat-filtered spectrum and result of fit using reference spectra.

under identical experimental conditions. A top-hat ﬁlter of width w ¼ 5 was used. Table 6 shows how the spectra were divided in regions of interest to produce the reference spectra. Because considerable x-ray attenuation is present in brass, separate references were created for the Ka and Kb of Cu and Zn. This was not done for Fe, Ni, and Pb because these elements are only present as minor constituents in the brass sample. Figure 16B shows the ﬁltered brass spectrum and the resulting ﬁt using the seven (ﬁltered) reference spectra. The region below CuKa is expanded 100 times, and the region above ZnKa is expanded 10 times. As can be seen, the agreement between the ﬁltered brass spectrum and the ﬁt is very good. The reduced w2 value is 8.5. This high value is probably due to small peaks shifts in the reference spectra compared to brass spectrum. Table 7 compares the results of the ﬁlter ﬁt with the results obtained by nonlinear least-squares ﬁtting using analytical functions (see Sec. VII). Although the w2 value of the nonlinear ﬁt is slightly better (2.7), one observes an excellent agreement between the two methods for the analytical important data (i.e., intensity ratios). The uncertainties for small peaks are slightly higher with ﬁlter-ﬁt method, as explained previously. The ﬁlter-ﬁt method is fast and relatively easy to implement. It can produce reliable results when the spectrometer calibration can be kept constant within a few electronvolts and suitable standards for each element present in the sample are available. The method

272

Van Espen

Table 6 Data on the Reference Spectra and the Unknown Spectrum Used in the Filter-Fit Procedure (Fig. 16) Spectrum

Region of interest (keV)

Pure Fe Pure Ni Pure Cu

4.258.12 4.739.57 5.708.46 8.469.81 6.599.09 9.0910.30 8.3613.68 5.2211.26

Pure Zn Pure Pb SRM 1103

Used as FeKa þ Kb reference NiKa þ Kb reference CuKa reference CuKb reference ZnKa reference ZnKb reference PbLa reference Unknown

Table 7 Comparison of Spectrum Evaluation Results Using Filter-Fit Method and Nonlinear Least-Squares Fitting; Ratios of the Intensity from the SRM 1130 Standard to the Pure Element Given Ratio of intensity in SRM 1104 to pure element Element Fe Ni Cu Zn Pb

Filter ﬁt

Nonlinear ﬁt

% Diﬀ

0.0053±0.0002 0.0019±0.0002 0.546±0.001 0.390±0.001 0.0310±0.0006

0.0047±0.0001 0.00201±0.00008 0.551±0.001 0.3912±0.0008 0.0308±0.0003

11 6 0.9 0.31 0.65

performs well when one has to deal with a diﬃcult to model continuum. If information on trace elements and major elements is required (very large peaks next to very small ones), the method might not be optimal. This ﬁlter-ﬁt method is frequently used to process x-ray spectra obtained with electron microscopes (SEM-EDX), often in combination with a ZAF or Phi-Rho-Z correction procedure. B.

Partial Least-Squares Regression

Spectrum evaluation as discussed in most of this chapter aims to obtain the net intensity of the ﬂuorescence lines. These net peak intensities are then used to determine analyte concentrations using one of the many empirical, semiempirical, or fundamental approaches detailed in various other chapters. In this, sub-section, we discuss an approach that is relatively new for the XRF community and is based on multivariate calibration. We will concentrate our discussion on partial least squares (PLS), a chemometrical technique that is extensively used in infrared and relates spectrometric methods. The basic idea of the method is to ﬁnd a (multivariate linear) model that directly relates the spectral data to the analyte concentrations, thus avoiding the explicit evaluation of the spectrum in terms of the net peak areas. The method involves a calibration step where a large number of standards are used to build and validate the model and the actual analysis step where the model is applied to spectra of unknown samples.

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273

1. Theory Multivariate spectroscopic calibration attempts to predicts properties of samples (concentration of analytes) based on measured spectral data via the following relation: Y ¼ XB þ F

ð70Þ

The Ynm matrix holds concentration of m analytes in the n samples. The Xnp matrix represents the spectral data, with n measured spectral data each having p x-ray intensities (channels) and Fnm is the matrix the residuals, part of the Y matrix not explained by the model. The regression coeﬃcients B can be calculated in several ways. The most straightforward approach is the use of multiple linear regression (see Sec. IX). The leastsquares solution is given then given by B ¼ ðX0 XÞ1 X0 Y

ð71Þ

When the number of x variables exceeds the number of samples ( p > n) and=or when there is a high degree of correlation between the x variables, the least-squares solution becomes unstable or cannot be obtained due to the fact that the covariance matrix X0 X cannot be inverted. This is exactly the fact when we attempt to apply multivariate calibration in XRF. The number of channels in the spectrum (1024 or more) largely exceeds the number of samples and the intensities of neighboring channels in each peak are very strongly correlated. To overcome this problem, the method of singular value decomposition can be used. The X matrix is decomposed in a number of linearly independent variables, know as principal components. In principle, there are as many principal components as there are original variables (channels), but only the most signiﬁcant [i.e., having the highest variance (or eigenvalue)] are retained. Using the matrix of this smaller number of principal components scores, rather than the entire X matrix in Eq. (70), is known as principal components regression (PCR). The disadvantage of PCR is that the selection of the principal components is based on how much variance they explain of the X matrix. The ﬁrst few principal components may have little relation with the concentrations that need to be predicted by the model. Partial least-squares regression (PLSR) is a variant of PCR that largely overcomes this problem. The methods uses two outer relations and one inner relation. The outer relations describe the decomposition of the X and Y matrices: X ¼ TP0 þ E ¼ Y ¼ UQ0 þ F ¼

A X a¼1 A X

ta p0a þ E

ð72Þ

ua q0a þ F

ð73Þ

a¼1

TnA and UnA are the score matrices, the values of the AðA pÞ latent variables. P0Ap and Q0Am are the loading matrices describing the relation between the latent variables (T and U) and the original variables (X and Y). The number of latent variables A in the model is of crucial importance and its optimum value must be found by cross-validation. The matrices E and F contain the residuals, part of the original spectral data and the concentration data, respectively, not accounted for when using A latent variables. The inner relation is written as: ua ¼ ba ta ;

a ¼ 1; . . . ; A

ð74Þ

from which the regression coeﬃcients B can be obtained. This operation can be seen as a leastsquares ﬁt between the X block and Y block scores. The ﬁnal PLS model can thus be written as

274

Van Espen

Y ¼ TBQ0 þ F

ð75Þ

A graphical representation of the PLS model is given in Figure 17. In the normal PLS algorithm Y is a vector of concentrations for one element and a separate model is build for each element. If all Y-variables are predicted simultaneously, as in the case of Equation (70), the PLS2 algorithm is used. This method performs better when the concentrations in the samples are highly correlated. The quality of the calibration model can be judged by the root mean square error of prediction (RMSEP): " #1=2 n 1X 2 RMSEP ¼ ð^ yi yi Þ ð76Þ n i¼1 where y^i is the concentration in sample i predicted by the model yi is the true concentration in the standard. To determine the optimum number of latent variables, the RMSEP is calculated using PLS models with diﬀerent numbers of latent variables A. The RMSEP values are plotted against A and the value where a minimum or a plateau is reached is taken. For small datasets, the calculation of the RMSEP is done using cross-validation. When a large number of standards are available, they are split in a training set (approximately two-thirds of the samples) and a prediction set (approximately one-third of the samples). The training set is used to build the PLS model and the RMSEP is calculated based on the concentrations of the prediction set. Alternatively, for smaller calibration sets, leave-one-out cross-validation can be used. Each sample is excluded once from the dataset and predicted by the model built with the remaining samples. This is repeated until all samples have been excluded once. Geladi and Kowalski (1986) published a tutorial on PLS and its relation to other regression techniques. A standard work on the PLS method is the book by Martens and

Figure 17

Graphical representation of the PLS algorithm.

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275

Naes (Martens and Naes, 1989). Theoretical and statistical aspects of the method can be found in chemometrics literature (Manne, 1987; Lorber et al., 1987; Pratar and De Jong, 1997; Hoskuldsson, 1988). 2. Application The PLS method is illustrated with the analysis of aqueous solution containing Ni, Cu, and As with concentration in the ranges 1850, 0.545, and 520 g=L, respectively, whereas Zn, Fe, and Sb were present in more or less constant amounts. Spectra were acquired for 1000 s from 5-mL solution using an EDXRF spectrometer equipped with a Rh x-ray tube operating at 30 kV and a Si(Li) detector having 160 eV resolution at MnKa. From the 22 samples, 10 were used to build the PLS mode and the remaining were used as the prediction set. Table 8 gives the composition of the samples in the calibration set. Figure 18 shows part of the spectrum between 5 and 14 keV of sample number 9. The CuKa line is considerably interfered by the NiKb line. Absorption eﬀects can be expected because the NiK absorption edge (8.33 keV) is just above the CuKa line (8.04 keV) and the AsK lines can cause secondary ﬂuorescence of Cu and Ni. A PLS 1 model is build for Cu. The X matrix consists of 451 x-ray intensities (variables) between 5 and 14 keV (channels 250 to 700) of the 10 samples. The Y matrix contains the Cu concentration of those samples. In Figure 19, the RMSEP based on leave-one-out cross-validation is plotted as function of the number of latent variables. A minimum error of 0.57 g=L is obtained for three latent variables so that the PLS model with three latent variable is retained. Figure 20 compares the true Cu concentration with the predicted concentration for the calibration set. The PLS model predicts very accurately the Cu concentrations in the range from 1 to 40 g=L. The so-called regression coeﬃcients of the PLS model show which variables (channels) are used in the model. They are plotted in Figure 21. As could be expected, the Cu concentrations are predicted from the channel content corresponding to the CuK lines. The inﬂuence of absorption and enhancement results in a small negative contribution from the Ni peak and a small positive contribution from the As peaks, respectively. The PLS model thus handles both the problem of spectral interference and matrix eﬀects. The Cu concentration predicted by the PLS model for the test set is given in Table 9. Except for the two samples with the highest concentrations, the Cu concentration

Table 8

Composition of Samples Used to Build the PLS Model Concentration (g=L)

Sample No. 1 2 3 4 5 6 7 8 9 10

Ni

Cu

Zn

18.0 18.1 18.5 18.1 53.6 53.8 57.3 56.9 51.4 17.9

6.7 5.0 3.5 8.4 1.8 0.8 1.1 1.5 17.7 41.5

6.1 6.0 6.1 5.7 20.1 19.8 19.3 20.0 20.9 5.3

276

Figure 18

Van Espen

X-ray spectrum of sample number 9 in the calibration set used to build the PLS model.

Figure 19 RMSEP values versus the number of latent variables for the prediction of the Cu concentration.

in the ‘‘unknown’’ samples is very well estimated. The predicted of the test set is generally somewhat worse that the prediction of the calibration set. To build an accurate model, a large number of standards spanning the concentration range of interest for each element is required. This is certainly the major drawback of PLS for its application in XRF,

Spectrum Evaluation

277

Figure 20 Comparison of ‘‘true’’ and predicted Cu concentrations for the samples in the PLS calibration set.

Figure 21 Regression coefficients of the PLS model for Cu, showing which variables (channels in the x-ray spectrum) are used to predict Cu.

278

Van Espen

Table 9

Comparison of ‘‘True’’ and Predicted Cu Concentration Using the PLS Model Cu concentration (g=L)

Sample No. 11 12 13 14 15 16 17 18 19 20 21 22

‘‘True’’

Predicted

Diﬀerence

6.30 7.20 6.30 7.50 6.30 0.90 1.00 1.20 1.80 2.50 17.60 44.70

6.38 7.34 6.32 7.40 6.09 1.27 0.75 1.55 2.11 2.86 16.51 40.59

0.08 0.14 0.02 0.10 0.21 0.37 0.25 0.35 0.31 0.36 1.09 4.11

especially when solid samples are considered. This problem can to some extend be overcome by building a calibration set via Monte Carlo simulation. Just as with the ﬁlterﬁt method, standards and unknowns need to be measured under strictly identical spectrometer conditions. Changes in gain or resolution will cause systematic errors in the calculated concentrations. Swerts and Van Espen (1993) demonstrated the use of PLS for the determination of S in graphite using a Rh x-ray tube excitation XRF equipped with a Si(Li)detector. Because of diﬀraction eﬀects, least-squares ﬁtting of the spectra was nearly impossible. Using PLS, the sulfur content could be determined in a concentration range 260%, with an accuracy of better than 5% relative standard deviation. Urba´nski and Kowalska applied the PLS method to the determination of Sr and Fe in powdered rock samples and to the determination of the S and ash content in coal using a radioisotope XRF system equipped with a low-resolution gas proportional counter. They also demonstrated the usefulness of this method for the determination of the thickness and composition of Sn-Pb and Ni-Fe layered structures (Urba´nski and Kowalska, 1995). Molt and Schramm (1987) compared principal components regression (PCR), PLS for the determination of S, exhibiting strong interference from Mo, in aqueous and solid samples. The results were also compared with quantitative analysis using the method developed by Lucas-Tooth and Price (1961). Equally good results were obtained with all three methods. Similar results were obtained by Lemberge and Van Espen for the determination of Ni, Cu, and As in liquid samples. They demonstrated that taking the square root of the data improves the PLS model and that the PLS method extracts information from the scattered excitation radiation to describe the matrix eﬀects (Lemberge and Van Espen, 1999).

VII.

LEAST-SQUARES FITTING USING ANALYTICAL FUNCTIONS

A widely used and certainly the most ﬂexible procedure for evaluating complex x-ray spectra is based on least-squares ﬁtting of the spectral data with an analytical function. The method is conceptually simple, but not trivial to implement and use.

Spectrum Evaluation

A.

279

Concept

In this method, an algebraic function, including analytical importantly parameters such as the net areas of the ﬂuorescence lines, is used as a model for the measured spectrum. The object function (w2) is deﬁned as the weighted sum of squares of the diﬀerences between this model y(i) and the measured spectrum yi over a region of the spectrum: n2 X 1 w2 ¼ ½ yi yði; a1 ; . . . ; am Þ2 ð77Þ 2 s i i¼n1 where s2i is the variance of data point i, normally taken as s2i ¼ yi , and aj are the parameters of the model. The optimum values of the parameters are those for which w2 is minimal. They can be found by setting the partial derivatives of w2 with respect to the parameters to zero: @w2 ¼ 0; @aj

j ¼ 1; . . . ; m

ð78Þ

If the model is linear in all the parameters aj, these equations result in set of m linear equations in the m unknowns aj, which can be solved algebraically. This is known as linear least-squares ﬁtting. If the model is nonlinear in one or more of its parameters, a direct solution is not possible and the optimum value of the parameters must be found iteratively. An initial value is given to the parameters and they are varied in some way until a minimum for w2 is obtained. The latter is equivalent to searching for a minimum in the m þ 1-dimensional w2 hypersurface. This is known as nonlinear least-squares ﬁtting. The selection of a suitable minimization algorithm is very important because it determines to a large extent the performance of the method. A detailed discussion of linear and nonlinear least-squares ﬁtting is given in Sec. IX. The most diﬃcult problem to solve when applying this least-squares procedure is the construction an analytical function that accurately describes the observed spectrum. The model must be capable of describing accurately the spectral data in the ﬁtting region. This requires an appropriate model for the continuum, the characteristic lines of the elements and for all other features present in the spectrum such as absorption edges, escape, and sum peaks. Although the response function of the energy-dispersive detector is, to a very good approximation, Gaussian, deviation from the Gaussian shape needs to be taken into account. Failure to construct an accurate model will result in systematic errors, which under certain conditions may lead to gross positive or negative errors in the estimated peak areas. On the other hand, the ﬁtting function should remain simple, with as few parameters as possible. Especially for the nonlinear ﬁtting, the location of the w2 minimum becomes problematic when a large number of parameters is involved. In general, the ﬁtting model consists of two parts: X yP ðiÞ ð79Þ yðiÞ ¼ yB ðiÞ þ P

where y(i) is calculated content of channel i and the ﬁrst part describes the continuum and the second part the contributions of all peaklike structures. Because the ﬁtting functions for both linear and nonlinear least-squares ﬁtting have many features in common, we treat the detailed description of the ﬁtting function for the most general case of nonlinear least squares. Moreover, if the least-squares ﬁt is done using the Marquardt algorithm, the linear least-squares ﬁt is computationally a particular case of the nonlinear least-squares ﬁt. Programs based on this algorithm can perform

280

Van Espen

linear and nonlinear ﬁtting using the same computer code. A large part of the discussion given here is based on the computer code AXIL, developed by the author for spectrum ﬁtting of photon-, electron-, and particle-induced x-ray spectra (Van Espen et al., 1977a, 1977b, 1979b, 1986). B.

Description of the Continuum

To model the continuum, various analytical expressions are in use, depending on the excitation conditions and on the width of the ﬁtting region. Except for electron microscopy, it is virtually impossible to construct an acceptable physical model that describes the continuum, mainly because of the large number of processes that contribute to it. For this reason, very often some type of polynomial expression is used. 1. Linear Polynomial A linear polynomial of the type yB ðiÞ ¼ a0 þ a1 ðEi E0 Þ þ a2 ðEi E0 Þ2 þ þ ak ðEi E0 Þk

ð80Þ

is useful to describe the continuum over a region 23 keV wide. Wider regions often exhibit too much curvature to be described by this type of polynominal. In Eq. (80), Ei is the energy (in keV) of channel i [see Eq. (84)] and E0 is a suitable reference energy, often the middle of the ﬁtting region. Expressing the polynomial as a functin of (Ei 7 E0) rather than as a function of the channel number is done for computational reasons; (Ei 7 E0)3 is, at most, of the order of 103, whereas i3 can be as high as 109. Most computer programs that implement a polynomial model for the continuum allow the user to specify the degree of the polynomial; k ¼ 0, 1, and 2 produce respectively a constant, a straight line, and a parabolic continuum. Values of k larger than 4 are rarely useful because such high-degree polynomials tend to have physical nonrealistic oscillations. Equation (80) is linear in the ﬁtting parameters a0, . . . , ak, so that this function can be used in linear as well as in nonlinear least-squares ﬁtting. 2. Exponential Polynomial A linear polynomial cannot be used to ﬁt the continuum over the entire spectrum or to ﬁt regions of high positive or negative curvature. Higher curvature can be modeled by functions of the type yB ðiÞ ¼ a0 exp½a1 ðEi E0 Þ þ a1 ðEi E0 Þ2 þ þ ak ðEi E0 Þk

ð81Þ

where k is the degree of the exponential polynomial. A value of k as high as 6 might be required for an accurate description of a continuum from 2 to 16 keV. This function is nonlinear in the ﬁtting parameters a1, . . . , ak and requires a nonlinear least-squares procedure and some initial guess of these parameters. Initial values for these nonlinear parameters can be determined by ﬁrst estimating the shape of the continuum using one of the procedures described in Sec. IV followed by a linear ﬁt of the logarithm of this continuum. These initial guesses are then further reﬁned in the nonlinear ﬁtting procedure. 3. Bremsstrahlung Continuum The exponential polynomial is not suitable for describing the shape of the continuum observed in electron- and particle-induced x-ray spectra, mainly because of the high

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curvature at the low-energy region of the spectrum. The continuum results from photons emitted from the sample by the retardation of fast electrons. The slope of the emitted continuum is essentially an exponentially decreasing function according to Kramer’s formula. At low energies, the emitted photons are strongly absorbed by the detector windows and by the sample. A suitable function to describe such a radiative continuum is an exponential polynominal multiplied by the absorption characteristics of the spectrometer: yB ðiÞ ¼ a0 exp½a1 ðEi E0 Þ þ þ ak ðEi E0 Þk Ta ðEi Þ

ð82Þ

A detailed discussion of the function Ta(E) is given on page 288. To be physically correct, the absorption term must be convoluted with the detector response function, because the sharp edges due to absorption by detector windows (Si or Au) or elements present in the sample are smeared out by the ﬁnite resolution of the detector. 4. Continuum Removal An alternative to an algebraic description of the continuum is to estimate the continuum ﬁrst using one of the procedures outlined in Sec. IV and to substract this continuum from the measured spectrum before the actual least-squares ﬁtting. To correctly implement the least-squares ﬁt after subtraction of the continuum, the weights 1=s2i [Eq. (77)] must be adjusted. If y0i represents the spectral data after subtraction of the continuum, y0i ¼ yi yB ðiÞ, the variance of y0i is given by s0i 2 ¼ s2i þ s2yB ðiÞ . A reasonable approximation for s2yB ðiÞ is yB ðiÞ itself, so that the total variance becomes s0i 2 ¼ yi þ yB ðiÞ. If this adjustment of the weights is not made, the uncertainty in the net peak areas are understimated, especially for small peaks on a high continuum. It is rather diﬃcult for an inexperienced user to select the appropriate continuum model for a given spectrum. The following might serve as a general guideline. For ﬁtting regions 23 keV wide, a linear polynomial continuum is often adequate. To ﬁt large regions of XRF spectra, the exponential polynomial provides the most accurate results, with k typically, between 4 and 6. The same holds for the bremsstrahlung continuum of SEMEDX and PIXE spectra. The simplest method from the users’ point of view is the continuum stripping, but this method does not provide optimum results. A slight underestimation or overestimation might occur, resulting in large relative errors in the area determination of small peaks (Vekemans et al., 1995). C.

Description of Fluorescence Lines

Because the response function of most solid-state detectors is predominantly Gaussian, all mathematical expressions used to describe the ﬂuorescence lines involve this function. When dealing with the K lines of high-atomic-number elements, such as Pb or U, the inﬂuence of the natural line shape becomes appreciable and the use of the more complicated Voigt proﬁle is required (Wilkinsin, 1971; Gunnink, 1977; Pessara and Debertin, 1981). 1. Single Gaussian A Gaussian peak is characterized by three parameters: the position, width, and height or area. It is desirable to describe the peak in terms of its area rather than its height because the area is directly related to the number of x-ray photons detected, whereas the height depends on the spectrometer resolution. The ﬁrst approximation to the proﬁle of a single peak is then given by

282

Van Espen

" # A ðxi mÞ2 pﬃﬃﬃﬃﬃﬃ exp 2s2 s 2p

ð83Þ

where A is the peak area (counts), s is the width of the Gaussian expressed in channels, and m isp the location of the peak maximum. The often used FWHM is related to s by the ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ factor 2 2 ln 2 or FWHM ¼ 2.35s. In Eq. (83), the peak area is a linear parameter; the width and position are nonlinear parameters. This implies that a nonlinear least-squares procedure is required to ﬁnd optimum values for the latter two parameters. Using a linear least-squares method assumes that the position and width of the peak are know with high accuracy from calibration. To describe part of a measured spectrum, the ﬁtting function must contain a number of such functions, one for each peak. For 10 elements and 2 peaks (Ka and Kb) per element, we would need to optimize 60 parameters. It is highly unlikely that such a nonlinear leastsquares ﬁt will terminate successfully at the global minimum. To overcome this problem, the ﬁtting function can be written in a diﬀerent way as shown in the next subsection. 2. Energy and Resolution Calibration Function The ﬁrst obvious step is to drop the idea of optimizing the position and width of each peak independently. In x-ray spectrometry, the energies of the ﬂuorescence lines are know with an accuracy of 1 eV or better. The pattern of peaks observed in the spectrum is directly related to elements present in the sample. Based on those elements, we can predict all x-ray lines that constitute the spectrum and their energies. The peak function [Eq. (83)] is therefore written in terms of energy rather than channel number. Deﬁning ZERO as the energy of channel 0 and expressing the spectrum GAIN in electronvolts=channel, the energy of channel i is given by EðiÞ ¼ ZERO þ ðGAINÞi and the Gaussian peak can be written as " # GAIN ½Ej EðiÞ2 Gði; Ej Þ ¼ pﬃﬃﬃﬃﬃﬃ exp 2s2 s 2p with Ej the energy (in eV) of the x-ray line and s the peak width given by NOISE 2 2 s ¼ þ3:58ðFANOÞEj 2:3548

ð84Þ

ð85Þ

ð86Þ

In this equation, NOISE is the electronic contribution to the peak width (typically 80100 eV FWHM) with the factor 2.3548 to convert to s units, FANO is the Fano factor ( 0.114), and 3.85 pﬃﬃﬃﬃﬃ ﬃ is the energy required to produce an electronhole pair in silicon. The term GAIN=s 2p in Eq. (85) is required to normalize the Gaussian so that the sum over all channels is unity. For linear least-squares ﬁtting, ZERO, GAIN, NOISE, and FANO are physically meaningful constants. In the case of nonlinear least squares, they are parameters to be reﬁned during the ﬁtting. The advantage of optimizing the energy and resolution calibration parameters rather than the position and width of each peak is a vast reduction of the dimensionality of the problem. The nonlinear ﬁt of 10 peaks would now involve 14 parameters compared to 30. Even more importantly, all information available in the spectrum is used to estimate ZERO, GAIN, NOISE, and FANO and thus the positions and the widths

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of all peaks. Imagine a small, severely overlapping doublet with a well-deﬁned peak on both sides of this doublet. These two peaks will contribute most to the determination of the four calibration parameters, virtually ﬁxing the position and the width of the two peaks in the doublet. As a consequence, their areas can be determined much more accurately. Referring to our discussion on information content in Sec. II, we did not obtain this extra performance for nothing. We have supplied extra information: the energy of the peaks and the two calibration relations [Eqs. (84) and (86)]. Fitting with Eq. (85) requires that the extra information we supply is indeed correct. With modern electronics, the linearity of the energy calibration [Eq. (84)] holds very well in regions above 2 keV. Fitting the entire spectrum, including the low-energy region, might require more complex energy calibration functions. To ﬁt PIXE spectra from 1 to 30 keV, Maenhaut and Vandenhaut (Maenhaut and Vandenhaut, 1986) suggested the following function: i ¼ C1 þ C2 E þ C3 expðC4 EÞ. The relation between the square of the peak width and the energy [Eq. (86)] is based on theoretical considerations. The relation holds very well if the doublet splitting of the x-ray lines is taken into account. The Ka1Ka2 separation increases from a negligible value for Ca (3.5 eV) to nearly 100 eV for Mo. The observed peak shape of a K line is actually an envelope of two peaks. This envelope can be represented rather well by a single Gaussian, but failing to take this doublet splitting into account (i.e., ﬁtting with a single Gaussian where doublets are required) will result in peak widths that do not obey Eq. (86). To illustrate this, the observed width of a number of Ka lines as function of the x-ray energy is presented in Figure 22. The dotted line represents the width of the Ka doublet ﬁtted as one peak. The solid (straight) line shows the width of the individual lines in the doublet. 3. Response Function for an Element A second modiﬁcation to the ﬁtting function that will reduce the number of ﬁtting parameters is modeling an entire element, rather than single peaks. A number of lines can be considered as logically belonging together, such the Ka1 and Ka2 of the above-mentioned doublets or all K lines of an element. This group can be ﬁtted with one area parameter A representing the total number of counts of all the lines in the group. The spectrum of an element can then be represented by yP ðiÞ ¼ A

NP X

Rj Gði; Ej Þ

ð87Þ

j¼1

where G are the Gaussians for the various lines with energy Ej and Rj the relative inP Rj ¼ 1. tensities of the lines. The summation runs over all lines in the group (NP) with The transition probabilities of all lines originating from a vacancy in the same (sub) shell (K, LI, LII, . . .) are constants, independent of the excitation. However, the relative intensities depend on the absorption in the sample and in the detector windows. To take this into account, the x-ray attenuation must be included in Eq. (87). The relative intensity ratios are obtained by multiplying the transition probabilities with an absorption correction term: Rj Ta ðEnj Þ R0j ¼ P j Rj Ta ðEnj Þ

ð88Þ

Contributions from various subgroups (i.e., between K and L, between LI and LII, etc.) depend on the type of excitation (photons, electrons, protons) and on the excitation energy. General values cannot be given and must be determined for the particular excitation

284

Figure 22

Van Espen

FWHM of various Ka lines, ﬁtted as single peak and as a Ka1Ka2 doublet.

if one wants to combine lines of diﬀerent subgroups. Transition probabilities of lines in various groups, determined experimentally and calculated from ﬁrst principles, can be found in the literature (McCrary et al., 1971; Salem et al., 1970; Salem and Wimmer, 1970; Scoﬁeld 1970, 1974a, 1974b). In Figure 23, the ﬁt of a tungsten L line spectrum using 24 transitions from the 3 L subshells is shown. The relative intensities of the L lines within each subshell were taken from the literature (Scoﬁeld, 1974b). The ﬁt was thus done with one peak-area parameter for the L1, the L2 and the L3 sublevels. Fitting elements rather than individual peaks enhances the capability of the method to resolve overlapping peaks. The area of the CrKa peak, interfered by a VKb peak, can be obtained with higher precision (lower standard deviation) because the area of the VKb peak is related to the area of the (interference free) VKa peak. Again, we have introduced more a priori knowledge into our model. If this information (the relative intensity ratio) is not correct, we will introduce systematic errors (e.g., the CrKa peak area, although having a low standard deviation, will not be correct). In practice, there will be a trade-oﬀ between the gain in accuracy and the gain in precision. Errors in the values of the transition probabilities and in the value of the absorption correction term are suﬃciently small so that small to moderate high peaks (up to 105 counts peak area) can be ﬁtted as one group. 4. Modified Gaussians When ﬁtting very large peaks with a Gaussian, the deviation from the pure Gaussian shape becomes signiﬁcant. In Figure 24, a MnK spectrum with 107 counts in the MnKa peak is shown. One observes a tailing on the low-energy side of the peaks and a continuum that is

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Figure 23 Fit of a complex L line spectrum of tungsten. In total, 24 transitions, divided over the 3 L subshells, are required for the description on the spectrum.

Figure 24 MnK line spectrum with very good counting statistics. The diﬀerence from the Gaussian response is obtained by subtracting all Gaussian peaks. From this, the peak-shape correction is calculated.

286

Van Espen

higher at the low energy side of the Ka peak than at the high energy side of the Kb peak. The observed peak shape is partially caused by nonideal behavior of the detector. For lowenergy lines ( < 10 keV), incomplete charge collection and other detector artifacts play an important role. For higher energetic lines, Compton scattering in the detector also contributes. Another part of the deviation from the Gaussian shape is attributed to phenomena taking place in the sample itself. X-ray satellite transitions such as KLM radiative Auger transitions on the low-energy side of the Ka and peak and KMM transitions on the low-energy side of the Kb contribute to the overall shape of the spectrum. In the literature, considerable attention has been given to the explanation and to the accurate mathematical description of the observed peak shape (McNelles and Compbell, 1975; Van Espen et al., 1977b; Wielopolski and Gardner, 1979; Campbell et al., 1985, 1987, 1997, 1998, Gardner et al., 1986; Yacout et al., 1986; Campbell, 1996). Failure to account for the deviation from the Gaussian peak shape causes a number of problems when ﬁtting x-ray spectra. Small peaks sitting on the tail of large ones (e.g., NiKa in front of CuKa) cannot be ﬁtted accurately, resulting in large systematic errors for the small peak. Because the least-squares method seeks to minimize the diﬀerence between the observed spectrum and the ﬁtted function, the tail might become ﬁlled up with peaks of elements that are not really present. Also, the continuum over the entire range of the spectrum becomes diﬃcult to describe. This is illustrated in Figure 25a, where a spectrum

Figure 25 Fit of the spectrum of a brass sample (NIST SRM 1106) (a) fitted with a simple Gaussians and (b) fitted with Gaussians including tail and step functions.

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of a brass sample (NIST SRM 1106) is ﬁtted with simple Gaussians and a linear continuum. An unrealistically high area for the NiKa peak is obtained in this way. A number of analytical functions have been proposed to account for the true line shape. Nearly all of them include a ﬂat shelf and an exponential tail, both convoluted with the Gaussian response function. Similar functions are also used to describe the peak shape of g-ray spectra (Phillips and Marlow, 1976). To account for the deviation from the Gaussian peak shape, the Gauss function G(i, Ej) in Eq. (85) can be replaced by Pði; Ejk Þ ¼ Gði; Ejk Þ þ fS Sði; Ejk Þ þ fT Tði; Ejk Þ

ð89Þ

where G(i, Ej) is the Gaussian part given by Eq. (85) and S and T are the step and tail functions, respectively, GAIN EðiÞ Ejk pﬃﬃﬃ ð90Þ Sði; Ejk Þ ¼ erfc 2Ejk s 2 GAIN EðiÞ Ejk EðiÞ Ejk 1 pﬃﬃﬃ þ pﬃﬃﬃ exp Tði; Ejk Þ ¼ ð91Þ erfc 2 2gs exp½1=ð2g Þ gs g 2 s 2 In these equations, s represents spectrometer resolution [Eq. (86)] and g is the broadening of the exponential tail. The parameters fS and fT in Eq. (89) describe the fraction of the photons that arrive in the step and the tail, respectively. The complement of the error function used to convolute the step and the tail is deﬁned as Z u 2 2 et dt ð92Þ erfcðuÞ ¼ 1 erfðuÞ ¼ 1 pﬃﬃﬃ p 0 and can be calculated via series expansion (Press et al., 1988). Figure 25b shows a ﬁt of a brass spectrum when these step and tail functions are included. In this case, a much more realistic area for the NiKa is obtained and the continuum is described more correctly. Campbell et al. (1987) used a Gaussian, a short-tail, and a long-tail exponential to ﬁt the peak in energy-dispersive spectra obtained from Ka1 and La1 lines selected by Bragg reﬂection from a curved crystal. In this way, the inﬂuence of the doublet structure and the satellite lines is eliminated. Excellent ﬁts with reduces chi-square values between 1.02 and 1.16 were obtained for peaks having 106 counts. Fitting real x-ray spectra with modiﬁed Gaussians as given by Eq. (89) increases dramatically the number of parameters that need to be optimized ( fS, fT, g) for each peak. The phenomena contributing to the deviation from the Gaussian shape depend on the energy of the detected x-ray line, so that these parameters can be expressed as smooth functions of energy (Wielopolski and Gardner, 1979; Yacout et al., 1986). These relations can be established using pure element spectra with very good counting statistics. An additional problem is that the tailing of the Kb peaks tend to be larger due to the more intense KMM radiative transition, thus requiring a separate relation for Ka and Kb tails. A further modiﬁcation of the ﬁtting function involves the use of a Voigt proﬁle to account for the natural line width of the x-ray lines. This becomes important when ﬁtting K lines of higher-Z elements (Ba and above) where the natural linewidth becomes considerable ( > 20 eV) compared to the detector resolution. The peak proﬁle, Gði; Ej Þ in Eq. (89) must then be replaced by a Voigt proﬁle (the convolution of a Lorenzian and a Gaussian): GAIN EðiÞ Ejk aL pﬃﬃﬃﬃﬃﬃ K pﬃﬃﬃ ; pﬃﬃﬃ ð93Þ s 2p s 2 2s 2 with K the real part of the complex error function and aL the natural width of the x-ray line:

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Van Espen

Kðx; yÞ ¼ Rebez erfcðizÞc; 2

z ¼ x þ iy

ð94Þ

Voigt proﬁles can be calculated using numerical approximations (Schreier, 1992). The imaginary part of the complex error function can be used to calculate the derivatives necessary in the least-squares-ﬁtting algorithm. An alternative procedure to describe the peak shape is used in the AXIL program (Van Espen et al., 1977b). The deviation from the Gaussian shape is stored as a table on numerical values, representing the diﬀerence of the observed shape and the pure Gaussian. The table extends from zero energy up to the high-energy side of the Kb peak and is normalized to the area of the Ka peak. The deviation is obtained from pure element spectra having very good counting statistics. (Ka 107 counts). Preferably, thin ﬁlms are used to keep the continuum as low as possible and to avoid absorption eﬀects. The area, position, and width of all peaks in the spectrum are determined by ﬁtting Gaussians on a constant continuum over the full width at tenth maximum (FWTM) of the peaks. The Gaussian contributions are then stripped from the spectrum. The resulting non-Gaussian part as shown in Figure 24 is further smoothed to reduce the channel to channel ﬂuctuations and subsequently used as a numerical peak-shape correction. The ﬁtting function for an element is then given by ( ) NP X 0 0 Rj Gði; Ej Þ ð95Þ yP ðiÞ ¼ A RKa ½Gði; EKa Þ þ CðiÞ þ j¼2

where CðiÞ is the numerical peak-shape correction at channel i. Values in the table are interpolated to account for the diﬀerence between the energy scale of the correction and the actual energy calibration of the spectrum. Similar to the parameters of the nonGaussian analytical functions, the shape of the numeric correction seems to vary slowly from one element to another. This allows us to interpolate the peak-shape correction for all elements from a limited set of experimentally determined corrections. A major disadvantage of this method is that it is quite diﬃcult and laborious to obtain good experimental peak-shape corrections. Although they are, in principle, detector dependent, experience has proven that the same set of corrections can be used for diﬀerent detectors with reasonable success, proving the fundamental nature of the observed non-Gaussian shape. Another disadvantage is that the peak-shape correction for the Kß becomes underestimated if strong diﬀerential absorption takes place because the peak-shape correction is only related to the area of the Ka peak. Also, it is nearly impossible to apply this method to the description of L-line spectra. A mayor advantage however is the computational simplicity of the method and the fact that no extra parameters are required in the model. 5. Absorption Correction The absorption correction term Ta ðEÞ, used in the Eqs. (82) and (88), includes the x-ray attenuation in all layers and windows between the sample surface and the active area of the detector. For high energetic photons also, the transparency of the detector crystal needs to be taken into account. In x-ray ﬂuorescence, the attenuation in the sample, causing additional changes in the relative intensities, can also be considered, providing the sample composition is known. The total correction term is thus composed of a number of contributions: Ta ðEÞ ¼ TDet þ TPath þ TSample The detector contribution for a Si(Li) detector is given by

ð96Þ

Spectrum Evaluation

TDet ðEÞ ¼ emBe ðrdÞBe emAu ðrdÞAu emSi ðrdÞSi ð1 emSi ðrDÞSi Þ

289

ð97Þ

where m, r, and d are the mass-attenuation coeﬃcient, the density, and the thickness of the Be window, the gold contact layer, and the silicon dead layer. In the last term, D is the thickness of the detector crystal. Any absorption in the path between the sample and the detector can be modeled in a similar way. For an air path, the absorption is given by TPath ðEÞ ¼ emair ðrdÞair

ð98Þ

The mass-attenuation coeﬃcient of air can be calculated assuming a composition of 79.8% N2, 19.9% O2, and 0.3% Ar. The density of air in standard conditions is 1.261073 g=cm3. In PIXE setups, it is common practice to use additional absorbers between the sample and the detector, with one absorber sometimes having a hole (funny ﬁlter). The absorption behavior of such a structure can be modeled by TPath ðEÞ ¼ emf ðrdÞf ½h þ ð1 hÞemff ðrdÞff

ð99Þ

where the ﬁrst term accounts for the absorption in (all) solid ﬁlters and the second term accounts for the absorption of the ﬁlter having a hole. The fraction of the detector solid angle subtended by the hole is denoted by h. The sample absorption correction in the case of x-ray ﬂuorescence, assuming excitation with an (equivalent) energy E0 , is given by Tsample ðEÞ ¼

1 ews ðrdÞs ws ðrdÞs

ð100Þ

and the sample attenuation coeﬃcient is given by ws ¼

ms ðEÞ ms ðE0 Þ þ sin y1 sin y2

ð101Þ

This sample attenuation coeﬃcient can only be calculated if the weight fractions of the constituting elements are known, which is often not the case because the aim of the spectrum evaluation is to obtain the net peak areas from which the concentrations are to be calculated. Although x-ray attenuation in solid samples might become very large, it is important to realize that not the absolute values of the absorption corrections but their ratio is of importance in Eq. (88). This ratio changes less dramatically especially because the energy diﬀerence of the lines involved is small. For an inﬁnitely thin Fe sample, the Kb=Ka intensity ratio is 0.134. In an inﬁnitely thick Fe matrix, the absorption correction term Ta ðEKb Þ and Ta ðEKa Þ are respectively 161 and 134, assuming MoKa excitation, causing the Kb=Ka ratio to change to 0.161. Therefore, a rough estimate of the sample composition is often suﬃcient. Van Dyck and Van Grieken (1983) demonstrated the integration of spectrum analysis with quantitative analysis. The sample absorption term is recalculated based on the estimated sample composition, and the spectrum ﬁtting is repeated. 6. Sum and Escape Peaks As indicated in Sec. II, Si escape peaks can be modeled by a Gaussian with an energy 1.750 keV below the parent peak. The area relative to the area of the parent peak can be calculated from the escape fraction [Eq. (4)]:

290

Van Espen

Z¼

Ne f ¼ Np 1 f

ð102Þ

Including the escape peaks, the description of the ﬂuorescence of an element becomes yP ðiÞ ¼ A

NP X

R0j ½Gði; Ej Þ þ ZGðði; Ej 1:750Þ

ð103Þ

J¼1

Also, various polynomial type functions expressing the escape ratio as a function of the energy of the parent peak are in use. The coeﬃcients of the function can be determined by least-squares ﬁtting from experimental escape ratios. For spectra obtained with a Ge detector, one needs to account in a similar way for both the GeKa and GeKb escape peak for elements above arsenic. The incorporation of the sum peaks in the ﬁtting model is more complex. The method discussed below was ﬁrst implemented by Johansson in the HEX program (Johansson, 1982). Initially, the spectrum is ﬁtted without considering pileup peaks. The peaks are then sorted according to their height and the n largest peaks are retained. Peaks that diﬀers less than 50 eV are combined to one peak. Using Eqs. (5) and (6), the relative intensities of all possible nðn þ 1Þ=2 pileup peaks and their energies are calculated and the m most intense are retained. Knowing the relative intensities and the energies of these m pileup peaks, they can be included in the ﬁtting model as one ‘‘pileup element.’’ In the next iteration, the total peak area A of this pileup element is obtained. The construction of the pileup element can be repeated during the next iterations as more reliable peak areas become available. In Figure 26, part of an PIXE spectrum is shown ﬁtted with and without sum peaks included. D.

Special Aspects of Least-Squares Fitting

1. Constraints When using nonlinear least-squares ﬁtting, it can be advantages to impose limits on the ﬁtting parameters to eliminate physically meaningless results. Some illustration of what can happen if too much freedom is given to the ﬁtting parameters is provided by Statham (1978). The incorporation of the energy and resolution calibration functions [Eqs. (84) and (86)] in the ﬁtting model already places a severe constraint on the ﬁt, avoiding the situation in which two peaks that are very close together swapping their positions or becoming one broader peak. The ﬁtting parameter can be eﬀectively constrained by deﬁning the real physically meaningful parameter Pj [e.g., the GAIN in Eq. (84)] as an arctangent function of the ﬁtting parameter aj (McCarthy and Schamber, 1981; Nullens et al., 1979): 2Lj arctan aj ð104Þ Pj ¼ P0j þ p where P0j is the expected value (initial guess) of the parameter Pj , and Lj deﬁnes the range. As a result of this transformation, the parameter Pj will always be in the range P0j Lj . Apart from signiﬁcantly adding to the mathematical complexity, such transformation has the disadvantage that the w2 minimum, although lying within the selected interval, cannot be reached because the path toward the minimum passes a ‘‘forbidden’’ region. Also, this constraint makes no distinction between the more probable values of the parameter lying near the center of the interval and the unlikely values at the limits.

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Figure 26 Fit of part of a PIXE spectrum having very high count rates for Fe and Zn. (Top: without considering sum peaks; bottom: sum peaks included in the fitting model.)

An alternative approach, proposed by Nullens et al. (1979), relies on the modiﬁcation of the curvature of the w2 surface. The ﬁtting parameters, such as ZERO, GAIN, NOISE, and FANO, are considered as random variables with an expected value equal to the initial guess a0j and having an uncertainty Daj . They can be included in the expression of w2 , just as the observed data points yi si : X 1 X 1 w2 ¼ ½ yi yðiÞ2 þ ½a0 aj 2 ð105Þ 2 2 j s Þ ðDa i j j i Using this expression in the Marquardt nonlinear least-squares-ﬁtting algorithm results in modiﬁed equations for the diagonal terms of the a matrix and for the b vector (see Sec. IX): X 1 @y0 ðiÞ2 1 þ ð106Þ ajj ¼ 2 @a si j ðDaj Þ2 i and bj ¼

X 1 @y0 ðiÞ a0j aj ½ y y ðiÞ þ i 0 @aj s2i ðDaj Þ2 i

ð107Þ

If no signiﬁcant peaks are present in the ﬁtting interval, the derivatives of the ﬁtting function with respect to the position and width of the peaks are zero. Therefore, the

292

Van Espen

second term in bj will dominate, causing the parameter estimate in the next iteration to be such that aj tends toward a0j (the initial value). If well-deﬁned peaks are present, however, the second term in b will be negligibly small compared to the ﬁrst term and the method acts as if no constraints were present. Another way to look at this is by considering the curvature of the w2 surface near the minimum. Figure 27 shows a cross section through the w2 surface along the peak position parameter. The variation of the w2 values, with and without constraints, are shown for a large (10,000 counts) and a small (100 counts) peak. The true peak position is at channel 100 and the FWHM of the peak is eight channels. The constraint, Da, on the peak position is one channel. From Figure 27, it is evident that the w2 minimum is much better deﬁned for a small peak when the constraint is applied, whereas this has no inﬂuence for a large peak. This method has been implemented in the AXIL program to constraint the energy and resolution calibration parameters. 2. Weighting the Fit The weights wi used in the least-square methods [Eq. (77)] are deﬁned in terms of the ‘‘true’’ (population) standard deviation of the data yi , which can be obtained directly from the data itself: wi ¼

1 1 2 yi si

ð108Þ

This approximation is valid for moderate to good statistics. When regions of the spectrum with very bad statistics (low channel contents) are ﬁtted, estimating the weight from the measured channel content causes a systematic bias leading to an underestimate of the peak areas. To overcome this problem, Phillips suggested estimating the weights from the

Figure 27 Effect of the use of a constraint on the shape of the w2 response surface. (Left: marginal effect for a large peak; right: important contribution for a small peak.)

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average of three channels (Phillips, 1978). In the AXIL program, the weights are initially based on the measured channel content, but when the overall w2 value of the ﬁt falls below 3, the weights are calculated from the ﬁtted channel content. This is based on the idea that when the calculated spectrum approached the measured spectrum, the calculated channel contents are a better estimate of the ‘‘true’’ channel content than the measured values. The eﬀect of the weighting on the ﬁt is considerable, as can be seen in Figure 28. E.

Examples

To illustrate the working of the nonlinear least-squares-ﬁtting method, an artiﬁcial spectrum with CuK and ZnK lines is ﬁtted with four Gaussian peaks on a constant continuum. Using the Marquardt algorithm, the position, width, and area of each peak are determined. The ﬁtting function thus is " # 4 X ði ajþ1 Þ2 ð109Þ aj exp yðiÞ ¼ a0 i þ 2a2jþ2 j¼1 with i the channel number (independent variable) and aj the parameters to be determined, 13 in total. In Table 10, the values of the parameters to generate the spectrum (true values), the initial guesses for the nonlinear parameters, and the ﬁnal ﬁtted values are given. The initial guesses were deliberately taken rather far away from the true values. Figure 29 (top) shows the ﬁtted spectrum after the ﬁrst and second iterations. During the second iteration, the Marquardt algorithm evolved into a gradient search, drastically changing the position and width of the peaks. Even after ﬁve iterations, the calculated spectrum deviates considerably from the ‘‘measured’’ spectrum, as can be seen from Figure 29 (bottom).

Figure 28 The effect of weighing the least-squares fit is shown on part of an PIXE spectrum with a very small number of counts per channel.

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Table 10 True Value of Spectrum Parameters Used to Generate the Artiﬁcial Spectrum in Figure 29, Initial Guesses and Fitted Values of the Peak Area, Position, and Width Obtained by Nonlinear Least-Squares Fitting Parameter Area (counts) CuKa CuKb ZnKa ZnKb Position (channel number) CuKa CuKb ZnKa ZnKb Width (channels) CuKa CuKb ZnKa ZnKb

True value 100,000 13,400 30,000 4,106 402.05 445.25 431.55 478.60 3.913 4.033 3.995 4.123

Initial guess

Fitted value

0 0 0 0

100,134±321 13,163±169 30,092±213 4138±83

395 450 435 485

402.03±0.01 445.35±0.06 431.59±0.04 478.68±0.09

3 3 3 3

3.91±0.01 3.99±0.05 4.02±0.03 4.06±0.08

Finally, after 10 iterations, a perfect match between them is obtained, with a reduced w2 value of 0.96. From Table 10, it is evident that the ﬁt was quite successful, with all peak areas, positions, and widths estimated correctly within the calculated standard deviation. One observes that the uncertainties in the peak areas are approximately equal to the square root of the peak area and that the position and the width of the peaks are estimated very precisely (within 0.01 channel or 0.2 eV), especially for the larger peaks. By observing how the iteration procedure changes the peak width and position parameters, one can imagine that something might go wrong. Especially if the spectrum is more complex, chances are high that the iteration stops in a false minimum or even drifts away completely. In both cases, physically incorrect parameter estimates will be obtained. In practice, it is course possible to give much better initial estimates for the peak position and width parameters than used in this example. In a next example, a complex spectrum (geological reference material JG1, excited with MoK x-rays from a secondary target system) is evaluated using nonlinear leastsquares ﬁtting. In Figure 30, the spectrum and the ﬁt are shown together with the residuals of the ﬁt (see p. 298). Due to the large number of overlapping lines, the method used in the ﬁrst example (ﬁtting the position and width of each peak independently) is not applicable in this case. For the description of the spectrum from 1 to 18 keV, the ﬂuorescence lines of 21 elements were used. Au, Hg, Pb, and Th were treated each as one L group (L1 þ L2 þ L3 ) and Al, Si, Ti, V, Cr, Mn, Ni, Cu, Zn, Ga, Al, Br, Sr, Rb, Y as one K group (Ka þ Kb); K, Ca, and Fe were ﬁtted with individual Ka and Kb peaks. The coherently scattered MoKa radiation was ﬁtted with a Gaussian and the incoherent scattered MoKa radiation is ﬁtted with a Voigt proﬁle. Including escape and sum peaks, this amounts to well over 100 peak proﬁles. Step and tail functions [Eq. (89)] were included for all peaks, using expressions to relate the step and tail heights and the tail width to the energy of the peak. The continuum is described by an exponential function [Eq. (81)] with six parameters. The least-squares ﬁt thus performed required the reﬁnement of 37 parameters

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Figure 29 Artificial CuK and ZnK line spectrum ﬁtted with a nonlinear least-squares procedures to optimize peak area, position, and width. (Top: Fitted spectrum after ﬁrst and second iteration; bottom: after ﬁfth and ﬁnal iterations.)

(4 calibration parameters, 27 peak areas, 6 continuum parameters, 3 step and tail parameters, and 1 Voigt parameter). The minimum w2 value of 1.47 is obtained after 16 iterations. The residuals indicate an overall good ﬁt with most of the residuals in the 73 to þ3 interval, without any systematic patterns. Its interesting to note that the ﬁtted continuum is well below the base of the peaks, especially in the region from channel 200 to 600. The continuum describes correctly the small scattered bremsstrahlung contribution from the x-ray tube above 12 keV (channel 600) and the Compton scattering in the detector at the low-energy side. Most of the apparent continuum in this secondary-target EDXRF spectrum is due to incomplete charge collection and tailing phenomena of the scattered Mo excitation radiation and the ﬂuorescence lines. F.

Evaluation of Fitting Results

In order to understand and appreciate the capabilities and limitations of least-squares ﬁtting, either using library spectra, linear, or nonlinear analytical functions, it is useful to study the eﬀect of random and systematic errors in some detail. Random errors are associated with the uncertainty si of the channel content yi . As will be seen, these uncertainties inﬂuence the precision of the net peak areas and determine the ultimate resolving power of the least-squares method. Systematic errors, on the other hand,

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Figure 30 Example of the fit of a complex x-ray spectrum (thin film deposit of geological reference material JG1 excited with a Mo secondary-target system). The residuals are plotted as an indicator of the quality of the fit.

are caused by discrepancies between the ﬁtting model and the observed data and cause inaccuracies in the net peak areas. 1. Error Estimate Section IX explains how the least-squares-ﬁtting method (linear as well as nonlinear) allows the estimation of the uncertainties in the ﬁtted parameters. These uncertainties result from the propagation of the statistical ﬂuctuations in the spectral data into the parameters. Intuitively, one could come to the conclusion that the standard deviation of the peak area should be close to the square root of the peak area. This is indeed the case for a large, interference-free peak on a low continuum, but if the peak is sitting on a high continuum and=or is seriously overlapped by another peak, the uncertainty in the estimated peak area will be much larger. A properly implemented least-squares method not only correctly estimates the net peak areas but also their uncertainty, taking into account the continuum and the degree of peak overlap, providing, of course, that the ﬁtting model is capable to describe the measured spectrum. The closer together the peaks are, the higher the uncertainties in the two peak areas will become. (Theoretically, in the limit of complete overlap, the uncertainty will be inﬁnite and the area of the two peaks will have complete erratic values, but their sum will still represent correctly the total net peak area of the two peaks; in practice, the curvature matrix a will be singular so that the matrix inversion fails.)

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The uncertainties in the net peak areas can be used to decide if a peak is indeed present in the spectrum and to calculate the detection limit. A peak area is statistically not signiﬁcant diﬀerent from zero if its value is in the range ±3s. Any value above 3s gives clear evidence that the peak is present; any value below 73s would indicate that there is something wrong with the model because truly negative peak areas are physically meaningless. Because the uncertainty in the net peak area includes the inﬂuence of continuum and peak overlap, it can be used to calculate the a posteriori detection limits of the elements (peaks) present in the spectrum. Three situations can occur: 1. Estimated peak area > 36 standard deviation ? report: area±standard deviation 2. 736 standard deviation estimated area 36 standard deviation ? report: detection limit equal to 36 standard deviation 3. Area < 736 standard deviation ? revise the ﬁtting model 2. Criteria for Quality of Fit From the deﬁnition of the least-squares method, it follows that the w2 value [Eq. (75)] estimates how well the model describes the data. The reduced w2 value, obtained by dividing w2 by the number of degrees of freedom; 1 1 w2 w2n ¼ w2 ¼ n nm

ð110Þ

has an expected value of 1 for a ‘‘perfect’’ ﬁt. The number of degrees of freedom equals the number of data points (n) minus the number of parameters (m) estimated during the ﬁt. Because w2 is also a random variable, the observed value will often be slightly larger or smaller than 1. Actually, w2 follows (approximately) a chi-square distribution and the 90% conﬁdence interval is given by w2n;0:95 w2n w2n;0:05

ð111Þ

where w2n;0:95 and w2n;0:05 are the tabulated w2 values at the 95% and 5% conﬁdence intervals, respectively. For n ¼ 20, the conﬁdence interval is 0.5431.571, and for n ¼ 200, it is 0.8411.170. An observed value of w2n in this interval indicates that the model describes the experimental data within the statistical uncertainty of the data. In other words, all remaining diﬀerences between the data and the ﬁt can be attributed to the noise ﬂuctuations in the channel contents and are statistically not signiﬁcant. When ﬁtting complex spectra with good statistics, much higher chi-square values will be obtained due to small imperfections in the continuum or peak model. This does not mean that the estimated peak areas are not useful any more, but high reduced w2 values ( > 3) might indicate that the ﬁtting model needs improvement. The reduced chi-square value as deﬁned in Eq. (110) gives an estimate of the overall ﬁt quality over the entire ﬁtting region. Locally, in some part of the spectrum, the ﬁt might actually be worse than indicated by this value. It is therefore useful to deﬁne the chi-square value for each peak separately: w2P ¼

m 1 X 1 ½yi yðiÞ2 n1 n2 i¼n2 s2i

ð112Þ

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where n1 and n2 are the boundaries of the peak at FWTM and n2 n1 approximates the number of degrees of freedom. High values of w2P indicate that the peak is ﬁtted poorly and the resulting peak area should be used with caution. In this case ðw2P > 1Þ, it is advisable to give a conservative estimate for the uncertainty in the net peak area, by multiplying the calculated uncertainty with the square root of the w2 value: qﬃﬃﬃﬃﬃ sA0 ¼ sA w2P ð113Þ Although the w2 values gives an indication of the goodness of ﬁt, visual inspection of the ﬁt is highly recommended. Because of the large dynamic range of the data, a plot of the spectrum and the ﬁt on a linear scale nearly always give the impression of a perfect ﬁt. A plot of the logarithm or the square root of the data is more appropriate. The best method is to plot the residuals of the ﬁt as in done in Figure 30. The residuals are deﬁned as ri ¼

yi yðiÞ si

ð114Þ

It is the sum of the squares of these residuals that were minimized by the least-squares method. Values in excess of þ3 or 73 and especially the presence of a pattern in the residuals then indicate poorly ﬁtted regions. G.

Available Computer Codes

In the literature, a number of computer programs for spectrum evaluation based on the least-squares method are reported. Without attempting to be complete, the main characteristics of a number of programs are summarized in the following paragraphs. Wa¨tjen made a compilation of the characteristics of eight computer packages for PIXE analysis (Wa¨tjen, 1987). An intercomparison of ﬁve computer program for the analysis of PIXE spectra revealed a very good internal consistency among the ﬁve programs (Campbell et al., 1987). PIXE spectra of biological, environmental, and geological samples were used and their complexity put high demands of the spectrum analysis procedures. The following programs were tested: AXIL, University of Gent, Belgium; HEX, University of Lund, Sweden; SESAM-X (Marburg, FRG), the Guelph program; PIXAN, Lucas Heights. It was concluded that the most serious disagreement occurred for small peaks on the low-energy tails of very large peaks, pointing to a need to a more accurate description of the tail functions. Also, very good agreement between the linear (SESAM-X) and the nonlinear least-squares approach was observed. The Los Alamos PIXE Data Reduction software (Duﬀy et al., 1987) contains three components. The K and L relative x-ray intensities of the elements making up the sample are computed taking into account the detector and sample absorption. Using Gaussian peak shape, the energy and resolution calibrations of the spectrum are calculated. With the relative peak areas and the calibration functions obtained is this way, the spectrum is ﬁtted using a Gaussian peak shape and a polynomial continuum. Escape and pileup peaks can be included. A linear least-squares ﬁt is done with the relative elemental concentrations and the polynomial continuum coeﬃcient as unknowns. The continuum and the relative concentrations are constrained to non-negative values and all elements having x-ray lines in the spectrum interval considered are included. The PIXASE computer package (Zolnai and Szabo´, 1988) performs spectrum analysis using nonlinear least-squares ﬁtting. Elements are represented by groups of lines with

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ﬁxed, absorption-corrected, relative intensities and including escape peaks. Each peak is modeled by a Gaussian with the addition of a exponential tail and an error function as step. The square of the peak width is a ﬁrst-order polynomial of the peak energy and the position is a second-order polynomial of the peak energy. The continuum is described as the sum of an exponential polynomial and two simple exponentials. Pileup eﬀects are treated as one pileup element. The nonlinear least-squares ﬁtting is done using a simple grid search technique. The search space of each parameter is limited by user-supplied minimum and maximum values. Fitting a large series of similar spectra linear least-squares ﬁtting using a library of calculated spectrum components can be done. Bombelka et al. (1987) described an PIXE analysis program based on linear least squares. The peak shape includes a Gaussian low-energy tail function to account for the incomplete charge collection and the escape peak. The position and the square of the width of the peaks are given by ﬁrst-order linear function of the energy. The ﬂuorescence lines of an element are modeled as a sum of those peak shapes with relative intensity ratios corrected for absorption in the detector windows and absorbers. The continuum is composed of a fourth-order exponential polynomial multiplied by the x-ray attenuation term (bremsstrahlung continuum) and a second-order linear polynomial. Pileup is taken into account as a pileup element. The energy and resolution calibration parameters are obtained from selected peaks in the spectrum. The parameters of the exponential continuum are calculated from continuum spectra. The parameters obtained by the linear leastsquares ﬁt are the amplitude parameters of each element and of one pileup element, the amplitude parameter of the bremsstrahlung continuum, and the linear polynomial continuum parameters. The computer implementation for PIXE is called SESAMX and is highly interactive with graphical representation of the spectral data. Other computer program for tube-excited (Breschinsky et al., 1979) and synchrotron radiation- (Petersen et al., 1986) excited XRF were developed based on this code. SAMPO-X (Aarnio and Lauranto, 1989) is intended for the analysis of electron induced x-ray spectra and is based on the well-known SAMPO code original developed for g-ray spectroscopy (Routti and Prussin, 1969). A Gaussian with two exponential tails, as in the original SAMPO program, is used to represent the peaks. The height and the position parameters are obtained by the nonlinear least-squares ﬁt. The peak width and the tail parameters are obtained from shape-calibration tables by interpolation. The continuum is modeled by the semi-empirical electron bremsstrahlung intensity function proposed by Pella et al. (1985). The thickness of the detector beryllium window and the atomic number of the sample, which occur in this formula, are adjusted by the leastsquares ﬁt. The program also includes an element identiﬁcation based on the energy and intensity of the ﬁtted peaks and a standard ZAF matrix correction. Jensen and Pind (1985) described a program for the analysis of energy-dispersive xray spectra. The program uses a sum of Gaussians, one for each ﬂuorescence line. The continuum is subtracted ﬁrst using a linear, parabolic, or exponential function ﬁtted from peak-free regions in the spectrum. The peak width is obtained from a calibration function which expressed the log of the peak width as a linear function of the peak position. The width calibration is done using nonoverlapping peaks in a calibration spectrum or in the spectrum to analyze. Peak positions are determined using a peak search method or entered by the operator with the aid of a graphical display of the spectrum. The peak heights are then determined using linear least-squares ﬁtting. The computer code developed at the technical University of Graz (Marageter et al., 1984a) is primary intended for the evaluation of x-ray ﬂuorescence spectra. A Gaussian response function with an low-energy tail is used to describe the peaks. The square of the

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Van Espen

peak width is a linear function of the peak energy. A straight-line equation relates the peak position to the energy of the peaks. A parabola is used to describe the continuum, whereas absorption edges are modeled by a complementary error function. The ﬁtting parameters are the peak heights, the three continuum parameters, the height of the absorption edges, and the two energy calibration parameters. Nonlinear least-squares ﬁtting is done with the Marquardt algorithm using a tangent transformation to constrain the ﬁtting parameters to physical meaningful values (Nullens et al., 1979). Provision for escape peaks and Auger peaks is also made (Marageter et al., 1984b). The AXIL program (Van Espen et al., 1977a) was originally developed for the analysis of x-ray ﬂuorescence spectra and later modiﬁed to allow the evaluation of electronand particle-induced x-ray spectra (Van Espen et al., 1979b, 1986). It uses the Marquardt algorithm for nonlinear least-squares ﬁtting with a modiﬁed (constrained) chi-square function. Linear, exponential, and bremsstrahlung polynomials can be used to model the continuum as well as continuum stripping. X-ray lines are described by Gaussian functions with an optional numerical peak-shape correction. Escape and sum peaks can be included in the model. The peak position and the square of the peak width is related to the x-ray energy by linear functions. Provision is made to correct for the absorption in detector windows, ﬁlters, and the sample. In a later version, an orthogonal polynomial continuum model was added, as well as step and tail functions to describe the deviation from the Gaussian peak shape (Vekemans et al., 1994). The code was implemented as a Windows application. An example of the screen output of this program is given in Figure 31.

VIII.

METHODS BASED ON THE MONTE CARLO TECHNIQUE

Monte Carlo techniques for the simulation of x-ray spectra are becoming more and more popular, particularly because of the fast computers available today. These simulated spectra are useful for studying the behavior and performance of various spectrum processing methods. The Monte Carlo technique also can be used in quantitative analysis procedures, as will be discussed in Sec. VIII. B. A.

Simulation of X-ray Spectra

During the development and test phase of a spectrum processing method, it is often advantageous to use computer-simulated spectra. For these spectra, such features as position, width, and area of peaks are exactly known in advance. They can be generated to any desired complexity. In order to make any real use of them, the simulated spectra must possess the same channel-to-channel variation according to a Poisson distribution as experimental spectra. A simple and adequate procedure consist of ﬁrst calculating, over the channel range of interest, the ideal spectrum y0 using, for example, a polynomial for the continuum and a series of Gaussians as given by Eq. (109). More complex functions, including a more physically realistic model for the continuum and tailed peaks, can be used if desired. The next step is to add or subtract some number of counts from each channel content so as to obtain data with the desired counting statistical noise. In other words, the ‘‘true’’ content y0i needs to be converted into a random variable, yi , so that is obeys a Poisson distribution [Eq. (1)] with N0 ¼ y0 . Poisson-distributed random variables can be generated by various computer algorithms. An example (Press, 1988) is given in Sec. X. For y0 > 30 the Poisson-distributed

Figure 31

Screen capture of the spectrum analysis program WinAxil, showing the fitted spectrum and the results obtained.

Spectrum Evaluation 301

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Van Espen

random variable can be approximated by the much easier to calculate normal-distributed random variable. The probability to observe y counts in a channel, assuming a normal distribution, is given by ðymÞ2 1 ð115Þ PðyÞ ¼ pﬃﬃﬃﬃﬃﬃ e 2s2 s 2p with m ¼ s2 ¼ y0 . For large y0 the normal distribution is a very good approximation of the Poisson distribution. Even for small channel contents this approximation is quite satisfactory. The probability of observing 6 counts, assuming the true value is 4, is 0.10 according to a Poisson distribution, whereas it is 0.12 according to a normal distribution. The problem of adding counting statistics to the calculated spectrum is thus reduced to calculating a normal-distributed random variable y with mean value m ¼ y0 and variance s2 ¼ y0 . Starting from a uniformly distributed random variable U in the interval (0, 1), which can be generated by a pseudo-random number generator, normal distributed random variables with zero mean and unit variance can be obtained with the BoxMuller method (Press, 1988). n1 ¼ 2Ui 1 n2 ¼ 2Uiþ1 1 r¼

n21

þ

ð116Þ

n22

if r < 1

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 ln r=r pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ z2 ¼ n2 2 ln r=r

z1 ¼ n 1

n1 and n2 and r are calculated from two uniformly distributed random numbers Ui and Uiþ1 . If r is less than 1, two normally distributed random numbers z1 and z2 can be calculated. If r 1; n1 ; n2 ; and r are recalculated using a new set of uniform random numbers. The normally distributed random number y, with mean m ¼ y0 and variance s2 ¼ y0 , is then obtained by simple scaling: pﬃﬃﬃﬃﬃ y ¼ m þ zs ¼ y0 þ z y0 ð117Þ Applying this to all channels will produce the desired counting statistics. Since z is normally distributed with mean 0 and unit variance, z can be negative as well as positive. The count p rate in each channel will thus be increased or decreased randomly and proportional to y0 . The ﬁnal step in the computer simulation is to convert the real numbers that were used during the calculation to integers. Another interesting procedure is to generate artiﬁcial spectra from parent spectra. The parent spectrum is a spectrum acquired for a very long time so that it exhibits extremely good counting statistics (high channel content). A large number of child spectra with lower and varying counting statistics can be generated by the procedure explained below. This method is useful to study the eﬀect of counting statistics on spectrumprocessing algorithms (Ryan, 1988). From the parent spectrum yi , which might be ﬁrst smoothed to reduce the noise even further providing not too much distortion is introduced, the normalized cumulative distribution function Yj is calculated Pj yi ð118Þ Yj ¼ Pi¼0 n i¼0 yi

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n being the number of channels in the spectrum and Yj in the interval (01). To generate the child spectrum we select N times a channel i according to the equation i ¼ Y1 ðUÞ

ð119Þ

where Y1 is the inverse cumulative distribution and U is a uniformly distributed random number. Each time channel i is selected, one count is added to that channel. Since N is the total number of counts in the child spectrum, the counting statistics can be controlled by varying N. However, Y1 cannot be expressed as an analytical function, but for each random number U we select the channel i for which Yi U < Yiþ1 . Figure 32 shows an EDXRF spectrum from a 0.187 mg=cm2 pellet of IAEA Animal Blood reference material acquired for 1000 s with a Tracor Spectrace 5000 instrument, which is used as a parent spectrum. The cumulative distribution function and a child spectrum simulated with N ¼ 36104 are also shown. The total number of counts in the original spectrum is 36106. The child spectrum is equivalent to a spectrum that would have been acquired for 10 s. Some computer routines useful for simulation experiments are given in Sec. X.

Figure 32 Simulation of a ‘‘child’’ spectrum from a ‘‘parent.’’ (Top: original spectrum; middle: the cumulative distribution; bottom: generated child.)

304

B.

Van Espen

Spectrum Evaluation Using Monte Carlo Techniques

Another interesting application of the Monte Carlo technique is the simulation of the complete response of an x-ray ﬂuorescence setup. Given an excitation spectrum and the excitation-detection geometry the interactions of the primary photons with the sample are simulated and the events giving rise to detectable phenomena are registered. With such a Monte Carlo simulation, the intensity of the characteristic lines and the scattered excitation spectrum are estimated, taking primary and secondary eﬀects (absorption, enhancement, and single and multiple scattering) into account. The obtained spectrum is one as seen by an ideal detector with inﬁnite resolution. This spectrum can then be convolution with the response function of a real detector to mimic a typical observed pulse-height spectrum. Apart from a detailed Monte Carlo simulation code, this technique requires detailed knowledge of the excitation spectrum and a very accurate detector response function. Doster and Gardner were among the ﬁrst to apply this technique to simulate the complete spectral response of an EDXRF system (Doster and Gardner, 1982a; Doster and Gardner, 1982b; Yacout and Dunn, 1987). More recently, Janssens and coworkers developed highly eﬃcient computer codes for the simulation of conventional and synchrotron energy-dispersive x-ray ﬂuorescence systems, allowing such conditions as polarized radiation and heterogeneous samples to be taken into consideration (Janssens et al., 1993; Vincze et al., 1993; Vincze et al., 1995a, 1995b; Vincze et al., 1999). Figure 33 shows the results of the application of this Monte Carlo simulation for a NIST SRM 1155 steel sample excited with a ﬁltered Rh-anode x-ray tube. At the top the simulated spectrum as seen by a perfect detector is shown, at the bottom the spectrum after convolution with a suitable detector response function is compared with a real measured spectrum of this sample. Except for some sum peaks that occur in the measured spectrum, an excellent agreement between the simulated and measured ﬂuorescence lines, scattered peaks, and continuum is obtained. These simulated x-ray spectra can be used in various ways in quantitative analysis. Using the complete spectral response of spectrometer, one can try to ﬁnd a sample composition that minimizes (in the least-squares sense) the diﬀerence between the simulated and the measured spectrum. The analysis involves the following steps: (1) simulation of the X-ray intensities over the expected composition range of the unknowns; (2) convolution with the detector response function to obtain spectra; (3) construction of a w2 map (weighted sum of squares of diﬀerences between simulated spectra and experimental spectrum) as a function of the sample composition; (4) interpolation of w2 for the composition corresponding to the minimum. An interesting aspect of this method is that all the information present in the spectrum (characteristic lines, scattered radiation, continuum) is considered. Doster and Garnder demonstrated an analytical accuracy of the order of 2% absolute for the analysis of Cr-Fe-Ni alloys with a 109Cd radioisotope system (Doster and Gardner, 1982a). Based on this work, Yacout and Dunn (Yacout and Dunn, 1987) demonstrated the use of the inverse Monte Carlo method, which requires in principle only one simulation to analyze a set of similar samples. Another method is called Monte Carlolibrary least-squares analysis (Verghese et al., 1988). Starting from an initial guess of the composition of an unknown sample, a spectrum is simulated taking into account all the interactions in the sample. During the simulation one keeps track of the response of each element to construct library spectra. After the simulation these library spectra are used to obtain the elemental concentrations by linear least-squares ﬁtting (see Sec. VI). In the concentration in the unknown samples diﬀer too much from the initial assumed concentration, the simulation is repeated.

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Figure 33 Monte Carlo simulation of the spectral data from an NIST SRM 1155 sample excited with a Rh-anode x-ray tube. (Top: simulated spectrum as seen by an ideal detector; bottom: simulated spectrum after convolution with the detector response function and comparison with measured spectrum.)

In contrast to the normal library least-squares method, this method has the advantage that the library spectra are simulated for a composition close to the composition of the spectrum to analyzed, rather than measured from standards. This eliminates the necessity of applying the top-hat ﬁlter and problems related to changes in Kb=Ka ratios, and again the method combines spectrum evaluation with quantitative analysis. Finally the ability to simulate x-ray spectra that agree very well with real measured spectra opens the possibility to used them as ‘‘standards’’ for the quantitative analysis based on partial least-squares regression (see Sec. VI.B). Indeed, as this method only functions correctly if the PLS model is built using a large number of standards covering the entire concentration domain, it seems advantageous to use simulated spectra for this. All the inter-element interactions can be accounted for by the simulation and only a few real standards are required to scale the simulated spectra.

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IX. THE LEAST-SQUARES-FITTING METHOD The aim of the least-squares method is to obtain ‘‘optimal’’ values for the parameters of a function that models the dependence of experimental data. The method has its roots in statistics but is also considered part of numerical analysis. The least-squares parameter estimate, also known as curve ﬁtting, plays an important role in experimental science. In x-ray ﬂuorescence, it is used in many calibration procedures and it forms the basis of a series of spectrum analysis techniques. In this section, an overview of the least-squares method with emphasis on spectrum analysis is given. Based on the type of ﬁtting function, one makes a distinction between linear and nonlinear least-squares ﬁtting because one requires numerical techniques of diﬀerent complexity to solve the problem in the two cases. The linear least-squares method deals with the ﬁtting of functions that are linear in the parameters to be estimated. For this problem, a direct algebra solution exists. If the ﬁtting function is not linear in one or more of the parameters, one uses nonlinear least-squares ﬁtting and the solution can only be found iteratively. A group of linear functions of general interest are the polynomials, the straight line being the simplest case. The special case of orthogonal polynomials will also be considered. If more than one independent variable ðx1i ; x2i ; . . . ; xmi Þ is associated with each measurement of the dependent variable yi , one speaks of multivariate regression. Spectrum analysis using the library function (e.g., ﬁlter-ﬁt method) belongs to this category. If analytical function (e.g., Gaussians) are ﬁtted to a spectrum, the method of linear or nonlinear least square is used, depending on whether nonlinear parameters, such as the peak position and width, are determined. A.

Linear-Least Squares

Considered the problem of ﬁtting experimental data with the following linear function: y ¼ a1 X1 þ a2 X2 þ þ am Xm

ð120Þ

This function covers all linear least-squares problems. If m ¼ 2; X1 ¼ 1, and X2 ¼ x, the straight-line equation y ¼ a1 þ a2 x is obtained. For m > 2 and Xk ¼ xk1 , Eq. (120) is a polynomial y ¼ a1 þ a2 x þ a3 x2 þ þ am xm1 to be ﬁtted to the experimental data points fxi ; yi ; si g i ¼ 1; . . . ; n. If X1 ; . . . ; Xm represents diﬀerent independent variables, the case of multiple linear regression is dealt with. Because of this generality, we will discuss the linear least-squares method based on Eq. (120) in detail. Assume a set of n experimental data points: fx1i ; x2i ; . . . ; xmi ; yi ; si g;

i ¼ 1; . . . ; n

ð121Þ

with xki the value of the kth independent variables Xk in measurement i, assumed to be known without error, and yi the value of the dependent variable measured with uncertainly si . The optimum set of parameters a1 ; . . . ; am that gives a least-squares ﬁt of Eq. (120) to these experimental data are those values of a1 ; . . . ; am that minimize the chisquare function: w2 ¼

n X 1 ðyi a1 X1 a2 X2 am Xm Þ2 2 s i i¼1

ð122Þ

The minimum is found by setting the partial derivatives of w2 with respect to the parameters to zero:

Spectrum Evaluation n X @w2 1 ¼ 2 ðyi a1 X1 a2 X2 am Xm ÞXk ¼ 0; 2 @ak s i i¼1

307

k ¼ 1; . . . ; m

ð123Þ

Dropping the weights 1=s2i temporarily for clarity, we obtain a set of m simultaneous equations in the m unknown ak : X X X X yi X 1 ¼ a1 X1 X1 þ a2 X2 X1 þ þ am Xm X1 X X X X yi X 2 ¼ a1 X1 X2 þ a2 X2 X2 þ þ am Xm X2 ð124Þ .. . X X X X yi X m ¼ a1 X 1 X m þ a2 X2 Xm þ þ am Xm Xm where the summations run over all experimental data points i. These equations are known as the normal equations. The solution—the values of ak —can easily be found using matrix algebra. Because two (column) matrices are equal if their corresponding elements are equal, the set of equations can be in matrix form as X X 3 2 X 3 2X a1 X1 X1 þ a2 X 2 X 1 þ þ am Xm X1 yi X1 X X X X 7 6 7 6 6 6 yi X2 7 X1 X2 þ a2 X 2 X 2 þ þ am Xm X2 7 7 6 a1 7 6 ð125Þ 7¼6 7 6 .. .. 6 7 6 7 4X . 5 4 X 5 . X X yi Xm X 1 X m þ a2 X2 Xm þ þ am Xm Xm a1 The right-hand column matrix can be written as the product of a square matrix a and a column matrix a: X X 3 2 X 3 2X X1 X1 yi X1 X2 X1 Xm X1 2 a1 3 X X 7 6 X 76 7 6X 6 6 yi X2 7 X1 X2 X2 X2 Xm X2 7 a2 7 7 6 76 6 6 . 7 ¼ ð126Þ 7 7 6 6 6 . .. 7 6 74 .. 7 6 5 5 4X 5 4 X .. . X X a2 yi Xm X1 Xm Xm Xm X2 Xm or b ¼ aa

ð127Þ

This equation can be solved for a by premultiplying both sides of the equation with the inverse matrix a1 , a1 b ¼ a1 aa ¼ Ia

ð128Þ

or, I being the identity matrix, a ¼ a1 b Introducing the weights again, the elements of the matrices are given by n X 1 bj ¼ y X ; j ¼ 1; . . . ; m 2 i j s i i¼1 n X 1 Xk Xj ; j ¼ 1; . . . ; m; k ¼ 1; . . . ; m ajk ¼ s2i i¼1 and

ð129Þ

ð130Þ ð131Þ

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Van Espen

aj ¼

m X

a1 jk bk ;

j ¼ 1; . . . ; m

ð132Þ

k¼1 1 where a1 jk are the elements of the inverse of the matrix a . The uncertainty in the estimate of aj is due to the uncertainty of each measurement multiplied by the eﬀect that measurement has on aj : 2 n X @aj s2aj ¼ s2i ð133Þ @yi i¼1

Because a1 jk is independent of yi , the partial derive is simply m @aj 1X ¼ 2 a1 Xk ðiÞ @yi si k¼1 jk

After some rearrangements, it can be shown that " # m X m n X X 1 2 1 1 saj ¼ ajk ajl Xk Xl s2i i¼1 k¼1 l¼1 the term in the brackets being akl and XX 1 1 s2aj ¼ a1 jk ajl akl ¼ ajj k

ð134Þ

ð135Þ

ð136Þ

l

This results in the simple statement that the variance (square of uncertainty) of a ﬁtted parameter aj is given by the diagonal element j of the inverse matrix a1 . The oﬀ-diagonal elements are the covariances. For this reason, a1 is often called the error matrix. Similarly, a is called the curvature matrix because the elements are a measure of the curvature of the w2 hypersurface in the m-dimensional parameter space. It can easily be shown that X 1 1 @ 2 w2 ¼ Xk Xj ¼ ajk 2 @aj @ak s2i i

ð137Þ

If the uncertainties in the data points si are unknown and are the same for all data points ðsi ¼ sÞ, these equations can still be used by setting the weights wi ¼ 1=s2i to 1. Assuming the ﬁtting model is correct, s can be estimated from the data: 1 X s2i ¼ s2 s2 ¼ ðyi a1 X1 a2 X2 am Xm Þ2 ð138Þ nm i The uncertainties in the parameters are then given by s2aj ¼ s2 a1 jj

ð139Þ

If the uncertainties in the data points are known, the reduced w2 value can be calculated as a measure of the goodness of ﬁt: 1 X 1 1 w2 w2n ¼ ðyi a1 X1 a2 X2 am Xm Þ2 ¼ ð140Þ n m i s2i nm The expected value of w2n is 1.0, but due to the random nature of the experimental data values slightly smaller or greater than 1 will be observed even for a ‘‘perfect’’ ﬁt. w2n follows a chisquare distribution with n m degrees of freedom, and a 90% conﬁdence interval is deﬁned by

Spectrum Evaluation

309

w2 ðn; P ¼ 0:95Þ w2n w2 ðn; P ¼ 0:05Þ

ð141Þ

where w2 ðn; PÞ is the (tabulated) critical value of the w2 distribution for n degrees of freedom at a conﬁdence level P. Observed w2n values outside this interval indicate a deviation between the ﬁt and the data that cannot be attribute to random statistical ﬂuctuations. B.

Least-Squares Fitting Using Orthogonal Polynomials

A special group of linear functions are orthogonal polynomials. Orthogonality means that the polynomial terms are uncorrelated, and this has some distinct advantages. Let Pj ðxi Þ be an orthogonal polynomial of degrees j; a function can then be constructed as a sum of these orthogonal polynomials of successive higher degree: m X yðiÞ ¼ Cj Pj ðxi Þ ð142Þ j¼0

The least-squares estimates of the coeﬃcient Cj are determined by minimizing the weighted sum of squares: n X wi ðyi yðiÞÞ2 ð143Þ w2 ¼ i¼1

which results in a set of m þ 1 normal equations in the m þ 1 unknown. Because the Pj ðxi Þ are a set of orthogonal polynomials, they have the property that n X wi Pj ðxi ÞPk ðxi Þ ¼ gk djk ð144Þ i¼1

with gk a normalization constant and djk ¼ 0 for j 6¼ k. Because of this property, the matrix of the normal equations is diagonal and the polynomial coeﬃcients are directly obtained from n X wi yi Pj ðxi Þ ð145Þ Cj ¼ gj i¼1 and the variance of the coeﬃcient is given by 1 s2cj ¼ gj

ð146Þ

Another advantage of the use of orthogonal polynomials is that the addition of one extra term Cmþ1 Pmþ1 does not change the values of the already determined coeﬃcients C0 ; . . . ; Cm . Further, if the yi are independent, then also the Cj are independent; that is, the variancecovariance matrix is also diagonal. As a result, much higher-degree orthogonal polynomials can be ﬁtted compared to ordinary polynomials without running into problems with ill-conditioned normal equations and oscillating terms. Orthogonal polynomials can constructed by following recurrence relation: Pjþ1 ðxi Þ ¼ ðxi aj ÞPj ðxi Þ bj Pj1 ðxi Þ

j ¼ 0; . . . ; m 1

aj and bj are constants independent of yi given by Pn wi xi ½Pj ðxi Þ2 aj ¼ i¼1 ; j ¼ 0; . . . ; m gj

ð147Þ

ð148Þ

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Van Espen

Pn bj ¼

i¼1

wi xi Pj ðxi ÞPj1 ðxi Þ ; gj1

j ¼ 0; . . . ; m

ð149Þ

Further, the normalization factor is given by gj ¼

n X

wi ½Pj ðxi Þ2

ð150Þ

i¼1

and b0 ¼ 0 and

P0 ðxi Þ ¼ 1

ð151Þ

Thus, an example of a ﬁrst-order orthogonal polynomial is C0 P0 þ C1 P1 ¼ C0 þ C1 ðxi a0 Þ

ð152Þ

with a0 ¼

n X

w i xi

.X

ð153Þ

wi

i¼1

C.

Nonlinear Least Squares

In this part, we consider the ﬁtting of a function that is nonlinear in one or more ﬁtting parameters. Examples of such functions are a decay curve, yðxÞ ¼ a1 expða2 xÞ

ð154Þ

or a Gaussian on a linear background, ðx a4 Þ2 yðxÞ ¼ a1 þ a2 x þ a3 exp 2a25

! ð155Þ

Equation (154) is nonlinear in a2 ; Eq. (155) is nonlinear in the parameters a4 and a5 . Equation (154) is representative for a group of functions for which linear least-squares ﬁtting can be applied after suitable transformation. Fitting with Eq. (155) implies the application of truly nonlinear least-squares ﬁtting with iterative optimization of the ﬁtting parameters. 1. Transformation to Linear Functions Taking the logarithm of Eq. (154), ln y ¼ ln a1 þ a2 x

ð156Þ

and deﬁning y0 ¼ ln y; and a01 ¼ ln a1 , a linear (straight line) ﬁtting function is obtained, y0 ¼ a01 þ a2 x

ð157Þ

and the method discussed earlier can be applied, but not without making the following important remark. We have transformed our original data yi to y0i ¼ ln yi . Consequently, also the variance s02 i has been changed according to the general error propagation formula: 0 2 @y s0i 2 ¼ s2i ð158Þ @y

Spectrum Evaluation

311

or in this particular case, s0i 2 ¼

s2i yi

ð159Þ

Thus, even if all original data points had the same uncertainty ðsi ¼ sÞ and unweighed linear least-squares ﬁtting could have been used, after the transformation a weighted linear least-squares ﬁt is required. The results of the ﬁt are the parameters and the associated uncertainties of the transformed equation, and to obtain the original parameter, we must perform a back transformation with the appropriate error propagation, 0

a1 ¼ e a 1 s2a1 ¼ s2a0

1

@a1 @a01

ð160Þ

2 ¼ a1 s2a0

1

ð161Þ

2. General Nonlinear Least Squares For the general case of least-squares ﬁtting with a function that is nonlinear in one or more of its ﬁtting parameters, no direct solution exists. Still, we can deﬁne the object function w2 : X 1 ½y yðxi ; aÞ2 ð162Þ w2 ¼ 2 i s i i whose minimum will be reached when the partial derivative with respect to the parameters are zero; however, this will result in a set of m equations for which no general solution exists. The other approach to the problem is then to consider w2 as a continuous function of the parameters aj (i.e., w2 will take a certain value for each set of values of the parameters aj for a given dataset fxi ; yi ; s2i gÞ. w2 thus forms a hypersurface in the m-dimensional space, formed by the ﬁtting parameter aj . This surface must be searched to locate the minimum of w2 . Once found, the corresponding coordinate values of the axes are the optimum values of the ﬁtting parameters. The problem of nonlinear least-squares ﬁtting is thus reduced to the problem of ﬁnding the minimum of a function in an m-dimensional space. Any algorithm that performs this task should operate according to the following: 1. Given some initial set of values for the parameters aini evaluate w2 : w2old ¼ w2 ðaini Þ 2. Find a new set of values anew such that w2new < w2old . 3. Test the minimum of w2 value: if else

w2new is the (true) minimum accept anew as the optimum values of the ﬁt w2old ¼ w2new and repeat Step 2.

From the scheme, the iterative nature of the nonlinear least-squares ﬁtting methods becomes evident. Moreover, it shows some other important aspects of the method: Initial values are required to start the search, we need a procedure to obtain a new set of parameters which preferably are such that w2 is decreasing, and we need to be sure that the true minimum, not some local minimum, is ﬁnally reached. A variety of algorithms has been proposed, ranging from brute-force mapping procedures dividing the m-dimensional parameter space in small cells and evaluating w2 in each point, to more subtle simplex search procedures (Fiori et al., 1981). The most

312

Van Espen

important group of algorithms is nevertheless based on the evaluation of the curvature matrix. The gradient method and the ﬁrst-order expansion will be discussed brieﬂy, as they form the basis of the most widely used LeverbergMarquardt algorithm (Marquardt, 1963; Bevington and Robinson, 1992; Press et al., 1988). a. The Gradient Method Having a ﬁtting function y ¼ yðx; aÞ and w2 deﬁned as a function of the m parameters aj , n X 1 w2 ¼ w2 ðaÞ ¼ ½y yðxi ; aÞ2 ð163Þ 2 i s i i¼1 the gradient of w2 in the m-dimensional parameter space is given by Hw2 ¼

X @w2 j @aj j

ð164Þ

where j is the unit vector along the axis j and the components of the gradient are given by X 1 @w2 @y ¼ 2 ½yi yðxi ; aÞ 2 @aj @aj si i

ð165Þ

It is convenient to deﬁne bj ¼

1 @w2 2 @aj

ð166Þ

The gradient gives the direction in which w2 increases most rapidly. A method to locate the minimum can thus be developed on this basis. Given the current set of parameters aj , a new set of parameters a0j is calculated (for all j simultaneously): a0j ¼ aj þ Daj bj

ð167Þ

which follows the direction of steepest descent and guarantees a decrease of w2 (at least if the appropriate step sizes Daj are taken). The gradient method works quite well away from the minimum, but near the minimum, the gradient becomes very small (at the minimum, even zero). Fortunately, the method discussed next behaves in the opposite way. b. First-Order Expansion If we write the ﬁtting function yðxi ; aÞ as a ﬁrst-order Taylor expansion of the parameters aj around y0 , yðx; aÞ ¼ y0 ðx; aÞ þ

X @y0 ðx; aÞ daj @aj j

ð168Þ

we obtain an (approximation) to the ﬁtting function which is linear in the parameter increments daj . y0 ðx; aÞ is the value of the ﬁtting function for some initial set of parameter a. Using this function, we can now express w2 as " #2 X 1 X @y0 ðxi ; aÞ 2 w ¼ yi y0 ðxi ; aÞ daj ð169Þ @aj s2i j i and we can use the method of linear least squares to ﬁnd the parameters daj so that w2 will be minimal. We are thus ﬁtting the diﬀerence y0i ¼ yi y0 ðxi ; aÞ with the derivatives as

Spectrum Evaluation

313

variables and the increments daj as unknowns. With reference to the section on linear least-squares ﬁtting [Eq. (122)], Xj ¼

@y0 ðxi Þ @aj

ð170Þ

and [Eq. (130) and (131)] n X 1 @y0 ðxi Þ ½y y0 ðxi Þ 2 i @aj s i i¼1 n X 1 @y0 ðxi Þ @y0 ðxi Þ ajk ¼ @ak s2i @aj i¼1

bj ¼

ð171Þ ð172Þ

deﬁning a set of m normal equations in the unknowns daj , b ¼ ada

ð173Þ

with solution daj ¼

m X

a1 jk bk

ð174Þ

k¼1

It is not very diﬃcult to prove that bj ¼

1 @w20 2 @ak

ð175Þ

(i.e., the component of the gradient of w2 at the point of expansion) and ajk

1 @ 2 w20 2 @aj dak

ð176Þ

Thus, ajk in Eq. (172) is the ﬁrst-order approximation to the curvature matrix whose inverse is the error matrix. The ﬁrst-order expansion of the ﬁtting function is closely related to the ﬁrst-order Taylor expansion of the w2 hypersurface itself: w2 ¼ w20 þ

X @w2 0 daj @a j j

ð177Þ

where w20 is the w2 function at the point of expansion: w20 ¼

n X 1 ½y y0 ðxi ; aÞ2 2 i s i i¼1

ð178Þ

At the minimum, the partial derivation of w2 with respect to the parameter ak will be zero: @w2 @w20 X @ 2 w20 ¼ þ dak ¼ 0 @ak @ak @aj @ak j

ð179Þ

This results in a set of equations in the parameters dak : m X @w20 @w20 ¼ dak @ak @aj @ak j¼1

ð180Þ

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Van Espen

bk ¼

X

ajk dak

ð181Þ

which is the same set of equations, except that in the expansion of the ﬁtting function, only a ﬁrst-order approximation of the curvature matrix is used. Because near the minimum the ﬁrst-order expansion of the w2 surface is a good approximation, we can conclude that also the ﬁrst-order expansion of the ﬁtting function (which is computationally more elegant because only derivatives of functions and not of w2 are required) will yield parameter increments daj which will direct us toward the minimum. For each linear parameter in the ﬁtting function, the ﬁrst-order expansion of the function in this parameter is exact and the calculated increment daj will be such that the new value aj þ daj is optimum (for the given set of nonlinear parameters which might not be at their optimum value yet). c. The Marquardt Algorithm Based on the observation that away from the minimum the gradient method is eﬀective and near the minimum the ﬁrst-order expansion is useful, Marquardt developed an algorithm that combines both methods using a scaling factor l that moves the algorithm either in the direction of the gradient search or into the direction of ﬁrst-order expansion (Marquardt, 1963). The diagonal terms of the curvature matrix are modiﬁed as follows:

ajk ð1 þ lÞ; j ¼ k 0 ajk ¼ ð182Þ j 6¼ k ajk ; where ajk is given by Eq. (172) and the matrix equation to be solved for the increments daj is X bj ¼ a0jk dak ð183Þ k

When l is very large (l 1), the diagonal elements of a dominate and Eq. (183) reduces to bj a0jj dak or dak

1 1 @w2 b j a0jj a0jj @ak

ð184Þ ð185Þ

which is the gradient, scaled by a factor a0jj . On the other hand, for small values of l (l 1), the solution is very close to ﬁrst-order expansion. The algorithm proceeds as follows: 1. 2. 3. 4.

Given some initial values of the parameters aj , evaluate w2 ¼ w2 ðaÞ and initialize l ¼ 0.0001. Compute b and a matrices using Eqs. (171) and (172). Modify the diagonal elements a0jj ¼ ajj þ l and compute da. If w2 ða þ daÞ w2 ðaÞ increase l by a factor of 10 and repeat Step 3; If w2 ða þ daÞ < w2 ðaÞ decrease l by a factor of 10 accept new parameters estimates a a þ da and repeat Step 2.

The algorithm thus performs two loops: the inner loop incrementing l until w2 starts to decrease and the outer loop calculating successively better approximations to the optimum values of the parameters. The outer loop can be stopped when w2 decreases by a negligible absolute or relative amount.

Spectrum Evaluation

315

Once the minimum is reached, the diagonal elements are an estimate of the uncertainty in the ﬁtting parameters just as in the case of linear least squares: s2aj ¼ a1 jj

ð186Þ

which is equal to ða0jj Þ1 providing the scaling factor l is much smaller than 1. In Sec. X, a number of computer programs for linear and nonlinear least-squares ﬁtting are given. Further information can be found in many textbook (Press et al., 1988). The book by Bevington and Robinson (1992) contains a very clear and practical discussion of the least-squares method.

X.

COMPUTER IMPLEMENTATION OF VARIOUS ALGORITHMS

In this section, a number of computer routines related to spectrum evaluation are listed. The calculation routines are written in FORTRAN. Some example programs, calling this FORTRAN routines, are written in C. The programs were tested using Microsoft FORTRAN version 4.0A and C version 5.1. Most of the routines are written for clarity rather than optimized for speed or minimum space requirement.

A.

Smoothing

1. Savitsky and Golay Polynomial Smoothing The subroutine SGSMTH calculates a smoothed spectrum using a second-degree polynomial ﬁlter (see Sec. III.B.2). Input:

Output

Y NCHAN ICH1, ICH2 IWID S

Original spectrum Number of channels in the spectrum First and last channel number to be smoothed Width of the filter (2m þ 1), IWID < 42 Smoothed spectrum, only defined between ICH1 and ICH2

SUBROUTINE SGSMTH (Y, S, NCHAN, ICH1, ICH2, IWID) INTEGER*2 NCHAN, ICH1, ICH2, IWID REAL*4 Y(0:NCHAN1), S(0:NCHAN1) REAL C(20:20) C - - Calculate filter coefficients IW ¼ MIN(IWID, 41) M ¼ IW=2 SUM ¼ FLOAT( (2*M 7 1)*(2*M þ 1)*(2*M þ 3) ) DO 10 J ¼ M, M C(J) ¼ FLOAT( 3*(3*M*M þ 3*M 7 1 7 5*J*J) ) 10 CONTINUE C - - Convolute spectrum with filter JCH1 ¼ MAX( ICH1, M ) JCH2 ¼ MIN( ICH2, NCHAN 7 1 7 M ) DO 30 I ¼ JCH1, JCH2 S(I) ¼ 0.

316

20 30

Van Espen DO 20 J ¼ 7 M, M S(I) ¼ S(I) þ C(J)*Y(I þ J) CONTINUE S(I) ¼ S(I)=SUM CONTINUE RETURN END

2. Low Statistics Digital Filter The subroutine LOWSFIL smooths a spectrum using the low statistics digital ﬁlter algorithm (see Sec. III.B.3). Input:

Output:

Y NCHAN ICH1, ICH2 IFWHM S

Original spectrum Number of channels First and last channel to be smoothed FWHM in channels of a peak in the middle of the smoothing region Smoothed spectrum, only defined between ICH1 and ICH2

SUBROUTINE LOWSFIL (Y, S, NCHAN, ICH1, ICH2, IFWHM) INTEGER*2 NCHAN, ICH1, ICH2, IFWHM REAL Y(0:NCHAN 7 1), S(0:NCHAN 7 1) LOGICAL NOKSLOPE, STOOHIGH REAL AFACT, FFACT, MFACT, RFACT PARAMETER (AFACT ¼ 75., FFACT ¼ 1.5, MFACT ¼ 10., RFACT ¼ 1.3) C - - Adjust smoothing region IW ¼ NINT( FFACT*IFWHM ) JCH1 ¼ MAX(ICH1 7 IW, IW) JCH2 ¼ MIN(ICH2 þ IW, NCHAN 7 1 7 IW) DO 100 I ¼ JCH1, JCH2 IW ¼ NINT( FFACT*IFWHM ) SUML ¼ 0. SUMR ¼ 0. DO 20 J ¼ I 7 IW, I 7 1 SUML ¼ SUML þ Y(J) 20 CONTINUE DO 30 J ¼ I þ 1, I þ IW SUMR ¼ SUMR þ Y(J) 30 CONTINUE C - - Adjust window 50 CONTINUE SUMT ¼ SUML þ Y(I) þ SUMR IF( SUMT .GT. MFACT ) THEN SLOPE ¼ (SUMR þ 1.)=(SUML þ 1.) NOKSLOPE ¼ SLOPE.GT.RFACT .OR. SLOPE .LT. 1.=RFACT STOOHIGH ¼ SUMT .GT. AFACT*SQRT(Y(I)) IF( NOKSLOPE .OR. STOOHIGH .AND. IW .GT. 1 ) THEN SUML ¼ SUML 7 Y(I 7 IW) SUMR ¼ SUMR 7 Y(I 7 IW) IW ¼ IW 7 1 GOTO 50

Spectrum Evaluation

C -100 C --

110

120

317

ENDIF ENDIF Smoothed value S(I) ¼ SUMT=FLOAT (2*IW þ 1) CONTINUE Copy data points that could not be smoothed DO 110 I ¼ ICH1, JCH1 7 1 S(I) ¼ Y(I) CONTINUE DO 120 I ¼ JCH2 þ 1, ICH2 S(I) ¼ Y(I) CONTINUE RETURN END

For each channel i in the spectrum, two windows, one on each side of the channel of width f6FWHM(Ei ) channels are considered. In both windows, the channel contents are summed, yielding a left sum L and a right sum R. Both windows are subsequently reduced in width until either the total sum S ¼ L þ yi þ R falls below some constant minimum M or until two conditions are met: pﬃﬃﬃﬃ 1. S is less than a cutoﬀ value N ¼ A yi , with A a constant. 2. The slope ðR þ 1Þ=ðL þ 1Þ lies between 1=r and r, with r a constant. The minimum constant M sets the base degree of smoothing in a region of vanishing counts. The ﬁrst condition ensures that smoothing is conﬁned to the low statistics region of the spectrum; the second condition avoids the incorporation of the edges of the peaks in the averaging. When the above conditions are satisﬁed, the average S=(2f6FWHM þ 1) is adopted as a smoothed channel count. The following parameters were found to yield good results when treating PIXE spectra: f ¼ 1.5, A ¼ 75, M ¼ 10, and r ¼ 1.3.

B.

Peak Search

The subroutine LOCPEAKS locates peaks in a spectrum using positive part of tophyhat ﬁlter (see Sec III.C). Input:

Output:

Y NCHAN R IWID MAXP NPEAK IPOS

Spectrum Number of channels in the spectrum Peak search sensitivity factor, typical 2 to 4 Width of the filter, approx. equal to the FWHM of the peaks Maximum number of peaks to locate (size of array IPOS) Number of peaks found Peak positions (channel number)

SUBROUTINE LOCPEAKS (Y, NCHAN, IWID, R, IPOS, NPEAKS, MAXP) INTEGER*2 NCHAN, IWID, NPEAKS, MAXP INTEGER*2 IPOS (MAXP) REAL Y (NCHAN), R C - - Width of filter (number of channels in the top)

318 C

C --

C --

20

22 C --

C C

C

100

Van Espen must be odd, and at least 3 NP ¼ MAX (IWID=2)*2 þ 1, 3) NPEAKS ¼ 0 Calculate half width and start and stop channel N ¼ NP=2 I1 ¼ NP I2 ¼ NCHAN-NP Initialize running sums I ¼ I1 TOTAL ¼ 0. TOP ¼ 0. DO 20 K ¼ 7N*2, NP TOTAL ¼ TOTAL þ Y(I1 þ K) CONTINUE DO 22 K ¼ 7N, N TOP ¼ TOP þ Y(I1 þ K) CONTINUE Loop over all channels LASTPOS ¼ 0 SENS ¼ R*R FI ¼ 0. FNEXT ¼ 0. SNEXT ¼ 0. DO 100 I ¼ I1 þ 1, I2 TOP ¼ TOP 7 Y(I-N-1) þ Y(I þ N) TOTAL ¼ TOTAL 7 Y(I-NP) þ Y(I þ NP) FPREV ¼ FI FI ¼ FNEXT SI ¼ SNEXT FNEXT ¼ TOP þ TOP 7 TOTAL SNEXT ¼ TOTAL Significant? IF( FI.GT.0. .AND. (FI*FI.GT.SENS*SI) ) THEN Find maximum IF( FI .GT. FPREV .AND. FI .GT. FNEXT ) THEN IF( FPREV.GT.0. .AND. FNEXT.GT.0. ) THEN and store (ch# is array index 1 and FI refers to I1) NEWPOS ¼ I2 IF( NEWPOS.GT.LASTPOS þ 2 ) THEN NPEAKS ¼ NPEAKS þ 1 IPOS(NPEAKS) ¼ NEWPOS LASTPOS ¼ NEWPOS IF( NPEAKS .EQ. MAXP ) RETURN ENDIF ENDIF ENDIF ENDIF CONTINUE RETURN END

The routine is optimized for speed and requires no other arrays than the spectrum and a table to store the peak maxima found. This is achieved by using a variant of the top-hat

Spectrum Evaluation

319

ﬁlter; that is, for a ﬁlter width of 5, the coeﬃcients are 71, 71, þ1, þ1, þ1, þ1, þ1, 71, 71, 71. The next point in the ﬁltered spectrum can be calculated from the current by subtracting and adding and only the value of the previous, the current, and the next points in the ﬁltered spectrum are retained. This makes the routine quite cryptic, but it works very fast and is reliable. C.

Continuum Estimation

1. Peak Stripping The subroutine SNIPBG, a variant of the SNIP algorithm, calculates the continuum via peak stripping (see Sec. IV.A). INPUT:

Output: Comment:

Y NCHAN ICH1,ICH2 FWHM NITER YBACK Uses subroutine

Spectrum Number of channels in the spectrum First and last channels of region to calculate the continuum Width parameter for smoothing and stripping algorithm, set it to average FWHM of peaks in the spectrum, typical value 8.0 Number of iterations of SNIP algorithm, typical 24 Calculated continuum in the region ICH1ICH2 SGSMTH

SUBROUTINE SNIPBG (Y, YBACK, NCHAN, ICH1, ICH2, FWHM, NITER) INTEGER*2 NCHAN, ICH1, ICH2, NITER REAL*4 Y(0:NCHAN-1), YBACK(0:NCHAN-1), FWHM PARAMETER (SQRT2 ¼ 1.4142, NREDUC ¼ 8) C - - Smooth spectrum IW ¼ NINT (FWHM) I1 ¼ MAX (ICH1 7 IW, 0) I2 ¼ MIN (ICH2 þ IW, NCHAN-1) CALL SGSMTH (Y, YBACK, NCHAN, I1, I2, IW) C - - Square root transformation over required spectrum region DO 10 I ¼ I1, I2 YBACK(I) ¼ SQRT (MAX(YBACK(I), 0.)) 10 CONTINUE C - - Peak stripping REDFAC ¼ 1. DO 30 N ¼ 1, NITER C.. Set width, reduce width for last NREDUC iterations IF( N .GT. NITER-NREDUC ) REDFAC ¼ REDFAC=SQRT2 IW ¼ NINT (REDFAC*FWHM) DO 20 I ¼ ICH1, ICH2 I1 ¼ MAX (I 7 IW, 0) I2 ¼ MIN (I þ IW, NCHAN-1) YBACK(I) ¼ MIN(YBACK(I), 0.5* (YBACK(I1) þ YBACK(I2))) 20 CONTINUE 30 CONTINUE C - - Back transformation DO 40 I ¼ ICH1, ICH2 YBACK(I) ¼ YBACK(I)*YBACK(I)

320 40

Van Espen CONTINUE RETURN END

2. Orthogonal Polynomial Continuum The subroutine OPOLBAC ﬁts the continuum of a pulse-height spectrum using an orthogonal polynomial. Continuum channels are selected by adjusting the weight of the ﬁt (see Sec. IV.C). Input:

Output:

Workspace:

NPTS X Y R YBACK W A,B C SC FAILED

Number of data points (channels) Array of channel numbers Array of spectrum Adjustable parameter [Eqs. (51) and (52)], typical value 2 Array of fitted continuum Array of weights Coefficients of the orthogonal polynomial Fitted parameters of the orthogonal polynomial Uncertainties of C Logical variable TRUE if no convergence after MAXADJ weight adjustments RCHISQ Reduced chi-square value of the fitted continuum WORK1, WORK2 of size NPTS

The routine calls ADJWEIG to adjust the weights. Further, the subroutine ORTPOL is used to ﬁt the polynomial. The iteration (adjustment of weights) stops when all coefﬁcients cj change less than one standard deviation or when the maximum number of iterations is reached.

SUBROUTINE OPOLBAC (NPTS, X, Y, W, YBACK, WORK1, WORK2, > NDEGR, A, B, C, COLD, SC, FAILED, RCHISQ, R) INTEGER NPTS, NDEGR REAL*4 X(NPTS),Y(NPTS),W(NPTS),YBACK(NPTS) REAL*4 WORK1 (NPTS), WORK2 (NPTS) REAL*4 A (NDEGR), B(NDEGR), C(NDEGR), COLD(NDEGR), SC(NDEGR) REAL*4 RCHISQ, R LOGICAL*2 FAILED PARAMETER (MAXADJ ¼ 20) LOGICAL*2 NEXT C -- Initialize DO 10 J ¼ 1, NDEGR COLD(J) ¼ 0. 10 CONTINUE DO 20 I ¼ 1, NPTS YBACK(I) ¼ 0. 20 CONTINUE C - - Main iteration loop DO 100 K ¼ 1, MAXADJ C .. Calculate weights CALL ADJWEIG (NPTS, Y, W, YBACK, R, NBPNTS)

Spectrum Evaluation

321

C .. Fit orthogonal polynomial CALL ORTPOL (NPTS, X, Y, W, YBACK, WORK1, WORK2, > NDEGR, A, B, C, SC, SUMSQ) RCHISQ ¼ SUMSQ=FLOAT (NBPNTS 7 NDEGR) S ¼ SQRT (RCHISQ) C .. Test if further adjustments of weights is required NEXT ¼ .FALSE. DO 30 J ¼ 1, NDEGR SC(J) ¼ S * SC(J) IF( ABS(COLD(J)-C(J)) .GT. SC(J) ) NEXT ¼ .TRUE. COLD(J) ¼ C(J) 30 CONTINUE C .. convergence IF( .NOT.NEXT ) THEN FAILED ¼ .FALSE. RETURN ENDIF 100 CONTINUE C - - No convergence after MAXADJ iterations FAILED ¼ .TRUE. RETURN END SUBROUTINE ADJWEIG (NPTS, Y, W, YFIT, R, NBPNTS) C * * Adjust weights to emphasize the continuum INTEGER NPTS, NBPNTS REAL*4 Y(NPTS), W(NPTS), YFIT(NPTS), R NBPNTS ¼ 0 C - - Loop over all data points DO 10 I ¼ 1, NPTS IF (YFIT(I) .GT. 0.) THEN IF (Y(I).LE. YFIT(I) þ R*SQRT (YFIT(I))) THEN C .. Point is considered as continuum W(I) ¼ 1./YFIT(I) NBPNTS ¼ NBPNTS þ 1 ELSE C .. Point is NOT considered as continuum. W(I) ¼ 1./(Y(I) 7 YFIT(I))**2 ENDIF ELSE C .. Continuum <¼ 0, weight based original data C (initial condition) W(I) ¼ 1./MAX(Y(I),1.) NBPNTS ¼ NBPNTS þ 1 ENDIF 10 CONTINUE RETURN END

D.

Filter-Fit method

The C program FILFIT is a test implementation of the ﬁlter-ﬁt method (see Sec. VI). This program simply coordinates all input and output, allocates the required memory, and calls two FORTRAN routines that do the actual work. The subroutine TOPHAT returns the

322

Van Espen

convolute of the spectrum with the top-hat ﬁlter or the weights (the inverse of the variance of the ﬁltered spectrum). The general-purpose subroutine LINREG is called to perform the multiple linear least-squares ﬁt. The output includes the reduced w2 value, the parameters of the ﬁt aj (which is an estimate of the ratio of the intensity in the analyzed spectrum to the intensity in the standard for the considered X-ray lines), and their standard deviation. The routine ‘‘GETSPEC’’ reads the spectral data and must be supplied by the user. #include #include #include #include

<stdio.h> <malloc.h> <math.h>

void fortran TOPHAT( ); void fortran LINREG( ); float spec[2048]; main( ) { int nchan, first_ch_fit, last_ch_fit, width, ierr; int i, first_ch_ref, last_ch_ref, ref, num_ref, num_points; int filter_mode ¼ 0, weight_mode ¼ 1, ioff; float meas_time, ref_meas_time, *scale_fac; float *x, *xp, *y, *w, *yfit, *a, *sa, chi; double *beta, *alpha; char filename [64]; // input width of tophat filter scanf(‘‘%hd’’, &width); // input spectrum to fit and fitting region scanf(‘‘%s’’, filename); scanf(‘‘%hd %hd’’, &first_ch_fit, &last_ch_fit); nchan ¼ Getspec(spec, filename, &meas_time); num_points ¼ last_ch_fit 7 first_ch_fit þ 1; // filter spectrum and store in y[ ] y ¼ (float *)calloc(num_points, sizeof(float)); TOPHAT(spec, y, &nchan, &first_ch_fit, &last_ch_fit, &width, &filter_mode); //calculate weights of fit and save in w[ ] w ¼ (float *)calloc(num_points, sizeof(float)); TOPHAT(spec, w, &nchan, &first_ch_fit, &last_ch_fit, &width, &weight_mode); //read reference spectra, filter and store in x[ ] scanf(‘‘%hd’’, &num_ref); scale_fac ¼ (float *)calloc(num_ref, sizeof(float)); x ¼ (float *)calloc(num_points*num_ref, sizeof(float)); for(ref ¼ 0; ref < num_ref; refþþ) { scanf(‘‘%s’’, filename); nchan ¼ Getspec(spec, filename, &ref_meas_time); scale_fac[ref] ¼ ref_meas_time/meas_time; scanf(‘‘%hd %hd’’, &first_ch_ref, &last_ch_ref); if (first_ch_ref < first_ch_fit) first_ch_ref ¼ first_ch_fit; if(last_ch_ref > last_ch_fit) last_ch_ref ¼ last_ch-fit; ioff ¼ ref*num_points þ first_ch_ref 7 first_ch_fit; xp ¼ x þ ioff; TOPHAT(spec, xp, &nchan, &first_ch_ref, &last_ch_ref, &width, &filter_mode); }

Spectrum Evaluation

323

// perform least squares fit yfit ¼ (float *)calloc(num_points, sizeof(float)); a ¼ (float *)calloc(num_ref, sizeof(float)); sa ¼ (float *)calloc(num_ref, sizeof(float)); beta ¼ (double *)calloc(num_ref, sizeof(double)); alpha ¼ (double *)calloc(num_ref* (num_ref þ 1)/2, sizeof(double)); LINREG (y,w,x &num_points, &num_ref, &num_points, &num_ref, yfit, a, sa, &chi, &ierr, beta, alpha); if( ierr ¼ ¼ 0 ) { printf(‘‘Filter fit: Chi-square ¼ %f\n’’, chi); printf(‘‘Standard Int. in analyse spectrum/Int. in standard\n’’); for(i ¼ 0; i < num_ref; i þ þ ) printf(‘‘ %hd %f n%f\n’’, i þ 1, a[i]*scale_fac[i], sa[i]*sacle_fac[i]); for(i ¼ 0; i < num_points; i þ þ ){ printf(‘‘%4hd %7.0f %9.2f’’, first_ch_fit þ i, y[i],yfit[i]); for (ref ¼ 0;ref < num_ref; ref þ þ ) printf(‘‘%7.0f’’, x[ref * num_points þ i]); printf(‘‘\n’’); } } }

SUBROUTINE TOPHAT (IN, OUT, NCHAN, IFRST, ILAST, IWIDTH, MODE) INTEGER*2 NCHAN, IFRST, ILAST, IWIDTH, MODE REAL*4 IN (NCHAN), OUT(1) C ** Tophat filter of width IWIDTH, MODE ¼ 0 calculate filtered specturm, C MODE ! ¼ 0 calculate weights (1/variance of filtered spectrum) C - - Calculate filter constants. IW ¼ IWIDTH IF(MOD(IW,2).EQ.0)IW ¼ IW þ 1 FPOS ¼ 1./FLOAT(IW) KPOS ¼ IW/2 IV ¼ IW/2 FNEG ¼ 7 1./FLOAT(2*IV) KNEG1 ¼ IW/2 þ 1 KNEG2 ¼ IW/2 þ IV N¼0 C - - Loop over all requested channels. DO 30 I ¼ IFRST þ 1, ILAST þ 1 C . . Central positive part, YPOS ¼ 0. DO 10 K ¼ 7 KPOS, KPOS IK ¼ MIN(MAX(I þ K,1),NCHAN) YPOS ¼ YPOS þ IN(IK) 10 CONTINUE C . . Left and right negative part, YNEG ¼ 0. DO 20 K ¼ KNEG1, KNEG2 IK ¼ MIN(MAX(I 7 K,1),NCHAN) YNEG ¼ YNEG þ IN(IK) IK ¼ MIN(MAX(I þ K,1),NCHAN) YNEG ¼ YNEG þ IN(IK)

324 20

30

E.

Van Espen CONTINUE N¼Nþ1 IF(MODE.EQ.0) THEN OUT(N) ¼ FPOS * YPOS þ FNEG * YNEG ELSE VAR ¼ FPOS*FPOS*YPOS þ FNEG*FNEG*YNEG OUT(N) ¼ 1./MAX(VAR,1.) ENDIF CONTINUE RETURN END

Fitting Using Analytical Functions

The C program NLRFIT is an example implementation of the nonlinear spectrum ﬁtting using an analytical function (see Sec. VII). The program only coordinates input and output. The actual ﬁtting is done using the Marquardt algorithm with the FORTRAN subroutine MARQFIT. The ﬁtting function consists of a polynomial continuum width NB terms and NP Gaussians. The continuum parameters and the area, position, and with of each Gaussian are optimized during the ﬁt. The ﬁtting function is calculated using the routine FITFUNC. The derivatives of the ﬁtting function with respect to the parameters are calculated by the routine DERFUNC. // Program NLRFIT #include <stdio.h> #include <malloc.h> #include #include <math.h> #define MAX_PERKS 10 #define MAX_CHAN 1024 void fortran MARQFIT(); float spec [MAX_CHAN]; // Fortran common block structure COMMON/FITFUN/NB, NP struct common_block {short NB, NP;}; extern struct common_block fortran FITFUN; main( ) { char specfile[64]; int nchan, first_ch_fit, last_ch_fit, nb, np; int i, j, n, num_points, num_param, ierr, max_iter; float ini_pos[MAX_PEAKS], ini_wid[MAX_PEAKS]; float *x, *xp, *y, *w, *yfit, *a, *sa, chi, lamda, crit_dif; float *b, *beta, *deriv, *alpha; double *work; // Input of parameters and spectral data scanf(‘‘%s’’, specfile); scanf(‘‘%hd %hd %hd %f’’, &first_ch_fit, &last_ch_fit, &max_iter, &crit_dif); scanf(‘‘%hd %hd’’, &np, &nb); for(i ¼ 0; i < np; i þ þ ) scanf(‘‘%f %f’’, &ini_pos[i], &ini_wid[i]);

Spectrum Evaluation

325

nchan ¼ GetSpec(spec, specfile); num_points ¼ last_ch_fit 7 first_ch_fit þ 1; num_param ¼ nb þ 3*np; //Allocate memory for y[ ], w[ ], x[ ] y ¼ (float *)calloc(num_points, sizeof(float)); w ¼ (float *)calloc(num_points, sizeof(float)); x ¼ (float *)calloc(num_points, sizeof(float)); // Store independent variable (spectrum), weights and dep. var (channel #) for(i ¼ first_ch_fit, n ¼ 0; i < ¼ last_ch_fit; i þ þ , n þ þ ) { y[n] ¼ spec[i]; w[n] ¼ (spec[i] > 0.) ? 1./spec[i] : 1.; x[n] ¼ (float)i; } // allocate memory for other arrays required yfit ¼ (float *)calloc(num_points, sizeof(float)); a ¼ (float *)calloc(num_param, sizeof(float)); sa ¼ (float *)calloc(num_param, sizeof(float)); b ¼ (float *)calloc(num_param, sizeof(float)); beta ¼ (float *)calloc(num_param, sizeof(float)); deriv ¼ (float *)calloc(num_param, sizeof(float)); alpha ¼ (float *)calloc(num_param*(num_param þ 1)/2, sizeof(float)); work ¼ (double *)calloc(num_param*(num_param þ 1)/2, sizeof(double)); // initialize, all linear parameters to zero, peak position and width // to their initial guesses lamda ¼ 0.001; for(i ¼ 0; i < np; i þ þ ){ a[nb þ np þ i] ¼ ini_pos[i]; a[nb þ 2*np þ i] ¼ ini_wid[i]; } // perform least squares fit FITFUN.NP ¼ np; FITFUN.NB ¼ nb; MARQFIT(&ierr, &chi, &lamda, &crit_dit, &max_iter, x, y, w, yfit, &num_points, a, sa, &num_param, b, beta, deriv, alpha, work); if( ierr ¼ ¼ 0 ) { printf(‘‘\nNon-linear fit: Chi-square ¼ %f\n’’, chi); printf(‘‘Polynomial continuum parameters\n’’); for(i ¼ 0; i < nb; i þ þ ) printf(‘‘%hd %f n %f\n’’, i þ 1, a[i], sa[i]); printf(‘‘Peak parameters Area Position Width\n’’); for(i ¼ 0; i < np; i þ þ ){ printf(‘‘%hd %10.0f n % 7 10.0f’’, i þ 1, a[nb þ i], sa[nb þ i]); printf(‘‘%10.3f n % 7 10.3f’’, a[nb þ np þ i], sa[nb þ np þ i]); printf(‘‘%10.3f n % 7 10.3f\n’’, a[nb þ 2*np þ i], sa[nb þ 2*np þ i]); } for (i ¼ 0; i < num_points; i þ þ ) printf(‘‘%4hd %7.0f %9.2f\n’’, first_ch_fit þ i, y[i], yfit[i]); } }

326

Van Espen

SUBROUTINE FITFUNC(X, YFIT, NPTS, A, NTERMS ) REAL*4 X(NPTS), YFIT(NPTS), A(NTERMS) COMMON /FITFUN/ NB, NP C ** Fitting function, polynomial continuum and NP gaussians C with position, width and area as parameters PARAMETER(SQR2PI ¼ 2.50663) C - - Loop over all channels DO 100 I ¼ 1, NPTS C . . continuum YFIT(I) ¼ A(1) DO 20 J ¼ 2, NB YFIT(I) YFIT(I) þ A(J) * X(I)**(J 7 1) 20 CONTINUE C . . Peaks DO 30 K ¼ 1, NP AREA ¼ A(NB þ K) POS ¼ A(NB þ NP þ K) SWID ¼ A(NB þ 2*NP þ K) Z ¼ ((POS 7 X(I))/SWID)**2 IF( Z.LT.50.) THEN G ¼ EXP( 7 Z/2.)/SWID/SQR2PI YFIT(I) ¼ YFIT(I) þ AREA*G ENDIF 30 CONTINUE 100 CONTINUE RETURN END SUBROUTINE DERFUNC(X, NPTS, A, NTERMS, DERIV, I) REAL*4 X(NPTS), A(NTERMS), DERIV(NTERMS) COMMON/FITFUN/NB, NP C ** Derivatives of fitting function: polynomial continuum and NP Gaussians C with postion, width and area as parameters PARAMETER (SQR2PI ¼ 2.50663) C -- Derivatives of function with respect to the continuum parameters DERIV(1) ¼ 1. DO 10 J ¼ 2, NB DERIV(J) ¼ X(I)**(J 7 1) 10 CONTINUE C -- Derivatives of function with respect to the peak parameters DO 30 K ¼ 1, NP AREA ¼ A(NB þ K) POS ¼ A(NB þ NP þ K) SWID ¼ A(NB þ 2*NP þ K) Z ¼ ((POS 7 X(I))/SWID)**2 IF(Z.LT.50.) THEN G ¼ EXP( 7 Z/2.)/SWID/SQR2PI C . . Peak area DERIV(NB þ K) ¼ G C . . Peak position DERIV(NB þ NP þ K) ¼ 7 AREA*G*(POS 7 X(I))/SWID/SWID C . . Peak width DERIV(NB þ 2*NP þ K) ¼ AREA*G*(Z 7 1.)/SWID ELSE

Spectrum Evaluation DERIV(NB þ K) ¼ 0. DERIV(NB þ NP þ K) ¼ 0. DERIV(NB þ 2*NP þ K) ¼ 0. ENDIF CONTINUE RETURN END

30

F.

327

Monte Carlo Methods

1. Uniform Random-Number Generator The function URAND is a FORTRAN function returning uniform distributed random numbers in the interval 0 U < 0. The random-number generator is based on Knuth’s ‘‘subtractive’’ method (Press et al., 1988) (see Sec. VIII. A) Input: Output:

ISEED URAND

Set to any negative number to initialize the random generator Uniform random number in the interval 0 URAND < 1

REAL*4 FUNCTION URAND (ISEED) INTEGER*2 ISEED REAL*4 UTABLE(56) REAL*4 UBIT, USEED PARAMETER (UBIG ¼ 4000000., USEED ¼ 1618033.) SAVE I1, I2, UTABLE, INIT C -- Initialize table IF (ISEED.LT.0 .OR. INIT.EQ.0) THEN U ¼ USEED þ FLOAT(ISEED) U ¼ MOD(U,UBIG) UTABLE(55) ¼ U UTMP ¼ 1. DO 10 I ¼ 1, 54 II ¼ MOD(I*21, 55) UTABLE(II) ¼ UTMP UTMP ¼ U 7 UTMP IF(UTMP.LT.0.) UTMP ¼ UTMP þ UBIG U ¼ UTABLE(II) 10 CONTINUE DO 30 K ¼ 1, 4 DO 20 I ¼ 1, 55 UTABLE(I) ¼ UTABLE(I) 7 UTABLE(1 þ MOD(I þ 30,55)) IF(UTABLE(I).LT.0) UTABLE(I) ¼ UTABLE(I) þ UBIG 20 CONTINUE 30 CONTINUE I1 ¼ 0 I2 ¼ 31 ISEED ¼ 1 INIT ¼ 1 ENDIF C -- Get next ‘‘random’’ number I1 ¼ I1 þ 1 IF(I1.EQ.56) I1 ¼ 1

328

Van Espen I2 ¼ I2 þ 1 IF(I2.EQ.56) I2 ¼ 1 U ¼ UTABLE(I1) 7 UTABLE(I2) IF(U.LT.0.)U ¼ U þ UBIG UTABLE(I1) ¼ U URAND ¼ U/UBIG RETURN END

2. Normal Distributed Random Deviate The function NRAND returns normal distributed random number with zero mean and unit variance, using the BoxMuller method (see Sec. VIII. B). Input: Output:

ISEED NRAND

Set to any negative number to initialize the random sequence Normal distributed random deviate with zero mean and unit variance.

REAL*4 FUNCTION NRAND (ISEED) INTEGER*2 ISEED SAVE NEXT, FAC, V1, V2 IF (NEXT.EQ.0 .OR. ISEED.LT.0) THEN 10 CONTINUE V1 ¼ 2. * URAND(ISEED) 7 1. V2 ¼ 2. * URAND(ISEED) 7 1. R ¼ V1*V1 þ V2*V2 IF(R.G.E.1. .OR. R. EQ. 0.) GOTO 10 FAC ¼ SQRT( 7 2. * LOG(R)/R) NRAND ¼ V1*FAC NEXT ¼ 1 ELSE NRAND ¼ V2*FAC NEXT ¼ 0 ENDIF RETURN END

3. Poisson Distributed Random Deviate The function ‘‘PRAND’’ can be used to produce approximately Poisson distributed random deviates. For small numbers ( < 20), the direct method is used; for larger numbers, the Poisson distribution is approximated by the Normal distribution. Input: Output:

Y ISEED PRAND

(population) mean of deviate Set to any negative number to initialize the random sequence Poisson distributed random deviate with mean Y

Spectrum Evaluation

329

REAL *4 FUNCTION PRAND(Y, ISEED) INTEGER*2 ISEED REAL*4 Y REAL*4 NRAND, URAND IF(Y.LT.20.) THEN C -- Use direct method G ¼ EXP( 7 Y) PRAND ¼ 7 1. T ¼ 1. 10 CONTINUE PRAND ¼ PRAND þ 1. T ¼ T * URAND(ISEED) IF(T.GT.G) GOTO 10 ELSE C -- Approximate by normal distribution PRAND ¼ Y þ SQRT(Y) * NRAND(ISEED) ENDIF RETURN END

G.

Least-Squares Procedures

1. Linear Regression Subroutine LINREG is a general-purpose (multiple) linear regression routine (see Sec. IX. A). Input:

Output:

Workspace:

Y W X N M NMAX,MMAX YFIT A SA CHI IERR BETA

Array of dependent variable Array of weights 1/s2i ) Matrix of independent variables Number of data points Number of independent variables (columns of X) Size of X matrix Array of fitted Y values Estimated least-squares parameters Standard deviation of A Chi-square value Error condition, 1 if fit failed (singular matrix) of size M

SUBROUTINE LINREG (Y, W, X, N, M, NMAX, MMAX, > YFIT, A, SA, CHI, IERR, BETA, ALPHA) INTEGER*2 N, M, NMAX, MMAX, IERR REAL*4 Y(N), W(N), YFIT(N), A(M), SA(M), CHI REAL*4 X(NMAX, MMAX) REAL*8 BETA(M), ALPHA(1) c Accumulate BETA and ALPHA matrices JK ¼ 0 DO 10 J ¼ 1, M BETA(J) ¼ 0.0D0 DO 2 I ¼ 1, N

330

Van Espen

BETA(J) ¼ BETA(J) þ W(I)*Y(I)*X(I, J) CONTINUE DO 6 K ¼ 1, J JK ¼ JK þ 1 ALPHA(JK) ¼ 0.0D0 DO 4 I ¼ 1, N ALPHA(JK) ¼ ALPHA(JK) þ W(I)*X(I,K)*X(I,J) 4 CONTINUE 6 CONTINUE 10 CONTINUE Invert ALPHA matrix CALL LMINV (ALPHA, M, IERR) IF(IERR .EQ. 7 1) THEN RETURN ENDIF Calculate fitting parameters A DO 20 J ¼ 1, M A(J) ¼ 0. JJ ¼ J*(J 7 1)/2 DO 12 K ¼ 1, J JK ¼ K þ JJ A(J) ¼ A(J) þ ALPHA(JK)*BETA(K) 12 CONTINUE DO 14 K ¼ J+1, m JK ¼ J þ K* (K 7 1)/2 A(J) ¼ A(J) þ ALPHA(JK)*BETA(K) 14 CONTINUE 20 CONTINUE Calculate uncertainties in the parameters DO 30 J ¼ 1, M JJ ¼ J*(J þ 1)/2 SA(J) ¼ DSQRT(ALPHA(JJ)) 30 CONTINUE Calculate fitted values and Chi-square CHI ¼ 0. DO 40 I ¼ 1, N YFIT(I) ¼ 0. DO 32 J ¼ 1, M YFIT(I) ¼ YFIT(I) þ A(J)*(I, J) 32 CONTINUE CHI ¼ CHI þ W(I)*(YFIT(I) 7 Y(I))**2 40 CONTINUE CHI ¼ CHI/FLOAT(N 7 M) RETURN END 2

c

c

c

c

2. Orthogonal Polynomial Regression Subroutine ORTPOL ﬁts an orthogonal polynomial to a set of data points [xi, yi, wi] (see Sec. IX.B).

Spectrum Evaluation

Input:

Output:

Workspace:

NPTS X Y W NDEGR A, B C SC SUMSQ PJ, PJMIN

331

Number of data points Array of independent variable Array of dependent variable Array of weights (wi ¼ 1s2i ) Degree of orthogonal polynomial to be fitted Parameters of the orthogonal polynomials Fitted orthogonal polynomial coefficients Standard deviation of C Chi-square value of size NPTS

SUBROUTINE ORTPOL (NPTS, X, Y, W, YFIT, PJ, PJMIN, NDEGR, A, B, C, SC, > SUMSQ) INTEGER NPTS, NDEGR REAL*4 X(NPTS), Y(NPTS), W(NPTS), YFIT(NPTS) REAL*4 PJ(NPTS), PJMIN(NPTS) REAL*4 A(NDEGR), B(NDEGR), C(NDEGR), SC(NDEGR), SUMSQ C - - Initialize DO 10 I ¼ 1, NPTS PJ(I) ¼ 1. PJMIN(I) ¼ 0. YFIT(I) ¼ 0. 10 CONTINUE GAMJMIN ¼ 1. C - - Loop over all polynomial terms DO 100 J ¼ 1, NDEGR C .. Accumulate normalization factor, A and B constants for term j GAMJ ¼ 0. A(J) ¼ 0. B(J) ¼ 0. DO 20 I ¼ 1, NPTS GAMJ ¼ GAMJ þ W(I)*PJ(I)*PJ(I) A(J) ¼ A(J) þ W(I)*X(I)*PJ(I)*PJ(I) B(J) ¼ A(J) þ W(I)*X(I)*PJ(I)*PJMIN(I) 20 CONTINUE A(J) ¼ A(J) / GAMJ B(J) ¼ B(J) / GAMJMIN C .. Least squares estimate of coefficient C C(J) ¼ 0. DO 30 I ¼ 1, NPTS C(J) ¼ C(J) þ W(I)*Y(I)*PJ(I) 30 CONTINUE C(J) ¼ C(J)=GAMJ SC(J) ¼ SQRT(1.=GAMJ) C .. Contribution of this term to the fit DO 40 I ¼ 1, NPTS YFIT(I) ¼ YFIT(I) þ C(J)*PJ(I) 40 CONTINUE C .. Next polynomial term IF (J .LT. NDEGR) THEN

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Van Espen

DO 50 I ¼ 1, NPTS PJPLUS ¼ (X(I) 7 A(J))*PJ(I) 7 B(J)*PJMIN(I) PJMIN(I) ¼ PJ(I) PJ(I) ¼ PJPLUS 50 CONTINUE GAMJMIN ¼ GAMJ ENDIF 100 CONTINUE C -- Weighted sum of squares value SUMSQ ¼ 0. DO 110 I ¼ 1, NPTS SUMSQ ¼ SUMSQ þ W(I)*(Y(I) 7 YFIT(I))**2 110 CONTINUE RETURN END

3. Nonlinear Regression The subroutine MARQFIT performs nonlinear least-squares-ﬁtting according to the Marquardt algorithm (see Sec. IX.C). Input:

Output:

Workspace:

MAXITER X Y W NPTS NTERMS A IERR CHISQR FLAMDA YFIT A SA CRIDIF B, BETA, DERIV ALFA, ARR

Maximum number of iterations Array of independent variable Array of dependent variable Array of weights (wi ¼ 1=s2i ) Number of data points Number of parameters Array of initial values of the parameters Error status, 7 1 indicates failure of fit Reduced chi-square value Marquardt control parameter Array of fitted data points Least-squares estimate of the fitting parameters Standard deviation of A Minimum percent difference in two chi-square values to stop the iteration of size NTERMS of size NTERMS (NTERMS þ 1)=1

The routine requires two user-supplied subroutines: FITFUNC to evaluate the ﬁtting function y(i) with the current set of parameters a and the DERFUNC to calculate the derivatives of the ﬁtting function with respect to the parameters. SUBROUTINE MARQFIT (IERR, CHISQR, FLAMDA, CRIDIF, MAXITER, > X, Y, W, YFIT, NPTS, A, SA, NTERMS, > B, BETA, DERIV, ALFA, ARR) INTEGER*2 IERR, NPTS, NTERMS REAL*4 CHISQR, FLAMDA, CRIDIF REAL*4 X(NPTS), Y(NPTS), W(NPTS), YFIT(NPTS) REAL*4 A(NTERMS), SA(NTERMS) REAL*4 B(1), BETA(1), DERIV(1), ALFA(1) REAL*8 ARR(1) PARAMETER (FLAMMAX ¼ 1E4, FLAMMIN ¼ 1E 7 6)

Spectrum Evaluation C -- Evaluate the fitting function YFIT for the current parameters C and save the Chi-square value NITER ¼ 0 CALL FITFUNC(X, YFIT, NPTS, A, NTERMS) CHISQR ¼ CHIFIT(Y, YFIT, W, NPTS, NTERMS) FLAMDA ¼ 0. C -- Set ALFA and BETA to zero, save the current value of the parameters A 100 CONTINUE NITER ¼ NITER þ 1 CHISAV ¼ CHISQR DO 110 J ¼ 1, NTERMS B(J) ¼ A(J) BETA(J) ¼ 0. 110 CONTINUE DO 112 J ¼ 1, NTERMS*(NTERMS þ 1)=2 ALFA(J) ¼ 0. 112 CONTINUE C -- Accumulate Alpha and Beta matrices DO 120 I ¼ 1, NPTS D ¼ Y(I) 7 YFIT(I) C .. Calculate derivatives at point i CALL DERFUNC (X, NPTS, A, NTERMS, DERIV, I) DO 120 J ¼ 1, NTERMS BETA(J) ¼ BETA(J) þ W(I)*D*DERIV(J) JJ ¼ J*(J 7 1)=2 DO 120 K ¼ 1, J JK ¼ JJ þ K ALFA(JK) ¼ ALFA(JK) þ W(I)*DERIV(J)*DERIV(K) 120 CONTINUE C -- Test and scale ALFA matrix DO 140 J ¼ 1, NTERMS JJ ¼ J*(J 7 1)=2 JJJ ¼ JJ þ J IF(ALFA(JJJ) .LT. 1.E 7 20) THEN DO 130 K ¼ 1, J JK ¼ JJ þ K ALFA(JK) ¼ 0. 130 CONTINUE ALFA (JJJ) ¼ 1. BETA(J) ¼ 0. ENDIF SA(J) ¼ SQRT (ALFA(JJJ)) 140 CONTINUE DO 160 J ¼ 1, NTERMS JJ ¼ J*(J 7 1)=2 DO 150 K ¼ 1, J JK ¼ JJ þ K ALFA(JK) ¼ ALFA(JK)=SA(J)=SA(K) 150 CONTINUE 160 CONTINUE C -- Store ALFA in ARR, modify the diagonal elements with FLAMDA 200 CONTINUE DO 210 J ¼ 1, NTERMS

333

334

Van Espen

JJ ¼ J*(J 7 1)=2 DO 205 K ¼ 1, J JK ¼ JJ þ K ARR (JK) ¼ DBLE(ALFA(JK)) 205 CONTINUE JJJ ¼ JJ þ J ARR(JJJ) ¼ DBLE(1. þ FLAMDA) 210 CONTINUE C -- Invert matrix ARR CALL LMINV (ARR, NTERMS, IERR) IF (IERR .NE. 0) RETURN C -- Calculate new values of parameters A DO 220 J ¼ 1, NTERMS DO 220 K ¼ 1, NTERMS IF(K .GT. J) THEN JK ¼ J þ K*(K 7 1)=2 ELSE JK ¼ K þ J*(J 7 1)=2 ENDIF A(J) ¼ A(J) þ ARR(JK) = SA(J)*BETA(K) = SA(K) 220 CONTINUE C -- Evaluate the fitting function YFIT for the new parameters and Chi-square CALL FITFUNC(X, YFIT, NPTS, A, NTERMS) CHISQR ¼ CHIFIT(Y, YFIT, W, NPTS, NTERMS) IF (NITER .EQ. 1) FLAMDA ¼ 0.001 C -- Test new parameter set IF (CHISQR .GT. CHISAV) THEN C .. Iteration NOT succesful, increase flamda and try again FLAMDA ¼ MIN(FLAMDA * 10., FLAMMAX) DO 300 J ¼ 1, NTERMS A(J) ¼ B(J) 300 CONTINUE GOTO 200 ENDIF C .. Iteration succesful, decrease LAMDA FLAMDA ¼ MAX(FLAMDA=10., FLAMMIN) C .. Get next better estimate if required PERDIF ¼ 100.*(CHISAV 7 CHISQR)=CHISQR IF (NITER .LT. MAXITER .AND. PERDIF .GT. CRIDIF) GOTO 100 C -- Calculate standard deviations and return DO 320 J ¼ 1, NTERMS JJ ¼ J*(J þ 2)=2 SDEV ¼ DSQRT(ARR(JJ)) = SA(J) SA(J) ¼ SDEV 320 CONTINUE RETURN END FUNCTION CHIFIT(Y, YFIT, W, NPTS, NTERMS) REAL*4 Y(NPTS), YFIT(NPTS), W(NPTS) C ** Evaluate chi-square CHI ¼ 0. DO 300 I ¼ 1, NPTS

Spectrum Evaluation

300

335

CHI ¼ CHI þ W(I) * (Y(I) 7 YFIT(I))**2 CONTINUE CHI ¼ CHI = FLOAT(NPTS 7 NTERMS) CHIFIT ¼ CHI RETURN END

4. Matrix Inversion Subroutine LMINV is a general-purpose routine to invert a symmetric matrix. Input:

ARR

Output:

N IERR

Upper triangle and diagonal of real symmetric matrix stored in linear array, size ¼ N(N þ 1)=2. Order of matrix (number of columns) Error status, IERR ¼ 0 inverse obtained, IERR ¼ 1 singular matrix

SUBROUTINE LMINV ( ARR, N, IERR ) INTEGER*2 N, IERR REAL*8 ARR(1) REAL*8 DIN, WORK, DSUM, DPIV INTEGER*2 I, IND, IPIV, J, K, KEND, KPIV, L, LANF, > LEND, LHOR, LVER, MIN KPIV ¼ 0 DO 10 K ¼ 1, N KPIV ¼ KPIV þ K IND ¼ KPIV LEND ¼ K 7 1 DO 4 I ¼ K, N DSUM ¼ 0.DO IF (LEND .GT. 0) THEN DO 2 L ¼ 1, LEND DSUM ¼ DSUM þ ARR(KPIV 7 L) * ARR(IND 7 L) 2 CONTINUE ENDIF DSUM ¼ ARR(IND) 7 DSUM IF (I .EQ. K) THEN IF (DSUM .LE. 0.D0) THEN IERR ¼ 7 1 RETURN ENDIF DPIV ¼ DSQRT(DSUM) ARR(KPIV) ¼ DPIV DPIV ¼ 1.D0 = DPIV ELSE ARR(IND) ¼ DSUM * DPIV ENDIF IND ¼ IND þ I 4 CONTINUE 10 CONTINUE IERR ¼ 0 IPIV ¼ N*(N þ 1)=2

336

12

14

20

22

24 30

Van Espen IND ¼ IPIV DO 20 I ¼ 1, N DIN ¼ 1.DO = ARR(IPIV) ARR(IPIV) ¼ DIN MIN ¼ N KEND ¼ I 7 1 LANF ¼ N 7 KEND IF (KEND .GT. 0) THEN J ¼ IND DO 14 K ¼ 1, KEND WORK ¼ 0.D0 MIN ¼ MIN 7 1 LHOR ¼ IPIV LVER ¼ J DO 12 L ¼ LANF, MIN LVER ¼ LVER þ 1 LHOR ¼ LHOR þ L WORK ¼ WORK þ ARR(LVER) * ARR(LHOR) CONTINUE ARR(J) ¼ 7 WORK * DIN J ¼ J 7 MIN CONTINUE ENDIF IPIV ¼ IPIV 7 MIN IND ¼ IND 7 1 CONTINUE DO 30 I ¼ 1, N IPIV ¼ IPIV þ I J ¼ IPIV DO 24 K ¼ I, N WORK ¼ 0.D0 LHOR ¼ J DO 22 L ¼ K, N LVER ¼ LHOR þ K 7 I WORK ¼ WORK þ ARR(LVER) * ARR(LHOR) LHOR ¼ LHOR þ L CONTINUE ARR(J) ¼ WORK J¼JþK CONTINUE CONTINUE IERR ¼ 0 RETURN END

REFERENCES Aarnio PA, Lauranto H. Nucl Instrum Methods A276:608 (1989). Bertin EP. Principles and Practice of X-ray Spectrometric Analysis. New York: Plenum Press, 1970. Bevington PR, Robinson DK. Data Reduction and Error Analysis for the Physical Sciences. New York: McGraw-Hill, 1992. Black WW. Nucl Instrum Methods 71:317, 1969.

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Bloomﬁeld DJ, Love G. X-Ray Spectrom 14:8, 1985. Bombelka E, Koenig W, Richter FW. Nucl Instrum Methods B22:21, 1987. Breschinsky R, Krush E, Wehrse R. Diplomarbeit, Fachbereich Physik. Bremen: Universita¨t Bremen, 1979. Brook D, Wynne RJ. Signal Processing Principles and Applications. London: Edward Arnold, 1988. Campbell JL. X-Ray Spectrom 24:307, 1995. Campbell JL. Nucl Instrum Methods B109=110:71, 1996. Campbell JL, Perujo A, Millman BM. X-Ray Spectrom 16:195, 1987. Campbell JL, Maxwell JA, Papp T, White G. X-Ray Spectrom 26:223, 1997. Campbell JL, Millman BM, Maxwell JA, Perujo A, Teesdale WJ. Nucl Instrum Methods B9:71, 1985. Campbell JL, Maenhaut W, Bombelka E, Claytan E, Malmqvist K, Maxwell JA, Pallon J, Vandenhaute J. Nucl Instrum Methods B14:204, 1986. Campbell JL, Cauchon G, Le´py M-C, McDonald L, Plagnard J, Stemmler P, Teesdale WJ, White G. Nucl Instrum Methods A418:394, 1998. Cirone R, Gigante GE, Gualtieri G. X-Ray Spectrom 13:110, 1984. Clayton E, Duerden P, Cohen DD. Nucl Instrum Methods B22:64, 1987. Doster JM, Gardner RP. X-Ray Spectrom 11:173, 1982a. Doster JM, Gardner RP. X-Ray Spectrom 11:181, 1982b. Duﬀy CJ, Rogers PSZ, Benjamin TM. Nucl Instrum Methods B22:91, 1987. Enke CG, Nieman TA. Anal Chem 48:705A, 1976. Fiori CE, Myklebust RL, Gorlen K. NBS Spec Pub 604, 233, 1981. Gardner RP, Yacout AM, Zhang J, Verghese K. Nucl Instrum Methods A242:299, 1986. Geladi P, Kowalski BR. Anal Chim Acta 185:1, 1986. Gertner I, Heber O, Zajfman J, Zajfman D, Rosner B. Nucl Instrum Methods B36:74, 1989. Gunnink R. Nucl Instrum Methods 143:145, 1977. Heckel J, Scholz W. X-Ray Spectrom 16:181, 1987. Hertogen J, De Donder J, Gijbels R. Nucl Instrum Methods 115:197, 1974. Hoskuldsson A. J Chemometr 2:211, 1988. Jansson PA. Deconvolution with Applications in Spectroscopy. New York: Academic Press, 1984. Janssens K, Van Espen P. Anal Chim Acta 184:117, 1986. Janssens K, Dorrine´ W, Van Espen P. Chemometr Intell Lab Syst 4:147, 1988. Janssens K, Vincze L, Van Espen P, Adams F. X-Ray Spectrom 22:234, 1993. Jenkins R, Gould RW, Gedcke D. Quantitative X-Ray Spectrometry. New York: Marcel Dekker, 1981. Jensen BB, Pind N. Anal Chim Acta 117:101, 1985. Johansson GI. X-Ray Spectrom 11:194, 1982. Joy DC. Rev Sci Instrum 56:1772, 1985. Kajfosz J, Kwiatek WM. Nucl Instrum Methods B22:78, 1987. Lemberge P, Van Espen PJ. X-Ray Spectrom 28:77, 1999. Lorber A. Wangen LE, Kowalski BK. J Chemometr 1:19, 1987. Lucas-Tooth HJ, Price BJ, Metallurgia 64: 149, 1961. Maenhaut W, Vandenhaute J. Bull Soc Chim Belg 95:407, 1986. Manne R. J Chemometr Intell Lab Syst 2:187, 1987. Marageter E, Wegscheider W, Mu¨ller K. Nucl Instrum Methods B1:137, 1984a. Marageter E, Wegscheider W, Mu¨ller K. X-Ray Spectrom 13:78, 1984b. Marquardt DW. J Soc Ind Appl Math 11:431, 1963. Martens H, Naes T. Multivariate Calibration. Chichester: John Wiley, 1989. Massart DL, Vandeginste BGM, Deming SN, Michotte Y, Kaufman K. Chemometrics: A Textbook. Amsterdam: Elsevier, 1988. McCarthy JJ, Schamber FH. NBS Spec Publ 604:273, 1981. McCrary JH, Singman LV, Ziegler LH, Looney LD, Edmonds CM, Harris CE. Phys Rev A4:1745, 1971.

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McCullagh H. Report EGG-PHYS-5890, Idaho National Engineering Laboratory, Idaho Falls, ID, 1982. McNelles LA, Campbell JL. Nucl Instrum Methods 127:73, 1975. Molt K, Schramm R. Fresenius J Anal Chem 359:61, 1997. Nielson KK. X-Ray Spectrom 7:15, 1978. Nullens H, Van Espen P, Adams F. X-Ray Spectrom 8:104, 1979. Nunez J, Rebollo Neira LE, Plastino A, Bonetto RD, Gue´rin DMA, Alvarez AG. X-Ray Spectrom 17:47, 1988. Op De Beeck JP, Hoste J. Atomic Energy Rev 13:151, 1975. Pella PA Feng L, Small JA. X-Ray Spectrom 14:125, 1985. Pessara W, Debertin K. Nucl Instrum Methods 184:497, 1981. Petersen W, Ketelsen P, Kno¨chel A. Nucl Instrum Methods A245:535, 1986. Phillips GW. Nucl Instrum Methods 153:449, 1978. Philips GW, Marlow KW. Nucl Instrum Methods 137:525, 1976. Pratar A, De Jong S. J Chemometr 11:311, 1997. Press WH, Flannery BP, Teukolsky SA, Vetterling WT. Numerical Recipes in C, The Art of Scientiﬁc Computing. Cambridge: Cambridge University Press, 1988. Reed SJB, Ware NG. J Phys E5:582, 1972. Robertson A, Prestwich WV, Kennett TJ. Nucl Instrum Methods 100:317, 1972. Routti JT, Prussin SG. Nucl Instrum Methods 72:125, 1969. Ryan CG, Clayton E, Griﬃn WL, Sie SH, Cousens DR. Nucl Instrum Methods B34:396, 1988. Salem SI, Wimmer RJ. Phys Rev A2:1121, 1970. Salem SI, Saunders BG, Melson C. Phys Rev A1:1563, 1970. Savitzky A, Golay MJE. Anal Chem 36:1627, 1964. Schamber FH. In: Dzubay T, ed. X-Ray Fluorescence Analysis of Environmental Analysis. Ann Arbor, MI: Ann Arbor Science, 1977, p. 241. Schreier F. J Quant Spectros Radiat Transfer 48:743, 1992. Schwalbe LA, Trussell HJ. X-Ray Spectrom 10:187, 1981. Scoﬁeld JH. Phys Rev 179:9, 1970. Scoﬁeld JH. Phys Rev A9:1041, 1974a. Scoﬁeld JH. Phys Rev A10:1507, 1974b. Sherry WM, Vander Sande JB. X-Ray Spectrom 6:154, 1977. Smith DGW, Gold CM, Tomlinson DA. X-Ray Spectrom 4:149, 1975. Statham PJ. X-Ray Spectrom 5:16, 1976a. Statham PJ. X-Ray Spectrom 5:154, 1976b. Statham PJ. X-Ray Spectrom 7:132, 1978. Statham PJ, Nashashibi T. In: Newbury DE, ed. Microbeam Analysis. San Francisco: San Francisco Press, 1988, p. 50. Steenstrup S. J Appl Crystallogr 14:226, 1981. Swerts J, Van Espen P. Anal Chem 65:1181, 1993. Urba´nski P, Kowalska E. X-Ray Spectrom 24:70, 1995. Van Dyck P, Van Grieken R. X-Ray Spectrom 12:111, 1983. Van Espen P, Adams F. X-Ray Spectrom 5:123, 1976. Van Espen P, Nullens H, Adams F. Anal Chem 51:1325, 1979a. Van Espen P, Nullens H, Adams F. Nucl Instrum Methods 142:243, 1977a. Van Espen P, Nullens H, Adams F. Nucl Instrum Methods 145:579, 1977b. Van Espen P, Nullens H, Adams F. X-Ray Spectrom 9:126, 1980. Van Espen P, Janssens K, Nobels J. Chemometr Intell Lab Syst 1:109, 1986. Van Espen P, Nullens H, Maenhaut W. In: Newbury DE, ed. Microbeam Analysis. San Francisco, San Francisco Press, 1979b, p. 265. Van Espen P, Nullens H, Adams F. Anal Chem 51:1580, 1979c. Vekemans B, Janssens K, Vincze L, Adams F, Van Espen P. X-Ray Spectrom 23:275, 1994. Vekemans B, Janssens K, Vincze L, Adams F, Van Espen P. Spectrochim Acta 50B:149, 1995.

Spectrum Evaluation Verghese K, Mickael M, He T, Gardner RP. Adv X-Ray Anal 31:461, 1988. Vincze L, Janssens K, Adams F. Spectrochim Acta 48B:553, 1993. Vincze L, Janssens K, Adams F, Jones KW. Spectrochim Acta 50B:1481, 1995a. Vincze L, Janssens K, Vekemans B, Adams F. Spectrochim Acta 54B:1711, 1999. Vincze L, Janssens K, Adams F, Rivers ML, Jones KW. Spectrochim Acta 50B:127, 1995b. Wa¨tjen U. Nucl Instrum Methods B22:29, 1987. Wilkinson DH. Nucl Instrum Methods 95:259, 1971. Wielopolski L, Gardner RP. Nucl Instrum Methods 165:297, 1979. Yacout AM, Dunn WL. Adv X-Ray Anal 30:113, 1987. Yacout AM, Gardner RP, Verghese K. Nucl Instrum Methods A243:121, 1986. Yacout AM, Gardner RP, Verghese K. Adv X-Ray Anal 30:121, 1987. Yule PH. Nucl Instrum Methods 54:61, 1967. Zolnai L, Szabo´, Gy. Nucl Instrum Methods B34:118, 1988.

339

5 Quantiﬁcation of Inﬁnitely Thick Specimens by XRF Analysis Johan L. de Vries* Eindhoven, The Netherlands

Bruno A. R. Vrebos Philips Analytical, Almelo, The Netherlands

I.

INTRODUCTION

Quantitative x-ray ﬂuorescence (XRF) analysis involves the conversion of measured ﬂuorescent intensities to the concentration of the analytes. In most of the literature on this subject, the measured intensities at this stage of the analytical procedure are assumed to be corrected for background and line overlap. The same assumptions will be made here, although some comments will be made on these topics throughout this chapter. Furthermore, energy-dispersive and wavelength-dispersive x-ray spectrometers diﬀer signiﬁcantly with respect to the counting channel and the resolution or resolving power that can be achieved. This has a direct inﬂuence on, for example, the line overlap correction procedures that need to be applied (if any) to obtain net intensities. Wavelength-dispersive x-ray ﬂuorescence spectrometry has been discussed in Chapter 2; energy-dispersive systems are the subject of Chapters 3 (for x-ray tube excitation) and 7 (for radioisotope excitation). The software provided by the vendors of x-ray spectrometry systems for their customers compensates to some extent for the peculiarities of each design and considerable diﬀerences in the software packages oﬀered with the instruments are to be expected. It is therefore diﬃcult to give guidelines that are generally applicable. In some cases, most notably those where the intensity of the background is essentially constant (between specimens), the correction for background can be neglected. This is usually the case when the diﬀerence in terms of composition of the standard specimens and the unknowns is rather small [i.e., the concentration ranges (not necessarily the concentration levels) of all components are rather small]. This condition of similarity between specimens is suﬃcient to ensure constant background, but it is not a requirement, as, in certain cases, the intensity of the background is (nearly) constant even when the concentration range of one or more analytes is quite large. Note that the only requirement

*Retired. 341

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de Vries and Vrebos

here is for the background to be constant; at this point, no restrictions on the magnitude of the background have been imposed. Under these circumstances, correction for background is not required and the gross count rate (or intensity) can be used with equal success as the net intensity for the determination of the specimen’s composition. The situation with line overlap is somewhat more complicated, as it is almost never constant. This is due to the intensity of the overlapping line(s): It is very rarely a constant for diﬀerent specimens. There are several methods to correct for line overlap, and the preference of the individual analyst as well as the type of spectrometer and its associated software (if any) are some of the factors inﬂuencing the ﬁnal choice. The basic principles of quantitative analysis have not changed much since the early years of x-ray ﬂuorescence analysis, especially for the wavelength-dispersive systems. Energy-dispersive systems have seen more changes due to, for example, development of diﬀerent solid-state detectors and primary optics, such as capillaries (see Chapter 11), and the increasingly more easily available computing power which allowed for more comprehensive spectral treatment programs at the ﬁngertips of the analysts (see also Chapter 4). In general, the method has always involved some calibration step, during which the intensities of selected elements in a suite of standard specimens are measured according to a scheme or a recipe that has been optimized earlier. These intensities are combined with the composition of the standard specimens, and the calibration curves for the elements of interest are then constructed. These calibration curves might include a correction for matrix eﬀects of some sort. The resulting calibrations could then be used for the analysis of unknown specimens with compositions similar to the standards. This method is rather time-consuming, but it yields, in general, the highest possible analytical accuracy. Also, approximate knowledge of the composition of the specimens prior to the analysis is assumed. Such assumptions can be made very implicitly (e.g., when selecting the calibration curve to be used for analysis). If the method is routinely used and the number of unknown specimens to be analyzed is rather large, the beneﬁts of this instrumental technique become very clear, when compared to ‘‘classical’’ wet chemistry methods, which tend to be much more time-consuming. Since the second half of the 1980s, interest has been growing for a diﬀerent kind of quantitative analysis on wavelength-dispersive spectrometers, in which the elements present in the specimens and their approximate concentration levels are not known prior to the analysis. Semiquantitative analysis consists of an intimate combination of qualitative and quantitative analyses. During the qualitative analysis, the presence of the elements present in the specimen is established. This can be done in a variety of ways. Qualitative analysis is based on the collection of a spectrum of the specimen, followed by peak search and peak identiﬁcation. Alternatively, the intensities of a rather large and ﬁxed set of characteristic lines are collected. The lines are selected by the manufacturer and allow for the determination of most of the commonly occurring elements. In both cases, the net intensities are then determined and a quantiﬁcation is made, involving matrix correction and general calibration lines for each of the elements. These calibration lines have been determined earlier from a set of standard specimens, although some manufacturers insist on calling their method ‘‘standardless.’’ The packages that are available commercially diﬀer on the basis of the qualitative analysis (accumulating a spectrum or measurement on ﬁxed wavelength positions), the number of standard specimens used, the ﬂexibility of the spectrometer conﬁguration that they can cope with, and the method used to correct for matrix eﬀect. The focus of energy-dispersive systems has always been somewhat more qualitative and more oriented toward research, ﬂexibility, and the analysis of small batches of

Quantification of Infinitely Thick Specimens

343

unknown specimens. The software allowed for quantitative analysis without the need to have standards similar to the unknowns. These systems are thus more ﬂexible in their use in that respect. This does not mean that such systems are unsuitable for routine analysis in industry; many of them actually are used for that purpose. In this chapter, several methods for the quantiﬁcation of inﬁnitely thick specimens are discussed. For all methods described here, it is assumed that the specimens are inﬁnitely thick: The intensity of the radiation from the specimen is constant and is not aﬀected by increasing the thickness of the specimen. The analysis of specimens with lessthan-inﬁnite thickness is discussed in Chapter 6. Also, the specimens are assumed to be homogeneous: Their composition is the same throughout the specimen. This assumption is valid for a large range of material types found in many applications. The most important exception are specimens with one or more thin coatings.

II.

CORRELATION BETWEEN COUNT RATE AND SPECIMEN COMPOSITION

A.

Introduction

In quantitative analysis, the measured x-ray ﬂuorescent intensity of a given element is converted into its weight concentration in the specimen. As a ﬁrst approximation, one would expect a linear relationship. Each atom of the analyte element i has the same probability of being excited by the primary photons and emitting its characteristic photons with wavelength li. Indeed, if separate atoms or ions (e.g., in a gas or in a very diluted solution) are considered, the following relation holds: I i ¼ K i Wi

ð1Þ

where Ii is the measured intensity of the ﬂuorescent radiation of the analyte i and Wi is the weight fraction of the analyte i in the specimen. The proportionality constant Ki consists of many physical and instrumental factors such as the following: The intensity and the distribution of wavelengths of the photons in the primary beam The probability that an atom i emits its characteristic radiation li The probability that these photons li pass through the measuring channel: collimators, diﬀracting crystal, pulse-height window The probability that these photons are being detected and registered. For a given instrument, the voltage and power on the x-ray tube remain constant for the analyte i and they can be determined by measuring the ﬂuorescent intensity of the pure element i. However, we are generally dealing with compact specimens where the atoms are arranged into chemical compounds. Both the primary x-rays and the ﬂuorescent x-rays will be absorbed by the diﬀerent atoms in the specimen. B.

General Relationship Between Intensity and Concentration

1. Primary Fluorescence by Monochromatic Radiation First, excitation by monochromatic radiation will be considered. Let the intensity of the incident beam with wavelength l0 at the surface of the specimen be given by I0(l0). This beam strikes the surface of the specimen at an angle c0 (see also Fig. 1). A parallel beam is assumed here and the specimen is considered to be extending to inﬁnity in all three dimensions. The incident radiation is gradually absorbed by the specimen, and at a layer at

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Figure 1 Schematic representation of the geometry involved in the calculation of primary fluorescence emission.

depth t below the surface, the remaining fraction of the intensity It(l0) is given by the Lambert–Beer law: It ðl0 Þ ¼ I0 ðl0 Þ exp½ms ðl0 Þrs t ðcsc c0 Þ

ð2Þ

ms(l0) is the mass-attenuation coeﬃcient of the specimen for photons with wavelength l0 (the subscript s refers to the specimen) and rs is the density of the specimen. Note that due to the angle of incidence, the path length traveled is given by t csc c0 . The mass-attenuation coeﬃcient ms(l0) in this equation is thus calculated for the specimen; this is done by adding the mass-attenuation coeﬃcients for all elements j present in the specimen, each multiplied with its mass fraction Wj: ms ðl0 Þ ¼

n X

½mj ðl0 ÞWj

ð3Þ

j¼1

where n is the total number of elements present in the specimen. The fraction of the incident beam absorbed by the analyte i in the layer between t and (t þ dt) is given by Wi mi ðl0 Þrs csc c0 dt

ð4Þ

It is assumed here that the composition of the specimen is uniform throughout. In other words, Wi and Wj are independent of the position of the layer (t, t þ dt) within the specimen. Only a fraction of the photons absorbed creates vacancies in the K shell; this fraction is given by (riK 1)=riK, where riK is the absorption jump ratio of the K shell of element i. The fraction of K vacancies emitting x-rays is given by the ﬂuorescence yield oiK. The fraction of Ka photons in the total of x-rays emitted for the analyte is given by the transition probability fiKa. These factors can be combined in a factor designated Qi(l0, li): Qi ðl0 ; li Þ ¼ oi

riK 1 fiKa riK

ð5Þ

In the above derivation, it is assumed that the characteristic line of interest is a Ka. If another line is considered, the relevant changes need to be made. The characteristic photons thus generated are isotropically emitted in all directions, without a preferential direction. Only a fraction is emitted toward the detector. If O is the solid angle as viewed by the collimator – detector system, expressed in steradians, then the fraction is given by O=4p. The angle O should be small enough so that the beam can be considered to be a parallel beam leaving the specimen at a single well-deﬁned angle c00 with the surface. The fraction of characteristic photons with wavelength li not absorbed between the layer (t, t þ dt) and the surface is given by

Quantification of Infinitely Thick Specimens

345

exp½ms ðli Þrs t csc c00

ð6Þ

All photons reaching the surface and propagating in the direction indicated by the angle c00 are assumed to be detected. If the detector has absorbing elements (such as windows) or detection eﬃciencies diﬀerent from unity, then these can also be taken into account. Also, the attenuation by the medium (e.g., air) between specimen and detector can be calculated using similar expressions. The intensity of element i, as excited by the incident beam with wavelength l0, is labeled Pi(l0) (explicitly denoting the primary ﬂuorescence eﬀect) and is given by Pi ðl0 Þ ¼ I0 ðl0 Þ exp½ms ðl0 Þrs t csc c0 mi ðl0 ÞWi Qi ðl0 ; li Þrs cscc0 dt

O exp½ms ðli Þrs t csc c00 4p

ð7Þ

Combining factors leads to O 4p exp½ðms ðl0 Þ csc c0 þ ms ðli Þ csc c00 Þrs trs dt

Pi ðl0 Þ ¼ I0 ðl0 Þmi ðl0 ÞWi Qi ðl0 ; li Þ ðcsc c0 Þ

ð8Þ

The contribution of all layers between the surface and the ‘‘bottom’’ of the specimen have to be summed. This can be done by integrating the above expression over dt, from 0 (the surface) to the ‘‘bottom.’’ In practice, for x-rays, the thickness of bulk specimens can be considered to be inﬁnite, so the upper limit is 1 . Note that t is always used in combination with rs, so the integration will be done over rst. Taking all constant factors outside the integral, one obtains Pi ðl0 Þ ¼ I0 ðl0 Þmi ðl0 ÞWi ðcsc c0 ÞQi ðl0 ; li Þ Z1

O 4p

exp½ðms ðl0 Þ csc c0 þ ms ðli Þ csc c00 Þrs trs dt

ð9Þ

0

From a textbook on calculus, Z1 exp½ax dx ¼

1 a

ð10Þ

0

Taking a ¼ (ms(l0) csc c0 þ ms(li) csc c00 ) and noting that rs dt ¼ d(rst), the following expression is obtained for the primary ﬂuorescence of the analyte i in the specimen s: Pi ðl0 Þ ¼

I0 ðl0 Þmi ðl0 ÞWi ðcsc c0 ÞQi ðl0 ; li ÞðO=4pÞ ms ðl0 Þ csc c0 þ ms ðli Þ csc c00

ð11Þ

Often, the element speciﬁc factors, given by Eq. (5), are combined with the instrument speciﬁc factor O=4p. This leads to a simpler expression for the primary ﬂuorescence: mi ðl0 Þ Wi Pi ðl0 Þ ¼ Ki I0 ðl0 Þ ð12Þ ms ðl0 Þ þ Gms ðli Þ where Ki ¼ ðO=4pÞQi ðl0 ; li Þ and G ¼ csc c00 =csc c0 ¼ sin c0 = sin c00 . In the above derivation, the following assumptions have been made: (a) The specimen is completely homogeneous. (b) The specimen extends to inﬁnity in three dimensions.

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(c) The primary rays are not scattered on their way to the layer dt. (d) No enhancement eﬀects occur. (e) The characteristic radiation is not scattered on its way to the specimen surface. The simplest case is excitation of Ka (or Kb) photons. For the characteristic photons associated with the L lines, other eﬀects, such as Coster–Kronig transitions and so forth have to be taken into account when describing the fraction of absorbed primary photons that give rise to characteristic photons. For a more detailed discussion on these eﬀects, refer to Chapter 1. 2. Secondary Fluorescence Excited by Monochromatic Radiation Under certain conditions, the characteristic photons of an element j can excite atoms of the analyte i. This will lead to additional characteristic photons of element i. The term ‘‘additional’’ is used here in the sense that these photons are not considered in the above derivation for the primary ﬂuorescence and, consequently, are not predicted by Eq. (12). It is beyond the scope of this chapter to derive the mathematical expressions following a rigorous approach, but certain aspects of secondary ﬂuorescence emission are readily seen from the above derivation and from Figure 2. In Figure 2, two layers are now considered. The layer on the left, indicated by j, at a depth of t1 with a thickness of dt1, is where the primary ﬂuorescence of element j is excited. This is described by Eqs. (2)–(5). Obviously, the element considered at this stage is the enhancing element j, so the fundamental parameters for j need to be used instead of these for i in Eqs. (2)–(5). Once the ﬂuorescent radiation is created, it will be absorbed on its way through the specimen. The precise direction is of no concern, as we are now no longer considering the fraction that travels toward the detector. The attenuation along its path is described by equations such as Eq. (2) or Eq. (6). Keep in mind that the angles and the distances involved are now different and that the wavelength l for which the mass attenuation needs to be calculated [Eq. (3)] is now lj instead of l0. At the second layer, indicated by i, at a depth of t2 with a thickness of dt2, part of this (primary ﬂuorescence) radiation will be absorbed. Again, this eﬀect is described by an equation such as Eq. (4), with the angle c0 replaced by the angle of this path. The factors to be applied in order to lead to characteristic radiation of the analyte i are given by Qi(lj, li); Eq. (6) describes the attenuation (wavelength is now li) on the way toward the detector. These factors can easily be recalculated. The complication is in the description of the geometry, especially in the description of the distance between the two layers: The position of the two layers relative to one another is not restricted, the path of the photon lj does not have to be in the plane of the ﬁgure, nor is the second layer

Figure 2 Schematic representation of the geometry involved in the calculation of secondary fluorescence emission.

Quantification of Infinitely Thick Specimens

347

always at a larger depth than the ﬁrst layer. The angle X 0 in Figure 2 can take any value between 0 and 2p, and the three paths shown are not necessarily in one plane. This aspect has been discussed by Li-Xing (1984). Anyway, after integration, the ﬁnal result Sij ðl0 ; lj Þ is given by Sij ðl0 ; lj Þ ¼

I0 ðl0 Þðcsc c0 Þmj ðl0 ÞWj Qj ðl0 ; lj Þmi ðlj ÞWi Qi ðlj ; li ÞðO=4pÞ 2ðms ðl0 Þ csc c0 þ ms ðli Þ csc c00 Þ sin c0 ms ðl0 Þ sin c00 ms ðlj Þ ln 1 þ ln 1 þ þ ms ðl0 Þ ms ðlj Þ ms ðlj Þ sin c0 ms ðlj Þ sin c00

ð13Þ

where Sij ðl0 ; lj Þ describes the secondary ﬂuorescence (enhancement) of the analyte i by characteristic photons with wavelength lj; these photons have been excited by primary photons with wavelength l0. The photons to be considered for the enhancement of the analyte are not limited to Ka only, as the Kb lines and other characteristic lines that have suﬃcient energy to excite the shell of interest of the analyte i must also be taken into account. For this reason, all individual contributions need to be added in order to calculate the total enhancement of the analyte i by element j: X Sij ðl0 Þ ¼ Sij ðl0 ; lj Þ ð14Þ i

There are two criteria to be satisﬁed for a characteristic photon lj to be able to cause secondary ﬂuorescence: 1. It must be excited by the incident photon l0 (this means that the energy of the incident photon must be higher than the energy of the absorption edge associated with lj). 2. The energy of the photon lj must be higher than the energy of the absorption edge for the analyte. The derivation of the intensities of characteristic x-rays as a function of specimen composition has ﬁrst been done by Sherman (1955). The resulting equations, however, were rather unwieldy. Equations (9) and (13) have been published later by Shiraiwa and Fujino (1966) and Sparks (1976). In practice, many computer programs only consider a few lines for enhancement, and lines that have a low transition probability f are usually neglected. Data regarding these transition probabilities for a line within a series can be found, for example, in Appendix II of Chapter 1. The enhancement phenomenon will be more pronounced if the x-rays of the enhancing elements are only slightly more energetic than the energy of the absorption edge of the element i. The enhancement may contribute up to 40–50% of the total ﬂuorescent radiation Ii , especially where the concentration of the enhancing elements is much greater than the concentration of the analyte. This applies even more for the light elements, where the primary spectrum may not be very eﬀective as the most intense wavelengths are far away from the absorption edges of these light elements. The eﬀect of scattered radiation is generally ignored, although its contribution can also be calculated (Pollai et al., 1971). If the characteristic line of the analyte considered is one of its L lines, then it is possible that other lines of the analyte itself enhance the characteristic line considered. The energy of the K edge of La (atomic number Z ¼ 57) is 38.9 keV (see Chapter 1), so the K lines of all elements with Z < 57 are excited if, for example, an x-ray tube is used at

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Table 1

Data for Selected Absorption Edges and Characteristic Line of Pb

Absorption edge K L1 L2 L3

Energy (keV)

Characteristic line

Energy (keV)

88.04 15.86 15.20 13.04

La1(L3–M5) Lb1(L2–M4) Lg1(L2–N4)

10.55 12.61 14.76

Source: Appendix I and Appendix II of Chapter 1 (this volume).

voltages above 40 keV, as is commonly the case with wavelength-dispersive spectrometers. Under these conditions, the L lines of these elements are enhanced by the K lines. The situation for elements with high atomic numbers is more complex. Let us consider Pb (Z ¼ 82). Lead is very often determined through its La1 or Lb1 line because the K lines of Pb are too energetic to be measured with reasonable eﬃciency by scintillation or Si(Li) detectors. Also, excitation voltages higher than 88 keV are required in order to excite the K lines. Thus, one would not readily expect enhancement by Pb on its own L lines. From the data in Table 1, however, it follows that the PbLa1 line (L3 –M5 ) is enhanced by PbLg1 (L2 –N4 ) because its energy (14.76 keV) is higher than the energy of the L3 edge (13.04 keV). The total excited characteristic intensity is then given by adding the primary and the secondary ﬂuorescence contributions: X Sij ðl0 ; lj Þ ð15Þ Ii ðl0 Þ ¼ Pi ðl0 Þ þ j

Also, tertiary ﬂuorescence is possible, where the incident photon excites element k (primary ﬂuorescence), whose radiation excites element j (secondary ﬂuorescence), whose radiation, in turn, excites element i, causing tertiary ﬂuorescence. This contribution is generally lower than 3% of the total ﬂuorescence and is commonly ignored, as shown by Shiraiwa and Fujino (1967, 1974). 3. Excitation by Continuous Spectra If the incident beam is polychromatic rather than monochromatic, Eq. (15) needs to be calculated for each wavelength. Wavelengths longer than the wavelength of the absorption edge ledge,i of the analyte cannot excite ﬂuorescence, so these need not to be considered. If JðlÞ is the function representing the tube spectrum, then Eqs. (11) and (13) can still be used, provided that I0 ðl0 Þ is replaced by JðlÞ, representing the intensity of the incident spectrum at wavelength l: lZedge; i

Ii ¼

lZedge; i

Ii ðJðlÞÞ dl ¼ lmin

" Pi ðlÞ þ

lmin

X

# Sij ðl; lj Þ dl

ð16Þ

j

This equation allows the calculation of the intensity of characteristic radiation of a given analyte in an inﬁnitely thick specimen. Equations such as Eqs. (11), (14), and (16) are often referred to as fundamental parameter equations because they allow the calculation of the intensity of ﬂuorescent radiation as a function of the composition of the specimen (weight fractions Wi ), the incident spectrum [JðlÞ], and the conﬁguration of the spectrometer used (c0 ; c00 , and O). All other variables used are fundamental constants, such as the mass-attenuation coeﬃcients for a given element at a given wavelength or its ﬂuorescence yield and so forth.

Quantification of Infinitely Thick Specimens

C.

349

Some Observations

The integration over t in Eq. (9) is taken from zero to inﬁnity. It is obvious that the ﬁrst layers contribute more to the intensity of li than the more inward layers. Theoretically, even at large values of t, a very minor contribution to the intensity is still to be expected. Often, the (minimum) inﬁnite depth is deﬁned arbitrarily as that thickness t where the contribution of the layer ðt; t þ dtÞ is 0.01% of that of the surface layer. In this case, it is deﬁned relative to the surface layer. Alternatively, it is deﬁned as the thickness where the contribution to the total intensity is less than 1%. In this case, it is relative to the total intensity from a ‘‘truly inﬁnitely’’ thick specimen. The value of the inﬁnite thickness depends on the value of the absorption coeﬃcients and the density of the specimen. In practice, it may vary from a few micrometers for heavy matrices and long wavelengths to centimeters for short wavelengths and light matrices, as in solutions. For a given element and a ﬁxed geometry, an eﬃciency factor Cðl0 ; li Þ can be introduced: mi ðl0 Þ Pn ð17Þ j¼1 Wj mj ðl0 Þ þ G j¼1 Wj mj ðli Þ P It is obvious that the terms Wj mj are the origin of nonlinear calibration lines, as variations in Wj inﬂuence the value of the denominator. For a pure metal, this reduces to Cðl0 ; li Þ ¼ Pn

Cðl0 ; li Þ ¼

mi ðl0 Þ mi ðl0 Þ þ Gmi ðli Þ

ð18Þ

In the general case, the wavelengths in the primary spectrum close to the absorption edge are the most eﬀective in exciting the analyte i. The eﬃciency factor Cðl0 ; li Þ is thus a combination of the absorption curve of analyte i as a function of l and the spectral distribution. In a ﬁrst approximation, an ‘‘eﬀective wavelength’’ le can be introduced, which has the same eﬀect of excitation of element i as the total primary spectrum. The exact value of this le will be inﬂuenced by the characteristic tube lines if they are active in exciting i. Otherwise, le can, in general, be assumed to have a value of approximately two-thirds of the absorption edge ledge. Its actual value is, however, dependent on the chemical composition of the specimen. For instance, for Fe, the wavelength of the K edge ledge is 0.174 nm (7.11 keV) and an eﬀective wavelength le of 0.116 nm is obtained using this rule of thumb. In the ZnO–Fe2O3 system, le was found to vary from 0.130 nm for 100% Fe2O3 to 0.119 nm for 10% Fe2O3 in ZnO. The estimated value of 0.116 nm is in good agreement with the experimental one for Fe2O3. Another interesting case is the analysis of a heavy element i in a light matrix. In the summation in the denominator of Eq. (12), the terms Wi mi ðl0 Þ and Wi mi ðli Þ are then the most important. The other terms can, in ﬁrst approximation, be neglected if Wi is not small. However, that means that the terms Wi in the numerator and the denominator cancel and the measured intensity becomes independent of Wi ; thus, the analysis becomes impossible in this extreme case. A solution to this problem is found by making the inﬂuence of the terms Wi mi ðlÞ in the denominator less dominating by adding a large term Wa ma ðlÞ. This can be done be making Wa large (e.g., diluting) or ma ðlÞ large, by adding heavy absorber. Equation (17) enables one to calculate beforehand how large this term should be to eliminate ﬂuctuations in the concentration of the other elements. In the derivation of the ﬂuorescence of the analyte i in the preceding paragraphs, the following simpliﬁcations were made:

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1.

First, it was assumed that the primary rays follow a linear path to the layer dt at depth t. However, the primary rays may also be scattered. In general, the loss in intensity of the primary beam of photons due to scattering may be neglected. These scattering eﬀects become more important when the primary x-rays are more energetic and the average atomic number of the matrix decreases. This scatter may give a higher background in the secondary spectrum, thus leading to a poorer precision of the analysis. On the other hand, the excitation eﬃciency may be enhanced, as the primary rays ‘‘dwell longer’’ in the active layers, thus having a higher probability to encounter atoms of element i. This eﬀect may overrule the increase in intensity of the background radiation. A case in hand is the determination of Sn in oils which gives better results using the SnK lines at a high x-ray tube voltage, than using the SnL lines at moderate voltages. Incidentally, this scattering of the primary radiation makes it possible to check the voltage over the x-ray tube. According to Bragg’s law, the intensity of the primary spectrum is zero at an angle y0, given by nlmin 1 y0 ¼ sin ð19Þ 2dcrystal where 2dcrystal is the 2d spacing of the crystal used and n is an integer number. lmin (in nm) is given by

1:24 ð20Þ V where V is the voltage on the x-ray tube, expressed in kilovolts. In practice, a lower value for V will be found when Compton scattering dominates over Rayleigh scattering and thus lmin found has a value too low by the Compton shift, which is about 0.024 nm for most spectrometers; the actual value depends on the incidence and exit angle (see also Sec. V.C.1.) 2. The integral in Eq. (9) was taken from zero to inﬁnity; further, it was assumed that the specimen is completely homogeneous. This, of course, is never realized in practice, as we are dealing with discrete atoms in chemical compounds. In powders, the diﬀerent compounds may have a tendency to cluster. The particles will, in general, have diﬀerent sizes and shapes. Putting the sample into solution, either aqueous or solid (melt) may overcome this problem. lmin ¼

It was stated formerly that inﬁnite thickness may vary from 20 mm to a few centimeters. However, the most eﬀective layers are much thinner. Thus, the number of discrete particles actually contributing to the ﬂuorescent radiation may be rather small.

III.

FACTORS INFLUENCING THE ACCURACY OF THE INTENSITY MEASUREMENT

A.

Introduction

The total uncertainty of the analysis consist of many errors whose source may be the following: The measurement of the intensity The reproducibility of the specimen preparation The conversion of intensity into concentration

Quantification of Infinitely Thick Specimens

351

The uncertainty of a single determination may be found from n determinations of the same analysis, given a mean result xmean, of the all individual results xi. If n is suﬃciently large, the standard deviation s may be found from sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Pn 2 i¼1 ðxi xmean Þ ð21Þ s¼ n1 In this total standard deviation s, random and systematic uncertainties are combined. Random uncertainties give an indication of the precision of an analysis (the scatter of results around the mean value), whereas systematic errors are the reason for deviations of the mean value from the ‘‘true’’ value. An analysis may thus be precise, but not very accurate, if systematic errors are present, whereas accurate values could be found from the mean of widely scattered measurements if only large random uncertainties were present. The total error of a measurement is composed of all the separate errors. If only random errors s1, s2, and so forth are considered, the resulting standard deviation is given by s2 ¼ s21 þ s22 þ s23 þ þ s2n

ð22Þ

where s1, s2, and so on are the errors associated with, for example, intensity measurements, specimen preparation, instrumental settings, and so on. In practice, it is often found that s is dependent on the concentration of the analyte Wi (Johnson, 1967; Hughes and Hurley, 1987): pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ s ¼ K Wi þ Wb ð23Þ where Wb is a small concentration oﬀset (typically 0.001 in weight fraction). Thus, K, rather than s, becomes an indication of the accuracy of the determination. B.

Random Errors

1. Counting Statistics If an x-ray measurement consisting of the determination of a number of counts N is repeated n times the results N1,N2, N3, . . . ,Nn would spread about the true value N0. If n is large, the distribution of the measurements would follow a Gaussian distribution, " # 1 ðN N0 Þ2 WðNÞ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ exp ð24Þ 2N 2pN pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ provided N is also large. The standard deviation s of the distribution is equal to Nmean , again if n and N are large, where Nmean is the mean of n determinations. From the properties of the Gaussian distribution, the following hold: 68.3% of all values N will be between N0s and N0 þ s. 95.4% of all values N will be between N02s and N0 þ 2s. 99.7% of all values N will be between N03s and N0 þ 3s.

pﬃﬃﬃﬃ Similarly,pthere ﬃﬃﬃﬃ is a certain probability that the true result N0 will lie between N N and N þ N, assuming the same distribution for N and N0. Measurement results are commonly expressed as a count rate (intensity per unit time) instead of an intensity, which gives the number of counts collected in the counting interval. This allows an easier comparison of measurements made with diﬀerent counting times, but the measuring time needs to be speciﬁed in order to be able to assess the counting statistical error.

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The determined concentration is dependent on the net count rate, which is the peak count rate Rp minus the background count rate Rb. The total measuring time T equals tp þ tb , where tp and tb are the measurement times for peak and background, respectively. In modern equipment, there is no signiﬁcant statistical error in the measurement of t. We can thus assume that R follows the same Gaussian distribution as N with the same relative standard deviation eN. eN is deﬁned as sN ð25Þ eN ¼ N Hence,

and

pﬃﬃﬃﬃ N 1 1 ¼ pﬃﬃﬃﬃ ¼ pﬃﬃﬃﬃpﬃﬃ ¼ eR eN ¼ N N R t

ð26Þ

pﬃﬃﬃﬃ R sR ¼ eR R ¼ pﬃﬃ t

ð27Þ

It is obvious that the relative counting error decreases as t increases. When a net count rate has to be determined the peak, Rp and the background, Rb, have to be measured; there are thus two independent variables. The standard deviation of the net intensity, sd, is given by sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Rp Rb sd ¼ s2p þ s2b ¼ ð28Þ þ tp tb and the relative standard deviation ed by pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Rp =tp þ Rb =tb ed ¼ Rp Rb

ð29Þ

When using a sequential wavelength-dispersive x-ray ﬂuorescence spectrometer and when the count rates are rather low and time is limited, it is of interest to divide the total counting time available over tp and tb in the best possible way. In principle, there are three methods to split up the total counting time T: 1.

2.

Fixed time. Peak and background are measured for the same time; tp ¼ tb ¼ T=2. The resulting standard deviation in the diﬀerence between the peak and the background count rate is then given by rﬃﬃﬃﬃqﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 ðRp þ Rb Þ ð30Þ sd ¼ T Fixed count. The same number of counts, N, is collected on the peak and on the background: Np ¼ Nb or tp Rb ¼ tb Rp The resulting standard deviation for the diﬀerence is given by rﬃﬃﬃﬃqﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃsﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Rp Rb 1 ðRp þ Rb Þ þ sd ¼ T Rb Rp

ð31Þ

ð32Þ

Quantification of Infinitely Thick Specimens

353

3. Fixed time optimal. The optimum division of the total measurement time T over tp and tb can be found by diﬀerentiating Eq. (29) with respect to tp, where tb ¼ T tp. It is found that for optimal results, the ratio of the measuring times is sﬃﬃﬃﬃﬃﬃ tp Rb ¼ ð33Þ tb Rp The resulting standard deviation is given by rﬃﬃﬃﬃ 1 pﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ sd ¼ ð Rp þ Rb Þ T and the relative standard deviation by 1 1 ed ¼ pﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ T Rp Rb

ð34Þ

ð35Þ

It can easily be demonstrated that sFTO sFT sFC

ð36Þ

where sFTO refers to the method of ﬁxed time optimized, sFT is for ﬁxed time, and sFC is for ﬁxed counts. When Rp is very large compared with Rb, sFTO is close to sFT. Thus, the ﬁxed time method is often used in practice because Rp and Rb are not known beforehand. If there is a great diﬀerence between peak and background count rate, then usually all the available time is spend counting the peak. Thus, the background is not measured. A constant value for the background can be assumed or it can be ignored altogether. Compared with method of ﬁxed time, the following observation can be made: If, Rb is not measured, the standard deviation in the net count rate (calculated as the diﬀerence between Rp and Rb) is given by rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Rp þ s2b ð37Þ sd ¼ T pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ This approximation pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ is allowed if this value is smaller than ð2=TÞ ðRp þ Rb Þ. This is the case if sb 5 Rb =T. This applies if the background is more or less constant between specimens and a ﬁxed value for the background can be deducted. If the background pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃis altogether ignored, sb ¼ Rb . This is only allowed when Rb is much smaller than Rp =T. follows that when one aims for optimum instrumental conditions, the expression pﬃﬃﬃﬃﬃﬃ It p ﬃﬃﬃﬃﬃﬃ This is often used as a ﬁgure of merit Rp Rb is a good quality function. p ﬃﬃﬃﬃﬃﬃ parameter pﬃﬃﬃﬃﬃﬃ (FOM). Obviously, the highest value of R gives the best result. When it can be R p b pﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ assumed that Rp Rb approximates Rp, for low intensities and high background, optimizing this term, equals optimizing M pﬃﬃﬃﬃﬃﬃ Rb

ð38Þ

where M (the slope of the calibration line in counts per second per percent) is proportional to Rp Rb. If the ratio of two count rates R1 and R2 has to be determined, the methods of ﬁxed time and ﬁxed count give the same result, whereas the method of optimum division, where rﬃﬃﬃﬃﬃﬃ t1 R2 ¼ ð39Þ t2 R1 always gives the best result.

354

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The total counting uncertainty is the combination of instrumental error and counting statistics. When the count rate is very high, the counting uncertainty is small and it may be worthwhile to apply a ratio method to reduce possible instrumental uncertainties. However, if the count rate is low, it is better to spend all the available time in analyzing the specimen to reduce the counting error. 2. Instrumental Errors If the instrumental and counting uncertainties are random and independent variables, then qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ etot ¼ e2instr þ e2count ð40Þ einstr ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ e2tot e2count

ð41Þ

Although the counting error is inﬂuenced by the instrumental error, Eq. (41) is still a good approximation. stot can be found from a series of repeated results of one measurement; thus, scount is known. To check the instrumental instability, all the functions should be measured separately. A radioactive source 55Fe, for example, can be used to check the detector and electronic circuitry. The x-ray tube can be checked by repeating the measurements with specimen and goniometer in a ﬁxed position, eliminating errors stemming from the mechanics of the spectrometer. In another series of experiments, recycling between diﬀerent angles checks the reproducibility of the goniometer, whereas repositioning the specimen between measurements checks the specimen holder, the reproducibility of the specimen loading mechanics, and so forth. A comprehensive series of tests for wavelength-dispersive x-ray ﬂuorescence spectrometers is described in the Australian Standard 2563-1982 (1982). Sometimes the error stot found is smaller than expected, or even smaller than scount . This may indicate that an unexpected systematic error is involved, or there may be an uncorrected deadtime in the equipments, (i.e., the true counting rate is higher than measured, which means that the relative error is smaller). 3. Detection Limit A characteristic line intensity decreases with decreasing concentration of the analyte and ﬁnally disappears in the background noise. The true background intensity may be constant, but the results of the measurements ﬂuctuate around a mean value Rbmean . To be signiﬁcantly diﬀerent from the background, a signal Rp must, although it is larger than Rbmean , be distinguished from the spread in Rb . In other words, if we measure a signal Rp larger than Rb and we assume the analyte is present, what is the probability that our assumption is correct? If the results of the measurements are random and follow a Gaussian distribution, then this probability is determined by sRb . If the measurement Rp is higher than Rb þ 2sRb , then the probability that our assumption is correct is approximately 95% if a higher certainty is required (e.g., 99.7%), then Rp should be larger than Rb þ 3sRb . Thus, the net intensity is 3sRb and the detection limit, DL, would be DL ¼

3sRb M

ð42Þ

where M is the sensitivity in counts per second per percent. So, the detection limit in the above equation is the concentration corresponding to a net peak intensity of 3sRb . However, in x-ray spectrometry, the background signal is specimen dependent and cannot

Quantification of Infinitely Thick Specimens

355

be measured independently, as in radioactivity measurements. Hence, Rb has to be measured in an oﬀ-peak location in the spectrum. The resultﬃ Rb found in measuring time t s is pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ assumed to be Rbmean and sRb is assumed to be ðRb =tÞ. Thus, two measurements have to be made: Rp and Rb , each in time t s. The detection limit thus becomes pﬃﬃﬃ rﬃﬃﬃﬃﬃﬃ 3 2 Rb ð43Þ DL ¼ M T where T ¼ 2t. If we are satisﬁed with a 95% probability that our assumption is correct, then pﬃﬃﬃ rﬃﬃﬃﬃﬃﬃ 2 2 Rb ð44Þ DL ¼ M T which is roughly equal to rﬃﬃﬃﬃﬃﬃ 3 Rb DL ¼ M T

ð45Þ

It is obvious that the detection limit decreases if the counting time increases. However, the total error in Rb contains the instrumental error as well. Thus, there is no sense to increase the counting time when the instrumental error dominates. Ingham and Vrebos (1994) have shown that the detection limit can be improved by carefully selecting a primary ﬁlter. If the application of such a primary beam ﬁlter reduces the intensity of the (scattered) continuum from the tube more than it aﬀects the sensitivity, the detection limit is improved. The loss in sensitivity M needs to be more than compensated for by the reduction in background intensity; as from Eq. (45), the detection limit is proportional to the square root of Rb and inversely proportional to the sensitivity M. 4. Variation in X-ray Spectrum The ﬂuorescent intensity of the analyte Ii is, in ﬁrst approximation, dependent on the primary spectrum according to Ii ¼ KiðV0 Vc Þp

ð46Þ

where K is a constant, i is the current of the x-ray tube, V0 is the working voltage, Vc the excitation voltage, and p varies between 1 and 2, depending on the ratio of excitation by characteristic tube rays and white continuum. In modern instruments, the tube voltage is not dependent on the mains cycle, but they run on constant potential, which still may ﬂuctuate. If the working voltage V0 , or the region or line of highest excitation probability is rather close to Vc , then small ﬂuctuations in V0 will introduce a considerable error in Ii . For instance, if V0 ¼ 1:5 Vc , then a 1% error in V0 gives an error in Ii of 6% when p ¼ 2. It is therefore better to run the tube at three to ﬁve times the excitation voltage of the analyte. A too high voltage, however, might introduce an unproportionally high background. 5. Other Instrument Errors Other possible random instrumental errors include positioning the specimen and setting the goniometer in wavelength-dispersive x-ray spectrometers. These errors have to be checked by repeated measurements of one specimen in a systematic way:

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Repeated Repeated Repeated Repeated

counting counting counting counting

with with with with

stationary stationary stationary stationary

specimen and goniometer specimen and repositioning goniometer goniometer and repositioning specimen goniometer and reloading specimen holder.

Some diﬀracting crystals have a rather high coeﬃcient of thermal expansion; therefore, their d value may ﬂuctuate with ﬂuctuations in temperature. This results in a setting of the goniometer, slightly oﬀ-peak, which introduces a change in measured intensity. Therefore, most modern equipment are operating at a stabilized temperature. 6. Particle Statistics Only a limited volume of the specimen can actually contribute to the ﬂuorescent radiation. As long as this active volume is the same in standards and actual specimens and the atomic distribution is completely homogeneous, this poses no problem. However, the atoms are bound into chemical compounds, forming ﬁnite particles with diﬀerent chemical compositions. The analyte may only occur in particles with a certain chemical composition and not in other particles. Then, only these speciﬁc particles can contribute to the ﬂuorescent radiation of the analyte i. Therefore, the count rate Ri measured depends on the number of those particles present in the active volume, where, evidently, the ﬁrst layers contribute most of the ﬂuorescent radiation. Table 2 gives an indication of the penetration depth of radiation of various wavelengths into matrices with varying absorption power. It is evident that for most solid specimens, the ﬂuorescent radiation originates within 20 mm or less from the surface. To get an idea of how many particles can actually contribute to the ﬂuorescent intensity of analyte i, let us assume that the irradiated area of 10 cm2 is covered with cubic particles of 100 mm dimension in a random fashion. Assuming a ﬁlling factor of 0.8 and assuming that the analyte i is only present in 10% of the particles, then 10 108 104 101 0:8 ¼ 8000 particles could be actually contributing. Assuming pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ a Gaussian distribution, then this number would have a standard deviation of 8000 ¼ 90 particles, or a relative standard deviation of approximately 1.1%. If the concentration of analyte is only 1%, then this relative standard deviation would be roughly 3.3%. In practice, these errors might even be larger, as the radiation of the specimen is not homogeneous because the primary spectrum originates in a rather small anode and passes through a large window and is, thus, conically shaped. Spinning the specimen in its own plane during the analysis will reduce this error. Furthermore, the ﬁrst layers are the most eﬀective, having

Table 2

‘‘Inﬁnite’’ Thickness (in mm for Certain Analytical Lines as a Function of the Matrix)

Analytical line SnKa MoKa NiKa CrKa AlKa NaKa CKa

Fe base

Mg base

H2O solution

Borate

Borate La2O3 10%

300 100 12 33 1.5 0.7 0.3

10,000 3,700 340 120 4 20 0.3

100,000 30,000 2,400 900 10 9 4

70,000 30,000 2,000 800 15 6 1

10,000 2,600 300 250 5 5 0.4

Note: Both the incidence and exit angles are 45 and excitation is by a Rh tube at 60 kV. The inﬂuence of elementspeciﬁc absorption can be seen from, for example, the values for NiKa and CrKa in the Fe-base matrix.

Quantification of Infinitely Thick Specimens

357

only a small number of particles containing the analyte with corresponding larger relative errors. In extreme cases, it is evident that the eﬀective layer is very thin, less than 1 mm. Care should thus be taken that the specimen surface is as smooth as possible; as surface irregularities (e.g., grooves, ridges) will introduce a considerable error. Spinning the specimen will, again, reduce this error. It is therefore vital that the specimen be completely homogeneous. If this is not possible and powders have to be analyzed, care should be taken that the particles are very small, less than a few micrometers in diameter. Specimen preparation will be discussed in full detail in Chapter 14. C.

Systematic Errors

1. Dead T|me After an x-ray photon is detected in the counter and accompanying electronics, it takes a certain time before the counting circuit is ready to accept the next photon. Any photon entering the counter within this period, called the dead time of the counter circuit, is simply not registered and is thus lost. This dead time is of the order of a few microseconds. The counting losses are thus dependent on the actual count rate. The measured count rate Rm is always lower than the true count rate RT . Their relation can be approximated by the expression RT ¼

Rm 1 td Rm

ð47Þ

where td is the dead time. For instance, if td ¼ 1 ms and Rm ¼ 105 counts per second, the dead-time loss is approximately 10%. With modern equipment, very high count rates can be handled, to reach suﬃcient precision in a short time. It is therefore necessary to reduce these losses. In most wavelength-dispersive instruments, an automatic dead-time correction circuit is included. Energy-dispersive instruments, on the other hand, tend to collect counts for the speciﬁed time; thus, the measurement takes longer because the total time required in this case consists of the speciﬁed measuring time (‘‘lifetime’’) and the dead time. 2. Matrix Effects The ﬂuorescent intensity of the analyte i is, as discussed earlier, not only dependent on its concentration but can also be strongly dependent on the composition of the specimen itself. The primary rays will be absorbed and scattered and secondary ﬂuorescence may occur. All of these eﬀects depend on the chemical composition of the specimen. The importance of these eﬀects depends on the concentration of these matrix elements and their inﬂuence (e.g., their absorption of primary and secondary x-rays). These matrix effects may introduce large systematic errors when they are not properly accounted for, as discussed in Sec. V. D.

Choice of Optimal Conditions

1. Selecting the Analytical Line Some considerations in choosing the analytical line are the following: 1. High sensitivity, thus preferably the strongest line in the emission spectrum is used; this is commonly the Ka

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2. 3. 4.

Low intensity of background radiation Constant angle of Bragg diﬀraction (wavelength-dispersive instruments) No coincidence (line overlap) with lines of other elements

Each of these items will be discussed in more detail below. 1.

2.

3.

4.

For the light- and medium-Z elements, the Ka line is by far the strongest line in their spectrum and is therefore often chosen as the analytical line. For the elements with K-edge energies exceeding 40 keV, the L lines are preferred because the K lines, in general, cannot be used, the maximum voltage for most spectrometers being limited to 100 kV or less. Typically, a voltage of three times Vc or more is needed to get a high characteristic intensity. Furthermore, the resulting K lines are very energetic and are only detected with mediocre eﬃciency by the Si(Li) detectors or NaI scintillation detectors. Ge detectors have a higher eﬃciency for that energy range and these are the preferred detectors. Another reason the L lines are preferred for the heavier elements is that in direct tube-excited XRF, the background due to scattering of the primary x-rays is much lower in the L region of the spectrum. The wavelength of some analytical lines may shift slightly with the valence state of the elements, especially for the light elements and the L lines of the transition metals (Wood and Urch, 1978); thus, the standard used in setting the goniometer to the analytical line should correspond to the specimen in this respect. Another reason for an apparent shift in angle may be the change in d value of analyzing crystal with temperature. The analytical line should ideally be completely free of any disturbing lines.

However, there are many sources of distributing inﬂuences; some of these are described next. The following sections deal with line overlap in the case of wavelength dispersive spectrometers. For energy dispersive spectrometers, please refer to Chapter 4. 2. Spectral Overlap Two or more characteristic lines may not be completely separated from the analytical line. This separation may be improved by using a crystal with better dispersion (e.g., a lower d value). However, the choice must often be made between high intensity and high dispersion. If the disturbing line is due to a high-order crystal reﬂection, its inﬂuence may be strongly reduced by the proper setting of the pulse-height selector. However, in some cases, the escape peak of the interfering line may be within the pulse-height selector window. For instance, the third-order reﬂection using a penta erythritol (PE) crystal, of the characteristic tube lines of a Sc anode, scattered by the specimen will slightly interfere with the analysis for Al in a light matrix, as the energy of the ScKa escape peak in an Arﬁlled gas detector is very close to the energy of the AlKa line. Often, the overlap is due to a diagram line of an element of which another diagram line is free of overlap. As, in general, two diagram lines of one element have a constant intensity ratio, the measured intensity of the nonoverlapped line of the disturbing element multiplied by a constant factor (experimentally determined) may be subtracted from the measured intensity of the analytical line to give the characteristic intensity. Absorption eﬀects can, however, strongly inﬂuence the ratio. This is most clearly the case when there is an absorption edge of a major element between the two diagram lines considered.

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3. Primary Radiation Scattered by the Specimen Photons of all wavelengths present in the white spectrum of the incident beam are scattered by the specimen, including the characteristic lines of the tube anode, giving rise to a continuous background. If, however, the specimen consists of rather coarse grains, it may happen that a crystallite is in a favorable position for Bragg diﬀraction for a wavelength of the continuum; thus, a sharp peak will be found in the spectral analysis. The inﬂuence of primary tube lines, coherently or incoherently scattered, may be eliminated by the proper choice of the anode. 4. Spurious Reflections by the Analyzing Crystal LiF(110) is a common analyzing crystal. The second order of reﬂection is used because the ﬁrst order is crystallographically forbidden; therefore, the actual planes used for the diffraction are the 220 planes. Hence, some manufacturers refer to it as the LiF(220) crystal; the same applies to the LiF(100) and LiF(200) denominations. However, when a very, very high intensity is observed, a ﬁrst-order reﬂection will still be found, due to asymmetry of the electronic cloud. Similarly, a second order may be observed using a Ge crystal. 5. Satellite Lines The common wavelength tables give only characteristic K, L, and M lines of most elements. The M lines of heavy elements may interfere with the K lines of light elements. Nondiagram satellite lines may also occur, often giving rise to an unexpected background level. One of the tables that includes such lines is provided by NIH (Bethesda, MD) (Garbauskas and Goehner, 1983).* The lines that are most obvious to observe are some of the satellites lines of Al, Si, and P with a wavelength-dispersive spectrometer. In Figure 3, a spectrum over aluminum is shown, on which some of these lines have been annotated. IV.

CALIBRATION AND STANDARD SPECIMENS

A.

Introduction

As shown earlier, standard specimens must cover the concentration range of interest, be stable over time, and have a certiﬁed composition. They are, however, not the only specimens required to set up a calibration for routine use and to maintain it over extended periods of time. Quality control specimens (also called quality assurance specimens) are used to assess the quality of the analysis obtained over time, whereas drift-correction monitors are used to correct for long-term drift of the equipment. Finally, recalibration standards can be used if the calibration graph must be reconstructed. B.

Quality Control Specimens

The use of at least one sample with known composition to assess the accuracy at the time of the calibration is highly recommended. The sample(s) used for this purpose should be typical for the unknowns and should not be used for calibration to avoid biases and overly optimistic estimates or accuracy. Furthermore, it is a sound practice to select at least one specimen, with a composition similar to the unknowns, as a quality control specimen to verify the repeatability over extended periods of time. Should the results on the quality *Database made available through C. Fiori, National Institutes of Health, Bethesda, Maryland.

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Figure 3 Spectrum of aluminum-containing specimen. The peaks are the following emission lines: (1) AlKb1, Kb3 doublet; (2) AlSKb0 ; (3) AlSKa7; (4) AlSKa5; (5) AlSKa4; (6) AlSKa3; (7) AlKa1, Ka2 doublet. The satellite lines are explicitly labeled with S. Note the logarithmic scale on the intensity axis. Conditions: wavelength-dispersive spectrometer with PE crystal, Rh tube.

control specimen fall outside a predetermined range, then adequate measures must be taken. The ﬁrst step is usually to perform a correction for drift. DeGroot (1990) has described how the use of statistical process control (SPC) can be beneﬁcial in this respect. The SPC charts that can be maintained in this way allow one to check the performance of the spectrometer system. C.

Drift-Correction Monitors

Correction for drift can be made by measuring selected specimens for each of the analytes and calculate the ratio between the observed intensities and those obtained when the calibration was performed. The ratios can then be applied as a correction to the slope of the calibration graphs or, as is done more often, the measured intensities of the unknowns are corrected for drift, prior to the conversion to concentration. If drift correction is performed regularly (e.g., once a day), the drift between subsequent measurements is very small and, thus, measurements with high precision are required, otherwise the counting statistical error of the measurement would become dominant. Drift correction should not even be applied unless the correction is signiﬁcant. Because the precision of an intensity measurement is determined by the number of counts collected, drift-correction monitors are, ideally, specimens on which high count rates can be obtained. Drift-correction monitors do not have to be specimens with a composition similar to the unknowns. As diﬀerent components of the spectrometer (such as the x-ray tube) age, not only will the sensitivity be aﬀected (this is generally a downward trend), but in some cases, the background will vary also. To correct for this change in background intensity, measurements of the background must be performed. This poses a problem, inasmuch as the count

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361

rate on the background is usually low and thus measurements are not vary precise or they require a long measuring time. However, the considerations on the counting statistical error made earlier (Sec. III.B.1) oﬀer some suggestions. First, the intensity of the background is irrelevant when the intensity of the analyte peak is much higher. Obviously, this has to apply for all specimens to be measured. If that is the case, then variations in the background intensity are also negligible. When the background matters, it should already be measured anyway and then drift correction on the background intensity is not required, as the net count rate is corrected for drift. The only case that is not covered here is the contribution of spectral contamination from the x-ray tube (e.g., Cu, W, Ag, Fe, etc.). In that case, the background, including the contamination, must be measured on peak. This implies that measurements must be done for specimens with zero analyte concentration. Again, two cases can be distinguished: If the background including the contamination is not important compared to the count rate observed, then no corrections are required. On the other hand, if one of the analytes is present at low concentration, then the contribution to the background due to the contaminant must be checked periodically and taken into account. After the drift correction is performed, a quality control specimen should be measured to verify the procedure. D.

Recalibration Standards

Sometimes (e.g., after a major maintenance on a spectrometer), drift correction does not bring the quality control specimens back in line with the expectations. In those cases, the calibration curve must be reconstructed. This can be done by measuring all the standard specimens again and repeating the complete calibration procedure. Because calibrations often use many standards and validating each calibration is required, this can be a timeconsuming process, even if the validation is limited to a quick visual inspection of the calibration graphs. In such cases, a recalibration can be performed based on only a few standards. The idea of recalibration is to reconstruct the calibration graph, without having to measure all the standards again. This is done by selecting a few standard specimens for each analyte and measuring these (the top and bottom point in Fig. 4). Subsequently, when determining the parameters (such as slope and intercept) of the regression, for the concentrations of these specimens in the calibration the certiﬁed values are no longer used, but the values as found on the calibration line at the time of the original calibration (the x-ray values) are. These x-ray values have been found based on all standard specimens used, and the idea is to ‘‘ﬁx’’ the calibration line again through these points. As a result, the statistical data are now skewed, but the values for the slope and the intercept are very close to the original ones; the small diﬀerences between old and new values are due to the counting statistical errors in the measurements and these are also present when unknowns are measured. The specimens used for recalibration have also been used for the calibration. The only requirement is that the recalibration specimens have count rates that are diﬀerent enough so that the determination of the slope is accurate enough. For each of the calibration lines, the number of selected specimens must be at least the same as the number of parameters to be determined. If the slope and intercept are determined, at least two recalibration standards are required. However, if in this case three or four standards are used, it is possible to detect gross counting artifacts (e.g., caused by mislabeled standards, incorrect loading, etc.). The root mean square error or the correlation coeﬃcient on the calibration line (which can only be calculated if more specimens are used than parameters determined) has no relationship with the accuracy of the analysis. In fact, a near-perfect correlation should be obtained. As in the case of drift correction, it is recommended to

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Figure 4 The original calibration line, based on seven data points (h) can be reconstructed using only two data points (in this case, top and bottom), with concentrations (x-ray values) modified to those obtained on the calibration (j).

measure, after recalibration, the quality control specimen(s), as gross errors might thus be identiﬁed before unknown specimen are analyzed. E.

Conclusion

Setting up a calibration to be used over extended periods of time requires considerable amounts of work and preparation. It involves not only the selection and procurement of standard specimens but also drift correction monitors and recalibration standards. Also, the specimen preparation method is an essential part of the whole procedure. The result, however, is the ability to produce quantitative results to a previously assessed degree of accuracy and precision over extended periods of time with minimal work, once the specimen preparation procedure is set up and the initial calibration is performed.

V.

CONVERTING INTENSITIES TO CONCENTRATION

A.

Introduction

The simplest equation relating intensity to concentration is Ii ¼ Ki0 Wi Ki0

ð48Þ

is assumed to be a constant. The equation holds in general for cases where the where total eﬀect of the matrix on the analyte i is constant (e.g., elements at minor and trace concentration levels in low-alloy steels or thin-ﬁlm specimens). The intensity Ii in Eq. (48) is a net intensity: measured intensity corrected for background, line overlap, and so forth.

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363

In practice, the measured intensity is often used directly without the subtraction of background, leading to a more general equation: Ii ¼ Bi0 þ Ki0 Wi

ð49Þ

where Bi0 is the measured intensity when Wi ¼ 0. If there is no uncorrected line overlap, Bi0 is the background. This equation can be rearranged to Wi ¼ Bi þ Ki Ii

ð50Þ

The constant Ki is called the sensitivity and is expressed in counts per second per unit concentration (e.g., percent, mg=L, etc.). The most common method of determining the constants Bi and Ki (or Bi0 and Ki0 ) is linear regression on a number of standard specimens. Linear regression can be done by minimizing the sum of squared residuals of W, or I; see Figure 5. In theory, the method of least squares assumes that the errors of the dependent variable [W in Eq. (50)] are normally distributed. The two lines obtained (one by minimizing DW, the other by minimizing DI ) are not the same. Because, for the analysis, the intensity Ii is measured, it is recommended to minimize for Wi . Also, in general, the relative error of the measured intensities is smaller than the relative error of the concentrations in the standard specimens. This is especially true for the determination of trace elements. The following formulas can be used for the determination of the values of the parameters Ki and Bi : Wi ¼ Bi þ Ki Ii Pn Pn Iij Þ=n j¼1 Wij Iij ð j¼1 Wij Pn 2 Pn j¼1 Pn j¼1 Iij ð j¼1 Iij j¼1 Iij Þ=n

ð50Þ

Pn Ki ¼

ð51Þ

Figure 5 Straight lines through the data points can be determined by minimizing either the sum of the squares of the residuals DI or DW.

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and

Pn Bi ¼

j¼1

Wij Ki

Pn

j¼1 Iij

n

ð52Þ

where the sums are over the standard specimens, j ¼ 1; 2; . . . ; n, with n the number of standard specimens used for analyte i. Equally important, however, are the variances on the parameters determined. Formulas to calculate the variances can be found in the literature [Draper and Smith (1966)], and many commonly used computer programs such as spread sheets and statistical packages include the relevant calculations as well. Some important conclusions are as follows: 1. 2. 3.

The concentrations of the standard specimens must cover the expected range of concentrations. The calculated concentration is more accurate at the center of the line than at the extremities; the estimated variance for Wx increases with (Wx Waverage). Because a calibration line is derived using data from several standards, the analysis of the unknown can sometimes be more accurate than the accuracy of the individual standard specimens; this is due to the eﬀect of averaging.

When the background is properly subtracted from the gross count rate, the background Bi is equal to zero and Eq. (50) reduces to Wi ¼ K i I i In this case, the value for the slope Ki is found from Pn Pn Iij j¼1 Wij Pn 2j¼1 Ki ¼ j¼1 Iij

B.

ð53Þ

ð54Þ

Matrix Effect

Applying Eq. (50) or (53) requires that all standards must be similar to the unknown in all aspects considered: matrix eﬀect, homogeneity, and so on. This would lead to the use of standard specimens with a very limited concentration range. Such a requirement is in disagreement with the observation that the variance on the slope factor is smaller with increasing range: The use of standard specimens covering only a small concentration range will lead to a calibration graph with large uncertainty on the slope and intercept. On the one hand, this advocates the use of a set of standards with a wide range of concentrations; on the other hand, the requirement of similarity in matrix eﬀects tends to limit the range. Obviously, a compromise must be made. Equation (50) is a simpliﬁcation of the more general equation describing the relationship among analyte concentration Wi , specimen homogeneity Si , measured intensity Ii , and matrix eﬀect Mi : Wi ¼ Ki Ii Mi Si

ð55Þ

The term ‘‘specimen homogeneity’’ also includes the grain size eﬀect and the mineralogical eﬀect. These are notoriously diﬃcult to treat mathematically; in fact, most methods describing the grain size eﬀect rigorously assume, for example, the dispersed phase to be perfect spheres of a given diameter or an arrangement of cubes (Bonetto and Riveros, 1985). Other methods allow more variability, but these also require a priori more information about the specimen, such as the composition of the individual granular phases,

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365

the average shape and size of the phases, and so forth (Hunter and Rhodes, 1972; Lubecki et al., 1968; Holynska and Markowicz, 1981). The fact that the specimen homogeneity is as yet not described by a single successful method is one of the reasons that Eq. (55) is commonly reduced to Wi ¼ K i I i Mi

ð56Þ

Fortunately, by using adequate specimen preparation methods the eﬀect of Si between specimens (standards as well as unknowns) can be rendered constant. This constant factor is then absorbed by the sensitivity Ki . As a ﬁrst approximation, the degree of variation in matrix eﬀect between two specimens for a given analyte i can be estimated by calculating, for both compositions, the following parameter: Ii ¼ Pi

mi ms ðl0 Þ þ Gms ðli Þ

ð57Þ

where ms ðl0 Þ and ms ðli Þ are the mass-attenuation coeﬃcients of the specimen considered for wavelengths l0 and li , respectively, and G is the geometrical factor for both compositions. The relative diﬀerence between these expressions should not exceed a few percent, otherwise the matrix eﬀects become too important to ignore. If the specimens include ﬂux or a binding agent, then these must also be considered in Eq. (57). With increasing range of concentrations, deviations from linearity will be observed due to variations in matrix eﬀects between specimens and standards. The analyst must then resort to other methods to obtain accurate results. Matrix eﬀects are studied most easily by considering binary systems (i.e., specimens with only two elements or compounds). In the case of absorption (both primary and secondary absorption must be considered), three cases can be distinguished (see Fig. 6): 1. A simple, linear relationship between relative intensity R and weight fraction W (Curve 1 in Fig. 6). In this case, there is no matrix eﬀect: The analyte and the matrix have very similar (in principle, the same) attenuation coeﬃcients for the incident and the characteristic radiations. 2. Curve 2 in Figure 6 is obtained when the matrix has a higher attenuation coeﬃcient for the analyte’s characteristic radiation than the analyte itself: The characteristic radiation is primarily absorbed by the matrix element. This is usually called positive absorption. 3. Curve 3 in Figure 6 is obtained when the matrix absorbs less than the analyte itself: The matrix has a smaller value for the attenuation coeﬃcient for the characteristic radiation than the analyte itself. This can be the case, for example, if the matrix has a much lower atomic number than the analyte, such as Mo in Al. This eﬀect is called negative absorption. Enhancement will generally lead to a calibration graph like curve 4 in Figure 6. The eﬀect of enhancement is usually smaller than that of positive or negative absorption, as indicated by the position, relative to curve 1, of curves 2 and 3 and curve 4. It can be shown that the behavior of the calibration curves can be explained in terms of attenuation coeﬃcients only if absorption is the only matrix eﬀect. Furthermore, if monochromatic excitation is used, a single constant (calculated from attenuation

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Figure 6 Calibration curves for binaries. Curve 1: no net matrix-effect; curve 2: net absorption of the analyte’s radiation by the matrix (positive absorption); curve 3: net absorption of the analyte’s radiation by the analyte (negative absorption); curve 4: enhancement of the analyte’s radiation by the matrix.

coeﬃcients) suﬃces to express the eﬀect of one element on the intensity of another. In the following section, various methods to deal with matrix eﬀects will be discussed. C.

Elimination or Evaluation of the Total Matrix Effect: Compensation Methods

1. Scattered Radiation: Compton Scatter (in cooperation with Mark N. Ingham, British Geological Survey, Keyworth, UK) If the variation in matrix eﬀects is mainly due to absorption, scattered x-rays can be used to obtain an estimate of the absorption coeﬃcient of the specimen at a certain wavelength ls . The intensity of the scattered radiation can be shown to be inversely proportional to the mass-attenuation coeﬃcient ms of the specimen: Is ðls Þ ﬃ

1 ms ðls Þ

ð58Þ

where ls is the wavelength of the scattered radiation and ms ðls Þ is the mass-attenuation coeﬃcient of the specimen for wavelength ls . This is illustrated in Figure 7. The intensity of the ﬂuorescent radiation, Ii ; is also inversely proportional to the mass-attenuation coeﬃcient [Eq. (57)], but at a diﬀerent wavelength: Ii ﬃ

Wi Wi ¼ ms ðl0 Þ þ Gms ðli Þ ms

ð59Þ

where ms ¼ ms ðl0 Þ þ Gms ðli Þ

ð60Þ

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367

Figure 7 The intensity of Compton-scattered radiation is inversely proportional to the massattenuation coefficient (mac) of the specimen. Conditions: Rh tube at 60 kV.

Mass-attenuation coeﬃcients at two diﬀerent wavelengths are virtually proportional, independent of matrix composition, provided there are no signiﬁcant absorption edges between the two wavelengths considered (Hower, 1959). Hence, the ratio Ii =Is is proportional to the concentration of the analyte. Both coherently and incoherently scattered primary radiation, such as tube lines for tube excited x-ray ﬂuorescence (XRF), as well as the scattered continuous radiation can be used. The method corrects also, to some degree, for surface ﬁnish, grain size eﬀects, and variations in tube voltage and current, but it does not correct for enhancement, thus limiting its use to analytes that are inﬂuences by absorption only. Furthermore, no absorption edges of major elements may be situated between the two wavelengths considered. If that is the case, the ratio between the mass-attenuation coeﬃcients for the wavelengths considered depends to a large degree on the concentration of the major element(s) and the ratio is no longer constant between specimens. This reduces the range of analytes that can be covered using the scattered tube lines. The use of scattered continuum radiation close to the analyte peak may be disadvantageous due to limited intensity, leading to either long measurement times or poor counting statistics. The intensity of the scattered Compton radiation is higher for specimens mainly consisting of low-atomic-number elements than for specimens with a higher average atomic number. This is illustrated in Figure 8, where the scattered radiation is plotted as a function of wavelength for two specimens: iron and magnesium. In this experiment, the spectrometer was equipped with a Rh anode x-ray tube. The two sharp peaks that can be observed are the RhKb (at 0.0546 nm) and the RhKa (at 0.0613 nm), respectively. The other two, much broader peaks are the Compton-scattered RhKb and the RhKa. The Compton-scattered peaks are much broader than Rayleigh-scattered

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Figure 8 Intensity of scattered radiation as a function of wavelength. Two different specimens have been used: iron and magnesium. A Rh anode tube was used in this experiment. Note the large difference in intensity between the specimens. For the specimen consisting of the element with the higher atomic number (Fe), the intensity of the Compton (incoherent)-scattered radiation is lower than the intensity of the Rayleigh (coherent)-scattered radiation, compared to the specimen with the lower-atomic-number element (Mg).

characteristic lines. They are also shifted toward longer wavelengths by an amount Dl, which is given by Dl ¼ 0:00243ð1 cos cÞ

ð61Þ 0

00

where c is the angle through which the radiation is scattered: c ¼ c þ c and Dl is expressed in nanometers. In the spectrometer used for the recording of the spectra in Figure 8, c is 100 . The Compton shift, Dl, is thus 0.0029 nm for this conﬁguration. The maxima of the Compton peaks (at 0.0576 nm and 0.0644 nm) in Figure 8 are in good agreement with the theoretical values: 0.0546 nm þ 0.0029 nm ¼ 0.0575 nm and 0.0613 nm þ 0.0029 nm ¼ 0.0642 nm, respectively. The intensity of the Compton-scattered radiation is higher for shorter wavelengths and for specimens consisting of elements with low atomic numbers. For a given wavelength (e.g., the characteristic radiation of a tube line), the intensity of Compton scatter decreases as the specimen consists of more and more elements with higher atomic numbers. For specimens made up of oxides, the scattered intensity is usually so intense that it can be measured with suﬃcient precision in a relatively short time. On the other hand, for specimens made up predominantly of heavier elements, such as steel and even more so for brasses and solders, the intensity of the scattered radiation is very low (see Fig. 8) and counting the statistical error can preclude precise analysis in a reasonable amount of time. The most common application where this approach (or a variant thereof) is used is in the determination of trace elements in specimens of geological origin. This is illustrated in Figure 9 for the determination of Sr in specimens of widely

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varying geological origin. In Figure 9a, the net count rate for SrKa is plotted against the concentration for a large number of specimens. There is a considerable spread around the calibration line established. The scatter is greatly reduced when the net count rate of

Figure 9 (a) Net count rate of SrKa as a function of Sr concentration for a large number of specimens of varying geological origin. There is considerable spread around the calibration line. (b) The ratio of the count rates of the SrKa radiation and the Compton-scattered tube line is plotted against the concentration of Sr. The spread of the data points around the calibration line is now much reduced compared to (a); this is especially the case for the point labeled A.

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SrKa is divided by the count rate of the RhKa Compton-scattered tube radiation, as indicated in Figure 9b. Feather and Willis (1976) have shown that the intensity of the Compton peak can also be used as an estimate for the background under characteristic lines; this eliminates measuring the intensity of the background near the peak. With this method, not only is the measurement time per specimen reduced (using WDS) but the diﬃcult task of ﬁnding interference-free background positions is no longer required. 2. Internal Standard In this method, an element is added to each specimen in a ﬁxed proportion to the original sample. This addition has to be made to the standard samples as well as to the unknowns. The characteristic radiation of the element added should be similar to the characteristic radiation of the analyte in terms of absorption and enhancement properties in the matrix considered. Such an element is called an ‘‘added internal standard’’, or ‘‘internal standard’’ for short. In practice, the method works equally well if a pure element or a pure compound is added, or if a solution with the internal standard element is used. If a solution is used, care must be taken that the solution itself does not contain any elements that are to be analyzed. The composition of the solution used as the additive must be constant, otherwise it might aﬀect the matrix eﬀect. The intensity of the internal standard is aﬀected by matrix eﬀects in much the same way as the intensity of the analyte, provided there are no absorption edges (leading to diﬀerence in absorption) or characteristic lines including scattered tube lines (leading to diﬀerence due to enhancement) between the two wavelengths considered. Because Wi ¼ K i I i Mi

ð62Þ

for the analyte i and Ws ¼ Ks Is Ms

ð63Þ

for the internal standard s, the following ratio can be obtained by dividing Equation (62) by Equation (63): Ii ¼ Kis Wi Is

ð64Þ

where Kis ¼

K s Ms K i Mi W s

ð65Þ

Because the same amount of internal standard is added to all specimens, Ws is essentially a constant and can be included in the constant Kis. It should be noted that Mi (and Ms) is not a constant over the concentration range of interest (otherwise linear calibration would suﬃce) but depends on the matrix elements. However, if both Mi and Ms vary in a similar manner with the matrix elements, the ratio Mi=Ms is less sensitive to variation in the matrix eﬀect and, in practice, can be considered a constant. In practice, the constant Kis is determined using linear regression. The main advantage of the internal standard method over the scattered-radiation method is its ability to correct eﬀectively for enhancement as well as for absorption. It also corrects—at least partially—for variations in density of pressed specimens. The requirement that the intensity of the characteristic radiation of both the analyte and the internal standard element vary in the same manner with the

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matrix eﬀects imposes that there should be no absorption edges and no characteristic radiation from other elements between the measured line of the analyte and that of the internal standard. Furthermore, ideally, the analyte should not enhance the internal standard or vice versa. If Ka radiation is measured and if the atomic number of the analyte is Z (with Z > 23), then, very often, the elements with atomic number Z 1 or Z þ 1 are very good candidates. This assures that there are no K absorption edges and no K emission lines of other elements between the two elements considered. The element with atomic number Z is not enhanced by Ka radiation from an element with Z þ 1, but only by the much weaker Kb radiation, whereas for elements with atomic number Z þ 2 and higher, both Ka and Kb contribute to enhancement. In practice, some enhancement between the internal standard element and the analyte or vice versa is allowed, as the concentration of the internal standard is constant and the ratio is based on intensities. The absence of major elements in the specimens with L absorption edges and emission lines, however, must be checked for. The situations that must be avoided are (1) the case where a major line of a major matrix element is between the absorption edges of the analyte and the internal standard and (2) the case where a major absorption edge of a major matrix element is situated between the measured characteristic lines of the analyte and the internal standard. In the ﬁrst case, the matrix element would enhance either the analyte or the internal standard element, but not both; in the second case, the matrix element absorbs strongly either the radiation from the analyte or the internal standard, but not both. In both of these cases, varying concentrations of the matrix element will lead to variable and diﬀerent eﬀects on the intensities of the analyte and of the internal standard, and the ratio used in Eq. (64) will not compensate for such events. The method, however, has some important limitations: The specimen preparation is made more complicated and is more susceptible to errors. The addition of reagents and the requirement of homogeneity of the specimen tends to limit the practical application of the method to the analysis of liquids and fused specimens, although it sometimes ﬁnds application in the analysis of pressed powders. Although the rule Z þ 1 or Z 1 can serve as a rule of thumb, it is quite clear that for samples where many elements are to be quantiﬁed, a suitable internal standard cannot be found for every analyte element. Sometimes, more elements are used in one internal standard solution to provide suitable internal standards for more analytes. Also, the fact that the internal standard method is easier to apply to liquids can generate some problems. Heavier elements (e.g., Mo) are more diﬃcult to determine using this method, because liquid specimens are generally not of inﬁnite thickness for the K wavelengths of these heavier elements. In such cases, the L line can be used, with an appropriate internal standard. The method will, however, also provide some compensation for the eﬀects of noninﬁnite thickness, especially if the wavelength of the internal standard selected is very similar to that of the analyte line. Theoretically, L lines of a given element can be used as internal standards for K lines of other elements and vice versa if these wavelengths are reasonably close to each other and neither interfering lines nor edges occur between them. In principle, the method allows the determination of one or two elements in a specimen without requiring analysis (or knowledge) of the complete matrix.

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The range of concentration over which this method is suitable can be quite large, up to 10–20 % in favorable situations, but the internal standard technique is most eﬀective at low concentrations (or high dilutions). The method ﬁnds, for instance, application in the determination of Ni and V in petroleum products, where MnKa is used as an internal standard (ISO, 1995). The internal standard method allows an accurate determination of these elements in a much wider variety of petroleum products than the method based on linear calibration. 3. Standard Addition Methods Another method of analysis involves the addition of known quantities of the analyte to the specimen and is referred to as the standard addition method. If the analyte element is present at low levels, and no suitable standards are available (e.g., the matrix is unknown), standard addition and=or dilution may prove to be an alternative, especially if the analyst is interested in only one analyte element. The principle is the following: Adding a known amount of the analyte i ðDWi Þ to the unknown specimen will give an increased intensity ðIi þ DIi Þ. Assuming a linear calibration, the following equations apply: Wi ¼ K i I i

ð62Þ

for the original specimen and Wi þ DWi ¼ Ki ðIi þ DIi Þ

ð66Þ

for the specimen with the addition. Thus, the method assumes that linear calibration is adequate throughout the range of addition, because it assumes that an increase in the concentration of the analyte by an amount DWi will increase the intensity by Ki DIi .

Figure 10 Standard addition method. The net intensity is plotted versus weight fraction of the element added to the sample and a ‘‘best-fit’’ line is determined. The intercept of that line with the concentration axis is Wi.

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These equations can be solved for Wi. To check the linearity of the calibration, the process can be repeated by adding diﬀerent amounts of the analyte to the specimen and plotting the intensity measured versus the concentrations added (Fig. 10). The intercept of the line on the concentration axis equals Wi. Note that the concentration in the unknown is actually found through extrapolation of the linear calibration toward zero intensity. The intensities used for calibration must be corrected for background and line overlap. If this correction is not performed (or not performed accurately), the value of the concentration determined will be overestimated by an amount proportional to the intensity of the background (or line overlap) and inversely proportional to the sensitivity. The method is suitable mainly for determination of trace and minor concentration levels, as the amount DWi added to the sample must be in proportion to the amount Wi in the sample itself. The extrapolation error can be quite large if the slope of the line is not known accurately. Adding signiﬁcant amounts of additives to the sample might, however, lead to nonlinearity, as it will alter the matrix eﬀect. Compounds and solutions can be used for the standard addition. If the analyte i in the original sample is in a diﬀerent phase than in the additive, care must be taken in the calculation of the concentration sought. The relevant stoichiometric or gravimetric factors must be included. This also applies if the analyte is present under elemental or ionic form in, for example, the original sample and in a compound phase in the additive, or vice versa. Another way to alter the concentration of the analyte is by diluting the liquid or solid solution of the sample. By diluting several times by known amounts, a line can be established. By repeating this procedure with a standard solution containing a known amount of i, the unknown concentration can be found. 4. Dilution Methods Dilution methods can also eliminate or reduce the variation of the matrix eﬀect, rather than compensating for such variation. The dilution method can be explained using Eq. (17): mi ðl0 Þ Pn j¼1 Wj mj ðl0 Þ þ G j¼1 Wj mj ðli Þ

Cðl0 ; li Þ ¼ Pn

ð17Þ

which can be rewritten as Cðl0 ; li Þ ¼

mi ðl0 Þ ms ðl0 Þ þ Gms ðli Þ

ð67Þ

where ms ðl0 Þ and ms ðli Þ are the mass-attenuation coeﬃcients of the specimen for the primary wavelength l0 and analyte wavelength li, respectively. Apparently, deviations from linearity are due to variations in ms ðl0 Þ and=or ms ðli Þ. Enhancement is ignored at this stage. If one adds, to the sample, D grams of a diluent (d ) for each gram of sample, the denominator or Eq. (67) becomes 1 D ½m ðl0 Þ þ Gms ðli Þ þ ½m ðl0 Þ þ Gmd ðli Þ 1þD s 1þD d

ð68Þ

If the term D=ð1 þ DÞ½md ðl0 Þ þ Gmd ðli Þ is much larger than 1=ð1 þ DÞ½ms ðl0 Þ þ Gms ðli Þ, the factor Cðl0 ; li Þ becomes essentially a constant and variations due to varying matrix eﬀects between samples become negligible. This can be done in two ways. (a)

Making D=ð1 þ DÞ large by diluting each sample by adding a large, known amount of a diluent.

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(b)

Adding a smaller quantity of diluent than in the previous case, but with a much larger value for md ðl0 Þ þ Gmd ðli Þ. This is called the technique of the heavy absorber.

Both of these procedures, however, require the addition of reagents to the sample. This can easily be done for dissolved samples, either in liquids or fused samples, but it is more diﬃcult for powdered samples (homogeneity!). These methods do not eliminate the matrix eﬀects completely, but reduce their inﬂuence. On the other hand, they also reduce the line intensity of the analyte; thus, a compromise must be sought. Dilution methods also have the advantage of reducing the enhancement eﬀect if one uses a nonﬂuorescing diluent (e.g., H2O or Li2B4O7). In this case, the eﬀect is reduced by the fact that the concentrations of both the enhancing element and the analyte are reduced. If the diluent contains elements whose characteristic x-rays can excite the analyte, as well as some other matrix elements, then the contribution of those unknown quantities of matrix elements to the total enhancement is reduced: The enhancement of the analyte by the diluent would then be determining and can be considered to be constant. This method allows the determination of all measurable elements in the sample, as opposed to the standard addition method, where an addition must be made for each element of interest. D.

Mathematical Methods

1. General The term ‘‘mathematical methods’’ refers to those methods that calculate rather than eliminate or measure the matrix eﬀect. Mathematical methods are independent of the specimen preparation in the sense that specimen preparation is taken into account if the composition of the specimen presented to the spectrometer has been changed (e.g., by fusion), but mathematical methods do not prescribe the specimen preparation method as is done, for example, by the standard addition method. The actual calculation method used to convert intensities to concentrations does not aﬀect the choice of the specimen preparation method. The aim of the specimen preparation is limited to the presentation to the spectrometer of a specimen that is homogeneous (with respect to the XRF technique) and that has a well-deﬁned, ﬂat surface representative for the bulk of the specimen. Mathematical methods usually require knowledge of all elements in the standard specimens and allow determination of all measurable elements in the unknowns. In practice, trace elements can be neglected in the calculations for the analytes present at higher concentrations, as these trace compounds are neither subject to an important (and variable) matrix eﬀect nor do they contribute signiﬁcantly to the matrix eﬀect of other elements. Their concentrations are often found by straightforward linear regression. The mathematical methods are divided in two main categories: the fundamental parameter method and the methods using inﬂuence coeﬃcients. 2. The Fundamental Parameter Method a. Introduction The fundamental parameter method is based on the theory that enables one to calculate the intensity of ﬂuorescent radiation, originating from a specimen of known composition. The equations used usually consider both primary and secondary ﬂourescence (enhancement).

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Higher-order eﬀects and eﬀects due to scattered radiation are usually neglected. Formulas to calculate intensities of ﬂuorescent radiation were proposed very shortly after the introduction of the commercial XRF spectrometers, in the early 1950s (Gillam and Heal, 1952). The equation describing the intensity of the ﬂuorescent radiation as a function of specimen composition, spectrometer conﬁguration, and incident spectrum was derived earlier (see Sec. II.B) and is repeated here: lZedge;i "

Ii ðJðlÞÞ ¼

Pi ðlÞ þ lmin

X

# Sij ðl; lj Þ dl

ð69Þ

j

where Pi ðlÞ is the contribution of the primary ﬂuorescence caused by incident photons with wavelength l, and Sij ðl; lj Þ is the contribution of the secondary ﬂuorescence (enhancement) by characteristics photons lj which have been excited by primary photons l. The summation in Eq. (16) or (69) is over all elements j that have characteristic lines that can excite the analyte i. For each of these elements j, all characteristic lines must be considered. This is quite simple if none of the L lines or M lines of element j can excite the analyte. In that case, only the Ka and Kb lines are to be considered. If the L lines of an element j are energetic enough for enhancement of the analyte, the sheer number of L lines (e.g., W has more than 20 characteristic L lines that can be considered) would make the calculation very time-consuming. Therefore, most programs consider only three to ﬁve L lines for each element. A similar reasoning holds for the M lines. Application of these formulas was originally limited to the prediction of intensities for specimens with given composition. The application of those formulas for analysis was not pursued until the 1960s. The method of analyzing specimens by fundamental parameter equations has been developed independently around the same time by Criss and Birks (1968) and by Shiraiwa and Fujino (1966). Due to the large amount of calculations involved (especially the integration of the incident spectrum and the calculation of the contribution of enhancement), these programs initially ran on mainframes and mini computers. A PC version of the same program was proposed by Criss in 1980 (Criss, 1980a). The application of a fundamental parameter method for analyzing specimens consists of two steps: calibration and analysis. Both steps will be discussed in more detail in the following subsections. b. Calibration The fundamental parameter equation is used to predict the intensity of characteristic lines for a composition identical to that of the standard used. If more than one standard specimen is used, the calculations are repeated for each of the standards. The calculations are performed using the appropriate geometry (i.e., incidence and take-oﬀ angles are taken in agreement with those of the spectrometer used) and the parameters determining the tube spectrum (such as the anode material, voltage, thickness of beryllium window, and so on) correspond to the ones used in the spectrometer for the measurements. The intensities predicted are (almost always) net intensities, void of background, line overlap, crystal ﬂuorescence, and so on. Hence, the measured intensities must be corrected for such spectral artifacts. These theoretically predicted intensities are then linked to the actually measured ones. If only one standard specimen is used, the ratio between the measured intensity and the calculated intensity is calculated. If more than one standard is used, the net intensities obtained from the measurements are plotted versus the calculated intensities and a straight line can be determined for each characteristic line measured. The slope of

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such a line is the proportionality factor between predicted (calculated) and measured intensities. In general, this relationship will be determined more accurately if more standards are used. A special case of calibration ensues when one uses pure elements as standards. Dividing each of the measured (net) intensities of the corresponding pure element gives the relative intensity. This relative intensity can be calculated directly by some fundamental parameter programs. In fact, some programs use equations that express the intensity of characteristic radiation directly in terms of relative intensity. The relative intensity in this respect is thus deﬁned as the intensity of the sample, divided by the intensity of the corresponding pure element (or compound, if the concentration of the analyte is deﬁned in compound concentration), under identical conditions for excitation and detection. Basically, the calibration function, which is determined using the measured intensities of the standards and the calculated intensities, accounts for instrument-related factors only; matrix eﬀects are accounted for by using the physical theory as described in the fundamental parameter equation. The instrument-related parameters for a wavelength-dispersive spectrometer are as follows: Collimation Crystal reﬂectivity Eﬃciency of detector(s) Fraction of emergent beam, allowed into the detectors, after Bragg reﬂection (is also dependent on Bragg angle) For energy-dispersive spectrometers, the instrument-related parameters are collimation and detector eﬃciency. The eﬀect of the windows of the detector, the dead layer, and so forth can also be taken into account. c.

Analysis Step 1. For every unknown specimen, a ﬁrst estimate of the composition is made. There are several ways to obtain such a ﬁrst estimate. They vary from using a simple, ﬁxed composition (e.g., equal to 100%, divided by the number of elements considered; in the ﬁrst estimate, all the concentrations of each of the elements are thus taken equal to one another) to the composition derived from the measured intensities in combination with the calibration curves. Using the calibration data, it is possible to estimate for each element the intensity that would be obtained if the pure elements were measured. These numbers are then used to divide the intensity of the corresponding element, measured in the unknown specimen. The resulting fractions are scaled to 100% and used as the ﬁrst estimate. Step 2. For this estimate of composition, the theoretical intensities are calculated. These are converted to measured intensities, using the calibration data, so that these two sets of intensities for the same specimen can be compared. Step 3. The next estimate of composition is obtained based on the diﬀerence between the measured and calculated intensities. Again, there are diﬀerent methods available: 1.

2.

The simplest method is based on linear interpolation. If, for a given element, the measured intensity is 10% higher than the calculated intensity, the concentration of that element is increased by 10%. Rather than a linear relationship, some authors (Criss and Birks, 1968) use an interpolation based on three points. This is done because the relationship between concentration and intensity is usually non linear over a wider range. If the specimen is a pseudobinary, hyperbolic relationships have proven to be

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better approximations. For more complex specimens, it still works out quite well, because the concentrations of the other elements are considered ﬁxed at this stage. The hyperbolic equation requires a minimum of three points for its parameters to be determined. The points requiring the least additional calculation time are the following: The origin (net intensity zero, at concentration zero). The pure element, W ¼ 100%; the corresponding intensity has already been calculated. The point W ¼ current estimate, the intensity has already been calculated. These three points allow an hyperbolic relation to be established around the current estimated composition. From this curve and the measured intensity, the new concentration estimate is derived. This approach is repeated for every analyte element. This method usually provides a faster convergence than using the simple straight line. 3. It is also possible to use gradient methods to determine the next composition estimate. The formula for the ﬁrst derivative, with respect to concentration, of the fundamental parameter equation have been published (Shiraiwa and Fujino, 1968), but the calculation is cumbersome and time-consuming. Also, the derivatives can be obtained by a ﬁnite-diﬀerence method, where the eﬀect of a small change in composition on the intensity is observed. Step 4. The process, starting at Step 2, is now repeated until convergence is obtained. Diﬀerent convergence criteria exist. The calculation can be terminated if one of the following criteria is satisﬁed for all the elements (compounds) concerned: 1. The intensities, calculated in Step 2, do not change from one step to another, by more than a present level. 2. The intensities, calculated in Step 2 agree, to within a preset level, with the measured intensities 3. The compositions, calculated in Step 3, do not change from one step to another by more than a present level (e.g., 0.0005 or 0.0001 by weight fraction). One or more of these criteria might be incorporated in the program. These criteria, however, are no guarantee that the ﬁnal result is accurate to within the level, speciﬁed in the convergence criteria. Furthermore, especially with convergence criteria based on concentrations (such as criteria 3), it must be realized that a convergence criterion of 0.0005 is unacceptable when determining elements at levels below 0.0005. d. Extensions to the Method More complex scenarios are possible. The most common one includes the calculation of a set of inﬂuence coeﬃcients (based on theoretical calculations) to obtain a composition quickly, close to the ﬁnal result. Next, the fundamental parameter method is applied (Rousseau, 1984a). This should yield faster convergence in terms of computation time, because the calculation by inﬂuence factors of the preliminary composition is very fast. This method reduces the number of evaluations of the fundamental parameter equation. Also, the diﬀerent programs available diﬀer quite markedly in their treatment of the intensities of the standards measured. Some programs use a weighting of the standards, stressing the standard(s) closest (in terms of intensity) to the unknown (Criss,

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1980a). Such programs use a diﬀerent calibration for each unknown specimen. The unknown specimen dictates which standards will be given a high weighting factor and which standards will be used with less weighting. Other programs use all standards with equal weighting. e. Typical Results Early results of the fundamental parameter method on stainless steels are given by Criss and Birks (1968). The average relative diﬀerences between x-ray results and certiﬁed values were about 3–4%. Later, Criss et al. (1978) reported accuracies of about 1.5% relative for stainless steels, using a more accurate fundamental parameter program. Typical results for tool steel alloys are given in Table 3. Often, the fundamental parameter method is considered to be less accurate than an inﬂuence coeﬃcient algorithm. This is caused primarily by the fact that the fundamental parameter method has extensively been used and described as a method that allows quantitative analysis with only a few standards. This is obviously an advantage, but it does not imply that fundamental parameter methods cannot be used in combination with many standards similar to the unknown. As a matter of fact, on several occasions and with a variety of matrices, the authors have obtained results of analysis with an accuracy similar to that of inﬂuence coeﬃcient algorithms when using the same standards in both cases. f. Factors Aﬀecting Accuracy The accuracy of the ﬁnal results is determined by the following: The The The The

measurement specimen preparation physical constants used in the fundamental parameter equation limited description of the physical processes that are considered in the fundamental parameter equations The standard and the calibration In the following discussion, the eﬀect of measurements and specimen preparation will not be considered.

Table 3 Analysis of Seven Tool Steels with a Fundamental Parameter Program [XRF11, from CRISS SOFTWARE, Largo, MD (Criss, 1980)] Element W Co Mn Cr Mo S P Si C

Minimum conc. (%)

Maximum conc. (%)

Standard deviation (%)

1.8 0.0 0.21 2.9 0.2 0.015 0.022 0.14 0.65

20.4 10.0 0.41 5.0 9.4 0.029 0.029 0.27 1.02

0.52 0.20 0.01 0.013 0.04 0.003 0.003 0.03 0.16

Note: One standard has been used. The minimum and maximum concentrations refer to the minimum and maximum concentrations in the set of analyzed specimens, respectively. The standard deviation is calculated from the difference between concentration values found and certiﬁed. Source: Data courtesy of Philips Analytical, Almelo, The Netherlands.

Quantification of Infinitely Thick Specimens

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379

Physical Constants

The physical constants used in the fundamental parameter equations are as follows: Incidence and exit angles Spectrum of incident beam Mass-attenuation coeﬃcients Flourescence yields Absorption jump ratios Ratios of intensity of diﬀerent lines within a given series (e.g., Ka=Kb ratio) Wavelengths (or energies) of absorption edges and emission lines Incidence and exit angles. The incidence angle in most wavelength-dispersive (WD) and energy-dispersive (ED) spectrometers is, in fact, deﬁned by a relatively wide cone with a diﬀerent intensity at the boundaries compared to the center. This incident cone is neglected and the incident radiation is considered parallel, along a single, ﬁxed direction. A similar observation holds for the exit angle. The eﬀect is far less pronounced if diﬀraction from a plane crystal surface is used for dispersion, as is done, for example, in most sequential WD spectrometers. This has been studied to some extent by Mu¨ller (1972). To our knowledge, none of the fundamental parameter programs available takes this eﬀect into account. Its inﬂuence, however, is, to some extent, compensated for by calibration with standard specimens. Spectrum of incident beam. The spectrum of the incident beam from an x-ray tube spectrum requires more attention. Parts of the primary spectrum might excite an element B that, in turn, excites element A very eﬃciently. In such cases, this enhancement may make the intensity of element A sensitive to small errors in the tube spectrum representation, which would not be compensated for if the pure A was used for calibration. This can arise, for example, in the analysis of silica–zirconia specimens with a Rh tube (Criss, 1980b). Pure silica is relatively insensitive to the intensity of the characteristic K lines of Rh. In combination with Zr, however, the situation is diﬀerent. Indeed, the RhK lines are strongly absorbed by Zr. Zr then emits K and L lines that enhance Si. As a result, the Si intensity is more sensitive to the RhK lines in SiO2–ZrO2 mixtures than it is in pure SiO2. Tube spectra have been calculated using, for example, the algorithm of Pella et al. (1985). Mass-attenuation coeﬃcients. There are several compilation of mass-attenuation coeﬃcients, published in the literature. A continuing eﬀort to compile the most comprehensive table has been undertaken by the National Institute of Standards and Technology (formerly National Bureau of Standards), Gaitherburg, MD. When selecting a table of mass-attenuation coeﬃcients for use in a fundamental parameter program, the following question must be addressed: Does the table cover all the analytical needs? (In practice, does it cover the complete range of interest from the longest wavelength considered, down to the excitation potential of the tube?) The analyst should be aware that the use of formulas to generate mass-attenuation coeﬃcients can lead to values that can be signiﬁcantly diﬀerent from the corresponding table values. Presently, for applications in XRF, the complications of McMaster et al. (1969), Heinrich (1966), Leroux and Thinh (1977), or Veigele (1974) are most often used. A short discussion on the agreement between some of these compilations has been presented by Vrebos and Pella (1988). A more recent compilation has been published by de Boer (1989). Fluorescence yields. A comprehensive reference to ﬂuorescence yields, including Coster–Kronig transitions, can be found in the work of Bambynek et al. (1972). (see also Chapter 1, and Appendix VI).

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Absorption jump ratios. These can be derived from the tables of attenuation coefﬁcients. Ratios of diﬀerent ﬂuorescent lines within a family. Data for the K spectra can be found in the work by Venugopalo Rao et al. (1972) (see also Chapter 1). Wavelengths of absorption edges and emission lines. A comprehensive table was published by Bearden (1967) and is also presented in the appendices to Chapter 1. Because attenuation coeﬃcients are wavelength dependent, an error in a wavelength of any characteristic line will automatically lead to a bias in the corresponding attenuation coeﬃcients. h.

Limited Physical Processes Considered

The fundamental parameter equation [Eq. (16)] does not consider all physical processes in the specimens. Three of the most obvious that are missing are described here. Tertiary ﬂuorescence. Although the formula for tertiary ﬂuorescence has been derived by, for example, Shiraiwa and Fujino (1966) and Pollai and Ebel (1971), it is not included in most fundamental parameter programs. Usually, the tertiary ﬂuorescence eﬀect is considered small enough to be negligible. Shiraiwa and Fujino (1967, 1974) have presented data showing a maximum contribution of tertiary ﬂuorescence of about 3% relative to the total intensity of Cr in Fe–Cr–Ni specimens. Therefore, even in Fe–Cr–Ni specimens whose characteristic lines and absorption edges are ideally positioned relative to one another to favor enhancement, the eﬀect of tertiary ﬂuorescence is quite limited. Higher-order enhancement is also possible, but it is even less pronounced than tertiary ﬂuorescence. Scatter. Other processes not considered in most of the fundamental parameter methods are coherent and incoherent scatter of both the primary spectrum and the ﬂuorescent lines. This is usually justiﬁed by pointing out that the photoelectric eﬀect is, by far, the major contribution to the total absorption. It is believed that the contribution by scattered photons to the excitation of characteristic photons is negligible. However, in some cases the scattered primary spectrum may have a considerable inﬂuence, as illustrated earlier in this chapter. The equations describing the contribution of scatter to ﬂuorescent intensity have been derived by Pollai et al. (1971). These equations have obviously many similarities to those for secondary ﬂuorescence. Photoelectrons. The processes that are probably the must unknown in the fundamental parameter method are related to the contributions of the photoelectrons and of the Auger electrons that are produced as a result of absorption of the primary and ﬂuorescent x-ray photons. These electrons have suﬃcient energy to excite other atoms and thus create additional ﬂuorescence. This is especially important in the case of low-atomic-number elements, as has been described by Mantler (1993) and has been illustrated, for example, by Kaufmann et al. (1994). i.

Standards and Calibration

The use of good standards (similar to the unknown) will almost always lead to more accurate results, compared to a situation where the standards used have a widely diﬀerent composition from that of the unknown. This is because most of the uncertainties, caused by inaccuracies in the physical constants, cancel. The degree of similarity between standards and unknown has an important eﬀect on the accuracy of the analysis. 3. Influence Coefficient Algorithms Another class of mathematical methods calculates the matrix eﬀect by means of coeﬃcients, rather than by evaluating the fundamental parameter equation for each unknown. It will be shown that these coeﬃcients can also be calculated from theory, using funda-

Quantification of Infinitely Thick Specimens

381

mental parameters. Many of such inﬂuence coeﬃcient algorithms have been proposed and they have been divided and subdivided in diﬀerent ways (Lachance, 1979). It is not the intention to discuss all of the algorithms here; only a few selected ones will be discussed. This selection is based on the popularity of the methods and=or on some interesting characteristics of their underlying theory. Some of these algorithms use only one single coeﬃcient per interfering element; others use more than one. The distinction used here, however, depends on whether the inﬂuence coeﬃcients is considered to be a constant for a given application or whether the value of the coeﬃcient varies with composition. The latter methods will be discussed in Sec. V.5. Only two algorithms that use constant inﬂuence coeﬃcients will be discussed here: the Lachance–Traill and the de Jongh algorithms. The practical application of the resulting equations (i.e., calibration and analysis) will be treated separately in Sec. V.7. One has to emphasize that the approach based on constant inﬂuence coeﬃcients has many common aspects with all the inﬂuence coeﬃcient algorithms discussed in Secs. V.4 and V.5. All of the inﬂuence coeﬃcient models express the total matrix eﬀect Mi for a binary mixture ij as follows: Mi ¼ ½1 þ mij Wj

ð70Þ

where mij indicates the true binary inﬂuence coeﬃcient describing the matrix eﬀect of j on the analyte i in binaries ij. More generally, 3 2 n X 7 6 mij Wj 7 Mi ¼ 6 5 41 þ

ð71Þ

j¼1 j6¼e

for a multielement specimen, with n being the total number of elements or compounds. In most of the inﬂuence coeﬃcient algorithms, one element is eliminated from the summation {i.e., one inﬂuence coeﬃcient is used when dealing with binaries, as in Eq. (70), and n1 coeﬃcients deal with a specimen consisting of n compounds [Eq. (71)]}. In Eq. (71) this is explicitly indicated by the j 6¼ e under the summation sign. The eliminated compound e can be any of the ones present in the specimens; however, most authors eliminate the analyte. The expression for Mi is then used as follows: 2 3 n X 6 7 Wi ¼ R i 6 1 þ mij Wj 7 4 5

ð72Þ

j¼1 j6¼e

which links the relative intensity and the inﬂuence coeﬃcients to the composition of the specimen. The relative intensity Ri for a given analyte is deﬁned as the ratio of the net measured intensity Ii in the specimen and the intensity that would have been measured on the pure analyte IðiÞ under identical conditions: Ri ¼

Ii IðiÞ

ð73Þ

In practice, the relative intensity is often derived indirectly from measurements on standards, and the pure element (or compound) is not required. Equation (56) can be rewritten in terms of Ri : Wi ¼ R i Mi

ð74Þ

382

de Vries and Vrebos

By deﬁnition, Mi ¼

Wi Ri

ð75Þ

The ratio of the weight fraction of the analyte Wi and its relative intensity Ri is the matrix eﬀect Mi . When the analyte radiation is absorbed (or when the absorption eﬀects are dominating over enhancement), Mi is larger than 1. On the other hand, when enhancement is dominant, Mi is smaller than 1. Also, Mi is smaller than 1 in the absence of enhancement, but when the absorption by the analyte is signiﬁcantly higher than that of the matrix elements, as indicated by curve 3 in Figure 6. One of the consequences of Eq. (74) is that for the pure analyte Wi ¼ 1, Ri ¼ 1 and, thus, Mi is also equal to 1. This implies that the matrix eﬀect as introduced here should be viewed as relative to the pure element and not in absolute terms. Even in the pure element specimen, x-ray photons are subject to matrix eﬀect. It is possible, even in the pure analyte, to be subjected to enhancement eﬀects. This is, for example, the case when L lines are analyzed if the K lines of the same analyte are also excited. Furthermore, the L lines of elements with large atomic numbers can be ﬂuoresced by other L lines, as indicated in Table 1. However, it is customary to refer to the situation as a situation without matrix eﬀect. Also, if Mi ¼ 1, Eq. (74) reduces to Wi ¼ R i

ð76Þ

In other words, in the absence of matrix eﬀect, the concentration of the analyte is equal to the relative intensity. For a specimen containing, for example, 25% (by weight) of the analyte, an intensity will then be measured that is 25% of that of the pure analyte. Comparing Eqs. (56) and (74) and considering the deﬁnition of the relative intensity given by Eq. (73) yields Ki ¼

1 IðiÞ

in other words, the sensitivity is the reciprocal of the intensity of the pure analyte. So, the intensity of the pure analyte can also be obtained without making measurements for a specimen of the pure element. It can be obtained from the slope of the calibration line or even from the measurement on a single specimen: IðiÞ ¼

Ii M i Wi

ð77Þ

where Mi is calculated and Ii is measured. Mi can be calculated using Eq. (75), where Ri is calculated from theory for the standard specimen of known composition. Therefore, although many inﬂuence coeﬃcient algorithms are presented using the format of Eq. (72), involving the relative intensity, there is no real need to perform measurements of the pure analyte, as Eq. (72) can be written as 2 3 n X 6 7 mij Wij 7 Wi ¼ K i I i 6 41 þ 5

ð78Þ

j¼1 j6¼e

where Ki is then determined during the calibration phase. Furthermore, if the background is not subtracted, Eq. (78) can be written as

Quantification of Infinitely Thick Specimens

3

2 6 Wi ¼ Bi þ Ki Ii 6 41 þ

383

n X j¼1 j6¼e

7 mij Wij 7 5

ð79Þ

where Bi is the background expressed as a concentration equivalent. The constants Ki and Bi are then determined during regression analysis. 4. Algorithms with Constant Coefficients a.

The Lachance–Traill Algorithm Formulation. In 1966, Lachance and Traill proposed a correction algorithm based on inﬂuence coeﬃcients (Lachance and Traill, 1966). The equations are, for a ternary consisting of the elements (or compounds) A, B, and C: WA ¼ RA ½1 þ aAB WB þ aAC WC

ð80aÞ

WB ¼ RB ½1 þ aBA WA þ aBC WC

ð80bÞ

WC ¼ RC ½1 þ aCA WA þ aCB WB

ð80cÞ

RA , RB , RC are the relative intensities of A, B, and C, respectively. The coeﬃcients aAB , aAC , aCA , and so forth are called inﬂuence coeﬃcients. A more general notation of the Lachance–Traill algorithm is, for analyte i, 2 3 n X 6 7 Wi ¼ R i 6 1 þ aij Wj 7 4 5

ð81Þ

j¼1 j6¼i

where the summation covers all n elements (or compounds) in the specimen, except the analyte itself. Hence, there are n1 terms in the summation. This is common to all currently used algorithms. Most of the algorithms, developed earlier [such as Sherman’s (1953) and Beattie and Brissey’s (1954)], used to have n terms, rather than n1, for specimens with n elements. Equations (80a), (80b), and (80c) are linear equations in the concentrations of the elements WA , WB , and WC , respectively. Note that there are only two coeﬃcients for each analyte element. Consider, for example, the ﬁrst equation of the set 80, namely Eq. (80a): Element A is the analyte and its concentration is equal to the relative intensity RA , multiplied by the matrix correction factor (1 þ aAB WB þ aAC WC ). This matrix correction factor has only two coeﬃcients: one (aAB ) to describe the eﬀect of element B on the intensity of A and, similarly, one to describe the eﬀect of element C on the intensity of A. The value of the coeﬃcient aAA , which would correct for the eﬀect of A on its own intensity (sometimes—but incorrectly—referred to as self-absorption) is zero. Similarly, aBB and aCC are also zero. The eﬀect of A on A, however, is taken into account, as will be shown in the next subsection. Calculation of the coeﬃcients. Lachance and Traill also showed that the inﬂuence coeﬃcients, ij can be calculated for monochromatic excitation by photons with wavelength 0 (assuming absorption only) from the expression aij ¼

mj ðl0 Þ cscðc0 Þ þ mj ðli Þ cscðc00 Þ 1 mi ðl0 Þ cscðc0 Þ þ mi ðli Þ cscðc00 Þ

ð82Þ

384

de Vries and Vrebos

When secondary ﬂuorescence (enhancement) is involved, the coeﬃcients are calculated in the same way. Thus, enhancement is being treated as negative absorption. This assumption is not valid when enhancement is quite severe. Diﬀerences in primary absorption may easily be confused with enhancement (see Fig. 6). From Eq. (82), it follows clearly that aii is always zero. Therefore, it is not included in the summation of Eq. (81). Also, from Eq. (82), it follows that the value of the coeﬃcients for the Lachance–Traill algorithm cannot be less than 1. It must be stressed that Eq. (82) is only valid strictly for monochromatic excitation and for those analytes that are only subject to absorption (no enhancement). In this case, the inﬂuence coeﬃcient is concentration independent: It is a constant, even for the complete concentration range from 0% to 100%. It does, however, depend on parameters, such as the wavelength of the primary photons and the incidence and exit angles. In all other cases (polychromatic excitation and=or enhancement), Eq. (82), strictu sensu, cannot be used. A polychromatic beam (from, e.g., an x-ray tube) can be ‘‘replaced’’ by a monochromatic one, by resorting to the eﬀective wavelength. The eﬀective wavelength, however, is composition dependent (see Sec. II.C). The value of the coeﬃcients, calculated using Eq. (82), is also dependent on composition, although Wi nor Wj ﬁgure explicitly in Eq. (82). If enhancement is dominant, another method must be applied to calculate the coeﬃcients. Inﬂuence coeﬃcients for the algorithm of, for example, Lachance–Traill can also be calculated based on actual measurements. Rewriting Wi ¼ Ri ½1 þ aij Wj

ð83Þ

to aij ¼

ðWi =Ri Þ 1 Wj

ð84Þ

yields an expression that can be used to obtain aij , based on the composition of the binary and the relative intensity Ri . The drawbacks associated with this method are as follows: 1.

2. 3.

4.

The calculation of Ri requires the measurements of the intensity on the pure i (element or compound). This could lead to large errors if the intensity of i in the binary is much lower than that of the pure, due to, for example, nonlinearity of the detectors. The pure elements (or compounds) are not always easily available or could be unsuitable to present to the spectrometer (e.g., pure Na or Tl). Equation (84) is very prone to error propagation when Wi is close to 1. The numerator is then a diﬀerence between two quantities of similar magnitude, and the denominator is then close to zero, magnifying the errors. Also, the availability of suitable binary specimens can present problems: Some alloys tend to segregate and homogeneous specimens are then diﬃcult to obtain.

The coeﬃcients aij can also be calculated from theory: Calculate Ri for the binary with composition ðWi ; Wj Þ rather than obtain it from measurements and substitute in Eq. (84). This method eliminates drawbacks 1, 2, and 4. However a better method—without the problem associated to error propagation—is to use Eq. (75) directly with the values for Ri and Wi . Lachance (1988) has also presented methods to calculate the values of the coeﬃcients from theory.

Quantification of Infinitely Thick Specimens

385

The Lachance–Traill algorithm assumes the following: 1.

b.

The inﬂuence coeﬃcients can be treated as constants, independent of concentration; this limits the concentration range in cases where the matrix eﬀects change considerably with composition. 2. The inﬂuence coeﬃcients are invariant to the presence and nature of other matrix elements. So aFeCr , determined for use in Fe–Cr–Ni ternary specimens, is the same as aFeCr in Fe–Cr–Mo–W–Ta or Fe–Cr specimens. The de Jongh Algorithm

Formulation. In 1973, de Jongh proposed an inﬂuence coeﬃcient algorithm (de Jongh, 1973), based on fundamental parameter calculations. The general formulation of his equation is " # n X Wi ¼ E i R i 1 þ aij Wj ð85Þ j¼1 j6¼e

where Ei is a proportionality constant (and is usually determined during the calibration). The summation covers n1 elements (as is the case with the algorithm of Lachance and Traill), but the eliminated element, e, is the same for all equations. If, for a ternary specimen, element C is eliminated, the following equations are obtained: WA ¼ EA RA ½1 þ aAA WA þ aAB WB

ð86aÞ

WB ¼ EB RB ½1 þ aBA WA þ aBB WB

ð86bÞ

WC ¼ EC RC ½1 þ aCA WA þ aCB WB

ð86cÞ

Note that in order to obtain the concentration of elements A and B, the concentration of C, WC , is not required. This is diﬀerent from Lachance and Traill’s algorithm: In order to calculate the concentration of A, using Eq. (80a), the concentrations of both B and C are required. If the user is not really interested in element C (e.g., element C is iron in stainless steels), Eq. (86c) need not be considered and the analysis of the ternary specimen can be done by measuring RA and RB and solving Eqs. (86a) and (86b). Calculation of the coeﬃcients. de Jongh also presented a method to calculate the coeﬃcients from theory. The basis is an approximation of Wi =Ri by a Taylor series around an ‘‘average composition’’: Wi ¼ Ei þ di1 DW1 þ di2 DW2 þ þ din DWn Ri where Ei is a constant given by Wi Ei ¼ Ri average

ð87Þ

ð88Þ

and DWi ¼ Wi Wi;average

ð89Þ

dij are the partial derivatives of Wi =Ri with respect to concentration: dij ¼

@ðWi =Ri Þ @Wj

ð90Þ

386

de Vries and Vrebos

In practice, these derivatives are calculated as ﬁnite diﬀerences. Wi =Ri is calculated for a specimen with the average composition W1;average ,W2;average , . . . ,Wn;average (symbol ½Wi =Ri average Þ. Then, the concentration of each element j in turn is increased by a small amount [e.g., 0.1% (0.001 in weight fraction)] and Wi =Ri is calculated for that composition (symbol [Wi =Ri Wj þ0:001 ). Substituting in Eq. (90) yields dij ¼

@ðWi =Ri Þ ½Wi =Ri Wj þ0:001 ½Wi =Ri average ¼ @Wj 0:001

ð91Þ

This process is repeated for each of the elements j to calculate all the coeﬃcients for analyte i. This is also repeated for the other analyte elements. The coeﬃcients, calculated from Eq. (91), can be used in Eq. (87) for analysis. Equation (87), however, has n factors rather than n1 and uses DWj rather than Wj . Using the fact that n X

DWj ¼ 0

ð92Þ

j¼1

or DWe ¼ DW1 DW2 DWn

ð93Þ

one element e can be eliminated. The resulting equation is similar to Eq. (87), but has only n1 terms: Wi ¼ Ei þ bi1 DW1 þ bi2 DW2 þ þ bin DWn Ri

ð94Þ

bi1 ¼ di1 die

ð95Þ

with

for all bij except bie , which is equal to zero. Equation (94) has n1 terms, but they are still in DW, rather than W. Transformation of DW to W is done by substituting Eq. (89) in Eq. (94): n n X X Wi ¼ Ei bij Wj;average þ bij Wj Ri j6¼e j6¼e

ð96Þ

which can be rearranged to Eq. (85) with aij ¼

Ei

bij j6¼e bij Wj;average

Pn

ð97Þ

Combining Eq. (85) with Eqs. (75) and (88) yields 3 2 n X 7 6 aij Wj 7 Wi ¼ Mi;average Ri 6 5 41 þ

ð98Þ

j¼1 j6¼e

indicating that the weight fraction of the analyte is calculated from its relative intensity, a matrix correction term, and the matrix correction term for the average composition (which for a given composition is a constant). The value for Mi,average can be calculated using the inﬂuence coeﬃcients calculated and taking the composition (Wi, Wj) equal to the average

Quantification of Infinitely Thick Specimens Table 4

387

Analysis of Stainless Steels with Theoretical Inﬂuence Coefﬁcients (de Jongh)

Element Mn Cr Ni

Min.a (%)

Max.a (%)

Std. dev.b (%)

0.64 12.40 6.16

1.47 25.83 20.70

0.015 0.06 0.06

a

Min., Max.: minimum and maximum concentration in the set of analyzed specimens, respectively. Std. dev.: standard deviation, calculated from the difference between concentration values found and certiﬁed. Source: Data courtesy of Philips Analytical, Almelo, The Netherlands. b

composition. This equation is very similar at ﬁrst sight to the Lachance–Traill equation [Eq. (81)], except for the term Mi,average. Typical results. Tables with de Jongh’s coeﬃcients have been used for a wide variety of materials, including high-temperature alloys, brass, solders, cements, glasses, and so forth. An example for stainless steels is given in Table 4. Results are shown for Mn, Ni, and Cr only. For these analytes, the matrix eﬀects are the most important. Other elements, such as Si, P, S, and C, are also present at trace level. The coeﬃcients are calculated at a given composition [see Eq. (91)]. The practical range of concentration over which these coeﬃcients yield accurate results varies from 5% to 15% in alloys to the whole range from 0% to 100% in fused oxide specimens. Comparison between the algorithms of de Jongh and Lachance–Traill. The following points can be noted: 1. The basis of the de Jongh algorithm is a Taylor series expansion, around an average (or reference) composition. The values of the coeﬃcients calculated depend on this composition. 2. De Jongh can eliminate any element; Lachance and Traill eliminate the analyte itself: aij is zero. De Jongh eliminates (i.e., ﬁxes the coeﬃcient to zero) the same element for all analytes. Eliminating the base material (e.g., iron in steels) or the loss on ignition (for beads) generally leads to smaller numerical values for the coeﬃcients and avoids the necessity to determine all elements. 3. De Jongh’s coeﬃcients are calculated at a given reference composition. They are composition dependent and take into account all elements present. A coeﬃcient aij represents the eﬀect of element j on the element i in the presence of all other elements: They are multielement coeﬃcients rather than binary coeﬃcients. This is seen in Table 5, where the values of the coeﬃcients aCrCr and aCrNi are shown for several diﬀerent specimens, but with identical concentrations for the analyte

Table 5 Values for Inﬂuence Coefﬁcients aCrCr and aCrNi, Calculated According to the Algorithm of de Jongh, for Selected Specimens WCr

WFe

WNi

WTi

WW

aCrCr

aCrNi

0.18 0.18 0.18

Bal. Bal. Bal.

0.08 0.08 0.08

0 0 0.08

0 0.08 0

2.06 2.30 2.49

0.750 0.817 0.868

Note the variation of the values of the coefﬁcients, depending on the presence of Ti or W.

388

de Vries and Vrebos

and the interferent. In all cases, Fe has been eliminated and W and Ti are either present at 0.08 or not. 4. The calculation of the coeﬃcients is based on theory, treating both absorption and enhancement eﬀects. Hence, the coeﬃcients are susceptible to the errors, described earlier (Sec. V.D.2). However, the calculation of the coeﬃcients involves a division of the matrix correction terms for the ‘‘slightly aﬀected’’ composition by the corresponding term of the reference composition [Eq. (97)]. This compensates to a large degree for some of the biases introduced by the fundamental parameters. 5. For a specimen, containing n elements, there are n equations (one for each of the elements) if one uses the Lachance–Traill algorithm. Each of these equations has n1 inﬂuence coeﬃcients; the coeﬃcient of the analyte in each of the equations has been set to 0. For the same specimen, de Jongh only requires n1 equations, using n1 coeﬃcients per equation. One element has been eliminated throughout. This element (or compound) is usually one that is of no or little interest to the analyst (e.g., iron in steels or the loss on ignition in fused beads) (de Jongh, 1979). The nth equation (for the eliminated element) can also be written: Its form is identical to the others, it has also n1 terms, and the coeﬃcients can be calculated following exactly the same procedure as for the other coeﬃcients. 6. The coeﬃcients in the Lachance–Traill equation have been calculated empirically, using measured data from many more standards than used in the de Jongh’s algorithm, which always used theoretical coeﬃcients. These coeﬃcients could be obtained from Philips, eliminating the need for a large computer at each users’ site. With increased computer capabilities at the disposition of every analyst, the calculation of the coeﬃcients from theory has now been feasible for some years. It is possible to calculate the coeﬃcients for the Lachance–Traill algorithm from theory as well (Lachance, 1988). 5. Algorithms withVariable Coefficients a. Introduction Both algorithms discussed so far use a single, constant coeﬃcient for each (except one) interfering element: The expression from Lachance and Traill uses coeﬃcients, expressing the eﬀect of one element on the characteristic intensity of the analyte, relative to the analyte, ignoring all other elements. Such coeﬃcients are therefore referred to as binary coeﬃcients. The algorithm of de Jongh calculates multielement inﬂuence coeﬃcients effectively. Such coeﬃcients predict the eﬀect of one element on the intensity of another in a given matrix. This distinction between binary and multielement coeﬃcients can clearly be seen in, for example, a ternary specimen. Algorithms based on binary coeﬃcients add interelement eﬀects from each of the constituent elements. They calculate the matrix eﬀect using inﬂuence coeﬃcients that have been calculated for binaries. Assume a Ni–Fe–Cr specimen. The total matrix eﬀect of Cr is accounted for using a coeﬃcient expressing the inﬂuence of Ni on Cr (aCrNi) and a similar coeﬃcient for Fe on Cr (aCrFe). Both of these coeﬃcients are calculated for the corresponding binaries (Ni–Cr and Fe–Cr, respectively). In a ternary specimen (e.g., Ni–Fe–Cr), however, the eﬀect of, for example, Ni on Cr is aﬀected by the presence of Fe (and Ni similarly aﬀects the eﬀect of Fe on Cr). This eﬀect is called the crossed eﬀect and will be discussed in a following section.

Quantification of Infinitely Thick Specimens

389

The original expression presented by Lachance and Traill (1966) to calculate the inﬂuence coeﬃcients required the use of monochromatic excitation. An equivalent wavelength was used to calculate the coeﬃcients when the polychromatic excitation is applied. The equivalent wavelength, however, has been shown to vary with composition, as treated earlier in this chapter. The theoretical coeﬃcients, calculated according to de Jongh, are also composition dependent, as the reference composition is used explicitly in the calculations. This variation is due to the fact that the composition of the matrix varies considerably if analysis is required over a wider range of concentrations. This has been recognized early in the development of inﬂuence coeﬃcient algorithms, and many diﬀerent algorithms with variable coeﬃcients have been proposed. A variable inﬂuence coeﬃcient in this respect is an inﬂuence coeﬃcient that varies explicitly with concentration of one or more components in the specimen. Some of these will be discussed in the subsequent subsections. b. The Claisse–Quintin Algorithm Formulation. Claisse and Quintin (1967) extended Lachance and Traill’s algorithm by considering a polychromatic primary beam. The resulting equation for WA can be expressed as " # n n n n X X X X 2 ð99Þ aAj Wj þ aAjj Wj þ aA j k W j W k WA ¼ RA 1 þ j6¼A

j6¼A

j6¼A k6¼A;k>j

where the summation over j has n1 terms (all n elements, except i), and the summation over k has (n2)=2 terms (all n elements, except the analyte i and element j; furthermore, if aAj k is used, then aAkj is not). For a binary specimen, Eq. (99) reduces to WA ¼ RA ½1 þ a0AB WB

ð100Þ

a0AB ¼ aAB þ aABB WB

ð101Þ

with clearly showing that the inﬂuence coeﬃcient a0AB varies linearly with composition (i.e., WB ). For binaries, WA ¼ 1 WB ; hence, Eq. (101) can also be rearranged to a0AB ¼ aAB þ aABA WA

ð102Þ

Equations (101) and (102) are, at least theoretically, identical. It has been shown, however, that Eq. (102) is preferable to Eq. (101) if specimens with more than two elements (or compounds) are analyzed (Lachance and Claisse, 1980). This will be discussed in more detail in Sec. V.D.6. Note that the value of aAB in Eq. (101) is diﬀerent from its value in Eq. (102). Cross-product coeﬃcients. For a ternary specimen, the Claisse–Quintin algorithm can be written WA ¼ RA ½1 þ aAB WB þ aABB W2B þ aAC WC þ aACC W2C þ aABC WB WC The terms aAB WB þ aABB W2B and aAC WC þ aACC W2C

ð103Þ

390

de Vries and Vrebos

are the matrix corrections, due to B and C, respectively. The term aABC WB WC corrects for the simultaneous presence of both B and C and is referred to as a cross-product coeﬃcient. Calculation of the coeﬃcients. Claisse and Quintin (1967) also published methods to calculate the coeﬃcients from measurements on binary and ternary mixtures or from theory. These methods, however, are now generally superseded by theoretical calculations, such as discussed in Sec. V.C.4.c and V.C.4.d. Rousseau (1984b) has presented a calculation method for the coeﬃcients in the Claisse–Quintin algorithm, and Wadleigh (1987) has commented upon this approach. c. The Rasberry–Heinrich Algorithm Following a systematic study of the Fe–Cr–Ni ternary system, Rasberry and Heinrich (1974) concluded that the two phenomena—absorption and enhancement—are to be described by two diﬀerent equations. They introduced the following algorithm: " # n n X X Bik ð104Þ Aij Wj þ Wk Wi ¼ R i 1 þ 1 þ Wi j6¼i k6¼i where only one coeﬃcient is used for each interfering element. The coeﬃcients Aij are used for cases where absorption is the dominant eﬀect. In this case, the coeﬃcient Bik is taken equal to zero. If, for a given analyte, all Bik coeﬃcients are zero, Eq. (104) reduces to the Lachance–Traill expression. When enhancement by element k dominates, a Bik coeﬃcient is used. The corresponding Aij coeﬃcient is then taken equal to zero. Hence, the total number of terms in both summations is n 1. The correction factor for enhancement by element k can be rewritten as aik ¼

Bik 1 þ Wi

ð105Þ

showing that aik varies with concentration in a nonlinear fashion. The algorithm is very popular when analyzing stainless steels and steels in general. Among the disadvantages of the Rasberry–Heinrich algorithm are the following: 1.

2.

3.

It is not always clear which interfering elements should be assigned a B coeﬃcient and which one an A. In Pb–Sn alloys, the SnLa line is ﬂuoresced (enhanced) by both SnK and PbL lines. Yet, the calibration curve for SnLa clearly shows that absorption is dominant (Fig. 11). Furthermore, Eq. (105) suggests that the value of Bik at Wi ¼ 0 is twice the value at the ‘‘other’’ end of the calibration range when Wi ¼ 1. This is not generally valid. Mainardi and co-workers (1982) have therefore suggested replacing the 1 in the denominator by an additional coeﬃcient. Rasberry and Heinrich did not publish a method for calculating the coeﬃcients from theory.

Some of the disadvantages of calculating empirical coeﬃcients have been discussed in Sec. V.D.4.a. For these reasons, the Rasberry–Heinrich algorithm is not generally applicable. However, the concept of a hyperbolically varying inﬂuence coeﬃcient has been incorporated in the three-coeﬃcient algorithm of Lachance. d.

The Three-Coeﬃcient Algorithm of Lachance

Formulation. In 1981, Lachance (1981) proposed a new approximation to the binary inﬂuence coeﬃcient B ij given by

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391

Figure 11 Calibration curves for PbLa (solid line), SnKa (dashed line), and SnLa (dotted line) in Pb–Sn binaries. The SnLa is apparently dominated by absorption, rather than by enhancement of SnK and PbL lines.

mij ﬃ aBij ¼ aij1 þ

aij2 Wm 1 þ aij3 ð1 Wm Þ

ð106Þ

with Wm ¼ 1 Wi

ð107Þ

Wm is the concentration of all matrix elements. It has been shown by Lachance and Claisse (1980), as well as by Tertian (1976), that variable binary coeﬃcients must be expressed in terms of Wm (or 1Wi). For binary specimens, Eq. (106) can be rewritten using Wj for Wm and Wi for (1Wm): mij ﬃ aBij ¼ aij1 þ

aij2 Wj 1 þ aij3 Wi

ð108Þ

For specimens with more than two compounds, however, the diﬀerence between Eqs. (106) and (108) becomes clear. The value for the inﬂuence coeﬃcient mij is approximated over the complete concentration range for the binary by the function in Eq. (106), which relies on three coeﬃcients only. The excellent agreement between the true inﬂuence coeﬃcient mij and the approximation of Eq. (106) is shown in Figure 12 for Fe in FeNi (severe enhancement) and for Fe in FeCr (pronounced absorption). For multielement specimens, cross-product coeﬀcients aijk are used to correct for the crossed eﬀect, similar to Eq. (99). The general equation for a multielement specimen is " # n n n X X X aij2 Wm aij1 þ aijk Wj Wk Wi ¼ R i 1 þ ð109Þ Wj þ 1 þ aij3 ð1 Wm Þ j6¼i j6¼i k6¼i;k>j

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Figure 12 The binary influence coefficient mFeNi in FeNi binary systems (j, enhancement, top) and mFeCr in FeCr (j, absorption, bottom) and the approximation by the hyperbolic threecoefficient algorithm for Lachance (COLA). Note the excellent agreement in both cases. Conditions: W tube at 45 kV, in a spectrometer with an incidence angle of 63 and 33 take-off angle.

where the summation over j has n 1 terms (all n elements, except i) and the summation over k has (n 2)=2 terms (all n elements, except the analyte i and element j; furthermore if aijk is used, then aikj is not). Vrebos and Helsen (1986) have published some data on this algorithm, clearly showing the accuracy of the algorithm, using theoretically calculated intensities. The use of

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393

Table 6 Composition of the Specimens Used for the Calculations of the Coefﬁcients for Lachance’s Three-Coefﬁcient Algorithm, in Weight Fraction Specimen No. 1 2 3 4 5 6 7

Wi

Wj

Wk

0.999 0.001 0.5 0.999 0.001 0.5 0.30

0.001 0.999 0.5 0.0 0.0 0.0 0.35

0.0 0.0 0.0 0.001 0.999 0.5 0.35

theoretically calculated intensities has the advantage that it avoids errors due to specimen preparation and measurement errors associated with actual measured data. Pella and coworkers (1986) have presented a comparison of the algorithm with several others and with a fundamental parameter method using experimental data. Calculation of the coeﬃcients. The coeﬃcients aij1 , aij2 , and aij3 are calculated using fundamental parameters at three binaries i j. The cross-product coeﬃcients are calculated from a ternary. The compositions of the specimens concerned are listed in Table 6. The ‘‘specimens’’ referred to in Table 6 are hypothetical specimens. The intensities are calculated from fundamental parameters and require no actual measurements on real specimens. Step 1. Calculate the relative intensity Ri for the ﬁrst composition in Table 6. If the analysis of interest has more than three elements, then the system is divided in combinations of three elements i, j, k at a time. The analyte is element i, and j and k are two interfering elements. If the system considered is with compound phases, such as oxides, then the compositions in Table 6 are assumed to be for the oxides. Step 2. Using Eq. (84), the corresponding inﬂuence coeﬃcient aBij can be calculated. Step 3. For this composition, Wm ¼ 1 Wi ¼ Wj ¼ 0:001, which is small enough to be considered zero. Hence, Eq. (106) reduces to aBij ¼ aij1

ð110Þ

aBij has been calculated in Step 2, so aij1 can be computed. Step 4. Calculate the intensity for the second composition of Table 6 and use Eq. (84) to calculate aBij . In most cases, this value will be diﬀerent from the one found in Step 2 because the compositions involved are diﬀerent. Step 5. 1 Wm ¼ Wi ¼ 0:001 is small enough to be considered zero; hence, Eq. (106) reduces to aBij ¼ aij1 þ aij2

ð111Þ

aij1 and aBij are known so aij2 can be calculated. Step 6. Calculate the intensity for the third composition of Table 6 and use Eq. (84) to calculate aBij . In most cases, this value will be diﬀerent from the one found in Step 2 or 4 because the compositions involved are diﬀerent. Step 7. Using Wm ¼ 1 Wi ¼ 0:5 ¼ Wi , Eq. (106) reduces to aBij ¼ aij1 þ

aij2 ð0:5Þ 1 þ aij3 ð0:5Þ

ð112Þ

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which can be rearranged to aij2 aij3 ¼ B 2 aij aij1

ð113Þ

all coeﬃcients on the right-hand side are known, so aij3 can be calculated. Step 8. Repeat Steps 1–7 for Wi and Wk , to compute the coeﬃcients aik1 , aik2 , and aik3 . Step 9. Calculate the intensity Ri for the ternary (composition 7 in Table 6). Calculate aBij and aBik , using Eq. (106) and the coeﬃcients determined earlier. Step 10. Eq. (109) combined with Eq. (106) for a ternary specimen i–j–k reduces to Wi ¼ Ri ½1 þ aBij Wj þ aBik Wk þ aijk Wj Wk

ð114Þ

which can be rearranged to solve for aijk: aijk ¼

ðWi =Ri Þ 1 aBij Wj aBik Wk Wj Wk

ð115Þ

All variables on the right-hand side of Equation (115) are known, so aijk can be calculated. Step 11. Repeat for other interfering elements ( j and k) and repeat for other analytes i. Tao et al. (1985) published a complete computer program illustrating the method and allowing the calculation of the coeﬃcients and analysis of unknowns. This program suﬀers from an oversimpliﬁcation in that only the measured analytical lines are considered for enhancement. This would generate erroneous values for the coeﬃcients in cases such as Cu–Zn alloys, where the ZnKa line cannot ﬂuoresce the K shell of Cu, but the ZnKb can do so. If the ZnKa line is used for analysis, the eﬀect of the ZnKb line (enhancement of Cu) is not taken into account by the program. In practice, however, the only lines that are considered for enhancement are the characteristic lines used for the analysis of the other elements. This can be seen, for example, by calculating the coeﬃcients twice for Fe–Si with identical conditions; once indicating FeKa and SiKa lines are to be used and once indicating that the FeLa be used (Table 7). The value of the coeﬃcients for Fe will change; this is quite obvious because the magnitude and the sort of the matrix eﬀects on the FeKa and the FeLa characteristic lines are quite diﬀerent. The value for Si, however, should not change: in both cases, the same elements are present, and using the same excitation conditions, there is no reason why the coeﬃcients should be diﬀerent as the matrix eﬀects are the same. e.

The Algorithm of Rousseau

Formulation. Rousseau and Claisse (1974) used a linear relationship to approximate the binary coeﬃcients and cross-product coeﬃcients: " # n n n X X X Wi ¼ R i 1 þ ðaij1 þ aij2 Wm ÞWj þ aijk Wj Wk ð116Þ j6¼i

j6¼i k6¼i;k>j

The binary inﬂuence coeﬃcients are thus approximated by mij ﬃ aBij ¼ aij1 þ aij2 Wm

ð117Þ

This model can be used as a stand-alone inﬂuence coeﬃcient algorithm, but it has also been proposed as the starting point for a fundamental parameter algorithm (Rousseau, 1984a). The degree of agreement between the inﬂuence coeﬃcient mij and the approxi-

Quantification of Infinitely Thick Specimens Table 7

395

Values for the Coefﬁcients for Eq. (109) for Si in Fe–Si Binaries

aij1 aij2 aij3

SiKa(FeKa)

SiKa(FeLa)

5.396 1.890 0.409

6.284 0.015 0.846

Note: In the second column, Fe is measured using the Ka line and in the last column, the La line is used. Conditions: W tube at 45 kV, in a spectrometer with an incidence angle of 63 and a 33 take-off angle.

mation is shown in Figure 13 for the FeNi and the FeCr binaries. The agreement for the straight line is obviously not as good as with the COLA algorithm, especially in those cases where the value of the true inﬂuence coeﬃcient varies markedly, as is the case for Fe in FeCr (absorption). Equation (117) has been compared to the three-coeﬃcient algorithm of Lachance by Vrebos and Helsen (1986). They show that the accuracy is somewhat less than for Lachance’s method, but for most practical purposes, the Rousseau algorithm should give acceptable results. Calculation of the coeﬃcients. Rousseau has shown that the fundamental parameter equation can be rearranged to " # n X aij Wj Wi ¼ R i 1 þ ð118Þ j6¼1

and he also proposed a method to calculate the a coeﬃcients directly from fundamental parameters, without calculating the intensity ﬁrst (Rousseau, 1984a). As a matter of fact, Rousseau ﬁrst calculates the coeﬃcients for a given composition and then calculates the intensity, using Eq. (118). The coeﬃcients in Eq. (116) are calculated in a way very similar to the method described in Sec. V.D.5.d. The compositions involved are given in Table 8. The ‘‘specimens’’ referred to in Table 8 are hypothetical specimens. The intensity is calculated from fundamental parameters and requires no actual measurements on real specimens. For the ﬁrst two binaries of Table 8, the inﬂuence coeﬃcient is calculated [symbol aij(0.20,0.80) and aij(0.80,0.20), respectively]. Then the corresponding values are substituted in Eq. (117): aij ð0:20; 0:80Þ ¼ aij1 þ aij2 ð0:80Þ

ð119aÞ

aij ð0:80; 0:20Þ ¼ aij1 þ aij2 ð0:20Þ

ð119bÞ

These equations can be solved for aij1 and aij2 . Similarly, using compositions 3 and 4 from Table 8, the corresponding coeﬃcients for i–k can be calculated. The cross-product coeﬃcients aijk are calculated using Eq. (115). 6. Specimens with More thanTwo Compounds The methods described by Lachance [Eq. (106)] and Rousseau [Eq. (117)] explicitly describe algorithms to calculate the value of the binary inﬂuence coeﬃcient by a rather simple, hyperbolic or linear, relationship. Combining these binary coeﬃcients to describe the matrix eﬀect for specimens with more than two elements (or compounds) is described in this subsection. The ternary system FeNiCr is taken here as an example. Figure 14 gives

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Figure 13 The binary influence coefficient mFeNi in FeNi binary systems (j, enhancement, top) and mFeCr in FeCr (j, absorption, bottom) and the approximation by the straight line as suggested by Rousseau and Claisse. Note the rather large deviations, especially at the high concentration ranges in FeCr. Conditions: W tube 45 kV, in a spectrometer with an incidence angle of 63 and 33 take-off angle.

the relative intensity of FeKa as a function of the weight fraction of Fe in FeNiCr specimens. There is considerable spread of the intensity of FeKa, even for a constant weight fraction of Fe. For specimens with a weight fraction of 0.10 Fe, the relative intensity of FeKa varies between 0.036 and 0.16 (points marked 1 and 2 in Fig. 14). This is due to the

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397

Table 8 Composition of the Specimens Used for the Calculations of the Coefﬁcients for the Linear Approximation According to Rousseau’s Algorithm, in Weight Fraction Specimen No.

Wi

Wj

Wk

1 2 3 4 5

0.20 0.80 0.20 0.80 0.30

0.80 0.20 0.0 0.0 0.35

0.0 0.0 0.80 0.20 0.35

rather diﬀerent eﬀect that Ni and Cr have on Fe:Cr is an absorber for FeKa radiation, whereas the NiK radiation can enhance FeK radiation through the process of secondary ﬂuorescence (enhancement). For these specimens, the matrix eﬀect MFe can be calculated from Eq. (75). The total matrix eﬀect on Fe, MFe(FeNiCr), in these specimens, at a ﬁxed Fe concentration of 0.1, for example, varies from 0.63 (for 0.1 Fe in FeNi, point 2) to 2.8 (for 0.1 Fe in FeCr, point 1). Now, the problem is how to calculate the matrix eﬀect in this case, based on inﬂuence coeﬃcients. Assume a specimen with the following composition: WFe ¼ 0.1,

Figure 14 The relative intensity of FeKa as a function of the concentration of Fe, in the presence of Ni and Cr. For every given weight fraction of Fe, the highest value of the intensity is obtained for the binary system FeNi (enhancement), whereas the lowest intensity is for the binary FeCr (absorption). The intermediate values are for ternary specimens, where the concentrations of Ni and Cr vary in steps of 0.1 weight fraction. At a weight fraction of Fe ¼ 0.7, the four data points labeled a, b, c, and d represent the following specimens (WFe, WNi, WCr): a ¼ (0.7, 0.0, 0.3), b ¼ (0.7, 0.1, 0.2), c ¼ (0.7, 0.2, 0.1), and d ¼ (0.7, 0.3, 0.0). Points labeled 1 and 2: see text. Experimental conditions: W tube at 45 kV, 1-mm Be window, incidence and take-off angles 63 and 33 , respectively.

398

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WCr ¼ 0.3, and WNi ¼ 0.6 (again, all concentrations are expressed as weight fractions). The total matrix eﬀect on Fe will be caused by both Ni and Cr and its magnitude will be between MFe(FeNi; WFe ¼ 0.1) ¼ 0.63 (for Fe in FeNi) and MFe(FeCr; WFe ¼ 0.1) ¼ 2.8 (for Fe in FeCr). It is assumed that the total eﬀect MFe(FeNiCr; WFe ¼ 0.1) is proportional to the concentrations of Ni (0.6) and Cr (0.3) in this example. Applying the law of weighted averages, the total matrix eﬀect is given by WNi WNi þ WCr WCr ¼ 0:1Þ WNi þ WCr

MFe ðFeNiCr; WFe ¼ 0:1Þ ﬃ MFe ðFeNi; WFe ¼ 0:1Þ þ MFe ðFeCr; WFe

ð120Þ

There is not a strict derivation indicating the validity of Eq. (120) in the general case. For cases involving absorption only, the derivation is rather straightforward and based on the additivity law for absorption. For now, let it suﬃce to indicate that the matrix eﬀect will change gradually when adding an element or when changing the composition of the specimen slightly; this is described by Eq. (120). By substituting the numerical values, MFe ðFeNiCr; WFe ¼ 0:1Þ ﬃ 0:63

0:6 0:3 þ 2:8 ¼ 1:35 0:6 þ 0:3 0:6 þ 0:3

ð121Þ

a value of 1.35 is obtained. This is in good agreement with the theoretical value of 1.31. Equation (120) is based on the availability of binary inﬂuence coeﬃcients calculated at specimen compositions given by WFe ¼ 0.10; WNi ¼ 0.90 and WFe ¼ 0.10; WCr ¼ 0.90. Using the more general expressions for matrix eﬀects, MFe ðFeNiÞ ¼ ½1 þ mFeNi;bin WNi;bin

ð122aÞ

MFe ðFeCrÞ ¼ ½1 þ mFeCr;bin WCr;bin

ð122bÞ

and

the following is obtained: WNi WNi þ WCr WCr þ ½1 þ mFeCr;bin WCr;bin WNi þ WCr

MFe ðFeNiCrÞ ﬃ ½1 þ mFeNi;bin WNi;bin

ð123Þ

where WNi,bin is the concentration of Ni in the binary FeNi (for example, CNi,bin ¼ 0.90). Because WNi;bin ¼ WCr;bin ¼ 1 WFe ¼ WNi þ WCr

ð124Þ

Equation (123) can be rearranged to MFe ðFeNiCrÞ ﬃ ½1 þ mFeNi;bin WNi þ mFeCr;bin WCr

ð125Þ

stressing the point again that the inﬂuence coeﬃcients are to be calculated for binaries ij with the composition Wi ; Wj ¼ 1 Wi . This is the reason why Eqs. (106) and (117) use Wm instead of Wj. However, applying binary coeﬃcients to multielement specimens leads to an incomplete matrix correction because we are trying to describe the eﬀect of Cr and Ni on Fe in FeNiCr based on the matrix eﬀects in the corresponding binaries only. This eﬀect

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399

is referred as the crossed eﬀect and has been described by Tertian (1987), who also proposed a method to correct for this. The proposed method (Tertian, 1987) involves the use of weighting factors based on the reciprocals of the relative intensities of the binaries involved. It is a rather cumbersome method, but theoretically valid, and it does not imply any approximation whatsoever; a discussion is outside the scope of this work. An easier method is to use the cross-product coeﬃcients as used in Eq. (99). The derivation from Tertian and Vie le Sage (1977) oﬀers some insight in this matter. Tertian and Vie le Sage (1977) assume that a multielement inﬂuence coeﬃcient aM ij can be approximated as the sum of the binary coeﬃcient aBij and a linear variation with the other elements: B aM ij ¼ aij þ tijk Wk

ð126Þ

where tijk is a coeﬃcient expressing the eﬀect of element k on the inﬂuence coeﬃcient aM ij . Similarly, B aM ik ¼ aik þ tikj Wj

ð127Þ

Substituting Eqs. (126) and (127) in M ½1 þ aM ij Wj þ aik Wk

ð128Þ

(the superscript M is used to explicitly indicate the use of multi element inﬂuence coeﬃcients) yields ½1 þ aBij Wj þ aBik Wk þ aijk Wj Wk

ð129Þ

aijk ¼ tijk þ tikj

ð130Þ

with

It is to be realized that crossed eﬀect is introduced by the use of binary coeﬃcients; use of multi element coeﬃcients would not lead to crossed eﬀect. Thus, the equation expressing the matrix eﬀect using binary inﬂuence coeﬃcients for specimens with more than two compounds is " # n n1 X n X X mij;bin Wj þ aijk Wj Wk Mi ¼ 1 þ ð131Þ j¼1 j6¼i

j¼1 k¼jþ1 j6¼i k6¼i

and is based on cross-product coeﬃcients to correct for crossed eﬀect introduced by the use of binary inﬂuence coeﬃcients. The use of the cross-product coeﬃcients is not mandated by the concentration range to be covered (the binary coeﬃcients as calculated by, for example, the algorithm of Lachance are more than adequate) but is a consequence of the use of binary coeﬃcients. 7. Application In Secs.V.D.4 and V.D.5, several inﬂuence coeﬃcient algorithms have been discussed. Application of the resulting equations for calibration and analysis will be discussed here and is equally valid for any of the inﬂuence coeﬃcient algorithms. a.

Calibration

Step 1. It is assumed that the coeﬃcients have been calculated from theory, for example, using Eq. (84) or (97).

400

de Vries and Vrebos

Step 2. Calculate the matrix correction term [the square brackets in Eq. (81), (85), (99), (104), and (109)] for all standard specimens and for a given analyte. The coeﬃcients are known (Step 1), and for standard specimens, all weight fractions Wi and Wj are known. Step 3. Plot the measured intensity of the analyte, multiplied by the corresponding matrix correction term against analyte weight fraction. Then, determine the ‘‘best’’ line, Wi ¼ Bi þ Ki Ii ½1 þ

ð132Þ

by minimizing DWi (see Sec. V.A). Note that Eq. (132) is more general than Eq. (50), which does not correct for matrix eﬀects. This process is repeated for all analytes. Other methods are also feasible. The most common variant is the one where W ¼ Bi þ Ki Ii ½1 þ

ð133Þ

is used. This is nearly equivalent to Eq. (132) but with brackets: Wi ¼ ðBi þ Ki Ii Þ½1 þ

ð134Þ

The term ðBi þ Ki Ii Þ is related directly to the relative intensity Ri. Corrections for line overlap should only aﬀect this term. b. Analysis For each of the analytes, a set of equations has to be solved for the unknown Wi, Wj and so forth. If the matrix correction term used is the one according to Lachance and Trail [Eq. (81)] or de Jongh [Eq. (85)], then the set of equations can be solved algebraically (n linear equations with n unknowns for Lachance and Traill and n1 equations with n1 unknowns for de Jongh). Mostly, however, an iterative method is used. As a ﬁrst estimate, one can simply take the matrix correction term equal to 1. This yields a ﬁrst estimate of the composition Wi, Wj, and so on. This ﬁrst estimate is used to calculate the matrix correction terms for all analytes. Subsequently, a new composition estimate can be obtained. This process is repeated until none of the concentrations changed between subsequent iterations by more than a preset quantity. If the matrix correction is done using algorithms which use more than one coeﬃcient {e.g., Claisse and Quintin [Eq. (99)] or Rasberry and Heinrich [Eq. (104)]}, then the equations are not linear in the unknown concentrations and an algebraic solution is not possible. An iterative method, such as described earlier can be used. 8. Algorithms with Empirical Coefficients Empirical coeﬃcients are coeﬃcients that are not calculated from theory but from actually measured specimens using regression analysis (Anderson et al., 1974). They were the basis of the earliest correction methods, but now they are largely superseded by more theoretical ones. Such empirically determined coeﬃcients tend to mix the matrix correction with the sensitivity of the spectrometer. On the one hand, the matrix eﬀect is determined by the composition of the sample and ‘‘physical’’ parameters such as take-oﬀ and incidence angles and tube anode and voltage. These are the same for spectrometers of similar design. The sensitivity of the spectrometer, on the other hand, depends on the reﬂectivity of the crystals, the eﬃciency of the detectors, and so on. These parameters are unique for each spectrometer. Also, if one of the analyte lines is overlapped by another x-ray line, some of this eﬀect can also aﬀect the value of the inﬂuence coeﬃcients. The

Quantification of Infinitely Thick Specimens

401

coeﬃcients thus determined are instrument-speciﬁc and are not transferable to other instruments. Stephenson (1971) has noted that the regression equations involved in the determination of the coeﬃcients in such an empirical way become unstable as the degree of correlation between the independent variables increases. This mandates careful planning of the experiment, including the composition of the synthetic standards. Klimasara (1994, 1995) has illustrated the use of standard spreadsheet programs for the calculation of the values of empirical inﬂuence coeﬃcients and composition. a.

The Sherman Algorithm

Sherman (1953) was among the ﬁrst to propose an algorithm for correction of matrix eﬀects. For a ternary system, the algorithm can be represented by the following set of equations: ðaAA tA ÞWA þ aAB WB þ aAC WC ¼ 0 aBA WA þ ðaBB tB ÞWB þ aBC WC ¼ 0 aCA WA þ aCB WB þ ðaCC tC ÞWC ¼ 0

ð135Þ

where aij represents the inﬂuence coeﬃcient of element j on the analyte i and ti is the time (in s) required to accumulate a preset number of counts. The constants aij are determined from measurements on specimens with known composition. Determination of the composition of an unknown involves the solving of the above set of linear equations [Eq. (135)]. This set, however, is homogeneous: Its constant terms are all equal to zero. So, only ratios among the unknown Wi can be obtained. In order to obtain the weight fractions Wi, an extra equation is required. Sherman proposed using the sum of all the weight fractions of all the elements (or components) in the specimen, which ideally, should be equal to unity. For a ternary specimen, WA þ WB þ WC ¼ 1

ð136Þ

Using Eq. (136), one of the equations in the set of Eqs. (135) can be eliminated. The solution obtained, however, is not unique: For a ternary, any one of the three equations can be eliminated. This yields three diﬀerent combinations. Furthermore, any of the three elements can be eliminated in each of the combinations. Hence, a total of 363 ¼ 9 diﬀerent sets can be derived from Eqs. (135) and (136), and each of these sets will generate diﬀerent results. In general, the algorithm yields n2 diﬀerent results for a system with n elements or compounds. This is clearly undesirable, because it is hard to determine which set will give the most accurate results. Another disadvantage is the fact that the sum of the elements determined always equals unity, even if the most abundant element has been neglected. Furthermore, the numerical values of the coeﬃcients depend, among other parameters such as geometry and excitation conditions, also on the number of counts accumulated. Nonquantiﬁable parameters, such as reﬂectivity of the diﬀracting crystal used in wavelength-dispersive spectrometers, or tube contamination will also aﬀect the value of the coeﬃcients. The coeﬃcients determined on a given spectrometer cannot be used with another instrument; they are not transferable. The other algorithms discussed use some form of a ratio method: The Lachance and Traill algorithm, for example, uses relative intensities. The measurements are then done, relative to a monitor; this reduces, or eliminates, the eﬀect of such nonquantiﬁable parameters.

402

b.

de Vries and Vrebos

The Algorithm of Lucas-Tooth and Price

Lucas-Tooth and Price (1961) developed a correction algorithm, where the matrix eﬀect was corrected for, using intensity (rather than concentration) of the interfering elements. The equation can be written as " # n X Wi ¼ Bi þ Ii k0 þ kij Ij ð137Þ j6¼i

where Bi is a background term and k0 and kij are the correction coeﬃcients. A total of n þ 1 coeﬃcients have to be determined, requiring at least n þ 1 standards. Usually, however, a much larger number of standards is used. The coeﬃcients are then determined by, for example, a least-squares method. The corrections for the eﬀect of the matrix on the analyte are done via the intensities of the interfering elements; their concentrations are not required. The method assumes that the calibration curves of the interfering elements themselves are all linear; the correction is done using intensities rather than concentrations. The algorithm will, therefore, have a limited range. Its use will be limited to applications where only one or two elements are to be analyzed (it still involves measurements of all interfering element intensities) and where a computer of limited capabilities is used (although calculation of the coeﬃcients involves much more compute capabilities than the subsequent routine analysis of unknowns). The advantages of the method are as follows: The method is very fast, because the calculation of composition of the unknowns requires no iteration. Analysis of only one element is possible; this requires, however, the determination of all relevant correction factors. Very simple algorithm, requiring very little calculation. c.

Algorithms Based on Concentrations

Algorithms similar to Eq. (137) have been proposed, using corrections based on concentrations rather than intensities. The values of the coeﬃcients were then to be derived from multiple-regression analysis on a large suite of standards. The main aim was to obtain correction factors that could be determined on one spectrometer and used, without alteration, on another instrument. In practice, the coeﬃcients still have to be adjusted because of the intimate and inseparable entanglement of spectrometer-dependent factors with matrix eﬀects. Furthermore, compared to the algorithms based on intensities, some of the advantages of the latter are not retained: A calibration for all elements present is now required, calculation of the composition of unknowns requires iteration, and so forth. In principle, methods based on theoretically calculated inﬂuence coeﬃcients are recommended.

VI.

CONCLUSION

Among the advantages of XRF analysis are the facts that the method is nondestructive and allows direct analysis involving little or no specimen preparation. Analysis of major and minor constituents requires correction for matrix eﬀects of variable (from one specimen to another) magnitude. If the matrix varies appreciably from one specimen to the next, then even the intensity of elements present at a trace level can be subject to matrix eﬀects and a correction is required. Several methods for matrix correction have been

Quantification of Infinitely Thick Specimens

403

described. Each of these methods have their own advantages and disadvantages. These, by themselves, do not generally lead to the selection of ‘‘best’’ method. The choice of the method to use is also determined by the particular application. From the previous sections, it may appear that the mathematical methods are more powerful than the compensation methods. Yet, if only one or two elements at a trace level in liquids have to be determined, compensation methods (either standard addition or the use of an internal standard) can turn out to be better suited than, for examples, rigorous fundamental parameter calculations. Compensation methods will correct for the eﬀect of an unknown, but constant, matrix. Also, they do not require the analysis of all constituents in the specimen. The mathematical methods (fundamental parameters as well as methods based on theoretical inﬂuence coeﬃcients), on the other hand can handle cases in which the matrix eﬀect is more variable from one specimen to another. In this respect, they appear to be more ﬂexible than the compensation methods, but they do require more knowledge of the complete matrix. All elements contributing signiﬁcantly to the matrix eﬀect must be quantiﬁed (either by x-ray measurement or by another technique) even if the determination of their concentrations is not required by the person who submits the sample to the analyst. Once a particular algorithm is selected, it is customary to use for all analytes. However, it must be stressed that this is not a requirement. There is only one requirement for adequate matrix correction: Each analyte should be corrected adequately, by whatever method. If complete analysis (covering all major elements) is required, the analyst has the choice between the fundamental parameter method and algorithms, based on inﬂuence coeﬃcients. Commonly, fundamental parameter methods are (or were) used in research environments rather than for routine analysis in industry. This choice is more often made on considerations such as the availability of the programs and computers than on differences in analytical capabilities. Inﬂuence coeﬃcient algorithms tend to be used in combination with more standards compared to fundamental parameter methods, because their structure and simple mathematical representation facilitates interpretation of the data (establishing a relationship between concentration and intensity, corrected for matrix eﬀect). The ﬁnal choice, however, has to be made by the analyst.

REFERENCES Anderson CH, Mander JE, Leitner JW. Adv X-Ray Anal 17:214, 1974. Australian Standard 2563-1982, Wavelength Dispersive X-ray Fluorescence Spectrometers— Methods of Test for Determination of Precision. North Sydney, NSW: Standards Association of Australia, 1982. Bambynek W, Crasemann B, Fink RW, Freund HU, Mark H, Swift CD, Price RE, Venugopala Rao P. Rev Mod Phys 44:716, 1972. Bearden JA. Rev Mod Phys 39:78, 1967. Beattie HJ, Brissey RM. Anal Chem 26:980, 1954. Bonetto RD, Riveros JA. X-Ray Spectrom 14:2, 1985. Claisse F, Quintin M. Can Spectrosc 12:129, 1967. Criss JW. Adv X-Ray Anal 23:93, 1980a. Criss JW. Adv X-Ray Anal 23:111, 1980b. Criss JW, Birks LS. Anal Chem 40:1080, 1968. Criss JW, Birks LS, Gilfrich JV. Anal Chem 50:33, 1978. de Boer DKG. Spectrochim Acta 44B:1171, 1989. de Jongh WK. X-Ray Spectrom 2:151, 1973.

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de Jongh WK. X-Ray Spectrom 8:52, 1979. DeGroot PB. Adv X-Ray Anal 33:53, 1990. Draper NR, Smith H. Applied Regression Analysis. New York: Wiley, 1966. Feather CE, Willis JP. X-Ray Spectrom 5:41, 1976. Garbauskas MF, Goehner RP. Adv X-Ray Anal 26:345, 1983. Gillam E, Heal HT. Br J Appl Phys 3:353, 1952. Heinrich KFJ. In: McKinley TD, Heinrich KFJ, Wittry DB, eds. The Electron Microprobe. New York: Wiley, 1966, p 296. Holynska B, Markowicz A. X-Ray Spectrom 10:61, 1981. Hower J. Am Mineral 44:19, 1959. Hughes H, Hurley P. Analyst 112:1445, 1987. Hunter CB, Rhodes JR. X-Ray Spectrom 1:107, 1972. Ingham MN, Vrebos BAR. Adv X-Ray Anal 37:717, 1994. ISO, Determination of Nickel and Vanadium in Liquid Fuels—Wavelength-Dispersive X-Ray Fluorescence Method, ISO 14597. Geneva: ISO, 1995. Johnson W. International Report BISRA MG=D=Conf Proc=610=67, 1967. Kaufmann M, Mantler M, Weber F. Adv X-Ray Anal 37:205, 1994. Klimasara AJ. Adv X-Ray Anal 37:647, 1994. Klimasara AJ. Workshop on the Use of Spread Sheets in XRF Analysis, 44th Annual Denver X-Ray Conference, Colorado Springs, CO, 1995. Lachance GR. X-Ray Spectrom 8:190, 1979. Lachance GR. International Conference on Industrial Inorganic Elemental Analysis, Metz, France, 1981. Lachance GR. Adv X-Ray Anal 31:471, 1988. Lachance GR, Claisse F. Adv X-Ray Anal 23:87, 1980. Lachance GR, Traill RJ. Can Spectrosc 11:43, 1966. Leroux J, Thinh TP. Revised Tables of Mass Attenuation Coeﬃcients. Quebec: Corporation Scientiﬁque Claisse, 1977. Li-Xing Z. X-Ray Spectrom 13:52, 1984. Lubecki A, Holynska B, Wasilewska M. Spectrochim Acta 23B:465, 1968. Lucas-Tooth HJ, Price BJ. Metallurgia 64:149, 1961. Mainardi RT, Fernandez JE, Nores M. X-Ray Spectrom 11:70, 1982. Mantler M. Adv X-Ray Anal 36:27, 1993. McMaster WH, Delgrande NK, Mallet JH, Hubbel JH. Compilation of X-Ray Cross Sections, UCRL 50174, Sec II, Rev 1, 1969. Mu¨ller RO. Spectrochemical Analysis by X-Ray Fluorescence. New York: Plenum Press, 1972, chap 9. Pella PA, Feng LY, Small JA. X-Ray Spectrom 14:125, 1985. Pella PA, Tao GY, Lachance GR. X-Ray Spectrom 15:251, 1986. Pollai G, Ebel H. Spectrochim Acta 26B:761, 1971. Pollai G, Mantler M, Ebel H. Spectrochim Acta 26B:733, 1971. Rasberry SD, Heinrich KFJ. Anal Chem 46:81, 1974. Rousseau RM. X-Ray Spectrom 13:121, 1984a. Rousseau RM. X-Ray Spectrom 13:3, 1984b. Rousseau RM, Claisse F. X-Ray Spectrom 3:31, 1974. Sherman J. The Correlation Between Fluorescent X-Ray Intensity and Chemical Composition. ASTM Special Publication No 157. Philadelphia: ASTM, 1953, p 27. Sherman J. Spectrochim Acta 7:283, 1955. Shiraiwa T, Fujino N. Jpn J Appl Phys 5:886, 1966. Shiraiwa T, Fujino N. Bull Chem Soc Japan 40:2289, 1967. Shiraiwa T, Fujino N. Adv X-Ray Anal 11:63, 1968. Shiraiwa T, Fujino N. X-Ray Spectrom 3:64, 1974. Sparks CJ. Adv X-Ray Anal 19:19, 1976.

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Stephenson DA. Anal Chem 43:310, 1971. Tao GY, Pella PA, Rousseau RM. NBSGSC, a FORTRAN program for quantitative X-Ray Fluorescence Analysis. NBS Technical Note 1213. Gaithersburg, MD: National Bureau of Standards, 1985. Tertian R. Adv X-Ray Anal 19:85, 1976. Tertian R. X-Ray Spectrom 16:261, 1987. Tertian R, Vie le Sage R. X-Ray Spectrom 6:123, 1977. Vrebos BAR, Helsen JA. X-Ray Spectrom 15:167, 1986. Vrebos BAR, Pella PA. X-Ray Spectrom 17:3, 1988. Veigele WJ. In: Robinson JW, ed. Handbook of Spectroscopy. Cleveland, OH: CRC, 1974, p 28. Venugopala Rao P, Chen MH, Crasemann B. Phys Rev A 5:997, 1972. Wadleigh KR. X-Ray Spectrom 16:41, 1987. Wood PR, Urch DS. J Phys F: Metal Phys 8:543, 1978.

SUGGESTIONS FOR FURTHER READING Bertin EP. Principles and Practice of X-Ray Spectrometric Analysis. 2nd ed. New York: Plenum Press, 1975. Jenkins R, Gould RW, Gedcke D. Quantitative X-Ray Spectrometry. 2nd ed. New York: Marcel Dekker, 1995. Tertian R, Claisse F. Principles of Quantitative X-Ray Fluorescence Analysis. New York: Wiley, 1982.

6 Quantiﬁcation in XRF Analysis of Intermediate-Thickness Samples Andrzej A. Markowicz Vienna, Austria

Rene´ E. Van Grieken University of Antwerp, Antwerp, Belgium

I.

INTRODUCTION

A number of approaches have been developed for quantitation in x-ray ﬂuorescence (XRF) analysis of intermediate-thickness samples whose mass per unit area m fulﬁlls the relation mthin < m < mthick

ð1Þ

where mthin and mthick are the values of mass per unit area for thin and thick samples [for a deﬁnition, see Chapter 1, Eqs. (93) and (95)]. Intermediate samples can be preferable to thick specimens because remaining uncertainties about mass-attenuation coeﬃcients have a smaller eﬀect on the analysis results, less material is required, the sensitivity is more favorable for low-Z elements, and secondary enhancement eﬀects are less important. Historically, the oldest correction method applied in quantitative XRF analysis of intermediate-thickness samples is the emission–transmission (E–T) method in which the speciﬁc x-ray intensities from a sample are measured successively with and without a target positioned adjacent to the back of the sample in a ﬁxed geometry. Recently, the E–T method has thoroughly been evaluated, and both the advantages and limitations of the technique are well identiﬁed. A number of modiﬁcations of the E–T method, developed in recent years, allowed an essential extension of its applicability range. To avoid additional measurements that are inevitable in the emission–transmission method, some alternative correction procedures based on the use of scattered primary radiation were also developed. In recent years, however, only a few papers have been published in this ﬁeld. The underlying principles as well as the ranges of applicability and the limitations of the correction procedures applied to XRF analysis of both homogeneous and heterogeneous intermediate-thickness samples are outlined here.

407

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II.

Markowicz and Van Grieken

EMISSION---TRANSMISSION METHOD

In the absence of enhancement eﬀects and assuming monochromatic excitation, the mass per unit area of the element i; mi , for homogeneous intermediate-thickness samples, can be calculated from [see Eq. (90) in Chapter 1]: mi ¼

Ii ðEi Þ Abcorr Bi

ð2Þ

where GI0 ðE0 Þ"ðEi Þi0 ðE0 Þ!i pi ð1 1=ji Þ sin 1 and Abcorr is the absorption correction factor given by Bi ¼

Abcorr ¼

½ðE0 Þ csc 1 þ ðEi Þ csc 2 m 1 expf½ðE0 Þ csc 1 þ ðEi Þ csc 2 mg

ð3Þ

ð4Þ

[The symbols used in Eqs. (2)–(4) are explained in Chapter 1, Eqs. (88) and (89).] The value of the constant Bi (sometimes called the sensitivity factor) can be determined either experimentally as the slope of the straight calibration line for the ith element obtained with thin homogeneous samples or semiempirically based on both the experimentally determined GI0 ðE0 Þ value and the relevant fundamental parameters (Yap et al., 1987; Markowicz et al., 1992a). The absorption correction factor Abcorr represents the combined attenuation of the primary and ﬂuorescent radiations in the whole specimen and can be determined individually for each sample by transmission experiments (Leroux and Mahmud, 1996; Giauque et al., 1973). These are done by measuring the x-ray intensities with and without the specimen from a thick multielement target located at a position adjacent to the back of the specimen, as shown in Figure 1. If ðIi ÞS ; ðIi ÞT ; and ðIi Þ0 are the intensities after background correction from the sample alone, from the sample plus target, and from the target alone, respectively, then the combined fraction of the exciting and ﬂuorescent radiations transmitted through the total sample thickness is expressed by expf½ðE0 Þ csc

1

þ ðEi Þ csc

2 mg

¼

ðIi ÞT ðIi ÞS H ðIi Þ0

ð5Þ

After a simple transformation, Eq. (2) can be rewritten as

Figure 1 Experimental procedure used in the emission–transmission method for the correction of matrix absorption effects.

Quantification of Intermediate-Thickness Samples

409

Ii ðEi Þ ð ln HÞ ð6Þ Bi 1 H The emission–transmission method can only be applied in the quantitative XRF analysis of homogeneous samples of which the mass per unit area is smaller than the critical value, mcrit ; deﬁned by ln Hcrit ð7Þ mcrit ¼ ðE0 Þ csc 1 þ ðEi Þ csc 2 where Hcrit is the critical value of the transmission factor deﬁned by Eq. (5); in practice, Hcrit ¼ 0:1 ðor 0:05Þ: To minimize possible absorption correction errors resulting from enhancement of the specimen radiation by scattered target radiation, targets that yield a high ratio of scattered to ﬂuorescent radiation should not be used. Giauque et al. (1979) developed a modiﬁed version of the emission–transmission method. Using data from the attenuation measurements and Eq. (5), the values of ½ðE0 Þ csc 1 þ ðEi Þ csc 2 m ð¼ T mÞ are calculated for the energies of characteristic x-rays of all elements present in a thick multielement target. If these values are plotted versus the ﬂuorescence x-ray energy on a log–log scale (Fig. 2), an approximate value for mi ¼

Figure 2 Curves of [m(E0) csc 1 þ (Ei) csc 2]m and mm(Ei) csc 2 values versus fluorescence x-ray energy for an NBS SRM 1632 coal specimen. (From Giauque et al., 1979. Reprinted with permission from Analytical Chemistry. Copyright American Chemical Society.)

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ðE0 Þ csc 1 can be obtained by extrapolation of the curve to the energy of the excitation radiation. In turn, values for mðEi Þ csc 2 can be calculated, a curve for these values drawn, and a new value for mðE0 Þ csc 1 established. This last step is iterated several times. Using data from the latter curve, the absorption correction factors for all radiations of interest can be calculated from Eq. (4). If some elements to be determined are major or minor constituents, a few separate curves for mðEi Þ csc 2 values should be plotted between the preselected x-ray energies corresponding to the relevant absorption edges. In the emission–transmission method of Giauque et al. (1979), the incoherent scattered radiation corrected for matrix absorption is used as the internal standard to compensate for variations in sample mass, x-ray tube output, and sample geometry. In practical applications of the emission–transmission method, the values of the transmission factor H [Eq. (5)] determined for a few energies of the characteristic x-rays of some elements present in a multielement target are used to construct a curve ln lnðH1 Þ ¼ lnðT mÞ versus ln E; which enables one to calculate the absorption correction factor Abcorr for any energy. In some cases, however, the relationship of ln lnðH1 Þ versus ln E exhibits some discontinuities which correspond to the absorption edges of minor (major) elements present in the unknown samples. In such cases the approach proposed by Giauque et al. (1979), based on the construction of separate curves for the predeﬁned energy regions, can only be used if at least two experimental points are available for each energy region. When only one or no experimental point is available, a modiﬁed version of the emission–transmission method can be used (Markowicz and Haselberger, 1992). To explain the modiﬁed procedure, let us assume that a multielement target consists of Ca, Ti, Fe, Zn, Sr, Zr, and Pb, and the material to be analyzed contains some minor elements, such as Fe and Ca. From the E–T measurements, the values of A ðA ¼ H1 Þ are easily obtained for higher energies (e.g., for the characteristics x-rays of Fe, Zn, Sr, Zr, and Pb in this case) (see Fig. 3). First, a straight line is ﬁtted to the points corresponding to Zn, Pb, Sr, and Zr (region I). Second, a straight line of the same slope and passing through a point for Fe is constructed (region II). If no experimental point is available for Ca, the discontinuity for the K absorption edge of Ca has to be taken into account by applying a computation routine. In the ﬁrst step, the product 1 ¼ WCa ðabove below Þ is calculated, where WCa is the weight fraction of Ca obtained by using a direct extrapolation of curve II and above and below are the mass absorption coeﬃcients just above and below the K absorption edge of Ca, respectively. Next, the corrected value of the total mass absorption coeﬃcient cor T ¼ ðln AÞ=m 1 csc 2 is calculated for the energy of the CaKabs edge, where A is taken from curve II and 2 is the emerging angle for the characteristic x-rays. Based on the cor T value, a straight line passing through point C and having the same slope as that one in region II is constructed (the coordinates of the point C are x ¼ ln CaKabs and y ¼ ln½ln A m1 csc 2 ). In the next step, the values of ln T m and the absorption correction factor Abcorr are calculated for the energy of Ca characteristic x-rays based on the straight line for region III. Because the true value of the weight fraction for Ca is unknown, all calculations for the absorption edge of Ca (region III) are iterated until the following convergence is obtained: ðWCa Þn ðWCa Þn1 0:001 ðWCa Þn

ð8Þ

where ðWCa Þn and ðWCa Þn1 are the values of the weight fractions for Ca obtained in the subsequent iterations.

Quantification of Intermediate-Thickness Samples

411

Figure 3 Graphical explanation of the idea for the modified version of the E–T method: see text for details. (From Markowicz and Haselberger, 1992. Reprinted with permission of Pergamon Press Ltd.)

A calculation procedure applied for the Ca absorption edge can easily be repeated for any other absorption discontinuity at a lower energy and, ﬁnally, a total absorption curve ln ln A versus ln E is obtained. The importance of the additional correction for the discontinuities in the absorption properties of the samples to be analyzed was demonstrated, among others, for the analysis of coal samples. Neglecting the absorption edges correction for Fe and Ca at the concentration level of around 1% resulted in considerable errors of the analysis, up to 60% for the determination of Ca (Markowicz and Haselberger, 1992). Van Dyck et al. (1980) developed a correction method that allows calculations of the absorption coeﬃcients (and absorption correction factors as well) at any energy for intermediate-thickness samples, without additional measurements, by using the ratio of the x-ray signals from a Zr wire positioned in front of the sample and from a Pd foil placed behind the sample, both in a ﬁxed geometry, as shown in Figure 4. The Zr wire provides an external reference signal ðZrKÞ; which is applied for normalization of all measured ﬂuorescent intensities to reduce considerably the eﬀect of variations in exciting x-ray tube intensity and of dead-time losses. The coeﬃcients for higher energies are calculated with an iterative program from the experimentally measured absorption coeﬃcient at the PdL energy (2.9 keV), ðEPd Þ. In the ﬁrst step, the total attenuation coeﬃcient at the PdL energy, caused exclusively by the lowZ elements (e.g., Z < 17) in the sample that show no characteristic peak above 3.0 keV in the spectrum, is calculated from the normalized measured intensities, ðIPd ÞT and ðIPd Þ0 and

412

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Figure 4 Measurement geometry applied in an automatic absorption correction method. (From Van Dyck et al., 1980. Reprinted by permission of John Wiley & Sons, Ltd.)

the diﬀerent characteristic peaks recorded in the spectrum. This attenuation coeﬃcient due to low-Z elements, low-Z ðEPd Þ, is given by (Van Dyck et al., 1980) low-Z ðEPd Þ ¼

ln½ðIPd Þ0 =ðIPd ÞT =m csc

2

ðE0 Þ csc 1 =csc P0 1 nj¼1 Wj

2

Pn 0 j¼1

Wj j ðEPd Þ

ð9Þ

where n0 is the number of characteristic peak in the XRF spectrum and j ðEPd Þ is the mass attenuation coeﬃcient for the PdL energy in the element j (McMaster et al., 1969), giving rise to a characteristic peak in the XRF spectrum. The weight fraction of the element j in the sample, Wj , is correlated with the recorded characteristic peak through the sensitivity factor, routinely obtained by measuring thin single or multielement standards. As a ﬁrst approximation, the characteristic peak intensities are not corrected for absorption. In a second step, based on the low-Z ðEPd Þ value, the absorption coeﬃcients of the low-Z matrix for other energies, low-Z ðEÞ, are calculated quantitatively by assuming nearly parallel properties of the logarithmic absorption curves ln ðEÞ versus ln E. The total mass attenuation coeﬃcient ðEÞ for the characteristic x-rays of an element in the whole sample can now be calculated, taking into account the contributions j ðEÞ of the high-Z elements: ! n0 n0 X X ðEÞ ¼ 1 Wj low-Z ðEÞ þ Wj j ðEÞ ð10Þ j¼1

j¼1

The overall procedure is summarized and schematically represented in Figure 5. Better ðEÞ values [Eq. (10)] are obtained in the second and following loops by carrying out appropriate absorption corrections to the characteristic intensities from which the Wj values are derived using ðEÞ and ðE0 Þ values from the previous loop and by including in Eq. (9) the ðE0 Þ values, taken as zero in the ﬁrst loop. The iteration is stopped when the diﬀerence in ðEÞ between two loops is negligible. Subroutine ENHANC is applied to evaluate the enhancement eﬀect of the PdL x-rays caused by all elements in the sample of which the x-ray energy is higher than the L1 ; L2 and L3 absorption edges of Pd. The intensity caused by this enhancement eﬀect can be considered to result in an apparent increase in ðIPd Þ0 . The enhancement contributions, IenhPd , are added to the ðIPd Þ0 in Eq. (9). This total ðIPd ÞOeff value and the detected ðIPd ÞT signal allow us to calculate the correct values via Eqs. (9) and (10). A comprehensive discussion of the inﬂuence of secondary enhancement of the PdL x-rays by the samples as well as of the inﬂuence of two other complicating factors, grain

Quantification of Intermediate-Thickness Samples

413

Figure 5 The calculation steps. From the experimentally measured total absorption coefficient at the PdL energy, m(EPd), the calculated contribution from high-Z elements (giving a characteristic peak in the spectrum) is substracted; through this low-Z matrix contribution at the PdL energy, the total low-Z absorption curve (dashed line) is calculated; the contribution from high-Z elements (e.g., Ca and Fe) is then added, to yield the total absorption curve (thick line). (From Van Dyck et al., 1980. Reprinted by permission of John Wiley & Sons, Ltd.)

size eﬀects, and a heterogeneous sample load is presented in the work of Van Dyck et al. (1980). The grain size and sample heterogeneity eﬀects induce inaccuracies on the absorption coeﬃcient determinations that may well reach 20% for particulate samples, such as intermediate-thickness deposits of geological materials. Thus, this approach has the same limitations as all emission–transmission methods applied to heterogeneous samples. A.

Accuracy and Limitations of the Emission--Transmission Method

The overall (total) absolute uncertainty of the mass per unit area of the ith element, mi , can be calculated from the law of error propagation applied to Eq. (2) (Markowicz et al., 1992b): " #1=2 Abcorr 2 Ii ðEi Þ 2 Ii ðEi ÞAbcorr 2 2 2 2 mi ¼ ½Ii ðEi Þ þ ðAbcorr Þ þ ðBi Þ ð11Þ Bi Bi B2i where Ii ðEi Þ is the absolute uncertainty of the characteristic x-ray intensity Ii ðEi Þ (usually provided by a computer routine applied for spectrum evaluation), Abcorr is the absolute uncertainty of the absorption correction factor, and Bi is the absolute uncertainty of the sensitivity (or calibration) factor Bi (usually provided by a computer calibration routine, as a mean percent diﬀerence between the experimental and ﬁtted values for the calibration factors). The total relative uncertainty of the mass per unit area mi =mi is calculated from " #1=2 mi Ii ðEi Þ 2 Abcorr 2 Bi 2 ¼ þ þ ð12Þ Ii ðEi Þ mi Abcorr Bi

414

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In order to calculate Abcorr , one has to apply the law of error propagation to Eqs. (4) and (5): Abcorr ¼

A 1 ln A ðA 1Þ2

A

ð12aÞ

with A ¼ H1 [Eq. (5)] and " A ¼

½ðIi Þ0 2

½ðIi ÞT ðIi ÞS 2

þ

½ðIi ÞT 2 ½ðIi Þ0 2 ½ðIi ÞT ðIi ÞS 4

þ

½ðIi ÞS 2 ½ðIi Þ0 2 ½ðIi ÞT ðIi ÞS 4

#1=2 ð12bÞ

where ½ðIi Þ0 and ½ðIi ÞT are the absolute uncertainities of the characteristic x-ray intensities obtained from the target alone and from the sample plus target, respectively. After a simple transformation, mi =mi can be given by " #1=2 mi Ii ðEi Þ 2 ðA 1 ln AÞ2 Bi 2 2 ¼ þ ðAÞ þ ð13Þ Ii ðEi Þ mi Bi ðA 1Þ2 A2 ðln AÞ2 It is obvious that the total relative uncertainty of the weight fraction of the ith element Wi =Wi is equal to mi =mi . Equations (13) and (12b) can be used for calculating the total uncertainities of the emission–transmission method as well as the contribution from various sources of the uncertainties ½Ii ðEi Þ; Abcorr ; Bi . It has been demonstrated that in the determination of trace elements, the dominant contribution to the total uncertainty is from the uncertainty of the peak-area calculation for the characteristic x-rays of the element of interest. In this case, the contribution from the uncertainties of the absorption correction factor and sensitivity can practically be neglected. In the determination of major elements, the largest contribution to the total uncertainty is from the uncertainty of the absorption correction factor and=or sensitivity; in this case, the contribution of the uncertainty of the peak-area calculation can safely be neglected. Such a detailed analysis of the contribution of various sources of uncertainties can be useful when there is a need to identify the most critical point of the whole analytical procedure and some reduction of the total uncertainty of the analysis is required. Another factor which inﬂuences the accuracy of the emission–transmission method is the heterogeneity of the sample loading. Markowicz and Abdunnabi (1991) derived general expressions describing the accuracy of the E–T method for various types of sample loading heterogeneities (including incompletely loaded samples) within a wide range of sample thicknesses and characteristics x-ray energies. The results, conﬁrmed by experiments, clearly show that inhomogeneities of sample loading may critically aﬀect the accuracy of the E–T method, in particular in the analysis of intermediate-thickness samples with strong inhomogeneities and large values for the ðE0 Þðcsc 1 Þm and ðEi Þðcsc 2 Þm factors. The most critical inﬂuence of sample loading heterogeneity on the accuracy of the analytical results is observed for incompletely loaded samples of intermediate thickness. The method proposed in the work of Markowicz and Abdunnabi (1991) gives a possibility for evaluating the errors arising from some types of sample loading heterogeneities as well as for evaluating the range of applicability of the simplest version of the E–T method (Markowicz et al., 1992b) if a certain value for the total uncertainty Wi =Wi is accepted.

Quantification of Intermediate-Thickness Samples

III.

415

ABSORPTION CORRECTION METHODS VIA SCATTERED PRIMARY RADIATION

The use of scattered primary radiation in XRF analysis provides an alternative to the common problem of matching standards of similar composition to samples to be analyzed. The backscatter peaks are sometimes treated as ﬂuorescent peaks from internal standards because they suﬀer matrix absorption similar to that of ﬂuorescent peaks and behave similarly with instrumental variations. They also provide the only direct spectral measure of the total or average matrix of geological, biological, or other materials containing large quantities of light elements, such as carbon, nitrogen, and oxygen, usually not observed by their characteristic x-ray peaks. X-ray ﬂuorescence matrix correction methods based on the use of scattered radiation have mostly been applied in quantitative analysis of inﬁnitely thick samples under a wide variety of experimental conditions, including discrete and continuum primary radiation sources and detection by both wavelength- and energy-dispersive systems (Andermann and Kemp, 1958; Kalman and Heller, 1962; Taylor and Andermann, 1971, 1973; Leonardo and Saitta, 1977; Livingstone, 1982; Kikkert, 1983; Markowicz, 1984). The scattered-radiation methods utilize the incoherent (Compton) and=or coherent (Rayleigh) scatter peaks from line excitation sources or the intense high-energy region from continuum sources.

A.

Absorption Corrections Based on Incoherent Scattered Radiation

For specimens of less than inﬁnite thickness, the intensity ICom of Compton-scattered radiation can be expressed by (Meier and Unger, 1976) ICom ¼

k0 I0 Com ðE0 Þ½1 expfððE0 Þ csc 1 þ ðECom Þ csc ðE0 Þ csc 1 þ ðECom Þ csc 2

2 Þmg

ð14Þ

where k0 is a constant for a given measurement geometry and detection eﬃciency, Com ðE0 Þ is the Compton mass-scattering coeﬃcient of the sample material for the primary radiation of energy E0 ðcm2 =gÞ; ðECom Þ is the total mass-attenuation coeﬃcient of the sample material for Compton-scattered primary radiation of energy ECom ðcm2 =gÞ [see Eq. (52) in Chapter 1]. Equation (14) is valid for monochromatic excitation. Assuming that the relation of the atomic number to the mass number is constant for every element to be found in the sample and that ðE0 Þ ﬃ ðECom Þ, the following simpliﬁed formula for the intensity of Compton-scattered radiation from a multielement sample can be obtained: ICom ¼

k01 I0 f1 exp½ðE0 Þ csc 1 ð1 þ csc 2 =csc ðE0 Þ cscð 1 Þð1 þ csc 2 =csc 1 Þ

1 Þmg

ð15Þ

where k01 is a constant for given geometry of measurement and the energy of the incident radiation. The value of the product k01 I0 is determined experimentally from a reference scatterer and is valid as long as the reference scatterer has a matrix that is not too diﬀerent from that of the specimens. Kieser and Mulligan (1979) worked out a method based on the use of the incoherent scatter radiation, which gives accurate mass absorption coeﬃcients for a limited average Z range. The mass absorption coeﬃcient ðE0 Þ for specimens of intermediate thickness is

416

Markowicz and Van Grieken

found from Eq. (15) after a numerical solution. To obtain a value of the mass absorption coeﬃcient ðEÞ at any energy E, Kieser and Mulligan (1979) assumed that the slope of the curve log ðEÞ versus log E is constant for all elements (approximately 2.7) over a range of x-ray energies. The proposed Compton-scattered method for the determination of an intermediate specimen’s mass absorption coeﬃcient at any energy can be applied as long as no absorption edge of a major or minor element intervenes. When the values of the ðEÞ are determined, calculation of the absorption correction factor [Eq. (4)] is straightforward if, of course, the mass of the sample to be analyzed is known. A modiﬁed ﬂuorescent Compton correction method for quantitative XRF of intermediate specimens was developed by Holynska and Markowicz (1979). The method is based on the use of the measured x-ray ﬂuorescent intensities of all determined elements and the intensity of Compton-backscattered radiation. The authors derived the following expression for the determination of the mass per unit area of the element i, mi: mi ¼

1þ

Pn0 l¼1

ai I i

ð16Þ

ail Il þ bi ICom =m

where ai ; ail ; and bi are constant coeﬃcients obtained experimentally on the basis of standard samples and n0 is the number of the elements to be determined, including the ith element. As is seen from Eq. (16), the absorption matrix correction is carried out via the intensities of characteristic x-rays of all elements determined and the intensity ICom, reﬂecting, for the most part, the variations in the composition of light matrix. To apply the absorption matrix correction, Eq. (16), the total mass per unit area of the specimen must be evaluated, for example, by sample weighing. This ﬂuorescent–Compton correction method can be used in the XRF analysis of homogeneous intermediate-thickness samples in a limited range of mass per unit area (m < 10mthin).

B.

Absorption Corrections Based on Both Coherent and Incoherent Scattered Radiations

Several absorption correction methods based on both coherent and incoherent scattered radiations have been developed, mostly in the 1970s and 1980s, and applied in quantitative XRF analysis of intermediate samples. This group of correction methods is represented either by relatively simple approaches (Bazan and Bonner, 1976; Markowicz, 1979) or by very sophisticated fundamental parameter procedures (Nielson, 1977; Van Dyck and Van Grieken, 1980; Nielson et al., 1982; Nielson and Rogers, 1984) providing superior analytical ﬂexibility. For specimens of less than inﬁnite thickness, the intensity Icoh of coherent scattered radiation can be calculated from Icoh ¼

k00 I0 coh ðE0 Þ ðE0 Þðcsc 1 þ csc

2Þ

f1 exp½ðE0 Þðcsc

1

þ csc

2 Þmg

ð17Þ

where k00 is a constant for a given measurement geometry and detection eﬃciency for the primary x-rays of energy E0 , and coh ðE0 Þ is coherent mass-scattering coeﬃcient of the sample material for the primary radiation (cm2=g) [see Eq. (73) in Chapter 1]. Bazan and Bonner (1976) showed, for the ﬁrst time, a linear relation between the eﬀective absorption coeﬃcient (deﬁned as the sum of the sample absorption coeﬃcients for exciting and characteristic x-rays) and the ratio of incoherent to coherent scattering.

Quantification of Intermediate-Thickness Samples

417

However, the coeﬃcients of the calibration line varied somewhat with the matrix, and this hampered practical applications of this simple approach. Markowicz (1979) found that, theoretically the sensitivity of the absorption correction via the incoherent=coherent scattered x-ray intensities ratio is better than that of the absorption procedure involving each of the scattered radiations individually. For intermediate-thickness samples, in a limited range of rather small values of mass per unit area, the intensities of the Compton-scattered radiation ICom and the coherent scattered radiation Icoh are diﬀerent functions of the total mass-attenuation coeﬃcient of the incident radiation ðE0 Þ; the intensity ICom is a linearly decreasing function and the intensity Icoh appears to be a linearly increasing function of the ðE0 Þ. For a limited range of ðE0 Þ values, the following simple expression can be used to evaluate ðE0 Þ (Markowicz, 1979): ðE0 Þ ¼ C1 þ C2 m þ C3 r þ C4 mr

ð18Þ

where r¼

ICom Icoh

and C1–C4 are constants calculated by the least-squares ﬁt on the basis of experimental results for standard samples. The values of the total mass-attenuation coeﬃcient of the ﬂuorescent radiation in a whole sample, ðEi Þ, is obtained from the simple dependence of the ðEi Þ=ðE0 Þ ratio on the values ðE0 Þ, calculated separately for each element to be determined. Finally, sample weighing provides the value of mass per unit area, m, and the calculation of the absorption correction factor Abcorr via Eq. (4) can be simply performed if, of course, the values of the eﬀective angles 1 and 2 are evaluated experimentally or theoretically. The applicability of the proposed matrix correction method (Markowicz, 1979) involving both incoherent and coherent scattered primary radiations is limited to XRF analysis of intermediatethickness samples of mass per unit area smaller than about 10mthin. A backscattered fundamental parameters (BFP) method for quantitative XRF analysis of intermediate samples of variable composition and thickness was developed by Nielson (1977). The method utilizes thin-ﬁlm multielement calibration of the spectrometer and mathematical matrix correction in which the samples are modeled as a composite of heavy elements, which are quantiﬁed through their characteristic radiation, and light elements, estimated through the coherent and incoherent x-ray scatter peaks. Figure 6 schematically illustrates the basis for analyzing the heavy elements (Z > 13) and the light elements (H, C, N, O, Na, and others), which must be estimated by the diﬀerence from the scattered x-ray peaks. The BFP method utilizes coherently and incoherently scattered x-rays to identify and estimate the quantities of two light elements representative of the light-element portion of sample matrix. The quantities of the two light elements a and b are estimated by solving for Wa and Wb in the simultaneous equations gIcoh

n0 X

Wj cohj ðE0 Þ ¼ Wa coha ðE0 Þ þ Wb cohb ðE0 Þ

ð19Þ

j¼1

and hICom

n0 X j¼1

Wj Comj ðE0 Þ ¼ Wa Coma ðE0 Þ þ Wb Comb ðE0 Þ

ð20Þ

418

Markowicz and Van Grieken

Figure 6 Light-element contributions to x-ray scattering, from which absorption corrections are computed. (From Nielson, 1977. Reprinted with permission from Analytical Chemistry. Copyright American Chemical Society.)

where g and h are the geometry-dependent calibration factors determined experimentally by using any standard of known total composition. Because several light-element pairs may satisfy Eqs. (19) and (20), the pair is chosen whose incoherent=coherent scattering cross-sectional ratios lie immediately on either side of the ratio of the observed scatter attributable to light elements (Nielson, 1977): Pn 0 Coma ðE0 Þ hICom j¼1 Wj Comj ðE0 Þ Comb ðE0 Þ < < P0 coha ðE0 Þ cohb ðE0 Þ gIcoh nj¼1 Wj cohj ðE0 Þ

ð21Þ

The heavy- and light-element concentrations are used in computing the absorption correction factor Abcorr [Eq. (4)] and the enhancement correction factor [Eq. (91) in Chapter 1]. Because the concentrations and corrections are interdependent, all calculations are carried out by iteration [for more details see (Nielson, 1977)]. To improve the sensitivity of the determination of low-Z elements, Sanders et al. (1983) extended the previously described BFP method. The new method utilizes the coherent and incoherent backscatter intensities to compute matrix corrections (Nielson and Sanders, 1982) from the combined results of two separate energy-dispersive XRF (EDXRF) data from diﬀerent (TiK and ZrK) excitation sources. The Ti-excited spectrum allows a more sensitive determination of elements in the Al–Ca range. The coherent=incoherent scatter ratio is also applied in an absorption correction procedure developed by Van Dyck and Van Grieken (1980) for monochromatic x-ray excitation. In this method, coherent and incoherent scattered radiations are used to calculate, ﬁrst, the eﬀective mass of the sample and, second, the absorption coeﬃcients for x-rays of interest and, hence, the absorption correction factors. The eﬀective thickness is the sample thickness weighted at every point for the excitation-detection eﬃciency, in the same way as the measured characteristic radiation is weighted.

Quantification of Intermediate-Thickness Samples

419

Assuming that the major elements of the sample do not diﬀer too greatly in atomic number, the eﬀective thickness meﬀ can be calculated from (Van Espen et al., 1979) meff ¼

ICom fb0 Abcorr ðECom Þ½b0 Icoh Abcorr ðECom Þ=a0 ICom Abcorr ðEcoh Þb1 gða1 b1 Þ1

ð22Þ

where a0 ; a1 ; b0 ; and b1 are experimental constants obtained by ﬁtting the results of measured standards; for mixtures or compounds, the coherent and incoherent scatter factors Scoh and SCom , (in fact, the relevant mass-scattering coeﬃcients), are given by Scoh ¼ a0

n X

Wj Zaj 1

j¼1

and SCom ¼ b0

n X

Wj Zbj 1

ð23Þ

j¼1

respectively. Abcorr(ECom ) and Abcorr(Ecoh ) are the absorption correction factors for the incoherent and coherent scatter radiation, respectively, as deﬁned in Eq. (4). A reasonably accurate eﬀective mass is obtained by modeling the sample as a composite of high-Z elements, calculated from their characteristics peaks using Eq. (2), and of a light matrix with mass per unit area, mlow-Z , evaluated from the coherent and incoherent scatter peaks after subtraction of the high-Z element contribution. The method for the determination of the eﬀective thickness allows the analysis of samples of heterogeneous thickness and irregular shape. More recently, Arau´jo et al. (1990) have developed a very similar procedure for eﬀective sample mass assessment of intermediate thickness samples via the coherently and incoherently scattered radiation, as a ﬁrst step in the automated matrix eﬀect evaluation, for the case of ﬁltered polychromatic continuum excitation with a Rh tube. Validation with geological standards and deposited slurries on Mylar foils and Nuclepore ﬁlters gave very satisfactory results. In the method proposed by Van Dyck and Van Grieken (1980), calculation of the mass-attenuation coeﬃcient for x-rays of interest is preceded by an evaluation of the mass-attenuation coeﬃcient m (2.956 keV) at the ArKa energy (2.956 keV). This energy is preferred because it is at the lower end of the energy range that can safely be used in conventional EDXRF analysis and because, when working under vacuum, it is situated in a peak-free part of an XRF spectrum. The value of m (2.956 keV) is derived from the ratio coherent to incoherent scatter intensities R, based on the relationship (see Fig. 7) of the calculated mass-attenuation coeﬃcient at 2.956 keV versus the measured R ratio: ð2:956 keVÞ ¼ g0 þ g1 R þ g2 R2

ð24Þ

where g0 ; g1 ; and g2 are constant coeﬃcients derived by means of a least-squares ﬁt based on the experimental results with standard samples. To improve the accuracy of the method, the mass absorption coeﬃcient of the low-Z matrix at 2.956 keV, mlow-Z (2.956 keV), must be calculated from the measured ratio of coherent to incoherent scatter intensities, corrected for the high-Z elements contribution using their characteristic x-ray intensities. Through this low-Z matrix contribution at 2.956 keV, the total low-Z absorption curve is calculated (in full analogy to the low-Z matrix contribution at the PdL

420

Markowicz and Van Grieken

Figure 7 Calculated mass absorption coefficient at 2.956 keV versus measured coherent=incoherent scattered intensity ratios for pure elements and compounds (circles) and a graphite–sulfur mixture (crosses). (From Van Dyck and Van Grieken, 1980. Reprinted with permission from Analytical Chemistry. Copyright American Chemical Society.)

energy in Fig. 5). Finally, the mass absorption coeﬃcients for the diﬀerent x-ray energies E are calculated by adding the low-Z and high-Z absorption contributions [see the total absorption cure, Eq. (10) and Fig. 5]. The proposed procedure for the automatic determination of the mass-attenuation coeﬃcient, based on the coherent=incoherent scatter ratio, has several obvious merits compared with the emission–transmission absorption correction method. First, there is no supplementary measurement or work needed [apart from experimentally obtaining the semiempirical dependence of the (2.956 keV) on the R ratio (Fig. 7) and sensitivity factors Bi , Eq. (2), on the basis of thin standard samples], because the additional information on a sample composition is present in the spectrum itself. Second, because the energies of coherently and incoherently scattered primary radiation, from which the information is extracted, are higher than that of the incident radiation used in transmission experiments, secondary eﬀects (i.e., grain size eﬀects, inhomogeneous thickness of the sample, and irregular sample surface) are less important. A third positive point

Quantification of Intermediate-Thickness Samples

421

of the described procedure (Van Dyck and Van Grieken, 1980) is the independence, within certain limits, of the analytical results on the mass of the specimen. Also, it is worth emphasizing the capability of determining reasonably accurate mass absorption coeﬃcients when the mean atomic number of the sample varies drastically. Most of the existing correction procedures in XRF analysis of intermediate-thickness samples ignore the enhancement eﬀect. It appears, however, that for some special cases, the enhancement eﬀect should be taken into account. Van Dyck et al. (1986) derived theoretical formulas for secondary ﬂuorescent x-ray intensities in medium-thickness samples based on the Sherman equations. Their computer routine for enhancement corrections was incorporated into an overall program for evaluation of x-ray spectra and calculation of x-ray absorption correction factors from scatter peaks. The methods, based on coherent and incoherent scatter radiation, developed by Van Dyck and Van Grieken (1980) for overall matrix eﬀect corrections in the case of monochromatic excitation and by Arau´jo et al. (1990) for eﬀective sample mass determination in case of polychromatic radiation have later been expanded by He and Van Espen (1991) into a general and versatile procedure for quantitative EDXRF with polychromatic excitation. The comprehensive method uses the scattered radiation to estimate the composition and mass of the low-Z element matrix (with optimal use of the fundamental parameter approach) and also uses the characteristic x-ray peaks to estimate the concentrations of the higher-Z elements. An iterative process is then executed. Using the initially estimated sample mass and composition, the weight fraction of each element is calculated until the calculated composition converges. The absorption and enhancement corrections are calculated from the latest iteration. Better sample mass, low-Z matrix composition, and analyte concentrations are computed after this. This overall procedure is repeated again until subsequent overall iterations do not yield signiﬁcant diﬀerences. The method is, in principle, applicable to samples of any thickness and composition and to polychromatic excitation. The procedure has been implemented as part of the popular software package AXIL-QXAS (Van Espen et al., 1997), which includes spectrum acquisition and spectrum analysis and runs on a PC. Excellent quantitative results have been obtained with it. Wegrzynek et al. (1993) have recently developed a direct correction procedure for the enhancement eﬀect in intermediate thickness samples, which is based on the results of the emission–transmission measurements and does not require any iterative calculations. The enhancement term ENHi is calculated for the intermediate-thickness samples from (Wegrzynek et al., 1993) ENHi ¼

1 ½ðD þ 1ÞSUM1 þ DðSUM2 þ SUM3Þ 2i ðE0 Þ

ð25Þ

with 1 ðE0 Þ ðEi Þ D ¼ exp þ m 1 sin 1 sin 2 SUM1 ¼

X

j;q

sin 1 ðE0 Þ=ðEq Þ ln þ 1 sin 1 ðE0 Þ sin 2 ðEi Þ=ðEq Þ þ ln þ 1 ðEi Þ sin 2

ð25aÞ

i ðEq ÞKjq ðE0 ÞWj

ð25bÞ

422

Markowicz and Van Grieken

SUM2 ¼

X

j;q

SUM3 ¼

X j;q

(

sin 1 ðE0 Þ=ðEq Þ ln 1 sin 1 ðE0 Þ sin 2 ðEi Þ=ðEq Þ þ ln 1 sin 2 ðEi Þ

i ðEq ÞKjq ðE0 ÞWj

ð25cÞ

i ðEq ÞKjq ðE0 ÞWj

sin 1 sin 2 ðE0 Þm þ exp sin 1 ðE0 Þ ðEi Þ

ðEi Þm ðE0 Þ ðEi Þ þ exp þ Ei ðEq Þm exp m sin 2 sin 1 sin 2

sin 1 ðE0 Þ sin 2 ðEi Þ Ei ðEq Þþ Ei ðEq Þþ m þ m sin 1 sin 2 ðE0 Þ ðEi Þ

sin 1 ðE0 Þ sin 2 ðEi Þ Ei ðEq Þ Ei ðEq Þ m m sin 1 sin 2 ðE0 Þ ðEi Þ

ð25dÞ Zx EiðxÞ ¼ 1

expðtÞ dt ¼ þ ln jxj þ t

1 X n¼1

n

ðxÞ Exponential integral n n!

¼ 0:577215664 . . . Euler constant where i ðEq Þ is the photoelectric mass absorption coeﬃcient for the ith element at the energy of the qth characteristic line of the jth enhancing element (Eq ), ðEq Þ is the total mass absorption coeﬃcient for the sample at the energy Eq , ! 1 Kjq ðE0 Þ ¼ j ðE0 Þ 1 j !j pqj Jq Jqj and pqj are the jump ratio for the relevant absorption edge and probability of the emission of the qth characteristic x-ray of the jth element, respectively. The calculation of the enhancement eﬀect correction is included in a complete procedure based on the emission–transmission measurements (Holynska et al., 1994). In the ﬁrst step, the calibration of the XRF system is performed [to ﬁnd the values of GI0 ðE0 Þ"ðEi Þ], and sin 1 and sin 2 are determined from the x-ray measurements for single-element standard samples of known mass per unit area. In the next step, the concentrations of the elements not enhanced by any other element present in the sample are calculated (with the absorption eﬀect correction based on the emission–transmission measurements). In the ﬁnal step, the correction for the enhancement eﬀect is made; the values of the fundamental parameters are taken from the published tables, whereas the values of ðE0 Þ; ðEi Þ, and ðEq Þ are derived from the results of the emission–transmission measurements. The enhancement factor for the intermediate thickness samples, ENHi, can also be assessed by using another simple formula (Markowicz et al., 1992b; Austrian Matrix Correction Routine, 1990): X ENHi ¼ ðENHi Þthick Wj ½1 expfðEj Þmg ð26Þ j

Quantification of Intermediate-Thickness Samples

423

with (ENHi)thick is the enhancement factor for the thick sample of the same composition as the intermediate thickness sample to be analyzed: ðENHi Þthick ¼

1 1 j ðE0 Þ 1 i ðEj Þ !j p j 2 Jj i ðE0 Þ ðE0 Þ= sin 1 ln 1 þ ðsin 1 Þ½ðE0 Þ1 ðEj Þ ðEi Þ= sin 2 ðsin 2 Þ½ðEi Þ1 þ ln 1 þ ðEj Þ

ð26aÞ

where pj is the transition probability for a given spectral line in an x-ray series of the enhancing jth element. Also in this case, the emission–transmission measurements can be used to support the enhancement eﬀect correction. The enhancement factor is calculated either for each spectral line of the enhancing element separately or for a weighted average energy of the whole x-ray series. The diﬀerences in the results obtained by using the two approaches are practically negligible (Markowicz et al., 1992b). Equation (26), although very simple, can also be applied in order to (1) deﬁne the region of sample thickness where the enhancement eﬀect is negligible (or smaller than a certain predeﬁned level) and (2) to determine the minimum sample thickness for which the bulk enhancement correction can already be used.

IV.

QUANTITATION FOR INTERMEDIATE-THICKNESS GRANULAR SPECIMENS

The fundamentals of the quantitative XRF analysis of granular specimens involve typical problems of radiation physics (i.e., the interaction of photons with a specimen of ﬁnite size). A.

Particle Size Effects in XRF Analysis of Thin and Intermediate-Thickness Specimens

The analysis of granular materials by XRF, when traditional methods of fusing or grinding cannot be used, requires careful consideration of the so-called particle size eﬀects. These eﬀects exist in XRF analysis of any granular materials, irrespective of the mass per unit area of the specimen, and may constitute a major source of error in quantitative analysis. The size of particles aﬀects not only the intensity of characteristics x-rays but the intensity of both backscattered and transmitted x-rays and low-energy g-radiation as well (Van Dyck et al., 1980; Berry et al., 1969). Diﬀerent models have been proposed to account for the inﬂuence of particle size on the characteristic x-ray intensity leaving the sample (Berry et al., 1969; Claisse and Samson, 1962; Lubecki et al., 1968; Hunter and Rhodes, 1972; Rhodes and Hunter, 1972; Hawthorne and Gardner, 1978; Krasnopolskaya and Volkov, 1986). Many of these models involve relatively complex calculations, particularly when the particle size eﬀects for thick granular samples must be evaluated. This section is limited to the particle size eﬀects in the quantitative XRF analysis of thin and intermediate-thickness (monolayer) samples. For granular intermediate-thickness samples, the mass per unit area of the element, i; mi , is given by (Rhodes and Hunter, 1972)

424

Markowicz and Van Grieken

mi ¼

Ii ðEi Þ Bi Fi

ð27Þ

where Bi is the sensitivity factor as in Eq. (1) and Fi is the heterogeneity factor which for a certain discrete particle size is deﬁned by Fi ¼

1 exp½ðf ðE0 Þ csc ðf ðE0 Þ csc

1

þ

1 þ f ðEi Þ csc 2 Þar f ðEi Þ csc 2 Þar

ð28Þ

where f ðE0 Þ and f ðEi Þ are the linear attenuation coeﬃcients (cm1) for primary and ﬂuorescent radiation in ﬂuorescent particles, respectively, and ar is the radiometric particle diameter. The radiometric diameter, introduced by Claisse and Samson (1962), represents the mean geometrical path of x-rays through one particle. These and many other authors (Berry et al., 1969; Lubecki et al., 1968; Hunter and Rhodes, 1972; Rhodes and Hunter, 1972) have taken ar as simply equal to the volume of the grain divided by the particle area presented to the radiation, averaged over all possible orientations of the grain. Hence, for spherical particles, the radiometric diameter is equal to 0.67a (with a ¼ geometric diameter). One can easily visualize, however, that such an approach assumes equal weighting of the contribution from all possible radiative paths to the average. In view of the diﬀerent absorption eﬀects themselves, this is untrue. Markowicz et al. (1980) introduced, instead of the commonly used radiometric diameter approach, the concept of an eﬀective absorption-weighted radiometric diameter for ﬂuorescent radiation, depending on both the geometry and absorption eﬀects, and provided a comparison of these two approaches for single spherical particles for two excitation-detection geometries ( and /2). This alternative approach allows quantitative evaluation of the discrepancies resulting from the concept ar ¼ particle volume=average area. It was concluded that for the geometry, both approaches give practically the same results (maximum relative diﬀerences amount only to 5%), but for /2 geometry, the radiometric diameter approach can safely be applied only for very small particles and=or at relatively high energies of primary radiation (Markowicz et al., 1980). For samples with a certain particle size distribution described by a function fðar Þ, the heterogeneity factor Fi can be calculated according to the formula (Rhodes and Hunter, 1972) aZr max

Fi ¼

fðar Þð1 exp½ar ðf ðE0 Þ csc

ar min

ðf ðE0 Þ csc

1

þ

1

þ f ðEi Þ csc

f ðEi Þ csc

2 Þar

2 ÞÞ

dar

ð29Þ

where ar min and ar max are the smallest and the largest radiometric particle diameters, respectively, and fðar Þ ¼

dVf ðVf Þt dar

ð30Þ

where dVf is the volume of ﬂuorescent particles having a size between ar and ðar þ dar Þ and ðVf Þt is the total volume of the ﬂuorescent particles. The theoretical predictions given by Eqs. (28) and (29) have been compared with the experimental results for the heterogeneity factors obtained for samples with some discrete particle sizes and various particle size distributions (Holynska and Markowicz, 1981). For granular samples of copper sulﬁde, a satisfactory agreement has been obtained; for samples of iron oxide, some discrepancies due to agglomeration of the particles have been observed.

Quantification of Intermediate-Thickness Samples

425

An in-depth study of the inﬂuence of sample thickness, excitation energy, and excitation-detection geometry on the particle size eﬀects in XRF analysis of intermediatethickness samples was carried out and the results are presented in the work of Van Dyck et al. (1985). B.

Correction Methods for the Particle Size Effect in XRF Analysis of Intermediate-Thickness Specimens

As already mentioned, the heterogeneity factor Fi describing the magnitude of the particle size eﬀect in XRF analysis of thin and monolayer samples can simply be calculated if the particle size or the function of the particle size distribution is known. This occurs in XRF analysis of air particulates (or aerosols), for example, when special sampling techniques, involving a cascade impactor, are applied. At diﬀerent stages of the cascade impactor, the particles of deﬁnite sizes are collected (Katz, 1977) and calculation of the heterogeneity factor Fi may be straightforward if, of course, the kind of ﬂuorescent particles is known. A simple empirical particle size correction factor ð1 þ baÞ2 was proposed by Criss (1976), in which a is the particle diameter b is a coeﬃcient that depends on particle composition and experimental conditions. The author has provided a table of the values for b for the determination of 48 diﬀerent elements in 200 compounds using either a Cr- or W-target x-ray tube. Another correction for the particle size eﬀect based on the model of Berry et al. (1969), also requiring evaluation of the particle size in a sample, is due to Nielson (1977). All the particle size corrections mentioned are of limited applicability because evaluation of a particle size or particle size distribution function must be done before XRF analysis. Moreover, in many cases it may be necessary to consider how the indirectly determined particle sizes relate to true sizes. However, even when there is some uncertainty in the sizes and compositions of the particles, which are input parameters in the particle size correction, it is better to make an appropriate correction than no correction at all. 1. Empirical Particle Size Correction Method Using Dual Measurements To overcome the problems encountered when calculated particle size correction factors are applied in the quantitative XRF analysis of granular samples, a particle size correction method based on dual measurements of characteristic x-rays excited by x- or g-radiation of two diﬀerent energies was developed (Holynska and Markowicz, 1982). The method utilizes the diﬀerence in the particle size eﬀect for two excitation energies and oﬀers the possibility of experimental detection and correction of this eﬀect. In general, these two excitation energies should be chosen so that the eﬀect of particle size is small for one of them and large for the other. Thus, the measured intensities of the characteristic x-rays of the element to be determined are diﬀerent functions of the particle size. The ratio of these two intensities is sensitive to the particle size and it can be used for obtaining the particle size correction factor Ki ¼ 1=Fi . First, a calibration curve giving the relationship between the correction factor Ki2 and the ratio ðIi1 =Ii2 Þrel must be plotted. The ratio ðIi1 =Ii2 Þrel is given by Ii1 Ii1 =Ii2 ¼ ð31Þ Ii2 rel ðIi1 =Ii2 Þhom where Ii1 =Ii2 is the ratio of the intensities of the characteristic x-rays of the ith element excited in a granular sample by primary x-rays of two diﬀerent energies (indexes 1 and 2, respectively) and ðIi1 =Ii2 Þhom is the same for a thin homogeneous sample. Taking into

426

Markowicz and Van Grieken

account that in the thin-sample technique, interelement eﬀects may be neglected, singleelement standard samples can be used to obtain the ðIi1 =Ii2 Þhom ratio. The calibration curve mentioned earlier can be obtained either theoretically or experimentally with the use of calibration samples of known discrete particle size fractions. It has been shown (Holynska and Markowicz, 1982) that there is also a possibility of applying such a calibration curve for granular samples with various particle size distribution functions. For a p geometry, the calibration curves for the determination of the correction factor Ki2 are described by the equation (Markowicz, 1983) Ii1 t0 ar Ki2 ¼ ð32Þ Ii2 rel 1 et0 ar where t0 ¼

f ðE01 Þ þ f ðEi Þ f ðE02 Þ þ f ðEi Þ

¼ f ðE02 Þ þ f ðEi Þ

ð33Þ ð34Þ

f ðE01 Þ and f ðE02 Þ are the linear absorption coeﬃcients for primary radiation of two diﬀerent energies (indexes 1 and 2, respectively) in ﬂuorescent particles (cm1). The particle size correction factor Ki2 is given by ar ð35Þ Ki2 ¼ 1 ear The proposed method of particle size correction was veriﬁed experimentally for the determination of copper, applying 238Pu and 241Am radioisotopes as sources of primary radiation, and a satisfactory agreement between theoretical predictions and experimental results was reported (Holynska and Markowicz, 1982). 2. Applicability of the Particle Size Correction Method Accurate determination of the correction factor Ki2 is mainly aﬀected by ﬂuctuations resulting from counting statistics in all measured intensities of the characteristic radiation. The absolute error Ki2 in determining the correction factor Ki2 can be calculated from the formula Ki2 ¼

t0 ar Sc 1 et0 ar

ð36Þ

where Sc is standard deviation resulting from counting statistics for the ratio ðIi1 =Ii2 Þrel . The relative error in determining the particle size correction factor Ki2 is given by Ki2 Sc ¼ ¼ Sr Ki2 ðIi1 =Ii2 Þrel

ð37Þ

where Sr is relative standard deviation resulting from counting statistics for the ratio ðIi1 =Ii2 Þrel . Equations (32) and (37) enable us to estimate the applicability of the particle size correction method. This can be done with the aid of Figure 8. From two families of curves in Figure 8, one can estimate the maximum value of the particle size correction factor ðKi2 Þmax that can be determined with the particle size correction method for given values of Sr and t0 . Figure 9 presents the theoretical relationship of the maximum value of the correction factor ðKi2 Þmax with the parameter t0 for diﬀerent values of the relative standard

Quantification of Intermediate-Thickness Samples

427

Figure 8 Theoretical relationship of the particle size correction factor Ki2 with the ratio of the intensities ðIi1 =Ii2 Þrel for different values of the parameter t0 (family curves) and with the error Ki2 for different values of the relative standard deviation Sr (family of straight lines). (From Markowicz, 1983. Reprinted by permission of John Wiley & Sons, Ltd.)

deviation Sr ; Figure 10 shows the theoretical relationship of ðKi2 Þmax with Sr for diﬀerent values of t0 . The families of curves shown in Figures 9 and 10 allow us to determine the application limits of the particle size correction method [i.e., to determine ðKi2 Þmax in various conﬁgurations]. In practice, however, it is more interesting to know the maximum value of the radiometric particle diameter for which the particle size eﬀect can still be corrected. This can be determined from Eq. (32). 3. Selection of the Optimum Measurement Conditions When the ith element is present in a given chemical compound and the maximum values of the radiometric diameter of the particle in a sample are known, it is possible to deﬁne the optimum value t0opt of the parameter t0 for a given energy of exciting radiation E02 .

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Figure 9 Theoretical relationship of the maximum value of the particle size correction factor 0 is the ðKi2 Þmax with the parameter t0 for different values of the relative standard deviation Sr : topt 0 optimum value parameter t (see text for details). (From Markowicz, 1983. Reprinted by permission of John Wiley & Sons, Ltd.)

The value of t0opt can be determined from Figure 9 for a given value of the relative standard deviation Sr . Thus, one can obtain the following inequality, which should be fulﬁlled by the parameter t0 : t0 t0opt

ð38Þ

This means that the value of the energy of the exciting radiation E01 of the ‘‘correction’’ source should fulﬁll the inequality E01 ðE01 Þopt

ð39Þ

where ðE01 Þopt is the energy of the exciting radiation E01 for which t0 ¼ t0opt . Taking into account that the eﬃciency of photoexcitation depends on the energy of the primary

Quantification of Intermediate-Thickness Samples

429

Figure 10 Theoretical relationship of the maximum value of the particle size correction factor ðKi2 Þmax with the relative standard deviation Sr for different values of the parameter t0 : ðSr Þ0 is the maximum acceptable value of the relative standard deviation Sr (see text for details). (From Markowicz, 1983. Reprinted by permission of John Wiley & Sons, Ltd.)

radiation, the practical conclusion can be drawn that the values of both the parameter t0 and the energy of the exciting radiation E01 should be as close as possible to the values of t0opt and ðE01 Þopt , respectively. On the other hand, for a given pair of sources of primary radiation (i.e., for a given value of the parameter t0 ), it is possible to estimate, using Figure 10, the maximum acceptable value ðSr Þ0 of the relative standard deviation of the ratio ðIi1 =Ii2 Þrel . In consequence, the appropriate measurement times and=or activities of the sources of primary radiation can be selected. Although the correction method may look complicated, it is currently the only real correction procedure dealing with the particle size eﬀect in XRF analysis of granular intermediate-thickness specimens. The idea of applying diﬀerent excitation x-ray energies to estimate particle size corrections was also exploited by Nielson and Rogers (1986).

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REFERENCES Andermann G, Kemp JW. Anal Chem 30:1306, 1958. Arau´jo MF, Van Espen P, Van Grieken R. X-ray Specrom 19:29, 1990. Austrian Matrix Correction Routine, Description of the Program, Technical University of Graz, Austria, 1990. Bazan F, Bonner NA. Adv X-ray Anal 19:381, 1976. Berry PF, Furuta T, Rhodes JR. Adv X-ray Anal 12:612, 1969. Claisse F, Samson C. Adv X-ray Anal 5:335, 1962. Criss JW. Anal Chem 48:179, 1976. Giauque RD, Garrett RB, Goda LY. Anal Chem 51:511, 1979. Giauque RD, Goulding FS, Jaklevic JM, Pehl RH. Anal Chem 45:671, 1973. Hawthorne AR, Gardner RP. X-ray Spectrom 7:198, 1978. He F, Van Espen PJ. Anal Chem 63:2237, 1991. Holynska B, Markowicz A. X-ray Spectrom 8:2, 1979. Holynska B, Markowicz A. X-ray Spectrom 10:61, 1981. Holynska B, Markowicz A. X-ray Spectrom 11:117, 1982. Holynska B, Ptasinski J, Wegrzynek D. Appl Radiat Isot 45:409, 1994. Hunter CB, Rhodes JR. X-ray Spectrom 1:107, 1972. Kalman ZH, Heller L. Anal Chem 34:946, 1962. Katz M (ed), Methods of Air Sampling and Analysis. Washington, DC: American Public Health Association, 1977, p 592. Kieser R, Mulligan TJ. X-ray Spectrom 8:164, 1979. Kikkert J. Adv X-ray Anal 26:401, 1983. Krasnopolskaya NN, Volkov VF. X-ray Spectrom 15:3, 1986. Leonardo L, Saitta M. X-ray Spectrom 6:181, 1977. Leroux J, Mahmud M. Anal Chem 38:76, 1966. Livingstone LG. X-ray Spectrom 11:89, 1982. Lubecki A, Holynska B, Wasilewska M. Spectrochim Acta B23:465, 1968. Markowicz A. X-ray Spectrom 8:14, 1979. Markowicz A. X-ray Spectrom 12:134, 1983. Markowicz A. X-ray Spectrom 13:166, 1984. Markowicz A, Abdunnabi AA. X-ray Spectrom 20:97, 1991. Markowicz A, Haselberger N. Appl Radiat Isot 43:777, 1992. Markowicz A, Haselberger N, Mulenga P. X-ray Spectrom 21:271, 1992a. Markowicz A, Van Dyck P, Van Grieken R. X-ray Spectrom 9:52, 1980. Markowicz A, Haselberger N, El Hassam HS, Sewando MSA. J Radioanal Nucl Chem 158:409, 1992b. McMaster WH, Delgrande M, Mallett JH, Hubbell JM. University of California. Lawrence Radiation Laboratory Report, UCPL-50174, 1969. Meier H, Unger E. J Radioanal Chem 32:413, 1976. Nielson KK. Anal Chem 49:641, 1977. Nielson KK, Rogers VC. Adv X-ray Anal 27:449, 1984. Nielson KK, Rogers VC. Adv X-ray Anal 29:587, 1986. Nielson KK, Sanders RW. The SAP3 Computer Program for Quantitative Multielement Analysis by Energy-Dispersive X-ray Fluorescence, US DOE Report PNL-4173, 1982. Nielson KK, Sanders RW, Evans JC. Anal Chem 54:1782, 1982. Rhodes JR, Hunter CB. X-ray Spectrom 1:113, 1972. Sanders RW, Olsen KB, Weiner WC, Nielson KK. Anal Chem 55:1911, 1983. Taylor DL, Andermann G. Anal Chem 43:712, 1971. Taylor DL, Andermann G. Appl Spectrosc 27:352, 1973. Van Dyck PM, Van Grieken RE. Anal Chem 52:1859, 1980. Van Dyck P, Markowicz A, Van Grieken R. X-ray Spectrom 9:70, 1980.

Quantification of Intermediate-Thickness Samples Van Dyck P, Markowicz A, Van Grieken R. X-ray Spectrom 14:183, 1985. Van Dyck PM, To¨ro¨k SB, Van Grieken RE. Anal Chem 58:1761, 1986. Van Espen P, Janssens K, Nobels J. J Chemom Intell Lab Syst 1:109, 1997. Van Espen P, Van’t dack L, Adams F, Van Grieken R. Anal Chem 51:961, 1979. Wegrzynek D, Holynska B, Pilarski T. X-ray Spectrom 22:80, 1993. Yap CT, Kump P, Tang SM, Wijesinghe L. Appl Spectrosc 41:80, 1987.

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7 Radioisotope-Excited X-ray Analysis Stanislaw Piorek* Niton Corporation, Billerica, Massachusetts

I.

INTRODUCTION

Radioisotope x-ray ﬂuorescence (XRF) and x-ray preferential absorption (XRA) techniques are used extensively for the analysis of materials, covering such diverse applications as analysis of alloys, coal, environmental samples, paper, waste materials, and metalliferous mineral ores and products (Rhodes, 1971; Rhodes and Rautala, 1983; Watt, 1978 and 1983; Watt and Steﬀner, 1985; Piorek 1997). Many of these analyses are undertaken in the harsh environment of industrial plants and in the ﬁeld. Some are continuous on-line analyses of material being processed in industry, where instantaneous analysis information is required for the control of rapidly changing processes. Radioisotope x-ray analysis systems are often tailored to a speciﬁc but limited range of applications. They are similar and often considerably less expensive than analysis systems based on x-ray tubes, but these attributes are often gained at the expense of ﬂexibility of use for a wide range of applications. Operators making analyses in the ﬁeld or in industrial plants are usually less skilled than those working in the laboratory with x-ray tube systems. Manufacturers of radioisotope x-ray analysis systems compensate for this by producing simple semiautomated or fully automated systems whose output, calibrated for the speciﬁc application, is given directly in terms of concentrations of elements required or in terms of a simple pass=failtype decision. Radioisotope x-ray techniques are preferred to x-ray tube techniques when simplicity, ruggedness, reliability, and cost of equipment are important, when minimum size, weight, and power consumption are necessary, when a very constant and predictable x-ray output is required, when the use of high-energy x-rays is advantageous, and when short x-ray path lengths are required to minimize the absorption of low-energy x-rays in air. Also of signiﬁcant analytical importance is the fact that the radioisotope excitation is usually monoenergetic (monochromatic) as opposed to polyenergetic (polychromatic) excitation characteristic for x-ray tubes.

*Previous aﬃliation: Metorex Inc., Princeton, New Jersey. 433

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X-ray ﬂuorescence techniques based on the x-ray tubeBragg crystal spectrometer are considerably more sensitive than those based on radioisotope sources. This high sensitivity is due to the excellent x-ray resolving power of the crystal spectrometer, which is superior to that of the gas-ﬁlled or solid-state detector typically used with radioisotopes. Radioisotopes cannot be used with crystal spectrometers because of the low geometrical eﬃciency of this spectrometer coupled with the fact that the x-ray photon output of radioisotope sources is relatively very low, about six orders of magnitude less than that of x-ray tubes used with crystal spectrometers. Presently, the bulk of use of XRF analysis is in portable and on-line equipment. Portable systems outnumber on-line installation, and the gap increases from year to year. A signiﬁcant number of radioisotope benchtop XRF systems have been installed for quality control applications, as oﬀ-line auxiliary instruments. These, however, are very quickly yielding the ﬁeld to XRF systems based on low-power x-ray tubes. For some applications, x-ray preferential absorption (XRA) and x-ray scattering (XRS) techniques are preferred to XRF techniques, particularly when coarse particulate material is to be analyzed. Radioisotopes are the only practical source of x-rays for these applications, because to penetrate deep into the material, high energy, usually above 100 keV, x-rays are required. The most important applications of XRA and XRS techniques are the on-line analysis of particulate material on conveyors. Some of the terminology used in this chapter is now brieﬂy deﬁned. The element whose concentration in the sample is to be determined is the analyte and the other elements of the sample are the matrix elements, or simply the matrix. Sensitivity of an analytical method for a given analyte is deﬁned as net change (increase) of the measured signal of the analyte per unit concentration change (increase) of that analyte. This term is notoriously, but wrongly, used in place of minimum detectable level. A minimum detectable level (DL) for a given analyte is understood as that amount of analyte in a sample that produces spectral signal equal to or greater than three standard deviations of a signal obtained on a sample with no analyte present. The DL improves when sensitivity of analysis improves. The common link between all techniques and applications discussed here is the dependence of the analysis primarily on the absorption of x- and g-rays. Compton and coherent scattering are the other important interactions taking place in the sample. The terms x-ray and g-ray can often be used interchangeably. The term ‘‘x-ray’’ is always used when discussing ﬂuorescent x-rays. Radioisotope sources emit either g-rays directly from the nucleus or ﬂuorescent x-rays emitted following the ejection of an atomic electron. g-Rays emitted by radioisotopes usually have energies greater than 50 keV. X-ray ﬂuorescence analysis depends on both x-ray and g-ray excitation, but most XRA and XRS analyses are based on the use of g-rays. The term ‘‘high-energy g-ray’’ is used when the g-ray interaction in the sample is essentially entirely due to Compton scattering (typically above 300 keV) which is being used to determine either the bulk density or mass per unit area of the sample. The term ‘‘low-energy g-ray’’ is used when photoelectric interactions are important to the analysis. This chapter reviews radioisotope-excited x-ray ﬂuorescence, preferential absorption, and scattering techniques. The characteristics of radioisotope sources and x-ray detectors are described, and then the x-ray analytical techniques are presented. The choice of radioisotope technique for a speciﬁc application is discussed along with major factors aﬀecting the overall accuracy of analysis. This is followed by a summary of applications of these techniques, with a more detailed account given of some of the most representative applications, particularly those of considerable industrial importance.

Radioisotope-Excited X-ray Analysis

II.

435

BASIC EQUATIONS

The basic equations for x-ray analysis are given in Chapter 1. Some additional equations used for XRF, XRA, and XRS analyses are presented here. The typical geometries (Watt and Steﬀner, 1985; Jenkins et al., 1981 b) of the radioisotope source, sample, and detector used are shown in Figures 1 and 2.

Figure 1 The three geometries for radioisotope-excited x-ray fluorescence analysis: (a) annular source, (b) central source, and (c) side source. (From Jenkins et al., 1981b.)

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Figure 2 Arrangement of radioisotope g-ray source, collimators, and scintillation detector used in x-ray preferential absorption analysis. (From Watt and Steffner, 1985.)

A.

Absorption of X-rays

The intensity, I, of a narrow beam of monoenergetic x- or g-rays transmitted through a sample (already shown in Chapter 1) is given by I ¼ I0 emrt

ð1Þ

where m¼

n X i¼1

ðWi mi Þ

ð2Þ

Radioisotope-Excited X-ray Analysis

437

and n X

Wi ¼ 1

ð3Þ

i¼1

I0 is the intensity of x-rays detected without the sample. m and t are the mass absorption coeﬃcient and path length of x-rays in the sample, respectively, r is the bulk density of the sample, and mi and Wi are the mass absorption coeﬃcient and weight fraction of the ith element in the sample, respectively. Equation (1) also holds for broad beams of x-rays when the cross section for photoelectric absorption is much greater than that for Compton and coherent scattering [i.e., particularly for low-energy x-rays and high-atomic-number (Z) elements]. This assumption can be veriﬁed by reference to Appendices VIVIII of Chapter 1. B.

Fluorescent X-ray Intensity

When a monoenergetic beam of x-rays excites the K shell x-rays of the analyte i in an inﬁnite-thickness sample and both the incident and emitted x-rays are normal to the sample surface, the detected intensity Ii of the Ka x-rays of analyte i is given approximately by Eqs. (94) and (90) in Chapter 1, namely Ii ¼

GeðEi Þai ðE0 ÞI0 ðE0 Þ mðE0 Þ þ mðEi Þ

ð4Þ

where G eðEi Þ ai ðE0 Þ I0 ðE0 Þ mðE0 Þ; mðEi Þ t0i ðE0 Þ oi ri ji

Geometrical constant Intrinsic eﬃciency of the detector to the x-rays of the analyte i Wi t0i ðE0 Þoi ri ð1 1=ji Þ The source emission (photons=s) Mass absorption coeﬃcients for the exciting radiation with energy E0 and the characteristic radiation with energy Ei , respectively, in the sample (cm2/g) Total photoelectric mass absorption coeﬃcient for the ith element at energy E0 (cm2/g) The K shell ﬂuorescent yield for the analyte i Relative transition probability for Ka lines of analyte i Jump ratio

Enhancement eﬀects (discussed in Chapter 5, Sec. II.B.2) have been assumed to be negligible. The intensities of L and M shell ﬂuorescent x-rays can be calculated from equations similar to Eq. (4). For radioisotopes emitting x-rays of more than one energy, Ii can be separately calculated for each emitted energy and the total ﬂuorescent x-ray intensity determined by summing the products of Ii and the probability of emission of the given x-ray energy from the radioisotope. C.

Scattered X-ray Intensities

X-rays are scattered from the sample and its surroundings to the detector by a mechanism of coherent and Compton (or incoherent)-scattering interactions. There is no loss of

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Piorek

energy of incident radiation during coherent scattering. The energy E of the Comptonscattered x-ray is given by [Eq. (52) of Chapter 1] E0 ð5Þ 1 þ gð1 cos yÞ where E0 is the energy (in the units of keV) of the incident x- or g-ray, g ¼ E0 =511, and y is the scattering angle. The scattering angle y, measured from the direction of the incident x-ray, in most radioisotope XRF systems (Fig. 1) is in the range 90 150 . The loss in energy of the incident x-ray due to Compton scattering at 90 , 120 , and 150 is shown in Figure 3 and can be seen to be relatively very small at energies below 20 keV. The detected intensity Is of x-rays scattered from an inﬁnitely thick sample to the detector is given by P GI0 ðE0 ÞTs es ðmsi ðyÞWi Þ P ð6Þ Is ¼ ½ðmi þ msi ÞWi E¼

where G; I0 ðE0 Þ, and Wi are the same as in Eq. (4), Ts is the transmission of the scattered x-rays through the ﬁlter and the detector window; es is the eﬃciency of the detector for the scattered x-rays; msi ðyÞ is the diﬀerential scattering cross section for the x-rays scattered by the ith element toward the detector, and msi is the mass absorption coeﬃcient of the scattered x-rays for the ith element of the sample. Equation (6) holds for both coherent and Compton scattering when the appropriate scattering cross section [Eqs. (73) and (65) of Chapter 1] is used. It assumes that the photoelectric absorption cross section of the x-rays in the sample is much greater than that for scattering and that incident and emergent x-rays are normal to the sample surface. The diﬀerential and total Compton-scattering cross sections per atom are proportional to Z=A, where A is the atomic weight of the atom, and, except for hydrogen, are almost independent of the atomic number of the atom. Hence, the scattered intensity is approximately inversely proportional to the sum of the mass absorption coeﬃcients in the sample of the incident and emerging x-rays.

Figure 3

Loss in energy of x-rays in Compton scattering at angles 90 , 120 , and 150 .

Radioisotope-Excited X-ray Analysis

439

The cross section for coherent scattering is highest for small scattering angles, lowenergy x-rays, and high-atomic-number atoms. For angles greater than 90 , the cross section is low and varies by only a factor of about 2. D.

X-ray Fluorescence Analysis

The concentration of the analyte in the sample is determined from measurement of the intensity of its ﬂuorescent x-rays, often combined with measurements of the intensities of the ﬂuorescent x-rays of matrix elements and=or the Compton-scattered x-rays. Often the denominator of Eq. (4) is proportional to that of Eq. (6); that is, X mðE0 Þ þ mðEi Þ / ½ðmi þ msi ÞWi ð7Þ Hence, Eqs. (4) and (6) can be combined to give Wi ﬃ k

Ii ICom

ð8Þ

where k is a constant. Note that the intensity of scattered radiation, Is , is replaced here by ICom , because Eq. (6) holds for both coherent and Compton scattering when the appropriate scattering cross section is used. If a major matrix constituent, a, has an absorption edge energy between the energy of the characteristic x-rays of the analyte and the energy of the incoherently (Compton) scattered primary radiation, then 1 þ k1 ma Wa Wi ﬃ kIi ð9Þ ICom where k1 is a constant and the subscript a refers to the major matrix constituent. Equations (8) and (9) can be checked for accuracy for any speciﬁc application by substituting mass absorption coeﬃcients (see Appendices VIIIX of Chapter 1) and elemental concentrations into Eqs. (4) and (6). Relation (8) has a great analytical signiﬁcance; in many applications, it linearizes the calibration curve for a given analyte, thus making analysis more accurate and robust. Analysis of Eqs. (4) and (3) reveals that the intensity of the characteristic x-rays of a given analyte is a function not only of this analyte concentration but also of the concentrations of all the other elements in the sample. This poses diﬃculty in solving this equation directly for Wi . However, by adopting, for example, a simple approximation proposed by Lucas-Tooth and Price (L-TP) (1962), this problem can be solved. The L-TP approximation states that because the x-ray intensities of elements are functions of their respective concentrations, one can substitute their measured x-ray intensities for concentrations of matrix elements. In this way, Eq. (4), via Eq. (3), can be solved for Wi , which is now expressed, for each analyte, by all measured x-ray intensities of the elements in the sample: ! j¼n X kij Ij W i ¼ b1 þ I i k 0 þ ð10Þ j6¼n

The I ’s are the x-ray intensities of the elements measured, kij are correction coeﬃcients determined by multivariable linear least-squares ﬁtting, and Wi is concentration of the analyte i. There are as many of this equations as there are analytes to be measured. The L-TP model allows for the calibration of an x-ray analyzer with the suite of calibration samples without the necessity of knowing assays for any element but the

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Piorek

desired analyte. Due to its simplicity and ruggedness, this approach is often used in the calibration of benchtop and portable x-ray analyzers; the analyzer measures x-ray intensities of the important interfering elements in calibration samples and then develops a calibration equation for the analyte(s) by multivariable, linear least-squares ﬁtting. E.

X-ray Preferential Absorption Analysis

X-ray preferential absorption analysis is based on the measurement of the intensities of two or more monoenergetic x-rays transmitted through the sample (Fig. 2). Sensitivity of analysis depends on the selective absorption of the transmitted x-rays by the analyte compared with absorption by the sample matrix. The greater the diﬀerence (or contrast) between the mass absorption coeﬃcients of the analyte and matrix for the transmitted x-rays, the better the sensitivity of analysis for that analyte. From Eqs. (1)(3), the concentration of the analyte is given by Wi ¼

lnðI0 =IÞ=rt mM mi mM

ð11Þ

where mi is the mass absorption coeﬃcient of the x-rays in the analyte and mM is the absorption coeﬃcient of the matrix, given by mM ¼

n X

ðmj Wj Þ

ð12Þ

j6¼i

and the subscript j refers to the matrix elements and S Wj ¼ 1 Wi . The concentration of the analyte can thus be determined if the product of the bulk density and thickness of the sample is known and the mass absorption coeﬃcients of the matrix elements are approximately equal or the composition of the matrix does not vary. In practice, XRA analysis usually involves measurements of transmission of narrow beams of x-ray, at two x-ray energies, through the sample (Watt and Steﬀner, 1985). This is called dual-energy (x- or g-ray) transmission (DUET) analysis. The beams are usually coincident and thus not diﬀerently aﬀected by heterogeneity of the sample in the beam path (Fig. 2). From Eq. (11), the concentration Wi of the desired analyte is Wi ¼

ðm0i

ðm0M Rm00M Þ m0M Þ Rðm00i m00M Þ

ð13Þ

where R¼

lnðI0 =IÞ0 lnðI0 =IÞ00

ð14Þ

R¼

m0M þ ðm0i m0M ÞWi m0 ¼ m00M þ ðm00i m00M ÞWi m00

ð15Þ

and

The prime and double prime refer to the ﬁrst and second x-ray energies, respectively. The concentration is thus determined independently of the density and thickness of the sample through which the coincident x-ray beams pass.

Radioisotope-Excited X-ray Analysis

441

The sensitivity of analysis is high when m0i m0M and when rt is large. The analysis is accurate when the ratio m0M =m00M is constant, independently of variations in composition of the sample matrix. This ratio is approximately constant when the x-ray energies of the transmitted x-ray are just above and below the K shell absorption edge energy of the analyte and at higher energies when, at each x-ray energy, the mass absorption coeﬃcients of all matrix elements are about the same. In the latter case, the energy of the higherenergy x-ray is usually chosen so that the mass absorption coeﬃcients of the analyte and matrix elements are the same, the transmission measurement thus determining the mass per unit area of sample in the x-ray beam. The uncertainty in determination of R [Eq. (14)] caused by counting statistics is sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 R ðdI=IÞ0 ðdI=IÞ00 ð16Þ þ dR ¼ rt m0 m00 where 0 2 0 2 0 2 dI dI dI ¼ 00 þ 0 I I0 I

ð17Þ

and 00 2 00 2 00 2 dI dI0 dI ¼ þ 00 00 I I0 I

ð18Þ

and where dI=I is the relative counting statistical uncertainty and m is the mass absorption coeﬃcient of the x-ray in the sample. The corresponding uncertainty in determining the concentration of the analyte can be found by substituting R þ dR for R in Eq. (13). The uncertainty in the determination of the concentration of the analyte, caused by an increase in the concentration (dC) of one matrix element k replacing another matrix element l, can be calculated by increasing the mass absorption coeﬃcient of the sample matrix by mM ½new ¼ mM ½old þ ðmk ml ÞdC

ð19Þ

and substituting the new mass absorption coeﬃcient into Eq. (13). These equations accurately predict all aspects of XRA analysis (Watt and Steﬀner, 1985) except when the sample is so highly heterogeneous that within the beam of x-rays, there are signiﬁcant diﬀerences in absorption of the x-rays.

F.

X-ray Scattering Analysis

Of the two x-ray scattering methods of analysis, one relies on comparison of the detected intensities of the Compton-scattered and coherent scattered x-ray (Schatzler, 1979), whereas the other one is based on determination of the intensity of the Compton-scattered x-rays (Fookes et al., 1975). The former method is essentially a measure of the ratio of the diﬀerential scattering cross sections of the two components, which is proportional to between Z and Z2 (Siegbahn, 1965). The latter methods depends on the absorption of x-rays in the sample, which, in the photoelectric region, is proportional to between Z4=A and Z5=A (Siegbahn, 1965). Thus, the method is very similar to XRA analysis and is considerably more sensitive than the Compton-scattered to coherent scattered ratio x-ray method. Both methods are accurate only when the changes in detected x-ray intensities

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caused by changes in the concentration of the analyte are much greater than those caused by changes in the concentration of the matrix elements. The sensitivity of both techniques and uncertainties due to variations in concentrations of matrix constituents can be predicted using Eq. (6), where the photoelectric absorption cross section in the sample is much greater than the scattering cross section. Dual-energy scattering techniques (Outokumpu Mintec, 1986), analogous to dualenergy preferential absorption techniques, are used to minimize the eﬀects of sample heterogeneity. The x-ray scattering techniques are used in applications in which only one side of the sample is accessible and thickness of the sample is too great to allow suﬃcient penetration of x-rays. Compared with DUET analysis, the main disadvantage is that narrow beams of x-rays cannot often be used because of the lower geometrical eﬃciency of the source, sample, and detector. Hence, multiple scattered x-rays are detected with a consequent loss in accuracy of analysis.

III.

RADIOISOTOPE X-RAY SOURCES AND DETECTORS

The characteristics of radioisotope x-ray sources and detectors are described here. A full understanding of the diﬀerent characteristics of scintillation, proportional, and solid-state detectors is essential because of the need to tailor their use to speciﬁc applications and to environmental conditions in the ﬁeld and in industrial plants. A.

Radioisotope Sources

There are only a few radioisotope sources that are used frequently for x-ray analysis; these are listed with their most important characteristics in Table 1. Also included are two radioisotopes that emit high-energy x-rays and they are used most frequently with the x-ray sources to correct for changes in sample mass per unit area, thickness, or bulk density. The activity of radioisotopes is speciﬁed in terms of the rate of disintegration of the radioactive atoms [i.e., decays per second, or becquerel (Bq)]. One becquerel (Bq), an SI unit, is deﬁned as one disintegration per second. The unit of becquerel replaces the non-SI unit, the curie (Ci), which equals 3.761010 becquerel. Unfortunately, the unit of becquerel is not very practical. For example, a typical, useful activity of 100 mCi has to be expressed in gigabecquerels (GBq). A practical conversion relation between the two units is 100 mCi ¼ 3:7 GBq The typical number of x- or g-rays emitted per disintegration by the given radioisotope is listed in Table 1 so that the essential parameter of the radioisotope source, the number of x-rays or g-rays emitted per second, can be calculated. The emission rate of radioisotope decreases with time according to the law of natural decay, the number of radioisotope atoms decaying from N0 to N after an elapsed time t being given by N ¼ N0 e0:693t=T1=2

ð20Þ

where T1=2 is the so-called half-life of the radioisotope. The source decays to half of its original emission rate during the time equal to its half-life. The radioisotope source is usually replaced after one to two half-lives. The physical size of radioisotope x-ray sources is small. Figure 4 shows the encapsulations of typical cylindrical and annular sources of 109Cd (Amersham, 1986). Cylindrical sources used in portable analyzers are usually 8-mm-diameter by 5-mm-thick

Radioisotope-Excited X-ray Analysis

443

Table 1 Properties of Radioisotope Sources Used for XRF, XRA, and XRS Analysis and Determination of Bulk Density r, Mass per Unit Area (rt), and Thickness t in X-ray Analysis Radioisotope 55

Fe Pu 244 Cm 109 Cd 238

125

I

241

Am Gd

153

57

Co

Half-life (years)

X- or g-ray energy (keV)

Photons per disintegration

2.7 88 17.8 1.3

MnK x-rays (5.9, 6.5) UL x-rays (1330) PuL x-rays (1421) AgK x-rays (22, 25) g-rays at 88 g-rays at 35 TeK x-rays (2732) g-rays at 59.5 EuK x-rays (4148) g-rays at 97 g-rays at 103 g-rays at 122 g-rays at 136 g-rays at 81 g-rays at 276 g-rays at 303 g-rays at 356

0.28 0.13 0.08 1.02 0.04 0.07 1.38 0.36 1.10 0.30 0.20 0.86 0.11 0.34 0.07 0.18 0.62 0.09 0.85

0.16 433 0.66

0.74

133

Ba

10.8

137

Cs

30.2

a

g-rays at 662

Dose at 1 m from 1 GBq (27 mCi) (mSv=h)

Analytical technique

—a —a —a —a

XRF XRF XRF XRF

2.7

XRF

3.6 27

XRF, XRA, XRS XRA

24

XRF, XRA, XRS

65

XRA, XRS, rt

83.7

rt

It is difﬁcult to assign a radiological protection meaning to the dose of low-energy x-rays.

capsules and are often referred to as ‘‘pellets.’’ When economics justiﬁes it, a special source may be designed for a particular type of the analyzer. A so-called ‘‘lollipop’’ source, designed speciﬁcally for light-element analysis probes made by Metorex Int. is such an example (Amersham IEC.600 series). It is made in a form of a ﬂat copper ring, 1 mm thick, 15 mm in diameter, with a 8-mm opening. One side of this ﬂat copper annulus is electroplated with a 55Fe isotope, over which a few-micrometer-thick Ni ﬂashing is applied. Such source geometry allows for a very close coupling between sample and a proportional detector window. Consequently, the quantitative analysis of light elements, down to A1, is possible without the nuisance of helium purge or vacuum. There are international codes for the safe use of radioisotopes, and a simple introduction to radiation protection has been published (Martin and Harbison, 1986). Each organization using radioactive substances is required to hold a license, issued in most countries by a government health department or atomic energy authority. The International Commission on Radiological Protection (ICRP, 1985) recommends that, for members of the public, it would be prudent to limit exposures to radiation on the basis of a lifetime average annual dose of 1 millisievert (mSv). Table 1 lists the approximate, typical dose rates at 1 m from each radioisotope source, assuming no absorption of the emitted radiation within the source or by air. The x-ray dose is inversely proportional to the square of the distance from the source. X-ray doses received during the operation of x- and g-ray instrumentation and gauges are trivial compared with the maximum permitted doses because of the low x-ray output of radioisotope sources, careful design of operating techniques, and x-ray shielding.

444

Figure 4 Encapsulation of disk and annular (From Amersham, 1986.)

Piorek

109

Cd g-ray sources. Dimensions are in millimeters.

Radioisotope-Excited X-ray Analysis

445

The International Organization for Standardization (ISO, Geneva) has produced a system for classifying sealed radioisotope sources based on safety requirements for typical uses (Amersham, 1986). Prototype sealed radioisotope sources undergo temperature, external pressure, impact, vibration, and puncture tests (Table 2), which increase in severity as the class designation increases from 1 to 6. The ISO classiﬁes the test requirements for speciﬁc types of application of the sealed sources. The classiﬁcation for low-energy g-ray gauges and XRF analysis instruments used in industry is C33222 [i.e., from Table 2, the ﬁrst classiﬁcation 3 is temperature ( 740 C and 180 C), the second classiﬁcation 3 is 25 kPa to 2 Mpa, and so on]. This classiﬁcation, which meets the minimum requirements, is marked in Table 2 by shaded areas. Most radioisotope sources are designed and manufactured to have a greater integrity than required by this classiﬁcation. For example, the 109Cd sources (Fig. 4) are coded C64344 and C33344 according to the ISO classiﬁcation, compared with C33222 required. The use of radioisotope-containing devices is regulated in each country by the appropriate government agency, which may require the user to just register the device with it or obtain special license to posses and use the device. B.

X-ray and g-ray Detectors

Scintillation, proportional, and solid-state detectors are—in this ascending order of importance—extensively used in radioisotope x-ray analysis. The important characteristics of these detectors are x-ray energy resolution, eﬃciency, the ratio of the full energy peak to total detection eﬃciency, the spectrum of x-rays not in the peak, the sensitive area and thickness of the detector, the count rate capability of detector and associated electronics, the complexity of the detector and associated electronics, the robustness of the overall system, and—last but not least—its cost. An excellent source on all aspects of detectors of nuclear radiation can be found in the work of Knoll (1999). The complexity and associated cost of equipment is greatest for solid-state detectors and least for scintillation detectors. The need to use liquid nitrogen (LN2) with solid-state detector systems and their relative complexity and cost have proven to be a cost penalty but not a limiting factor, even for applications of on-line analysis in industry. Successful application of thermoelectric cooling (Peltier eﬀect) for semiconductor detectors, speciﬁcally those based on the structure of p-i-n junction (diode), allowed the abandonment of LN2 cooling without compromising analytical performance of those detectors (Amptek, 1977; EPA, 1997; Shefsky, 1997). 1. Energy Resolution of the Detector Energy resolution is the detector parameter by which diﬀerent detectors are compared to each other. For detectors of x-rays and low-energy g-rays, the energy resolution is deﬁned as the full width of the MnKa peak measured at half of its maximum (a so-called FWHM), when the detector is irradiated directly by the collimated beam from the 55Fe radioisotope, at a total count rate in the whole spectrum not exceeding 1000 counts=s (see Fig. 5). Usually, also a shaping time constant of the ampliﬁer is speciﬁed to be not less than 10 ms. For semiconductor detectors, their energy resolution is expressed directly in units of energy (eV). For gas-ﬁlled proportional and scintillation detectors, their energy resolution is customarily expressed as percent relative to the energy of the MnKa peak (5895 eV). For example, the best gas-ﬁlled detectors may reach FWHM of about 700 eV, which, when related to 5895 eV of the peak energy, equals about 12% relative. On the other hand, semiconductor detectors such as Si(Li) crystals feature energy resolution better than 3%

No test

Punctured

740 C (20 min.) þ180 C (1 h)

3

4

5

6

740 C (20 min.), 740 C (20 min.), 740 C(20 min.), þ 400 C (1 h) and þ 600 C (1 h), þ 800 C (1 h) thermal shock thermal shock from thermal shock from from 400 C to 20 C 600 C to 20 C 800 C to 20 C 25 kPa absolute to 25 kPa absolute to 25 kPa absolute 25 kPa absolute 25 kPa absolute to atmospheric pressure 2 MPa absolute to 7 MPa absolute to 70 MPa absolute 170 MPa absolute 50 g from 1 m 200 g from 1 m 2 kg from 1 m 5 kg from 1 m 20 kg from 1 m 90 min, 2580 Hz 30 min, 25500 Hz at 30 min, 2550 Hz at at 1.5 mm amplitude 5gn peak amplitude 5gn peak amplitude; peak-to-peak; 5090 Hz at 0.635 mm 802000 Hz at 20gn amplitude peak-to-peak; 90500 Hz at 10gn 1 g from 1 m 10 g from 1 m 50 g from 1 m 300 g from 1 m 1 kg from 1 m

740 C (20 min.), þ80 C (1 h)

2

b

Details of the testing procedures are given in ISO.2919 and BS.5288. A further class X can be used when a special test procedure has been adopted. External pressure 100 kPa ¼ 1 atm (approximate). c The source, positioned on a steel anvil, is struck by a steel hammer of the required weight; the hammer has a ﬂat striking surface, 25 mm in diameter, with the edges rounded. d The source, positioned on a hardened steel anvil, is struck by a hardened pin, 6 mm long and 3 mm diameter, with a hemispherical end, ﬁxed to a hammer of the required weight. Note: The shaded cells of the table when read from top to bottom will correspond to class C33222, which is the minimum requirement for low-energy g-ray and x-ray sources used in XRF instrumentation and g-ray gauges.

a

No test No test

External pressureb Impactc Vibrations

No test

Temperature No test

1

Class

Classiﬁcation of Performance Standards for Sealed Radioisotope Sources as per ISO.2919a

Test for resistance to

Table 2

446 Piorek

Radioisotope-Excited X-ray Analysis

447

Figure 5 Definition of energy resolution, FWHM, of an x-ray detector, which is measured at the half height of MnKa peak. Note the shaded area, which indicates the width of background window.

relative, and its is therefore more convenient and accurate to express their FWHMs in units of energy rather than percent. Energy resolution, expressed as the full width at half(peak)-maximum (FWHM) and shown for each type of detector as continuous line in Figure 6, was calculated from equations given by Jenkins et al. (1981d). It is important to remember that practical energy resolution for any detector is always worse than that quoted by 10% to sometimes 20% relative. This is because the spectra measured from real samples usually contain large amounts of backscattered radiation, and, additionally, the intensities of the analytes may be very high. As mentioned earlier, the energy resolution of solid-state detectors is superior to that for proportional and scintillation detectors (see Fig. 26 in Chapter 2). Figure 7 shows the calculated energy spectrum for the detection of 8-keV x-rays in each detector and also the energies of the Ka x-rays in the 69-keV energy range. Figure 8 shows the diﬀerence in energy of Ka x-rays between adjacent atomic number elements. Table 3 compares the diﬀerence in Ka x-ray energies with the energy resolution for aluminum, iron, and tin based on data given in Figures 6 and 8. Solid-state detectors are the only detectors that can fully resolve Ka x-rays of adjacent Z elements. The factors aﬀecting their resolution are discussed in detail in Chapter 3 (Sec. III.D). Proportional detectors have an energy resolution less than twice the energy diﬀerence in Ka x-rays of adjacent Z elements. Hence, their energy-resolving power is useful even if there are adjacent Z elements in the sample. Scintillation detectors (see Chapter 2, Sec. III.F.2) have such limited resolving power that other techniques must be used to discriminate between adjacent Z elements, such as balanced ﬁlters. This achieved, however, at the expense of some loss in sensitivity of analysis. Figure 9 illustrates the resolving capabilities of detectors by showing calculated energy spectra for identical concentrations of Fe and Ni in the same sample.

448

Piorek

Figure 6 X-ray energy resolution (FWHM) of scintillation, proportional, and solid-state detectors. The continuous lines are calculated (data from Jenkins et al., 1981d), the silicon detector results being based on a small detector (10 mm263 mm): The (diamonds) resolutions are for various solid-state detectors (EG&G Ortec, 1986); (þ) typical and (6) best for the high-resolution proportional detectors (Metorex, 1986); and (open circles) typical and (solid circles) best resolutions for specific NaI scintillation detectors (Harshaw).

2. Energy Resolution of Detector and Minimum Detectable Level The energy resolution of the detector not only determines its ability to resolve the x-rays of adjacent elements but also decides about the minimum quantity of the element that can be detected by the detector, a so-called minimum detection limit, or the detection limit (DL). The better the resolution (i.e., the narrower the x-ray peak), the better (i.e., smaller) the minimum detectable quantity of element. The detection limit is frequently deﬁned by 3 DL ¼ S

rﬃﬃﬃﬃﬃﬃﬃﬃ Ibgd t

ð21Þ

where S is the sensitivity (in counts=s per unit concentration of analyte), Ibgd is the background intensity measured in the analyte’s integration window, and t is the measurement time.

Radioisotope-Excited X-ray Analysis

449

Figure 7 The calculated energy spectra for the detection of 8 keV x-rays in scintillation, proportional, and solid-state detectors. (From Watt, 1983.)

The background intensity, Ibgd , is measured in the same window as the intensity of a given x-ray peak. The width of the window is usually set equal to the peak FWHM. If a sample containing a certain percentage of the analyte is measured with a detector of energy resolution E1 (where E1 ¼ FWHM), it will generate in its FWHMwide window a certain intensity, I1. If the detector resolution now degrades to, say, E2 (i.e., the analyte’s x-ray peak is now wider), the intensity I2 will still be equal to I1 (as long as the measurement window is FWHM-wide). This is because the energy deposited by the photon in the detector is represented by the area of the whole peak. Therefore, if the peak is wider, it has to be smaller, and if it is narrower (better resolution), it has to be taller—for the area of the peak to remain constant (see Fig. 5). However, the background intensity will be larger in the case of a worse energy resolution, E2, than in the case of E1, because the integration window for the background intensity is wider in the case E2 than in the case E1. Then, it follows from Eq. (19) that the DL for the E2 case will be larger (worse) than in the case of E1 energy resolution. Therefore, for the best DLs, it is of paramount importance to use the detector with the best energy resolution. 3. Detector Efficiency for X-rays and Detector-Sensitive Area Figure 10 shows the calculated eﬃciencies of scintillation and solid-state detectors over the energy range 1150 keV (see also Chapter 3). At low energies, the decrease in efﬁciency is due to the absorption of x-rays in the beryllium window at the front of the detector.

450

Piorek

Table 3 Difference in Energy of the Ka X-rays of Adjacent Atomic Number Elements, and the Energy Resolution of Three Types of Detector Energy resolution of detector, FWHM (in eV) Atomic No. Energy of Diﬀerence in of element Ka x-rays (eV) Ka energies (eV) 13 (Al) 26 (Fe) 50 (Sn)

1,490 6,400 25,300

253 527 1087

Solid state

Gas-ﬁlled proportional

Scintillation

117 160 275

425 660 1,750

3,000 6,200 12,200

The eﬃciency of the detector for registering the x-rays at high x-ray energies is determined by the probability that the x-ray interacts with atoms in the sensitive volume of the detector. The most eﬃcient detectors are those with a high atomic number and a high mass per unit area. For x-ray energies above about 40 keV, solid-state detectors made of germanium are preferred to those made of silicon. Also, of the two detectors made of the same material, the thicker one will have a better eﬃciency for high-energy x-rays. This is why even the smallest Si(Li) detector (usually at least 3 mm thick) is superior to the siliconp-i-n diode detector, which is usually not thicker than 0.5 mm. The eﬃciency of

Figure 8

The energy difference between the Ka x-rays of adjacent atomic number elements.

Radioisotope-Excited X-ray Analysis

451

Figure 9 Comparison of energy resolution of Si(Li) (FWHM ¼ 160 eV) and gas proportional detector (FWHM ¼ 660 eV). The solid line is spectrum of the 1-to-1 ratio of iron and nickel collected with the Si(Li) detector. The dotted spectrum is generated with a gas proportional detector for the same sample.

Figure 10 Calculated efficiencies of NaI scintillation detector and silicon and germanium solidstate detectors used in XRF analysis.

452

Piorek

proportional detectors depends on the type and pressure of the ﬁlling gas and the diameter of the detector (Fig. 11). The sensitive area for scintillation detectors is usually from 1000 to 2000 mm2; for proportional detectors, it ranges from about 500 to 1000 mm2; for silicon solid-state detectors, it is 10100 mm2; and for the commercially available silicon p-i-n diode detectors,

Figure11 X-ray detection efficiencies of Metorex International (formerly Outokumpu Electronics) proportional detectors with different types of gas fillings. The number code below each graph is gas pressure, bar (1), window thickness (in mm) (2), gas mixture (3), low background (4), high efficiency (5) long lifetime (6), high count rate (7), and high resolution (8). The ratings shown are excellent (xxx), good (xx), fair (x).

Radioisotope-Excited X-ray Analysis

453

it is currently 313 mm2. Hence, in general, count rates are highest for the poorer-resolution detectors. The sensitive area of the detector determines the type of measurement geometry that can be used with each detector type. Central source geometry (Fig. 1b) is normally used with scintillation and proportional detectors, because the large sensitive area compensates for the shadowing eﬀect of the centrally located source. An annular source geometry (Fig. 1a) is characteristic for Si(Li) solid-state detectors and especially for their stationary installations. A side source geometry (Fig. 1c) is the only choice for the small-area detectors, speciﬁcally the most recent semiconductor detectors such as HgI2, silicon p-i-n-diode, CdTe, CdZnTe, and so forth. The eﬃciency of solid-state detectors is discussed in some depth in Chapter 3 (Sec. III.F). 4. Ratio of Full-Energy Peak toTotal Spectrum The ratio of the area of the full-energy peak to the total area of the x-ray spectrum is critical to the sensitivity of XRF analysis. The spectrum outside the full-energy peak is caused by many factors. The ﬁrst and most important is the occurrence of the so-called ‘‘escape peak’’ (Jenkins et al., 1981c), resulting from incomplete photoelectric absorption of the incident x-ray in the detector material followed by escape of some of its ﬂuorescent x-rays from the detector. For example, if an x-ray photon of iron energy 6.4 keV enters the active volume of the detector, its energy is being absorbed by the detector material also by the mechanism of excitation of characteristic x-rays of the detector material, such as Si, Ar, and so forth. If, in turn, the excited x-ray photon of, say, silicon, of energy 1.74 keV, is not absorbed in the detector but wanders outside of it, then the total energy deposited in the detector by the original iron photon will be smaller by the energy of silicon x-ray photon which ‘‘escaped’’ from the system. The energy left in the detector equals the diﬀerence in energy of the x-ray entering the detector and that of the escaping ﬂuorescent x-ray. This will give rise to the small peak always located left to the original x-ray peak, at a distance equal to the energy of characteristic x-rays of the detector material. Thus, for example, for silicon-based detectors, the escape peak may be observed 1.74 keV left to any original photopeak. The escape peak is greatest for the higher-Z detectors. The ratio of x-rays in the escape and full-energy peaks is highest for proportional detectors with gas ﬁllings of Xe (Ka x-ray of 29.7 keV), Kr (12.6 keV) and Nal (iodine Ka of 28.5 keV) scintillation detectors. However, even silicon (1.74 keV) has about 12% of the detected counts in the escape peak when excited by 2 keV or higherenergy x-rays. The magnitude of the escape peak is also strongly dependent on the size, shape, and geometry of the detector itself. The smaller the detector, the more likely the escape of the characteristic x-ray of detector material is, because it is more likely for the photon to be generated at the wall of the detector. Also, escape peaks are more intense for more intense original photopeaks. Other factors that lead to incomplete absorption of the energy of the x-ray in the detector are Compton scattering of the incident x-ray in the detector, with the scattered x-ray or Compton electron escaping from the sensitive volume; alternatively, the incident x-ray may be photoelectrically absorbed in the detector, but the photoelectron escapes from the sensitive volume of detector before losing all its energy. These phenomena, which are diﬀerent manifestations of the ‘‘escape’’ mechanism, result in the contribution to the spectrum at energies diﬀerent from the energy of the original x-ray photon. The full-energy peak to total spectrum is highest for the high-Z detector materials and for low-energy x-rays. It is lowest for the low-Z gases used in some proportional detectors.

454

Piorek

Apart from the phenomena taking place within the detector itself, the total spectrum is also a function of the instrument design. For example, typically most of the radiation reaching the detector is not characteristic x-ray radiation from elements in the sample, but primary radiation from the source scattered on sample and its environment. It is, therefore, very important that the design of the instrument be focused on minimizing and optimizing of measurement geometry in order to reduce to absolute minimum any parasitic radiation reaching the detector. 5. Comments on the Characteristics of Proportional Detectors The characteristics of proportional detectors vary considerably with type of ﬁlling gas and its pressure and are much more variable than the characteristics of scintillation and solidstate detectors. The best energy resolution is obtained by using Penning mixtures as gas ﬁllings of these detectors (Jarvinen and Sipila, 1984b). Although the improvement in energy resolution is relatively small (Fig. 6), it is critically important for applications in the atomic number range 2630 (iron to zinc). The low average ionization energy of Penning mixtures also leads to other important advantages (Jarvinen and Sipila, 1984b): the voltage required is lower, hence the gas pressure can be higher. This leads to a higher eﬃciency of detection, fewer wall eﬀects, and smaller escape peaks and, consequently, to a higher ratio of peak to total spectrum. The life of the detector is also increased to more than 1013 counts because of the use of only noble gases. The characteristics of proportional detectors supplied by Metorex International (formerly Outokumpu Oy) are summarized in Figure 11. The recommended gas ﬁllings for proportional detectors depend on the speciﬁc analysis application. The eﬃciency of detection of low-energy x-rays is limited by the absorption of the x-rays in the beryllium window. Proportional detectors with lower gas pressures are used in the detection of lowenergy x-rays because the thinner windows do not withstand high pressures. For those low-energy x-rays, the detectors ﬁlled with neon gas and ﬁtted with Be windows as thin as 13 mm are commercially available. 6. Developments in Solid-State Detectors The solid-state detector (SSD) is the best type of x-ray detector for XRF analysis, but, until recently, its potential has not been fully realized, particularly in industrial and ﬁeld use, because of the need for liquid-nitrogen cooling. The only successful exceptions have been pioneering and notable Kevex designs, known as AnalystTM Model 6700—a factory ﬂoor model—and X-SITETM—a very ﬁrst portable, solid-state detector alloy analyzer (Spiegel and Horowitz, 1981; Kevex Corp.). Both of these models featured a traditional Si(Li), liquid-nitrogen-cooled detector and either annular or capsule-type radioisotope source(s). Perhaps the best testimonial to the success of these designs of the early eighties is the fact that owners of quite a few still working systems spare no eﬀort and ingenuity to keep them alive, despite a total abandonment by the original manufacturer. There has been much promising research into mercuric iodide, cadmium telluride and gallium arsenide SSDs (Cuzin, 1987) which can operate at or near ambient temperatures. The ﬁeld-eﬀect transistor (FET) of the low-noise charge preampliﬁer associated with these detectors must be cooled to at least 720 C and stabilized at this lower temperature in order to reduce electronic noise. This is achieved with the Peltier element; the one-stage Peltier cryostat can reduce the temperature of cooled object by about 20 C. Over the last decade, the Peltier-cooled mercuric iodide detector has been used in the commercially available, portable x-ray analyzers (Berry and Voots, 1989; Piorek, 1997).

Radioisotope-Excited X-ray Analysis

455

However, production yields for these detectors are still problematic, and the same applies to other semiconductor detector materials such as CdZnTe. It is reasonable to say that should the demand for those detectors be as high as for silicon in the electronics industry, we would most likely had mastered the technology of these promising semiconductor materials a long time ago. Developments have led to the production of silicon-lithiumdrifted detectors that can be operated at temperatures much higher than that of liquid nitrogen (195 C). Madden et al. (1986) used these silicon detectors cooled in a Peltier cryostat. The front-end assembly, mounted in the cryostat, contains a silicon detector and a FET and is mounted on a four-stage Peltier cooling cell. With the assembly under high vacuum, a temperature of 774 C is achieved with a cell power of 4.3 W. For a 16-mm262-mm-thick detector, an energy resolution of 190 eV at 5.9 keV was achieved. As the ﬁrst commercial manufacturer, Kevex Corp. announced the development of a Peltiercooled silicon detector system with x-ray energy resolutions of 155, 180, and 240 eV (at 6 keV) respectively for 10-, 30-, and 80-mm2 detectors, which is as good as have been achieved for liquid nitrogen cooling. Other organizations have developed silicon detectors and Peltier cryostats that also give good energy resolutions for x-rays (Tractor X-Ray, now Spectrace). Systems based on these detectors are available as either laboratory or benchtop versions. It should not come as surprise that the most recent ‘‘room-temperature’’ detector is a silicon p-i-n diode, manufactured using well-known technologies of silicon wafer and chip fabrication. Two U.S. companies manufacture these detectors commercially. These truly silicon (no lithium doping) detectors are supplied in a TO-8 package, containing a detector element, a FET, a feedback loop, and a Peltier element. Typically, the detector is a tiny, 713-mm2 area by 0.3-mm-thick silicon diode, with energy resolution currently reaching below an impressive barrier of 190 eV (Amptek, 1997). These small detectors are predominantly used in portable designs. Their main drawback is relatively high power consumption (12 W on average) by Peltier cooler, which in the case of a portable, battery-operated device is not desirable. The small thickness of these detectors (0.5 mm appears to be maximum these days) makes them ineﬃcient for the detection of x-rays above 25 keV. However, these Peltier-cooled systems did indeed replace liquid-nitrogencooled systems and opened the way for the widespread use of silicon detectors in industry. A very good insight into the current status of research in solid-state detectors is provided by Schlesinger and James (1995). From this reference, one may infer that the next major development in detector technology may come from gallium arsenide. This semiconductor material is very important for semiconductor electronic and optoelectronic industry and, therefore, is a subject of serious research. The windfall of this research may be a new detector material for x-rays. C.

Electronics

The electronics used with the various detectors are discussed in Chapters 2 (Sec. III.F) and in 3 (Sec. IV) and are also covered in detail by Jenkins et al. (1981a). The limits to accuracy and sensitivity of XRF analysis are usually determined by the limitations of the detector in energy resolution, eﬃciency of detection, and maximum count rate, for example, rather than of the electronics. With the excellent gain stabilization electronics now available and the reliability of low-power, surface-mount components, it is rare that the electronics system is a signiﬁcant limiting factor even in the harsh environmental conditions of industrial plants.

456

IV.

Piorek

X-RAYAND g-RAY TECHNIQUES

The range of radioisotope x- and g-ray techniques used for analysis is far more extensive than the range based on x-ray tube techniques. Almost all x-ray tube systems are based on the high-energy resolution of wavelength-dispersive (the crystal spectrometer) or energydispersive (the solid-state detector) devices. With this high resolving power, there is less need to tailor a technique to the speciﬁc application. Radioisotope x-ray systems, especially those involving scintillation or proportional detectors, usually must be carefully matched to the speciﬁc application. This disadvantage is more than compensated for by such attributes as mechanical ruggedness, simplicity, and portability, which are so important in industrial and even more so in ﬁeld applications. The selection of the radioisotope source to analyze diﬀerent elements depends on many factors, including whether the energy of the radioisotope x- or g-rays is suﬃcient to excite the element, the energies of the x-rays scattered by the sample, and the energy resolution of the detector. Figure 12 is an approximate guide and, although prepared for proportional detectors (courtesy of Metorex Int., formerly Outokumpu Oy), can be used for solid-state detectors and to a more limited extent for scintillation detectors. The isotopes of 238Pu and 244Cm, emitting x-rays similar in energy, can be used interchangeably although curium is preferred for safety reasons. A 57Co isotope can be used with scintillation detectors or germanium solid-state detectors for the K shell XRF analysis for highZ elements, such as uranium, gold, and lead. This section reviews the radioisotope XRF techniques used with solid-state, proportional, and scintillation detectors, x-ray preferential absorption techniques that are normally based on the use of scintillation detectors, and x-ray scattering techniques that are often based on use of scintillation detectors. An example of the application of each technique is also given. A.

XRF Techniques Based on Solid-State Detectors

The analysis of samples of copper ores taken from the process streams of three diﬀerent mineral concentrators is used to illustrate XRF analysis with a solid-state detector (Gravitis et al., 1974). The ore samples were excited by x-rays from a 3.3-GBq (or about 90 mCi) 238Pu source and the ﬂuorescent x-rays were detected by a 28-mm263-mm-thick silicon-lithium-drifted [Si(Li)] detector. The x-ray spectrum (Fig. 13) of one of the samples shows well-resolved peaks of Ka x-rays from iron, copper and zinc, and La; Lb, and Lg x-rays from lead and the complex spectra of the Compton-scattered and coherently scattered x-rays. The count rates of the copperKa x-rays (Fig. 14a) lie within three bands, separated from each other because of the large diﬀerence in absorption of x-rays in the sample matrix caused by the widely diﬀerent iron concentrations (5, 20 and 50 wt%) of the diﬀerent ores. The use of the scattered x-ray component in the calibration equation for copper corrects for matrix absorption and reduces the overall uncertainty to better than 0.15 wt% copper (Fig. 14b). Solid-state detectors are the only type of detector for which the x-ray energy resolution (Fig. 6) is suﬃcient to resolve the ﬂuorescent x-rays of adjacent Z elements (Fig. 8). There are minor problems of overlap in some cases in which the energy of the Ka x-ray of the analyzed, Z element, overlaps the energy of the Kb x-ray of another, usually Z 7 1, element in a sample. These overlaps can easily be identiﬁed from the energies of ﬂuorescent x-rays as a function of Z (see Appendix II of Chapter 1). Fluorescent x-rays can also overlap slightly if the concentration of an interfering, adjacent Z element is much

Radioisotope-Excited X-ray Analysis

457

Figure12 Appropriate radioisotope x-ray source for use with proportional or solid-state detectors to gain a high sensitivity of XRF analysis for elements in a specified atomic number range. The principal energies of the x-rays emitted by each source are indicated above the x axis. The diagonal lines are the K and L shell absorption-edge energies. K and L shell excitation is required to gain a wider coverage of atomic number elements using the same radioisotope source and, in some cases, for high-atomic-number elements.

Figure13 The spectrum of x-rays from a copper, lead, and zinc ore sample excited by 238Pu x-rays and detected by a Si(Li) solid-state detector.

458

Piorek

Figure 14 XRF analysis of copper in flotation feed samples from three mineral concentrators, based on the intensity of (a) copper Ka x-rays and (b) the ratio of the intensities of copper Ka and scattered x-rays. The analysis was based on a 238Pu source and silicon solid-state detector.

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higher than that of the analyte. The extent of this overlap can be calculated using the ﬂuorescent x-ray energies and x-ray energy resolution (FWHM) of the detector. The overlaps of this type, which are known as ‘‘spectral overlaps’’ or ‘‘spectral interferences,’’ can be easily corrected either mathematically or empirically. The small sensitive area and, consequently, small solid angle are the main limitations of solid-state detector compared with proportional and scintillation detectors. With solidstate detectors, it takes longer to obtain the same counting statistics. It is not always possible to use higher-activity sources to overcome this limitation because of self-absorption of x-rays in the source and, for some radioisotopes, the cost of the source. On the other hand, the relatively small solid angle limits the amount of scattered x-rays reaching the detector, which reduces the spectral background considerably and improves the detector peak-to-background ratio. Figure 15 shows the 3s minimum detection limits for low concentrations of various elements in a low-Z matrix (Hoﬀmann, 1986; Spatz and Lieser, 1977). The counting time was 600 s. The measurements with 109Cd (185 MBq or 5 mCi), 241Am (370 MBq or 10 mCi), and 57Co (370 MBq or 10 mCi) were made using a 30-mm265-mm-thick siliconlithium-drifted, solid-state detector (FWHM of 250 eV at 6.4 keV), and the sample matrix was silica gel. The measurements with 125I isotope (185 GBq or 5 Ci) were made using a 50mm263-mm silicon detector (FWHM 250 eV) and a matrix of average atomic number 10. The measurements with 133Ba isotope (370 GBq or 10 Ci) were made using an 800mm2613-mm germanium detector (FWHM 590 eV at 122 keV) and a water matrix. The detection limits using one source vary greatly with atomic number; hence, to maintain low detection limits over a wide atomic-number range, several radioisotope sources, emitting x-rays of diﬀerent energies, must be used. In this case, detection limits less than 10 mg=g are achieved for many elements. Iron-55 can be used to extend the range of sensitive detection down to Z ¼ 15. Similar detection limits for elements in a soil matrix were reported using a portable, Si(Li)-detector-based x-ray analyzer (Piorek, 1994). B.

XRF Techniques Based on Proportional Detectors

The main advantages of proportional detectors over classical, Si(Li) solid-state detectors are the larger sensitive area, simpler equipment, and no need for cooling the detector to a very low temperature. The higher count rates possible with these detectors lead to shorter analysis times, except for those applications in which the energy resolution (Fig. 6) limits the sensitivity of analysis. Many important XRF applications are not limited by the poorer energy resolution, and many do not involve adjacent Z elements. Selection of the appropriate type of proportional detector for the speciﬁc XRF application is very important. The best type of detector is determined by optimizing the various characteristics summarized in Figure 11 for a speciﬁc application. In practical terms, it means proper matching of the detector gas ﬁll, its pressure, and type and thickness of detector window, with given analytical requirements. For example, in order to analyze the fraction of percent of zinc and phosphorus in oil, one can select a single probe with a detector, which will usually compromise eﬃciency for both analytes. On the other hand, a much higher eﬃciency of detection and, hence, better analytical sensitivity for these two analytes can be achieved by using two diﬀerent probes, each with the detector matched to the analyte; neon ﬁlling and thin window detector for phosphorus and a highpressure argon detector for zinc. Figure 16 shows the spectra of x-rays, taken with 4-mm-thick samples of pure water and water containing 100 mg=g of both iron and zinc, measured with a 244Cm source and a

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Figure 15 Detection limits (3s) determined with various radioisotope x-ray sources and a solidstate detector. [Data from Hoffman (1986) and Spatz an Lieser (1977).]

proportional counter ﬁlled with a neonargon Penning mixture gas to a pressure of 7 bar (Jarvinen and Sipila, 1984b). The detection limits are comparable to those obtained with a silicon solid-state detector with x-ray tube excitation (Jarvinen and Sipila, 1984b). If other elements in the atomic number range 2630 had been present, however, there would have been incomplete resolution of the K x-rays emitted and, hence, poorer sensitivity of analysis. Figure 17 shows detection limits for the elements in water (Jarvinen and Sipila, 1984a; other data courtesy of Asoma Instruments), normalized to a common counting time of 100 s. The low detection limits were obtained by careful choice of ﬁlling gas and pressure in relation to the exciting x-ray energy used. The issue of proper match of the detector with the energy of exciting radiation is much more important for gas-ﬁlled proportional detectors than for the solid-state ones. This is because of the much greater uniform detection eﬃciency of the solid-state detector for x-rays than that of the gas-ﬁlled one. For most, except low-atomic-number elements, the detection limits were less than 10 mg=g, which is similar to the results for solid-state detectors (Fig. 15); however, they were achieved in a time six times shorter than before. These low detection limits were achieved using diﬀerent proportional-counter gas ﬁllings, and, as a consequence, the technique is less ﬂexible than that based on the solid-state detector. If adjacent Z elements had been present, the detection limits for the proportional counter would have been considerably worse, but for the solid-state detector, they would have been much less changed.

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Figure 16 Spectrum of x-rays from a pure water sample and from water spiked with 100 mg/g of iron and zinc as determined using a (Ne þ Ar)-filled proportional detector and excited by x-rays from a 244Cm source. (From Jarvinen and Sipila, 1984b.)

The relatively modest improvement in energy resolution of the high-resolution gas-ﬁlled detector over standard proportional detectors leads to considerable improvement in the accuracy of analysis in some applications. Hietala and Viitanen (1978) indicated that an improvement in resolution from 16% to 10% at 8 keV results in the relative standard deviation for determination of zinc in copperzinc tailing, containing 0.1 wt% of both copper and zinc, to be reduced from 0.40% to 0.05% relative. C.

XRF Techniques Based on Scintillation Detectors

The energy resolution of scintillation detectors is so poor (Fig. 6) that the detector cannot be used in most applications to resolve the K x-rays of the analyte and matrix elements. Selectivity to the analyte is obtained with ﬁlters and radiators and by a proper choice of the energy of the x-rays exciting the sample (Watt, 1983). Scintillation-detector XRF systems have been used extensively in ﬁeld work and in industrial plants because of their simplicity, high x-ray detection eﬃciency, portability, ruggedness of the detector and electronics, and low cost. These systems are simpler than those based on proportional detectors and hence, if suﬃciently sensitive and selective to the desired analyte, are the preferred system. They are best used for applications requiring the determination of the concentration of one or two elements only. Applications involving the determination of

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Figure 17 Minimum detectable levels (3s) for low concentrations of elements in water using proportional detectors and 100 s counting time. The measurements with the argon detector (gas pressure, 5 bar) were made with 9-mm-thick water samples and 111 MBq of 109Cd, 2.2 GBq of 244 Cm, and 1.67 GBq of 241Am, with the neon detector (1 bar pressure and a 0.05-mm-thick beryllium window), with 20-mm-thick water samples and 3.7 GBq of 55Fe, and with the xenon detector (1 bar) with 110 MBq of 109Cd. (Data from Jarvinen and Sipila, 1984a; Asoma Instruments.)

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more elements are best undertaken with proportional and solid-state detector systems. The detection limits of scintillation techniques are at least a factor of 10 higher (i.e., worse) than those for solid-state detectors. Three types of head unit are used with scintillation detectors: direct excitation, grayx-ray source excitation, and detectorradiator (Fig. 18). Filters can be used with all three assemblies. 1. Filters Filters placed between the sample and detector (Fig. 18) increase the sensitivity of analysis by ﬁltering out a higher proportion of ﬂuorescent x-rays of matrix elements than those of the analyte. Zinc, for example, may be the analyte in samples also containing iron. Calculations based on Eq. (1) and mass absorption coeﬃcients [see Eq. (76) in Chapter 1] show that a 27-mg=cm2 aluminum ﬁlter transmits 27% of the Ka x-rays of 8.6 keV but only 4.5% of the iron Ka x-rays of 6.4 keV (Fig. 19); that is, it reduces the intensity of iron K x-rays relative to the zinc K x-rays by a factor of 6. If the sample also contains lead, however, about 55% of the lead La x-rays (10.5 keV) would be transmitted, twice that of the zinc Ka x-rays. In this case (Fig. 19), an absorption-edge-type ﬁlter of copper (22.4 mg/ cm2) could be used to reduce the lead La x-ray transmission to only 1 % and also to reduce the iron K x-ray peak. This selective enhancement of the zinc compared with the iron and lead x-ray components partly compensates for the limitation of the poor resolution of the scintillation detector. Although some iron and lead ﬂuorescent x-rays will still be detected within the pulse-height channel set about zinc Ka x-ray peak, in many applications this component will have been suﬃciently reduced to make the analysis possible. If the measurement with one absorption-edge ﬁlter does not give suﬃcient selectivity to the desired element’s ﬂuorescent x-rays, balanced ﬁlters are used (Fig. 20). The intensities of x-rays in the ﬂuorescent x-ray channel are measured separately, ﬁrst with one ﬁlter and then with the other (with ‘‘up’’ and ‘‘down’’ ﬁlter). The atomic numbers of the two ﬁlters are chosen so that their K shell absorption-edge energies are just above and below the energy of the Ka x-rays of the analyte. The masses per unit area of the two ﬁlters are chosen so that the product mrt [Eq. (1)] is practically identical for both ﬁlters, except

Figure 18

Three types of radioisotope XRF assembly based on scintillation detectors.

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Figure 19

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X-ray transmission through aluminum and copper filters.

Figure 20 X-ray transmission through balanced filters of copper (6) and nickel (solid circle). Count-rate measurements made first with one and then with the other filter are subtracted to give a count rate proportional to the intensity of zinc Ka x-rays from a sample.

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within the energy window enclosing the Ka x-ray energy of the measured element. Hence, the diﬀerence in the count rates using the two ﬁlters is proportional to the intensity of ﬂuorescent x-rays of the analyte. An excellent example of the balanced-ﬁlter application has been ‘‘MetallurgistTM,’’ Model 9266, by TN Technologies (Spiegel and Horowitz, 1981). The portable analyzer used a series of balanced ﬁlters to quantitatively analyze stainless steels, nickel, copper, and cobalt alloys. It needs to be mentioned here that ﬁlters are also used with some low-energyresolution gas proportional detectors and also with x-ray tube excitation (Oxford Instruments). A rather unique application of a single absorption-edge ﬁlter has been reported for analysis of Mn in Hatﬁled steels using a gas-ﬁlled proportional detector (S. Piorek, personal communication, 1987). There, a MnKa peak is wedged between intense CrKa and FeKa peaks. By placing a Cr ﬁlter over the detector window, the FeKa xray intensity is reduced by a factor of 6 relative to MnKa x-ray, thus enabling quantitative analysis of Mn. However, because manufacturing of the ﬁlters and necessary mechanics is rather expensive, the practice of using ﬁlters becomes less and less common, in favor of using systems with better energy resolution. Figure 20 shows the transmission of x-rays by copper and nickel ﬁlters, which are chosen when the analyte is zinc (Rhodes, 1966). Except in the energy window enclosing the zinc Ka x-rays, the transmission is the same for the two ﬁlters. The balanced-ﬁlter technique is thus highly selective to the zinc Ka x-rays. The count rates, measured separately in the ﬂuorescent x-ray channel with the two ﬁlters, are usually high and the diﬀerence in count rates can be quite small, often resulting in high relative uncertainties of measurement. This is one of the main disadvantages of the balanced-ﬁlter technique. There are two other disadvantages of ﬁlter techniques: The sensitivity of analysis is poor when the ﬂuorescent x-rays of the main interfering matrix element have an energy just below that of ﬂuorescent x-rays of the analyte (Fig. 19), and the sensitivity is considerably less than the obtained with detectors that have the inherent resolving power to isolate the ﬂuorescent x-rays of desired and matrix elements. These losses in sensitivity result from the only partial absorption of interfering x-rays in the ﬁlter and, for absorption-edge ﬁlters, from the detection of ﬁlter K x-ray in the channel of the analyte. The latter is a direct consequence of the use of the broad-beam geometries of radioisotope XRF system. The ﬁlter K x-rays are mainly excited by the source radiation scattered from the sample. 2. DirectExcitation Assemblies The directexcitation technique (Fig. 18) is the most widely used of the three scintillation detector assemblies. The energy of the radioisotope x-ray is usually chosen so that the ﬂuorescent and backscattered x-rays are resolved by energy analysis (Fig. 21). The intensity of the backscattered x-rays is used to correct for the absorption of the analyte’s ﬂuorescent x-rays by the sample matrix [Eq. (8) or (9)]. The ﬁlter enhances the sensitivity and selectivity of analysis. Direct-excitation assemblies are used extensively in industry (e.g., in laboratory and portable elemental analyzers (Rhodes, 1971) and in on-line (in-stream) analysis of mineral slurries (Watt, 1983). 3. DetectorRadiator Assemblies The detectorradiator assembly (Fig. 18) discriminated well against interfering x-rays of energy just below that of the ﬂuorescent x-rays of the measured element (Watt, 1972). The

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basis of this discrimination is that the atomic number of the radiator element can be chosen so that, of the two x-ray components with nearly similar energies, only the higher of the two has suﬃcient energy to excite the K x-rays of the radiator element. The detector is shielded from the sample and, hence, sees only the x-rays emitted by the radiator. Balanced-radiator techniques, analogous to balanced-ﬁlter techniques, can also be used to improve selectivity to the analyte if there is another matrix element, in this case emitting ﬂuorescent x-rays of energy higher than those of the analyte. The count rates obtained with detectorradiator assemblies are about 5% of those obtained with direct-excitation assemblies, using a source of the same activity, because of the additional excitation stage of the radiator. The intensity of higher-energy x-rays scattered by the sample can be measured simultaneously in the one assembly by use of a second radiator element of atomic number considerably higher than that of the ﬁrst radiator. The x-ray energies of the two components are well resolved and similar to that shown in Figure 21. Detectorradiator systems are much less widely used than balanced-ﬁlter techniques. They are less versatile than balanced-ﬁlter systems. Applications include the determination of lead in zinc concentrates, in which the zinc Ka x-rays (8.6 keV) from the high concentration of zinc (e.g., 50 wt%) swamp the lead L x-rays (10.514.8 keV) from the low concentration of lead (e.g., 0.5 wt%). A radiator of zinc (absorption-edge energy of 9.66 keV) is excited by the lead L but not by the zinc K x-rays. This radiator technique

Figure 21 Typical spectrum of x-rays from copper ore slurry excited by a measured using a scintillation detector.

238

Pu source and

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467

improves the sensitivity to lead to that of zinc by a factor of about 20. However, the absolute signals from the analytes when using the radiator technique may be up to an order of magnitude smaller than when excited directly. 4.

-Ray-Excited X-ray Assemblies

A limited number of x- or g-ray energies are emitted by radioisotope sources (Table 1). A secondary x-ray source, in which g-rays from a radioisotope source excite the ﬂuorescent x-rays of a target material, can be used to obtain essentially monoenergetic x-rays of energy determined by the atomic number of the target element (Fig. 18). Hence, the energy of the x-rays incident on the sample can be chosen to suit the speciﬁc XRF application. The g-ray-excited x-ray assembly (Watt, 1983) makes use of ﬁlters, including balanced ﬁlters, similarly to the direct excitation assembly. The count rates using the secondary excitation source assembly are only about 5% of the count rate of a direct excitation assembly using the same activity source. A balanced-energy technique, in which separate measurements are made with two targets (in the secondary source) whose ﬂuorescent x-rays straddle the K shell absorption-edge energy of the wanted element, can be used to obtain more selectivity to the analyzed element. The g-ray-excited x-ray assembly is used as an alternative to direct excitation when no suitable energy x-ray is emitted by radioisotope source. One application is in the determination of the coating mass of tin on steel by the detection of the tin K x-rays. If 60-keV g-rays from 241Am isotope are used to excite the tin, K x-rays from both sides of the steel are detected; by choosing the energy of the exciting radiation to be just above that of the tin K shell absorption edge, tin K x-rays from only the one side are detected because of the high absorption of the lower-energy excitation radiation in the steel. D.

X-ray Preferential Absorption Techniques

X-ray preferential absorption analysis is often based on the dual-energy g-ray transmission technique (Fig. 2) because of important industrial applications involving the analysis of coarse and heterogeneous materials, such as coal (Fookes et al., 1983; Gravitis et al., 1987) and metalliferous mineral ores. Low-energy g-rays must be used in these applications to obtain suﬃcient transmission through the material, so that the only practical approach is to use radioisotope sources. Scintillation detectors are used to ensure eﬃcient detection of the g-rays, with pulse-height analysis to separate the two energies. Figure 22 illustrates the results using the DUET technique to determine the lead content of zinc concentrate and residue (tailings) samples (Ellis et al., 1969). The radioisotopes 241Am and 153Gd were used. Their g-ray energies, 60 keV of 241Am and 97 and 103 keV of 153Gd, envelope the K shell absorption-edge energy of lead (88 keV). A common calibration curve is obtained despite the great diﬀerence in absorption by the matrix of the concentrates (with about 50 wt% zinc) and of the tailings, also called residues (with about 0.6 wt% zinc). This technique becomes more complicated when the g-ray transmission measurements are made on material on fast-moving conveyors, as for the on-line determination of the ash content of coal (Fookes et al., 1983; Gravitis et al., 1987). Equations (13) and (14) hold only for time intervals during which there is little change in mass per unit area, whereas there would be a linear summing of count rates in time although the correct response is logarithmic [Eq. (14)]. This problem can be addressed by counting for shorter intervals during which the mass per unit area changes little and summing the logarithms of the counts during many of these intervals (Gravitis et al., 1987).

Figure 22 g-Ray preferential absorption analysis for lead, showing (left) the transmission of 153Gd g-rays ( 100 keV), and (right) the combination of the separate transmission measurements for 153Gd and 241Am (59.5 keV) g-rays.

468 Piorek

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E.

469

X-ray Scattering Techniques

The single most important applications of x-ray scattering techniques are in the on-line systems, such as for the continuous analysis of particulate material on conveyors and sample bylines in industrial plants. There is often no need to crush the material before analysis because of the penetration of the g-rays in the material. Americium-241 g-ray has been used to determine the ash content of coal both in a sample byline (Fauth et al., 1986) and, when suﬃcient thickness of coal is available, directly on a conveyor belt (Cierpicz 1986). Dual-energy g-ray scattering has been used in ore-sorting applications (Outokumpu Mintec Oy, 1986), in which the thickness of the ore lumps is too great for XRA techniques. F.

Count Rates and Calibration

The count rates, or intensities, of ﬂuorescent and backscattered x-rays from the sample are determined by many factors [Eqs. (4) and (6)]. Activities of the radioisotope sources used range from 100 MBq to 10 GBq (or 3300 mCi), the higher activities being used mainly with solid-state detectors, which have a smaller eﬀective area than scintillation and proportional detectors. For x-ray preferential absorption analysis, higher activities of 110 GBq (30300 mCi) are used for industrial on-line applications in which high count rates are essential with the rapidly changing mass per unit area and where distances between analyzed medium and detector are relatively large. The source activities used for x-ray scattering analysis are approximately the same as those for XRA analysis. X-ray ﬂuorescence analysis is, as is the vast majority of other analytical methods, a comparative analytical technique. Radioisotope x-ray systems are usually calibrated by comparing the measured count rates with analyses of the same samples by more conventional techniques. The coeﬃcients linking the count rates and chemical analysis are determined by linear multiple regression. This empirical in nature method of calibration is essential in most applications, especially when the materials to be analyzed have an unknown particle size. For the calibration to be valid, it is essential to calibrate with materials covering the full range of variations in elemental composition and particle size. In some industrial applications, these factors may change slowly with time and, hence, the calibration must be regularly updated. The widespread use of room-temperature solid-state detectors as well as availability of microcomputers made so-called standardless methods of calibration a realistic alternative to the empirical ones. These ‘‘standardless’’ methods are variations of the fundamental parameters (FP)-based approaches, in which the concentrations of analytes are deduced iteratively by combining spectral information from the sample with that of one known standard and=or with fundamental equations of x-ray physics. It is to be noted, however, that any empirical calibration based on a good set of standards will always be more accurate than that derived with the FP algorithm.

V.

FACTORS AFFECTING THE OVERALL ACCURACY OF XRF ANALYSIS

The choice of a radioisotope x-ray analysis technique for a speciﬁc application depends on several interacting factors: the overall accuracy of the sampling and analysis required, the time available to achieve this accuracy, technique available to obtain a suﬃciently representative sample of the material being analyzed, and the sample preparation

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requirements, such as grinding, pressing, and so forth. The inﬂuence of each of these factors on the accuracy of XRF analysis is discussed here.

A.

Overall Accuracy and Time for Analysis

The overall accuracy of analysis depends on uncertainties in calibration, sampling, sample preparation, and x-ray analysis. The requirements of a good accuracy of analysis stay in striking contrast with the time allotted for analysis. The maximum acceptable time must include not only the time for the x-ray analysis but also the time for sampling and sample preparation. Uncertainties due to nonrepresentative sampling are too often underestimated in industrial and ﬁeld applications, resulting in the accuracy of the overall analysis often being compromised. The total uncertainty of analysis, stot , can be expressed as the sum of all contributing uncertainties: qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ stot ¼ s2instr þ s2stat þ s2het þ s2cal þ s2samp:plac: þ s2matrx þ s2part:size þ þ s2?? ð22Þ where Instrumental component such as thermal drift Uncertainty due to statistics of counting Uncertainty reﬂecting sample heterogeneity Reﬂects uncertainties contributed by reference analysis and calibration ﬁt Uncertainty due to sample placement Uncertainty associated with matrix eﬀects Uncertainty caused by varying particle size of analyzed material Any other potential, unidentiﬁed uncertainty

sinstr sstat shet scal ssamp:plac: smatrx spart:size s??

This formula assumes that all uncertainties included are random and normally distributed and that systematic uncertainty (bias) is negligible. Under these assumptions, the stot can be regarded as a good measure of the accuracy of analysis. The ﬁrst two uncertainty components, sinstr and sstat , are often treated together as main constituents of instrument precision (repeatability), whereas all the other uncertainties identiﬁed in Eq. (22) can be regarded as contributing to user or application related uncertainty. Thus, Eq. (22) can be written as follows: stot ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ s2instr:precis: þ s2applic:

where sinstr:precis: ¼ and sapplic: ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ s2instr þ s2stat

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ s2het þ s2cal þ s2samp:plac: þ s2matrx þ s2part:size þ þ s2??

ð23Þ

ð24Þ

ð25Þ

Some of the uncertainties can be identiﬁed and easily controlled or minimized by the analyst, such as uncertainty due to statistics of counting, or calibration uncertainty. For example, the statistical uncertainty of counting, sstat , can be easily reduced by extending

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the measurement time, whereas the uncertainty associated with calibration, scal , can be reduced by using more accurately analyzed calibration samples and by selection of a better calibration equation. Sample placement uncertainties, ssamp:plac: , can be reduced by always following the same procedure for preparing and loading the sample into the measuring chamber of the analyzer, or—partially—by spinning the sample during measurement. Uncertainties due to matrix variability, smatrx , and particle size, spart:size , can be signiﬁcantly reduced by proper sample preparation techniques such as matrix dilution (Jenkins et al., 1981a) and by grinding, pelletizing, or fusing the sample with spectrometric ﬂux, respectively. Although all these uncertainty minimization techniques are valid and readily applicable in laboratory environments, they are an unattainable luxury in ﬁeld and industrial applications where grinding, fusing or any type of sample preparation is simply out of the question. This is why the accuracy of analysis in ﬁeld and industrial environments tend to be governed by the application-related uncertainties rather than by the usually much smaller contribution from the instrument’s precision. It is also for this reason that the measurement time may often be selected from the point of view of process throughput rather than overall uncertainty of analysis; usually, good instrumental precision allows the analyst to shorten the measurement time without sacriﬁcing the former. Nevertheless, the measurement time is always a compromise between the throughput of the process and analytical requirements. In the coating industry, the time for the determination of the coating mass of tin and zinc on steel must be less than a few seconds if the result is to be useful; hence continuous analysis directly on the main coating line is essential. In mineral concentrators, rapid changes in the grade of ore entering the plant and the time taken for the ore to pass through the plant (about 15 min) make it essential for the process slurries to be analyzed within 5 min. This can be achieved by continuous analysis either directly in stream or on slurries in sampling loops (bylines). In scrap yards, alloys must be sorted into diﬀerent types in short periods: otherwise the sorting operation is not economical. The lower the price per pound of scrap, the faster the sorting has to be. In this case, it is essential to have analysis equipment that is both portable and capable of producing rapid results. It is important to realize that XRF analysis does require some ﬁnite time period within which a meaningful measurement can take place. The consequence of this fact is that XRF analysis results in process control applications are, most of the time, the average rather than point estimates of the process parameters such as coating thickness, sulfur concentration of diesel fuel, and so forth. B.

Uncertainties of Sampling and Sample Presentation

Uncertainties originated by sampling processes are probably one of the most signiﬁcant contributors to the overall analytical uncertainty—the point we attempted to make in the previous subsection. This is especially true for ﬁeld and industrial applications of XRF methods, where sample preparation and conditioning is nonexistent, and which, as we stated in the Introduction to this chapter, are mostly radioisotope based. That is also why we devote so much space in this section to this issue. All analyses involve sampling, some more than others. In the continuous on-line measurement of material on conveyors or in process slurry streams, the analysis is

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averaged over large quantities of material, but still has the potential for sampling uncertainties because not all of the material on the belt or in the slurry stream is viewed. If material is continuously sampled from the main stream and fed through a sampling loop past the x-ray analyzer, the uncertainties in sampling the main stream and in viewing only part of the sample loop must both be considered. A mathematical approach to sampling of a particulate media has been detailed by Gy (1982) and Pitard (1993). The most readily accessible information on sampling is given in the various international standards for the sampling of process material from moving conveyors and from slurry streams (ISO, Geneva). These describe preferred sampling practices and estimate the accuracies of sampling, including the eﬀects of diﬀerent top sizes and the number and mass of sample increments taken. These standards are regularly amended. Sampling uncertainties are of critical importance to the exploration for and assessment of deposits and the mining of both metalliferous ores and coal. After all, it is possible to analyze only a very small proportion of the ore. There are established practices for choosing where to sample (e.g., on a regular spaced grid), the number of samples to be taken, and the weight of each sample to be taken. This complex ﬁeld of sampling geostatistics is thoroughly covered in textbooks (David, 1971; Ingamells and Pitard, 1986; Isaaks and Srivastava, 1989). 1. Uncertainty Due to Sample Heterogeneity Samping uncertainties, speciﬁcally in analysis of particulate type of material, are mainly caused by heterogeneity of the measured medium. Should the analyte be uniformly distributed throughout the mass=volume of the measured material, any sample taken for analysis would contain the same concentration of the analyte. However, even this ideal condition would be compromised by the morphology of the sample. If the sample is liquid, then the dissolved analyte can, indeed, be homogeneously dispersed in a liquid matrix. However, if the analyzed medium is ore or soil, or any other powder, the chances of nonuniform distribution of the analyte in matrix increase dramatically. Not only might the grains of the analyte not mix well with grains of the matrix, but any handling of the sample will promote tendencies for segregation of one type of grain from the other. A very important contributing factor is the size of the sample taken for XRF analysis. The smaller sample is likely to emphasize its heterogeneity as compared to a larger sample. The main consequence of heterogeneity of the material analyzed is the fact that the sample of this material taken for analysis may not be representative of the material. If the uncertainty cannot be minimized, the next best way of dealing with it is to estimate its magnitude. It is rather easy to evaluate the contribution of material heterogeneity factor to the total uncertainty. The only requirement is the availability of a sufﬁcient quantity of analyzed material. Two series of measurements are required. In the ﬁrst, a sample of material to be analyzed is measured in x-ray analyzer for at least seven times, without being disturbed in any way. This series of measurements will yield a standard deviation, which will correspond to stot , as per Eq. (20), except for the shet , that being equal to zero because only one sample was measured. Next, at least seven diﬀerent samples of the same material are measured under the same conditions as before (i.e., the same measuring cup, counting time, etc.). This second series of results will yield a standard deviation, which will correspond to stot , exactly as per Eq. (20), now including the heterogeneity factor, shet . By subtracting the squared standard deviation of the ﬁrst series

Radioisotope-Excited X-ray Analysis

473

from that of the second and taking the square root of the diﬀerence, a numerical value of shet is readily obtained. For example, in soil analysis for lead, the two standard deviations obtained were 25 and 65 mg=kg, respectively. Therefore, after subtraction, the shet was 60 mg=kg, a signiﬁcant contribution to the total uncertainty. It is very important to realize the fact that in the majority of analyses of granular or particulate material, its heterogeneity determines the overall accuracy of analysis. This is true not only for XRF, which by its nature is a surface measurement method, but also for other instrumental techniques such as atomic absorption or inductively coupled plasma, as long as an aliquot analyzed is not representative of the material from which it was taken. It is therefore of paramount importance that good homogenization is applied to analyzed material and that the amount of sample taken is suﬃcient to be representative of it, especially if veriﬁcation or validation of XRF results with other analytical techniques is required. 2. Sample Heterogeneity and Detection Limits A detection limit (DL) for a given analyte is understood to be that amount of analyte in a sample that produces a spectral signal equal to or greater than three standard deviations of a signal obtained on a sample with no analyte present. The author’s own data and values cited in literature indicate that a typical DL for lead in soil is about 50 kg=kg when a 109Cd isotope and semiconductor detector are used for analysis. The 109Cd isotope x-rays reach to a depth of about 1.3 cm into a silica-based soil. However, if there is lead in this sample, the excited lead L series x-rays can reach the surface of soil only if they are generated at a depth not greater than 0.20.3 cm, as measured from the soil surface. A 50-mg=kg DL for lead means, in practice, about 650 mg of total lead as potentially seen by the analyzer in a soil ‘‘cylinder’’ of 13 g mass (cylinder base area of 5 cm2 by 1.3 cm high by 2 g=cm3 of assumed speciﬁc density of silica soil). However, the lead x-rays reach the detector from a depth of only 0.3 cm and, therefore, the eﬀective mass of total lead ‘‘seen’’ is only 150 mg. This mass of lead translates into a 0.030-cm (or 12 mils)-diameter sphere of pure lead. Even in a form of an oxide, the size of the lead oxide grain would not be much more. If this small, by all means, lead grain happens to be within the ﬁrst 3 mm of soil, the XRF analyzer will be able to measure it. However, if the lead grain ends up at deeper layer of soil, it will never be noticed. Obviously, depending on its characteristic x-rays energy and excitation source, any other element will be similarly aﬀected by its location in the sample. This perhaps extreme but simple example illustrates how illusionary and misleading the concept of detection limit may be in case of granular material. C.

Choice of Radioisotope X-ray Technique

The choice of the most appropriate radioisotope x-ray technique for a speciﬁc application depends on the requirements of accuracy of sampling and x-ray analysis and the time available for the analysis. The simplest radioisotope x-ray technique that satisﬁes these requirements is usually chosen. In the laboratory, where many diﬀerent types of analyses can be performed, XRF analysis with a solid-state detector is the most ﬂexible method. For industrial and ﬁeld applications, there is often a more restricted range of analyses paired with a greater need for simple and reliable equipment, the cost of which may also be an important consideration. X-ray ﬂuorescence analysis based on scintillation or proportional detectors is often, but not always, the best approach. These considerations lead to the use of a much wider range of radioisotope x-ray techniques in industry than in the laboratory. The choice

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of technique is more complicated. Should an ore sample be ground before analysis and L shell XRF be used for a high-Z element, such as uranium, or can K shell XRF techniques determine the concentration with suﬃcient accuracy despite the heterogeneity of the sample? Should the analysis technique be chosen to make a direct measurement on-line, in a sampling loop of the process stream, or on a sample taken to a laboratory? Should XRF techniques even be used in an on-line application, given the inaccuracies introduced by the heterogeneity of the material to be analyzed? The practical alternative may be to use XRA or XRS techniques or a nuclear technique, based on more penetrating radiation, such as high-energy g-rays or neutrons. The applications discussed in the next section indicate the preferred solutions to some important analysis applications, particularly in industry.

VI.

APPLICATIONS

Table 4 lists some important applications of radioisotope x-ray techniques based on XRF equipment, usually referred to as laboratory or portable elemental analyzers. The analysis techniques involve the use of scintillation, proportional, and semiconductor detectors. These analyzers are in widespread use in many application areas and in many industries (Rhodes, 1971; Rhodes and Rautala, 1983). The total number used worldwide probably is about 10,000. Some major suppliers of radioisotope based instruments are listed in Table 5. A photograph of a portable model of an x-ray analyzer for alloys, build around a silicon p-i-n diode detector is shown in Figure 23 and a laboratory model based on gas-ﬁlled proportional detectors is shown in Figure 24. Many solid-state detector systems are also in routine use in laboratories. Table 6 lists types of on-line analysis systems in routine use. These include systems for the on-line analysis of mineral slurries, ﬂowing powders, coal, coal slurries, paper, determination of sulfur in oil and petroleum products, and analysis of coatings. Most of the analysis systems tend to be based on scintillation and proportional detectors, but some of the more recently developed systems also use solid-state detectors. Table 6 is also an attempt to list some commercial suppliers of the on-line equipment. Example of the application of these techniques are now discussed in details, with emphasis given to applications of industrial importance. Some examples illustrate the interaction of sampling and sample presentation with the selection of the appropriate radioisotope technique. A.

Identification of Alloys

Historically, the ﬁrst applications of portable XRF analyzers were in mining and prospecting. It was only with the advent of the on-board memory and microprocessors that the portable XRF analyzers found wider acceptance and use for analytically more demanding alloy identiﬁcation and analysis. Since then, several thousand of these analyzers have been sold, making alloy sorting and analysis a ‘‘ﬂagship’’ application for portable XRF analyzers. These analyzers can assay the alloy and=or identify it by its grade or common name. Rapid sorting of alloys is required in many areas of the metals industry, such as smelting, fabrication, inventory and incoming material control, and the sorting of scrap (Piorek and Rhodes, 1986; Berry, 1981; Piorek, 1989). Some common alloy groups include nickel alloys, copper alloys, stainless and high-temperature steels, and carbon and chromiummolybdenum steels. Although 4050 elements are involved in the alloying process, in any given alloy there are only 1020, and of these, only about 10 are required for the identiﬁcation of speciﬁc alloy.

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Table 4 Some Typical Applications of Radioisotope Based Laboratory and Portable Elemental Analyzers Application Alloy sorting and identification Mining and mineral

Pulp and paper Environmental

Fibers, films and coatings Chemicals and process control Plastics

Agricultural

Cosmetics Pharmaceutical Petroleum products

Typical examples Low-alloy steels; stainless steels; nickel alloys; high-temperature alloys; titanium, aluminum alloys; specialty alloys; metal scrap Copper, lead, zinc, tin, arsenic, molybdenum, nickel, iron, chromium, bismuth, and uranium in commercial-grade ores, concentrates, and tailings; titanium and iron in silica sand; silicon, potassium, titanium, and iron in clays; phosphate rock Thickness of silicone coatings on paper and polymer membranes; calcium, titanium, filler in paper Soil screening for metals (Cr, Cu, Ni, Pb, Zn, As, Cd, Hg, Sb); hazardous materials (e.g., lead, arsenic, chromium, or cadmium in waste sludge); trace elements in wastewater discharge; metals in air particulates on filters; chlorine (halogens) in waste oil; sulfur in diesel fuel Copper, zinc, tin, gold, silver and chromium plating thicknesses; metals in plating solutions; silver in photographic film; manganese coating thickness on magnetic tape; titanium on glass; ruthenium on electrodes Lead, titanium, and zinc in paint; sulfur, iron, alumina, silica, and calcium in cement; vanadium in catalysts; palladium and gold coatings on silica spheres uses as catalysts; zinc, chromium, nickel in plating baths Calcium, lead, tin, and chlorine in PVC; zinc and bromine in polystyrene; chlorine in urethane rubbers; bromine and chlorine in butyl rubbers; silicon in polythene; TiO2 in nylon; bromine in Styrofoam Fertilizers (calcium, phosphates, potassium); copper, chromium, and arsenic in wood preservatives and treated wood; bromine in almonds; ironzinc ratio in meat for grading; minerals in cattle feed; titanium in fillers Titanium, iron, lead in powders Metals in vitamin pills; zinc in insulin Lead, calcium, sulfur, vanadium, and chlorine in gasoline or oil; sulfur in petroleum coke; sulfur and ash in coal; lubricating oils additives

The main requirements of analytical equipment for alloy identiﬁcation are portability, speed and reliability of identiﬁcation, and an ability to be used by unskilled operators. Balanced-ﬁlter techniques have two main disadvantages. Concentrations of at least several elements must be determined; hence, separate measurements must be made with several sets of balanced ﬁlters. The sensitivity of analysis is insuﬃcient for the lower concentrations of some speciﬁc elements in the alloys. Piorek and Rhodes (1986) showed that by using XRF analysis based on a 111-MBq (3 mCi) 109Cd source and a high-resolution proportional detector, many alloys can be identiﬁed in one measurement. Measurement is ﬁrst made to identify the alloy by group. The spectrum of the unknown alloy is then compared with the key features of spectra of known alloys in the group, which are stored in a memory chip in the equipment. This socalled grade-identiﬁcation mode of analysis is particularly useful because it does not require the operator to be proﬁcient in the speciﬁcs of alloy composition. In the gradeidentiﬁcation mode, the analyzer uses a pattern-recognition algorithm to compare the x-ray intensities of the measured sample with those of the stored references of alloy standards. This approach is very fast, taking about 35 s to identify a single alloy. In

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Table 5 Manufacturers and Suppliers of Radioisotope-Based Laboratory and Portable Elemental Analyzers Manufacturera

Analyzer type

Detector type

Amdel ASOMAb BRGM Metorex Int.b

Laboratory Laboratory Laboratory Portable, laboratory

Niton Corp.

Portable

Outokumpu Electronics, Oy

Portable, laboratory

Oxford Instrumentsb Radiation Monitoring Devices TN Technologies

Laboratory Portable Portable

Scintillation Gas proportional Solid-state Si(Li) High-resolution gas proportional; solid state: Si(Li) or silicon p-i-n diode Solid state: silicon p-i-n diode, CdZnTe High-resolution gas proportional; solid state: Si(Li) Gas proportional Solid state: CdTe Solid state: mercuric iodide, HgI2

a b

See the list of manufacturers and their addresses in the Appendix of this chapter. These manufacturers reacted to the dynamics of the market and industry by switching from isotope-based instrumentation to x-ray-tube-excited devices.

Figure 23 The Metorex Int. portable alloy analyzer based on high-resolution silicon p-i-n diode detector, MetalMasterTM 2000. (Courtesy Metorex Int.)

Radioisotope-Excited X-ray Analysis

477

Figure 24 A laboratory, benchtop, elemental analyzer based on a high-resolution proportional counter, X-METTM 820. (Courtesy of Outokumpu Electronics.)

comparison, assaying the alloy may take up to 4 min in order to maintain low measurement uncertainty. Table 7 shows the identiﬁcation results for diﬀerent alloys by group, obtained with the analyzer pictured in Figure 23. The probability of correct identiﬁcation is satisfactory for all alloys except the carbon steels. The results in Table 7 were generated by measuring each alloy sample for 5 s for at least 10 times and comparing its spectral features against the library of about 145 diﬀerent alloys. The number of correct identiﬁcations ratioed to the total number of trials for a given alloy group was then recorded in Table 7 as a measure of probability of correct identiﬁcation. Some of the identiﬁcation failures are for alloys very close in composition, for which the main alloying elements diﬀer in concentrations by less than 1%. The most diﬃcult identiﬁcation is for carbon steels in which the concentrations of alloying elements are very low in the presence of almost 100% iron, and the diﬀerence in concentrations of the same elements between two grades approaches the sensitivity of the XRF technique. Therefore, the XRF is not a method of choice for identiﬁcation of carbon steels, for which better techniques, such as optical emission spectrometry, exist (Piorek et al., 1997). By using a silicon p-i-n diode detector in place of a gas proportional one, it has been possible to extend positive identiﬁcation to such speciﬁc alloys as B1900 and B1900Hf, which diﬀer from each other by less than about 1% Hf, in the presence of several percent of W and Ta and a balance of nickel. Overall, the portable alloy analyzers oﬀer a much simpler approach to identiﬁcation of alloys, with a reliability of identiﬁcation as good as that for more complex techniques. Despite the signiﬁcant diﬀerences in design, the portable alloy analyzers exhibit similar performance in the assay mode; that is, they oﬀer measurement precisions ranging from 0.01% to 0.5% absolute depending on the element, its concentration range, and the

478 Table 6

Piorek On-line Analysis Systems Based on Radioisotope X-ray Sourcesa

Application

Analytical technique, detectorb

Manufacturerc

Metal content of mineral slurries

XRF/XRA, s and ss, in-stream XRF, p, sample line XRF, ss, sample line XRF, ss, in-stream XRF, ss, sample line XRF, p, sample line XRF, ss, sample line Dual-energy XRS, s, on-line

Amdel Asoma Outokumpu Mintec, Oy Texas Nuclear Ramsey Asoma Outokumpu Mintec, Oy Outokumpu Mintec, Oy

Dual-energy XRS, s, on-line Dual-energy XRS, s, on-line XRS, s, on-line XRS, s, sample line XRS, p, sample-line XRF, neutron, and g transmission, s, on-line XRS, p

Outokumpu Mintec, Oy MCI; Harrison Cooper; SAI EMAG Humboldt-Wedag Sortex Amdel

XRF, s

Rigaku; Outokumpy Mintec, Oy Yokogawa

Metal content of clay and mineral powders Iron and chromium in ore on converyors Ore sorting Ash in coal on conveyor

Solids weight fraction and ash in coal in slurries Tin content of galvanizing solutions Calcium in cement raw mix Sulfur in oil, diesel fuel, gasoline Lead in gasoline Metals in plating bath solutions (Ni, Cu, Cr, Ta, etc.) Cement analysis for Ca, Si, Mg, Al, S, Fe Corrosion products (Cr, Fe) in steam generator feedwater of nuclear power plants Ash content and/or mineral filler material in paper Coating mass of: Zn, Sn/Cr, Sn/Ni, Zn/Fe, Sn/Pb on steel and other substrates

Rigaku

XRF, ion chamber, sample line XRF, p, sample-line XRF, p, sample-line XRF, p, sample-line

Mitsubishi; Metorex Int. Metorex Int. Asoma; Metorex Int.

XRF, p, sample-line

Metorex Int.

XRF, p, sample-line

Detora Analytical

XRA, p, b-rays transmission, on-line XRF, p, on-line XRF, p, on-line

Sentrol; Yokogawa; Paul Lippke Sentrol Data Measurement; FAG; Gammametrics

a

X-ray tube techniques can be used in some of these applications. Detectors: s ¼ scintillation, p ¼ gas proportional, ss ¼ solid state [usually Si(Li)]. c See the list of manufactures and their addresses in the Appendix of this chapter. b

matrix. Several references treat this subject in greater detail (Fookes et al., 1975; Jenkins et al., 1981c, 1981d; Piorek and Rhodes, 1986; Piorek 1989; Spiegel and Horowitz, 1981; Berry, 1981). Table 8 shows typical performance data for a contemporary portable XRF alloy analyzer, such as illustrated in Figure 23.

Radioisotope-Excited X-ray Analysis Table 7

479

Performance of a Portable X-ray Analyzer in Alloy Identiﬁcation

Alloy group

Identiﬁcation results (% feasible)

Measured elements

Nickel and cobalt alloys Copper alloys Stainless steels and high-temperature alloys Cr/Mo steels Low-alloy steels Titanium alloys Aluminum alloys

Ti, Cr, Fe, Co, Ni, Cu, Nb, Mo, W, Hf, Ta Mn, Fe, Ni, Cu, Zn, Pb, Sn Ti, Cr, Mn, Fe, Co, Ni, Cu, Nb, Mo

100 90100; 98100 90100; 100

Cr, Fe, Ni, Mo Cr, Mn, Fe, Ni, Cu, Nb, Mo Ti, V, Mn, Cr, Zr, Mo, Sn Mn, Fe, Cr, Cu, Zn

90100; 100 6580; 90100 95100; 90100; 95100

Note: If two results are given, the ﬁrst refers to a gas-ﬁlled proportional detector, and the second to a solid state, silicon p-i-n diode detector. Source: From Piorex et al., 1997.

B.

Determination of Uranium and Gold in Ore

During the exploration of mining of metalliferous ores, large numbers of ore samples must be analyzed to compensate for the inherent variability of expression of the ore. The ore may be analyzed in the laboratory or, for higher-Z elements, at the mine face or in situ in boreholes. K shell XRF techniques are often preferred for the analysis for uranium; because of the penetration of the uranium K x-rays in the ore, little or no crushing of samples is required and, additionally, the uranium concentration is averaged over much larger samples. Uranium can be determined down to 20 mg=g or about 0.7 oz.=ton (1s) in 30 s using a 222-MBq (6 mCi) 57Co and a 28-mm265-mm germanium detector (EG&G Ortec, 1986). This technique was used routinely to survey samples for uranium and other Table 8 Standard Performance Data for a Typical, Contemporary, Commercially Available Portable XRF Alloy Analyzer Alloy group

Nb/Mo

Sn

Pb

0.006 0.01

0.15

0.15

Stainless, hi-temp steels 0.015 0.20 0.10 0.20 0.20 0.20 0.06 0.20 0.03 0.30 0.20 0.30 0.30 0.10

0.01 0.03

0.30

0.05 0.30

Ni/Co alloys

0.15

Low-alloy steels

Ti 0.01 0.02

Cr

Mn

0.04 0.1 0.1 0.2

Fe

Co

Ni

Cu

Zn

0.25 0.25 0.10 0.05 0.10 0.50 0.25 0.15

0.10

0.20 0.10 0.12 0.10 0.20 0.05 0.30 0.50 0.30 0.50 0.50 0.50 0.30

0.02 0.08

0.30

Cu alloys, brass/bronze 0.02

0.10 0.02 0.02 0.05 0.05 0.15 0.07 0.06 0.06 0.08 0.40 0.30

0.01

0.008 0.20 0.20 0.30

Aluminum alloys

0.02

0.05 0.10 0.05 0.05 0.04 0.05 0.06 0.20 0.10

0.003 0.005

0.005 0.01 0.20 0.02

Titanium alloys

0.40

0.10 0.10 0.06 0.05 0.05 0.02 0.02

0.008

0.005 0.01

Notes: The values listed are typical precision ranges in percent absolute for total assay time per sample not longer than 60 s. The differences are due to the analyzer model and/or the radioisotope(s) used. Precision data listed are obtained with a 185-MBq (5 mCi) 109Cd and 1.48-GBq (40 mCi) 55Fe sources. Source: From Piorek et al., 1997.

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high-Z elements (Z 40) in the laboratory by the Australian company Geopeko in the extensive exploration programs that found the large Ranger uranium deposit at Jabiru in the Northern Territory, Australia (G. Sherrington, personal communication, 1987). It is also used for borehole logging, with detection limits of 0.04 wt% (1s) for uranium, tungsten, and lead and 6 mg=g for gold in 120 s counting time using a 518-MBq (14 mCi) 57 Co source and a small silicon detector in a 32-mm-diameter borehole probe (Scitech Corp. undated). An analysis system has been developed for the in situ determination of gold in ore at the mine face (Hill and Garber, 1978). The hand-held probe consists of a 4.4-GBq (120 mCi) 109Cd (88-keV g-rays) source, a 200-mm267-mm-thick germanium detector, and a small Dewar ﬂask containing liquid nitrogen, which must be replenished after 6 h of operation. The sensitivity to gold has been optimized by careful choice of the incident g-ray energy and by measuring gold Kb x-rays whose energy is greater than the energies of most of the Compton-scattered g-rays. The precision for a 30-s scan time is 20 mg=g (1s) and 2 mg=g for 100630-s scans. The technique is suitable for use in high-grade gold mines but has insuﬃcient sensitivity to be applied widely in gold mining, in which 0.3 mg=g or 0.01 oz.=ton (1s) is normally required. C.

On-Line Determination of Coating Mass

Tin on steel (tinplate), zinc on steel (galvanized iron), zinc and aluminum on steel (zincalum), iron oxide on plastic (magnetic tape), and silicon coating on paper are only a few examples of coated products manufactured in large quantities. The coatings are applied at high speed. Accurate control of the coating mass per unit area is essential to economize the operation. Two XRF techniques can be used to determine the mass of the coating. Fluorescent x-rays of the coating element can be excited and their intensity measured. The intensity of the coating element x-rays increases with coating thickness. Alternatively, ﬂuorescent x-rays of element in the base material can be excited. Their intensity decreases with the increase in coating mass because of the absorption of the incident and excited x-rays in the coating. Both radioisotopes and x-ray tubes are used as the source of x-rays, with radioisotope sources preferred, except for those applications requiring very fast response, such as 0.1 s. Radioisotope XRF techniques for the on-line determination of coating mass are based on the use of proportional detectors. These are preferred to scintillation detectors because of their better energy resolution and because they can be used at the relatively high temperatures that occur above the hot tin and galvanized iron coating processes. In commercially available systems (Table 6), the analysis head unit continuously scans across the width of the strip so that coating mass can be controlled across the whole strip. These commercial systems are used worldwide in most high-throughput coating operations. They can also be used to determine the separate coating masses of multiple coatings. Coated products are usually sold with a speciﬁed minimum coating mass. The accurate coating mass determination has led to coatings being controlled much closer to the minimum speciﬁcation. The variations in coating mass obtained on the zinc galvanizing line of John Lysaght Pty. Ltd., Port Kembia, Australia, corresponding to no gauge, gauge with manual control, and gauge with automatic control are shown in Figure 25. In 1977, improved control of zinc coating mass led to savings of A$300,000 per year per line; similarly, at the nearby Australian Iron and Steel Pty. Ltd., savings of A$1 million per year (Watt, 1978) were made for tinplate.

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481

Figure 25 The frequency distribution for product zinc coating mass before and after installation of an on-line zinc coating mass gauge at John Lysaght Pty. Ltd. Product minimum specification is 300 g/m2. (From Watt, 1978).

Another typical coating mass application for XRF instruments is measurement of zinc phosphate coating on steel (Johnson et al., 1989), where phosphorus x-ray intensity is used rather than zinc to increase the sensitivity of analysis. A precision of 2 mg=cm2 was achieved in a 30 s with gas-ﬁlled proportional detector and a 1.5 GBq (40 mCi) 55Fe source, for up to 450 mg=cm2 coating mass. D.

On-Stream Analysis of Metalliferous Mineral Slurries

Most metalliferous minerals are concentrated from their ores by froth ﬂotation. The grade of ore fed to the concentrator can vary rapidly; hence, to control the ﬂotation process, the concentrations of valuable minerals in the plant process slurries should be determined continuously. The concentrations, in the slurry solids, of such base metal minerals as nickel, copper, zinc, and lead are usually in the range of 0.315 wt% for feed streams, tens of weight % in concentrate streams, and 0.030.3 wt% in tailings (residue) streams. The

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solids weight fraction is in the range of 1550 wt%. The time for analysis (less than 5 min) is too short for laboratory analysis of samples taken from the process streams. A radioisotope x-ray based on the use of scintillation detectors was introduced in 1973 (Watt, 1985). Cesium-137 g-ray transmission is used to determine the bulk density of the slurry and, hence, the solid weight fraction. X-ray ﬂuorescence techniques based on direct excitation and detectorradiator assemblies (Fig. 18) are used to determine the concentration of all but some high-Z elements, such as lead, which are determined by 153 Gd g-ray (about 100 keV) transmission. In each case, the x- or g-ray measurements are combined with the solids determination to obtain the concentration of elements in the slurry solids. These techniques are suﬃciently sensitive for all but a few tailing streams containing very low concentrations of valuable mineral. This limitation was overcome later by the introduction of a solid-state detector probe. The radioisotope x-ray system (Fig. 26) is based on probes, each containing a radioisotope source and a detector, which are immersed directly into the plant process slurries (Watt, 1985). Electrical signals from the probes are fed to an analyzer unit and its output to a central computer. Thus, not only is there no need for sampling from the plant process streams, but all streams are analyzed continuously rather than sequentially. This radioisotope x-ray system is very diﬀerent from x-ray tube and crystal spectrometer systems for on-stream analysis. These systems were developed in the 1960s to scan slurries sequentially in up to 14 simple sampling loops (Leppala et al., 1971). Analysis of the slurries involved sampling from the main process stream, running slurries through long sections of sampling loops to the central analyzer, and subsampling from the sampling loops before presentation to the analyzer. This is a complex and expensive system. X-ray tube analysis systems developed in the 1980s sequentially route the slurries from up to ﬁve process streams through a common ﬂow cell viewed by an XRF analyzing unit (Saarhilo, 1983). The analyzer is mounted in the plant near the process streams and thus overcomes much of the mechanical complexity of the 14-stream system just discussed. Both the x-ray tube crystal spectrometer system and the radioisotope XRF solid-state detector system are routinely used with this new sample loop system. Both systems can also be used for the continuous analysis of ﬁne powders. The radioisotope system is capable of determining elements of atomic number as low as 14 (silicon), because with the short x-ray path length, the absorption of the low-energy x-rays in air is minimized. The development of diﬀerent systems for the on-stream analysis of mineral slurries illustrates the strong inﬂuence of sampling and sample presentation on the type of XRF analysis system used. Various radioisotope (Table 6) and x-ray tube systems are now in use, with about several hundred plant systems being installed in concentrators throughout the world. The radioisotope systems, both in stream (Watt, 1985; Berry et al., 1983) and using short sample loop, and the ﬁve-stream x-ray tube system, are preferred for new installations because they cost less and are mechanically less complex. Improvements in plant control based on this analysis information have led to an increase in the recovery of valuable minerals (Fig. 27), decreased reagent addition, and reduced need for assay and sampling staﬀ (Watt, 1985). Total savings per concentrator vary in the range from US$100,000 to several million dollars a year. E.

On-Line Determination of the Ash Content of Coal

Continuous on-line determination of the ash content of coal on conveyors is required for the control of coal mining, blending, sorting, and preparation operations. The coal is carried by the conveyors, usually at the rate of 100600 tons=h, the speed of the conveyor

Figure 26 A system for the on-stream analysis of mineral slurries. The microcomputer outputs solids weight fraction and concentrations of valuable minerals in the solids of each stream. (Courtesy of AMDL.)

Radioisotope-Excited X-ray Analysis 483

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Figure 27 Improvements in recovery of mineral concentrators after installation of on-stream analysis systems. The symbols refer to tin (diamonds), lead (square), zinc (circles) and copper (asterisks).

is about 3 m=s, and the coal particle top size is in the range 10150 mm, depending on the application. The process requirement for analysis time varies from 1 or 2 s for the fast sorting of coal to 10 min for the steady-state control used in coal preparation plants. Three x-ray or low-energy g-ray techniques have been developed: backscatter of x-rays (H. Fraenkel, private communication, 1987), backscatter of low-energy g-rays (Fauth et al., 1986; Cierpicz, 1986), and dual-energy g-ray transmission (Fookes et al., 1983; Gravitis et al., 1987). Each relies on the photoelectric eﬀect, which depends on atomic number, and on the fact that ash (mainly SiO2 and Al2O3 with smaller concentrations of Fe2O3) has an eﬀective atomic number, Z, greater than that of the coal matter (carbon, hydrogen, nitrogen and oxygen). The x-ray technique (H. Fraenkel, private communication, 1987) depends on the scatter of approximately 17-keV x-rays from 238Pu (or 244Cm) source in the coal and, at the same time, excitation of iron K x-rays in the coal to correct for the high absorption per unit mass by Fe2O3 compared with Al2O3 and SiO2. Because the low-energy x-rays penetrate only thin layers of coal, the coal is sampled from the conveyor, subsampled, and ground to 5-mm top size particles, partially dried, and then presented in a moving stream of controlled geometry to the radioisotope x-ray analysis system. This system compensates for the eﬀect of variations in Fe2O3 in the ash, a signiﬁcant source of uncertainty in some applications. However, it involves complex sampling, sample handling and processing, and blockages occur when the coal is very wet. The low-energy g-ray technique, using an 241Am (60 keV) source, depends on measurement of the intensity of g-ray scattered from thick layers (< 15 cm) of coal. It was ﬁrst used on a high-throughput sample line (Fauth et al., 1986). Although coal must be sampled, there is no need for the coal to be subsampled and crushed because of the high

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penetration of g-rays in the coal. The technique has been adapted for use on-line (Cierpicz, 1986), the analysis head unit riding on a raft that is spring loaded so that it is always touching the top of the coal on the conveyor. Its use is restricted to conveyor speeds of less than 2 m=s and to minimum thickness of 15 cm of coal on the conveyor compared to the normal practice of 520 cm. The dual-energy g-ray transmission technique (Fookes et al., 1983; Gravitis et al., 1987) measures coal directly on the conveyor belt (Fig. 28), eliminating the need for sampling the coal. The ash content is determined independently of vertical segregation of coal on the belt, and if segregation across the belt occurs, the narrow beam of g-rays can be made to scan across the belt to obtain a representative sample. The coal mass per unit area in the g-ray beam must be < 3 or 4 g=cm2 to achieve suﬃcient sensitivity of analysis. Variations of iron in the ash limits the accuracy of ash determination in some applications. The choice of a suitable x- or g-ray analysis technique is highly inﬂuenced by the complexities in sampling of the coal on the conveyor and the subsequent subsampling and grinding. Radioisotope techniques that measure directly on-line are preferred to those involving sampling, and if sampling is necessary, preference is given to those that minimize sampling and sample presentation. Dual-energy g-ray transmission is now the preferred technique for the on-line determination of ash content of coal, except for applications in which unacceptable uncertainties in ash are caused by variations in iron in the ash. In this case, a high-energy g-ray technique, which is based on the pair production interaction and is much less sensitive to variations in iron, is preferred. Its main disadvantages are cost and that it must operate on a sample line (Sowerby, 1986). F.

On-Line Analysis of Paper

Continuous on-line analysis of paper is required for control of the production process. Paper consists of cellulose, water, and mineral ﬁllers. The characteristic and the quality of various types of paper are, to a great extent, dependent on the quality, quantity, and distribution of ﬁller materials. These ﬁller materials occupy the spaces between ﬁbers and improve the printing properties of the paper. Filler materials include CaCO3, kaolin, talc, and TiO2, and the concentrations of each may vary. The analysis may be achieved by a combining XRF, XRA, and b-ray transmission techniques (Kelha et al., 1983). The mass absorption coeﬃcients of x-rays in the 110-keV region are shown in Figure 29, with abrupt changes in the K shell absorption-edge energies of calcium (4.04 keV) and titanium (4.96 keV). X-ray preferential absorption measurements are made at x-ray energies of 5.9 (55Fe), 4.51 (TiKa x-rays), and 3.69 keV (CaKa x-rays), the latter two energies being contained by exciting a secondary target material with 55Fe x-rays (Fig. 30). The total mass per unit area of the paper is determined by b-ray transmission. The distribution of the ﬁller material through the paper is determined by making XRF measurements, using 55Fe x-rays to excite calcium K x-rays, to give the diﬀerence in CaCO3 concentrations near the surface on each side of the paper. These techniques can be used to determine the concentrations of cellulose, kaolin or talc, CaCO3 and TiO2. G.

Determination of Sulfur and Chlorine in Oil

Sulfur in oil is a source of air pollution. Strict environmental controls are applied to limit sulfur release into the atmosphere. Oil in sample lines from main pipelines is monitored

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Figure 28 System for on-line determination of the ash content of coal based on dual-energy g-ray transmission techniques.

routinely by one of two radioisotope x-ray techniques. One technique combines XRF (55Fe and a proportional detector) and b-ray transmission. The other combines XRA, using 22-keV silver K x-rays that are absorbed equally per unit weight by both carbon and hydrogen, with a nonnuclear technique to determine the density of the oil. In Japan alone, more than 100 of the former gauges and several hundred of the latter are used (Table 6). More recently, this application received an application boost from regulatory agencies such as U.S. Environmental Protection Agency. Since October of 1993, no on-highway diesel fuel the United States can contain more than 500 mg=kg of sulfur. This regulation forced sulfur monitoring for compliance at reﬁneries, distributorships, and local suppliers. A proportional detector with neon gas ﬁlling and a thin, 13-mm Be window, coupled with a 1.48-GBq (40 mCi) 55Fe isotope allows sulfur analysis in diesel fuel down to 50 mg=kg, with a precision of 8 mg=kg (1s), in 300 s measurement time (Piorek and Piorek, 1993).

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Figure 29 Mass absorption coefficients of the most frequently used filler materials, water and cellulose, relating to the manufacture of paper. (From Kelha et al., 1983.)

A factor of 5 better results may be obtained by using low-power x-ray tube in the place of the isotope. The issue of sulfur measurement in petroleum products for compliance with the EPA regulations has been addressed by American Society for Testing and Measurement (ASTM), with two ASTM Standard Test Methods, D2622 and D4294. Similar regulations for sulfur in petroleum products for on-highway use are being implemented in Europe and Asia. Spent engine oil is burned as fuel in industrial boilers and home heating furnaces. Because chlorinated solvents have routinely been mixed with much of this oil, there is a danger that hazardous levels of these compounds or their derivatives will be released into the atmosphere. The U.S. Environmental Protection Agency has banned the sale of used oil for fuel if the total halogen level (interpreted as total chlorine level) exceeds 1000 mg=g. If the total halogen level is more than 4000 mg=g, such oil has to be considered as

Figure 30 1983.)

Measuring principle used in the on-line analysis of paper: (a) XRF, (b) XRA, and (c) b-ray absorption. (From Kelha et al.,

488 Piorek

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hazardous waste and cannot be reused. However, if the total halogen level is greater than 1000 but less than 4000 mg=g, it may be diluted with less contaminated oil to reduce the halogen content below 1000 mg=g, after which it can be burned as fuel. Piorek and Rhodes (1987) and Gaskill et al. (1987) showed that a portable XRF analyzer, with a highresolution proportional detector and a 0.74-GBq (20 mCi) 55Fe source, can be used to analyze for chlorine in oil. The used oil also contains phosphorus, sulfur, potassium, and calcium, and the intensities of the K x-rays from all these elements were incorporated into the calibration so that the measured chlorine K x-ray intensity could be properly corrected for spectral overlap. The analysis time was 30 s per sample, and the overall accuracy achieved was quite adequate. H.

Analysis of Environmental Samples

Perhaps the second most important group of applications well served by XRF is in the analysis of a variety of environmental samples. This is especially true of portable XRF analyzers, which, in a very natural way, drifted from mineral prospecting to soil screening for metals; the two applications virtually diﬀer only by name. Successes of XRF in analysis of thin ﬁlms and coatings have been parlayed into measurement of air particulates collected on ﬁlter media. Even much more analytically demanding analysis of water can also be performed with XRF using ion-exchange membranes. All these applications will be discussed. 1. Air Particulates in Ambient Air The concentrations of particulates suspended in ambient air are determined to identify sources of air pollution. The air particulates are collected by drawing large volumes of air through a ﬁlter paper, which is then submitted to multielements analysis. Rhodes et al. (1972) and Florkowski and Piorek (1974) showed that energy-dispersive XRF analysis based on radioisotope sources and a silicon solid-state detector is a simple and costeﬀective method for determining the elemental concentrations on the ﬁlter paper. The advantage of the solid-state detector is that simultaneous multielement analysis is achieved with excellent sensitivity and short analysis time. Analyses were made using 4.44 GBq (120 mCi) 55Fe, 14.8 GBq (400 mCi) 238Pu, and 444 MBq (12 mCi) 109Cd, a 80-mm264-mm-thick silicon detector (energy resolution of 180 eV at 5.9 keV), and counting for 10 min per ﬁlter paper with 238Pu and 109Cd and 5 min with 55Fe. The detection limit (3s of the background under the relevant peak) for the 19 elements measured varied in the range from 0.03 to 0.24 mg=cm2. This is more than adequate for levels required in environmental pollution analysis. Rhodes and Rautala (1983) have since shown that the same radioisotopes, with activities of 111370 MBq (310 mCi), can be used with a high-resolution proportional detector to determine the same elements with detection limits (3s) between 0.4 and 12 mg=cm2 in 4 min. The main competitive techniques are energy-dispersive techniques using x-ray tubes and solid-state detectors (Chapter 3) and particle accelerators in proton-induced x-ray emission (PIXE) techniques (Chapter 12). The energy-dispersive XRF have been used for the last 20 years by various governmental and international institutions for routine monitoring of ambient air pollution. 2. Air Particulates Monitoring in Stack Emissions and Workplace Air The diﬀerence between ambient air monitoring and stack emission or workplace air is only in the levels of concentrations to be measured and frequency of sampling. As in analysis of ambient air by ﬁltering the air onto a thin membrane substrate, a signiﬁcant

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preconcentration of the sample is realized, resulting in improved sensitivity of the XRF analysis. Although the concentrations of metals in stack emissions and in workplace air are much higher than in ambient air, they can only be sampled and analyzed for short periods of time. Nevertheless, despite these restrictions, the portable x-ray analyzer can still provide suﬃcient sensitivity of measurement. The typical application is analysis of metals in deposits obtained with personal samplers for breathing zone air used by workers occupationally exposed to polluted air. Samples of air particulates are collected for 8 h by pumping air through a 37-mm-diameter membrane, at a rate of about 2 L=min. These deposits can then by analyzed by the same portable XRF analyzer which was used for soil screening during remediation eﬀort. Within the last 3 years, the issue of metals monitoring in stack emissions from industrial incinerators has been the subject of regulatory action in the United States and Europe. Regulations in the United States are pending which will limit emissions of toxic metals from incinerators stacks and will require continuous on-line monitoring for compliance. Of the several attempts at Continuous Emission Monitoring (CEM) Systems, XRF proved to be a viable and relatively inexpensive alternative. The proof of principle experiment using XRF was performed by Piorek et al. in 1995 (Piorek et al., 1995) during the EPA-sponsored ﬁeld test in the EPA’s own hazardous waste incinerator in Little Rock, Arkansas. The test showed that particulate deposits obtained by ﬁltering stack exhaust for 10 min on ﬁlter membrane can be readily analyzed by the portable XRF system with radioisotopes as excitation sources (Makov et al., 1968). Table 9 lists the capabilities of the portable analyzer for metals ﬁltered from air on a membrane substrate (Piorek, 1997). As can be seen, the detection limits for air particulates collected on the ﬁlter are on the order of a microgram per cubic meter of air. Further improvement of the performance is possible using x-ray tubes as excitation sources and integrating the analyzer into an continuous on-line monitor.

Table 9 Performance of a Si(Li)-Detector-Based Field Portable X-ray Analyzer in Monitoring Air Particulates in Workplace Air and Industrial Incinerator Exhaust Element Cr Mn Fe Ni Cu Zn As Se Pb Cd Sb Sn a

Detection limit (mg/cm2)

Detection limita (mg/cm3)

PELbin air (mg/cm3)

Detection limit CEMc (mg/cm3)

0.43 0.45 0.45 0.29 0.29 0.27 0.16 0.16 0.24 0.07 0.06 0.05

3.0 3.2 3.2 2.0 2.0 2.0 1.1 1.1 1.7 0.5 0.5 0.4

1000 50 104 (as Fe2O3) 1000 1001000 (15) 103 10 200 50 1003000 500 2000

1.5 ND ND ND ND ND 1.5 ND 3.1 3.1 1.5 ND

For one cubic meter of air (gas) pumped through a Millipore SMWP membrane of 9 cm2 area. The OSHA Permissible Exposure Limit in work place air, PEL (as per 29 CFR 1910.1000, 1 Jan. 1977). c Hazardous waste combustors, revised standards, proposed rule, 40 CFR Part 60 et al., April 19, 1996. ND ¼ not determined; CEM ¼ continuous emission monitor. Source: From Piorek, 1997. b

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The same technique for analyzing thin-layer samples can be used for measurement of metallic ions in water, by cycling the water through an ion-exchange membrane. Piorek and Pasmore (1993) reported the detection limits for transition metals in water at less than 0.1 mg=L if 250 mL of water was cycled seven times through a 37-mm-diameter ionexchange membrane, followed by a 200-s analysis with a 0.37-GBq (10 mCi) 109Cd source. 3. Soil Screening for Inorganic Contaminants On-site analysis of metals in soils, sediments, and mine wastes is required as part of rehabilitation studies at hazardous waste sites. The distribution of the hazardous metals is heterogeneous owing to the exigencies of dumping and possible leaching by rainwater. Many samples must be taken over the surface area and at depths to deﬁne the zones of metal accumulation and metal depletion. Chappell et al. (1986) and Piorek and Rhodes (1987) demonstrated that the portable XRF analyzers based on a high-resolution proportional detector and a 3.7-GBq (100 mCi) 244Cm source can be used to determine concentrations of arsenic, copper, zinc, and lead in soils in the range of approximately from 10 to 10,000 mg=g, with an accuracy that is adequate for this type of investigation. Rapid, on-site analyses can be incorporated immediately into a ﬁeld investigation program, making it possible to change the density of sampling at any spot, depending on the results of previous analyses. Since then, portable XRF analyzers have been improved to include high-resolution semiconductor detector probes and extensively used to measure in situ metals in soil, such as lead, zinc, copper, arsenic, nickel, chromium, and cadmium. In the majority of reported cases, the correlation between the XRF and the conﬁrmatory method results [such as Atomic Absorption Spectrometry (AAS) or Inductively-Coupled Plasma Atomic Emission Spectrometry (ICP-AES)] was very good (correlation coeﬃcient greater than or equal to 0.9) (Piorek, 1994). Cole et al. (1993) report on an extensive lead study conducted during the summer of 1991 in Leadville, Colorado, where over 3700 samples of soil were analyzed for lead using portable XRF analyzers. The accuracy of the lead determination by the XRF in that study was estimated at about 8% relative. Another very important observation from that study is that the accuracy of the determination of spatial distribution of lead in the soil does not depend so much on the accuracy of the individual result as on the total (large) number of samples. In other words, raising the analytical quality of the individual data has very little eﬀect on the total uncertainty; however, increasing the number of samples will considerably improve the accuracy of the determination of spatial distribution of lead. This conclusion has been conﬁrmed by other researchers as well (Raab et al., 1991). The portable XRF analyzer has been successfully used during remediation eﬀorts at the cleanup of soil contaminated with chromium (Waligora, 1997). More than 3000 samples were measured in the ﬁeld, providing an immediate feedback to an excavating crew which was working literally ‘‘on the heels’’ of the ﬁeld analytical team. Figure 31 illustrates typical performance parameters of the contemporary portable XRF analyzer with a Si(Li) detector probe for metals in soil matrix. As it can be seen, the typical detection limits are in the range 50100 mg=kg. The success of portable XRF in soil screening led to coining of a new term of ﬁeld-portable x-ray ﬂuorescence (FPXRF), probably to distinguish this application from the others. The FPXRF analyzer oﬀers an attractive, in situ alternative to the traditional laboratory analysis of contaminated soil. The operator equipped with the portable XRF analyzer can perform a large number of multielement measurements in situ by placing the probe directly on the soil surface, thus allowing for the decisions to be made in real time. Moreover, the measurements can also be performed at any stage of the remediation

Figure 31 Typical detection limits (DLs) for metals in soil obtained with field portable X-ray analyzer equipped with a Si(Li) detector probe. Conditions: Si(Li) detector, 30 mm2 area; measurement time 200 s; sample matrix: silica sand. K series excitation with 109Cd and 241Am isotopes. Open rectangles correspond to 1s uncertainty of counting; solid rectangles mark the DL. The DLs for heavy metals determined via their L series x-rays, such as Pb, Hg, Bi, are numerically similar to those from Zn to Br. (From Piorek, 1994.)

492 Piorek

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process to either verify that the cleanup is accomplished or that the work has to continue. The results obtained with the XRF analyzer will reveal relative variations of contaminant(s), their spatial distribution, and pinpoint locations with elevated contamination levels (so-called ‘‘hot spots’’), thus making sampling plans realistic and eﬃcient. This is particularly important because any contaminated site is likely to have an extremely heterogeneous distribution of hazardous chemicals in the soil. For that reason, the sampling strategies will mandate that many densely distributed samples are taken, in order to ensure that ‘‘hot spots’’ are not missed. This type of analysis, which is referred to as ‘‘screening,’’ is allowed by the EPA to have uncertainties as large as 50% relative (EPA, 1987). Most recently, the oﬃcial recognition of the FPXRF as a valid soil characterization method has materialized in a form of a new EPA Method 6200, which has been published in June of 1997 (EPA, 1997). 4. Analysis of Liquid Hazardous Waste The analysis of liquid hazardous waste (LHW) is required prior to its treatment in order to properly set up the treatment process and, also, after the treatment to determine the ﬁnal disposition of the treated waste. Liquid waste is a mixture of solids suspended in a liquid medium. Solid particles vary considerably in particle size and composition, whereas the liquid phase is often a mixture of immiscible liquids. The sludge remaining after the treatment of liquid waste is either transported to a waste dump or, preferably, utilized for soil amendment in agriculture and forestry because it is often rich in nutrients. However, depending on the origin of the waste, it may also contain a whole range of heavy metals, which even at moderately elevated concentrations, are very toxic to plants and animals. In order to avoid the accumulation of toxic metals in soil, the concentration of these metals in the original waste, as well as in postprocess sludge, has to be known. Traditionally, the liquid waste samples are analyzed by a ﬂame or graphite furnace AAS or ICP-AE. Those methods, while oﬀering low detection limits, are slow and labor intensive. Additionally, they call for rather involved sample preparation (digestion) and are relatively expensive. Energy-dispersive x-ray ﬂuorescence spectrometry was considered as an alternative analytical method for liquid waste due to its known features of speed, multielement character, wide dynamic range, and little to no sample preparation requirement. However, the diversity of form and composition makes LHW an extremely diﬃcult medium to analyze with XRF as a result of the severe matrix eﬀects present in a sample of LHW. The severity of matrix eﬀects in liquid hazardous waste precludes the use of many known calculational models correcting for absorption=enhancement phenomena because mathematical models are valid only for physically and chemically homogeneous materials. To overcome these diﬃculties, a novel empirical approach was proposed for stabilization of the sample matrix by dilution (ASTM, 1997; Piorek et al., 1997), For the purpose of XRF analysis of liquid hazardous waste, a calcined alumina powder (Al2O3) was used as a diluent. This powder is characteristic for its highly adsorptive properties toward liquids. Thus, after adding a small amount of liquid to an aliquot of Al2O3, the latter instantly absorbs the liquid. After brief mixing and shaking, the originally liquid sample is converted to a homogeneous powder. The mass ratio of the diluent to sample has to be maintained constant from sample to sample and should not be smaller than 3:1, otherwise the sample=diluent mixture will behave more like a slurry than a powder. This kind of sample matrix stabilization is eﬀective for liquids as well as liquids with suspended solids. By diluting and mixing, both the liquid as well as the suspended solids are now ‘‘suspended’’ in alumina powder, which does not allow waste solids to settle. Another important function of alumina is that it dominates the matrix of the sample and, therefore,

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the composition diﬀerences between the original samples of waste are reduced to easily manageable levels. For calibration, a set of samples of alumina powder is prepared by spiking with a known amount of standard solutions. The details of the technique are described in the ASTM Standard Method D6052-97 (ASTM, 1997). Using a portable Si(Li)-detector-based probe with a 0.74-GBq (20 mCi) 109Cd source and a 1.11-GBq (30 mCi) 241Am source, the 200-s measurement time detection limits obtained with this method for elements ranging from Cr to As were between 35 and 10 mg=kg, respectively. For heavy elements such as Hg, Pb, Cd, and Sb, detection limits were 12, 8, 4, and 4 mg=kg, respectively. 5. Analysis of Lead in Wall Paint A prolonged exposure to low levels of lead may lead to serious health problems in young children, including impairment of the central nervous system, behavioral disorders, and mental retardation. Paint, dust, soil and, to a lesser extent, food, water, and air are all potential sources of exposure to lead. After banning the use of leaded gasoline in the U.S., lead-based paint is recognized as the principal source of lead in dust and soil. From here it can easily enter a child’s system as the result of hand-to-mouth activity, typical for young children. These concerns were addressed with federal regulations in the early seventies. The essence of those regulations is that any applied lead-based paint exhibiting more than 1.0 mg of lead per square centimeter must be removed, as well as any applied to surface paint that contains more than 0.5% lead by dry weight. In recent years the lead-based paint danger became the focus of attention in European countries. France, for example, introduced guidelines for lead levels in paint and followed with enforcement regulations (Law No. 99-483 of June 9, 1999). It is expected that other countries will follow France’s example in coming years. The preferred method for testing applied paint for lead is FPXRF (McKnight et al., 1989; Piorek et al., 1995). This is because XRF is nondestructive and fast and oﬀers overall good performance. Additionally, lead x-rays, either K or L series, have high penetrating power (especially the Ks), and are not interfered with by the other elements which make up the paint. As the result, the XRF method is quite selective for lead in leaded paint applied on any substrate. A number of accounts on this subject has been published over the years, and a number of portable XRF devices was designed for just lead in wall paint measurements. The majority of the instruments used lead K shell x-rays, excited either with 57 Co or 109Cd (the 88-keV g-rays of this isotope), but some have utilized L shell excitation of lead. The latter approach has been reported to be able to measure lead with a proportional detector and a 2.1-GBq (60 mCi) 244Cm source with a precision of 0.05 mg Pb=cm2. Probably the most successful overall lead in paint analyzer has been Model LPA-1 from Radiation Monitoring Devices. The self-contained instrument in a shape of the lightweight pistol uses a 57Co isotope and CdTe detector to determine the lead concentration in applied paint at a 2s conﬁdence level in 5 s measurement time. The analyzer automatically adjusts the measurement time to reach a preset level of total measurement uncertainty. The instrument is an example of well thought out ergonomical design, responding well to the needs of the market. The analyzer can be preset for a speciﬁc, local lead abatement action level and it can generate a hard-copy analytical report. Another interesting entry in the lead in paint market is the Model XL from the Niton Corporation (1999). Although a marvel of miniaturization, its shape seems less conforming to the requirements expected of a hand-held device operated continuously for several hours. This is also the only commercially available analyzer which uses L shell lead x-rays for analysis and the ratio of the lead La to Lb lines for determination of the depth at

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which leaded paint is present under overcoats of other paint (Shefsky et al., 1997); in those instances in which leaded paint may be buried under overcoats of paint, the instrument uses K shell lead lines for measurement, as there are very little aﬀected by top layers of paint. I.

Analysis of Corrosion Products

An interesting on-line application has recently been reported by Connolly and Millet (1994) and Connolly and Harvey (1995) for corrosion product monitoring in nuclear power plant feedwater. By coupling the XRF analyzer probe to a specially designed sample ﬂow chamber, the authors realized continuous measurement of incremental accumulation of corrosion products such as iron. The feedwater is circulated through the sample ﬂow chamber, where it passes through the membrane ﬁlter. The solids present in the water are collected on the ﬁlter surface and then measured with an XRF probe. The measurement is incremental, so that the ﬁlter does not have to be removed for analysis after each sampling cycle. A gas-ﬁlled proportional detector and 3.7-GBq (100 mCi) 244Cm source allow one to routinely achieve the detection limits for iron of 1 ppb, at a 300-s sampling period followed by 300 s of measurement time. Further work is under way to improve these performance and to extend the applicability of this method to other elements and liquid media. VII.

FUTURE OF RADIOISOTOPE-EXCITED XRF ANALYSIS

As we have stated in the Introduction, radioisotope x-ray analysis is mainly useful in industrial and ﬁeld applications. This trend will continue, although the proportion of online installations for process control to number of ﬁeld-portable x-ray analyzers will constantly decrease. First, it will be the result of a more general contemporary trend in the instrument industry of bringing the analytical method to the site. Second, the number of applications that can beneﬁt from portable XRF analysis is much greater and more diversiﬁed than the number of installations for process control. Third, radioisotope-based on-line installations will gradually be displaced by systems based on x-ray tubes. Last but not least, on-line installations will face ﬁerce competition from new analytical technologies based on lasers, which, in many cases, oﬀer instantaneous results at a high repetition rate. As a result, in the very near future, radioisotope x-ray analysis may become synonymous with ﬁeld-portable XRF and, therefore, the future of radioisotope X-ray analysis will most likely depend on the developments in ﬁeld-portable XRF. The next designs of the portable XRF analyzers will feature new, high-resolutionsemiconductor, room-temperature detectors. Most likely these will be improved detectors based on silicon diodes (‘‘p-i-n’’ detector) or composite semiconductors such as ZnCdTe, CdTe, GaAs, and so forth. A recently introduced x-ray detector (Xﬂash Detector) built on a silicon chip as a solid-state drift chamber oﬀers count-rate capabilities well into 50,000 cps at an energy resolution still better than 200 eV (Ro¨ntec 1997). The portable XRF analyzer of the future will incorporate a GPS (Global Positioning System) module, which—if needed—will instantly locate the measurement site on the map and—if properly equipped—will transmit the measurement results via a wireless link to a central data collection station. An instrument with a GPS chip installed will also be very easy to track down if misplaced or stolen, a very desirable feature for a radioisotope-containing device. The FPXRF will travel underground, as an analytical module built into the push-in probe or in a cone penetrometer. The ﬁrst trials were very encouraging (Elam et al., 1997).

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Technological progress will also bring us truly small x-ray sources, which will gradually replace radioisotopes in applications requiring less than 35-keV-energy x-rays (Moxtek Inc., private information, 1998). We should also expect an increased number of dedicated, ‘‘smart’’ XRF analyzers customized to speciﬁc applications, such as the already mentioned measurement of lead in wall paint. New portable instruments will be smaller, lighter, and ‘‘smarter’’ than their predecessors. Perhaps the ﬂavor of the future direction of development of portable, radioisotope excited XRF is best exempliﬁed by Model XL by Niton Corp. (Fig. 32), a one-piece, miniature XRF analyzer with built-in ADC=MCA, and weighting only about 1.6 kg (Niton Corp., 1999).

VIII.

CONCLUSIONS

X-ray ﬂuorescence spectroscopy is perhaps the ﬁrst spectroscopic technique to successfully enter the ﬁeld and plant environments. Radioisotope x- and g-ray techniques have been widely used for analysis in the laboratory, in industrial plants, and in the ﬁeld. Because of their simplicity, they are preferred to x-ray tube analysis techniques in many applications, particularly for more routine analyses involving a limited range of sample compositions. The techniques are less sensitive, however, than those based on x-ray tubecrystal spectrometer systems and less ﬂexible to widely diﬀerent analysis applications. Radioisotope x- and g-ray techniques have had great impact on industrial and ﬁeld applications. Their use

Figure 32

Niton II, XRF Alloy Analyzer based on a silicon p-i-n diode. (Courtesy Niton Corp.)

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for the continuous on-line analysis in the coating, mineral, coal, paper, and petroleum industries has led to better control of the industrial processes. This, in turn, has led to products that more closely meet speciﬁcations, with consequently large savings in production costs. Portable XRF analyzers have been around for about 30 years, and during that period, their design has been reﬁned and their performance greatly improved. A contemporary, high-resolution, portable XRF analyzer brings to the ﬁeld site not only an excellent performance often matching that of the laboratory instrument but also unsurpassed savings in time and labor, contradicting the popular conviction about the inherent inferiority of portable instrumentation. The use of portable instruments in the ﬁeld (e.g., for the sorting of alloys and for detection of hazardous waste materials) has led to a much more rapid analysis of materials and, hence, to the much wider use of x-ray techniques in ﬁeld analysis. Field-portable x-ray ﬂuorescence is an example of a well-balanced compromise among portability, ruggedness, reliability, and analytical performance. There are not that many analytical techniques that can rival FPXRF in simplicity, speed of operation, and relaxed requirements of sample preparation. Whenever a fast and frequent elemental analysis for inorganics is required, there is a place for the FPXRF analyzer. This applies in particular to scenarios in which a quick veriﬁcation type of analysis is needed. APPENDIX: LIST OF COMPANIES THAT MANUFACTURE RADIOISOTOPE-BASED X-RAYANALYZERS AND SYSTEMS AMDL (Australian Mineral Development Laboratories), P.O. Box 114, Eastwood, SA 5063, Australia ASOMA Instruments, 11675 Jollyville Road, Austin, TX 78759, USA BRGM, BP 6009-45060 Orleans, Cedex 02, France Data Measurement Corporation, P.O. Box 490, Gaithersburg, MD 20877, USA Detora Analytical, P.O. Box 2747, Alliance, OH 44601-0747, USA Electrical Engineering and Automation (EMAG), Katowice, Poland FAG Kugelﬁscher Georg Schafer KG Auf Aktien, Tennenloher Strasse 41, Erlangen, Germany Harrison Copper Systems, Inc., AMF Box 22014, Salt Lake City, UT 84122, USA Harshaw Radiation Detectors, Harshaw=Filtrol Partnership, 6801 Cochran Road, Solon, OH 44139, USA Humboldt Wedag, P.O. Box 2729, 4630 Bochum, Germany Kevex Corporation, P.O. Box 4050, Foster City, CA 94404, USA Metorex International Oy, P.O. Box 85, SF-02631 Espoo, Finland Mineral Control Instrumentation Pty. Ltd., P.O. Box 64, Unley, SA 5061, Australia Mitsubishi Corporation, 6-3, Marunouchi 2-chone, Chiyoda-Ku, Tokyo 100, Japan Niton Corporation, 900 Middlesex Turnpike, Bldg. 8, Billerica, MA 01821, USA Outokumpu Oy, PO Box 85, SF-02201, Espoo, Finland Oxford Instruments, 20 Nuﬃeld Way, Abingdon, Oxon OXI4 1TX, UK Paul Lippke GmbH & Co. KG, Postfach 1760, 5450 Neuwied 1, Germany Radiation Monitoring Devices, 44 Hunt Street, Watertown, MA 02172, USA Ramsey Ltd., 385 Enford Rd., Richmond Hill, Ontario, L4C 3G2, Canada Rigaku Denki, 14-8 Akaoji, Takatsuki-shi, Osaka, Japan Science Applications Inc., 1257 Tasman Drive, Sunnyvale, CA 94089, USA Sentrol Systems Ltd., 4401 Steeles Avenue West, North York, Ontario, Canada, M3N2S4

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Sortex Ltd., Pudding Mill Lane, London E15 2PJ, UK TN Technologies (Texas Nuclear Corporation), 2555 North Interstate Hwy 35, Round Rock, TX 78689, USA Tracor X-ray, Inc., 345 East Middleﬁeld Rd., Mountain View, CA 94043, USA Yokogawa Hokushin Electric Corporation, 9-32 Nakacho 2-chome, Musashino-shi, Tokyo, 180, Japan

ACKNOWLEDGMENTS The author thanks the many scientists, organizations, and companies who generously supplied information used in this chapter and to the following for permission to reprint ﬁgures used in this chapter: Pergamon Press (Figs. 2, 14, 18, 21, 22, and 28); EG&G Ortec (Fig. 1); The Radiochemical Centre, Amersham (Fig. 4); the International Atomic Energy Agency (Fig. 7); The Analyst (Fig. 20); Outokumpu Oy (now Metorex Int.) (Figs. 11 and 23); IEEE (Fig. 16); Automatica (Fig. 25); Australian Mineral Development Laboratories (Fig. 26); The Australasian Institute of Mining and Metallurgy (Fig. 27); and Acta Polytechnica Scandinavia (Figs. 29 and 30).

REFERENCES Amersham. Industrial Gauging and Analytical Instrumentation Sources, Radiochemical Centre, UK, Catalogue: 1986, pp 15, 5960. Amptek Inc. Product Summary Catalogue. Bedford MA: Amptek Inc., 1997. ASTM. Method D6052-97, Standard Test Method for the Preparation and Elemental Analysis of Liquid Hazardous Waste by Energy Dispersive X-ray Fluorescence. Philadelphia: ASTM, 1997. Berry PF. Developments in Design and Application of Field-Portable XRF Instruments for On-Site Alloy Identiﬁcation and Analysis. 40th ASNT National Fall Conference, Atlanta, 1981. Berry PF, Garber W, Blake K. Proceedings of 11th Annual Mining and Metallurgy Industries Symposium, Tucson, AZ, 1983, pp 3142. Berry PF, Voots GR. In Boogaard, J, Van Dijk GM, eds. Non-Destructive Testing (Proceedings of the 12th World Conference on Non-Destructive Testing, Amsterdam, The Netherlands, April 2328, 1989). Amsterdam: Elsevier Science Publishers, 1989, p 737. Chappell RW, Davis AO, Olsen RL. Proc. National Conf. On Management of Uncontrolled Hazardous Waste Sites, Washington, DC, 1986, p 115. Cierpicz S. In Gamma, X-ray and Neutron Techniques for the Coal Industry. Vienna: IAEA, 1986, p 149. Cole WH, Kuharic CA, Singh AK, Gonzales D, Melon AB. An X-Ray Fluorescence Survey of Lead Contaminated Residential Soils in Leadville, Colorado, Case Study. EPA Report EPA=600=R-93=073, Washington, DC: Oﬃce of Research and Development, 1993. Connolly D, Millett P. Ultrapure Water J 2:61, 1994. Connolly D, Harvey S. Ultrapure Water J 11:28, 1995. Cuzin M. Nucl Instrum Methods A253:407, 1987. David M. Geostatistical Ore Reserve Estimation. Amsterdam: Elsevier, 1977. EG&G Ortec. Catalogue: Nuclear Instruments and Systems. EG&G Ortec, 1986, pp 316, 320. EG&G Ortec. Portable Assay Instruments for Detection and=or Measurement of Ore Values, EG&G Ortec, undated. Elam WT, Adams J, Hudson KR, McDonald B, Eng D, Robitaille G, Aggarwal I. In Proceedings of the Fifth International Symposium on Field Analytical Methods for Hazardous Wastes and Toxic Chemicals, Pittsburgh: Air and Waste Management Association, 1997, p 681.

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Ellis WK, Fookes RA, Gravitis VL, Watt JS. Int J Appl Radiat Isot 20:691, 1969. EPA. US EPA Data Quality Objectives for Remedial Response Activities—Development Process, EPA=540=G-7=003. Washington, DC: EPA, 1987. EPA, Method 6200, Field Portable X-Ray Fluorescence Spectrometry for the Determination of Elemental Concentrations in Soil and Sediment. Washington, DC: EPA, 1997. Fauth G, Leininger D, Ludke H. In Gamma, X-ray and Neutron Techniques for the Coal Industry. Vienna: IAEA, 1986, p 165. Florkowski T, Piorek S. Nukleonika 19 (10):891, 1974. Fookes RA, Gravitis VL, Watt JS. Anal Chem 47:589, 1975. Fookes RA, et al. Int J Appl Radiat Isot 34:63, 1983. Gaskill A, Estes ED, Hardison DL. Evaluation of Techniques for Determining Chlorine in Used Oils, Vol. 1. Research Triangle Institute Project Number 472U-3255-05, 1987. Gravitis VL, Greig RA, Watt JS. Australas Instit Min Metall Proc 249:1, 1974. Gravitis VL, Watt JS, Muldoon LJ, Cochrane EM. Nucl Geophys 1:111, 1987. Gy PM. In Developments in Geomathematics, Vol. 4. New York: Elsevier, 1982. Hietala M, Viitanen J. In Rood CO et al., eds. Advances in X-ray Analysis, Vol. 21. New York: Plenum Press, 1978, p 193. Hill RF, Garber W. IEEE Trans Nucl Sci NS-25:790, 1978. Hoﬀmann P. Fresenius Z Anal Chem 323:801, 1986. ICRP. Quantitative Bases for Developing a Uniﬁed Index of Harm. Annals of the ICRP Publication 45, Oxford: Pergamon Press, 1985. Ingamells CO, Pitard FF. Applied Geochemical Analysis. New York: WileyInterscience, 1986. Isaaks EH, Srivastava RM. An Introduction to Applied Geostatistics. New York: Oxford University Press, 1989. Jarvinen M-L, Sipila H. In Cohen et al., eds. Advances in X-ray Analysis, Vol. 27. New York: Plenum Press, 1984a, p 539. Jarvinen M-L, Sipila H. IEEE Trans Nucl Sci NS-31:356, 1984b. Jenkins R, Gould RW, Gedcke D. Quantitative X-ray Spectrometry. New York: Marcel Dekker, 1981a. Jenkins R, Gould RW, Gedcke D. Quantitative X-ray Spectrometry. New York: Marcel Dekker, 1981b, p 94. Jenkins R, Gould RW, Gedcke D. Quantitative X-ray Spectrometry. New York: Marcel Dekker, 1981c, p 132. Jenkins R, Gould RW, Gedcke D. Quantitative X-ray Spectrometry. New York: Marcel Dekker, 1981d, pp 120, 129, 192. Johnson G, Kalnicky D, Lass B. Ind Finishing 1, Jan. 1989. Kelha V, Luukkala M, Tuomi T. Acta Polytech Scand Appl Phys Ser 138:90, 1983. Knoll GF. Radiation Detection and Measurement. 3rd ed. New York: Wiley, 1999. Law No. 99-483 of June 9, 1999, J.O., June 11, 1999, p 8544 (French Law). Leppala A, Koskinen J, Leskinen T, Vanninen P. Trans Soc Mining Eng. AIME 250:261, 1971. Lucas-Tooth HJ, Price BJ. Metallurgia 54(363):149, 1962. Madden NW, Hanepen G, Clark BC. IEEE Trans Nucl Sci NS-33:303, 1986. Makov VM, Losev NF, Pavlinski GV. Zavod Lab 34(12):1459, 1968. Martin A, Harbison SA. An Introduction to Radiation Protection. London: Chapman & Hall, 1986. McKnight ME, Byrd WE, Roberts WE, Lagergren ES. Methods for Measuring Lead Concentrations in Paint Films. NIST Report NISTIR 89-4209, NIST, December 1989. Niton Corp. Niton II Alloy Analyzer, Product Brochure. Billerica, MA: Niton Corp., 1999. Outokumpu Mintec Oy. Beltcon 100 GS: Technical Description. Espoo, Finland: Outokumpu Mintec Oy, 1986. Piorek S. Adv X-ray Anal 32:239, 1989. Piorek S. In Proceedings of the 1994 Symp. on Radiation Measurements and Applications, 1994, p 528. Piorek S. Field Anal Chem Technol 1(6):317, 1997.

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Piorek S, Rhodes JR. In Proceedings of ISA-1986, Houston, 1986, p 1355. Piorek S, Rhodes JR. In Proceedings of 15th Environmental Symposium, Long Beach, CA, 1987, pp 292297. Piorek S, Pasmore JR. In Proc. of the 3rd Int. Symp. on Field Screening Methods for Hazardous Wastes and Toxic Chemicals. Pittsburgh: Air and Waste Management Association, 1993, p 1135. Piorek S, Piorek E. Measurement of Sulfur in Diesel Fuel Oil in the Range Below 1000 mg=kg Using Bench Top Energy Dispersive X-ray Analyzer. 1993 Gulf Coast Conference, Corpus Christi, TX, 1993. Piorek S, Pasmore JR, Lass BD, Koskinen J, Sipila H. In Breen JJ, Stroup CR, eds. Lead Poisoning, Exposure, Abatement, Regulation. Boca Raton, FL: CRC Press, 1995, p 127. Piorek S, Piorek E, Johnson G. In Proceedings of the Fifth International Symposium on Field Analytical Methods for Hazardous Wastes and Toxic Chemicals. Pittsburgh: Air and Waste Management Association, 1997, p 842. Piorek S, Ojanpera J, Piorek E, Pasmore JR. In Green RE, ed. Nondestructive Characterization of Materials VIII, New York: Plenum Press, 1998, p 461. Pitard FF. Pierre Gy’s Sampling Theory and Sampling Practice. 2nd ed. Boca Raton, FL: CRC Press, 1993. Raab GA, Enwall RE, Cole III WH, Faber ML, Eccles LA. In Simmons MS, ed. Hazardous Waste Measurements. Ann Arbor, MI: Lewis Publishers, 1991, p 159. Rhodes JR. Analyst 91:683, 1966. Rhodes JR. In X-ray and Electron Probe Analysis. Philadelphia: ASTM, 1971, p 243. Rhodes JR, Rautala P. In Clayton CG, ed. Nuclear Geophysics Oxford: Pergamon Press, 1983, p 333. Rhodes JR, Pradzynski AH, Sieberg RD. ISA Trans 11(4):337, 1972. Ro¨ntec. Xﬂash Detector, Ro¨ntec Info, No 4. Berlin: Ro¨ntec GmbH, 1997. Saarhilo K. Experiences of a New On-Stream X-ray Analyzer in a Metal Reﬁnery. In Proceedings of IFAC Automation in Mining, Mineral and Metal Processing, Helsinki, 1983, pp 357367. Schatzler HP. Int J Appl Radiat Isot 30:115, 1979. Schlesinger TE, James RB, eds. Semiconductors for Room-Temperature Nuclear Detectors, Semiconductors and Semimetal Series. Vol. 43. New York: Academic Press, 1995. Scitech Corporation, MAP Portable Assayers: Speciﬁcations and Technical Information, Kennewick, WA: Scitech Corporation, undated. Shefsky S. Proceedings of the Fifth International Symposium on Field Analytical Methods for Hazardous Wastes and Toxic Chemicals. Pittsburgh: Air and Waste Management Association, 1997, p 195. Shefsky S, Pesce J, Martin K. Proceedings of the Fifth International Symposium on Field Analytical Methods for Hazardous Wastes and Toxic Chemicals. Pittsburgh: Air and Waste Management Association, 1997, p 490. Siegbahn K. ed. Alpha-, Beta- and Gamma-ray Spectroscopy. Amsterdam: North-Holland, 1965. Vol. 1, p 38. Sowerby BD. Gamma, X-ray and Neutron Techniques for the Coal Industry. Vienna: IAEA, 1986, p 131. Spatz R, Lieser KH. Fresenius Z Anal Chem 288:267, 1977. Spiegel FX, Horowitz E. Instruments for the Sorting and Identiﬁcation of Scrap Metal. Baltimore, MD: The Johns Hopkins University, Center for Material Research, 1981. Waligora MK. Proceedings of the Fifth International Symposium on Field Analytical Methods for Hazardous Wastes and Toxic Chemicals. Pittsburgh: Air and Waste Management Association, 1997, p 815. Watt JS. Int J Appl Radiat Isot. 23:257, 1972. Watt JS. Practical Aspects of Energy Dispersive X-ray Emission Spectrometry. Vienna: IAEA, 1978, p 135. Watt JS. In Clayton CG, ed. Nuclear Geophysics. Oxford: Pergamon Press, 1983, p 309. Watt JS. Proc Aust IMM 290:57, 1985. Watt JS, Steﬀner EJ. Int J Appl Radiat Isot 36:867, 1985.

8 Synchrotron Radiation-Induced X-ray Emission Keith W. Jones Brookhaven National Laboratory, Upton, New York

I.

INTRODUCTION

Elemental analysis using the emission of characteristic x-rays is a well-established scientiﬁc method. The success of this analytical method is highly dependent on the properties of the source used to produce the x-rays. X-ray tubes have long existed as a principal excitation source, but electron and proton beams have also been employed extensively. The development of the synchrotron radiation x-ray source that has taken place during the past 40 years has had a major impact on the general ﬁeld of x-ray analysis. Even after 40 years, the science of x-ray analysis with synchrotron x-ray beams is by no means mature. Improvements being made to existing synchrotron facilities and the design and construction of new facilities promise to accelerate the development of the general scientiﬁc use of synchrotron x-ray sources for at least the next 10 years. The eﬀective use of the synchrotron source technology depends heavily on the use of high-performance computers for analysis and theoretical interpretation of the experimental data. Fortunately, computer technology has advanced at least as rapidly as the x-ray technology during the past 40 years and should continue to do so during the next decade. The combination of these technologies should bring about dramatic advances in many ﬁelds where synchrotron x-ray science is applied. A short summary of the present state of the synchrotron radiation-induced x-ray emission (SRIXE) method is presented here. Basically, SRIXE experiments can include any that depend on the detection of characteristic x-rays produced by the incident x-ray beam from the synchrotron source as they interact with a sample. Thus, experiments done to measure elemental composition, chemical state, crystal structure, and other sample parameters can be considered in a discussion of SRIXE. It is also clear that the experimentalist may wish to use a variety of complementary techniques for study of a given sample. For this reason, a discussion of computed microtomography (CMT) and x-ray diﬀraction is included here. It is hoped that this present discussion will serve as a succinct introduction to the basic ideas of SRIXE for those not working in the ﬁeld and possibly help to stimulate new types of work by those starting in the ﬁeld as well as by experienced practitioners of the art.

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The topics covered include short descriptions of (1) the properties of synchrotron radiation, (2) a description of facilities used for its production, (3) collimated microprobes, (4) focused microprobes, (5) continuum and monoenergetic excitation, (6) detection limits, (7) quantitation, (8) applications of SRIXE, (9) CMT, and (10) chemical speciation using x-ray absorption near-edge structure (XANES) and extended x-ray absorption ﬁne structure (EXAFS). An eﬀort has been made to cite a wide variety of work from diﬀerent laboratories to show the vital nature of the ﬁeld. There are many review articles and books that cover all aspects of the production and use of synchrotron radiation (Kim, 1986; Kunz, 1979; Margaritondo, 1988; Winick and Doniach, 1980). The early article on SRIXE by Sparks (1980) is a useful source of information on various aspects of the use of SRIXE as part of a high-resolution x-ray microscope (XRM) system. Several more recent review articles have covered new developments (Chen et al., 1990; Jones and Gordon, 1989; Kno¨chel, 1990; Vis, 1990; Vis and van Langevelde, 1991). Microscopy using x-rays can be carried out in several ways. A large eﬀort is aimed at producing extremely high-resolution maps of the linear attenuation coeﬃcient for low-energy x-rays in biological materials (Sayre et al., 1988). Another approach is based on the detection of electrons emitted from the specimen (Ade et al., 1990). Elemental detection can be accomplished by mapping above and below the absorption edge for the element. Rarback et al. (1987) have shown that this may be a preferable method for elements lighter than calcium because of the small values of the ﬂuorescence yield. This approach has been used by Kenney et al. (1985) to study the calcium distribution in a bone specimen using the Ca L absorption edge for the image formation. Image formation using ﬂuorescent x-rays is advantageous for the high-Z elements in terms of attaining the best possible values for minimum detection limits (DLs) combined with the best possible spatial resolution. It is not possible to use x-ray detection in some situations where the thickness of the specimen is much larger than the absorption depth of the ﬂuorescent x-rays. In this case also, the use of the above- and below-edge imaging using the techniques of computed microtomography is possible. In the discussion that follows, the use of all these methods is disscussed although the major emphasis is placed on SRIXE. Images may also be formed using other types of information obtained through probing the sample. X-ray diﬀraction can be used to determine crystal structure. Information on the chemical state of the elements and position in the lattice can be obtained by use of XANES or EXAFS. These are, of course, methods widely applied with both conventional and synchrotron x-ray sources. A merging of these methods with technologies for producing micrometer-sized beams is a natural direction for the development of the instrumentation to follow. This is beginning to happen now, and the rate of development is bound to increase in the future. No matter what mode is used for image formation or to characterize a single volume element in a specimen, the work is dominated by the fact that only limited numbers of atoms are present to produce a signal. As an example, consider the number of zinc atoms present in an organic material as a function of the specimen volume probed for a constant weight fraction of one part per million (ppm). Values are shown in Figure 1, and it can be seen that there are only 104 atoms present in a 1-mm3 volume. The detection of such a small number is a technical challenge no matter what mode of detection is used.

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Figure 1 The number of zinc atoms contained in a given volume element is plotted as a function of volume. A constant zinc concentration of 1 ppm contained in an organic matrix with a density of 1 g=cm3 is assumed. This shows that the detection of trace amounts of an element at a given concentration level becomes increasingly difficult as the probe decreases. (From Jones KW, Pounds JG. Biol Trace Element Res 12:3, 1987.)

II.

PROPERTIES OF SYNCHROTRON RADIATION

It has been known for almost 100 years that the acceleration of a charged particle will result in the radiation of electromagnetic energy. The development of the betatron and synchrotron electron accelerators 50 years later led to the experimental observation of radiation from electrons circulating in a closed orbit (Elder et al., 1947) and to naming it speciﬁcally synchrotron radiation after the accelerator used to produce it. The ﬁrst recognition of the unique properties of the synchrotron radiation by Tombulian and Harman (1956) then brought about an explosive development of activity in constructing improved sources for production of the radiation and using the radiation in experimental science. The original synchrotrons were designed for use in nuclear physics research. Later, facilities were designed to optimize the conditions for production of x-rays. The components of the new facilities include a source of electrons or positrons and an accelerator to produce high-energy beams. This might be done by the use of a linear accelerator to produce energies of around 100 MeV. These beams are then injected into a synchrotron and boosted to energies in the GeV range. Finally, the beam is stored in the same accelerator used to attain the ﬁnal energy or in a separate storage ring. An acceleration ﬁeld in the ring is used to supply energy to the beam to compensate for the radiation energy loss.

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The lifetime of the stored beam is many hours, so that, in practice, the synchrotron source is almost starting to become similar to a standard x-ray tube in use. The synchrotron-produced x-ray beams have unique properties that make them desirable for use. They have a continuous energy distribution so that monoenergetic beams can be produced over a wide range of energies. The photons are highly polarized in the plane of the electron beam orbit, which is extremely important for background reduction in SRIXE-type experiments in particular. The x-rays are emitted in a continuous band in the horizontal direction but are highly collimated in the vertical direction. It is therefore possible to produce intense beams with little angular divergence. The source size is small and, as a result, the production of intense beams of small area is feasible. The synchrotron source is a pulsed source because of the nature of synchrotron-type accelerators. The x-rays are produced in narrow bursts, less than 1 ns in width, and have a time between pulses of around 20 ns or more. The main parameters of interest in deﬁning the synchrotron source are as follows: 1.

2.

Magnitude of the stored electron=positron current. Typically, currents are in the range from 100 to 1000 mA. The lifetimes of the stored beam are many hours. The lifetimes for stored electrons at the National Synchrotron Light Source (NSLS) at Brookhaven National Laboratory, (Upton, NY) have typical values of around 24 h. At Laboratoire pour L’Utilisation de Radiation Electromagnetique (LURE), Orsay, France, where positron beams are used, the lifetimes are even longer because the positive beam does not trap positively charged heavy ions produced from the residual gases in the vacuum chamber. Assuming that the size and angular divergence of the electron beam are not important, the source brightness is deﬁned as [in units of photons=s=mr2=(0.1% bandwidth)]: d 2I o ¼ 1:327 1013 E2 ðGeVÞiðAÞH2 dy dc oc

ð1Þ

where E is the energy of the electron beam (in GeV), i the current (in A), H2 (o=oc) is a function tabulated in Kim (1986), o is the photon angular frequency, and oc is the critical frequency which splits the emitted power into halves and is given by the expression 3g3c=2r, where g is the electron energy in units of the electron rest mass, c is the velocity of light, and r is the radius of curvature of the electron path. The angles y and c are the angles of emission in the plane of the electron orbit and perpendicular to that plane, respectively. 3. The total photon emission is found by integrating over c and is given by [in units of photons=s1=mr1=(0.1% bandwidth)] dI o ¼ 2:457 1013 EðGeVÞ iðAÞ G1 dy oc

ð2Þ

For electron beams with nonzero emittance (ﬁnite area and angular divergence), it is necessary to deﬁne another quantity, the brilliance, which is the number of photons emitted into angular intervals dy and dc at angles y and c from an inﬁnitesimal source area (in units of photons=s1=mr2=mm2=(0.1% bandwidth)]. The values of brilliance and brightness are important in evaluating the performance expected for focused and collimated x-ray microscopes. Kim (1986) provides a detailed discussion of the emittance eﬀects, polarization, and performance of wiggler and undulator insertion devices.

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Values attained for these quantities at the 2.5-GeV x-ray ring of the NSLS are shown in Figures 2–4. Its x-ray beams are typical of those produced by second-generation synchrotron storage rings. The ring energy is high enough to produce x-rays over an energy range suﬃcient to produce K x-rays from elements to about Z ¼ 40 with good eﬃciency and L x-rays throughout the periodic table. Thus, it is highly suitable for use as the basis of a system for x-ray microscopy-based SRIXE. The brilliance obtained from two third-generation synchrotrons is shown in Figures 5 and 6 as a function of the photon energy for several types of devices. The brilliance obtained from an undulator insertion device at the 7-GeV Argonne National Laboratory Advanced Photon Source is shown in Figure 5. The brilliance for diﬀerent bending magnet, undulator, and wiggler devices at the 8-GeV Spring-8 facility in Japan is given in Figure 6. It can be seen that the third-generation storage rings have gained approximately three to four orders of magnitude in brilliance when compared to that found from bending magnets at the NSLS. Similar gains are found for third-generation rings operating at lower energies although the spectrum of x-rays is naturally aﬀected by the energy of the stored electron=positron beam. This increase translates to a gain in x-ray intensity that enables new types of x-ray ﬂuorescence experiments. The high degree of linear polarization of the x-rays from the synchrotron source is a major factor in making the synchrotron XRM a sensitive instrument. The physics describing the interaction of polarized x-rays with matter is therefore an important topic. Hanson (1990)

Figure 2 The brilliance of the NSLS x-ray ring is plotted as a function of the x-ray energy produced. The brilliance is important in defining limitations on production of x-ray images using focusing optics. (From Jones KW, et al. Ultramicroscopy 24:313, 1988.)

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Figure 3 The brightness [defined in Eq. (1)] of the NSLS x-ray ring as a function of the x-ray energy produced. The brightness is of importance in defining the usefulness of x-ray beams produced by use of a collimator. (From Jones KW, et al. Ultramicroscopy 24:313, 1988.)

has carried out an extensive examination of the scattering problem and given methods for assessing particular geometries used in the XRM. A second point is the continuous x-ray energy spectrum extending from energies in the infrared to hard x-rays with energies of several hundred kiloelectron volts. The tunable energy is important for many types of experiments. III.

DESCRIPTION OF SYNCHROTRON FACILITIES

There are now many synchrotron facilities located around the world that are suitable for use in various types of x-ray spectrometry measurement. They fall roughly into three classes. First-generation synchrotrons were built primarily as high-energy physics machines and were used secondarily for synchrotron radiation production. The Cornell High Energy Synchrotron Source (CHESS) at Ithaca, NY is an example. Second-generation synchrotrons were optimized as radiation sources, and as a result, produce x-ray beams with superior brilliance and brightness characteristics. The two rings of the Brookhaven NSLS fall in this category. Finally, the third generation of synchrotrons is now being designed. They will be the ﬁrst sources intended to incorporate insertion devices, wigglers, and undulators in the design phase. In some cases, the ring energy is increased to give better performance with the insertion devices. Synchrotrons designed with these features will begin operation in the latter part of the 1990s. The European Synchrotron Radiation Facility (ESRF) at Grenoble,

Synchrotron Radiation-Induced X-ray Emission

507

Figure 4 The x-ray flux produced by the NSLS x-ray ring is plotted as a function of the x-ray energy produced. The values are given after integration over the vertical opening angle of the beam. The definition of the flux is given in Eq. (2). (From Jones KW, et al. Ultramicroscopy 24:313, 1988.)

France, the Super Photon Ring-9 GeV (SPring-8) in Kansai, Japan, and the Advanced Photon Source (APS) at Argonne National Laboratory, Argonne, IL, illustrate this case. A listing of a number of synchrotron laboratories producing high-energy x-ray beams suitable for use in SRIXE is given in Tables 1–3 (Fuggle, 1990; Jackson, 1990; Winick, 1989, 1990). The number of XRM beam lines is growing rapidly and their employment for research as a result is becoming widespread.

IV.

APPARATUS FOR X-RAY MICROSCOPY

The apparatus that is used for x-ray microscopy measurements varies substantially from laboratory to laboratory. A schematic diagram of the components of a very comprehensive system is shown in Figure 7. All the components are not necessarily used in a speciﬁc instrument. The most important diﬀerences between systems lies in the treatment of the incident beam. The simplest approach is to use the white beam (full-energy spectrum) and a collimator. The more complex systems use focusing mirrors to collect more photons and demagnify the beam and monochromators to produce monoenergetic beams. At present, the performance of the various systems is quite comparable in terms of spatial resolution and detection limits (DLs). Thus, a versatile and ﬂexible approach to the

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Figure 5 The brilliance obtained from an undulator insertion device at the 7-GeV Argonne National Laboratory Advanced Photon Source. (From Advanced Photon Source Research, ANL=APL=TB-31 1(1):5, 1998. Argonne National Laboratory, managed and operated by The University of Chicago for the U.S. Department of Energy under Contract No. W-31-109-ENG-38.)

Figure 6 The brilliance for different bending magnet, undulator, and wiggler devices at the 8-GeV SPring-8 facility in Japan. (From Kitamura H. SPring-8 Annual Report 1994, JASRI, 1995, p 47.)

Synchrotron Radiation-Induced X-ray Emission Table 1

509

First-Generation Synchrotron Light Sourcesa

Storage ring (Lab)

Energy (GeV)

Location

ADONE (LNF) DCI (LURE) VEPP-3 (INP) BEPC (IHEP) SPEAR (SSRL)

1.5 1.8 2.2 2.2–2.8 3.0–3.5

Frascati, Italy Orsay, France Novosibirsk, USSR Beijing, China Stanford, USA

ELSA (Bonn Univ.) DORIS II (HASYLAB) VEPP-4 (INP) CESR (CHESS) Acc. Ring (KEK) Tristan (KEK)

3.5 3.5–5.5 5.0–7.0 5.5–8.0 6.0–8.0 25–30

Bonn, Germany Hamburg, Germany Novosibirsk, USSR Ithaca, USA Tsukuba, Japan Tsubuka, Japan

URL www.lure.u-psud.fr

www-ssrl.slac.stanford.edu= welcome.html www-elsa.physik.uni-bonn.de www-hasylab.desy.de www.chess.cornell.edu www.kek.jp

Note: Lists of synchrotron sites can be found from many of the home pages listed. a Most of these early facilities were built to do research with the primary electron or positron beams. Synchrotron radiation research was parasitic. Today, all have become at least partly dedicated as synchrotron light sources [see Winick (1989, 1990), Jackson (1990), Fuggle (1990)].

Table 2

Second-Generation Synchrotron Light Sourcesa

Storage ring (Lab) CAMD (LSU) DELTA (DU) NSLS x-ray (BNL) Photon Fac. (KEK) Synch. Rad. Source (SRS)

Energy (GeV)

Location

URL

1.4 1.5 2.5 2.5 2.0

Baton Rouge, USA Dortmund, Germany Upton, USA Tsukuba, Japan Daresbury, UK

www.camd.lsu.edu prian.physik.uni-dormund.de www.nsls.bnl.gov www.kek.jp www.dl.ac.uk

Note: Lists of synchrotron sites can be found from many of the home pages listed. a This is the ﬁrst generation of machines built to be dedicated synchrotron radiation facilities. The use of bending magnet ports is emphasized although some straight sections have insertion devices (e.g., undulators and wigglers) [see Winick (1989, 1990), Jackson (1990), Fuggle (1990)].

choice and design of the components is advisable. A number of the diﬀerent instruments are described brieﬂy in the following subsections. A.

Collimated X-Ray Microscopes

A highly eﬀective XRM system can be made by simply collimating the white beam (continuous energy) of x-rays produced by the synchrotron. This approach has been followed mainly by groups at the Hamburg storage ring (Kno¨chel et al., 1983), HASYLAB, and at the Brookhaven NSLS (Jones et al., 1988). The use of white radiation is feasible because the high brightness gives a high ﬂux of photons in a small area, and the high polarization of the synchrotron beams minimizes scattering from the sample into the detector. White radiation also makes possible eﬃcient multielement detection over a very broad range of atomic numbers. The collimators for the instrument can be made from a set of four polished tantalum strips. The strips can be spaced apart with thin plastic or metal foils to produce apertures usable to a beam size of 1 mm. Alternatively, the slits can be attached to individual stepper

510 Table 3

Jones Third-Generation Synchrotron Light Sourcesa

Light source

Energy (GeV)

Location

ALS ASTRID BSRF

1.0–1.9 0.58 1.4–1.55

Berkeley, USA Aarhus, Denmark Beijing, China

CLS SLS SRC SRRC INDUS II MAX II BESSY II ELETTRA LNLS PLS SIBERIA II ESRF APS SPring-8

2.5–2.9 2.1 0.8–1.0 1.3 1.4 1.5 1.5–2.0 1.5–2.0 2.0 2.0 2.5 6.0 7.0 8.0

Saskatoon, Canada Villigen, Switzerland Stoughton, USA Hsinchu, ROC Indore, India Lund, Sweden Berlin, Germany Trieste, Italy Campinas, Brazil Pohang, Korea Moscow, USSR Grenoble, France Argonne, USA Kansai, Japan

URL www-als.lbl.gov www.dfi.aau.dk solar.rtd.utk.edu= china=ins= IHEP=bsrf=bsrf.html sal.usask.ca=cls=cls.html www.psi.ch=sls www.src.wisc.edu 210.65.15.200=en=index.html www.maxlab.ln.se www.bessy.de www.elettra.trieste.it www.lnls.br

www.esrf.fr www.epics.aps.anl.gov www.spring8.or.jp

Note: Lists of synchrotron sites can be found from many of the home pages listed above. a This is the newest generation of dedicated synchrotron radiation facilities. The use of insertion devices (e.g., undulators and wigglers) is emphasized, but bending-magnet ports will also be available [see Winick (1989, 1990), Jackson (1990) Fuggle (1990)].

motors for producing variable-sized beams of size greater than about 10 mm. The collimation approach is ultimately limited by beam-spreading related to ﬁnite source and pinhole dimensions and to diﬀraction. The collimated XRM (CXRM) was ﬁrst operated at the now defunct Cambridge Electron Accelerator by Horowitz and Howell (1972). They used a pinhole made by the evaporation of a thick gold layer around a 2-mm quartz ﬁber, followed by subsequent etching away of the quartz. The x-ray energy was about 2.2 keV, and image contrast was achieved using determination of the linear attenuation coeﬃcient. The specimen was moved past the incident beam, and ﬂuorescent x-rays were detected with a proportional counter. A spatial resolution of 2 mm was measured with this pioneering apparatus. Later versions of the CXRM have been put into operation at Hamburg and Brookhaven. Some of the details of the operations are given to illustrate some of the more important operational details. It is often best to maximize the photon ﬂux on the sample by employing a white beam of x-rays. The ﬂux available at a point 9 m from a NSLS bending magnet source is shown in Figure 8. The ﬂux from this source, integrated over the entire energy spectrum, is about 36108 photons=s=m2 with a 100-mA stored electron current. For best operation of the CXRM, care must be taken in shaping the incident x-ray spectrum using ﬁlters. The inﬂuence of the beam ﬁlters has been studied by the Hamburg group (Kno¨chel et al., 1983). Their results are shown in Figure 9. By varying the eﬀective energy, it is possible to tailor the beam to give the best possible DL for a given atomic number. Filters on the detector can be used in some cases to reduce the eﬀect of a major

Synchrotron Radiation-Induced X-ray Emission

511

Figure 7 A comprehensive synchrotron beam line designed for use as a x-ray microscope. Not all the components would be utilized at a given time in practice. The rather varied uses of the system require a flexible approach to the design of the equipment.

element. In experiments that examine the distribution of trace elements in bone, a ﬁlter of polyimide can reduce the high rates caused by the calcium in the bone, but it will not have a large eﬀect on the x-rays from iron, copper, and zinc. An equally critical task is the alignment of the energy-dispersive x-ray detector. Kwiatek et al. (1990) have reported on this phase of the optimization procedure. The importance of aligning the detector in the horizontal plane can be seen by reference to Figure 10. In addition, the energy spectrum and degree of polarization change as a function of the vertical distance from the plane of the electron orbit in the storage ring. Figure 11 shows the relative photon ﬂux for the two polarization states. Examination of the curves shows that the alignment needs to be made to an accuracy of better than a few hundred micrometers to get the best reduction of scattered background. Results of the experimental background=peak ratio determination as a function of vertical displacement are shown in Figure 12 for elements from calcium to zinc in a gelatin matrix and for palladium in a pyrrhotite matrix. The dependence on scattering angle in the

512

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Figure 8 Energy distribution of photon flux produced by a bending magnet on the NSLS x-ray ring at a electron current of 100 mA. (From Jones et al., 1990b.)

Figure 9 Values for DLs obtained for a white-light XRM are shown as a function of atomic number. The change in the DLs as a function of filtering of the incident beam using aluminum filters is an important feature of this type of arrangement. (From Kno¨chel et al., 1983.)

Synchrotron Radiation-Induced X-ray Emission

513

Figure 10 Dependence of incoherent scattering cross sections for x-rays polarized parallel and perpendicular to the plane of the stored electron orbit on the scattering angle. Observation at a scattering angle of 90 gives optimum signal-to-background conditions. (From Jones KW, et al. Ultramicroscopy 24:313, 1988.)

Figure 11 Polarization of NSLS x-ray beams is given as a function of distance from the plane of the electron orbit for x-ray energies of 10, 20, and 30 keV. (From Hanson AL, et al. Nucl Instrum Methods Phys Res B24=25:400, 1987.)

514

Jones

Figure 12 The background-to-signal ratio is shown as a function displacement of the detector normal to the plane of the orbit of the stored electrons. The alignment becomes increasingly critical at the higher x-ray energies. The curves shown (a) were obtained for Ca, Fe, Zn, Br and, Sr contained in a gelatin matrix. The curve shown in (b) was obtained for Pd contained in a pyrrhotite matrix (From Kwiatek et al., 1990.)

horizontal plane is shown in Figure 13. The details of the methods used to make the alignments are given by Kwiatek et al. (1990). The Brookhaven work has shown that it is possible to achieve spatial resolutions below the 10-mm range using the CXRM. The spatial resolution of the instrument has been demonstrated by scanning a thin-evaporated gold straight edge through the beam and recording the intensity of the L x-rays as a function of distance. The resolution in this case was 3.5 mm. Some control over the resolution for a given collimator is obtained by inclining the collimator with respect to the incident beam. The Brookhaven device is routinely used for trace element measurements with a spatial resolution of less than 10 mm. The sensitivity and DLs obtained under these conditions have been reported by Jones et al. (1990b) and their results are displayed in Figures 14 and 15.

Synchrotron Radiation-Induced X-ray Emission

515

Figure 13 The background-to-signal ratio is plotted as a function of scattering angle in the horizontal plane. The curve was obtained by displacement of the detector and the equivalent angular range spanned was roughly 4 . (From Kwiaek et al., 1990.)

It is interesting to realize that an absolute determination of the elemental concentrations can be made based either on the theoretical estimates of photon ﬂux or from a direct determination using an ion chamber. Kwiatek et al. (1990) point out that the expected counting rate for a given element is given by Z1 dO exp ½mðEZ Þwd NðEÞ exp f½mðEÞwsfl ðEÞgdE ð3Þ YðEZ Þ ¼ NA Ek 4p Eab

where YðEZ Þ is the count rate for an element with a characteristic x-ray energy EZ ; NA is the number of target atoms within a beam spot, ek is the detector eﬃciency, dO=4p is the solid angle, NðEÞ is the photon ﬂux [number of photons=(s mm2 0.1-keV bandwidth)], mðEÞ is the linear attenuation coeﬃcient for Al and mðEZ Þ that for polyimide, w is the Al thickness, wd is the polyimide thickness, sfl ðEÞ is the ﬂuorescent cross section, Eab is the energy of the absorption edge, and E is the photon beam energy. Results of a comparison of measured and calculated rates for a section of gelatin with known amounts of iron and zinc are shown in Table 4 for three diﬀerent incident beam spectra. The results of the comparison are excellent for iron, less so for zinc. It is clear, however, that it is feasible to make determinations of the abundances of the elements without reference to standards. B. Focused X-Ray Microscopes Focused x-ray microprobes (FXRM) have been the subject of great interest over the years. The ﬁrst version appears to have been developed by Sparks (1980) and was used at

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Figure14 The sensitivity of the BNL collimated XRM is given as a function of atomic number for K and L x-ray detection. Values are given for a beam size of 8 mm68 mm. A 3-mm-diameter Si(Li) x-ray detector placed 40 mm from the beam was used as a detector. The curves were calculated from basic principles. (From Jones et al., 1990b.)

Figure 15 DLs measured for the BNL collimated XRM under the conditions described in the text and for Fig. 14. (From Jones et al., 1990b.)

Synchrotron Radiation-Induced X-ray Emission Table 4

517

Experimental and Predicted Count Rates for Different Beam Filtersa Fe (cps)

Zn (cps)

Beam ﬁlter thickness (mm A)

Mb

Tc

M

T

50 100 200

86.5 54.8 24.7

75.6 48.5 23.6

77.7 59.1 37.0

53.7 43.2 28.8

a

Experimental errors at 10–15%. Measured values. c Theoretical values. b

SPEAR. In an interesting application, it was used in a search for superheavy elements (Sparks et al., 1978). A variety of diﬀerent schemes have been used in the intervening years. A summary of recent approaches at diﬀerent laboratories is given here to illustrate the directions now being taken by this approach to the production of intense x-ray beams. Early estimates of DLs for speciﬁc conﬁgurations have been given by Gordon (1982), Gordon and Jones (1985), and Grodzins (1983). 1. LURE (1987) The equipment used for SRIXE at LURE has been described by Chevallier et al. (1987) and Brissaud et al. (1989). The white beam from the storage ring passes through a beryllium window and is incident on a curved crystal of mosaic pyrolytic graphite. The size of the incident beam is 2 cm, 1 cm before the monochromator, which produces a ﬁnal focused beam of 14-keV photons with a size of 1 mm61 mm. The ﬂuorescent x-rays are detected with a Si(Li) detector collimated using a 2.8-mm aperture. The DLs achieved are around 1 ppm for thick geological specimens and much less for organic matrices. The probe has only been used for measurements, which do not require high spatial resolution. 2. Photon Factory (1990) The group working at the Photon Factory in Japan has made seminal contributions to the development of SRIXE and of SXRM. An early instrument used Wolter-type focusing optics and achieved spatial resolutions of around 10 mm630 mm (Hayakawa et al., 1989). A later version was developed to provide higher photon ﬂux and give improved values for the DLs (Hayakawa et al., 1990). The design chosen was similar to one described by Jones et al. (1984), which has been partially implemented at the NSLS. A schematic diagram of the Photon Factory apparatus is shown in Figure 16. The platinum-coated ellipsoidal mirror was designed to produce a demagniﬁcation of the beam of 1:9.5. The mirror accepted 0.6 mrad of beam in the horizontal direction and 0.05 mrad in the vertical direction and was run at a grazing angle of 7 mrad. A gain in intensity of a factor of 170 was found for a monoenergetic beam of 8 keV. The system produced a photon ﬂux of about 1.4 103 photons=(s mm2 mA). The ability to locate the system in close proximity to the electron beam is a key factor in maximizing the photon ﬂux.

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Jones

Figure 16 Schematic diagram of the focussing XRM used at the Photon Factory. (From Hayakawa et al., 1990.)

Figure 17 Schematic diagram of the focusing XRM in operation at the Daresbury XRM. (From Van Langevelde et al., 1990c.)

3. SRS (1990) The instrumentation at the Daresbury SRS has been developed by a group of collaborators from the Free University in Amsterdam and Warwick University at Coventry. The latest device uses a silicon crystals as both a monochromator and focusing device (Van Langevelde et al., 1990a, 1990b, 1990c). A diagram of the instrument is shown in Figure 17. The crystal

Synchrotron Radiation-Induced X-ray Emission

519

is a 100-mm-thick silicon crystal bent with a radius of 100 mm in the sagittal plane and 5740 mm in the meridional plane. The beam is focused to a spot size of about 15 mm620 mm at an energy of 15 keV. The demagniﬁcation is 1000 in the horizontal plane and 15 in the horizontal plane, with a photon ﬂux increase of greater than 104. A ﬂux of 15-keV photons at the specimen of 104 photons=(S mm2 mA) is obtained. This is an impressive result when it is remembered that the device is located about 80 m from the electron beam. 4. Novosibirsk (1989) Baryshev et al. (1989) working at the Novosibirsk VEPP-3 synchrotron have used both monoenergetic and white beams of x-rays. A single-crystal pyrolytic graphite monochromator was used to produce monoenergetic beams with energies between 8 and 35 keV. The spatial resolution was 60 mm for the monoenergetic beam and 30 mm for the white beam. The DL for the monoenergetic beam was 10 ppm for elements from iron to strontium for a 1–3-s run. 5. LBL (1988) The group at the Lawrence Berkeley Laboratory has developed a focusing system based on the Kirkpatrick–Baez geometry (Giauque et al., 1988; Kirkpatrick and Baez, 1948; Thompson et al., 1987; Thompson et al., 1988; Underwood et al., 1988, Wu et al., 1990). A schematic diagram of their system is shown in Figure 18. The system uses a parallel beam of photons to produce an image which is demagniﬁed by about a factor of 100 to produce ﬁnal images of a few micrometers. The use of multilayer coatings of tungsten carbide on the mirrors gives a quasimonoenergetic beam with a bandwidth of about 10% at 10 keV and a high throughput. Much of the experimental work has been carried out at the NSLS in collaboration with the BNL group. It is thus possible to make comparisons in performance between the CXRM and FXRM on the same storage ring.

Figure 18 Schematic diagram of the focusing XRM of the Berkeley group used at the BNL X26 beam line. (From Thompson et al., 1988.)

520

Jones

The LBL FXRM has produced photon ﬂuxes of about 36105 photons=(s mm2 mA) at 10 keV energy. It is important to note that the use of a ﬁnal collimator is not required in this apparatus. Improvements in the device will make it possible to reach higher x-ray energies and provide easy tunability of the x-ray beam energy. 6. LBL at the Advanced Photon Source (ALS) (1988) A much more elaborate imaging capability has now been developed by LBL at the ALS (Warwick et al., 1998). A number of diﬀerent beam lines can be used to do experiments with energies ranging from 100 eV to 15 keV. The spatial resolutions vary from 0.1 mm to about 2 mm, depending on the beam energy and imaging mechanism. Some of the techniques used are Fourier transform infrared spectroscopy (FTIR), scanning transmission x-ray microscopy, photoemission electron microscopy, x-ray microdiﬀraction, and micro x-ray ﬂuorescence. A schematic diagram of the apparatus developed to cover the x-ray energy range from 2 to 15 keV is shown in Figure 19. The chosen hardware conﬁguration uses Kirkpatrick–Baez focusing mirrors, as did the earlier system described. Beam sizes as small as 0.8 mm have been achieved with an experimental setup. 7. Cornell High Energy Synchrotron Source (CHESS) Focusing of both monoenergetic and white x-ray beams can be accomplished by total external reﬂection of the x-rays in tapered glass capillaries (Bilderback et al., 1994a; 1994b; 1994c; 1995a; 1995b; Hoﬀman et al., 1994; Thiel et al., 1992). The Cornell group has been one of the leading proponents of this approach over the past 10 years. It has also been actively developed at other laboratories and installed on beam lines at several diﬀerent synchrotron facilities. One such installation at the ESRF is described in the following subsection. The experiments at CHESS have shown that it is possible to produce beams with a diameter as small as 0.05 mm (Hoﬀman et al., 1994). A scan across an edge demonstrating this spatial resolution is shown in Figure 20. The x-ray ﬂux is increased by factors of up to 1000. It seems fair to say that this experiment marks a major step forward in x-ray microscopy techniques. The initial experiments showed that the capillary system could be used for Laue diﬀraction measurements and for x-ray radiography at this size scale. Preliminary work done in 1994 at the BNL X26A beam line by Sutton, Hoﬀman,

Figure 19 Schematic layout of K–B mirrors and a four-crystal channel-cut monochromator for micrometer precision micro-x-ray absorption spectrometry and micro-diffraction developed by Padmore et al. at the ALS. (From Warwick et al., 1998.)

Synchrotron Radiation-Induced X-ray Emission

521

Figure 20 The experiments at CHESS have shown that it is possible to produce beams with a diameter as small as 0.05 mm. A scan across an edge demonstrating this spatial resolution is shown. (From Hoffman et al., 1994.)

Bajt, Jones, and Bilderback showed that SRIXE experiments can be successfully performed with the capillaries. SRIXE experiments are made diﬃcult by the large divergence of the x-ray beam after it emerges from the capillary tip. In order to achieve the best spatial resolution, it is necessary to place the sample within a few millimeters of the capillary and at the same time arrange optimal detection geometry for the x-rays from the sample at a 90 angle to the incident beam. 8. European Synchrotron Radiation Facility (1994) The microfocus beam line on ID 13-1 developed by Riekel (personal communication, 1995) and collaborators is intended mainly for microdiﬀraction experiments. It has also been used for SRIXE and ﬂuorescent computed microtomography experiments. The beam line uses an undulator x-ray source and is designed for operation between 6 and 16 keV. The optics use either silicon or multilayer monochromators and an ellipsoidal mirror to produce a focal spot in the experimental hutch. Collimation can be used to produce a 7-mm full width at half-maximum focal spot with a beam ﬂux of 261011 photons=s at 13 keV. The silicon channel-cut monochromator can be combined with capillary focusing optics to reduce the spot size to about 262 mm2. The photon ﬂux is also about 261011 photons=s in this conﬁguration. A schematic diagram of the beam line optics is shown in Figure 21. The photon ﬂux that can be used for SRIXE experiments is around four orders of magnitude higher than the values found for the NSLS X26A beam line. The expected detection limits at the ESRF are then about two orders of magnitude lower. The high ﬂux at the ESRF has made it possible to achieve detection limits around 100 ag for transition elements (K. Janssens, personal communication, 1998). 9. Advanced Photon Source (1998) The University of Chicago Center for Advanced Radiation Sources is operating several beam lines at the APS with capabilities for SRIXE, computed microtomography, EXAFS,

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Jones

Figure 21 ESRF schematic diagram of the beam line optics. (From C. Riekel, private communication, 1995.)

XANES, and high-pressure experiments. The SRIXE apparatus is located on an undulator beam line and uses cryogenically cooled Si monochromators to cover photon energy ranges from 4.5 to 21 keV and 15 to 80 keV. The monochromator design and arrangement is such that both can be used simultaneously. Focusing of both white and monoenergetic beams is done with two 1-m-long silicon Kirkpatrick–Baez-type mirrors that reduce the beam size by factors ranging from 3 to 10. Experiments on this beam line will be microprobe experiments to study various parameters of geological and environmental samples including compositions of ﬂuid inclusions and trace element partitioning. The experimental techniques include SRIXE, EXAFS=XANES, and computed microtomography.

10. Summary Beam Line Design Comments Beam lines and experimental apparatus used for SRIXE and related experiments during the past 25 years have been discussed here. It is clear that very major developments have taken place in the experimental capabilities during that time. The obvious foundation for the developments has been the rapid improvements in the synchrotron x-ray source through the introduction of the wiggler and undulator insertion devices. However, the parallel improvement in x-ray optics for focusing these beams and in x-ray detectors has been necessary to make possible the most eﬀective use of these beams for experiments. Indeed, future progress in the latter ﬁelds may be the most important factor for the actual application of these techniques in experiments.

C.

Experiments at a Distance

The eﬀects of the remarkable increase in computer communications through the Internet and World Wide Web are visible everywhere. In the past few years, there has been a development of interest in making use of these technologies to help make facilities useful to researchers at remote locations. One example has been the DOE 2000 initiative of the U.S. Department of Energy (US Department of Energy, 1997). There are two components of this initiative that are related to use of synchrotron beam lines. First, there is the

Synchrotron Radiation-Induced X-ray Emission

523

concept of national collaboratories. These are ‘‘laboratories without walls that unite expertise, instruments, and computers, enabling scientists to carry out cooperative research independent of geography.’’ The second is to set up pilot projects that are ‘‘virtual laboratories that give scientists the technology to collectively observe and attack problems using combinations of ideas, methodologies, and instrumentation that do not exist at any single location.’’ Examples of this idea exist for several types of instrumentation. An example of the application to a synchrotron beam line is that of the ‘‘Spectro-Microscopy Collaboratory at the Advanced Light Source: A Distributed Collaboratory Testbed’’ (Agarwal et al., 1997). This prototype of a collaboratory is supposed to enable operation of equipment on an undulator beam line at the ALS from the University of Wisconsin–Milwaukee. Some of the topics of concern for the collaboratory are security mechanisms, data dissemination, remote monitoring and control, safety mechanisms for beam line operation, resource arbitration to decide location of control, video conferencing and remote viewing, network needs, and shared electronic notebooks for data handling. Another example involves the University of Florida and the APS (CAT Communicator, 1998). This project is supposed to allow remote operation of a materials research beam line. The computer components needed are similar to those listed for the LBL–Wisconsin demonstration.

V.

CONTINUUM AND MONOCHROMATIC EXCITATION

Successful XRMs have been put into operation using both continuum and monoenergetic synchrotron-produced x-rays. The continuum radiation is extremely convenient to use because it is easy to construct a CXRM with a minimum of equipment and achieve excellent performance. Further, the broad-band excitation means that measurements can be made for essentially all elements in the periodic table in a single exposure. Monoenergetic radiation can be used at an energy optimized for production of the characteristic x-rays of a given element, thus reducing radiation damage in organic materials. Counting-rate limitations in energy-dispersive detectors are reduced because of the elimination of scattered x-ray events. The energy can be tuned to eliminate interferences (e.g., lead–arsenic) and to eliminate excitation of elements with Z higher than that of the element of interest. Maps can also be constructed by subtraction of images obtained above and below absorption edges. Successful XRMs have also been produced in collimated and focused modes employing either continuum or monoenergetic radiation. The brute-force-type collimated continuum radiation microprobe employed at Brookhaven and Hamburg has been comparable to the other types of probe in terms of spatial resolution and DL. The performance of the Brookhaven instrument has been compared with the performance of the LBL Kirkpatrick–Baez XRM operated on the same NSLS beam line (Giauque et al., 1988). A comparison of results obtained with the CXRM positioned at 10 m from the source with the FXRM at 20 m from the source is given by Rivers et al. (1992). Figure 22 shows the results of the comparison. The DL obtained with the FXRM is somewhat better than for the CXRM, but the wider energy range of the CXRM is a substantial advantage in some cases. The quality of the Kirkpatrick–Baez optics makes it feasible to dispense with the use of a collimator placed close to the specimen, which is an advantage in the design of the experiments in some cases. For ultimate performance in terms of the spatial resolution, FXRM might be the best choice because the construction of good collimators on the 1-mm2 size or smaller scale

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Figure 22 Sensitivity of the BNL collimated white-light XRM compared with the sensitivity obtained with the Berkeley focused XRM. (From Rivers ML, et al., 1992.)

is diﬃcult. However, it is feasible to fabricate capillaries that act essentially as a collimator. In that case, the beam divergence is reduced and the capillary can produce a small beam without the need to place the sample close to its end. The challenge is thus to the following: 1. 2.

To obtain better optics to improve the spatial resolution To improve the eﬃciency of the optics to get higher photon ﬂuxes

These goals can be addressed most eﬀectively by use of focusing optics. An alternative to the capillary approach has been demonstrated by Padmore et al. (1997), who used an elliptical focusing mirror and Kirkpatrick–Baez mirrors to obtain a white beam with a size of 0.8 mm. SRIXE systems with submicrometer spatial resolution are not now in use on a routine basis. However, it appears that the combination of improving x-ray optics and the high-ﬂux x-ray beams produced by third-generation synchrotron undulators will bring this advance about in the next few years.

VI.

QUANTITATION

Methods for making quantitative elemental determinations using x-ray ﬂuorescence have been developed over many years. These approaches are discussed for conventional x-ray sources in Chapter 5 and 6. Some of the approaches used at synchrotron sources are given here to show how the methods developed for use with conventional tubes have been used with the new radiation source.

Synchrotron Radiation-Induced X-ray Emission

525

Giauque et al. (1986) measured several U.S. National Bureau of Standards (NBS), now National Institute of Standards and Technology (NIST), and the Japan National Institute of Environmental Science (NIES) standard reference materials (SRM) at the 54pole wiggler beam line at the Stanford Synchrotron Radiation Laboratory. They referred their measurements to a copper standard prepared by evaporation where the weight was determined from gravimetric measurements. Multielement standards were prepared by dissolving known weights of an element in an acidic solution. Monoenergetic radiation was used for the work at energies of 10 and 18 keV. The beam size was deﬁned by an aperture 3 mm in size. The diﬀerences in sample mass were accounted for by normalizing to the intensity of the scattered radiation (Giauque et al., 1986). The results obtained for three NBS materials are shown in Table 5. The work of Giauque et al. (1986) addressed quantitation in thin biological specimens where matrix eﬀects were negligible and where the Compton scattering could be used for normalization of masses. It is necessary to extend this approach if thick specimens are to be investigated. There are many geological experiments where this situation holds. Methods for quantitation have been discussed by Brissaud et al. (1989) and by Lu et al. (1989). Corrections are made for attenuation of incoming and ﬂuorescent x-rays by the sample matrix and by any ﬁlters employed as well as for secondary ﬂuorescence. The composition of the major elements is generally known for geological materials; hence, concentrations can be referred to that of one of the major elements. Brissaud et al. (1989) compared the SRIXE results with several standards. The results are shown in Table 6. The table gives the recommended concentration and the SRIXE value. It can be seen that agreement is quite good. In this case, as noted earlier, a comparatively large 1-mm beam was used. Lu et al. (1989) used a microprobe with a beam size of 30 mm660 mm to analyze diﬀerent specimens of feldspars. They compared their SRIXE results with values obtained using an electron probe and atomic absorption spectroscopy. The agreement with the electron probe was good, but the atomic absorption values tended to be systematically lower than the SRIXE values. Results for the comparison with the electron probe are shown for iron and strontium in Figure 23. For thin biological specimens, concentrations can be established by use of a sensitivity curve such as the one displayed in Figure 14. Corrections for diﬀerences in thickness can be made by normalizing to the intensity of the scattered incoherent peak if a monoenergetic beam is used or to the continuum comprising both incoherent and coherent scattering if white beam is used (Giauque et al., 1979). Quantitation of SRIXE draws on many years of experience gained using tube-excited x-ray ﬂuorescence. The main diﬀerence between the x-ray sources occurs when SRIXE is used with microbeams with dimensions on the micrometer scale. In this case, strict attention must be given to the uniformity of the standards used and to the experience gained in calibration of the electron microprobe (Hren et al., 1979) and proton microprobe (Johansson and Campbell, 1988).

VII.

SENSITIVITIES AND MINIMUM DETECTION LIMITS

The related questions of sensitivites and DLs have been addressed by calculations based on the known physical parameters of the XRM systems and by empirical determinations. The detailed understanding of the x-ray production process using synchrotron radiation is helpful in assessing the sensitivities and DLs that can be achieved in SXRM. The results cited for the sensitivities and DLs should be taken as representative of the current

17,800 2,000 12,000 800 < 0.6 0.33 0.12 2.30 0.16 0.24 0.06 0.65 0.04 46.9 0.9 < 0.05 0.09 0.04 12.1 0.2 13.1 0.2 3.69 0.10 < 0.1 < 0.1

16,900 3,000 13,000 500 0.0026 0.0007 0.26 0.06 (2.1)

0.0008 0.0002 0.019 0.003

0.7 0.1 46.1 2.2 (0.0019) 0.11 0.01 (12) (11)

XRF

NBS

Nonfat milk powder SRM 1549 diska 51 mg=cm2 wtb 37 mg wtc 500 mg

b

0.001 0.0008 0.020 0.010

8.5 0.5 18.3 1.0 (0.18) 2.0 0.3 10.6 1.0 (0.006) 1.1 0.2 (9) (1)

1,360 40 190 10

NBS 1,220 130 174 10 < 0.3 8.2 1.8 17.1 4.8 0.11 0.06 1.88 0.12 10.3 0.4 < 0.03 0.92 0.06 8.5 1.4 0.94 0.06 0.82 0.04 < 0.06 < 0.1

XRF

Wheat ﬂour SRM 1567 diska 60 mg=cm2 wtb 43 mg wtc 400 mg

Results Determined for Three NBS Standard References Materials (mg=g)

Mass thickness of disks. Weight of area scanned. c Recommended sample weight. Source: From Giauque et al., 1986.

a

K Ca Cr Mn Fe Ni Cu Zn As Se Br Rb Sr Hg Pb

Element

Table 5

0.0060 0.0007 0.045 0.010

20.1 0.4 8.7 0.6 (0.16) 2.2 0.3 19.4 1.0 0.41 0.05 0.4 0.1 (1) (7)

1,120 20 140 20

NBS

1,360 160 158 14 < 0.4 22.1 2.8 9.1 1.2 0.18 0.06 2.21 0.22 21.9 1.8 0.42 0.09 0.38 0.04 1.19 0.17 8.4 0.9 0.19 0.04 < 0.08 0.10 0.09

XRF

Rice ﬂour SRM 1568 diska 60 mg=cm2 wtb 43 mg wtc 400 mg

526 Jones

Synchrotron Radiation-Induced X-ray Emission Table 6

16.6- and 21.7-keV SRXRF Analysis of Three International Geostandardsa GSN

BEN

SRIXE Element K, % Ca, % Ti, % V Cr Mn Fe, % Ni Cu Zn Ga Rb Sr Y Zr Nb Ba W Pb Th U

527

MICA-Fer

SRIXE

SRIXE

GST

16.6

21.7

GST

16.6

21.7

GST

16.6

21.7

3.84 1.78 0.41 65 55 433 2.63 34 20 48 22 185 570 19 235 23 1400 470 53 44 8

3.54 1.68 0.41 125 45 479 3.13 36 16 60 21 115 469 — — — — 500 51 11 —

3.34 1.66 0.38 188 117 551 3.12 35 25 66 27 189 905 22 296 32 1103 610 82 13 4

1.15 9.85 1.57 235 360 1540 8.99 267 72 120 17 47 1370 30 265 100 — 30 4 11 2.4

1.05 8.98 1.54 185 346 1662 9.66 320 62 117 10 27 1550 — — — — — 2 2 —

0.86 8.74 1.49 360 407 1679 9.20 278 85 129 21 61 1759 34 343 136 — 30 5 3 2

7.26 0.31 1.50 135 90 2695 17.96 35 4 1300 95 2200 5 25 800 270 — — 13 150 60

6.10 0.15 1.35 42 125 3006 18.95 39 — 1320 79 1013 1 — — — — — 23 51 —

5.89 0.07 1.47 257 134 3200 19.15 — 14 1501 118 2696 5 50 1058 375 — — 17 50 33

a

GST are the admitted values. Units are in ppm or %. The abbreviations, GSN, BEN, MICA-Fer are the names of standards as given in Gavindaraju K. Geostandards Newslett, 3:3, 1979; 4:49, 1980, 8:173, 1984. Source: From Brissaud et al., 1989.

situation. The actual values depend on the particular experimental conditions, synchrotron ring currents, spatial resolutions, and so forth so that exact comparisons are not terribly meaningful. Sparks (1980), Gordon (1982), Gordon and Jones (1985), and Grodzins (1983) have presented calculations of the minimum detection limits to be expected using the secondgeneration synchrotron radiation sources such as the NSLS to produce x-ray microbeams. Many experimental determinations have been made for the two quantities. The results obtained by diﬀerent groups using diﬀerent types of XRMs are given here. Figures 14 and 15 show the values obtained by Jones et al. (1990b) using the collimated microprobe at the NSLS. Their values are relevant to thin specimens with an organic matrix. Values for the calculated DLs extrapolated from the work of Gordon and Jones (1985) are included in Figure 15. It can be seen that the experimental values are in the same range as the calculations although the experimental system was not the same as the one considered theoretically. The sensitivities that are shown in Figure 14 indicate the agreement between predicted (solid curve) and measured (data points) counting rates. The curves were calculated from knowledge of the experimental arrangement, synchrotron energy spectrum, and specimen parameters. The experimental points show the measured values assuming the same specimen parameters.

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Figure 23 Comparison of concentrations for Sr and Fe in feldspar obtained using the NSLS XRM with those obtained using an electron microprobe. The solid lines show the values expected for exact agreement between the two methods. The good agreement validates the use of the XRM method for geological analyses. (From Lu et al., 1989.)

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Figure 24 shows the results obtained for the DLs by Giauque et al. (1988) using the Kirkpatrick–Baez XRM at the NSLS. The sensitivity curve has been previously discussed in comparing sensitivities for the collimated and focused instruments (see Fig. 22). Jones et al. (1988) compared the DL values for the collimated-white-beam and monoenergeticfocused-beam approaches and showed that the DL values were very similar. Ketelsen et al. (1986) have also compared DL values for white beams and monoenergetic beams. The results are shown in Figure 25 and also demonstrate that the two approaches give comparable results. The absolute values of the two experiments cannot be directly compared because of the beam size and ring operating conditions.

Figure 24 Relative sensitivity for determination of elements from K to Zn obtained using the LBL Kirkpatrick–Baez XRM at the NSLS X26 beam line. NIST (NBS) thin glass Standard Reference Materials 1832 and 1833 were used. (From Giauque et al., 1988.)

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Figure 25 Detection limits observed for aerosol samples using Hamburg synchrotron with a 70-mA stored beam of 3.7-GeV electrons for 300 s. The open triangles were obtained with a white beam filtered with 8 mm of Al, filled circles for a 31-keV monoenergetic beam, and open circles for a 12.5-keV monoenergetic beam. (From Ketelsen et al., 1986.)

Few experiments appear to have been devoted to measuring detection limits at the third-generation sources. Results from the ESRF have been discussed in comparison to the values found at the NSLS in the survey of beam lines given in Sec. IV.B. Improvements in spatial resolution and=or detection limits can be generally assumed. These improvements should be by about a factor of 100 when comparing the use of undulator beam lines at the new facilities with their bending magnet performance. VIII.

BEAM-INDUCED DAMAGE

Passage of a photon beam through a specimen results in energy deposition through the photoelectric eﬀect and Compton-scattering processes. This energy deposition results in a breaking of molecular bonds as the secondary electrons produced lose their energy by further ionization or scattering processes with other atoms. The eﬀects have been examined in great detail for electron beams used in electron microscopy. Much less has been done for the XRM. This is not surprising because the ﬁeld is much younger and less extensively developed than is the ﬁeld of electron microscopy. Another important reason is that the photon beam ﬂuences employed thus far have been substantially smaller than those employed in the electron microscope, and as a result, the magnitude of beaminduced damage has not been so important. Biological and other organic materials are generally more susceptible to beaminduced damage than are other materials. For this reason, the discussion is limited to these materials. In the future, when more intense synchrotron x-ray sources are available, it will be necessary to expand the list of materials considered.

Synchrotron Radiation-Induced X-ray Emission

531

A qualitative calculation can also be done to illustrate the energy deposition for photon beams of various energies. Assume that the photon ﬂux used for a typical ﬂuorescence experiment is 106 photons=(s mm2). Typical run times are 10 min or less. The maximum photon ﬂuence for current XRM instruments is thus about 66108 photons=mm2. It is interesting to note that these ﬂuences are now starting to approach the range of the ﬂuences found in use of the electron microprobe. In order to estimate the energy deposition, it is assumed that attenuation coeﬃcients can be represented by the photoelectric process attenuation coeﬃcient only and that all of the energy of the photoelectrons is absorbed in the volume considered. The Compton-scattering process is relatively small and can be neglected in a qualitative estimate of the dose. The results of the calculation are shown for photon beams with energies less than 20 keV. The x-ray dose needed to kill living biological systems has been examined in detail over the years (Kirz and Sayre, 1980; Sayre et al., 1977, 1978; Slatkin et al., 1984; Themner et al., 1990). A dose which exceeds 1 Gy is likely to cause serious damage to a biological cell or system. Thus, there are limitations to the use of x-ray beams for the examination of living systems, as there are, of course, for all other beams. The limitations depend on the absorbed dose. One way that this can be done is by measuring the linear attenuation coefﬁcients in CMT or in projection-radiography-type experiments. Spanne (1989) has calculated the dose given to an object in obtaining an image with a signal-to-noise ratio of 5 for a water phantom with a contrasting detail with a diameter 0.005 of the phantom diameter. The results of Spanne for objects with diﬀerent diameters and compositions are shown in Figure 26a and 26b. It can be seen that the dose is strongly energy dependent and

Figure 26 (a) Absorbed dose at the center of a circular water phantom for detection of an element of fat with a signal-to-noise ratio of 5 as a function of photon energy. The cylinder diameters are shown in the figure. (b) Absorbed dose at the center of 1-mm-diameter water phantoms for detection of elements of fat, air and calcium with a signal-to-noise ratio of 5. The element diameter is 0.005 of the phantom diameter or 5 mm. (From Spanne, 1989.)

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for relatively small objects (1 mm), the observed dose at the optimum energy is about 102 Gy. Examination of living systems using CMT with resolution on a level of 5–10 mm thus may not be practicable. However, relaxation of the resolution criterion would reduce the dose at a point where in vivo examination of living systems is feasible. It can also be seen from Figure 26 that the optimum energy for examining very thin specimens (cells) becomes very low. This represents a separate ﬁeld of research and is not considered here. The use of projection radiography methods should also be useful. The dose is much less because fewer photons are needed to form an image. In this case, the examination of living systems will be easier. If x-ray beams are used to make ﬂuorescence measurements on nonliving systems, a loss of mass and possibly of trace elements can occur. Slaktin et al. (1984) investigated changes in the morphology of human leukocytes and showed that severe damage resulted for ﬂuences of 15-keV photons of about 1017 to 1019 photons=cm2. Figure 27 shows a photomicrograph of the leukocytes after bombardment by ﬂuences from 0.461019 to 2.46109 photons=cm2. Damage to thin sections of the kidney (10 mm) was much less. Mass loss in other types of organic materials was measured by Themner et al. (1990). They determined mass loss by measuring the change in scattered radiation counting rate as a function of total dose. The results that they obtained for irradiation of a skin sample is shown in Figure 28. It can be seen that changes can be observed at dose values comparable to those used in many ﬂuorescence-type experiments. Mass loss, therefore, should be carefully measured if the scattered radiation is used as measure of the specimen areal density for quantitation purposes. The eﬀects discussed can lead to loss of trace elements and to errors in the assignment of concentrations, as is well known from the case of electron or proton microprobes. Just as in those cases, measurements need to be made of the yield of characteristic x-rays as a function of photon ﬂuence for ﬂuorescence measurements. One such examination shows the yield of chromium x-rays observed in the bombardment of a section of rat kidney which was investigated in a study of the toxic eﬀects of chromium. It can be seen that there are no indications of loss of the chromium under the conditions used. Diﬀraction experiments are possibly one type of experiment that will be very sensitive to radiation damage eﬀects. Protein crystallography experiments at the NSLS (Sweet et al., 1995) showed that focused bending magnet radiation from the NSLS was suﬃcient to render samples useless in a short period of time. Diﬀraction of 13-keV x-rays by thin sections of polyethylene was studied at the ESRF microfocus beam line. In this case, the diﬀraction patterns were essentially destroyed in 30–60 s (Jones et al., personal communication, 1995). In summary, although beam-induced damage can be observed, it does not seem to have been a problem in experiments conducted to date for experiments that are not sensitive to crystal structure. However, because much higher ﬂuences are to be expected as focusing methods are improved and as the synchrotron source itself becomes more powerful, the beam damage eﬀects will become of more central importance. Further studies of these eﬀects should be conducted.

IX.

APPLICATIONS OF SRIXE

There are now examples of the uses of the XRM in many diﬀerent ﬁelds. A brief description of the diverse applications is given here to illustrate the rapid development of the ﬁeld and to give an idea of the ways the XRM may be used in the future. The examples presented cover

Figure 27 Morphology of red blood cells is shown as a function of the fluence of incident 15-keV photons. The fluences for the different exposures are as follows: 0, 0.461019, 1.161019, 1.661019, 2.061019 and 2.461019 photons=cm2 for frames (a)–(f), respectively. (From Slatkin et al., 1984.)

534

Jones

Figure 28 Mass loss observed for skin sample irradiated with a 66-keV beam of photons shown as a function of the dose. The mass loss was determined by observation of the scattered radiation intensity. (From Themner et al., 1990.)

mainly activities during the past 10 years. The total number of experiments performed during this time is very large. The intent is to give an example of work covering a variety of topics. There is no intent to give an encyclopedic review of all applications. A.

Archaeology

Brissaud et al. (1989) made low-resolution measurements on a number of diﬀerent materials. One case involved examination of three Gallic coins using synchrotron radiation and a comparison with the results obtained using proton-induced x-ray emission and neutron-activation analysis. The x-ray and proton beams probe material close to the surface of the coin, whereas the activation approach gives the bulk concentration. The results given in Figure 29 show diﬀerences in the concentrations found with these methods and also show that the activation approach is not applicable to all elements. This straightforward example shows that the synchrotron can be used to good eﬀect in studying archaeological and other materials with spatial resolutions of the order of 1 mm. It is however, a type of experiment that should be viewed as a bridge between the use of conventional x-ray tube sources and the synchrotron sources with high brilliance. B.

Biology=Medical: Calcified Tissue Studies

The distribution of trace elements in bone and other calciﬁed tissues is generally of great interest since the concentrations of the essential trace elements are relevant to bone growth

Figure 29 Chemical compositions of coins determined using SRIXE, PIXE (solid bar), and neutron-activation analysis (open bar). The SRIXE work was done at 17 keV (diagonal cross-hatch) and 35 keV (horizontal cross-hatch). (From Brissaud et al., 1989.)

Synchrotron Radiation-Induced X-ray Emission 535

536

Jones

and disease. Therapeutic agents used to treat disease states may modify the trace element concentrations and will deposit in particular patterns in the tissue themselves; ﬁnally, toxic elements such as lead are stored in the bone in localized patterns. Several experiments can be cited to illustrate these points. Tros et al. (1990) have examined neonatal hamster tooth germs using both microPIXE and the XRM approach. In this work, neonatal hamsters were treated with a ﬂuoride compound commonly used to inhibit tooth decay. A tooth was obtained from the hamster after a 1-day interval and thin sections were measured using the two types of beam. The ﬂuorine concentrations were found using the proton microprobe at Amsterdam and the zinc distribution by XRM at Daresbury. The XRM was chosen for the determination of the essential trace elements because of its high sensitivity and low beaminduced damage. The proton beam was suited to the determination of the ﬂuorine because the cross sections for inelastic proton scattering are high. The combined use of the two methods shows that ion beams can be used eﬀectively in combination with the photon beams to cover all elements in the periodic table. The results of the XRM measurement of Zn and Ca are shown in Figure 30. Gallium nitrate is another therapeutic element that is used to treat accelerated bone resorption found in cancer patients. The mechanisms by which gallium interacts with the bone are as yet poorly understood. Bockman et al. (1990) used the NSLS X26 XRM to study sections of rat tibia obtained from animals treated with gallium nitrate. A scan across the tibia from periosteum to endosteum giving the distribution of Ca, Ga, and Zn in a 12-mm-thick section of bone is shown in Figure 31a. A two-dimensional map of the gallium and calcium distributions in a fetal rat ulna bone that was exposed to gallium (25 mm) in culture medium for 48 h is shown in Figure 31b. The bone structure is shown in an electron micrograph in the center section of the ﬁgure. The noncalciﬁed portion accumulates little gallium compared to the calciﬁed portion, where the gallium accumulates in the metabolically active regions where new bone matrix is being formed. The method has been used to study the kinetics of gallium absorption in the bone for diﬀerent dosed and for diﬀerent states of the bone metabolism. Changes in the concentrations of iron and zinc as a function of the gallium treatment indicate that it may be possible to infer particular enzymes, which are targets for the gallium. Lead can be used to show the uses of the XRM in the study of the eﬀects of toxic metals. Lead is a major public health problem in many countries. Most of the lead in the body is stored in the skeleton and can be released to cause serious health eﬀects under certain conditions. It is known that it causes neurological and other problems in children, and it is associated with kidney and cardiovascular disease in adults. Understanding the deposition patterns and kinetics of lead in bone are therefore of great importance from a particular standpoint. Jones et al. (1990b, 1992) have reported measurements on lead distributions in the human tibia and in sections of deciduous teeth. Figure 32 shows the distribution of lead in such a tooth. The end objective in this case will be to attempt to correlate the distributions with the blood-lead concentrations at birth from examination of the enamel (formed by the time of birth) and the dentine (later exposures). Knowledge of the timeintegrated lead exposure can then be related to neurological deﬁcits and other eﬀects. An initial experiment to investigate the kinetics for the accumulation and desorption of lead in bone was carried out in a controlled experiment that investigated chick tibia (Jones et al., 1997). The measurements were performed the proximal end of thin (60 mm) sections of tibia. Chicks raised on diets that contained normal amounts of Ca and Cadeﬁcient diets and lead were analyzed. Maps over the bone were made with a step size between points of 60 mm. Analysis of these maps showed that Pb and Ca depositions in

Synchrotron Radiation-Induced X-ray Emission

537

Figure 30 Scan over a 2-mm section of a neonatal hamster tooth germ showing the variation of zinc (a) and calcium (b). The zinc is located on the outside edge of the calcium distribution, but the reasons for difference remain to be determined. (From Tros et al., 1990.)

bone are similar and that the eﬀects on bone growth and mineralization caused by Pb inﬂuences the distribution of both cations. This experiment shows that SRIXE can be used as a valuable complement to more standard techniques used in biomedical investigations. The eﬀect of lead on the developing brain has been investigated by Cline et al. (1996) and Jones et al. (1997). They studied the retinotectal system of frog tadpoles as a function of exposure to various levels of lead during growth. The lead content in the optic tectum of the exposed tadpoles was measured using SRIXE at the NSLS X26 beam line with white light and a collimated beam. Lead levels of about 200 ppb were successfully detected. Use of SRIXE was advantageous because it can be used with the very small amounts of sample

538

Jones

Figure 31 (a) Scan over a thin section of the tibia diaphysis of a rat which had been treated with gallium nitrate showing the distribution of the Ca, Zn, and Ga. Some of the variations come from irregularities in specimen thickness. The spatial distribution is a function of the gallium nitrate treatment of the rat. The step size was 9 mm. (b) Map of the distribution of gallium (left) and calcium (right) in a fetal rat ulna bone after exposure to gallium (25 mM) in a culture medium. The light regions have the highest elemental concentrations. A scanning electron micrograph (center) shows the bone structure in the same region. The dark portions at the top of the figure are the noncalcified cartilaginous portions of the bone, which accumulate little gallium. Most of the gallium is distributed in metabolically active regions of the metaphysis and diaphysis. (From Bockman et al., 1990.)

material available. Atomic absorption spectroscopy could not be used for that reason. The results of the overall experiment demonstrated the impact of lead on the developing brain through the structure of the retinal axons. Further, work with the chelating agent 2,3dimercaptosuccinic acid showed that the eﬀect of lead on the neuronal structure could be reversed at a level superior to removal of the lead source alone. These conclusions are relevant to lead-induced cognitive deﬁcits in humans. The results demonstrate that SRIXE can be a useful measurement technique in many diﬀerent types of biomedical applications where it is necessary to measure trace concentrations of an element in thin tissue sections where sample size or handling preclude other conventional approaches. Lower detection limits at undulator beam lines will increase the scope of this type of analytical experiment. Osteoarthritic degeneration of articular cartilage is a degenerative disease of major importance which can be brought about by senescence and trauma. It causes changes in the cartilage and in the bone. The metacarpal joints of horses aﬀected by osteoarthritis can be used as an animal model for study of the disease. Rizzo et al. (1995) have studied 5-mmthick cartilage slices obtained from normal and arthritic horses using x-ray microbeams (10 mm in size) at the X26A beam line at the NSLS to map the spatial distribution of S, Ca, and Zn along a line through the section. It was found that there were diﬀerences in the

Synchrotron Radiation-Induced X-ray Emission

539

Figure 31

results for the normal and arthritic tissue for all three elements. It is argued that the zinc may be contained in the metalloenzyme alkaline phosphatase. This result may be of use in understanding the metabolic processes of importance in osteoarthritis. C.

Geology and Environmental Sciences

The XRM has been used for many diﬀerent types of geological and environmental experiments. A survey of geological and environmental applications of synchrotron radiation has been given by several authors (Smith, 1995; Jones, 2000). A few examples of diﬀerent applications are given here. Scientiﬁc questions related to the environment are generally very complex and require understanding on a number of spatial scales. Synchrotron radiation analytical techniques can be used to eﬀectively investigate many of these problems. This has been recognized explicitly at the ALS (Robinson, 1997), where they have considered how to integrate synchrotron techniques with environmental projects. A particular virtue of photon use is ease of analyzing wet samples as compared with electron spectroscopy. It was suggested that an appropriate name for the ﬁeld would be Molecular Enviromental Science (MES) in the soft X-ray Region, although it could be argued that the use of ‘‘molecular’’ is too restrictive. In any case, some of the experiments described here ﬁt well into such a conceptual framework. One topic that has been the subject of many investigations has been the uptake of metals by plants and trees from contaminated soils and water. Experiments have recently been carried out at the ESRF and at the NSLS, which are relevant to this topic. The ESRF experiments were carried out using high-resolution beams formed by capillary focusing to

540

Jones

Figure 32 The relative concentrations of Ca, Zn, and Pb are for a scan across a section of a child’s deciduous tooth which was thick compared to the absorption of the characteristic x-rays. The variation of the lead in the scan may be useful in the future in understanding the time dependence of lead exposure and uptake by the child. The spatial resolution and step size were both about 10 mm. (From Jones et al., 1992.)

determine wood density and elemental composition. Figure 33 shows results obtained for the distribution of Mn and the density in the region analyzed. The area examined is at the boundary between winter and onset of growth in the spring. Measurements at the cellular level make it possible to examine pathways for accumulation of toxic elements from the environment. The experiments at the NSLS have been aimed at looking at accumulations of metals over a long-term period at trees growing in close proximity to a metal smelter (Martin et al., 1998). The focus of this work was to attempt to determine the relationship between the time-varying environmental conditions and the composition of the wood. The results obtained using SRIXE and secondary ion mass spectroscopy (SIMS) to examine thick wood specimens showed that the distributions were very heterogeneous and that growth cycles were not easily distinguished from the trace element concentrations. In geology, many of the experiments used thin sections of rocks for studies of speciﬁc types of minerals in a heterogeneous matrix. Examples of this type of application are the study of zoned carbonate gangue cements found in Tennessee (Kopp et al., 1990). The XRM measurements were used in an attempt to interpret the eﬀects of trace amounts

Synchrotron Radiation-Induced X-ray Emission

541

Figure 33 Results obtained for the distribution of Mn and the density in the region analyzed in the ESRF experiments were carried out using high-resolution beams formed by capillary focusing to determine wood density and elemental composition. (Rindby A, Engstro¨m, From ESRF Highlights—1996=1997, p 80.)

of Mn and Fe on cathodoluminescence in carbonates. Further, the variations in the trace element concentrations should aid in deducing the source and direction of ﬂow of ﬂuids responsible for the formation of dolomites. Small particles can also be eﬀectively examined using XRM. Sutton and Flynn (1988) and Flynn and Sutton (1990) have carried out a series of experiments dealing with analyses of extraterrestrial particles. These analyses were carried out on particles of sizes less than 100 mm. Minimum detection limits less than 10 ppm were obtained for particles less than 20 mm in size using the BNL X26 XRM. Tuniz et al. (1991) also used this equipment for examination of ﬂy ash taken from diﬀerent types of power plants and incinerators. They measured elemental composition in individual ﬂy-ash particles with sizes down to a few micrometers and also made two-dimensional maps of the distributions of the elements in 10-mm-thick sections of particles produced by a lapping technique. The results may be useful in verifying models for production of toxic compounds in incinerators based on the presence of speciﬁc metals in the ash (Karasek and Dickson, 1987; Hagenmaler et al., 1987; Altwicker et al., 1990). D.

Materials and Chemical Sciences

Fluorescence can also be used to advantage in the material sciences. For example, Isaacs et al. (1991) have studied the concentration gradients produced in a solution during the localized corrosion of stainless steel. The combination of high spatial resolution and excellent detection sensitivity enabled them to study the variation in the nickel concentration above a stainless-steel surface immersed in a bulk chloride electrode. Figure 34 shows the electrochemical cell used in the work and the observed variation of nickel concentration above the stainless-steel surface. From a study of the concentration gradients, Isaacs et al. (1991) were able to identify eﬀects arising from silicon in the steel. In situ studies of kinetic eﬀects should be of increasing interest, not only in corrosion measurements, but also for other types of chemical reaction. SRIXE can also be applied to the measurement of relevant to supported catalysts. In particular, it can be used to measure the distribution of Cr catalyst distributions in polyethylene polymerization particles (Jones et al., 1997). The use of SRIXE makes it possible to study the catalyst distribution at high yields that are not possible by other methods. The experiments was done on microtomed sections of polyethylene particles using the NSLS X26A XRM. A map of the Cr distribution is shown in Figure 35. Sharp peaks observed at the periphery of the particle are consistent with observations made using computed microtomography. The presence of rather uniform concentration of the catalyst through the particle has not been observed previously. The observation shows that it is possible to investigate the fragmentation of the catalyst during the polymerization process by making sequential measurements as the polymerization process evolves. Use of higher

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Figure 34 The apparatus used for studying corrosion of stainless steel (S. S.) in a chloride environment is shown on the left (a). The spatial distribution of nickel observed above the stainless is shown as a function of distance above the metal–liquid interface (b). (From Isaacs et al., 1991.)

spatial resolution and improved detection limits feasible with new XRM equipment will make it possible to substantially improve this type of investigation.

X. TOMOGRAPHY Computed microtomography (CMT) is an important approach to nondestructive analysis and has been extensively developed using conventional x-ray sources. The synchrotron source gives substantial advantages because of its high brilliance and continuous x-ray spectrum. The superior properties of the synchrotron source have led to CMT instrumentation capable of superior spatial resolution and shorter data acquisition times. Almost all the work that has been done has concentrated on the use of CMT in the attenuation mode where determinations are made of the linear attenuation coeﬃcients. The absorption mode is a simple and eﬀective approach to CMT imaging. It can be used on samples with a variety of sizes and attenuation coeﬃcients by choosing the appropriate x-ray energy. The use of SRIXE is more restricted because the specimen must be small enough to allow the escape of the characteristic x-ray of interest. However, as is the case with EXAFS and XANES, the absorption and emission (SRIXE) approaches are complementary and both need to be available as part of the basic XRM. Reﬁnements to the method are necessary in order to get better information on the elemental composition of the materials. The absorption approach can be reﬁned by producing tomograms above and below an elemental x-ray absorption edge. Subtraction of the tomograms gives the concentration of that element. The diﬀerent technique is valuable for the study of major and minor elements to the 0.1% level. The imaging of trace elements often necessitates the use of SRIXE so that speciﬁc elements are selected through the detection of their characteristic x-rays. The detection system must be chosen for high eﬃciency and high counting-rate capability. The type of specimen, which can be investigated, is limited by the attenuation of the ﬂuorescent x-rays in the sample being investigated. The sample dimension is strongly constrained because of this.

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Figure 35 Map of the Cr distribution. The experiment was done on microtomed sections of polyethylene particles using the NSLS X26A XRM. Sharp peaks observed at the periphery of the particle are consistent with observations made using computed microtomography. (From Jones et al., 1997.)

Several groups have done CMT work with synchrotrons. Flannery et al. (1987) developed a third-generation type system at the NSLS. They used a x-ray magniﬁcation system with a scintillation detector coupled to an image intensiﬁer to produce images with a claimed resolution down to 1 mm. A similar approach was used by Bonse et al. (1986) and Kinney et al. (1988) at SSRL. Spanne and Rivers (1987) demonstrated at a ﬁrstgeneration system at the NSLS X26 beam line. Later work with the apparatus has produced images with a spatial resolution of 1 mm61 mm and a slice thickness of 5 mm, quite comparable to the results from the third-generation devices. More recently, a third-generation apparatus has also been put into operation at BNL (Dowd et al., 1998). It can be used for analyses with voxel sizes as small as 2.7 mm3. The ﬁrst-generation approach takes longer to produce images but has the advantage that beam scattering eﬀects in the sample are eliminated and SRIXE measurements can be performed to produce elemental maps. In the third-generation systems, elemental maps are made by subtracting images taken above and below the absorption edge of interest. The DL for such an approach is about 0.1%. Spanne (1990) has carried out a pilot study with the aim of evaluating the potential for mapping of light elements at the cellular level in the rat sciatic nerve using ﬂuorescence CMT. A comparison of the mean free path for characteristic x-rays from potassium and typical sciatic nerve sizes shows that it is feasible to make corrections for the attenuation of the potassium K x-rays in the nerve. The computed emission tomogram of the distribution of potassium in the epineurium of a rat sciatic nerve given in Figure 36 illustrates this point. Note that the short escape depth for the potassium x-rays that are observed

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Figure 36 Computed emission tomogram showing the distribution in the epineurium of a freezedried rat sciatic nerve. The pixel size was 3 mm63 mm and a slice thickness of 5 mm. The matrix size was 1756175 pixels. (From Jones et al., 1990a.)

necessitates special measures during the reconstruction of the image. Saubermann (personal communication, 1989) points out that ﬂuorescence CMT makes the studies of elemental distributions in unsectioned samples possible. Examination of unsectioned samples also makes in vitro analysis of sections of nerves several millimeters long feasible.

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Longitudinal distributions of elements can then be conveniently studied by scanning at diﬀerent heights, and it is even possible to go back to a previously mapped region for a more detailed examination if steep concentration gradients are discovered. Longitudinal concentration gradients have been demonstrated in the rat sciatic nerve, although with a very poor longitudinal resolution and may be of signiﬁcance in nerve injury (LoPachin et al., 1988; LoPachin et al., 1990). Exxon (Auzerais et al., 1996) and BNL (Spanne et al., 1994, Coles et al., 1996, 1998a, 1998b) groups have investigated topics related to the microgeometry of sandstones in parallel experiments at the NSLS. In addition to measuring properties of the sandstones such as porosity, premeability, tortuosity, and connectivity, attention was given to the displacement of oil by water and modeling of ﬂow through the experimentally determined structures. A structure of Fontainebleau sandstone measured in the experiment of Spanne et al. (1994) is shown in Figure 37. The structure was analyzed to give a theoretical representation of the structure that could be used for predicting ﬂuid ﬂow at larger sizes scales. Feng et al. (1999) have examined the structure of micrometeorites using the BNL equipment. Previous examinations of the particle have shown that there are void spaces and small nuggets of Pt in the interior of some of these particles. This information was gained by laborious sequential sectioning of particles with a diameter of about 400 mm. The use of CMT to gain the same information represents a major step forward in the analytical technique for the ﬁeld. A representation of a Pt nugget observed in a micrometeorite is displayed in Figure 38. The results show that it will be possible to systematically examine large numbers of specimens to obtain signiﬁcant information about the nuggets and about the history of the particle as it passes through the Earth’s atmosphere. Several groups at the ESRF and NSLS have been concerned with the development of improved methods for obtaining contrast between materials with similar x-ray absorption coeﬃcients (see, for example, Raven et al., 1997). Buﬃere et al. (1998) have applied the phase-contrast technique, based on diﬀraction patterns produced at discontinuities in the sample, to measure damage in metal matrix composites. This was an ingenious experiment that investigated the development of cracks in the composite as a function of tensile forces applied to the specimen in situ. The experiment revealed the cracking of SiC particles, decohesion between sample phases, and propagation of pores. The development of CMT at synchrotrons has been very rapid over the past 10 years from the standpoint of instrumentation. Applications have also been developed, but perhaps at a slower pace. This will change in the future, with an increase in the availability of the technique and improvements that can be made with the increasing use of third-generation synchrotron undulators. It is apparent that both absorption and emission approaches are needed and that both approaches are required at any synchrotron XRM facility. The emission technique will be greatly assisted by the use of undulator beam lines to obtain higher counting rates. XI.

EXAFS AND XANES

Extended x-ray absorption ﬁne structure and the related XANES have been widely applied to give information on the chemical state of elements in many diﬀerent materials (Winick and Doniach, 1980; Koningsberger and Prinz, 1988). Many of the experiments that have been carried out have used relatively large x-ray beams and thick specimens to make absorption measurements on a timescale of minutes possible. This approach is not useful for elemental concentrations less than about 0.1–1%. The use of EXAFS and XANES at lower concentrations can be achieved by use of ﬂuorescent x-ray detection. Cramer et al.,

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Figure 37 View of the pore space of Fontainebleau sandstone obtained by CMT. The pore structure is shown as white and the rock as black; that is, the pore structure is opaque and the rock transparent. (From Spanne et al., 1994.)

(1988) have developed at 13-element Si(Li) x-ray detector which gives a high eﬀective detection eﬃciency for this type of application. Other workers have developed means of acquiring EXAFS spectra on a millisecond timescale (Tolentino et al., 1990) and have made measurements of chemical state on a ﬁner spatial scale (Iida et al., 1989). The future development of x-ray microscopy can thus be seen to include the use of SRIXE with EXAFS and XANES and the development of new techniques to make it possible to work with improved DLs and beam sizes at the micrometer level. The NSLS X26 group has employed a simple channel-cut silicon monochromator with an energy resolution of about 1.1 eV for several demonstration XANES experiments. The beam was ﬁrst deﬁned with a four-jawed aperture whose size could be adjusted using computer-controlled stepping motor drivers. This was followed by the monochromator placed about 10 cm upstream from the target. The beam moved vertically on the target by about 60 mm during the scan. This was not important for the resolutions used, but could be easily compensated for by a correlated motion of the target to keep the same spot under the beam. The ﬁrst test used a thick specimen of NIST SRM 1570 spinach leaves, which contained 550 ppm iron. The iron x-rays were detected with the XRM equipment described earlier. The work was done with a beam size of 2 mm2. The results of a scan of a pure iron specimen (beam size 200 mm6200 mm) and the spinach leaves are shown in Figure 39. The spectra agree well with the results of a similar scan done on the NSLS X19 beam line using the 13-element Si(Li) detector. Extrapolation from these initial values showed that work with beam sizes down to 100 mm and perhaps lower was feasible with the existing equipment for this particular target.

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Figure 38 CMT three-dimensional image of the interior of type-I deep-sea spherule with parts of the surface cut away. The compact bright spot in the upper right is a platinum group metal nugget. The light mass in the foreground represents a hole in the interior of the spherule. (From Feng et al., 1999.)

A more stringent demonstration is the use of thinner specimens. For this purpose, measurements were made on the chromium contained in olivine and pyroxene components in a 30-mm section of lunar mare basalt 15555 from Apollo 15 and in a 10-mm section of a rat kidney. The lunar basalt study was undertaken because the oxidation state of the chromium could shed light on conditions existing at the time of the formation of the mineral studied (Sutton et al., 1991). The rat kidney measurement was needed to cast light on nephrotoxic eﬀects resulting from environmental exposures. It is hypothesized that the oxidation state of the chromium changed during its passage from lungs to kidneys with related implications for health eﬀects. The basalt specimen contained chromium at a level of about 1000 ppm and the kidney at a level of about 50 ppm. Figure 40 shows the XANES spectrum for chromium in pyroxene and olivine contained in the lunar basalt taken with a beam resolution of 200 mm6200 mm. Figure 41 shows the spectrum obtained for the rat kidney, but in this case with a 1-mm61-mm beam size. The beam was positioned over the medulla portion of the kidney. The XANES spectra show that chromium exists primarily in the 2þ and 3þ states in the lunar olivines but as the 3þ state in the kidney.

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Figure 39 Fluorescent XANES spectrum obtained for Fe at the 550-pm level in a thick NIST SRM 1570 (spinach) sample compared to the spectrum obtained under the same conditions for an iron foil.

The application of SRIXE to EXAFS and XANES experiments has expanded greatly on the X26 beam line since these initial experiments. This has been especially true in the geological and environmental ﬁelds. Delaney and co-workers have continued their examination of both geological and extra terrestrial materials (Delaney et al., 1996, 1998a, 1998b; Flynn and Sutton, 1990). A number of experiments have been performed that

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Figure 40 Fluorescent XANES spectrum obtained for Cr at the 1000-ppm level in a 30-mm-thick section of pyroxene and olivine contained in lunar basalt 15555. The spatial resolution was 200 mm6200 mm. (From Sutton et al., 1991.)

relate to speciation of contaminants in soils and sediments (Bertsch et al., 1994; Bajt et al., 1993; Sutton et al., 1994; Tokunaga et al. 1997, 1998). Ilman et al. (personal communication, 1996) have investigated the chemical state of Cr and As in wood treated with chromated copper–arsenate and the role of Mn in fungal diseases of wood. The work, taken as a whole, serves as an actual demonstration of the concept mentioned in Sec. IX for deﬁnition of a ﬁeld of Molecular Environmental Science (Robinson, 1997). There are several examples of the way that these experiments can be extended at thirdgeneration synchrotron facilities. Sarret et al. (1998) used a bending magnet beam line at the ESRF to do experiments with a spatial resolution of 300 mm6300 mm for examination of trace elements at concentrations as low as 100 ppm. These parameters are similar to those at the NSLS X26 beam line. Measurements were made to study uptake mechanisms of Pb and Zn in lichens. Lichens were of interest because they are commonly used as a biomonitor to assess environmental pollution. Analysis of these spectra made it possible to reach an understanding of how Pb and Zn are transported and absorbed in the plant. A demonstration experiment similar in intent to the work at the NSLS and ESRF has been performed at the ALS (Warwick et al., 1998). In this example, measurements were made with a much higher spatial resolution. The oxidation state of Cr in a soil sample was determined by ﬁrst scanning a region of 60 mm680 mm. One region of high Cr concentration was found and chosen for examination with XANES. The spectrum obtained

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Figure 41 Fluorescent XANES spectrum obtained for Cr at the 50-ppm level contained in a thin section of rat kidney. The spatial resolution was 1 mm61 mm. The spectrum shows that there is little Cr VI present in this portion of the kidney.

from this region with an area of a few square micrometers indicated that the Cr was in both Cr(III) and Cr(VI) oxidation states. The results of this experiment are shown in Figure 42. Another example is given by the work of Cai et al. (1998) at the APS. They describe a beam line using a undulator source and zone plate focusing to obtain a beam size of less than 0.25 mm and a photon ﬂux of 561010 photons=s at a bandwidth 0.01%. It is possible to do x-ray diﬀraction, x-ray ﬂuorescence, and computed microtomography

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Figure 42 Elemental mapping and chemical speciation of Cr in a soil specimen on the micrometersize scale. Measurements performed by R. Reeder et al. (Dept. of Geosciences, SUNY Stony Brook) at the ALS. (From Warwick et al., 1998.)

simultaneously on this beam line. Data are presented for spatial distributions of Mn, Fe, Cu, and Zn in a root, hydrated P. lanceolata, infected by a fungus, mycorrhizal fungus G. mosseae. A XANES spectrum for Mn obtained from a location in the root at a spatial resolution of 1 mm63 mm is shown in Figure 43. The examples of work that has been done at the NSLS, ESRF, ALS, and APS on soils and metal transport in plants during the past decade illustrates the usefulness of the synchrotron XRM for geological and environmental work. A similar collection of experiments can be assembled to illustrate other applications of SRIXE=XANES=EXAFS using both second- and third-generation facilities. It is clear that the scientiﬁc applications of these techniques will increase dramatically in the future.

XII.

FUTURE DIRECTIONS

What does the future hold for synchrotron XRM? Several diﬀerent developments can be predicted for the ﬁrst decade of the next millenium, 2000–2010, with some conﬁdence. First, continuation of the present types of experiments will go on at an expanded rate as the virtues of the method become better known. The applications will beneﬁt from enhancements in the facilities, which will bring spatial resolutions to 1 mm2 for two-dimensional maps or to 1 mm3 for CMT. Relatively minor improvements to the existing systems are needed to achieve this level of resolution. Detection-level (DL) values should also improve somewhat with the introduction of improvements in the x-ray detector systems, again using known techniques. There will be a further merging of XRM techniques with those developed for EXAFS and XANES. Second, development of new focusing methods will begin and this will lead to the routine achievement of spatial resolutions of the order of 0.1 mm by 2000–2005. An example is the production of zone plates suitable for focusing of 8-keV x-rays by Bionta et al. (1990) and the application of capillary focusing by Bilderback et al. (1994a,b,c).

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Figure 43 A XANES spectrum for Mn obtained from a location in the root at a spatial resolution of 1 mm63 mm. (Cai et al., 1998. Argonne National Laboratory, managed and operated by The University of Chicago for the U.S. Department of Energy under Contract No. W-31-109-ENG-38.)

Figure 44 The spatial resolution (expressed as the beam area) obtained with high-energy x-ray microscopes is plotted as a function of time. The best spatial resolutions in 1990 are around 1–4 mm2. If improvements had continued at the same rate, resolutions would have been close to atomic dimensions around the year 2000. This did not happen.

Figure 45 The average and peak spectral brightness as a function of photon energy for the LCLS as compared to the values for other operating and proposed facilities. (From Cornacchia, 1998.)

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Third, a new era in SRIXE research has been signaled by the operation of thirdgeneration synchrotron sources such as the Advanced Photon Source (APS) at Argonne National Laboratory and the European Synchrotron Radiation Facility (ESRF) at Grenoble. These systems produce beams with improvements of three to four orders to magnitude when using undulator insertion devices and with higher energies when using either bending magnets or wigglers. A general discussion of the impact of these new rings on XRM has been given by Sparks and Ice (1990), who found detection limits of 108 atoms in a solid and of 103 free atoms for ﬂuorescence detection with a 1-mm2 beam and a data acquisition time of 1 s. An early discussion of the impact of the APS on the geosciences was given by Sutton and Flynn (1988). It remains to be seen how close these predictions will approach reality. It is presently clear that SRIXE is now a major analytical method for the geological and environmental sciences and possibly to a lesser extent for other scientiﬁc ﬁelds. Figure 44 summarizes the time dependence for the synchrotron XRM spatial resolution. The area resolution has decreased exponentially over the past 15 years. Extrapolating into the future assuming the same rate of improvement would lead to an estimate of an area resolution corresponding to a beam size of roughly 1 nm by the year 2000. Looking at what has been accomplished for focusing low-energy x-rays with zone plates and at the high-energy end with capillaries, it is probable that the XRM resolutions will approach a constant value of about 0.025–0.05 mm. We have already seen the characteristic of x-ray sources improved by many orders of magnitude. There is a good probability that this rate of improvement is not about to come to an end. There has been an intense eﬀort to develop a free-electron laser at a number of laboratories. For instance, the Stanford Linear Accelerator Center (SLAC) has developed a design for a Linac coherent light source (LCLS) based on a 15-GeV linear accelerator (Cornacchia, 1998). Figure 45 shows the average and peak spectral brightness as a function of photon energy for the LCLS as compared to the values for other operating and proposed facilities. Improvements of three or more orders of magnitude should be attained. Many new types of experiments should be possible, considering the time structure of the photon beams and their intensity. It seems very safe to conclude that SRIXE experiments should continue to be an area of continuing high interest for the foreseeable future. This is based on the established value of the analytical techniques and the prospectus for major improvements in its capabilities from future improvements in the x-ray source, in x-ray optics and detectors, and in the interfacing of computational methods with the experiments. ACKNOWLEDGMENTS I am particularly indebted to my colleagues for simulating discussions and interactions, which have inﬂuenced my views of x-ray microscopy over the years. Among them are R. S. Bockman, R. D. Giauque, Y. Gohshi, L. Grodzins, A. L. Hanson, J. B. Hastings, K. Janssens, J. G. Pounds, C. Riekel, M. L. Rivers, A. J. Saubermann, J. V. Smith, P. Spanne, S. R. Sutton, A. C. Thompson, S. To¨rok, C. Tuniz, J. H. Underwood, and R. D. Vis. I am particularly saddened to note that P. Spanne died in the crash of Swissair Flight 111, September 1998. This work was supported in part by the Oﬃce of Basic Energy Sciences, U.S. Department of Energy for development and application of analytical techniques under Contract Nos. DE-AC02-76CH00016 and DE-AC02-98CH10886 and by the National Institutes of Health Biotechnology Research Resources Grant No. P41RR01838.

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Sparks CJ, Ice GE. X-Ray Microprobe-Microscopy. Proceedings 15th International Conference on X-Ray and Inner-Shell Processes, Knoxville, TN, 1990. Sparks CJ, Raman S, Ricci E, Gentry RV, Krause MO. Phys Rev Lett 40:507, 1978. Sutton SR, Flynn GJ. Stratospheric Particles: Synchrotron X-Ray Fluorescence Determination of Trace Element Contents. Proceedings of the 18th Lunar and Planetary Science Conference, Houston, TX, 1988, p 607. Sutton SR, Jones KW, Gordon B, Rivers ML, Smith JV. Lunar and Planetary Science XXII. Houston, TX: Lunar and Planetary Institute, 1991, p 1365. Sutton SR, Rivers ML, Bajt S, Jones KW, Smith JV. Nucl Instrum Methods Phys Res A 347: 412, 1994. Themner K, Spanne P, Jones KW. Nucl Instrum Methods Phys Res B49:52, 1990. Thiel DJ, Bilderback DH, Lewis A. SPIE 1740:248, 1992. Thompson AC, Wu Y, Underwood JH, Barbee TW Jr, Nucl Instrum Methods Phys Res A225:603, 1987. Thompson AC, Underwood JH, Wu Y, Giauque RD, Jones KW, Rivers ML. Nucl Instrum Methods Phys Res A266:318, 1988. Tokunaga TK, Brown GE Jr, Pickering IJ, Sutton SR, Bajt S. Environ Sci Technol 31:1419, 1997. Tokunaga TK, Sutton SR, Bajt S, Nuessle P, Shea-McCarthy G. Environ Sci Technol 32:1093, 1998. Tolentino H, Baudelet F, Dartyge E, Fontaine A, Lena A, Tourillon G. Nucl Instrum Methods Phys Res A289:307, 1990. Tombulian DH, Harman PL. Phys Rev 102:1423, 1956. Tros GHJ, Van Langevelde F, Vis RD. Nucl Intrum Methods Phys Res B50:343, 1990. Tuniz C, Zanini F, Jones KW, Nucl Instrum Methods Phys Res B56=57:877, 1991. US Department of Energy. DOE 2000. DOE Publication 797. Washington, DC: Department of Energy, 1997, pp 4–7. Underwood JH, Thompson AC, Wu Y, Giauque RD. Nucl Instrum Methods Phys Res A226:296, 1988. Van Langevelde F, Tros GHJ, Bowen DK, Vis RD. Nucl Instrum Methods Phys Res B49:544, 1990b. Van Langevelde F, Tros GHJ, Bowen DK, Vis RD. Non-imaging Optics for Photon Probe Microanalysis at the SRS, Daresbury (U.K.). In: Jasienska S, Maksymowicz LJ, eds. Proceedings of XIIth IXCOM, Krakow, Poland, 1989. Bristol, UK: Institute of Physics Publishing Ltd., 1990c, p 453. Van Langevelde F, Bowen DK, Tros GHJ, Vis RD, Huizing A, de Boer DKG. Nucl Instrum Methods Phys Res A292:719, 1990a. Vis RD. Fresenius Z Anal chem 337:622, 1990. Vis RD, Van Langevelde F. Nucl Instrum Methods Phys Res B54:417, 1991. Warwick T, Anders S, Hussain Z, Lamble GM, Lorusso GF, MacDowell AA, Martin MC, McHugo SA, McKinney WR, Padmore HA. Synchrotr Radiat News 11:5, 1998. Winick H. Synchrotr Radiat News 2:25, 1989. Winick H. Nucl Instrum Methods 291:401, 1990. Winick H, Doniach S, eds. Synchrotron Radiation Research. New York: Plenum Press, 1980. Wu Y, Thompson AC, Underwood JH, Giauque RD, Chapman K, Rivers ML, Jones KW. Nucl Instrum Methods Phys Res A291:146, 1990.

9 Total-Reﬂection X-ray Fluorescence Peter Kregsamer, Christina Streli, and Peter Wobrauschek Atominstitut, Vienna, Austria

I.

INTRODUCTION

The phenomenon of total reﬂection of x-rays had been discovered by Compton (1923). He found that the reﬂectivity of a ﬂat target strongly increased below a critical angle of only 0.1 . In 1971, Yoneda and Horiuchi (1971) ﬁrst took advantage of this eﬀect for x-ray ﬂuorescence (XRF). They proposed the analysis of a small amount of material deposited on a ﬂat totally reﬂecting support. This idea was subsequently implemented in the socalled total-reﬂection x-ray ﬂuorescence (TXRF) analysis which has spread out worldwide. It is now a recognized analytical tool with high sensitivity and low detection limits, down to the femtogram range. TXRF is basically an energy-dispersive technique in a special excitation geometry. This geometry is achieved by adjusting the sample carrier inclined under angles of about 1 mrad (0.06 ) to the primary beam. The incident beam thus impinges at angles below the critical angle of (external) total reﬂection for x-rays. Usually the liquid sample, with a volume of only 1–100 mL, is pipetted in the center of the reﬂector. As a result of the drying process where the liquid part of the sample is evaporated, the residue is irregularly distributed on the reﬂector (within an area of a few millimeters in diameter), forming a very thin sample. (See Figure 1.) Total-reﬂection XRF oﬀers the following advantages: If the adjusted incident angle of the primary radiation is below the critical angle, almost 100% of the incident photons are totally reﬂected, so the primary radiation scarcely penetrates into the reﬂector and the background contribution from scattering on the substrate is drastically reduced. The sample is excited by both the direct and the reﬂected beam, which results in a doubling of the ﬂuorescent intensity. Due to the geometry, it is possible to position the detector close to the surface of the reﬂector where, in the center, the sample is located. This results in a large solid angle for the detection of the ﬂuorescence signal. Consequently, the peak-to-background ratio in TXRF is high and the detection limits are drastically improved by several orders of magnitude as compared to conventional XRF. The very distinct angular dependence of the characteristic x-ray intensities close to the total-reﬂection regime can be used to investigate surface impurities, thin near-surface 559

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Figure 1

Scheme of TXRF.

layers, and even molecules absorbed on ﬂat surfaces. The analysis of the composition, thickness, and density of layers as well as the nondestructive in-depth examination of concentration proﬁles in the range of 1–500 nm are possible. Demands for TXRF are as follows: Higher photon ﬂux of excitation sources as compared to standard XRF Small source size (typically 10 mm640 mm for x-ray tubes for TXRF) Low divergence (in one direction below 1 mrad). As most widely spread excitation sources, high-power Mo x-ray diﬀraction tubes are in use. Depending on the chemical elements of interest, other anodes might be preferred. The insertion of a spectral modiﬁcation device in the beam path of the primary radiation improves the background; that is, a multilayer monochromator suppresses in the ideal case all photons, except the ones with an energy range which covers the most intense characteristic line of the anode material. It was demonstrated that the optimum excitation source for TXRF is a synchrotron storage ring. The routine determination of light elements (below Si) with TXRF is still a challenging task: Both the excitation as well as the detection of the ﬂuorescence radiation of interest are diﬃcult. In this context, several spectrometers were constructed, based on ultrathin entrance window and windowless detectors as well as synchrotron radiation and new prototype x-ray tubes applied for excitation. Typical applications of TXRF include analysis of drinking water, river water, rainwater, seawater, wastewater, body ﬂuids, tissue, purity of chemicals (acids, bases, solvents, etc.), oils and greases, aerosols, ﬂy ash, soils, forensic and art-historical objects, study of thin layers and depth proﬁles, and so forth.

II.

PHYSICAL PRINCIPLES

The theoretical fundamentals of TXRF can be deduced in a way analogous to that applied in light optics. One has to consider the interaction of an electromagnetic wave that hits the plane boundary between vacuum and a medium described by its refraction index n, which takes into account both scattering and absorption and can be calculated by quantum mechanics:

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n ¼ 1 d ib

561

ð1Þ

It can also be expressed as function of atomic scattering factors f1 and f2 (see Fig. 2), which are tabulated by atomic number dependent on the incident x-ray energy (Henke et al., 1993): n ¼ 1 NA

e 2 l2 r ð f1 þ if2 Þ m0 c2 2p A

ð2Þ

where NA is Avogadro’s number, l is the wavelength of the incident radiation, r is the density of the medium, A is the atomic mass, e and m0 are the electric charges and rest mass of an electron, respectively, and c is the velocity of light. By comparing Eqs. (1) and (2), we obtain r0 r ð3Þ d ¼ NA l2 f1 2p A where r0 is the classical electron radius: r0 ¼

e2 m0 c 2

ð4Þ

The real part of the complex refraction index (1 7 d) is slightly smaller than unity with d ¼ 10 7 5–10 7 6 in the energy range of x-rays. The minus sign reﬂects the fact that the bound electrons follow the exciting photons in the opposite phase. Consequently the refraction index for x-rays is always smaller than 1 (except sometimes for the energies close to absorption edges). The imaginary part b (usually smaller than d) is a measure of the absorption and is related to the photoelectric mass absorption coeﬃcient tm :

Figure 2 f1 and f2 for silicon as a function of the incident radiation energy. At the absorption edge energy of 1838 eV, discontinuities in the scattering and absorption behavior are found. For energies far above this energy, f1 approaches asymptotically the value of its atomic number Z ¼ 14.

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b ¼ NA

r0 2 r l l f2 ¼ rtm 2p A 4p

ð5Þ

According to Figure 3, the following rules are valid: The incident, the reﬂected, and the refracted beams span a plane that is perpendicular to the boundary plane. The angles of the incident beam j0 and the reﬂected beam jR are equal: jR ¼ j0 . The angles of the incident beam j0 and the refracted beam jT follow Snellius’ law: nvacuum;air cos jT ¼ nreflector cos j0

ð6Þ

For x-rays, the real part of nreflector is slightly less than 1, whereas the refraction index for vacuum is 1 (in good approximation also valid for air). Consequently, the reﬂector is the optically thinner medium than vacuum (air), and the reﬂected beam will be deﬂected toward the boundary, which is completely diﬀerent as compared to usual light optics. Due to the fact that the angles for which the total reﬂection eﬀect is observed for x-rays are small, sine functions can be replaced by their argument ðsin j0 ! j0 Þ, and cosine by 1 j20 =2. The electromagnetic waves for the incoming, reﬂected, and refracted beams can be deﬁned in the usual way for light optics, and Fresnel’s formulas will give the ratios of, for example, the reﬂected ðER Þ and incoming ðE0 Þ amplitudes: qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ j j20 2d 2ib 0 ER qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ ð7Þ E0 j þ j2 2d 2ib 0

0

For exact calculations, the polarization state of the primary radiation and the propagation of the refracted beam as the so-called inhomogeneous wave have to be considered. It appears, however, that the eﬀects for MoKa primary radiation and a quartz reﬂector are negligible and can be ignored in the calculation of reﬂection (R ) and transmission (T ) coeﬃcients as well as of refraction angles jT :

Figure 3 The incident (I0), the reflected (IR), and the refracted (transmitted, IT) beams at the interface of two media. The refraction index of the medium from the radiation comes (usually air or vacuum) is greater than the one of the reflector.

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2 ER 4x2 ðx0 xT Þ2 þ y2 R ¼ IR ¼ ¼ T E0 4x2T ðx0 þ xT Þ2 þ y2 2 ET xT 16x3T x0 ¼ T ¼ E0 j0 4x2T ðx0 þ xT Þ2 þ y2

563

ð8Þ ð9Þ

The parameters xT ; x0 , and y are deﬁned as follows: xT ¼

jT jcrit

ð10aÞ

x0 ¼

j0 jcrit

ð10bÞ

y¼

b d

ð10cÞ

The reﬂection and transmission coeﬃcients fulﬁll the following condition (as a direct consequence of energy conservation): RþT1 The refraction angle jT, given by ﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ h i pﬃﬃﬃ 2 jT ¼ d x20 1 þy2 þ x20 1

ð11Þ

ð12Þ

is a function of the incidence angle and coincides with it for the angles well above the critical angle. The refraction angle is very small, but does not vanish for an (hypothetical) incidence parallel to the interface ðj0 ¼ 0Þ (see Fig. 4).

Figure 4 The refraction angle jT as a function of the incident angle j0 for MoKa and a quartz reflector. The critical angle for this configuration is at 1.8 mrad. Even for very small incident angles close to 0 , the refraction angle is 5 mrad.

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The so-called critical angle of total reﬂection, jcrit , can be written as pﬃﬃﬃﬃﬃ jcrit ¼ 2d

ð13Þ

and is closed to the position of the respective inﬂection points of the transmission and reﬂection coeﬃcient curves and also of the refraction angle. For energies well above the reﬂector material’s absorption edges, a simple dimensional equation can be derived, sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 28:8 Zr ½g=cm3 ð14Þ jcrit ½mrad ¼ E ½keV A showing that the critical angle is inversely proportional to the energy of the incident radiation, which means that the higher the energy of the incident photons, the smaller the angle of the reﬂector has to be adjusted to observe total reﬂection. Z is the atomic number, r is the density, and A is the atomic mass. The dependence of the reﬂectivity on the critical angle of the reﬂector material is demonstrated in Figure 5: The lower the atomic number of the reﬂector, the lower its critical angle and the more step-function-like is its shape. For the combination of MoKa incident radiation reﬂected on quartz, jcrit ¼ 1.8 mrad ( ¼ 0.10 ). The Fresnel equations are based on classical dispersion theory and have been derived assuming a perfectly ﬂat and smooth interface between homogeneous media. Even though a real surface is, in general, rough on a microscopic scale, the experimental results have shown suﬃcient agreement. The penetration depth zP for total reﬂection, in accordance with the deﬁnition for conventional XRF, is the distance measured normally to the interface at which the

Figure 5 Calculated reflectivities of several reflector materials for MoKa radiation. The values of the critical angles jcrit for C, Si, Ge, Nb, and Ta (1.75, 1.8, 2.5, 3.1, and 4.2 mrad, respectively) are indicated by circles.

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565

intensity of the (refracted) beam is reduced by a factor of e (i.e., to 37% of its value). It is directly proportional to the refraction angle (for conventional XRF, this angle is equal to the angle of incidence): zP ¼

1 j rtm T

ð15Þ

As will be shown later, the penetration depth has no direct inﬂuence on the background intensity. Only some years after TXRF was introduced as a spectrometric tool, the interference eﬀect on top of the reﬂector between the incoming and reﬂected beam was recognized. The undisturbed coherent superposition of the plane electromagnetic waves results in a variation of the intensity pattern, depending on the distance above the surface, called a standing wave. Figure 6 displays the fundamental facts; the length D gives the distance between two maxima of the standing wave with wavelength l, D¼

l 2j0

ð16Þ

and is typically in the range of 10–100 nm. The intensity of the standing wave is a function of the height z above the reﬂector surface and the incident angle: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ z Iðj0 ; zÞ ¼ I0 1 þ Rðj0 Þ þ 2 Rðj0 Þ cos Fðj0 Þ 2p ð17Þ Dðj0 Þ This intensity can vary between zero and four times the primary intensity I0 . The phase factor Fðj0 Þ cos F ¼ 2x20 1

ð18Þ

was found by Bedzyk et al. (1989) for the case b ¼ 0 (i.e., when absorption was neglected). For standard TXRF where granular residues deposited on a reﬂector are analyzed, a problem might arise that samples with diﬀerent thickness (but smaller than D ) can give diﬀerent ﬂuorescence intensities. For homogeneous residues for which many of these maxima and minima of the standing waves occur within the sample thickness, this eﬀect will level out to a large extent. If an internal standard is homogeneously mixed with the sample material, the ratio of the respective ﬂuorescence intensities will lead to

Figure 6 The interference (standing wave) zone between the incident (I0) and reflected (IR) plane waves with wavelength l shows nodes and antinodes with a period of D.

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an acceptable precision in any case. Therefore, the use of an internal standard is inevitable for TXRF analysis of the residues. (See Figure 7.) For a broad range of elements, TXRF provides x-ray spectra characterized by high sensitivities and low background. The background generated by the sample support can be assessed from the transmitted part of the primary radiation. One of the main features of TXRF is a strong dependence of both the scattered and ﬂuorescence signals, which originate from the substrate itself, on the incidence angle. The intensity of the scattered radiation Iscatter(j0) is given by Iscatter ðj0 Þ / j0 Tðj0 Þ

ds 1 dO mm ðE0 Þ

ð19Þ

The following factors had been considered: For standard TXRF, the area on the reﬂector, seen by the detector, is generally smaller than the area hit by the primary radiation; therefore, a geometry factor proportional to sin j0 has to be used (de Boer, 1989) (and the sine can be replaced by its argument). Only those primary photons that are not totally reﬂected on the surface are able to penetrate and induce interactions. This leads to the transmission coeﬃcient T. Considering the phenomena of elastic and inelastic scattering, the diﬀerential scattering coeﬃcients ds=dO, which depend on the scattering angle (for TXRF in the range of 90 ) should be utilized (Kregsamer, 1991) (see Fig. 8).

Figure 7 Relative intensities of the standing waves for the incident angles of 0.9, 1.8, (jcrit), and 3.6 mrad (primary radiation: MoKa, silicon substrate). The standing-wave effect above the interface increases with decreasing incident angle. The dependence of the distance between maxima D on the incident angle [see Eq. (16)] is also seen. Inside the medium, the refraction angle is responsible for the intensity decrease in the direction normal to the surface; the propagation is very shallow below the surface for small incident angles.

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Figure 8 Differential coherent and incoherent scattering coefficients for the MoKa incident radiation, scattered by amorphous silicon. The scattering angle is the sum of the incident and the detection angles and is close to 90 for TXRF.

The attenuation of the refracted beam inside the substrate on the way to the location where an interaction takes place may not be neglected. However, the attenuation from this point to the surface toward the detector can usually be ignored. For the incident angles larger than the critical angle, the geometry factor is responsible for the increase in the spectral background, whereas the sudden reduction for the smaller angles originates from the almost perfect total reﬂection; the reﬂection coeﬃcient is nearly 1 in this range. (See Figure 9.)

III.

INSTRUMENTATION

A.

Excitation Sources for TXRF

The usual excitation source for TXRF is a high-power diﬀraction x-ray tube with a Mo anode with an electrical power of 2–3 kW. This type of x-ray tube is also available with Cr, Cu, Ag, and W targets. The line focus of the anode has to be used so that the emitted brilliance is in correlation with the slit collimation necessary to produce a narrow beam with the divergence less than the critical angles involved. A higher photon ﬂux on the sample can be achieved by using rotating anodes, which can stand up to 18 kW. In all cases, the focal size of the electron beam on the anode is a line with the dimensions of 0.468 mm2 (ﬁne focus) or 0.4612 mm2 (long ﬁne focus). The emission of the x-rays is observed under the angle of 6 to the anode surface, so that the width of the focus is reduced optically by the projection with sin 6 ( ¼ 0.1) to 0.04 mm.

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Figure 9 The angular dependence of the background intensity for TXRF measurements (MoKa primary radiation under total reflection on a quartz reflector; the critical angle is 1.8 mrad). The dashed line represents the case where no total reflection is observed.

The emitted spectrum consists of the continuum (bremsstrahlung) and superimposed are the characteristic lines of the anode material (e.g., MoKa and MoKb) (see Fig. 10). Insertion of spectral modiﬁcation devices in the path of the primary radiation beam improves the background (see Fig. 11). A so-called cutoﬀ reﬂector, acting as a low-pass ﬁlter, suppresses the high-energy photons (above the K lines of the anode material), whereas the low-energy part of the primary radiation, including the characteristic lines, should be reﬂected, improving the background—in particular, in the low-energy region. High-energy photons of the continuum (although not of direct inﬂuence to the background under the lines of interest) can get to the front side of the detector and can be inelastically backscattered (Compton backscattering). By-products of such events are the recoil electrons of the scattering atom, which are registered in the detector. Their energy is a function of the scattering angle and the primary energy. As a consequence, the spectrum shows a shelf (Compton edge), starting from zero up to several kiloelectron volts. The eﬀect of inserting a cutoﬀ reﬂector into the beam path between the x-ray tube (collimation unit) and the sample reﬂector can be seen from the diﬀerence for the highenergy parts of the spectra in Figures 12a and 12b; almost all photons above 20 keV are suppressed for the latter. A proper alignment is fairly critical, because, otherwise, either some high-energy photons can pass the device or, even worse, a part of the needed excitation spectrum is suppressed (see Fig. 12c, where MoKb is drastically reduced in intensity). A typical resulting sample spectrum for a well-adjusted TXRF spectrometer (sample reﬂector also adjusted) is displayed in Figure 13. Unknown samples can be quantiﬁed (referring to a given internal standard element) with sensitivity factors [for deﬁnition, see Eq. (22)] (Fig. 14).

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Figure 10 Measured primary spectrum of a fine-focus Mo diffraction x-ray tube (45 kV acceleration voltage) as typically used for TXRF. The characteristic MoKa and MoKb lines are superimposed on the bremsstrahlung background. In the energy range between 5 and 10 keV, several additional lines resulting from anode contaminations and so on were present for this particular tube.

Figure 11

Major components of a TXRF spectrometer.

Monochromators also can modify the primary radiation and they are usually set to the energy of the most intense characteristic line of the anode material. For a Mo-anode x-ray tube MoKa or for a W-anode x-ray tube WLb are selected, but a part of the continuum can be monochromatized as well. Commonly used crystal monochromators

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Figure 12 (a) Scattered spectrum of a Mo x-ray tube (40 kV acceleration voltage) as used in a TXRF module spectrometer; (b) scattered spectrum, under the identical conditions as above, but with a cutoff reflector inserted—adjusted to suppress primary photons between 20 and 40 keV. (c) scattered spectrum, under the identical conditions as in (a) and (b), but with the cutoff reflector slightly de-adjusted. As a consequence, the MoKb line is also suppressed.

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571

Figure 12

have the disadvantage of a very narrow energy band transmitted (usually in the range of few electron volts), whereas synthetic multilayer structures are characterized by higher DE=E and reﬂectivities of up to 75% for premium quality materials. (See Fig. 15.) The excitation of an element is the most eﬃcient when the energy of the exciting radiation is close to its absorption edge. The search for anode materials for special applications has led to the development of Al- or Si-anode x-ray tubes which are suitable for the determination of elements below Si, such as F, Na, Mg, and Al. A spectrometer has been developed with a variable double multilayer pair used as a tunable monochromator combined with an x-ray tube with an anode made of a homogeneous alloy of molybdenum and tungsten (Knoth et al., 1997). Three discretely adjustable excitation energies (9.7, 17.5, and 35 keV) are supplied by this combination and cover a much wider range of the elements than the detectable with a single spectrometer. The detection limits were determined to be 0.4 pg for Ni using WLb excitation, 0.6 pg for Pb with MoKa excitation, and 8 pg for Cd (determined by using its K lines) with the 35-keV bandpass excitation. The applicability of three conﬁgurations of curved multilayer mirrors as monochromatizing and focusing elements has been studied theoretically (Stoev et al., 1998) and the predicted theoretical results were compared with those obtained with a prototype spectrometer (tunable between 9.7 and 17.5 keV, with the above-mentioned W=Mo alloy anode) for the element Ni (Knoth et al., 1999). Currently, synchrotron radiation is the most brilliant photon beam for XRF and it has almost ideal features for TXRF. It is several orders of magnitude brighter than the output of x-ray tubes and is naturally collimated and linearly polarized in the plane of the orbit of the electrons of positrons. The spectral distribution is continuous; therefore,

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Figure13 Spectrum of a multielement standard (25 ng of each element) prepared as residue sample on a quartz sample reflector (well-adjusted cutoff reflector). Conditions: TXRF module, Mo tube, 50 kV, 40 mA, 100 s measuring time. SSr and DLSr are the sensitivity factor and detection limit for Sr, respectively.

monochromatization is necessary and selective excitation is possible. Because scattering and absorption of primary photons in air should be avoided, the use of a vacuum chamber is recommended. As a result, the ArK line disappears from the spectrum and the absorption of primary and ﬂuorescent photons is also avoided. If for any reasons the measurements in vacuum are not possible, ﬂushing the chamber with helium is advantageous. B.

Sample Reflectors

For the trace analysis of granular residues, a carrier with high reﬂectivity that serves as a totally reﬂecting sample support is required. Therefore, the mean roughness should be in the range of only a few nanometers and the overall ﬂatness should typically be less than l=20 (l ¼ 589 nm, the mean wavelength of the visible light). Furthermore, reﬂectors should be free of impurities so that the blank spectrum should be free from contamination peaks and the carrier material must not have ﬂuorescence peaks in the spectral region of interest. In addition, the carrier material must be chemically inert (also against strong chemicals, which are often used for the sample preparation), easy to be cleaned for repeated use. They should be commercially available and inexpensive. Typically, they are disk shaped with a 30-mm diameter and a thickness of 3–5 mm. The carrier materials commonly used are quartz glass in the synthetic form as Synsil, or Suprasil, fused silica, and elemental Si. Heavier elements like Ti, Nb, and Ta were

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Figure 14 Experimentally determined sensitivities (total) for the K lines of several elements for a TXRF spectrometer with a cutoff filter. The excitation can be split (theoretically) into the contributions from the characteristic MoK lines (MoKa Kb) and bremsstrahlung (continuum). The ratio of the two sensitivity factors depends, among others, on the applied high voltage (in this case 45 kV). Zr and Nb (Z ¼ 40, 41) have the K absorption edges already above the energy of MoKa.

recently tested and showed a good performance. Boron nitride, glassy carbon, Plexiglas, and Perspex (for single use) are suitable (e.g., for the determination of light elements). All components of a TXRF spectrometer need to be adjusted (at least once) by translation and rotation stages and tilters, either manually, or by using remote controllers. To visualize the eﬀect of total reﬂection, one can use a proper charge-coupled detector (CCD) camera where the x-rays are falling directly on the chip and become visible on a monitor. In a more traditional approach, the x-rays can be observed on a ZnS screen. C.

Detectors

Total-reﬂection XRF is an energy-dispersive XRF method (see also Chapter 3); the radiation is measured mainly by Si(Li)-detectors. A good detector oﬀers a high-energy resolution [Full width at half-maximum (FWHM) in the range of 140 eV at 5.89 keV], intrinsic eﬃciency close to 1 for the x-ray lines of interest, symmetric peak shapes, and low contribution to the background. Primarily, incomplete charge collection at the electrodes leads to low-energy tailing. The detector escape eﬀect creates escape peaks and thus an increased background in certain spectral regions. An inherent advantage of semiconductor detectors is the possibility of bringing the detector crystal very close to the sample, which results in a large solid angle. Light elements emit ﬂuorescent lines in the range from 100 to 1000 eV. The usually used Be entrance window would completely absorb them, so new window materials, oﬀering better transmission characteristics, are used instead.

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Figure15 Two spectra of a sample with a total mass of 700 ng, containing Mg, Cl, Sc, and so forth excited with a Cr tube, obtained with a cutoff reflector inserted into the beam path (Cut-off), as a spectrum-modifying device, and a multilayer monochromator (ML). The spectral background and the characteristic lines are smaller for the latter case; also the scattered CrKb line is suppressed.

The development of Peltier-element-cooled detectors is a step in the direction of being independent of liquid-nitrogen supply. Energy-dispersive detectors with a Si p-i-n diode and Peltier cooling are used as light hand-sized units with an acceptable resolution of 160–200 eV. D.

Manufacturers of TXRF Spectrometers Atomika Instruments (Germany), spectrometers for chemical analysis (ch ) and wafer surfaces (w ); e-mail: [email protected] Atominstitut (Austria) (ch ); e-mail: [email protected] Diﬀraction Technology (Australia) (ch ); e-mail: [email protected] Italstructures (Italy) (ch ); e-mail: [email protected] I.U.T. (Germany) (ch ); e-mail: [email protected] Rigaku (Japan) (w ); e-mail: [email protected] Technos (Japan) (w ); Fax: 0720-20 2002

IV.

CHEMICAL ANALYSIS

In 1971, Yoneda and Horiuchi (1971) published an article with the ﬁrst experimental data from an energy-dispersive x-ray ﬂuorescence measurement with a Ge detector in

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total-reﬂection geometry. The sample was a dried spot of 100 ng of Ni from a water solution of NiCl2 on an ‘‘optical ﬂat.’’ In 1974, Aiginger and Wobrauschek (1974) published results using a Si(Li) detector and a Suprasil reﬂector where 5 mL of Cr salts in aqueous solution were dried and analyzed with a Cu-anode x-ray tube. A more detailed publication (Wobrauschek and Aiginger, 1975) contained theoretical considerations, details of the setup, calibration curves, steps for the quantiﬁcation, and detection limits of 4 ng for the element Cr. In 1977, Knoth et al. (1977) presented their ﬁrst prototype of a total-reﬂection XRF spectrometer and analytical results for a blood serum analysis. From that time on, several groups worldwide participated in the further development of this technique. From 1986 on, regular biannual TXRF user meetings have been held (Proceedings of the TXRF Conference, 1989, 1991, 1993, 1995, 1997). The ﬁrst book dealing exclusively with TXRF was written by Klockenka¨mper (1997). A.

Sample Reflectors and Their Cleaning

Various sample carrier materials have been used for chemical analysis with TXRF, as mentioned in the previous chapter. Nonreﬂecting residues on top of suitable surfaces give a doubled ﬂuorescence intensity, when measurements are made at angles of incidence smaller than the respective critical angle for the substrate, owing to the twofold excitation—by the direct beam (1) and the reﬂected (2) one (see Fig. 16). The reﬂector must be long enough so that the latter can fully contribute. The high sensitivity makes cleaning of the sample carriers critical, particularly in routine operations when large numbers of them are in use. The following cleaning procedure has proven to be eﬀective. 1. Mechanical removal of sample residues from previous analysis using tissue or a brush. 2. Rinsing with water or, in case of organic residuals, with acetone. 3. Gentle boiling for 1 h in a detergent bath (e.g. diluted Extran neutral or acidic), preferably in special carrier supports from Teﬂon, in order to avoid scratching or even breaking. 4. Gentle boiling for 1 h in diluted nitric acid (p.a. grade). 5. Rinsing with ultrapure water and subsequent drying.

Figure 16 Twofold excitation of the sample for TXRF by the primary beam (1) and the reflected beam (2), for a case where the reflector is long enough so that beam (2) is observed.

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6.

7. B.

Nonhydrophobic surfaces, such as quartz, must be coated with a silicone solution (e.g., available from Serva) and subsequently dried at 100 C for about 1 h to keep aqueous samples within a small spot. After cleaning, a blank spectrum should be measured for each reﬂector.

Special Sample Preparation Techniques for TXRF

Numerous samples have been prepared for TXRF by using sample preparation techniques, which are basically the same as those applied for atomic absorption spectrometry (AAS) or ICP–OES. There are, however, special cases where suitable sample preparation techniques are required to take full advantage of the particular features of TXRF (e.g., the capability of analyzing small samples and the advantage of an inert sample carrier). An example is the direct collection of air particulates on the sample carrier with the aid of size-separating samplers (impactors). Plexiglas carriers can be used. If the collected air is wet, the aerosols are reliably deposited on the carrier; when the air is dry, they can be bounced oﬀ. This drawback can be prevented by coating the reﬂectors with a thin ﬁlm of suitable grease (Salva et al., 1993). The reﬂectors loaded with collected aerosols can be directly analyzed after adding an internal standard. Detection limits are in the range of 0.1 ng for a 1-h collection time and a sampling volume of 0.5 m3. Of course, aerosols also can be analyzed after collection on a ﬁlter and subsequent digestion of the loaded ﬁlter material (see Fig. 17). Due to direct analysis, systematic errors are avoided (Injuk and Van Grieken, 1995).

Figure 17 Spectrum of an aerosol sample, which was collected in a Batelle impactor (stage 1) directly on a polycarbonate sample carrier. On top 30 ng of Sc were pipetted as internal standard element (Cr anode, 30 kV, 30 mA, multilayer monochromator, 500 s measuring time).

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One of the inherent advantages of TXRF is the small sample volume required for analysis; Only a few nanograms of sample mass (or a few microliters of sample volume) are required, which might be considered as an advantage compared to some other sensitive analytical methods for trace analysis. For liquid samples, the easiest way of performing TXRF is the direct analysis. A few microliters (of the sample solution mixed with the internal standard) are pipetted on the sample reﬂector and dried, either in a desiccator, with an infrared lamp, on a hot plate, or in a (dedicated) oven. The sample support can be utilized to allow chemical reactions in microliter volumes, for example, in the following way: Up to 100 mL solution loaded with alkali and alkaline-earth salts are placed on the quartz carrier, previously rendered hydrophobic by means of a silicone solution. The droplet on the carrier is spiked with about 5 mL of a 1% sodium dibenzyldithiocarbamate solution. After allowing the sample to dry for about 30 min, the reﬂector is rinsed with ultrapure water. The insoluble metal carbamates, including that of the internal standard element, remain ﬁxed on the hydrophobic surface, whereas the soluble matrix is dissolved and removed by the water. The resulting specimen gives substantially improved detection limits compared to the unprocessed sample. The properties of the sample support in combination with the high detection performance of TXRF are also utilized in the analysis of thin sections of tissue (microtome section). A small piece of tissue is frozen and cut by a microtome in thin sections of about 15 mm thickness. A section is then placed on a sample reﬂector and spiked with an internal standard. This method was applied to tissue of kidney, liver, and lung. Detection limits of 10 ng=g have been reported (von Bohlen et al., 1988). C.

General Sample Preparation

The detection limits obtained for a special sample depend very much on the sample preparation. Figure 18 gives an overview of various common methods for sample preparation in TXRF, depending on the kind of sample to be analyzed. Of course, one has to be aware that sample preparation can cause loss of elements as well as contamination by other elements, and the sample taken for analysis must represent the whole specimen; therefore, homogenization might be required. Solid samples can be crushed and then ground to a ﬁne powder of micrometer grain size. This powder can be mixed with a liquid to produce a suspension, which can be pipetted after adding an internal standard on the sample reﬂector. The pulverized sample can also be dissolved in a suitable solvent, and after adding the internal standard, an aliquot is pipetted on the sample reﬂector and dried. For the decomposition of biological and environmental materials, various methods have been utilized (e.g. with a low-temperature oxygen plasma asher, followed by dissolving the ash in an acid). The most popular method of decomposition of biological and environmental samples like plants, tissue, sediments, and so forth is the wet digestion in Teﬂon vessels (Teﬂon bombs) with acids like HNO3, HF, HNO3 þ HCl, HNO3 þ H2O2, and so forth, in diﬀerent proportions. Using the hydroﬂuoric acid might be a problem if quartz glass reﬂectors are used. The use of a microwave oven for heating the Teﬂon bomb reduces the time of digestion to less than 1 h. The volume of some sample solutions or any sample containing water can be reduced by freeze-drying. The sample is frozen and the solvent is evaporated under vacuum conditions. The dried residue can be dissolved in small volume of acid or wet digested.

578

Figure 18

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Sample preparation methods for TXRF.

It is also possible to extract traces of certain elements by phase separation. To a given volume of sample water solutions at appropriate pH and spiked with an internal standard, an organic solvent is added and mixed thoroughly. Then, the two phases are separated. The traces of metal ions stay in the organic phase, whereas the matrix

5–50 pg=mL

Chemical matrix separation Open digestion Ashing

Preparation

Suspension

Solution

1 pg=mL

1–10 mg=g

3–25 mg=g

Digestion: 0.1–1 mg=g

0.6–20 ng=cm2 10–100 mg=g 10–100 mg=g 1–10 mg=g

1–15 mg=g

20–100 pg=mL 3–20 pg=mL 1–3 ng=mL Digestion: 20–80 ng=mL 40–220 ng=mL 2–30 ng=mL 5–200 mg=g 10–100 mg=g 0.1–3 mg=g

Source: Data from Klockenka¨mper, 1997.

High-purity acids Tissue, foodstuﬀ, biomaterial Mineral oil Mussel, ﬁsh High-purity water

Air dust, ash, aerosols Air dust on ﬁlter Suspended matter Sediment Powdered biomaterial Fine roots

0.1–3 ng=mL

Drying

Freezedrying

Inﬂuence of Sample Preparation Methods on Detection Limits

Rain, river water Blood, serum

Sample

Table 1

0.1–1 mg=g

15–300 mg=g 0.2–2 mg=g

0.2–6 ng=cm2

Pressure digestion

0.5–5 mg=g

Freezecutting

Total-Reflection XRF 579

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elements are left in the inorganic solution. The organic liquid can be directly pipetted onto the reﬂector. Also, the separation of traces by adding a chelating agent and precipitating the metal ions is a commonly used technique. The metal complexes are ﬁltered through a membrane ﬁlter and dissolved in a suitable organic solvent. Generally, the selected sample preparation method has a tremendous inﬂuence on the obtainable detection limits, which is documented in Table 1. Details on sample preparation procedures can be found in the works of Prange et al. (1989), Klockenka¨mper and von Bohlen (1999), Holynska et al. (1996), Dargie et al. (1997). D.

Some Applications

Three main advantages characterize TXRF: simultaneous multielement capability, low detection limits for many elements, and small sample volume. Additional advantages are the absence of matrix eﬀects, easy calibration, fast analysis, and comparatively low costs. Table 2 gives an overview of various kinds of sample that have been already analyzed with TXRF. Generally all kinds of aqueous or acidic liquids where the liquid matrix can be evaporated, leaving a small amount on a sample reﬂector, can be analyzed. Oils, alcohols, whole blood, and blood serum can be analyzed after special treatment. Typical for TXRF are water samples (Fig. 19). Pure water like rainwater or tap water can be directly analyzed (Holynska et al., 1998; Barreiros et al., 1997); for riverwater or seawater as well as wastewater, sample preparation is usually required to remove

Table 2

Applications of TXRF

Environment Water: Rain water, river water, sea water, drinking water, waste water Air: Aerosols, airborne particles, dust, fly ash Soil: Sediments, sewage sludge Plant material: Algae, hay, leaves, lichen, moss, needles, roots, wood Foodstuff: Fish, flour, fruits, crab, mussel, mushrooms, nuts, vegetables, wine, tea Various: Coal, peat Medicine=biology=pharmacology Body fluids: Blood, serum, urine, amniotic fluid Tissue: Hair, kidney, liver, lung, nails, stomach, colon Various: Enzymes, polysaccharides, glucose, proteins, cosmetics, biofilms Industrial=technical Surface analysis: Water surfaces Implanted ions Thin films Oil: Crude oil, fuel oil, grease Chemicals: Acids, bases, salts, solvents Fusion=fission research: Transmutational elements Al þ Cu, iodine in water Geology=mineralogy Ores, rocks, minerals, rare earth elements Fine arts=archeology=forensic Pigments, paintings, varnish Bronzes, pottery, jewelry Textile fibres, glass, cognac, dollar bills, gunshot residue, drugs, tapes, sperm, finger-prints

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Figure 19 Spectrum of 10 mL of the NIST water reference standard SRM 1634c, analyzed in an Atominstitut TXRF spectrometer (Mo anode, 40 kV, 50 mA, multilayer monochromator, vacuum conditions, 1000 s measuring time).

suspended matter or the salt content. Detection limits are in the low nanograms per milliliter range. To reduce the detection limits down to the 10-pg=ml level, freeze-drying with additional leaching of the residue with HNO3 has to be applied. To remove the salt matrix from seawater, complexation, chromatographic adsorption, and subsequent elution have to be performed. This method was developed by Prange et al. (1985). Suspended matter separated from ﬁltrates can be analyzed by digesting the ﬁlter with HNO3 (Prange et al., 1993). For the analysis of sediments, microwave digestion is recommended (Koopmann and Prange, 1991). Environmental monitoring can also be done by analyzing appropriate plants like moss or lichen (biomonitoring). The analysis of plants requires a proper sample preparation. After cleaning, shredding, freeze-drying, and ﬁnally pulverizing, the powder is mixed with HNO3 and digested in a Teﬂon bomb. Vegetable oil is recommended to be diluted with toluene and an oil-based standard should be added as internal standard (Reus, 1991). After pipetting an aliquot on the sample reﬂector, it is heated up to evaporate the volatile parts of the matrix. Detection limits are in the range of 3–20 ng=g. Also, low-temperature oxygen plasma ashing is suitable; the resulting residue has to be dissolved in HNO3. Both depletion and accumulation of trace elements can inﬂuence the biological functions of human beings. Whole blood and blood serum, amniotic ﬂuid, organ tissue, hair, and dental plaque have been analyzed by TXRF. Detection limits down to 20 ng=mL for body ﬂuids after microwave digestion (Prange et al., 1989) were reported. Quality control of ultrapure reagents, like acids, bases and solvents can be performed by using TXRF (Prange et al., 1991). Crude oils (Ojeda et al., 1993), lubricating oils (Bilbrey et al., 1987), motor oils (Freitag et al., 1989), and diesel fuel (Yap et al., 1988) have also been analyzed. Light oils can be diluted with chloroform or toluene, the volatile

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matrix then removed by evaporation, and detection limits in the range of micrograms per milliliter are achievable. If detection limits of nanograms per milliliter are required, lowtemperature oxygen plasma ashing is necessary. For forensic applications the microanalytical capability of TXRF is appreciated, because microsamples are usually available. In recent years, several speciﬁc problems have been investigated, including analysis of hair samples, glass particles, tape fragments, drug powder, semen traces, gunshot residues, and textile ﬁbers (Ninomiya et al., 1995; Prange et al., 1995). Oil paints were investigated by analyzing the pigments, which are characterized by a limited number of major elements. Cotton tips can be used to remove an amount of about 1 mg from the paint and deposit it on a sample reﬂector (Klockenka¨mper et al., 1993). E.

Quantification

One of the inherent advantages of TXRF is the fact that the sample forms a thin ﬁlm on the sample reﬂector; thus, no matrix eﬀects have to be considered and the so-called thinﬁlm approximation is applicable (both absorption and enhancement eﬀects can be neglected). The intensity of the ﬂuorescent radiation for, the Ka line, for example, of an element i with concentration ci in the sample (with mass m ) is then given by ZEmax IðEiKa Þ

¼

I0 ðEÞG

m i s ðEÞci f ðEiKa ÞeðEiKa Þ dE sin j Ka

ð20Þ

E¼Eiabs

where Eabs is the energy of the absorption edge of element i, Emax is the maximum energy of i is the energy of the Ka line of element i, I0 ðEÞ is the spectral the excitation spectrum, EKa distribution of the exciting radiation, G is the geometry factor, siKa is the ﬂuorescence cross i Þ is the absorption factor for the ﬂuorescence section for the K shell of element i, fðEKa i Þ is the relative detector eﬃciency radiation between the sample and the detector, and eðEKa i for the energy EKa . It is assumed that the sample is always completely irradiated by the primary radiation. In this special case, the relation between concentration and ﬂuorescence intensity is linear. The so called sensitivity [counts=(second)(sample mass)] can be deﬁned: Si ¼

Ii ci m

ð21Þ

and Si depends only on fundamental parameters and the measuring conditions, which usually can be assumed to be constant. When an element St is used as the internal standard the relative sensitivity (sensitivity factors) for elements i deﬁned as R Emax i fðEiKa ÞeðEKa Þ E¼E I0 ðEÞsiKa ðEÞ dE i i abs Srel ¼ ð22Þ R Emax St St fðESt Ka ÞeðEKa Þ E¼ESt I0 ðEÞsKa ðEÞ dE abs

can be established experimentally with artiﬁcially prepared standards (Fig. 20) or calculated theoretically. The determination of the concentration ci of the element i in an unknown sample spiked with the same internal standard element is then simple: ci ¼

Ii 1 cSt ISt Sirel

ð23Þ

There is a linear correlation between intensity Ii and concentration ci . The addition of one element as the internal standard of known concentration to the sample is necessary

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583

Figure 20 Fitted calibration curve (relative sensitivities) for K lines (Atominstitut TXRF module, Mo anode) with Ga as internal standard element. The circles represent the measured elements actually used for the calibration (standard solutions with various concentrations). Other elements in a sample can also be determined by interpolation (in this case, S, K, Mn, Br, and Rb).

primarily because of the inhomogeneous excitation due to the standing-wave ﬁeld above the sample reﬂector surface, where the sample is positioned. Also, geometric and volumetric errors can be canceled. A sample is ‘‘thin’’ if its thickness does not exceed the critical thickness (Klockenka¨mper, 1997), which is about 4 mm for organic tissue, 0.7 mm for mineral powders, and 0.01 mm for metallic smears. Under the assumption that the matrix absorption for the analyte diﬀers only slightly from that of the internal standard element, these values can generally be higher by a factor of 10–400. For the calculation of these values, the standing-wave ﬁeld was not taken into account. This eﬀect and the sample selfabsorption can lead to contradictory requirements for the sample thickness (de Boer, 1991a). Figure 21 shows a comparison of detection limits for various analytical methods in the trace element range, following the work of Klockenka¨mper (1997). Inductively coupled plasma–mass spectrometry (ICP–MS) and INAA are more macro than micro methods. In comparison, TXRF requires only very small sample volumes (in the range of microliters). ICP–MS provides lower detection limits, but both spectral interferences and matrix eﬀects make quantiﬁcation more complex.

V.

SURFACE ANALYSIS

In order to remain competitive, the semiconductor industry is being constantly forced to increase the performance and reduce the cost of integrated circuits by shrinking device

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Figure 21 Relative detection limits of INAA, TXRF, ET-AAS, and ICP–MS, applied to trace analysis of aqueous solutions. A 50 mL specimen was used for TXRF and ET-AAS; 3 mL were needed for INAA and ICP–MS. The individual values are approximated after the work of Klockenka¨mper (1997).

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585

dimensions and increasing the number of devices per unit area on chips. A strong correlation has been found between the presence of metal contamination on the wafer surface and process yields. Even very low concentrations of dispersed metals can create defects at the atomic scale, which lead to leakage current, gate insulator breakdown, or poor threshold voltage control. All of these can result in device failure or reliability problems. With decreasing size of devices, the acceptable level of metal contamination decreases as well. The ability to measure low levels of contamination is crucial for the development of techniques to be used to remove these contaminations. The Semiconductor Industry Association (SIA) 1997 National Technology Roadmap for Semiconductors (NTRS; http:==www.sematech.org) projects that the required sensitivities for the transition metals will be at the level of 36108 atoms=cm2 in the year 2001. One of the most important applications showing the analytical power of TXRF is the analysis of impurities on the surface of Si wafers (Na, Al, Ca, Ti, Fe, Ni, Cu, Zn, etc.) for the semiconductor industry (Knoth et al., 1989; Berneike et al., 1989; Weisbrod et al., 1991; Schwenke et al., 1992). The sample is the wafer itself, with its polished plane surface having already the quality required for total reﬂection of x-rays (sometimes except ﬂatness). With the attributes of TXRF [nondestructive analysis, multielement capacity, mapping ability (Fig. 22), and excellent detection limits], this technique dominates over

Figure 22 Wafer mapping for the element Na of 7610 spots (1-mm scan steps, relative coordinates). Synchrotron radiation excitation (SSRL, Beam line III-4), measuring time for each pixel: 10 s.

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Kregsamer et al.

others in this ﬁeld. Presently, the 109-atoms=cm2 level is reached by TXRF using standard tube excitation (Atomika TXRF8030W, http:==www.atomika.com) or rotating anode tube excitation (Philips TREX 630 T=S, http:==www.analytical.philips.com; Rigaku 3750 TXRF). The possibility to map a wafer [i.e., scan several points on the surface and determine the level of contamination as a function of the coordinates (Berneike, 1993)], can help to locate sources of contaminations uniquely. To measure surface contaminations on wafers, a special setup is required, without any front surface contact and with a possibility of making measurements at various angles (Fig. 23). Correct quantiﬁcation is strongly related to the glancing angle setting. Measuring the angle dependence of the ﬂuorescence signal allows one to distinguish the form of the contamination (i.e., if it is ﬁlm or particulate type). Figure 24 shows the ﬂuorescence intensities of Si, Sc, and Ni as a function of the incident angle around the critical angle. The steplike function (Sc) is obtained if the contamination is found in particles on the surface of the wafer (equivalent chemical analysis TXRF). The peaking curve (Ni) is obtained when the atoms are evenly distributed within a layer of a few nanometers thickness placed on the wafer surface. Most of the real samples do not show one of these extreme cases. Prange and Schwenke, (1992) ﬁrst gave some examples of applications of TXRF for surface analysis. If one does not want to measure the complete angle-dependent behavior, only a single measurement performed at the operating angle, where the two curves (Sc, Ni) are crossing, allows accurate quantiﬁcation. For completeness, the typical curve for the bulk material (silicon of the wafer) or bulk contamination is given. Usually, the bulk Si signal is used for control of the angular adjustment.

Figure 23 Typical components of a TXRF spectrometer for wafer analysis with all necessary degrees of freedom for surface mapping (x and y movement), height control (z movement), and rotation around the x axis (angle of incidence). The special detector and the evacuable sample chamber allow also the detection of low-Z elements.

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587

Figure 24 Fluorescence intensities of Sc, prepared as a residue sample (conventional TXRF), Si of the sample refractor substrate, as representative of bulk signal, and Ni, prepared as a thin-film sample. The angles of incidence are normalized to the respective critical angles.

A.

Quantification

For the case of granular residues on a substrate (particulate type), which is equivalent to chemical analysis by using TXRF, the intensity above the critical angle is constant because the thin, small ‘‘sample’’ is completely excited. The intensity doubles at the critical angle in a steplike fashion and remains at the twofold value down to very small angles due to total reﬂection, Iiparticle ðj0 Þ ¼ kparticle I0 ci ½1 þ Rðj0 Þ

ð24Þ

following the angular behavior of the reﬂection coeﬃcient Rðj0 Þ. This intensity is proportional to the primary intensity I0 and the interesting area-related concentration ci. For incidence angles j0 below jcrit , Eq. (24) can be simpliﬁed to ð24aÞ Iiparticle ﬃ 2 kparticle I0 ci For buried layers (ﬁlm like) in a substrate, the intensity far above the critical angle becomes constant. The asymptotic behavior of the intensities for the particulate and ﬁlmlike contaminations can even be equal if both concentration values are the same and the appropriate scaling factors kparticle and kﬁlm are chosen. However, the intensity for the buried layers steadily increases with decreasing angle and can reach (theoretically) the fourfold value at the critical angle. For the smaller incidence angles, the intensity is going to zero, according to j Iifilm ðj0 Þ ¼ kfilm I0 ci ½1 Rðj0 Þ 0 zP j0 film rtm ¼ k I0 ci Tðj0 Þ ð25Þ jT ðj0 Þ

588

Kregsamer et al.

ZP is the penetration depth as deﬁned in Eq. (15). The angular behavior is caused by the countereﬀect of the transmission coeﬃcient Tðj0 Þ and the refraction angle jT. The intensities of the characteristic x-rays for homogeneously distributed contaminations and major constituents(s) in an inﬁnitely thick and ﬂat substrate can be described by Iibulk ðj0 Þ ¼ kbulk I0 ci ½1 Rðj0 Þj0 ¼ kbulk I0 ci Tðj0 Þj0

ð26Þ

which is related to Eq. (19) with respect of the angular behavior. Due to the geometry factor, represented by the direct proportionality to j0, there is a monotonous increase in intensity for angles above the critical angle. For practical purposes, the divergence of the primary beam should be considered. The scaling (calibrating) factors k must be determined during the calibration of an instrument. Layer-type external standards produced by immersion or spin coating of a wafer with a spiked solution are recommended (Torcheux et al., 1994; Mori et al., 1995); however, particulate-type (Fabry et al., 1995) and bulk-type standards have been used (Gutschke, 1991; Schwenke and Knoth, 1995). All types of standards are commercially available. Calibration by internal standard is not permissible. A common preconcentration technique used for the analysis of Si-wafer surfaces is VPD (vapor-phase decomposition). The wafer is exposed to HF vapor, leading to dissolution of the SiO2 surface layer of the wafer (Neumann and Eichinger, 1991). The impurities can then be collected by scanning the surface systematically with a droplet of a special liquid or water from the whole surface. The droplet can then be dried and measured by TXRF. This leads to a detection capability of 108atoms=cm2; the improvement is given by the area of the wafer (100 mm, 200 mm, 300 mm ) divided by the area seen by the detector. The disadvantages of the wafer analysis with VPD preconcentration are the loss of the local information and the destructive character (Fabry et al., 1994). Care must be taken to get a residue within an acceptable small area and to place it just under the detector. VI. THIN FILMS AND DEPTH PROFILES In many technological applications, thin-layered materials and implantations have an increasing importance. A characterization of such structures can be achieved with angledependent x-ray ﬂuorescence analysis in the region around the critical angle of total reﬂection. As the penetration depth of the incoming x-rays changes from the nanometer to the micrometer region when the angle of incidence passes the critical angle, information can be obtained on depth distributions and thin ﬁlms in that range. A.

Depth Profiles

The ﬂuorescence intensity Ii ðj0 Þ of an element i, implanted in a ﬂat smooth substrate, with a depth proﬁle ci ðzÞ showing a variation of its concentration over depth z (normal to the surface), is a function of the incident angle j0 : j0 Ii ðj0 Þ / Tðj0 Þ jT ðj0 Þ

Z1 m1 þ m2 rz r dz ci ðzÞ exp jT ðj0 Þ

z¼0

ð27Þ

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Figure 25 Calculated NiKa fluorescence intensities as a function of the incidence angle for Ni depth profiles in quartz excited by MoKa (normalized to values at 10 mrad). As a depth distribution for the Ni atoms, a step function was assumed. The maxima of the angle distributions with less than 50 nm are in the vicinity of the critical angle of quartz ( ¼ 1.8 mrad).

which is the generalized form of Eq. (26). The mass-attenuation coeﬃcients m1 and m2 describe the absorption of the exciting and the ﬂuorescence radiation of the element i, respectively; r is the density of the substrate. The transmission coeﬃcient Tðj0 Þ takes into account the refracted part of the incoming radiation and jT is the refraction angle inside the substrate. Examples are given for Ni (Fig. 25) and Al (Fig 26) depth proﬁles. In practice, the general shape of a depth proﬁle has to be given and some parameters can be ﬁtted (Kregsamer et al., 1999) or diﬀerent functions can be tested (Weisbrod et al., 1991; Schwenke et al., 1997; Mori et al., 1997), although, theoretically, a Laplace transformation is possible. B. Thin Films The composition, density, and thickness of single as well as periodic and nonperiodic multiple ﬁlms can be analyzed (Weisbrod et al., 1991; Schwenke et al., 1992). The ﬂuorescence intensity Ii ðj0 Þ of an element i (with concentration ci ) embedded in a single thin ﬁlm with thickness d on top of a ﬂat smooth substrate, is a function of the incident angle j0 and is described by (de Boer, 1991a, b)

1 exp½ðm1 =jT þ m2 Þrd Ii ðj0 Þ / ci jET j2 m1 =jT þ m2 1 exp½ðm 1 =jT þ m2 Þrd þ jER j2 m1 =jT þ m2 1 exp½f½ið4pÞðjT =lrÞ þ m2 grd þ 2 Real part ET ER ð28Þ ið4pÞðjT =lrÞ þ m2

590

Kregsamer et al.

Figure 26 Angular dependence for the intensity of the AlK line obtained for 1016 atoms=cm2 Al ions implanted in silicon at 80 to the surface normal. Fitted with an asymmetric triangular depth distribution with several depths of the maximum as parameter (the best fit is for 10 nm).

The electric ﬁelds ET (transmitted by the ﬁlm surface) and ER (reﬂected from the substrate) are derived from Fresnel’s equations and are complex quantities. The wavelength of the exciting radiation is l. The formation of standing wave above a reﬂecting surface is already taken into account by this formalism. An example is given for a Ni thin ﬁlm in Figure 27. The layer(s) and the substrate must be homogeneous, ﬂat, and smooth, with perfectly sharp interfaces. With some eﬀorts, even a certain roughness of the interfaces can be taken into account in the calculations (van den Hoogenhof and de Boer, 1993; Nevot and Croce, 1980). A review on grazing incidence X-ray spectrometry and reﬂectometry had been given by Stoev and Sakurai (1999).

VII.

SYNCHROTRON RADIATION EXCITATION

There are diﬀerent strategies to improve the detection limits, either by increasing the sensitivity, by reduction of the spectral background, or extending the measuring time (which, however, is limited for practical reasons). One approach to reducing the spectral background it to use total-reﬂection geometry. A further possibility for reducing the background is the use of polarized primary radiation (Ryon and Zahrt, 1993; Aiginger and Wobrauschek, 1974). Due to the anisotropy in the intensity of scattered radiation, it is advantageous to place a detector in such a position that almost only the isotropic emission of the ﬂuorescence signal is detected. It is possible to combine TXRF with polarized primary radiation excitation. Moreover, the use of monochromatic primary radiation improves the background conditions, because only photons with one energy are scattered (incomplete charge

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Figure 27 Calculated NiKa fluorescence intensities as a function of the incidence angle for Ni films on quartz substrate, excited by MoKa (normalized to values at 10 mrad). For a thickness around 1 nm, the maximum of the distribution is found at the critical angle of quartz (1.8 mrad). For around 25 nm film thickness, the maximum position coincides with the critical angle of nickel (3.6 mrad).

collection still causes a background). Tunable monochromators enable the exciting energy to be adjusted slightly above the absorption edge of the element of interest (‘‘selective excitation’’). Sometimes, it is even possible to set this energy below the absorption edge of a matrix element with high concentration, with the drawback, however, of an increased background due to Raman scattering if it is too close to this edge. Highly intensive broad-band photon sources realized in the synchrotron radiation facilities can increase the sensitivity by orders of magnitude as compared to conventional x-ray tubes. Its outstanding properties oﬀer new possibilities for TXRF. The intense primary beam with a continuous spectral distribution from infrared to high-energy photons, the linear polarization in the orbit plane, the small source size, and its natural collimation are features best suited for the excitation in total-reﬂection geometry. For optimum excitation conditions, the spectral distribution can be modiﬁed by using elements like cutoﬀ mirrors, crystal and multilayer monochromators and ﬁlters, and so forth (Wobrauschek et al., 1997a, b). Multilayer monochromators are well suited for synchrotron radiation-excited XRF (Bilderback, 1982; Pianetta and Barbee, 1988). In comparison to crystal monochromators, they oﬀer a larger bandwidth (DE=E 0.01), which leads to higher photon ﬂux on the sample. The combination of TXRF with synchrotron radiation can be achieved by several geometrical arrangements for the entrance slit, sample reﬂector, and detector. Figure 28 shows three possibilities. For the uppermost geometry (Fig. 28a), the polarization eﬀect is fully utilized by positioning the detector axis in the plane of the orbit. Scattered radiation is not emitted in that direction. The beam eﬃciently and homogeneously illuminates the

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Three possible geometrical arrangements of synchrotron radiation-excited TXRF.

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sample with a certain width in the horizontal plane. There are no losses due to the entrance slits particularly, because the beam is naturally collimated in the vertical plane (0.1– 0.2 mrad, depending on the energy), whereas the detection of the ﬂuorescence signal is not optimum, because the detector must be side-looking (to utilize the polarization eﬀect). The ﬂuorescent radiation has a long path in the sample itself before it reaches the detector. The excitation conditions for the arrangement in Figure 28b are poor. Most of the photons in the horizontal plane are stopped in the collimation system. The vertical intensity distribution drops drastically and, therefore, the ﬂuorescence intensity also drops with the deviation of sample regions from the plane of reference. However, the geometrical detection eﬃciency is very good. Good excitation and detection conditions will be met with the arrangement in Figure 28c. Unfortunately, the polarization eﬀect is not used at all. Experiments have been performed at HASYLAB, Beamline L, with a bending magnet (Wobrauschek and Streli, 1997). The arrangement can be seen in Figure 29, following the geometry of Figure 28b, with the vertical reﬂector and the side-looking detector. The beam is collimated by a primary slit system and then impinges on a multilayer monochromator. The nonreﬂected part of the primary beam behind the monochromator is absorbed in a beam stopper. The shielding of the vaccum chamber is extremely important because of multiple scattering eﬀects for high-energy photons. The beam is totally reﬂected on a rectangular Suprasil reﬂector with a Ta plate at the front edge to prevent scattering from the front edge into the detector (Rieder et al., 1995; Wobrauschek et al., 1997a, b; Go¨rgl et al., 1997). For spectra, see Figures 30 and 31. Several other groups also performed synchrotron radiation-excited TXRF experiments, for example, in Japan (Iida, 1991; Kondo et al., 1997), at NSLS (Pella, 1988), at SSRL (Brennan et al., 1994) and at Frascati (Sanchez et al., 1994). The semiconductor industry requires the analysis of wafer surface contaminations at ultratrace-level detection limits in the 108-atoms=cm2 range (corresponding to femtograms in mass units). Synchrotron radiation-excited TXRF has therefore become an interesting tool for the task. Experiments have been performed at HASYLAB, leading to detection limits of 13 fg for Ni in a droplet sample, simulating the VPD method (or 1.36108 atoms=cm2—assuming an inspected area of 1 cm2) (Wobrauschek et al., 1997a, b). A group at SSRL (Pianetta et al., 1995, Fischer-Colbrie et al., 1997) established a spectrometer for routine wafer analysis at SSRL, Beamline 6-II (focused wiggler beam line

Figure 29 Experimental setup for TXRF experiments at HASYLAB, utilizing the radiation from the DORIS III positron storage ring. Several slits and diaphragms and beam stoppers are used to prevent the count-rate saturation of the detection system by direct and scattered radiation from primary beam, transmitted photons through the monochromator and so on.

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Figure 30 Spectrum of a 10-pg Ni sample on a Si-wafer substrate. Measured at HASYLAB, Beamline L, with monochromatic excitation at 10 keV, 77 mA ring current, and measuring time 100 sec. A 10-mm-thick Al detector filter suppresses the SiK lines. Ca, Fe, and Cu originate from contaminations. The extrapolated detection limits for Ni were determined as 13 fg.

Figure 31 Spectrum of a 200-pg Cd sample. Measured at HASYLAB, Beamline L, with monochromatic excitation at 31.3 keV, 70 mA ring current, and 100 s measuring time. The extrapolated detection limits are 150 fg.

with a double multilayer monochromator). The measurements are performed under a clean-room environment. The automatic handling for 150-mm and 200-mm wafers, the possibility of wafer mapping, and detection limits of 36108 atoms=cm2 for transition metals meet the requirements of the semiconductor industry. The wafer is held by a

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combined vacuum–electrostatic chuck in the vertical position, the detector is side-looking, and the measurements are performed under vaccum conditions. A project is being implemented at ESRF on an undulator beamline (Comin et al., 1998). A Si(111) double-crystal monochromator as well as a multilayer monochromator will be used. The wafer will be held by a chuck in the horizontal position and two detector arrays consisting of seven elements in a side-looking geometry will measure the ﬂuorescent radiation (down to Na). Detection limits below 108 atoms=cm2 are expected for 200- and 300-mm wafers.

VIII.

LIGHT ELEMENTS

There is a lack of analytical methods performing nondestructive and rapid multielement determinations of light elements at trace levels. TXRF can be suitable for these purposes (Streli et al., 1991, Streli et al., 1993a, b) if a special spectrometer adapted to the speciﬁc problems of the energy-dispersive detection of low-energy radiation is used. The detection limits achievable are mainly inﬂuenced by the kind of excitation source, which should provide a large number of photons with the energy near the K absorption edge of these elements (from 200 eV upward). Of determining inﬂuence is the integral over the intensity of the source spectrum I0 ðEÞ multiplied by the photoelectric absorption coeﬃcient ti ðEÞ of an element i Z Ii /

I0 ðEÞti ðEÞ dE

ð29Þ

The absorption coeﬃcient drops steadily as the energy E increases above an element’s absorption edge. Therefore, (diﬀraction) x-ray tubes with standard anode materials (Sc, Cr, Cu, etc.) are poor exciters for light elements, as their characteristic emissions are far above the absorption edges of the elements of interest. To improve the sensitivity for light elements, it is necessary to use either an x-ray tube which emits intensive characteristic radiation with an energy as close as possible to the absorption edge of the interesting element, or synchrotron radiation with its continuous spectral range down to low energies. The spectrometer used for the experiments described in Sec. VIII is characterized in detail in the works of Streli et al. (1993a, b). A special Ge(HP) detector which meets all the requirements for low-energy detection is used. It has a thin entrance window (diamond window, 0.4 mm thick, transmission for oxygen Ka ¼ 85%), an ion-implanted contact layer, a thin dead layer, and a low electronic noise contribution (FWHM at 5.9 keV ¼ 125 eV). Measurements were performed under vacuum conditions to reduce absorption. The radiation of a Cr x-ray tube (1.3 kW) was monochromatized with a W=C multilayer. Details are described in the work of Streli et al. (1995). To improve the excitation conditions, a windowless x-ray tube was used. Recently, various anode materials were tested after computational optimization of the takeoﬀ angle of the x-ray tube to obtain a high photon ﬂux on the sample. The calculation takes into account the required brilliance and the self-attenuation of the emitted photons in the anode. Al, Si (Fig. 32), and Mo were tested experimentally as anode materials. Si oﬀers the advantage of a high t value for Al, but a low one for Si itself, which is often the major constituent of the reﬂector substrate. For the analysis of impurities on Si-wafer surfaces, this is a great advantage, because the high ﬂuorescence intensity of

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Figure 32 Spectrum of 4.5 ng of Na, Mg, and Al each. The sample was measured in an evacuated TXRF spectrometer with a Si-anode windowless x-ray tube (20 kV, 25 mA, 200 s measuring time), cutoff reflector, and a Ge(HP) detector with a diamond window.

Si is reduced (Streli et al., 1997a, b). Detection limits obtained diﬀer very much with the excitation conditions. The standard Cr x-ray tube provides about 90 pg for Mg, with the windowless Si-anode tube, 7 pg were obtained. Synchrotron radiation as an excitation source oﬀers the advantage of an intensive, naturally collimated x-ray beam of a wide spectral range, also in the low-energy region; it is, therefore, the ideal source for light element’s determination by TXRF. Experiments have been performed at SSRL, Beamline III-4. This bending magnet beam line is equipped with a horizontally deﬂecting torroidal Au mirror to cut oﬀ photons with an energy higher than 3 keV. A diﬀerential pumping system oﬀers the possibility of connecting the measuring chamber directly, without any window. Various ﬁlters and a double multilayer monochromator can be inserted into the beam path. The evacuable sample chamber was connected to the monochromator chamber with a ﬂexible coupling and an interlock valve that closes the main valve upstream if the pressure exceeds 10 7 3 mbar. Due to the downlooking design of the used detector, the advantage of the linear polarization of synchrotron radiation for further background suppression could not be utilized (Streli et al., 1994, 1997a, b). In order to adapt the spectral distribution of the synchrotron radiation for the analysis of Si-wafer surfaces, a 12-mm-thick Si ﬁlter was inserted into the beam path, leading to ‘‘quasimonochromatic’’ radiation with a bandwidth of about 400 eV ( just below the absorption edge of Si). A promising application of low-Z TXRF is the quality control of Si-wafer surfaces. For the semiconductor industry, Al and Na are of special interest, because they inﬂuence the production yield of ICs negatively. It is important to have an analytical method sensitive enough for the determination of ultratraces of these elements. Droplet samples of 100 pg of Mg had been prepared on Si-wafer surfaces and analyzed with synchrotron radiation excitation. The best results were obtained for the ‘‘quasimonochromatic’’ mode with detection limits of 60 fg for Mg (Fig. 33) (Streli et al., 1997a, b).

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Figure 33 Spectrum of a 100-pg Mg sample on a Si wafer. Measured at SSRL, Beamline III-4 with 12-mm-thick Si filter inserted into the primary beam path (ring current 54 mA, measuring time 100 s). The extrapolated detection limits are 60 fg. Na and Al were contaminations.

Depth proﬁle and thin-ﬁlm analysis of low-Z elements, usually not detectable by conventional instruments, can be performed by synchrotron radiation excitation. An angular scan for a thin ﬁlm of carbon (25 nm) on silicon demonstrates the power of synchrotron-radiation-excitation XRF in total-reﬂection geometry—even for low-energy ﬂuorescent lines (Fig. 34). Currently, there is only one TXRF spectrometer commercially available for the determination of Na and Al, with a rotating W-anode x-ray tube (excitation by WM lines); detection limits of 861010 and 361011 atoms=cm2 for Na and Al, respectively, were obtained (Fukuda et al., 1997).

IX. A.

RELATED TECHNIQUES X-ray Reflectometry

X-ray reﬂectometry is widely used for the analysis of surfaces and determination of parameters of single-layer or multilayer thin ﬁlms. Usually a y–2y scan with a goniometer (monochromatic excitation) is performed in the regime of glancing incidence with a proportional counter in order to obtain the characteristic intensity proﬁles (Kiessig fringes) and Bragg peaks for periodic structures (Leenaers et al., 1997). The applications include measurements of layer thickness, surface and layer density, surface and interface roughness, and the characterization of periodic (and nonperiodic) multilayer structures. In the case that all parameters are well speciﬁed, x-ray reﬂectometry can be used for the determination of the atomic scattering factor (Stanglmeier et al., 1992).

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Figure 34 Carbon K fluorescence intensity as a function of the angle of incidence for a 25-nm thin-film sample prepared on a Si substrate. Measured at SSRL, Beamline III-3, monochromator set to 1.74 keV. The model used for fitting yielded a 18-nm thickness.

B.

Grazing Emission XRF

In grazing emission XRF, the uncollimated, polychromatic excitation radiation hits the sample (placed on a reﬂecting substrate) normally to the surface and the ﬂuorescence radiation (generated near the sample surface) is detected under exit angles equal to the critical angle of total reﬂection (Becker et al., 1983). According to Snellius’ law [Eq. (6)], the scattered radiation cannot be detected for very small detection angles, due to the fact that for x-rays, vacuum (air) is optically denser than the reﬂector material, which will result also in almost background-free spectra (de Bokx et al., 1997; Urbach and de Bokx, 1996; Claes et al., 1997a, b). If a collimated or focused primary beam can be used, a lateral resolution of a few micrometers can be obtained. The energy-dispersive detector (as used for TXRF) can be replaced by a wavelength-dispersive detector, enabling a more reliable detection of the characteristic x-rays of the light elements (because of the better energy resolution for low energies). On the other hand, sample self-absorption eﬀects become more severe, because of the great path length of the emerging ﬂuorescence radiation of the low energy. Grazing-incidence and grazing-exit XRF is another modiﬁcation of the angledependent XRF method. When both the incident and the detected ﬂuorescent beams are at grazing angles, the ﬂuorescent intensity can provide information on the chemical composition of thin ﬁlms and vertical concentration proﬁles. Both the incident and takeoﬀ angles have to be controlled with high accuracy and are strongly restricted in their divergence, which poses severe constraints to the intensity (Sasaki and Hirokawa, 1990; Tsuji et al., 1993, 1994, 1995, 1996, 1997, 1999a, 1999b; Sato et al., 1996).

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C. Total-Reflection Particle-Induced X-ray Emission Ions with energies of a few megaelectron volts have been used to excite target atoms. The de-excitation can result in the emission of ﬂuorescence radiation, which is known as particle-induced x-ray emission (PIXE). Experiments with proton and a-beams at small incident angles ( < 35 mrad) have been performed (van Kan and Vis, 1996a, 1996b, 1997). The determination of the angle at which such particles are being reﬂected from a surface requires the ion–surface interaction potential to be considered. For smaller angles, the impulse component of the incoming particle perpendicular to the target surface becomes too small to overcome the potential and the particle is reﬂected. There is no well-deﬁned angle below which reﬂection takes place.

D.

Grazing-Exit EPMA and Grazing Exit PIXE

A new method of grazing-exit electron probe microanalysis (EPMA) was developed (incident electron angle approximately 45 ) (Tsuji et al., 1999a). The x-rays emitted under grazing-exit conditions of samples with particles collected on a sample carrier from the carrier surface and the particles were measured. It was found that surface-sensitive analysis with low spectral background was possible. Both wavelength-dispersive and energydispersive detectors were tested for this purpose. The grazing exit technique was also applied to PIXE analysis of thin ﬁlms and aerosols deposited on Si wafers, where a 2.5-MeV proton beam bombarded the sample at an incident angle of 90 (Tsuji et al., 1999b).

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10 Polarized Beam X-ray Fluorescence Analysis Joachim Heckel Spectro Analytical Instruments, GmbH & Co. KG, Kleve, Germany

Richard W. Ryon Lawrence Livermore National Laboratory, Livermore, California

I.

INTRODUCTION

In this chapter, we explore the sources of spectral background and show how polarization eﬀects can yield improved detection thresholds. In 1963, Champion and Whittem pointed out that the detection of Co traces in aqueous solution can be improved signiﬁcantly due to polarization of the background in a wavelength-dispersive x-ray ﬂuorescence (WDXRF) setup. We see that the stationary arrangement of components used in energydispersive x-ray ﬂuorescence (EDXRF) is ideally suited for geometrical conﬁgurations that exploit polarization phenomena to reduce background and thereby improve signal-tonoise ratios. The ability of simultaneously measure a wide range of elements is one of the greatest advantages of EDXRF. This advantage is strongly reduced when the count-rate limitation of the ED detection electronics is taken into consideration. This is due to simultaneous recording of the entire primary source radiation scattered on the specimen and is especially true for examinations on samples with light matrices. Thus, the main purpose of this chapter is to show that the multiple-element character of EDXRF can be advantageously utilized through the application of polarized x-rays. Observed spectral background is caused by several interactions of radiation with system components, the specimen, and the detector. A principal cause of background is the scatter of source radiation by the specimen into the detector. The scattered radiation adds directly to the background under analyte lines when broad-band primary radiation is used to excite ﬂuorescence. Even when monochromatic radiation is used for excitation, the scatter of this primary radiation adds indirectly to the background because of incomplete charge collection in the detector. In addition, primary radiation carries with it ﬂuorescence from the x-ray tube anode contaminants and collimator materials, and the scattered radiation causes ﬂuorescence of collimator materials between the specimen and the detector, and the gold contact layer on the surface of the detector. Scattered radiation also causes low-energy background due to the residual electron kinetic energy when Compton scatter from the detector itself occurs. All of these sources of spectral noise and interference can be reduced by minimizing the scatter of source radiation into the detector by using polarized radiation to excite ﬂuorescence. The most important advantage of this technique in trace element analysis 603

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is that the intensity of the source radiation can be increased, thereby proportionally increasing the intensity of analyte lines without exceeding the count-rate limitations of the detection electronics due to the counting of unwanted scattered source radiation. The following thought experiment demonstrates how the spectral background reduction stemming from the use of a polarized x-ray source leads to improved sensitivities and detection limits. Polarized and nonpolarized radiation with the same spectral distribution are used to excite ﬂuorescence and the two spectra are compared. The source intensities are adjusted to give the same count rate in each case by adjusting the x-ray tube current. Then, ðN þ Bnp Þinp ¼ ðN þ Bp Þip

ð1Þ

where N is the spectral sum of all net peak counts per second (cps) and per milliampere, B is the spectral sum of all background counts per second (cps) and per milliampere, i is the tube current (in mA) and the subscripts np and p indicate the nonpolarized case and the polarized case, respectively. Using excitation with polarized radiation with a degree of polarization of P (0 P 1), the background [see also Eq. (18)] is reduced to Bp ¼ ð1 PÞBnp

ð2Þ

which leads to ðN þ Bnp Þ ¼

ip ½N þ ð1 PÞBnp inp

ð3Þ

The matrix-dependent ratio consisting of the net count rate and the background count rate, V ¼ N=Bnp , is a deciding factor for the required increase in the current F ¼ ip =inp : F¼

ip 1þV ¼ V þ ð1 PÞ i

ð4Þ

If one considers three practical examples, it is possible to recognize the advantages and also the limits of polarized EDXRF (EDPXRF). Example 1: Traces in polymers are measured using a polarization degree of 90% ðP ¼ 0:9Þ that can be achieved in practice with EDPXRF. For this application, V 0, which leads to F ¼ 10. The net count rate, N, and, with it, the sensitivity are improved by a maximum factor of 10 when polarization is used. The spectral background is the same for both types of excitation when the current is increased in the polarized case. In this example, the detection limits improve proportionally to the sensitivity, also by a maximum factor of 10. Example 2: Using synchrotron radiation with a polarization degree of approximately 97% ðP ¼ 0:97Þ, the sensitivity and the detection limits can be improved by a factor of about 33 for idealized applications from Example 1. Example 3: In practice, a polarization degree of 90% ðP ¼ 0:9Þ can be achieved with EDXRF and is used to measure elements in steel. For this application, V a, thus F ¼ 1, because the ﬂuorescence excitation is absolutely dominant compared to the scatter in the specimen. Therefore, there is no advantage in using polarized radiation for this example. Example 1 demonstrates that the spectral background for EDPXRF can be identical with the spectral background for unpolarized excitation. From this, it follows that an additional improvement in the detection limits can be achieved through the use of radiation ﬁlters with EDPXRF.

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605

Polarized x-rays may be produced by various interactions of radiation with matter (Howell and Pickles, 1974) or the source may be intrinsically polarized, as in synchrotron radiation. When using conventional x-ray tubes, radiation caused by a single collision of an electron with the anode is polarized. Examples are bremsstrahlung from thin targets and radiation near the maximum energy for thick targets. X-rays passing through crystals by Borrmann diﬀraction are also polarized. However, scattering of x-ray tube radiation from suitable materials has proven to be the most promising for EDPXRF facilities. With x-rays, the index of refraction is very near to unity, so the corresponding angle required for nearly complete polarization is p=2.

II. THEORY A.

Scattering of Nonpolarized Radiation

A decade after the discovery of x-rays by Roentgen in 1895, Barkla (1905) demonstrated that this newly discovered radiation could be polarized by scattering, thus supporting the hypothesis that x-rays are electromagnetic radiation (i.e., their wave nature). A few years later, Friedrich et al. (1913) demonstrated x-ray diﬀraction, which also substantiated the electromagnetic hypothesis. Thomson and Thomson (1933) found, based on the classical electromagnetic theory, that the intensity of radiation scattered by a free electron is 2 2 1 e IS ¼ I0 2 sin2 a ð5Þ r m0 c2 where I0 is the intensity of the incident beam at the x-ray tube window, IS is the intensity of the scattered beam at distance r, e is the charge of the electron ð1:6 1019 CÞ, m0 is the rest mass of the electron ð9:11 1031 kgÞ, c is the speed of light ð3:00 108 m=sÞ, r is the distance to the point of observation (m), and a is the angle between the scattering electron ~ S (Pointing vector) and the direction of acceleration of the electron. The acceleration vector is perpendicular to the direction of propagation of the incident radiation ~ S0 and S0 is in the z direction parallel to the electric ﬁeld vector ~ E0 of the incident radiation. If ~ and ~ S is in the xz plane, the situation is as shown in Figure 1. For standard x-ray sources, such as x-ray tubes, ~ E0 of the photons incident on the scattering electron is random in direction (but always perpendicular to the direction of propagation) so that on the average hE20;x i ¼ hE20;y i ¼ 12E20

ð6Þ

To obtain the scattered intensity at a general scattering angle Y in the xz plane, it is simply necessary to decompose the incident ~ E0 into components and sum their individual intensity contributions. Now, ~ E0;y causes the electron to oscillate in the y direction so that E0;x causes the electron to vibrate in the x direction, the angle a ¼ p=2 and sin2 a ¼ 1. ~ so the angle a ¼ p=2 þ Y and sin2 a ¼ cos2 Y, where Y is the indicated scattering angle. Combining these considerations of the dependencies of ~ E0;x and ~ E0;y on the scattering angle a and recalling that the intensity of a wave is equal to the squares of its amplitude ðI ¼ E20 ¼ E20;x þ E20;y Þ, the following result is obtained from Eqs. (5) and (6): 2 2 1 e IS ¼ I0 2 ð1 þ cos2 YÞ 2r m0 c2

ð7Þ

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Figure1 Geometry for Thomson scattering. E0;x is in the xz scatter plane and E0;y is perpendicular to it. Y is the angle between the direction of the incident beam S0 and the direction of the scattered beam S in the xz plane. a is the angle between the electric field vector E0 of the incident beam and the direction of scatter S. b is the angle between the electric field vector E0 of the incident beam and the electric field vector E of the scattered beam.

From small-angle solid geometry, the quantity r2 in Eq. (7) is equal to dA=dO, where dA is the surface area of the expanding wave front of the solid angle dO at a distance r from the point of scatter. Therefore, IS r2 dse dO dA ¼ e ð1 þ cos2 YÞ dO ¼ I0 2 dO

ð8Þ

If the intensity of the scattered radiation is summed over the surface area dA, the fraction of the incident photon ﬂux, which is scattered by the electron into the solid angle dO in the direction Y, where re ¼ e2 =m0 c2 ¼ 2:82 1015 m is deﬁned as the classical electron radius and dse =dO as the diﬀerential cross section for scattering by a single electron, dse r2e ¼ ð1 þ cos2 YÞ dO 2

ð9Þ

This type of scattering is often referred to as Rayleigh scattering but is more appropriately called Thomson Scattering. Note that, relative to the core atomic electrons, the valence electrons are ‘‘free’’ in the Thomson sense. Corrections for electrons that are neither core nor valence require the atomic structure factor.

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Compton scattering is often taken to have the same cross section as Thomson scattering, but Thomson cross sections are only low-energy approximations. At higher energies, the inclusion of quantum mechanical considerations leads to the Klein–Nishina formulation (Klein and Nishina, 1928), presented in Chapter 1, which is repeated here: ! dsKN r2e 1 þ cos2 Y g2 ð1 cos YÞ2 ¼ ð10Þ 1þ dO 2 ½1 þ gð1 cos YÞ2 ð1 þ cos2 YÞ½1 þ gð1 cos YÞ where g is the incident photon energy=m0 c2 ¼ 1:96 103 incident photon energy (keV). Equation (10) presents the collisional cross section, which is also the approach of most other authors. Because of the decrease in energy upon Compton scattering, the energy scattered in a particular direction as a fraction of the incident intensity (Evans, 1958) is needed. Thus, the diﬀerential Klein–Nishina scattering cross section is dsSKN =dO ¼ ðn0 =n0 ÞðdsKN =dOÞ. The wavelength shift due to Compton scattering is l0 l ¼ hð1 cos YÞm0 c, converting to frequency by the relationship ln ¼ c; n0 = n ¼ 1=½1þ gð1 cos YÞ is obtained. The fractional background reduction, even at a scattering angle of exactly p=2, is not quite zero because of the Klein–Nishina limitation. The limitation to achievable polarization is signiﬁcant when exciting K-line ﬂuorescence of heavy elements (Strittmatter, 1982). As g approaches zero in Eq. (10), the Klein–Nishina cross section approaches the classical Thomson value in Eq. (9). The transition to description of the scattering on the electron shell of an atom with the atomic number, Z, is carried out by introduction of the scattering amplitude. If the small Klein–Nishina eﬀect at lower energies is ignored, it follows that unpolarized incident electromagnetic radiation scattered through an angle of Y ¼ p=2 is linear ð~ S ¼ S~ ex Þ polarized, with only Ey surviving. B.

Scattering of Linear Polarized Radiation

Linear polarization of x-ray radiation is the predominant direction of the electromagnetic ﬁeld vector in space. For scattering of the incident radiation presented in Figure 1, the polarization P0 in the yz plane is deﬁned as P0 ¼

hE20;x i hE20;y i hE20;x i þ hE20;y i

ð11Þ

where hE20;x i and hE20;y i represent the mean intensity components I0;x and I0;y , respectively, for all of the observed photons. The dependency of the coherent and incoherent scattering cross sections for polarized radiation is exploited when using EDPXRF. The angle b (see Fig. 1 with b ¼ p=2 a) between the electric ﬁeld vector of the incident radiation ~ E0 and the electric ﬁeld vector of the scattered radiation ~ E is introduced to describe the polarization dependence of the scattering cross section. The following is obtained from Eq. (5) (Hanson, 1986): dse ¼ r2e cos2 b dO

ð12Þ

The polarization dependency of scattering is determined using cos2 b. This means that the component E of the electric ﬁeld vector of the scattered photon is proportional to the projection of ~ E0 onto the plane which is perpendicular to the direction of propagation ~ S of the scattered photon E2 E20 cos2 b

ð13Þ

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The orientation of ~ E corresponds to the projection of ~ E0 onto the plane perpendicular to ~ ~ ~ S lie in a single plane: S. Thus, E0 ; E, and ~ ð~ E0 ; ~ E; ~ SÞ ¼ 0 S0 Þ ¼ 0 ð~ E0 ~

ð14Þ

ð~ E ~ SÞ ¼ 0 The ﬁrst equation in Eq. (14) is also valid for Compton scattering, as g approaches zero in Eq. (10). Using Eqs. (13) and (14), the mean electric ﬁeld vector is deﬁned as hE2 i ¼ hE2x i þ hE2y i þ hE2z i as a function of E0;x ; E0;y ; j, and d (see Fig. 1) (Brumme, 1990): hE2 i ¼ r2e ðhE20;x i sin2 d þ hE20;y i½1 sin2 d cos2 jÞ

ð15Þ

With Eqs. (11) and (13), this leads to a description of the polarization-dependent scattering cross section: ds r2 ¼ e ð1 þ sin2 d sin2 j þ P0 ½sin2 d cos2 j cos2 dÞ ð16Þ dO d;j 2 Neglecting the atomic form factor and the scattering amplitude, the following is obtained from Eq. (16) for the scattering cross section sðP0 ; dmax Þ of a photon with the direction of propagation ~ S0 and the polarization P0 in the solid-angle component O ¼ 2pð1 cos dmax Þ (see also Fig. 1): r2e 1 sO ðP0 ; dmax Þ ¼ p 3ð1 cos dmax Þ ð1 cos3 dmax Þ 3 2 ð17Þ 3 P0 ðcos dmax cos dmax Þ When a series approximation of Eq. (17) is conducted around dmax ¼ 0 for a scattering angle Y ¼ p=2 to the second order in dmax , the following is obtained: sO ðP0 ; dmax Þ ¼

r2e pð1 P0 Þd2max 2

ð18Þ

drawing on Eq. (18) for the derivation of Eq. (2). For a polarization P0 ¼ 1, which according to Eq. (11) is equivalent to ~ E0 ¼ E0;x~ ex , and a scattering angle of Y ¼ p=2; there is ey ; scatno scattering in the x direction. With P0 ¼ 1, which is equivalent to ~ E0 ¼ E0;y~ tering in the x direction reaches its maximum. In addition, it follows from Eq. (18) that the radiation scattered in the solid-angle component O ¼ 2pð1 cos dmax Þ is proportional to the square of the scattering angle dmax . The scattering angle dmax is deﬁned by the collimator system. The polarization P of the scattered radiation measured in the xz plane according to Figure 1 is P¼

hEy2 i hEx2 i hEz2 i hEy2 i þ hEx2 i þ hEz2 i

ð19Þ

When scatter in the solid angle O is also considered, in accordance with the above observations the following is obtained:

Polarized Beam XRF Analysis

P1

d2max 1 P0

609

ð20Þ

In the ideal case of scatter exactly around p=2 ðdmax ¼ 0Þ, the polarization P of the scattered radiation is independent of the primary polarization and has a value of unity. The electric ﬁeld vector has exactly one component in the y direction. For greater scattering angles (i.e. greater collimator diameters), the polarization is reduced as described in Eq. (20). The primary polarization has a smaller inﬂuence than dmax . The best values of P ey Þ. E0 ¼ E0;y~ are achieved, as expected, with a primary polarization P0 ¼ 1ð~ C.

Cartesian Geometry

In Barkla’s experiment (Compton and Hagenow, 1924) a source of x-rays is incident on a scatterer S1, as shown in Figure 2. The scattered rays from S1 intercepted by S2 are nearly linearly polarized, as the angle S0–S1–S2 is p=2 plane. The electric ﬁeld vector is perpendicular to the S0–S1–S2 plane. Radiation from S1 scattered by S2 into Iz also undergoes p=2 scattering, but it is in the same plane as the ﬁrst scattering and no further annihilation takes place. However, scattering of the radiation from S1 by S2 into Iy (also through p=2) annihilates all remaining radiation. Thus, Iz receives radiation, by Iy does not. This discussion is idealized; the measured value Iy=Iz depends on the tightness of collimation of the paths S0 to S1, S1 to S2, S2 to Iz, and S2 to Iy, and also on the materials and thickness of S1 and S2. For the experimental determination of the degree of polarization (Brumme et al., 1990) according to Figure 2, it is necessary to realize that the value Peff calculated with Eq. (21) represents an eﬀective polarization.

Figure 2 Schematic geometry for polarized beam XRF (PXRF). S0 is the x-ray source, S1 is the polarizer, S2 is the specimen, and I y is the intensity at the detector. I z is the intensity at an alternate, in plane position for the detector, which does not eliminate source radiation.

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Peff ¼

Iz Iy Iz þ Iy

ð21Þ

Multiple scattering in the scatterer S1 and in the specimen S2, in particular, but also the ﬁnite apertures of the two collimators C3 and C4, lead to worsening of the degree of polarization P > Peff . If the ﬁrst scatter occurs at some angle other than 90 , there will be a component which can scatter a second time at a right angle relative to the initial incident beam. The degree of polarization of such multiple scattering events is very low (P580%). It is, therefore, necessary to consider the polarization unit (S0, S1, C1, and C2 ) as well as the excitation unit (S2, C4, and the detector in the y direction) during optimization of EDPXRF system. The scattering target, S1 in Figure 2 can, of course, be replaced by a secondary target. In this case, the x-ray radiation emitted form the source, S0, is also polarized by p=2 through scattering on the target, S1. Because the ratio between scattering on the target and emission of the characteristic radiation of the metal is very small for most metals, polarization of the source radiation plays as less decisive role than for typical scattering targets. When a low-Z Barkla scatterer is used in an orthogonal geometry, marked improvements in the detection limits are possible [detection in the xy plane instead of xz plane in Figure 2 (Bisga˚rd et al., 1981)]. An intermediate case is when Mg, Al, or Si secondary targets are used for trace analysis. The polarization of source radiation becomes more important in comparison to a planar excitation geometry, as the scattering eﬀectiveness of these metals cannot be neglected. In multiple-element analysis, however, secondary targets are disadvantageous, because only a limited number of elements can be detected with good sensitivity. In order to react ﬂexibly to analytical applications with Cartesian geometry, secondary targets are indispensable, in addition to Barkla and Bragg polarizers. The eﬀectiveness of this technique is shown in Figure 3.

III.

BARKLA SYSTEMS

Proof-of-principle experiments in the early 1970s (Dzubay et al., 1974; Howell et al., 1975), using single collimators to deﬁne the three orthogonal beams (Fig. 2), demonstrated improved signal-to-noise ratios, but at severely reduced intensities. A few years later, it was demonstrated (Ryon, 1977) that the way to maximize analyte intensities while minimizing background is to open the apertures of the collimators until maximum systems throughput is achieved while maintaining the orthogonal beam geometry. Such nonzero beam divergence causes a small decrease in the degree of polarization, which is more than compensated for by the increase in x-ray ﬂux. Detection limits are improved in comparison

"

Figure 3 Comparison of excitation methods. The sample is an oil standard which contains 21 elements (e.g., Ca, Ti, V, Cr, Mn, Fe, Ni, Cu, Zn, Mo, Ag, Cd, Sn, Ba, and Pb) with concentrations of 30 mg=g. The spectra (log scale) are (top) nonpolarized, direct excitation by radiation from rhodium anode x-ray tube, (middle) molybdenum secondary excitation, and (bottom) polarized excitation by scatter from a HOPG=Al2 O3 target. The measuring time amounts to 200 s for each spectrum. The counting rate was first adjusted by increasing the tube current to its maximum value and then increasing the collimator aperture to give a total count rate approximately 60,000 cps using a Si(Li) detector with an active area of 25 mm2 and a energy resolution of 166 eV for MnKa. In all cases, a Rh end-window tube was used.

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to either direct excitation or secondary ﬂuorescence (Ter-Saakov and Glebov; 1984, Bertin, 1975). Energy-dispersive x-ray ﬂuorescence analysis is ideally suited to measuring a wide range of elements simultaneously because all photon energies are separately binned. When the analytical goal is to exploit this capability of measuring many elements with good sensitivity and low detection limits, a polarized polychromatic source is preferred. This is in contrast to nonpolarized sources, for which the increased sensitivity due to eﬃcient excitation is oﬀset by the increased background under the peaks. To optimize polarization and scattering intensity, all inﬂuencing factors, such as the scattering material and its dimensions and form, multiple scattering in the polarizer and in the specimen, scope of the collimator, and the ﬁnite dimension of the x-ray source and the detector, must be taken into consideration. The primary intensity of the x-ray tube and the optimized polarization equipment must be tuned so that, depending on the required resolution, the highest possible input count rate is achieved. This means that the loss in intensity due to the geometry and scattering loss are balanced. Brumme (1990) examined the above-mentioned inﬂuencing factors using Monte Carlo simulations. A.

Scattering Material

The scattering eﬀectiveness, e, of a planar target, S1 (see Fig. 2) is deﬁned as sS ðEÞ rmðEÞd eðEÞ ¼ 1 exp mðEÞ sin 45

ð22Þ

where mðEÞ is the mass-attenuation coeﬃcient of the target material for the x-ray energy E (in cm2=g), sS is the scattering coeﬃcient (coherent þ incoherent) for the target material for a scattering angle of p=2 and an x-ray energy E (in cm2=g), r is the density of the target (in g=cm3), and d is the thickness of the target (in cm). The quotient s=m of scatter and absorption must be a maximum for the scattering target material. Materials with a very small average atomic number are especially suited for this. They have a relatively small mass-attenuation coeﬃcient (m < 1 cm3=g) in the energy ranges of interest, so that the thickness of the scattering target must be increased in order to obtain a suﬃciently high scattering intensity according to Eq. (22). Thick scattering targets enable great variations in the scattering angle, which logically leads to poor polarization (Table 1). Additionally, the probability of multiple-scattering processes increases by increasing the thickness of the scattering target and, with it, the depolarization (see Table 1). Polarization losses have been calculated by Zahrt and Ryon (1981).

Table 1 Scattering Target Effectiveness e and Polarization Calculated with Monte Carlo Simulations Without ðP0 Þ and With ðPMS Þ Consideration of Multiple-Scattering Processes in the Target as a Function of the Thickness, dT , of a Planar Carbon Target (E ¼ 19:6 keVÞ dT (mm) 0.5 1.0 2.0 5.0 10.0 Source: From Brumme, 1990.

e

P0 (%)

PMS (%)

0.011 0.021 0.040 0.084 0.131

94.1 93.7 93.2 91.3 90.4

92.8 91.3 89.8 86.3 84.2

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According to Table 1, the thickness of the scattering target should not be increased unlimitedly to get a high scattering eﬀectiveness. Therefore, the density r of the scattering material plays a decisive role. For a constant thickness, the materials with a higher density have the advantage of having a higher scatter eﬀectiveness [see Eq. (22)] for a nearly equal fraction of multiple scattering and degree of polarization. Materials with density r < 2 g=cm3 are not ideal Barkla polarizers. Because the scattering eﬀectiveness is a function of the energy, there energy ranges, E < 10 keV, 10 < E < 25 keV, and 25 < E < 60 keV, are considered in the following. In the energy range E < 10 keV, photoabsorption dominates compared with scattering so that the scattering eﬀectiveness is too small, even for poor polarizers (P < 0:7). Barkla polarization is therefore too ineﬀective even using x-ray tubes with anode materials that emit characteristic x-rays in this energy range (Cr, Cu, W, etc.). In the mid-energy range, it is advantageous to use Barkla polarizers if the characteristic radiation of the tube anode material lies in this energy range (Mo, Rh, Pd, and Ag). Boron carbide (Ryon, 1977), pyrolitic graphite (Kanngießer, 1990; Kanngießer et al., 1992), and boron nitride are particularly suitable as scattering target materials. Beryllium oxide and boron can also be used. In the upper energy range, the bremsstrahlung emitted from x-ray tubes with anode materials with Z > 40 can be used for polarization. Corundum (Svoboda et al., 1993), boron nitride, beryllium oxide, silicon nitride, and boron carbide are especially useful as scattering materials. Because the average penetration depth of the scatter radiation and the fraction of multiple scattering diﬀer in both energy ranges, special optimization examinations are necessary. Systems using layers of diﬀerent materials are suitable as a compromise for the entire energy range 10 < E < 60 keV. In addition to the above-mentioned parameters, other material properties are of practical interest: Purity: Because the ﬂuorescence radiation excited in the scattering target is nonpolarized, the smallest impurities in the target material lead to additional background components in the measured spectrum. Elements with atomic numbers Z > 20 should not be present with concentrations C > 10 mg=g. Resistance to high temperatures (x-ray tube exit window), air humidity, and x-ray hardness. Mechanical stability. Availability. B.

Geometry of the Polarization Unit

In order to minimize as much as possible the geometric loss in intensity, it is necessary to keep the distances between the tube and the target, the target and the specimen, and the specimen and the detector small. It is, however, very important to prevent the sample from directly irradiating by the primary radiation of the x-ray tube and also the detector from directly detecting of scattered x-rays from the target or other parts of the unit. The dimensions of the collimator system C1, C2, and C4 (see Fig. 2) exercise the largest inﬂuence on the properties of the polarization unit. As can be seen in Table 2, the scattering eﬀectiveness increases by a factor of approximately 50 with variation of the collimator diameter (d1 ¼ d2 ) of the primary collimator C1, d1, and the secondary collimator C2, d2, from 4 to 12 mm with a simultaneous loss in polarization from 0.938 to 0.8. The ratio between single and multiple scattering events and with it the degradation of P0 caused by multiple scattering (P0 PMS ) is not inﬂuenced signiﬁcantly by increasing collimator diameters (Brumme, 1990).

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Table 2 Inﬂuence of the Collimator Diameter d1 (Collimator C1) ¼ d2 (Collimator C2) on the Polarization Without ðP0 Þ and With ðPMS Þ Consideration of Multiple Scattering and on the Total Scattering Effectiveness G for a Planar Carbon Target with a Thickness of 2 mm d (mm) 4 5 6 8 12

G6106

P0 (%)

PMS (%)

0.31 0.65 1.38 4.21 17.1

93.8 92.9 90.7 87.8 79.6

90.3 89.6 87.4 84.6 76.7

Note: Point Source S0, Distance S0S1: 36 mm, collimator length C1, l1 ¼ 2 mm; distance S1S2: 35 mm, collimator length C2, l2 ¼ 12 mm. Source: From Brumme, 1990.

With a diameter d1 ¼ d2 ¼ d3 ¼ 5 mm for C1, C2, and C3 a tube voltage of 50 keV (Rh anode) and a tube power of 280 W, an input count rate at the detector of approximately 105 cps is obtained for the excitation of 4-g base oil (Conostan) specimen using a Barkla target. The detected radiation consists mainly of incomplete polarized radiation scattered at the sample. A further increase in the total scattering eﬃciency is possible through an increase in the target thickness [see Eq. (22)] or through an increase in collimator diameters. Monte Carlo simulations indicate (Brumme, 1990) that the use of larger collimator diameters is preferred to the use of thicker targets (see the diﬀerence (P0 PMS ) in Tables 1 and 2). The increase of the ratio between multiple-scattering events and singlescattering events within the target, and the depolarization (P0 PMS ), coupled with it, due to growing target thickness is the decisive factor. The Monte Carlo simulations demonstrate that for a given target thickness, the target area and, with it, the target volume can be extended beyond the volume, in which a single-scattering event of primary radiation from the x-ray tube into the direction of the sample can occur (Brumme, 1990). The increase of depolarization caused by the described extending of the target area is insigniﬁcant. C.

X-ray Tubes and Filters

All results of Monte Carlo simulations described so far are based on the assumption that the focal spot of the x-ray tube is inﬁnitely small. The consideration of a ﬁnite focal spot leads to a reduction of the polarization (see Table 3). When mounting a side-window x-ray tube, it should be noted that the bremsstrahlung is slightly polarized in the plane deﬁned by electron beam and x-ray beam from the focal spot to the Be window of the x-ray tube. For this reason, the electron beam and the specimen-detector axis should be parallel. When using line-focus x-ray tubes, the

Table 3 Inﬂuence of the Dimensions of a Quadratic Anode Focal Spot on the Polarization (Target Thickness: 1 mm) Focus (mm)

Point

161

363

666

868

P0

0.957

0.957

0.954

0.936

0.918

Source: From Brumme, 1990.

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615

line-shaped focal spot should lie in the scattering plane. For end-window x-ray tubes, the diameter of the ring-shaped focal spot should be as small as possible. In practice, a compromise among tube power, focal spot size, and distance between target and tube anode should be found. Except for the Compton shift, the spectral distribution of the tube primary spectrum is approximately unchanged after Barkla scattering. Thus, the form of the spectral background that is generated by incompletely polarized tube primary radiation approximately corresponds to the known background from direct excitation (see also Fig. 3). For this reason, x-ray ﬁlters can be used to optimize the excitation of a limited number of elements also in EDPXRF similarly to direct excitation EDXRF. Polarization measurements show that ﬁlters between the x-ray source and target do not lead to worsening of the eﬀective polarization. In contrast, ﬁlters in the radiation path between the target and the specimen cause a depolarization of up to 2%. Nonpolarized radiation that results from contamination in the Barkla target can be absorbed by ﬁlters between the target and the specimen. For example, Cr, Fe, and Ni contamination occurs in B4C due to the pressing procedure; the ﬂuorescence radiation of these elements can be absorbed by a 30-mm Rh ﬁlter on its way to the specimen. D.

Multiple Scattering in the Specimen

In Sec. II.C, it was mentioned that the eﬀective polarization, Peff , must be considered in the description of the suppression of polarization. The diﬀerences between PMS from Table 1 and Peff were caused either by x-ray ﬁlters between target and specimen or by multiplescattering processes in the specimen. The fraction of multiple scattering is dependent on the matrix. Monte Carlo simulations (Brumme, 1990) show that polarization is reduced by multiple scattering in a 0.5-mm-thick SiO2 specimen from PMS ¼ 1:0 to Peff ¼ 0:96. In the case of fully polarized radiation (PMS ¼ 1:0), the ratio between multiple-scattering events and single-scattering events within a 3-mm-thick SiO2 specimen is 40%. This ratio amounts to only 15–20% for a typical polarization of 0:8 PMS 0:9 and causes a decrease of polarization PMS Peff of 1–2%. Considering multiple scattering within the sample and target, the eﬀective polarization Peff amounts to 0.80–0.85 in the energy range E < 25 keV and to 0.76–0.83 in the energy range E > 25 keV. E.

Multiple-Layer Scatterers

As later indicated (see Table 4), if a single element is determined, a monochromatic source gives the lowest detection limits. However, a polychromatic source is desirable when multielement analysis is performed. If polychromatic radiation is to be polarized by scattering, high-Z scatterers absorb the low-energy portion of the incident spectrum, whereas low-Z scatterers do not interact eﬀectively with the high-energy components of the spectrum (Zahrt and Ryon, 1986). In both cases, this limits the intensity over a broad range of energies. In both of the examined energy ranges, E < 25 keV and 25 < E < 60 keV, B4C is well suited as the polarizer. However, optimized polarization systems for both energy ranges diﬀer from each other. Compared with B4C target, slight improvements can be achieved using a combination of HOPG (highly oriented pyrolytic graphite) on Al2O3. An HOPG target (d ¼ 1:2 mm) optimized for the energy range 8 < E < 25 keV is ineﬃcient in the upper energy range, due to its low density (r ¼ 2:2 g=cm3). For the collimator diameters of 5 mm used here, the Bragg reﬂection is negligible (Kanngießer et al., 1992). When the target is glued to an Al2O3 substrate,

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Table 4 Detection Limits (n.d. ¼ Not Detectable) for Some Elements in a Base Oil Using Direct Excitation, Monochromatic Excitation Using a Mo Secondary Target, and Direct Excitation with Linearly Polarized X-rays Detection limits (in mg=g) Element Ca Ti Cr Mn Cu Zn Mo Cd Sn Pb

Direct excit.

Mo second

Polarization

13.0 3.8 3.1 2.6 1.7 1.7 2.3 18.0* 12.0 3.9

8.8 2.9 2.0 1.2 0.31 0.30 n.d.a n.d. n.d. 0.31

4.1 1.6 0.78 0.51 0.34 0.33 0.95 1.6 2.0 0.79

Note: 175 W for a Measuring Time of 200 s and an Incident Pulse Density of About 60,000 cps. a Overlapping with RhKb2.

the polarization decreases imperceptibly due to multiple scattering in the lower energy range E < 25 keV. However, in the energy range E < 25 keV, the Al2O3 substrate improves the scattering eﬃciency at the cost of polarization. On balance, such a multiple-layer scatterer is useful for determining a wide range of elements simultaneously with very good detection limits and sensitivities. F.

Applications

Based on Eqs. (3) and (4), Barkla targets are especially suited for multiple-trace-element analysis in a light matrix. Typical applications include traces in organic substances [polymers, oils, pharmaceuticals, biological materials (Heckel, 1995), etc.], traces in silicates [rocks (Heckel et al., 1991), soils (Heckel et al., 1992), sludges, cinders, etc.], as well as traces in Mg and Al alloys. In a ﬁrst application, detection limits and sensitivities for traces in an oil matrix are compared among direct excitation, secondary target excitation, and polarization excitation (see Fig. 3). As mentioned several times earlier, the count-rate limitation of the detection system is the deciding factor for optimization of the polarization system. It must also be noted that the electronic noise and, with it, the energy resolution of the spectrometer are functions of the pulse peaking time of the pulse-processing electronics and, through it, of the maximum possible pulse throughput rate. For multiple-element analysis in the energy range between 1 and 50 keV, the energy resolution for MnKa should be better than 180 eV. For the spectra displayed in Figure 3, a detection system with an energy resolution of 166 eV at an input count rate of about 60,000 cps was used. It must be noted that no x-ray ﬁlters are used to keep the goal of a multielement analysis in a short measuring time. Especially in the case of polarized or direct excitation, the detection limits of a limited number of elements can be improved by using x-ray ﬁlters signiﬁcantly. The detection limits, DLs (see Chapter 2; conﬁdence level: 99.86%), for the excitation types displayed in Figure 3 are summarized in Table 4; the analytical sensitivities, are summarized in Table 5. A comparison of the numerical values in Table 4 indicates that

Polarized Beam XRF Analysis

617

Table 5 Analytical Sensitivity for Some Elements in a Base Oil Using Direct Excitation, Monochromatic Excitation Using a Mo Secondary Target, and Direct Excitation with Linearly Polarized X-rays Sensitivity [in cps=(mg=g)] Element Ca Ti Cr Mn Cu Zn Mo Cd Sn Pb

Direct excit.

Mo second

Polarization

0.064 0.29 0.49 0.64 1.71 1.73 1.89 0.28 0.22 0.9

0.037 0.11 0.16 0.25 1.35 1.54 0.0 0.0 0.0 1.57

0.25 0.49 0.98 1.70 4.25 4.28 5.56 2.52 2.15 2.90

Note: 175 W for a Measuring Time of 200 s and an Incident Pulse Density of About 60,000 cps.

the detection limits determined with excitation using polarized x-rays are, on average ﬁve times better than those determined with direct excitation. An improvement of only a factor of 2.4 was achieved for the element Mo. In contrast to direct excitation using RhK radiation, the Compton scattered RhKa line is unable to excite Mo. The detection limits for the elements 46 < Z < 60 are worse than those for the elements 23 < Z < 41, because these elements are only excited by bremsstrahlung. An improvement is possible when a voltage > 50 kV is used. The Rh tube is operated with a power of 175 W for polarization. If a 2-kW tube is used, the eﬀective polarization can be improved from Peﬀ 82% for the Compton-scattered RhKa line to Peﬀ 87% for the same detector input count rate through reduction of the collimator diameter. This clear increase in polarization results in a 40% improvement of the detection limits displayed for polarization in Table 4. If one compares the values for polarization equipment with those obtained with Mo secondary targets, an improvement in the detection limits of a factor up to 2.5 is obtained for all of the elements with absorption edges that are only slightly below the energy of the MoKa line (e.g., Pb in Table 4). Similar improvements are expected for the elements Sr and Zr. Detection limits of the polarization setup become better compared to those obtained with Mo secondary target if the energy diﬀerence between absorption edge of the elements and the MoKa line becomes greater. Of course, not all of the elements can be determined with the Mo secondary target. This reﬂects an additional advantage of Cartesian excitation geometry. A change in the targets is suﬃcient to switch from a multiple-element application with a polarizer to a ‘‘single-element application’’ when analyzing single elements, such as Pb in oil, or a couple of elements with absorption edges that are only slightly below the energy of the characteristic radiation of the secondary target. Changing of the targets is computer controlled in commercially available instruments. A comparison of the numerical values in Table 5 shows that, on the average, a factor-of-4 improvement in the sensitivity is obtained using the polarization unit as compared to direct excitation. The sensitivities are also improved by a factor of between 2 and 7 compared to those of the Mo secondary target.

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If the detection limits given in Table 4 are not suﬃcient for the elements with atomic numbers in the range 45 < Z < 61, for example, then an improvement can be obtained by optimizing the polarization unit for the respective energy range. For example, the determination of Cd, Sn, and Sb with a concentration range C > 0.1 mg=g in rocks represents an interesting geological application. From the above discussion, it follows that an additional improvement in the peak-to-background ratio can be achieved using an x-ray ﬁlter. The result of the optimization is displayed in Figure 4 for the spectrum of the standard granite AC-E. As can be seen in Figure 4, the peak-to-background ratios for Ag and Cd are suﬃcient for the determination of the elements in sub-ppm concentration range. The root-mean-square deviation for the Cd calibration [see Eq. (28)], which is based on 22 international rock standards, is 0.2 mg=g. The DLs are summarized in Table 6. In contrast to the analysis of oil, which has been discussed earlier, when analyzing geological samples with input count rates of 60,000 cps or more, it is necessary to take pileup peaks into consideration at energies E < 25 keV. This is because the concentration of the elements K, Ca, Ti, and Fe may amount to % levels. A pileup peak from FeKa þ FeKa at 12.8 keV near the PbLb line at 12.6 keV and near the ThLa line at 12.95 keV or a pileup peak from FeKa þ CaKa at 10.09 keV near the HgLa line at 9.99 keV are typical examples. These analytical problems can be contained using ﬁlters between the target and the specimen or between the specimen and the detector, by reducing the input count rate (see Fig. 5) or by improving the pileup rejection. With the given analytical conditions, detection limits at sub-ppm levels were obtained for most of the trace elements in geological specimens (see Tables 6 and 7). Because the iron content in many geological specimens amounts to the % levels, an increase in the background in the energy region E < 6.4 keV occurs due to incomplete charge collection in the Si(Li) detector. This background worsens the detection limits, particularly for V, Cr, and Mn. With a Co secondary target and an Fe ﬁlter between the target and the specimen, only the CoKa line excites the specimen. This line is not able to excite Fe. The detection limits of the elements V, Cr, and Mn are improved by using the Co secondary target in an Fe-rich matrix (e.g. Fe ores). The secondary target is again successfully combined with polarizers. G.

Limitations of Barkla Polarization

As established in Sec. III.A, the scattering eﬀectiveness in the energy region E < 10 keV continually decreases in favor of photoabsorption. Thus, assuming a Cartesian excitation geometry, it is necessary to examine other excitation possibilities. Using secondary targets, the number of elements that can be excited with a suﬃcient sensitivity decreases with falling energy. Therefore, many diﬀerent secondary targets must be used. In practice, the measuring times would be unacceptably high. Bragg polarizers are an alternative to secondary targets and are discussed in the following section.

IV.

BRAGG SYSTEMS

A.

Orthogonal System

Diﬀraction oﬀers an excellent means of obtaining both monochromatic and polarized radiation. Synthetic multilayer materials that have very high reﬂectivities can have a d-spacing chosen virtually at will to give a diﬀraction at 2y of p=2 for energies up to about 1.0 keV. For higher energies, crystalline materials must be used. Candidate materials for

Figure 4 Spectrum (log scale) of the standard specimen AC-E, excited using an Al2O3 polarizer. A 180-mm Ta filter was placed between the specimen and the target. Some of the certified concentrations are as follows: Ag, 0.1 mg=g; Cd, 0.6 mg=g; In, 0.1 mg=g; Sn, 12.0 mg=g.

Polarized Beam XRF Analysis 619

Figure 5 Spectrum (log scale) of the standard specimen AC-E, excited using an B4C polarizer. A 50-mm Pd filter was placed between the specimen and the target. Some of the certified concentrations are as follows: Ni: 1.5 mg=g; Cu: 4.0 mg=g; Sr: 3.0 mg=g; Tl: 0.9 mg=g.

620 Heckel and Ryon

Polarized Beam XRF Analysis Table 6 X-rays Element Mo Pd Ag Cd In Sn

621

Detection Limits for Various Elements in Rocks Using Excitation with Linearly Polarized DL (mg=g)

Element

DL (mg=g)

Element

DL (mg=g)

0.32 0.24 0.17 0.13 0.16 0.18

Sb Te I Cs Ba

0.25 0.38 0.55 0.82 1.1

La Ce Pr Nd U

1.4 1.9 2.4 3.3 0.8

Note: Voltage: 59 kV; current: 33 mA; measuring time: 900 s; Al2O3 polarizer; 180-mm Ta ﬁlter.

Table 7 X-rays Element Co Ni Cu Zn Ga Ge As

Detection Limits for Various Elements in Rocks Using Excitation with Linearly Polarized DL (mg=g)

Element

DL (mg=g)

Element

DL (mg=g)

1.1 0.9 0.8 0.6 0.3 0.2 0.2

Se Br Rb Sr Y Zr Nb

0.2 0.1 0.2 0.3 0.3 0.3 0.4

Hf Ta W Tl Pb Bi Th

1.9 1.5 1.1 0.6 0.5 0.5 0.6

Note: Voltage: 38 kV; measuring time: 900 s; B4C polarizer; 50-mm Pd ﬁlter.

use in a polarized-beam XRF experiment should have several characteristics. The ﬁrst criterion is a material that has a reﬂection of high diﬀraction power (integral reﬂectivity > 104 rad) at 2y near p=2 for the characteristic wavelength of the anode material used. In addition, the material should be stable, must be free of contamination with characteristic lines in the energy region 1 < E < 15 keV, easy to orient and cut to the desired orientation, obtainable at reasonable cost, and perhaps nontoxic. The orthogonal triaxial geometry for Bragg-diﬀraction polarization is similar to that for Barkla scattering, shown in Figure 2. What is important here is the physics of the diﬀraction process and the parameters of the diﬀraction crystal. There are two theories of x-ray diﬀraction; the wave kinematics (K theory) and the dynamic (D theory). The K theory is valid for small crystals or mosaic blocks. The D theory must be used when absorption and=or the interaction between incident and diﬀracted beams becomes important. According to the theory of Zachariasen (1967), the integrated reﬂectivity Ri of an ideal mosaic crystal in the symmetrical Bragg case has the following form: 2 r0 1 þ cos2 2yB 1 sin2 yB Ri ¼ jFhkl j2 l3 exp 2B ð23Þ 2m0 V 2 sin 2yB l2 where Ri r0 yB

Integrated intensity compared to the incident beam Classical electron radius Bragg diﬀraction angle

622

Heckel and Ryon

Fhkl un ; vn ; wn a; b; c fn V m0 B

PN

n¼1 fn exp½2piðhun þ kvn þ lwn Þ structure factor of the lattice plane hkl Fractional coordinates ¼ 1=axn ; 1=byn ; 1=czn of the nth atom in the unit cell Unit cell lengths Atomic scattering factor Unit-cell volume Linear absorption coeﬃcient Thermal Debye parameter at room temperature (B ¼ B0 K þ B293 K )

The atomic form factor fn is approximately half-Gaussian in shape with argument sin yB =l and fn ð0Þ ¼ Z, the number of electrons in atom n. To maximize Ri , we thus want V to be small and FH large, which often means taking sin(yB )=l to be small, which, for cubic crystals, implies low values for the Miller indices (h, k, l ), based on 1 h2 þ k2 þ l2 4 sin2 yB ¼ ¼ d2 a2 l2

ð24Þ

Equation (23) is only valid when the absorption and extinction eﬀects within the mosaic blocks are negligible. These eﬀects may markedly reduce the integral reﬂectivity. Taking into consideration critical values for primary and secondary extinction as well as true absorption, Beckhoﬀ et al. (1992) give a detailed overview of possible Bragg polarizers for MoKa, CrKa, CuKa, and AgKa radiation from the respective tube anode materials. The integral reﬂectivities of possible Bragg polarizers for MoKa, RhKa, and AgKa radiation are comparable with the integral (energy and solid angle) Barkla-scattering eﬀectiveness in the energy region E > 10 keV. On the other hand, experimental comparisons (Kanngießer et al., 1991; Kanngießer et al., 1992; Ryon et al., 1982) have shown that Barkla polarization is preferred in this energy region. At lower energies, 5 < E < 10 keV, excellent detection limits are achieved by using Bragg polarization of the CrKa and CuKa line (Aiginger et al., 1974; Aiginger and Wobrauschek, 1981; Wobrauschek and Aiginger, 1985). Wobrauschek et al. (1988) obtained a detection limit of 66 ppb for Co in an aqueous solution (0.45 mL volume) using excitation with the CuKa line (1.6 kW), for example, with a measuring time of 1000 s. The CuKa line was polarized with a HOPG (006). Working with a suitable spectroscopy or diﬀraction x-ray tube involves interesting, special applications (e.g., compared to AAS or ICP) for trace detection for a couple of elements. Assuming the possibility of simultaneous multielement analysis with EDXRF, Bragg polarization of the L radiation from one of the Mo, Rh, Pd, or Ag x-ray tubes (10–400 W) normally used in EDXRF is, in practice, of greatest analytical interest. In this energy range, the bremsstrahlung spectrum emitted from the tube has such a weak intensity that Barkla polarization of this radiation is not useful from the practical point of view. Thinner Be windows (d 75 mm) can be placed in end-window x-ray tubes than those used in side-window tubes, leading to the observation of clearly increased intensity for the emitted low-energy L radiation. Crystals with an integral reﬂectivity, R > 0.001 rad, and a diﬀraction angle, Y, near 45 5 for the L radiation from the above-mentioned end-window x-ray tubes are excellent Bragg polarizers. The calculation of the integral reﬂectivity and, with it, the selection of a suitable crystal become more diﬃcult due to the expected absorption and extinction eﬀects. The HOPG, which has already been referred to as a Barkla scatterer and Bragg polarizer, is also an outstanding Bragg polarizer for the (002) reﬂection of the RhLa radiation (y ¼ 43.2 ). Beckhoﬀ and Kanngießer

Polarized Beam XRF Analysis

623

(personal communication, 1996) have calculated an integral reﬂectivity of Ri ¼ 0:0032 rad. The spectrum of a Rh end-window x-ray tube linearly polarized with an HOPG crystal (FWHM mosaic spread: 1.7 ) and measured in air is shown in Figure 6. Because Barkla scattering prevails on the carbon target, the polarization spectrum from a carbon target (displayed for comparison) enables a direct comparison of Bragg and Barkla polarizations in a narrow beam geometry. As described in Sec. III, an increase in the Barkla-scattering intensity shown in Figure 6 is possible by enlargement of the collimator diameter, for example. However, the optimized Barkla-scattering intensities do not contribute greatly to the polarization spectrum in the energy region of the (002), (004), and (006) reﬂections. The sensitivity of the elements with a critical excitation energy below the RhLa energy, for example, may be improved by up to 30% when not only the RhLa line but also the high-energy RhLb lines can be used for excitation. To do this, the polarization geometry must be changed so that the diﬀraction angle yB ¼ 40:6 for reﬂection of the RhLb lines can be realized. The degree of polarization for Bragg polarization is reduced according to Eq. (25) from P 99% to P 94%. P

1 cos2 2yB 1 þ cos2 2yB

ð25Þ

However, as the degree of polarization worsens, there is an increased probability for overlapping of sample element x-ray lines with escape lines, because scattering of nonpolarized L lines from the tube anode material on the sample may occur. In addition to the polarization geometry, the focal spot size of the x-ray tube and the mosaic spread (standard deviation of the Gaussian-shaped distribution of the mosaic block orientations) of the mosaic crystal also pay a deciding role in optimizing the

Figure 6 Excitation spectrum of a flat HOPG crystal in comparison with graphite with a Bragg angle of 43.2 (distance, tube crystal: 26 mm; distance, crystal detector: 28 mm; collimator diameter: 1 mm each; measurement in air; tube voltage: 30 kV). Logarithmic scale!

624

Heckel and Ryon

polarization unit. Equation (23) is valid assuming that the mosaic spread Z is much greater than the critical value for secondary extinction: rﬃﬃﬃ 2 ð26Þ Z4 Ri p A greater mosaic spread is advantageous for the collimator diameter (C1, C2, and C4 in Fig. 2) 45 mm, normally used in EDXRF, as a larger angular acceptance is obtained. On the other hand, the peak reﬂectivity is reduced with increasing mosaic spread. In the case of an HOPG crystal (Ri ¼ 0.0032 rad), the FWHM mosaic spread according to Eq. (26) is much greater than 0.34 . The experimentally determined optimum for the FWHM mosaic spread lies in the range between 1.7 and 2.1 . The large angular acceptance desired for the HOPG (002) reﬂection leads to an increase in intensity of the (004) and (006) reﬂections, as bremsstrahlung spectrum of larger energy range can be reﬂected. An eﬀective excitation (K radiation) of the elements K to Fe is possible with these reﬂections so that the spectrum from Na to Fe, which can be simultaneously excited with the three reﬂections, becomes extremely interesting for many applications. The product Kn, j according to Eq. (27) is interesting for the analytical evaluation of the three reﬂections (n ¼ 1, 2, 3): r1 Kn; j ¼ Ri; n oj ð27Þ r j where Ri,n is the integral reﬂectivity of the nth reﬂection, n ¼ {1, 2, 3}, oj is the ﬂuorescent yield of the element j, excited with reﬂection n, and rj is the jump ratio for the element j. When observing the K values for typical elements from the three excitation regions, for example, Si, Ca, and Fe, similar values are found: K1;Si K2;Ca K3;Fe ; that is, the element speciﬁc sensitivities for the given excitations are similar. The very small ﬂuorescence yield for the light elements, such as Al and Si, is compensated for by higher integral reﬂectivity.

B.

Curved Crystals

To increase the intensity at the sample, one may bend (Johann) or bend and grind (Johansson) the crystals (see Chapter 2). With a ﬂat crystal in a narrow-beam geometry, the angular acceptance is essentially the rocking curve, generally 0.01 . With a bent crystal, an angular acceptance of 5 –10 is readily conceivable, with an increase in intensity proportional to the size of the fan beam. Aiginger and Wobrauschek (1981) report an intensity gain of 3.1 using Cu(113)–CuKa in Johann geometry compared to the ﬂat crystal. Calculations for Johann and Johannson geometries have been performed by Zahrt (1983, 1986). Although the calculations used the estimated values for a mosaic distribution function and its full width at half-maximum, the reported gain is also calculated. Beckhoﬀ and Kanngießer (personal communication, 1996) calculated the intensity of the reﬂected RhL radiation on a spherically bent HOPG(002) crystal with a 32 mm radius of curvature (C1 ¼ C2 ¼ C3 ¼ 12 mm according to Fig. 2) for various focal spot sizes of a Rh tube and crystal positions using Monte Carlo simulations. When compared to a ﬂat crystal, it is possible to have increases in the excitation intensity in a zone diameter excitation ¼ 20 mm (for example) on the sample by the factors of 17.2, 11.1, and 2.6 for focal spot diameters of 0.1 mm, 1 mm, and 6 mm of a Rh tube, respectively. Experimental examinations with a focal spot diameter of focus 1 mm have approximately conﬁrmed this improvement by a

Polarized Beam XRF Analysis

625

factor of 8.1 for the (002) reﬂection. Focal spot diameters of focus 1 mm can be realized with the end-window x-ray tubes only in conjunction with reduced powers. However, even tube powers of 50 W are suﬃcient for the most applications. C.

Applications

Because only a limited number of elements can advantageously be excited with Bragg systems, the range of applications of an EDPXRF instrument optimized exclusively for Bragg polarization is limited. The determination of the thickness of Al layers on diverse substrates (e.g., SiO2) is an interesting application of this kind. For a layer thickness of d 90 nm, for example, reproducibilities [relative standard deviation (RSD)] <1% (99.86% conﬁdence level) were obtained with a measuring time of 50 sec per measuring point using a spherically bent HOPG(002). Due to the monochromatic excitation of Al and Si, there were no disturbing peaks in the measured spectrum. The optimization criteria described in Sec. III, for the generation of highly intense Barkla polarized radiation, is crucial for optimization of the geometry of an EDPXRF system for Bragg as well as Barkla polarization. Typical applications for Bragg systems within the given framework include, for example, the determination of the elements from Na to Fe in ceramics, refractory materials, cements, ores, ferro-alloys, slag, or additives in oil. The detection limits for 12 selected elements in base oil displayed in Table 8 demonstrate that EDPXRF is also well suited to the simultaneous trace detection of lighter elements (see also Fig. 7). Protective foils, 4 mm prolene, were used in front of the detector and between crystal and sample for the measurements listed in Table 8. These easily replaceable protective foils are necessary in routine work. It is, however, possible to obtain much better detection limits for Na (120 mg=g) without the protective foils. The relatively poor detection limits for Na is caused in part by an overlap of the NaKa line with the escape line from nonpolarized PdLa radiation backscattered from the sample. Greatly improved detection limits and sensitivities can be obtained for Fe, Cu, and Zn by using a Barkla scatterer analogous to that in Sec. III. F. However, these improvements are not required for the given application, additives in oil; an additional measurement unnecessarily increases the measuring time. The detection limits for P, S, and Cl, in particular, open the range of possible applications for EDPXRF (e.g., the determination of S in diesel fuels and gasoline). The comparison shown in Table 9 demonstrates the brightness of curved crystals in combination with ﬁne focus x-ray tubes in EDPXRF.

Table 8 Detection Limits and Sensitivities, S, for Various Elements in an Oil Matrix Using Excitation with Polarized Radiation Element

DL (mg=g)

S [cps=(mg=g)]

Element

DL (mg=g)

S [cps=(mg=g)]

Element

DL (mg=g)

S [cps=(mg=g)]

Na Mg Si P

270.0 28.0 1.6 0.7

0.00288 0.0230 0.341 0.823

S Cl K Ca

0.6 0.5 5.5 0.7

1.81 3.38 0.243 0.424

Fe Cu Zn Ba

0.9 1.0 1.2 1.4

1.66 1.63 1.63 0.332

Note: Pd tube; voltage: 15 kV; current: 14 mA; measuring time: 300 s; ﬂat HOPG polarizer; He environment; 4-mm prolene ﬁlters in front of the detector and between crystal and sample.

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Heckel and Ryon

Figure 7 Oil sample spectrum (log scale) excited by using an HOPG polarizer. Some of the concentrations are as follows: Mg, 90 mg=g; P, 220 mg=g; S, 3650 mg=g; Cl, 70 mg=g; Ca, 480 mg=g; Cu, 80 mg=g; Zn, 250 mg=g; Ba, 190 mg=g.

The polarization geometry described here is also suited to the analysis of fusion samples. Preparation of lithium tetraborate glasses from oxidic samples, for example, is utilized to conduct precise determination of major and minor elements and also, in some cases, trace elements. Excellent sensitivity and reproducibility are required for simultaneous analysis of the elements from Na to Fe with dilution ratios from 1 : 6 to 1 : 20. RSD ¼ 0.12% for the reproducibility of major elements and root-mean-square deviations RMSs for the calibration lines [see Eq. (28)], presented in Table 10 conﬁrm the advantageous parameters of the analytical technique based on Bragg polarization:

Table 9 Comparison of Sensitivities, SN , Normalized to Tube Current (mA) and Detector Area (mm2 ) for Various Elements in Base Oil SN 102 [cps=(mg=g)=mA=mm2] Element Mg Si P

SN 102 [cps=(mg=g)=mA=mm2]

SN 102 [cps=(mg=g)=mA=mm2]

(1)

(2)

Element

(1)

(2)

Element

(1)

(2)

0.014 0.203 0.490

0.350 6.50 15.3

S Cl K

1.08 2.01 0.145

32.6 70.3 2.83

Ca Ba

0.252 0.198

5.73 4.50

Note: (1): Pd tube with an approximately round focal spot of 8 mm in diameter (15 kV, 14 mA); ﬂat HOPG polarizer; Si(Li) detector with an energy resolution of 164 eV for MnKa and a visible area of 12 mm2; (2): Pd tube with a focal spot of 1 mm in diameter (25 kV, 1 mA); spherically bent HOPG polarizer (26 mm radius of curvature); Si drift chamber with an energy resolution of 161 eV for MnKa and a visible area of 3 mm2. All measurements are done in a He environment and with 4-mm prolene ﬁlters in front of the detector and between crystal and sample.

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627

Table10 Root-Mean-Square Deviation and Range of the Calibration Lines for Different Elements in Lithium Tetraborate Glasses Using Excitation with Linearly Polarized Radiation Element

Range (g=kg)

RMS (g=kg)

Element

Range (g=kg)

RMS (g=kg)

Element

Range (g=kg)

RMS (g=kg)

Na2O MgO Al2O3 SiO2

0–70 0–970 0–890 0–990

0.34 1.1 1.0 1.1

P2O5 SO3 K2 O CaO

0–10 0–36 0–130 0–680

0.23 0.28 0.32 0.71

TiO2 Cr2O3 MnO Fe2O3

0–1000 0–160 0–50 0–560

0.75 0.58 0.3 0.95

Note: Rh tube; voltage: 15 kV; current 11 mA; measuring time: 300 s; ﬂat HOPG polarizer; no ﬁlter; vacuum.

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ PM certified Cicalibrated Þ2 i ðCi RMS ¼ Mf

ð28Þ

where M is the number of standards used (M ¼ 35) and f is the number of regression coeﬃcients determined.

V.

BARKLA^BRAGG COMBINATION SYSTEMS

The HOPG crystals are characterized as very good Barkla targets in Sec. III and as excellent Bragg polarizers in Sec. IV. Polarization of the entire spectrum emitted from a tube has been successfully performed with these crystals. Excitation by monochromatic Bragg reﬂections dominates the energy range E < 10 keV; polychromatic Barkla polarized radiation dominates in the energy range E > 10 keV. Optimization of this Barkla–Bragg combination system is matrix dependent due to the count-rate limitation of the detection system and possible pileup lines. Using modern Si(Li) detectors or Si-drift chambers, it is possible to process 104–105 cps so that optimization is facilitated by taking advantage of the ability to control the tube current. Additional targets are only required for special applications, because it is possible to excite all of the elements between Na and U with this combination target. Of course, it is possible to optimize the excitation conditions using radiation ﬁlters between the target and sample, analogously to direct excitation with nonpolarized primary tube radiation. Being able to use air-cooled low-power x-ray tubes (50 W) as the source of primary radiation is an important advantage of Barkla–Bragg combination targets. The spectrum shown in Figure 8 was measured with an integral input count rate of 25,000 cps (dead time: 18%) using a Peltier-cooled Si-drift chamber with an active area of 5 mm2 and an energy resolution of 159 eV for MnKa.

VI.

SECONDARY TARGETS

The basic principles when working with secondary targets are to generate nearly monochromatic radiation and to use it for excitation. Because the characteristic radiation of the target material is nonpolarized, the detection system is burdened by backscattered characteristic radiation from the target material, particularly when analyzing light matrices. The advantageous application of secondary targets in the Cartesian excitation geometry required for polarization is described by Bisga˚rd et al. (1981). The suppression of

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Figure 8 Spectrum (log scale) of a geological standard sample, GnA, excited with the polarized primary spectrum of a 50-W low-power Pd x-ray tube, measured with a Peltier-cooled Si-drift chamber. Some of the concentrations are as follows: Si, 334 mg=g; K, 21.8 mg=g; Ca, 4.4 mg=g; Ti, 120 mg/g; Mn, 1.3 mg=g; Fe, 41.4 mg=g; Cu, 18 mg=g; Zn, 78 mg=kg; Rb, 2020 mg=g; Zr, 70 mg=g; Nb, 94 mg=g; Mo, 100 mg=g; Sn, 1900 mg=g; Bi, 220 mg=g.

primary tube radiation scattered on the target plays an important role, especially when using secondary targets with low atomic numbers, such as Al. This is because radiation ﬁlters cannot be placed between the target and the sample in this case. The detection limits listed in Table 4 show that, compared to polarization targets, better results can be obtained with a given x-ray tube for a limited number of elements. For example, a detection limit of 0.2 mg=g was achieved for Hg in an organic solution using a measuring time of 500 s with a Zr secondary target. Secondary targets in the Cartesian excitation geometry are often utilized to determine trace elements with atomic numbers Ztrace < Zmajor without exciting the major element Zmajor. For example, a Co secondary target is used for the detection of Mn traces in an iron matrix; an Fe ﬁlter is used to absorb the CoKb line that can excite Fe. A Si secondary target is used for the trace detection of Na, Mg, and Al in a SiO2 matrix. A detection limit of 11 mg=g for Al in glass sand prepared as a pressed powder pellet is achievable. Thus, secondary targets are an ideal complement to Barkla and Bragg targets in a Cartesian excitation geometry.

VII.

CONCLUSION

In the fundamental publications in the 1980s and 1990s about the application of polarized radiation in EDXRF by Ryon and Zahrt for Barkla polarization and Aiginger and Wobrauschek for Bragg polarization, tube powers in kilowatt ranges are a condition for presentation of the advantages of EDPXRF over direct excitation with nonpolarized x-ray radiation. In recent years, the development of Cartesian excitation geometries has been

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propelled forward so that when coupled with powerful semiconductor spectrometers that enable the processing of about 100,000 cps with resolution FWHM < 200 eV, it is now advantageous to use 50-W tubes in PXRF. In practice, PXRF has asserted itself in many industrial laboratories and research institutes as a powerful method for elemental analysis. The following typical applications of EDPXRF instruments* have been identiﬁed: Determination of heavy metals in soils and sludges Major, minor, and trace element determination in geological samples Precise major and minor element determination in ceramics and refractory materials Trace elements determination in biological and organic samples Analysis of aerosols Analysis of additives in oils Determination of trace elements in polymers and pharmaceutical products Rapid determination of heavy metals and halogens in wastes Analysis of slag In the future, the adjustment of tube and crystal parameters will be additionally improved so that new applications with EDPXRF will be possible (e.g., major, minor and trace element determination within sample areas 1 mm2). It will be a long time before the developments in the analytical ﬁelds become completed, especially the use of curved crystals in EDXRF.

REFERENCES Aiginger H, Wobrauschek P. J Radioanal Chem 61:281, 1981. Aiginger H, Wobrauschek P, Brauner C. Nucl Instrum Methods 120:541, 1974; also in measurement, Detection and Control of Environmental Pollutants. IAEA: Vienna, 1976. Barkla CG. Trans Roy Soc (London) 204A:467, 1905; Barkla CG. Proc Roy Soc (London) 77:247, 1906. Beckhoﬀ B, Kanngießer B, Scheer J, Swoboda W. Adv X-ray Anal 35:1083, 1992. Bertin EP. Principles and Practice of X-ray Spectrometric Analysis. 2nd ed. Plenum Press: New York, 1975, p 117. Bisga˚rd KM, Laursen J, Schmidt Nielson B. X-ray Spectrom 10:17, 1981. Brumme M. Die BARKLA-Polarisationseinrichtung in der energiedispersiven Ro¨ntgenﬂuoreszenzanalyse. PhD dissertation, TU Dresden, Dresden, Germany, 1990. Brumme M, Heckel J, Irmer K. Isotopenpraxis 26:341, 1990. Champion KP, Whittem RN. Nature 199:1082, 1963. Compton AH, Hagenow CF. J. Opt Soc Am 8:487, 1924. Dzubay TG, Jarrett BV, Jaklevic JM. Nucl Instrum Methods 115:297, 1974. Evans RD. In: Handbuch der Physik. Springer-Verlag: Berlin, 1958, Vol 34. Friedrich W, Knipping P, von Laue M. Ann Phys. 41:971, 1913. Hanson AL. Nucl Instrum Methods A243:583, 1986. Heckel J. J Trace Microprobe Tech 13:97, 1995. Heckel J, Brumme M, Weinert A, Irmer K. X-ray Spectrom 20:287, 1991. Heckel J, Haschke M, Brumme M, Schindler R. J Anal Atomic Spectrom 7:281, 1992. Howell R, Pickles W. Nucl Instrum Methods 120:187, 1974. Howell R, Pickles W, Cate Jr J. Adv X-ray Anal 18:265, 1975.

*Manufactured by Spectro A.I. GmbH & Co. KG, Kleve, Germany.

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Kanngießer B. Die Anwendung von HOPG-Kristallen in der energiedispersiven Ro¨ntgenﬂuoreszenanalyse. Diploma thesis, University of Bremen, 1990. Kanngießer B, Beckhoﬀ B, Scheer J, Swoboda W. X-ray Spectrom 20:331, 1991. Kanngießer B, Beckhoﬀ B, Scheer J, Swoboda W. Adv X-ray Anal 35:1001, 1992. Klein O, Nishina Y. Z Phys 52:853, 1928. Ryon RW. Adv X-ray Anal 20:575, 1977. Ryon RW, Zahrt JD, Wobrauschek P, Aiginger H. Adv X-ray Anal 25:63, 1982. Strittmatter RB. Adv X-ray Anal 25:75, 1982. Svoboda W, Beckhoﬀ B, Kanngießer B, Scheer J. X-Ray Spectrom 22:317, 1993. Ter-Saakov AA, Glebov MV. Atomnaya Energiya 58:260, 1984. Thomson JJ, Thomson G. The Conduction of Electricity Through Gases. 3rd ed. Cambridge University Press: Cambridge, 1933. Wobrauschek P, Aiginger H. Adv X-ray Anal 28:69, 1985. Wobrauschek P, Aiginger H, Owesny G, Streli C. J Trace Microprobe Tech 6:295, 1988. Zachariasen WH. Theory of X-ray Diﬀraction in Crystals. Dover: New York, 1967. Zahrt JD. Adv X-ray Anal 26:331, 1983. Zahrt JD. Nucl Instrum Methods A242:558, 1986. Zahrt JD, Ryon RW. Adv X-ray Anal 24:345, 1981. Zahrt JD, Ryon RW. Adv X-ray Anal 29:435, 1986.

11 Microbeam XRF Anders Rindby Chalmers University of Technology and University of Go¨te¨borg, Go¨tebo¨rg, Sweden

Koen H. A. Janssens University of Antwerp, Antwerp, Belgium

I.

INTRODUCTION AND HISTORICAL PERSPECTIVE

Up to the end of the 1980s, elemental microanalysis based on the emission of characteristic x-rays was largely restricted to charged-particle-beam techniques (electrons, protons, and heavier nuclei). Although the use of primary x-rays for inducing the production of the ﬂuorescent radiation was widespread in bulk x-ray ﬂuorescence (XRF) instrumentation, the lack of brilliance of conventional x-ray tubes and the inherent diﬃculties of focusing x-rays in the 1–30-keV energy range hampered the development of the microscopic variant of XRF as a microanalytical tool. Nevertheless, Glockner and Schreiber (1928) quite early demonstrated the possibility of doing chemical analysis by ﬂuorescence spectroscopy with small x-ray beams. In the mid-1950s, Long and Cosslett (1957) reported detection limits at the picogram level from an x-ray microbeam device. Zeits and Baez (1957) predicted a theoretical detection limit from an optimal instrument to be below 1014 g. They concluded that compared to the electron microprobe, the x-ray microprobe was superior in relative sensitivity, but not in terms of absolute detectability. Localization was also poor for x-rays; however, the energy impact is much smaller in comparison to charged-particle beams. In the early 1980s, the availability of highly intense radiation beams from synchrotron sources increased the interest in using microscopic x-ray beams for performing m-XRF experiments (Chapter 8); also around this period, capillary optics were being used as simple, compact, and relatively inexpensive means of ‘‘concentrating’’ a beam of x-rays down to a smaller size. This innovation lead to the development of microbeam XRF instrumentation employing (conventional) x-ray tubes as the source of primary radiation. During the last two decades, monocapillary and polycapillary focusing devices have been incorporated into a variety of m-XRF spectrometers; an overview of their characteristics is provided in Table 1. More recently, the introduction of commercially available compact minifocus and microfocus tubes has considerably increased the potential of microbeam XRF, making possible the use of microbeams in small-sized and easily transportable XRF equipment. Another signiﬁcant development of recent years is the maturation of the techniques 631

n.s. n.s. n.s. n.s. n.s. n.s. 40 5 n.s. n.s. 40

40 5 70 70 40 22

n.s. n.s. n.s. n.s. n.s. < 0.25 0.25 0.4 12 0.030 0.4 8 0.25 0.25 0.4 8

n.s. 0.015 0.1

0.5 10 0.5 10 0.25 0.25 0.03 0.006

Distance anode–optics (mm)

10, 15 mm ell. con. cap. 160 mm polycap. 50, 100 mm polycap. 21 mm polycap.

10, 30, 100 mm coll. 200 mm str. cap 30 mm coll. 10, 30, 100 mm coll. 22, 7.5 mm par. cap. 10, 30, 100 mm coll. 200 mm str. cap. 10 mm str. cap. 18, 23, 27 mm str. cap. 70 mm coll. 18 mm con. cap. 7.5 mm ell. cap. 15, 65, 70 mm con. cap. 7.5 mm ell. cap. 15 mm con. cap. 4, 8, 29, 100 mm str. cap.

Opticsb (diameters given) n.s. n.s. n.s. n.s. n.s. > 0.8 0.4–0.6 0.5–0.7 n.s. 0.8 0.3 0.5 0.3 n.s. 1.0 0.9 3.6 3.6 0.96 67

1,500 1,000 n.s. 1.6 (Cu, W) 1.4 (Mo) 18,000 18,000 60 12

Brilliance (kW=mm2)

1,000 n.s. 2,500 500–1,000 n.s. 50 1,900–3,000 15–21 n.s. 50 1,000

Tube power (W)

Cu, Mo Mo Mo Mo

Mo Cr, Cu Mo Cu, W, Mo

Cu W Cu Cu, Mo Mo Mo Cr, W Mo, W Cu, W, Mo Mo Cr

Anode

(a) Nichols and Ryan, 1986; (b) Rindby, 1986; (c) Boehme, 1987; (d) Nichols et al., 1987; (e) Yamamoto and Hosokawa, 1988; (f ) Wherry et al., 1988; (g) Engstro¨m et al., 1989; (h) Rindby et al., 1989; (i) Carpenter, 1989; Carpenter et al., 1989; Carpenter and Taylor, 1991; (j) Furata et al., 1991; Furata et al., 1993; Hosokawa et al., 1997; (k) Pella and Feng, 1992; (l) Larsson et al., 1990; (m) Shakir et al., 1990; (n) Rindby, 1993; (o) Attaelmanan et al., 1995; (p) Holynska et al., 1995; (q) Carpenter et al., 1995b; (r) Janssens et al., 1996a; (s) Vekemans et al., 1998; (t) Bichlmeier et al., 2000; (u) Gao et al., 1997. b str.cap., con.cap., par.cap., ell.cap.: resp. straight, conical, paraboloidal, ellipsoidal capillary; coll.: collimator; polcap.: monolithic polycapillary lens; n.s.: not speciﬁed.

a

Diffraction tube (a) W tube þ Mo secondary target (b) Philips interchangeable tubes (c) Fine-focus diffraction tube (d) n.s. (e) Low-power tube (f) Long, fine-focus diffr. tube (g,h) Radiographic microfocus tube (i) Diffraction tube ( j) Low-power tube (k) Long, fine-focus diffraction tube (l,m) Idem (n) Idem (o) n.s. (p) Radiographic microfocus tube (q) Rotating anode tube (r) Rotating anode tube (s) Minifocus tube (t) Radiographic microfocus tube (u)

a

Focal size on anode (mm2)

Overview of Laboratory-Scale m-XRF Instruments Described in the Literature

X-ray source speciﬁcation

Table 1

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for manufacturing monolithic polycapillary x-ray lenses. In addition to being used for focusing tube-generated x-rays into microscopic spots, at synchrotrons, these devices are now also being employed for performing m-XAS (x-ray absorption spectroscopy), m-XRD (x-ray diﬀraction), and x-ray radiography experiments. The state of the art of m-XRF has recently been summarized in a special issue of the journal X-ray Spectrometry (Carpenter, 1997) and in a book (Janssens et al., 2000a). A.

Development of Microscopic X-ray Emission Analysis

With the availability of an intense electron microbeam and an eﬃcient detection system for x-rays, a scanning x-ray microscope was quite an obvious development; one of the very ﬁrst scanning x-ray microscopes was described by Duncumb (1957). The system used electron excitation and was constructed as a microfocused x-ray tube. A proportional counter was used for detecting the emitted x-rays. Proportional counters are very eﬃcient devices but are not able to separate the K x-rays of adjacent elements unless the counter is combined with some kind of Bragg-reﬂecting dispersion element. With the introduction of the Si(Li) detectors in the late 1960s, the construction of XRF spectrometers in general was simpliﬁed. One of the ﬁrst x-ray excitation scanning ﬂuorescent microbeams was constructed by Long and Cosslett (1957) using a microfocused transmission tube in combination with an aperture. Next, by using highly brilliant sources of hard x-rays as provided by synchrotron storage rings, microbeam spectroscopy became possible at a much more sophisticated level. The basic instrumental arrangement employed during m-XRF experiments is shown in Figure 1. The very ﬁrst synchrotron-based x-ray microbeam setup was described by Horowitz and Howell (1972). Tabletop m-XRF devices have developed rapidly over the last decade and the number of applications is growing. Today, m-XRF instruments are used in forensic science, industrial quality control, and environmental science. They are used in material science for analyzing polymers, composite materials, ﬁber materials, soft tissues from plants, and so forth. Several instruments are now commercially available and most of these setups have a similar set of components. A standard microbeam spectrometer consists of an x-ray source, some kind of focusing device (or just an aperture), a sample holder, and a detection system. The ﬁrst two

Figure 1

Basic experimental scheme employed during m-XRF investigations.

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components will deﬁne the spectral properties of the primary beam (large or narrow energy bandwidth). The sample holder should allow for an accurate remote control of the sample position and the detection system for intensity measurement of ﬂuorescent, transmitted, or=and diﬀracted radiation. A conventional microscope (or just a camera and a lens system) can be used as a monitoring system for identifying the part of the sample that is to be analyzed. Normally, ﬂuorescence spectroscopy can be done in diﬀerent modes: point analysis, line scan, and area scan. In addition to ﬂuorescence spectroscopy, absorption and diﬀraction measurements can be done in the same modes. Other types of spectroscopy can also be performed such as XANES (x-ray absorption near-edge spectroscopy) or EXAFS (extended x-ray absorption ﬁne structure spectroscopy) (Bertsch et al., 1994); however, these types of spectroscopy require the possibility to tune the primary energy to the absorption edges of element(s) of interest and, in most cases, are performed only at synchrotron beam lines. With the development of capillary optics, a considerable number of tabletop microbeam instruments have been developed and described in the scientiﬁc literature (see Table 1). These instruments normally use capillary optics in combinations with conventional ﬁne-focus x-ray tubes, where the virtual brilliance can be fully utilized, or with micro-focused tubes, where the (actual) brilliance is very high. The sensitivity of these instruments can be quite high: Larsson and Engstro¨m (1992) reported a detectable limit (DL) for such an instrument of about 40 fg (4 1014 g) of calcium in a paper specimen, which is in agreement with the prediction of Zeitz and Baez (1957). B.

Development of Microfocusing X-ray Optics

The reason why x-ray spectroscopy has not been applied on the microscopic level as has been the case with electron-probe x-ray microanalysis is mainly attributed to the diﬃculties of optically controlling an x-ray beam as compared to an electron beam. Although diﬀerent types of optics for focusing and imaging in the x-ray region have been developed, their eﬃciencies are still inferior to those of conventional electron optical systems. However, even with poor optical eﬃciency, the use of x-ray microbeams oﬀers many advantages in comparison to electron beam excitation. Also, for x-ray microscopy, there has been a great need for microfocusing optics. As early as 1936, Sievert (1936) used a conventional aperture of a few micrometers to generate a small x-ray source for x-ray microscopy. Although the resolution was good, the intensity was poor and von Ardenne (1939) proposed an electron lens to generate small x-ray sources for projection x-ray microscopy. A microfocus x-ray tube based on the idea of von Ardenne was actually constructed by Cosslett and Nixon (1952) for this purpose. Although the electron lens system constructed in the early 1950s was the starting point for a very rapid development of electron microscopy, x-ray microscopy and microbeam spectroscopy did not develop at the same rate due to the inability to generate eﬃcient optical systems for x-rays. With the introduction of synchrotrons in the late 1960s, a new interest in x-ray optics was created and several synchrotron-based x-ray microbeam setups are now in operation (Iida, 2000). In recent years, also tabletop x-ray microbeam setups have been constructed in which both conventional and nonconventional optics have been applied (Rindby, 2000). 1. First-Generation Focusing Systems One of the very ﬁrst focusing devices used for x-rays was probably the von Seeman–Bohlin (von Seeman, 1919) camera, which was a Rowland circle setup. The practical problem of

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x-ray optics was recognized early when Compton (1923) demonstrated that although x-rays have measurable optical properties, the refractive power of most materials in the x-ray region is very low. This means that a conventional lens system for x-rays was very impractical. However, in 1924, Larsson, et al. (1924) used the edge of a prism block to focus x-rays. The system, which was rather ineﬃcient and had severe chromatic aberration, was later used by Hink and Petzold (1958), but only to measure refractive indices. The technical possibilities left for x-ray optics were Bragg reﬂection and total external reﬂection. Bragg reﬂection was used in the Johan (1931) and Johansson (1933) cameras. An advantage with Bragg reﬂection is that normal incidence can be used, thus reducing astigmatism and aberrations. However, the system only works within a very small energy band width and the rest of the radiation energy is completely lost. In order to construct broad-band x-ray optics, total-reﬂecting devices need to be employed. Jentzsch (1929) established the essential geometrical principles for x-ray focusing from total-reﬂecting spherical surfaces. He pointed out the importance of using highly ﬁnished surfaces but also concluded that glancing-angle incidence will always generate severe astigmatism. Attempts were also made to form images from the inside of a glass tube. Experiments were carried out by Ehrenberg and Jentzsch (1929) and by Na¨hring (1930a, 1930b) and imaging properties were also discussed by Kellerman (1943), but no result was ever reported. One of the very ﬁrst practical focusing devices was a totalreﬂecting curved crystal developed by Ehrenberg (1947). In order to overcome some of the problems with astigmatism, another approach to the imaging requirements was investigated by Wolter (1952a) and Herrnring and Wiedner (1956). They suggested the use of cylindrical mirrors (i.e., the Wolter microscope). A more practical method for astigmatism correction is the crossing of two cylindrical mirrors at right angles. In general, large aberrations in cylindrical and spherical mirrors can always be reduced by a second mirror. Pioneering work with this kind of compound system was done by Kirkpatrick and Baez (1948). Other compound systems were proposed by Herrnring and Wiedner (1956) and Wolter (1952b), suggesting combinations of hyperboloid and paraboloid surfaces. 2. Second-Generation Focusing Systems Today, the Kirkpatrick–Baez compound system as well as the Wolter mirror are in use at diﬀerent synchrontron-based microbeam setups. Due to the improved technique for developing highly ﬁnished and perfect surfaces, ellipsoidal and paraboloidal mirrors can now be manufactured with high perfection. The multilayer technique has also improved the reﬂecting quality of modern mirrors. The LBL (Lawrence Berkeley Laboratory) microbeam setup uses a Kirkpatrick– Baez system with two mutually perpendicular spherical mirrors coated with a W–C (tungsten–carbon) multilayer. The system, when operating at the X-26C beamline at NSLS (Upton, NY), provides a 50-fold demagniﬁcation and generates a 10-mm2 x-ray focus. At the NSLS X26A beamline, an elliposoidal mirror is used for focusing purposes. A Wolter focusing mirror combined with a hyperboloidal and an elipsoidal surface is used at the Photon Factory microbeam setup (Tsukuba, Japan). A bent crystal is used for the SRS microprobe (Daresbury, UK). A comparison between diﬀerent microprobe setups and their sensitivity was given by Larsson and Engstro¨m (1992). Reviews of synchrotron microprobes are given by Jones and Gordon (1989) and Rivers and Sutton (1991); Rindby et al. (2000) provides an up-to-date overview of the technology that is currently used to focus x-rays. See also Chapter 8.

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3. X-ray Concentrators Although improvements have been made in the fabrication of optical elements such as mirrors and zone plates, the technique is limited by its basic imaging properties. Whatever imaging system is used, the focal spot is nothing else than the image of the source; thus, in order to produce an x-ray beam with small focal spot, one needs a small x-ray source. Variation and instability of the primary source can induce a corresponding instability of the focal spot, and small variation in the geometry of the reﬂecting mirrors, due to thermal gradients, can cause a large ﬂuctuation in the focal length and focal spot size. Many of these problems can be circumvented by going beyond the conventional imaging optics and apply nonimaging optics (NIO). If the optical system is only used to focus the beam, NIO technology is normally superior to conventional imaging optics. Nonimaging optics was born in the mid-1960s when Baranov, Ploke, and Winston (see Winston, 1991) independently designed the ﬁrst compound parabolic concentrator. In nonimaging optics, no focal plane can be deﬁned, but the optical elements are designed only to generate maximum (spatial) concentration of the radiation. These optical elements, which are normally called concentrators, can generate intensities far better than any conventional imaging system. A typical NIO technique for x-rays is the application of small conical glass capillaries used for generating micrometer-sized x-ray beams. The advantage of the x-ray concentrators (mostly referred to as ‘‘capillary optics’’) is their ‘‘source independence.’’ The beam size is deﬁned by the capillary inner diameter rather than the source size. As it is a NIO technique, a high virtual brilliance source (such as the point focal output from a line-focus x-ray tube) can be used in an eﬃcient way. Instability and ﬂuctuations in the source position will not aﬀect the position or the size of the focal spot. As the beam is deﬁned by the capillary opening, the beam position can easily be established by an ordinary optical microscope. Due to its cylindrical symmetry, thermal gradients will not aﬀect the beam position even at high radiation exposure. Because it is a total-reﬂection device, capillaries also have broad-band properties. They have submicrometer capabilities, and compared to conventional x-ray optics, are very inexpensive. One disadvantage, however, is that the highest intensity is achieved at the capillary opening so that the sample needs to be very close to the capillary output in order to utilize the high photon density completely. Although the divergence of a capillary beam is normally very small (it is of the order of the critical angles for total reﬂection), this problem can be severe for near-mircometer or submicrometer capillary beams. After the attempts of Ehrenberg and Jentzsch (1929) to form x-ray images from the inside of a glass-capillary tube, Jentzsch and Na¨hring (1931) used straight glass capillaries of diﬀerent dimensions, of length 10–75 cm and an opening of 0.1–2 mm in diameter, to generate x-ray microbeams. Practical applications of glass capillaries began at the end of the 1940s. Chesley (1947) designed a microcamera that used a short, straight glass capillary; in the same year, Kreger (see Hirsch, 1955) used a 20-mm beam from a 1-cm-long conical glass capillary to study the structure of the wax-rod coating of a sugar cane stem. Hirsch and Keller (1951) used photographs of the emergent beam from a 35-mm capillary to properly align the capillary. In the period 1952–1954, capillary-generated microbeams were applied to the study of the deformation, recovery, and recrystallization of metals (Hirsch, 1955). The smallest beam used was 8 mm. Mosher and Stephanakis (1976) described the eﬃcient transport of soft x-rays through hollow glass and metal tubing. A bundle of such tubes was used to remotely image a weak plasma source of soft x-rays. Comprehensive theoretical analysis of x-ray guides was conducted by Chung and Pantell (1977) and by Pantell and Chung (1978, 1979), who investigated the transmission

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and power enhancement of x-rays in cylindrical wave guides, the transmission of x-rays through curved wave guides, and also studied the inﬂuence of surface roughness on the propagation of x-rays through capillaries. Rindby (1986) measured the critical angle for total reﬂection for x-rays inside a 200-mm glass capillary. In the same year Nozaki and Nakazawa (1986) described a conical x-ray guide tube (XGT). The XGT was used with the smaller opening nearest to the x-ray tube to obtain an almost parallel x-ray beam. Carpenter et al. (1987) carried out several experiments to determine the variation in beam proﬁle, intensity, and spectrum with capillary size and position. The ﬁrst focusing XGT was reported by Yamamoto and Hosokawa (1988), who developed a parabolic inner-wall x-ray guide tube made of glass that focused the x-ray beam into a 5.7-mm spot. The ﬁrst scanning x-ray microprobe with glass-capillary collimation, which had a beam size of 10 mm, was described by Carpenter et al. (1988). The ﬁrst capillary-generated submicron x-ray beam was reported by Engstro¨m et al. (1991), from a synchrotron radiation source. Hoﬀman et al. (1994) reported a 50-nm x-ray beam from a conical capillary, which was measured with energies up to 8 keV, using synchrotron radiation. Kumakhov (1986, 1990) and Kumakhov and Komarov (1990) performed extensive research, both theoretical and practical, on the multireaction properties of glass capillaries, which led to the design of the Kumakhov or polycapillary x-ray lens, a type of device now manufactured by several laboratories worldwide (Kardiawarman et al., 1995; Ullrich et al., 1995; Kumakhov, 1998; Ding et al., 1998; Vekemans et al., 1998; Gao et al., 1998). This optical technology possesses real imaging properties and has found broad application areas in both pure and applied sciences. In combination with microfocus x-ray tubes, beam sizes down to 15–30 mm can now be achieved. C.

Chapter Overview

In this chapter, after providing some information on the basic x-ray–matter interactions which are employed for microfocusing x-ray and comparing the way in which x-ray and charged-particle microbeams interact with matter, the instrumentation required for performing microbeam XRF is discussed, with some emphasis on recent developments in this ﬁeld (tubes, optics, detectors). Next, data acquisition strategies for point, line, and image XRF analysis are described and the appropriate procedures for evaluating, segmenting, and quantifying (large amounts of) spectral data are discussed. Finally, an overview of applications of the microbeam XRF technique in various ﬁelds is presented: the latter include the use of m-XRF in the industrial, environmental, cultural, and forensic sectors of activity.

II. THEORETICAL BACKGROUND A.

Photon Versus Charged-Particle-Induced X-ray Emission

When considering the (dis)advantages of microbeam XRF relative to other microanalytical methods, in the ﬁrst instance, it is relevant to consider the factors determining the sensitivity of the microbeam XRF technique and its closest analogs, EPXMA (electron-probe x-ray microanalysis) and m-PIXE (proton-induced x-ray emission). For all x-ray emission techniques that employ some form of photon counting in the detection chain (mostly when energy-dispersive detectors are used), the lowest detectable concentration level cDL (DL: detection limit) for a given chemical element can be estimated

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from the net photopeak intensity N and the corresponding background intensity B, which are collected from a sample containing a concentration c of a given element, as: pﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ k B k sB 1 pﬃﬃﬃﬃ ¼ ð1Þ cDL ¼ c N sN It where k is usually a number in the range 1–5 (often k ¼ 3), sN ¼ N=cIt is the net x-ray yield for this element and sB ¼ B=It is the corresponding background yield. The sensitivity of any x-ray emission technique therefore can be increased by the following: 1. 2. 3.

Decreasing the background abundance sB Increasing the elemental yield sN Increasing the beam intensity I and=or the collection time t

It is useful to consider the relative and absolute magnitudes of the above-mentioned quantities when comparing the analytical characteristics of various x-ray emission techniques. Because, in the present context, diﬀerent microanlytical techniques are compared, the achievable beam size S and the beam ﬂux density F ¼ I=S (expressed, e.g., in photons, electrons, or protons per second and per square micrometer) are also important ﬁgures of merit. a.

Penetration Depth

In Table 2, the penetration depth of 20-keV x-ray photons and electrons and 2.5-MeV protons are compared. The much larger penetrative power of photons in light matrices (Z < 20) relative to charged particles is immediately clear; protons take an intermediate position between electron and photons. The penetration range for charged particles is determined by the gradual energy loss that occurs as a result of many inelastic scattering interactions with the sample material [quantitative described by the stopping power function SðEÞ] and by the curvature of the trajectories (mainly resulting from many elastic collisions between projectile and sample atoms). On the other hand, for x-ray photons in the range 1–50 keV, the most probable interaction mechanism (for many sample materials) is the photoelectric eﬀect. For example, for 10-keV photons in Al, the probabilities for photoionization and Compton (i.e., inelastic) and Rayleigh (elastic) scattering are 97.5%, 0.4% and 2.1%, respectively. Because the photoelectric eﬀect causes the primary photon to be annihilated, the majority of the primary photons will only be subject to a single

Table 2

Penetration Depth of Electrons, Protons, and X-ray Photons in Various Materials Penetration range (mm)

Element C Si Fe Ag Pb

Atomic number

Atomic mass

Density (g=cm3)

maL (cm 7 1)

Xb 20 keV

e7 c 20 kV

pþ c 2.5 MeV

6 14 26 47 82

12 28.1 55.9 107.9 207.2

1.9 2.3 7.8 10.5 11.3

0.79 10 200 189 971

12,000 1,000 50 50 10

5 5 1.6 1.7 2

55 68 27 28 37

mL ¼ linear mass absorption coefﬁcient. 1=e range. c Bethe range. a

b

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(scattering or photelectric) interaction; only a very small fraction will ﬁrst be (in)elastically scattered by a sample atom before ionizing a second atom. b. Phenomena Contributing to the Background The consequence of this diﬀerent behaviour is that samples irradiated with monochromatic x-ray photons of energy E0 , next to the characteristic radiation of the sample atoms, will only emit scattered photons in a limited energy window from a few kiloelectron volts below up to the primary energy E0 . In contrast, the irradiation of a material with charged particles of energy E0 will result in a bremsstrahlung continuum (primarily due to decelerating electrons) spanning the entire energy range from 0 to E0 . This continuum is especially noticeable in EPXMA spectra, where the continuum is generated at relatively shallow depths (Reed, 1975) (typically 5 mm; see Table 2 and Fig. 2a). In case of PIXE, the abundance of the continuum is supressed because it is caused by secondary electrons of relative low energy generated by the primary protons (Johansson and Campbell, 1988) (Fig. 2b). When a sample is irradiated with a polychromatic x-ray beam, also in XRF spectra, a scatter continuum spanning the entire energy range can be observed (Fig. 2c). In contrast to SR-based microbeam XRF instruments, in which linearly polarized radiation is employed in order to reduce the scatter background level (Fig. 3a), laboratory m-XRF instruments operate with unpolarized polychromatic radiation, giving rise to an appreciable background continuum in the EDXRF spectra (Fig. 3b). c.

Production of Characteristic Radiation

In Figure 4, the cross sections for characteristic x-ray production by x-ray photons, electrons, and protons are compared (Vis, 1990). Two important observations can be made: (1) for the three projectile types, the overall probability for ionization is comparable; (2) for the charged particles, the ionization cross sections decrease with increasing atomic number of the target atoms—the photoionization process favors heavier over lighter elements. Considering Eq. (1), it can therefore be concluded that for equal beam intensity I and collection time t, the lower background levels that are recorded with monochromatic photon-induced x-ray emission, together with the comparable or higher ionization cross sections, will result in lower detection limits (DLs) for m-XRF than for EPXMA and m-PIXE in the case of heavy elements (Z > 25). Unfortunately, whereas in conventional electron microprobes routinely currents of, for example, 1 nA (i.e., about 6 109 electrons=sec) are focused in an area smaller than 0.001 mm2 and whereas in m-PIXE apparatus, usually 10–1000 pA are concentrated into a 1–10-mm2 spot (resulting in ﬂux densities in the range of 106–109 protons=s=mm2), equivalent photon ﬂux densities cannot be straightforwardly achieved using conventional x-ray sources. The reasons for this are twofold: 1. The spot emitting x-rays on the anode of a conventional x-ray tube is comparatively large (one to a few tens of square millimeters), whereas the generated photon beam is emitted into a broad cone. 2. It is not possible to focus x-rays as easily as charged particles or as radiation of longer wavelengths. The problem is even aggravated by the fact that x-ray tubes (based on electron-induced x-ray emission) produce a continuum of x-rays rather than monochromatic radiation. Typical primary ﬂux densities on a sample surface achievable with a standard x-ray tube are situated in the range of 104–105 photons=s=mm2 (Cu tube, 30 mA, 45 kV, sample at 5 cm from the anode spot) when only the principal anode line intensity is considered.

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Figure 2 EPXMA, m-PIXE, and white beam SR-XRF (NSLS X26A station) spectra collected by irradiating NIST SRM K961 glass microspheres.

Therefore, in order to take advantage of the inherently better peak-to-background ratio of photon-induced x-ray emission relative to excitation by charged particles, either much more brilliant x-ray sources than conventional tubes must be used or the radiation from an x-ray tube must be focused into a microspot on the sample or both.

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Figure 3 ED-XRF spectra of NIST SRM 1577 Bovine Liver obtained using (a) an 8 8 -mm2 polychromatic SR beam (220 s counting time) at the NSLS beam line X26A station, (b) a Mo-anode x-ray tube using a conical capillary (300 s counting time). Dots: experimental spectra, lines: predicted spectral distributions obtained by Monte Carlo simulation (see text).

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Figure 4 Cross sections for K-shell x-ray production by x-rays, protons, and electrons as a function of target atomic number for various projectile energies.

In addition to the use of synchrotron sources (Chapter 8), in the laboratory, microfocus or minifocus x-ray tubes may be used, either in combination with an aperture or with focusing optics. Of the latter, in practice, only compact devices such as singlecapillary concentrators or monolithic polycapillary lenses are being used in laboratory m-XRF instruments. B.

X-ray^Matter Interactions Employed for Microfocusing

1. Optical Theory of X-rays The description of the propagation of x-rays in any refractive medium is based on the same concepts as used for describing ordinary optical light propagation. These types of models rely on the fact that most materials can be described in terms of a refractive index n which is related to its dielectric properties pﬃﬃﬃﬃ ð2Þ n ¼ er where er is the relatively dielectric constant. As in general, er is a complex number and the index of refraction can be written under the form: n ¼ 1 d where d ¼ a þ ib. a and b can be written as a¼

2pNe2 0 f ; me o2

b¼

2pNe2 00 f me o2

ð3Þ

where f 0 and f 00 are the real and imaginary parts of scattering factor, respectively, o is the frequency of the x-ray, N is the number of atoms per unit volume, me is the electron rest mass, and e is its charge. The imaginary part of the index of refraction is also related to the linear absorption coeﬃcient mL : b¼

cmL 2o

ð4Þ

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Figure 5 The real part of the atomic scattering factor times the number of atoms per cubic centimeter for silver and for SiO2 glass.

For a multielectron atom, f 0 can be written as a sum of scattering factors for all the electrons. For energies far below any absorption edge, the corresponding electrons will only contribute in a minor way to the atomic scattering factor. However, for energies far above the absorption edge, the corresponding electron will contribute with unity. Thus, for energies far above the K edge, the scattering factor will be equal to the atomic number. Close to any edge, the scattering factor will have a ‘‘resonance structure,’’ as shown in Fig. 5. 2. X-ray Reflectivity If a plane wave propagating in a medium characterized by the refractive index n1 ¼ 1 d1 is incident on a smooth-plane boundary of another medium with refractive index n2 ¼ 1 d2 then if d1 6¼ d2 , a reﬂected and a transmitted wave is generated. The reﬂectivity (R) as well as the transmittance (T ) can be calculated from the Fresnel formula that simply assumes the electric and magnetic ﬁeld components to be continuous at the boundary interface (Azaroﬀ et al., 1974). Using Snell’s law, the parallel and perpendicular components of the reﬂectivity Rp and Rs can be expressed in terms of the incident angle yi and the indices of refraction of the two media: rp ¼ rs ¼

ð1 d2 Þ2 sin yi ð1 d1 Þ½ð1 d1 Þ2 ð1 d1 Þ2 cos2 yi 1=2 ð1 d2 Þ2 sin yi þ ð1 d1 Þ½ð1 d2 Þ2 ð1 d1 Þ2 cos2 yi 1=2 ð1 d1 Þ2 sin yi ½ð1 d2 Þ2 ð1 d1 Þ2 cos2 yi 1=2 ð1 d1 Þ2 sin yi þ ½ð1 d2 Þ2 ð1 d1 Þ2 cos2 yi 1=2

;

;

Rp ¼ rp rp

Rs ¼ rs rs

ð5Þ

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By assuming that the glancing angle is small, the parallel and perpendicular components will become almost identical and, furthermore, by putting d1 ¼ 0 (incidence from air or vacuum) and d2 ¼ d, we can simplify the formulas for the reﬂectivity to rs ¼ rp ¼

yi ðy2i 2dÞ1=2 yi þ ðy2i 2dÞ1=2

ð6Þ

Note that this expression becomes unity if y2 ¼ 2d, provided that d is real. This situation corresponds to total reﬂection of the incident wave front and thus the critical angle yc ¼ ð2dÞ1=2 / 1=E, where E is the x-ray energy. Because d, in general, is complex (taking the absorption into account) the reﬂection coeﬃcient will not be exactly unity. However, for most of the optical materials considered here, the reﬂectivity will be close to unity within a small angular range from zero degrees up to a few minutes of arc and then suddenly drop to about d2 =4 (see Fig. 6). Because d is complex ðd ¼ a þ ibÞ, the following substitution will be convenient: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y2 2d ¼ y2 2a 2ib ¼ p þ iq ð7Þ Thus, R can be written as Rðy; EÞ ¼ where

ðy pÞ2 þ q2 ðy þ pÞ2 þ q2

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 p ¼ ð ðy2 2aÞ2 þ 4b2 þ y2 2aÞ; 2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 q2 ¼ ð ðy2 2aÞ2 þ 4b2 y2 þ 2aÞ 2

ð8Þ

2

ð9Þ

3. Geometrical Aberrations As the reﬂectivity for x-rays is extremely small except within a small range from zero up to critical angle yc , most x-ray optics are based on the principle of grazing incidence, although optical devices based on Bragg diﬀraction are also in use (Ice, 1997). The diﬀerent types of geometrical aberration that occur for such an optical system can be described in terms of spherical aberration, astigmatism, curvature of the ﬁeld, and coma. However, in contrast to the imaging system of light optics, there is no axial symmetry. Thus, the normal optical treatment cannot be applied and the classiﬁcation of aberration will be somewhat diﬀerent. Spherical aberration arises when rays (originating from a point source), which will be incident on the mirror surface at diﬀerent angle, are not intercepting at the same point. Thus, the image of a point object formed by such a nonideal reﬂector will become a disk on confusion rather then a point. Astigmatism will arise if the meridian and sagittal rays are not focussed at the same distance from the mirror. Coma is a kind of ﬁeld aberration that is due to diﬀerent magniﬁcations being produced by diﬀerent regions of the mirror. Curvature of the ﬁeld means the focal plane is slightly curved due to diﬀerences in focal lengths at diﬀerent position in the object plane. This type of aberration will determine the physical ‘‘depth of focus’’ for the system.

Figure 6 Left: The reflectivity for glass as a function of the glancing angle at E ¼ 5.411 keV; the absorption coefficient was chosen to be mglass (i.e., the normal value), 10mglass and 0.1mglass. Right: Calculated reflectivity of glass at two energies (CrKa and MoKa) while using the normal value for absorption coefficient.

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III.

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INSTRUMENTATION FOR MICROBEAM XRF

Table 1 summarizes the characteristic features of several tabletop instruments recently described in the scientiﬁc literature. A m-XRF setup essentially consists of four major parts: 1. 2. 3. 4.

An x-ray source A focusing or collimation device A motorized sample stage þ microscope One or more detectors

As we can see from Table 1, capillary optics is, by far, the most popular type of optics for the tabletop instruments. Most of the tabletop setups are equipped with either microfocused tubes in combination with collimators or straight capillaries or high-ﬂux tubes with conical capillaries. When simple pinhole or cross-slit systems are used to aperture the beam, the problem is to get close enough to the actual source in order to pick up the highest ﬂux density. Straight capillaries have been used by many authors as a means of doing so and guiding it out to the sample position without the 1=r2 geometrical loss that occurs when conventional collimators are employed. As shown in Figure 7, these setups may utilize specially designed microfocused tubes where the capillary optics have been integrated in the tube construction. By using conical capillaries, one can operate further away from the source, as the ‘‘squeezing’’ of the beam will compensate for the reduced ﬂux density. As a ‘‘nonimaging device,’’ the conical capillaries can utilize virtual brilliance that is achieved in standard diﬀraction tubes by the projection of long ﬁne-focal spots. For any focusing device, accurate alignment is important; thus, the focusing element has to be completely integrated and permanently ﬁxed to the source or be mounted on a ﬂexible stage so that realignment can be done. A more recent development is the use of polycapillary x-ray lenses in combination with compact low-power tubes for generating x-ray beams in the size range above 15–30 mm.

Figure 7 Microbeam XRF setup with the capillary focusing unit built into a microfocus x-ray tube. (From Carpenter et al., 1988.)

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647

X-ray Sources Suitable for Microbeam XRF

The diﬀerent types of x-ray tube commercially available at present are discussed elsewhere in this volume (Chapter 5). General information on the x-ray continuum produced in x-ray tubes can be found in Bethe (1930), Webster et al. (1993), Mott and Massey (1949), Metchnick and Tomlin (1963), Robinson (1974), and Go¨rgl et al. (1992). In the framework of microbeam XRF, basically two types of tube can be distinguished: high-power and lowpower tubes. High-power tubes have a water-cooled anode, are normally equipped with a side-looking x-ray exit window, and can operate up to several kilowatts. The limitation of the source brilliance is due to the heat load on the anode that is a function of the power load, focal spot size, cooling eﬃciency, thermal conductivity, and so forth. The standard type of diﬀraction tube can take a power load up to 0.6 kW=mm2. For diﬀraction tubes, the focal spot is formed as a ﬁne line allowing for both a line-focal output and a point focal output, where the virtual high brilliance, from the projection of the line, can be utilized. The rotating anode technique is a way to improve the cooling eﬃciency of the anode and is used for very high-power loads. Low-power tubes can be side-looking, end-window, or transmission tubes. The low power means that the total ﬂux of radiation is low; however, the brilliance can still be high when all photons emerge from a small focal spot. For most types of X-ray focusing optics (see next subsection), the brilliance is the vital parameter and not the total ﬂux. Thus, low-power tubes are also very interesting in m-beam XRF. For side-looking, microfocused tubes (50– 100 mm spot size and smaller), the brilliance is comparable or better than that of the highpower tubes, although the total power is only a few watts. In transmission tubes, the radiation is taken out in a direction parallel with the electron beam. Here, the anode is just a thin foil that acts as a radiation window. For this type of tube, the power load is further limited by the reduced heat conductivity in the anode. In Table 3, the characteristics of some microfocus and minifocus tubes are listed. A comparison of the brilliance values listed in Tables 1 and 3 shows that all of these newly available minifocus and microfocus sources are well (if not better) suited for micro-XRF setups than the more conventional sources used up to now. B.

X-ray Optics

As can be seen from Table 1, in laboratory m-XRF setups, monocapillary or polycapillary optics is employed almost exclusively for beam focusing or concentration. At synchrotron

Table 3

Characteristics of Some Commercially Available Minifocus and Microfocus Tubes

Manufacturer= model Kevex=PXS4-613 Oxford=XTF5011 Oxford=XTF5000HP Oxford=XTF5000HP Kevex=PSX5-724 Hamamatsu=L673101 Oxford Ultrabright

Nominal Distance Nominal Maximum Maximum Maximum maximum anode to brilliance Be window power current focal size voltage (kW=mm2) (W) (mA) (mm) (kV) Anode (mm2) Mo Mo Mo W W W W

250–250 150–150 100–100 50–50 10–10 8–8 (15–40)2

60 50 50 50 70 80 90

1 1 1 1 0.1 0.1 2

60 50 50 50 7 8 80

0.96 1.2 5 20 70 125 128

15.8 17 17 17 12.2 12 1.6

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beam lines (having lower natural divergence and higher intensity), in addition to these, more expensive and elaborate microfocusing X-ray optics are in use also. These devices were reviewed by Ice (1997). Capillary optics is one of the fastest growing x-ray optical technologies because of its capacity of generating high-ﬂux-density beams in the micrometer and submicrometer range. Due to its broad-band characteristics, nonimaging properties, and simplicity, it is a very attractive optical device for microbeam XRF and x-ray scanning microscopy. 1. Different Capillary Types There are basically four major types of X-ray capillary concentrator available today: straight, conical, and ellipsoidal (or parabolic) monocapillary concentrators and monolithic polycapillary lenses. The principle of x-ray propagation inside these diﬀerent types of capillary shapes is schematically shown in Fig. 8. Straight capillaries can be used for ‘‘transporting’’ the x-ray intensity from close to a point source to the sample thereby eliminating the 1=r2 loss from isotropic point sources. Radiation propagating inside such a device will have a constant angle of incidence and these types of capillary can easily be manufactured and are easy to apply to almost any kind of source. Conical capillaries operate in a similar way as the straight tubes but in addition to ‘‘transporting’’ the radiation from the source to the sample, they also squeeze the x-ray beam down to the dimension of the capillary output diameter. Upon each reﬂection, the

Figure 8 The principle of x-ray propagation inside straight, conical, and ellipsoidal capillaries and in monolithic polycapillary x-ray lenses. In the ellipsoidal and polycapillary cases, a true focus is formed by rays emitted from the focal point of the device. Any ray emitted outside the optical axis will propagate inside the ellipsoidal by multiple reflections similar to the conical capillary.

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angle of incidence will increase by an amount 2g, where g is the capillary cone angle. Thus, photons propagating inside the cone will gradually be absorbed by the walls, because their angle of incidence will approach the critical angle. Thus, for conical capillaries the maximum angular divergence will always be equal to the critical angle (at least for energies above a certain threshold). In this sense, conical capillaries always tend to maximize the output intensity. For conical capillaries, the transmission spectrum is more complicated with a typical bandpass structure, as compared with straight tubes that just have a cutoﬀ energy. By varying the length and cone angle, it is possible to optimize the design of the capillaries for various energies within a wide range. One drawback with conical shapes is that the maximum intensity is reached at the capillary opening, and after that, the beam divergence is of the order of y. For borosilicate glass, yc (mrad) 30=E (keV). Thus, the sample has to be put close to the capillary opening if the high intensity is to be retained. For medium-sized capillaries (50–10 mm), this is normally not a problem; however, for very small capillaries or irregularly shaped samples, this can be a problem. If the capillary is too close to the sample surface, it might block some of the ﬂuorescent radiation on its way to the detector. A more recent development is the ellipsoidal capillary where the divergence can be reduced due to the imaging properties of ellipsoidals. In a perfect ellipsoidal, any ray coming from one of the focal points will eventually be focused in the other focal point, thus ‘‘small’’ objects placed in one of the focal points will generate an image in the other point. If an ellipsoid-shaped capillary is placed so that the x-ray source is positioned in one focal point, part of that source will be focussed in the other focal point. However, for any source point far away from the optical axis, the radiation will propagate through the ellipsoidal capillary much the same as in a conical capillary. If such a capillary has its opening just in front of its focal plane, the radiation coming out will consists of two components, one ‘‘nonimaging’’ divergent beam and one ‘‘imaging’’ convergent beam. The sum of the two components will somewhat balance each other up to the focal point. After that point, both of them will be divergent. This means that there exists a short distance, for the capillary opening to the focal point of the ellipsoidal, where the divergence of the beam will be substantially reduced. Monolithic polycapillary x-ray lenses consists of a very large number of straight channels (from a few thousands up to 250,000) which are ﬁrst bundled into monolithic strands; then, the thick multiﬁber is pulled in such a way that radiation originating from a focal source point S, which is located typically 2–5 cm from the lens, is transported by the channels, changed in direction, and refocused into a focal spot F at the other side of the lens. In contrast to monocapillaries, the marked advantage of polycapillary devices is that a large solid angle of the x-ray source is captured and that the radiation is focused into a point which is several centimeters from the X-ray lens itself, allowing this type of optics to be used for noncontact=nondestructive types of investigations. Early approaches (Kumakhov, 1986, 1990; Kumakhov and Komarov, 1990; Kardiawarman et al., 1995) to the manufacture of polycapillary lenses consisted in the use of individual straight capillaries which were bent and kept in place by a series of perforated metal disks; this resulted in fairly large devices [e.g., see Kumakhov (1990) for details]. Monolithic polycapillary lenses are more compact and consists of one or more bundles of closely packed straight capillaries that, after consolidation into a monolithic bundle, have been shaped in the desired form. As these lenses are the result of a fairly sophisticated production process, only a limited number of laboratories in the world are able to produce them (Ullrich et al., 1995; Kumakhov, 1998; Ding et al., 1998; Vekemans et al., 1998; Gao et al., 1998).

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2. X-ray Transport Inside Capillaries X-rays entering into a capillary under a very small glancing angle can propagate inside the device by successive reﬂections without losing almost any intensity. If the intensity of x-rays from such a capillary is compared with the intensity when no reﬂections would occur (which can be determined by replacing the capillary with a collimator with the same diameter as the ﬁnal capillary crosssection), the reﬂectivity of the capillary wall material can be studied and the intensity gain can be determined. For a straight capillary, the gain factor can be deﬁned as the ratio between the effective solid angle DOeff seen by the x-ray source (corresponding to the critical angle) and the solid DO when no reﬂections occur (defned by the exit end), assuming the reﬂectivity to be unity up to the critical angle (Ig ¼ DOeff =DOÞ, as shown in Figure 9. The gain factor shows the behavior of a low-pass ﬁlter. In Figure 10, Emin corresponds to the smallest angle of yc or the angle that corresponds to the entrance opening, and Emax corresponds to the exit end of the capillary. Note that the gain factor is proportional to 1=E2 in the range from Emin to Emax . When calculating the transmittance of x-rays through the capillary, one has to know the grazing angle of incidence and the number of reﬂections. For a straight capillary, the angle of incidence for each reﬂection and hence the reﬂectivity will be the same. This is not

Figure 9

The capillary gain factor (Ig) of a straight capillary is defined by the ratio of DOeff to DO.

Figure 10 The capillary gain factor versus x-ray energy for a straight capillary without any absorption.

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the case for a conical geometry, where the angle will increase for each reﬂection, and one has to calculate the reﬂectivity for each reﬂection. In the model presented in next subsection, a perfectly shaped capillary and an x-ray point source situated at the capillary symmetry axis are assumed. All yi and g are suﬃciently small to allow for the approximations: tan yi yI and tan g g. For a conical capillary geometry (as deﬁned in Fig. 11), the glancing angle of the ith reﬂection can be written as (Stern et al., 1988) yi ¼ y0 þ ð2i þ 1Þg

ð10Þ

From the geometrical constraints shown in Figure 11, one can easily deduce that the number of reﬂections N is given by N¼

y0 ðl þ l0 Þ 1 þ d1 2

ð11Þ

The total reﬂection coeﬃcient Rtot can be written as the product of each individual reﬂection coeﬃcient Rðyi ; EÞ at each successive angle of incidence yi encountered by the ray when passing through the capillary: Rtot ðy0 ; EÞ ¼

N Y

Rðyi ; EÞ

ð12Þ

i¼1

As long as the ﬁnal angle of incidence is below the critical angle, the x-rays propagate inside the capillary by successive reﬂections without losing almost any intensity. If the intensity of the x-rays from such a capillary is compared with the intensity where no reﬂections occur, which can be determined by replacing the capillary with an aperture positioned at the site corresponding to the far end of the capillary, the reﬂectivity of the capillary wall material can be determined. The intensity gain factor, which can also be deﬁned as the ratio of the intensity with and without reﬂection, can then be written as 1 Ig ¼ DO

Figure 11

Zymax Rtot ðy0 ; EÞ2p sin y0 dy0 þ 1 ymin

Conical capillary geometry.

ð13Þ

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where DO is the solid angle seen by the x-ray when no reﬂections occur, corresponding to the far end of the capillary. The limits of integration, ymin and ymax , are the minimum and the maximum glancing angles, respectively, for the x-rays inside the capillary as deﬁned by the geometric dimensions of the capillary and the x-ray source–capillary distance (l0 ). 3. Ray-Tracing Models Although explicit expressions can be obtained for calculating the reﬂection of radiation inside a capillary (Engstro¨m, 1991; Lechtenberg, 1994; Vincze, 1995c), they are only valid for perfectly shaped conical capillaries and can only be applied for point sources. Any model that would intend to describe ‘‘real’’ capillaries applied to ‘‘real’’ extended sources have to rely on some kind of Monte Carlo model where all the source and capillary parameters are used as input data. There are many diﬀerent factors that have an inﬂuence on the capillary performance and it is hard to quantitatively assess the signiﬁcance of each individual factor by interpretation of simple test experiments. This problem is augmented by the fact that the manufacture of capillary optics is still very much an art rather than a well-controlled technique, making it diﬃcult to produce series of capillaries where one parameter is systematically varied. The need to gain insight into the relative importance of the various above-mentioned factors has prompted the development of a detailed ray-tracing code (Vincze, 1995c) that is able to simulate the beam-forming properties of realistic capillary devices assuming various experimental conditions. A number of diﬀerent factors such as capillary material, surface roughness, deviations in shape that real devices assume in comparison to the ‘‘ideal’’ straight, conical, or ellipsoidal shapes, need to be properly described in such a simulation calculation. In addition to that, the size, divergence, and distance of the x-ray source and the energy distribution of the photons that enter the capillary also strongly inﬂuence the performance of the capillary beam and thus also have to be speciﬁed in detail. Such a full-scale ray tracing simulation model has been described by Vincze et al. (1995c). Because the capillaries in such a simulation are treated as truly three-dimensional devices having numerically deﬁned shapes, the only restriction on the modeled shape is the assumed circular cross section in the plane perpendicular to the capillary axis. This allows overall distortions of the capillary shape, such as bending of the capillary axis, to be included in the model. Also, the composition of the capillary material as well as the roughness of the reﬂecting surface can be freely chosen. The implemented surface roughness model describes both the attenuation of the specular reﬂectivity and the eﬀects of diﬀuse scattering by the rough surface. An additional feature of the program is that the possible transmission of photons through the capillary wall is also taken into account. In the case of a perfectly conical capillary coupled to a point x-ray source, the simulation is in perfect agreement with the analytical models, as is demonstrated in Figure 12. However, the ray-tracing models, is also able to simulate all kinds of capillary–source combination which cannot be simpliﬁed so that they may be described by simple analytical models. As we can see from Figure 12, the transmission eﬃciency versus energy will appear as a steplike function, where the ‘‘steps’’ represents the maximum number of reﬂections that can occur inside the capillary before the critical angle is reached. This distinct feature disappears for an extended source, as the initial inclination angle will vary over a wide range corresponding to the source size. The graphs in Figure 12 also show that the eﬃciency goes down for an extended source and the ‘‘energy of maximum transmission eﬃciency’’ will be shifted to lower energies.

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Figure 12 Comparison of the analytical model and the ray-tracing model as applied to a perfect conical-shaped capillary and a point source. The solid curve shows the transmission efficiency (out going flux=ingoing flux) calculated for a capillary 5 cm away from a point source with the dimensions l ¼ 7 cm, d0 ¼ 31 mm, and d1 ¼ 5 mm. The symbols correspond to ray-tracing calculations for a point and mm extended x-ray source (100 100 mm2 ).

The two-dimensional distribution of the outcoming photons can also be simulated by the ray-tracing code. For a monochromatic beam, the diﬀerent reﬂection orders will appear as clearly visible rings in the intensity distribution observed at a distance from the capillary opening (see Fig. 13, left panel). However, even here, the impact of an extended source will blur this feature (Fig. 13, center panel). Geometrical distortions or misalignment of the capillary will be manifested by the appearance of asymmetric rings or crescents in the angular distributions. (The intensity distribution shown in the right panel of Figure 13 was recorded at a signiﬁcant distance from the capillary tip.) Intensity distributions recorded— or calculated—close to the opening will normally show a rather homogeneous and sharp distribution corresponding to the zero-order radiation, imaging the actual cross section of the capillary itself. Thus, for elemental maps, recorded close to the opening, the spatial resolution can be improved by the application of deconvolution methods. The technique and possibilities for this enhanced resolution have been discussed by Attaelmanan (1995). 4. Practical Considerations According to most models of capillary optics, almost any gain factor could be obtained if there are no geometrical constraints on the capillary dimensions. However, in practice, the capillaries have to be designed within a narrow range of geometrical limits. In most cases, the source characteristics and the beam diameter is ﬁxed or dictated by the analytical

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Figure 13 Intensity distribution from a conical capillary 5 cm away from the opening. left; ray tracing calculation for a point source, center; ray tracing calculation for an extended source (400 400 mm2) and right; recorded image from a synchrotron set-up. The central spot represents the direct beam and the rings correspond to different reflection modes (first ring ¼ one reflection, second ring ¼ two reflections etc.).

requirements and, thus, length, shape and capillary material can be optimized. In some cases, the source–sample distance is determined by the mechanical arrangement so that only the shape (i.e., the capillary cone angle) can be varied. Figure 14 shows an example

Figure14 Calculated effective solid angle as a function of entrance diameter for different capillary lengths. The calculation was performed at an energy of 7 keV and a fixed output opening of 5 mm assuming a point source and a source–capillary distance of 5 cm. The surface roughness was characterized by assuming a slope error of 30 A˚ (see Vincze, 1995c). The result was also obtained for an ideal surface.

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Figure 15 The effective solid angle for various x-ray source sizes. The calculation was performed for a 7-cm-long conical capillary with 5-mm opening placed 5 cm away from the source.

of how the lengths and cone angle will aﬀect the ultimate gain factor at a given set of experimental parameters. These calculations were performed with the introduction of a realistic surface roughness model. The impact of surface roughness is indicated in Figure 15 by showing the gain factor with perfect reﬂecting inner walls. Although the gain factor will be dependent of the capillary length, the ﬂux density will be about the same for all three capillaries in Figure 15 and they will all have an optimum at an entrance opening of about 30 mm. If the distance between the source and capillary is substantially longer than the capillary itself, then the optimum will be reached for a certain cone angle for any capillary length, but the ﬂux density obtained will depend on how long one can make the capillary. If the source is extended, then the eﬀective solid angle in Figure 14 will decrease, and the maximum values will be reached at a slightly higher entrance diameter as the capillary can pick up more of the ‘‘oﬀ-axis’’ rays emitted from the source (see Fig. 15). 5. Capillary Alignment For (mono) capillary optics, it is necessary to have two linear and two rotational stages for a complete alignment procedure (see Fig. 16). Most alignment procedures are monitored by the output ﬂux (normally recorded by some simple device such as an ion chamber, a diode, etc.) and the operator has to ﬁnd the position for the maximum throughout. This is not a straightforward task, as there always will be ‘‘secondary maxima’’ when operating the four diﬀerent stages. The ‘‘secondary maximum’’ problem can be avoided by combining the linear and rotational stages in such a way that rotational axes are intercepting the x-ray

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Figure 16 Photograph of the COX Analytical Instrument’s capillary holder and gimbal mechanics. The capillary rest in a highly viscous liquid that has a high cross section for x-ray absorption. The ‘‘semiliquid’’ efficiently damps out microvibration while preventing any x-rays from passing the holder outside the capillary. The capillary and the liquid are encapsulated into a stainless-steel tube which is put into the gimbal.

source. However, a simple systematic alignment procedure normally works well and realignment is a matter of a few minutes if there is some measurable ﬂux being transmitted through the capillary. If the beam is completely lost, realignment will take a longer time. In this case, prealignment of the optics holder with a wider capillary can be a solution. In general, the alignment of polycapillary lenses is easier and less critical than that of monocapillary tubes; this is due the larger acceptance angle of polycapillary lenses. Normally, only a stage having two transational stages is suﬃcient; in some cases, alignment of the lens can even be achieved by using plastic spacers of various thicknesses to appropriately position the lens relative to the anode focal point of the tube (Worley et al., 1999).

C.

Sample Movement and Visualization Equipment

Although x-rays can be focused into microbeams, they cannot be controlled in the same way as charged-particle beams. Thus, scanning has normally to be promoted by the movement of the sample rather than of the beam. For monocapillary focused x-ray beams, the maximum intensity is achieved at the outlet of the capillary; thus, in this case, the sample has also to be positioned very close to the capillary. The precise positioning of the sample is normally achieved by a sample holder which consists of two or three linear stages—and sometimes also a rotational stage—driven by computer controlled dc or step motors. The motor position is usually monitored by some kind of encoder. The minimum step size should be well below the minimum lateral resolution of the spectrometer as deﬁned by the beam size. The sample position can be monitored by an optical microscope with a camera or some simple lens system attached to a charge-coupled device (CCD) camera. The optical features and quality of the monitoring system will depend on the resolution required. In some setups, the camera is placed in a ﬁxed position aimed at a ﬁxed sample site. This is possible with a pinhole or polycapillary beam or any beam where the focal plane is far away from any bulky optical equipment. However, the sample has to be turned (away from the plane perpendicular to the beam) in order to coincide with the focal plane of the

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camera. In this position, the sample position is monitored directly by the camera. A drawback is that the camera has to operate with fairly long-focal-length optics (> 2 cm) that will limit the magniﬁcation; sometimes, the illumination can be diﬃcult to arrange in a proper way. For monocapillary focused beams, this direct inspection is often not feasible, as the capillary is shadowing a large part of the sample due to the short distance between capillary opening and sample surface. In these cases, the camera (or microscope) is placed a bit away from the measuring position and the sample has to be brought into the camera’s focal plane by the sample stage (see Fig. 17). By combining linear and rotational movements, the camera can be placed in almost any convenient position and operated with any range of focal length and magniﬁcation. In this way, the sample inclination in the measuring position is independent of the camera position. However, there will always be a time delay while moving the sample between the measuring position and the camera position; this may require a large-range linear stage and possibly an extra rotational stage (see Fig. 18).

Figure 17 Photograph of the microbeam XRF setup at Chalmers University of Technology in Go¨teborg. The picture shows a closeup of the capillary and sample stage with a rotational stage on the top. The optical microscope is used for positioning the sample in the beam (Photo courtesy of A. R.).

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Figure 18 Schematic drawing of the information flow in the microbeam XRF setup at Chalmers University of Technology. Here, the microscope is looking right into the capillary opening; thus, the sample has to be rotated 180 in order to see the front side. (From Engstro¨m, 1991).

For the ‘‘direct inspection’’ camera setup, it is necessary to have at least one (probably two) mechanical linear stage to adjust the camera focal plane so that it will coincide with the sample surface plane. For the ‘‘indirect inspection’’ system, the camera can be ﬁxed to the table, but it will be necessary to recalibrate the exact distance (in motor steps) between the central point in the camera focal plane and the location of the beam at the measuring position. The camera (or microscope) can be equipped with diﬀerent types of magniﬁcation lens, and optical micrographs from the sample are usually recorded through a framegrabber to a video screen.

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Detectors

Most microbeam XRF tabletop systems are equipped with a conventional solid-state x-ray detector such as Si(Li) or HP (high purity) Ge devices. For monocapillary-based microbeam XRF, it is important to have a large detector solid angle, as the primary photon ﬂux normally is very slow, especially for applications in trace element mapping. Thus, the detector area needs to be large and the detector crystal should be close to the entrance window (Fukumoto et al., 1999). It is important that the end-cap design is such that the detector entrance window can come close to the sample. Normally, the count-rate capability is not critical; however, for fast mapping of mineral samples (or any material with relatively high concentrations of heavy elements such as metal alloy samples), the count rate can induce variations in the electronic dead time, generating a distortion in the elemental maps. Deviations from the linear relation between the count rate of characteristic radiation and the image gray-scale level will occur in areas where local concentrations are high. This problem can be avoided by using the lifetime settings of the MCA (multichannel analyzer), but only when ‘‘static’’ scanning is used (see Sec. IV.A). With the introduction of the solid-state drift-chamber x-ray detectors (Bertucci, 1992; Lechter et al., 1996; Stru¨der et al., 1998), the linear range can be extended for count rates beyond 20,000 counts without any serious deviation and still with reasonable energy resolution and low dead time. Other types of detectors that can be integrated in the systems are x-ray p-i-n diodes used for monitoring the primary intensity and for recording the transmitted intensity. In this way, microradiographic images of the sample can be recorded and the sample thickness or density can be determined with high accuracy at each image pixel (Attaelmanan et al., 1994). A large-area p-i-n diode is also useful for alignment of capillary optics when the operator has to ﬁnd the transmission maximum. Standard CCD detectors have also been used as beam monitors, especially for studying the proﬁle of the primary beam (Attaelmanan et al., 1993). This is important for precise and accurate alignment of many types of x-ray optics. It is also necessary to have detailed information of the beam proﬁle in order to perform any kind of image deconvolution.

E.

Analytical Characteristics of l-XRF Setups

1. Monocapillary Setups a.

Estimated Beam Flux and Sensitivity

The sensitivity from a tabletop m-XRF instrument can be estimated from the ﬂux density, beam size, and detector eﬃciency. The ﬂux density is just a function of the source brilliance and the eﬃciency of the optical device used. The brilliance of characteristic radiation from a Cu-anode tube can be estimated from the Metchnik and Tomlin formula (1963). Assuming a voltage of 45 kV and a spot size of 100 100 mm2, the brilliance would be about 3.7 1012 Ka photons=s=mA=sr. Assuming that the anode is a large solid piece of metal cooled by water, the maximum load would be about 100 W before the melting temperature is reached. Thus, it is reasonable to assume that such a tube could operate up to 1 mA (less than half of the maximum current). The expected ﬂux density from a 7-cm capillary with a 5-mm opening, at 5 cm away from the focal spot, can be calculated to be 13.7 105 photons=s=mA, corresponding to about 2 104 Ka photons=s=mm2. This is approximately the number being reported for some of the monocapillary based tabletop instruments, although achieved with very different types of x-ray source (see Attaelmanan et al., 1994 and Carpenter et al., 1995b).

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Carpenter et al. (1995b) reported an absolute DL below 1013 g (corresponding to the ppm range) from a 4-mm-capillary focused Cu beam with a ﬂux density of 2 104 photons=s=mm2. Similar values were reported by Larsson and Engstro¨m (1992). b. Spectral Distribution Because the critical angle for total reﬂection is smaller for lower energies inside a (nonstraight) capillary, for higher-energy photons, there is a greater chance for absorption than for x-rays of lower energy (see Fig. 10). The wavelength dependence of the reﬂectivity decrease due to surface roughness even increases this trend (Vincze et al., 1995c). This energy-dependent transmission eﬃciency causes capillaries to act as highenergy ﬁltering devices which can introduce non-negligible changes in the polychromatic tube spectrum. Figure 19 illustrates this principle: Two EDXRF spectra obtained by irradiating NIST SRM 1833 using microbeams obtained by means of a 200 200mm2 pinhole and by using a conical capillary with approximate conical shape and a 40mm end diameter are plotted normalized to the Ti peak intensity. The high-energy eﬀect in the capillary spectrum can clearly be observed, and at the excitation energy (17.5 keV), it results in a decrease in relative intensity by about a factor of 4 relative to 4.5 keV (TiKa). It is obvious that during quantitative calculations based on fundamental parameters these changes in the excitation spectrum need to be taken into account. c.

Elemental Yields and Detection Limits

In Figure 20a, the elemental yields obtained by irradiating NIST SRM 1832 and 1833 (thin-glass standards) using Mo-anode excitation in the m-XRF setup at the University of

Figure 19 EDXRF spectra of NIST SRM 1833 irradiated with Mo-anode-derived x-ray microbeams.

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Figure 20 (a) Thin target elemental yields (in counts=s=(mg=cm2) and (b) absolute DLs (in pg) obtained for thin samples obtained by means of a conical capillary (15 mm effective beam diameter) and several polycapillary lenses (160 mm beam diameter) using a Mo rotating anode tube m-XRF setup and at the m-SRXRF station at NSLS beam line X26A; (c) relative DL (normalized to 1000 s) in mg=g derived from irradiation of a 100 mg=cm2 NIST SRM 1577a bovine liver sample. The absolute DL values for the synchrotron setup (not shown in Fig. 20b) situate themselves in the femtogram range.

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Continued

Antwerp (Janssens et al., 1996), employing a 10-mm tapered capillary (15-mm eﬀective beam diameter at the sample surface) are compared to values obtained for the NSLS beam line X26A m-SRXRF spectrometer (8-mm beam diameter) (Jones and Gordon, 1989). For the laboratory setup, net yield values which are lower by a factor of 30–50 than those obtained for the synchrotron setup. The absolute detection limits derived from the same spectra (Fig. 20b) indicate that a comparable absolute detectability to that obtained with other laboratory-scale m-XRF setups (Carpenter et al., 1995b) is obtained, situated at the 0.1–1.0-pg level. For thick samples, rather than the absolute DL values, the peak-tobackground ratios (P=Bs) found in experimental spectra determine, to a large extent, the lowest measureable concentration levels. Figure 3 compares the EDXRF spectrum of NIST SRM 1577a bovine liver obtained using the above-mentioned monocapillary setup to the spectral distribution obtained by using a polychromatic synchrotron (micro)beam of comparable cross section at NSLS (Jones and Gordon, 1989) on the same sample. In Figure 20c, the corresponding relative DLs corresponding to a 1000-s counting time are plotted. Clearly, in the spectrum of Figure 3a (polychromatic synchrotron excitation), a much better peakto-background ratio is obtained: overall, the laboratory m-XRF spectrum features P=B ratios which are about 10–30 times worse (Fig. 3b). The abundant scatter background in the region of 10–16 keV makes fairly long counting times (>1000 s) necessary in order to obtain information on elements present at or below the 10-ppm level (e.g., Br: 18 ppm). As can be seen in Figure 20c, for the elements Fe to Rb, DL values are situated in the 3–10 ppm range for the lab-scale spectrometer, whereas they are about an order of magnitude better in the case of polychromatic synchrotron excitation. One can conclude that by means of the laboratory m-XRF instrument involved, trace-level

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microanalysis at the 10-ppm=10-mm level is possible, provided that suﬃciently long counting times are employed.

2. Polycapillary Setups In contrast to monocapillary concentrators, polycapillary lenses have the ability to actually collect divergent radiation in a fairly large solid angle and focus it onto a point which is located several centimeters away from the other end of the lens (see Fig. 8). a. Beam Size and Divergence In Figure 21, the beam size by vertically scanning a 10-mm Cr wire through the beam produced by two polycapillary lenses is plotted for diﬀerent values of the lens-to-wire distance d. Both lenses had an input and output widths of 5.5 mm and lengths in the range of 50–64 mm. The distance of the lenses to the Mo anode of a minifocus tube was 4 cm so that they collect radiation in a solid angle of 0.015 sr. The lenses consist of 250,000 individual monocapillary channels, each with an internal diameter of 7 mm (35% open area) (Vekemans et al., 1998). Lens A yields a focal spot with a diameter of 60 mm at a distance d of 15.7 mm from the end of the lens; in the case of lens B, a focus of around 120 mm is reached at d 42 mm.

Figure 21 Beam FWHM obtained by perpendicular scanning of a 10-mm Cr wire through the beams produced by two polycapillary lenses of different output lengths. Lens A has an output focal length of 15 mm and focuses the radiation into a spot of 60 mm FWHM; lens B has an output focal length of 42 mm and produces a focal spot of around 120 mm in diameter.

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The beams produced by the polycapillary lenses are strongly convergent before and strongly divergent behind the focal plane. If we consider that the Cr atoms in the Cr–Fe–Ni wire will preferentially be excited by radiation at or just above the Cr K edge at 5.998 keV (say, with an average energy of 7 keV) and that this radiation leaves the individual glass tubes (with 7 mm diameter) on average with an angle y 1.6 mrad, it follows that at an average distance of 42 mm, the 7-mm beam will have spread to a beam of 7 mm þ 2 42 mm 1.6 103 ¼ 141 mm wide, a value which is quite consistent with the experimental beam size of 120 10 mm found by Cr-wire scanning at d ¼ 40–43 mm. Similarly, at d ¼ 15 mm, a divergence of 1.6 mrad corresponds to a focus of 7 mm þ 2 15 mm 1.6 103 ¼ 55 mm wide. It therefore appears that the errors in the orientation of the individual capillaries probably contribute only in a minor way to the overall spot size and that primarily the natural beam spreading of the individual cone beams deﬁnes the beam spot. One of the smallest polycapillary beam sizes reported thus far (2000) was 21 mm (Gao et al., 1998) and was obtained by using a combination of a 12-W microfocus tube (Mo anode) and a lens with 8.4 mm output focal distance. b.

High-Energy Transparency

Figure 22a shows scan proﬁles obtained by scanning a 10-mm bronze (Cu=Sn) wire through the beam produced by polycapillary B (Fig. 21). These data illustrate the transparency of the polycapillary lenses for the higher-energy radiation. Clearly, in both cases, a focused beam 200 mm wide is present (Fig. 22a), but at high energy (which excites the SnK level), this focused beam is surrounded by a halo several millimeters in diameter. The abundance of the halo is tube-voltage dependent (see Pd wire data in Fig. 22b) and becomes very important when tube voltages above 40 kV are employed. For practical purposes, it is therefore recommended to use the lenses at lower tube voltages and to employ them for local analysis of elements with ﬂuorescent lines situated in the 5–20-keV range. c.

Lens Transmission and Gain Factors

The gain factor Ig of a lens can be deﬁned as the increase in beam ﬂux density as a result of using the lens (index L) instead of an aperture (index A) of comparable diameter as the focal spot produced by the lens: Ig ¼

FL =dL2 lL2 FA =dA2 lA2

where FL and FA are the measured beam ﬂuxes at a distance lL and lA , respectively, and dL and dA are the beam FWHM produced by both devices. Because sometimes slightly diﬀerent deﬁnitions for the gain factor are used in the literature, is it diﬃcult to directly compare the individually reported values; however, under optimal conditions, values of a few hundred to several thousand have been measured (Gao et al., 1998). When due to external constraints, the x-ray source or lens characteristics are less than optimal, gain factors between 10 and 100 are obtained. An example of such situation was described by Worley et al. (1999), where a polycapillary lens was designed to ﬁt into an existing m-XRF spectrometer. The fact that the lens was only 29 mm long caused the gain factors to vary between 124 (at 4 keV) and 2.5 (at 16 keV). When using a rotating anode as the x-ray source (where the distance between anode spot and lens is of the order of 90 mm due to the size of the anode tower), gain factors of 7–20 were observed (Vekemans et al., 1998), despite the relatively high brilliance of the source (see Table 1). In a study to ﬁnd the

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Figure 22 Fluorescent intensity profiles derived from a lens by perpendicular scanning of (a) a 20mm Cu=Sn wire through the beam at 50 kV tube operating voltage and (b) a 250-mm Pd wire at various tube operating voltages in the range 30–50 kV. The importance of the transmission halo increases with tube voltage.

optimal match between various commercially available x-ray sources and polycapillary lenses, Bichlmeier et al. (2001) obtained gain factors between 30 and 300 for diﬀerent lenses coupled to a Mo-anode minifocus tube. d.

Analytical Characteristics

In Figure 20a, the elemental yields [in counts=s=(mg=cm2)] obtained by irradiating NIST SRM 1832 and 1833 (thin-glass standards) using Mo excitation at 40 kV excitation and three polycapillary lenses are compared to values obtained for the NSLS beam line X26A m-SRXRF spectrometer (8 mm beam diameter) and to those obtained by means of the monocapillary concentrator. For the polycapillary lenses, net yield

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values which are a factor of 3–5 lower are obtained than for the synchrotron setup. In the setup employed, the lenses yielded a beam size of 160 mm. Considering a ratio of (160=8)2 ¼ 400 in spot sizes, this indicates that by means of polycapillary lenses, a ﬂux density is obtained which is 1000–2000 lower than at the synchrotron beam line. The absolute detection limits derived from the same spectra (Fig. 20b) are situated in the 1–10-pg range, whereas the corresponding values for the monocapillary are one order of magnitude better. Similar values were found by Bichlmeier et al. (2001), whereas Gao et al. (1998) and Ding et al. (1998) reported absolute DL values in the 0.1–10-pg level. The relative detection limits (Fig. 20c) derived by irradiation of NIST SRM 1577a bovine liver sample by means of the three lenses are situated at the 2–3ppm level (i.e., intermediate between the monocapillary characteristics and those of the NSLS X-26A beam line). In a glass matrix (NIST SRM610), Bichlmeier et al. (2001) reported relative DL values in the 10–100-ppm range for the elements Ca to Rb. It can be concluded from these measurements that the polycapillary lenses are able to produce highly intense beams with moderate ﬂux density and relatively small size, permitting the determination of minor and trace constituents with a lateral resolution in the range 20–150 mm.

F.

Commercial Equipment

At present, four commercial companies market microbeam XRF instruments. KevexSpectrace Inc. (Sunnyvale, CA) has been oﬀering the Omicron instrument since 1986 (Nichols and Ryan, 1986; Yamamoto and Hosokawa, 1988), based on a 50-W minifocus tube (250 250-mm spot on a Rh anode) and 10-, 30-, and 100-mm apertures. The apertures are about 6 cm away from the anode spot and 3 mm from the sample; ﬂuorescent radiation is collected with a 50-mm2 Si(Li) detector. Such a device can also be retroﬁtted with a polycapillary lens (Worley et al., 1999). Hosokawa et al. (1997) of Horiba Ltd. (Kyoto, Japan) developed a scanning x-ray analytical microscope (the XGT 2000 instrument) based on capillaries with paraboloidal inner walls to focus radiation produced in a compact Rh-anode x-ray tube with a 200 200-mm2 spot size. A highpurity Si(Li) detector collected ﬂuorescent radiation. The capillary and detector nose are situated in a small vacuum chamber sealed with a thin resin foil so that the airpath that primary and ﬂuorescent x-rays need to pass on their way to and from the sample is less than 1 mm. Beams of 10 and 100 mm can be used in this device. The XGT 2000 is distributed in the United States and Europe by Oxford Instruments. COX Analytical Instruments is a Swedish company (Go¨teborg, Sweden) that is marketing the monocapillary-based ITRAX instrument oﬀering m-beam XRF capacity down 5 mm; the instrument is equipped with a high-ﬂux x-ray tube (Attaelmanan et al., 1995). COX Analytical Instruments also provides modular units for converting existing XRF equipment into microbeam spectrometers. EDAX (Tilburg, The Netherlands) has introduced the EAGLE m-probe device; this m-XRF instrument device uses a microfocus x-ray tube and a polycapillary focusing unit to generate a 30-mm x-ray beam. All commercial equipment includes a motorized scanning stage and optical microscopy system for sample viewing (Haschke et al., 1998). More speciﬁcally intended for thickness gauging and (other) applications in the microelectronics industry, Veeco Inc. (Plainview, NY) oﬀers several microscopic and small-spot XRF systems, the smallest beam size available being 50 mm. Seiko Instruments (Torrance, CA) also oﬀers ‘‘small-spot’’ XRF equipment for use in the microelectronics industry.

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COLLECTION AND PROCESSING OF l-XRF DATA

There are three diﬀerent types of measuring mode for the microbeam XRF instrument: point analysis, line scanning, and area scanning. The scanning procedure is promoted by a regular movement of the sample in the beam with a simultaneous recording of the intensities of diﬀerent characteristic x-ray energies by the x-ray detector. The operator normally controls the sample position by manually inducing precise movements of the diﬀerent sample stage motors while the camera is viewing the sample surface. When the desired sample position is found, the operator can normally select a point, line, or an area of interest. During line and area scanning, the important choices for the operator are the step size and exposure time. These parameters are interdependent and the choice is related to the type of analytical data required and the time available for the analysis. Normally, the step size is set equal to the eﬀective diameter of the beam and the exposure time is selected from the expected concentration of the elements being analyzed. However, in some cases, it might be wise to operate with smaller step sizes (increasing the number of pixels) and decreasing exposure times. Attaelmanan et al. (1994) described a scanning system where the step size is always set equal to the pixel size of the optical image from the monitoring microscope. The ﬂuorescent data can then be superimposed directly on the micrograph image (Rindby et al., 1992). This allowed the operator to check for correlations between the distribution of certain elements and well-deﬁned structures within the specimen. As the amount of information obtained by the scan is entirely determined by the time of analysis, no information is lost by scanning in steps smaller than the beam size. The small step size might also allow for an increased image resolution by applying deconvolution techniques if precise knowledge of the beam proﬁles can be implemented into the algorithms (Zahrt, 1989). The simplest way of implementing coordinated sample movement and spectrum collection is to employ a motorized stage and detector electronics, which are both controlled via PC plug-in cards or the equivalent. From within a scanning program, synchronized motor movement and data collection and storage commands can then be issued. A.

Static and Dynamic Scanning

When all necessary scan parameters are given, the scan starts by moving the sample into position. The movement scheme should be such that the maximum resolution of the motors is utilized and backlash problems are avoided. The software system can either move the samples in steps corresponding to the selected step size and wait for the spectrum recording, readout, and so forth (static scanning), or move with a predeﬁned constant velocity along a line while spectrum collection and readout are done ‘‘on the ﬂy’’ (dynamic scanning). The former method will always assure that the spectrum is recorded at the correct position and with the correct exposure time. However, it will generate a lot of dead time in the system as the sample movement and all other computer overhead time will occur when the MCA is closed. For a short exposure time, the dead time and computer overhead time can be up to 50% or more. With the latter method, no dead time is generated because the MCA is open (almost) all the time; however, there is no guarantee that the recording–readout process will always take the same time. Internal clock systems must be used to compensate for variations in the exposure time; several authors have described such systems (Carpenter et al., 1995b; Janssens et al., 1996a). By performing a number of such line scans in sequence, for example, a 50 50-pixel image (called a frame) can thus be recorded within, for example,

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Figure 23 Dynamically collected x-ray images from polished piece of Roman glass sample using two, four, and ten 0.25s=pixel frames and by static image collection (60s=pixel dwell time).

50 50 0.25 ¼ 625 s (i.e., in about 10 min). If after this period of time, (some of) the obtained images are too noisy, one or more additional frame scans can be performed and the collected x-ray distribution can be added to the one collected during the previous frame(s) until images of suﬃcient contrast and clarity are obtained. Usually, the number of required frames is diﬀerent for each elemental signal; as an illustration, Figure 23 shows dynamically collected x-ray images using two, four, and ten 0.25-s=pixel frames obtained from a sectioned piece of corroded Roman glass (see the electron micrograph shown in Fig. 24a). Also shown are the corresponding images obtained using the ‘‘static’’ mode of collection (60 s dwell time per pixel) to highlight the fact the both modes of operation in the end yield essentially the same information. Note that the images of Figure 23 result from integration of the spectral intensity within various windows positioned around characteristic peaks and that, therefore, no spectral background correction was performed; in Figure 24b, the complete set of

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Figure 24 (a) Backscatter electron micrograph of a polished section of Roman glass showing a cut through a hemispherical corrosion growth. The complex morphology of the corrosion layer is visible. The embedding resin and the original glass are visible in respectively the left bottom and right upper corners. Scale bar length is 100 mm. (b) Full set of background-corrected m-XRF images (see also Fig. 25).

(background corrected) elemental maps is shown. These maps are obtained by recording the intensity from characteristic lines in the spectra for each pixel and storing these values as two-dimensional image ﬁles. Thus, these images ﬁles can later be subjected to any kind of image processing (see the next subsection). The recording can be done directly by reading the counts from operator predeﬁned ROIs (regions of interest) in the spectra or by storing a complete spectrum from every pixel and doing peak-area evaluation and elemental map reconstruction in a second phase.

B.

Spectrum Processing

The data-reduction procedures associated with m-XRF spectrometry share a number of features with those for m-PIXE data. In contrast to their bulk equivalents, scanning m-PIXE and m-XRF are able to rapidly produce massive amounts of raw data: The collection of a 64 64-pixel x-ray image may involve, for example, the collection of 8 Mbyte of raw data and the processing of 4096 separate x-ray spectra. Although, in principle, the same spectrum evaluation procedures that are used for the bulk technique data can be

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Figure 24

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Continued

employed, the large size of these microscopic data sets makes the adaptation of these procedures mandatory so that they can operate in a reliable way with minimal user intervention on large numbers of spectra. In m-PIXE setups, which frequently employ list mode data acquisition strategies, x-ray line scans and x-ray maps for a particular element can be obtained by sorting the recorded events using an appropriate energy window, centered around the peak energy of the element in question. A similar procedure is also often employed during on- or oﬀ-line processing of m-XRF data. This simple procedure is very useful for explorative data analysis but implicitly assumes that within the energy window used, a single, nonoverlapped peak is present with a high peak-to-background ratio so that the integrated intensity within the window is a good estimate of the net intensity of the peak. Unfortunately, both for m-PIXE and m-XRF spectra, in general, these assumptions are not valid: peak overlap frequently occurs in energy-dispersive x-ray spectra, whereas, especially for peaks corresponding to trace constituents, the background intensity below the peak may be of the same order or larger than the net peak intensity. In these cases, the use of too simple spectrum evaluation procedures may negate all the eﬀorts that are made

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both during the data collection and during the further quantitative processing of the data to increase the reliability of the ﬁnal (trace) element concentration proﬁles or images. As an example, in Figure 25, the result of processing in various ways the elemental maps derived from the Roman glass sample of Figure 24 are shown. These images were obtained respectively by plotting the integrated ROI intensities (Fig. 25a), by using a simple ‘‘straight-line’’ background compensation interpolation approach between the ROI boundaries (Fig. 25b), and by using net peak intensities obtained via least-squares ﬁtting of the individual spectra (Fig. 25c) (Vekemans et al., 1994; Van Espen et al., 1986; Vekemans et al., 1995). See Chapter 4 for a detailed discussion on spectrum evaluation and nonlinear least-squares ﬁtting. During the latter processing, the evaluation of the spectra is largely unsupervised: the ﬁtting model (containing a list of photopeaks to be ﬁtted and a speciﬁcation of the background modeling function to be used) is made a priori using a few spectra from the set or by means of the sum spectrum. As a background compensation algorithm, the use of a robust procedure is recommended when processing is unsupervised. In this case, the background of each spectrum is described by linear combination of mutually orthogonal polynomials, typically of the order of 10–15 (Vekemans et al., 1994, 1995). This algorithm permits the calculation of the appropriate background shape of the spectra while this shape changes throughout the series of spectra (e.g., for spectra collected from diﬀerent phases in the scanned area) and also results in correct background values in case spectra with much noise are processed. In Figure 24b, the full set of x-ray maps obtained via method c is shown. The most striking diﬀerence between the results of method a (no background compensation) and methods b and c is that the high scatter background contribution in the Ni and Br ROIs is eliminated. For images with much contrast between phases (see, e.g., the Mn map), the three methods yield quasi-identical results. Method c resolves the overlap between the KKb and CaKa signals, which results in a more detailed Ca map. Similarly, instead of being a replica of the Mn distribution, the Fe-image appears quite diﬀerent in reality after proper processing. In the case of Ni, the use of the ﬁtting procedure results in a less noisy image than when the straight-line background (method b) is used for background correction. Finally, from the Br image, one can conclude that in the corroded area, in general, more Br is present than in the original glass but that the noise level does not permit to discern details. It can therefore be concluded that for m-XRF data, in addition to resolving peak overlap and correct background compensation in the neighborhood of intense matrix lines, especially the ability of the (unsupervised) software to do reliable background subtraction below the x-ray peaks of the heavier trace elements (atomic number >26) is valuable. Using current-day workstations or personal computers, the computational eﬀort associated with the nonlinear least-squares evaluation does not represent a large fraction of the total time required for acquisition and processing of the x-ray image data. As described by Janssens et al. (1996b), by proper synchronization of diﬀerent tasks, the evaluation of a spectrum corresponding to a particular pixel may be accomplished on-line while the spectrum of the next pixel is being collected. C.

Image Interpretation and Segmentation

1. Color Encoding For visualizing the distribution of major components in a specimen, the intensity of ﬂuorescent radiation is suﬃciently high for generating elemental maps with large dynamics within reasonable time. Structures within such an elemental map can normally be easily

Figure 25 Micro-XRF images obtained from a polished piece of corroded Roman glass shown in Figure 24. (a) Integrated ROI maps, (b) ROI maps after ‘‘straight-line’’ background subtraction, (c) net intensity maps obtained via least-squares fitting of individual EDXRF spectra.

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Figure 26 Image reconstructions from different trace element distributions derived from ballpoint ink on paper. The figure shows the reconstructed text ‘‘24’’ that has been overwritten by another ink quality. The reconstructions were done at the National Laboratory of Forensic Science in Linko¨ping, Sweden. (From Stocklassa and Nilsson, 1993.)

identiﬁed. However, for trace element distribution, the intensity is poor and only individual counts are recorded in each pixel, which makes it more diﬃcult to identify any structure in the sample. This is why it is so important to be able to correlate trace element maps directly to the optical micrograph image. For trace element distributions in general, the question is normally ‘‘where’’ rather than ‘‘how much.’’ Figure 26 shows a typical example of the identiﬁcation of a structure from the distribution of trace elements (Stocklassa and Nilsson, 1993). For many of these maps, it is diﬃcult to identify any recognizable structure, but if you know what to look for (‘‘24’’), then most of the maps show a signiﬁcant increase in trace element concentration for that particular structure. Although the contrast from trace mapping can be very poor, it can normally be enhanced by a standard type of image processing technique, such as dynamics optimization, gray-level histogram equalization, and so forth. With good pixel statistics, deconvolution or simple smoothing techniques can be used in order to achieve higher lateral resolution or to utilize the screen dynamics of a color screen in a more eﬃcient way. For example, by utilizing the full dynamic of a color screen, three diﬀerent elemental maps can be superimposed at the same time using the RGB channels as separate gray-scale images, as shown in Figure 27. In this way, correlations between diﬀerent elements can be studied in a convenient way; for noncorrelating elements, however, the color mixing will complicate the interpretation. 2. Automated Image Segmentation A m-XRF image scan generates an nchan ncol nrow data cube (nchan is the number of channels in the spectrum collected per image pixel; nrow and ncol are number of rows and columns in the image). Evaluation (‘‘ﬁtting’’) of this series of ncol nrow spectra (Vekemans

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Figure 27 A comparison between the scanning electron backscattered micrograph of a sample of malachite ore (left) and a color-combined microfluorescence image (right). (From Stocklassa and Nilsson 1993.)

et al., 1994) allows one to reduce this dataset (typically 1024 50 50 datapoints) to dimensions nel ncol nrow , where nel is the number of elemental intensities derived from each spectrum; nel typically has values in the range 5–20. In either case, the data cube can be considered as a multivariate dataset, consisting of a number of objects (the pixels in the image), each characterized by a number of properties (either nchan channel contents or a vector of nel x-ray intensities). Because during image scans, usually fairly short collection times are employed per pixel (typically in the range 1–100 s), the photopeaks in individual spectra are often buried in the background noise. This results in noisy images for the corresponding elements (see Figs. 24–26). For performing quantitative analysis with a precision of the same order as the accuracy oﬀered by the calibration model employed, usually a better signal-to-noise ratio is required. The heterogeneous materials that are investigated in many cases are built up out of a limited number of (quasi)homogeneous phases (Carpenter, 1997; Vekemans et al., 1997; Rindby, 1993). Thus, m-XRF elemental maps usually can be segmented into a limited number of regions, in which all pixels provided equal or similar compositional information, or in the multivariate sense, they display a similar pattern of properties. After segmentation of m-XRF images, it becomes possible to calculate the sum spectrum corresponding to each region (by simply adding all spectra corresponding to each of the pixels inside a region) having a much better signal-to-noise level than the spectra corresponding to individual pixels. A similar approach is also employed in m-PIXE analysis (Ryan et al., 1988; Svietlicki et al., 1993). Segmentation of individual images can be done in various ways, by using, for example, edge-enhancement ﬁlters or, on the basis of their corresponding, gray-level histograms (Russ, 1995; Bonnet, 1995a). In the case of multivariate datasets (produced by m-XRF or a similar technique), it is better to employ the information in all images simultaneously during the segmentation process (Bonnet, 1995b; Geladi, 1995). This can be done by employing appropriate mathematical procedures for (1) eliminating redundancies, (2) distinguishing the signiﬁcant information from random noise, and (3) splitting the information in mutually orthogonal (i.e., noncorrelated) components. Techniques for doing this have been reviewed by Bonnet (1995a, 1995b) and illustrated with datasets

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taken from electron probe microanalysis (Paque et al., 1990; Bright et al., 1988; Bright, 1995), auger electron spectrometry (Prutton et al., 1990), electron energy loss spectroscopy (EELS) mapping (Jeanguillaume, 1985), and secondary ion microscopy (Van Espen et al., 1992). For the speciﬁc case of m-XRF datasets, Cross et al. (1992) have described the use of principal-component analysis (PCA) (see Geladi and Kowalski, 1986; Johnson and Wickern, 1988; Press et al., 1992; Trebbia et al., 1995 for details) for colinearity removal and dimensionality reduction. By manually grouping pixels in the space of the resulting principal components [a procedure called interactive correlation partitioning (ICP); see Paque et al., 1990], semiautomated or supervised image segmentation was shown to be feasible for datasets in which a limited number (three) of principal components were present. A procedure for automated correlation partitioning (ACP) was proposed by Vekemans et al. (1997); this method can also be applied to datasets in which more than three principal components are present so that it is possible to automatically extract a small number of region-speciﬁc sum spectra from a m-XRF image dataset. Instead of attempting to reduce the dimensionality of the ncol nrow nel dataset by looking for collinearity (or covariance) between the various elemental signals within the images (as is done in PCA), in order to perform automated image segmentation, all (ncol nrow ) pixels can be considered as objects characterized by a vector of nel properties (a multivariate ‘‘ﬁngerprint’’) and by using an algorithm such as K-means clustering (KMC) (Bonnet, 1995a; Vekemans et al., 1997) to generate a limited number of pixel groups having similar ﬁngerprints. A more sophisticated approach is to use a combination of PCA and K-means pixel clustering: In a ﬁrst step, the eigenvalues and principal-component images (or eigenimages) of the original dataset are calculated using PCA; this serves to separate the meaningful structure in the data from the noise. Second, a limited number of these (noise-free) eigenimages are used as input to the KMC algorithm, allowing the original elemental maps to be segmented. A concise overview of the mathematical background of the PCA and KMC procedures can be found in Vekemans et al. (1997), in addition to a discussion on the behavior of this procedure with datasets of variable complexity. The usefulness of this combined procedure can be illustrated by again considering the x-ray images derived from the glass sample shown in Figure 24. The elemental maps and electron micrograph show quite a complicated picture: In addition to the Mn-rich areas where precipitation occured, several bands of diﬀerent composition can be distinguished. The top layer corresponds to the original glass (high Ca signal) and the bottom layer is the embedding resin. Many of the trace element maps (e.g., Ti, Cr, Ni, Zn, Br, Pb), however, have a noisy appearance. In Figure 28, the ﬁrst nine principal-component images, resulting from PCA processing of all the original maps of ﬁgure 24b are shown (step 1 in the procedure described earlier). Images PC1–PC7 are contrast-rich, whereas in the remaining PC’s noise dominates the pictures; PC1–PC7 explain more than 99% of the variance in the data. When considering score and loading scatterplots [see Vekemans et al. (1997) for details], it appears very diﬃcult, if not impossible, to ﬁnd a pair of principal components where all pixel clusters can be straightforwardly separated. By using the seven meaningful PCA images as input for the KMC procedure, the pixel grouping indicated in Figure 29 was obtained. The corresponding sum spectra are shown in Figure 30. Cluster 7, which is associated with a high Compton scatter signal and to none of the elemental signals, corresponds to the embedding resin; the corresponding

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Figure 28

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Score images obtained by PCA of the x-ray maps of Figure 24b.

sum spectrum (Fig. 30) shows (almost) only background and a high Compton=Rayleigh ratio. Sum spectrum 2 shows abundant Ca, Mn, and Fe signals, a clear Sr peak, and a much lower Compton=Rayleigh ratio; this image segment corresponds to the original glass. Cluster 3 shows a similar pattern but features clear Cu and Bi signals and probably corresponds to a Bi-rich precipitate in the crack between the original glass and the corroded layer. Clusters 6 and 9 contain only a few pixels of Bi ‘‘hot spots’’ but essentially show the same pattern. Clusters 4 and 1 corresponds respectively to the MnO2 precipitation area and the phase it originated from (high Mn and Fe signals), whereas clusters 8 and 10 refer to respectively the top part of the leached layer, parallel with the surface, and the precipitate crust formed on top of the surface (high Ca signal associated with a relatively low Sr signal).

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Figure 29 Individual and compound segmentation masks obtained by KMC on the first seven score images shown in Figure 28.

Comparison of the x-ray maps of Figure 24b and of the sum spectra in Figure 30 shows that although some of the x-ray maps show an excessively noisy distribution, most of the sum spectra feature (sometimes weak but) predominatly noisy-free photopeaks, allowing the corresponding major and trace constituents to be determined with an uncertainty which mainly derives from the errors in the calibration model and not from counting statistics.

D.

Quantitative Analysis

1. General Considerations X-ray ﬂuorescence is well known for its ability to yield reliable quantitative data and, as such, is used in many (industrial and research) laboratories for routine analysis of a variety of material types. In many cases, in order to ensure high accuracy, wavelength dispersive instrumentation and an empirical form of quantitative calibration is employed. Relative accuracies better than a few percent can be reached for speciﬁc matrix types (see Chapter 13). In laboratories which are more research oriented and=or where the type of material being analyzed is more diverse, qualiﬁcation models which (1) allow a rapid

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Figure 30

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Image and cluster sum spectra corresponding to the clusters shown in Figure 29.

changeover from one matrix type to another, (2) do not require many calibration standards, and (3) readily allow for quantitative analysis of samples of diﬀerent and intermediate thickness are preferred and the so-called ‘‘fundamental parameter’’ (FP) method is often employed (Shiraiwa and Fujino, 1966, 1968b). As explained in detail in Chapter 13, these models essentially attempt to correct mathematically for all phenomena (of ﬁrst and higher order) that inﬂuence the detected intensity of the characteristic radiation emitted by the sample. Because in these corrections

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a number of atomic constants of the elemental species involved are used (mass-absorption coeﬃcients, photoelectric cross section, ﬂuorescent yields, transition probabilities, etc.), the uncertainty on these fundamental parameters propagates into the uncertainty on the elemental concentrations calculated with these methods. In addition, as the mathematical expressions for the corrections are established for ideal(ized) geometrical conditions (e.g., parallel primary beam, perfectly ﬂat sample surfaces, etc.), systematic deviations may be introduced by using these models when the experimental reality does not conform to their implicit assumptions. When models of suﬃcient sophistication are employed and calibration is done with standards very similar to the unknown samples, accuracies of the order of 2–5% can be obtained (He and Van Espen, 1991) for inﬁnitely thick samples. For partially transparent samples, some of these models include the estimation of the sample mass (rd from the energy-dispersive spectrum; because this introduces additional errors, the accuracy in these cases is of the order of 5–10% (Shiraiwa and Fujino, 1966, 1968b). For quantiﬁcation of m-XRF spectra, which normally are collected from samples featuring some form of heterogeneity (either in sample shape, composition, or both) (Janssens et al., 1996c), the use of FP-based models rather than empirical calibration schemes is the most appropriate. Examples of heterogeneous sample types are, for example, particles on a ﬁlter backing, ﬂuid inclusions, in a geological material, multilayered samples, and interfaces between two minerals inside a thin rock section. 2. Information Depth As already mentioned (see Table 2), in most materials, x-rays have a much larger penetration range compared to charged particles. The absorption corrections for the characteristic x-rays are therefore much larger in the case of XRF than with EPXMA or m-PIXE, at least when intermediate or thick samples are concerned. To correct for matrix eﬀects in EPXMA and m-PIXE, corrections have to be applied for the (continuous) slowing down of the particles and for the attenuation of the x-rays on their way out of the sample toward the detector. For protons, the slowing down does not strongly depend on the matrix composition; also, the corrections due to this eﬀect are partly compensated by the self-absorption of the characteristic x-rays. In XRF, there is no such compensation, as both the primary and ﬂuorescent x-rays are attenuated. In EPXMA and PIXE, usually the penetration range of the primary projectiles determines the depth of information; in XRF, this is not the case: Because the characteristic radiation is usually more strongly absorbed than the primary radiation, the information depth in most nonthin samples is strongly element dependent. One way of deﬁning the information depth for element i (with characteristic) radiation of energy Ei) is to equate it to the samples thickness d99%,i that would yield 99% of the intensity that might be derived from an inﬁnitely thick target with the same composition. In the case of monochromatic excitation, d99%,i satisﬁes the equation dZ99%;i

e 0

wrz

Z1 dz ¼ 0:99 0

ewrz dz , rd99%;i ¼

lnð1 0:99Þ wðE0 ; Ei Þ

ð14Þ

where wðE0 ; Ei Þ ¼ mðE0 Þ cscðaÞ þ mðEi Þ cscðbÞ: mðE0 Þ and mðEi Þ represent the mass absorption coeﬃcients of the sample at the primary and ﬂuorescent energy. a and b are the incidence and takeoﬀ angles, respectively (see Fig. 31). In the case of polychromatic excitation, Eq. (14) needs to be integrated over the appropriate excitation energy range and a value of d99%,i may be obtained numerically.

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Figure 31 Size of area at beam impaction point required to be homogeneous to permit application of conventional calibration models to heterogeneous samples.

In Table 4, the information depth d99% is calculated for a few matrices, corresponding to monochromatic MoKa excitation (E0 ¼ 17.4 keV); in addition to the energy of the characteristic radiation, the major composition and density of the sample also inﬂuences d99% is a signiﬁcant way. This means that for a sample of given thickness, the self-absorption can also vary considerably, the same sample being inﬁnitely thick for low-energy radiation while being (partially) transparent for the more energetic lines. To illustrate this, in Table 5, the absorption factors Ai ¼ ½1 expðwi rd Þ=wi rd are listed for a 100-mm-thick foil of the same materials as in Table 4. Ai is the ratio of the elemental intensity derived from this foil to that when no self-absorption would take place in it. 3. Self-Absorption Correction in Heterogeneous Samples In heterogeneous samples, consisting of several phases of diﬀerent composition, instead of modeling the attenuation by simple expressions as in Eq. (14), the total attenuation along the path of the impinging and exiting x-rays must now be calculated as the line integral of the absorption coeﬃcient mL. For example, for a sample which features a variable concentration ci (z) in the z direction (depth) but is homogeneous parallel to the surface,

Table 4 Information Depth (d99 %) for Different Elements in a Few Matrices (E0 ¼ 17.4 keV, a ¼ b ¼ 45 ) (in mm) d99% ¼ lnð100Þ=wðE0 ; Ei Þr (mm)

Matrix Polypropylene Soda-lime glass Apatite, Ca5(PO4)3OH Malachite, Cu2CO3(OH)2 Brass, 70 % Cu, 30 % Zn Lead

Density (g=cm3)

CaKa (3.7 keV)

FeKa (6.4 keV)

PbLa (10.54 keV)

SrKa (14.16 keV)

MoKa (17.54 keV)

0.95 2.5 3.2 4 8.5 11.3

830 43 42 24 8 6

4,300 150 60 90 25 5

16,000 540 200 60 15 12

26,000 1,000 400 100 27 11

33,000 1,500 550 150 37 12

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Table 5 Absorption Factor (Ai ) for Different Elements in a Few Matrices (E0 ¼ 17.4 keV, a ¼ b ¼ 45 ) Ai ¼ ½1 expðwi rdÞ=wi rd; d ¼ 100 mm

Matrix Polypropylene Soda-lime glass Apatite (Ca5(PO4)3OH) Malachite Cu2CO3(OH)2 Brass (70=30 Cu=Zn) Lead a

Density (g=cm3)

CaKa (3.7 keV)

FeKa (6.4 keV)

PbLa (10.5 keV)

Sr-Ka (14.2 keV)

MoKa (17.4 keV)

0.95 2.5 3.2 4 8.5 11.3

0.76 0.09a 0.08a 0.05a 0.02a 0.01a

0.948 0.3 0.12a 0.19a 0.05a 0.01a

0.985 0.67 0.38 0.12a 0.03a 0.03a

0.991 0.8 0.56 0.22a 0.06a 0.02a

0.993 0.86 0.65 0.29 0.08a 0.02a

d99% d (i.e., sample inﬁnitely thick).

the concentration to the x-ray intensity of element i derived from an inﬁnitesimal sample volume with thickness dz situated at a depth z would become (in ﬁrst order) Z z dIi 0 0 0 / I0 ðE0 ÞrðzÞci ðzÞti ðE0 Þ exp ½mL ðE0 ; z ÞcscðaÞ þ mL ðEi ; z Þ cscðbÞdz dz Z0 z 0 0 ¼ I0 ðE0 ÞrðzÞci ðzÞti ðE0 Þ exp wL ðE0 ; Ei ; z Þdz ð15Þ 0

where mL ðE; zÞ ¼ mðE; zÞ rðzÞ is the (depth-dependent) linear absorption coeﬃcient. In the case that this sample consists of a number of discrete layers (index k from 1 to N ), each homogeneous and which its own concentration ci,k, mass absorption coeﬃcient mk ðEÞ, density rk , and thickness dk , the contribution dIi;k to the intensity of element i from a slice of thickness dz in the kth layer becomes " # k X dIi;k / I0 ðE0 Þri ci;z ti ðE0 Þ exp ½mj ðE0 Þ cscðaÞ þ mj ðEi Þ cscðbÞrj zj dz j¼1 " # k X ¼ I0 ðE0 Þri ci;z ti ðE0 Þ exp ð16Þ wj ðE0 ; Ei Þrj zj j¼1

with zj ¼

8 < dj :z

k1 P

if j < k dn

if j ¼ k

n1

For other heterogeneous samples with comparable (simple) heterogeneity, analogous expressions such as that of Eq. (16) may be established. As can be seen from this equation, even for samples with limited heterogeneity, fairly complex expressions relate the observed ﬂuorescent intensity Ii to the characteristics of each layer. In the general case (sample heterogeneous in three dimensions), the relation between Ii and the sample structure can only established by numerical integration of the sample concentration and absorption characteristics over the interaction volume of the primary and ﬂuorescent x-rays. What is important to note when considering Eqs. (15) and (16) (and their equivalents for other cases) is that the magnitude of the exponential absorption factors depends very

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strongly on how the total path of, for example, the ﬂuorescent x-rays [total length z csc(b) in Eq. (16)] is distributed over the various phases. Small variations in this repartition may cause large ﬂuctuations in the integrated absorption coeﬃcient along the total path. As an example, P in Table 6, for various thickness combinations the integrated attenuation exp½ j mj ðEi Þrj zj cscðbÞ is listed of several characteristic line energies Ei in a hypothetical three-layered sample of 100 mm total thickness consisting of parallel layers of KAlSi3O8, NaAlSi3O8, and SiO2 bottom layer), assuming that this radiation originates from a depth of 90 mm. The data in Table 6 illustrate that in order to reliably correct for strong self-absorption (CaKa, CrKa, and FeKa), the matrix composition and detailed spatial distribution of the various phases that make up a heterogeneous sample must be known with high accuracy. Whereas, in many cases, the matrix of these phases may be approximately known (e.g., from measurements on more homogeneous parts of the material or by using other methods), the details on the spatial distribution of the various phases inside natural samples (e.g., a rock sample composed of diﬀerent minerals) is hard to ﬁnd. Table 6 also shows that when the self-absorption is intermediate to weak (PbLa, SrKa, MoKa, in the example), only an approximate knowledge of the repartition of phases is required. In the former case (strong self-absorption), even when the sample geometry is known, the position of the microbeam relative to this heterogeneous structure must also be known in order to allow experimental spectra to be corrected for self-absorption. This is illustrated for a very simple case in Figure 32. This ﬁgure shows experimental data collected by scanning a NIST K961 glass microsphere of 32 mm diameter through a microbeam of 8 8 mm2 in the x and y direction( cf. Fig. 31). When the sample movement is done in the beam–sample–detector plane (x direction, see Figs. 32a and 32b), the characteristic radiation of K, Ti, and so forth must pass through the body of the glass sphere on its way to the detector and the intensity proﬁles are strongly and assymetrically deformed by selfabsorption in the glass. [see Voglis and Rindby (1995) for a more detailed discussion]. Note that the maxima of the K, Ti, and so forth proﬁles do not coincide with each other or with the geometrical center of the particle. Experimentally, it is therefore diﬃcult to position the beam exactly at the center of the particle because this position does not correspond with a maximum in the detected count rate. The situation is simpler when the sample movement is executed perpendicularly to the beam–detector plane ( y direction, see Figs. 32c and 32d); in this case, the ﬂuorescent radiation leaves the particle sideways toward, the detector and a

Table 6 Variation of the Attentuation of Characteristic Radiation upon Leaving a Three-Layered (SiO2, NaAlSi3O8, KAlSi3O8) Sample with Average Density of 2.5 g=cm3 Characteristic radiation (keV)

25 mm KAlSi3O8 45 mm NaAlSi3O8 30 mm SiO2

35 mm KAlSi3O8 35 mm NaAlSi3O8 30 mm SiO2

45 mm KAlSi3O8 25 mm NaAlSi3O8 30 mm SiO2

CaKa (3.7) CrKa (5.4) FeKa (6.4) PbLa (10.5) SrKa (14.2) MoKa (17.4)

1.3 10 7 5 0.02 0.089 0.57 0.79 0.88

0.9 10 7 5 0.017 0.08 0.55 0.78 0.87

0.7 10 7 5 0.015 0.072 0.54 0.77 0.87

Note: The radiation takeoff angle b is 45 .

Microbeam XRF

683

symmetrical response is obtained so that all proﬁle maxima occur at the same position (i.e., when the beam irradiates the center of the spherical particle). From the above considerations, it may be concluded that, with the exception of samples having a simple and well-known spatial distribution and matrix composition of the phases that constitute them, quantitative m-XRF analysis of truly heterogeneous

Figure 32 Experimental and simulated net intensity profiles resulting from scanning a 32-mmdiameter K961 glass microsphere through a 8 8-mm2 white SR beam (a, b) in the beam–sample– detector plane (horizontal sample movement). (c, d) perpendicular to this plane (vertical sample movement), (e) m-XRF spectra collected during scan (a) at 10 mm to the left (lower curve) and right (upper curve) of the geometrical center of the particle.

684

Figure 32

Rindby and Janssens

Continued

samples is prone to error, especially when low-energy characteristic radiation is involved for which the information depth is of the same order as the dimensions of the phases present. Nevertheless, in many practical situations, quantitative measurements can be performed provided these samples being analyzed are locally homogeneous.

Microbeam XRF

Figure 32

685

Continued

4. Conditions for Local Homogeneity=Factors Determining Lateral Resolution A sample may be deﬁned as being locally homogeneous at a given location when both the primary photons entering the sample at that point and the resulting ﬂuorescent x-rays while leaving the sample on their way to the detector will pass through the same phase. In this case, the spectral data derived from such a measurement can be quantitatively processed as if they were obtained from a completely homogeneous sample. In Figure 31, the three-dimensional size and shape of the area which must be homogeneous around the beam-impact point is schematically drawn. Clearly, this area is much wider in the beam–detector plane (Lx) than in the direction perpendicular to it (Ly). Lx is determined by the angles of incidence and takeoﬀ of the radiation and depends more on the information depth of the ﬂuorescent radiation being considered than on the actual beam size dbeam,x. For a parallel primary beam and a small detector far away from the sample, one can derive that Lx;i ¼ Lz;i ðcot a þ cot bÞ þ dbeam;x csc a Ly;i ¼ dbeam;y

ð17Þ

Lz;i ¼ minðdsample ; d99%;i Þ In the case of radiation for which the sample is inﬁnitely thick, Lz;i ¼ d99%;i ; for more penetrant radiation ðd99%;i > dsample Þ, the sample thickness determines the lateral size Lx of the interaction volume in the x direction and therefore also the lateral resolution in this direction. In the y direction and perpendicular to the beam–detector plane), only the beam size dbeam,y deﬁnes the size of the interaction volume, at least when the x-ray detector

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subtends a small solid angle, as is usually the case at sychrotron m-XRF setups. In laboratory instrumentation, sometimes large-area detectors are used (e.g., 80 mm2 at a fairly close distance to the sample (e.g., 2 mm) so that the detector solid angle O becomes quite large. In this case, the conditions for local homogeneity are somewhat stricter and the following expressions must used for Lx and Ly (with tan2 g ¼ O=p): Lx;i ¼ Lz;i ½cot a þ cotðb gÞ þ dbeam;x csc a Ly;i ¼ Lz;i ð2 tan gÞðcsc bÞ þ dbeam;y Lz;i ¼ minðdsample ; d99%;i Þ

ð18Þ

Degradation of the lateral resolution due to spreading of the primary beam as a result of scattering interactions (as is the case EPXMA) can be neglected in most cases. From Eqs. (17) and (18), it follows that in order to maximize lateral resolution in m-XRF, the highest possible incidence and takeoﬀ angles should be employed. Because such an arrangement also corresponds to the shortest attentuation paths of both primary and ﬂuorescent radiation and yields the smallest volume in which the sample must be homogeneous, it is also recommended for obtaining the highest quantitative accuracy. The only argument against the use of large incidence and takeoﬀ angles is that when their sum is no longer 90 , the scatter peak intensity and spectral background will be higher than in the 90 geometry. This is especially true for synchrotron setups employing linearly polarized radiation where the scatter background intensity can increase by a factor 10 or more when a nonoptimal detection geometry is used. In laboratory m-XRF, the implications will not be so important because unpolarized radiation is used and also because the radiation is usually collected in a larger solid angle. Hence, for sample types that do not excessively scatter the primary radiation, the use of perpendicular irradiation-detection geometries may have more beneﬁts than disadvantages. Up to now, however, no systematic studies evaluating the eﬀect of diﬀerent geometries on instrument sensitivity, lateral resolution, and accuracy of m-XRF instruments have been reported. 5. Prediction of the Spectral Response of m-XRF Spectrometers For simplicity, in the derivation of Eqs. (15) and (16), only ﬁrst-order beam–sample interactions were considered. In reality, also other phenonema such as photon scattering (Rayleigh and Compton) and higher-order interactions (e.g., enhancement) take place which complicate the analytical treatment (He and Van Espen, 1991; Shiraiwa and Fujino, 1966, 1968b; Criss and Birks, 1968; Van Dyck et al., 1986). One direct way of using all knowledge of the various interaction processes that can occur between an x-ray photon and the material in which it is traveling is to implement it in a Monte Carlo photon trajectory calculation program. Vincze et al. have published a number of articles describing such a detailed computer model for predicting in detail the spectral response of a generic EDXRF spectrometer (Vincze et al., 1993, 1995a, 1995b, 1999a, 1999b). This computer code takes as input parameter the excitation conditions, the detection geometry, and the characteristics of the (heterogeneous) sample being irradiated and generates the equivalent of experimentally collected EDXRF spectra. Using this code, it is possible to systematically study all parameters inﬂuencing experimentally recorded EDXRF spectra and evaluate their signiﬁcance. One can obtain an idea of the predictive power of this Monte Carlo model from Figures 3, 32e, 33a, and 33b in which experimental and predicted XRF spectra are compared corresponding to diﬀerent experimental conditions and sample materials. A

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Figure 33 Experimental and simulated EDXRF distributions of (a) NIST SRM 662 steel using monochromatic 15-keV excitation and (b) NIST SRM 620 glass using polychromatic white-beam excitation. Both experimental spectra were recorded at the NSLS X26A SR-XRF station.

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high degree of similarity between experimental and simulated spectral data can be reached, allowing the simulation to be used for quantiﬁcation purposes (Janssens et al., 1993b), for example, by varying the input parameters of the simulation program (primarily the sample characteristics) until the resulting simulated spectral distribution shows the highest overlap with the experimental spectrum that one wants to quantify (Vincze et al., 1993, 1995a, 1995b). In Table 7, some quantitative results obtained with the latter approach are summarized for diﬀerent excitation conditions and sample types (experimental data collected at the Hasylab m-SRXRF station). In general, root-mean-square deviations between certiﬁed and calculated concentrations in the range of 5–15% are observed; for laboratory m-XRF spectrometers, similar deviations are observed. An advantage of using this simulation program rather than the more conventional analytical approaches to fundamental parameter-based quantiﬁcation (see, e.g., He and Van Espen, 1991) is that it is relatively easy to extend it toward the case of heterogeneous samples (Janssens et al., 1996c; Vincze et al., 1999b, 1999c). As an example, in Figure 32, in addition to the experimental data obtained by scanning over a 32-mm glass microsphere, the corresponding Monte Carlo predictions (solid lines in Figs. 32b, 32d, and 32e) are plotted also. The Monte Carlo model is clearly capable of predicting the self-absorption in the glass sphere in a correct way. In Figure 32e, two experimental spectra collected during the horizontal scan (Figs. 32a and 32b), corresponding to locations at 10 mm left and right of the geometrical center of the sphere are shown. Table 8 lists the quantitative results obtained from these two spectra while (erroneously) assuming in the calculation model that they correspond to the being at center of the particle. When the beam strikes the particle at the þ10 mm position (see Table 8), the absorption path of the x-rays exciting the particle is smaller than in the center-position. Accordingly, the concentration of low-Z elements are seriously overestimated; because at this position, the amount of material being irradiated is lower than at the center-position, high-Z elements [with x-ray energies for which the glass is (nearly) transparent] will be slightly underestimated. In the reverse case (10 mm position) the absorption paths are longer than assumed in the calculation, resulting inan underestimation of the low-Z elements. Because in this position, less material is being irradiated than at the center-position, the higher-Z constituents are also underestimated. However, the availability of a Monte Carlo model that is capable of reliably predicting the spectral data that may be derived from a given heterogeneous sample in speciﬁc experimental circumstances is, in general, not suﬃcient to perform the reverse operation (i.e., to derive from a set of experimental m-XRF data the composition of the various phases that constitute such a heterogeneous sample). By means of such a model, it is only possible to verify whether a given hypothesis about the structure=composition of a particular sample is consistent with the experimental spectra data that were derived from it. In many cases, however, where external information is available on the structure and=or matrix composition of the sample (as in Fig. 32), this approach may be employed. 6. Analytical Model for m-XRFAnalysis of Particles For the simpler situation when the x-ray beam is larger than the particle (i.e., when the response is averaged out over the entire particle), Lankosz (1993), Lankosz et al. (1993), and Lankosz and Pella (1994a, 1994b, 1995, 1997) developed a calibration model for quantitative analysis that attempts to correct for the self-absorption eﬀects in these samples by using an analytical expression combined with the intensity of elastic and

87 258

14.8 22.7 14.5 9.2 12.7 35.2

48.9

12 25 14 (10) 12 37

45

1.41% 2.1%

Calc. (ppm)

91 300

1.47% 2.09%

Cert. (ppm)

NIST SRM 1571

0.34

18.3

193 130 0.055

10.3 270

0.97% 0.012

Cert. (ppm)

204 134 < DL 9 19.6

7.9 261

0.92% < DL

Calc. (ppm)

NIST SRM 1577a

Biological materials

10.3

0.46 11.55 53.3

2.94 19.7

Cert. (%)

10.4

0.41 11.9 53.4

2.74 18.4

Calc. (%)

NIST SRM 1206

5.38 3.24

19.2 0.82 51.9

0.46 17.5

Cert. (%)

5.30 3.35

19.16 – 50.8

0.52 17.3

Calc. (%)

NIST SRM 1208

Metallic materials

121 13.0 32.5 4.6 21.3

9 370 26 155 15

600 52 119 14 58

62 93 102 540 59

10 4 391 33 22 6 177 29 13 5 153 43 12 7 34 13 54 29 11

150

8

124 25

110

123 14

2.24% 1.20% 0.15% 4 (180)b 0.086% 2.59%

0.53 0.16% 7.1 0.2% 1.4 0.1% 417 148 0.18 0.03% (ref )

0.39% 7.95% 1.53% 280 0.137% 8.79%

110

Cert.a (ppm)

Calc. (ppm)

Cert.a (ppm)

592 16 57 3 125 5 13 2 72 3

65 4 99 5 105 4 557 15 52 3

153 10

1.64 0.17% 0.94 0.09% 0.11 0.02% 177 34 0.10 0.02% (ref )

Calc. (ppm)

ATHO Rhyolite

Geological materials

KL2 Basalt

Results of Quantitative Analysis of Various Types of Material by Means of the Monte Carlo Model

The geological materials were irradiated as fused glass; the certiﬁed concentrations refer to the parent basaltic rocks. Note: For details see Vincze et al., 1993, 1995a, 1995b.

b

a

K Ca Ti Cr Mn Fe Co Ni Cu Zn As Br Rb Sr Y Zr Nb Mo Ba La Ce Pr Nd Pb

Element

Material

Matrix

Table 7 Microbeam XRF 689

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Table 8 Quantitative Analysis Results (in % w=w) Obtained from Different Locations on a 32 mm Diameter Glass Microsphere Assuming the Beam to Point at the Particle Center Beam position At center

Element K Ca Ti Mn Fe Ni Zn Br Zr

þ10 mm Oﬀ-center

Actual conc.

Calculated conc.

2.49% 3.57% 1.2% 0.32% 3.5% 100 ppm 100 ppm 100 ppm 100 ppm

3.36% 4.51% 1.38% 0.34% 3.58% 100.7 ppm 98.6 ppm 97.0 ppm 96.3 ppm

Relative dev. % þ35 þ26 þ15 þ6 þ2 þ0.7 71.4 73 74

710 mm Oﬀ-center

Calculated conc.

Relative dev. %

1.49% 2.27% 0.88% 0.27% 3.0% 87.3 ppm 89.1 ppm 94.1 ppm 94.4 ppm

740 735 726 716 714 713 711 76 76

inelastic beam scatter. The eﬀective mass thickness of the particle is approximated by an iterative technique that employs experimentally detected elastic and inelastic scattered intensities. The method is restricted to particles with homogeneous element distributions. In addition, the beam size had to be greater than the size of the particle and the method worked best with light-element matrices. When used to analyze the NIST glass standards K-411 and K-3267, over the particle size range, 50–150 mm, the results for K-411 showed reasonable agreement with certiﬁed values. For K-3267, however, the results for Si and Ca were nearly 30% diﬀerent from the certiﬁed values. The poor results for K-3267 were attributed to the more irregular and elongated shape of those particles. 7. Detection of Systematic Variations in m-XRF Data Due to Topological Effects In the ‘‘fundamental parameter’’ approach, the ﬂuorescent intensity from an element in a material is dependent not only on the corresponding concentration of that element in the sample but also on a number of other parameters which can be divided into three groups: (1) spectrometer-dependent parameters such as primary intensity, beam size, detector eﬃciency, and so forth, (2) element-dependent parameters such as the photoionization cross section, ﬂuorescence yield, and so forth, and (3) sample-dependent parameters such as the sample thickness, attenuation coeﬃcients for primary and secondary radiation, and the sample geometry (e.g., the inclination of the sample surface plane relative to the beam and detector). In order to be able to apply the conventional FP approach, all of the sampledependent parameters have to appear constant over the interaction volume (sample volume aﬀected by the beam exposure). If this is not the case, it will be necessary to apply

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diﬀerent types of explicit models describing the variation of these parameters, at least over the exposed volume. It is convenient to distinguish between compositional and topological variations. Thus, in this context, the concept of topological variations is deﬁned as any variation (except compositional ones) of the sample-dependent parameters that will induce variation in the detected ﬂuorescent intensities. A topological model could be a mathematical function that describes the topology over a certain sample volume. From the general mathematical features of the FP theory, the topological models necessary for describing the variation in ﬂuorescence intensity can be reduced to a ﬁnite set of variables modeling the sample thickness variation, slope variation, and so forth. Thus, the intensity (versus elemental concentration) from any characteristic line can be written as a function f of these topological parameters {p} and the attenuation coeﬃcient for the corresponding characteristic energy: Ii ¼ f ðfpg; mi Þ ci Accordingly, in situations where there is only one independent topological parameter p (provided f is strictly monotonous), there will exists a well-deﬁned inverse function f 1 relating this single topological parameter with the intensity of a characteristic line from one element: Ii p ¼ f 1 ; mi ci Thus, the intensity from another element can be described as a function of the intensity from the ﬁrst element through its dependence of the topological parameter. In this case, it will be possible to write an explicit expression for how the intensity of element i will ‘‘depend’’ on the intensity from element j. These types of correlation between pairs of elements can be used to verify the chemical homogeneity in samples that have a simple and well-known topological structure (such as small spheres or wedge-shaped fragments of glass) or to identify speciﬁc topology structures in smples where the chemical homogeneity has already been established. In general, this kind of reduced topological parameter space (or interdependency between the parameters) cannot be expected and, normally, some of the topological parameters will appear explicity in the correlation functions. Thus, in order to determine the explicit relation between intensities from element i and j, the exact scan path (in terms of topological parameter values) has to be known. However, in some special cases, it will be possible to formulate explicit expressions of correlation functions entirely independent of the particular scan path taken. For example, the intensity from samples which are ‘‘inﬁnitely’’ thick will only depend on the slope of the surface and intensity from samples which are very thin will entirely depend on the thickness variation. In these types of single-topological-parameter samples, the correlation functions between diﬀerent elements can easily be demonstrated. Rindby et al. (1996) obtained explicit expressions for the correlation functions at diﬀerent conditions, showing that they could be written as second-order polynomial and that the linear term for thin samples was completely independent on any sample parameters. Somogyi et al. (2000a, 2000b) used binary correlation plots to distinguish between m-XRF data sets obtained by two-dimensional scanning of homogeneous and (partially)heterogeneous particles and to select the most appropriate mathematical model to be used during quantiﬁcation of the raw x-ray intensities.

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8. Microheterogeneity Characterization of Reference Materials Appropriate reference materials are essential for quantitative calibration of m-XRF spectrometers. Because during m-XRF experiments, only a very small sample mass (sub-microgram) is being analyzed, in principle, reference materials of high homogeneity are required. With the exception of glass, for most matrix types, highly homogeneous reference materials with certiﬁed levels of one or more trace elements are not available. In addition, due to a lack of suitable techniques that allow one to measure the microheterogeneity of (trace) constituents inside a material, information on the microheterogeneity of existing references materials is either not available or very approximate. In a few articles, the suitability of m-XRF for investigating the microheterogeneity of existing reference materials has been evaluated (Wegrzynek et al., 1999; Kempenaers et al., 2000). An important advantage of the use of m-XRF in this respect is that the method is nondestructive, so that surface enrichment or other changes in the microheterogeneity of the material that may be induced by the use of other microbeam methods can be avoided.

E.

Fluorescent X-ray Microtomography

As outlined earlier, in some situations, the very large penetration and sampling depth of a microscopic photon beam can limit the practical applicability of the m-XRF method. However, by using another data collection method, this penetrative character can also be exploited to gain information on the three-dimensional distribution of medium-to-heavy elements inside small samples. The conventional XY manner of scanning a sample through the microbeam yields two-dimensional images which actually are projections of the threedimensional compositional structure of this sample parallel to the direction of the primary beam. In order to obtain information on this three-dimensional structure, the same sample may be irradiated under various diﬀerent observation angles, thus creating diﬀerent twodimensional projections of the same three-dimensional structure. As an example, Figure 34 shows m-XRF maps collected from an oil ﬂy-ash particle (Janssens et al., 1998). Fly ash is formed when fossil fuels such as coal, lignite (brown coal), or oil are burned in power plants; it is a complex heterogeneous mixture of partially burned hydrocarbons (forming a hollow carbon skeleton) and the chemically and physically altered remains of the mineral fraction of the fuel. Oil ﬂy ash usually takes the form of spongelike hollow spheres, into and onto which (volatile) metals such as V, Ni, Cu, and Zn condense during the cooling of the gases in the power plant chimney stack. Although the large-sized particulates are too heavy to become airborne or inhaled and therefore do not present a direct health risk, the ash which is produced in a power plant may be subject to water-induced leaching during long-term storage, causing the metals to be released in the environment. It is therefore relevant to study where inside the particles the heavy metals are concentrated and what physical and chemical changes take place during leaching processes. As can be seen in Figure 34, ﬂy-ash particles, when investigated with a 4-mm x-ray beam, show a markedly heterogeneous structure: in addition to a relatively homogeneous S distribution, especially the metallic constituents of the particulate appear to be concentrated in speciﬁc areas. From one set of XRF maps, it is not possible to determine whether these ‘‘hot spots’’ are situated on the surface (i.e., the most prone to leaching) or are inside the particle (where they might be more shielded from chemical attacks by moisture or other chemical agents by the carbon skeleton). When another image scan is

Microbeam XRF

693

Figure 34 XRF maps of various elements in a large fly-ash particle obtained using two different sample-detector orientations: (a) 0 , (b) 90 . Image size: 40 35 pixels; pixels size: 4 4 mm, spectrum collection time per pixel: 1 s.

performed on the same particle after it was rotated over 90 relative to the beam, equivalent maps are obtained which show, as expected, the ‘‘hot spots’’ at other locations in the projection maps. In a simple case, two perpendicularly collected elemental ‘‘views’’ may be suﬃcient to ‘‘triangulate’’ the approximate position of the metal-rich areas inside the particulate (see, e.g., the Cr and Fe maps in Figures 34a and 34b). However, when the distribution of the element involved is more complicated, it is recommended to collect more than two views. If, instead of collecting XRF spectra while the samples is moved according to an XY scanning pattern, the sample height Y is ﬁxed and instead a XY scan is performed, during which the horizontal position and orientation angle of the particle relative to the beam are varied, instead of elemental maps, one obtains so-called ‘‘sinograms.’’ Figure 35 shows a photograph and schematic drawing of the sample environment during XRF tomography experiments. During tomographic data collection, the sample (mounted on a goniometer head and on a rotation stage) is translated in a stepwise manner through the microbeam (with a horizontal increment Dx of, e.g., 4 mm), after which the sample=goniometer head assembly is rotated (with angular increment Dy of, e.g., 3 ) around the vertical axis and the translation movement starts again until an angular range of 180 or 360 has been covered. In this manner, instead of performing a conventional XY scan, yielding multiple XY elemental maps, an XY-scanning movement of the sample through the beam is executed, which results in several elemental sinograms (see Fig. 36). Through the use of the appropriate reconstruction algorithms, the latter can be back-projected into XZ maps of the internal distribution of chemical elements in the horizontal plane through the sample at a particular vertical position Y on the sample.

694

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Figure 35 Photograph and schematic drawing of the sample environment during XRF tomography experiments: The sample is placed at the end of a thin glass tube mounted in a manually adjustable goniometer head. This allows the center of the sample to be aligned with the rotation axis of the motorized XYZY stage and with the capillary.

Figure 36 Conventional elemental XY maps, elemental sinograms (or XY maps), and elemental tomographs (or XZ maps) of the fly-ash particle shown in Figure 34. Size in sinograms: 30 120 pixels, pixel size: 5 mm 3 ; size of resulting tomograms: 30 30 pixels size: 5 5 mm. T.B. tomographic back-projection.

As an example, Figure 36 (middle panel) shows the sinograms obtained from the particle shown in Figure 34 at the indicated height by employing an angular increment of 3 and a horizontal translation increment of 5 mm while the sample was kept at constant height. In this way, the distribution of elements inside a plane situated at 35 mm above the lower edge of the particle can be determined. From the elemental sinograms, by means

Microbeam XRF

695

of a mathematical technique called ‘‘weighted back-projection,’’ the equivalent elemental tomograms (i.e., the distributions of the elements in the XZ plane through the particle at a speciﬁc vertical coordinate) may be obtained (Fig. 36, right panel). For a more detailed discussion on tomographic data collection strategies and back-projection theory and algorithms, see, for example, Herman (1980) and Russ (1995). As can be seen from the superposition of the S, V, and Cu tomograms, the metal-rich areas of the particulate appear to be located on the outside of the particle. From the S tomogram, it also appears that this particle is hollow and that the sulfur is distributed in a shell around a central cavity. This type of structure has also been observed during other experiments (Rindby, 1997a, 1997b). The ‘‘hot spots’’ apparently are small, quasi spherical metal-rich particles which have become attached to this S-rich shell. However, not all ‘‘hot spots’’ contain the heavy metals in the same relative amounts: for example, the three enriched areas visible in the tomograms of Figure 36 all contain the elements Ca, V, Cr, Ni and Mo, but only two show a detectable Cu signal. It is clear that by varying the height of the sample relative to the beam, equivalent tomograms at various vertical positions inside the particle may be obtained allowing a full three-dimensional visualization of the distribution of various chemical elements throughout the entire particle volume and this with a spatial resolution equal to the primary beam size. Whereas the example shown in Figure 36 highlights some of the analytical possibilities the use of small, penetrant photon beams may have for three-dimensional investigation of certain sample types, it is appropriate also to point to some of the present limitations of the technique in its current state. First, the technique is limited to fairly small samples: Because it is the intensity of the ﬂuorescent signals which is used to construct the elemental tomograms, obviously only samples which are smaller or of the same diameter as the sampling depth of the ﬂuorescent radiation can be studied in this way. In the case shown in Figure 36, this presents no problem, as the eﬀective density of the ﬂy-ash sample (mainly consisting of a carbon skeleton and hollow areas) is fairly low. However, in the metal-rich areas, although they are fairly small in diameter ( 10–15 mm), the local density and atomic number is much higher and considerable selfabsorption occurs, especially for the ﬂuorescent radiation of the lighter elements (e.g., S and Ca). This can be seen in the S sinogram in Figure 36 in which the sinuous bands that correspond to speciﬁc S-rich area in the XZ cross section under observation do not show the same intensity under all angles, as parts of the particle that absorb the SKa radiation to a diﬀerent extent rotate into the path of the SKa x-rays on their way to the detector. The self-absorption, which inﬂuences both the intensity of the primary beam on its way to a particular volume element inside the particle and the intensity of the emerging ﬂuorescent radiation, must be corrected for in order to obtain quantitatively reliable tomographic maps. In order to perform such a quantitative correction of the experimentally corrected sinograms, the variation of the linear absorption coeﬃcient mL(x, z, E) within the slice under consideration (x, z) and at the primary and various ﬂuorescent energies (E) involved must be known, or, equivalently, the (heterogeneous) matrix composition within the slice must be approximately known. Studies with simulated data indicate that under such conditions, via the use of an iterative correction procedure, self-absorption phenomena may be corrected and the initial compositional distribution iteratively reﬁned, provided that the self-absorption of the ﬂuorescent signals is not too severe (Nellessen, 1996; Vincze et al., 1999b, 1999c); considerable eﬀorts, however, still are necessary to solve this heterogeneous calibration problem in a general way (Simionovici et al., 2000). Because, at the moment, such correction

696

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algorithms are not (yet) available in validated form (Sakellariou et al., 1997a, 1997b), the tomograms in Figure 36 can only be used in a semiquantitative manner (by implicitely assuming either a nonexistent or a constant self-absorption inside the particle) (Vekemans et al., 2000). Note: The elemental XY projection images, sinograms, and XZ tomograms show in Figure 36 were obtained at the m-SRXRF setup at HASYLAB Beamline L by means of an ellipsoidal lead glass capillary, providing a beam of 4–5 mm diameter (Janssens et al., 1998). Similar measurements, but with less good lateral resolution, could also be performed by employing a monocapillary-based laboratory m-XRF spectrometer using at 15-mm x-ray beam from a Mo anode operated at 40 kV.

V.

APPLICATIONS

Although laboratory m-XRF instruments in their current conﬁguration, have limitations with respect to the net detectable count rate, the fact that elements in the range K to Zr can be determined down to the 10-ppm level makes it a valuable tool for investigating problems where trace element information is useful and=or required, where the in-air operation of the technique, its local analysis possibility, and=or the penetrative character of the primary and ﬂuorescent radiation can be advantageously applied. A.

Industrial Applications

X-ray ﬂuorescence has been used in the industrial laboratory and on the factory ﬂoor to analyze small specimens or small areas for many years (Adler et al., 1960; Miller, 1961; Shiraiwa and Fujino, 1968a; Stebel and Silverman, 1984). By simply restricting the x-ray beam with vendor-supplied apertures or other special attachments, spatially resolved quantitative data could be obtained without using expensive electron beam equipment. Special attachments for conventional XRF instrumentation reduced beam sizes to 100 mm for application to a variety of problems in the industrial laboratory. It became obvious that many materials problems such as plating and ﬁlm thickness measurement could be solved with the lower resolution of the ‘‘m-XRF’’ technique. In the mid-1980s, two developments occurred which led to increased industrial acceptance of m-XRF. The ﬁrst instrument speciﬁcally designed for energy-dispersive m-XRF was developed and marketed in 1986 (Wherry and Cross, 1986). Perhaps more signiﬁcantly, it was capable of element imaging, allowing the user, instead of simple point analysis, to superimpose elemental maps and optical images. The availability of specialized instrumentation led to applications involving microelectronic devices, industrial waste screening, forensics, wood, paper, and other problems in materials science. The other signiﬁcant innovation in the mid-1980s was the development of more intense laboratory sources using glass-capillary optics. 1. Plating Thickness=Composition Gauging The plating industry has used XRF to measure and control ﬁlm thickness for many years (Seaman, 1985; Bush and Stebel, 1985; Czechanski, 1986; Anderson et al., 1991; Asher and Ruiz, 1986). The m-XRF technique has largely replaced beta-backscatter in the measurement of ﬁlm thickness in small areas. Beta-backscatter requires contact with the surface; thus, the correct placement of small areas in the beam is diﬃcult. The standoﬀ

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in a m-XRF instrument allows the use of precise alignment techniques such as microscopes and=or lasers. In addition, for beta-backscatter, the substrate atomic number must be at least 20% diﬀerent from that of the substrate ﬁlm, whereas XRF can resolve signals from adjacent elements in the periodic table. XRF is also the only technique that can simultaneously measure the composition of a ﬁlm [e.g., Sn=Pb on Cu substrate (Stebel and Silverman, 1984)] or of high-temperature superconductor tape (Cross and Wherry, 1988) and its thickness. Individual ﬁlm layer thicknesses in multiple ﬁlms can be measured by XRF, because each element produces a distinctive signal and absorption corrections can be made from a knowledge of the order in which each ﬁlm is laid down. A common example in the microelectronics industry in Au over Ni on a Cu substrate, a combination which cannot be measured with beta-backscatter. The characterization of plating and coating layers in microelectronics, jewelry, and small coated wires requires measurements of ﬁlm thickness in areas on the order of several hundred microns or on curved parts where small beam sizes are needed to reduce the eﬀects of surface curvature. Examples of coatings are Au=Ni=Cu stacks in connector pins (Stebel and Silverman, 1984) or in microelectronic lead frames (Nakazawa, 1983), coated wires (Ni or Sn on Cu, Cd or Zn on steel) (Stebel and Silverman, 1984) or switch components (Au on Ag; Sn–Pb, Ag, Sn, Ni on Cu; Zn on steel), and Cd or Ni layers on steel fasteners. In heat sinks, integrated circuit strips, and alloy transistors (Cross and Wherry, 1988), Ni–P coatings on Al, Cu, and Fe are characterized by m-XRF. For this type of investigations, XRF instruments with apertured x-ray tube are used, with beams down to 100 mm in diameter. 2. Microelectronics Industry In the microelectronics industry, thin-ﬁlm technology is being used to solve a number of connector and packaging problems that have developed with the downsizing of integrated circuits. The lead counts on VLSI (very large-scale integration) circuits have increased in proportion to the increase in the number of components per chip. Close tolerances and narrow statistical quality control parameters have placed great demands on the characterization of these thin ﬁlms, m-XRF is the method of choice for the measurement of ﬁlm thickness and composition in the small areas involved in electronic devices. Wherry and Cross (1986) described a m-XRF instrument that had several advantages over conventional small-spot, proportional detector systems with respect to applications in the microelectronics industry. The instrument used a solid-state Si(Li) detector and had imaging capabilities for revealing hidden chemistry information and qualitative information on ﬁlm thickness homogeneity. Anderson and co-workers (1991) presented several examples of the utility of the new m-XRF analyzer to support the manufacture of integrated circuits. Tape Automated Bonding (TAB) is a method for bonding input=output leads to VLSI circuits. They accurately measured the Sn ﬁlm thickness on TAB tape by monitoring the SnKa line with a 100-s counting time. The m-XRF technique was said to be superior to Auger and SEM–EDXRF analyses because of the larger thickness range covered, lower detection limits for impurity elements, and simpliﬁed measurements procedure (no sample preparation needed and ability to operate in air, helium, and=or low vacuum). They also demonstrated the possibility of monitoring the Cu=Sn intermetallic formation by measuring the CuK and SnL lines via the use of a thinwindow solid-state detector.

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3. Waste Characterization The characterization of industrial wastes in an area of concern because of potential environmental pollution due to improper wastes disposal. The identity and quantity of hazardous elements present must be determined before disposal. In addition, new disposal methods must be developed to contain hazardous materials and prevent leaching into the soil. m-XRF has played a role in both of these areas. a.

Electronic Components

Tissot and Boehme (1994) suggested the use of m-XRF as a replacement for the Toxic Characterization Leaching Procedure (TCLP) in the classiﬁcation of waste from electronic components. TCLP is an expensive (US $2000=sample) wet-chemical process that is required by the United States Environmental Protection Agency as the basis for compliance with the Resource Conservation and Recovery Act (RCRA). It involves leaching the waste material with dilute acetic acid; if speciﬁc toxic element concentrations exceed as prescribed amount in the leachate, then sample and leachate must be disposed of in a hazardous landﬁll. In the m-XRF-based procedure, large areas of fractured waste are scanned in air; samples required essentially no special preparation. No other analytical imaging method could have been used to obtain the required data. Electronic components from dismantled nuclear weapons at Sandia National Laboratory were broken up using an industrial forge hammer to create samples having particle sizes in the 2–10 mm range. These samples were used for the TCLP and for m-XRF analysis. Sample chambers (4 in. 5 in., ﬁlled one layer deep with granulate) were scanned with a 100 mm x-ray beam from a Mo-target x-ray tube (50 kV). In a sample in which TCLP found Cd (10 1 ppm) and Pb (30 10 ppm) levels above the regulatory limit (1 and 5 ppm, respectively), these elements were also detected by means of m-XRF. In addition, Cr was found to be present by m-XRF, but this element was not present at hazardous levels in the leachate. Though present at relatively low concentration levels in the sample as a whole, the presence of these elements could readily be determined by m-XRF because in the crushed electronic components, they are concentrated in small areas that are still signiﬁcantly larger than the beam. Thus, one small particle containing a large concentration of Pb would yield a large XRF count rate and that could correspond to pm quantities for the entire sample. Furthermore, element associations in the electronic component waste, determined by comparing m-XRF element maps, were used to estimate the relative toxicity of the hazardous elements. For example, Cr, associated in stable form with Fe and Ni as stainless steel, was not expected to toxic and, therefore, was not leachable in the TCLP, Pb, on the other hand, was associated with Sn as solder and was expected to readily undergo acid attack. b. Evaluation of Industrial Waste-Disposal Methods Carpenter and co-workers described two studies of new waste-disposal methods in which m-XRF was used along with m-x-ray diﬀraction (m-XRD) to determine the phases resulting from procedures designed to prevent leaching of toxic elements into the soil and groundwater (Carpenter et al., 1995a). In the ﬁrst study, the goal was to determine the eﬀect of lead oxychloride phases on the volatility of lead from lead oxide in toxic-waste repositories. To determine what phases of lead oxychloride would form, a stimulation consisting of suspending PbO pellets in HCl gas was conducted. The reaction zone, an approximately 1 mm-thick surface layer, was relatively uniform except in regions in which the Cl concentration was relatively high. Atomic Cl=Pb ratios in the range 0.5 (inner edge of reaction zone) to 0.72 (edge of specimen) were found.

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Micro-XRD measurements revealed the presence of Pb2O2Cl. In the reaction zone, Cl=Pb ranged between 0.85 and 1.3. m-XRD data in this region revealed Pb5O2Cl6 as the major phase, along with PbO and Pb2O2Cl as intermediate to minor phases; the range of Cl=Pb values could be accounted for by changing proportions of these phases within the area included by the x-ray microprobe beam, at least up to Cl=Pb ¼ 1.2 (corresponding to Pb5O2Cl6). Although m-XRD patterns provide direct information on the mineralogical phases present, the value of the m-XRF quantitative line scans in this analysis was that detailed chemical information could be obtained in a shorter period of time (300 s per spectrum) than required by m-XRD (several hours per pattern). This analysis emphasized the complementary nature of the two microbeam techniques. In another study involving the development of new waste-disposal techniques, MgO powder was mixed with dilute HCl solutions of sludge from an industrial wastewatertreatment plant in order to form a hardened cement. It was anticipated that metal ions incorporated into the lattices of the cement phases would be contained in a stable environment for long-term storage. The material was studied using m-XRF, m-XRD, and SEM and was found to consist of large Mg(OH)2 particles and Mg3(OH)5Cl5H2O. Other particles contained signiﬁcant amounts of the metal constituents and were thought to be undissolved sludge. However, the diﬀuse distribution of Fe, Cr, and Ni throughout the matrix phase was indicative of the desired solid-solution formation. Presumbly, a greater eﬀort to more thoroughly dissolve the sludge might produce more complete encapsulation of the waste metals. The diﬀuse distribution could be easily seen in the Fe m-XRF map. On the other hand, in the corresponding SEM maps, only the Fe in the particles at the surface of the sample was visible and showed no Fe in association with the matrix phase. This diﬀerence illustrates one of the advantages of m-XRF over energy-dispersive x-ray analysis in the SEM: due to the larger penetration depth of the x-ray photons and the higher sensitivity to Fe by the CuKa exciting radiation, the lower concentration of Fe in the matrix was detectable by m-XRF and not by SEM. This complementary nature was recognized early in the development of modern m-XRF techniques. SEM analysis produces backscatter images, as well as light-element maps. Combining these data with m-XRF images of heavier elements results in a more complete analysis of, for example, ceramics, composites, and biological materials. 4. Homogeneity Testing Wide-area scans, using beam sizes ranging from a few hundred micrometers to 1 mm, are used in the steel industry to determine gross element segregation. Welfringer and coworkers described the m-XRF analysis of an area 150 150 mm2 on as cast 100C6 steel using a spot size of 1 mm2 (Welfringer et al., 1992). The authors pointed out that the easily detectable Cr and Mn tend to cosegregate with the minor elements P and S, which undergo extensive segregation during melting. The detection and characterization of segregation allows the metallurgist to predict the properties of the product and make adjustments to the continuous casting machine. Quantitative maps of Cr and Mn were produced from calibration curves. It was found that a milled surface ﬁnish was adequate for the m-XRF technique and that the data were as good as or better than that obtained by metallographic etching or the more expensive and complicated electron microprobe technique. 5. Materials Analysis in the Industrial Laboratory The use of m-XRF as an analytical tool in failure analysis and in the investigation of production problems is also becoming more prevalent in the industrial laboratory.

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The technique provides unique information that cannot be obtained with other analytical methods. As with any other technique, the data provided by m-XRF are best used in a complementary manner with data from other microanalytical tools, such as scanning electron microscopy, x-ray diﬀraction, and x-ray photoelectron spectroscopy. Carpenter (2000) describes (1) the use of a 10-mm x-ray beam for analysis of stainless-steel spray nozzles (1 mm in diameter), revealing that the investigated oriﬁce had undergone extensive corrosion (resulting in local Cr enrichment) during the spraying of molten ceramic material, (2) the use of m-XRF instead of SEM for analysis of volatile materials (which can contaminate the vacuum system of a SEM) e.g., for establishing the rate of amalgamation between Cu and Zn powders and Hg contaminated with radioactive waste, and (3) the analysis of Zr=Mo coating-hardened metal pins, allowing to qualitatively compare the thickness and composition of the coatings and of the base alloy of the pins used by various manufacturers. Pella and Feng described the analysis of a MnO2 coating on the wash-coat surface of a cylindrical automobile catalyst by m-XRF to determine the eﬀect of a fuel additive on the catalyst (Pella and Feng, 1992). The use of electron beam methods was not appropriate because thickness measurements were required. The m-XRF technique was also advantageous because no sample preparation was required to make the sample electrically conductive. After longitudinal sectioning, a scan parallel with the cylindrical axis revealed that the coating thickness was reduced by about a factor of 2 at about 8 mm from the inlet side of the catalyst. Worley and Havrilla (1998) characterized mixed-oxide (MOX) fuel feed material surrogates composed of Ga2O3 in CeO2. MOX fuel contains trace level of Ga2O3 that can potentially damage the fuel cladding. Because the majority of the Ga2O3 is extracted during MOX fuel feed processing by reduction to volatile and mobile Ga2O, spatially resolved spot analyses, line and image scans were performed by means of m-XRF to determine the gallium distribution in surrogate pellets. A CeO2 matrix was used because it is easier to handle than PuO2 and because it exhibits chemical properties similar to those of plutonium. The gallium (as a volatile suboxide) was found to migrate to the exterior of the pellets when reduced with hydrogen; it also followed a thermal gradient in the reducing oven and moved toward the pellet tops at high reduction temperatures. B.

Environmental Applications

An overview of environmental application of laboratory and synchrotron-based microbeam XRF can be found in Osan et al. (2000). In what follows, primarily the applications that can be realized with laboratory m-XRF setups are brieﬂy outlined. 1. Individual Particulate Analysis Because the atmosphere, the soil, and the groundwater are recipients of industrial and other antropogenic pollutants, the regulation of these emissions is based on the monitoring of their presence. Traditionally, XRF has been one of the prime and analytical methods for the determination of many elements at reasonable cost in a large series of samples. A considerable number of elements and a profusion of organic compounds are emitted into the atmosphere; of these, more than 50% is preferentially present in particulate matter form rather than in the gas phase. Many methods are in use for the elemental analysis of aerosols; most methods provide the average composition of the total amount of particulate matter collected on, for example, an air ﬁlter with or without size fractionation. In general, a distinction is made between the ﬁne (respirable) aerosol fraction (0.1–2.5 mm

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diameter) and the coarse (2.5–100 mm) fraction (sedimentation-type particles). By analyzing particles individually, after a classiﬁcation step, the contributions and composition of each particle source at the sampling location may be determined directly. Especially EPXMA (e.g., Van Borm et al., 1990) and to a lesser extent m-PIXE (Jaksic et al., 1991) and microbeam XRF (Janssens et al., 1992, 1993a; Rindby et al., 1997a, 1997b; To¨ro¨k et al., 1994; Osan et al., 1997) have been used in the last decade for characterization of individual particles. EPXMA has suﬃcient lateral resolution in the relevant size range to allow fast and automated major analysis of coarse- and ﬁne-mode aerosols; even for large particles, however, it is limited to the determination of the major components. To allow determination of trace concentrations (1–10-ppm level) in very small sample volumes (a 1-mm-radius aerosol typically has a mass 10 pg), a detection power in the (sub)femtogram range is required. Therefore, m-PIXE and m-SRXRF are mostly applied to coarse-mode aerosol. A comparison of sensitivities using glass microspheres of 20–30 mm, (Janssens et al., 1992; Gordon and Jones, 1993) revealed that synchrotron microbeam XRF (NSLS microprobe) provided the best relative detection limits (1–10-ppm range for the elements K to Zn in 100 s), that m-PIXE (University of Amsterdam setup) took up an intermediate position, and that EPXMA was limited to concentrations above 0.1–0.2%. However, when EPXMA and m-SRXRF were compared in terms of absolute detectability for particles of comparable size to the x-ray microbeam (4-mm radius), the performance of both methods was not very diﬀerent (DLs in the 0.05–0.5-pg range for m-XRF; 0.5–5 pg for EPXMA). It is expected that a third-generation synchrotrons, where absolute detection limits in the ag (1018 g) range were reported already (Rindby et al., 1997b), trace analysis of ﬁne-mode aerosols will be feasible. Another type of information that can be obtained from individuals aerosol particles by means of an x-ray microbeam of tunable energy is the chemical state of minor and trace constituents of environmental materials. Because of the changeover from nuclear to conventional fossile-fuel-burning power plants, interest in the characterization of ﬂy ash has increased in recent years in view of the possibility of direct public exposure to toxic metals enriched on the surface of ﬂy-ash particles or the released into the environment of these metals due to leaching. To obtain a better understanding of the toxicity of some of the metals, their oxidation state can be determined by ﬂuorescent-mode XANES (x-ray absorption near-edge structure spectroscopy). By means of a 100 100-mm2 tunable monochromatic beam, it was possible to establish that most of the As in single ﬂy-ash particles, collected from a Hungarian power plant, was present as As5þ and not as the more toxic As3þ (To¨ro¨k et al., 1994; Osan et al., 1997). More recently, Janssens et al. (2000c) used a polycapillary lens to focus bending magnet radiation down to 40–50 mm to collect XANES proﬁles of trace elements such as As and Se, present at the 40-ppm level in ﬂy-ash particles. Also, the local oxidation state of U and Ni in microscopic particles has been studied in this manner. 2. Tree Ring Analysis A wide range of living organisms can be used for biomonitoring of the quality of the environment (Burton, 1986; Markert, 1993). They can reﬂect the varying circumstances by growth reduction or decrease of biological activity. The immobilized chemical components incorporated in these organisms are historical records of the environment. Moreover, some of the trace pollutants can be enriched in their tissues, thus facilitating the analysis of lower concentrations. The history of the environment of a given area can be studied using

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biological materials spending their whole lifetime in the studied territory. Trees are ideal plants to study because their lifetime is long enough to range several decades but is usually too short to use other physical or chemical techniques for their dating. In the temperate zone, trees usually grow one ring during one vegetation period, which is easily identiﬁable by simple optical methods. Growth variations can reﬂect meteorological, ecological, and environmental changes. Microbeam XRF is an eﬃcient and nondestructive method for studying both the trace element composition and the density variations of wood; when the analysis of a single tree ring is aimed for, XRF analysis of suﬃcient lateral resolution or mechanical separation (cutting) of the sample is needed. Experiments in the last decade, attempting to use bulk XRF for the trace element determination in the tree rings, required chemical preconcentration and therefore were destructive (Nagj et al., 1987). The precise identiﬁcation of the rings and their physical separation is cumbersome and the obtained concentrations can hardly be correlated to the date. m-PIXE and EPXMA were also applied to obtain data on individual rings; theses methods are not very suitable in this case because energy deposition by the charged particles can cause loss of volatile components and damage to the plant material. Nagj et al. (1987) studied the same tree samples by both XRF and PIXE and claimed that PIXE sulfur values are ﬁve times lower than the XRF values, which is due to local heating of the sample caused by the proton beam. The concentration values for other elements are slightly higher when determined by PIXE, which is probably caused by the loss of matrix induced by charged-particle heating. Experiments aimed at establishing the trace element composition of tree rings were performed by several groups. Hayakawa et al. (1990) were the ﬁrst to show the applicability of SR–XRF for analysis of cedar tree rings using a collimated 200-mm beam of 10 keV energy. The Ca distribution could related to the structure of the tree ring, whereas Fe, Cu, and Zn maps (ppm level) showed that these elements are localized in small regions of the sample. They also concluded that (external beam) PIXE analysis causes strong sample damage due to the low thermal conductivity of the sample. Gilfrich et al. (1991) used both conventional and synchrotron x-ray sources for the analysis of the sassafras southern red oak. The conventional XRF measurements were carried out with a collimated and unﬁltered Rh-anode beam of 0.5 mm 0.7 mm in vacuum. Because the size of tree rings is larger than the beam size, the limitation of the experiment was the relatively low peak=background. The SR–XRF analysis was carried out at the NSLS X-23B beam line by using a 10-keV beam apertured to 50 mm 1 mm. The radial distribution of nine elements in the tree ring samples was established. The study showed that year-to-year variations of trace elements can be obtained by the SR–XRF analysis in reasonable time (long-range scans can be done within 1 h), but, from the environmental point of view, a great deal of research is still required in order to establish the correlation between the pollutants in the soil, water, and air to which the tree has been exposed. Selin et al. (1993) reported on a tabletop system which used a 0.1 1-mm beam from a Mo secondary target for trace element analysis of tree rings. The rings to be analyzed were identiﬁed optically. They showed that a minimum detection limit of the order of a few ppm for thick samples (tree biopsies) can be reached within the 300-s measurement time. Long-term variations of the Zn and Mn levels in spruce, pine, and birch from central Sweden were indicative of the occurrence of stress to all trees around 1980, attributed to changes of the general environmental conditions in the area caused by extensive cutting of wood. Some elements such as zinc, iron, and lead showed dramatic variations (four to six times) in the long-term period and also within 1 year. Lead and zinc were in high correlation, which could indicate a direct inﬂuence from air pollutants.

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a.

703

Density Variations Between and Within Individual Rings

In the temperature zone, the tree rings have dark and light areas resulting from seasonal diﬀerences in biological activity and growth rate. The rings therefore have diﬀerent densities and diﬀerent structures but are made of approximately the same organic material. A microscopic x-ray beam can be eﬃciently used to determinate the density variations of the wood material even within a single year ring. Microbeam-based densitometric analysis has two approaches; the density variations can be studied (1) by transmission or (2) by coherent scattering measurements. 1. Transmission P method. The intensity Itr of the transmitted beam is proportional to expð i wi mi rzÞ ¼ expðMrÞ, where M denotes the product of the total mass-attenuation coeﬃcient m and the sample thickness z. (Because the composition Pof the wood is approximately constant, the total mass-attenuation coeﬃcient i wi mi can be simply written as m, and z is assumed to be constant due to the sample preparation.) I is the intensity of the primary beam and Ir is the intensity of the transmitted radiation. As a result, the intensity of the transmitted x-rays is a simple function of the density r (Kuczumow et al., 1995). 2. Scatter method. Assuming constant thickness z, the scattered intensity is also a monotonous function of the density r. The scatter intensity can be used for studying density variations, or via a series of appropriate standards (Vincze, 1995), the density can be determined quantitatively. The most important advantage of the use of the scattering signal is that in addition to information on the density also the chemical composition can be obtained simultaneously and from the same ED XRF spectrum. Also, the sample preparation is not as demanding as for transmission measurements. Kuczumow et al. (1995) aimed to obtain density variational data within one individual growth ring (using 20 mm resolution) and to record long-range density changes (80 mm resolution). The primary intensity obtained from a conventional x-ray tube (Cu anode) with capillary focusing (7-, 18-, 23- and 79-mm openings) was suﬃcient for transmission and coherent scattering measurements. Core samples of two similar tree species (Pinus silvestris from Poland and Picea abies from northern Sweden) but growing in diﬀerent climatic conditions were analyzed. Ten-year scans showed that the transmission data and the temperature distribution were in good correlation; each increase in ambient temperature caused a density decrease. From the long-range densitometric measurements, year rings with extremely high mean density could be selected, as an evidence of low-temperature years (Jones et al., 1995) (e.g., as a result of volcanic explosions, causing a global temperature decrease due to dust emission into the atmosphere). By means of a 18-mm capillary beam (Kuczumov, 1995), within individual year rings signiﬁcant diﬀerences between the Polish and Swedish wood were found. In the former wood type, a typical yearly density variation of pine growing in the temperate zone was found showing a thick layer of dense ‘‘late’’ wood. The Swedish wood lacked this layer; its density rapidly decreased in spring and sharply increased in autumn. When the densitometric measurement was carried out in the scattering mode, Attaelmanan et al. (1994) demonstrated that some nutritional elements such as Ca and Mn followed the density variations, whereas others (K, Cl) were completely uncorrelated. Kuczumow et al. (1999a, 1999b) also studied diﬀerences in density and in composition of year rings in petriﬁed wood by means a combination of techniques.

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Laboratory x-ray tube systems have also been used for biomedical applications. This type of investigation was reviewed by Fukumoto et al. (1999). Already in 1990, Fukumoto et al. (1990) demonstrated the possibility of performing trace element mapping on tumor tissues and living plant leaves by means of a 200-mm beam extracted from a conventional x-ray tube. Trace element mapping on tree leaves was later demonstrated by Carpenter (1995b), whereas Voglis et al. (1993) performed high-resolution mapping of Ca in bone. C.

Applications in Art and Archaeometry

According to Lahanier et al. (1986), the ideal method for analyzing objects of artistic, historic, and archaeological value should be (1) nondestructive, (2) fast, (3) universal, (4) versatile, (5) sensitive, and (6) multielemental. Because XRF meets a number of these requirements, analysis of objects of this type with XRF is fairly common and is, in fact, one of the most applied ways of obtaining qualitative and semiquantitative information on the materials of which these objects are made. However, the use of conventional XRF for reliable quantitative analysis is severely hampered by the fact that the irradiated area usually is too large (preventing details of decorations, shapes, and so forth to be analyzed separately), whereas also the irradiation geometry and sample surface are usually nonideal and=or not well deﬁned, possibly introducing systematic errors in the quantiﬁcation. Some of the limitations can be circumvented by using smaller x-ray beams. Motorized sample movement also allows to extend the local analysis capability toward two-dimensional imaging of certain elements on the surface of (decorated) artifacts. Because many objects of artistic and=or archaeological nature are fairly large and bulky (e.g., statues, oil paintings, vases, treasury objects), instrumentation operating in air atmosphere that can accommodate various object shapes is very useful. More recently, polycapillary lenses have also been used for the local and nondestructive analysis of artistic objects. The fact that polycapillaries only produce beams of 100–200 mm diameter does not present a limitation in investigations where mostly submillimeter-sized (or larger) features of macroscopic objects are analyzed. The marked advantage polycapillaries have over monocapillaries is the fact that they form a focal spot at a considerable distance from the end of the lens, whereas with monocapillaries the smallest beam diameters are reached at the point where the beam leaves the capillary. This allows for a completely noncontact and nondestructive kind of investigation and makes polycapillary lenses ideal for local analysis of sensitive, precious, and=or unique objects of macroscopic size such as statues, miniature paintings, coins, utilitarian objects in glass or metal (goblets, vases, etc.), and the (often multicolored) decorations which are applied to them. In what follows, a few examples of the application of monocapillary- and polycapillary-produced submillimeter x-ray beams to the above-described type of problems are given (Janssens et al., 2000b). 1. Trace Element Mapping in Corroded Roman Glass The backscattered electron micrograph and x-ray maps shown in Figures 24–29 were collected from a fragment of Roman glass that had been buried in moist Jordanian soil for about 1900 years; details on the nature and history of this material can be found elsewhere (Janssens et al., 1996c, 1996d). As a result of the interaction with groundwater during this period, a leached or corroded layer was produced, in which a number of the original cations of the glass (mainly Naþ , Kþ , and Ca2þ ) were replaced by protons and other

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cations dissolved in the groundwater. Due to a number of other physical and chemical processes that have occurred inside the leached layer and in the surface crust on top of the original surface, a chemically and physically complex multilayered structure was formed. For example, in Figure 24a, a parallel corrosion front next to a hemispherical corrosion body can be distinguished. In order to obtain a better understanding of the mechanisms that lead to the formation of these complex structures, the distribution and migratory patterns of various (trace) elements (e.g., Mn, Bi, Sr, Zr, etc.) must be determined and correlated to their ionic size and bonding nature to the glass network structure in the various layers (Newton and Paul, 1980). A comparison of m-XRF- and m-PIXE-generated elemental maps obtained (Janssens et al., 1996a) showed that both techniques provide approximately equivalent information on the distribution of trace constituents such as Cr, Ni, Cu, Zn, Pb, Bi, Sr, and Mo, although, generally speaking, the noise level in the m-XRF images is higher. Also, the achievable resolution in state-of-the-art m-PIXE setups is signiﬁcantly better than what is now possible in lab-scale m-XRF instruments. In cases, however, where complementary information on matrix elements can be obtained by, for example, EPXMA and a lateral resolution of 10 mm is suﬃcient, the maps in Figure 24b suggest that lab-scale m-XRF can be considered as a valuable (and less expensive) alternative microanalytical trace-level technique. 2. Local Analysis of Decorations in Japanese18th-Century PorcelainVases An area in which authentiﬁcation of art(istic) objects is of considerable (economical) interest in that of Oriental china ware. Since until the 17th century, high-quality porcelain was exclusively manufactured in China and Japan; imported chinaware were highly valued and expensive. In the later periods up to now, an extensive production system of counterfeit Chinese and=or Japanese porcelain both in Europe and in Asia has come into existence. Some of these reproductions are very sophisticated and cannot easily be distinguished from the original counterparts by visual inspection alone. However, objective ways of distinguishing between original period pieces and more contemporary reproductions have up to now focused predominantly on the porcelain base material itself (Yap and Tang, 1984, 1985). An alternative approach is to concentrate on the variation of the composition of the enamel paints used to decorate the objects. In view of the size and shape of the pieces, this type of analysis is diﬃcult to perform using SEM–EDX (scanning electron microscopy equipped with energy-dispersive x-ray detection) and the intricacy of the decoration patterns usually preclude the use of conventional XRF except when only averaged analysis results are required (Yu and Miao, 1997, 1999); m-XRF analysis of individual decorations is feasible however. As detailed in Janssens et al. (1996a), by employing a monocapillary-base setup, it is possible to record large-scale m-XRF maps of these decorations and to determine which transition elements (e.g., Mn, Fe, Cu) were employed to color the PbO-based enamel paint; it was also possible to distinguish between originally present and add-on decorations of the same color on the basis of the local XRF spectra. Also, polycapillary lenses, yielding a beam size of approximately the same dimensions as the details of the decorations, have been used to perform local analysis of this type of material (Vekemans et al., 1998). 3. Local Analysis of Bronze Statues In the 19th century, a great interest arose for the ancient Egyptian civilization; many museums in the world currently own a collection of small Egyptian statues of religious

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nature in stone or bronze (Oost, 1995). To a varying extent, most metal statues have developed a corrosion layer which prevents reliable analysis of the Cu-based alloys of the objects were made of (Robbiola et al., 1988; Robbiola and Fiaud, 1993). Nevertheless, this information is of interest because it can be used to trace the evolution of metallurgy in the period 3000–0 BC (Wuttmann, 1986, 1996). Analyses of this type can also be used to distinguish between original pieces and contemporary forgeries (Vekemans et al., 1998) or to establish, for example, whether all parts of a statue originate from the same workshop. By employing microbeam or millibeam XRF, several parts of a statue can be examined in reproducible conditions (e.g., by avoiding strongly curved surfaces) whereas for quantitative analysis of the underlying metal, only small areas of the corrosion layer (e.g., on the base of a statue) need to be removed. As an example, Table 9 lists quantitative analysis results obtained from a statue of the god Nefer-Hotep represented in Figure 37 (about 40 cm high, from the collection of the Museum Vleeshuis, Antwerp, Belgium (Oost, 1995) of exceptional quality, dated to the XXII–XXIII dynasty ( 1000 BC) (Vittiglio et al., 1999)]. It was nondestructively analyzed by means of a simple milli-XRF setup consisting of a collimated low-power minifocus tube (250 mm focal spot on Mo anode), yielding a pencil beam of 300 mm in diameter and a 30-mm2 Si(Li) detector. A fundamental parameter method was used to obtain quantitative results; analyses of brass and bronze standards revealed that the elements Cu, Zn, Sn, and Pb (the elements most informative of Cu metallurgy in this period) could be determined with an accuracy of 10%. Whereas the body of the statue is copper-colored, has a very smooth surface, and shows only a very thin corrosion layer, the double crown and base of the statue are darker in color and have a rougher texture, suggesting the presence of a thicker corrosion crust and possibly the use of different alloy composition. Both crown and base were made separate from the body. Because the data in Table 9 suggest a difference in composition between the body of the statue (legs, front of body) and its crown and base, on the bottom side of the base, a few square millimeters of the corrosion layer were polished away, exposing the original metal. The results obtained after polishing are also shown in Table 9 and indicate that there is, in fact, no large difference in composition between base and body and that probably all parts of the statue were made using a similar alloy composition and in the same workshop. The compositions found after polishing are consistent with literature data on this type of material and period. It is clear that the nondestructive analysis possibilities of m-XRF in this case can contribute signiﬁcantly to a better understanding of ancient metallurgical techniques employed in the period involved. 4. Document Forgeries The ability to analyze materials in air and in a nondestructive manner makes m-XRF an excellent technique for the analysis of valuable documents (e.g., to determine their authenticity). Typically, diﬀerent inks, while having the same visible appearance, will have diﬀerent chemical compositions. Although m-PIXE and m-SRXRF has been used to examine documents such as the Gutenberg Bible and other early printed works (Cahil, 1981, 1984; Mommsen, 1996), the relative simplicity and nondestructive nature of m-XRF make it an attractive analytical alternative. Larsson (1991) and Stocklassa and Nilsson (1993) described analysis by m-XRF of an 500-year-old Swedish possession letter (dated April 1, 1499). The document, a sales contract for an estate, showed signs of alteration. It is suspected that the alteration was made in the 1530s when the Swedish king gave land back to certain nobles, who had been

0.12 81.2 0.05 13.7 4.8

% (w=w)

Fe Cu Ag Sn Pb

0.14 82 0.04 12.9 4.5

Front

0.25 76.5 0.12 14.8 7.6

Head band

Body

0.09 72.8 0.18 19.8 6.5

Lower 0.09 75.8 0.13 17.5 6.2

Upper

Crown

0.11 72.6 0.17 22.3 4.3

Front 0.11 70.7 0.19 21.6 7

Back 0.15 71.4 0.13 21 6.7

Bottom A

Base (unpolished)

0.09 80 0.07 15.3 4.6

Bottom A

Quantitative Results on Local Analysis of Different Locations on the Nefer-Hotep Statue Shown in Figure 37

Right leg

Table 9

0.07 84.9 0.06 12.2 2.5

Bottom B

Base (polished)

0.08 84.8 0.06 12.2 2.5

Bottom C

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Figure 37 Photograph of a bronze statue of the Egyptian god Nefer-Hotep (collection of the Museum Vleeshuis, Antwerp, Belgium), dated to the XXII-XXIII dynasty (ca. 1000 BC).

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stripped of their estates by an earlier ruler. Although alteration was suspected by visual inspection (the original name of the estate was removed by scraping), the original text was unreadable. By employing m-XRF-generated Zn maps (a trace constituent of the original ink), the original name could be established however. In Figure 38, m-XRF scans of Ca and Zn are shown across a selected area of the parchment. In the Ca map, the (falsiﬁed) visible text can be seen, featuring the family name ‘‘Ga¨smestad i Bo¨re’’; in the Zn map of the same area, however, a completely diﬀerent text reading ‘‘Bøtinge i Asbo’’ becomes visible. Obviously, the forger used a diﬀerent ink for the alteration, accounting for the diﬀerent chemical makeup. In a forensic (see next subsection) rather than a historically oriented study (Stocklassa and Nillson, 1993), a comparison analysis of 10 blue ball-point inks of similar color by m-XRF analysis revealed the presence in some inks of Fe and Rb, of W, of Fe and Cu, or of Fe, Cu, and Zn patterns, indicating the possibility to distinguish between the various inks used by diﬀerent manufacturers. A text (the number ‘‘24’’, see Fig. 26) was written on a piece of paper with one pen and completely overwritten with another ball-point pen of

Figure 38 The Ca (bottom panel), Mn, Fe, and Zn (top panel) maps of an area of a Swedish possession letter. In the Ca map, the family name ‘‘Ga¨smested i Bo¨re’’ is visible, corresponding to the visible (but falsified) writing; in the Zn map of the same area, however, a completely different name reading ‘‘Bøtinge i Asbo’’ is visible.

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similar color but diﬀerent brand. By using the appropriate elemental image speciﬁc for the ink of the ﬁrst pen, the original text could be easily visualized. D.

Forensic Applications

In view of the nondestructive nature of m-XRF and the minimal specimen preparation required, it is an ideal tool for the analysis of material found at crime scenes and for the investigation forgeries. The use of m-XRF permits to avoid the destructive of valuable evidence, as is the case for many other methods commonly employed in forensic analytical chemistry. In Stocklassa et al. (1992), the analysis of various forensic materials such as single hair ﬁbers, paint chips, glass particles, and ball-point pen ink using m-XRF is described. 1. Hair Analysis With respect to the analysis of trace elements in hair, m-XRF has the advantage of being nondestructive, quick, and able to map element distributions both along the length of the ﬁber and across its thickness. Trace elements in hair originate from both internal (endogeneous) and external (exogeneous) sources (Chatt and Katz, 1988). Exogeneous elements originate from environmental sources such as tap water or air pollution. The assumption is generally made that exogeneous elements will be concentrated nearer the surface of hair strands and the concentrations of those elements will increase with length, as long as the external deposition continues; in contrast, endogeneous elements should be distributed relatively uniformly throughout the thickness of the ﬁber, the relative amounts of each element determined by diet and individual metabolism. In a few studies, the claim is made that within a person’s hair, the compositional ﬁngerprint among individual hair is relatively constant and that this patterns is unique to a single individual (Toribara et al., 1982; Seta et al., 1982, Valkovic et al., 1973); thus, (m-XRF) analysis of a single hair strand could potentially provide the forensic scientist with unique information to match hair ﬁbers found at crime scenes. Engstro¨m et al. (1989) showed that detection limits of 1 ppm were possible in the analysis of trace elements in a single hair ﬁber using a m-XRF analyzer with a 200-mm beam. With the same instrument and a small (19 mm) Cr-anode beam, Larsson et al. (1990) obtained self-absorption corrected radial distributions of sulfur and calcium from single hair ﬁbers without cross sectioning the ﬁber. The Ca distribution was maximum at the surface of the ﬁber and showed a minimum in the center of the ﬁber, consistent with other work indicating the presence of signiﬁcant amounts of exogenous Ca in hair from washing in tap water and with PIXE observations by Cookson and Pilling (1975). Toribara (1987) constructed an XRF analyzer for the analysis of single hair ﬁbers in 1–2-mm steps along the length of the hair. The authors pointed out that the data obtained from multiple hair samples from a single individual did not appear to vary signiﬁcantly and that forensic applications were possible. Detection limits were found to be similar to those for PIXE, or less than about 30 ppm. The longitudinal distribution of Hg in the hair of an Iraqi woman who had consumed wheat produced by seed that had been treated with methyl mercury insecticide was determined. From the fact that hair grows at an average rate of about 1 cm per month, the period of ingestion of the Hg could be determined. This is similar to PIXE data used to show slow poisoning of Napolean by As, except that it was done faster and with higher spatial resolution (Forshufvud et al., 1961;

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Smith et al., 1962). The same instrument was also used to determine trace elements in ﬁbrous materials: chemical diﬀerences could be found in carpets made by diﬀerent manufacturers, pointing to the possibility to match=identify single textile ﬁbers found on victims and=or criminals. 2. Individual Particle Analysis The analysis of microscopic particles found on a suspect can potentially place the suspect at the scene of a crime. Particles may originate from bullets passing through clothing, from gun shot residue or from glass breakage. a.

Glass Fragment Analysis

The elements present in glass vary depending on the type of glass involved and the manufacturing lot. A group of European forensic laboratories (Sto¨cklein et al., 1997) compared various analytical techniques for discriminating between various types of ﬂoat glass (as used, e.g., in shop windows). Float glass is one of the major types of evidence encountered in crimes such as burglary, road accidents, and vandalism. A rapid and widespread manner of distinguishing between the diﬀerent ﬂoat glass manufacturers is to determine the index of refraction of the glass (usually in the range 1.517–1.520). However, due to the globalization of trade and because narrow limits are set for the major components used in the manufacturing process of ﬂoat glass (i.e., Na2O, K2O, CaO, and SiO2), the ability of this method to discriminate between glass samples of diﬀerent origin in the future will be reduced. Two varieties of m-XRF (employing a unﬁltered, collimated Rh tube and capillary focused Cu-anode beam, respectively), SEM–EDX and ICP–MS (inductively couple plasma–mass spectrometry) were used to analyze various standards glasses, some of which with the same refractive index. Of these, only capillary m-XRF and ICP–MS were able to distinguish between glass samples made in the same furnace but produced in an interval of 9 months; these methods also detected minor inhomogeneities within individual pieces of glass. Both varieties of m-XRF, with DL values in the range 1–100 ppm, were able to diﬀerentiate between glass manufactured in diﬀerent plants, by making use of the transition element ﬁngerprints of the glass. Other quoted advantages of the use of m-XRF for glass fragment ﬁngerprint were its multielemental character, the simple sample preparation, and its nondestructive character (as compared to ICP–MS). Disadvantages are its limitations with respect to light-element determination and (for irregularly shaped fragments) its semiquantitative character. b.

Gunshot Residue Analysis

When a gun is ﬁred, a pattern of residue is distributed on the person ﬁring the weapon and on the target. Analysis of particles (taken from the clothing or skin of the suspect=victim) in this case is typically done using scanning electron microscopy and automated routines may be used to search over large areas, identify particles, and carry out quantitative analysis on them (Nilsson and Stocklassa, 1993), provided the particles are on the surface of the material. m-XRF has been used to directly map the residue distribution on fabrics where the particles have penetrated into the material. In a case where a man was shot during hunting, the same authors describe the automated search and analysis of metallic particles deposited into textile fabric by passage of a bullet; the bullet itself was not available for analysis. All particles were found to contain the same concentrations of Ni, Cu, and Zn; it could be concluded that the bullet fragments could only have come from one of two weapons suspected to have ﬁred the bullet.

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Robinson JW. Handbook of Spectroscopy. Boca Raton, FL: CRC Press, 1974, Vol 1. Russ JC. The Image Processing Handbook. 2nd ed. Boca Raton, FL: CRC Press, 1995. Ryan CG, Clayton E, Griﬃn WL, Sie SH, Cousens DR. Nucl Instrum Methods Phys Res B34:396, 1988. Sakellariou A, Cholewa M, Saint A, Legge GFJ. Meas Sci Technol 8:746, 1997a. Sakellariou A, Cholewa M, Saint A, Legge GFJ. Nucl Instrum Methods Phys Res B131:253, 1997b. Seaman A. Finishing 9:18, 1985. Selin E, Standzenieks P, Boman J, Teeyasoontranont V. X-Ray Spectrom 22:281, 1993. Seta S, Sato H, Yoshino M, Miyasaka S. Scan Electron Microsc 1:127, 1982. Shakir N, Larsson S, Engstro¨m P, Rindby A. Nucl Instrum Methods Phys Res B52:194, 1990. Shiraiwa T, Fujino N. Jpn J Appl Phys 5:886, 1966. Shiraiwa T, Fujino N, Adv X-ray Anal 11:63, 1968a. Shiraiwa T, Fujino N, Adv X-ray Anal 11:95, 1968b. Sievert RM, Acta Radiol Stockh 17:299, 1936. Simionovici A, Chukalina M, Schroer Ch, Drakopoulos M, Snigireva I, Snigirev A, Schroer Ch, Lengeler B, Janssens K, Adams F. IEEE Trans Nucl Sci 47:2736, 2000. Smith H, Forshufvud D, Wassen A, Naute 194:725, 1962. Somogyi A, Vincze L, Janssens K, Vekemans B, Rindby A, Adams F. Spectrochim Acta B 55:75, 2000a. Somogyi A, Vincze L, Janssens K, Vekemans B, Rindby A, Adams F. Spectrochim Acta B 55:1037, 2000b. Spiller E, Segmuller A. Appl Phys Lett 27:2105, 1974. Stebel MD, Silverman WM. XRF programmable plating thickness measurement instrumentation for automated testing of small parts and printed wiring boards. Proceedings of the ICWA 1984 Coil Winding Conference, 1984, p 75. Stern EA, Kalman Z, Lewis A, Lieberman A. J Opt Soc Am 27:5135, 1988. Stocklassa B, Nilsson G. Document examinations using the Intrax. Proceedings of the IAFS (International Association of Forensic Societies), Du¨sseldorf, 1993. Stocklassa B, Nilsson G, Paulsson N. Paper presented at the European Conference on EDXRS, Myconos, 1992. Sto¨cklein W, Gunaratnam L, Hicks T, Kidfelt G, Rindby A, Warman G, Widstedt I. The analysis of ﬂoat glass using elemental analysis: Comparison of techniques. Proceedings of the EAFS (European Association of Forensic Societies) 1997 Conference, Lyon, 1997. Stru¨der L, Fiorini C, Gatti E, Hartmann R, Holl P, Krause N, Lechner P, Longoni A, Lutz G, Kemmer J, Meidinger N, Popp M, Soltau H, Weber U, von Zanther C. J. Synchrotr Radiat 5:268, 1998. Svietlicki E, Larsson NP, Yang C. Nucl Instrum Methods Phys Res B77:195, 1993. Tissot RG Jr, Boehme DR. The novel use of XRMF spectroscopy for environmental toxicity evaluation and recyclable potential in electronic components. Proceedings Denver X-ray Conference, 1994. Toribara TY. Adv X-Ray Anal 30:293, 1987. Toribara TY, Jackson DA, French WR, Thompson AC, Jaklevic JM, Anal Chem 54:1844, 1982. To¨ro¨k Sz, Faigel Gy, Jones KW, Rivers ML, Sutton SR, Bajt S. X-ray Spectrom 23:3, 1994. Trebbia P, Wulveryk JM, Bonnet M. Microbeam Anal 4:85, 1995. Ullrich JB, Huang KG, Owens SM, Aloisi DC, Hoﬀmann FA, Gao N, Klotzko IL, Gibson WM. Nucl Instrum Methods Phys Res A 364:362, 1995. Valkovic V, Miljanic D, Wheeler RM, Liebert RB, Zabel T, Phillips GC. Nature 243:543, 1973. Van Borm W, Wouters L, Van Grieken R, Adams F. Sci Total Environ 90:55, 1990. Van Dyck PM, To¨ro¨k SzB, Van Grieken RE. Anal Chem 58:1761, 1986. Van Espen P, Janssens K, Nobels J. Chemom Intell Lab Syst 1:109, 1986. Van Espen P, Janssens G, Vanhoolst W, Geladi P. Analusis 20:81, 1992. Vekemans B, Janssens K, Vincze L, Adams F, Van Espen P. X-ray Spectrom 23:278, 1994. Vekemans B, Janssens K, Aerts A, Adams F, Hertogen J. X-ray Spectrom 26:333, 1997.

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Vekemans B, Janssens K, Vincze L, Adams F, Van Espen P. Spectrochim Acta 50B:149, 1995. Vekemans B, Janssens K, Vittiglio G, Adams F, Andong L, Yiming Y. Adv X-ray Anal 41:278, 1998. Vekemans B, Vincze L, Szaloki I, Feng H, Riekel C, Van Grieken R, Adams F, Jones KW. Micro Xray ﬂuorescence tomography on single sediment particles from New York/New Yersey Harbor. Book of Abstract of European Conference on Energy Dispersive X-ray Spectrometry, Krakow, 2000, p106. Vincze L. Monte Carlo Simulations of conventional and synchrotron x-ray ﬂuorescence spectrometers. PhD dissertation, University of Antwerp (UIA), Antwerp, 1995. Vincze L, Janssens K, Adams F. Spectrochim. Acta 48B:553, 1993. Vincze L, Janssens K, Adams F, Jones K. Spectrochim Acta B50:1481, 1995b. Vincze L, Janssens K, Adams F, Rindby A. X-ray Spectrom 24:27, 1995c. Vincze L, Janssens K, Vekemans B, Adams F. Spectrom Acta B54:1711, 1999a. Vincze L, Janssens K, Vekemans B, Adams F. J Anal Atomic Spectrom 14:529, 1999b. Vincze L, Janssens K, Vekemans B, Adams F. Monte Carlo simulaiton of x-ray ﬂuorescence and scattering tomography experiments. Proc SPIE (Soc Photo-optical Inst Eng) Conference on Developments in X-ray Tomography II, Denver, CO, 1999c, p 328. Vincze L, Janssens K, Adams F, Rivers M, Jones K. Spectrochim. Acta B50:127, 1995a. Vis R. In: Catlow C, Greaves R, eds. Applications of Synchrotron Radiation. London: Blackie, 1990, p 311. Vittiglio G, Janssens K, Adams F, Oost A. Spectrochim Acta B54:1697, 1999. Voglis P, Rindby A. J Trace Microprobe Technol 13:177, 1995. Voglis P, Attaelmanan, Engstrom P, Larrson S, Rindby A, Bostro¨m K, Helander CG. X-ray Spectrom 22:229, 1993. von Ardenne M. Naturwissenschaften 27:485, 1939. von Seeman H. Ann Phys 59:455, 1919. Webster DL, Hansen WW, Duveneck FB. Phys Rev 44:258, 1933. Wegrzynek D. A feasibility study: Computer micro tomography using a laboratory x-ray ﬂuorescence micro-beam spectrometer. Book of Abstract of European Conference on Energy Dispersive X-ray Spectrometry, Krakow, 2000, p 25. Wegrzynek D, Holynska B, Ostachowicz J. X-ray Spectrom 28:209, 1999. Welfringer J, Bennoit P, Guyon M, Adv X-ray Anal 35:1301, 1992. Wherry D, Cross B. Kevex Analyst 12:8, 1986. Wherry DC, Cross B, Briggs TH. Adv X-ray Anal 31:93, 1988. Winston R. Sci Am 52:March, 1991. Wolter H. Ann Phys Lpz 10:94, 1952a. Wolter H. Ann Phys Lpz 10:286, 1952b. Worley CG, Havrilla GJ. Anal Chem 70:2957, 1998. Worley C, Havrilla G, Gao N, Xiao QF. Adv X-ray Anal 42:26, 1999. Wuttmann M. In: Valloggia M, Balat I, Le Mastaba de Medoe-Nefer, Annexe III: Analyse et e´tude du me´tal cuivreux de certains objets. FIFAO 31=1, Le Caire, 1986, p 215. Wuttmann M. The beginnings of copper and copper alloys metallurgy in Egypt. The need for a better characterisation of the artefacts. Proceedings First Int. Conference on Ancient Egyptian Mining and Metallurgy and Conservation of Metallic Artefacts, Cairo, 1996. Yamamoto N, Hosokawa Y. Jpn J Appl Phys 27L:2203, 1988. Yap CT, Tang SM, Archaeometry 26:78, 1984. Yap CT, Tang SM. Archaeometry 26:61, 1985. Yu KN, Miao JM. Appl Radiat Isot 48:959, 1997. Yu KN, Miao JM. Appl Radiat Isot 51:279, 1999. Zahrt JD. Adv X-ray Anal 32:89, 1989. Zietz L, Baez AV. In: Cosslett VE, Engstro¨m A, Pattee HH, eds. X-ray Microscopy and Microradiography, New York: Academic Press, 1957.

12 Particle-Induced X-ray Emission Analysis Willy Maenhaut Ghent University, Ghent, Belgium

Klas G. Malmqvist Lund University and Lund Institute of Technology, Lund, Sweden

I.

INTRODUCTION

In 1970, Johansson et al. (1970) demonstrated that the bombardment of a specimen with protons of a few megaelectron volts (MeV) gives rise to the emission of characteristic x-rays and that this can form the basis for a highly sensitive elemental analysis. This landmark article formed the starting point of the x-ray emission analysis technique which became known as particle-induced x-ray emission analysis (PIXE). Although PIXE is sometimes considered a variant of x-ray ﬂuorescence (XRF), such a classiﬁcation is not correct in a strict sense, because the technique does not rely on excitation of the sample by x-rays. Instead, heavy charged particles (i.e., protons, a-particles, or heavy ions) are used in PIXE to create inner-shell vacancies in the atoms of the specimen. As in XRF and electron-probe microanalysis (EPMA), the characteristic x-rays produced by deexcitation of the vacancies can be measured by either a wavelength-dispersive or an energy-dispersive detection system. However, whereas the two detection systems are employed in both XRF and EPMA, an energy-dispersive spectrometer with a Si(Li) detector is almost exclusively used in PIXE. The incident charged-particle beams in PIXE are invariably generated by particle accelerators. For the great majority of PIXE work, protons of 1–4 MeV, which can be produced by small accelerators (e.g., Van de Graaﬀ accelerators or compact cyclotrons), are employed. Such small accelerators are also used in other analytical techniques that utilize ions of a few megaelectron volts per mass unit, such as Rutherford backscattering spectrometry (RBS), nuclear reaction analysis (NRA), charged-particle activation analysis (CPAA), and accelerator mass spectrometry (AMS). Because of their common use of ion beams, these techniques and PIXE are often jointly referred to as ion beam analysis (IBA) techniques. Furthermore, because the same incident particle type and energy may be used in several of the techniques, the simultaneous analysis of a sample by PIXE and some other IBA techniques (particularly RBS and NRA) is often feasible, so that the elemental coverage may be increased down to the very light elements and=or information on the depth distribution may be obtained. Compared to x-rays, protons or other heavy charged particles have the advantage that they can be focused by electrostatic or (electro)magnetic lenses and may be transported over large distances without loss in beam intensity. As a result, incident ﬂuence 719

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densities (expressed as the number of impinging particles per square centimeter and per second) are generally much higher in PIXE than in ordinary tube-excited XRF. Moreover, focusing of particle beams down to micrometer sizes is possible, so that PIXE allows an analysis with high spatial resolution. The microbeam variant of PIXE has become known as micro-PIXE, whereas the common variant, which makes use of a millimeter-sized beam, is now often referred to as macro-PIXE. Focusing to micrometer sizes is also possible with electrons and has given rise to the powerful EPMA technique, but heavy charged particles have the clear advantage that they give rise to much lower continuum background intensity in the x-ray spectrum. As a result, the relative detection limits (micrograms per gram) are typically two orders of magnitude better in micro-PIXE than in EPMA. After the initial experiment by Johansson et al. (1970), the favorable characteristics of PIXE were rapidly realized by many researchers, particularly within the nuclear physics community. Its applicability and potential for solving numerous trace element analytical problems were extensively examined and abundantly demonstrated. In addition to the traditional bombardment in vacuum, external beam approaches (with the specimen either in the laboratory air or in a nitrogen or He atmosphere) were also attempted and were found to be useful, particularly in examining delicate and=or large objects. The progress of PIXE over the years can be followed from the proceedings of the eight international conferences exclusively dedicated to the PIXE technique and its applications (Johansson, 1977, 1981; Martin, 1984; Van Rinsvelt, 1987; Vis, 1990; Uda, 1993; Moschini and Valkovic´, 1996; Malmqvist, 1999) and of the ﬁve conferences on nuclear microprobe technology and applications (Grime and Watt, 1988; Legge and Jamieson, 1991; Lindh, 1993; Yang et al., 1995; Doyle et al., 1997), which contain numerous papers on microPIXE. By now, PIXE has evolved into a rather well-used and mature technique, as is demonstrated by the increasing numbers of research papers in which PIXE provided the analytical results and by the publication of the two textbooks on the technique (Johansson and Campbell, 1988; Johansson et al., 1995). For a comprehensive treatment of PIXE and its applications, a reading of these books is highly recommended.

II.

INTERACTIONS OF CHARGED PARTICLES WITH MATTER, CHARACTERISTIC X-RAY PRODUCTION, AND CONTINUOUS PHOTON BACKGROUND PRODUCTION

A.

Interaction of Charged Particles with Matter

1. Slowing Down of Charged Particles in Matter: Stopping Power When a beam of heavy charged particles of a few megaelectron volts per atomic mass unit penetrates into matter, it loses its energy gradually with depth, until it is ﬁnally stopped. The energy loss occurs mainly through inelastic Coulombic encounters with bound electrons, and in contrast to the case of electron beams, the direction of travel of an ion beam is scarcely altered during the slowing-down process. The stopping power S(E ) of an ion with energy E is deﬁned as the energy loss per unit mass thickness traversed: SðEÞ ¼

1 dE r dx

ð1Þ

where r is the density of the stopping material and x is the distance. As deﬁned here, S(E ) is expressed in units of kiloelectron volt per gram per square centimeter.

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Numerous experimental measurements of stopping powers are available. They formed the database to ﬁt the parameters of semiempirical equations (Anderson and Ziegler, 1977; Ziegler et al., 1985), which are now commonly used to obtain the stopping powers for all elements of the periodic table. For the energy range of 1–4 MeV, which is most important in PIXE, the uncertainty of the values calculated with the semiempirical equations is estimated at less than 3%. The stopping power for compounds or more complex matrices is obtained from those of the constituent elements through the Bragg–Kleemann additivity rule: Smatr ðEÞ ¼

n X

wi Si ðEÞ

ð2Þ

i¼1

where wi and Si ðEÞ are the mass fraction and stopping power of constituent element i, respectively. The total path length R of an ion may easily be obtained by integration of the stopping powers: Z0 R¼

dE SðEÞ

ð3Þ

E0

where E0 is the incident ion energy. Although the total path length is larger than the projected range, the diﬀerence between the two is smaller than 1% for incident protons of a few megaelectron volts. 2. Inner-Shell Vacancy Creation: Ionization and X-ray Production Cross Sections Many of the Coulombic interactions between protons or heavier ions and matter result in the ejection of inner-shell electrons. It is those interactions and their cross sections that are of importance in PIXE. Three basic theoretical approaches have been used to calculate the cross sections for inner-shell vacancy creation: the binary encounter approximation (BEA), the semiclassical approximation (SCA), and the plane-wave Born approximation (PWBA). The PWBA model, which applies perturbation theory to a transition from an initial state (plane-wave projectile and bound atomic electron) to a ﬁnal state (plane-wave projectile and ejected continuum electron), has been most elaborated, particularly by Brandt and co-workers. These researchers incorporated a series of modiﬁcations in the model to correct for its inherent approximations, and this resulted in the so-called ECPSSR treatment (Brandt and Lapicki, 1979, 1981) of K and L shell ionization cross sections. The ECPSSR treatment deals with the deﬂection of the projectile due to the nuclear Coulomb ﬁeld (C ), perturbation of the atomic stationary states (PSS) by the projectile, relativistic eﬀects (R), and energy loss (E ) during the collision. Cohen and Harrigan (1985) used the ECPSSR model to produce an extensive tabulation of K and L subshell ionization cross sections for most target elements and for protons and helium ions between 100 keV and 10 MeV. As in the original ECPSSR version of Brandt and Lapicki (1981), Cohen and Harrigan (1985) employed nonrelativistic hydrogenic wave functions to describe the atomic electrons. More elaborate relativistic Dirac–Hartree–Slater (DHS) wave functions within a Brandt and Lapicki (1979) formalism were used by Chen and Crasemann (1985, 1989) to produce K, L, and M shell ionization cross sections for protons of a few selected energies from 100 keV to 5 MeV and for a narrow range of selected target elements.

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For K shell ionization with protons, the cross sections as predicted by the ECPSSR and other theories were thoroughly compared with experimental data by Paul and coworkers (Paul and Muhr, 1986; Paul, 1987, 1989; Paul and Sacher, 1989) and a set of reference cross sections was derived (Paul and Sacher, 1989). Although no theory emerges that will predict the experimental data within a few percent for all target elements and energies (Paul, 1989), it is generally agreed (Johansson and Campbell, 1988; Cohen, 1990; Campbell et al., 1990) that both the tables of Cohen and Harrigan (1985) and of Chen and Crasemann (1985, 1989) are adequate for most K shell proton PIXE work. For the case of the L ionization, the situation is less satisfactory, however, as is discussed in detail by Johansson and Campbell (1988), Campbell (1988), Cohen (1990), and Johansson et al. (1995). Orlic´ and co-workers compared a large number of experimental L shell ionization cross section data for proton impact with the ECPSSR theoretical predictions (Orlic´ et al., 1994) and derived reference cross sections (Smit and Orlic´, 1994) in a similar manner as was done for the K shell by Paul and Sacher (1989). For practical purposes, one requires x-ray production cross sections for individual x-ray lines (i.e., sX p with p the x-ray line used for analysis). The x-ray production cross sections are related to the ionization cross sections through the following equations. For K x-rays, GKp I ¼ s o ð4Þ sX Kp K K GK where the index K refers to the K shell, sIK is the K shell ionization cross section, oK is the ﬂuorescence yield and GKp =GK is the fractional radiative width for line p. For L x-rays, X GLi;p ¼ s ð5Þ sX Li;p Li GLi where Li (with i ¼ 1; 2; 3) refers to the ith L subshell, sX Li is the x-ray production cross section for the subshell Li whose vacancy is ﬁlled up by the radiative transition, and GLi;p =GLi is the fractional radiative width. The x-ray production cross sections sX Li are related to the L subshell ionization cross sections sILi through I sX L1 ¼ sL1 o1

ð6Þ

I I sX L2 ¼ ðsL2 þ sL1 f12 Þo2

ð7Þ

I I I 0 sX L3 ¼ ½sL3 þ sL2 f23 þ sL1 ð f13 þ f12 f23 þ f13 Þo3

ð8Þ

0 where oi denotes the ﬂuorescence yield, fij denotes the Coster–Kronig probability, and f13 is the small radiative intrashell vacancy transfer probability. Calculation of the x-ray production cross section from the ionization cross sections thus involves additional atomic parameters; that is, ﬂuorescence yields and fractional radiative widths in the case of the K shell and ﬂuorescence yields, Coster–Kronig yields, and fractional radiative widths for the L shell. As to the K fractional radiative widths, either the experimental data of Salem et al. (1974) or the theoretical values of Scoﬁeld (1974a) that are derived from Dirac–Hartree–Fock (DHF) calculations are considered accurate within 1%. There is, however, a small failure of the theory in the atomic number region 22 < Z < 30, where experimenters (Perujo et al., 1987) agree on a deviation of a few percent. The K ﬂuorescence yields are usually taken from Krause (1979). These data

PIXE Analysis

723

are quite accurate for atomic numbers above Z ¼ 20, but the situation for the lighter elements is less clear (Maenhaut and Raemdonck, 1984; Paul, 1989). For these lighter elements, one currently prefers the data from Hubbell (1989). For the case of the L x-rays, Cohen and Clayton (1987) and Cohen (1990) suggest employing the fractional radiative widths of Salem et al. (1974) and the Krause (1979) ﬂuorescence and Coster–Kronig yields, and Cohen and Harrigan (1986) used this approach to convert their table of ECPSSR L ionization cross sections (Cohen and Harrigan, 1985) into a very useful table of production cross sections for up to 16 individual L x-ray lines. Campbell (1988), however, advocates the use of the DHF radiative widths of Scoﬁeld (1974b) and of the DHS ﬂuorescence and Coster–Kronig yields of Chen et al. (1981) in combination with the tabulated DHS L ionization cross sections of Chen and Crasemann (1985, 1989). A very practical alternative to tabulated theoretical ionization and x-ray production cross sections, particularly with computer calculations in mind, are parameterized or analytical formulas which are obtained by ﬁtting polynomial expressions to theoretical or empirical cross-sectional data. The equations derived by Johansson and Johansson (1976) have been widely used by PIXE analysts in the past, but it is now known that they progressively underpredict cross sections with increasing Z of the target element. More accurate formulas, in which similar functions were often used as in the Johansson and Johansson equations, have been presented by Paul (1984), Cohen and Clayton (1987), Miyagawa et al. (1988), Johansson and Campbell (1988), and Sow et al. (1993). For the proton energy range of 1–4 MeV typically used in PIXE, the ionization (and x-ray production) cross sections increase with increasing proton energy and decrease with

Figure 1 K and L shell ionization cross sections in barns (1 barn ¼ 10 7 24 cm2) as a function of proton energy and target atom. The values are theoretical ECPSSR predictions (Cohen and Harrigan, 1985). (From Johansson and Campbell, 1988. Copyright John Wiley & Sons, Ltd. Reproduced with permission.)

724

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increasing atomic number of the target atom. This is illustrated in Figure 1, which displays theoretical K and total L shell ionization cross sections (Cohen and Harrigan, 1985) for selected target elements. The very steep fall in the K ionization cross section with Z is particularly notable. 3. Elastic Encounters Protons or other heavy charged particles that pass near atomic nuclei can be scattered elastically (i.e., without causing nuclear or atomic excitation). The cross section for the elastic scattering is strongly dependent on the scattering angle and on the atomic number (Z ) of the scattering nuclide; it decreases with increasing angle and increases with increasing Z. The energy of the scattered particles also increases with Z. By measuring the elastically scattered particles, information can be obtained on the elemental composition of the sample and on the distribution of the elements with depth. This has given rise to the widely used technique of Rutherford backscattering spectrometry (Chu et al., 1978), which can often be elegantly employed in combination with PIXE to provide complementary information. On the other hand, the energetic scattered particles can also cause problems in PIXE. Indeed, when they enter the Si(Li) detector, their large energy is deposited there, and this gives rise to serious resolution deterioration and other electronic problems in many x-ray pulse processing systems (Mingay, 1983). A solution to this problem is to interpose a light-element absorber (preferably Be) between specimen and Si(Li) detector (e.g., Maenhaut and Raemdonck, 1984), but this invariably results in loss of sensitivity for the light elements. 4. Nuclear Reactions In addition to elastic encounters between incident particles and target nuclei, various inelastic interactions or nuclear reactions are possible. They include ( p, g), ( p, p0 g), and ( p, ag) reactions when protons are used as incident particles. The cross sections for those reactions vary in a rather irregular way with target nuclide and with incident particle energy. They generally increase with increasing energy but may exhibit intense resonance peaks at particular energies. Also, because of the Coulomb barrier, the cross sections are smaller for the heavier target elements than for lighter ones. By detecting the promptly emitted g-rays or charged particles for these nuclear reactions, elemental analysis and depth proﬁling of certain elements is possible. The analytical technique that employs these possibilities is referred to as nuclear reaction analysis. A comprehensive treatment of this technique can be found in the Handbook of Modern Ion Beam Materials Analysis (Tesmer et al., 1995). The technique is also brieﬂy presented in Sec. VII.B. 5. Other Interactions As indicated earlier, the interactions of charged particles and matter occur mainly through inelastic Coulombic encounters with bound electrons. This results in electron excitation and ionization but also in secondary phenomena that contribute to the continuous photon background in the PIXE spectrum. Another interaction which contributes to that background is projectile bremsstrahlung. All these interactions are discussed in some detail here. B.

Continuous Photon Background Production

1. Electron Bremsstrahlung The characteristic x-ray lines in a PIXE spectrum are superimposed on a continuum background that has a strong resemblance to that observed in EPMA. In both cases,

PIXE Analysis

725

electron bremsstrahlung is the major background component. Although this originates from primary electron interactions in EPMA, secondary phenomena are at its origin in PIXE. In the mid-1970s, Folkmann and various co-workers made very signiﬁcant contributions to our understanding of the continuum background in PIXE [e.g., Folkmann et al. (1974a, 1974b), Folkmann (1976)]. This work was later complemented and extended by Ishii and Morita (1984, 1987, 1988, 1990). It is by now clear that the electron bremsstrahlung in PIXE originates essentially from three processes: quasi-free electron bremsstrahlung (QFEB), secondary electron bremsstrahlung (SEB), and atomic bremsstrahlung (AB). SEB is formed by a two-step process: the incident particle ﬁrst ejects an electron from a target atom, and the secondary electron is subsequently scattered in the Coulomb ﬁeld of a target nucleus, thus producing the bremsstrahlung. The photon spectrum of SEB is characterized by an ‘‘end-point’’ energy Tm ¼ 4me Ep =Mp , with me and Mp the electron and projectile masses and Ep the projectile energy. Above Tm , the intensity of SEB decreases rapidly. QFEB is emitted when an electron of a target atom is scattered by the Coulomb ﬁeld of the projectile (this is a process in the projectile frame). The QSEB endpoint energy Tr is equal to Tm =4. The process AB occurs when a bound target electron is excited to a continuum state by the projectile and, returning to its original state, emits a photon. The relative contributions of QFEB, SEB, and AB to the electron bremsstrahlung background are schematically shown in Figure 2. As can be seen from Figure 2, AB predominates in the high-energy part of the spectrum (i.e., for photon energies above Tm ), whereas QFEB becomes the prevailing component at low photon energies (below Tr ). As both Tr and Tm increase linearly with increasing projectile energy Ep , the PIXE spectrum (which typically extends from 0 at about 20–30 keV) has a quite diﬀerent appearance depending on the value of Ep . In addition to Ep , the matrix composition of the target also plays a critical role in both the shape and intensity of the electron bremsstrahlung background. The intensity of this background per unit mass thickness is roughly proportional to the average Z of the matrix. The electron bremsstrahlung is emitted anisotropically and is lower at forward and backward angles than at 90 . For this reason, the Si(Li) detector is frequently positioned at an angle of 135 in an experimental PIXE chamber.

Figure 2 Schematic representation of the relative contributions of QFEB, SEB, and AB to the electron bremsstrahlung background. For an explanation of the acronyms and symbols, see the text. (From Ishii and Morita, 1987; with permission from Elsevier Science.)

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2. Projectile Bremsstrahlung While passing through the target, the incident charged particles may be decelerated in the Coulomb ﬁeld of atomic nuclei and thereby give rise to the emission of projectile bremsstrahlung (PB). Although this process is negligible within the context of the projectile energy loss, it contributes to some extent to the continuum background in the PIXE spectra. For a detailed treatment on PB in PIXE we refer to the papers by Folkmann and co-workers (Folkmann et al., 1974a, 1974b; Folkmann, 1976). The PB is much less important than the electron bremsstrahlung and becomes only signiﬁcant at photon energies above 10–20 keV.

3. g-Ray Background The prompt g-rays emitted as a result of nuclear reactions between the projectiles and the target atoms are generally quite energetic and are therefore far outside the 0–30-keV energy range typically observed in PIXE. However, through Compton interactions of these g-rays with the Si(Li) detector, a wide spectrum of electron energies is generated, and this, in turn, gives rise to a slowly varying continuum in the PIXE spectrum. This Compton-scattering background is not predictable in the sense that the bremsstrahlung background is predictable, as it does not depend on the matrix composition of the target alone but rather on the presence of particular elements with large cross sections for nuclear reactions yielding g-rays. Na and, particularly, F are examples of elements that are often responsible for a noticeable or even signiﬁcant g-ray background. A large fraction of the g-ray background in the PIXE spectrum may originate from nuclear reactions that do not take place in the specimen but instead in the beam collimators, Faraday cup, or various other parts of the PIXE irradiation chamber [even the x-ray absorber placed in front of the Si(Li) detector]. As discussed by Johansson and Campbell (1988) and Johansson et al. (1995) and brieﬂy indicated in Sec. III.B.1, the materials for the collimators and various other parts of the chamber should therefore be selected with care.

4. Other Sources of Background In many cases, intense characteristic x-ray peaks are present in the PIXE spectrum. This is particularly true when the matrix consists of element(s) with atomic number above Z ¼ 11. These intense peaks and their escape and pileup peaks seriously hamper (or even preclude) the detection of other elements with characteristic x-rays of similar energy. Furthermore, incomplete charge collection in the Si(Li) detector and other processes (Johansson and Campbell, 1988; Johansson et al., 1995) have the eﬀect that each x-ray peak exhibits low-energy tailing, which typically consists of two major components, namely an exponential tail and a ﬂat shelf with height up to 1% of the peak height. Consequently, tailing associated with very intense peaks forms a substantial component of the total background. Because of the intense peaks and associated tails for higher-Z matrices, and because, in addition, the electron bremsstrahlung intensity increases with increasing Z of the matrix, PIXE is much more suitable for analyzing trace elements in light-element matrices than in heavy-element matrices. When heavy ions are used as projectiles, processes other than those discussed thus far also contribute to the continuum background. According to Folkmann (1976), radiative electron capture and molecular orbital x-rays are important.

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5. Enhancement of Signal to Background inTotal-Reflection PIXE The angle of incidence y (i.e., the angle between the incident proton beam and the specimen surface) in a normal PIXE setup is typically either 45 , 67.5 , or 90 . When a much smaller angle (0–35 mrad) is used, the MeV proton (or a-particle) beam can be totally reﬂected from the surface (Vis and Van Langevelde, 1991). This phenomenon forms the basis of total-reﬂection PIXE (TPIXE). In TPIXE, characteristic x-rays and bremsstrahlung (and g-ray) background are only produced from a very shallow surface layer. This feature can advantageously be used to increase the ratio of the characteristic X-ray signal intensity to the background when analyzing thin layers on a thick substrate, such as top layers on Si wafers or dried solutions on a backing ﬁlm. In normal PIXE, the bulk of the Si wafer or the backing ﬁlm will also provide a substantial contribution to the background intensity, whereas this contribution is absent or at least much reduced in TPIXE. The principle and applicability of TPIXE are similar to those of totalreﬂection XRF (TXRF). Pioneering work on TPIXE was done by the Amsterdam PIXE group and further details on the technique can be found in several publications from that group (Vis and Van Langevelde, 1991; van Kan and Vis, 1995, 1996, 1997; van Kan, 1996).

III.

INSTRUMENTATION

A.

Accelerators

The ion beams used for PIXE analysis are produced in an accelerator. The energy range required, 1–4 MeV=u, means that relatively small accelerators are suﬃcient. Electrostatic accelerators or cyclotrons are suitable sources (Scharf, 1989). Electrostatic accelerators are most commonly used. They may be of a regular or modiﬁed Van de Graaﬀ type, in which a high-voltage terminal is charged by a belt or a metal chain. A second type of electrostatic accelerator uses a high-frequency voltage multiplication stage to charge the terminal, the ions being accelerated in the electrostatic ﬁeld created by the terminal. Cyclotrons have been employed at some laboratories. These are particularly suitable for PIXE at higher ion energies (Durocher et al., 1988; McKee et al., 1989; Peisach and Pineda, 1990) and in certain combinations of PIXE with complementary techniques, such as particle elastic scattering analysis (PESA) or forward alpha scattering (FAST) (Cahill et al., 1984). In most analytical situations in which PIXE is applied, the choice of accelerator is governed by access to a particular machine rather than by free choice. When a dedicated accelerator can be selected, however, usually a single-ended or tandem electrostatic machine of 2–3 MV is chosen. Such accelerators may also favorably be employed for a combined use of ion implantation and ion-beam analysis, including PIXE. Certain types of analysis, such as analyses by microbeam techniques, impose special requirements on ion source brightness, energy stability, beam emittance, and minimum scattering of the ion beam. Such requirements may play an important role in selecting an accelerator. It is often stated that the need for an accelerator in PIXE is a major drawback, for reasons or both technical complexity and economy. Although such objections contain some truth, it should be pointed out that the prices of small electrostatic accelerators are of the same order as those for complex ‘‘traditional’’ analytical equipment, such as mass spectrometers or electron microscopes, and that a comparable technical skill is required of the operator. It is therefore unjustiﬁed to rule out the PIXE method for such reasons.

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Beam Handling and Target Chambers

1. Macro-PIXE inVacuum Commercially available vacuum chambers used as scattering chambers in nuclear physics experiments are generally not well suited for PIXE. Their shapes and dimensions hamper obtaining large solid angles for detection and introducing multispecimen holders and sample changers. As a consequence, few existing PIXE chambers are of commercial origin. There are exceptions to this rule, however, and certain chambers have been built entirely from commercially available components. Figure 3 shows the design and principal components of a hypothetical ‘‘typical’’ setup for macro-PIXE analysis. In the following, the details of such a system are outlined. The ion beam emerging from the accelerator ﬁrst passes through an analyzing magnet, which sorts out the ions of the correct mass and velocity, and is then normally focused by electrostatic or magnetic quadrupole lenses onto the specimen. A typical experimental facility for macro-PIXE analysis employs a system for producing a homogeneous beam so that the specimen is evenly irradiated. This is required for quantitative analysis of heterogeneous samples. There are various methods of achieving this: 1.

2.

3.

Focusing the beam onto a scattering foil, which distributes the ions homogeneously over the cross section selected by a diaphragm. The foil material should be able to withstand the ion beam for long times. It is often made of thin metal, such as gold, which is an excellent scatterer with low stopping power for the ions. Rastering the beam over the sample by electrostatic deﬂection, for example. This does not give a genuinely homogeneous distribution and may, therefore, produce high instantaneous count rates in heterogeneous samples. On the other hand, this method has the advantage of relatively low loss of beam intensity, whereas such a loss may be more than 90% in the case of method 1. After the initial alignment of the beam, straightforward symmetric defocusing by the quadrupole magnets.

After homogenization, the beam is deﬁned by a pair of collimators (diaphragms), which are normally circular with diameters between 1 and 10 mm and placed approximately

Figure 3 Typical experimental arrangement for routine PIXE, with detectors for x-rays and g-rays. Normally, the collimators can be changed between 1 and 10 mm diameter, and the x-ray absorber holder would contain 5–10 different filters.

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100–200 mm apart, just before the entrance to the irradiation chamber. The collimator material should preferably withstand bombardment by the ions over long periods and produce low background in the PIXE spectra. This can be achieved by using a material containing nuclei with low cross sections for producing g-rays at the ion energy interval in question. Common choices of collimator materials are tantalum and graphite (Johansson et al., 1995). To measure the beam ﬂuence on the target accurately, the collimators should be grounded and electrically shielded from the irradiation chamber in which the target is placed, and secondary electrons generated at these collimators should be prevented from entering the target chamber. A target holder deﬁnes the proper position of the sample during analysis. In a typical setup for PIXE analysis, in vacuo, several samples are simultaneously inserted into the vacuum, so that a high sample throughput is facilitated. The normal slide frame format (565 cm2) is a standard target frame in many laboratories, and often 36–80 target frames are contained in vacuum. The samples are irradiated at an angle suitable to detect the x-rays, and often the detector views the sample from a backward angle to 135 or more, as this produces higher peak-to-background ratios in the PIXE spectra (see Sec. II.B.1). In contrast to energy-dispersive XRF (EDXRF) spectra, PIXE spectra have most of their x-ray intensity in the low-energy region, so that the spectral shape can advantageously be modulated by placing an x-ray absorber between sample and detector. The aim of absorbers invariably is to reduce or eliminate unwanted continuum background and=or intense x-ray peaks and their associated pileup peaks and at the same time to allow bombardments at higher beam intensities, so that the elements of interest can be measured in shorter bombardment times and with fewer spectral interferences. The absorbers are usually made from organic material (e.g., Mylar) or from light-element metal foils (Be and Al). Plain absorbers are employed if complete elimination of the low-energy part of the spectrum is desired. In many cases, however, it is preferable to allow a certain fraction (e.g., a few percent) of the low-energy x-rays to pass on onto the detector. This can be realized elegantly by resorting to pinhole absorbers (usually called ‘‘funny ﬁlters’’ within the PIXE community). Also, more sophisticated designs consisting of a combination of various layers with diﬀerent thicknesses and pinhole diameters have been used (Carlsson and Akselsson, 1981). Figure 4 illustrates the eﬀect of some absorbers on the appearance of the PIXE spectrum for a biological reference material (Maenhaut, 1990a). In the acquisition of the top spectrum, a funny ﬁlter, consisting of a 52-mm-thick Be foil and a 324-mm Mylar ﬁlter with 5.54% hole, was placed in front of the detector, and even at a beam current of 10 nA, the count rate exceeded 2000 cps. For acquiring the bottom spectrum of Figure 4, a 660-mm Mylar absorber was interposed between specimen and detector. This had the eﬀect a allowing a beam current of 150 nA, and the count rate remained below 1000 cps. If one wants to analyze samples with high concentrations of elements whose lines severely interfere with nearby lines of the elements of interest, it is useful to resort to ‘‘bandpass’’ absorbers. For instance, in the PIXE analysis of steel, a chromium absorber selectively suppresses the strong FeK lines (Ahlberg et al., 1975b). Other examples in which such complex absorbers are advantageous are the analysis of metal alloys in archaeology [e.g., of bronzes (Swann, 1983; Swann and Fleming, 1990)] and various analyses in material studies. When one is interested in measuring the light element (Z < 13) by PIXE, only a very thin x-ray absorber may be used, and for the very light elements, a windowless detector or a detector with an ultrathin polymer window (Quantum X-ray detector, Kevex Corp., 1987) would even be advisable. However, as indicated in Sec. II.A.3, bombardment of a sample with charged particles also gives rise to backscattered particles, and a fraction of

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Figure 4 PIXE spectra for a 5-mg=cm2 NIST bovine liver specimen. Incident proton beam of 2.4 MeV. Top spectrum: taken with funny filter (see text); beam current 10 nA, x-ray count rate 2200 cps, and preset charge 20 mC. Bottom spectrum: taken with a 660-mm Mylar absorber; beam current 150 nA, count rate 900 cps, and preset charge 200 mC. The ordinate scale applies to the top spectrum. All marked peaks are K lines (Ka and Kb). Most of the unmarked peaks in the top spectrum are sum peaks. (From Maenhaut, 1990a.)

these may penetrate into the Si(Li) detector and cause problems. Hence, it is advisable to use an absorber that is suﬃciently thick to stop all scattered particles, but this hampers or precludes the detection of those light elements. According to Musket (1986) and Nejedly et al. (1995), placing a magnetic deﬂection trap between the sample and the detector crystal is a viable alternative to an absorber for removing the scattered ions. However, the installation of such system also results in a lower solid angle of detection. The Si(Li) detectors used for PIXE are the same as those in EDXRF (see Chapter 3). They typically have a sensitive area from 10 to 80 mm2, but this area is often reduced by inserting an x-ray collimator in front of the detector. The aim of such collimator is to minimize the low-energy peak tailing that results from incomplete charge collection at the edge of the detector crystal. In TPIXE, the special geometry with a very small angle of incidence allows one to position the x-ray detector very close to the specimen (van Kan and Vis, 1997), similarly as in TXRF. Consequently, a very large solid angle can be obtained, thereby increasing the sensitivity. This increased sensitivity in combination with the reduced bremsstrahlung background is at the origin of the low limits of detection in TPIXE (van Kan and Vis, 1997). The ampliﬁers=pulse processors in energy-dispersive x-ray spectroscopy require large time constants for optimum energy resolution (see Chapter 3). Unfortunately, large time constants also imply that pulse pileup already becomes a serious problem at relatively low count rates. It is therefore common practice to incorporate an electronic pileup rejector

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and its associated dead-time correction circuitry in the pulse processing chain. Such solution reduces the throughput of signals to the analyzer (i.e., it decreases the ratio of output signal count rate to beam current) and this forms a drawback when analyzing samples that deteriorate during bombardment. To minimize beam-induced damage during PIXE analysis, an on-demand beam excitation system is recommended. Such a system employs a set of deﬂection plates placed in the beam path upstream of the sample; when an x-ray event is detected, the beam is deﬂected immediately and then held oﬀ until the event has been processed (Malmqvist et al., 1982). It should be indicated here that pileup peaks (or sum peaks) are not entirely eliminated by pileup rejectors or on-demand beam excitation, because such systems have a ﬁnite pulse pair resolving time or beam-switching time. In PIXE of thin specimens, the ions pass through and are dumped in a Faraday cup (see Fig. 3) for charge integration. To minimize the g-ray background originating from nuclear reactions in the Faraday cup, long cups in which the particles are collected far from the detector are preferable. Furthermore, the escape of secondary electrons from the cup as well as the entrance of secondary electrons from the specimen should be prevented by placing a negatively biased ring in from of the cup. The charge integration itself is accomplished by connecting the Faraday cup to a sensitive current integrator=digitizer. When the samples are thick enough to stop the ions, the beam current must be measured either on the whole irradiation chamber or through some indirect approach. One possible method is to place a thin foil in the beam path upstream of the target and to measure the intensity of the particles scattered from it with a surface barrier detector (Mitchell et al., 1980). Another, related method uses a beam chopper (e.g., a thin metal strip, which periodically passes through the beam) (Volvo et al., 1983). The bombardment of an insulating thick (or semithick) specimen in vacuum generally gives rise to charge buildup, and the specimen may reach a positive potential of up to several tens of kilovolts before breakdown and sparking. The high potential accelerates electrons up to tens of kiloelectron volts, and as a consequence, a huge bremsstrahlung background is produced in the PIXE spectrum. The peak-to-background ratios then decrease signiﬁcantly (Ahlberg et al., 1975a). To avoid this, various methods can be used: 1. Increased pressure in the chamber (Ahlberg et al., 1975a) 2. Thin carbon coating of the specimen (Cabri et al., 1985), as is commonly done in EPMA 3. Placing a thin carbon foil just in front of the sample (Chaudri and Crawford, 1981) 4. Spraying with electrons from an electron gun (Ahlberg at al., 1975a) 5. Using strong permanent magnets (Mingay and Barnard, 1978) These methods either avoid the buildup of a high potential by conduction of the positive charge or by producing electrons to neutralize the positive charge or avoid the eﬀects from the electrons impinging on the sample (magnets). In addition to the equipment used for quantitative PIXE analysis, target chambers also often include equipment necessary for the complementary IBA techniques presented in Sec. VII, such as surface barrier detectors of measuring charged particles or Ge detectors for measuring prompt g-rays. 2. Nonvacuum Macro-PIXE As has already been indicated, it is sometimes advantageous to use atmospheric pressure or moderate vacuum instead of high vacuum during analysis (Williams, 1984). The advantages

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are that the heat conductivity is increased and the target temperature decreased, the charge is conducted from the target, and the vacuum requirements are less, thus making it easier to design a low-cost chamber. Because the accelerator requires a good high vacuum, the ion beam must be extracted into the moderate vacuum or atmospheric pressure region through a thin exit foil. The beam eventually deteriorates this foil, and, therefore, its material has to be carefully selected. The best choice is the polyimide foil Kapton, which withstands high intensities and a high radiation dose before mechanical breakdown. Supporting the foil by a carbon grid and direct ﬂow of liquid-nitrogen-cooled helium gas allows the use of high beam intensities over extended time intervals (Hyvonen-Dabek et al., 1982). The chamber gas is normally helium or nitrogen. In addition to its better cooling properties, helium produces less bremsstrahlung than nitrogen. Some x-ray detector windows are not leak proof to helium, however, and this has disastrous consequences for the cryostat vacuum unless the detector is separated from the chamber gas by an additional window. The choice of chamber atmosphere is determined by the objectives of the analysis and by the samples to be analyzed. A rather low pressure of helium gas, for example, suﬃces to improve the heat conductivity and to reduce thermal losses of elements or compounds (Martinsson, 1987). On the other hand, an external beam in air is needed when large objects are to be analyzed without sectioning of subsampling. This is very important in archaeology and art science, for example. The disadvantages of bombardment in air are that there is a danger of sample oxidation during irradiation and interfering argon x-ray lines are present in the spectrum. The strong argon lines may also be used to monitor the beam ﬂuence, however, which is otherwise diﬃcult to do in nonvacuum PIXE. In fact, one generally relies on some indirect method of beam current measuring (Mitchell et al., 1980; Volkov et al., 1983); although with special precautions, it is possible to measure the beam current directly (Wookey and Rouse, 1987). The advantages of using higher pressure during irradiation have been demonstrated in various applications. For example, in studies on volatile organic compounds, using a combination of PIXE and complementary techniques (PESA), it was shown that signiﬁcant losses of certain constituents occurred when bombarding in high vacuum, whereas the losses were insigniﬁcant for bombardments of similar beam intensity in helium at a pressure of 100 torr (Martinsson, 1987). When using external beams, some safety precautions are necessary. Besides the obvious hazard of direct exposure, there is the potential exposure to the scattered beam and the activation of the air to form positron-emitting radionuclides. These hazards are discussed in detail by Doyle et al. (1991) in a review of external beam work with various analytical techniques. 3. Nuclear Microprobes In nuclear microprobe or micro-PIXE analysis, the particle beam is collimated and=or focused down to dimensions in the range of 1–50 mm. With the regular equipment used in accelerator-based ion-beam analysis, beam sizes down to typically a few tenths of a millimeter diameter are easily obtained, but to produce a genuine microbeam, specially designed equipment is required. The simplest way of producing a microbeam is to employ a pinhole collimator (Horowitz and Grodzins, 1975). For very small collimator sizes, however, the beam intensity obtainable is much too low for practical use. In addition, a substantial fraction of the ions are scattered at the edge of the collimator, and this gives rise to a halo (with an intensity of several percent of the beam current) around the central beam. In most micro-

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733

PIXE systems developed since the early seventies, collimation is therefore combined with an electrostatic or magnetic demagniﬁcation system, such as quadrupole doublets, triplets, quadruplets, or superconducting solenoids (Legge, 1997). The initial collimation in an object aperture normally provides a beam with dimensions of 5–100 mm, and this beam is then demagniﬁed by a factor of 5–100. New-generation microprobes with enhanced demagniﬁcation power (200–3006) are under development in a few laboratories (Breese et al., 1999). The best systems currently available are able to produce a spatial resolution of about 0.5 mm at the specimen while maintaining an ion current that is useful for PIXE analysis ( > 100 pA). Figure 5 shows the layout of a nuclear microprobe. Although the exciting particles form the only diﬀerence between micro-PIXE and energy-dispersive EPMA, much better peak-to-background ratios and, consequently, lower detection limits are obtainable by micro-PIXE. This is illustrated in Figure 6, where x-ray spectra for a biological specimen obtained by both techniques are compared. The much better peak-to-background ratio, particularly for the heavier elements (Z > 20), justiﬁes the use of the much more complex analytical equipment of the nuclear microprobe. In addition, the high resolution of the nuclear microprobe makes it possible to combine PIXE with several other analytical techniques for imaging and quantitative analysis. As shown in Figure 5, the normal components of a nuclear microprobe comprise the following: 1. 2. 3. 4.

A particle accelerator (normally electrostatic) with a very bright ion source Precision collimators Magnetic or electrostatic quadrupoles for focusing A scanning system to raster the beam over the sample, as in a scanning electron microscope

The detection system is, in principle, identical to that for macro-PIXE but usually includes complementary surface barrier detectors for scattered particles and for particles emitted in nuclear reactions to extend the analytical arsenal (Malmqvist, 1995a). Sometimes, a detection system for secondary electrons (Kneis et al., 1982) and=or a detection system for near-visible light, ionoluminescence (Yang et al., 1993) are also included, for imaging of the specimen (see Sec. VII.C). Facilities for accurate positioning of the beam on the specimen area of interest are also required. Hence, an optical viewing system with high magniﬁcation and good resolution is needed, as well as a precision sample holder controlled by stepping motors or piezoelectrical crystals with a position accuracy below 1 mm. The data are acquired in dedicated computer systems that can produce both quantitative results and qualitative elemental maps. Various types of object collimators can be used. The collimator design should minimize the contribution of scattered ions to the specimen. Such devices with precisionpolished cylinder surfaces (Fischer, 1988) to deﬁne the ion beam and local or remote control of the collimator width are commercially available and are employed in most nuclear microprobe setups. Other laboratories prefer to use ﬁxed apertures of the same design as for electron probes (Legge et al., 1982). The precision parts of the collimators are normally protected against beam damage and excessive heating by adding a slightly larger precollimator in which most of the energy is dissipated. Each conﬁguration of demagniﬁcation devices has its advantages and drawbacks. The rather complex Russian quadruplet, which was used in the ﬁrst micro-PIXE system at Harwell (Cookson et al., 1972), produces a symmetric image but is diﬃcult to align mechanically and has an increased risk of parasitic aberrations. Because of its simplicity, some laboratories use the doublet conﬁguration, although it requires rectangular

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Figure 5 Base components (not to scale) of a typical nuclear microprobe system: (a) electrostatic particle accelerator; (b) primary object aperture; (c) secondary collimator; (d) focusing system; (e) scanning system; (f ) video camera and microscope; (g) surface barrier detector for scattered particles; (h) x-ray detector; (i) specimen; ( j) surface barrier detector for transmitted particles (STIM); (k) front-end CAMAC with data bus; (l) main computer and display with elemental map.

collimation for obtaining a symmetric image. The most common system, at present, is the commercially available triplet system from Oxford MicrobeamsTMLtd. (Oxford, UK). When more than two quadrupoles are used, two magnets are connected in series to the same current supply. Whatever conﬁguration is used, the current supplies should be very stable ( < 10 7 4), to reduce distortions due to chromatic aberrations. The image size at the specimen is easily determined from ﬁrst-order calculation. For small apertures, however, the ﬁrst-order calculation does not suﬃce and second- and third-

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Figure 6 X-ray spectra from a thin biological specimen (human brain) obtained with electron (a) and nuclear microprobe (b) excitation. Note the very large difference in peak-to-background ratio, particularly from about 3 keV (channel 100) up. (From Johansson and Campbell, 1988. Copyright John Wiley & Sons, Ltd. Reproduced with permission.)

order calculations are required. The lens aberrations can be calculated if the details of the conﬁguration are known. For more information on this subject, the reader is referred to the book by Grime and Watt (1984), which provides a comprehensive compilation of various magnetic quadrupole systems. It should be noted here, that among the intrinsic aberrations, the third-order, spherical (angular) aberrations are the dominating ones in most nuclear microprobes. Hence, some systems make use of octupole magnets to reduce the spherical aberrations (Jamieson and Legge, 1988). Other important factors are the mechanical precision (to reduce parasitic aberrations) and, particularly, the rotational defects in the magnetic ﬁelds (these can severely distort the image). To allow full use of its powerful analytical capabilities, a nuclear microprobe setup should include a scanning system for rastering the ion beam over the specimen surface.

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The scanning system could be installed before focusing ( predeﬂection), but such a system deﬂects the beam out of the optical axis and could therefore, for large deﬂection angles, signiﬁcantly increase the aberrations. By proper selection of the position of deﬂection, however, the beam can be made to pass through the optical center of the lens system so that the aberrations are minimized. The scanning system can also be installed after the demagniﬁcation lenses. Because of the short distance between lens and specimen, however, scanning systems using electrostatic deﬂection plates require a high electric ﬁeld (with concomitant risk of electric discharges) to obtain a suﬃciently large deﬂection amplitude. An alternative is to employ magnetic scanning. By using scanning coils with ferrite cores, a reasonably large amplitude and a scanning frequency of more that 5 kHz can be obtained (Tapper et al., 1988). Such systems provide a good compromise between scanning speed and amplitude. The scanning system is usually computer controlled and connected to the data acquisition system (see Sec. III. C.2). The detailed design of the irradiation chamber for the nuclear microprobe is beyond the scope of this chapter. We will therefore limit ourselves to giving some recommendations for the essential components. The sample-positioning system may be a commercially available x-y-z precision translator as designed for scanning electron microscopes. The sample holder should preferably take many samples to avoid the need for frequent opening of the vacuum chamber. The microscope used for viewing the specimen should have a magniﬁcation of 200–4006 and preferably be equipped with a zoom lens. In addition to the x-ray detector, surface barrier detectors should be entered in the forward and backward directions. Such detectors are needed for extending the elemental coverage and for determination of the specimen thickness. The specimen thickness is of paramount importance for PIXE quantiﬁcation in thin and semithick samples (see Secs. IV.B. and IV.C). The specimen thickness may also be provided by measuring the energy loss in each specimen pixel, as is done in scanning transmission microscopy (STIM) (Bench et al., 1992). Finally, it is recommended to install a secondary electron detector and to allow for detection of visible light to facilitate imaging of the specimen surface. A good high vacuum is crucial to reduce residual gas scattering that would degrade the beam quality. A high vacuum close to and in the specimen chamber is maintained by direct pumping with oil diﬀusion, turbomolecular, or cryopumps. Although they have many advantages, the two latter types may transmit vibrations and magnetic disturbance to the demagniﬁcation lens system and to the specimen holder. Hence, it is crucial to arrange those pumps to minimize such eﬀects. The focusing of small beams is adversely aﬀected by these or other vibrations. It is therefore common to place the whole microprobe system on a rigid optical bench on a fundament with good damping. The accurate positioning of the optical elements on this bench is realized by high-precision mechanical controls. Furthermore, to avoid the eﬀects of the Earth’s magnetic ﬁeld and stray ﬁelds from surrounding equipment, beam tubes are sometimes shielded by m-metal foils. As is the case for macro-PIXE, micro-PIXE may also be carried out under nonvacuum conditions (see Sec. III.B.2). Because of scattering of the beam by the gas, however, nonvacuum micro-PIXE is only feasible for moderately small beam sizes (20–100 mm). After collimation and=or focusing, the beam is passed to the nonvacuum region through a pinhole or an exit foil. In the ﬁrst approach, the high vacuum in the beam line is maintained by means of diﬀerential pumping. If the spatial resolution requirements are not too high, the nonvacuum micro-PIXE technique is rather straightforward and simple to use. It is very useful when examining large samples or sensitive art objects, such as bronze ﬁgures and ancient documents, and may, hence provide unique information (Swann, 1983).

PIXE Analysis

C.

737

Data Acquisition

1. Macro-PIXE The data acquisition systems used in PIXE show great similarity with the EDXRF data acquisition systems discussed in Chapter 3. For x-ray detection, a high-resolution solidstate detector is virtually always employed. This could be either a Si(Li) or high-purity Ge (HPGe) detector, but the former is highly preferable in most PIXE work. Ge detectors give rise to much more intense escape peaks than Si(Li), and this can render the spectra unacceptably complex. The higher detection eﬃciency of Ge for energies above 20 keV is not very useful in 1–4-MeV PIXE because of the very rapid decrease in K ionization cross sections with increasing Z (see Sec. II.A.2). On the contrary, the higher detection eﬃciency may be a disadvantage, as the background contribution resulting from Compton-scattered g-rays is larger than with Si. Most PIXE chambers contain only one Si(Li) detector for x-ray detection. However, to improve the sensitivity for the heavier elements while retaining the capability of measuring the light elements during the same bombardment, the use of two Si(Li) detectors has been advocated (Wa¨tjen et al., 1990), and such a system is now employed in an increasing number of PIXE facilities (Johansson et al., 1995). The second Si(Li) detector, used for measuring the heavier elements, is provided with a thick absorber to cut down the high count rate from the light elements and has a larger solid angle of detection than the ﬁrst Si(Li). As an alternative to such a second Si(Li), a Ge x-ray detector is well worth considering. In addition to the Si(Li) detector(s) for x-rays, PIXE chambers generally also contain detectors for the complementary IBA techniques (i.e., surface barrier detectors for scattered particles or for particles resulting from nuclear reactions and a Ge detector for measuring prompt g-rays). Hence, the acquisition system must include several analog-to-digital converters. Furthermore, because the spectral intensity in PIXE is proportional to the number of incident particles, the measurements are generally carried out for a preset charge (or some parameter related to it in the case of indirect beam current measurement) instead of for a preset live time. The charge (or related parameter) is usually measured by an external counter, and this unit forces the data collection to stop when its preset is reached. For acquiring the spectra, either a personalcomputer (PC)-based multichannel analyzer (MCA) or a classical MCA may be used, but the latter should be interfaced to a computer so that the spectra can be saved on disk and evaluated by appropriate computer programs.

2. Micro-PIXE The data acquisition in scanning nuclear microprobe analysis is more complex than in macro-PIXE and is invariably controlled by computers. As in more advanced systems for macro-PIXE, signals from several detectors must be handled, but, in addition, the positional information must be recorded. In Figure 7, a typical modern design of a data acquisition system is outlined (Elfman et al., 1997). Two main principles are employed: on-line display of elemental maps and event-by-event acquisition with oﬀ-line sorting of data. Sometimes, a combination of both is used. Elemental maps on-line are obtained by setting energy windows for the characteristic x-ray lines of interest (and for background regions) and reading out the count rate within each window for each position of the beam. The line intensity values (with or without background subtraction) for each beam position (pixel) are stored in a computer, and maps are generated on-line with intensity modulation by gray or color codes. This technique gives good feedback, so, that, during analysis, one can concentrate on the more interesting regions of the specimen. This is quite important

Figure 7 Schematic outline of a comprehensive data acquisition system for a nuclear microprobe including several detectors. A CAMAC system is used as front end and several personal computers interact with the system control and the data acquisition. (Courtesy of M. Elfman.)

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739

in micro-PIXE analyses, because they are often very time-consuming. However, a disadvantage of on-line elemental mapping is that truly quantitative information on the elemental concentrations is normally not provided. However, in a further development of this technique, dynamic analysis, on-line quantitative elemental maps are produced (Ryan and Jamieson, 1992). Using a spectral decomposition transform, that closely approximates the time-consuming nonlinear least-squares method, quantitative PIXE analyses can be performed in live time, continuously updating as the data accumulate. This dynamic analysis approach is able to accumulate on-line PIXE elemental maps that are inherently overlap-resolved and background-subtracted. In the event-by-event or list-mode type of analysis, data acquisition is normally handled by the same computer that controls the beam scanning. When an event (x-ray, scattered particle, g-ray, or secondary electron) is registered in a detector, the computer is triggered, and the detector label, the energy of the radiation, and the coordinates of the pixel where the event occurred are recorded on disk. The same procedure is repeated for each event detected. It is also possible to record the dead-time losses and the accumulated charge for each pixel. The data obtained for all events can be sorted or analyzed oﬀ-line in any selected manner. The use of modern very powerful personal computers or workstations has facilitated rapid evaluation of spectra generated from the data in each pixel, group of pixels, or whole maps, so that quantitative results of high accuracy may be obtained.

IV.

QUANTITATION, DETECTION LIMITS, ACCURACY, AND PRECISION

A.

Analysis of PIXE Spectra

Once a PIXE spectrum has been acquired, the ﬁrst step in the quantitation is the extraction of the net peak intensities for the elements of interest. This task is similar to that to be carried out in all other x-ray emission analysis techniques with energy-dispersive detection, and we therefore refer to Chapter 4 for a detailed discussion on the subject. By far the most common spectrum analysis approach in PIXE is to model the spectrum by an analytic function. This function includes modiﬁed Gaussians to describe the characteristic x-ray peaks and a polynomial or exponential polynomial to represent the underlying continuum background (Campbell et al., 1986). An alternative to analytical background modeling is to use some kind of mathematical background removal method (Maxwell et al., 1989, 1995). A third approach is to employ a peak-clipping algorithm to remove all peaks from the measured spectrum, thus generating a numerical background spectrum. The model spectrum is then built by adding Gaussians to this background (whose overall height is a single variable) (Ryan et al., 1990a, 1990b). Because of the low continuum background in PIXE (particularly when compared to energy-dispersive EPMA), the range of peak heights in a PIXE spectrum can be up to ﬁve to six orders of magnitude. This leads to PIXE spectra that often exhibit ﬁne details, such as escape and sum peaks, and low-energy tailing for intense peaks. Whereas escape peaks and low-energy tailing may also be quite important in EDXRF spectra, sum peaks tend to be a minor problem in the latter, because the most intense peaks are generally located in the upper part of the spectrum. In PIXE, however, the most intense peaks are usually situated in the lower part of the spectrum (this results from the fact that cross sections increase with decreasing Z), and sum peaks therefore show up when high count rates are used during spectrum acquisition. An illustration of the importance of sum peaks is given in Figure 8. The presence of the many ﬁne details in the PIXE spectra places stringent

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Figure 8 PIXE spectrum obtained for a NIST orchard leaves specimen [incident proton energy 2.4 MeV, funny filter in front of Si(Li) detector, x-ray count rate 2200 cps]. The marked peaks are K lines, except where indicated otherwise. The unmarked peaks between FeKb and CuKa are sum peaks. The dots represent the experimental spectrum, the curve through the dots is the modeled spectrum, as obtained from a nonlinear least-squares fit, and the dashed line is the modeled background. (From Maenhaut, 1990a.)

requirements on the spectrum model. As far as the modeling of the sum peaks is concerned, this is generally done by representing them by a single pileup element, according to an approach ﬁrst proposed by Johansson (1982). Despite the many ﬁne details, accurate modeling of PIXE spectra is quite feasible, as was demonstrated in an intercomparison exercise of ﬁve diﬀerent PIXE spectrum analysis programs (Campbell et al., 1986). It is also illustrated by the good agreement between modeled and experimental spectrum in Figure 8. B.

Quantitation for Thin Specimens

When protons of 1–4 MeV are used in PIXE, elements with Z up to about 50 are generally determined through their K x-rays (typically the Ka line), and the heavier elements are measured through their L x-rays (La line). The basis for a quantitative analysis is that there is a relationship between the net area of an element’s characteristic K or L x-ray line in the PIXE spectrum and the amount of element present in the sample. For proton bombardment and an inﬁnitely thin specimen (by this is meant a specimen that is suﬃciently thin so that matrix eﬀects become negligible), the relation is given by Yp ðZÞ ¼

N0 s X pZ ðE0 ÞEp NCZ rt AZ sin y

ð9Þ

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741

where Yp ðZÞ is the number of counts in a characteristic x-ray line p of the analyte element with atomic number Z, N0 is Avagadro’s number, sX pZ ðE0 Þ is the x-ray production cross section for line p at the incident proton energy E0 , Ep is the absolute detection eﬃciency (including the solid angle) for x-ray line p, N is the number of incident protons, CZ is the concentration of the analyte element in the specimen, r is the specimen density, t is the specimen thickness, AZ is the atomic mass of the analyte element, and y is the angle between the incident proton beam and the specimen surface. In formulating Eq. (9), it is implicitly assumed that the specimen is uniform and that the beam size is smaller than the specimen area. For a formulation of the situation where the beam size is larger than the specimen area, the reader is referred to Johansson and Campbell (1988). In deriving the analyte concentration from its x-ray yield, several approaches are possible (Johansson and Campbell, 1988). One can solve Eq. (9) for CZ , and thus employ the absolute or fundamental parameter method. This requires accurate knowledge of all parameters involved. The most critical parameters are the x-ray production cross sections and the absolute detection eﬃciency. The accuracy of the x-ray production cross sections was addressed in Sec. II.A.2. The absolute detection eﬃciency of a Si(Li) detector has been the subject of numerous research papers. Its determination poses no problems for photon energies in the 5–30 keV region (Campbell and McGhee, 1986), in which use can be made of accurate long-lived radionuclide standards and the relative detection eﬃciency can be accurately modeled. Unfortunately, no such standards are available for the 1–5 keV range, where there is a steep decrease in eﬃciency. The best solution for characterizing this region is to use a strong radionuclide source in combination with secondary ﬂuorescers, and some laboratories (Denecke et al., 1990; Lepy et al., 1992) have developed carefully calibrated devices for this purpose. Because of the diﬃculties with the absolute quantitation method, many PIXE workers prefer to rely on a relative approach, and they calibrate their experimental PIXE setup using thin ﬁlm standards [e.g., Johansson et al. (1981), Maenhaut and Raemdonck (1984), Borbe´ly-Kiss et al. (1985)]. This method yields so-called thin-target sensitivities kp ðZÞ which combine several of the quantities of Eq. (9): kp ðZÞ ¼

N0 s X pZ ðE0 ÞEp AZ sin y

ð10Þ

The units of kp ðZÞ are x-ray counts per unit proton charge (usually mC) and per mg=cm2. Both the absolute and relative quantitation methods generally require the knowledge of the specimen mass thickness if the results are to be obtained as concentrations in the specimen material. As discussed by Johansson and Campbell (1988), there are a variety of ways to determine the specimen thickness. They include direct weighing, thickness measurement via ancillary photon transmission measurements, thickness measurement via energy loss of transmitted protons, and energy loss determination by means of a beam stop. The requirement for knowing the specimen mass thickness can be avoided by spiking the sample with a known amount of an internal standard element before specimen preparation. Such spiking is easily done for liquid samples and is also feasible when one deals with powdered solid materials, but it is, of course, impossible in the nondestructive analysis of solid samples. Another advantage of spiking is that the number of incident particles (beam ﬂuence or preset charge) need not be measured accurately. Indeed, when a spike is used, the quantitation involves division of Eq. (9) for the analyte element by the same equation for the internal standard, so that the number of incident particles N as well as the mass thickness rt cancel out.

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Quantitation for Specimens of Intermediate Thickness and for Infinitely Thick Specimens

In practice, specimens are rarely thin enough that matrix eﬀects are entirely negligible. For example, in 2.5-MeV-proton PIXE of a 0.5-mg=cm2-thick US National Institute of Standards and Technology (NIST) bovine liver specimen, and when basing the analysis on the Ka x-ray line, the matrix correction factor is 1.03–1.05 for the elements K to Sn, and it increases strongly with decreasing Z for the light elements (e.g., it is 1.1 for S, 1.2 for Si, and 1.5 for Mg). For specimens of intermediate thickness and for inﬁnitely thick specimens (the latter are specimens that are thicker than the particle range), Eq. (9) has to be replaced by N0 Ep NCZ Yp ðZÞ ¼ AZ

ZEf

sX pZ ðEÞTp ðEÞ SðEÞ

dE

ð11Þ

E0

where E0 ad Ef are the incident proton energy and the energy of the protons after passage through the target (Ef ¼ 0 for an inﬁnitely thick specimen), respectively, E is the proton energy, Tp ðEÞ is the transmission of the x-rays from successive depths in the specimen, and SðEÞ is the matrix stopping power. Tp ðEÞ is itself given by 0 Bmp sin y Tp ðEÞ ¼ exp@ sin f

ZE

1 dE C A SðEÞ

ð12Þ

E0

with mp the mass attenuation coeﬃcient for line p in the sample matrix, and f the angle between the specimen surface and the specimen–detector axis (i.e., the x-ray takeoﬀ angle). It should be noted here that the relation between the analyte element line intensity Yp ðZÞ and the concentration CZ , as expressed by Eqs. (11) and (12), does not include secondary or tertiary ﬂuorescence enhancement eﬀects. Detailed treatments of these eﬀects were given by Campbell et al. (1989, 1993). Although enhancement eﬀects are less pronounced in PIXE than in XRF and are, in fact, often negligible, secondary ﬂuorescence should be accounted for when the analyte elements are lighter than the matrix elements. For example, in 3-MeV-thick target PIXE of stainless steel (with y and f both equal to 45 ), the CrKa intensity is raised by about 50% as a result of the secondary ﬂuorescence from the FeK x-ray lines (Campbell et al., 1989). As for thin specimens, several quantitation approaches are possible for thick specimens. If one relies on the fundamental parameter approach and thus solves Eqs. (11) and (12) for CZ , other parameters are needed in addition to those already required for the thinspecimen case. These additional parameters are mp and SðE Þ, the mass attenuation coeﬃcient for line p in the sample matrix and the matrix stopping power, respectively. The values of those parameters for the sample matrix can be obtained from those for the matrix constituents by employing Bragg’s additivity rule, as already indicated in Sec. II.A.1 for the matrix stopping power. Such calculations require knowledge of the matrix composition and databases for the mass-attenuation coeﬃcients in the various elements and for the elemental stopping powers. The stopping power database and its accuracy were dealt with in Sec. II.A.1. The problem of selecting an accurate database for the massattenuation coeﬃcients is the same as in all other x-ray emission techniques and will not be discussed here. The matrix elemental composition places the major burden on the calculations and usually contributes most to the uncertainty in the calculated mp and SðEÞ

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743

values for the sample matrix. For heavier-element matrices, in which all matrix elements are detected in the PIXE spectrum, iterative procedures can be applied to obtain the matrix composition, but for light-element matrices, one must resort to a priori information (e.g., obtained by other techniques) or certain assumptions have to be made (e.g., that the elements are present as oxides). In any case, this problem of the matrix composition is common to all x-ray emission analysis techniques. When specimens of intermediate thickness are analyzed, the transmitted proton energy Ef (or rather the energy loss E0 Ef ) is also needed for evaluating the integral in Eq. (11). This implies knowledge (or determination) of the specimen mass thickness, because the energy loss is related to the latter through the matrix stopping power. Alternatively, the energy loss can be measured experimentally. It is evident that any uncertainty in the specimen mass thickness (or in the experimental energy loss) will also be transmitted to the value of CZ . The magnitude of this uncertainty transmission increases with decreasing specimen thickness and, ultimately, a given relative uncertainty in the specimen mass thickness produces an identical relative uncertainty in the value of CZ , as is, in fact, also the situation for the inﬁnitely thin specimens just discussed. The uncertainty transmission from the matrix composition, from the databases, and for intermediately thick specimens also from the specimen mass thickness can be much reduced by the use of an internal standard element. Indeed, the uncertainty transmitted in the integral of Eq. (11) is, to a large extent, in the same sense for the analyte and the spike, so that a signiﬁcant uncertainty reduction occurs when dividing the two integrals. An alternative to the pure fundamental parameter quantitation approach is to make use of experimental thin-target sensitivities kp ðZÞ, as deﬁned by Eq. (10), so that Eq. (11) can be written as kp ðZÞNCZ Yp ðZÞ ¼ X spZ ðE0 Þ= sin y

ZEf

sX pZ ðEÞTp ðEÞ dE SðEÞ

ð13Þ

E0

By solving this equation for CZ , one basically uses a relative method (relative to thin-ﬁlm standards), but the correction for matrix eﬀects is made by a fundamental parameter approach. As in the relative quantitation method for thin specimens, this mixed approach requires no knowledge of the absolute detection eﬃciency or of the radiative transition probabilities and ﬂuorescence yields [the latter two parameters cancel out in the ratio of the x-ray production cross sections in Eq. (13)], but ionization cross sections and Coster– Kronig yields are still required. However, the division of the x-ray production cross sections also has the eﬀect that the impact of the Coster–Kronig yields is marginal and that for ionization cross sections, essentially only their dependence on proton energy is needed, which has a much smaller uncertainty than the absolute value of the ionization cross section. In the analysis of inﬁnitely thick specimens, one can also utilize experimental thicktarget calibration factors instead of relying on the fundamental parameter approach or on experimental thin-target sensitivities. The thick-target calibration factors incorporate the integral of Eq. (11) and are usually expressed in x-ray counts/mC and per mg=g. They are commonly derived from PIXE measurements on samples with known trace element composition (standards). In a strict sense, the thick-target factors are only valid for the analysis of unknown samples with identical (matrix) composition to the standards, but in practice, some variability in composition can be tolerated or corrected for. The necessary correction factor is, in this case, the ratio of the integral of Eq. (11) for the standard to the corresponding integral for the unknown.

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As discussed by Johansson and Campbell (1988) and Johansson et al. (1995), still other quantitation approaches are possible, for example, making use of thick singleelement standards. Before closing this section on quantitation for semithick and inﬁnitely thick specimens, it should be warned that Eqs. (11)–(13) are, in a strict sense, only valid for perfectly ﬂat homogeneous samples and that, for specimens made up of particulate material, the particle size should be as small as possible (ideally below 1 mm when quantifying light elements, such as Na through Si). The surface roughness eﬀects in PIXE were dealt with by Cookson and Campbell (1983). With regard to particle size eﬀects in PIXE, these are especially of concern when analyzing deposits of atmospheric particulate material (aerosols) on ﬁlter or cascade impactor samples. Model calculations for proton-induced x-ray generation in a monolayer of spherical particles having realistic matrix compositions were presented by Jex et al. (1990). In aluminosilicate particles of 2 mm diameter, the attenuation is over 20% for the elements Na to P. D.

Detection Limits in Thin- and Thick-Target PIXE

As in other spectrometric techniques, the detection limits in PIXE are determined by the sensitivity (calibration) factors, on the one hand, and by the spectral background intensity where the analyte signal (x-ray line) is expected, on the other hand. Various deﬁnitions for the limit of detection (xL ) were proposed in the early years of PIXE (Johansson and Campbell, 1988), but it is now general practice to deﬁne xL as that amount (or concentration) of analyte element that gives rise to a peak area which is equal to three times the standard deviation (square root) of the background intensity NB in the spectral interval of the principal x-ray line. The spectral interval for integration of NB is usually deﬁned in terms of the full width at half-maximum (FWHM) of the principal x-ray line, but regions of one, two, and three FWHMs have been used in the PIXE literature; this range of choice introduces a variation of 1.7 in detection limits deduced from the same dataset. For thin specimens, the relationship between line intensity Yp ðZÞ and analyte concentration CZ was given by Eq. (9). If we represent the probability for the production of continuum background radiation per unit of x-ray energy by sB and if we further assume (to keep the formulation simple) that the background originates from a single matrix element with atomic mass AB , the following relation can be written for the background intensity NB : NB ¼

N0 sB ðE0 ÞnFWHMEp Nrt AB sin y

ð14Þ

where nFWHM indicates the spectral interval used for summation of the background, and 1=2 all other symbols have the same meaning as in Eq. (9). By setting Yp ðZÞ equal to 3NB and solving Eqs. (9) and (14) for the xL value of CZ , one obtains sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3AZ sB ðE0 ÞnFWHM sin y xL ðZÞ ¼ X ð15Þ N0 AB Ep Nrt spZ ðE0 Þ It thus appears that xL is proportional (or inversely proportional) to the square root of the experimental parameters FWHM, Ep ; N, and the specimen mass thickness. Hence, to optimize xL , a detector with very good resolution has to be used, and, rather obviously, the solid angle of detection should be made as large as possible, but improvement in this

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745

parameter is limited by the area of the detector and by the fact that the detector can only get to about 2 cm from the specimen. Much more ﬂexibility is provided by the number of incident protons N (preset charge), which can be increased either by a longer measurement time or by an increase in beam current. Both theoretical calculations and experimental measurements have been performed to obtain xL values for thin specimens [e.g., Folkmann (1976), Johansson and Johansson (1976), Ishii and Morita (1988)]. In most of this work, one adopted (or employed) a lightelement matrix (typically carbon or an organic polymer) and a specimen mass thickness of 1 or 0.1 mg=cm2. Johansson and Johansson (1976) produced a very useful contour plot of xL values as a function of incident proton energy and atomic number of the analyte element for the case of 0.1-mg=cm2-thick carbon matrix. Their plot, reproduced here in Figure 9, was based on experimental measurements of the continuum background, and it was further assumed that elements with atomic number up to about 50 are determined through their Ka x-ray line and the heavier elements through their La line. As can be seen in Figure 9, there is a valley of optimum detection limits for both the K and L cases, with the best K xL values (less than 0.5 mg=g) obtained at lower proton energy than the best L xL values (0.5–1 mg=g). Furthermore, within either the K or L case, the bombarding energy for optimum detection limits depends on the atomic number of the analyte elements of interest, with higher bombarding energies favoring the heavier elements. Selection of the energy should thus be made with the objective of the analysis in mind, but, in practice, some compromise is necessary. Johansson and Johansson (1976) concluded from their contour plot that the optimum proton energy is about 2 MeV for the analysis of trace elements in biological and environmental samples. Such bombarding energy also has the advantage that the xL values show rather little variation (about one decade only) for the analyte elements with Z between 15 and 90. More recently, Ishii and Morita (1988) produced a contour plot similar to that of Johansson and Johansson (1976), but they based it solely on theoretical calculations and adopted a pure oxygen matrix (which was considered representative for biological samples). The conclusion of this study was that the

Figure 9 Contour plot of the limit of detection (xL) as a function of incident proton energy and atomic number of the analyte element for the case of a 0.1-mg=cm2-thick carbon matrix. Experimental conditions: detector FWHM 165 eV, solid angle of detection 38 msr, collected charge 10 mC. The background interval selected for calculating xL was equal to one FWHM. (From Johansson and Johansson, 1976, with permission from Elsevier Science.)

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best detection limits were obtained with about 3 MeV protons. Considering these investigations on xL values, it is thus no surprise that most PIXE laboratories employ proton energies in the range of 2–3 MeV. Several laboratories actually consistently use the same bombarding energy (e.g., 2.5 MeV) for their typical applications. Detection limits have also been examined for other projectiles than protons, but the results were generally not very encouraging. A noteworthy exception is given in the detailed study by Johansson (1992) on the use of 5-MeV helium ions (a-particles). It was found that these oﬀer signiﬁcantly improved detection limits (about three times better than protons) for thin organic specimens. The detection limits in Figure 9 are expressed in relative units (concentrations), but they are easily converted into absolute units (masses) by multiplying by the specimen mass probed by the beam. Because this mass is invariably small (e.g., for a 0.1-mg=cm2-thick specimen as in Figure 9 and a typical beam size of 0.2 cm2, the probed mass is 0.02 mg), it follows that the absolute detection limits are in the range of 10–100 pg for the case of 2-MeV protons and the experimental parameters of Figure 9. As indicated by Eq. (15), the concentration limit of detection improves with the square root of the mass thickness. However, this relation remains only valid as long as the thin-specimen criterion holds. Because of the existence of matrix eﬀects (i.e., the decrease in sX pZ and in the cross sections for bremsstrahlung production with decreasing proton energy and the attenuation of the characteristic x-rays and continuum radiation by the sample matrix), xL rapidly approaches its optimum value. In general, the improvement in xL is quite limited above a few mg=cm2. Depending on the origin of the continuum background and the energy dependence of its production cross section, xL may actually deteriorate somewhat beyond a certain specimen thickness. Such situations may occur for analyte lines in the spectral region where prompt g-radiation forms the major background source (typically above about 10 keV). Whereas xL studies for thin specimens have concentrated on light-element matrices, in similar investigations for inﬁnitely thick targets, heavy-element matrices were also considered. Teesdale et al. (1988) conducted a comprehensive experimental study of xL values for 1–5-MeV proton bombardment of pure single-element matrices of carbon, aluminum, silicon, titanium, iron, germanium, molybdenum, silver, tin, ytterbium, and lead. They paid particular attention to the choice of appropriate x-ray ﬁlters for suppression of the matrix characteristic x-ray lines and their pileup peaks and to the choice of the optimum proton energy. It was found that increase of energy up to 3 MeV is proﬁtable, but that a further increase confers only small beneﬁts. Increasing absorber thickness were suggested for increasing atomic number of the matrix up to Z ¼ 40. Under these conditions and using a beam charge of 1 mC, the xL values were a few micrograms per gram for the light-element matrices (C and Al) and 10–50 mg=g for the intermediate matrices Ti through Ge. By using an optimized PIXE setup and a preset charge of 100 mC (which corresponds to a 15-min bombardment with 100 nA current), xL values down to a few tenths of microgram per gram may be obtained for light-element matrices. The detection limit, expressed as concentration in the original sample, can be further improved by a physical or chemical separation of the material of interest or by a chemical separation of the analyte element(s) from the bulk of the sample. Drying or freeze-drying is an obvious preconcentration step for natural water samples, but also for biological tissues (which typically contain 80% water). For natural waters with high mineral content, such as seawater, chemical preseparation schemes are advantageous, and detection limits of 1 ng=L (or 1 pg=g) have been obtained by such an approach (Johansson and Johansson, 1984). For dried

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biological tissues, further preconcentration may be realized by resorting to high- or lowtemperature ashing. As demonstrated by Pallon and Malmqvist (1981) and Maenhaut et al. (1984), however, the gain in detection limits remains limited to a factor of about 2. In this entire discussion on xL values, it was assumed that the PIXE analyses were carried out in a vacuum chamber. In external beam or nonvacuum PIXE (in air or in a helium or nitrogen atmosphere), poorer xL values are expected because of the background contribution from interactions in the beam exit foil material and in the air or chamber gas and, for the light elements, also because of the substantial attenuation of their soft x-rays by the same gases. However, practical xL values in nonvacuum PIXE appear to be rather comparable to the xL values in vacuum PIXE, at least for analyte elements with atomic number above 25 (Ra¨isa¨nen, 1986). E.

Precision and Accuracy in Thin- and Thick-Target PIXE

As in any other analytical technique, high precision and accuracy should be aimed for in PIXE. It is therefore essential that careful attention be given to all stages of the analysis. These include sample and specimen preparation, specimen bombardment, spectral data processing, quantiﬁcation, and correction for matrix eﬀects. For a discussion of the critical facets in the various stages, the reader is referred to the speciﬁc section dealing with each stage. As far as the specimen bombardment stage is concerned, it should be added here that one should be aware of the danger of radiation- or heat-induced losses during PIXE bombardment. Such losses are particularly feared for volatile analyte elements (e.g., the halogens, S, As, Se, and Hg) and, in the case of organic or biological specimens, also for certain matrix elements (mainly H and O). The current density applied during analysis plays a major role, and the danger for losses is therefore more severe in micro-PIXE than in macro-PIXE. In any case, it should be determined which irradiation conditions are safe for a particular application. More information on this subject can be found in a tutorial paper by Maenhaut (1990a) and in research papers of Legge and co-workers (Legge and Mazzolini, 1980; Cholewa and Legge, 1989; Kirby and Legge, 1991), Themner and coworkers (Themner et al., 1990; Themner, 1991), and Van Lierde et al. (1995, 1997). The reproducibility (precision) of an entire PIXE analytical procedure (including the contribution from sample processing and specimen preparation) can be examined by preparing several specimens from the same material, subjecting these to PIXE, and calculating a standard deviation (s) from the spread in the results obtained. Under optimum conditions, this standard deviation should be the same as that expected on the basis from counting statistics alone (sc). However, when the percentage standard deviation from counting statistics (%sc) approaches values smaller than about 1–2%, diﬀerences between s and sc are often unavoidable because of the limitations in sample and specimen homogeneity. It should indeed be realized that, even in macro-PIXE of inﬁnitely thick specimens, the probed sample mass is at most a few milligrams, so that only nanogram amounts of analyte elements are actually examined for concentration levels of a few micrograms per gram. Particularly in the analysis of biological, geological, and atmospheric aerosol samples, a %s of 1–2% is often the ultimate practical limit of precision. Such precisions were obtained by Maenhaut et al. (1987), for example, in PIXE analysis of biological reference materials. The accuracy of a PIXE procedure should be evaluated by analyzing (certiﬁed) reference materials or through comparisons with other analytical techniques. Ultimately, the accuracy will depend on the extent of spectral interferences and matrix eﬀects and on how well these can be controlled or corrected for. Several PIXE analyses of reference materials

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and other accuracy investigations have been reported in the literature, and selected studies dealing with the analysis of trace elements in biological, environmental, and geological samples were reviewed by Maenhaut (1987). The book of Johansson et al. (1995) also reports on several accuracy studies and presents some ﬁne examples of investigations in which micro-PIXE was used. As an example of a macro-PIXE accuracy investigation on biological materials, the study by Maenhaut et al. (1987) can be cited. A total of 18 elements were measured in up to 14 (certiﬁed) reference materials, and from a comparison of the PIXE results with the reference values (when available), it was concluded that the accuracy was better than 5%. For assessing the accuracy in macro-PIXE of atmospheric aerosol samples, one has to rely on comparisons with other techniques. Good examples of such intercomparisons are the work by Wa¨tjen and co-workers (Wa¨tjen et al., 1983; Bombelka et al., 1984) and more recently by Nejedly et al. (1995).

V.

SAMPLE COLLECTION AND SAMPLE AND SPECIMEN PREPARATION FOR PIXE ANALYSIS

A.

General

In this section, general aspects of sample collection (sampling) and sample and specimen preparation are discussed. Methods or procedures which only apply to samples of a speciﬁc type (e.g., biological and environmental) are touched upon in Sec. VI, which deals with the applications of PIXE. Furthermore, the present section discusses aspects that are of general importance in both macro- and micro-PIXE. Points that are relevant for microPIXE only are dealt with in Sec. V.B. Particle-induced x-ray emission can, in principle, be applied to any type of sample. Considering that the bombardments are normally done in vacuum, however, it is evident that the technique is more suitable for analyzing solids than liquids. PIXE analysis of liquids normally involves some preconcentration by drying (which can be as simple as drying a drop of the liquid on a suitable backing ﬁlm) or some other physical or chemical separation of the analyte elements from the liquid phase. As far as the analysis of solids is concerned, it should be kept in mind that even in macro-PIXE of inﬁnitely thick samples, the mass actually probed by the beam is at most a few milligram. Determination of the bulk composition of a solid sample without any preliminary sample preparation is therefore only possible for samples that are homogeneous in all three dimensions. Hence, before deciding on analyzing a sample by PIXE, it should be carefully examined whether the analytical problem is not much better solved by some other technique. The analysis of liquid samples and the bulk analysis of large heterogeneous samples are clearly problems for which PIXE may not be most appropriate. However, as concluded in a paper by Maenhaut (1990b) in which PIXE and various nuclear and atomic spectrometric techniques were compared, there are numerous analytical problems for which PIXE is the most suitable technique or, at least, among the more suitable. Examples of such are the multielemental analysis of milligram-sized samples consisting of a light-element matrix (e.g., biomedical and atmospheric aerosol samples), the nondestructive analysis of millimetersized areas on a large sample or of thin superﬁcial layers on a bulk sample, and various problems that require sensitive analysis with high spatial resolution. When it has been decided to tackle an analytical problem by PIXE, full use should be made of the inherent characteristics of the technique, particularly of its nondestructive and instrumental character. Therefore, if possible, sampling should be done in such a way that subsequent sample preparation can be avoided or kept to a strict minimum. The collection

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of suspended atmospheric particulate material (atmospheric aerosols) is a good example where such strategy should be adopted. For unique samples or samples of high commercial or historical value, sample preparation or subsampling may even not be allowed because the sample generally must be returned unaltered after the analysis. Examples of such samples are historical documents, various objects of art, and extraterrestrial dust particles. In many situations, however, some sample preparation is required. This may vary from simple cleaning of the sample (to remove surface contamination), polishing (to eliminate surface roughness eﬀects), and powdering (to homogenize the sample and to reduce the particle size), to digestion or physical or chemical preconcentration or separation. Furthermore, the last step in the sample preparation usually consists of preparing specimens that are suitable for PIXE bombardment. Such specimen preparation may involve depositing a drop of a liquid (e.g., of an acid digest) or a few milligrams of powdered material on a clean, strong substrate ﬁlm (for thin and intermediate specimens) or pressing a certain amount of sample into a pellet (for inﬁnitely thick specimens). The prepared specimen is often mounted or held in a target frame (e.g., a 25-mm-diameter plastic ring or a square target frame that ﬁts in a standard 35-mm slide tray holder). Overall, the sample and specimen preparation procedures in PIXE are quite similar to those in the other x-ray emission analysis techniques (see Chapter 14). As far as the backing ﬁlms for thin and semithick specimens are concerned, an additional requirement in PIXE is that such ﬁlms should be able to withstand the irradiation by the particle beam. More detailed information on specimen backing ﬁlms for PIXE is given by Johansson and Johansson (1976) and Russell et al. (1981). Considering that PIXE analyses often aim at measuring microgram per gram levels of trace elements and that the absolute amounts of analyte elements actually examined are then in the nanogram region or below, contamination control is very important. Hence, acid-cleaned plastic (e.g., polyethylene or Teﬂon) or quartz containers and tools should be employed during sampling and sample processing. The chemicals, acids, and water used in sample preparation (e.g., for digestion or dissolution) should be of high purity. Also, all critical manipulations should be done in a clean bench with laminar airﬂow. When applying thin-specimen procedures, realistic blank specimens should always be prepared. This should be done by applying the same procedures and using the same substrate ﬁlms as for the actual sample specimens. Another point of concern is potential losses of analyte elements during sample storage, sample processing, and specimen preparation. During storage of aqueous samples, analyte elements may be deposited on the container walls. Perhaps more important is that some analyte elements (e.g., the halogens, S, As, Se, Hg, and Pb) may be volatilized by drying of the sample at elevated temperature and especially in sample preparation methods which involve high- or low-temperature ashing or acid digestion in an open vessel. For more information about general aspects of sample and specimen preparation for PIXE, the tutorial paper by Mangelson and Hill (1990) can be recommended. This paper also provides a ﬁne overview of the various physical and chemical methods of sample preparation. B.

Specimen Preparation for Micro-PIXE

Because of the similarity between micro-PIXE and EPMA, the specimen preparation techniques developed for EPMA are generally also applicable in micro-PIXE. However, the diﬀerence in ionizing particles (typically 10–20-keV electrons in EPMA versus MeV protons in PIXE) has the eﬀect that the depth probed in the analysis is signiﬁcantly greater

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in micro-PIXE. For example, this depth amounts to several tens of micrometers for 3MeV protons. In order to obtain meaningful results, the specimen should be homogeneous throughout the depth analyzed, and for optimum use of the lateral capability of the nuclear microprobe, the specimen thickness should preferentially be of the same order as the size of the focus of the microbeam. On the other hand, the specimen has to be suﬃciently thick (0.1–1.5 mg=cm2) in order to obtain a high x-ray yield. It should also be realized that the support material (backing ﬁlm or other support material) may cause interference in the x-ray spectrum, and it should therefore be selected with care. The actual specimen preparation will, to a large extent, be determined by the material to be studied. For biological materials, cryosectioning of frozen samples followed by freeze-drying and mounting of the material on a thin clean plastic foil is often the method of choice. For minerals, the specimens may consist of thin, polished, or ion-milled disks mounted on a glass plate or an electron microscope grid. Finally, the preparation of thin or semithick specimens should be done in such a way that the sample mass examined in each pixel can be determined. This mass thickness is required in order to allow expressing the results as elemental concentrations (see Secs. IV.B and IV.C).

VI.

APPLICATIONS

The applicability of PIXE to various analytical problems has been amply demonstrated in many publications. Furthermore, numerous studies have been carried out in which PIXE provided part or all of the requested trace element concentration data. This section presents a brief selection of the applications of PIXE and is mainly based on publications from the period 1992 through mid-1998. Many more examples can be found in the proceedings of the international conferences on PIXE (Johansson, 1977, 1981; Martin, 1984; Van Rinsvelt, 1987; Vis, 1990; Uda, 1993; Moschini and Valkovic´, 1996; Malmqvist, 1999) and nuclear microprobe technology (Grime and Watt, 1988; Legge and Jamieson, 1991; Lindh, 1993; Yang et al., 1995; Doyle et al., 1997), in the proceedings of the international symposia on Bio-PIXE (Ishii et al., 1992; Zheng et al., 1996), in two textbooks on the PIXE technique (Johansson and Campbell, 1988; Johansson et al., 1995), and in the chapter on PIXE in the ﬁrst edition of this Handbook of X-Ray Spectrometry (Maenhaut and Malmqvist, 1992). A.

Biological and Medical Samples

Most samples of biological origin are composed of essentially organic material, so that its matrix elements are light and do not give rise to characteristic x-ray lines in the PIXE spectrum. Furthermore, the electron bremsstrahlung background is lower for an organic matrix than for a matrix of heavy elements. Biomedical samples are therefore well suited for trace element determinations by PIXE. The disadvantage is that the organic matter is sensitive to radiation damage and to heating eﬀects. As discussed in Sec. IV.E, the light matrix elements, especially H and O, but also some analyte elements may be lost to some extent, in particular in micro-PIXE. The elements of interest in biological materials are either ‘‘essential’’ minor or trace elements (e.g., K, Ca, Mn, Fe, Cu, Zn, and Se) or ‘‘toxic’’ trace elements (e.g., Cd and Pb). A comparison of the detection limits in PIXE (see, e.g., Fig. 9) with the levels of the minor or trace elements in biological tissues [e.g., Iyengar et al., (1978), Iyengar (1989)] reveals that most elements of interest in physiology or pathology, with a few exceptions, such as

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Cd, can suitably be determined by PIXE in most tissue types. Consequently, ever since the development of PIXE started, biological and medical applications have been prominent. In the last decade, however, competition from new ultratrace techniques and the need for high-sensitivity measurements in bioscience have resulted in growing diﬃculties to promote PIXE and other x-ray spectrometric techniques. Furthermore, in recent years, the emphasis has shifted from ‘‘total’’ element determination toward elemental speciation (i.e., measuring of the chemical compound or biological molecule to which the element is bound or associated). PIXE and the other x-ray techniques lend themselves much less to this speciation work than some of the new atomic spectrometric techniques. In applying PIXE to biomedical problems, one should therefore look for cases where one can fully utilize the special advantages of PIXE, such as high spatial resolution, accurate quantitative analysis, and small samples. Particularly micro-PIXE, as used in the nuclear microprobe, is invaluable in biomedical trace element research. 1. Sample and Specimen Preparation The preparation of biomedical samples for PIXE analysis depends on the type of sample, its composition, the information looked for (bulk concentrations or spatially resolved data), the elements of interest, and the mode of irradiation (vacuum or nonvacuum). For nuclear microprobe analyses, special requirements apply, as indicated in Sec. V.B. Many elements exhibit suﬃciently high concentrations in biological material that a simple physical sample preparation method may suﬃce in order to obtain the requested concentration data. Several of the purely physical sample preparation methods are discussed in detail by Mangelson and Hill (1990). They include drying or freeze-drying, homogenizing and pulverizing, and cutting of thin sections. In some cases, however, particularly for natural levels of toxic elements and for levels of some essential elements in certain tissue types, preconcentration by destruction of the organic matrix or some other chemical preconcentration or preseparation is required. This may be performed in various ways (Mangelson and Hill, 1990). The most common methods are (1) wet digestion in acids, either in open or closed vessels, (2) dry ashing in an oven, (3) low-temperature ashing in a plasma asher, and (4) biochemical separation techniques. In wet digestion, the organic sample matrix is decomposed by concentrated strong acids, normally nitric and=or hydrochloric acid. As the acid has to be removed by evaporation prior to PIXE analysis, the use of hygroscopic acids (i.e., sulfuric, perchloric, and phosphoric acid) should deﬁnitely be avoided. Moderate heating in an oven or microwaveassisted digestion signiﬁcantly increases the decomposition rate. The digestion rate may be further increased by performing the digestion in a closed pressurized vessel [e.g., in a Teﬂon bomb in a regular oven (Duﬂou et al., 1987) or in a closed vessel in a microwave oven (Pinheiro et al., 1990)]. Such a procedure has the additional advantage that losses of volatile analyte elements are greatly reduced. Unfortunately, losses of certain elements (particularly of the halogens) may still occur while opening the vessel and during the drying of the digest on the backing ﬁlm while preparing the PIXE specimen. Dry ashing provides a greater mass reduction factor than wet ashing, with hightemperature ashing even being better in this respect than low-temperature ashing. By dry ashing at 550 C, preconcentration factors of up to 500 have been obtained for certain biological materials (Saurela et al., 1995). However, the very severe risk of loosing volatile analyte elements may prohibit the use of a high temperature. The more complex lowtemperature ashing in an oxygen plasma is therefore often preferred, but also this technique involves a serious risk of loss (Maenhaut et al., 1984; Maenhaut, 1990a).

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Biochemical separation techniques are used in research on elemental speciation. Pallon et al. (1987) developed a gel ﬁltration method to separate a blood serum sample into 100 fractions for subsequent analysis by macro-PIXE in order to examine the association of trace elements with serum proteins. Szo¨kefalvi-Nagy et al. (1990, 1993) performed several studies using gel electrophoresis to localize and quantify metals in various enzymes. They worked with polyacrylamide and cellulose acetate gels and separated the proteins according to electrical charge. Polyacrylamide gel electrophoresis (PAGE) was also used by Weber and co-workers (Weber et al., 1996; Strivay et al., 1998). Vogt et al. (1996) coupled PIXE on-line with capillary-zone electrophoresis (CZE). A Japanese group (Hirokawa et al., 1991; Hu et al., 1992) applied isotachophoretic separation. After the physical or chemical sample preparation, or even when no sample preparation is used, specimens for the actual PIXE bombardment generally have to be prepared, as indicated in Sec. V.A. When preparing specimens by pipetting a drop of a liquid (e.g., of an acid digest) on a backing ﬁlm, the uniformity of the deposit can be enhanced by pretreatment of the ﬁlm (Mangelson et al., 1981; Duﬂou et al., 1987), by adding lecithin or some other suitable additive to the liquid prior to pipetting (Campbell et al., 1985; Mangelson and Hill, 1990), and by rapid drying of the pipetted solution (e.g., by placing the targets in a vacuum desiccator) thereby favoring the formation of ﬁne crystallites (Maenhaut, 1990a). 2. Examples of Macro-PIXE The macro-PIXE application examples presented here were chosen somewhat arbitrarily. However, their selection was guided by the aim of demonstrating the particular potential of the PIXE technique. Furthermore, most examples were taken from the medical ﬁeld. a. Body Fluids Blood, blood plasma, and serum have always been popular study objects for PIXE, despite the fact that only a few real trace elements (Fe, Cu, Zn, Se, Br, Rb) can be measured in these sample types. Compared to the early days of PIXE, there is now increasing competition from other highly sensitive analytical techniques, such as inductively coupled plasma–mass spectrometry (ICP–MS), which are more suitable for analyzing liquid samples. Therefore, it becomes hard to justify the use of PIXE. The motivation often is that one has a good-running and cost-eﬀective bio-PIXE research program set up at a small accelerator or baby cyclotron and that the blood, plasma, or serum samples form part of a wider variety of sample types which are studied in the research project. When working with blood or blood components from small experimental animals (e.g., mice or rats), the small sample size provides additional justiﬁcation for the use of PIXE. Shenberg et al. (1995) used a combination of PIXE and instrumental neutron activation analysis (INAA) for measuring K, Fe, Zn, Se, Br, and Rb in blood and blood fractions (plasma and red cells) of colorectal patients. It was concluded that the two techniques complement each other. The mean values for K, Fe, Se, Br, Rb, and the Br=Rb ratio were signiﬁcantly lower for cancer cases than for healthy individuals, and it was suggested that this may be applicable as additional information for diﬀerentiating malignant cases from normals. It was also found that the Br=Rb ratio was much lower in Belgium than in Israel, which was explained by the inﬂuence of diﬀerences in dietary habits and environmental factors on the Br level in the blood of both populations. Miura et al. (1996) used PIXE for measuring 7 elements in serum of patients with acute myocardial infarction (AMI), and they examined the correlations between the elements and the vascular cell adhesion molecule-1 (VCAM-1), which is closely related to various types of

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inﬂammation. Soluble VCAM-1 was markedly enhanced in the sera of the patients with respect to controls, and two of the elements measured (i.e., Ca and Zn) were signiﬁcantly correlated with it. The authors suggested that there was an interaction between the divalent cations Ca2 þ and Zn2 þ and VCAM-1, but concluded that further investigations are needed to elucidate the actual mechanism of the alterations. Blood plasma samples from experiments with animal models were analyzed by PIXE by Sato et al. (1995) and Tamanoi et al. (1995). The study of Sato et al. (1995) dealt with the eﬀects of diethyleneaminepentaacetic acid (DTPA). This chelating agent is recommended for eliminating several kinds of radionuclides from persons contaminated with them. It is generally administered as calcium-trisodium salt (Ca-DTPA) or zinc-trisodium salt (Zn-DTPA), but both compounds have been reported to induce injuries, be it that Zn-DTPA is less toxic than Ca-DTPA. Male Wistar rats were intraperitoneally (i.p.) injected with the salts, and blood was collected from the tail vein just before and at 3, 6, and 24 h after the injection. Na, K, Ca, Fe, Cu, and Zn were measured in the plasma. CaDTPA signiﬁcantly lowered the concentration of Zn, whereas no signiﬁcant changes were observed for the other metals. Zn-DTPA did not lower any metal concentrations, but a signiﬁcant increase of plasma Zn was observed. Tamanoi et al. (1995) used PIXE to measure Cl, K, and Ca in blood plasma of mice in which EL-4 lymphatic tumor cells had been transplanted. The motivation for employing PIXE was that blood samples were taken at several intervals after the transplantation and that it was desirable to collect samples as small as possible in order to induce no stress. A sample size of 5 mL plasma was used for the PIXE analysis. Certain body ﬂuids, such as cerebrospinal ﬂuid (CSF), exhibit very low concentrations of trace elements, even for Fe, Cu, and Zn. PIXE of CSF requires special sample preparation and analysis procedures. Kupila-Rantala et al. (1996) used ultrathin (20– 30 mg=cm2) Formvar ﬁlms as sample support and a 625-mm-thick Kapton foil with a pinhole as the x-ray absorber. The detection limits for Fe, Cu, Zn, and Br were 6, 4, 8, and 18 mg=L, respectively, for a CSF sample size of 50 mL. b.

Soft Tissues and Organs

The fact that PIXE only requires a small amount of sample makes it possible to investigate the regional distribution of trace elements in large heterogeneous tissues, such as human brain. Following up on earlier work by Duﬂou et al. (1989), Hebbrecht et al. (1994a) measured the concentrations of two minor (K, Ca) and six trace elements (Mn, Fe, Cu, Zn, Se, Rb) and the dry-to-wet weight ratio in 50 diﬀerent structures of additional normal brains. The datasets were combined and examined with chemometric techniques (Maenhaut et al., 1993a). Mn and Cu were higher in the gray matter of the cerebellum than in the gray matter of the cerebrum. Both elements and also Fe were elevated in the nucleus caudatus, putamen, and globus pallidus. The substantia nigra exhibited elevated levels of Fe and, particularly, Cu. Hierarchical cluster analysis indicated that morphologically similar regions or structures involved in the same physiological function often conglomerated in the same cluster. This strongly suggests that there is some relationship between the trace element proﬁle of a brain structure and its function. Hebbrecht and colleagues also examined alterations in trace element levels and water content in hemorrhagic and nonhemorrhagic cerebral infarcts (Hebbrecht et al., 1994b) and they studied the inﬂuence of neurotropin for human brains that were aﬀected by recent middle cerebral artery infarcts (De Reuck et al., 1993). The most striking ﬁnding of the latter study was a less signiﬁcant increase in the water content in the white matter of the infarcts of patients who had been treated with neurotropin. In addition, decreases in intracytoplasmic

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elements, such as Rb and Cu, and increases in Ca were more clearly present in the infarcts of patients who had been treated with placebo compared to neurotropin. In comprehensive study, Vandenhaute and Maenhaut (1994a, 1994b) used acute and chronic intoxication by i.p. injection of CCl4 to induce liver injuries (liver necrosis, steatosis, and cirrhosis) in female Wistar rats. Liver, kidneys, and blood serum were collected from the experimental animals and from controls and analyzed for up to 12 elements (i.e., K, Ca, Mn, Fe, Cu, Zn, As, Se, Br, Rb, Sr, and Mo). The acute intoxication (leading to necrosis and steatosis) caused deﬁnite alternations of many trace element levels. As expected, the alternations were most pronounced in the liver. In this organ, Ca exhibited a strongly increased concentration. Important alterations for the elements K, Zn, and Se were also observed. For virtually all elements, however, the concentration alterations showed an idiosyncratic character. This means that the intensity and sometimes the direction of the alterations are hard to predict. Furthermore, it turned out that the concentration diﬀerences resulting from the liver disorders were often of the same order as those between various populations of the control animals. As a result, it is very hard to assess the physiological condition of an organ solely on the basis of its elemental concentrations. This ﬁnding also seems to preclude the use of trace element alterations for diagnostic or prognostic purposes in liver injuries in humans. c.

Human Hard Material

Hard human material includes materials as varied as hair, nails, human stones, calciﬁed tissues, teeth, and bone. As solid materials, they have in common that direct analysis using in vacuo or external-beam analysis can normally be used. It should be taken into account, though, that the direct irradiation of a solid sample gives concentration data that are only representative for the superﬁcial layer of 100 mm at most. If bulk data are required, the samples are usually crushed and pulverized, followed by pelletizing or depositing a small amount of the powder on a backing ﬁlm. Shizuma et al. (1991) used in vacuo millibeam PIXE to examine the distribution of Cl, K, Ca, Mn, Fe, Cu, and Zn in ﬁve diﬀerent types of gallstone. They sandwiched sections of the stones between two aluminized Mylar ﬁlms in order to eliminate charging eﬀects and scanned the sections with a beam of 0.5 or 1 mm diameter. The Ca concentration was high in the central region of nearly all gallstone samples and it was also high in the outer layers of the combination stones. Lane and Duﬀy (1996) analyzed dental enamel from human teeth collected from the Oxfordshire area of the United Kingdom. Elements heavier than Ca were correlated to a number of parameters, including the health of the teeth and age and gender of the donor. The concentrations of Fe, Sr, and Cu were found to correlate with dental health, and the mean concentration of Pb was found to be 7 mg=g for those under 25 years of age and increased with age. Archaeological samples of human teeth originating from Peru, Venice, and a Roman site were examined by Buoso et al. (1992). The depth proﬁles from enamel to the tooth interior were determined for elements heavier than Ca in order to understand their postmortem diﬀusion with time. Solis et al. (1996) analyzed human teeth from preColumbian, colonial, and contemporary population groups in Mexico in order to evaluate elemental changes. For Mn, Fe, Cu, Zn, and Sr, no diﬀerences between the three populations were detected. Pb, in contrast, was higher in colonial teeth than in contemporary ones, and it was below the detection limit for pre-Columbian teeth. The high Pb in the colonial period was attributed to the introduction of glaze pottery by the Spaniards. Bone acts as a ‘‘repository’’ for many trace elements, so that it may provide information on the long-term trace element status of an individual. In a study on the role of

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trace elements in Alzheimer disease (AD), Robertson et al. (1992) used external-beam thick-target PIXE for analyzing cortical and trabecular bone autopsy samples from 4 AD patients and 12 age-matched controls. The results indicated a possible imbalance in Zn, Br, and Rb for the AD patients. d. Zoology, Botany, and Environment Fish is a very important food in many coastal regions and in several national ‘‘cuisines’’ (e.g., in Japan). Changing environmental conditions may aﬀect both the quantity and the quality of the ﬁsh stocks. Arai and colleagues (Arai and Sakamoto, 1993; Arai et al., 1994) used PIXE for analyzing otoliths (i.e., ear stones) in several species of teleost ﬁsh, and they examined relationships between trace element concentrations and environmental parameters, such as water temperature and salinity. It was found that Sr and Zn concentrations increased with increasing seawater temperature. In addition, there were signiﬁcant differences in trace element composition between otoliths of reared red sea bream and rockﬁsh and otoliths of wild ones (Arai et al., 1994). Furthermore, the Sr=Ca ratio for red sea bream otoliths varied among rearing stations (Arai et al., 1995), so that it can be used for ﬁsh stock discrimination. Similar investigations, but on statoliths from squids, were performed by Ikeda et al. (1995). Kakuta et al. (1995) studied the changes of blood and urine parameters of carp that was exposed to diluted (20%) raw sewage for 30 min. Blood and urine samples were taken before and just after the exposure, and at 4, 12, and 36 h after the exposure. The changes in trace element levels were largest at 4 h after exposure, with signiﬁcant increases in blood S, Cl, and Br and in urine S, Cl, K, Ca, Zn, and Br. After 36 h, the levels had essentially returned to normal. The PIXE technique is very well suited for studying the distribution of trace elements in small botanical samples. Yokata et al. (1994) applied it to study alterations in Al, P, K, and Cl of alfalfa root tip that was exposed to low pH or aluminum stress. One or two pieces of root tips were suﬃcient for the analysis. Short-term (within 4 h) decreases in K=P and Cl=P ratios were observed under the stress conditions. Yukawa et al. (1995) studied the distribution of elements in soybean seedlings (beam sprouts). Soybeans were germinated in a dark box using pure water without any mineral supplement. After 20, 40, 65, and 137 h, the seedlings were harvested and each sprout was separated into cotyledons, hypocotyls, leaves, and root. Each bean sprout part was freeze-dried and analyzed by PIXE. It was observed that K, Mn, Fe, Cu, and Zn moved from the cotyledons and the primary root to the hypocotyls and ﬁrst leaf, whereas Ca in the seed hardly moved. Naturally growing moss and lichens are often used as biomonitors (or bioindicators) of atmospheric deposition of trace elements. They are normally collected at diﬀerent sites within a large region or country in order to study regional diﬀerences in that deposition. The basis for their use is that biomonitors obtain most of their supply of chemical substances directly from precipitation or from dry deposition of airborne particles. Williams et al. (1996) analyzed lichens from several sites in the intermountain western United States for up to 21 elements by PIXE. Certain samples had high concentrations of S, Cu, As, and some other elements. It was concluded that some wilderness areas are impacted by emissions from industrial plants. Reis et al. (1996) applied a combination of PIXE and INAA for measuring 43 elements in 250 lichen samples from Portugal. The dataset was subjected to Monte-Carlo-aided target-transformation factor analysis and the geographical distribution of the factor values was assessed. Acidiﬁcation of forests is a major ecological threat as well as important economic factor in forestry. By stepping or scanning a millimeter-sized beam over a core radially bored from the stem of a living tree and following the changes in the trace element content

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along the core, it is expected that changes in the environment can be followed. However, due to diﬀusion before and during this kind of analysis, there have been some doubts about the reliability of data; hence, micro-PIXE studies have been used for a more detailed understanding (Malmqvist, 1995b). Examples of trace element analyses of trees by macroPIXE in recent years are those Glass et al. (1993), Harju et al. (1996), and Aoki et al. (1998). Glass et al. (1993) extracted four cores (at two heights and two positions) from a 60-year-old water oak. Substantial diﬀerences were found in elemental distributions for the diﬀerent cores. The diﬀerences were not due to sample preparation, but possibly to diﬀerential deposition in the tree. The results indicate that caution must be exercised when attempting to link trace element studies of tree rings to pollution chronology of the surrounding environment. Harju et al. (1996) used thick-target PIXE in air to study the seasonal variation of trace elements within tree rings of Norway spruce and Scots pine. The samples were scanned with a 1-mm-diameter beam. The highest concentrations for most elements were obtained for earlywood in the beginning of the growth season and the lowest values for latewood, thus reﬂecting the biological activity. 3. Examples of Micro-PIXE The fact that one can focus the ion beam to small groups of cells or even single cells as well as to diﬀerent cell strata in various tissue, in combination with the high analytical sensitivity and accuracy of the PIXE technique, makes the nuclear microprobe a strong analytical tool in biomedicine. Its high lateral resolution represents an obvious advantage for studies on the role of elements in cellular functions. By selecting some good examples, we will try to illustrate the great importance of micro-PIXE in biomedicine. a. Single Cells The elemental composition of single cells can be studied on cells that are either present in their natural surrounding tissue or are grown free in cell cultures. In most studies of modern biology, human or animal isolated cells are used as experimental models. The advantage of analyzing ‘‘normally’’ grown cells is that one can assume that they are not perturbed elementally. Cultured cells, due the diﬃculties of exactly simulating normal growing conditions, may display some elemental deviations. However, the study of individual cells in a living organism is impracticable because of the diﬃculty in identifying the revelant cells, in manipulating their behavior in a controlled manner, and in separating eﬀects due to intrinsic properties of the cells from eﬀects due to the interaction among the many cell types present in the tissue. Hence, the use of cell cultures has increased during the last decades, in particular for studies with x-ray microanalysis, including micro-PIXE by using the nuclear microprobe. The purpose of working with isolated cells as a experimental models is to check, in vitro, consequently in reproducible conditions, a cellular function or the eﬀect of any molecule with a biological activity (Moretto and Llabador, 1997). One limitation of PIXE, other x-ray techniques, or most other elemental analysis techniques is that no information on the ionic form or speciation of the element is obtained. For instance, analysis of calcium ions in cells is important for the understanding of cell physiology. Also, although micro-PIXE is more sensitive than traditional x-ray microanalysis as used in an electron microprobe, both techniques provide information only on total Ca and not on the Ca2 þ concentration, which is the relevant physiological parameter. In the studies by Pa˚lsga˚rd et al. (1994) and Pa˚lsga˚rd and Grime (1996) on the role of Ca2 þ as the second messenger, this problem has been overcome by using a Ca analog. In secretory cells, such as the insulin-synthesizing b-cells, a rise in the Ca2 þ level triggers the exocytosis of secretory vesicles and the release of insulin. To discriminate between

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endogenous Ca2 þ and Ca2 þ entering as a result of stimulation, Sr was used as a tracer. Due to its chemical aﬃnity, Sr2 þ is known to enter the Ca2 þ channels and to mimic the role of Ca2 þ in the secretory process (Wroblevski et al., 1989). This elegant approach makes use of the multielemental character and high sensitivity of PIXE. To assure that this analog substitution works as assumed, Pa˚lsga˚rd et al. (1995) performed another study in which they examined whether a diﬀerent analog (Ba2 þ ) behaves the same. When one is interested in measuring trace metals rather than electrolytes in cells, micro-PIXE has particular relevance. Examples are research on heavy-metal intoxication, but also studies on elemental changes induced by the physiological or pathological state of the cells. In a study of Menkes disease, the Melbourne nuclear microprobe group investigated the intracellular Cu concentration in cultured Menkes ﬁbroblasts by means of individual cell microanalysis (Allan et al., 1994). There was a highly signiﬁcant diﬀerence of a factor of 6 between the Cu levels of Menkes and normal ﬁbroblasts. By using the individual cell analysis, it was also found that there was large biological variability between the individual cells, despite the fact that they were grown under uniform conditions. Some transition metals delay important regulatory roles in gene expression. The disturbance of their cellular levels could be involved in oncogene expression and tumorigenesis. In a study by the Bordeaux nuclear microprobe group (Ortega et al., 1997), micro-PIXE was used to measure Mn, Fe, Cu, and Zn in cultured human neuroblastoma cells. The study indicated that the nuclear microprobe is a useful tool for investigating transition metal concentrations in cultured cells and that further research is required to investigate the detailed role of these trace elements. Another interesting example of single-cell analysis is that of individual marine plankton. The occurrence of annual marine phytoplankton blooms is becoming a global problem. Within a European Union-funded project, it is investigated if the unbalanced nutrient composition of the water promotes the dominance of harmful phytoplankton species. A combination of new advanced methods is used to allow simultaneous determination of toxin content, elemental composition, genetic expression, and bio-optical properties of toxic phytoplankton at the species level in natural communities. One of the tasks is the determination of the elemental composition of single phytoplankton cells. This is carried out with the Lund nuclear microprobe, using both micro-PIXE and nuclear reaction analysis (NRA) with deuterons, with a special focus on C, N, P and K. The method used to isolate single living cells while reducing their salt environment is an important part of the analytical procedure. Figure 10 shows elemental maps obtained by nuclear microprobe analysis of a phytoplankton. Also, the results from STIM measurements and an optical micrograph are shown (J. Pallon, personal communication, 1999). b.

Soft Tissues

With the nuclear microprobe, one can raster the focused ion beam over selected areas on various biological structures and form elemental or structural maps which are displayed on-line. This facilitates the combination of overall analysis and detailed spot investigations. By overlaying a video image or an optical micrograph, it is possible to localize and identify particularly interesting regions or cell strata. Because of the stringent requirements on the sample preparation in order to preserve biological trace element information, the tissue specimens are normally not stained but only freeze-dried. Hence, the lack of contrast in optical mode of such specimens makes it necessary to combine all means of imaging and analysis available (Malmqvist, 1995a). The central nervous system has been extensively studied using the nuclear microprobe. Starting in Oxford and continuing at the National University of Singapore (NUS),

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Figure 10 Elemental maps and an off-axis STIM map from a single phytoplankton, Dinophysis sp., obtained by nuclear microprobe analysis with 2.5-MeV protons. P to Ca are determined by micro-PIXE, and C and N with backscattering. The scan size is 64 mm. (Courtesy of J. Pallon.)

Watt and co-workers have carried out several studies on brain tissue taken postmortem from diseased patients diagnosed as suﬀering from Alzheimer disease. The presence of elevated levels of certain elements, especially Al, in neuroﬁbrillary tangles and neuritic plaques from such patients has been the subject of great controversy (Jacobs et al., 1989; Landsberger et al., 1992). In a recent study on neuroﬁbrillary tangles, Makjanic´ et al. (1997) used freeze-dried samples for analysis by both micro-PIXE and STIM. The results indicated that the elevated Al concentration measured is an artifact introduced during sample preparation. Using animal models, the Lund group performed several studies on rat brains in order to understand the mechanisms behind cell death in special cell strata of the brain as a consequence of stroke or epileptic seizures. The micro-PIXE results for groups of cells in the substantia nigra indicated that the onset of the cell death was characterized by a change in Ca concentration, and this before any histopathological effects could be observed (Themner et al., 1988; Inamura et al., 1989; Tapper et al., 1991). Atherosclerosis is a major cause of death in the Western world. The understanding of mechanisms behind the calciﬁed depositions on the artery wall is of paramount importance. Several studies using micro-PIXE and the nuclear microprobe have been carried out on this subject (Rokita et al., 1991; Pinheiro et al., 1996). The detailed information obtained, including data on several trace elements, provides clues on the atherogenesis. Being a very well-stratiﬁed tissue, skin lends itself very easily to microprobe analysis. Many studies have been carried out with the Lund nuclear microprobe by Forslind and coworkers (Pallon et al., 1992, 1996, 1997). Both normal skin and pathological states (e.g.,

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atopic and psoratic skin) have been extensively investigated. The trace elemental proﬁles across the various skin strata are also useful in the basic understanding of the skin barrier function. This is a very essential mechanism of epidermal skin to protect the body while still being permeable. Perturbations of the barrier have recently been investigated using a nuclear microprobe by the Lawrence Livemore Laboratory group (Mauro et al., 1998). Figure 11 shows the Ca distribution before and after acetone-induced barrier disruption. Another type of tissue that has been studied frequently using the nuclear microprobe is kidney cortex. In an attempt to understand the etiology of Menkes disease, Kirby et al. (1998) used an animal model (mice) and studied the distributions of Cu, Fe, and Zn in kidney cortex. The tissue specimens were prepared by cryosectioning and freeze-drying. Localization was facilitated by enzyme histochemical staining of adjacent sections. The micro-PIXE technique clearly showed how the three elements in question were distributed. c.

Hard Tissue

Due to the convenience in sampling (hair, nails), the elemental long-term stability (mineralized tissues), ‘‘time-recorder capability’’ (hair, nails, shells), and stability under the ion beam, hard tissues have been rather popular specimens for nuclear microprobe studies. Besides ﬁsh scales, various aragonite structures in ﬁsh and squid have been analyzed, in particular for Ca and Sr (Tang et al., 1997). The elemental variations in the slowly growing statoliths or otoliths (ear stones) reﬂect variations in living conditions. In most countries with extended coastlines, ﬁshing is of great economical importance, and hence the migration patterns of ﬁsh are obviously of great economical as well as ecological interest. Lipinski et al. (1997) studied squid statoliths using micro-PIXE. The results were

Figure 11 Calcium distribution in skin as measured by micro-PIXE with and without acetone provocation. A 3-mm microbeam of 3-MeV protons was used for analysis. (From Mauro et al., 1988, by permission of Blackwell Science, Inc.)

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not easily interpreted, and further investigations were suggested using also techniques that can reveal the crystal structure of the aragonite. In studies of ﬁsh migration, one has already made extensive use of nuclear microprobe analysis of otoliths (Halden et al., 1996). The good retention of absorbed elements in otoliths has been demonstrated, and hence they can serve as reliable ‘‘life-history records.’’ The Sr concentration will be dependent on the salinity of the water in which the ﬁsh is living and migration from freshwater to bracken water or seawater, as, for instance, salmon might carry out, can then be seen as variations in Sr concentration. Figure 12 shows two elemental maps obtained by micro-PIXE analysis of a brown trout (Salmo trutta) otolith. The maps clearly show how diﬀerent the elements Zn and Sr are taken up during the otolith growth. Zn shows clear seasonal ring patterns, whereas Sr has a strong peak in the center, at the time of birth of the ﬁsh. The interpretation would be that a marine (saltwater) mother has grown the egg at sea and thereby transferred a high Sr amount to it, and then migrated to freshwater where the egg was laid and the trout remained until it was caught. The Zn is incorporated with highest concentrations during the cold season (K. Limburg, Stockholm University, personal communication, 1999). The shells from several other species (e.g., pearl mussels, abalone, and oysters) have also been tried as bioindicators of environmental and climatic changes. The success in doing so has been limited, mainly because of the complexity of the mechanisms involved and the diﬃculties in interpreting obtained data. Human bone and teeth represent another material that has been investigated in great detail with nuclear microprobes (Mitani et al., 1997). Bone and teeth dentine act as good and relatively stable recorders of exposure and have therefore been used to assess the exposure to Pb. More recently, however, more interest has been directed toward basic growth patterns and the eﬀects of metal implants (Pa˚lsga˚rd et al., 1997). Trace element detection is used in the investigations of leaching of metal from, for instance, titanium implants (Ektessabi et al., 1997). In this kind of work, electron and nuclear microprobes can complement each other well, as was recently demonstrated in a study on implanted artiﬁcial hips (Jallot et al., 1998).

Figure 12 Elemental maps (Zn and Sr) from an otolith from brown trout. Analysis was done by 2.5-MeV protons and the scan size was 4 mm. Dark areas indicate high concentration and light areas indicate low concentration. (Courtesy of K. Limburg.)

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Botany and Environment

The relatively high, but still limited, lateral resolution of the nuclear microprobe makes it ideal for studying plant material. The cells are often rather large and it is possible to examine subcellular structures even with the modest beam resolution. Przybylowicz and co-workers performed several studies on plants, with special emphasis on the accumulation of metals, and they also made a comprehensive review of the botanical applications of the nuclear microprobe (Przybylowicz et al., 1997). They are using a combination of micro-PIXE, backscattering spectrometry, and STIM, and they use the dynamical quantitative elemental mapping technique to produce accurate distribution maps on-line. They conclude that the nuclear microprobe is almost ideal for extensive characterization of plant tissues, but that there are some limitations due to beam-induced damage of the specimen. The Oxford scanning proton microprobe (SPM) has also been used for studies of hyperaccumulation of metals in plants (Kra¨mer et al., 1997). A special plant (Alyssum lesbiacum) was selected because it is known to concentrate heavy metals in the leaves and because such species might be used for the cleanup of contaminated soil. It was found that the plant clearly concentrated Ni in the epidermal trichomes. The nuclear microprobe, with micro-PIXE as the most important tool, also has great potential in studies of environmental problems (Malmqvist, 1996). A few selected examples are studies of Al proﬁles in plant roots (Schoﬁeld et al., 1998), metal uptake in lichens (Watkinson and Watt, 1992), and heavy-metal uptake in bracken (Watt et al., 1991). B.

Atmospheric Aerosols

The multielement analysis of airborne particulate material (atmospheric aerosols) has been a very popular and highly successful application of PIXE since the early days of the technique. Actually, the pioneering paper of Johansson et al. (1970) already presented a PIXE spectrum of such material. A more recent spectrum, obtained from the bombardment of an urban aerosol sample (Campbell et al., 1986), is shown in Figure 13. Considering that atmospheric aerosols are often collected as a thin-sample layer on some thin ﬁlter or substrate ﬁlm, that such samples can be analyzed nondestructively by PIXE without sample preparation, that the sample matrix consists of light elements, and that a 5–10 min bombardment suﬃces to detect up to 20 elements, including interesting anthropogenic elements such as S, V, Ni, Cu, Zn, As, and Pb, the analysis of aerosol samples forms almost an ideal application of PIXE. Compared to EDXRF, which shares several favorable characteristics, PIXE oﬀers sensitivities (expressed as characteristic x-ray count rate per microgram of element actually exposed to the analysis) that are typically at least one order of magnitude better (Maenhaut, 1989), so that it requires much less sample mass and thus allows the use of compact samplers with high time and size resolution. Another advantage of PIXE over EDXRF is that it can be complemented with other ion-beam analysis techniques (Cohen, 1993, 1998; Cahill, 1995; Swietlicki et al., 1996a), so that the light elements (H, C, N, and O) that make up most of the aerosol mass can be measured as well. Motivation for the study of atmospheric aerosols and of their physical and chemical characteristics is that they reduce visibility, aﬀect human health, contribute to acidiﬁcation of terrestrial and aquatic ecosystems, and cause damage to structures and buildings (Elsom, 1992; Boubel et al., 1994; Cahill, 1995; Seinfeld and Pandis, 1997). During the past decade, renewed interest in the health risks of aerosols has been generated by the ﬁnding of strong correlations between increased mortality and the concentrations of (ﬁne) airborne particles in metropolitan areas in the United States (Pope et al., 1995). Furthermore, it has become clear that aerosols play a much more important role in climate than previously thought

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Figure 13 PIXE spectrum of an urban aerosol sample. (From Campbell et al. 1986, with permission from Elsevier Science.)

(Charlson and Heintzenberg, 1995). At the same time, one has come to realize that there remain serious gaps in our knowledge of atmospheric aerosols. With regard to the observed correlations between aerosols and increased mortality, it is uncertain whether these are due to the particles as a whole or to certain speciﬁc components, such as ﬁne sulfate, the carbonaceous aerosol (organic carbon and elemental carbon), or heavy metals. There is consensus, though, that the ﬁne particles are more harmful to human health than the coarse ones and this has led to the replacement in 1997 in the United States of the PM10 (particulate matter smaller than 10 mm) standard by the PM2.5 (particulate matter < 2.5 mm) standard for aerosol measurements in atmospheric monitoring networks. As to the climatic eﬀect of aerosols, this stems from the fact that they physically aﬀect the heat balance of the Earth, both directly by reﬂecting and absorbing solar radiation and by absorbing and emitting some terrestrial infrared radiation and indirectly by inﬂuencing the properties and processes of clouds, and, possibly, by changing the heterogeneous chemistry of reactive greenhouse gases (e.g., O3) (Charlson and Heintzenberg, 1995; Schwartz, 1996). Changes in the heat balance due to anthropogenic or externally imposed changes are referred to as forcings. There is a very large uncertainty associated with the aerosol forcing estimate, and this is essentially due to our poor state of knowledge regarding the sources, spatial and temporal variability, and chemical, physical, and optical properties of atmospheric aerosols. As for the impact on human health, ﬁne (submicrometer-sized) particles are more important than the coarse ones for both the direct and indirect eﬀect on climate. However, there is still much unknown about the formation, evolution, and composition of the ﬁne particles. Quite often, all measured aerosol constituents in ﬁne-aerosol characterization projects do not add up to the measured ﬁne-particulate mass (PM), and a major fraction of it remains unexplained. The recent ﬁndings about the eﬀects of aerosols together with the realization of the serious gaps in our knowledge have given a strong new impetus to aerosol

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research. The PIXE technique can continue to provide an invaluable contribution to this research by giving data for major, minor, and trace elements in large numbers of ﬁne and highly size-resolved aerosol samples. The data for the major elements (S, Na, Cl, Al, Si, Fe) are needed for assessing the climatic eﬀect of aerosols, for estimating the contributions of important aerosol types (e.g., sea salt, crustal material), and for arriving at chemical mass closure, whereas the multielemental data set as a whole (which comprises data for various anthropogenic tracers) can be used for resolving the contributions from diﬀerent source categories. On the other hand, one should realize that PIXE only provides part of the desired information. With regard to the chemical composition, it is highly desirable to perform also measurements for important ionic species (e.g., ammonium, nitrate) and for organic carbon (OC) and elemental carbon (EC). Furthermore, in order to arrive at a complete aerosol characterization, the chemical measurements have to be complemented with physical and optical (radiative) aerosol measurements. Recent insights in the importance and role of atmospheric aerosols were provided by Andreae and Crutzen (1997). The chapter by Cahill (1995) in the textbook of Johansson et al. (1995) gives a comprehensive overview of the applications of PIXE to aerosol research up to the early 1990s and gives valuable information on various aspects of aerosols. In a review paper by Maenhaut (1992), PIXE applications of the early 1990s to aerosols and other environmental samples are discussed. 1. Sampling Devices and Collection Surfaces Sampling of atmospheric aerosols for chemical analysis is usually carried out by means of ﬁlters, cascade impactors, cyclones, or a combination of these devices [e.g., Hinds (1982), Spurny (1986), Lodge (1991), Willeke and Baron (1993)]. Similar to XRF (see also Chapter 14), samplers that collect the particulate material on the surface of a ﬁlter or substrate ﬁlm are also most suitable for PIXE. Although it is highly preferable for XRF that the particles be present as a uniform layer, PIXE can also easily handle nonuniform samples, such as those collected by single-oriﬁce or some other cascade impactors. The collection of the aerosol by ﬁltration is done in single-ﬁlter samplers (high-volume or low-volume samplers) which may be provided with a PM10, PM2.5, or PM1 inlet, but also in certain devices that fractionate the aerosol in two size fractions, such as the dichotomous sampler or virtual impactor (Dzubay and Stevens, 1975; Loo et al., 1976) and the stacked ﬁlter unit (SFU) (Parker et al., 1977; Cahill et al., 1977). A recent example of the latter type is the Gent PM10 SFU sampler (Maenhaut et al., 1994a; Hopke et al., 1997). This device utilizes double NILU-type ‘‘open face’’ 47-mm-diameter ﬁlter cassettes [Norwegian Institute for Air Research (NILU), Kjeller, Norway], operates at a ﬂow rate of 16 L=min, and separates the aerosol in two size fractions of 2–10-mm equivalent aerodynamic diameter (EAD) and < 2 mm EAD, respectively. In most types of ﬁlter devices, membrane or ﬁbrous ﬁlters may be used, but the former, in particular Nuclepore polycarbonate ﬁlters, are preferred for analysis by PIXE. With ﬁbrous ﬁlters and, in fact, with certain membrane ﬁlters, a large fraction of the aerosol particles penetrates into the ﬁlter material, so that cumbersome corrections for matrix eﬀects are required (Kemp, 1977). For certain applications, Teﬂon membrane ﬁlters may be preferable to Nuclepore ﬁlters. Unfortunately, their high ﬂuorine content gives rise to a pronounced prompt g-ray background in the PIXE spectrum. Furthermore, some Si(Li) detection systems, when exposed to high rates of such prompt g-rays, may exhibit resolution deterioration and other electronic problems similar to those encountered when scattered particles penetrate the detector. As it is impossible to ﬁnd a ﬁlter type that is suitable for every kind of

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chemical analysis, systems have also been developed, such as the IMPROVE modular aerosol monitor (Eldred et al., 1990), whereby four PM2.5 samplers with diﬀerent ﬁlter types are operated in parallel. Single-ﬁlter samplers, dichotomous samplers, and stacked-ﬁlter units allow collecting samples that may be analyzed by several techniques, including PIXE, but they do not make full use of the favorable characteristics of PIXE, in particular its small mass requirement. Several PIXE groups have therefore spent considerable time and eﬀort in designing innovative samplers that combined several of the following features: light weight, battery powered, automated, and providing good time and size resolution. Nelson and co-workers (Nelson et al., 1976; Nelson, 1977; Baumann et al., 1981) developed the linear streaker and its successors: the circular and two-stage circular streaker. In these devices, a Nuclepore polycarbonate ﬁlter surface is continuously moved over a sucking oriﬁce, so that the particulate material is collected as a linear or circular streak on the ﬁlter material. The rate of movement is typically 1 mm=h, so that subsequent analysis of the Nuclepore strip with a 2-mm-wide beam provides a time resolution of 2 h. In recent years, Annegarn et al. (1996) developed a variant of the streaker that can be deployed on a small aircraft. Another example of a compact sampler is the SMART (solar monitoring of aerosols in remote terrain), which can operate for 2 weeks at a time unattended, giving PM10 and=or PM2.5 size cuts (Malm et al., 1994a). In order to obtain ﬁner size resolution than with the above samplers, the 1 L=min single-oriﬁce Battelle-type cascade impactor, as modiﬁed and commercialized by PIXE International Corporation (Baumann et al., 1981), can be used. This unit diﬀerentiates the aerosol in up to 10 size fractions and can be operated from small battery-powered pumps. A more sophisticated design that combines both good time and size resolution is the DRUM impactor, developed by the University of California at Davis (Raabe et al., 1988). Cascade impactors that are widely deployed in conjunction with other analytical techniques, such as the Berner low-pressure impactor (Berner, 1984) and the micro-oriﬁce uniform deposit impactor (MOUDI) (Marple et al., 1991), can also be used to collect samples for subsequent analysis by PIXE (Maenhaut et al., 1993b, 1993c). These devices have the advantage that they have several stages in the submicrometer size range and that they operate at a ﬂow rate of 25 or 30 L=min. However, as the aerosol particles are deposited on a rather large area, the PIXE detection limits (in nanograms per cubic meter and per stage) are not much better than with the 1-L=min PIXE Int. impactor (Maenhaut et al., 1993b). An impactor which combines high size resolution with improved PIXE detection limits is the small-deposit-area low-pressure impactor (SDI) (Maenhaut et al., 1996d). The SDI was especially designed for aerosol collections with subsequent PIXE analysis. It provides 12 size classes (down to about 50 nm), operates at a ﬂow rate of 11 L=min, and is a multinozzle device, but the total aerosol deposit for each stage remains conﬁned to an area with diameter of less than 8 mm, so that it can be entirely enveloped by the PIXE beam. A cross section of the SDI is shown in Figure 14. The collection surfaces in cascade impactors are typically thin polyester (Mylar) or polycarbonate (Kimfol) ﬁlms. To reduce particle bounce-oﬀ eﬀects during sampling, the ﬁlms are commonly coated with vaseline or paraﬃn. 2. Examples There is a wealth of examples on the application of PIXE to atmospheric aerosol samples. It is comforting to note that publications in which PIXE was used are not only published in key journals on PIXE and the IBA techniques, such as Nuclear Instruments and Methods

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Figure 14 Cross section of the SDI. Only 4 of 12 impaction stages are shown. (From Maenhaut et al., 1996d.)

in Physics Research B and the International Journal of PIXE, but also in atmospheric and environmental chemistry journals, such as Atmospheric Environment, Journal of Geophysical Research (section Atmospheres), and others. The aerosol samples analyzed by PIXE range from those collected indoors (e.g., in work environments) or near speciﬁc pollution sources to samples dealing with urban, regional, and global air-pollution problems, and to samples collected in areas as remote as the Arctic and Antarctic. a. Work Environments and Near Speciﬁc Sources An example of a study in a work environment is that of Formenti et al. (1998b). These researchers collected aerosols inside an arc-welding shop with a single-stage streaker sampler, using a time resolution of 1 h. By examining the temporal variability and the correlations among the airborne particulate elements, they extracted an arc-welding source proﬁle (i.e., the composition of the emitted particles). It was also found that the welding shop activities gave rise to concentrations of up to 100 mg=m3 for Fe and up to 7 mg=m3 for Cr. Studies on heavy metals in the atmosphere near copper, zinc, and steel smelters were conducted in Chile (Romo-Kro¨ger et al., 1994), China (Zhu and Wang, 1998), and Italy (Prati et al., 1998). The study near the copper smelter in Chile (Romo-Kro¨ger et al., 1994) was done over a period of 1.5 months, but because of a major labor strike, there was a total shutdown of the smelter during 22 days. It was found that S, Cu, Zn, and, especially, As were quite enriched in the air during normal working periods relative to the strike period. The average ﬁne ( < 2.5 mm EAD) As level at the sampling site (at 13 km from the smelter) during normal operation was 240 ng=m3. Also, the aerosol emissions from nonindustrial and larger-scale sources, such as the 1991 Kuwaiti oil ﬁres and tropical forest and savanna biomass burning, have been investigated using PIXE [e.g., Cahill et al. (1992), Reid et al. (1994), Echalar et al. (1995), Gaudichet et al. (1995), Maenhaut et al. (1996c)]. However, in most of these studies, the PIXE measurements were complemented with other analyses [e.g., for black carbon (BC)] in order to arrive at a more complete characterization of the emitted particles. Some of the collections for the Kuwaiti oil ﬁres (Cahill et al., 1992) were done with SFU samplers that were installed on an aircraft. The samples were subjected to optical, gravimetric, scanning electron microscopy (SEM), and PIXE analyses, yielding information on the morphology, mass, and composition of the aerosols. It was found that the mass in the coarse size fraction was dominated by soil-derived particles. In the ﬁne fraction, organic matter and

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ﬁne soils each accounted for about one-fourth of the mass, whereas salt and sulfates contributed about 10% and 7%, respectively. The work on the aerosol emissions from tropical forest and savanna biomass burning (Echalar et al., 1995; Gaudichet et al., 1995; Maenhaut et al., 1996c) resulted in source proﬁles which can be used in chemical massbalance receptor (CMB) modeling work on ambient aerosol datasets in order to apportion the contribution from biomass burning to the PM and various aerosol constituents (Maenhaut et al., 1996a, 1996c). b.

Urban Areas

Numerous PIXE application papers have dealt with the study of atmospheric aerosols in urban areas. Considering the renewed interest in the health risks of aerosols in such areas (Pope et al., 1995), PIXE may continue to ﬁnd an important application niche in this ﬁeld. Miranda (1996) presented an overview of the studies in large urban areas in which PIXE was used and which were published up to 1995. Besides assessing the airborne levels of the heavy metals and of important elements such as S, major objectives in most studies are to identify the major sources (source types) of the heavy metals and elements and to apportion them to the sources. For this source identiﬁcation and apportionment, one mostly relies on multivariate techniques, such as absolute principal-component analysis (APCA), absolute principal factor analysis (APFA), and multiple regression analysis. Occasionally, these approaches are complemented or compared with CMB analysis and wind direction analysis. Several urban aerosol studies with PIXE were carried out in Latin America, particularly in Mexico City, Santiago de Chile, and Sao Paulo city (Miranda et al., 1994, 1998; Andrade et al., 1994; Cahill et al., 1996; Aldape et al., 1996). Andrade et al. (1994) applied APCA to their coarse and ﬁne aerosol data from Sao Paulo city and also to a dataset of various meteorological variables, and they subsequently performed a correlation analysis between the various matrices with ‘‘absolute principal-component scores’’ (APCS). This indicated, for example, that the industrial components of both size fractions were positively correlated with the northeast wind direction. For Mexico City and Santiago de Chile, the aerosol data were compared with those from Los Angeles, California (Cahill et al., 1996). A rather surprising result was the rough equivalence among the three cities in terms of several classes of ﬁne particles. Coarser particles were, however, far more prevalent in Mexico City and Santiago than Los Angeles. In Europe, extensive work with PIXE has been done on urban aerosols from Hungary (Molna´r et al., 1993; Ali et al., 1994; Borbe´ly-Kiss et al., 1996; Salma et al., 1998). In a study by Molna´r et al. (1993), data for downtown Budapest were compared with those for suburban Budapest and for rural air, and selected suburban and rural data were subsequently used in a source–receptor model for estimating the dry deposition for a number of elements. Urban aerosol studies in which PIXE was used were also done in several other European countries (Climent-Font et al., 1994; Luis-Simon et al., 1995; Braga Marcazzan, 1996; Swietlicki et al., 1996b; Harrison et al., 1997; Maenhaut and Cafmeyer, 1998; Wrobel and Rokita, 1998). Swietlicki et al. (1996b) combined their PIXE data for Lund, Sweden with results of gaseous species (SO2, NO2, O3) and examined the combined dataset with APCA. About one-third of the SO2, most of the ﬁne particulate S, and about half of the ﬁne Zn and Pb were attributed to a regional background source originating from ferrous and nonferrous smelters. Also, Harrison et al. (1997) used PIXE aerosol data together with results from other measurements and examined the combined datasets with receptor modeling. In addition to two European cities (Birmingham, UK, and Coimbra, Portugal), Lahore, Pakistan was included in this study. It was found that

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there were considerable similarities between the two European cities, but large contrasts with Lahore, with its dryer climate and poorly controlled pollution sources. Other cities in Asia where aerosol studies with PIXE have been performed are Singapore (Bao et al., 1995) and Kyoto, Japan (Kasahara and Gotoh, 1995). An example of a comprehensive study on urban aerosols from Australia is the work by Chan et al. (1997). PM10 samplers were used during 1 year at ﬁve sites in Brisbane (and a dichotomous sampler at one of the sites), and the samples were analyzed by a variety of techniques, including PIXE, other IBA techniques, ion chromatography (IC), and atomic spectrometry. The major components in the PM10 aerosols were crustal matter (25% by mass), organics (17%), sea salt (12%), elemental carbon (10%), and ammonium sulfate (7%). Furthermore, apart from signiﬁcant local inﬂuences at some of the sites (such as a cement factory), most anthropogenic emissions appeared to be rather evenly and widely distributed in Brisbane. Also, urban aerosols from Africa have been analyzed by PIXE [e.g., from Khartoum, Sudan (Eltayeb et al., 1992, 1993), from Lagos and Ile-Ife, Nigeria (Ogunsola et al., 1993), and from Soweto, South Africa (Formenti et al., 1998a)]. c.

Rural Areas and Regional Aerosol Studies

Particle-induced x-ray emission has been extensively employed for analyzing aerosols from rural areas and in regional aerosol studies. Similar to in urban and local air pollution studies, the investigations on rural and regional scales often aim at identifying and assessing the sources of the particulate matter. However, whereas the urban=local-scale studies deal with individual sources and=or source types, the emphasis in the rural= regional-scale studies is placed on the source regions. To aid in this work, one often relies heavily on air mass back-trajectories and wind sector data. In the United States, on the other hand, the major objective in the regional-scale studies has been the examination of the relationship between ﬁne aerosol constituents and visibility degradation. Scandinavia is a receptor region for long-range transported gaseous and particulate pollutants from various European source regions. Research on this subject is ongoing for several decades, and PIXE has contributed to much of our knowledge about aerosols for Denmark and southern Sweden [see, e.g., Maenhaut and Malmqvist (1992)]. Also in the 1990s, PIXE has continued to be used for work in Denmark (Kemp, 1993, 1996). In addition, it has been utilized in both campaign-type and long-term aerosol studies at some sites in southern Norway (Pakkanen et al., 1996; Maenhaut et al., 1997b). In the long-term study (Maenhaut et al., 1997b), Gent SFU samplers were employed for the continuous collection of coarse and ﬁne aerosols at Birkenes and Skrea˚dalen during a 4-year period. The samples were analyzed for the PM, BC, and over 40 elements (by a combination of PIXE and INAA). Seven air mass transport sectors were deﬁned, and the air mass trajectories for each individual sample served as the basis to classify the sample in one of the sectors (or to leave it unclassiﬁed). For each sector, median coarse and ﬁne concentrations were then calculated for each element. Fine PM, S, BC, and various metals (V, Mn, Ni, Zn, As, Sb, Pb) were highest for air masses coming from sectors E, SE, and S (Russia, Poland, Germany, Benelux). For air masses from sector WSW (UK), the medians were lower by a factor of 2 or more, and for air from the sectors WNW and NNW (Atlantic Ocean and Norway), the median levels were 5–10 times lower. Substantial diﬀerences were also found in the multielemental proﬁles of the diﬀerent sectors. For example, the ratios to noncrustal V (an indicator for heavy oil burning) varied by over a factor of 4 for As and Se and by over a factor 10 for noncrustal Mn. Somewhat related to the research on the longrange transport from continental Europe to Scandinavia is that on the transport to and over the North Sea. PIXE was used for analyzing aerosol samples that were collected

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above the North Sea using a small aircraft or aboard ships (Injuk et al., 1992; Rojas et al., 1993; Franc¸ois et al., 1993). As part of the Great Dun Fell Cloud Experiment 1993 in the United Kingdom, Swietlicki et al. (1997) collected ﬁne aerosols on ﬁlters with a 2-h time resolution and performed analyses for soot, for elements by PIXE, and for anions and cations by IC. Examination of the resulting dataset by APCA revealed three major source types, longrange transport (LRT), sea spray, and a local source. LRT accounted for most of the soot, S, Zn, sulfate, and ammonium. The results were largely consistent with those from APCA on a separate cascade impactor dataset and with the other observations used in the source identiﬁcation. Several rural=regional aerosol studies with PIXE were conducted in central and eastern Europe (Molna´r et al., 1995; Swietlicki and Krejci, 1996; Jagielski et al., 1996; Amemiya et al., 1996). In the study by Molna´r et al. (1995), total ﬁlter PIXE datasets for three Hungarian sites were complemented with Cd data obtained by atomic absorption spectrometry (AAS), with AAS data from cascade impactor samples, and with inductively coupled plasma–atomic emission spectrometry (ICP–AES) data from precipitation samples in order to derive the atmospheric budget of several particulate elements over Hungary. It was found that the country is a net source for elements that are mostly produced by fossil fuel combustion (V, Ni), whereas it is a net sink for elements released during industrial processes and automotive transport. As indicated earlier, the regional-scale studies in the United States have concentrated on the relationship between ﬁne aerosol constituents and visibility (Eldred and Cahill, 1994; Malm et al., 1994a, 1994b, 1996; Cahill, 1995; Malm and Pitchford, 1997). Using a large network of IMPROVE samplers that were placed in remote areas, national parks, and monuments, long-term collections of ﬁne (< 2.5 mm EAD) aerosol were made. PIXE was used for measuring S, Si, K, Ca, Ti, Fe, Na, Cl, and other elements, and the data obtained were employed to estimate the concentrations of sulfate, mineral dust, and some other aerosol types. Another IBA technique [i.e., particle elastic scattering analysis (PESA)] provided data for H, from which organic matter was derived. In addition, PM was obtained by weighing, OC and EC were measured with a thermal optical reﬂectance (TOR) technique, anionic and cationic species were determined by IC, and the optical absorption of the aerosol collected on ﬁlters was measured with the Laser Integrating Plate Method (LIPM). At a number of the sites, the normal IMPROVE work was complemented with measurement of the light scattering of the ambient (in situ) aerosol by an integrating nephelometer or with DRUM aerosol collections for obtaining the detailed size distribution of sulfate (derived from PIXE S). In examining the relationship between light scattering, absorption, or extinction (the latter is the sum of scattering and absorption) and the concentrations of the various aerosol types, two approaches were used. The ﬁrst (multivariate) approach relied on factor analysis or on multiple linear regression of the optical data on the concentrations of the aerosol types (or species). In the second approach, the particle scattering (or absorption) is estimated from the species concentrations and published mass-scattering (or absorption) eﬃciencies and then compared with the directly measured scattering (or absorption). From their examination of the relationship in the Grand Canyon area, Malm et al. (1996) concluded that using EC, as derived from TOR, to estimate the aerosol absorption may signiﬁcantly underestimate absorption. In addition to EC, OC and soil provided substantial contributions. Finally, as examples of rural=regional studies outside of Europe and the United States, those of Romo-Kro¨ger and Llona (1993) and of Liu et al. (1996) can be mentioned. Romo-Kro¨ger and Llona (1993) employed battery-powered SFUs on the slopes of the

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Los Andes mountain range to collect aerosols for subsequent PIXE. Liu et al. (1996) examined the spatial patterns of ﬁne-particle S and Pb concentrations in New South Wales, Australia. The data for the latter study were obtained from a network of 24 ﬁneparticle monitoring sites. d.

Tropical, Subtropical, and Polar Regions, and Very Long-Range Transport

In the past decades, there has been an increasing recognition that equatorial and tropical regions, and in particular the South American and African continents and the large Southeast Asian=Australian region, play a very important role in regional and global atmospheric chemistry. Especially the atmospheric eﬀects of tropical biomass burning (ﬁres in savannas, deforestation, the burning of fuel wood and of agricultural waste) have received much attention [e.g., Crutzen and Andreae (1990)]. The emission of particulate matter during biomass burning occurs mainly in the form of submicrometer, accumulation-mode particles (smoke). These ﬁne particles are eﬃcient scatterers of solar radiation, and, as a consequence, they have often a large impact on local and regional visibility and contribute to the planetary albedo, thus aﬀecting regional and global climate. Moreover, many of the pyrogenic particles can act as cloud condensation nuclei (CCN) and thereby change the radiative properties of clouds. Outside of the burning season, natural biogenic emissions from the vegetation provide a very substantial contribution to the atmospheric aerosol burden, and a signiﬁcant fraction of the biogenic aerosol in the ﬁne-size range may be produced by oxidation and gas-to-particle conversion of volatile organic compounds which are emitted by the vegetation. During the 1990s, several aerosol studies were conducted in Brazil, in which one relied heavily on PIXE (Artaxo et al., 1994; Artaxo and Hansson, 1995; Gerab et al., 1998a; Maenhaut et al., 1996a). Artaxo et al. (1994) presented results from a long-term study on the ﬁne-aerosol composition at three sites in the Amazon Basin. The samples were obtained with SFUs and analyzed for the PM, BC, and up to 20 elements. APFA on the resulting ﬁne datasets identiﬁed four components: soil dust, biomass burning (with BC, K, Cl), natural primary biogenic particles, including a gas-to-particle component (with K, S, Ca, Mn, and Zn), and marine aerosol. The biogenic and biomass burning components provided the largest contribution to the ﬁne PM. In southern Africa, aerosol studies with PIXE (Salma et al., 1994; Maenhaut et al., 1996c) were conducted as part of the Southern Africa FireAtmosphere Research Initiative (SAFARI-92), for which the ﬁeld work took place in August–October, during the dry season. In the study by Maenhaut et al. (1996c), aerosols were simultaneously collected with Gent SFU samplers at three sites in the eastern Transvaal, South Africa. In addition to PIXE, INAA was employed for the multielement analysis, and the resulting data sets were examined by APCA and CMB receptor modeling. Four components were identiﬁed in the ﬁne size fraction (i.e., mineral dust, sea salt, biomass burning products, and sulfate). The pyrogenic component was the dominant contributor to the atmospheric concentrations of the ﬁne BC, K, Zn, and I. About 40% of the ﬁne PM was, on the average, attributed to the pyrogenic particles and about one-third of it to the sulfate component. The relation of the time trends of the various components with three-dimensional air mass back-trajectories indicated that elevated levels of pyrogenic products were mostly found in air masses arriving from the north. The original stimulation for intensive aerosol research in the Arctic came from work on the sources of Arctic haze in the 1970s (Rahn et al., 1977). In the 1980s, the ﬁnding of an inverse relationship between non-sea-salt Br and surface O3 during polar sunrise (Barrie et al., 1988) provided an additional impetus. More recently, the increasing recognition that the Arctic is a very sensitive environment and is highly important for regional and global

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atmospheric chemistry and climate is giving further motivation for continued aerosol research in the region. During the 1990s, long-term aerosol studies involving PIXE were continued in Alaska (Polissar et al., 1996, 1998a, 1998b). It was found that the ratio of the aerosol optical absorption coeﬃcient to the S concentration can be used as an indicator of smoke from forest ﬁres (Polissar et al., 1996). Furthermore, the long-term ﬁne-particle datasets from seven sites were examined using a new type of factor analysis, positive matrix factorization (PMF) (Polissar et al., 1998b). Four main sources contributed to the observed concentrations at the more northerly locations in Alaska: long-range transported anthropogenic aerosol (Arctic haze aerosol), sea salt, local soil dust, and aerosol with high BC concentrations from regional forest ﬁres or local wood smoke. Also in the European Arctic, long-term aerosol research involving PIXE has been conducted. Results from several years of aerosol collections at the Zeppelin mountain station in Spitsbergen were presented (Maenhaut et al., 1994b, 1997c; Havranek et al., 1996). The relative contributions from natural and anthropogenic sources to the non-sea-salt (nss) sulfate were assessed, and about one-third of the nss sulfate during summer was attributed to natural marine biogenic emissions (emission of dimethylsulﬁde by phytoplankton) (Maenhaut et al., 1994b). The PIXE technique also contributed to the characterization of the atmospheric aerosol during the International Arctic Ocean Expedition 1991 (Maenhaut et al., 1996b). Particle-induced x-ray emission was extensively used for measuring the detailed size distribution of particulate S, sea salt, crustal and other elements above the Greenland Ice Sheet (Hillamo et al., 1993; Jaﬀrezo et al., 1993; Bergin et al., 1995; Kerminen et al., 1998). The major objective in these studies was to obtain information that is needed for calculating the dry deposition velocities of the elements and to contribute to a better understanding of the air-to-snow transfer. At Dye 3, an 8-stage PIXE Int. cascade impactor was used for the aerosol collections (Hillamo et al., 1993; Jaﬀrezo et al., 1993), but for the subsequent work at Summit, 12-stage SDIs were deployed (Bergin et al., 1995; Kerminen et al., 1998). The contributions of snow, fog, and dry deposition to the summer ﬂux of a number of elements and species at Summit were assessed. For S, and the crustal and sea salt elements, dry deposition accounted for around 15% of the summer ﬂux (Bergin et al., 1995). Antarctic aerosol studies involving analysis by PIXE were also done (Artaxo et al., 1992; Mittner et al., 1996; Correia et al., 1998). That gaseous air pollutants and also natural and anthropogenic aerosols can be transported over several thousands of kilometers fascinates atmospheric scientists for more than three decades. During SAFARI-92, it was documented by PIXE and other techniques that pyrogenic, biogenic, and mineral aerosols from southern Africa can be transported to the central tropical South Atlantic (Swap et al., 1996). Perhaps a more surprising ﬁnding derived from PIXE aerosol data is that desert dust from North Africa may, under certain conditions, be transported to central Illinois and the eastern United States (Gatz and Prospero, 1996; Perry et al., 1997). e.

Micro-PIXE

Whereas normal macro-PIXE is very extensively used for analyzing atmospheric aerosols, the application of micro-PIXE and complementary nuclear microprobe techniques to this sample type has been much more limited. The emphasis in contemporary aerosol research is clearly placed on the submicrometer-sized particles, and to examine individual aerosol particles of such sizes, a nuclear microbeam of 0.1 mm with an intensity of 100 pA would be very welcome. However, progress in the development of such beam sizes has been slow. Also, atmospheric aerosols consist of a heterogeneous population of particles of widely variable size and composition. In order to obtain results that are representative for the

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aerosol as a whole, thousands of individual particles should preferably be analyzed. However, due to the fact that the nuclear microprobe analysis is rather time-consuming, at most a few hundred particles are analyzed in micro-PIXE analyses of aerosols. For the fast multielemental analysis of large numbers of submicrometer-sized particles, EPMA is certainly to be preferred over micro-PIXE, and this is despite the fact that its detection limits (in micrograms per gram) are about two orders of magnitude worse and that accurate quantiﬁcation in it is much more problematic. Rather than trying to compete with EPMA in the analysis of aerosols, micro-PIXE should be used as a complementary technique, for example, for obtaining additional information and=or for special aerosol samples or problems where accurate quantitative results and=or data for real trace elements are needed. Orlic´ (1995) reviewed the nuclear microprobe work on atmospheric aerosols that was carried out up to about 1994. Examples of micro-PIXE applications on ambient (outdoor) aerosols since then are the studies by Orlic´ et al. (1995) on particles from Singapore, of Maenhaut et al. (1997a) on aerosols from Israel, of Gerab et al. (1998b) on Amazon Basin aerosols, and of Rajta et al. (1998) on aerosols from Debrecen, Hungary. In the study by Maenhaut et al. (1997a), micro-PIXE was used in combination with micro-Rutherford back-scattering (RBS) to determine the quantitative composition of individual particles and to diﬀerentiate, for calcareous aerosols, between particles consisting of pure compounds and of simple mixtures. For the latter type, the contribution from each compound was quantitatively determined. In the study by Rajta et al. (1998), a quantitative PIXE analysis was performed on 412 single particles (of 2–12 mm in diameter). The resulting dataset was examined by hierarchical cluster analysis, and several types of soil particles were identiﬁed. De Bock et al. (1996) employed micro-PIXE as a complement to EPMA in their study on indoor aerosols from the Correr museum in Venice, Italy. They used the Oxford SPM to produce elemental maps for giant particles ( > 8 mm). Because of the better detection limits, micro-PIXE allowed the detection of elements such as Ti, Cr, Fe, and Zn. C.

Other Environmental and Earth Science Applications

In addition to atmospheric aerosols, various other types of environmental sample have been analyzed by PIXE. The applications to biological samples for environmental purposes, including the use of bioindicators, were already addressed in Sec. VI.A. Here, we will present examples of applications to samples collected from various combustion units, such as ﬂy-ash particles, and samples from the aqueous environment, including rainwater, river water, and river sediments. In addition, the potential of PIXE and in particular of the nuclear microprobe for mineral prospecting and for improving our understanding of basic geological processes will be illustrated. 1. Combustion Sources The physicochemical characterization of particles from combustion units is of importance from a purely environmental viewpoint, but also to understand and improve the combustion technology. The environmental impact comes from particles that are emitted through the stack into the atmosphere, but also from the particulate material that is collected by electrostatic precipitators (ESPs) or other cleaning devices, or is recovered as bottom ash. For ESP ash and bottom ash, the levels of heavy metals are of critical importance for making decisions on their use as construction material, their distribution to agricultural or other land, or their disposal as waste.

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Several studies involving micro-PIXE have been done on particles from coal combustion (Jaksˇ ic´ et al., 1993; Caridi et al., 1993; Bogdanovic´ et al., 1994, 1995; Cereda et al., 1995a, 1995b, 1996; Rousseau et al., 1997; Hickmott et al., 1997). The PIXE groups from Zagreb and Milan, in cooperation with the Oxford SPM group, have been especially active in this area. The ﬂy-ash and other ash particles usually had sizes of several micrometers, which makes them more appropriate objects for nuclear microprobe investigations at the individual particle level than, for example, submicrometer-sized atmospheric aerosols. Bogdanovic´ et al. (1994) used STIM in combination with micro-PIXE in order to obtain information on the areal density and topology of individual ﬂy-ash particles and to be able to convert the elemental intensity maps into actual concentration distributions. Cereda et al. (1995b) examined about 100 particles that were collected at the outlet of a pilot ESP of a coal-ﬁred power plant. As a result of the combustion of the diﬀerent components of the mineral matter in the parent coal, seven particle classes with characteristic matrix compositions and diﬀerent trace element concentrations were observed. The trace elements were found to be nonuniformly distributed over the particles. Elements which are associated with volatile compounds, such as S, Ni, Zn, and Ga, were clearly more concentrated in the smaller particles ( < 2 mm) than in the larger ones. Macro-PIXE in combination with other techniques has been used for the analysis of cascade impactor samples that were collected at various locations in the ﬂue gas stream of coal burners (Maenhaut et al., 1993c; Lind et al., 1994, 1996) and recently also of biomass combustion units. In their study on the volatilization of heavy metals (Cu, Zn, Cd, Pb) during circulating ﬂuidized bed combustion of forest residue, Lind et al. (1999) found that none of these metals were enriched in the ﬁne particles at the inlet of the ESP. In order to obtain source proﬁles from the combustion of herbaceous and wood fuels, Turn et al. (1997) conducted biomass burning experiments in a wind tunnel, collected the emitted particles with a variety of samples, including DRUM impactors, and analyzed the samples by PIXE, XRF, and some other techniques. 2. Aqueous Environment Liquid samples are not very suitable for direct analysis by PIXE. They are often analyzed after pipetting a fraction on a suitable backing ﬁlm, letting it dry, and then bombarding the residue. However, in order to obtain detection limits that can compete with those in ICP–MS or AAS, one has to resort to preconcentration techniques. Savage et al. (1995) experimented with dried algae for preconcentrating metallic trace elements from water for subsequent PIXE analysis. For the preconcentration of precipitation samples, Tschierch et al. (1996) built a spray-drying apparatus that was similar to that developed by Hansson et al. (1988). Examples of applications of PIXE to rainwater are the work of Ghorai et al. (1993) in Alabama (USA) and of Kasahara et al. (1996) in Kyoto, Japan. In the latter study, rainwater was collected with 0.1 mm of rainfall from the start of each rain, and the samples were ﬁltered through a 0.2-mm pore-size Nuclepore ﬁlter, so that the soluble and insoluble components could be determined. In accordance with earlier studies, it was found that the concentrations of the elements in the rain decreased quickly until about 0.5 mm of rainfall. Si, Ti, and Fe were mainly present in the insoluble component of the rain, whereas virtually all the S and Cl was soluble. River water from India and Portugal was analyzed by PIXE (Kennedy et al., 1998; Araujo et al., 1998). Cecchi et al. (1996) gave an overview of their comprehensive study on the Venice Lagoon. Somewhat special water samples are ice cores drilled from glaciers or inland ice. Such samples are collected to investigate variations in their composition and trace element content with depth

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(e.g., along the core) and, by doing so, to obtain information on past environmental or climatic changes or to assess the impact of major volcanic eruptions. PIXE oﬀers the possibility of analyzing ice cores with millimeter resolution, so that the seasonal and annual elemental variation can be studied in great detail (Hansson et al., 1993). Samples for PIXE may be prepared by placing ice sections on a thin backing ﬁlm and removing the ice by sublimation. Hansson et al. (1993) used this approach for samples from a deep core, which was drilled at Dye 3, South Greenland. The spatial (and thus temporal) variations in the elements measured by PIXE were used to assist in the dating of some sections from the core. Laj et al. (1996) coupled macro-PIXE with EPMA for characterizing atmospheric aerosol particles and soluble species in ice cores from Summit, Central Greenland. Several PIXE studies have been devoted to the study of sediments in rivers, estuaries, or the sea (Randle et al., 1993; Limic´ and Valkovic´, 1996; Valkovic´ and Bogdanovic´, 1996; Martı´ n et al., 1996, 1998a, 1998b; Benyaich et al., 1997; Dutta et al., 1998; Tang et al., 1998), mostly to assess the anthropogenic input of heavy metals and other elements. Randle et al. (1993) compared INAA and PIXE for the determination of heavy elements in estuarine sediments and concluded that the two techniques should be regarded as complementary rather than competitive. Valkovic´ and Bogdanovic´ (1996) used a combination of PIXE and XRF. A total of 19 elements were measured by PIXE and 12 by XRF. For elements that were determined by both techniques, reasonable agreement was observed. Also, Benyaich et al. (1997) made a comparison of PIXE and XRF, and they concluded that the two techniques are complementary. Dutta et al. (1998) analyzed ferromanganese oxide deposits from diﬀerent locations of the Indian Ocean by PIXE. Based on the Mn=Fe ratio, the deposits were classiﬁed into hydrogenous and hydrothermal types, and the distribution of various minor and trace elements in both types were studied. Several elements exhibited higher concentrations in the hydrogenous deposits than in the hydrothermal ones. This was attributed to suitable physicochemical oceanic conditions prevailing at the depositional sites and the growth rate of the deposits. The concentrations of Co, Ni, and Cu were dependent on ocean depth. It was also found that the hydrothermal processes seem to be more controlling in the uptake of V and As than the hydrogenous processes. 3. Mineral Prospecting and Geology In addition to sediments in aqueous systems, there exists a very large variety of other geological materials. During the past decade, macro-PIXE sometimes in combination with complementary IBA techniques has been applied for analyzing soils (Cruvinel and Flocchini, 1993), mineral grains (Vogt et al., 1992; Pillay et al., 1993), microspherules in bedded chert (Miono et al., 1993), emeralds (Ma et al., 1993), and vein quartz from the Tasmania gold mine (Russell et al., 1996). In the latter study, it was found that goldbearing quartz has higher Ge, Li, As, Mn, Fe and Al contents and lower K, S, Na and Cl than barren quartz samples. Because of its unique ability to provide quantitative trace element information on a micrometer scale, micro-PIXE oﬀers great potential for mineral prospecting and for improving our understanding of basic geological processes. Actually, the nuclear microprobe could serve as an excellent complement to the electron microprobe in these areas. The capabilities of micro-PIXE are increasingly recognized by scientists in mineralogy and geology, and some major research programs have been running over a long period (Cabri et al., 1985; Sie, 1997). The mineral sample preparation techniques used for other microscopic techniques (e.g., for the electron microprobe) are, to a large extent, also suitable for micro-PIXE.

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Great care should be taken to reduce surface roughness, however, as otherwise accurate quantitative data cannot be obtained. For realizing a smooth surface, grinding or polishing are normally used, but in applying such procedures for PIXE, contamination control is much more stringent than for the case of the electron microprobe. Hence, it is recommended to use diamond powder for polishing, because this contains only carbon. Furthermore, to take full advantage of the spatial resolution of the microbeam, it is necessary to prepare specimens that are less than a few micrometers thick. This can be done by using an ‘‘ion-milling’’ technique. The complex matrix composition of many geological specimens means that accurate correction for projectile slowing and x-ray attenuation is quite complicated. Also, the PIXE spectra of such specimens are often complex, so that special evaluation codes are required (Ryan et al., 1990a). Finally, it should be realized that because of the relatively large range of protons in matter, underlying layers of possibly quite diﬀerent composition may also contribute to the PIXE spectrum for certain specimens. To improve the accuracy of nuclear microprobe analysis, it is almost always preceded by electron microprobe analysis from which the matrix composition can be inferred. However, in a study to check the quantiﬁcation of the matrix elements, Campbell et al. (1997) investigated silicates and glasses using only micro-PIXE. They used two x-ray detectors simultaneously to be able to also measure the very light elements, and they showed that, with good calibration, one can obtain the matrix composition with adequate accuracy. The micro-PIXE analysis can be done either point-by-point or by performing a twodimensional scan over a selected area. The latter method provides ﬁne maps for the major and minor elements but is normally not sensitive enough at the trace element level. The two-dimensional image can be used for selecting interesting spots for subsequent point-bypoint analysis. Alternatively, the pixels in arbitrary regions of interest can be added and quantiﬁed oﬀ-line. As will be discussed in Sec. VII.C.2, a complementary technique, ionoluminescence, is preferably combined with micro-PIXE for imaging and for analysis of the geologically very important rare earth elements. Figure 15 shows elemental maps (from micro-PIXE) and the distribution of the ionoluminescence (IL) obtained by scanning over a 0.2560.25-mm2 area of a zircon. The combination of optical viewing through a microscope and the IL imaging technique signiﬁcantly facilitates orientation on the specimen and guides the geologist to ﬁnd the relevant spot for detailed microanalysis. One important ﬁeld of application for micro-PIXE is ore mineralogy. In particular, microanalysis of sulﬁdes has been extensively used. Griﬃn et al. (1996) employed microPIXE to analyze polished tourmaline sections collected in massive sulﬁde deposits and demonstrated the usefulness of trace element data for mineral exploration. A combined electron and nuclear microprobe study of ﬂuid inclusions in quartz crystal from a porphyry copper deposit has contributed to the understanding of the metallogeny of the deposit, even though only qualitative micro-PIXE data were obtained (Damman et al., 1996). Other examples of successfully using micro-PIXE are studies on meteorites (Vis, 1997), patina layer analysis on rock artifacts in South Africa (Pineda et al., 1997), analysis of platinum-group metals and gold (Maetz et al., 1997), and studies of fossil ﬂuids in rocks (Volﬁnger et al., 1997). The great potential and future of nuclear microprobes in geoscience have recently been comprehensively discussed by Sie (1997). As indicated in Sec. III.B.3, micro-PIXE can also be done in an external beam setup in air and this possibility is also useful for analyzing some geological specimens. Accurate elemental maps for areas of several hundreds of square millimeters can be produced by mounting the specimen, which may be very large, on a simple x–y plotter table controlled by the normal beam-scanning electronics (Lo¨vestam and Swietlicki, 1989).

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Figure 15 Nuclear microprobe analysis of a natural zircon using 2.5-MeV protons. Elemental maps obtained from micro-PIXE (Zr, Hf, Fe) and a map of ionoluminescence (IL). Scan size: 250 mm.

The most spectacular use of PIXE in the past few years has undoubtedly been the deployment of the portable a-proton x-ray spectrometer (APXS) on Mars (Rieder et al., 1997a, 1997b). The APXS was a compact (600 g) device installed on board the Sojourner rover of the Mars Pathﬁnder. Alpha-particles from a 50-mCi 244Cm source were used for the excitation, and the x-rays were detected with a thermoelectrically cooled Si p-i-n detector with a resolution (FWHM) of 250 eV at 6.4 keV. Besides PIXE, RBS and NRA [using (a, p) reactions] were employed with the APXS. The PIXE results indicated that the Martian rocks were high in Si and K, but low in Mg compared to Martian soils and Martian meteorites. The analyzed rocks were similar in composition to terrestrial andesites and close to the mean composition of the Earth’s crust. Addition of a maﬁc component and reaction products of volcanic gases to the local rock material is necessary to explain the soil composition. D.

Applications in Arts and Archaeology

One application of PIXE and, in particular, of micro-PIXE that has been unexpectedly successful is the analysis of archaeological and art objects. Recently, Swann (1997) gave a comprehensive review of the applications of nuclear microprobes in this ﬁeld. Recognition of the potential of PIXE and other IBA techniques for art investigations has led to the installation of an ion-beam analysis facility in the scientiﬁc laboratory of the Louvre Museum in Paris (Menu et al., 1990), with a subsequent inclusion of a nuclear microprobe. Such probes are currently used in several other laboratories specializing in archaeology.

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Because nondestructiveness is more important here than in other applications, there must often be a trade-oﬀ between beam size and intensity in order not to damage an object. Larger probe sizes and lower intensities than in other ﬁelds of application are normally used. External-beam microprobes are particularly useful. Because of the heterogeneous structure of many archaeological artifacts, using a highly collimated=focused ion beam is somewhat problematic as the elemental data obtained depend very much on the position of the beam. Many objects may contain inclusions and grains which are not representative of the bulk. Consequently, the data must be very carefully scrutinized and one should be careful about drawing conclusions from a few single-spot analyses. Although metallic objects are not very sensitive to radiation damage, problems for non-noble metals may arise from the presence of a patina or thick layers of corrosion. The shallow analytical depth of the nuclear microprobe requires the removal of any such layers if one wants to obtain results that are representative for the bulk material. Removal may be prohibitive in some cases, but the situation is generally even worse when using most other analytical methods. Several large-scale studies have been performed on copper-based alloys, mainly antique bronzes. In such studies, especially great care has to be taken to avoid erroneous analytical data due to surface eﬀects caused by corrosion. The Schonland nuclear microprobe in Johannesburg was used to characterize Iron Age tin artifacts following bulk trace element analysis by INAA and ICP-MS (Przybylowicz et al., 1993). The heterogeneity revealed by the nuclear microprobe provided a plausible explanation for the large discrepancies between the bulk analysis techniques. In addition, the microanalytical results yielded information on smelting, crafting, and mineral composition of the local ores (Przybylowicz et al., 1993). Noble metals, such as gold and platinum, are much less aﬀected by deep corrosion over long time spans, so that problems of surface layers are less pronounced. Demortier and colleagues (Demortier et al., 1991; Demortier and Morciaux, 1994) used the external nuclear microprobe at Namur, Belgium in extensive systematic investigations of gold jewelery with the aim of determining the age and manufacturing processes. For ancient artifacts with a gold surface, the nuclear microprobe is an ideal tool to characterize narrow regions, in order to understand the skill of ancient goldsmiths. The presence of Cd in ancient soldering processes is discussed, with many details on the interpretation, in old metallurgical handbooks. In the 19th century, Cd was used as an additive to lower the melting point when soldering gold. Its presence has therefore been regarded as a sign of a modern object. However, the correlation between Cu and Cd is reversed in the newly manufactured material. By he multielemental character of the PIXE method, forgeries of ‘‘ancient’’ gold items can thus be revealed. Also, the Bartol group (Weldon et al., 1996) performed studies of gold jewelery using an external nuclear microprobe, with the aim of dating the manufacturing of some golden pendants. Microanalytical techniques allow one to study pigments in paintings in greater detail, for instance, for authentication of suspect forgeries or for understanding ancient painter technologies. If samples can be collected without interfering with the visible parts of a painting and there is a strong enough interest to make a detailed study, the nuclear microprobe is a good complement to nondestructive techniques and to electron microprobes. By removing small pieces of pigments from a lead–tin yellow region of a painting, Grime et al. (1991) were able to produce elemental maps both by PIXE and RBS. The conclusion drawn from this work was that, due to the deep penetration of the ions and the risk of reaching layers below those relevant, micro-PIXE work on pigments should, whenever possible, be combined with the depth-resolving ion-backscattering technique.

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In the ﬁeld of osteology, the Oxford SPM group has carried out some studies, both looking at the uptake mechanisms of trace elements by bone under laboratory conditions and studying the eﬀects of environmental chemistry on human bone from a variety of sites (Elliot and Grime, 1993). Ca and Fe distributions were determined within the matrix of an unprovenanced human clavicle from the ‘‘Mary Rose’’ shipwreck of 1563. The results (see Fig. 16) clearly show an accumulation of Fe at the bone surfaces that is attributed to Fe-ﬁxing marine micro-organisms. Another ﬁnding of the study was the presence of a surface enrichment of Pb. These results and others from terrestrial burials indicate that the interactions between buried bone and its surroundings are extremely complex and that it is diﬃcult to distinguish the uptake of trace elements during a lifetime from the alterations that occurred after burial. Another nuclear microprobe study on a skeleton was performed at the Pierre Su¨e laboratory in Gif sur Yvette (Boscher-Barre and Trocellier, 1993). A 6thcentury woman’s skeleton buried in a lead sarcophagus was excavated near Lyon. Femoral transverse sections were analyzed using micro-PIXE, NRA, and particle-induced g-ray emission analysis (PIGE). A comparison was made with similarly prepared samples from a newly deceased female. Carbon, O, Ca and Pb proﬁles were determined across the sections. This study was made possible by the high lateral resolution of the nuclear microprobe. The dark black ink used in the Gutenburg bibles has fascinated other printers and scientists through the centuries. Even after 500 years, the printing is very clear and distinct. At the University of California Davis, the milliprobe PIXE system was initially designed with the main purpose of characterizing the ink used in the Gutenburg bible (Kusko et al., 1984), and it was later used for studying the ‘‘Vinland’’ map (Cahill et al., 1987). The Davis group has improved the system over the years, providing enhanced security systems. The results from analyzing large numbers of individual letters show that the main reason for the high printing quality is the high concentration of Pb and Zn in the pigment. In addition to determining the ink composition, the composition of individual paper sheets was determined also. At Lund, an external semimicroprobe was used to examine portions

Figure 16 Elemental maps of an archaeological human bone sample collected in a marine environment. The deposition of iron (Fe map) on the surface in the voids of the bone structure (Ca map) is clearly shown. Dark areas indicate high concentration and light areas indicate low concentration. (From Elliot and Grime, 1993, with permission from Elsevier Science.)

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of handwriting on a 2000-year-old papyrus. By scanning the microbeam, maps of several elements were produced for regions with faded writing. The multielement dataset was then treated by a multivariate statistical technique in order to improve the contrast, and by doing so, a two-dimensional pattern was obtained which revealed the actual letters (Lo¨vestam and Swietlicki, 1990). More recently, several groups have followed with similar studies. In Florence, a system for document analysis using an external nuclear microbeam has been set up (Lucarelli and Mando`, 1996). The emphasis has been on detailed characterization of ancient inks in order to support theories on technology and trade routes of raw material. Systematical methodological examinations were performed which showed that relevant information can be obtained for medieval inks (Cambria et al., 1993). Among the documents investigated were some from the famous father of modern experimental physics, Galileo (Giuntini et al., 1995). An external nuclear microprobe with special arrangements for document analysis has also been set up at Oxford (Grime, personal communication, 1999). In these systems, it is crucial that the precious old documents are not damaged in any way by the ion beam. Furthermore, it is important to aim the nuclear microbeam accurately at the region of interest, which is normally facilitated by laser beams. In addition to micro-PIXE, macro-PIXE has also been extensively used in arts and archaeology and found to provide very useful information. Actually, the distinction between micro- and macro-PIXE is diﬃcult to make in this ﬁeld of application, as much work is carried out with (external) miniprobes or milliprobes, which have beam sizes ranging from a few tens of micrometers to about 1 mm. Also, one often applies both micro- and macro-PIXE analyses on the same samples. The multielement character of PIXE and the speed of analysis are very useful in provenance studies of pottery and earthenware [e.g., Lin et al. (1992), Hamanaka et al. (1994), Pio et al. (1996a, 1996b), Cheng et al. (1996), Gosser et al. (1998), Zucchiatti et al. (1988)]. Cheng et al. (1996) found that the trace elements Rb, Sr, and Zr can be used to trace the place of production of ancient porcelain. By measuring major, minor, and trace elements in white glaze by PIXE, they could distinguish a precious Chinese Qing dynasty porcelain from a fake. Also, obsidians and prehistoric ﬂint tools have been examined by PIXE and other IBA techniques (Murillo et al., 1998; Smit et al., 1998). Murillo et al. (1998) characterized obsidian samples from diﬀerent mineral sites in Mexico. Oxygen was determined by means of the 16 O(d, p)17O reaction, and the elements from K to Br by PIXE using an external-beam facility. The light elements Na, Al, and Si were measured by PIGE, and RBS, using aparticles, was employed to determine O, Si, and Fe. Smit et al. (1998) performed a study on the usewear-induced deposition of polish on prehistoric ﬂint tools. They concluded that the concentration of the elements Ca and P in the polish can be used to determine whether the tool was used for work on bone or wood. Cast iron, ancient iron slag, archaeological bronzes, ancient Greek copper coins, precious metals, and gold artifacts have also been the subject of studies with macro-PIXE and other IBA techniques (Kallithrakas-Kontos et al., 1993, 1996; Narayan et al., 1996; Miono et al., 1996; Climent-Font et al., 1998; Guerra et al., 1998; Salamanca et al., 1998). Narayan et al. (1996) used external-beam PIXE to examine cast-iron sow bars which had been produced in the mid-1600s at Saugus, Massachusetts. Saugus is one of the few sites where colonial Americans manufactured iron at that time, and a local rock gabbro served as replacement for the limestone ﬂuxing agent traditionally used in England. The study showed that gabbro introduced P and Ca in the iron, so that the levels of these elements can be used as a ‘‘ﬁngerprint’’ for Saugus iron. Climent-Font et al. (1998) employed a combination of PIXE, PIGE, RBS, and Auger electron spectroscopy (AES) for examining

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archaeological bronzes. They found that the joint use of these techniques was quite useful, as complementary information was obtained on elemental composition and chemical state. The possible disturbance of the patina as a mask for the determination of the original underlying bronze composition was also studied. Kallithrakas-Kontos et al. (1996) applied external-beam PIXE for analyzing ancient Greek copper coins. In this work, the patina was removed prior to analysis, as described in detail in an earlier publication (KallithrakasKontos et al., 1993). Gold jewelery artifacts of Tartesic origin (700–500 BC) were examined by external-beam PIXE by Salamanca et al. (1998). Collimators along the incident proton beam path allowed the artifacts to be irradiated in narrow regions down to 350 mm in diameter. Special attention was paid to the procedure of soldering in various narrow regions of the bindings of ﬁligrees, twisted wires, narrow strips, and granulations on ﬁnely decorated items. The relative concentrations of Au, Ag, and Cu in such regions were determined. The results seemed to indicate that the solderings were made by local fusion and brazing. No procedure of solid-state diﬀusion bonding like in Etruscan jewelery was identiﬁed. E.

Materials Analysis

Materials analysis is a vast ﬁeld of application for ion-beam analysis (IBA) techniques. A comprehensive treatment of the various IBA techniques (with the exception of PIXE) in the ﬁeld of materials analysis can be found in the Handbook of Modern Ion Beam Materials Analysis (Tesmer et al., 1995). Rutherford backscattering spectrometry (RBS) in particular is very commonly used for examining surface layers and determining elemental depth distributions in various materials. PIXE generally serves only as a complementary technique. Because most matrices in this area are made up of elements with relatively high atomic number, the characteristic x-rays of the matrix elements show up in the PIXE spectra, thereby worsening the detectability of the trace elements. When the matrix is rather light, however, as in the case of polymers, some semiconductor materials, and cellulose ﬁbers, PIXE can provide concentration data down to the microgram per gram level. For microscale characterization of materials, the nuclear microprobe has emerged as an important technique (Breese et al., 1996), in particular when micro-PIXE is combined with complementary IBA techniques. It has therefore also been implemented in the industrial environment (e.g., in development and production plants in the microelectronics industry). Although the micro-PIXE technique provides essentially only elemental maps and=or concentration data, the combination with various imaging techniques makes the nuclear microprobe an invaluable tool in materials analysis. It forms a ﬁne complement to the vast array of surface and near-surface characterization techniques, such as Auger electron spectroscopy, photoelectron spectroscopy, scanning electron microscopy, secondary ion mass spectrometry (SIMS), and many other techniques (Brune et al., 1997). 1. Semiconductor Materials The majority of analyses on this type of material are performed as routine PIXE and RBS measurements in semiconductor plants, with the aim of monitoring wafer processing. However, PIXE and nuclear microprobes are also frequently used in research and development environments for the characterization of silicon-based and other semiconductor materials, such as GaAs. Examples of macro-PIXE studies on the latter material are the work by Kuri et al. (1996) and Wendler et al. (1996). Kuri et al. (1996) used PIXE and XRF to study the eﬀect of MeV Au ion implantation on the composition of GaAs upon vacuum annealing. Practically no As loss was observed from the implanted

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region, whereas such loss was signiﬁcant for the unimplanted region. Optical micrographs taken on both regions showed the formation of Ga droplets on the unimplanted region above 600 C annealing. No such features were observed in the implanted region up to 850 C. This indicated that MeV Au ion implantation inhibits the As release from GaAs. For a given annealing temperature and duration, the amount of As loss was dependent on the dose of implanted Au. Wendler et al. (1996) measured displacements in 2-MeV Seþ implanted GaAs by RBS and PIXE channeling experiments. It is also possible to combine micro-PIXE and micro-RBS with channeling. Under channeling conditions (i.e., when the projectiles are ‘‘guided’’ along crystal axes or crystal planes), the interaction probability is decreased and this leads to a reduced yield of induced x-rays or scattered particles. By comparing the yields obtained for channeling and for random orientation, information on the crystal structure and on interstitial atoms may be obtained (King et al., 1993; Tesmer et al., 1995). To visualize the results from micro-PIXE and micro-RBS channeling experiments, channeling contrast microscopy (CCM) can be used. The Microanalytical Research Centre (MARC) in Melbourne is very active in studies on frontier materials, mainly semiconductors (Jamieson, 1997). It applied micro-PIXE, micro-RBS, and CCM in a study on diamond ion implantation and annealing. A diamond specimen was implanted with 4-MeV P þ ions, whereafter the implanted region was laser annealed and regrown, and examined with the nuclear microprobe. Figure 17 shows the channeling angular yield curves from the center of the laser-annealed and regrown region, together with the CCM images. The reduction in the yield of PKa x-rays in the channeling

Figure 17 Yield of PIXE and RBS versus the tilt angle from the channeling direction in diamond. CCM images are shown for both scattered particles and P x-rays. The CCM images cover an area of 1006100 mm2. (From Jamieson, 1997, with permission from Elsevier Science.)

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orientation indicates that at least 50% of the implanted P atoms are substitutional on the lattice sites. 2. High-Temperature Superconductor Materials Ever since the discovery of the high critical temperature (high-Tc ) superconductors, ion beams have been widely applied to study this new class of materials. For composition analysis, RBS and non-Rutherford backscattering are commonly used. Tang et al. (1993) explored the potential of deuteron-induced x-ray emission (DIXE) in combination with the simultaneous measurement of the prompt g-rays from the 16 Oðd; pgÞ17 O reaction for the complete stoichiometric analysis of Y–Ba–Cu–O superconductors. They found that the technique was very sensitive to particle size eﬀects, but concluded that good results should be obtained for high-quality bulk specimens and thin-ﬁlm samples. Sˇandrik et al. (1993) applied macro-PIXE in combination with RBS for the determination of the composition of high-Tc superconducting ﬁlms of Y–Ba–Cu–O and compared the results with those obtained by ICP–AES and XRF. It was concluded that the PIXE=RBS combination yields concentration data for the matrix elements with 5% accuracy and a precision better than 3%. In a subsequent study, Sˇandrik et al. (1996) examined the stoichiometry and lateral homogeneity of thin ﬁlms deposited from aerosol on MgO substrates. RBS and PIXE analyses distinguished areas that were found to diﬀer in the critical temperature values. The PIXE results conﬁrmed the diﬀerences in composition between those areas. Variations in layer thickness were found by means of RBS. Also, Ishii and Nakamura (1993) used a combination of macro-PIXE and RBS in their analyses of thin ﬁlms of oxide superconductors, but in addition to the matrix elements, some impurity elements such as Fe, Cr, and Ni were determined. Ecker et al. (1996) performed channeling studies of YBaCuO thin ﬁlms with combined RBS and PIXE and they concluded that valuable additional information was obtained on the properties of epitaxially grown ﬁlms. This is of particular interest for the thicker ﬁlms needed in microwave devices. Ahmed et al. (1997) performed nondestructive depth proﬁling on a small single crystal (1.5 mm6 1.3 mm60.1 mm) of an YBaCuO superconductor by scanning its thickness with a proton microbeam and analyzing the micro-PIXE spectra obtained. Elemental composition spectra, two-dimensional elemental distribution maps, and one-dimensional line-scan spectra from several parts of the thickness were produced. 3. Metals In most of the nuclear microprobe studies on metallurgical samples, use was made of NRA in order to determine the low-Z elements (Doyle et al., 1991; Breese et al., 1991). Because metals and metal alloys consist generally of high-Z elements, they are far from ideal specimens for PIXE. Nevertheless, by optimizing experimental conditions, it is possible to study, for instance, corrosion processes and diﬀusion proﬁles in such materials. One example of a micro-PIXE study is that done on proposed superconducting wires for magnets in the International Thermonuclear Experimental Reactor (ITER). These wires contain Sn and Nb in a matrix of ultrapure Cu, and they are plated with Cr. Micro-PIXE was used to study the diﬀusion of Cr into the Cu matrix as a function of the temperature (Morse et al., 1997). Already at rather low temperature, unexpectedly high Cr concentrations were found in the Cu, and this was interpreted as being due to the formation of precipitates. Other examples of metal analysis using micro-PIXE are the study of nodular cast in order to draw conclusions regarding the nodularization process (Songlin et al., 1995) and the investigation of lead in a chill-cast aluminum ingot of a bearing alloy (Breese et al., 1992).

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In the latter case, a combination of PIXE, backscattering spectroscopy, and STIM was used in addition to traditional techniques, such as an electron microprobe. The lead precipitates could be identiﬁed and localized. There are also examples of the application of macro-PIXE to metals and alloys. Feng et al. (1996) used it to determine the antinodularizing elements Al, Pb, and Bi in nodular cast iron. Wa¨tjen et al. (1996) used a combination of PIXE, RBS, and resonant elastic backscattering to determine the depth proﬁles and doses of Y implants in Cr and NiCr alloys, the mass contents of Cr and Ni, and the oxygen concentration in surface layers of the alloys before and after implantation. Only through the simultaneous use of PIXE was a severe error in the alloy preparation revealed. One of the nominal binary alloys was found to contain about 5% Fe, which inﬂuences the corrosion behavior of this material signiﬁcantly. Nakae et al. (1994) used a combination of PIXE channeling and RBS channeling for characterizing single crystals of type 304 stainless steel. He þ particles were used as incident ions for both types of measurements. The study indicated that a solution annealing process is absolutely necessary for producing a good single crystal. It was further found that the P atoms were mostly on substitutional sites of the face-centered cubic (fcc) structure and that the MeV He þ ion irradiation induced segregation of Si and S atoms to the (110) surface. In a subsequent study (Kawatsura et al., 1996), the radiationinduced segregation was further examined. 4. Cellulose Fibers The quality and properties of paper used for printing and various other purposes are highly variable, depending on the particular demand from the industry which uses the produced paper. One ﬁeld of great economical importance worldwide is the newsprint industry, where extensive use is made of recycled ﬁber material and where very tight time constraints exist in the printing phase. There is great demand for an enhanced understanding on the right combination of printing technique and paper quality. At Lund, a project involving the nuclear microprobe, and partially funded directly from industry, was initiated as part of a larger project on physical characterization methods for newsprint

Figure18 Copper and STIM map from investigation of newsprint paper by a nuclear microprobe. The Cu distribution reflects the pigment distribution and the STIM reflects the mass areal density of the cellulose fibers. Dark areas indicate high concentration and light areas indicate low concentration. Scan size: 0.5 mm. (Courtesy of P. Kristiansson.)

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paper (Kristiansson et al., 1997). The basic problem of interest is the transfer of pigment to the paper material in the printing process. The micro-PIXE technique can be used to produce elemental maps of printed dots on various quality papers, and by interpreting the patterns, it is possible to provide suggestions for appropriate modiﬁcations in the production process of the printed paper. The interpretation of the micro-PIXE results is facilitated when combined use is made of STIM to characterize the ﬁbrous structure of the newsprint paper. This is illustrated in Figure 18, where an elemental map of Cu and a corresponding STIM map are shown. The Cu intensity is proportional to the pigment concentration, and the STIM map shows the paper density. VII.

COMPLEMENTARY ION-BEAM-ANALYSIS TECHNIQUES

On several occasions in this chapter, it has already been indicated that the MeV ion beams employed for PIXE are also very useful for other ion-beam analysis (IBA) techniques. The main interaction of MeV ions are close encounters, but also ‘‘shallow’’ encounters leading to recombinations in the outer electron shells can be used by detecting the near-visible light. In this section, the various IBA techniques are brieﬂy presented. In particular, it is shown through examples of applications that the other IBA techniques can provide information that is complementary to that obtainable by PIXE (e.g., by extending the elemental coverage down to hydrogen) and that such analyses may often be done simultaneously with the PIXE analysis. This represents an important advantage of PIXE over other x-ray spectrometric techniques. For comprehensive accounts on the various complementary IBA techniques, we refer to a number of textbooks (Chu et al., 1978; Ziegler, 1975; Thomas and Cachard, 1978; Bird and Williams, 1989; Tesmer et al., 1995; Breese et al., 1996) and the proceedings of the biennial IBA conferences [see Gyulai et al. (1994), Culbertson (1996), and da Silva et al. (1998) for the last three conferences in the series]. A.

Elastic Scattering Spectrometry and Related Techniques

Elastic scattering spectrometry involves the detection of the elastically scattered incident particles or the elastically recoiled target nuclei. The ﬁrst approach is used in such techniques as Rutherford backscattering spectrometry (RBS), Rutherford forward-scattering spectrometry (RFS), particle elastic scattering analysis (PESA), and forward a-scattering (FAST). Strictly speaking, the terms RBS and RFS imply that the scattering process is purely Coulombic (i.e., according to the Rutherford law), but they are often used in a broader sense and also include non-Rutherford scattering. RBS is a very prominent technique in materials analysis, in which one generally applies 2-MeV a-particles and large scattering angles. With such particles, deviations from the Rutherford cross section are small for all target elements. In scattering spectroscopy using protons with an energy of a few megaelectron volts, as is often done when combined with PIXE, quite substantial deviations occur for the lighter target elements (up to about Ti) because of nuclear interference. As the incident particle energy increases, the deviations become greater both for protons and a-particles. The energy of the scattered particles depends on the scattering angle, the energy and mass of the incident ion, and the mass of the target element. When the specimen is not inﬁnitely thin, the scattered particle energy also depends on the interaction depth. Incident particles lose energy as they penetrate into the specimen and the scattered particles, in turn, lose energy on their way out to the detector. Elastic scattering spectrometry therefore allows us to obtain information on both the elemental composition of a sample and the distribution of the elements with depth, which is a very important

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asset for materials research. However, the dependence of scattered particle energy on the interaction depth also has the eﬀect that RBS spectra, in contrast to PIXE spectra, may contain broadened peaks (in the case of semithick samples), consist of a staircase of nearly rectangular shelves (for inﬁnitely thick specimens of homogenous composition), or exhibit a more complicated appearance. Furthermore, even for inﬁnitely thin specimens, the scatter peaks from analyte elements (nuclei) of neighboring mass become less and less resolved with increasing mass. For complex specimens, such as aerosol samples or biomedical or geological samples, RBS can only provide information on the lighter elements (up to about Si), and extraction of the requested concentrations is generally diﬃcult. The reasons for the successful application of RBS for examining medium and heavy elements in materials research is that substantial a priori information about the specimen is generally available and only a very limited number of medium- or high-Z constituents are present in detectable concentrations. To improve the elemental mass discrimination (and at the same time, ease the spectral analysis) in elastic scattering spectrometry of the light elements in atmospheric aerosol samples, Nelson and Courtney (1977) resorted to 16-MeV protons as incident particles and called this variant of the technique Particle Elastic Scattering Analysis (PESA). The disadvantage of using 16-MeV protons for PESA is that the measurements cannot be performed concurrently with the PIXE measurements. A separate ion bombardment and also a higher-energy accelerator are required. Other groups have therefore developed PESA techniques that are more compatible with PIXE and with 1– 4 MeV accelerators. Martinsson (1986) performed careful investigations on the dependence of the scattering cross section on incident proton energy and detection angle, with the measurement of H, C, N, and O in thin aerosol samples in mind, and he proposed employing 3.58-MeV incident protons with detectors set up at 170 (for C, N, and O) and at an angle between 29 and 59 (for H). Whereas elastic scattering spectrometry is a very useful tool in aerosol analysis, its applicability as a complement to macro-PIXE for analyzing biological and geological samples is more limited. As far as biological samples are concerned, scattering analysis is generally only employed to obtain information on the mass thickness of the specimen and=or to determine the beam ﬂuence. These types of applications are especially useful in nuclear microprobe investigations, in which the local mass thickness of, for instance, a microtome slice must be measured to convert the x-ray intensities from the PIXE spectrum into concentration values (Heck and Rokita, 1984; Themner and Malmqvist, 1986). As indicated, the depth-proﬁling capability of RBS is an important asset for materials research. This characteristic is also very useful in investigations related to art and archaeology. For example, RBS can be employed to measure depth proﬁles in samples where surface uniformities may be of high signiﬁcance, as in patina, corrosion and surface segregation processes, glass aging, and polishing procedures (Amsel et al., 1986). For very light elements, in particular hydrogen, it is advantageous to detect the elastically recoiled target nuclei instead of the scattered projectiles. The technique using this approach is termed elastic recoil detection (ERD). Because of the kinematics of the scattering process, the particle detector has to be placed in the forward direction in ERD, and it therefore implies that thick specimens must be bombarded at a glancing angle. Time-of-ﬂight techniques are very suitable for energy determination of the recoiling particle (Rijken et al., 1992). ERD is particularly useful in materials research, but it has also been employed in other applications, such as for depth proﬁling of hydrogen in obsidians (volcanic glasses) (Pretorius et al., 1988), with the aim of dating them.

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In addition to the elastically scattered particles, the transmitted beam may also be employed to extract valuable information on thin specimens. Indeed, by placing a surface barrier downstream from the sample and measuring the energy loss of the incident particles, the local mass thickness may be derived. Because of the small beam currents required, this approach is particularly useful in nuclear microprobe work, where it has given rise to the technique of scanning transmission ion microscopy (STIM). This technique is employed for high-resolution imaging of a specimen, and because an extremely low beam dose is suﬃcient, eﬀects of beam induced damage can also be studied in the virgin and damage state (Saint and Legge, 1997). By judicious optimization, it is possible to obtain images with a spatial resolution of better than 100 nm. B.

Nuclear Reaction Analysis

Nuclear reaction analysis is based on the detection of the prompt g-rays or prompt particles emitted as a result of nuclear reactions between the incident beam and the target nuclei (Tesmer et al., 1995). As indicated in Sec. II.A.4, the cross sections for such reactions vary in a rather irregular way with target nuclide and with incident particle energy. When using light ion beams of only a few megaelectron volts, nuclear reaction cross sections are only important for light- and medium-weight target nuclei. Of the two forms of NRA, the one in which the prompt g-rays are detected is by far the most common. It is usually referred to as particle-induced g-ray emission analysis (PIGE). The prompt g-ray measurement has the advantage over the detection of the promptly emitted charged particles that it allows for greater ﬂexibility in the experimental setup, and for thick samples, the PIGE spectra are much less complicated and far more easily analyzed than the prompt particle spectra. Proton-induced g-ray emission analysis, which employs ( p, g), ( p, p0 g), or ( p, ag) reactions, lends itself easily to concurrent use with PIXE for virtually all specimen types. Moreover, it is able to provide good detection limits for several elements that are not accessible by PIXE, so that it is a truly complementary technique. Ra¨isa¨nen (1987) carried out extensive investigations on the applicability of PIGE for analyzing thick biomedical specimens and reported that, under favorable conditions, the detection limits are down to the submicrogram per gram level for Li, B, F, and Na, about 10–25 mg=g for N, Mg, and P, and 100–300 mg=g for C, O, and Cl. Unfortunately, however, the optimum proton energies for all these elements are not the same. Moreover, the detection limits in PIGE depend strongly on the sample composition and may be much worse when some of the elements for which the sensitivity of the method is highest (e.g., Li, B, F, and Na) are present in elevated concentrations. The most popular application of PIGE on biomedical samples is unquestionably the determination of ﬂuorine. Either the reaction 19 Fð p; p0 gÞ19 F or 19 Fð p; agÞ16 O may be used for this purpose, but the former oﬀers about ﬁve times better detection limits (Ra¨isa¨nen, 1987). Particularly favorite study objects for PIGE ﬂuorine analyses are teeth. For example, Coote et al. (1997) used a nuclear microprobe to measure Ca and F in teeth from sheep by a combination of PIGE and PIXE. The enhanced F concentrations due to exposure in conjunction with a volcanic eruption were investigated, and in Figure 19, results from these measurements are shown. Nuclear microprobes were also used in several studies on human teeth. It was possible to determine the ﬂuorine proﬁle across the entire thickness of the enamel layer (Svalbe et al., 1984a) and across precarious and artiﬁcially induced lesions (Svalbe et al., 1984b; Coote and Vickridge, 1988).

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Figure19 Two-dimensional distributions of fluorine, as determined by the 19F( p, ag)16O reaction, for sections of sheep incisor, with the enamel to the left: (a) nonexposed sheep; (b) sheep with teeth in developing phase during fluorine exposure; (c) sheep with teeth in the mature state during exposure. Dark areas indicate high concentration and light areas indicate low concentration. (From Coote et al., 1997, with permission from Elsevier Science.)

Particle-induced g-ray emission is also useful as a complementary technique to PIXE in the analysis of atmospheric aerosol samples. Several groups have examined the optimum bombarding energy and=or presented procedures or systems for measuring several light elements in such samples by PIGE (Robaye et al., 1985; Asking et al., 1987; Boni et al., 1989). Asking et al. (1987) conclude that two proton energies, 2.64 and 2.96 MeV, are most suited for measuring Na in thin (< 0.25 mg=cm2) aerosol samples, and they report

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a detection limit of 100 ng=cm2. To smooth out the variations in the cross-section curves, Boni et al. (1989) advocate spreading out the beam energy, and they propose an incident proton beam of 3.2–3.5 MeV, with rectangular energy distribution. The detection limits obtained with their setup are 1 ng=cm2 for Li, 3 ng=cm2 for F, 25 ng=cm2 for Na, and of the order of 100 ng=cm2 for B, Mg, Al, Si, and P. Simultaneous PIGE=PIXE analyses of thick geological samples have been done by Carlsson and Akselsson (1981) and by Carlsson (1984). Carlsson (1984) used PIGE for measuring Li, F, Na, Mg, and Al, whereas heavier elements were determined by PIXE. Archaeological specimens are often examined by a combination of PIGE and PIXE. Such combination was, for example, used for provenancing obsidian artifacts (Duerden et al., 1986). The determination of 11 elements by PIXE and 3 by PIGE provided a very good characterization of the samples, as concluded from applying principal-components analysis on the dataset obtained. Tuurnala et al. (1986) applied external ‘‘milliprobe’’ PIGE=PIXE to check the authenticity of oil paintings. A relatively high beam energy of 4 MeV was used to raise the penetration depth, and Na, Mg, and Al were measured by PIGE. Still other examples of applications of PIGE in art and archaeology are given by Demortier (1989). For analyses in this ﬁeld, the depth-proﬁling capacity of PIGE is also quite useful. Although PIGE is almost exclusively performed with proton beams, deuterons are more commonly used as incident particles when the nuclear reaction analysis is based on the detection of the promptly emitted charged particles. Consequently, this form of NRA is less suitable for concurrent use with PIXE. Furthermore, as indicated earlier, prompt particle spectra are often much more complex than PIGE spectra, thus making the technique less attractive for routine use. As far as biomedical materials are concerned, NRA with particle detection can be used for measuring nitrogen (an indicator of protein content). For example, Go¨nczi et al. (1982) applied the 14 Nðd; pÞ15 N nuclear reaction, with 6-MeV deuterons, to measure the nitrogen depth proﬁle in 1000 individual wheat grains. The nitrogen distributions showed striking correlations with parameters describing the nitrogen content of the fertilizer, the time of harvesting, the grain position in a head, and the analyzed variety. The (d, p) reactions on 12C and 14N were used in combination with deuteron-induced x-ray emission (DIXE) measurements to determine the N=C and S=N ratios along a single hair (Varga et al., 1984). Some general aspects of NRA with particle detection using protons as incident particles have been discussed by Ra¨isa¨nen (1987). As an example of the application of (d, p) reactions for analyzing atmospheric aerosol samples, the work by Braga Marcazzan et al. (1987) can be cited. In this work, oxygen, nitrogen, and carbon were measured in particulate matter collected on a silver ﬁlter. In nuclear microprobe investigations on meteorites and cosmic dust particles, a (d, p) reaction was used to measure the carbon content (Vis et al., 1987). The applicability of particle detection NRA in art and archaeology was addressed by Amsel et al. (1986) and Demortier (1989). Similarly to RBS, NRA (including PIGE) is particularly useful in materials analysis. In such applications, the depth-proﬁling capacity of NRA (e.g., by making use of strong resonance peaks in the cross-sectional curves) is often quite valuable. As indicated in Sec. VI.E, however, PIXE, if applied at all, generally serves as the complementary technique, whereas the essential information is obtained by RBS or NRA. For examples of applications of NRA (including PIGE) to materials research, the reader is referred to the textbooks and proceedings mentioned at the beginning of this section.

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To improve the detection limits in NRA, Martinsson and Kristiansson (1993) and Kristiansson and Martinsson (1997) have developed a new technique, photon-tagged NRA (pRNA). By using fast coincidence electronics and applying stringent timing criteria between photons and charged particles, the background in the particle energy spectra is much reduced and better detection limits are obtained for several elements. In subsequent work, the principles of the pRNA technique were implemented in a nuclear microprobe setup (Sjo¨land et al., 1997d; Kristiansson et al., 1998). Figure 20 shows the schematic design of such a system. The technique was further developed for the microanalysis of special light elements (e.g., H, Li, and F) (Sjo¨land et al., 1997a, 1997b, 1997c). C.

Chemical and Structural Information

In biomedical and environmental research, considerably more insight may often be gained by determining the chemical species or chemical association of minor or trace elements than merely their elemental concentrations. For example, to understand the source and transformation processes that are responsible for particulate nitrogen in the atmosphere, it is of interest to ﬁnd out what fraction of total nitrogen is present as ammonium, nitrate, or some other N-containing species, and whether the ammonium is associated with the nitrate or sulfate, for example. Similarly, chemical and=or structural data on crystalline specimens in mineralogy are of great importance to geoscientists. The close-encounter reactions used in PIXE and most other IBA techniques do not provide any chemical or structural information. Such information can, however, be obtained by making use of other interactions or by employing special experimental conditions.

Figure 20 Schematics of experimental arrangements for photon-tagged nuclear reaction analysis in a nuclear microprobe. P: the incoming ion beam; SBD: surface barrier detector; PMT: photomultiplier tube. (From Sjo¨land et al., 1997d, with permission from Elsevier Science.)

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1. Ion-BeamThermography To obtain speciation information with IBA techniques and IBA-compatible aerosol samplers, Martinsson and Hansson (1988) developed the technique of ion-beam thermography (IBT). This technique consists of an ingenious use of controlled heating of the aerosol sample, whereby its temperature is gradually raised, with concurrent analysis by PIXE and PESA. The technique has been further developed and integrated with a data acquisition and specimen control computer system (Mentes et al., 1996). In principle, also simultaneous PIGE measurements are possible. To reduce or eliminate vacuum- or beaminduced losses of compounds or analyte elements from the sample, the measurements are performed at a prevacuum pressure ( p < 102 mbar) or in a 10-mbar He atmosphere. The sample is heated by passing an electrical current through the sample substrate, which consists of a 0.8-mm-thick aluminum foil. The temperature of the sample is monitored by determining the change in relative resistivity (deﬁned as the ratio of the resistance of the heated sample to that of the unheated sample). To achieve time resolution in the data acquisition so that the course of sample deterioration during the thermographic treatment can be followed, a special data acquisition program has been developed. The IBT results are presented as a set of thermograms, one for each element, which show the amount of element remaining in the sample as the temperature increases. From these thermograms, the chemical compounds and their concentrations are inferred. Figure 21 shows the set of thermograms obtained for a mixture of equal amounts of NH4NO3 and (NH4)2SO4. The ﬁrst 10 data points, at relative resistivities between 0.9 and 1, were taken without heating

Figure 21 Thermograms from a mixture of NH4NO3 and (NH4)2SO4. Amounts inferred: m(N) ¼ 4.1 mg=cm 2 , m(O) ¼ 4.3 þ 3.6 mg=cm 2 , m(S) ¼ 1.8 mg=cm 2 , m(NH 4 NO 3 ) ¼ 7.2 mg=cm 2 , m((NH4)2SO4) ¼ 7.4 mg=cm2. (From Martinsson and Hansson, 1988, with permission from Elsevier Science.)

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the sample and are listed in chronological order. The elemental amounts inferred from the thermograms are given in the legend to Figure 21 and are in good agreement with those expected on the basis of stoichiometry. The especially interesting features of IBT are that chemical compounds (not ions) are determined, that it is a multicompound technique, that it requires no pretreatment of the sample, and that some compounds may be measured down to trace levels under favourable conditions. 2. Ionoluminescence When a signiﬁcant amount of energy is transferred to a crystalline material, be it by electrons, photons, or charged particles, near-visible light is emitted due to luminescence in the material. The well-known techniques of photoluminescence (PL) and cathodoluminescence (CL) are widely used in material science and geology in order to study the chemistry, structure, or electrical properties of the material. The exact character of the impinging radiation does not matter very much for the basic interaction mechanisms involved in luminescence. As a consequence, much of the physical understanding from the PL and CL techniques can be directly incorporated in the ionoluminescence technique (IL) that was introduced and developed in a collaboration between the nuclear microprobe groups of Lund and Melbourne (Yang et al., 1993). The intensity and wavelength of the ionoluminescent light provide information concerning the nature of luminescence centers, such as trace substituents and structural defects, present in the matrix. This makes IL a useful complement to other IBA techniques, such as PIXE and, to a some extent, also RBS. Luminescence can be divided into two subgroups (intrinsic and extrinsic) according to its origin. The intrinsic luminescence, which usually contributes in emission through crystal structural defects, is not related to impurities but to crystal-lattice properties. The extrinsic luminescence depends on the impurities in a crystal through the processes activation, sensitization, and quenching. Luminescence phenomena are common in many solids, yet the use of the light produced is restricted because of the complex physical mechanisms behind the emission. For insulators, to which many minerals belong, crystalﬁeld theory can be used to explain luminescence (Henderson and Imbusch, 1989), whereas for semiconductor crystals, band theory is often employed (Yacobi and Holt, 1990). As long as no secondary eﬀects such as lattice damage or crystal modiﬁcation occur, the general luminescence properties of a material are not dependent on the nature of the excitation source. For excitation by energetic particles, the processes leading to luminescence take place in three steps. First, the energy of the MeV particles is dissipated into the excitation volume mainly through electronic stopping and partially through nuclear stopping processes. Ionization takes place in the excitation volume. Recombination of the electrons and the excited ions allows the crystal lattice to absorb the energy released and the optical system becomes highly excited. The second stage involves the de-excitation of the states of high excitation through radiationless transitions. The third stage, luminescence emission, occurs when the atoms de-excite from a low-excitation state to the ground level. The term activation is used when trace impurities cause a material to produce extrinsic luminescence. Transition metal ions, with an electron conﬁguration of 3d (1–9), can interact strongly with the crystal ﬁeld. This results in changes in the energy level structure of the free ion. Usually, the luminescence activated by transition metal ions is characterized by a broad peak width and a peak position strongly dependent on the host matrix. The centroid and width of the peak are sensitive to the chemical surroundings in the crystal and may be used to distinguish the host matrices involved. For example, with

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Mn2þ (3d5) ions located in a crystal with a strong crystal ﬁeld, the luminescence produced tends to be of longer wavelength. The luminescence of Mn2þ-activated calcite (Mn2þ ions in octahedral coordination), for example, is orange, whereas the luminescence of Mn2þ activated willemite (Mn2þ ions in tetrahedral coordination) is greenish. Trivalent rare-earth element ions and most divalent rare-earth element ions, except for those of Ce, Gd and Lu, have the electron conﬁguration 4f (k)[5s25p6]. These special electron conﬁgurations give rise to characteristic narrow luminescence emission bands for rare earth element (REE) ions with the conﬁguration 4f (k1)f 1* through a f ! f transition. The narrow luminescence emission band of the REE ion (trivalent or divalent) originates in the 4f subshell, which is partially shielded by [5s25p6] electrons. Therefore, for REE ions with a 4f ðk1Þ f 1 conﬁguration, the structure of the energy levels in the free REE ions is basically the same in diﬀerent host matrices. Figure 22 shows an IL spectrum from a natural zircon grain, where three peaks from REE are superimposed on a broad intrinsic luminescence band. Experimentally, the IL light is detected by using mirrors or lenses and either directly counting the photons in a PM tube or analyzing them in a spectrometer. The detailed spectrum (see Fig. 22) provides information on the crystal structure and can be used to determine the REE. A scanning grating spectrometer is a very slow system. Variation in beam current, beam-induced lattice damage, and so forth can adversely aﬀect the acquired spectrum. The situation can be improved by implementing a photodiode-array detector with, for instance, 1024 detector elements, each with a size of a few tens of micrometers. Such a system allows the simultaneous recording of all wavelengths and can signiﬁcantly improve the analytical capacity, speed up the analysis, and reduce beam damage eﬀects. Furthermore, because ion-beam bombardment of a solid material modiﬁes its properties and changes its luminescence characteristics, the rapid spectrum acquisition makes it possible to perform a detailed study of these transformations. The most common experimental facility for IL is the nuclear microprobe, where the imaging capability of IL is an invaluable asset, especially in studies on geological samples.

Figure 22 Ionoluminescence spectrum from a natural zircon with peaks, emanating from various rare earth elements, superimposed on a broad yellow intrinsic luminescence band.

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The very high cross sections for IL processes make it possible to employ extremely low beam doses. Synergistic eﬀects of the simultaneous use of PIXE and IL enhance the capability of both techniques (Malmqvist et al., 1996). Because of L-line interferences, PIXE is not very suitable for trace determination of the REE which are of great importance in geoscience. The elemental information obtained by IL (on the REE) and by PIXE (on many other elements) complement each other and the PIXE data can facilitate the interpretation of the IL spectrum, so that both techniques enhance each other. VIII.

CONCLUSIONS

Particle-induced x-ray emission is undoubtedly an invaluable and very powerful x-ray emission spectrometry technique. It has truly multielemental capability, covering a large part of the periodic system (from Na to U), with detection limits that vary smoothly as a function of atomic number. The use of x-ray absorbers can improve this Z dependence further. The sensitivity of PIXE is high and detection limits are low, although they do depend on the particular material being analyzed. The most favorable situation is the determination of trace elements in a light-element matrix (Z < 11), where detection limits are of the order of 0.1–1 mg=g. For samples with appreciable amounts of heavy and medium–heavy elements, the detection limits are somewhat higher; for example, typically a few micrograms per gram for elements between Z ¼ 20 and Z ¼ 50 in geological materials. In absolute terms, the detection limits are even more pronounced, down to 1012 g in macro-PIXE and down to 1017 g in micro-PIXE. Other favorable features of PIXE are the ability to analyze tiny samples (1 mg or less in macro-PIXE, and much less in microPIXE), the speed of the analysis (1–10 min bombardment time per specimen), and the possibility for automation. The degree of specimen preparation needed in PIXE varies from zero at one extreme (e.g., aerosol samples and archaeological materials) through modest (e.g., mineralogical samples and materials science) to signiﬁcant at the other extreme (e.g., some biomedical specimens). The specimens are normally in solid form, and the irradiated target material is, in general, not aﬀected by the particle beam at the current densities needed for conventional macro-PIXE, so that the technique is nondestructive. Even delicate materials such as paper or parchment are unaﬀected by so-called millibeams, although in these cases, special precautions such as low beam intensity and=or external beams are necessary. In micro-PIXE, the larger current densities incident on the specimen can cause damage and alteration, but, in practice, analyses are often conducted at current densities that do not cause such problems. However, as beam diameters become smaller and current densities higher, damage will become more important. Particle-induced x-ray emission is extremely versatile in terms of the size of sample that it can accommodate. Depending on the size, diﬀerent experimental arrangements may be used: a conventional macrobeam in vacuum (with beam size of the order of several millimeters), an external macrobeam or millibeam, or a genuine microbeam. The variety of specimens analyzed to date is enormous, ranging from a large painting to a single blood cell. They have in common, though, that they are normally in solid form. Micro-PIXE as used in the nuclear microprobe makes it possible to perform analyses with excellent spatial resolution (down to 1 mm or better), high sensitivity (detection limits of the order of 1 mg=g), and good accuracy. The importance of this can hardly be overestimated. For example, microscopically small grains in minerals or single cells in biological tissue can be analyzed. If the microbeam is used in the scanning mode, as is often done, elemental maps can be produced, giving a much more detailed picture of trace element distributions than measurements made at single points.

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Another favorable feature of PIXE is that it can be complemented with other IBA techniques, such as RBS and other forms of elastic scattering spectrometry, NRA, and PIGE. As a result, a simultaneous measurement of the light elements (H, Li, B, C, N, O, and F), which cannot be measured by PIXE, is feasible. This is an important asset when analyzing atmospheric aerosol samples, as discussed in detail by Cohen (1993, 1998), Cahill (1995), and Swietlicki et al. (1996a), but also for various other sample types. Some of the other IBA techniques, such as RBS and NRA, predate PIXE, and in certain ﬁelds of application (e.g., materials analysis), RBS is the major IBA technique, and PIXE generally serves only as a complementary technique. Some of the ancillary IBA techniques (e.g., oﬀaxis STIM and ionoluminescence) have been developed in the last decade as adjuncts to micro-PIXE, demonstrating the continued development of the IBA ﬁeld overall. Particle-induced x-ray emission combines a particular and unique set of advantageous features, but as any technique, it has also it limitations. Limitations which are also shared by XRF are that it suﬀers from spectral interferences and matrix eﬀects and that it does not allow the direct measurement of ultratrace elements that are present at nanograms per gram levels. Unlike in some other techniques, however, the matrix eﬀects are well understood and can be corrected for. Consequently, PIXE can provide results with an accuracy better than 5% relative. The most serious drawbacks of the technique are that it requires a MeV particle accelerator and that commercial PIXE apparatus are not readily available (the great majority of PIXE laboratories have built their own setup, and commercial systems are usually custom-built). Most PIXE laboratories have been set up in nuclear physics institutes, and although there have been interesting developments during the past decade, with the installation of dedicated PIXE laboratories and nuclear microprobes, the growth in PIXE has remained slow. There are currently more than 100 PIXE groups worldwide. Part of the reason for the slow growth of PIXE is, of course, that the technique has to compete with a wide plethora of other analytical techniques and that there have been substantial advances in several of these techniques over the past 10–15 years, for example, in ICP–MS. It is therefore worthwhile to compare the characteristics and capabilities of PIXE with those of the other techniques and to assess for what sample types and=or problems PIXE is most appropriate. Maenhaut (1990b) made such an evaluation for macro-PIXE 10 years ago, which was retained in the Conclusions section in the chapter on PIXE in the ﬁrst edition of Handbook of X-Ray Spectrometry (Maenhaut and Malmqvist, 1992). Although many of the conclusions still stand, that evaluation is now somewhat outdated. A more recent evaluation is that of Johansson and Campbell (1995) in the chapter ‘‘Comparison with Other Methods: Future Prospects’’ in the textbook ParticleInduced X-Ray Emission Spectrometry (PIXE). In the present Conclusions section, we have drawn heavily on that excellent and insightful chapter of Johansson and Campbell (1995). As indicated by them, any comparison with other techniques is fraught with risk. The range of techniques available means that it is not an easy matter to select the most appropriate for a particular task. It is often the case that proximity, familiarity, and ease of access play just as large a role as the matching between task and technique. Furthermore, because of the rapid progress in analytical techniques, any comparison becomes dated after a few years. The recent progress in x-ray emission spectrometry techniques can be judged from the various chapters in this handbook. For continued updates on this, the biennial fundamental review on x-ray spectrometry in the June issue (of every even year) of the journal Analytical Chemistry can be recommended [see To¨ro¨k et al. (1998) for the most recent one in the series]. That same June issue of Analytical Chemistry also contains fundamental reviews on many other analytical techniques. Furthermore, some other

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analytical chemistry journals, such as Journal of Analytical Atomic Spectrometry, also publish technique-oriented reviews on a regular basis. A natural starting point in a comparison of PIXE with other analytical techniques is conventional tube-excited XRF. In actual numbers, the relative detection limits of PIXE for thin samples and a light-element matrix vary within the range 0.1–1 mg=g for the elements of greatest interest (the transition metals and the heavy metals around Pb). With careful optimization (a-particle excitation, tight geometry, and heavy shielding), detection limits as low as 0.02 mg=g can even be reached in PIXE (Johansson, 1992). In EDXRF, the corresponding detection limits are of the order of 1–10 mg=g. Even more pronounced is the diﬀerence in detection limits in absolute terms. To obtain optimum relative detection limits in XRF, the sample weight has to be at least 10–100 mg, which means that the absolute detection limits are of the order of some hundred nanograms. In macro-PIXE, the amount probed by the proton beam can be 0.1 mg or even lower, and the corresponding absolute detection limits are of the order of a few picograms. However, as discussed by Johansson and Campbell (1995), besides sensitivity and detection limits, several other factors have to be considered. There are certainly many sample types and=or problems where EDXRF is to be preferred over macro-PIXE as the technique of choice. One substantial diﬀerence between PIXE and XRF is that with micro-PIXE, the spatial distribution of trace elements can be studied down to the submicrometer level, whereas conventional XRF does not oﬀer this possibility. In the mid-1980s, the use of glass capillaries for the concentration of the primary radiation from an x-ray tube was introduced into XRF (see Chapter 11). This and subsequent developments have led to micro-XRF and desktop x-ray microprobes, whereby a spatial resolution down to 10 mm with reasonable incident x-ray intensity may be obtained. These are important technical advances, conferring on tabletop XRF some, although not all, of the abilities of microPIXE. Another signiﬁcant step forward in XRF was the introduction of TXRF (see Chapter 9). However, this technique is much more suitable for the analysis of dilute aqueous solutions than of solid samples. It tends to compete more directly with other techniques (e.g., optical atomic spectrometry) than with PIXE. There appears, for example, to be little potential for high-throughput nondestructive analysis of aerosol particulate samples. A third far-reaching change in the XRF technique has occurred through the introduction of synchrotron radiation (SR) as the primary source, which has led to SRXRF (see Chapter 8). The main limitation of this technique is the same as for PIXE, but to a much greater extent, namely the availability. In the case of PIXE, it is quite feasible to set up an accelerator laboratory to be used solely for PIXE and related IBA work, whereas it is clearly out of the question to use a SR source mainly for elemental analysis. One is therefore limited to using existing facilities, whose number is small. The potential of SRXRF undoubtedly lies in its microbeam capability. However, it seems unlikely, for example, that the continued development and success of SRXRF will aﬀect the rapid growth of routine micro-PIXE analysis of mineral grains or the growing use of micro-PIXE and associated IBA techniques in art and archaeometry. Perhaps the main competition oﬀered to micro-PIXE will be for in situ problems where the energy deposition in the irradiated specimen is an important issue. Thus, microbeam SRXRF can complement micro-PIXE by extending in situ microbeam analysis to specimens that cannot withstand micro-PIXE. Beyond doubt, the greatest competition for the nuclear microprobe and micro-PIXE comes from electron probe microanalysis (EPMA) and other electron microscopic techniques (see Chapter 13). Micro-PIXE has the advantage over EPMA that it oﬀers relative detection limits that are typically two orders of magnitude better, that it is easier to

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quantify, and that it can be complemented with other IBA techniques. EPMA and other electron microscopic techniques, on the other hand, have as advantages that the instrumentation is more compact and is commercially available and that they oﬀer a much higher speed of analysis, better spatial resolution, and better imaging capabilities. It is clear that EPMA will be preferred over the nuclear microprobe in many studies. However, rather than being competitors of each other, EPMA and the nuclear microprobe can very well complement each other, whereby micro-PIXE is used for problems where its excellent detection limits and other speciﬁc advantages are required or at least highly desirable. The majority of bulk element analyses are carried out by optical atomic spectrometry or atomic mass spectrometry, in particular ICP–MS. In most of these techniques, the sample material has to be introduced into the instrument as a liquid, and in ICP–MS, a dilute solution is even preferred. Solid samples must therefore be dissolved, which means an extra preparation stage with the risk of incomplete dissolution, losses, or contamination. The dissolution also implies that the analysis is destructive. Although some techniques allow one to analyze powdered materials [e.g., electrothermal atomization atomic absorption spectrometry (ETAAAS)] or can handle various types of solid samples (e.g., laser ablation mass spectrometry), and continued research is done on the introduction of powders in other techniques, there are overall serious limitations with the analysis of solid samples by optical atomic spectrometry or atomic mass spectrometry. Such analyses suﬀer from severe matrix eﬀects and diﬃculties in obtaining accurate quantitative results for a wide range of elements, and they are invariably destructive. Also, certain techniques, such as ETAAAS, can only measure one element at a time and require the addition of an appropriate matrix modiﬁer. It is clear that the optical atomic spectrometry and atomic mass spectrometry techniques diﬀer very much from PIXE. Also, the optical atomic spectrometry techniques do not oﬀer any equivalent of microbeam capability. Such capability does exist in some atomic mass spectrometry techniques [e.g., in secondary ion mass spectrometry (SIMS)] and laser ablation inductively coupled plasma–mass spectrometry [LA–ICP–MS], but their spatial resolution is clearly worse than in micro-PIXE and accurate quantiﬁcation is diﬃcult. One way to answer the question of which sample types or analytical problems are better handled by PIXE and micro-PIXE than by other techniques is to survey the various applications. Section VI provides such a survey, from which one can identify certain applications where it appears diﬃcult to ﬁnd alternatives. To a large extent, these are applications involving solid and particulate sample material and having minimum or zero sample preparation. Because of its inherent characteristics, macro-PIXE has been very much applied for measuring trace elements in various types of biomedical samples. However, for several of such samples and for ultratrace determinations, the optical atomic spectrometric and ICP– MS techniques are now better suited. Also, in recent years, the emphasis has shifted from ‘‘total’’ element determination toward elemental speciation. PIXE and the other x-ray techniques lend themselves much less to this speciation work than some of the other techniques. In applying PIXE to biomedical problems, one should therefore look for cases in which one can fully utilize the special advantages of PIXE, such as spatial resolution, accurate quantitative analysis, and small samples. Particularly micro-PIXE, as used in the nuclear microprobe and complemented with other IBA techniques such as STIM, is invaluable in biomedical trace element research. One area of application where macro-PIXE has been and continues to be highly successful is in the analysis of airborne particulate material (atmospheric aerosols). For the analysis of very large numbers of small aerosol deposits, as collect by compact samplers

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which provide good time or size resolution, or both, there is virtually no competition to PIXE. Conventional tube-excited EDXRF cannot be used because its absolute sensitivity is too low. TXRF has the absolute sensitivity but is not physically appropriate and SRXRF is much more expensive and not really applicable for routine analysis of large numbers of samples. The optical atomic spectrometric techniques and ICP–MS require dissolution, which is time-consuming and may be incomplete for certain important matrix elements. Conventional ICP–MS also suﬀers from spectral interferences, causing the problem that some important elements cannot be measured. PIXE provides at the same time data for the major elements (S, Na, Cl, Al, Si, Fe), from which the concentrations of important aerosol types (sulfate, sea salt, crustal material) can be estimated, and for several anthropogenic and natural minor and trace elements (P, V, Mn, Ni, Cu, Zn, As, Se, Br, Rb, Pb) that can be used in source type identiﬁcation and apportionment. Furthermore, by complementing PIXE with other IBA techniques, concentrations can be obtained for the light elements (H, C, N, O) that make up most of the aerosol mass, and the hydrogen concentration can then be used to estimate the concentration of the important organic aerosol type (Cahill, 1995). Considering the current interest in the eﬀects of ﬁne particles on human health and in the role of aerosols in climate, it is expected that the analysis of atmospheric aerosol samples will remain a successful application for PIXE for many years to come. However, as indicated in Sec. VI.B, one should realize that PIXE provides only part of the desired information. With regard to the chemical composition, it is highly desirable to also perform measurements for important ionic species (e.g., ammonium, nitrate) and for organic carbon (OC) and elemental carbon (EC). Also, in order to arrive at a complete aerosol characterization, the chemical measurements have to be complemented with physical and optical (radiative) aerosol measurements. PIXE researchers should try to complement their work with these various other measurements or otherwise cooperate with other groups who are involved in such research, and they should try to become integrated in larger atmospheric aerosol and atmospheric chemistry research projects. Fortunately, this is increasingly realized within the PIXE community. Earth science is another ﬁeld in which the use of PIXE or speciﬁcally micro-PIXE has increased dramatically during the past decade. Here, however, there are some powerful competing techniques (e.g., the x-ray spectrometric techniques of micro-XRF and SRXRF). Furthermore, other physically based analytical techniques continue to develop, such as SIMS and LA–ICP–MS. SIMS provides spatial resolution of a few micrometers and relative detection limits that are frequently below 1 mg=g; it is clearly superior to micro-PIXE for the rare earth elements. LA–ICP–MS routinely oﬀers a spatial resolution of 20–40 mm and detection limits of 0.5 mg=g. These two techniques also oﬀer isotopic discrimination, which is not an option with PIXE. However, in each of these two techniques, the matrix eﬀects are much more complex and quantiﬁcation more diﬃcult than in PIXE. Micro-PIXE still maintains its advantage as a truly multielemental, in situ, nondestructive technique with detection limits of a few micrograms per gram, 1 mm spatial resolution, and straighforward matrix corrections based on simple, well-understood physics. It can be used for surveys involving a large number of samples for ore prospecting, and for the detailed study of minerals. In the latter case, complementary use of EPMA and micro-PIXE has turned out to be very fruitful. EPMA is a standard technique in mineralogy for the determination of major and minor elements, and micro-PIXE allows these studies to be extended in seamless fashion to trace elements. In studies of extraterrestrial materials, such as micrometeoritics and interstellar particles, the high sensitivity and nondestructiveness of micro-PIXE are a prerequisite. Similarly, the study of ore body emanations gives samples of extremely small mass and any other technique than PIXE

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seems to be excluded. Micro-PIXE is ideal for investigations of mineral and melt inclusions in many materials (e.g., diamonds). Meanwhile, in the quite diﬀerent context of multielement bulk rock analysis with sub-microgram per gram detection limits, INAA, ICP–AES, and ICP–MS remain the techniques of choice. Because PIXE and related IBA techniques allow the analysis of delicate samples without giving any visible damage and without introducing any radioactivity, they can also very advantageously be used in studies in art and archaeology or for analyzing other unique samples of which subsampling is impossible. XRF is often the natural choice for the nondestructive analysis of for example, archaeological artifacts, but PIXE has other advantages due to its somewhat better sensitivity and imaging capability. A situation in which the greater sensitivity is desired is the determination of the trace element proﬁle in various items, which can help in the characterization and identiﬁcation of material for addressing questions such as provenance, manufacturing procedure, and trade routes. Perhaps, the greatest asset of PIXE in these ﬁelds is its versatility. Bulk analysis can be carried out with macro-PIXE, and microscopic details can be studied with a microbeam. A very useful arrangement is an external beam with a cross section of 0.1–1 mm, called a millibeam. This can be used for studies of details in paintings or the ink of single letters in books and other documents. However, the developments in micro-XRF and microbeam SRXRF will give rise to strong competition for PIXE in this type of application. Finally, PIXE will certainly ﬁnd further application in the ﬁeld of materials research. However, as indicated earlier, RBS is the major IBA technique in this ﬁeld, and PIXE generally serves only as a complementary technique. The complementarity of PIXE resides in the fact that it is able to detect elements that are present at levels that are too low for RBS or for which RBS does not oﬀer suﬃcient Z discrimination.

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13 Electron-Induced X-ray Emission John A. Small, Dale E. Newbury, and John T. Armstrong National Institute of Standards and Technology, Gaithersburg, Maryland

I.

INTRODUCTION

The current trend in many technology-related ﬁelds such as electronics and materials science is toward miniaturization and increased spatial resolution well below the micrometer scale. The corresponding requirements for microanalytical instruments needed to meet the demands of this trend require analytical instruments that are capable of analyzing micrometer and submicrometer regions of samples. A schematic diagram outlining the generic features of a microanalytical instrument is shown in Figure 1. In such an instrument, the primary radiation (ions, photons, or electrons) is focused to form a beam from about 1 mm down to about 0.5 nm in size. The interaction of the primary beam with the specimen results in the emission of secondary radiation that is then analyzed with a spectrometer system to provide information on the structure and composition of the sample. For the analysis of micrometer and submicrometer domains, the electron-probe microanalyzer (EPMA), scanning electron microscope (SEM), and analytical electron microscope (AEM) use electron beams as their primary radiation source and incorporate electron-induced x-ray emission as one of the primary spectroscopies for obtaining analytical information. In electron beam instruments, the electron beam is generated from a tungsten ﬁlament, a lanthanum hexaboride (LaB6) electron source, or a ﬁeld-emission electron source. The emitter serves as the cathode in the electron gun and is maintained at a negative potential with respect to ground as shown in Figure 2 for a tungsten-ﬁlament source. In systems employing a tungsten ﬁlament, the source of choice for quantitative electron probe analysis due to its stability, the electron beam is produced by the thermal emission of electrons from a ‘‘hairpin’’ ﬁlament consisting of a tungsten wire bent in a V shape. The tip of the V is approximately 200 mm in diameter, as shown in Figure 3. The tungsten wire is directly heated to a temperature of 2700–2900 K, resulting in an electron emission current density of about 10 A=cm2.

Note: Certain commercial equipment, instruments, or materials are identiﬁed in this report to specify adequately the experimental procedure. Such identiﬁcation does not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the materials or equipment identiﬁed are necessarily the best available for the purpose. 811

812

Figure 1

Small et al.

A generic microanalytical instrument.

Figure 2 Schematic of a W-filament electron gun. (From Goldstein et al., 1975a. Reproduced with permission of Plenum Press.)

Electron-Induced X-ray Emission

Figure 3

813

Photomicrograph of a W filament.

In systems that employ the LaB6 electron source, the cathode consists of a rod of solid LaB6 that is milled on one end to a tip with a diameter of approximately 10 mm. The LaB6 source is heated to a temperature of about 1950 K. At this temperature, the emission current density for the LaB6 source is on the order of 100 A=cm2, a factor of 10 greater than the current density for the tungsten ﬁlament. The higher current density of the LaB6 enables the use of smaller primary beams for a given current than the tungsten ﬁlament. In the past, the LaB6 emission source has found only limited use in EPMA, primarily because of instability in the emission current. Recent advances in electronics and gun design have increased the stability of the electron emission from the LaB6 source to a level that allows the analyst to perform relatively high-quality x-ray analysis. The LaB6 source, however, is rapidly being replaced by ﬁeld-emission sources. In systems that employ a ﬁeld-emission electron source, the emitter consists of a single-crystal tungsten rod, the end of which has been formed into a sharp tip with a radius less than 100 nm. In a cold ﬁeld-emission source, a strong electric ﬁeld is applied through a series of extraction lenses to the tip, causing the emission of electrons without heating. In a thermally assisted ﬁeld-emission source, the tip is heated to 1800 K, increasing the beam stability and eliminating the need to ‘‘ﬂash’’ or clean the tip prior to each use. The main advantage of the ﬁeld-emission source is its inherent brightness on the order of 105 A=cm2, a factor of 104 greater than the conventional tungsten wire that allows the formation of a very small diameter probe on the order of 0.5 nm. This makes the ﬁeld-emission source particularly useful for x-ray analysis with low acceleration voltages and soft x-rays. After emission, the electrons in an electron probe with a W or LaB6 ﬁlament are focused through an initial crossover by the presence of the Wehnelt cap, which surrounds the ﬁlament as shown in Figure 2. The cap is biased several hundred volts negative compared to the ﬁlament, which creates an immersion ﬁeld that focuses the electrons to a crossover with a diameter d0 of approximately 50 mm for a conventional tungsten ﬁlament. In a ﬁelid-emission instrument, the crossover is done with the extraction lenses. After passing through the initial crossover, the electrons are accelerated by an anode plate

814

Small et al.

biased from 1 to 400 kV positive with respect to the cathode. Next, the initial electron crossover or spot is demagniﬁed by a series of apertures and electron optical lenses, including both condenser and objective lenses. The ﬁnal probe diameter used to interrogate the sample is dependent on ﬁlament material, accelerating voltage, and emission current. Figure 4 shows a plot of probe diameter versus probe current for diﬀerent ﬁlaments at 30 kV acceleration potential. As the acceleration potential decreases, the probe diameter increases as shown in Figure 5 (which shows a plot of probe diameter versus probe current for 1kV acceleration potential). A schematic of an electron-probe microanalyzer is shown in Figure 6. An excellent discussion of electron sources, optics, and electron probe formation is given by Goldstein et al. (1992b). Castaing developed the ﬁrst successful EPMA and outlined the fundamental physical concepts of quantitative analysis (Castaing, 1951). The electron microprobe that he developed made use of a focused beam of electrons to excite x-rays from a microscopic domain on a sample surface. In classical electron-probe microanalysis, the acceleration potential is on the order of 10–30 kV and the samples are polished ﬂat, homogeneous, and opaque with respect to the electron beam. The characteristic x-rays emitted as a result of the primary electron beam interaction with the atoms in the specimen are analyzed by either wavelength-dispersive spectrometry (WDS) or energy-dispersive spectrometry (EDS) to determine elemental compositions. The volume of the specimen that is excited depends on the specimen composition and the energy of the primary electron beam. Because the absorption path lengths for x-rays are considerably greater than those for secondary electrons used for electron imaging, the spatial resolution for classical x-ray microanalysis is 1–2 mm, compared to 3–5 nm for electron imaging. This is shown schematically in Figure 7, in which the x-ray emission volume is compared to the electron beam diameter of 1–5 nm. Specimens that are inhomogeneous at dimensions below the x-ray resolution cannot be readily analyzed by conventional microprobe analysis. These samples are best analyzed with low

Figure 4 Probe diameter versus probe current for different electron gun designs, plotted for a 30-keV accelerating potential. (From Goldstein et al., 1992a. Reproduced with permission of Plenum Press.)

Electron-Induced X-ray Emission

815

Figure 5 Probe diameter versus probe current for different electron gun designs, plotted for a 1-keV accelerating potential. (From Goldstein et al., 1992a. Reproduced with permission of Plenum Press.)

Figure 6 Schematic of the modern electron-probe microanalyzer: G ¼ electron gun, An ¼ anode plate, CL ¼ condenser lens, Ap ¼ column aperture, OL ¼ objective lens, S ¼ specimen, Cr ¼ diffraction crystal, and D ¼ x-ray detector. (From Heinrich, 1981a. Reproduced with permission of Van Nostrand Reinhold.)

816

Small et al.

Figure 7 Relative dimensions of the primary electron beam, the electron excitation and emission volumes, and the x-ray emission volumes in a Cu target. (From Goldstein et al., 1981a. Reproduced with permission of Plenum Press.)

acceleration voltages and soft x-rays, or in the analytical electron microscope at high voltages, 100 kV or more, as thin specimens.

II.

QUANTITATIVE ANALYSIS

In this seminal paper on quantitative probe microanalysis, Castaing (1951) proposed that quantitative elemental analysis could be carried out in the electron probe by comparing the x-ray intensity generated from a given element i in an unknown to the x-ray intensity of the same element i generated in a standard containing a known amount of the element. The ratio of the intensity of i in the sample to i in a pure element standard, Eq. (1), was referred to as the k ratio by Castaing and forms the basis for quantitative analysis: Iisam ¼ ki ð1Þ Iistd In this equation, the intensities must be corrected for background, peak interference, and dead time diﬀerences by the methods described in Chapters 2 and 3. Ideally, X Ci ¼ Ci since Cj ¼ 1 ð2Þ ki ¼ P all j Cj all j where C refers to the elemental concentrations, expressed as weight fractions, and the subscript j refers to all of the elements in the sample. Equation (2) only applies to a system in which the sample and the standard are identical and have been measured under identical experimental conditions. In practice, as the similarities between the sample and the standard decrease, Eq. (2), even as an approximation, fails and a series of corrections must be applied to the k ratio to obtain an

Electron-Induced X-ray Emission

817

accurate quantitative analysis. The corrections that must be applied to the k ratio include the following: 1. The atomic number correction for the diﬀerence between the electron scattering and penetration in the sample and the standard 2. The absorption correction for the diﬀerence in the absorption of the x-rays as they pass through the sample or standard before reaching the detector 3. The ﬂuorescence correction for the ﬂuorescence of x-rays induced by the absorption of characteristic and continuum x-rays that are generated in the sample by the primary electron beam and which propagate out from this source These corrections are applied to the various k ratios as part of theoretical or empirical correction procedures in order to obtain quantitative results. A.

Analytical Corrections for Quantitative Electron-Probe Microanalysis

1. ZAF Corrections In the ﬁrst correction procedures that were used to obtain quantitative elemental analysis from electron-probe data, the various corrections were applied to the k ratios as separate multiplicative terms: Ci ¼ kZ kA kF ki

ð3Þ

where the terms kZ, kA, and kF refer to the atomic number, absorption, and ﬂuorescence corrections, respectively. The methods based on Eq. (3) were appropriately referred to as ‘‘ZAF’’ methods. In the remainder of this section, each of the corrections will be discussed separately. a.

Atomic Number Correction, kZ

The atomic-number correction in electron microprobe analysis is applied to the k ratio to compensate for the diﬀerence in the electron retardation and electron backscattering between the sample and standard. Both the electron retardation and backscattering are Eq. (4). Therefore, any difdependent on the average atomic number of the sample Z, ference between the average atomic number of the sample and the standard should be addressed by this correction. As a general rule, ignoring the eﬀects of the atomic-number correction will result in an underestimation of the concentration of high-Z elements in low-Z matrices and the overestimation of concentrations of low-Z elements in high-Z matrices (Goldstein et al., 1981b). The average atomic number Z for the sample is given by X Ci Zi ð4Þ Z ¼ i

The general formulation of kZ for element i is given in Eq. (5): R E0 Q i Ec S sam dE Rsam ðkZ Þi ¼ istd R i E Qi 0 Ri std dE Ec

ð5Þ

Si

where the R and the S terms refer to the electron backscattering and the electron stopping power, respectively, and Q is the ionization cross section. The limits on the integral are from the incident electron energy, E0, to the critical excitation energy, Ec for the x-ray line of interest.

818

Small et al.

Electron stopping power. The electron stopping power S is deﬁned in Eq. (6) as the energy lost per unit electron path length in material of density r (Widdington, 1912): 1 dE ð6Þ S¼ r dX One of the most commonly used terms for S is the approximation by Bethe (1930) and Bethe and Ashkin (1953), which assumes a continuous function for the electron energy loss: Zi 1 1:166E S ¼ 78;500 ln ð7Þ Ji Ai E where S is in keV cm2=g and E is in keV. The value for the mean ionization potential, J, in Eq. (7) is not directly measured and several diﬀerent expressions have been used in the literature for the calculation of J. Various literature values for J are listed in Table 1 and are plotted as a function of Z in Figure 8. The various expressions for J all yield similar results for elements above Z ¼ 10 with relatively large discrepancies among expressions for Z < 10. The Berger–Seltzer expression for J is one of the most widely used in quantitative analysis procedures (Berger and Seltzer 1964). There are also several models in the literature that have been used to calculate Q, all of which have the general form described by Bethe (1930): Q¼C

ln U UEc2

ð8Þ

where U is the overvoltage deﬁned as E0 =Ec . Heinrich and Yakowitz (1970), however, have shown that the diﬀerence in the various models resulted, in negligible changes in the ﬁnal elemental concentrations for elements with Z > 10. Duncumb and Reed (1968) simpliﬁed the integration in Eq. (5), eliminating the need for the numerical integration Eq. (5) and the evaluation of Q. In their procedure, they assumed that the values for (1=S) dE were constant for the sample and standard over the electron range used in electron-probe analysis and, therefore, could be removed from the integral. As a result of this assumption, the integration of Q is unnecessary because it appears in both the numerator and denominator and is the same for the unknown and the standard. The stopping power factor can then be expressed as

Table 1

Different Values for J

Equation (in eV) J=Z ¼ 13 J=Z ¼ 11:5 J=Z ¼ 9:76 þ 58:82Z1:19 J=Z ¼ 9:0ð1 þ Z0:67 Þ þ 0:03Z J=Z ¼ 12:4 þ 0:02Z J=Z ¼ 14:0ð½1 e0:1Z þ 75:5=Z0:13Z Z=ð100 þ ZÞÞ J=Z ¼ 10:04 þ 8:25eZ=11:22 Source: From Heinrich, 1981b.

Reference Bloch (1993) Wilson (1941) Berger and Seltzer (1964) Springer (1967) Heinrich and Yakowitz (1970) Duncumb et al. (1969) Zeller (1975)

Electron-Induced X-ray Emission

819

Figure 8 Different formulations for the mean ionization potential, J, plotted as a function of the atomic number of the target. (From Goldstein, 1981c. Reproduced with permission of Plenum Press.)

S¼

Z 1:166E ln A J

ð9Þ

where A is atomic weight, and J, from Duncumb et al. (1969), equals J 75:5 ¼ 14½1 expð0:1ZÞ þ Z=7:5 Zð100 þ ZÞ Z Z

ð10Þ

Equation (9) is one of the most commonly used formulations for classical ZAF microanalysis. Duncumb and Reed (1968) have shown from experimental work that the stopping power for a multielement specimen can be expressed as a weighted sum of the stopping power factor for each element: Si ¼

j X 0

Cj Sij

ð11Þ

820

Small et al.

where Si is the stopping power for element i; Sij is the stopping power for element i in the presence of element j, and Cj is the weight fraction of element j. In addition to Eq. (9), alternate formulations for S have been proposed by Philibert and Tixier (1968), Pouchou and Pichoir (1991a), and Love et al. (1978). Philibert and Tixier derived an exact analytical solution to Eq. (6) that deﬁnes S in terms of a logarithmic integral: 1 1 ln w ¼ U0 1 hLiðwU0 Þ LiðwÞi ð12Þ S m w where 1:1666Ec J X Zi Ci m¼ Ai allðiÞ

w¼

ln J ¼

X ½Ci ðZi =Ai Þ ln Ji m allðiÞ

U0 ¼

E0 Ec

The logarithmic integral Li of a variable Y expressed as an inﬁnite series is LiðY Þ ¼ ln j ln Y j þ

1 X ðln Y ÞF þD FðF !Þ F¼1

ð13Þ

where D is Euler’s constant, which equals 0.577. Love et al. (1976) noted that the Bethe expression (Bethe, 1930) for S is valid only if E J and, as a result, modiﬁed the Bethe expression ‘‘ . . . to give better limiting behavior as E approaches J.’’ Their work results in the following formulation for S: 2 11 !1:07 30 1=2 1=2 X U0 1 1 4 Ci Zi A J 5@ ¼ 1 þ 16:05 ð14Þ S Ec Ai U0 1 allðiÞ Similarly, Pouchou and Pichoir (1991), to avoid problems with the limiting behavior as E approaches J in the Bethe expression, deﬁned S as 2 3 X 1 4 Ci Zi 5 1 ð15Þ S¼ f ðV Þ Ai J allðiÞ J is same as Philibert and Tixier (1968), and Ji is from Zeller (Table 1). where V ¼ E=J, Pk¼3 2 f ðV Þ ¼ k¼1 Dk Vpk with D1 ¼ 6:6 106 , D2 ¼ 1:12 105 ½ð1:35--0:45ÞJ , D3 ¼ 2:2 p1 ¼ 0:78, p2 ¼ 0:1, and p3 ¼ ½ð0:50:25ÞJ . 106 =J, Electron backscatter factor. The electron backscatter factor R in Eq. (5) is deﬁned as Ib R¼1 ð16Þ It where Ib is the x-ray intensity lost due to backscattered electrons and It is the x-ray intensity if there are no electrons backscattered.

Electron-Induced X-ray Emission

821

The fraction of electrons that are backscattered from a sample is known as the electron backscatter coeﬃcient Z, and is given by Z1 dZ Z¼ dw ð17Þ dw 0

where w is the ratio of the energy of the backscattered electron, Eb , to the beam energy E0 ði:e:; Eb =E0 Þ. An empirical expression for Z, Eq. (18), was obtained by Reuter (1972) from a ﬁt to Heinrich’s data (Heinrich 1966): Z ¼ 0:0254 þ 0:016Z 1:86 104 Z2 þ 8:3 107 Z3

ð18Þ

The number of ionizations generated in a sample by an electron with energy E is given, for a characteristic X-ray line with critical excitation energy Ec , by ZEc Q dE ð19Þ dE=drs E

Ib can then be obtained by multiplying Eqs. (17) and (19): Z1 Ib ¼

dZ dw

w0

ZEc

Q dE dw dE drs

ð20Þ

E

where the integration limit for Z is w0 ¼ Ec =E0 because electrons with energies less than Ec cannot excite the x-rays of interest. Similarly, It can be calculated from Eq. (21) with integration limits of E0 to Ec : ZEc It ¼

Q dE dE=drs

ð21Þ

E0

Finally, substituting Eqs. (20) and (21) into Eq. (16) results in the following formulation for R: 0 11 ZEc Z1 ZEc dZ Q Q B C dE dw@ dEA R¼1 ð22Þ dw dE=drs dE=drs w0

E

E0

Several tabulations of R have been made for pure elements as a function of Z and U (Duncumb and Reed, 1968); Green, 1963; Springer, 1966. Duncumb and Reed produced a table of R values for various elements and several diﬀerent overvoltage values. Their values were determined indirectly from Bishop’s (1966) measurements of the energy distributions of backscattered electrons and were in agreement with the direct measurements of R made by Derian and Castaing (1966). Figure 9 shows a plot of the Duncumb–Reed R values versus Z at diﬀerent overvoltages. Duncumb derived an algebraic expression for R in terms of wq and Z from his calculated values; Eq. (23) is still used in many analytical procedures (Heinrich, 1981c): R ¼ 1 þ ð0:581 þ 2:162wq 5:137w2q þ 9:213w3q 8:619w4q þ 2:962w5q Þ 102 Z þ ð1:609 8:298wq þ 28:791w2q 47:744w3q þ 46:540w4q 17:676w5q Þ 104 Z2 þ ð5:400 þ 19:184wq 75:733w2q þ 120:050w3q þ 110:700w4q þ 41:792w5q Þ 106 Z3

822

Small et al.

Figure 9 Duncumb backscatter correction factor R versus Z and overvoltage (From Heinrich, 1981c. Reproduced with permission of Van Nostrand Reinhold.)

þ ð5:725 21:645wq þ 88:128w2q 136:060w3q þ 117:750w4q 42:445w5q Þ 108 Z4 þ ð2:095 þ 8:947wq 36:510w2q þ 55:694w3q 46:079w4q þ 15:851w5q Þ 1010 Z5 ð23Þ where wq ¼ Ec =E0 . Yakowitz et al. (1973) obtained a simpliﬁed expression for R from a ﬁt of the Duncumb and Reed values: Rij ¼ R01 R02 lnðR03 Zj þ 25Þ

ð24Þ

where R01 ¼ 8:73 103 U 3 0:1669U 2 þ 0:9662U þ 0:4523 R02 ¼ 2:703 103 U 3 5:182 102 U 2 þ 0:302U 0:1836 R03 ¼ ð0:887U 3 3:44U 2 þ 9:33U 6:43Þ=U 3 In this Eq. (24) i represents the element being measured and j represents the elements in the specimen, including i. Rij is therefore the backscatter correction for element i in the presence of element j.

Electron-Induced X-ray Emission

823

Myklebust (1984) further simpliﬁed the expression for R: R ¼ 1 0:0081512Z þ 3:613 105 Z 2 þ 0:009582Z expðU Þ þ 0:00114E0

ð25Þ

This expression represents a ﬁt of the R values obtained from the NBS Monte Carlo program (Myklebust et al., 1975). Figure 10 shows the behavior of various R values from diﬀerent authors as a function of Z for selected x-ray lines. Myklebust and Newbury (1991) suggest that the diﬀerent formulations of R produce only small diﬀerences in the quantitative results. However, they recommend the formulation of R in Eq. (26), which uses the cross section from Fabre delta Ripelle (1949) with Eq. (22), as the best formulation, including the extreme conditions of very high or very low overvoltages: 0 10 11 ZwE0 Z1 ZE0 dZ A A C B CB dE dwA @ dEA R¼1@ ð26Þ dw B B w0

Ec

Ec

where A ¼ lnðE=Ec Þ and B ¼ ð1=Ec þ ð1:32=EÞÞ lnðð1:166=J ÞE Þ. In a multielement system, the factor R for element i can be calculated from Eq. (27), which was proposed by Duncumb and Reed (1968): X Cj Rij ð27Þ Ri ¼ j

Myklebust and Newbury (1988) compared results from Monte Carlo calculations to results from Eq. (25) for a 10% Cu–Au alloy to determine the accuracy of mass concentration averaging in multielement targets. Their results indicate that Eq. (27) is valid for the alloy studied. A detailed discussion of the R factor can be found in Myklebust and Newbury (1991). Evaluation of kZ . The results of analysis for copper in 2 wt% Cu–Al alloy have been used by Goldstein et al. (1981d) to demonstrate the magnitude of the atomic-number correction. The authors used both pure elements and a 46% Al–Cu alloy as standards for the analysis. The results, given in Table 2, indicate that the atomic-number correction for this analysis at 30-keV incident electrons is as high as 16% for the pure-element standards and is reduced to 8% when the alloy is used as the standard. b.

Absorption Correction, kA

The primary electron beam generates x-rays at varying depths within the sample. As a result, the x-rays must pass through that portion of the specimen that lies between the x-ray generation point and the detector before they escape the sample and are measured. As shown Figure 11, the distance A–B is referred to as the absorption path length because a percentage of the generated x-rays undergo photoelectric absorption, interacting with specimen atoms prior to escape. The eﬀect of this absorption is an attenuation of the generated x-ray intensity that is measured by the detector. Note that inelastic scattering of the x-rays is not a signiﬁcant process over the path lengths involved. The energy of the unabsorbed characteristic x-ray is not modiﬁed during its passage through the specimen. The magnitude of the attenuation is dependent on the composition of the specimen and a correction must be considered when the sample and the standards used for the analysis are dissmilar. Castaing (1951) described the characteristic x-ray intensity (without absorption) generated in a layer of thickness dz, at a depth z below the specimen surface, in a sample of density r as dI ¼ fðrzÞ dðrzÞ

ð28Þ

Figure10 Comparison of different functions for R versus Z for K (a), L (b), and M (c) x-ray lines. (From Myklebust and Newbury, 1991.)

Electron-Induced X-ray Emission Table 2 Alloy

825

kZ Correction for Copper in a 2 wt% Copper in Aluminum

Standard Elemental Cu Alloy

15 keV

30 keV

kZ (Cu) 1.16 1.08

kZ (Cu) 1.11 1.05

Source: From Goldstein et al., 1981d.

where fðrzÞ is the distribution of characteristic x-rays as a function of depth [density 6 distance (mg=cm2)] in the sample. A typical curve for CuKa radiation is shown in Figure 12. The total generated x-ray intensity for a given line can be obtained by integrating the area under the curve over the entire x-ray range: Z1 fðrzÞ dðrzÞ ð29Þ I¼ 0

The introduction of x-ray absorption into Eq. (29) results in the following expression for the x-ray intensity after absorption, I 0 : 0

Z1

I ¼

ð30Þ

fðrzÞ exp½ðm=rðrzÞ cscðCÞÞ dðrzÞ 0

where ðrzÞ cscðCÞ is the absorption path length for the x-rays in the specimen, C is the detector emergence angle, and m=r is the mass absorption coeﬃcient of the specimen for the characteristic line of the element of interest. The absorption term for EPMA is referred to as fðwÞ or fp , where w ¼ ðm=rÞ cscðCÞ and fðwÞ ¼ I 0 =I. From Eqs. (29) and (30), fðwÞ can be expressed in terms relating to the specimen as 8 1 9 Z1
Figure 11

X-ray absorption path length.

0

826

Small et al.

Figure 12 A f(rz) curves for CuKa radiation. The solid line are measurements by Castaing and Descamps and the dashed line is based on Philibert’s f (w) with the Duncumb–Shields s. (From Criss, 1968.)

The absorption correction for EPMA can then be expressed as kA ¼

fðwÞstd fðwÞsam

ð32Þ

The basic formulation of the fðwÞ term was derived by Philibert (1963), who considered the number of ionizations produced in a layer of element i with thickness dz at some depth z within a specimen: fðwÞ ¼

fð0Þhw 1 þ ð4þfð0ÞhÞs

ð1 þ w=sÞf1 þ ½h=ð1 þ hÞðw=sÞg

ð33Þ

In this equation, fð0Þ is the surface ionization function, s is Lenard’s (1895) coeﬃcient for a given incident electron energy, and h is given by h ¼ 1:67 106

A sE 2 Z2 0

ð34Þ

Electron-Induced X-ray Emission

827

Noting that fðwÞ was not sensitive to either fð0Þ or s, Philibert (1963) simpliﬁed Eq. (34) by setting fð0Þ ¼ 0 and h ¼ 1:2A=Z2 . This results in the following expression for fðwÞ: 1 w h w ¼ 1þ 1þ ð35Þ fðwÞ s 1 þ hs E1:65 where s ¼ 4:5 105 =ðE1:65 c Þ. 0 Duncumb and Shields (1996) modiﬁed s to take into account that electrons with energies less than Ec cannot generate x-rays from the line associated with Ec . They proposed the following expression for s: s¼

2:39 105 1:5 E1:5 0 Ec

ð36Þ

Heinrich (1970) ﬁt experimental values of fðwÞ from Green (1962) and proposed the following formulation for s: s¼

4:5 105 E1:65 E1:65 c 0

ð37Þ

The most common form of the absorption term found in the various analytical schemes is Eq. (35) with h ¼ 1:2A=Z2 and Heinrich’s s. This form of Eq. (35) is often referred to as the Philibert–Duncumb–Heinrich equation. Heinrich et al. (1972) empirically derived a simpliﬁed absorption term that was based on experimental data and the Philibert equation. In the formulation of this term, it was noted that the compositional dependence in Philibert’s h term was small compared to the scatter in the available experimental data. The resulting formulation for fðwÞ, given in Eq. (38), is referred to by Heinrich as the quadratic model. In this equation, the compositional dependence in the h term has been eliminated, making the model independent of the atomic weight and atomic number of the target: 2 1 ¼ 1 þ 1:2 106 gw fðwÞ

ð38Þ

where g is the quantity E1:65 E1:65 c . 0 In Eq. (35), h is dependent on target composition and must be averaged for the various elements in multielement targets. A compositionally weighted average for h, Eq. (39), is generally employed in analytical procedures: X ht ¼ Cj hj ð39Þ j

In addition, the mass absorption coeﬃcient, m=r, for the characteristic line of element i in a multielement target is the weighted sum over all elements in the target including i: sam X j m m ¼ Cj ð40Þ r i r i j where ðm=rÞ Ij is the mass absorption coeﬃcient for the line of element i in element j and Cj is the concentration of element j. The calculation of x-ray absorption from Eq. (35) is most accurate when the value of fðwÞ is greater than 0.7. Toward this end, it is important in setting up an experiment to minimize the degree of x-ray absorption by the following:

828

Small et al.

1. 2.

3.

Minimizing the absorption path length by selecting a low overvoltage E0 =Ec Selecting high-energy x-ray lines that have low-absorption cross sections. It is important to note that selecting a high-energy x-ray line requires a high acceleration potential and will, therefore, increase the absorption for elements in the sample that have relatively low-energy analytical lines Measuring the intensities at the highest possible x-ray emergence angle (which typically is ﬁxed for any given instrument)

A study was done by Small et al. (1991) comparing experimental measurements of fðwÞ with four diﬀerent absorption models. The experimental data were collected for Si at both 20 and 30 kV and for Fe and S in FeS2 at 20 kV acceleration potential. The models studied included the following: 1. 2. 3. 4.

Heinrich’s quadratic model, Eq. (38) (Heinrich, 1972) Heinrich’s duplex model, Eq. (41) (Heinrich, personal communication, 1987) Philibert’s model, Eq. (35) (Philibert, 1963) Pouchou and Pichoir’s model, Eq. (42) (Pouchou and Pichoir 1991a)

Duplex model: ð1 þ 1:65 106 gwÞ2 fðwÞ ¼ ð41Þ 1 þ að1:65 106 gwÞ p ﬃﬃﬃ ﬃ where a ¼ 0:18 þ 2=g þ ð8 109 ÞEc þ 0:005 Z with Ec in electron volts. Puchou and Pichior model: F1 ðwÞ þ F2 ðwÞ ð42Þ fðwÞ ¼ F where the formulation for F, F1, and F2 are given in Pouchou and Pichoir (1991a). A comparison of the diﬀerent models and the experimental data is shown in Figure 13. The results of the study indicated that no one model is best for all targets and conditions. The analyst interested in the highest accuracy should select the absorption correction that works the best for the particular type of sample and experimental conditions. c.

Fluorescence Correction, kF

The characteristic ﬂuorescence correction is necessary when the analysis involves the following conditions: 1. 2.

The characteristic x-ray peak from element j at energy Ej is greater than the critical excitation potential Ec;i for element i. The energy diﬀerence ðEj Ec;i Þ < 5 keV.

When these conditions exist, the characteristic line of element j will excite the characteristic line of element i in the specimen as shown in Figure 14. This ﬂuorescence results in an increased intensity for element i that must be taken into account in obtaining a quantitative analysis. In this case, the excitation is caused by x-rays and not electrons; hence, the generation of the secondary x-rays will originate from a much larger volume of the sample that if it were generated directly by electrons. In addition, the measured intensity of the ﬂuorescence radiation is proportional to the x-ray emergence angle. The basic formulation of the ﬂuorescence correction is given by ! !1 X If; j X If; j Fi ¼ 1 þ 1þ ð43Þ Ii Ii j j std

sam

Figure 13 Plots of the experimental data and literature models for f (w), where the residuals ¼ 100f½ f ðwÞmodel f ðwÞexperimental = f ðwÞexperimental g and high absorption ¼ w>10,000, medium ¼ 1000<w<10,000, and low ¼ w<1000. (From Small et al., 1991.)

Electron-Induced X-ray Emission 829

830

Figure 14

Small et al.

Secondary fluorescence in a specimen.

The factor If =Ii is the ratio of the x-ray-excited intensity for the characteristic line of element i to the electron-excited intensity. The summation over j is required because the total correction must be summed over all the elements in the specimen. The most commonly used ﬂuorescence correction factor was originally developed by Reed (1965) and is given in Eq. (44) [from Reed (1997)]: ! mij If ri 1 Ai ðUj ln Uj Uj þ 1Þ lnð1 þ mÞ lnð1 þ nÞ þ ¼ 0:5Cj oj ð44Þ ri ðUi ln Ui Ui þ 1Þ m n Ii mj Aj where Cj is the concentration of element j; m is the mass absorption coeﬃcient, r is the absorption jump ratio deﬁned as the ratio of m=r on the high-energy side of the absorption edge to m=r on the low-energy side, the factor ðri 1Þ=ri is 0.88 for K line ﬂuorescence and 0.74 for L line ﬂuorescence, oj is the ﬂuorescent yield of element j, Ai is the atomic weight, U is the overvoltage, m ¼ mi ðcosec cÞ=mj and n ¼ s (Lenard’s coeﬃcient)=mj. In practice, when calculating the ﬂuorescence correction from Eq. (43), the standard is either a pure element or there is no signiﬁcant ﬂuorescence of element i by other elements in the standard. For this situation, Eq. (43) reduces to !1 X If; j ð45Þ Fi ¼ 1 1 þ Ii j sam

which is the most common form of the correction found in analytical procedures. Goldstein et al. (1981e) demonstrated the magnitude of the ﬂuorescence correction with the analysis of a 10 wt% Fe–90 wt% Ni alloy. The results of the analysis are summarized in Table 3 and show that the intensity of the ﬂuoresced iron, If =IFe ranges from 16.8% to 34.6% of the observed iron x-ray intensity. The magnitude of the correction is lower at the smaller accelerating potential and detector takeoﬀ angle.

Electron-Induced X-ray Emission

831

Table 3 Fluorescence of FeKa by Ni in 10 wt% Fe–90 wt% Ni alloy C (deg) 52.5 15.5 52.5 15.5

E0 (keV)

If =IFe

FFe

15 15 30 30

0.263 0.168 0.346 0.271

0.792 0.856 0.743 0.787

Source: From Goldstein et al., 1981e.

In addition to the secondary excitation by characteristic x-rays, x-rays can also be excited by the continuum x-rays produced in the sample. The continuum x-rays are the result of the deceleration of beam electrons in the Coulombic ﬁeld of the specimen atoms. This radiation forms an x-ray background that is slowly varying with energy and ranges from E0 to zero. Although there has been some eﬀorts by Heinrich (1987) to simplify the calculations for determining continuum ﬂuorescence, the basic formulation for the added intensity from the continuum ﬂuorescence, Ic , was derived by Henoc (1968): m Ic ¼ f Z; o; r; ; C ð46Þ r The calculation of the continuum ﬂuorescence is relatively complicated, involving integration over the range of E0 to Ec for each element in the specimen. Because the correction can be as large as 2–4%, it should be included for highest accuracy. Myklebust et al. (1979), investigating the continuum ﬂuorescence, determined the following experimental conditions for which a correction for the ﬂuorescence by the continuum is required: fðwÞ > 0:95 Ci < 0:5 Z from the standard much diﬀerent from Z of the sample These requirements translate to a signiﬁcant continuum ﬂuorescence correction in the analysis of a small amount of an element with a high-energy x-ray line such as zinc, in a light matrix such as carbon. In Figure 15, Heinrich compared the ﬂuorescence intensities caused by the x-ray continuum to the intensities generated by the primary electron beam for Al and Cu at diﬀerent electron beam energies and takeoﬀ angles. 2. Analysis Method Based on Integration of fðrzÞ Curves In the classical ZAF procedures, the absorption correction, as mentioned earlier, was based on Philibert’s parameterization of a limited set of fðrzÞ curves, including several simpliﬁcations. The shape of his curve, although not correct, works well for higherenergy x-rays for which the shape of the curve is not critical due to the relatively long absorption path lengths. However, for light elements in specimens with high absorption, Al in Cu, or elements with low-energy x-rays, such as B, C, N, and O, where absorption is signiﬁcant, the shape of the fðrzÞ curve is critical. As a result, the classical ZAF routines based on the simpliﬁed Philibert model produce large errors in the concentrations for these elements.

832

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Figure15 Comparison of x-ray intensities fluoresced by the x-ray continuum to the intensities generated by the primary electron beam. I 0c =I 0p refers to the relative emergent fluorescent intensity at a given angle and Ic =Ip refers to the relative generated fluorescent intensity. (From Heinrich 1981d. Reproduced with permission of Van Nostrand Reinhold.)

Several researchers, including Love (1983), Love and Scott (1988), Bastin et al. (1986), Pouchou and Pichoir (1988), and Sewell et al. (1985a) have made experimental measurements of x-ray depth distributions, fðrzÞ curves, in a large number of targets, including pure elements, alloys, and oxides. In addition, other researchers [e.g., Heinrich et al. (1988), Gauvin et al. (1995), Murata et al. (1983), and Karduck and Rehbach (1988)] have made Monte Carlo calculations of x-ray depth distributions. The increased knowledge regarding the behavior of these curves for diﬀerent materials has made it possible to empirically develop corrections based on the integration of fðrzÞ curves. These procedures are particularly attractive for the analysis of low-energy x-ray lines from the elements boron through ﬂuorine, where fðwÞ is much less than 0.7 and the accuracy of the ZAF method is low. In these procedures, fðrzÞ refers to the depth distribution of x-rays in a bulk sample normalized to the x-ray intensity produced in an inﬁnitely thin, unsupported ﬁlm of the same composition. In principle, the fðrzÞ curve as described by Castaing (1951) contains the information necessary to correct experimental data for both the atomicnumber correction, Z (integrated area under curve), and absorption correction, A (shape of the curve). The Gaussian model proposed by Packwood and Brown (1981) combines the Z and A corrections in their model for fðrzÞ producing a single emitted intensity. It should be noted, as pointed out by Armstrong (1988a), that the use of fðrzÞ expressions

Electron-Induced X-ray Emission

833

for the correction of x-ray absorption requires only that a given expression produce the correct shape for the fðrzÞ curve. The use of the fðrzÞ expression for the atomic-number correction is risky because this requires ‘‘that the thickness of the tracer ﬁlms and the normalizing thin ﬁlms in the fðrzÞ experiments for diﬀerent matricies be known to a high degree of accuracy’’ (Armstrong 1988a). In practice, several of the fðrzÞ methods use of the shape of the fðrzÞ curve for the absorption correction and combine that with separate Z and F corrections. Independent of the procedure used, the analysis of specimens using the fðrzÞ correction method is dependent on the derivation of an accurate expression that describes the experimental fðrzÞ curves and the degree to which that expression can be universally applied to systems with unknown fðrzÞ curves. a. Analytical Expressions for fðrzÞ Listed below are a series of parameterizations that have been used by various authors to represent the shape and area of fðrzÞ curves. Packwood and Brown. Packwood and Brown (1981) proposed a modiﬁed Gaussian function to describe the shape of fðrzÞ curves. In their procedure, both the atomic-number correction and the absorption correction are parametrized by the model. The Gaussian is centered at the sample surface and modiﬁed by a transient function that was introduced to model the near-surface x-ray distribution. Figure 16 is a plot of the modiﬁed Gaussian function showing the inﬂuence of the various parameters a; b; g0 , and f0 . In the model given in Eq. (47) [from Packwood (1991)], the a term related to the width of the Gaussian and g0 relates to the amplitude. The b term in the transient is related to the slope of the curve in the near-surface region (i.e., the rate at which the focused electron beam is randomized through scattering in the target and the y intercept, f0 , is related to the surface ionization potential): fðrzÞ ¼ g0 exp½a2 ðrzÞ2 f1 ½ðg0 f0 Þ=g0 expðbrzÞg |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} Gaussian term

ð47Þ

Transient term

The formulas for the diﬀerent terms can be found in Packwood (1991).

Figure 16 Modified Gaussian function for fitting f(rz) curves. (From Packwood and Brown, 1981. Copyright John Wiley & Sons Limited. Reproduced with permission.)

834

Small et al.

Pouchou and Pichoir. Pouchou and Pichoir (1991b) proposed a fðrzÞ procedure, PAP that was designed to provide a broad range of analytical capabilities including bulk and stratiﬁed sample analysis. In this procedure, the authors separate the parameters that deﬁne the shape of the fðrzÞ curve (i.e., the absorption correction) from the parameters that deﬁne the total integral of the fðrzÞ curve (i.e., the generated primary intensity or atomic-number eﬀect). The authors refer to their procedure as ‘‘neither like a ZAF structure with separate Z, A, and F corrections nor like a global structure such as the model of Packwood and Brown (1981) in which the emerging intensity is obtained directly from the parametric description of fðrzÞ.’’ Area parameter: In the PAP procedure, the authors deﬁne the integral of the fðrzÞ curve from zero to inﬁnity, Eq. (29), as F. The analytical expression for F which is used for the atomic-number correction is given in Eq. (48), where R is the backscatter factor based on the mean-backscatter coeﬃcient and the meanreduced backscattered electron energy for the electrons, 1=S is the deceleration factor, and QA 1 ðE0 Þ is the cross section for x-ray production at level 1 for element A and voltage E0 : R F¼ ð48Þ QA 1 ðE0 Þ S A detailed description of the analytical expression used to calculate F is given in Pouchou and Pichoir (1991b). Shape parameter: For their procedure, Pouchou and Pichoir placed the following conditions on their analytical expression for the shape of the fðrzÞ curve: The integral of the shape function must be equal to F. The shape function must have a ﬁnite value for fð0Þ. The parameterization of the function must have a maximum, designated as Rm . The function must vanish with a horizontal tangent at a predetermined range, Rx . The shape of the fðrzÞ curve is deﬁned by a pair of intersecting parabolas. For this model, the curve is deﬁned by the following parameters: fð0Þ ¼ the surface ionization function Rm ¼ depth at which the maximum in the fðrzÞ curve occurs Rx ¼ the x-ray range The area under the fðrzÞ curve (i.e., the total generated intensity) The equation for the ﬁrst parabola nearest the target surface is given in Eq. (49) for values of rz between 0 and Rc and the second parabola by Eq. (50) for values of rz between Rc and Rx : f1 ðrzÞ ¼ A1 ðrz Rm Þ2 þ B1

ð49Þ

f2 ðrzÞ ¼ A2 ðrz Rx Þ2

ð50Þ

where the parameters A1 ; A2 , and B1 are functions of Rm ; Rx , and fð0Þ, and Rc is the crossover point of the parabolas. The equation for Rc is a function of the generated X-ray intensity, I, which is the integral of the area under the fðrzÞ curve, Equation (29). Figure 17 is the plot of the double-parabola curve with the various parameters located on the curve. Pouchou and Pichoir (1991b) introduced a simpliﬁed version of their model based on

Electron-Induced X-ray Emission

835

Figure 17 PAP double-parabola function for fitting f(rz) curves. (From Pouchou and Pichoir, 1991b. Reproduced with permission of Plenum Press.)

fðrzÞ ¼ A expðarzÞ þ ½Brz þ fð0Þ A expðbrzÞ ð51Þ The coeﬃcients A; B; a, and b can be determined from the following parameters: Z 1 A fð0Þ A B 1. þ 2 fðrzÞdðrzÞ ¼ þ a b b 0 2. fð0Þ is the surface ionization. 3. R ¼ mean ionization depth, A fð0Þ A 2B þ 2 R ¼ 2 þ a b2 b 4. The initial slope ¼ B aA b½fð0Þ A. Merlet. Merlet (1994) developed a parameterization of fðrzÞ based on a double partial Gaussian expression for the description of the fðrzÞ curve as shown in Figure 18. The parameterization is dependent on deﬁning the depth distribution at three points of the fðrzÞ curve, the surface at ½0; fð0Þ, the curve maximum at ½rzm ; fðmÞ, and the x-ray range at ½rzx ; fðmÞ=100. Equation (52) deﬁnes the partial Gaussian f1 for rz values between the surface and rzm , and Eq. (53) deﬁnes the partial Gaussian f2 for rz values from rzm to rzx : ( ) rz rzm 2 f1 ¼ fm exp ð52Þ b where rzm b¼ ½lnðfðmÞ=fð0ÞÞ0:5 and

rz rzm 2 f2 ¼ fm exp ð53Þ a where a ¼ 0:46598ðrzx rzm Þ A detailed description of this procedure is given in Merlet (1994).

836

Small et al.

Figure 18 Double partial Gaussian for fitting f(rz) curves. (From Merlet, 1994). Data points are taken from the work of Castaing and Henoc. (From Castaing and Henoc, 1965; Reproduced with permission of Springer-Verlag.)

Bastin. Bastin and Heijligers (1991) modiﬁed the Gaussian model of Packwood and Brown combining the a and g terms with a b term based on the PAP F term or area parameter. In Bastin’s PROZA program (Bastin et al., 1984), the b term is adjusted, for given a and g, so that the integral under the fðrzÞ curve is equal to the PAP area parameter F, and the peak of the fðrzÞ curve is optimized with respect to position and height. The a term is a function of the beam energy E0 , the critical excitation energy Ec , the overvoltage for the x-ray line of interest U0 , the atomic number Z, atomic weight A, and ionization potential J of the matrix. The g term is determined from one of two equations, depending on whether U0 is greater than 6. Both equations are functions of U0 and Z. The b term is derived from a; g and F and is divided into nine diﬀerent segments based on the value of Rðb=2aÞ. The equation f0 is taken from the PAP model. The various equations for the parameterization of a; b, and g are given in Bastin and Heijligers (1991). In addition to PROZA, Bastin et al. (1998) have developed a new parameterization based on the double partial Gaussian model of Merlet (1994), called PROZA96. The formulation of the two Gaussian functions are as follows: For 0 rz rzm fl ðrzÞ ¼ fm exp½b2 ðrz rzm Þ2

ð54Þ

For rzm rz 1 fr ðrzÞ ¼ fm exp½a2 ðrz rzm Þ2

ð55Þ

Electron-Induced X-ray Emission

837

In this case, the b term in Eq. (54) refers to the Gaussian rate of increase from 0 to rzm and is diﬀerent from the b term in the PROZA formulation. Likewise, the a term in Eq. (55) is the Gaussian rate of decrease from rzm out to inﬁnity. Bastin et al. adopted the new PROZA96 formulation for fðrzÞ because of the constraints placed on the position and height of the maximum in the fðrzÞ curve by the g term in the PROZA formulation. The area under the fðrzÞ curve in PROZA is ﬁxed by the F term. The main advantage of the new formulation is in absorption. The shape of the fðrzÞ curve in PROZA96 results in an increased ﬂexibility for analyzing ‘‘ultralight matrices’’ such as Be, B, and C, particularly for thin-ﬁlm calculations where the shape of the fðrzÞ curve is critical, and at high overvoltages, the maximum will be shifted deeper into the sample. Armstrong. Armstrong (1991) developed a fðrzÞ model, Eq. (56), based on the Gaussian equation from Packwood and Brown: 2

fðrzÞ ¼ g0 ea ðrzÞ ð1 qebrz Þ 2

where

ð56Þ

Z1:05 lnð1:166E0 =J Þ 0:5 a ¼ 2:97 10 1:25 ðE0 Ec Þ AE0 5 2 8:5 10 Z b¼ 2 AE ðg0 1Þ 5

0

5pU0 ½lnðU0 Þ 5 þ 5U00:2 ðU0 1Þ lnðU0 Þ g fð0Þ q¼ 0 g0

g0 ¼

Z and A are the atom-concentration weighted atomic number and atomic weight, respectively. The surface ionization function, fð0Þ, can be determined from the expression of either Reuter (1972) or Love and Scott (1978). Love and Scott. Love et al. (1984) proposed a fðrzÞ method in which they introduced a separate atomic-number correction. With the separate treatment of the atomicnumber correction, the fðrzÞ curve is used only for target absorption. As mentioned earlier, target absorption requires only that the shape of the curve be correct and not the absolute height. The authors proposed a quadrilateral proﬁle, Figure 19, to ﬁt the shape of the curve. The quadrilateral is deﬁned by the y intercept fð0Þ, the position and amplitude of the peak, rzm ; fðmÞ, and the x-ray range, rzr . From the quadrilateral model, the analytical expression for fðwÞ can be written as fðwÞ ¼ 2½A1 ðB þ CÞ where A ¼ ðrzr rzm Þ½rzm þ hðrzr Þw2 B ¼ expðwrzm Þ þ h expðwrzr Þ þ wðrzr rzm Þ h þ 1 expðwrzm Þðrzr hrzr Þ þ hrzr rzr rzm fðmÞ h¼ fð0Þ

C¼

ð57Þ

838

Small et al.

Figure 19 Quadrilateral function for fitting f(rz) curves. (From Scott and Love, 1991. Reproduced with permission of Plenum Press.)

To constrain the quadrilateral model when fðwÞ values exceed 0.5, the authors expressed (Scott and Love, both rzm and rzr in terms of the mean depth of x-ray generation, rz 1991). In conjunction with the quadrilateral ﬁt to the fðrzÞ curve for target absorption, the atomic-number correction combines the stopping power factor from Eq. (14) with the backscatter factor R, Eq. (58), from Love et al. (1978): 1 R 0:6 ¼ IðU0 Þ þ ZGðU0 Þ ð58Þ Z where U0 is the overvoltage, Z is the backscatter coeﬃcient, 1 0:25 IðU0 Þ ¼ 0:3 þ expð1:5 1:5U0 Þ U0 and GðU0 Þ ¼ ð0:368 0:075 ln U0 Þ expð1 2:3U04 Þ A full description of the quadrilateral model can be found in Scott and Love (1991). 3. Evaluation of ZAF and fðrzÞ Analytical Correction Procedures The relationship describing a ZAF or fðrzÞ correction procedure gives the impression that the concentration of a given element can be calculated directly from the k ratio. This, however, is not the case because the parameters used in the diﬀerent correction schemes are dependent on the composition of the sample, including the element of interest. For a simple binary system, the analyst could construct a calibration curve relating measured x-ray intensity to concentration. However, this does not apply to more complicated systems and the primary method used in correction procedures is an iterative method. A general ﬂowchart for a typical ZAF or fðrzÞ correction method is shown in Figure P20. The measured k ratio is used as the initial estimate of composition by taking Ci ¼ i ki =Ci . The diﬀerent ZAF or fðrzÞ corrections are then applied to the initial estimate of concentration to obtain the corresponding estimate of k. Next, a hyperbolic iteration, Eq. (59), is used to obtain a value for a:

Figure 20

General flowchart for a typical ZAF or f(rz) correction method.

840

Small et al.

ai C i ki ¼ P aj C j

ð59Þ

The a value is then used with the measured k ratio to obtain a new estimate of concentration that is then normalized and the iteration continued. Convergence with the hyperbolic function is rapid and seldom requires more than three iterations. The practical lower limit for quantitative x-ray microanalysis is on the order of 0.01% by weight for an element homogeneously distributed in an inﬁnitely thick, polished specimen. The x-ray excitation cross-sections and the signal-to-noise ratio of the x-ray detection systems primarily set this limit. The accuracy of a classical ZAF electron-probe analysis is shown in Figure 21, which is a histogram of the relative errors from 264 analyses of binary metal alloys. The accuracy of several diﬀerent analytical models, including both ZAF and fðrzÞ models is shown in Figures 22–24, which are adapted from a table compiled by Pouchou and Pichior (1991b) for 826 analyses of binary compounds. In these ﬁgures, the horizontal line for a given model is ±1 standard deviation for the mean value of k (calculated)=k (experimental), which is represented by a ﬁlled circle. The length of the horizontal line for a given model is an indicator of the variation in the accuracy for the performance of that model on the given dataset. Similarly, the proximity of the mean with respect to unity is an indicator of any systematic bias in the model. Figure 22 displays the results from the entire 826 analyses. The fðrzÞ models 1 and 2 have the lowest variations in accuracy, both less than 2%. fðrzÞ models 3 and 4 and ZAF model 5 are similar with variations less than 3%. The remaining two ZAF models (6 and 7) have variations greater than 4%. The systematic biases for the techniques are small and fall within ±1%, except for the ZAF method 5, which has a systematic bias of about 1.5%. Figure 23 displays the results of 577 of the 826 analyses involving samples where the atomic-number eﬀect is greater than the x-ray absorption eﬀect. The variances for all the models are similar generally less than 2%, except the fðrzÞ model 4, which has a variance of about 3%. The biases are also small, except for model 4, which has a positive systematic bias of about 1.5%. Figure 24 shows the result for 242 analyses involving samples where x-ray absorption is the predominate correction. For these samples, there is a signiﬁcant diﬀerence between the ZAF and fðrzÞ models. Three of the fðzÞ models (1, 2 and 4), have variances in accuracy less than 3% and biases less than 1%. ZAF model 5 and fðrzÞ model 3 are similar for both variation, about 4%, and systematic bias, about 2%. The remaining two ZAF models (6 and 7) have signiﬁcantly larger variances, about 6%, and systematic biases of around þ 4%. In addition to the above dataset, Pouchou and Pichior (1991b) also compiled data for a series of boride analyses, the results of which are given in Figure 25. The variance in accuracy for the diﬀerent methods range from about 4% for several of the fðrzÞ methods to about 25% for the worst ZAF method. Five of the seven methods show a slight systematic bias of þ1–2% with models 7 and 8 having signiﬁcant positive biases of 8% and 15%, respectively. Finally, Table 4 lists the results from the analyses of compounds containing ultralight elements with PROZA96 from Bastin et al. (1998). B.

Empirical Approach to Quantitative Analysis

Ziebold and Ogilvie (1964) developed a quantitative analysis procedure for emitted x-rays that is based on Castaing’s third approximation for generated x-rays given in Eq. (59) (Castaing, 1951). In this empirical procedure, the relationship between the k ratio and

Electron-Induced X-ray Emission

841

Figure 21 Histogram of the relative errors for a classical ZAF analysis from Goldstein et al. (From Goldstein et al., 1975b. Reproduced with permission of Plenum Press.)

Figure 22 Results from the analysis of 826 binary compounds with several different quantitative analysis procedures. (Adapted from Pouchou and Pichoir, 1991b.)

842

Small et al.

Figure 23 Results for 577 of the 826 binary compounds where the atomic-number effect is greater than the x-ray absorption effect. (Adapted from Pouchou and Pichoir, 1991b.)

Figure 24 Result for 242 analyses involving samples where x-ray absorption is the predominate correction. (Adapted from Pouchou and Pichoir, 1991b.)

Electron-Induced X-ray Emission

Figure 25 1991b.)

843

Results from the analysis of a series of borides. (Adapted from Pouchou and Pichoir,

concentration, expressed in Eq. (2), can be expanded to include real samples by the introduction of an eﬃciency factor a for element a in a sample: aa Ca ka ¼ Sai Ci where the summation is over all elements in the sample. Ziebold and Ogilvie, using Eq. (59), expressed the relationship between C and k for a binary system containing elements i and j as

or

Ci ¼ a þ ð1 aÞCi ki

ð60Þ

ki 1 1 ¼ þ 1 ki a Ci a

ð61Þ

where a for the binary is referred to as the a factor and equals aj =ai . Table 4 Performance of PROZA96 on Ultralight Element Data: Comparison of the Calculated k ratio (k 0 ) with the Measured One (k) Element

No. of measurements

Mean k 0=k

Root mean square deviation (%)

Boron Carbon Nitrogen Oxygen

192 117 144 294

1.0033 1.0001 0.9989 1.0004

3.3114 3.2092 3.6820 2.0090

Source: From Bastin et al., 1998.

844

Small et al.

Finally, solving for a yields Eq. (62) which is the formulation for the hyperbolic approximation because the plots of the calibration curves of C versus k described by this equation are hyperbolas: C 1k a¼ ð62Þ 1C k Equations (60) and (61) indicate a linear relationship between C=k and C, and k=C and k so that a plot of C=k versus C or k=C versus k are straight lines. The slope of the line will be positive if a is greater than 1 and negative if a is less than 1. Experimental conﬁrmation of the hyperbolic approximation is shown in Figure 26, which is a plot of k=C versus k for experimental measurements on AgLa x-rays from four diﬀerent Ag–Au alloys at several diﬀerent beam energies. Except for the 5-keV line, the plots for the diﬀerent alloys as predicted by the hyperbolic approximation are straight lines. The concentrations Ci and Cj , for elements i and j in an unknown binary, can be determined from Eq. (62) if the appropriate values for ai ; aj ; ki , and kj are known. In theory, the k values do not present a problem because they are experimentally measured. The determination of ai and aj , however, requires the analysis of a standard binary containing the same elements as the unknown. In the procedure for determining the a factors, ki and kj are obtained for the standard binary by comparison to pure-element standard of i and j. Because the values of Ci and Cj , in the standard binary are known, it follows that ai

Figure 26 k=C versus k for Ag–Au alloys at various electron beam energies. (From Goldstein et al., 1981f. Reproduced with permission of Plenum Press.)

Electron-Induced X-ray Emission

845

and aj can be calculated from Eq. (59). In the case of a binary, Eq. (63) can be used to calculate the concentration for element i in the unknown binary: Ci ¼

ki a 1 þ ki ða 1Þ

ð63Þ

The values for a are speciﬁc for a given experimental setup, and if the analysis parameters are changed, the a values must be recalculated. As long as the analyst is concerned with measurements on a given binary system under constant conditions, then Eq. (63) provides a very rapid and easy method for the determination of elemental composition with an accuracy that is comparable to that of the full ZAF procedure. In addition to their work on binary systems, Ziebold and Ogilvie (1964) also derived the analytical expression for a ternary system. The basic equation for the ternary remains Eq. (63), except that the combined a factor for element 1 in a ternary compound containing elements 1, 2, and 3 is given by the following expression: a1;2;3 ¼

a1;2 C2 þ a1;3 C3 C2 þ C3

ð64Þ

An important aspect of the ternary derivation is that the combined a factor for the ternary system in Eq. (64) can be expressed as a combination of the individual a factors for the diﬀerent binaries. Bence and Albee (1968), building on the work of Ziebold and Ogilvie, extended the application of the hyperbolic approximation to multielement systems containing six to eight elements. They were interested in obtaining a rapid, accurate procedure for the analysis of specimens of geological origin. In such specimens, it was not practical in 1968 to apply the full ZAF corrections due to the limited computational resources available. Even with modern computer capabilities, the empirical approach oﬀers a simple, straightforward approach to quantitative analysis that can be programmed and executed on a small pocket calculator. In addition, because of its simplicity and speed, it can be used for real-time processing of spectra image data (see, Sec. V). The general formula for element i in a system of n components is given by Ci ¼ ki bi

ð65Þ

where bi ¼

k2 ai2 þ k3 ai3 þ þ kn ain k2 þ k3 þ þ kn

In Eq. (65), the ai2 factor refers to the a factor for element i in a binary of element 2 and element i. Similarly, k2 refers to the intensity ratio of element 2 in the specimen to the element 2 in the standard. In practice, the analyst is required to measure nðn 1Þ values for a, which requires nðn 1Þ binary standards. The ﬂowchart for the empirical method of quantitative analysis is shown in Figure 27. The analyst, having determined the needed a values, measures the appropriate k values for the unknown and calculates the b term from Eq. (65). The b values are then used to calculate a ﬁrst approximation of the elemental concentrations. From Eq. (65), a second set of b values are determined with the approximate concentrations used for the k’s. The iteration loop is continued until the diﬀerence in successive b terms is below a predetermined limit. As previously mentioned, the analysis of multielement systems requires nðn 1Þ standards. Because the large number of standards for the determination of a factors in a complex system may not be available, a combination ZAF, fðrzÞ, or Monte Carlo a factor

846

Figure 27

Small et al.

An empirical analysis procedure.

approach has been developed. In this procedure, the analyst calculates the necessary a factors by assuming an appropriate composition and then running a ZAF, fðrzÞ, or Monte Carlo procedure to back-calculate the corresponding k ratios. The a factors can then be determined from Eq. (60). It is important that the analyst realize that the total uncertainty in this combined approach includes the uncertainties of both the empirical and the ZAF methods. Albee and Ray (1970) have used this procedure to determine the a factors for 36 elements relative to simple oxides. Laguitton et al. (1975) and Bence and Holzwarth (1977) proposed additional corrections along with more elaborate polynomial representations of the a factors. Armstrong (1988b) calculated a large number of oxide a factors using a combination of ZAF, fðrzÞ, and Monte Carlo methods combined with a second-order polynomial for the a factors, Eq. (66): 2 Ci Ci i ai; j ¼ c þ d ð66Þ þe Ci þ Cj Ci þ Cj where Ci and Cj are the oxide weight fractions of elements i and j in the binary and c; d, and e are the coeﬃcients of the polynomial ﬁt. The results indicated that the C=k values

Electron-Induced X-ray Emission

847

from the second-order polynomial a factor were essentially identical to the ZAF, fðrzÞ, or Monte Carlo method they were based on and the results can be expected to have the same accuracy and precision as the conventional procedures. The empirical method of analysis is most accurate when the diﬀerence between the sample and standards is mainly the result of x-ray absorption. The procedure is less accurate when there are large diﬀerences in atomic number and is least accurate when there is a signiﬁcant contribution from ﬂuorescence. Given these limitations, the empirical procedure works best for the analysis of materials such as geologic specimens and oxide systems that have relatively low mean atomic numbers and minimal secondary ﬂuorescence. C. ‘‘Standardless’’ Quantitative Electron Probe X-ray Microanalysis with EDS An increasing trend in recent years in performing quantitative electron probe x-ray microanalysis with energy-dispersive x-ray spectrometry has been the substitution of ‘‘standardless’’ methods in place of the traditional approach of measuring standards containing the elements of interest on the same analytical instrument under the same excitation and detection conditions (Russ, 1974). In standardless methods, the appropriate standard intensities necessary for quantiﬁcation are either calculated from ﬁrst principles, considering all aspects of x-ray generation, propagation through the solid target, and detection (‘‘true standardless’’), or else standard intensities are derived from a suite of experimental measurements performed remotely and adjusted for the characteristics of the local instrument actually used to measure the unknowns (‘‘ﬁtted standards’’). With either route to ‘‘standard intensities,’’ the resulting ‘‘k values’’ (unknown=standard) are then subjected to matrix corrections with one of the usual approaches [ZAF, fðrzÞ, etc.]. The apparent advantages of standardless analysis are considerable. Instrument operation can be extremely simple. There is no need to know the beam current, and, indeed, it is not even necessary for the beam current to be stable during the spectrum accumulation, a real asset for instruments such as the cold-ﬁeld-emission gun scanning electron microscope (SEM), FEG-SEM, where the beam current can be a strong function of time. Moreover, the detector solid angle is of no consequence to the quantitative procedure. Both the beam current uncertainty and the detector solid angle uncertainty are eﬀectively hidden by forcing the analytical total to a predetermined value (e.g., unity when all constituents are measured). When the spectrum has been accumulated to the desired level of statistical precision, the analyst needs to specify only the beam energy, the x-ray takeoﬀ angle, and list of elements to be quantiﬁed (of this list can be derived from an automatic qualitative analysis). The software then proceeds to calculate the composition directly from the spectrum, as described below, and in the resulting output report, the concentration is often speciﬁed to three or four signiﬁcant ﬁgures. The measurement precision for each element, calculated from the integrated peak and background counts, is also reported. The precision is really only limited by the patience of the analyst and the stability of the specimen under electron bombardment, so that precision values below 1% relative (1s can be readily achieved, even for minor constituents. Although such excellent precision is invaluable when comparing diﬀerent locations on the specimen, the measurement precision is independent of the accuracy of the analysis. For the traditional approach of measuring standards and calculating matrix corrections, there exists an extensive literature describing testing of the quantitative procedures. These test results are usually presented in the form of error histograms, which the analyst can use to make an estimate of the accuracy of the result. The relative error is deﬁned as

848

Small et al.

Measured True Rel: error ¼ ð100%Þ True

ð67Þ

An example of such an error histogram is shown in Figure 21, from which it can be deduced that, under the particular conditions used [binary alloys measured against pure element standards with NIST–ZAF matrix corrections (1975 formulation)], 95% of the analyses lie within ±5% relative of the correct value. Unfortunately, such error histograms are virtually never available for standardless analysis procedures as implemented in commercial EDS systems. Recent work indicates that these standardless analysis procedures may, in fact, be subject to large errors (Newbury et al., 1995; Newbury, 1998). 1. First-Principles Standardless Analysis Calculating an equivalent standard intensity from ﬁrst principles requires solution of 2 3 ZEc Q 6 rN0 7 dE5R fðwÞe ð68Þ Istd ¼ 4 o ðdE=dsÞ A E0

where the terms in brackets represent the excitation function: r is the density, N0 is Avogadro’s number, A is the atomic weight, o is the ﬂuorescence yield, Q is the ionization cross section, dE=ds is the stopping power, E is the electron energy, E0 is the incident beam energy, and Ec is the critical excitation energy. The other terms correct for the loss of x-ray production due to electron backscattering (R), the self-absorption of x-rays propagating through the solid ½ fðwÞ, and the eﬃciency of the detector, e. It is useful to consider the conﬁdence with which each of these terms can be calculated. a.

Excitation

Three terms are critical in the excitation function: the ionization cross section, the ﬂuorescence yield, and the stopping power. 1.

Ionization cross section: Several parameterizations of the K shell ionization cross section are plotted in Figure 28. The variation among these choices exceeds 25%. Although we cannot say that any of these is correct, it is certain that they cannot all be correct. Moreover, because of the continuous energy loss in a solid target, the cross section must be integrated from E0 to Ec , through the peak in Q and the rapid decrease to U ¼ 1. This region of the cross section is poorly characterized, so that it is diﬃcult to choose among the cross-section formulations based on experimental measurements. The situation for L and M shell cross sections is even more unsatisfactory. 2. Fluorescence yield: Various experimental determinations of the K shell ﬂuorescence yield are plotted in Figure 29. Again, a variation of more than 25% exists for many elements. The situation for L and M shell transitions is substantially more incomplete. 3. Stopping power: The classic Bethe formulation of the stopping power becomes inaccurate at low beam energies (< 5 keV), and eventually with decreasing energy, it becomes physically unrealistic with a sign change. The accuracy of the stopping power matters for calculating Eq. (68) because the cross section must be integrated to Ec , which for low-energy x-rays (e.g., C, N, O, F) involve electron energies in this regime. As discussed earlier in this chapter, several authors have suggested modiﬁcations to the Bethe formulation to correct for the

Electron-Induced X-ray Emission

Figure 28 authors.

849

Ionization cross section as a function of overvoltage U, as formulated by different

low beam energy regime. Unfortunately, the base of experimental measurements necessary to select the best choice is just being developed. b.

Backscatter Loss

The backscatter-loss correction factor R was initially formulated based on experimental measurements of the total backscatter coeﬃcient and the diﬀerential backscatter coeﬃcient energy. Although careful, extensive measurements of total backscatter were available in the literature, the database of the diﬀerential backscatter coeﬃcient as a function of energy, a much more diﬃcult experimental measurement, was limited to a few elements and was available at only one emergence angle. The development of advanced Monte Carlo simulations has permitted the rigorous calculation of R over all scattering angles and energy losses so that this factor is probably known to an accuracy within a few percent across the periodic table and the energy range of interest. c.

X-ray Self-absorption

The self-absorption of x-rays in the hypothetical standard is calculated with the formulation of the absorption factor used in matrix corrections. Fortunately, the absorption correction is generally small for the x-rays of a pure element, so that at least for higherenergy characteristic x-rays (e.g., greater than 3 keV), there is little uncertainty in this factor. However, the self-absorption increases both as photon energy decreases and in-

850

Small et al.

Figure 29

K Shell fluorescence yield as a function of atomic number.

cident electron energy increases, so that the error in calculating the intensity emitted from an element emitting low-energy photons (e.g., carbon) could be signiﬁcant. d. Detector Eﬃciency The last term in Eq. (69) is one of the most diﬃcult with which to deal: m m m rwin twin þ rAu tAu þ r tSiDL e ¼ exp r win r Au r SiDL Si m m m þ rcon tcon þ rice tice 1 exp rSi tSi r con r ice r Si

ð69Þ

In the traditional k-value approach, the detector eﬃciency cancels quantitatively in the intensity ratio because the same x-ray peak is measured for the unknown and the standard under identical (or at least accurately reproducible) spectrometer conditions. When standardless analysis is performed, this cancellation cannot occur because x-ray peaks of diﬀerent energies are eﬀectively being compared and, therefore, accurate knowledge of the detector eﬃciency becomes critical. Detector eﬃciency is mainly controlled by absorption losses in the window(s) and detector structure. The expression for detector eﬃciency, Eq. (69), consists of a multiplicative series of absorption terms for each component: detector window (win), gold surface electrode (Au), semiconductor ‘‘dead layer’’ (DL, actually a partially active layer below the electrode and the source of incomplete charge phenomena), and a transmission term for the detector thickness. Additionally, for most practical measurement situations, there may be absorption contributions from contamination on

Electron-Induced X-ray Emission

851

the detector crystal, usually arising from ice buildup due to pinhole leaks in the window or support (ice) and from contamination on the detector window usually deposited as hydrocarbons from the microscope environment (con). An example of the detector eﬃciency as a function of photon energy for several window materials is shown in Figure 30. The choice of the window material has a strong eﬀect on the detector eﬃciency for photon energies below 3 keV, and accurate knowledge of the window and detector parameters is vital for accurate interelement eﬃciency correction across the working range of the detector, typically 100 eV to 12 keV. The change in the detector eﬃciency with the accumulation of ice is illustrated in Figure 31. The buildup of ice and other contaminants and the resulting loss in eﬃciency is referred to as ‘‘detector aging.’’ Detector aging can result in a signiﬁcant loss of the low-energy photons (< 3 keV) relative to the higher-energy photons (3–12 keV) (Fiori and Newbury, 1978). 2. ‘‘Fitted Standards’’Standardless Analysis The ‘‘ﬁtted standards’’ technique is the more widely used approach for implementing ‘‘standardless’’ analysis on commercial computer-assisted EDS analyzer systems. In the ﬁtted standards technique, a suite of pure-element standards covering K-, L-, and M- family x-rays is measured at one or more beam energies on an electron beam instrument equipped with an EDS detector whose eﬃciency is known from independent measurements or at least can be estimated. An example of such a measurement for a portion of the K series from pure

Figure 30 Detector efficiency for various windows (Al window coating ¼ 0.02 mm; Au electrode ¼ 0.01 mm; Si dead layer ¼ 0.03 mm): (a) beam energies 0–5 keV; (b) beam energies 0–1 keV.

852

Small et al.

Figure 30

Continued.

elements measured at 20 keV is shown in Figure 32. Missing elements can be calculated by simply ﬁtting the available peaks and interpolating (e.g., in Figure 32, the intensity for gallium could be estimated by ﬁtting the smoothly varying data and interpolating). From the smooth change in peak height with atomic number seen in Figure 32, such an interpolation should be possible with reasonable accuracy. The situation is not as satisfactory in the L and M families, as illustrated in Figures 33 and 34, respectively, because the ﬂuorescence yield is a much more complicated function of atomic number. If the analysis must be perfomed at a beam energy other than that of the spectral database, then Eq. (68) must be used to shift the intensities appropriately. Similarly, if a diﬀerent EDS detector is used, the detector eﬃciency must be corrected using Eq. (69). The ‘‘ﬁtted standards’’ standardless procedure is expected to be more accurate than the ﬁrst-principles standardless procedure because it is tied to actual experimental measurements which directly incorporate the eﬀects of the cross section and the ﬂuorescence yield, at least over the range of the elements actually measured. a.

Testing the Accuracy of Standardless Analysis

The accuracy of standardless analysis procedure has been tested by carrying out analysis on microhomogeneous materials of known composition: NIST Microanalysis Standard Reference Materials (SRM), NIST Microanalysis Research Materials (RM), stoichiometric binary compounds (e.g., III–V compounds such as GaAs and II–VI compounds such as SrTe), and other materials such as ceramics, alloys, and minerals for which the

Electron-Induced X-ray Emission

Figure 31

853

Effect of ice buildup on the detector.

composition was available from independent chemical analysis and for which microhomogeneity could be established. Compositions were carefully chosen to avoid serious spectral overlaps (e.g., PbS, MoS2). Light elements such as B, C, N, O, and F were also eliminated from consideration because of large errors due to uncertainties in mass absorption coeﬃcients. In oxidized systems, the oxygen was calculated by means of assumed stoichiometry, but the resulting oxygen values were not included in the error histograms because of their dependence on the cation determinations. Figure 35 shows an error histogram for the ‘‘ﬁrst-principles’’ standardless analysis procedure embedded in the National Institute of Standards and Technology–National Institutes of Health Desktop Spectrum Analyzer (DTSA) x-ray spectrometry software engine (Fiori et al., 1992). The error distribution shows symmetry around 0% error, but in comparing this distribution with that for the conventional standards=ZAF procedure shown in Figure 21, the striking fact is that the error bins are 10 times wider for the ‘‘ﬁrstprinciples’’ standardless analysis procedure. Thus, the 95% error range is approximately ±50% relative rather than ±5% relative. The error distribution for a commercial standardless procedure based on the ‘‘ﬁtted standards’’ approach is shown in Figure 36. This distribution is narrower than the ﬁrstprinciples standardless approach, but the error bins are still ﬁve times wider than those of the conventional standards=ZAF procedure, so that the 95% error range is ±25% compared to ±5%. It must be emphasized that this distribution represents a test of only one of the many implementations of standardless analysis in commercial software systems and more extensive testing is needed.

854

Figure 32

Small et al.

K-Family peaks from transition elements; E0 ¼ 20 keV.

3. Using Standardless Analysis Given the width of these error distributions for standardless analysis, it is clear that reporting composition values to three or four signiﬁcant ﬁgures can be very misleading in the general case. At the same time, the error distributions show that there are signiﬁcant numbers of analyses for which the errors are acceptably small. Usually, these analyses involve elements of similar atomic number (e.g., Cr, Fe, and Ni in stainless steel) for which the x-ray peaks are of similar energy and are therefore measured with similar eﬃciency. The user of standardless analysis must be wary that any conﬁdence obtained in such analyses does not extend beyond those particular compositions. The most signiﬁcant errors are usually found when elements must be measured involving a mix of K, L, and M shell x-rays. The errors in the tails of the distributions are so large that the utility of any such concentration values is limited in terms of solving problems. Standardless analysis does have legitimate value. If a microhomogeneous material with a known composition similar to the unknown specimen of interest is available, then the errors due to standardless analysis can be assessed and included with the report of analysis. For example, if the Fe–S system is to be studied, then the minerals pyrite (FeS2) and troilite (FeS) would form a good test to challenge the standardless analysis procedure. The analyst should never attempt to estimate relative concentrations merely by inspecting a spectrum. There are simply too many complicated physical eﬀects of relative

Electron-Induced X-ray Emission

Figure 33

855

L-Family peaks; E0 ¼ 20 keV.

excitation, absorption, and eﬃciency of detection to allow a casual inspection of a spectrum. Standardless analysis incorporates enough of the corrections to allow a sensible classiﬁcation of the constituents of the specimen into broad categories: Major: greater than 10 wt% Minor: 1–10 wt% Trace: less than 1 wt% In the absence of a known material to test a standardless analysis procedure, it is recommended to use broad classiﬁcation categories instead of numerical concentration

856

Figure 34

Small et al.

M-Family peaks; E0 ¼ 20 keV.

Figure 35 Error distribution for the ‘‘first-principles’’ standardless analysis procedure embedded in the NIST–NIH Desktop Spectrum Analyzer x-ray spectrometry software system.

Electron-Induced X-ray Emission

857

Figure 36 Error distribution for a ‘‘fitted standards’’ standardless analysis procedure embedded in the commercial x-ray spectrometry software system.

values, which may imply far more apparent accuracy than is justiﬁed and which may lead to a loss of conﬁdence in quantitative electron-probe microanalysis when an independent test is conducted.

III.

MICROANALYSIS AT LOW ELECTRON BEAM ENERGY

A. What Is to Be Gained at Low Beam Energy? 1. Lateral and Depth Resolution Throughout the history of electron beam x-ray microanalysis, analysts have relied upon the strong dependence of the electron range (R0 ) on the incident energy to control the spatial resolution of analysis, which is critical in such problems as characterizing inclusions in a matrix or a thin layer on a substrate: R0 ¼ kðE0n Ecn Þ

ð70Þ

where k depends on matrix parameters [atomic number (Z), atomic weight (A), and density (r)], E0 is the incident beam energy, Ec is the critical excitation energy (‘‘critical ionization potential’’), and the exponent n is in the range 1.5–1.7. The ‘‘conventional’’ energy range for quantitative electron beam x-ray microanalysis can be thought of as beginning at 10 keV and extending to the upper limit of the accelerating potential, typically 30–50 keV, depending on the instrument. The lower limit of 10 keV for the conventional operating range is selected because this is the lowest incident beam energy for which there is at least one satisfactory analytical x-ray peak excited from the K, L, or M shells for every element in the periodic table that is accessible to x-ray spectrometry (Goldstein et al., 1992c). Only H, He, and Li are excluded due to a lack of x-ray emission or, in the case of Li, because the photon energy is so low (54 eV) that it is completely reabsorbed in the

858

Small et al.

target or the components of the x-ray spectrometer. Electron-excited x-ray microanalysis thus extends from Be (EK ¼ 0.116 keV) to the transuranic elements. This elemental range is based on establishing the minimum acceptable overvoltage as U ¼ E0 =Ec 1:25. At E0 ¼ 10 keV, this overvoltage criterion involves the use of K lines for 4 (Be) Z 27 (Co) L lines for 28 (Ni) Z 67 (Ho), and M lines for 68 (Er) Z to the transuranics, as illustrated by the periodic table in Figure 37, shaded according to the appropriate shell for analysis with E0 ¼ 10 keV and U 1.25. This broad elemental coverage of electron beam x-ray microanalysis is one of its most important features. No strict deﬁnition exists as to what energy value constitutes the beginning of the low-beam-energy microanalysis regime, but from the previous discussion, the low-energy regime can be considered those beam energies below 10 keV, and especially below 5 keV. Figure 38 shows the excitation range for several characteristic x-rays in a silicon matrix as a function of incident beam energy from 0 to 10 keV, as calculated with the Kanaya– Okayama formulation of the excitation range (Kanaya and Okayama, 1972): 0:0276A 1:67 RðmmÞ ¼ ð71Þ E0 E1:67 c Z0:89 r Near E0 ¼ 10 keV, the excitation range is substantially greater than 1 mm for the various elemental lines plotted, whereas at 5 keV and below, the range for these x-rays is below 500 nm. The excitation range as a function of overvoltage is shown in Figure 39. The range is less than 150 nm for AlK in Si for an overvoltage of 2 and diminishes sharply as the overvoltage is reduced, so that for an overvoltage of 1.25, the excitation range is approximately 20–30 nm.

Figure 37 Selection of x-ray peaks available for analysis with E0 ¼ 10 keV for conventional EDS. Note that elements in black squares either do not emit x-rays or else are too low in energy for practical detection.

Electron-Induced X-ray Emission

859

Figure 38 Range of excitation of various lines (AlK, FeL, and AuM) in a silicon matrix as a function of the beam energy, as calculated with the Kanaya–Okayama range.

Low-electron-beam-energy microanalysis below E0 ¼ 5 keV has been possible for many years with scanning electron microscopes (SEMs) based on tungsten and LaB6 ﬁlament electron guns. However, the relatively low brightness of these sources severely limits resolution in the low-beam-energy regime. The emergence of the high-performance FEG–SEM which can still achieve nanometer-scale beam diameters at low energy, has made it possible to actually make use of the potential analytical resolution given by Eq. (71), and this has greatly increased interest in low-energy x-ray microanalysis. The typical analytical FEG–SEM as currently delivered is equipped with a semiconductor (Si or Ge) energy-dispersive x-ray spectrometer with a limiting spectral resolution of 125– 150 eV. Although useful analyses can certainly be performed with the semiconductor EDS, the limited resolution of EDS (approximately 50–100 times the natural x-ray linewidth) and aspects of the physics of x-ray generation with low-energy excitation impose some severe constraints. As the electron beam energy is reduced, x-ray microanalysis becomes insensitive to certain elements because of these factors.

B. What Are the Negative Aspects of Low-Beam-Energy Analysis? 1. Limits of Detection At high overvoltage ðU 3Þ, experience with semiconductor EDS x-ray spectrometry has shown that the concentration limit of detection, CMDL , for most elements in most matrices is approximately 0.001 mass fraction in the absence of peak interference (Goldstein et al.,

860

Small et al.

Figure 39

Range of excitation of AlK in silicon.

1992d). The limit of detection depends strongly on the characteristic peak-to-bremsstrahlung background (P=B) and upon the characteristic x-ray peak counting rate (P) (Ziebold, 1967): CMDL ¼

3:29a ½ntPðP=BÞ1=2

ð72Þ

where ‘a’ is the ziebold–Ogilvie factor in the expression relating concentration to measured intensity ratio [unknown sample to pure-element standard (Ziebold and Ogilvie, 1964); ‘a’ can be taken as approximately unity for general estimation purposes], n is the number of measurements, t is the integration time per measurement, P is the peak counting rate, and B is the background counting rate. As the overvoltage is lowered, both the peak counting rate and the peak-to-background decrease. Figure 40a shows the experimentally measured behavior of P=B and CMDL as a function of overvoltage for SiKa from pure silicon.

" Figure 40 (a) Experimental measurement of the P=B for copper as a function of overvoltage and calculation of the concentration limit of detection for SiKa in a matrix of similar atomic number (e.g., Al). Detector: Si EDS (150 keV); 100 s integration. The vertical arrow designates the ‘‘lowkeV SEM’’ regime. (b) Calculated limit of detection (mass fraction), derived from experimental measurements of P=B and peak counting rate, as a function of overvoltage, U. Si EDS (150 eV FWHM) operated at 30–40% dead time. Three different accumulation times (nt) of 100 s, 200 s, and 1000 s are plotted. The vertical arrow designates the ‘‘low-keV SEM’’ regime.

Electron-Induced X-ray Emission

861

862

Small et al.

Measurement conditions were 100-s accumulation times (nt product) with a beam current and detector solid angle chosen so that the EDS system dead time was approximately 30% (total spectrum) for a 30-mm2 Si detector with a resolution of 150 eV (MnKa). As can be seen in Figure 40a, operation at low overvoltage results in restrictions on the concentration limit of detection. Although major (arbitrarily deﬁned as C > 0.1 mass fraction) and minor (deﬁned as 0.01 C 0.1 mass fraction) constituents can still be detected with overvoltages as low as U ¼ 2 at an accumulation time of 100 s, trace constituents (deﬁned as C < 0.01 mass fraction) cannot be reliably measured with U below 9.0. To reduce the overvoltage requirement to U ¼ 3 for trace levels above 0.001 mass fraction requires an accumulation time of 1000 s, as seen in Figure 40b. A higher-resolution semiconductor EDS detector (e.g., 130 eV at MnKa) would improve the situation by a factor of 10–20%. Any peak interferences would serve to make the detection situation much more unfavorable. 2. Limitations Imposed by FluorescenceY|eld When an energetic electron scatters inelastically with a bound atomic electron resulting in the ejection of the atomic electron, the ionized atom is left in an excited state. The atom returns to the ground state through processes that involve electron transitions. The diﬀerence in energy between the electron states can be released in the form of an x-ray photon, or the energy can be transferred to another atomic electron, which is ejected as an Auger electron. The fraction of ionizations that leads to photon emission is called the ﬂuorescence yield, o, which is related to the Auger yield, a: oþa¼1

ð73Þ

For low-electron shells with low binding energy, the Auger process is strongly favored over x-ray emission. For example, for carbon K shell ionization (EK ¼ 0.284 keV), the ﬂuorescence yield is 0.00198. Table 5 lists some typical values of the ﬂuorescence yield for low-ionizaiton-energy shells. Examination of these data reveals that an important consequence of having to select low-energy L and M shells to analyze for intermediate and heavy atoms is that the ﬂuorescence yields are substantially lower than those of similar K shells, a factor of 3–5 poorer for L shells and 10–30 poorer of M shells. Thus, the x-ray spectra of these elements with low-beam-energy excitation are likely to have characteristic peaks with low intensity relative to the background (x-ray bremsstrahlung), a problem that is further exacerbated by the poor resolution of the semiconductor EDS, which acts to spread the available peak information over a wide range of background. Table 5

Fluorescence Yields Shell

Edge energy (keV)

o

C Na Mg Si

K K K K

0.284 1.080 1.303 1.848

0.00198 0.0192 0.0265 0.0603

Zn Ga Ge

L L L

1.022 1.117 1.217

0.00736 0.00875 0.0103

Sm Eu

M M

1.080 1.130

0.00133 0.00137

Element

Electron-Induced X-ray Emission

863

The combination of low overvoltage and the low yield from L and M shell x-ray peaks can result in spectra which look extremely unfamiliar to analysts who are used to spectra excited in the conventional beam energy range. As an example, Figure 41 shows the situation for (a) SiO2 and (b) SiO excited with an incident beam energy of 5 keV and measured with a Si EDS (150 eV FWHM at MnKa). Both O and Si are measured with K lines, and peaks for both are prominent in the spectrum, with the O peak reduced relative to the Si due to stronger absorption eﬀects. By comparison, Figure 42 shows the spectrum for GeO2. The GeL family of peaks (unresolved) is much reduced in intensity relative to the OK and SiK peaks due to the relative ﬂuorescence yield diﬀerences, despite the fact that the overvoltage is actually greater for the GeL edge (ELIII ¼ 1:217 keV) than the SiK edge (EK ¼ 1:848 keV). Similarly, Figure 43 shows the situation for CaF2 excited with an electron beam energy of 3 keV, which is below the CaK edge (4.038 keV). Note that the FK peak is very prominent, but the Ca can only be detected through the CaL peak, which is a factor of 10 lower relative to FK, due to both the lower ﬂuorescence yield from the L edge and the increased absorption of the lower-energy CaL radiation. An extreme example of the eﬀects of excitation and ﬂuorescence yield is shown in Figure 44, which presents the spectrum of BaCO3 excited at 5 keV and measured with a Si EDS (150 eV FWHM at MnKa). The BaM family of peaks is just visible above background. In fact, the series of BaM ‘‘peak structures’’ is partially a result of the BaM absorption edges, which exist as discontinuities in the x-ray continuum (bremsstrahlung) and which are broadened into peaklike features as a result of the action of the detector broadening function. An overall assessment of the impact of both low excitation and the ﬂuorescence yield on elemental accessibility with a 5-keV incident beam is presented in Figure 45 and the situation with 2.5 keV is shown in Figure 46. At 5 keV, a few intermediate and heavy elements are inaccessible (black squares), whereas at 2.5 keV, the situation is much worse, with extensive sections of the periodic table eﬀectively lost to SEM=semiconductor EDS measurement. 3. Quantitative Analysis at Low Beam Energy Quantitative analysis procedures for the low-beam-energy regime is a subject of some controversy and ongoing development. The classic ZAF methods are based on a combination of empiricism and physical theory which were developed in the conventional highbeam-energy regime and should not be expected a priori to work well in the low-energy regime (Goldstein et al., 1992e). Very often, the analyst must employ x-ray peaks for which there are little data on mass absorption coeﬃcients and ﬁts to the available data are subject to signiﬁcant error. The more recently developed fðrzÞ quantitation methods have the advantage that at least some of the database of experimental measurements on which the algorithms are based were indeed measured with low incident beam energy for lowenergy photons, so that these methods should be inherently better for quantitation in the low-energy regime. An example of the analysis of transition metal sulﬁdes at 5 keV with conventional ZAF matrix corrections is given in Table 6. Signiﬁcant relative errors (overestimates) are found in the measurement of the metal components; consequently, the totals are anomalously high. By comparison, under conventional beam energy analysis (20 keV), the relative errors are less than 2% for this system. A particular problem in performing analysis at low beam energy is the actual structure of the specimen being measured. Most specimens have a surface layer that is diﬀerent from the interior due, for example, to surface oxidation, a conductive coating applied to minimize charging, or contamination. At high beam energy, this surface layer

864

Small et al.

Figure 41 (a) SiO2 excited with a beam energy of 5 keV and measured with a Si EDS (150-eV FWHM at MnKa). (b) SiO excited with a beam energy of 5 keV and measured with a Si EDS (150 eV FWHM at MnKa).

Electron-Induced X-ray Emission 865

Figure 42

GeO2 excited with a beam energy 5 keV and measured with a Si EDS (150 eV FWHM at MnKa).

866 Small et al.

Figure 43 CaF2 excited with an incident electron beam energy of 3 keV and measured with a Si EDS (150 eV at MnKa).

Electron-Induced X-ray Emission 867

Figure 44 BaCO3 with minor Sr ( 0.05) excited at 5 keV and measured with a Si EDS (150 eV at MnKa). The BaM and BaL lines are marked. Note the low peak intensity for the BaM series; the BaL lines are not excited.

868 Small et al.

Electron-Induced X-ray Emission

869

Figure 45 Selection of x-ray peaks available for analysis with E0 ¼ 5 keV for conventional EDS. Note that elements in black squares are not sufficiently excited to be practically measured.

Figure 46 Selection of x-ray peaks available for analysis with E0 ¼ 2.5 keV for conventional EDS. Note that elements in black squares are not sufficiently excited to be practically measured.

870 Table 6

Small et al. Low Beam Energy (5 keV) Analysis of Metal Sulﬁdes

FeS (meteoritic troilite) Fe S CuS Cu S ZnS Zn S

True

Analyzed

Rel. error (%)

0.635 0.365

0.667 0.365

þ5.0 þ0

0.665 0.335

0.730 0.334

þ9.8 0.2

0.671 0.329

0.732 0.320

þ9.1 2.7

thickness is a small fraction of the excitation range, so the eﬀect on quantitation is negligible. However, at low beam energy, the total sampling depth becomes so shallow that a surface layer constitutes a signiﬁcant fraction of the analyzed depth. The specimen should thus be considered as a case of ‘‘special geometry,’’ that of an overlayer of C (or oxide, etc.) on a substrate that is actually the specimen composition of interest. Rigorous testing for the accuracy of low-beam-energy microanalysis is similarly aﬀected by the issue of proper test specimens. The situation is diﬃcult enough at high beam energy, where the lateral homogeneity is the key issue. At low beam energy, the measurement becomes profoundly sensitive to the surface region of both the specimen and the standards, especially if these are diﬀerent. Thus, any testing of quantitation procedures can only sensibly proceed when a suitable series of test specimens and standards has been produced that do not suﬀer from limitations imposed by the quality of the surface. C. What Future Developments Are Possible? The physics of electron-excited x-ray production and propagation is determined by the choice of beam energy, the element(s) to be measured, and the matrix composition. Once the analyst’s room to maneuver is restricted through the choice of low beam energy, the only possible route to improve the measurement process is to improve the x-ray spectrometry. From Eq. (72) and the spectra presented in Figures 41–44, it is clear that improving the spectral resolution will increase the peak-to-background ratio, thereby improving the visibility of low-ﬂuorescence-yield peaks, reducing interferences, and lowering the limit of detection. Increasing the detector eﬃciency (i.e., solid angle) and reducing the pulse processing time will improve the peak counting rate, also contributing to improving the limit of detection. 1. Improvements to Wavelength-Dispersive Spectrometry Wavelength-dispersive x-ray spectrometry, based on Bragg diﬀraction of x-rays from a crystal, is the oldest form of x-ray spectrometry and is commonly incorporated in the instrument conﬁguration referred to as the electron-probe x-ray microanalyzer. The WDS is a high-resolution (10 eV or less), ‘‘single-channel,’’ narrow-bandpass spectrometer that must be mechanically scanned to cover a range of x-ray wavelengths (energy). The higher resolution of the WDS results in a higher P=B, yielding a lower limit of detection than semiconductor EDS. Typically, the WDS limit of detection is 1074–1075 mass fraction.

Electron-Induced X-ray Emission

871

The high spectral resolution has the added advantage of diminishing the likelihood of spectral interferences, although interferences may still occur for certain combinations of x-ray lines. However, the quantum eﬃciency of the WDS is signiﬁcantly lower than the EDS due to the relative ineﬃciency of the diﬀraction process. Consequently, beam currents of the order of 100 nA or more are normally used to achieve adequate peak counting rates. Such current requirements are incompatible with small beam size, even with the thermally assisted ﬁeld-emission gun. Additionally, when low-beam-energy excitation is considered, another problem becomes signiﬁcant. Low-energy x-rays are emitted as a result of electron transitions involving electron energy levels whose energy and population can be altered due to chemical bonding eﬀects. These chemical bonding eﬀects are manifested in the x-ray spectrum as shifts in the peak position and shape (Bastin and Heijligers, 1991). Because the WDS only detects a narrow band of energy, for an accurate measurement of a peak that is subject to alteration due to chemical bonding eﬀects, it is necessary to scan across the peak to accurately determine its position and the total intensity under the peak above background. This requirement for scanning further reduces the measurement eﬃciency and increases the need for beam current. Thus, WDS at low beam energy is normally performed under conditions that result in poorer spatial resolution than EDS. Two developments may lead to improved WDS eﬃciency, which can beneﬁt lowbeam-energy x-ray microanalysis. First, the WDS is a focusing device, wherein the x-rays diﬀracted at various points along a curved crystal are brought to a focus within the entrance slit of a gas proportional detector placed on the focusing circle of the spectrometer (‘‘Rowland circle’’). If the focused rays are allowed to diverge beyond the Rowland circle, they form a spatially dispersed image of the peak (with a ‘‘single focusing’’ geometry WDS). By placing an imaging detector array oﬀ the Rowland circle in the dispersion plane, a wider energy window could be simultaneously measured, providing a direct image of the full peak and adjacent background regions (Fiori et al., 1991). Second, the eﬃciency of the WDS can be improved by increasing its solid angle of collection. For low-energy x-rays (0–1 keV), a parabolic reﬂective x-ray optic has been described that gathers a wide solid angle and presents a parallel beam to a ﬂat crystal diﬀractor that scatters x-rays into a large-window proportional gas detector. With this augmented WDS, limits of detection as low as 105 mass fraction have been reported for light elements, such as C (Agnello et al., 1997). 2. Microcalorimetry Energy-Dispersive Spectrometry The ideal x-ray spectrometer would have the resolution of the WDS combined with the energy-dispersive character of EDS over the useful analytical range, 100 eV–10 keV. This seemingly impossible requirement has actually been realized with the development of a practical ‘‘fast’’ microcalorimeter EDS at NIST, Boulder (Wollman et al., 1998). In the microcalorimeter spectrometer (see also Sec. III.F.4 of Chapter 3), the energy of an individual photon is measured as the temperature rise in an absorbing target maintained near 100 mK. The NIST microcalorimeter design incorporates critical features such as the use of an absorber maintained in the normal conducting state and a temperature sensor based on a superconducting transition sensor which both contribute to a faster pulse response than earlier microcalorimeters. A novel superconducting quantum interference device (SQUID) is used as a stable, fast, high-gain ampliﬁer. With regard to analysis, the chief operating characteristics of the NIST microcalorimeter EDS are the following:

872

Small et al.

1.

2. 3.

Energy resolution < 10 eV. As an example of the energy resolution applied in a diﬃcult analytical situation, Figure 47 shows the separation of the BaL lines from the TiK lines in BaTiO3 achieved with the NIST microcalorimeter EDS and compares it with the unresolved peaks in the spectrum obtained with a semiconductor EDS. Limiting count rate (with beam blanking) 1 kHz. Photon energy coverage: 250 ev–10 keV. An example of this energy coverage is given in Figure 48, which gives a spectrum for YBa2Cu3O7 obtained with the microcalorimeter EDS, showing L line detection for Cu, Y, and Ba and K line detection for O. Carbon K shell x-rays at 0.282 keV have been successfully measured, as shown in Figure 49.

The NIST microcalorimeter EDS represents the leading edge of a revolution in x-ray spectrometry. In particular, microcalorimeter EDS technology should have a major and immediate impact on the application of the FEG–SEM to microanalysis at low beam

Figure 47 Comparison of spectra of BaTiO3 obtained with the microcalorimeter EDS (FWHM ¼ 10 eV) and with the semiconductor EDS (FWHM ¼ 140 eV).

Electron-Induced X-ray Emission

Figure 48

873

Microcalorimeter EDS spectrum of YBa2Cu3O7; E0 ¼ 10 keV.

Figure 49 Microcalorimeter EDS spectrum of carbon; logarithmic scale. Note the carbon coincidence peak, and the discontinuities in the background due to absorption in the spectrometer materials (Al window thermal protection coating and AG absorber).

874

Figure 50

Small et al.

Microcalorimeter EDS spectrum of WSi2; E0 ¼ 5 keV.

energy, for example, in the analysis of particles during defect review in simiconductor processing. As an example, Figure 50 shows the spectrum obtained at 5 keV from WSi2 with a 10-eV microcalorimeter EDS. The separation of the SiKa line from the WM family is virtually complete, thus eliminating the severe interference encountered with semiconductor EDS. The high P=B ratios evident in the spectra in Figures 47–50 will result in limits of detection similar to WDS. Finally, the resolution of the microcalorimeter EDS will make it possible to actually detect low-energy L and M lines that have low ﬂuorescence yield. As an example, Figure 51a demonstrates the detection of several of the lowyield peaks of the titanium L family and the barium M family in BaTiO3, which can be compared to Figure 51b, obtained with semiconductor EDS. With conventional EDS, the TiL peaks are unresolved from the OK peak, and the BaM peaks, although visible, are actually convolved with the BaM edge discontinuities in the background.

" Figure 51 (a) Microcalorimeter EDS spectrum of BaTiO3, low-energy region. Note detection of BaM-family members; E0 ¼ 10 keV. (b) Si EDS spectrum of BaTiO3, low-energy region. Note detection of BaM-family members; E0 ¼ 10 keV.

875

876

IV.

Small et al.

ANALYSIS OF SAMPLES WITH NONSTANDARD GEOMETRIES

As mentioned earlier, in classical electron-probe microanalysis, the samples are polished ﬂat, homogeneous, and opaque with respect to the electron beam. Although these constraints on sample geometry enable the analyst to perform high-quality quantitative analysis on a wide variety of samples, there are many samples which cannot be prepared with the classical geometry. These samples include, among others, layered samples, thin sections from samples that have inclusions ore phase domains smaller than the excitation volume of a 10–30-keV electron beam, particles, and samples with rough surfaces. To obtain some degree of quantitative information from these samples, the analyst must modify the analytical schemes outlined in the previous sections. A.

Quantitative Analysis of Layered Specimens

In the early days of electron microbeam analysis, characterization of specimens that were compositionally nonuniform within the analytical volume (the volume ‘‘excited’’ by the primary electron beam), such as particles, thin ﬁlms, or layered specimens, was either not attempted or involved use of simple empirical correction procedures. For example, it was noted that the thickness of thin ﬁlms (particularly single-element ﬁlms) on a common substrate could be determined by measuring two or more ﬁlms of the same composition as the unknown with diﬀerent known thicknesses and then using the results to construct a linear calibration curve [e.g., Sweeney et al. (1960), Bolon and Lifshin (1973)]. This procedure has reasonable accuracy as long as the ﬁlms are quite ‘‘thin’’ with respect to x-ray absorption and electron penetration and if the standards’ substrate compositions are identical or very similar in atomic number to those for the samples. As microprobe instrumentation and correction procedures continued to develop, the importance of considering the eﬀect of surface coatings (such as surface oxide layers or the thin conductive ﬁlms evaporated onto insulating samples) on analytical results obtained at low accelerating potentials was noted [e.g., Sweatman and Long (1969)]. At the same time, thin-ﬁlm analysis progressed by using variable accelerating potential analysis to estimate the thickness of the ﬁlms and by using low accelerating potential analysis conditions to analyze only the surface ﬁlm [e.g., Hutchins (1966), Colby (1968), Warner and Coleman (1973)]. With the development of increasingly more accurate algorithms expressing the relative production of x-ray intensity as a function of depth in the sample, it is possible to derive theoretical correction methods that enable accurate quantitative analysis of thinﬁlm or layered samples. Today, a number of laboratories are actively engaged in routinely analyzing complicated layered specimens and in developing and reﬁning the correction procedures that enable the quantitative determination of both layer thicknesses and compositions for such materials. A variety of through review papers have been recently written on layered-specimen analysis [e.g., Pouchou and Pichoir (1991a), Packwood and Remond (1992), Willich (1992), Pouchou (1993), Karduck, 1997]. One particularly detailed and cogent treatment of the subject can be found in the textbook by Scott et al. (1995). The two most commonly used approaches to correct for electron scattering, electron energy loss, and x-ray absorption in the quantitative analysis of thin ﬁlms and layered specimens are those based on (1) analytical expressions of the relative production of x-rays as a function of depth, f(rz) and (2) Monte Carlo modeling of electron scattering. In their typical development, neither of these approaches predicts the contribution of secondary

Electron-Induced X-ray Emission

877

x-ray ﬂuorescence, either by characteristic lines or the continuum. Several authors have separately described corrections for secondary ﬂuorescence in layered specimens that can be incorporated with either of the two approaches. The following subsections will outline the current status of these approaches and give examples of the types of results that can be obtained by their use. 1. Methods Based on Calculation of f(rz) The most commonly used procedure used to analyze thin ﬁlms or layered specimens is to measure the emitted x-ray intensity from the line(s) of interest in the sample and compare them with the emitted intensities of those lines from one or more thick, polished standards. In the simplest case, the standards employed are pure elements. In that case, when ﬂuorescence by characteristic lines and the continuum can be ignored, the relative intensity can be related to concentration and layer composition by the equation 0 1 11 Z Zrt2 Ithin kA ¼ ¼ CA f0A ðrzÞ expðw0A rzÞ dðrzÞ@ fA ðrzÞ expðwA rzÞ dðrzÞA Ibulk std rt1

0

ð74Þ fA0

fA

where and are the x-ray depth distribution functions for the layer of interest in the specimen and the bulk standard, respectively, w0A and wA are the respective absorption factors, and rt1 and rt2 are the depths from the surface of the top and bottom, respectively, of the layer of interest (i.e., rt1 ¼ 0 for the topmost layer, rt2 ¼ 1 for a bulk substrate). Only in a case in which the layer of interest is very similar in average atomic number both to its underlying (and overlying) layer(s) and to the bulk standard will f0A and fA be similar. In most cases, f0A needs to be modiﬁed from expressions developed for bulk specimens to account for the variable composition within the excited volume of the electron beam. Similarly, w0A and wA are seldom identical, and wA for all underlying layers needs to be modiﬁed to account for the diﬀerent degrees of absorption in the various overlying layers. If the order, thicknesses, and compositions of the various layers in the specimen are known, it is simple algebra to determine the absorption factor [cf. Pichou and Pichoir (1991a), Scott et al. (1995)]. However, even if the sequence of thickness and compositions were known, conventional parameterizations of f(rz) for bulk specimens cannot be used for layered specimens without signiﬁcant modiﬁcation. The fundamental diﬀerence between f0A ðrzÞ and fA ðrzÞ lies in the number and energy distribution of the electrons that scatter through the layer rz DðrzÞ; these, in turn, depend on the average atomic number, distance, and thickness of the layer(s) that lie above and below rz DðrzÞ. For example, at the top of a reasonably thick surface layer (with respect to the electron beam depth of penetration) on a substrate of considerably diﬀerent Z, both the upward traveling and downward traveling electron components of the f(rz) are similar for both fA0 ðrzÞ and fA ðrzÞ. At the bottom of a relatively thin surface layer over a bulk substrate, the downward-traveling electron component of fA0 ðrzÞ is similar to a bulk specimen of the surface ﬁlm composition, whereas the upward-traveling component is similar to a bulk specimen of the substrate composition. The situation gets increasingly complex as the layers get thinner and=or more numerous. A number of investigators have tried to come up with simpliﬁed expressions to account for the diﬀerences in electron scattering between layered samples and thick specimens, with varying degrees of success. Hutchins (1966) tried to modify the original

878

Small et al.

Castaing expression for bulk with expressions involving the initial slope of fðrzÞ curves and the backscatter coeﬃcient, whereas Colby (1968) modiﬁed Castaing’s expression with functions relating to the diﬀerences in the backscatter coeﬃcients, backsatter correction factors, and stopping powers between ﬁlm and substrate compositions [cf. Scott et al. (1995)]. Warner and Coleman (1973) modiﬁed Hutchin’s methods to apply it to biological materials, and Duzevic and Bonefacic (1978) modiﬁed his procedure to measure alloy ﬁlm compositions on elemental metal substrates. Oda and Nakajima (1973) modiﬁed Colby’s procedure and showed good results in analyzing Ag–Cu alloys on Fe–Ni substrates. None of these early attempts, however, resulted in widespread application. As more realistic models for fðrzÞ for bulk specimens were developed, further attempts were made to apply them to thin-ﬁlm and layered-specimen analysis. Reuter (1972) modiﬁed the fðrzÞ parameterization of Philibert (1963) to better ﬁt experimentally determined fðrzÞ curves and applied it to the determination of the thickness of elemental ﬁlms on single-element and multielement substrates. His fðrzÞ parameterization was a function of the electron backscatter coeﬃcient, Z. In order to account for the eﬀects of variable average atomic number in the diﬀerent layers, he introduced an eﬀective backscatter coeﬃcient, Zeff , which varied between the limiting values of the backscatter coefﬁcient of a bulk specimen of the composition of the ﬁlm, ZF , and the backscatter coeﬃcient of the bulk substrate, Zsub : nrz nrz Zeff ¼ Zsub þ ZF 1 ð75Þ n0 n0 Where nrz =n0 is the ratio of the number of electrons transmitted through layer rz compared to the number of electrons passing through an inﬁnitely thin ﬁlm at the surface of a sample and is expressed by nrz ¼ expðsrzÞ n0

or

1

4 104 Z 0:5 rz E1:7 0

ð76Þ

whichever is larger (the Lenard coeﬃcient s ¼ 4:5 105 =E1:65 0 ; Z is the average atomic number of the layer, E0 is the accelerating potential in kiloelectron volts). As noted by the author, this approximation accounts for diﬀerences between the number of backscattered electrons from ﬁlm and substrate, but not for diﬀerences in their energy distributions. Nevertheless, the author showed good agreement between calculated and experimental results for elemental ﬁlms on substrates, both having a wide range in Z. A number of investigators implemented various versions of this correction [e.g., Armstrong and Buseck (1975) in their original fðrzÞ expressions for thin ﬁlms and particles]. Love and Scott (1978) modiﬁed the ‘‘square’’ fðrzÞ model of Bishop (1974) for application to thin ﬁlms on substrates: kA ¼

Ithin 1 expðwrtÞ ¼ 1 expð2wrzm Þ Istd

ð77Þ

where Ithin and Istd are emitted intensities from a thin ﬁlm and bulk standard of the same material, t is the ﬁlm thickness, and zm is mean depth of x-ray penetration which Scott et al. (1995) proposed should be expressed by the equation of Sewell et al. (1985b): rzm ¼

rsm ½2:4 þ 0:07Z þ ð1:04 þ 0:48ZÞ= ln U0

where, in turn, rsm is the mean electron range given by

ð78Þ

Electron-Induced X-ray Emission

879

7:87E 6J 0:5 E1:5 þ 7:35E 7E02 P 0 ð79Þ Ci Zi =Ai Bushby and Scott (1993) reported results of analyses of oxide ﬁlms on metallic substrates using these corrections. The authors noted that this procedure is limited to analysis of ﬁlms on substrates of similar atomic number, as the procedure does not correct for diﬀerences in electron scattering. This correction, however, is particularly easy to calculate and is thus useful for ‘‘back of the envelop’’ calculations during experiment design. The various forms of the ‘‘modiﬁed, surface-centered Gaussian’’ fðrzÞ expression [see Eq. (46)], as originally developed by Packwood and Brown (1981), have been widely adapted for thin-ﬁlm and multilayer-specimen analysis. As described earlier, these equations have the general form rsm ¼

fðrzÞ ¼ g0 exp½a2 ðrzÞ2 ½g0 fð0Þ exp½a2 ðrzÞ2 brz

ð80Þ

with a variety of functions proposed for a; b; g0 and fð0Þ. Packwood et al. (1987) noted that for very thin ﬁlms on substrates, the eﬀect of the ﬁlm(s) on the x-ray distribution would be negligible and, thus, to a ﬁrst approximation, a fðrzÞ expression for a sample of the average atomic number of the substrate could be used for all layers. Similarly, for very thick homogeneous ﬁlms on substrates (or ﬁlms analyzed at an accelerating potential low enough so that few, if any, energetic electrons penetrate to the substrate), the fðrzÞ expression for a sample of the average atomic number of the ﬁlm could be used for both ﬁlm and substrate. The problem lies in the formulation of fðrzÞ expressions for layers of intermediate thickness between the two extremes. Packwood and co-workers (Packwood et al., 1987; Packwood, 1991; Packwood and Remond, 1992) note that most expressions for the g0 term in the Gaussian fðrzÞ equation are not functions of sample atomic number and so do not need to be modiﬁed for intermediatelayer analysis. They further note that most expressions for the a term are only weakly dependent on the atomic number of the specimen and that, in most cases, simply using an a term calculated for a sample of the average atomic number of the substrate is suﬃcient (assuming most electrons in the analysis penetrate into the substrate). They propose two alternate formulations of the b and fð0Þ terms and one formulation of the backscatter coeﬃcient, Z, to account for the change in electron scattering across layer boundaries: rd 1 rd bspecimen ¼ blayer þ bsubstrate ð81Þ 1=2a 1=2a and

fð0Þspecimen ¼ fð0Þlayer

rd 1 rd þ fð0Þsubstrate 1=2a 1=2a

ð82Þ

(where any layer of thickness greater than 1=2a is considered inﬁnite), or bspecimen ¼ blayer erf ðardÞ þ bsubstrate erfcðardÞ

ð83Þ

fð0Þspecimen ¼ fð0Þlayer erf ð2ardÞ þ fð0Þsubstrate erfcð2ardÞ

ð84Þ

Zspecimen ¼ Zlayer erf ð2ardÞ þ Zsubstrate erfcð2ardÞ

ð85Þ

and

(where the layer and substrate contributions are weighted under the electron distribution curve). In these equations, d is the thickness of the layer, r is the density, and a is the term

880

Small et al.

in Eq. (80). Using Eqs. (83)–(85), Packwood et al. (1987) [based on the original work of Packwood and Milliken (1985)] proposed the following general expressions for the emitted x-ray intensities for the various types of layered specimens: w 2 n h w i w o sqrtðpÞ g0 exp erf Isurface layer ¼ CA erf ad þ 2a 2a 2a 2a 2 wþb wþb wþb erf ad þ erf ð86Þ ½g0 fð0Þ exp 2a 2a 2a w 2 n h w i h w io sqrtðpÞ erf ad þ g0 exp erf ad0 þ Iburied layer ¼ CA 2a 2a 2a 2a 2 wþb wþb wþb 0 erf ad þ erf ad þ ½g0 fð0Þ exp 2a 2a 2a exp½d0 ðw0 wÞ w 2 n h w io sqrtðpÞ Icovered substrate ¼ CA 1 erf ad0 þ g0 exp 2a 2a 2a 2 wþb wþb 0 1 erf ad þ ½g0 fð0Þ exp 2a 2a

ð87Þ

exp½d0 ðw0 wÞ

ð88Þ

compared to their expression for conventional thick, ﬂat specimens:

w 2 h w i sqrtðpÞ Ibulk specimen ¼ CA 1 erf g0 exp 2a 2a 2a 2 wþb wþb ½g0 fð0Þ exp 1 erf 2a 2a

ð89Þ

where d is the mass thickness of the analyzed layer, d0 is the mass thickness of the overlying layer, w is the absorption term (¼ mA by specimen csc cÞ for x-rays in the analyzed area, and w0 is the absorption term by the overlying ﬁlm(s). Bastin and co-workers [e.g., Bastin et al. (1992, 1998), Bastin and Heijligers (1991)] proposed modifying their ‘‘PROZA’’ formulation of the surface-centered Gaussian f(rz) model to enable layered-specimen analysis in a manner similar to that done by Packwood and co-workers. They noted, in agreement with Packwood et al. (1987), that the f(rz) expression for a thin layer lies between the two extremes of a very thin layer, that has a f(rz) expression consistent with a composition with the surface layer elements ‘‘dissolved’’ in the substrate, and a very thick layer (with respect to electron penetration), whose f(rz) is similar to a bulk specimen having the ﬁlm’s composition. Similar to Packwood et al., they proposed modifying the expressions for the a, b, g0, and f(0) terms in the f(rz) equation for surface layers of intermediate thickness by empirical smoothing functions accounting for the diﬀerences in electron scattering among the diﬀerent layers and substrate: a ¼ abulk layer þ ðadissolved abulk layer Þ exp½1:5ðabulk layer T 106 Þ2

ð90Þ

6

b ¼ bbulk layer þ ðbdissolved bbulk layer Þ expð1:5abulk layer T 10 Þ

ð91Þ 6 2

g0 ¼ g0;bulk layer þ ðg0;dissolved g0;bulk layer Þ exp½2:0ðabulk layer T 10 Þ

ð92Þ 6 2

fð0Þ ¼ fð0Þbulk layer þ ½fð0Þdissolved fð0Þbulk layer exp½2:5½abulk layer T 10 Þ

ð93Þ

Electron-Induced X-ray Emission

881

where ‘bulk layer’ refers to a bulk specimen with the same composition as the analyzed layer, dissolved refers to the composition that would be calculated from an analysis assuming the elements in the surface layer(s) were dissolved in the substrate, and T is the layer thickness (in mg=cm2). Unlike Packwood et al., Bastin and co-workers assume that the composition of the surface layer(s) has no eﬀect on the f(rz) expression for the substrate. This latter assumption is certainly a source of error in cases of surface layers and substrates of dissimilar average atomic number analyzed under conditions where a substantial fraction of the electron trajectories occur in the surface layers. The f(rz) model of Pouchou and Pichoir [e.g., Pouchou and Pichoir (1985, 1987, 1991a), Pouchou (1993)] was initially designed to include the analysis of layered specimens. As discussed earlier, the model is treated in two parts, each of which is expressed by a parabolic function: fðrzÞ1 ¼ A1 ðrz Rm Þ2 þ B1 fðrzÞ2 ¼ A2 ðrz Rx Þ

2

ð94Þ ð95Þ

where Rm is the depth at which f(rz) is at a maximum and Rx is the x-ray range. These terms need to be modiﬁed for layered specimens, along with F, the total integral under the f(rz) curves, and f(0), the surface ionization potential. The expressions given by Pouchou and Pichoir for these terms were brieﬂy discussed earlier and can be found extensively derived in their articles. To account for layered samples of intermediate thickness, they suggested modifying these expressions in order to calculate the f(rz) for a ﬁctitious homogeneous composite sample having an appropriate average atomic number to produce electron scattering similar to that which occurs in the real multiphased sample. They utilize a weighting law of the form fðrzÞ ¼ Nðrz LÞ2 ðrz RÞ2

ð96Þ

where L and R are the low and high rz intercepts of the f(rz) expression with L < 0 and N is a normalizing factor so that the area under the f(rz) curve equals F. The various parameters are sequentially calculated using various estimates of L and R, and compositions of the diﬀerent layers are then calculated by an iterative process. For the details of this weighting procedure, see Pouchou and Pichoir (1991a). Willich (1992) proposed modifying the weighting procedure of Pouchou and Pichoir with a simple ‘‘eﬀective atomic number,’’ Zcombination, for a ﬁctitious homogeneous sample with the same electron scattering properties as the real multilayered specimen. Based on the work of Willich and Overtop (1988), he proposed rTfilm 0:65 Zcombination ¼ Zsubstrate þ ðZfilm Zsubstrate Þ ð97Þ 2rzmax where Zsubstrate and Zﬁlm are the average atomic numbers of the substrate and ﬁlm, rTﬁlm is the mass thickness of the ﬁlm, and rzmax is the x-ray range. In all of the above procedures, the layer and substrate thicknesses and compositions are calculated by an iterative procedure. Typically, elements are assigned to each of the layers; ﬁrst, estimates of the layer compositions are determined by running conventional bulk corrections and normalizing to 100%. Next, the thin-ﬁlm expressions are evaluated to determine the thickness at which the corrected k-ratio data would yield concentrations summing to 100%. Finally, using this thickness for a new estimate of layer compositions, followed by reiteration until convergence in both thickness and composition is achieved.

882

Small et al.

If elements are present in more than one layer, it is necessary to either assign proportionality of the element among the various layers on some basis or to collect data at more than one accelerating potential and try either by trial and error or by simultaneous solution of the f(rz) expressions at the various E0’s. Public-domain and commercial computer programs exist that utilize these corrections [e.g., Waldo (1988), Bastin et al. (1992), Pouchou (1993)]. 2. Methods Based on Monte Carlo Calculations The Monte Carlo procedures that have been used by numerous investigators to calculate electron deceleration trajectories in compositionally homogeneous bulk materials can be readily adapted to simulate x-ray emission from multilayer samples. The algorithms employed in conventional Monte Carlo calculations for thick specimens [e.g., Joy (1995)] do not have to be modiﬁed in any signiﬁcant way for thin ﬁlms or layered specimens. Only the boundary conditions need to be changed. In a layered specimen, each time an electron trajectory is calculated, the program need to be changed. In a layered specimen, each time an electron trajectory is calculated, the program needs to determine in which layer(s) the electron is moving and then use the appropriate scattering, emission, and absorption data for the assumed composition of that layer. The nature of the equations used for scattering, energy loss, ionization probabiliity, and so forth does not change. Unlike the case for the analytical expressions for f(rz) described earlier, no assumptions need be made regarding diﬀerences in the nature and amount of electron backscattering from the various layers; these are directly and independently calculated for each electron in the Monte Carlo method. Thus, at least to a ﬁrst approximation, the analytical accuracy of thin-ﬁlm and layeredspecimen analysis using Monte Carlo methods is comparable to that for bulk specimens. Calculations of x-ray emission from Monte Carlo calculations are made by determining a f(rz) curve based on the number, directions (path lengths), and energies of electrons that are calculated to pass through the layer rz D(rz): fðrzÞ ¼

CA oA NAV AA

ZEc

nðE; rzÞQA ðEÞ

dðrzÞ dðrsÞ dE dðrsÞ dE

ð98Þ

E0

[e.g., Armstrong (1991)]. For a multilayer specimen, a series of f(rz) functions are calculated, one for each element of interest in each layer. These are constructed by keeping track of the arrays of summed electron path lengths through each rz layer corresponding to the diﬀerent array components of energy. For details of the procedures, consult Joy (1995) or Armstrong (1991). One of the ﬁrst groups to utilize Monte Carlo calculations for thin-ﬁlm analysis was Bolon and Lifshin (1973). Utilizing the multiple scattering Monte Carlo method of Curgenven and Duncumb (1971), they produced calculated calibration curves of relative intensity versus thickness for thin elemental ﬁlms on elemental substrates (e.g., Au on Si) and showed good agreement with experimental data. Kyser and Murata (1974, 1976) used the single-scattering Monte Carlo model of Murata et al. (1971) to construct similar calibration curves for both elemental and compound (alloy) ﬁlms on elemental and compound targets. Using the examples of MnxBiy and CoxPty ﬁlms on SiO2, they were able to show self-consistent agreement in both concentration and ﬁlm thickness between measured and Monte Carlo data. Numerous groups more recently have used Monte Carlo calculations to predict emitted x-ray intensities from a wide variety of layered samples, including thin surface

Electron-Induced X-ray Emission

883

coatings, thin and thick multiple-layer structures, surface layers containing implanted species, surface and depth diﬀusion proﬁles, and surface-layer analysis during ion sputtering [e.g., Amman and Karduck (1990), Amman et al. (1992), Karduck and Rehback (1991), Armstrong (1991, 1993), Armigliato et al. (1996), Valamontes and Nassiopolos (1996), Karduck (1997), Chan and Brown (1997a, 1997b)]. These studies have emphasized the advantage of analyzing the same specimens at diﬀerent analytical conditions (such as electron beam accelerating potential and incidence angle) in order to elucidate their structure. If there are no elements in common in the diﬀerent analyzed layers, then each of these measurements is an independent estimate of layer thickness and composition. If there are elements that are present in more than one layer, then multiple measurements are essential to deconvolve the composition and structure. Joy (1995) and co-workers and Gauvin and co-workers (e.g., Hovington et al., 1997a,b; Drouin et al., 1997) have developed public-domain Monte Carlo programs that are easily adapted to the analysis of layered specimens. 3. Correction for Secondary Fluorescence All of the above discussion of quantitative analysis of layered specimens has only considered the case of electron excitation of inner-shell ionization resulting in production of characteristic x-rays. However, characteristic x-rays can also be produced by ﬂuorescence from any absorbed x-ray photons (characteristic or continuum=bremsstrahlung) having energies greater than the inner-shell ionization potential. These ﬂuorescing x-rays can be produced from any of the layers in the specimen. Because the range of energetic x-rays typically greatly exceeds that of the beam electrons, x-rays may be ﬂuoresced in layers (e.g., the substrate) that are considerably deeper than the electron range. As noted above, ﬂuorescence by characteristic or continuum x-rays can contribute up to 60% or 70% of the total x-ray emission from bulk specimens in extreme cases. Typically, secondary ﬂuorescence contributes to 10% or less of the emitted intensity. In the case of layered specimens, the amount of secondary ﬂuorescence will depend on the thicknesses and relative positions of the layers containing the ﬂuoresced and ﬂuorescing species. The typical x-ray mean free path between absorption=ﬂuorescence events is much greater than the typical electron mean free path between inelastic scattering=inner-shell ionization events. As a result, the relative contribution from secondary ﬂuorescence events will be small if either the ﬂuorescing or ﬂuoresced x-ray line is from even a major element in a very thin layer. The greatest contribution from secondary ﬂuorescence occurs for a minor element in a thick layer from a major element in that or another thick layer or substrate. Thus, in many applications, just as in the case of particles, correction for secondary ﬂuorescence can be safely ignored. Several investigators have proposed corrections for ﬂuorescence by characteristic lines and=or the continuum in layered specimens. Armstrong and Buseck (1985) derived a rigorous, theoretically based general expression for a variety of layer–substrate conditions. This expression requires numerical integration in multiple dimensions and is too lengthy to show here. The reader is referred to the original paper for the details. Waldo (1991) simpliﬁed these equations and adapted them for use in his public-domain thin-ﬁlm analysis software (Waldo, 1988). Cox et al. (1979) developed a simpliﬁed characteristic ﬂuorescence correction for multielement thin ﬁlms on multielements substrates. Their method was based on the assumptions that the ﬁlm was thin with respect to the mean depth of x-ray generation, and that the x-rays were generated isotropically from a point source located in the substrate at a distance from the ﬁlm–substrate boundary equal to the mean depth of

884

Small et al.

x-ray production. The details of the procedure can be found in the original paper and a good summary can be found in the original paper and a good summary can be found in Scott et al. (1995). Armigliato et al. (1982) incorporated this procedure into there thin ﬁlm correction and reported improved results in selected cases. However, by its nature, it is inappropriate for the analysis of thick ﬁlms on substrates or multiple ﬂuorescing layers. More recently, Pouchou and Pichoir (1991b) proposed corrections for ﬂuorescence by both characteristics lines and the continuum in layered specimens. The authors make some simplifying assumptions regarding the depth distribution of the primary x-rays and relative line intensities. Their derivation is beyond the scope of this book, and the reader is once again referred to the original paper for the details. Their corrections are employed in the commercial versions of their program. The continuum ﬂuorescence correction has been integrated as well into the public-domain program of Walso (1988, 1991). a. Analytical Results: Testing the Methods X-ray emission analysis, particularly involving wavelength-dispersive detectors at ‘‘microprobe’’ beam currents (i.e., 1–1000 nA) can produce thin-ﬁlm analysis results of very high precision. Detection limits of less than 10% of an atomic monolayer can be measured [e.g., Packwood and Remond (1992)]. Film thickness variations can be measured at a level of 0.1 nm. Compositional variations of 0.2% for major and minor element concentrations at a level of 100 ppm can be detected, under favorable conditions, in ﬁlms as thin as 10 nm. (For examples of the measurement sensitivities and detection limits, see the review papers of Pouchou and Pichoir (1991b), Packwood and Remond (1992), Willich (1992), Pouchou (1993), and Karduck (1997)]. Thus, x-ray emission analysis provides an important complementary tool to the more ‘‘conventional’’ techniques used in thin-ﬁlm and layeredspecimen characterization, such as Auger electron spectrometry (AES), Rutherfordbackscattering spectrometry (RBS), and secondary ion mass spectrometry (SIMS) [e.g., Cazaux (1992)] and, in certain circumstances, approaches or exceeds their detectability. X-ray emission analysis has the potential to be a more accurate, routine technique for quantitative determination of thin-ﬁlm=layered-specimen composition (and, in some cases, layer thickness) than techniques like AES, RBS, and SIMS. However, a serious problem in evaluating the comparative accuracy of thin-ﬁlm analysis with any of these techniques is the lack of certiﬁed thin-ﬁlm standards (both in terms of composition and thickness). Most of the papers cited above that have proposed quantitation schemes for ﬁlm analysis or reviewed the applications of thin ﬁlm=layered specimen analysis showed examples of the accuracy’’ or consistency=plausibility of the technique by results of analyses of laboratory samples of unknown composition and thickness. In some cases, the samples were analyzed and thicknesses were determined by another analytical technique. In other cases, an expected range of composition and thickness was stated and the results compared to that range. In no case that we have discovered, have thin ﬁlm or layered specimens of known, certiﬁable pedigree (i.e., having been analyzed and had thicknesses measured by more than one analytical procedure that also included analyses of certiﬁed reference materials for which limits of accuracy could be given) been used to evaluate the proposed correction procedures. In most instances, the ﬁlm=layer composition is assumed from the starting materials and the thickness may be measured by wet chemistry or RBS. Many of the papers containing raw thin-ﬁlm and layered-specimen data that have been used to formulate and evaluate the analytical procedures discussed above date back to when RBS was a new technique with relatively large uncertainties and when evaporation and sputtering techniques for thin ﬁlm manufacture did not have nearly the precision and accuracy that exists today.

Electron-Induced X-ray Emission

885

Although the necessary published data does not exist to make a precise evaluation of the relative accuracy of the various proposed thin-ﬁlm analysis techniques, we present several examples of published raw data from the literature processed through various of the correction procedures we have discussed to give an idea of the variation of the processed results. Three types of thin ﬁlm=layered specimen analysis are given. Table 7 presents the results of thickness determinations for monoelemental thin ﬁlms deposited on monoelemental substrates, taken from the raw data of Reuter (1972) and processed through various thin-ﬁlm correction procedures. The procedures used for this table are the f(rz) correction procedures of Reuter (1972), Pouchou and Pichoir (1991a) [PAP], Bastin et al. (1986), Bastin and Heijlingers (1990) [both using implementation of

Table 7

Experimental Data of Reuter (1972) Processed Through GMRFilm and CASINO

Film and x-ray line: Substrate:

AlKa

AlKa

B

B

Accelerating potential (keV) 12 12 Film thickness (mg=cm2) Chemical measurement 14.5 4.4 Thin-film corrections Reuter (1972) phi(PZ) 14.2 4.3 Monte Carlo (CASINO) PAP w contin. fluor. corr. 13.0 4.2 PAP w=o contin. fluor. corr. 13.0 4.2 Bastin (1986) 12.3 3.7 Bastin (1990) 13.2 4.2 Packwood and Brown 11.6 3.9 Mean 12.9 4.0 Std. dev. 0.9 0.2 % Sigma 6.8% 5.7% Mean–chem. 11.1% 8.2%

Be

Fe

AlLa Au

30

30

30

30

4.4

73.0

73.0

73.0

77

4.3

75.0

83.0

85.0

193

94 198 100 207 4.2 81.3 79.0 73.5 109 233 4.2 81.3 79.0 73.5 103 217 4.4 79.2 80.4 79.9 102 224 4.7 82.1 79.1 59.6 107 225 4.8 72.5 74.7 74.2 (138) (303) 4.5 79.3 78.4 72.1 104 221 0.3 3.9 2.2 7.5 3.7 9.8 6.3% 5.0% 2.8% 10.4% 3.6% 4.4% 1.4% 8.6% 7.5% 1.2% 35.3% 14.6% CuLa

Au

Pt

4

CuKa

Film and x-ray line:

AlKa

SiO2

AuLa

Substrate:

SiO2

Au

Pt

Accelerating potential (keV) Film thickness (mg=cm2) Chemical measurement Thin-film corrections Reuter (1972) phi(PZ) Monte Carlo (CASINO) PAP w contin. fluor. corr. PAP w=o contin. fluor. corr. Bastin (1986) Bastin (1990) Packwood and Brown Mean Std. dev. % Sigma Mean–chem.

30

30

10

10

15

18

22

30

83.0

83.0

83.0

83.0

26.0

26.0

26.0

26.0

72.0

85.0

71.0

82.0

23.0

23.0

24.0

23.0

89.2 85.3 82.9 86.5 84.7 83.4 6.0 7.2% 0.5%

82.4 82.3 89.1 82.1 96.0 86.4 6.1 7.1% 4.1%

77.6 77.6 78.1 86.1 80.8 80.0 3.6 4.5% 3.6%

86.4 86.4 93.0 73.8 95.9 87.1 8.5 9.8% 4.9%

26.4 25.0 26.5 26.1 30.6 26.9 2.1 8.0% 3.5%

24.5 23.4 24.9 24.2 26.5 24.7 1.1 4.6% 5.0%

26.6 25.4 26.6 25.9 26.8 26.3 0.6 2.2% 1.0%

26.1 25.1 26.0 25.2 24.1 25.3 0.8 3.2% 2.7%

886

Small et al.

Bastin et al. (1992)], Packwood (1991), and the Monte Carlo procedure of Gauvin and co-workers (Hovington et al., 1997a, 1997b; Drouin et al., 1997) (CASINO). The results of the Reuter procedure were taken from the article. The Monte Carlo results were determined by developing calibration curves of kel versus thickness from simulations using the public-domain CASINO program of Gauvin et al. The other procedures were tested using a modiﬁed version of the public-domain GMRFilm program of Waldo (1991). Examples are chosen to include cases of elements of very dissimilar atomic number as ﬁlm and substrate, cases of both soft and hard x-ray lines, and cases of variation of accelerating potential. None of the cases shown involve a large correction for ﬂuorescence either by the continuum or by characteristic lines. The results show (with the exception of a few outliers) an agreement of about 10–15% relative among the thicknesses calculated by the various correction procedures and an agreement of about 5–10% relative between the mean corrected value and the thickness determined by wet chemistry. Table 8 shows the results of the next more complicated type of thin-ﬁlm analysis, multielement thin ﬁlms on a multielement substrate of dissimilar average atomic number. The data were taken from the work of Kyser and Murata (1974). The data were processed through the same correction procedures as used for Table 7, except the Monte Carlo results of Kyser and Murata were used instead of the CASINO program and the early f(rz) expression of Reuter was not evaluated. The alloy ﬁlms deposited on SiO2 substrates were of composition MnxBiy and CoxPty. To conserve space, only the Mn and Co concentrations are shown in the Table 8; the Bi and Pt concentrations can be determined by difference from 100%. As was the case in Table 7, the PAP expression was evaluated with and without a continuum ﬂuorescence correction; all other expressions were evaluated without correction for continuum ﬂuorescence. The data in Table 8, indicates only a minor diﬀerence in calculated thickness and composition (typically 1%) due to continuum ﬂuorescence in these cases. The agreement among the various corrections in the alloy compositions is better than 5% relative; the agreement in thickness is better than 8%. The agreement in the ﬁlm composition between the mean of the correction procedures and the RBS measured value is better than 5%; the agreement in thickness is typically better than 15%, perhaps reﬂecting the larger uncertainties in these early RBS thickness measurements. Table 9 gives the results of a still more complicated type of analysis, two multielement layers of dissimilar average atomic number over a monoelemental layer with an atomic number signiﬁcantly diﬀerent from both of the layers. The data were taken from the work of Pouchou (1993). The data were processed through the same corrections as employed for Table 8, except the STRATA data from the article is given instead of results from the Monte Carlo expressions of Kyser and Murata. (STRATA, as described in the article, is a commercial program that does a best-ﬁt determination of data collected at multiple accelerating potentials using the PAP correction method.) The case studied is a Nl–Cr ﬁlm deposited on a layer of Fe–Gd–Pt that has been deposited on Si. There are small corrections for characteristic ﬂuorescence but no signiﬁcant continuum ﬂuorescence eﬀects. Data have been collected at three diﬀerent accelerating potentials to show the eﬀect of electron range. In most cases, agreement among the various models for data collected at a given E0 exceeds that for the same correction procedure at varying E0 ’s. Agreement in both calculated composition and thickness among the various procedures at a given accelerating potential is typically similar to those measured in Table 8. The agreement between RBS and calculated compositions and thickness is typically better than that observed in Table 8, possibly reﬂecting improvements in the accuracy of RBS measurements in this later work.

Electron-Induced X-ray Emission Table 8

887

Experimental Data of Kyser and Murata (1974) Processed Through GMRFilm Mn–Bi ﬁlms X115-3

rt

C–Mn RBS measurement Thin-film corrections: Monte Carlo (Kyser & Murata) PAP w continuum fluorescence correction PAP w=o continuum fluorescence correction Bastin (1986) Bastin (1990) Packwood and Brown Mean Std. dev. % Sigma Mean–RBS

X115-5

X115-7

rt

C–Mn

C–Mn

rt

26.6

48.2

53.5

33.0

80.1

26.2

25.5 26.5 26.3 26.8 26.8 26.4 26.4 0.5 1.8% 0.8%

53.0 52.1 51.4 49.0 51.8 53.5 51.8 1.6 3.0% 7.5%

51.0 52.1 51.8 52.7 52.5 51.7 52.0 0.6 1.2% 2.9%

38.0 37.2 36.5 35.4 36.7 37.6 36.9 0.9 2.5% 11.8%

80.0 80.0 79.9 80.5 80.3 79.7 80.1 0.3 0.4% 0.0%

29.0 28.6 27.9 27.6 28.3 28.5 28.3 0.5 1.8% 8.1%

Co–Pt Films 10-2 C–Co RBS measurement Thin-film corrections: Monte Carlo (Kyser & Murata) PAP w continuum fluorescence correction PAP w=o continuum fluorescence correction Bastin (1986) Bastin (1990) Packwood and Brown Mean Std. dev. % Sigma Mean–RBS

13-2

rt

19.4

49.0

18.5 19.0 18.6 19.0 19.0 19.6 18.9 0.4 2.1% 2.4%

53.0 52.5 51.7 49.5 52.3 54.0 52.2 1.5 2.9% 6.5%

C–Co 9.6 10.1 10.0 9.7 10.1 9.9 10.3 10.0 0.2 2.0% 4.3%

Mn ﬁlm

Bi ﬁlm

rt

rt

rt

33.6

40.1

48.6

40.0 39.5 39.1 37.6 39.0 40.4 39.3 1.0 2.5% 16.9%

42.0 41.2 40.1 39.6 41.3 41.2 40.9 0.9 2.2% 2.0%

54.0 54.7 54.1 51.3 54.3 56.6 54.2 1.7 3.1% 11.5%

Note: Concentration in element weight %, ﬁlm thickness (rt) in mg=cm2. Analyses performed at 20 keV, with psi ¼ 52.5. MnKa and CoKa, BiMa and PtMa lines measured. Films are on SiO2 substrates.

Close examinations of Tables 7–9 reveals diﬀerences in the levels of agreement of the various correction schemes, both with respect to the independent measurement data and with respect to ‘‘internal consistency’’ (e.g., giving the same answer for data collected at diﬀerent accelerating potentials) that might be used to choose one procedure over another as providing superior accuracy. However, due to the absence of independently analyzed and certiﬁed reference standards as well as to the absence of comparable raw data collected by more than one laboratory on these samples, no ‘‘best’’ correction scheme can be unambiguously determined. The internal agreement of the various procedures, however, is generally reasonably good—typically at a level of about 5% relative. This level is approximately twice what might be expected for these types of materials if they were

888 Table 9

Small et al. Experimental Data of Pouchou (1993) Processed Through GMRFilm Layers on Si Layer 1

E0

20 keV

25 keV

30 keV

Means

RBS meas. STRATA pgm Strata–RBS PAP Bastin (1990) Bastin (1986) Packwood Mean Std dev. % Sigma Mean–RBS PAP Bastin (1990) Bastin (1986) Packwood Mean Std dev. % Sigma Mean–RBS PAP Bastin (1990) Bastin (1986) Packwood Mean Std dev. % Sigma Mean–RBS PAP Std dev. % Sigma Mean–RBS Bastin (1990) Std dev. % Sigma Mean–RBS Bastin (1986) Std dev. % Sigma Mean–RBS Packwood Std dev. % Sigma Mean–RBS

Layer 2

C–Ni

C–Cr

T (nm)

C–Fe

C–Gd

C–Pt

T (nm)

14.4 14.7 2.1% 15.0 15.2 15.5 16.1 15.5 0.5 3.1% 7.3% 14.6 15.4 14.7 15.5 15.1 0.5 3.1% 4.5% 14.3 13.5 14.4 15.0 14.3 0.6 4.3% 0.7% 14.6 0.4 2.4% 1.6% 14.7 1.0 7.1% 2.1% 14.9 0.6 3.8% 3.2% 15.5 0.6 3.5% 7.9%

85.6 85.4 0.2% 85.0 84.8 84.5 83.8 84.5 0.5 0.6% 1.3% 85.4 84.6 85.3 85.5 85.2 0.4 0.5% 0.5% 85.7 86.5 85.6 85.0 85.7 0.6 0.7% 0.1% 85.4 0.4 0.4 0.3% 85.3 1.0 1.2% 0.4% 85.1 0.6 0.7% 0.5% 84.8 0.9 1.0% 1.0%

68.3 67.1 1.8% 70.8 72.2 69.8 73.1 71.5 1.5 2.0% 4.6% 69.6 68.3 69.3 67.4 68.7 1.0 1.5% 0.5% 68.8 67.2 68.3 64.0 67.1 2.2 3.2% 1.8% 69.7 1.0 1.4% 2.1% 69.2 2.6 3.8% 1.4% 69.1 0.8 1.1% 1.2% 68.2 4.6 6.7% 0.2%

51.4 52.0 1.2% 51.0 51.4 52.3 50.6 51.3 0.7 1.4% 0.1% 50.8 51.8 51.0 49.9 50.9 0.8 1.5% 1.0% 51.1 51.9 51.2 49.9 51.0 0.8 1.6% 0.7% 51.0 0.2 0.3% 0.8% 51.7 0.3 0.5% 0.6% 51.5 0.7 1.4% 0.2% 50.1 0.4 0.8% 2.5%

28.6 28.7 0.3% 29.1 29.3 27.8 30.4 29.2 1.1 3.7% 1.9% 29.8 28.9 29.9 31.3 30.0 1.0 3.3% 4.8% 29.3 28.5 29.4 30.7 29.5 0.9 3.1% 3.1% 29.4 0.4 1.2% 2.8% 28.9 0.4 1.4% 1.0% 29.0 1.1 3.8% 1.5% 30.8 0.5 1.5% 7.7%

20.0 19.3 3.5% 19.9 19.3 19.9 19.0 19.5 0.5 2.3% 2.4% 19.6 19.6 19.4 19.4 19.5 0.1 0.6% 2.5.% 19.6 19.6 19.4 19.4 19.5 0.1 0.6% 2.5.% 19.7 0.2 0.9% 1.5% 19.5 0.2 0.9% 2.5% 19.6 0.3 1.5% 2.2% 19.3 0.2 1.2% 3.7%

24.6 24.2 1.6% 24.6 26.3 22.7 27.2 25.2 2.0 7.9% 2.4% 24.1 22.5 24.8 25.0 24.1 1.1 4.7% 2.0% 24.2 22.7 24.4 24.0 23.8 0.8 3.2% 3.2% 24.3 0.3 1.1% 1.2% 23.8 2.1 9.0% 3.1% 24.0 1.1 4.7% 2.6% 25.4 1.6 6.4% 3.3%

Note: Concentration in element weight %, layer thicknesses in nm. Continuum ﬂuorescence correction applied in all cases. Lines analyzed: NiKa, CrKa, FeKa, GdLa, PtMa.

Electron-Induced X-ray Emission

889

prepared and analyzed as bulk specimens. The level of agreement between the corrected electron microbeam data and that determined by other techniques ranges from about 5% to 15% relative and certainly includes a considerable component of uncertainty from the comparative technique. In both cases, the level of agreement in the layer composition appears to be somewhat better than the level of agreement in the layer thickness. The data certainly suggest that electron microbeam analysis has the potential for determining thinlayer compositions and thickness with comparable accuracy to the more conventionally employed ‘‘surface’’ techniques, particularly when appropriate certiﬁable standards become available for correction procedure evaluation.

B.

Quantitative Analysis of Thin Specimens at High Voltages

The quantitative analysis of ‘‘thin samples’’ such as thin ﬁlms, thin sections of samples that are inhomogeneous on a submicrometer scale, or small particles with dimensions less than about 0.5 mm represents a unique class of analysis that must be considered separately from conventional probe analysis. Unlike conventional electron-opaque samples, this class of sample is best analyzed in the analytical electron microscope by a high-energy electron beam with an accelerating potential in excess of 100 kV. If the sample is thin enough, there is minimal electron backscattering and the electrons lose very little of their energy in the sample (Goldstein, 1979). As a result, the atomic number correction kZ can be neglected and the generated x-ray intensity, fDrt a , from a characteristic line of element a in an isolated thin ﬁlm of mass thickness Drt can be calculated from Eq. (99) according to Williams and Goldstein (1991): oa aa Qa ¼ N ð99Þ fDrt Ca ðDrtÞ a Aa where N is Avagadro’s number, Ca is the weight fraction of element A, oa is the ﬂuorescence yield for the analytical line of element A, aa is the fraction of total K, L, or M line intensity measured, Aa is the atomic weight of element A, Qa is the ionization cross section of the analytical line of element a, and t is the ﬁlm thickness. In addition to a simpliﬁed generation expression, if the sample can be approximated as an inﬁnitely thin ﬁlm, the eﬀects of x-ray absorption and ﬂuorescence can be neglected. This condition is often referred to as the ‘‘thin-ﬁlm approximation’’ and makes it possible to express the relationship between the measured Ia and the generated x-ray intensities for element a by Eq. (100): Ia ¼ fDrt a ea

ð100Þ

where ea is an eﬃciency factor related to the overall eﬃciency of the Si(Li) detector for the detection of x-rays from element a (Goldstein et al., 1986). Because the determination of sample thickness at each analytical point is impractical and the value of ea is not constant, the analysis schemes for thin samples involve the measurement of elemental ratios in which the relative concentration of one element to another can be expressed by the following equation: Ca Ia ¼ kab ð101Þ Cb Ib The sensitivity factor approach to analysis is common to many analytical techniques and the factor kab in Eq. (101) is referred to as the Cliﬀ–Lorimer factor or kab factor (Cliﬀ and

890

Small et al.

Lorimer, 1975; Nasir, 1976). It is related to the generation and eﬃciency terms for element b ratioed to those for element a. In many analytical schemes, the analyst determines, from binary standards, a set of kab factors. Then, for a binary system containing unknown concentrations of elements a and b, Ca and Cb can be determined from Eq. (100) and the knowledge that Ca þ Cb ¼ 1. In ternary and higher-order systems, the relative concentrations for the various elements can be determined if the kab factors are known for combinations of various binary compounds as shown in the following equations for a ternary system: Ca Ia ¼ kab ð102Þ Cb I b Cc Ic ¼ kcb ð103Þ Cb Ib ð104Þ Ca þ Cb þ Cc ¼ 1 In analytical electron microscopy, particularly in analyses related to geological specimens, the convention is to express the kab factors relative to silicon (i.e., kaSi ). In this form, they are referred to as simply k factors and are related to the kab factors as shown in kab ¼

KaSi Ka ¼ KbSi Kb

ð105Þ

Figure 52 shows the kaSi values from the work of Wood et al. (1984) and Schreiber and Wims (1981). In addition to the k factors relative to silicon, several researchers, particularly those involved in metallurgy, have found it useful to report the k factors relative to iron. Table 10 lists the kaFe for the K and L lines of several elements. It is also possible to calculate k factors from Eq. (99). A detailed discussion of the equations used to calculate the various terms in Eq. (99) can be found in Williams and Goldstein

Figure 52 Measured kaSi factors for thin-sample analysis from Wood et al. (From Wood et al., 1984. Reproduced with permission of Blackwell Science Ltd.)

Electron-Induced X-ray Emission Table 10

891

Experimental kaFe Factors for Various Elements K Lines

Element Na Mg Al Si P S K Ca Ti Cr Mn Co Ni Cu Zn Nb Mo Ag

Lorimer et al. (1976, 1977) (100 keV)

McGill and Hubbard (1981) (100 keV)

Wood et al. (1984) (120 keV)

1.16 0.8 0.71 — — 0.77 0.75 — — — — — — — —

0.96 ± 0.03 0.86 ± 0.04 0.76 ± 0.004 0.77 ± 0.005 0.83 ± 0.03 0.86 ± 0.014 0.88 ± 0.005 0.86 ± 0.03 0.90 ± 0.006 1.04 ± 0.025 0.98 ± 0.06 1.07 ± 0.06 1.17 ± 0.03 1.19 ± 0.04 2.14 ± 0.06 3.80 ± 0.09 9.52 ± 0.03

2.46 1.23 ± 0.08 0.92 ± 0.08 0.76 ± 0.08 — — 0.79 0.81 ± 0.05 0.86 ± 0.05 0.91 ± 0.05 0.95 ± 0.05 1.05 1.14 ± 0.05 1.23 ± 0.05 1.24 — 3.38 6.65 L Lines

Sra Zra Nba Aga Sn Ba W Au Pb

Wood et al. (1984) (120 keV)

Goldstein et al. (1977) (100 keV)

1.21 ± 0.06 1.35 ± 0.01 0.90 ± 0.06 1.18 ± 0.06 2.21 ± 0.07 — — 3.10 ± 0.09 —

— — — 1.04 2.39 2.18 2.43 3.27 4.14

a

k Factors for these elements are for combined La and Lb lines. Source: From Goldstein et al., 1986.

(1991) and Goldstein et al. (1986). Figures 53 and 54 show a comparison of the calculated and measured k factors for K and L line radiation from various elements (Wood et al., 1984). Sheridan (1989) has determined a large set of experimental kaSi factors for a wide range of Ka, La, Ma lines. The measurements were performed on a series of submicrometer particles ground from NBS multielement research glasses. After applying an absorption correction, the experimental results were compared to theoretical kaSi factors determined with several diﬀerent ionization cross sections. As the thickness of the sample increases, the electron transparency of the thin sample or small particle decreases, eventually reaching a thickness for which the sample no longer

892

Small et al.

Figure 53 Calculated and measured kaFe factors for K line radiation from several elemental thin films. Solid lines represent range of theoretical values. (From Wood et al., 1984. Reproduced with permission of Blackwell Science Ltd.)

Figure 54 Calculated and measured kaFe factors for L-line radiation from several elemental thin films. Solid lines represent range of theoretical values. (From Wood et al., 1984. Reproduced with permission of Blackwell Science Ltd.)

Electron-Induced X-ray Emission

893

conforms to the ‘‘thin-ﬁlm criteria.’’ Under these circumstances, corrections for x-ray absorption should be included in the analysis scheme. According to Williams and Goldstein (1991), the limits for the failure of the thin-ﬁlm criterion are an x-ray absorption correction, A, < 0.97 or > 1.03 and=or a ﬂuorescence correction > 5%. For these cases, Eq. (101) can be expanded to include absorption and ﬂuorescence: Ca Ia 1 ¼ kab FðabsÞ ð106Þ 1 þ ðFfl Þ Cb Ib where FðabsÞ is the absorption factor given in Eq. (107) and Ffl is the ratio of the ﬂuoresced to primary intensity, Ia =I0 , given in Eq. (108): 0 a 1 m a B r sam C 1 exp ½w ðrtÞ FðabsÞ ¼ @ b A ð107Þ 1 exp ½wb ðrtÞ m r sam

# ! ! " b I ra 1 Aa m b ½Ec a lnðUÞb rt m 0:932 ln ¼ Cb ob rt sec c ra r sam I0 Ab r a ½Ec b lnðUÞa 2 a

ð108Þ

where ra is the absorption jump ratio for element a and Aa and Ab are the atomic numbers for a and b, respectively. The absorption factor was derived by Goldstein et al. (1986) from the work of Tixier and Philibert (1969) and Konig (1976). Williams and Goldstein (1991) calculated the thicknesses of materials above which the absorption of x-rays was outside the 0.97–1.03 limits. The results of these calculations are given in Table 11. As pointed out by Williams and Goldstein (1991), the eﬀect of ﬂuorescence must be investigated in cases where the concentration of the ﬂuoresced element is at a minor to trace level, < 10 wt%, in a sample composed mainly of the exciting element. In a study of Fe–10 wt% Cr (Nockolds et al., 1980), the ﬂuorescence correction was only about 5%. The equations for the absorption and ﬂuorescence involve the measurement of the ﬁlm or particle thickness and the calculation of the mass absorption coeﬃcients for elements a and b in the specimen. As the result, it is necessary to measure the thickness at each analysis location and to calculate the concentrations in an iterative loop similar to that used for the conventional ZAF schemes. C.

Quantitative Analysis of Particles with Energy-Dispersive X-ray Spectrometry

Although some initial particle studies were done on the electron probe using wavelengthdispersive spectrometers (WDS), the variablity and complexity of many particle compositions coupled with the high currents necessary for WDS made elemental analysis Table 11 Calculated Maximum Thickness of Thin Samples for Which the Absorption Correction Is Less Than ± 3% Element of interest=x-ray line AlKa PKa MgKa, OKa

Sample

tmax ðnmÞ

NiAl Fe3P MgO

9 22 25

Source: Data from Williams and Goldstein, 1991.

894

Figure 55

Small et al.

Images of a particle taken at the angles for three different WD spectrometers.

of particles by WDS diﬃcult at best (Armstrong and Buseck, 1978). In addition, the use of multiple spectrometers, each with a diﬀerent view of the particle and, therefore, a diﬀerent particle geometry, as shown in Figure 55, limited the quantitative capabilities of the technique. The evolution of the various quantitative procedures resulted in the separation of particles into roughly three size categories that are based on the procedures and instruments used for analysis (see Fig. 56). The particles less than 0.5 mm in diameter are best analyzed in the analytical electron microscope using the procedures described in the previous subsection. The particles with dimensions below about 0.1 mm require very little, if any, particle-geometry corrections for elements with atomic numbers greater than about 11 and can be analyzed as inﬁnitely thin ﬁlms using kAB factors. As the particle size increases, approaching 0.5 mm and larger, the eﬀects of particle geometry must be included in the analysis procedure (Goldstein, 1979). Once the particle dimensions are greater than approximately 0.5 mm, the particles are too large to be analyzed in the AEM and are best analyzed in the electron-probe or scanning electron microscope. From 0.5 to about 20 mm in diameter, particle geometry may have a pronounced eﬀect on the generation and emission of x-rays and can signiﬁcantly aﬀect quantitative analytical results. Many diﬀerent analytical schemes have been developed for the analysis of the particles in this size range. Finally, when the diameter of the particle is greater than about 20 mm, the particle is large enough with respect to the electron beam than it is essentially a bulk sample and classical analytical corrections apply. The remainder of this subsection is concerned with the quantitative analysis of particles in the 0.5–10-mm size range.

Figure 56

Particle size ranges and associated analytical techniques.

Electron-Induced X-ray Emission

895

In classical electron-probe analysis, schemes employing either a ZAF, bulk-sample f(rz), or Bence–Albee approach, both sample and standard must be inﬁnitely thick with respect to the penetration of the electron beam and have ﬂat surfaces. By controlling sample dimensions and shape, the corrections for the interaction of the electron beam with the sample and the subsequent x-ray emission can be calculated from simple geometric relationships. In the quantitative analysis of particles, the shape and thickness of the specimen often cannot be controlled or measured. the diﬃculties in quantitative analysis of particles result from three diﬀerent eﬀects that inﬂuence the generation and measurement of x-rays from these samples (Small, 1981). The ﬁrst eﬀect is the result of the ﬁnite size (mass) of the sample. The mass eﬀect is related to the elastic scattering of the electrons and is strongly aﬀected by the average atomic number of the sample. The mass eﬀect is important when the sample thickness or particle size is smaller than the range of the primary electron beam so that a fraction of the beam escapes the sample before exciting x-rays. This is shown in Figure 57, which is a Monte Carlo simulation of electron trajectories. The majority of the primary electron trajectories terminate within the boundaries of larger sphere, but very few terminate within the boundaries of the smaller sphere. As the size of the sample decreases, the mass of material from which x-rays are excited drops and results in a reduction of x-ray intensities from the specimen compared to a bulk sample of the identical composition. The mass eﬀect can be demonstrated by comparing the x-ray emissions from a bulk target to the emissions from a particle of the same composition. This eﬀect can be seen in Figure 58, which is a plot of the BaLa x-ray intensity from particles normalized to the intensity from a bulk material of the same composition plotted versus particle diameter. The energy of the Ba x-rays is 4.47 keV, which is high enough so that the absorption eﬀects are minimal. The mass eﬀect is demonstrated by the decrease in the intensity measured on the particles compared to the bulk for particles less than 3 mm in diameter. The net result of not

Figure 57 Monte Carlo simulation of electron trajectories in a large and small particle. (From Small, 1981.)

896

Figure 58 1981.)

Small et al.

Normalized BaLa x-ray intensity plotted as a function particle diameter. (From Small,

correcting for the mass eﬀect in the analysis of particles less than about 3 mm in size will be an underestimation of the composition for all elements analyzed in the sample. The second eﬀect that must be corrected for is the absorption eﬀect. In the analysis of most particles, the x-ray emergence angle and, therefore, the absorption path length cannot be predicted accurately as it can for polished specimens. The magnitude of this eﬀect is largest when there is high absorption, as is typically the case for soft x-rays from elements like Al or Si that have energies less than 2 keV. The diﬀerence between the absorption path length in a particle and a bulk ﬂat sample can result in widely diﬀerent values of emitted x-ray intensities. For the particle shown in Figure 59, the path length A– B in the particle is less than the path length A–C in the polished sample. The path length A*–D in the particle, however, is greater than the path length A–C in the polished sample. The result of the varying absorption path lengths is shown in Figure 60, which is the SiKa x-ray intensity from particles normalized to the intensity from a bulk material of the same composition plotted versus particle diameter. For these spherical particles, the absorption path lengths in the particles are less than the bulk material, resulting in a higher emitted x-ray intensity form the particles compared to bulk. This eﬀect is detectable primarily for the lower-energy x-ray lines, such as that of aluminum at 1.49 keV, which are highly absorbed. This increased intensity for the low-energy x-rays results in an overestimation of the concentration for these elements and a corresponding underestimation of elements with high-energy x-rays for which absorption is not signiﬁcant. The third eﬀect is caused by the ﬂuorescence of x-rays by either the continuum or other characteristic x-rays. Because x-ray absorption coeﬃcients in solids are relatively small compared to electron attenuation, the secondary x-ray ﬂuorescence occurs over a much

Electron-Induced X-ray Emission

897

Figure 59 X-ray absorption in a particle compared to the absorption in bulk material. (From Small, 1981.)

larger volume than the primary electron excitation. In bulk samples and standards, the x-rays, for the most part, remain in the specimens. In the case of particles however, the particle volume may be only a small fraction of the x-ray excitation volume. As a result, the exciting x-rays will ﬂuoresce relatively few x-rays before leaving the particle. In those samples where the ﬂuorescence is important, the eﬀect of comparing a particle to a bulk standard may be signiﬁcant. This eﬀect is shown in Figure 61, which is a plot of the range for NiKa x-rays causing ﬂuorescence of FeKa x-rays in a Ni–Fe alloy. The net eﬀect of not correcting for the secondary excitation of x-rays in particle analysis is as follows:

Figure 60 1981.)

Normalized SiKa x-ray intensity plotted as a function particle diameter. (From Small,

898

Small et al.

Figure 61

1.

2.

Range of secondary x-ray fluorescence in a Ni–Fe alloy. (From Small, 1981.)

An underestimation of the concentration for elements that have a signiﬁcant contribution to their characteristic line or lines from excitation by other characteristic x-rays In the case of continuum ﬂuorescence, an underestimation of the concentrations for all elements, particularly those with higher-energy lines, which are excited by the higher-energy, longer-range continuum

1. Normalization One of the simplest methods for the quantitative analysis of particles is to normalize to 100% the concentrations, determined with bulk standards, from a conventional procedure such as ZAF Eq. (109) (Wright et al. 1963): Ca Can ¼ P i Ci

ð109Þ

where Can is the normalized concentration ofPelement a, Ca is the concentration of a determined from the analytical procedure, and i Ci is the sum of the concentrations for all elements in the sample. The analyst, in selecting this method of correction, makes the assumption that x-ray absorption and ﬂuorescence are the same for the particle as for a bulk specimen and that the mass eﬀect is the same for all elements. No elements can be determined by diﬀerence; that is, oxygen must be analyzed or calculated from an assumed stoichiometry. In addition, the analyst cannot determine, by obtaining an analysis total of less than 100%, the presence of any undetermined elements. In practice, normalization of results is most eﬀective for the correction of the mass eﬀect because the decrease in intensity as a function of particle size is nearly the same for all elements. Figure 62 shows that the diﬀerent elemental curves merge together for particle diameters less than about 2 mm. Because this

Electron-Induced X-ray Emission

899

Figure 62 Normalized x-ray intensities from several elements plotted as a function of particle diameter. (From Small, 1981.)

procedure does not accurately compensate for the absorption and ﬂuorescence eﬀects, the most accurate results will be obtained on particle systems that meet the following conditions: (1) systems for which all the analytical lines for the elements are above 4 keV where the absorption eﬀects are minimal; (2) if any of the analytical lines are below about 4 keV in energy, then the lines for all the elements should be as close together in energy as possible so that the matrix absorption is approximately the same in all cases; (3) systems for which there is no signiﬁcant ﬂuorescence. Table 12 lists the results from the analysis of lead silicate glass particles. The ﬁrst set of results are taken from the analysis of the PbMa line at 2.3 keV, which is close in energy to the SiKa line at 1.74 keV. Because the absorption and mass corrections are similar for these two lines, the lead and silicon concentrations are in good agreement with the true values. The second set of results were determined by analyzing the PbLa line at 10.6 keV. In this case, the two analytical lines have very diﬀerent energies and the particle absorption eﬀect is not similar in magnitude. As expected, the errors associated with this analysis are considerably higher than those associated with the PbMa analysis. The lack of an eﬀective absorption correction by simple normalization can also be seen in Figure 63, which shows the relative error distributions, by element, for the analysis of K-411 glass microspheres (Roberson et al., 1995) with a bulk K-411 glass standard. The error distributions show that the concentration of Mg with the lowest energy x-ray, 1.25 keV, is overestimated, Si at 1.74 keV is centered at about zero error, and the elements with harder x-ray lines (Ca, 3.7 keV and Fe, 6.4 keV) are underestimated. 2. Particle Standards The analyst can use a conventional analysis scheme and substitute particle standards for the normal polished standards (White et al., 1966). In this procedure, the assumption is

900

Small et al.

Table 12 Analysis

Analysis of Lead Silicate Glass K-229 by Normalization of ZAF Results Si (wt%)

% Error

Pb (wt%)

% Error

0.620 0.643 0.658 0.653 0.675 0.588 0.657

4.6 1.0 þ1.2 þ0.5 þ3.8 9.5 þ1.1

Analysis done with PbLa line (does not meet conditions) 1 0.134 4.5 0.663 2 0.177 26.3 0.578 3 0.159 þ14 0.612 4 0.166 þ18 0.602 5 0.017 þ88 0.894 6 0.100 29 0.731 7 0.157 12.3 0.616

þ1.9 11 5.8 7.4 þ37 þ12.4 5.2

Analysis done with PbMa line (meets conditions) 1 0.155 þ10.7 2 0.144 þ2.9 3 0.136 2.7 4 0.138 1.1 5 0.127 9.0 6 0.170 þ22 7 0.137 2.5

Note: Nominal composition: Si ¼ 0.140; Pb ¼ 0.650.

that the particle eﬀects, particularly the absorption eﬀect, will be approximately the same for the sample and standard. This assumption is reasonably valid, providing the sample and standard are close in composition and shape, and the particle diameter is above about 2 mm. Below 2 mm, as shown in Figure 60, any diﬀerence in size and shape between unknown and standards will be critical because a small change in eﬀective diameter will result in a large change in x-ray intensity. Figure 64 shows the results of using particle standards for the analysis of a series of K-411 glass microspheres. Figure 64a is the error distribution for the normalized analysis of a population of spheres ranging in size from about 1 to 10 mm analyzed with a piece of bulk K-411 glass as the standard. The error distribution in Figure 64a shows three distinct modes that correspond to the overestimation of underestimation of diﬀerent elemental concentrations because of the lack of an eﬀective absorption correction (see Sec. IV.C.1). In contrast, Figure 64b is the error distribution for the normalized analysis of the same particle spectra used in Figure 64a, except a 5-mm sphere was used as the standard rather than a piece of the bulk glass. The distribution in Figure 64b has a single maximum, is centered on zero, and is narrower, ± 0.1, than the distribution in Figure 64a. 3. Geometric Modeling of Particle Shape The procedure for the quantitative analysis of particles by geometric modeling of particle shape is based on Eq. (110), which deﬁnes the emitted x-ray intensity of element a from a layer at depth z in the sample, Ia0 ðrzÞ (Armstrong, 1991): Ia0 ðrzÞ ¼ fa ðrzÞ exp ðma csc CrzÞ

ð110Þ

where r is the density, fa ðrzÞ is the generated primary x-ray intensity from a layer at depth z in the sample, ma is the mass absorption coeﬃcient for element a, and C is the take-oﬀ angle.

Figure 63 dard.

Relative error distributions, by element, for the analysis of K-411 glass with a bulk K-411 glass stan-

Electron-Induced X-ray Emission 901

902

Small et al.

Electron-Induced X-ray Emission

903

Modifying Eq. (110) to take into account the particle absorption eﬀect and the loss of x-ray intensity due to the mass eﬀect, this equation becomes b2 ðry;rz Þ

aZ 2 ðrzÞ

Z

Ia ðrzÞ ¼

fa ðrx ; ry ; rz Þ exp ½ma gðrx ; ry ; rz Þdrx dry

ð111Þ

y¼a1 ðrzÞ x¼b1 ðry;rz Þ

for particles, where gðrx ; ry ; rz Þ is the distance from the point of x-ray generation to the particle surface in the direction of the detector and fa ðrx ; ry ; rz Þ is the generated x-ray intensity in the particle. Calculation of fa ðrx ; ry ; rz Þ requires the determination of the number of x-rays, DI of element a, product per electron path length Ds as shown in Eq. (112), which is identical to Eq. (99) except Ds is substituted for D(pt): DI ¼

Ca oa pia N Aa Qa Ds

ð112Þ

Introducing the integration limits for the particle volume and electron energy into Eq. (112) yields Eq. (113) for the calculation of fðrx ; ry ; rz Þ: fðrx ; ry ; rz Þ ¼

Ca N Aa oa pia

ZEc Z2p Z2p E0

nðE; rx ; ry ; rz ; y; gÞ Qa ðEÞ dg dy dE ð113Þ dE=drs drs=dV

y¼0 g¼0

where nðE; rx ; ry ; rz ; y; gÞ is the number of electrons of energy E, scattering at angle y, the angle relative to the beam axis, and at angle g, the azimuthal angle in plane normal to beam axis, dE=drs is the mean electron energy loss over distance drs, and drs=dV is the distance traveled by the electron going through the volume element dV at point x; y; z. The equation for including secondary ﬂuorescence is complex and is given in Armstrong (1991). The emitted x-ray intensity for element a, corrected for particle mass and adsorption, can be determined from Eqs. (111) and (112) if the function deﬁning the particle shape= volume is known. For irregular particles, this is not the case and the method developed by Armstrong and Buseck (1977, 1978) is based on the determination of a simple geometric shape or combination of shapes such as a square or pyramid which deﬁnes the boundaries of the particle of interest. The various particle eﬀects are then calculated for the chosen geometric shapes deﬁning the shape of the particle. The procedure back-calculates the x-ray production from the particle to an appropriate value for an inﬁnitely thick sample for which all primary electrons remain in the sample and which has the same composition as the unknown. The various mechanisms responsible for x-ray are corrected as follows: 1. Electron transmission: The amount of primary radiation lost as a result of electron transmission through the particle and sidescatter requires an analytical expression for fðrx ; ry ; rz Þ. This expression is calculated from the fðrzÞ expression developed by Armstrong (1991) (see Sec. II.2 in this chapter). In the modiﬁed form, the Armstrong fðrzÞ expression can be used to calculate x-ray production as a function of position within a particle by setting the integration limits of x; y, and z.

3 Figure 64 (a) Error distribution for the normalized analysis of a population of spheres ranging in size from about 1–10 mm analyzed with a piece of bulk K-411 glass as the standard. (b) Error distribution for normalized analysis of the same particle spectra used in (a), except a 5 mm sphere was used as the standard rather than a piece of the bulk glass.

904

Small et al.

2.

3.

4.

X-ray absorption: The correction for x-ray absorption is done by numerical integration of Eq. (111), where x, y, and z are determined for the geometric shape or shapes which deﬁne the overall particle. The shapes which have been included in the program include cube or rectangular prism, tetragonal prism, triangular prism, square pyramid, cylinder, hemisphere, and sphere. Atomic number: The expression of Duncumb and Reed (1968) can be used to calculate the loss of x-rays as a result of electron backscatter. The standard equation for bulk materials can be used because this correction is applied at a point where the eﬀects of particle shape have been removed. Secondary ﬂuorescence: Armstrong and Buseck (1985) showed that the ratio of the emitted, ﬂuoresced x-ray intensity to the primary x-ray for a particle relative to the ratio for a thick polished sample of the same composition was about 0.5 for a 10-mm particle, 0.33 for a 5-mm particle, and 0.1 for a 1-mm particle. In addition, they also found the ratio to be insensitive to beam energy and mostly dependent on absorption of the excited and exciting x-rays.

Based on the series of polynomial ﬁts to calculated particle data, Armstrong and Buseck (1985) proposed Eq. (114) to calculated the ratio of the emitted, ﬂuoresced x-ray intensity to the primary x-ray intensity for a particle relative to the ratio for a thick polished sample of the same composition, Rf : ð114Þ Rf ¼ A þ Bx þ Cx2 where X ¼ 1 expðmbj rrÞ, A ¼ 0.0260, B ¼ 1:1409 þ 0:2012y; C ¼ 0:2471 þ 0:2741y 0:01315y2 ; y ¼ wak =mbj , r is the particle radius, r is the density, mbj is the mass absorption coeﬃcient for the exciting element by matrix, and wak is the mass absorption coeﬃcient for the excited element by matrix. Armstrong (1991) and Myklebust (1975) (see Figs. 57, 58, and 60) have also used Monte Carlo procedures to study the generation and emission of x-rays from particles. These calculations involve setting boundary limits for the Monte Carlo calculations based on particle shapes. Armstrong has also proposed using Monte Carlo calculations based on the particle shapes in his ZAF program to develop analytical procedures for the analysis of the particles based on working curves or on a-factor approach (Armstrong, 1982, 1991). For this procedure, the particles are assumed to be homogeneous and are analyzed by scanning over the electron beam rather than positioning the beam at a single spot on the particle. The required input includes the following: Elements and lines analyzed Beam voltage and takeoﬀ angle Standards and compositions Standard morphology (i.e., bulk polished, thin ﬁlm, or particle of a particular shape as described earlier) Size, estimated shape (as described earlier), an estimated density of the unknown particle Measured standard and unknown x-ray intensities for the analyzed elements The program will also accept elemental k ratios. The performance of the geometric modeling procedure is shown in Figure 65. Figure 65a is the relative error distribution for particles analyzed with a thick-specimen ZAF procedure (results normalized to 100%). The distribution is quite broad with the majority of the analyses falling between ±30% relative error. In contrast, Figure 65b calculated from the geometric-modeling program is much narrower with almost the entire set of analyses falling within ±10% relative error.

Electron-Induced X-ray Emission

905

Figure 65 (a) Relative error distribution for particles analyzed with a thick-specimen ZAF procedure (results normalized to 100%). (b) Relative error distribution for particle concentrations calculated from the geometric-modeling program. (From Armstrong, 1991.)

906

Small et al.

4. Peak-to-Background Ratios A fourth method for the quantitative analysis of particles was developed by Small et al. (1978) and Statham and Pawley (1978). This method, derived from work on biological specimens by Hall (1968), is based on the following observation: To a ﬁrst approximation, the ratio of a characteristic x-ray intensity to the continuum intensity of the same energy for a ﬂat, inﬁnitely thick target is equivalent to the ratio from a particle or rough surface of the same composition. In the form of an equation, this observation can be expressed as I I ¼ ð115Þ B particle B bulk where I is the background-corrected peak intensity and B is the continuum intensity for the same energy window as the peak. It is assumed that the spatial distribution of characteristic x-ray excitation is identical to the distribution for continuum x-ray excitation. As a result, the eﬀects of particle shape and size on measured x-ray intensity will be the same for the continuum and the characteristic x-rays. It therefore follows that by taking the ratio of the two intensities, the particle mass and absorption eﬀects will cancel each other. In the procedure developed by Small et al. (1978), Eq. (115) is rearranged and the peak intensities for the particle or rough surface are scaled up to values similar to a bulk material of the identical composition: Iparticle ¼ Ibulk ¼

Iparticle Bbulk Bparticle

ð116Þ

The values of I for each element in the unknown can then be used as input for a standard quantitative analysis scheme. In practice, a bulk material will not exist for particle of unknown composition and the value of Bbulk at any given characteristic peak energy must be estimated as part of the analysis scheme. In the procedure, Bbulk is determined from Eq. (117), and for each characteristic peak energy, it must be summed over all elements in the unknown: Bbulk ¼

X FðZbulk Þ fðwÞbulk std Þ Bstd fðwÞstd Fð Z i

ð117Þ

terms describe the continuum intensity as a function of the concentrationThe FðZÞ weighted average atomic number for the hypothetical bulk material and standards. For the simplest case, Kramers’ relationship (Kramers, 1923), the ﬁrst part of Eq. (117) would be Zbulk =Zstd , where the current estimation of concentration from the iteration loop in the ZAF procedure is used for the calculation of Zbulk Bstd is the measured continuum intensity for a given standard at the energy of the analyzed x-ray. For multielement standards, Bstd must be multiplied by the weight fraction of the element of interest. Once the values for Bbulk are determined, they can be used in Eq. (116) to obtain the ﬁrst estimates for I . A set of k ratios can then be calculated and used as input to the ZAF routine. The set of concentrations from each iteration is used to calculate new values of Bbulk and the sequence is repeated until successive iterations agree within a predetermined limit. Various mineral particles have been analyzed with the peak-to-background method, FRAME P, (Small et al., 1979a). The results of these analyses are reported in Table 13 along with the result from the conventional ZAF routine, FRAME C (Myklebust et al., 1979). In all cases, the analyses with the peak-to-background routine are within 10% and usually better than 5% relative error of the stoichiometric values. In contrast, the errors

Electron-Induced X-ray Emission Table 13

907

Analysis of Mineral Particles by the Peak-to-Background Method Talc (Mg3[Si4O10](OH)2)

Nominal Normalized FRAME C P–B FRAME P

Wt% Mg

% Errora

Wt% Si

% Error

19.3 19.4 18.5

þ0.5 4.0

29.8 29.7 29.7

0.3 0.3

% Error

Wt% Fe

% Error

1.3 0.9

46.6 47.3 46.4

þ1.5 0.4

% Error

Wt% Fe

% Error

þ7.8 þ9.0

67.1 64.5 67.7

3.8 0.7

FeS2 Wt% S Nominal Normalized FRAME C P–B FRAME P

53.4 52.7 52.9

ZnS2 Wt% S Nominal Normalized FRAME C P–B FRAME P

32.9 35.4 36.0

Note: The results are the average from seven analyses. a % Error ¼ {[(nom.calc.)]=nom.}6100. Source: From Small, 1979a.

with the conventional ZAF routine range from 7.9% for S in ZnS to 47% for Mg and Si in talc. In addition, standard deviations for individual measurements are less for the peak-tobackground routine than they are for the conventional ZAF routine. In the analysis of particles, the assumption that the generation volumes for characteristic and continuum x-rays are identical is only valid for particles larger than about 2 mm in diameter. Below this size, the anisotropic generation of the continuum results in a signiﬁcantly diﬀerent excitation volume for the continuum compared to the isotropically generated characteristic x-rays (Albee and Ray, 1970). This eﬀect is shown in Figure 66, in which isotropic and anisotropic cross sections have been used to calculate peak-tobackground ratios from K-309 glass particles normalized to bulk glass. The composition of K-309 is 7.9% Al, 18.7% Si, 10.7% Ca, 13.4% Ba, and 10.5% Fe. These plots show that the introduction of an anisotropic cross section for the continuum results in signiﬁcantly higher peak-to-background ratios for the smaller particles. As a result, it is necessary to introduce a correction for anisotropic generation of the continuum for quantitative analysis with peak-to-background procedures. In the peak-to-background method developed by Statham and Pawley (1978), the peak-to-background ratio from a given element is compared to the ratio from a second element: Ca ðP=BÞa ¼ fab Cb ðP=BÞb

ð118Þ

In this case, fab is a correction factor that should have a minimal dependence on particle size and can be calculated or determined empirically from standards. Table 14 lists the fab

908

Small et al.

Figure 66 Peak-to-background ratios from glass spheres normalized to bulk ratios showing the effects of isotropic and anisotropic cross sections for continuum generation. (From Small, 1981.)

values for ﬁve particles of K-961 glass, the composition of which is listed in Table 15. These results show that the fab values do not exhibit any noticeable trend with particle size and the standard deviation of fab is less than 8% relative for elements with a concentration greater than 1 wt%. For the elements Mn and P, with concentrations less than 1%, the standard deviation in fab is small considering the relatively poor counting statistics.

Electron-Induced X-ray Emission Table 14

909

fab Values for K-961 Microspheresa

Diameter (mm)

Mg

Al

Si

P

K

Ca

Ti

Mn

Fe

2 3 4 6 9

1 1 1 1 1

1.24 1.36 1.30 1.48 1.27

1.74 1.68 1.66 1.83 1.51

1.72 1.57 1.95 1.43 1.74

2.43 2.44 2.63 2.96 2.72

3.12 3.46 3.36 3.56 3.17

3.01 3.13 3.16 3.14 3.06

3.77 2.56 3.29 3.81 3.28

3.06 2.90 3.21 3.06 3.33

Ave. s (%)

1 0

1.33 6.8

1.69 7.1

1.68 12.

2.64 8.3

3.33 5.7

3.10 2.2

3.34 15.

3.11 5.1

a Na was not included in the results because of poor statistics and high ion mobility. Source: From Statham and Pawsley, 1978.

Table 15

Nominal Composition of Glass K-961

Element

Wt%

Na Mg Al Si P K Ca Ti Mn Fe

2.97 3.02 5.82 29.9 0.22 2.49 3.57 1.20 0.32 3.50

D.

Analysis of Rough Surfaces

Results from the FRAME P analysis of rough surfaces show a similar improvements in accuracy with the peak-to-background method compared to conventional ZAF. Table 16 lists the results of analyses on fracture surfaces of Au–Cu alloys and a Fe–3.22% Si alloy. A micrograph of one of the Au–Cu fracture surfaces is shown in Figure 67. The use of the peak-to-background method for the analysis of rough surfaces can lead to signiﬁcant improvement in accuracy and precision for the analysis of these types of samples.

V.

SPATIALLY RESOLVED X-RAYANALYSIS

A.

x–y Mapping

In spatially resolved x-ray analysis, the position of the electron beam on the sample is couple to the output of an x-ray spectrometer. The x-ray spectrometer, in turn, is coupled to the output of the display on a cathode-ray tube, photographic plate, or computer memory such that when the beam is interrogating a point on the sample, the output from x-ray spectrometer is displayed or stored at the corresponding point on the storage medium. In this way, multiple analyses can be taken on a sample and the spatial

910 Table 16

Small et al. FRAME P Analysis of Fracture Surfaces Actual (wt%)

FRAME C (wt%) (normalized)

Rel. error

FRAME P (wt %)

Rel. error

SRM 482, 60% Au–40% Cu Location 1 Au Cu Location 2 Au Cu

60.3 39.6

49.6 50.4

18 27

58.0 44.0

4 11

29.1 70.8

52 79

52.0 41.0

14 3.5

SRM 482,80% Au–20%Cu Location 1 Au Cu Location 2 Au Cu

80.1 19.8

73.8 26.2

8 32

76.9 19.1

4 3.5

69.3 30.6

13 55

76.7 20.1

4.2 1.5

SRM 483, Fe–3.22% Si Location 1 Fe Si Location 2 Fe Si

96.8 3.22

97.0 2.9

0.2 8.2

100.0 3.2

3.3 0.3

96.4 3.6

0.4 11

97.7 3.5

0.9 7.4

Source: From Small et al., 1979b.

Figure 67 Secondary electron image of a fracture surface of Standard Reference Material 482. (From Small, 1979b.)

Electron-Induced X-ray Emission

911

relationship between each analysis location is retained. The analyst can then manipulate the stored data and construct an image of the sample based on its elemental composition. One of the earliest methods of obtaining spatially resolved analysis was to deﬂect the analog scan of electron beam in a single direction, usually x, across the specimen surface and simultaneously record and display the output from a WDS x-ray spectrometer on a CRT. In this way, a line proﬁle for the intensity of a chosen element can be overlaid on an image of the sample. An example is shown in Figure 68. The ﬁrst method of x-ray analysis to display elemental information in both the x and y directions was the x-ray dot map or x-ray area scan (Cosslett and Duncumb, 1956). Initially, this procedure, like the line scan, made use of the analog scan of the electron beam. As the beam was scanned slowly over the sample, the x-ray intensity of a given element as determined by a spectrometer was compared to a preselected threshold value. If the x-ray counts are above the threshold value, then a signal is sent to the recording CRT on the instrument, which records a full intensity spot at that location. The entire sample area selected by the instrument’s magniﬁcation is scanned in this manner and the resulting x-ray dot map is constructed. Figure 69 shows the result of such an analysis of grainboundary migration of Zn in a Zn–Cu system. In this procedure, only the presence or absence of a given element, as determined by the preselected thereshold value can be displayed, Minimal information is provided that reﬂects the amount or concentration of that element present. In addition to displaying black and white maps, the analyst could produce color overlays of up to three diﬀerent elements by separately photographing the dot maps using red, green, and blue ﬁlters. Most instruments are now equipped with digital beam control, where the electron beam is deﬂected such that it interrogates the sample in two-dimensional array consisting of n6n spot analyses, where n is usually 16–512 for x-ray mapping. In an instrument with stage control, the electron beam can be static and the stage stepped in the array pattern. Each beam or stage position in the two-dimensional array corresponds to a pixel on a map

Figure 68 Line profile of SKa x-ray intensity overlain on a target current image of a specimen from a failed pressure vessel.

912

Figure 69 1982.)

Small et al.

X-ray dot map of a grain-boundary migration in a Zn–Cu system. (From Piccone et al.,

of the specimen. By storing WDS or EDS spectra at each position, the analyst can obtain a full quantitative analysis at each pixel. Once the results from the analyses are stored in the array, the variations in elemental compositions can be coupled to the gray-scale or color output of a computer monitor. The completed array of pixels provides the analyst with a spatially resolved map of the elemental composition across the specimen. In addition, because the array is stored in the computer, the analytical information can be postprocessed in several ways. For example, a row of pixels can be selected providing the analysts with elemental x-ray line proﬁles across the sample in any direction or pattern desired. This information can be displayed as a simple line plot or can be used to construct contour maps of a given x-ray distribution. The uncertainty associated with a quantitative analysis obtained for compositional mapping is essentially identical to the uncertainty associated with a conventional analysis obtained at the same electron dose. For maps where high accuracy is required on minor constituents, the dwell time per pixel and=or beam current must be suﬃciently large to provide the appropriate x-ray intensity in the peak of interest. Compositional maps under these circumstances may require several hours to accumulate and care must be taken to ensure instrument stability over the required time interval. As previously mentioned, both EDS and WDS analysis systems can be used for compositional mapping. WDS detection of x-rays, however, is superior to EDS detection for compositional mapping, particularly for samples requiring the analysis of constituents at the minor and trace levels (Marinenko et al., 1987). One of the most important advantages of WDS detection is that the pulse processing time for WDS is at least an order of magnitude shorter than EDS. WDS detectors have a limiting count rate

Electron-Induced X-ray Emission

913

of about 105 counts per second (cps) compared with a limiting count rate of 104 for EDS detectors operating at maximum resolution. In addition, the WDS systems use diﬀraction crystals so that only a narrow energy band of x-rays are detected at any given time. The energy is adjustable so that most of these counts are the characteristic counts from the element of interest. This count rate compares to a limiting rate of about 102–103 cps in the peak of interest on an EDS detector because the limiting count rate of 104 is distributed over the entire energy range of the spectrum. In addition to its higher limiting count rate, the WDS detector has a factor of 10 higher peak to background ratio than the EDS detector. Taking into account the higher limiting count rate and peak-to-background ratio of the WDS detector, the WDS detector has a lower detection limit of about 100 ppm compared to a lower limit of 1% or greater for mapping on the EDS system. One limitation of the WDS system compared to EDS systems is in the mapping of samples which contain a large number of elements of interest. In WDS systems, the number of elements mapped during a given digital scan is limited to the number of WDS spectrometers. Typically, in EDS systems, all elements, with characteristic x-ray energies above about 1 keV can be detected simultaneously. Two important aspects that must be considered in the interpretation of compositional maps are the counting statistics and resulting uncertainties associated with the various point analyses at each pixel location. Under normal mapping conditions, it is impractical to obtain the level of accuracy associated with conventional probe analysis where 100 s or more are used for data accumulation. Marinenko et al. (1987) have shown that for a 1286128 map at a magniﬁcation below 5006, each pixel represents an area of about 2 mm in diameter on the specimen surface. This means that the spot size of the beam can be on the order of 1 mm in diameter and carry a current of about 1 mA. At this current density, the limiting count rate of 105 cps can be obtained on pure-element samples. Assuming a dwell time per pixel of 0.1 s, a 1286128 map would require 1600 s to accumulate and have 104 counts per pixel. From Poisson statistics, the 1s counting uncertainty per pixel is 1%. In the case where mulitelement standards are required, the dwell time per pixel must be increased to obtain the same counting statistics, as each element is present at less than 100%. In general, dwell times of about 0.4 s are suﬃcient for multielement standards and require less than 2 h per 1286128 map. As in conventional analyses, the standard maps can be archived and used indeﬁnitely, provided the instrument conditions are constant. In the analysis of samples where one or more of the constituents are at the minor or trace level (1% or less by weight), digital maps may require 10 h or more to accumulate. A 10-h scan on a 1286128 map translates to 2 s dwell time per pixel. If the element of interest is at the 1% level by weight, then the x-ray intensity in the peak would be 2000 counts, assuming a limiting count rate of 105 cps as stated above. The associated counting uncertainty is 2.2% for 1s. The actual uncertainty will be larger than this due to ﬂuctuations in the intensity of the x-ray background, which must be taken into account for quantitative analysis. In general, it should be possible to obtain 2s conﬁdence intervals of 10% or better for constituents at the 1% weight concentration in an unknown sample. The power of compositional mapping can be seen in Figure 70, which is a compositional map of the same Zn grain-boundary migration shown in Figure 69. The brighter areas on the map correspond to a Zn concentration of 10% by weight. The faint band extending to the left in the bottom of Figure 70 corresponds to a Zn concentration as low as 0.2% by weight.

914

Small et al.

Figure 70 Compositional map of the same Zn grain-boundary migration shown in Figure 69. (From Piccone et al., 1982.)

One of the major diﬃculties in obtaining quantitative x-ray maps with crystal spectrometers is the defocusing of the diﬀraction crystal as the electron beam is moved oﬀ the crystal axis. The magnitude of the defocusing eﬀect is proportional to the distance the beam is from a point centered on the optical focus of the electron probe. The defocusing is most severe at the low magniﬁcations of 200–8006 routinely used in digital x-ray maps. Corrections for the defocusing fall into four categories, (Marinenko et al., 1987). The ﬁrst method to avoid spectrometer defocusing is to employ, as mentioned above, the digital movement in the microscope stage rather than the electron beam. In this situation, the electron beam position remains static at the optical focus of the diﬀraction crystal and the sample is moved in a raster pattern under the beam. For WDS mapping, it is critical that the sample remains at the focus of the optical microscope during the x–y movement of the stage. This procedure has been used successfully by Mayr and Angeli (1985). The success of this method, since it involves the mechanical movement of the stage, is dependent on the reproducibility of the stage motion. If the reproducibility of the stage movement is poor, then the accuracy of the quantitative maps will be poor except for extremely low magniﬁcations. In the more modern instruments, which have optical encoding and a guaranteed stage positioning of 0.1 mm, the results from moving the stage will be comparable to the other corrections methods. The second method used to correct for spectrometer defocusing is to move or rock the diﬀraction crystal in synchronization with the raster of the electron beam, (Heinrich, 1981e). By slightly rocking the crystal, the entire sample area under study will be maintained at the focus of the x-ray crystal. The problems associated with crystal rocking are similar to those involving stage motion. Because beam rocking requires mechanical movement of the crystal, the results are dependent on the reproducibility of the crystal rocking mechanism. This procedure has been described for multiple spectrometer by Swyt and Fiori (1986). The third correction method is to collect a series of standard maps in conjunction with the collection of a map from an unknown. If the standard and the unknown are

Electron-Induced X-ray Emission

915

collected under the same experimental conditions, then the standard maps can be used to construct k ratios at each pixel location, thus normalizing out the eﬀects of spectrometer defocusing. Although this method does not require any mechanical movement, it does require that the instrument remain stable over the period of time necessary to collect all the maps from the standards and unknown. This is particularly important because a pixel-by-pixel comparison of the unknown to the standard is made which requires a constant crystal orientation with respect to the beam raster. Unless great care is taken during an analysis, the use of archival standard maps is not recommended because of the possibility of introducing large errors as a result of spectrometer drift. This procedure was used by Marinenko et al. (1987) for the analysis of the Au–Ag standard reference material 481 with the results shown in Figures 71 and 72. Figure 71 presents the uncorrected maps from the alloys and the pure-silver standard, which show the characteristic banding from the defocusing of the spectrometers. Figure 72 contains the puresilver map and the subsequent alloy maps that have been corrected for defocusing by normalization to the standard map. The normalized maps show very little gradation or structure in the image gray levels indicating the artifacts from defocusing have been removed. Table 17 contains a comparison of the average composition obtained from the digital map and the results from a conventional point analysis. In all cases, ‘‘the average composition from each map compares quite well with the conventional point analysis as well as with the NIST certiﬁed values’’ (Marinenko et al., 1987). The fourth method used is to model the spectrometer defocusing. Marinenko et al. (1988) observed that the defocusing artifacts (i.e., banding) were equivalent to an intensity proﬁle from a wavelength scan across the elemental peak of interest. This is shown in Figure 73, which compares a wavelength line scan across the CrKa1,2 peaks to a line proﬁle taken from a Cr map perpendicular to the defocusing bands. The similarity of these two traces implies that the intensity can be measured in the center band of a map and, from a

Figure 71 Elemental maps from Au–Ag alloys showing the effects of spectrometer defocusing (upper left) pure Ag, (upper right) Au20–Ag80, (lower left) Au60–Ag40, and (lower right) Au80– Ag20. The analytical line is AgLa. (From Marinenko et al., 1987.)

916

Small et al.

Figure 72 Elemental maps, corresponding to those in Figure 72, corrected for spectrometer defocusing by normalization to a standard map. (From Marinenko et al., 1987.)

model of the line proﬁle, the intensity at any other pixel location can be calculated. This procedure eliminates the need for measuring a standard map and greatly reduces the time required for obtaining a quantitative x-ray map. Figure 74 is a schematic showing the relationship between the x-rays emitted from a defocused point on the sample and the corresponding angular deviation from the line of focus for a vertical crystal spectrometer. The electron probe is designed such that the maximum x-ray intensity obtained on a crystal corresponds to the point A in Figure 74, which is centered on the electron optic axis with the aid of an optical microscope. The distance between point A and the crystal, S0, is S0 ¼ 2R sin y

Table 17

ð119Þ

Quantitative Analysis of Gold–Silver Alloys, SRM 481

Alloy Au20–Ag80

Au60–Ag40

Au80–Ag20

Element line

Certiﬁed value

AgL AuM AuL AgL AuM AuL AgL AuM AuL

77.58 22.43 22.43 39.92 60.05 60.05 19.93 80.05 80.05

Point beam analysisa 77.33 21.93 22.69 39.44 59.59 61.26 19.59 80.56 81.09

(±0.4%)b (±2.6%) (±2.3%) (±0.5%) (±2.4%) (±2.3%) (±0.6%) (±2.5%) (±2.0%)

Digital mapping 77.63 20.81 22.94 39.96 57.59 60.87 19.72 77.26 80.30

Note: Concentration are in weight %. a Excitation potential ¼ 20 kV, Faraday cup current ¼ 38 nA, point beam, ﬁve randomly selected samplings averaged for each alloy. b One relative standard deviation of a single measurement is in parentheses.

Electron-Induced X-ray Emission

917

Figure 73 Comparison of a WDS line scan across the CrKa peak to a line profile taken from a Cr map perpendicular to the defocusing bands of the spectrometer. (A) Line scan across the CrKa peak with a LiF crystal; (B) CrKa intensity profile from the compositional map, taken along a series of pixels perpendicular to the maximum intensity band. (From Marinenko et al., 1980.)

where R is the spectrometer radius and y is the Bragg angle. At a defocused point, point B, which is a distance DS from the optic axis, the angular deviation from the exact Bragg angle is Dy. Given that Dy is small, less than about 0.01 radians, which is the case for most mapping applications, it can be approximated as DS 0 ð120Þ Dy ¼ sin Dy ¼ S0 where DS 0 ¼ DSðsin CÞ for a vertical spectrometer and C is the x-ray takeoﬀ angle for the electron probe.

918

Small et al.

Figure 74 Relationship between the x-rays emitted from a defocused point on the sample and the corresponding angular deviation from the line of focus for a vertical crystal spectrometer. (From Marinenko et al., 1988.)

In calculating the background value for a given pixel in the compositional map, it is necessary to determine DS, which is the distance of the pixel from the maximum focus line of the spectrometer. The orientation and equation of the line of maximum focus for a given spectrometer can be deﬁned by the coordinates of two pixel points [i.e., (x1 ; y1 ) and ðx2 ; y2 Þ] which lie on the line. The orientation of the line can be determined from any raw intensity map by increasing the image theshold as shown in Figure 75. The line has the general form Ax þ By þ C ¼ 0 and it follows that DS for any pixel point, represented by coordinates ðx3 ; y3 Þ, in the x-ray map can calculated from DS ¼

Dx3 þ Y3 E ðD2 þ 1Þ1=2

ð121Þ

where D ¼ ðy2 y1 Þ=ðx2 x1 Þ and E ¼ y1 Dx1 . DS is positive for points located above the line of focus and negative for points below the line of focus. DS as deﬁned in Eq. (120) is deﬁned in units of pixel elements which must be converted to centimeters in order to obtain the linear distances for the calculation of Ds.

Electron-Induced X-ray Emission

919

Figure 75 Line of maximum x-ray intensity on a crystal spectrometer determined by increasing the image threshold. The threshold is low in the top image and high in the bottom image. (From Marinenko et al., 1988.)

Electron-Induced X-ray Emission

Figure 77

921

Diagram describing the CCH.

For this purpose, the scaling factor F is used, which is deﬁned in terms of the magniﬁcation M, the linear dimension of the CRT display L, and the number of pixel points in the matrix N: F¼

L MN

Dy can then be expressed as 0 F Dy ¼ Ds S0

ð122Þ

ð123Þ

The accuracy of this procedure is demonstrated in Figure 76. Figure 76A is an experimentally measured intensity proﬁle taken perpendicular to the focal axis and Figure 76B is the modeled proﬁle along the same axis. The intensity proﬁles shown together in Figure 76C are identical within the limitations of the experiment. The use of standard maps or modeling to characterize spectrometer defocusing is magniﬁcation limited. At magniﬁcations of about 1506 or less the deviation in intensity from the edge to the center of a map is about a factor of 2 and cannot be accurately

3 Figure 76 Comparison of experimental and modeled x-ray intensity profiles for TiKa x-ray: (A) Experimental x-ray profile for TiKa overlain on the corresponding Ti map; (B) modeled intensity profile; (C) profiles from (A) and (B) together. (From Marinenko et al., 1988.)

922

Small et al.

corrected by these methods. For very low-magniﬁcation maps where quantitative results are important, stage mapping or crystal rocking are preferred. Recently, Mott et al. (1995) have developed a form of spectral imaging that couples the measurement, by EDS, of x-ray photons emitted from a sample with the x–y position of the electron beam. Unlike conventional mapping techniques, where the electron beam is stepped relatively slowly from point to point and the full data collected for each point, this mapping procedure, known as ‘‘Position-Tagged’’ spectrometry, operates under digital beam control with the beam rastered at a rapid scan speed. The result is the real-time

Figure 78 Compositional maps for an aligned Cu–Ti eutectic: (A) Cu map; (B) Ti map. (From Bright et al., 1988.)

Electron-Induced X-ray Emission

Figure 79

923

CCH for the Cu–Ti aligned eutectic shown in Figure 78. (From Bright et al., 1988.)

collection of elemental images from preselected energy regions of interest, a frame-averaged electron image, and x-ray spectra from preselected areas (i.e., phases) of the sample where the spectrum is a summation of the spectra from all pixels within the area. The spectrum image is compressed and streamed in real time to disk storage. At the completion of the ‘‘Positioned-Tagged Spectrometry’’ (PTS) run, the spectrum image consists of complete x-ray spectra for each pixel in the digital image. This enables the analyst to construct additional elemental maps, or spectra as needed from the stored spectrum image. In addition, the spectra for a given image can be processed by an appropriate quantitative routine to obtain elemental concentration maps. It is important to note that the statistical considerations for PTS compositional mapping are the same as those mentioned previously in this section and the time requirements for high-quality compositional maps will be similar to those mentioned above. B.

Composition^Composition Histograms

The ﬁnal aspect of electron-probe analysis that will be considered is composition–composition histograms. Compositional maps provide the analyst with a visual method of interpreting the results from the elemental analysis of some 4000 individual points on a sample (Prutton et al., 1987, Browning, 1987, Bright et al., 1988). By associating color and intensity, the analyst is able to spatially relate the various elements and their compositions within the analyzed region of the sample. One problem in the interpretation of compositional maps is that it is often diﬃcult to visualize the compositional ranges and resulting interelement correlations from a color composite image, particularly in the case of minor or trace constituents. An alternative method of displaying the analytical information is in the form of a composition–composition histogram (CCH) (Bright et al., 1988). The CCH provides the analyst with an image that can be used to interpret the numerical relationships among the various components in the sample. Figure 77 is a schematic diagram describing the CCH. The concentration of element a at each pixel in the compositional map of a, is associated with the corresponding pixel

924

Small et al.

Figure 80 CCH and corresponding map areas for the high-Cu region of the eutectic: (A) outline of the high-Cu region of the CCH; (B) pixels in the compositional map that correspond to the outlined area of the CCH. (From Bright et al., 1988.)

and concentration of element b in the map of b. This is represented by the top two blocks in Figure 77. The associated concentrations for a and b at each pixel in the compositional image are then plotted in the CCH, bottom block, as a single point in a scatter diagram. Associations between elements are visible in the CCH as features, such as lines or areas, which have a detectable density of points above background. In addition, the overlapping of multiple-image pixels at one CCH point can be coupled to the intensity of the recording CRT and displayed in an appropriate gray-level or color scale.

Electron-Induced X-ray Emission

925

An example of the type of information conveyed by the CCH is shown in Figure 78A, 78B, and 79, which are the elemental maps and the CCH for an aligned Cu–Ti eutectic alloy. The CCH (Fig. 79) shows two lobes that have a high density of pixels and correspond to the two diﬀerent phases of the alloy (Bright et al., 1998). In addition to the two lobes in Figure 79, there is a distribution of points between the lobes which has a lower pixel density compared to the density in the lobes.

Figure 81 CCH and corresponding map areas for the region of the eutectic containing both Cu and Ti: (A) outline of the connecting pixels between the high-Cu and high-Ti lobes; (B) pixels in the compositional map that correspond to the outlined area of the CCH. (From Bright et al., 1988.)

926

Small et al.

A useful feature of the CCH is the ability to relate a given region or feature on the CCH to the corresponding pixels in the compositional map. An example of this ‘‘back trace’’ capability is shown in Figure 80 where the high-Cu lobe of the CCH is outlined in Figure 80A and the corresponding pixels in the compositional map are highlighted as the bright areas in Figure 80B. Applying the ‘‘back trace’’ method to the points between the lobes (Fig. 81) allows the analyst to determine readily that the connecting line between the two lobes corresponds to the boundary regions between the Ti and Cu phases.

REFERENCES Agnello R, Howard J, McCarthy J, O’Hara D. Microsc Microanal, 3(suppl 2):889, 1997. Albee AL, Ray L. Anal Chem 42:1408, 1970. Ammann N, Karduck P. In: Michael JR, Ingram P, eds. Microbeam Analysis—1990. San Francisco: San Francisco Press, 1990, p 150. Ammann N, Lubig A, Karduck P. Mikrochim Acta 12(suppl):213, 1992. Armigliato A, DeSalvo A, Rosa R. J Phys. D 1:529, 1982. Armigliato A, Lewis T, Rosa R. Mikrochim Acta 13(suppl):241, 1996. Armstrong JT. In: Heinrich KFJ ed. Microbeam Analysis—1982. San Francisco: San Francisco Press, 1982, p 175. Armstrong JT. In: Newbury DE, ed. Microbeam Analysis—1988. San Francisco: San Francisco Press, 1988a, p 239. Armstrong JT. In: Newbury DE, ed. Microbeam Analysis—1988. San Francisco: San Francisco Press, 1988b, p 469. Armstrong JT. In: Heinrich KFJ, Newbury DE, eds. Electron Probe Quantitation. New York: Plenum Press, 1991, p 261. Armstrong JT. Microbeam Anal 2:S13, 1993. Armstrong JT. Microbeam Anal 4:177, 1995. Armstrong JT, Buseck PR. Anal Chem 47:2178, 1975. Armstrong JT, Buseck PR. Development of a characteristic ﬂuorescence correction for thin ﬁlms and particles. Proceedings of the Twelfth Annual Conference of the Microbeam Analysis Society, Boston, 1977, p 42A. Armstrong JT, Buseck PR. In: Electron Microscopy and X-ray Applications to Environmental and Occupational Health Analysis. Ann Arbor, MI: Ann Arbor Press. 1978, p 211. Armstrong JT, Buseck PR. X-ray Spectrom 14:172, 1985. Bastin GF, Heijligers HJM. Quantitative electron probe microanalysis of carbon in binary carbides. Technical Report, University of Technology, Eindhoven, 1985. Bastin GF, Heijligers HJM. Scanning 12:225, 1990. Bastin GF, Heijligers HJM. In: Heinrich KFJ, Newbury DE, eds. Electron Probe Quantitation. New York: Plenum Press, 1991, p 145. Bastin GF, Dijkstra JM, Heijligers HJM., X-ray Spectrom 27:3, 1998. Bastin GF, Van Loo FJJ, Heijligers HJM. X-Ray Spectrom 13:91, 1984. Bastin GF, van Loo FJJ, Heijligers HJM. Scanning 8:45, 1986. Bastin GF, Dijkstra JM, Heijligers HJM, Klepper D. Mikrochim Acta 12(suppl):93, 1992. Bence AE, Albee A. J Geol 76:382, 1968. Bence AE, Holzwarth W. Proc. 12th MAS Conference, 1977, p 38. Berger MJ, Seltzer SM. Studies of Penetration of Charged Particles in Matter. NRC Publication 1133. Washington, DC: National Academy of Sciences, 1964, p 205. Bethe H. Ann Phys (Leipzig) 5:325 1930. Bethe HA, Ashkin J. In: Segre E, ed. Experimental Nuclear Physics. New York: Wiley, 1953, p 252.

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14 Sample Preparation for X-ray Fluorescence Martina Schmeling Loyola University Chicago, Chicago, Illinois

Rene´ E. Van Grieken University of Antwerp, Antwerp, Belgium

I.

INTRODUCTION

The preparation of samples presents, for any analytical method, one of the most important steps for a reliable analysis. It is critical with regard to errors, but better precision as well as lower detection limits can also be achieved by suitable sample pretreatment. The X-ray ﬂuorescence (XRF) techniques [energy-dispersive XRF (EDXRF) and wavelength dispersive XRF (WDXRF)] allow the analysis of almost all types of sample with usually less eﬀort of sample preparation. However, for some materials, the sample preparation step is even more time-consuming than the analysis itself; therefore, suitable procedures should be developed and introduced, leading to reliable results. Usually, the specimens are presented to the XRF spectrometer as solids, powders, briquettes, fusion products, liquids, or ﬁlms, and for most of them, the pretreatment can be kept to a minimum. Especially for metallic specimen, liquids and atmospheric particles need only small eﬀort. If required, possible sample preparation steps may include simple procedures like cutting, grinding, milling, and mixing, or a combination of them. The preparation step may also involve a conversion from the present state of the sample into another one in order to receive a homogenous sample. It is important that the treatment must be practicable for both samples and standards. Metals and metallic alloys often need only to be polished to obtain a suitable surface for analysis. Liquid samples, containing high enough concentrations for a direct analysis, can be transferred into special cups or pipetted onto a surface target and dried to obtain a thin ﬁlm. Atmospheric particles, collected onto ﬁlters, need practically no sample pretreatment. However, if sample preparation is required, as for geological and/or inhomogeneous samples, the procedure should be fast and reproducible, require small eﬀort, and avoid contaminations. A consultation of the literature is very important before the application of special sample preparation procedures because usually the problem occurred already earlier and some proposals or possible solutions can be found. Extensive sample preparation should always be avoided because with each additional step, the risk of losses and contaminations increases.

933

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A more extensive sample preparation is necessary when the sample is very inhomogeneous, the surface layer is not representative for the whole sample, the average particle size or the particle size distribution varies among the samples, and the amount of porosity or pore size varies among samples. As a condition for a reliable analysis by XRF, a ﬂat and even surface of the specimen is required. Particularly for the determination of light elements, the surface should be prepared mirrorlike, as scattering eﬀects due to a rough texture inﬂuence the results. An appropriate surface can be normally obtained by polishing, melting, or fusion of the samples. If absorption-enhancement eﬀects are severe, an internal standard or a masking agent must be added. Sometimes, it is also necessary to dilute the sample for the reduction of such eﬀects. For liquid samples with a low trace element amount, preconcentration or trace–matrix separation might yield satisfying results. A rare application is still the analysis of gaseous samples. Kno¨chel et al. (1983) used synchrotron-radiation XRF (SRXRF) for the determination of nitrogen in the presence of small amounts of xenon. Special cups were designed for this purpose and the results were quite promising. Therefore, it is surprising that this technique is not more frequently used for the determination of elements in gases. In the following section, the most common techniques of sample preparation are discussed, with emphasis on more recent applications and developments. Sample preparation for special techniques, like SRXRF, total-reﬂection XRF (TXRF), photon-induced x-ray emission (PIXE), and electron-probe microanalysis (EPMA), is only brieﬂy mentioned in some cases, because in earlier chapters of this book, extensive relevant discussions have already been given. It should be mentioned that several classical books about XRF include chapters about conventional sample preparation procedures (Bertin, 1970; Bertin 1798; Jenkins et al., 1981; Tertian and Claisse, 1982; Jenkins, 1988; Injuk and Van Grieken, 2000) and, recently, an entire book was dedicated to practical sample preparation in XRF and x-ray diﬀraction (Buhrke et al., 1998).

II.

SOLID SAMPLES

X-ray ﬂuorescence is a traditional analytical method for the analysis of solid samples. They may be presented to the spectrometer as such, as powders, as briquettes, or as fusion products. In principle, it is possible to determine the bulk composition of solid samples directly if the element distribution inside the solid specimen is homogeneous in all three dimensions and the specimen shows a satisfying ﬂat surface without scratches and striations. Often such samples need only to be polished at the surface with suitable abrasive materials and to be cut into the appropriate size for the sample holder of the spectrometer. If the sample is too small for the sample holder, the positioning might be critical, and it is useful to prepare a wax mold that ﬁts into the holder. Such a mold permits a series of samples and standards and can be replaced reproducibly. This technique especially allows the analysis of small manufactured parts (Bertin, 1970). Often solid samples to be analyzed show surface roughness and=or damages, which cause problems due to absorption eﬀects. Such specimens must be ﬁnished at the surface before analysis in order to reduce these damages. Several diﬀerent techniques are available for ﬁnishing a sample surface. The most common and eﬀective one is mechanical polishing with special abrasives like Al2O3 or SiC, or with coarse and ﬁne abrasive paper. A ﬁnal ﬁnish of 100 mm surface roughness for short x-ray wavelengths and down to 10 mm for long wavelengths may be satisfactory for a reliable analysis. Other polishing techniques

Sample Preparation for XRF

935

are electropolishing or etching. However, these are not as universal as mechanical polishing because the sample might be destroyed or the surface composition changed. A.

Metallic Specimens

X-ray ﬂuorescence is one of the most common techniques for the analysis of samples in the metal production and, therefore, the preparation of such a specimen should be fast, simple, and reproducible. Usually, the metallic specimens are prepared as solid disks by conventional methods like cutting and milling. Hard alloys can be ground and pelletized prior to analysis. However, for most metallic samples, it is required to polish the surface in order to avoid striations, which give rise to the so-called shielding eﬀects (Jenkins and Hurley, 1965; Mu¨ller, 1972). For a reliable analysis, the surface roughness should not exceed the path length at which 10% absorption on the radiation will occur (Jenkins et al. 1981). In Table 1, some 10% absorption path lengths (presented as x90% in the table) for common analytical situations of metallic specimens are indicated. For measurements at short wavelength, a ﬁnal ﬁnish of 100 mm might be satisfactory, which can easily be reached with abrasive paper or diamond paste. If light elements are determined, a 10–50-mm surface roughness should be claimed and ﬁne abrasives are required to obtain an almost mirrorlike surface. Striations, even after polishing, cause a reduction of the ﬂuorescence intensities, which is more serious for long wavelengths than for short ones (Bertin, 1978; Tertian and Claisse, 1982). Another aspect which should be taken into account is the orientation of the specimen inside the spectrometer. By simple rotating of the sample with 45 or 90 in the spectrometer and repetitive measurements, the grooves decrease the ﬂuorescence intensity more when oriented in the perpendicular direction to the incident beam than in the parallel direction. In Figure 1, a plot of the ﬂuorescent intensity versus the groove size for Al, Fe, Cu, Pb, and Mo is presented (Mu¨ller, 1972). Also, here it is obvious that the reduction of the ﬂuorescence intensity is much more severe for elements with low Z numbers. However, today, most spectrometers are provided with a spinning mechanism, which averages this eﬀect. For a spectrometer without a spinning mechanism, the sample should be placed in such a way that the grooves appear parallel to the incident radiation in order to minimize the eﬀect. Several mathematical models have been developed to determine the relationship between surface roughness and emergent analyte line intensity (Berry et al., 1969; Mu¨ller, 1972). However, simple mechanical polishing is not always possible or even advisable. Particularly, when such elements as Si, Al, or Fe are to be determined, it is sometimes diﬃcult to ﬁnd a suitable abrasive which does not contain one of these elements and does not introduce contaminations into the sample. SiC and Al2O3, for instance, are very Table 1 Measured radiation MgKa TiKa CuKa CuKa ZrKa

Mass Absorption Coefﬁcients and 90% Transmission Path Lengths for Some Elements Matrix composition

m-Matrix (cm2=g)

r (g=cm3)

x90% (mm)

Fe Cr Al U U

5430 144 50 306 57

7.87 7.19 2.70 18.7 18.7

0.02 1.02 7.7 0.18 0.98

936

Schmeling and Van Grieken

Figure 1 Change in relative fluorescent intensity of massive samples as a function of groove size a for different elements, calculated for pure elements. (From Ref. 8. Reprinted by permission of A. Hilger, Ltd.)

common and eﬀective abrasives, but both contain elements of high analytical interest. In such cases, electrolytic polishing or etching might be more useful. When the surface of the specimen is too rough (e.g., in the case of turnings) and suitable polishing and milling is not practicable, the samples can be prepared properly by pressing them into pellets with a hydraulic press at several hundreds of mega Pascals. With such a procedure, most of the turning are compacted into satisfactory smooth surfaces. If the sample is composed of soft, malleable multiphase alloys, smearing eﬀects of softer components like Pb or Al cause problems. The soft elements will be enriched onto the surface and the ﬂuorescent intensities of the softer phases increase, whereas those of the harder ones decrease (Tertian and Claisse, 1982). In such cases, the sample should be ﬁnished by electrolytical polishing or etching, instead of mechanical polishing. In the case of metallic specimens with irregular shape, it is possible to embed the piece in a special wax resin (e.g., acrylic resin and methyl acrylic resin). The resulting block can be polished and ground to appropriate surface smoothness with abrasives like SiC and diamond paste before analysis (Wybenga, 1979). If the surface of the sample is not representative for the bulk composition, due to heterogeneity, corrosion, smearing, and so forth, it is necessary to treat the sample in diﬀerent ways. Bronze alloys can be prepared, for example, by etching with a solution of alcoholic ferric chloride (120 mL C2H5OH, 30 mL HCl, and 10 g FeCl3), and brass alloys by a solution of aqueous ammonium persulfate [100 mL H2O and 10 g (NH4)2S2O8]. For archeological objects, however, it is not recommended to treat them by etching because many inclusions could be dissolved and lost. Another possibility is the complete decomposition or dissolution of the sample. However, such a radical method of sample preparation should only be carried out when other, less aggressive preparation techniques are not suitable for obtaining satisfactory results. It should always be taken into account that a digestion or dissolution step contributes an additional source of contaminations or losses and is quite time-consuming (Tscho¨pel et al., 1980). The procedure and the necessary reagents must be carefully selected in order to get the highest beneﬁt. An eﬀective and common method is the digestion

Sample Preparation for XRF

937

of the sample with aqua regia (HCl:HNO3, 3:1). Most of the alloys and metals, even Pd and Pt, can be attacked by this reagent. The strength of this combination is based on the formation of chloro complexes during the reaction and to the catalytic eﬀect of Cl2 and NOCl. The eﬀectiveness can be improved additionally when the solution is allowed to stand for 10–20 min before heating to 60 C. So-called inverted aqua regia (HNO3:HCl, 3:1) is an eﬀective combination to oxidize sulfur- and pyrite-containing samples. Another agressive decomposition reagens is the combination of HNO3=HF. Here, the complexing eﬀect of the ﬂuoride ion is utilized and, with this combination Si, Nb, Ta, Zr, Hf, W, Ti, Nb–Sn alloys, FeNb, Al–Cr alloys, FeSi, FeZr, Cu–Si alloys, FeP, FeW, as well as Al–Si and Ca–Si alloys can be dissolved. Table 2 shows some possible combinations and their applications (Bock, 1979). For the determination of traces in Nb, Mo, W, Ti, and Ta, Eddy and Balaes (1988) dissolved the metallic samples in HF=HNO3, and extracted the Ta by reversed-phase extraction chromatography using tributyl phosphate. They pipetted the sample solution either directly onto a ﬁlter paper or coprecipitated the trace elements with In. The detection limits were between 0.05 and 0.7 mg, depending on the element to be determined. Semiconductors as well as PbTe and GeTe can be destroyed by a combination of concentrated HNO3 and 10% sodium oxalate solution (1:2) (Fano and Zanotti, 1974). However, some elements may be lost by application of a digestion step. Especially elements like Se, Hg, and Sn form volatile components with various kinds of acid and their combinations. Samples treated with hot mixtures of HNO3=HF=HClO4 for example, lose Se and Cr completely, and Hg, As, Ge, Te, Re, Os, and Ru show particular losses (To¨lg, 1962). HCl alone is often not aggressive enough to attack metals and metallic alloys but might show satisfactory results in some cases. For the determination of Cu in W alloys, the samples are dissolved with 50% HCl, and the solution decanted from the undissolved W powder. The undissolved W is washed with water, and both, the received solution and the wash, are combined, diluted, and adsorbed on a piece of ﬁlter paper. Subsequently, the dried ﬁlter is presented to the instrument (Ning and Zhao, 1995). Sometimes, less common dissolution reagents are useful in XRF. Trace levels of Zr, Mo, Hf, and W in Ta metal can be determined after removal of the matrix elements with diantipyrylmethane and collection of the impurity elements onto ion-exchange paper (Knote and Krivan, 1982). For the determination of Co in Fe, Kato (1990) dissolved the sample with acid peroxide, mixed it with citrate, and percipitated Co from the solution of pH 7.0 with 2-nitroso-1-naphthol.

Table 2

Some Examples of Dissolution Reagents and Their Applications

Reagents

Reaction

Attacked elements

Non-attacked elements

HNO3

Oxidation

Cu, Ag, Bi, Tl, Te, Pb, Hg, Cd, V, U, alloys

Au, Pt metals, Al, B, Cr, In, Nb, Ta, Th, Ti, Zr, Hf

HNO3=HCl (1:3)

Oxidation with complexation Oxidation with complexation

Au, Pt, Pd, Mo, stainless steel, most alloys Nb, Ta, Mo, W, Ti, Sn, Sb, Si, Zr, Hf, Fe-alloy steel, Fe alloys

Rh, In, Ru, Ti

HNO3=HF

HCl

Solution

Au, Pt metals

938

Schmeling and Van Grieken

Other techniques are the extraction of elements from the metallic compound. Lobanov et al. (1991) extracted Al from Cu-, Ni-, and Zn-based alloys with C17–C20 carboxylic acids. An almost complete extraction of Al (99%) at 60–80 C was obtained. Another, relatively new technique, is the remelting of the sample (Coedo and Dorado, 1994; Coedo et al., 1994; Cobo et al., 1995). This method is particularly useful for the determination of Cr in Mo- and Nb-ferroalloys. Such alloys are not suitable for direct analysis by XRF, due to their inhomogenity and brittleness. With application of the remelting technique, the original ferroalloy is ground to grain sizes of 2 mm with a steel disk mill, followed by a W mill, after previously crushing with a jaw beaker. Subsequently the grained particles are mixed fairly with a portion of pure Fe (15 g FeMo alloy þ 25 g pure Fe, and 12 g FeNb alloy þ 28 g pure Fe) and this ﬁnal mixture is melted inside an induction furnace. Afterward, the melted metal is centrifuged and cast in an appropriate mold prior to analysis. Calibration samples can easily be obtained in the same way. The analysis of semiconductors is mainly carried out by TXRF or SRXRF; however, there is also an interesting way to apply XRF for the determination of Ni in Si wafers. Bubert et al. (1991) measured the Ni implantation of the wafers directly by XRF, but the calibration standards were prepared in an unconventional way by spiking the gelatincontaining Ni standard on parts of the Si wafer. The XRF analysis was performed after drying of the sample. B.

Powdered Specimens

A common technique for geological, industrial, and biological materials is the preparation of powders and pellets. Powdered specimens are prepared when the original sample is too heterogeneous for a direct analysis or too brittle to form self-supporting disks or when a suitable surface ﬁnishing is not possible. Powders can be derived from diﬀerent materials: solids like minerals, rocks, ores, slags, and so forth, metal ﬁllings, metal oxides, precipitates, and residues from solutions, ground fusion products, ion-exchange resins, and ashed or freeze-dried biological materials. Some of the materials are already available as powders, but most of them must be pulverized by crushing, grinding, and milling. The powders are presented to the spectrometer directly as loose powders, packed in cells or spread out on ﬁlm materials like Mylar1 or other foils and as pellets or briquettes. Sometimes, fusions (see Sec. III) of the powdered material may be a more suitable alternative, especially when particle size eﬀects are severe and need to be overcome. However, the preparation of a powdered sample requires much less time than the preparation by using a fusion technique. The analysis of a powdered specimen shows several advantages over bulk materials: Powdered samples allow the addition of a standard and low- and high-absorption dilution. The heterogeneity of the material will be reduced to a certain level, and standards can be prepared easily in the same way. However, if the sample consists of elements with diﬀerent grinding properties, which is often the case for mineralogical and geological samples, additives must be added to overcome such a problem. In general, 2–10% additive to aid in the grinding, blending, and briquetting process is suﬃcient for most of the materials. To fulﬁll the condition of a homogeneous sample surface with inﬁnite thickness, on which most of the correction algorithms are based (see Chapter 5), the sample must be converted into a very ﬁne powder, otherwise particle size eﬀects can occur and interfere with the quantitative analysis. For very ﬁne particles, the ﬂuorescent intensity is constant, but it decreases with increasing particle size. This eﬀect is more signiﬁcant for long wavelengths than for short or intermediate ones. Furthermore, the degree of packing of

Sample Preparation for XRF

939

the specimen inﬂuences the ﬂuorescence intensity additionally. However, this can be reduced by the preparation of pellets and by briquetting the sample; a higher pressure and a longer pressing time for the pellets give an additional improvement. Segregation and, hence, heterogeneity eﬀects can arise when the particles are not of the same size, shape, or gravity. This problem occurs particularly when the sample is analyzed in the form of a loose powder. Claisse (Claisse, 1957; Claisse and Samson, 1972) proposed a qualitative interpretation for the origin of such heterogeneity eﬀects in some publications. The eﬀects can be a major source of errors because large variations in ﬂuorescence intensity may occur and the inﬂuence becomes more and more detrimental with increasing wavelength; sometimes, errors of about 50% are possible (Berry et al., 1969; Rhodes and Hunter, 1972; Madlem, 1976; Holynska and Markowicz, 1982). To avoid severe errors, it is always recommended to tap the powdered sample for better mixing of the powder prior to analysis. Diﬀerent mathematical methods for the correction of particle size eﬀects have been developed. However, with only few exceptions (see Chapter 6), they are not very useful, as most of them require an a prior knowledge of the particle size distribution in the sample, which is usually unknown. 1. Grinding For the preparation of ﬁne powders, the original sample should usually be split and milled into smaller pieces, which are then suitable for further grinding. Splitting in smaller subsamples can be carried out by various equipments, like jaw mills or crushers. The main objective at this stage is to obtain a representative subsample from the original material and to avoid sampling errors due to wrong handling or treatment. For further grinding of the sample to appropriate small sizes, various methods and devices are available. The most common equipments are disk mills or shatterboxes, in which samples of smaller sizes than 6 mm can be ground further. However, both steps present a source of contaminations, as most of the grinding tools contain elements, which might be of interest in the sample. These elements can be introduced easily as blank values during the grinding process and, therefore, the material of the grinding tools should be selected carefully for each particular application. Agate, zircon, steel, alundum, and WC are the most common materials for such vessels. All of them can introduce various blank values. Agate, for instance, introduces traces of SiO2, Mg, and Ca to the specimen. This might be of less interest for geological materials, which often contain these elements as main compounds, but for biological material, this represents a major source of errors (Van Grieken et al., 1980). However, agate and zircon equipments insert still lower blank values to the sample in comparison to other materials. Unfortunately, agate is very expensive and brittle, and for this reason, it is mainly used for the preparation of small sample amounts by manual grinding in a mortar. With zircon devices, only blanks of Zr and Hf occur, which are negligible for most of the applications. Despite its good mechanical properties, the use of WC vessels is only partially recommended. During the sample preparation, substantial amounts of W, Nb, Ta, and Co can be introduced to the sample (Hergt and Sims, 1994). Especially the W contamination can give serious problems because the L and M lines of this element interfere with the elements of Z numbers between 28 and 35. However, despite this drawback, WC is the most favorable material for the preparation of hard alloys due to its material properties. Steel is also a frequently used material, and the preparation of rock samples with bakelite phenolic resin in a steel mill showed good results (Longerich, 1995). However, a contamination with Fe, Cr, Ni, and Mn is regular and can cause severe errors.

940

Schmeling and Van Grieken

When grease and oil-forming materials are analyzed, the selection of a suitable vessel material should be done carefully, as well. The preparation of these samples should be carried out in vessels with smooth, nonporous surfaces, like agate. Otherwise, parts of the sample can cling to the surface of the device and losses will occur. Sieber (1993) obtained homogenized samples of greases by mixing them with a chemplex binder in a high-speed rotar. The mixture was heated up during the procedure and was analyzed directly after cooling down as a cake. Oil samples can be ground with MgO and baked afterward at a temperature of 270 C. The received solid cake is suitable for direct analysis (Liu et al., 1992). Reagents or additives are used during the preparation procedure for diﬀerent reasons. Fine abrasives like Al2O3 or SiC are often added as grinding aids to accelerate the process (Adler and Axelrod, 1995). The addition of wet grinding aids like ethyl or isopropyl alcohol leads to a better homogeneity and higher eﬃciency. Calcination in an airstream at 800 C and subsequent grinding to a ﬁne power yields good results for materials like Pt or Pd (Gokhale and Wuensche, 1990; Norrish and Thompson, 1990). Materials that are diﬃcult to treat, like rocks or ores, can be prepared much faster and more eﬃciently after addition of sodium stearate. Normally, grinding is very fast and takes only a few minutes, but it should always be taken into account that the determination of light elements requires a ﬁne-particle size. The eﬀect of grinding time or particle size on the ﬂuorescent intensities of the elements Si, S, K, Ca and Fe in coal samples prepared by the powder technique (5-g sample was ground in WC rotary swing mill together with a 1-g boric acid binder and 100 mg sodium stearate) is shown in Figure 2a (Wheeler, 1983). From this ﬁgure, it is clear that a grinding time of around 6 min will reduce the particle size in a coal sample to the order of 50 mm, but no further increase in intensity is obtained for longer grinding times. 2. Loose Powders After grinding, the specimen can be analyzed directly as loose powder or further pressed into a pellet. For the irradiation as a powdered specimen, the ground sample is transferred into a special cup and covered by Mylar foil. The cup should be ﬁlled in such a way that inﬁnite thickness is assured, which means 75% in most of the cases. The covering ﬁlm should be made of a material with homogenous density in order to avoid scattering eﬀects. If a sample will be irradiated several times, as in the case of standards, such ﬁlms tend to become brittle and may break inside the XRF unit. To avoid this, the ﬁlm should be changed regularly. Another possibility for the analysis of a powdered specimen is the preparation of a thin ﬁlm by dusting the powder on adhesive tape (e.g., Scotch1) or onto foils. This can be carried out manually or, with higher beneﬁt, by using the so-called ‘‘puﬀ technique’’ (Bertin, 1978). A special device was developed for such purposes: The device is ﬁlled with the powdered sample and subsequently the powder is dispersed into ﬁne dust by blasting small air volumes inside the fairly evacuated chamber of the equipment. This dust is then deposited onto a ﬁlter or frit by further evacuation of the chamber. A further approach is the performance of a slurry analysis. By mixing the ﬁnepowdered sample with a solution of 2–5% (w=v) nitrocellulose in 1 mg=L acetate, a suspension is formed which can be transferred on a microscope slide or other supports (Bertin, 1978). A simpler approach is slamming of a slurry solution with water and ﬁne powder and sucking this suspension through a ﬁlter (Nuclepore1, Mylar, etc.) with ﬁne pore sizes, to obtain a thin and very homogenous layer (Wybenga, 1979; Araujo et al. 1990). A major drawback of this method is the loss of water-soluble elements. To avoid these problem, nonpolar solvents, like hexane, are sometimes used. An alternative

Sample Preparation for XRF

941

Figure 2 (a) Grinding time versus intensity. (b) Pelletizing pressure versus intensity. (From Ref. 30. Reprinted by permission of Plenum Press.)

procedure includes mixing the samples ( 1 g) with a few milliliters of double-distilled water and grinding to a ﬁne suspension. From the suspension, a 0.5-mL fraction is immediately pipetted onto Mylar foil. Careful evaporation at 80 C results in a target containing about 2 mg=cm2 of sample material. Microscopic photographs showed that the sample is quite homogenous with a grain size below 10 mm (Sauer et al., 1979; Van Grieken et al., 1979).

942

Schmeling and Van Grieken

3. Pelletizing Pelletizing of the powdered samples is often required for the reduction of surface effects and to yield better precision than with loose powders. The process can be carried out very fast and easily. The required tools are only a hydraulic press and a suitable die set, including a die body, base, a plunger, and two polished metal disks. When the disks are made of WC, they also allow the pressing of hard alloys and abrasive materials. For pelletizing, the powdered sample is transferred into such die and pressed under pressures of mega Pascals for several seconds up to 1 or 2 min. For most of the materials, this will be suﬃcient to reach a mirrorlike surface and to smooth scratches and turnings. The eﬀect of the pelletizing pressure on the ﬂuorescent intensities for a coal sample is displayed in Figure 2b. The coal powders were pelletized with a boric acid backing at 250 MPa. With ﬂuorescence measurements, the optimum pelletizing pressure was determined. The light elements such as Si and Ca show a much higher dependence on the pressure than, for example, Fe. However, other light elements like K and S seem to have much less dependence on pressure than Si and Ca. This might be an indication that signiﬁcant diﬀerences in the particle size of these elements are present in the coal sample (Wheeler, 1983). Binders are often added to the mixture to help form stable pellets and prevent caking of the sample at the die surface. However, in the selection of a suitable binder, some regulations should be carefully considered. First, binders should show low contamination, particularly for the elements of interest, have low absorption for all wavelengths of interest, and be stable under air, vacuum, and x-ray irradiation. The necessary amount of binding material must be chosen carefully as well, because binders are normally composed of light elements or organic materials and scatterred x-rays increase the background of the spectrum. Binders also dilute the sample, which might be a serious problem for the determination of trace elements. A recommended mixture is the addition of 2–10% binder to the sample. In some special cases, higher concentrations of the binder can be accepted. Bassari and Kumru (1994), for example, used a combination of 50% soil sample and 50% cellulose binder in their work. Diﬀerent types of binders are available; most of them also act as grinding aids and can be added during the grinding procedure. Typical binders are cellulose, starch, lucite, urea, boric acid, graphite, KCl, and so forth. (Bertin, 1978; Frechette et al., 1979; Tertian and Claisse, 1982). Liquid binders like ethyl alcohol or diethyl ether can be mixed with the sample manually, whereas it is recommended that binders be mixed mechanically for the formation of a homogenous mixture. Liquid binders have the advantage of evaporation after the mixing process. Finally, a stable pellet is formed and only minor amounts of binding material are still present in the specimen, which can almost be neglected. To accelerate the evaporation process, careful heating by an infrared (IR) lamp can be performed. If the heating process is carried out too fast, cracks may develop inside the pellet and this can weaken the whole structure. In the last years, several new binders, mainly polymers, have been recommended; for example, Pb-containing pulp samples were mixed with poly(vinyl acetate) binder in order to form a stable disk (Volodin and Uranov, 1990). Other polymers are poly(vinyl alcohol), poly(vinyl pyrrolidone) (PVP), or methyl methacrylate (Bettinelli and Taina, 1990; Domi, 1992; Wilson et al., 1995; Watson, 1996). Watson (1996) developed a simple and fast method for pellet preparation by mixing PVP and microcrystalline cellulose binder in a

Sample Preparation for XRF

943

new polyethylene bag. The obtained moist mixture is then pressed to a pellet and dried. For the determination of traces in geochemical samples, elvacite (a copolymer of methyl n-butyl methacrylate) was selected as the binder (Ingham and Vrebos, 1994). In one recent application, provenance studies were performed of coastal and inland archeological pottery in South Africa. Here, the ground samples were mixed with movariol as binder and analyzed (Punyadeera et al., 1997). For pressing a pellet, the powder–binder mixture is poured into a metal support, which has only the analysis surface open to prevent cross-contamination and to extend the life time of a pellet. The support is usually made of Al and painted on the sides. The exterior paint acts as an antifriction substance in the press mold, reducing scoring of the die. Before pressing, the container is tapped several times in order to reduce air pockets. Another alternative for obtaining a stable pellet is pressing after the addition of boric acid. In this case, the powder mixture is transferred into the die and ﬂattened by a plunger. The die has been ﬁlled earlier with a small cylinder, which acts as a spacer and prevents backing of the powder at the rim. The used plunger is made of Plexiglas to allow the observation of the ﬂattening process. The inset cylinder is then removed carefully, which leaves a gap between sample and die. Boric acid crystals are poured in this gap and on top of the sample, and the whole assembly is pressed. In some cases, heating of the sample during the mixing procedure with binders can occur and volatile components may evaporate. Special mixing machines with cooling mechanisms or slower milling procedures are in use instead of ring and puck mills for prevention of such losses (Domi, 1992). Another procedure can be applied when only small amounts of the sample are available. The powdered sample is placed as a thin layer on top of a backing material layer and the pellet can be subsequently pressed in an easy way. To check whether a specimen is inﬁnitely thick, the sample is covered by a small disk made of Cu or other metals, depending on the elements to be analyzed, and then pressed. The presence of Cu lines in the spectrum indicates that the sample is not inﬁnitely thick and the pressing procedure should be repeated with more material. For pressing of the ﬁnal pellet, some precautions need to be considered. Normally, the used pressure is about 15 tons, which is suﬃcient for the formation of stable pellets for a long time. The pressure should be released carefully to avoid breaking or backing of the pellet, and the removal of the pressed pellet from the press should be carried out with caution for the same reason. If the pellet is too brittle, it may be broken or pulverized again during rough handling. However, if oily or waxy samples are pressed, they may ﬂow under the pressure and form coatings at the surface of the pressing tools. This can be avoided by the addition of components, which stabilize such pellets. Sometimes, the sample structure itself might be destroyed by unsuitable pressures or too long exposure. This is often not visible immediately after pressing, but after storage of the pellet for longer times, problems might occur. Also, conical fractures can occur inside the pressed pellet and weaken it. Unfortunately, such fractures are also not always visible and breaking out of the pellet inside the instrument during the evacuation process might be the result. For the prevention of these unwanted eﬀects, the pellet can be exposed to vacuum in a desiccator before transferring it into the XRF unit. In their study, Novosel-Radovic and Malijkovic (1993) monitored the deterioration of a pressed sample as a function of time. They found that briquets ‘‘age’’ with time, resulting in the formation of surface cracks.

944

III.

Schmeling and Van Grieken

FUSED SPECIMEN

In 1956, Claisse proposed preparing fused samples for analyses by XRF in order to overcome the problems arising with powders and pellets (Claisse, 1956, 1957). This technique has become one of the most successful methods for the preparation of specimens in x-ray ﬂuorescence and can be applied to almost all types of samples (Feret, 1990, 1993). With a fusion, all particle and mineralogical eﬀects, which cause serious problems for analysis of powdered and pelletized specimen, become negligible and the analysis shows a better accuracy (Banajee and Olson, 1978; Frechette et al., 1979; Tertian and Claisse, 1982; Metz and Davey, 1992). The basic procedure consists of heating a powdered sample together with the ﬂux to such high temperatures that the ﬂux melts, dissolves the sample, and reacts with it by the formation of a one-phase disk. Agitation during the fusion process is essential for fusions with low ﬂuidity, in order to avoid the formation of bubbles and to receive a homogenous sample (LeHouillier and Turmel, 1974). The reaction time varies between 5 and 30 min depending on the sample composition. For a 30-mm disk 6–7 g material (ﬂux and sample) is required. For most applications, the ﬂux=sample ratio varies between 2:1 and 10:1 (Bower and Valentine, 1986; Eastell and Willis, 1990; Kvaratskheli and Kornienko, 1992; Ossaka et al., 1994; Spangenberg et al., 1994). In some special applications this ratio may be diﬀerent (Haukka and Thomas, 1977; Fink et al., 1993; Sato and Kohori, 1995). One of the major advantages of the fusion technique is the possibility of adding a heavy absorber for the reduction of matrix eﬀects, especially for the determination of light elements. Other advantages are the relatively simple preparation of synthetical standards and the wider calibration range (Feret, 1990, 1993). The main disadvantage of this technique presents the dilution eﬀect of the sample, which renders the determination of traces, especially of light elements, more diﬃcult, as their ﬂuorescence intensity is decreased. However, in the last years, several attempts were carried out to overcome this disadvantage [e.g., by application of the low dilution fusion technique (LDF) (Haukka and Thomas, 1977; Thomas and Haukka, 1978; Lee and McConchie, 1982; Eastell and Willis, 1990; Maruta et al., 1992; Yamamoto et al., 1995) or a semilow dilution technique (Malmqvist, 1998)]. Further disadvantages include the higher costs in comparison to the powder methods and the more complicated preparation procedure of the fusion. Some experience is required and most of the analysts have their special ‘‘tricks’’ to obtain the best results. Today, several automated ﬂux preparers are commercially available with more or less sophisticated technology (Stephenson, 1969; Schroeder et al., 1980; Kvaratskheli and Kornienko, 1992). For some elements like S, however, the briquetting technique is preferred and oﬀers better accuracy, because high losses due to volatilization during the fusion procedure occur. A.

Fluxes and Additives

Several ﬂuxes, with diﬀerent properties, are in use for the dissolution of the samples and the formation of glass disks. The ﬂuxes are mainly composed of light elements, which do not interfere strongly with the elements to be determined and form stable glass beads (Muia and Van Grieken, 1991). Borate ﬂuxes, containing metaborate or tetraborate, dissolve almost all materials except metallics, sulﬁdes, and organic materials. The choice for each application is dependent on the sample composition. Common ﬂuxes are lithium tetraborate (Li2B4O7), lithium metaborate (LiBO2), sodium tetraborate (Na2B4O7), and sodium phosphate (NaPO3)6. Low-temperature ﬂuxes are potassium pyrosulfate (K2S2O7) and sodium carbonate (Na2CO3).

Sample Preparation for XRF

945

Usually, the Li ﬂuxes are preferred to the Na ones. They produce more ﬂuid ﬂuxes and have lower mass absorption coeﬃcients; this inﬂuences the ﬂuorescence intensity less and is essential for light elements. Their disadvantages are the higher reaction temperatures and higher hygroscopicity. Lithium tetraborate is an eﬀective reagent for attacking basic oxides, like carbonates or Al oxide. However, its glass beads are cracking easily and the disks should be handled with care. Lithium metaborate forms glass beads with excellent mechanical properties and attacks specimen containing high concentrations of oxidic acids (e.g., silica). Often, mixtures of the lithium ﬂuxes are used, which can be more eﬀective, as they combine the properties of the single components. The addition of Li2CO3 or LiF increases the acidity or basicity as well as the reaction speed and ﬂuidity. With a mixture of sodium metaborate and sodium nitrate (NaNO3), samples which are diﬃcult to attack by lithium reagents (e.g., chrome-containing ores and refractories) can be dissolved (Sear, 1997). NaNO3 and boric acid react with geological materials like casserite (SnO2) and chromite (FeO Cr2O3) (Couture et al., 1993). However, Na ﬂuxes tend to wet and stick to the crucible, which makes it necessary to clean the crucibles regularly. In Table 3, some examples for ﬂux compositions and ﬂux=sample ratios are given.

Table 3 Flux base

Some Examples for Flux Compositions and Typical Applications Flux composition

Properties

LiBO2

LiBO2=Li2B4O7 4:1

LiBO2

LiBO2=silica 22:3

LiBO2

LiBO2

Li2B4O7

LiB4O7=LiBO2 12.5:10 Aggressive

Li2B4O7

Applications

Good mechanical Silica or alumina major components properties (Istone et al., 1991), CaO (Alvarez, 1990) Ceramics with lead zirconate (Sato and Kohori, 1995) High fluidity

Coal ash, rock samples (Sweileh and Van Peteghem, 1995) Cr-containing refractories (Giles et al., 1995)

Bead cracks easily Metal oxides, basic oxides, basic metal sulfides, rocks, carbonates, silicates, zeolithes (Luke, 1963; Le Houillier and Turmel, 1974; Eastell and Willis, 1990; Oishi et al., 1995, Rutherford, 1995)

Li2B4O7

Li2B4O7=Li2CO3

Ferro-alloys, paper pigments (LeHouillier et al., 1976; Alvarez, 1990; Muia and Van Grieken, 1991)

Li2CO3

Li2CO3=NH4NO3 20:1

Feores (Sato and Kohori, 1995)

Na2B4O7

Metal oxides, rocks volcanic ash, MgO, chrome-containing ores (Dow, 1981; Kvaratskheli and Konienko, 1992)

Na2B4O7 Na2B4O7=Na2CO3 3:1

Slags, sinters, metal oxides (Muia and Van Grieken, 1991)

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To overcome the problem of attacking the crucible material, especially by sulﬁdes and partially oxidized organic compounds, the sample can be preignited or an oxidant added (Martin and Richardson, 1992). Slags from Cu-smelting processes can be analyzed after dissolution in sodium tetraborate combined with lanthanum oxide (La2O3) as a heavy absorber and NaNO3 as an oxidant to prevent the attack of the crucible by Cu (Le Houillier and Turmel, 1974). Other examples for oxidants are BaO2, CeO2, KNO3, and LiNO3. Sulfur-, chlorine-, and ﬂuorine-containing samples tend to stick in the crucible and shorten their lifetimes. Preliminary roasting of the samples minimizes this problem (Dow, 1981). To avoid losses of elements, like sulfur and organic compounds, the sample can be calcinated for several hours in a muﬄe oven at 750–800 C prior to fusion (Pella et al., 1982; Norrish and Thompson, 1990; Muia and Van Grieken, 1991). Also, for the determination of sulfur in fusion samples, the compound should be oxidized completely to sulfate (Baker, 1982), as diﬀerent oxidation stages show slightly diﬀerent element lines in the spectrum. However, if the sample contains more than 1% sulfur, the loss of volatilization is too high to get reliable results. Under a highly oxidizing atmosphere during fusion, sulfur can be retained quantitatively; for example, with ﬂux compositions like Na2B4O7=NaNO3 (80:20) or Li2B4O7=LiNO3 (80:20) (Staats and Strieder, 1993). Diﬃcultto-attack sulﬁde-bearing ores can be dissolved by a combination of NaNO3=NaKCO3 and Li2B4O7 (Birch et al., 1995; Rao and Govil, 1995). Sometimes, La2O3, barium peroxide (BaO2), barium sulfate (BaSO4), or cerium oxide (CeO2) is added as a heavy absorber in order to minimize matrix eﬀects, which are particularly serious for long-wavelength x-ray lines (Pella et al., 1982; Norrish and Thompson, 1990; Muia and Van Grieken, 1991). Barium, for example, behaves as a heavy absorber in the analysis of rock samples for the major elements in glass beads (Bennett and Olivier, 1976). Some of them, like barium peroxide or barium nitrate [Ba(NO3)2], also act as oxidants. Catalysts as manganese oxide (MnO2) may be added to accelerate the oxidation procedure as well. However, the addition of any supporting reagent should be carefully estimated before analysis because each additional compound increases the risk of contamination and decreases the amount of elements, which could be determined without line interferences. As mentioned, one of the major drawbacks of the fusion technique is the dilution eﬀect of the sample. This makes it often diﬃcult or even impossible to determine trace elements in the specimen. For this reason, the low-dilution fusion technique was developed (Haukka and Thomas, 1977; Thomas and Haukka, 1978; Lee and McConchie, 1982; Eastell and Willis, 1990; Maruta et al., 1992; Yamamoto et al., 1995). Here, the ﬂux-tosample ratio is 2:1 (ﬂux:sample) (i.e., lower than usual). Most often, LiBO2 is applied as a ﬂux, as it is more reactive and forms ﬂuxes with higher ﬂuidity, which ensures better homogeneity. Especially in geological samples, relevant trace elements are often present with concentrations of less than 1000–1500 ppm. These low concentrations cause problems with the conventional fusion process because of its high dilution. On the other hand, most of these samples are diﬃcult to grind and pelletize. With the application of the low-dilution fusion technique and selection of a suitable ﬂux, these problems can be overcome. B.

Fusion Procedures

For the fusion procedure, the sample should be ground with at least 150-mesh grain sizes to yield a fast and homogenous fusion. For samples that are diﬃcult to fuse, like

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947

refractories, grain sizes down to 200 mesh are recommended in order to obtain a homogenous fusion and accelerate the entire fusion process (Norrish and Hutton, 1969; Hutton and Elliot, 1980). The sample is weighed out and mixed fairly. The ﬂux, if consisting of diﬀerent compounds, is also mixed well and transferred into the crucible. The sample is placed on top of the ﬂux and both are mixed. A selection of diﬀerent materials for the crucible is available (Giles et al., 1995); common ones are Pt, Pt–Au alloys (95%:5%), Pt–Rh–Au alloys (85%:15%:5%), zirconium grain-stabilized platinum (ZGS), and Ir or C vessels. The latter ones act also as oxidants, but C vessels are easily attacked by sulﬁdes and other reducing compounds. Preferable to Pt is the Pt–Au alloy, as it is harder and more diﬃcult to degrade than the pure metal. This combination is nonwetting and relatively long lasting. Recently, Ir crucibles became available, which show good properties, such as corrosion resistance to all metals, insolubility in acids, even aqua regia, and maintenance of their properties under normal atmosphere up to 1600 C. The fusion procedure can be carried out in high-frequency induction furnaces, gas burners, or muﬄe ovens. The ﬁrst two show the advantages of higher temperatures and faster heating. The sample is previously heated until the glass-forming agent is melted; with a further increase of the temperature to 1050–1100 C for 10–15 min, the ﬂux reacts with the sample, forming a glass bead. Agitation during the fusion process is recommended for fusions with low ﬂuidity in order to avoid bubble inclusions (Luke, 1963; Harvey et al., 1973; LeHouillier and Turmel, 1974; Pella et al., 1982; King and Vivit, 1988; Norrish and Thompson 1990). The melted fusion is casted into a preheated mold, for the formation of a disk to be analyzed, and allowed to cool down. For easier removal of the fusion from the crucible and mold, special ‘‘antiwetting’’ agents can be added. Iodides and bromides of light alkali and alkaline earth elements are favorable for this purpose. With application of these agents, the crucible shows a much longer lifetime and its cleaning is much easier. Only a few droplets of these agents are required to be added on top of the sample=ﬂux mixture before heating. Sometimes, the fusion is annealed at 200 C for some minutes in order to avoid thermal shocks (Bower and Valentine, 1986), but, usually, the mold is preheated to the same temperature and the melt can be transferred without suﬀering. For a reliable analysis, it may be required to clean the disk with alcohol and polish it with abrasives. The resulting disks are stable over a long time but should be stored in a desiccator under vacuum or in other moisture-free environments. Borates are hygroscopic and extended exposure to ambient air may destroy the disk’s surface. Various additives may be used for diﬀerent reasons, according to the sample behavior during the fusion procedure. The addition of glass-forming agents like SiO2 or Al2O3 is required to obtain a more stable disk when a higher amount of alkaline earth elements is present in the sample. Transparency-increasing agents (e.g., LiF, NaF, or KF) are added for higher ﬂuidity of the fusion and to assure better homogeneity. For the fusion of organic material, preignition for several hours at 450 C is necessary to volatize all CO2. Otherwise, a weight loss will occur during the fusion, which cannot be compensated for by mathematical calculation. Evaporation of moisture or crystal water can also introduce some weight loss. If this loss is similar for both the sample and ﬂux, it can be calculated using the method of Hutton and Elliot (1980) or compensated for by the addition of SiO2, if less than 20% (Harvey et al., 1973). Any additional mathematical correction procedure is then superﬂuous. The losses of volatiles and=or ﬂux usually increase with increasing temperature. King and Vivit (1988) developed the following simple method for the prevention of losses in geological samples: The powdered sample was dried for 2 h at 105 C and 2 g was

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weighed exactly and calcinated in a muﬄe furnace at 925 C until a constant weight was achieved. Cleaning of the mold can be carried out by reheating with a nonwetting agent or inserting the crucible into a warm HNO3 bath (Tertian and Claisse, 1982). Sometimes, the fused disks are ground and pelletized. For the determination of Nb and Ta in geological samples, the fused disks were pelletized with cellulose powder for a reliable analysis (Basu Chaudhury et al., 1987). Also, for the analysis of submarine polymetallic nodules, the samples were fused by the application of the low-dilution fusion technique and later pelletized without binder (Cai et al., 1992). The high-dilution technique of Tertian and Claisse (1982) was investigated for the determination of SiO2 in powdered plant materials. In this case, a 100-mg sample was dissolved in 9.9 g lithium metaborate ﬂux (Garivait et al., 1997) and satisfactory results could be achieved. Applications were also carried out in the ﬁeld of ceramic materials. One study dealt with the determination of Si as SiO2 in Si3N4. Here, the ground material was ﬁrst pyrolized at 1500 C before fusing with LiBO2 or a mixture of LiBO2 and B2O3 (Kaiser et al., 1995). When only small sample amounts are available, as in the case of air-dust samples, microbeads might be formed with LiF and Li2B4O7 (Moore, 1993).

IV.

LIQUID SPECIMEN

Liquid samples represent an almost ideal specimen for analysis by x-ray ﬂuorescence. They are homogenous, particle eﬀects are eliminated, and the obtained analytical results are representative for the whole sample. Furthermore, normally the matrix consists of elements with low atomic numbers, where absorption eﬀects are small and enhancement of the analyte lines is negligible. High absolute sensitivity is observed because the penetration depth of the primary beam is high and low absorption of the emitted secondary radiation exists. For the analysis of liquid-specimen special cells, made of stainless steel, polyethylene, or polytetraﬂuorethylene (PTFE or Teﬂon), are in use, into which the liquid sample is poured. These cells are usually covered by Mylar foils with thicknesses of 3– 6 mm; a lower thickness ensures less background scattering but is mechanically less stable than the thicker ones. Sometimes, also microporous ﬁlms are used, which allow the equalizing of pressure diﬀerences between the liquid cell and the environment by being permeable for air only. For the quantiﬁcation of liquid samples, internal standardization can be applied, but also calibration standards in each concentration range and blank samples for the background evaluation are easily prepared. Solids, which are problematic to analyze, such as powders, pellets or fusions, can be converted into liquids by dissolution and digestion or, if trace analysis is required, the traces might be extracted with a suitable solvent from the bulk material. The major drawback of the analysis of liquid samples is the highly scattered background. This makes it diﬃcult, or sometimes impossible, to determine light elements at low concentrations and it increases the detection limits. Another disadvantage is the deterioration of standards with time. In contrast to standards prepared by fusion or pelletizing, liquid standards change their concentration with time due to adsorption and=or precipitation eﬀects. Therefore, they should be prepared fresh or renewed each few days in order to avoid systematic errors in quantitative XRF analysis. Also, the samples and the standards may be altered by irradiation. During irradiation, the sample can be heated up

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949

and evaporation might occur, which inﬂuences the concentrations of the elements in the sample. Bubbles can be formed by heating of the sample and inﬂuencing the analyte-line intensity. Sometimes, precipitation occurs during the irradiation process, and the precipitate can be deposited at the bottom and change the properties of the sample. Typical detection limits for the energy-dispersive XRF analysis of liquid samples with 30 min counting time are in the ppm range (Van Grieken et al., 1976), which is satisfactory for several applications. Chlorine and iodine were determined by Morse (1992 and 1994) directly in photographic developer and ﬁxer, respectively, with cadmium standard solutions in polypropylene cups. Relatively high concentrations in the microgram per gram level of Fe, Ni, and V in crude oils can be determined easily by using XRF (Shay and Woodward, 1991). In the last few years, the ﬁeld of direct applications expanded, particularly in the oil sector (Kendall et al., 1995; Kira et al., 1995; Lyamina et al., 1995; Untenecker et al., 1995). It appears that XRS is an excellent technique for the analysis of various kinds of oil from highly raﬃnated greases (Untenecker et al., 1995) to petroleum products and used oils (Kendall et al., 1995; Kira et al., 1995; Lyamina et al., 1995). However, in most cases, the elemental concentrations in liquid samples (e.g., in environmental waters) are too low for a direct analysis of traces and the samples must be preconcentrated prior to analysis. In general, all preconcentration techniques, which are applied to other analytical methods, are also practicable for XRF analysis. Preconcentration has the additional advantage that possible matrix eﬀects can be reduced or even avoided (Van Grieken, 1982), and the accuracy of the results is improved. Unfortunately, a preconcentration step is often time-consuming, involves the risk of contamination and losses, and may not always be speciﬁc for each elemental species. Ideal for the presentation to XRF are thin-ﬁlm targets, in which the x-ray intensity is proportional to the mass of the element on the ﬁlm. To obtain a thin-ﬁlm specimen from suspended matter in liquid samples, the liquid sample is ﬁltered through a membrane ﬁlter—preferable Nuclepore or Millipore—and dried. Vanderstappen and Van Grieken (1976) found that ﬁltration through a Nuclepore ﬁlter leads to interference-free detection limits of 5–10 ng=cm2 for Mn, Ni, Cu, Zn, As, and Br, 10–20 ng=cm2 for Ti, V, and Cr, and about 50 ng=cm2 for K and Ca by using EDXRS and counting times of 2000 s. Recently, Civici (1994) used the ﬁltration method for the separation of suspended material from seawater. For the internal standardization, the standard element is added before ﬁltration and deposited together with the suspension onto the ﬁlter surface in order to get a homogenous distribution. X-ray ﬂuorescence also oﬀers an ideal technique for on-line process control in the industry using a continuous ﬂow of the sample through a sample chamber. Ceasy (1994) applied this for the on-line analysis of molten metal samples with a portable EDXRF instrument, equipped with two radioisotope sources, namely 55Fe and 244Cm, and a ﬂow cell. Another on-line application was carried out by Davidson (1994): he determined As, Cu, and S in solutions from a lab-scale Cu electrolyte puriﬁcation cell. In fact, the XRF analysis of liquids increases from year to year. Numerous applications are cited in the review articles by Ellis et al. (1996, 1997) and Bacon et al. (1995). A.

Physical Preconcentration

Physical preconcentration methods are practicable for the analysis of rainwater, sewage, and wastewater, and several diﬀerent approaches have been carried out to remove the liquid matrix in various ways. A simple and fast process of preconcentration is the

950

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evaporation of the liquid, leaving the elements to be determined as a residue on a surface, which can be analyzed directly. This technique shows several advantages because of its easy handling and direct approach. The probability of contaminations is low because no complicated and extra sample preparation steps are required. The costs are reasonable: only suitable carrier materials are necessary and all nonvolatile compounds remain quantitatively on the surface. Even large amounts of water can be evaporated or freezedried, and the residue is later mixed with binding materials, pelletized, and analyzed. Freeze-drying of 250 mL wastewater on 100 mg graphite followed by grinding and pelletizing of the residue leads to detection limits of 5 mg=L for many elements with accuracies around 10% with EDXRF (Smits and Van Grieken, 1997). Several attempts have been made to deposit analyte solutions on diﬀerent surfaces. In general, all materials with a smooth surface texture composed of low-atomic-number elements are suitable. Especially, Nuclepore ﬁlters and Mylar foils show low-background and blank values for XRF analysis. However, there are also some drawbacks to this technique. Volatile species and elements get lost during the evaporation step. In the case of high salinity or hardness, a formation of ﬁnite crystals will occur, which introduces particle size eﬀects. Special background correction methods need to be applied for diﬀerent types of water sample. Despite these problems, evaporation is still a popular technique for the preconcentration of various liquids. Often, the samples are pretreated or predissolved in solvents in order to increase their capability of spotting. For the determination of U and Ce in nuclear fuel, the sample was ﬁrst dissolved in nitric acid solution and, subsequently, microdroplets of this solution were spotted onto a ﬁlter. The XRF analysis of the dried ﬁlter showed an accuracy of better than 2% (Hanif et al., 1995) for the elements of interest. Contents of less than 1 mg=g of Ca, K, S, P, Cl, Mg, and Na could be determined in liquids, by spotting such a nitric acid solution on a highly pure polycarbonate membrane (Go¨diker et al., 1990). Mylar foil is one of the most popular substrates for the deposition of samples (Baryshev et al., 1995). Several interesting studies were carried out by use of this medium. Oil sample, for example, can be deposited after homogenization with an organic solvent, leaving a thin ﬁlm glued onto the Mylar (Civici, 1995). Other substrate materials are polypropylene and polycarbonate ﬁlms. Cations (Ca, K, Mg, Na) and anions (Cl and SO2 4 ) of drinking water were preconcentrated onto polypropylene foil. The addition of a polysaccharide (GELERITE) reduced the hydrophobic eﬀect of the foil and, thus, detection limits from 0.12 mg=L for Ca to 1.78 mg=L for SO2 4 were claimed (Warner et al., 1995). The advantage of this technique is the possibility of a simultaneous determination of cations and anions. In another application in this direction, the sample is transferred as a droplet onto a polycarbonate ﬁlm, which is cemented on a 35-mm plastic photographic slide mount. After evaporation, the resulting salt deposit is more or less circular. Detection limits in the microgram per liter region could be reached (Meltzer and King, 1991). Particularly, thin-ﬁlm supports (like the type A2) have recently been developed by Process Analytics (Orem, UT). The ﬁlm can accumulate a small amount of solution in a small spot because the ﬁlm is treated to have a hydrophilic center in a hydrophobic ﬁeld or the ﬁlm has a dimple in its center. These ﬁlms are clean and resistant to high temperatures. Wastewater samples with low salinity can be preconcentrated by a special vapor ﬁltration procedure (Luke, 1963; Rickey et al., 1977) in which the sample is deposited into a container, with a cellophane foil bottom. The foil is permeable for water vapor, but not for liquid water or dissolved material. Subsequent evacuation of the container leads to deposition of the dissolved solids onto the membrane or foil, which is then presented for analysis. In combination with particle-induced x-ray emission (PIXE), detection limits in

Sample Preparation for XRF

951

the range of 0.1–3 mg=L were claimed. However, this method shows several disadvantages: It is time-consuming and samples with high salinity cannot be applied successfully. Crystal formation can occur, even with very low salinity, which has to be corrected by special matrix correction procedures (Van Grieken, 1982). Sometimes, metal surfaces can also be used for the deposition of extracted solutions. A thin-ﬁlm technique was applied for the determination of SO2 4 and Cl in small volumes of saline samples, deposited onto suitable machined Cu disks (Dhir et al., 1995). Other procedures are a nebulization technique (Giauque et al., 1977) and a multidroplet spotting technique (Camp et al., 1974), which result in a fairly homogeneous distribution of the sample onto the surface. The evaporation of larger samples can be carried out by impregnation of the ﬁlter material (Whatman 41) with a sample solution. A wax ring 29 mm in diameter keeps the spotted sample solution inside the area to be irradiated when the water is evaporated by passing an airstream from underneath. The reported detection limits were below 50 mg=L for 2000 s analyzing time, with optimum secondary target excitation (Giauque et al., 1977). Water samples can be properly analyzed by TXRF without any preconcentration step. Some microliters of the sample are pipetted directly onto a highly polished quartz glass carrier and evaporated. The detection limits achieved with this techniques are one to sometimes two orders of magnitude better than for conventional XRF. However, if samples with higher salinity are analyzed, a separation step is also required. Probably the best solution, when high precision and low detection limits are necessary, is a sample preparation involving a chemical preconcentration or separation step. B.

Chemical Preconcentration

Chemical preconcentration may be grouped into three main classes: 1. (Co)precipitation 2. Ion exchange 3. Chelation and sorption immobilization Numerous applications to diﬀerent water samples have been established in these three groups (Camp et al., 1974; Rickey et al., 1977; Bruninx and Van Ebergen, 1979; Andrew et al., 1982; Chakravorty and Van Grieken, 1982; Van Grieken, 1982; Becker et al., 1985; Hirayama and Leyden, 1986; Shan et al., 1988) and most of them include a ﬁnal step in which the preconcentrated sample is deposited onto a surface (ﬁlter or foil) by sorption or ﬁltration. This surface serves as sample carrier and is subsequently exposed to the x-radiation. 1.

(Co)precipitation Methods

Precipitation methods are quite popular for the trace–matrix separation and they have been used in analytical chemistry for a long time. They can be carried out very quickly and with little eﬀort. Selective precipitation of one or two species is usually the main task and much of the time is dedicated to ﬁnding suitable reagents for this purpose. However, in the case of XRF as a multielement technique, it is much more important to obtain a precipitate containing all elements of interest. Many attempts were carried out in this direction and numerous reagents with diﬀerent properties are practicable for this. A simple and traditional method is the application of inorganic reagents like Fe or Al hydroxide, which coprecipitate many elements at selected pH values. Sulﬁdes are less common,

952

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because of the higher solubility of their precipitates. However, most of the inorganic reagents show an incomplete precipitation and a strong dependence on accurate pH values. More common and eﬀective are organic components, which form complexes of very high stability. Often a spike element is added, acting as a carrier and an internal standard at the same time. The most suitable organic reagents are the carbamates (Van Grieken, 1982). They form very stable and strong metal chelates. Luke (1968), for example, investigated, using his ‘‘coprex’’ technique, the coprecipitation of trace elements by sodium diethyldithiocarbamate (NaDDTC) in the presence of a suitable metal ion spike. In a further work, Kessler and Vincent (1972) improved the detection limits, obtained by Luke (1968), by ﬁltration of the received NaDDTC=hydroxide precipitate and analyzing of the residue with a highly collimated XRF setup. Toxic elements were successfully determined in plastic compounds for toys by applying NaDDTC. The samples were ﬁrst mineralized with molten NaOH. Then Cd, Ba, Sb(III), Cr(III), Hg, Pb, and As(III) were coprecipitated at pH 8.5 with NaDDTC and sodium rhodizonate and Fe3 þ ions as the carrier. The results achieved by XRF were comparable with the AAS results (Gimeno-Adelantado, 1993). When a high sample throughput is required and all samples are of the same kind, automation might be possible. For such purposes, Tanoue et al. (1979) developed an automated device for the preconcentration of trace metals in wastewater samples with DDTC and subsequent XRS measurement. Ammonium pyrrolidine dithiocarbamate (APDC) acts more eﬃciently, especially for concentrations below 10 mg=L at a pH of 4. APDC is the most successsful nonspeciﬁc reagent for the preconcentration of traces at this pH in comparison to other reagents. Quantitative recoveries are obtained for a dozen elements in water, independent of the alkaline earth element content. Various elements can be also determined in seawater, using APDC as reagent. A small amount of Mo as carrier ion is added to the ﬁltered seawater sample, then the pH is adjusted at 4, and a portion of the 1% APDC solution is added. After 15–20 min the precipitate is ﬁltered oﬀ, dried, and analyzed. Ulrich and Hopke (1977) achieved detection limits between 4 ng=cm3 for Br and 80 ng=cm3 for Ca by application of this method. In another approach, traces of soluble metals were separated from estuarine waters. In the ﬁrst step, the water was ﬁltered and acidiﬁed to pH 3. Fresh APDC solution (2%) was added and then the complexes ﬁltered. The obtained detection limits were between 2 and 5 m=L (Mazo-Gray et al., 1997). Quantitative recoveries are also received by application of dibenzyl dithiocarbamate (DBDTC) for Mn, Fe, Co, Ni, Cu, Zn, Se, Sb, Hg, Tl, Ag, Cd, and Pb with detection limits around 1 mg=mL for 100-mL samples (Watanabe et al., 1986). The very low solubility of this reagent also eliminates the need for a metal carrier (Lindner et al., 1978; Ellis et al., 1982). A detection limit of 10 mg=L could be reached for Mo in water samples by preconcentration with DBDTC at pH 3 (Saitoh et al., 1984). 1-(2-Pyridylazo)-2-naphthol (PAN) represents another attractive reagent for several water samples. This organic compound is soluble in hot water and also in ethanol, but not in cold water. This behavior can be utilized for the separation of special elements. In one study, 2 L of water were heated to 70 C and a 20-mg portion of PAN, dissolved in ethanol, was added. After cooling down, the precipitate was ﬁltered oﬀ and the obtained ﬁlter target was transferred to the XRF unit. With this procedure, about 15 cations could successfully be preconcentrated and detection limits about 0.5 mg=L achieved (Vanderstappen and Van Grieken, 1978). Other organic reagents, which are less known than the carbamates, are rhodizonate and tannin. Bauer-Wolf et al. (1993) applied these reagents for the coprecipitation of rare

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953

earth elements (REE) in geological samples. The samples were previously decomposed and the alkaline earth elements removed by an ion-exchange procedure. Subsequently, the REE were precipitated at pH 13 with rhodizinate and tannin, the residue collected onto a ﬁlter, and analyzed. Tannin, in combination with methyl violet, was used for the determination of Hf, Nb, Ta, and Zr in geological samples after digestion with HCl. The detection limits of the ﬁltered samples were in the range of 0.1–0.4 mg (Zhang and Ma, 1994). Oxine or 8-hydroxychinoline is a very common reagent for the complexation of elements in water, but its chelates are much less stable than the carbamate chelates. Despite this, it can be successfully applied for the preconcentration of Cu, Fe, Mn, Ni, and Zn in sea salt. Therefore, the salt is ﬁrst dissolved in HNO3 and ﬁltered through a membrane ﬁlter. After addition of Al as the carrier element, and oxine and thionalide solutions, the sample is ﬁltered again and subsequently analyzed. Iwatsuki et al. (1996) achieved detection limits from 0.01 mg=g for Mn to 0.05 mg=g for Fe with this procedure. Less eﬃcient than the organic chelates, and therefore less popular, for the application with XRF are inorganic reagents (e.g., Fe hydroxide and Al hydroxide). Aluminium hydroxide might be selected for the coprecipitation of trace metals in environmental waters. The optimum procedure involves the addition of a 1-mg Al(OH)3 carrier to a 200mL water sample at pH 7.3. Quantitative recoveries are obtained for Ti(IV), Cr(II), As(V), Pb(II), and Th(IV). The detection limits lay in the range between 0.2 and 0.8 mg=L (Eltayab and Van Grieken, 1992). In one study, a combined technique of organic and inorganic precipitants was used for the determination of As(III) and As(V) as well as the total arsenic content in plant material (peach tree). The samples were ﬁrst treated by acid digestion and then the As was separated and preconcentrated by DBDTC. Finally, the arsenic components were precipitated Fe(OH)3. Detection limits lower than 0.1 mg=g and a precision better than 5% were achieved (Pascual Brugues and Cortazar Castilla, 1995). Sometimes, other inorganic compounds are also applied for the multielement (co)precipitation and subsequent analysis by XRF; many references of earlier studies can be found in Van Grieken (1982). If only a few or even a single element should be separated, reagents, which form selective complexes of low solubility with the elements of interest, are required. The main problem, which must be solved, is the prevention of coprecipitation of other, interfering elements; accurate chemical conditions play an important role for such selective procedures. Tanaka et al. (1987) determined arsenic and antimony quantitatively after their reduction to As(III) and Sb(III). First, the elements were treated with KBr and HCl at 80 C for 1 h. Both elements were then coprecipitated at pH 9 with a solution of ZrOCl2. Detection limits of 0.3 mg=L for As and 6.1 mg=L for Sb were achieved. Sometimes, even a speciation is possible by careful adjustment of the reaction conditions. Trace amounts of V(V) and V(IV) can be quantitatively recovered from water at pH 1.8 and 4.0, respectively, by reaction with NaDDTC. However, the concentration of Fe, Co, Ni, Zn, and Pb should not exceed 100 mg=L in order to avoid interferences and receive a quantitative recovery (Hirayama and Leyden, 1986). Microgram quantities of Cr(II) and Cr(VI) were successfully separated and determined by Pera¨niemi and Ahlgren (1995). In the ﬁrst step, Cr(VI) is isolated from the original sample by collection onto zirconium-activated charcoal at pH 3.9 and the charcoal is ﬁltered. The ﬁltrate is spiked in a second step with FeCl3 and the pH is increased to 9. Subsequently, the formed precipitate is mixed with activated charcoal and ﬁltered. Detection limits better than 0.05 mg=L are reached for both species. PdCl2 serves as the precipitation agent for the determination of iodine in urine and water samples after digestion with chromic acid. The iodate was converted into iodite by Na2SO3 and ﬁnally precipitated as Pbl2 (Mwaura et al., 1994).

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However, it should always be kept in mind that inferences can occur by the presence of organic material or high concentrations of alkaline elements. Most of the organic material should be destroyed by acid treatment, but, often, stable components (e.g., humic substances) are diﬃcult to digest and still present in the solution. Sometimes, the organic content is so high that it is recommended to decompose the sample prior to further treatment. For the separation of high alkaline earth element contents, an ion-exchange or masking step might be useful before further precipitation. Another problem arises from elements which are present in the sample in diﬀerent chemical states. Some of the element species may be incompletely enriched or even escape from the enrichment procedure. Cr3þ, for example, is collected quantitatively by Fe hydroxide (Osaki et al., 1983; Mullins, 1984; 3þ species can Ahern et al., 1985; Leyden et al., 1985), whereas Cr6þ escapes as CrO2 4 . The Cr be then quantitatively collected by DBDTC or APDC in the presence of Co as the carrier (Ahern et al., 1985; Leyden et al., 1985). This fact, however, can also be utilized for the species analysis of Cr. In general, some drawbacks are present for each preconcentration procedure and therefore a selection of the optimum method requires a careful consideration. 2. Ion-Exchange Methods The most popular techniques for the enrichment of traces from liquid samples are the diﬀerent ion-exchange methods. A very large number of studies dealt with various types of ion exchangers, applied as resins packed in columns, in liquid form or impregnated onto ﬁlters. Ion-exchange resins are available in the form of macroporous beads, membranes, impregnates onto ﬁlters, or foamed plastics. The analyte, from which the traces should be separated, can be taken up by diﬀerent processes. One easy and fast solution is the batch procedure, where the analyte solution is mixed with the ion exchanger in one container. The whole solution is ﬁltrated or decanted after some reaction time, and the remaining residue is washed several times to eliminate possible impurities. The dried powder might be briquetted for analysis or spread out onto a support material. Another common application of ion-exchanger materials is the ﬁltration of a liquid sample through a ﬁlter, previously impregnated with an ion exchanger or the column technique. However, conventional ion exchangers are of less interest for the enrichment of elements from liquid samples, as they show only limited capacity and are relatively unspeciﬁc by the means of elements. In the presence of high alkali or alkaline earth element content, for example, the resins can easily be overloaded and most of the required elements pass either through the column or are only partially enriched. Despite of these drawbacks, the acid or basic resins are still quite popular for the enrichment of traces from samples with low alkali and alkaline earth element content. The most well-known chelating resins are Dowex A-1 and Chelex 100, containing iminodiacetate functional groups for the complexation. Both have a wide range for the collection of transition elements, and Co, Cu, Mn, Ni, Pb, Rb, Sr, and Zn are adsorbed quantitatively from water samples. Chelex 100 also shows a rather high tolerance of alkali and alkaline earth element content in natural water samples (Florkowski et al., 1976; Clanet and Deloncle, 1980). An anion-exchange separation of gold with Dowex 2-X10, after acid digestion of gold containing ores, enabled the quantitative removal of the interfering elements Hg and Zn for this application. The analysis of the dried and homogenized resin by XRF, using the AuLa line for quantiﬁcation, gave good results (Cojocaru and Spiridon, 1993). Also, other Dowex resins (e.g. Dowex 168 with 100–200-mesh grain size) are applied for the separation of rare earth elements from alkaline earth metals in geological

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samples. The removal of interfering elements, after digestion of the original sample by HF, HNO3, and HCl, is achieved by mixing the solvent with the anion exchanger. The rare earth elements are retained on the ion-exchange column while the alkaline earth elements pass through. The column is washed several times with HNO3 and propanol-2 to remove residuals and, ﬁnally, the rare earth elements are eluted by HCl (Bauer-Wolf et al., 1993). Rock samples with low concentrations of REE are enriched successfully by using ﬁbres impregnated with m-acetylchlorophophonazo. The rock sample is ﬁrst acid digested and then a masking agent containing EDTA, CDTA and EGTA, and ascorbic acid is added. Portions of this solutions are allowed to pass though the impregnated ﬁber in the form of an ion exchange column. Subsequently, the loaded ﬁber is digested with HNO3 and HClO4 and, ﬁnally, the precipitate is obtained with MgCl2 in aqueous ammonia. Gong et al., (1995) reached detection limits of 0.05–0.21 mg=g with this procedure. Speciation and preconcentration can be performed by using a mixture of Ag1-X8 (in Cl 7 form) and AG5OW-X8 (in Naþ form) and Chelex 100 beads, in order to separate Cr(III) and Cr (VI) (Prange and Kno¨chel, 1985). For a quantitative separation of transition metals from most water samples the ionexchange material should show less aﬃnity to alkaline and alkaline earth metals, but should be selective for the trace metals of interest. Several attempts have been carried out in this direction (Leyden and Luttrell, 1975; Burba and Lieser, 1977; Ducos-Frede et al., 1995). Polyamine–polyurea resin columns, for example, were prepared from tetraethyepentamine and toluene diisocyanate for the preconcentration of Ni, Cu, and Zn from seawater at neutral pH by Leyden and Luttrell (1975). Burba and Lieser (1977) used Cellulose hyphan eﬀectively as the complexing agent for the enrichment of Cu and U from seawater, freshwater, and mineral water samples. The reactive agent was received by immobilization of 1-(2-hydroxyphenylazo)-2-naphtol on cellulose powder and by diazotization of o-aminophenolcellulose and subsequent coupling with b-naphthol. Uranium traces might be also trapped in another way—by treatment of a Dowex 168 resin with aqueous KMnO4 solutions. In this case, amorphous manganese oxides were generated in the reticulated structure of the resin. After elution, the sample is presented on a thin Mylar foil to the XRF unit. Furthermore, the resin has interesting properties for the separation of inorganic micropollutants, which are frequently present in water, without enrichment of macrocomponents (Ducos-Frede et al., 1995). However, all preconcentration procedures based on ion-exchange columns suﬀer from several drawbacks. The resins or eluates obtained by the enrichment step need to be treated further before they can be presented in a suitable form to the spectrometer. Resins are usually ground, mixed with binder, and pelletized prior to analysis. Eluates must be ﬁxed by ﬁltration through a membrane or even a coprecipitation step is required. All of these procedures prolong the time of the sample preparation and include a risk of contamination and losses. Of much higher interest are ﬁlters, which are impregnated with an ion-exchange resin or speciﬁc agent for the enrichment of elements. The loaded ﬁlters can be then presented directly to the spectrometer, and any further sample pretreatment is superﬂuous. Therefore, the risk of contaminations and losses can be kept very low and time-consuming preparation steps are avoided. 3 2 2 2 2 3 2 AsO3 3 ; AsO4 ; CrO4 ; MoO4 ; SeO3 ; SeO4 ; VO2 ; and WO4 from diﬀerent water samples were successfully preconcentrated onto cellulose ﬁlters, containing 2,2-diaminodiethylamine as functional groups. Unfortunately, samples with high salinity cannot be enriched quantitatively, as more than 0.01 M NaCl disturbs the enrichment capability of the ﬁlters (Smits and Van Grieken, 1981).

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Filters impregnated with ion exchangers, containing sulﬁte and quarternary ammonium as functional groups, were used in a study for monitoring lake water. The water was ﬁltered through the ﬁlter membrane and the loaded membrane was directly exposed to the x-radiation. Such membranes might be also useful in providing general ‘‘ﬁngerprint’’-type information about trace element contents for river systems (Edwards et al., 1993). Uranium in natural water can be quantitatively separated by a simple cellulose phosphate disk (Minkkinen, 1977). Toxic elements from water samples are collected on sorption ﬁlters, pretreated with carboxylic acid and dithiocarbamate as complexing agents (Tsizin et al., 1993). For the multielement determination in natural waters, an interesting method of ﬁlter impregnation was investigated by Varshal et al. (1994). So-called DETATA ﬁlters, containing conformationally ﬂexible aminocarboxylic groups as highly effective sorbents bound to cellulose matrix, were established for the preconcentration of trace elements of natural waters. The ﬁlters were ﬁrst loaded with NHþ 4 for preconcentration and activation of the ﬁlter surface. The sample, consisting of snow, rainwater, or even seawater of high salinity, was dissolved in HCl solution before presenting to the ﬁlter surface. A monitoring of trace elements in underground and wastewater might be also possible with this technique. The use of a cascade of ﬁlters under dynamic ﬂow conditions renders it possible to preconcentrate suspended substances and trace elements from the solution in only one step. Earlier studies, dealing with the enrichment of trace elements from environmental samples onto ﬁlter surfaces, were mainly focused on ion-collecting papers (Van Grieken, 1982). However, most of these applications suﬀer from unselectivity and low capacity for the preconcentration of transient elements in large sample amounts, because alkaline and alkaline earth elements are also trapped. Moreover, their capacity of enrichment for small samples is too low to receive relevant detection limits for environmental samples. Hyphan prepared by pelletizing of cellulose ﬁber material as thin layers is of more interest for large volumes. Good recoveries for trace metals are found from large volumes of waters at pH 7. The detection limits usually lay in the range of 1 mg=L (Burba and Lieser, 1979). Selective analysis for Co, for example, can be performed, after washing the Hyphan-loaded ﬁlter with 3 M HCl. The washing procedure removes all other metals, except Co. The amount of 0.4 mg=L Co can be determined, even in the presence of 1 mg=L Fe (James and Lin, 1982; Coetzee and Leiser, 1986). 3. Chelation and Sorption Immobilization Methods For most of the analytical methods, the extraction techniques are popular, as they supply solutions which can be often transferred directly into the sample chamber of the instrument. For XRF, however, this is normally not possible because most of the element concentrations in the extracted solutions are still too low for a reliable analysis. Reversed-phase extractions, in which organic chelation agents are ﬁxed to a solid phase, are much more convenient for the measurement by XRF. For such a method, the adsorbent and the sample solution are mixed together in a vessel. Agitation of the solution during the reaction process ensures faster adsorption. Subsequently, the sorbent is separated from the solution by ﬁltration or decantation. For the removal of undesired traces, the sorbent can be washed with appropriate solutions several times and then dried or pelletized. The dried ﬁlter or pellet is then directly presented to the spectrometer. Activated charcoal is well-known and often used as adsorbent. This material is a good adsorbent for organic and colloidal materials and for the trace metal species bound

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to such materials. With the addition of a chelating compound to charcoal suspensions, free ions can be adsorbed quantitatively as well. Vanderborght and Van Grieken (1977), for example, applied 8-quinoline as a chelating agent. The optimized preconcentration procedure consists of the addition of 10 mg 8-quinolinol per liter of water sample at pH 8 and the application of 100 mg precleaned activated carbon to this solution. After ﬁltration of the suspension, quantitative recoveries were achieved for 20 ions from diﬀerent media, with enrichment factors near 10,000, more or less independently of the alkali and alkaline earth element content. Pera¨niemi et al. (1994) investigated zirconium-loaded activated charcoal as an adsorbent for As, Se, and Hg. The activated coal was added to an aqueous solution, containing these three elements; then the solution was stirred, allowed to stand, and, ﬁnally, ﬁltered through membranes. The ﬁlters were carefully dried at room temperature in order to avoid losses, placed between Mylar sheets, and irradiated. A selective determination of Se(IV) in various water samples can be carried out after the reduction of Se(IV) with ascorbic acid to elemental Se and subsequent adsorption onto activated carbon. After ﬁltration, the ﬁlter is analyzed by EDXRS. Detection limits of 0.05 mg=L and a precision of 6% were found by Robberecht and Van Grieken (1980). Foams represent another attractive extraction medium for the separation of traces from diﬀerent bulk materials. Carvalho et al. (1995) determined Ga from bauxite or Albased alloys as GaCl4, with polyurethane foam serving as sorption medium. The material was ﬁrst dissolved with HF–HCl, then evaporated to dryness and dissolved again in 6 mol=L HCl mixed with TiCl3. Finally, polyurethane foam, pretreated with HCl, was added and the solution ﬁltered after some minutes of reaction time. In combination with organic chelation agents, polyurethane foam is even more eﬃcient. Mercury was determined in diﬀerent matrices and chemical states by Braun et al. (1984) after preconcentration on DDTC-loaded polyurethane foam disks. The preparation of the disks was carried out by the addition of a 4% solution DDTC in chloroform (assuring a 20% concentration of the regent on the disk) and the addition of 0.2 mL dinonyl phthalate plasticizer. After adjustment to pH 5, 25 mL of the sample solution were shaken with the foam disk for 1 h to separate Hg. Extraction eﬃciencies between 88% and 100% were reported for 0.2 and 2 mg=L concentrations. Gold can be extracted with tributyl phosphate as solid extractant from ore samples. Therefore, the element is enriched from pulps after aqua regia leaching. For the determination by XRF, a back-extraction with thiourea solutions at 100 C is required; to obtain ﬁnally a thin sample, the thiourea solution can be placed inside a polyethylene cylinder, supplied with a ﬁlter paper disk covering the bottom, and evaporated. For a 25-g sample, the detection limits were at 10 ng=g (Dmitriev et al., 1991). The extraction of diﬀerent elements from coal samples is carried out in three steps, and each step is based on the diﬀerent solubilities of the diﬀerent trace elements. In the ﬁrst step, the ground coal sample was treated with deionized water 24 h, the residue removed and vacuum dried. In the second step, ammonium acetate is added to the sample and this suspension is heated to 70 C for 24 h. The resulting solution is ﬁltered and rinsed with water, and the obtained residue freeze-dried. In the last step, the remaining part of the coal sample is mixed with HCL and heated again to 70 C for 24 h. After vacuum drying, all received residues are pressed into pellets and analyzed with XRF (O’Keefe and Erickson, 1994). Total-reﬂection XRF has proven to be one of the most attractive techniques for the analysis of liquid samples. Its strength is in the determination of very small sample volumes with detection limits down to the ppt level (To¨lg and Klockenka¨mper, 1993). In the

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last years, several applications were carried out with preconcentration of trace elements and elution of small portions directly onto the sample carrier for analysis (Prange et al., 1985; Barreiros et al., 1997). In one recent application, the mineral water sample was ﬁrst preconcentrated with APDC, subsequently ﬁltered through a Nuclepore membrane, and dried and small portions of the powder transferred to a carrier for analysis (Kump et al., 1997).

V.

BIOLOGICAL SAMPLES

The determination of trace metals and inorganic compounds in biological materials is more and more important in analytical chemistry, in view of its potential for diagnostics or monitoring. Several plant materials can be used as monitoring systems for pollution and deposition of heavy metals in the environment. The interest in clinical applications of XRS increases from year to year as well. In vivo XRS is almost an accepted technique and overcomes all of the diﬃculties of laboratory treatment of the samples, like digestion, separation, or preconcentration. The material can be analyzed in situ without any manipulations or delay. For the patient, it is much more convenient, as often painful and protracted procedures can be avoided. Several studies dealt with the in vivo analysis, and the main interest is focused on the determination of heavy metals in bones. Particularly, Pb is of high interest and Rosen et al. (1993) showed that XRF represented a unique capability for a safe, accurate, and noninvasive quantiﬁcation of Pb in bones of children. The direct determination of Pb in bone gives an indication of the ongoing accumulation, whereas conventional blood studies only reﬂect the recent exposures. Several studies on the determination of Pb in tibia or other bones were carried out. They mainly dealt with the monitoring of Pb concentrations in bones of industrial workers, under consideration of the working conditions, duration of employment in this sector, lifestyle of the person, and so forth. (Bleecker et al., 1995; Cake et al., 1995; Roels et al., 1995; Ryde et al., 1995). Other heavy metals of interest were Cd and U, as studied by Bloch and Shapiro (1995) and Nilsson et al. (1995). Further in vivo investigations dealt with the concentration of various elements in diﬀerent organs. Here, the main eﬀort was in the area of kidney and lung research (Skerﬃng and Nilsson, 1992; Homma et al. 1995; Gerhardsson et al., 1995). However, the sensitivity and detection limits of the in vivo applications are still rather poor, and the treatment of the patient is not always easy. Furthermore, a highly focused and energetic x-ray beam is required for a direct exposure of the sample and to penetrate through the tissue material, whereas, normally, such applications were carried out using synchrotron radiation as the excitation sources. More conventional is the determination of traces in organic samples, like plant materials, tissues, and so on. For such applications, it is often required to separate the organic matrix from the traces, and several diﬀerent separation methods were proposed for solving this problem. They range from open digestions in the traditional way to fully automated and expensive equipments. Sometimes, only parts of the sample were separated and used as representative targets for the whole sample; also, separation of the sample in diﬀerent fractions, such as blood in various cellular and subcellular fractions, are used (Weber et al., 1980; Mangelson and Hill, 1981). However, in most cases, the amount of a sample available for the analysis is rather small and high sensitivity is required (Kim et al., 1987). Numerous articles and several reviews dealing with the sample preparation and analysis of biological materials were published (Bacon et al., 1990; Crews et al., 1990;

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Maenhaut, 1990; Mangelson and Hill, 1990; Taylor et al., 1994, 1995). In this section, the main and most convenient preparation techniques will be presented and discussed. A.

Physical Methods of Sample Preparation

In some cases, a suitable sample preparation is possible with little or almost no eﬀort. All biological samples which are self-supporting and of appropriate size can be often presented in their original form to the spectrometer. Examples of such applications are bones, hair, teeth, and nails. Also for plant samples, such as leaves, needles, tree rings, or wood, the sample preparation is relatively simple and fast. Specimens of this category can often be irradiated after simple procedures like cutting, grinding, or just drying. Hair samples present good examples with little or almost no preparation eﬀorts (Kubo, 1981; Toribara et al., 1982; To¨ro¨k et al., 1984). The main intention of studying hair samples lays in the deposition or enrichment of heavy elements in the whole specimen or in single segments. In the latter case, the hairs should be cut into the required segments and the segments directly mounted on a target frame for irradiation. Thinner samples, which do not have enough stability to be ﬁxed directly inside the target frame, might be placed ﬁrst on a suitable support. The main disadvantages of the direct measurement are the lack of possibility to add an internal standard and severe matrix eﬀects. Direct determination in a more sophisticated way was performed by Toribara (1995). He built a special EDXRF sample system for the determination of Hg in 1-mm hair segments of a single hair. More conventionally, Basco and Uzonyi (1995) studied the concentration of Ca, Cl, K, and S in hair by mounting the hair onto a sample support. However, this method seems to suﬀer from quantiﬁcation problems. Diﬀerences could be reported between two types of hair, but they were considered to be negligible in comparison to the measured biological variability of the material. Leaves and needles are samples which can be also analyzed almost directly. Moreover, these are samples of special interest, because they can be used as bio-indicators for monitoring pollution of air, soil, or groundwater, and together with the determination by XRS, an almost ideal combination is available (Kitsa et al., 1992; Marques et al., 1993; Vincze et al., 1993; Calliari et al., 1995; Ostachowicz et al., 1995; Somogyi and Pazsit, 1995). Pine needles are analyzed for trace elements after pulverizing and pressing into a tablet. This tablet is placed on Mylar foil and irradiated. The detection limits vary from 70 mg=g for the light elements such as K and Ca to 0.3 mg=g for the transition metals (Boman et al., 1996). In an unusual way, algae material was collected directly onto a quartz carrier from the sea. The mounted sample was washed, dried, and irradiated. However, the obtained results were more suitable for a ﬁrst screening as for a reliable analysis, because the organic matrix caused a high background signal (Boman et al., 1993). Sometimes, botanical samples are less stable and need to be stabilized by mixing with binding material (Bassari and Kumru, 1994; Omote et al., 1995; Wilson et al., 1995). Another quite common technique for ﬁxing of unstable materials is the embedding of the sample inside a resin or wax. For the determination of S in rapeseed for example, a single rapeseed was embedded into wax and analyzed by WDXRF (Schnug et al., 1993). Biological liquids, like blood and serum, might be spotted directly onto a suitable substrate without any pretreatment (Robberecht et al., 1982). Blood samples, for example, can be pipetted in 100-mL portions onto a simple paper ﬁlter for the determination of Br in blood (Hurst et al., 1994). Serum samples are treated almost in the same way, with the

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exception that instead of a paper ﬁlter, polypropylene foil should be selected as a carrier (Hurst, 1993). Urine samples as well as blood samples were pipetted onto Mylar foil and dried in a refrigerator at temperatures about 2–4 C in order to avoid losses of volatile components. Detection limits between 1532 mg=L Ca in urine and 6 mg=L for Sr in blood were claimed (Viksna et al., 1995; Hong and Ha, 1996). Several materials can be analyzed with XRF, even for trace metals, after freezedrying (lyophilization). For this procedure, the sample is kept under vacuum in liquid nitrogen for at least 1 day. Then, the residue is pulverized and pressed into a pellet for analysis. This technique has the advantage that the sample mass is reduced by a factor of 5, which results in a reduction of the spectrum background and improves the sensitivity. Soft organic material can also be stabilized in the same way. Maenhaut et al. (1984) used this technique for the analysis of human and animal tissue by PIXE. Often PIXE, TXRF, or SRXRF is applied for the analysis of biological material, as they show lower detection limits and require less eﬀort for the sample preparation than in the case of conventional XRF analysis. Lung tissue, for example, can be analyzed directly by TXRF as a microtome section mounted on the sample carrier. The sample is in a frozen state and can be cut to the required thickness for the analysis (von Bohlen et al., 1988). For a quantitative analysis, it is suﬃcient to determine the dry weight or the sample and add an internal standard. Especially for very small sample amounts of a few microliters, which are often only available for blood, serum, and plant liquid, TXRF is a very powerful technique. Blood serum can be pipetted directly onto the carrier for analysis with TXRF without any preparation (Gu¨nther et al., 1992) or after dilution with water containing the internal standard for the simultaneous determination of Br, Cu, Fe, and Zn in human serum (Yap, 1988). Xylem sap from cucumbers was also mounted directly to the carriers for analysis of Pb contaminations (Zaray et al., 1997). The Ni and Mo content of enzymatic material was determined by TXRF after dilution with 5% HNO3 for the partial destruction of the organic matrix (Fischer et al., 1996). B.

Chemical Methods of Sample Preparation

In most cases, the determination of trace elements at low levels by XRS is not possible without separation of the organic matrix. Several diﬀerent digestion, ashing, and other separation procedures were developed for this reason; they can also be applied for the analysis of such samples with other comparable analytical techniques, like AAS or ICPOES. A sample preparation method with minimum eﬀort is simple ashing of the material in a muﬄe furnace at temperatures between 400 C and 600 C. Raghavaiah et al. (1996) determined Zn in human head hair after ashing at 600 C in an electric furnace. The hair samples were ﬁrst cleaned with a mixture of acetone and deionized water for 30 min and, after drying, ashed and pulverized. The powder was spread over an adhesive foil for subsequent analysis. For the determination of Mn, Fe, Sr, and traces of Cr, Ni, and Zn as pollution elements in lake mussel shells, the collected shell samples were cleaned and separated into two diﬀerent valves before further pretreatment. Each valve was then baked in a furnace at 350 C in order to destroy the protein material. Subsequently, the material was pulverized and pressed into a pellet, which was presented to the spectrometer (Maddox et al., 1990). Sometimes only partial ashing is recommended. Havranek et al. (1986) performed this for the analysis of hair. An 1-g aliquot of hair is partially ashed in a muﬄe furnace at 200 C, the cool ash was homogenized, and a 0.3-g portion was treated with a few drops of

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polystyrene solution in chloroform and pressed into a pellet. Urine samples are analyzed as pellets as well after evaporation and partial decomposition at 220–450 C (Nielson and Kalkwarf, 1978). However, treatment of samples at high temperatures suﬀers from several drawbacks and is therefore rarely employed. The major drawback is the loss of volatile components of the sample. The addition of an ashing aid may minimize it, but the elements cannot be retained completely (Nielson and Kalkwarf, 1978; Hang and Ha, 1996). Another more sophisticated but even more expensive technique is the ashing of the biological material with an oxygen plasma. For this procedure, the ﬁnely pulverized sample is placed onto the bottom of a specially designed plasma tube made of quartz. The tube is then connected to the plasma unit and evacuated. The sample is ashed by passing oxygen through the tube, and continous cooling by a cooling ﬁnger assures that most of the volatile components remain in the sample. Selenium and arsenic can be kept almost quantitatively, but mercury shows high losses. The remaining residue is then taken up by HNO3 or a combination of HNO3 and HCL. Amounts of 50–100 mL of the solution are pipetted on a foil target of Mylar or Nuclepore and dried prior to analysis. Unfortunately, organic materials often contain high concentrations of Ca and K salts, which crystallize on the foil and cause a high-background signal. Sometimes, it is necessary to separate such salts by suitable extraction methods or to select an alternative way of sample preparation. In general, ashing in an oxygen plasma is rather fast and easy to handle. However, for complete ashing in an aﬀordable time, the sample should be pulverized to small grain sizes in order to obtain a suﬃciently large surface for the reaction with the oxygen gas. Furthermore, the reaction time depends on the material to be digested and might vary between several minutes and several hours for materials that are diﬃcult to digest (Raptis et al., 1983; Knapp, 1984). A traditional and relatively fast method for sample preparation is wet ashing with HNO3 or a mixture of HNO3 with H2SO4, or HClO4. With these mixtures, most of the organic material can be attacked (Bock, 1979). Cu, Fe, Rb, Se, and Zn in serum samples were determined after digestion with HNO3. For quantiﬁcation, Ga was added as an internal standard and a 20-mL aliquot was pipetted and dried onto a sample carrier for analysis by TXRF (Raptis et al., 1983). Even more eﬀective is a combination of HNO3 and HF, as it dissolves also silicates in organic materials. A microdigestion method was developed for the trace element determination in biological materials and human serum. For this, 500 mL of human serum were treated with 500 mL HF and dropped onto a Si-wafer chip. After drying, the chip was analyzed by XRF (Sayama, 1995). In general, a digestion procedure should be carried out in a closed vessel, as working under open conditions might cause contaminations and losses which inﬂuence the results (Tscho¨pel, 1983). Using HClO4 requires special care. This acid tends to explode when organic material is not completely oxidized. A better and safer method is the application of closed PTFE vessels or quartz tubes, which can also be used for the digestion under high pressure. However, the selection of quartz tubes is highly recommended, because the smooth surface of this material avoids adsorption eﬀects of the elements. Moreover, it can be cleaned easily by exposure to nitric acid fumes in order to avoid the introduction of contaminations. PTFE or Teﬂon might be easier to handle, but it shows higher contaminations and is more diﬃcult to clean compared to quartz because of rough surface. Zinc for example, is a well-known contaminant of Teﬂon vessels. Another problem that might occur with Teﬂon is diﬀusion from and into the walls. This is particularly serious for volatile elements and can lead to signiﬁcant mistakes in the ﬁnal data.

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For the determination of trace elements in lichen material, 200 mg of the collected samples are digested with 3 mL HNO3, 0.2 mL, H2O2, and 0.05 mL HF in a PTFE vessel. From the digested sample, targets can be prepared by pipetting 300 mL of the solution onto a Mylar ﬁlm and drying (Calliari et al., 1995). A very eﬀective method for the digestion of organic materials is the high-pressure digestion in specially designed bombs. Usually, a small amount of the pulverized sample is placed in a quartz or Teﬂon vessel and some milliliters of acid are added. The tube is then placed inside a high-pressure chamber and exposed to pressures of about 200 bar under nitrogen atmosphere, at temperatures of about 260 C. Under such high pressures, the organic material should be completely decomposed. Several drops of the resulting solution can be pipetted onto a foil or target, dried, and analyzed. The main drawbacks of this procedure are the substantial eﬀort and the requirement of a suitable instrument. Usually, one digestion needs about 3 h and includes slow heating and cooling down of the sample chamber. Furthermore, such instruments, working under high pressure, should be kept in special rooms in order to avoid accidents by possible explosions (Tscho¨pel, 1983; Schmeling et al., 1997). The destruction of biological materials with reactive gases is rarely carried out, although it seems to be quit eﬃcient for some materials. In one study for example, 90% of plant material was oxidized with HNO3 vapor within a short time and only a simple allglass apparatus was used for the vapor-phase oxidation (Thomas and Smythe, 1973). The addition of HClO4 can accelerate the reaction and ensure complete oxidation. Almost the same technique was applied by another group for the determination of Zn in brain tissue by digestion in PTFE vessels under pressure (Klitenick et al., 1983). The sample digestion by microwave plays a more and more important role (AbuSamra et al., 1975; Kingston and Jassie, 1986; Skelly and di Stefano, 1988), and the available digestion systems become better equipped. Automatic ventilation to remove acid fumes is already available in all new equipments. The interior of the oven is, in comparison to the conventional household devices, usually acid resistant to prevent rusting and to increase the lifetime of the device. Systems of 1000 W and greater power are in use for a fast and complete destruction of materials that are diﬃcult to decompose. Several metalfree materials ensure that contaminations from the vessels are kept low. With digestion vessels of tetraﬂuorometoxil (TFM), the memory eﬀects due to adsorption and diﬀusion are almost absent after two predigestion cycles of 30 min with concentrated HNO3 (Noltner et al., 1990). For vessels made of perﬂuoroalcoholoxil (PFA), a 1 day HNO3 vapor exposure is enough to reduce the blank values under the detection limits of XRS (Knapp, 1985). In using microwave digestion systems, care should be taken as well, because the pressure inside the vessels increases rapidly during the digestion procedure. Opening the vessels too soon after ﬁnishing the procedure, without careful releasing of the inside pressure, can create hazards. Especially when working with highly concentrated acids, special care is required. The vessels should be smoothly cooled down and stand for a while before removal of the sample. Usually, HNO3, in concentrated or diluted form, is recommended for the microwave digestion of organic materials. Neither mixtures of HNO3 with H2SO4 nor HNO3 with HClO4 are advised for use for decomposition by microwave. H2SO4 has a quite high evaporation point in comparison to HNO3 and a substantial increase of the pressure inside the vessel can occur during the digestion event, which might lead to explosions. HClO4 reacts explosively with incompletely reduced organic material, which can cause dangerous injuries. In several studies, explosions were reported resulting from overpressurizing the digestion bombs (Matthes et al., 1983; Fernando et al., 1986).

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Also, the relation between acid amount and sample as well as the complete reagent amount should be selected carefully. For safety reasons, it is better to start with small amounts of material and acid and increase this by careful observation of their behavior. A detailed discussion about working with microwave digestion systems can be found in the book by Kingston and Haswell (1997). The major advantage of the microwave technique is the short time required for a complete reaction. By digestion with microwaves, the sample is directly attacked and the used vessels or container do not need to be heated up. A destruction biological material can be performed within a minimum of time compared to other technique, where containers and sample chambers are included in the heating process. The wet digestion methods (by microwave or high-pressure ashing) are considerably more rapid, with reaction times of about 2–3 h, compared to at least 8 h necessary for a complete decomposition by dry ashing. A comparison of diﬀerent sample preparation methods for the analysis of organic material showed that slurry preparation needs 5 min, microwave digestion 45 min, dry ashing 24 h, and open wet digestion 36 h (Miller-Ihli, 1988). In combination with high pressure, the microwave digestion technique shows quantitative recovery for elements, which would be lost by volatilization during an open digestion. Also, traces of rare earth elements show good recoveries at the ppb level, even in the presence of organic material that is extremely diﬃcult to digest. Several studies dealt with the destruction and analysis of certiﬁed biological reference materials. Hay (V-10, IAEA) samples, for example, were digested in a microwave oven with HNO3 and H2O2. Five hundred milligrams of the powdered material were mixed with 4 mL concentrated HNO3 and 1 mL H2O2 and heated three times for 1 min each at 300 W. After each heating cycle, the vessel was allowed to cool down and the pressure released. Finally, the sample was exposed for 7 min at 300 W and 2 min at 600 W. The complete procedure needed no more than 30 min and showed good agreement with the certiﬁed results (Noltner et al., 1990). Very short digestion times of 1 min were found to be suﬃcient for the decomposition of NIST standard material Oyster Tissue (SRM 1566a) and Bovine Liver (SRM 1577a) in a closed PFA bomb with HNO3 (Stripp and Bogen, 1989). Lichen samples were studied after collection from diﬀerent places and were proposed as biomonitors. The collected material was ﬁrst dried and then acid-digested. For this procedure, 200 mg of lichen material were treated with 3 mL of HNO3, containing the internal standard, 0.2 mL H2O2, and 0.05 mL HF in closed Teﬂon vessels by microwave digestion. The ﬁnal targets were prepared by pipetting 300 mL of the decomposed sample onto Mylar foil. The detection limits were reported between 0.1 mg=L for Cu and 10 mg=L for S (Calliari et al., 1995). For the determination of Cr in seed material (e.g., barley seedlings), the samples are acid-digested using a microwave oven with 1200 W power. Then, 300–400 mg of the roots and leaves are placed into a PTFE vessel and decomposed with 3–4 mL HNO3. The sample is then presented as a thin ﬁlm mounted on Mylar foil to the EDXRF instrument (Calliari et al., 1993). A special energy-dispersive miniprobe multielement analyzer (EMMA) was developed for the determination of Pb and other traces in peats. The samples can be analyzed directly or after acid digestion in Teﬂon bombs in a microwave oven with 3 mL H2O2, 4 mL HNO3, and 1 mL HF. The comparison of both methods showed satisfying results (Cheburkin and Shotyk, 1996). Also, for the determination of animal and plant tissue by PIXE, a microwave digestion procedure might be applied. In this case, the digestion can be even performed in a

964

Schmeling and Van Grieken

conventional household microwave oven in closed Teﬂon vessels with high-purity HNO3 as the reagent. A high acid concentration of 14 M should be applied in order to reduce the dilution eﬀect of the sample and ensure suﬃcient sample material for the analysis of traces. The reaction vessels are ﬁlled with vegetable oil (7.5 mL to 100 mg of the sample) to prevent damage of the antenna. After digestion, 10 mL of the resulting solution are pipetted onto a 1.5-mm-thick Kimfol polycarbonate ﬁlm. The ﬁlm was previously treated with 14 M HNO3 and 0.05% polyvinylpyrrolydone solution to make it hydrophilic and to achieve a small spot. The results obtained by Pinheiro et al. (1989) showed an accuracy of better than 5% and a matrix reduction factor of 5. In body ﬂuids with very low trace element content, it is often recommended to enrich the traces with a preconcentration step and to separate the matrix. Cell ﬂuids and blood serum, for example, show a relatively thick layer of a low-Z-element matrix after evaporation, which increases the detection limits. In principle, all preconcentration methods, which are applicable for water analysis, are also practicable for organic ﬂuids. However, it should be considered that the amounts of such samples are usually much smaller, and working in microscale is often required. Very low contents of Cr (0.3 ng=mL) in plasma can be determined after complexation with APDC and extraction with methyl isobutyl ketone. After evaporation of the organic solvent, the residue is dissolved in acid and deposited on a thin polycorbonate foil (Simonoﬀ et al., 1985). APDC can be also applied for the analysis of hair samples. For that, the hair samples are ﬁrst digested with a combination of HClO4 and HNO3 (1:5 v=v), later distilled, and then precipitated with ADPC, whereas Y acts as the internal standard and coprecipitation agent. The resulting precipitate is ﬁltered through a Nuclepore membrane, dried, and measured. The detection limits are around 0.4 mg=g for Pb and Ni and around 2.2 mg=g for Fe (Eltayeb and Van Grieken, 1989, Eltayab and Van Grieken, 1990). For the determination of Se in serum of patients with liver cirrhosis by PIXE, Te was chosen as a coprecipitant and internal standard, and good results were reported (Cesaril et al., 1989). As for the preconcentration of traces from water samples, various kinds of ion-exchange resins have been proposed for biological samples as well. With the application of cellulose-hyphan, for example, the detection limits might be reduced by about a factor of 10, by separating the trace elements (Agarwal et al., 1975). Protein material can be preferably isolated by gel electrophoresis (Szo¨kefalvi-Nagy et al., 1987). C.

Sample Preparation for Analysis with Spatial Resolution

For the analysis of microsamples with spatial resolution special sample preparation methods were developed. They are well known in the ﬁeld of electron microprobe analysis and several books dealing with this subject are available (Hayat, 1970; Hall et al., 1974; Echlin and Galle, 1975; Reed, 1975; Erasmus, 1978; Hayat, 1980; Revel, 1984). By application of microprobe analysis, some regulations should be strongly considered regarding collection and handling of the sample in order to avoid systematic errors. For biological material, the time of sample collection is often of high importance; for example, it should be considered that by working with tissue samples, enrichment of Na, Cl, and Ca is taking place after the death of the cell and decreasing concentrations of Mg and K will inﬂuence the results also (von Zglincki et al., 1985). To avoid problems in this direction, the time delay between sampling and analysis should be kept as short as possible.

Sample Preparation for XRF

965

Furthermore, during analysis of the sample, the specimen will be transformed into another state. Usually, a high vacuum is required for the analysis, and to avoid evaporation of tissue water, these must be withdrawn or immobilized by suitable procedures. For ultrastructural studies with transmission electron microscopy, ‘‘wet chemical techniques’’ might be applied. For such procedures, the sample is chemically ﬁxed, the cellular processes are arrested, and the cell contents are immobilized. The tissue water can then be removed by immersion in an organic solvent. The dehydrated specimen is then inﬁltrated with a suitable resin for sectioning purposes. A more detailed discussion about such procedures can be found in Chapter 13 and in the works of Glauert (1974), Steinbrech and Zierold (1987), and Hayat (1989). Synchrotron-radiation XRF might be a suitable technique for microanalysis as well (see Chapter 8). Gilfrich et al. (1991) used synchrotron-radiation XRF for the determination of elements in tree rings, which are good indicators for the geological and atmospheric conditions during the growth of the tree. SRXRF oﬀers the advantage that the primary beam can be focused on small parts of the sample and sample preparation requires only polishing of the tree slices. The concentrations of manganese in living leaves were determined by Fukumoto et al. (1992). In this study, the leaf was directly placed into the instrument and analyzed. The inﬂuence of acid rain and the exposure to severe x-ray radiation of the leave was also monitored in the same study.

VI.

ATMOSPHERIC PARTICLES

The amount of publications dealing with the analysis of atmospheric particles by x-ray ﬂuorescence increased in the last 5 years and showed the suitability of this technique for such samples. XRS is still the most popular technique for this kind of specimen, especially for routine analysis, in the ﬁeld of x-ray analysis. Several studies were also carried out with TXRF or PIXE, but easy handling of the sample and less eﬀort make the conventional XRF more attractive, despite its higher detection limits. Heavy metals are nearly always present in the particulate phase of air and can be easily separated by ﬁltration or impaction. High enrichment factors are reached by ﬁltration of large air volumes, and homogeneously loaded ﬁlters are an ideal target for XRS. Nevertheless, there are several requirements, which should be considered carefully before reliable results can be obtained. The main problems which occur are bound to particle size and x-ray absorption eﬀects due to the ﬁlter media (‘‘ﬁlter penetration eﬀect’’). If the particle size distribution of a particular element is known, the particle size eﬀect can rather easily be corrected. However, in most cases, the particle size distribution is unknown or diﬃcult to measure. If this is the case, the particle size eﬀect should be kept low by selection of suitable ﬁlter materials, which collect particles on the surface like Nuclepore or Millipore ﬁlters. The absorption of x-rays by the ﬁlter medium depends on the depth distribution of the particles, which can be calculated approximately by measuring the ﬂuorescence intensities from the back and front sides of the loaded ﬁlter. The mass absorption coeﬃcient of the particles can be evaluated by transmission measurements and, later, the particle attenuation can be calculated (Adams and Van Grieken, 1975). If thin membranes, like Nuclepore or Millipore, are selected for the collection, the absorption eﬀects in the ﬁlters are small and often negligible above 5 keV, because,

966

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normally, all particles should stay at the ﬁlter surface. However, for light elements such as Si, P, S, and as well as for elements having very small particle sizes, like Pb and S, the absorption eﬀect needs to be corrected by using a suitable correction procedure. When choosing thick ﬁlters, for the collection of large particle amounts, an easy and proper way to minimize serious absorption eﬀects is the so-called sandwich geometry (Van Grieken and Adams, 1975). This technique is based on simple inward folding of the loaded ﬁlter to obtain a kind of sandwich. The heterogeneous distribution of the particles is minimized, despite the higher absorption eﬀect for the ﬁlter in this geometry, as it is assumed that all particles are on the surface and therefore in the center of the sandwich, which also averages the unknown deposition depth of the particles. Both methods (the back–front measurement and sandwich geometry) show accurate results when the absorption cross section of the ﬁlter is not excessively high. The particle size distribution on the ﬁlter surface inﬂuences the accuracy of the correction more than the depth distribution inside the ﬁlter (Davis et al., 1977). For the establishment of ﬁlter reference material for aerosol particles, diﬀerent types of ﬁlter were tested with a high-volume aerosol sampler. As conditions for the reference material were claimed, low particle size eﬀects, the ﬁlter thickness should not exceed 0.5 mg=cm2 and the lateral uniformity of the ﬁlter must be within certain limits. Fiber ﬁlters were generally excluded from this study, as they collect particles in their bulk. Only two types of ﬁlter (polycarbonate and cellulose nitrate membranes) fulﬁlled these conditions and showed an almost uniform deposition onto the surface, which was determined by analysis of diﬀerent spots from diﬀerent places at the loaded surface (Wa¨tjen et al., 1993). With ﬁltration of the atmospheric particles, usually all particle sizes present in the aerosol down to the ﬁlter cutoﬀ recorded and particle size eﬀects can easily occur. With stacked-ﬁlter units, operating with diﬀerent ﬁlters of diﬀerent pore sizes, and dichotomous samplers, some kind of size fractionation can be established. With such collection methods, the particle size eﬀects can almost be overcome, as the particles are divided into two or more fractions. For particles smaller than 2 mm, the particle size eﬀects are negligible in general, and, therefore, the determination of elements with a low Z number is also possible. If impactors with several stages are applied for the aerosol collection, the size distributions of the particles become more uniform and fulﬁll the requirements. Another possibility for overcoming eﬀects bound to diﬀerent particle sizes and to obtain reliable results is the analysis of ﬁlters as pressed pellets. For this, the loaded ﬁlters are pelletized and sometimes also previously mixed with binder and analyzed (Vigayan et al., 1997). Coal ﬂy ash, for example, was ground, mixed with poly(vinyl alcohol) binder, and pelletized (Bettelini and Taina, 1990). In one study, it was shown that pressing the sample into a pellet after grinding to a powder yields better reproducibility than applying the sample as a thin ﬁlm (Matherny and Balgava´, 1993). The preparation of fusions is another way to eliminate particle size eﬀects (Pella et al., 1978; Balcerzak, 1993). Good results were obtained by fusion of 70 mg of small dust samples with lithium ﬂuoride and lithium tetraborate (20:80). To yield a homogenous microbead, the sample was treated at 1050–1100 C for 7 min in a muﬄe oven. Subsequently, the fusion was poured into a specially designed mold and ﬁnally analyzed in a normal sample holder, modiﬁed with a spacer in order to ﬁx the sample (Moore, 1993). However, it should always be considered that each additional sample preparation step might introduce blank values to the material or dilute the sample. Especially by working in the

Sample Preparation for XRF

967

trace element region, which is usually the case for atmospheric particle samples, this can cause serious problems. One point, which should not be underestimated, is the partial or complete loss of volatile components of the sample. Especially in the wavelength-dispersive mode, voltaile compounds such as Cl, Br, and, in particular, S may be lost during evacuation of the sample chamber. This eﬀect is even more severe when working with PIXE. Hereby, the sample is irradiated with a highly focused charged particle beam under high vacuum and losses of about 50% of sulfur may occur. In addition to the losses of sulfur due to volatilization, chemical reactions can take place and lead to signiﬁcant losses of S when 2-MeV protons or 18-MeV a-particles are used (Hansen et al., 1980). For the collection of aerosols onto ﬁlters, a variety of ﬁlter materials is available. Each ﬁlter material provides special properties with respect to collection eﬃciency, mechanical stability, hygroscopy, and so forth and every selection is usually a compromise among the ﬁlter properties, the collection purpose, the available costs, and the compatibility with the analytical technique to be applied. Before choosing a ﬁlter material for the collection of atmospheric particles, the blank values of the material itself must be determined. The ﬁlter should consist of materials with low blank values for the required elements. Furthermore, the spectral background should be reasonably low in order to allow a reliable trace element determination. Cellulose membrane ﬁlters, for example, especially those made of cellulose nitrate and acetate, are unsuitable for the determination of P, S, K, and Ca, as they show high blank values of these elements (Krivan et al., 1990). Very practicable are Nuclepore membranes because their blank values are very low and the particles are collected only on the surface. Unfortunately, the price for these ﬁlters is quite high. Whatman 41 ﬁlters have a high collection eﬃciency and are mechanically stable, clean, and reasonable in price. However, the high hygroscopicity requires a controlled humidity for weighing and handling of the ﬁlter. They also show a much higher background signal in comparison to the Nuclepore membranes. The interpretation of the ﬁlter eﬃciency is often complicated, as the test data are usually based on unloaded or ‘‘clean’’ ﬁlters. However, during a sampling event, the ﬁlters change their properties depending on the particle load. With increasing amounts of collected particles, the eﬃciency increases as well, despite the higher ﬂow resistance of the loaded ﬁlter. In general, the real eﬃciency of the ﬁlter is higher than in the published data. Furthermore, the collection eﬃciency increases with higher surface velocity and larger particle size. For the Whatman 41 ﬁlter, for example, the collection eﬃciency is quite low for particle sizes of 0.26 mm and surface velocities of 1.5 cm=s. With surface velocities of 100 cm=s, the collection eﬃciency increases to 95% for both small and large particles. A 10-mm pore-size Teﬂon ﬁlter shows collection eﬃciencies of 60–90% for low surface velocities and particle sizes between 0.003 and 0.1 mm. For 0.5- and 1-mm pore-size Teﬂon ﬁlters, the eﬃciency is always > 99.99%, independent of surface velocity and particle size (Lippmann, 1978). Membranes with diﬀerent pore sizes can be used in stacked-ﬁlter units for sizefractionated collection or with dichotomous samplers. Quartz-ﬁber ﬁlters present another alternative. They show high collection eﬃciencies and are very clean and less hygroscopic than cellulose ﬁlters. Particles within the range of 100–0.1 mm are retained on this ﬁlter. Furthermore, they are heat resistant. Haupt et al. (1995) tested diﬀerent techniques for the preparation of standards for the direct analysis of quartz-ﬁber ﬁlters. One preparation technique was based on the generation of aerosols and their deposition onto the ﬁlter surface. The other one required spotting of a multielement

968

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standard solution onto the ﬁlter and subsequent air-drying. The aerosol generation procedure showed much better accuracy and is, therefore, preferable for the preparation of standards. Glass-ﬁber ﬁlters are not recommended for trace analysis of inorganic components in aerosols. They are much less expensive than the quartz ﬁlters, but very high and variable blank values of diverse elements are their major drawback. However, they are quite useful for the collection and determination of organic compounds. Sometimes, these ﬁlters are also applied for the analysis of one single component in the aerosol sample. Lead, for example, can be determined in total suspended air particulates, after collection onto glass-ﬁber ﬁlters without problems. LaFisca et al. (1991) cut disks of 32-mm diameter from the ﬁlters and irradiated them successfully. Koutrakis et al. (1992) collected household dust onto Teﬂon ﬁlter with a Harvard impactor. The ﬁne-particle mass was gravimetrically determined and, subsequently, the concentrations of elements associated with the ﬁne mass determined by XRS. In some cases, ﬁlters may also be used for the collection of reactive gases. For this purpose, the ﬁlter is impregnated with special chemicals which retain selected components of the gases. Hydrogen sulﬁde, for example, can be trapped by coating the ﬁlter with ferric ion solution. The excess Fe is removed from the ﬁlter surface by washing with appropriate solvents, whereas the Fe sulﬁde precipitation stays on the ﬁlter. Sulfur can be determined directly or indirectly by measuring the Fe x-radiation. Suitable standards are available for such measurements (Leyden et al., 1984). Because XRF is a well-established method, which shows good reliability and easy handling, the technique is widespread for the determination of aerosols. The use of radioisotope sources for the excitation makes it independent of the laboratory placement and suitable for ﬁeld application. The samples can be analyzed in situ and contaminations and losses, which might occur during transport and handling, are avoided or kept at a minimum (Gilfrich and Birks, 1978). In addition to conventional XRF, TXRF is more frequently applied for the analysis of aerosols. To utilize the low detection limits and to reduce the matrix eﬀects originating from the collection material, ﬁlter samples need to be digested. With this procedure, element concentrations between 0.2 ng=m3 for Cu and 1 ng=m3 for Mn were determined (Einax et al., 1994; Schmeling and Klockow, 1997). By collection with suitable impactor systems (e.g., Battelle impactor), the aerosols are separated directly onto the polished sample carriers required for the instrument, and detection limits of 2 ng for Cr and 15 ng for Ca were obtained (Injuk and Van Grieken, 1995; Klockenka¨mper et al., 1995).

VII.

SAMPLE SUPPORT MATERIALS

In addition to the specimen preparation procedure, the position of the sample inside the spectrometer plays an important role for a reliable and precise analysis. It should be ensured that the sample and the calibration standards are in the same form and position for analysis, in order to avoid deviations and ensure reliable quantiﬁcation. Usually, the sample size is determined by the size of the sample holder, in which the specimen is presented to the spectrometer. In some cases (e.g., for customized instruments), the sample chamber is adapted for the nondestructive analysis of special objects like archeological and art specimens. Usually, the sample holders of commercial instruments are cylindrical with a diameter of 5.1 cm and allow placing samples with a maximum thickness of 4 cm. Smaller samples can be ﬁxed by a special mask, which is inserted to the conventional holder. The bottom of the sample holder is normally

Sample Preparation for XRF Table 4

969

Degradation Resistance Properties of Selected Thin-Film Materials

Material

Contaminants

Poor degradation resistance for

Mylar

P, Ca, Zn, Sb

Strong, mineral acids (HCl, HNO3)

Polypropylene

Al, Si, Ti, Cu, Fe

Oxidizing, concentrated acids; aqua regia

Polyethylene

Oxidizing, concentrated acids; alcohols; esters; ketones

Polycarbonate

Oxidizing, concentrated acids; alcohols; esters; ketones; aliphatic and aromatic hydrocarbons; mineral, vegetable and animal oils

Polystyrene

Esters; ketones; aliphatic and aromatic hydrocarbons; mineral and vegetable oils

Kapton

Strong, mineral acids; alkalines

Formvar

Acids

covered with a thin ﬁlm to prevent losses of the sample and contamination of the spectrometer. As ideal targets for the XRS analysis, samples prepared as thin ﬁlms are used. For inﬁnitely thin samples all interelement and mass absorption eﬀects are negligible. However, in practice, inﬁnitely thin means that for most x-rays, the sample thickness should be between 10 and 200 mm, which is diﬃcult to obtain.

Figure 3 Analyte line transmittance for various thin-film substances and thicknesses. (From Ref. 182. Reprinted by permission of American Laboratory.)

970

Schmeling and Van Grieken

With the use of thin-ﬁlm supports, special conditions for the materials are required. The ﬁlms should be stable under measuring conditions and have low impurities, especially for the desired elements. The ﬁlm should be thin enough to provide the highest degree of transmittance, particularly for low concentration levels and low-energy photons. Furthermore, the material should show a high resistance for degradation, which means that the specimen is retained safely in an XRF cup during the measurement and shows chemical resistance against acids, organic materials, thermal softening, tearing, and stretching. For some materials these properties, are listed in Table 4. Figure 3 illustrates the eﬀect of thin-ﬁlm thickness on analyte-line transmittance for some substances (Solazzi, 1985). Except Teﬂon, almost all of the presented materials show transmission better than 90% for photons of 3 keV or more, whereas in the low-energy region, absorption eﬀects are considerably higher. Teﬂon should be used only in the higher-energy region because its transmittance for x-rays decreases rapidly below 4 kev. Thin Mylar ﬁlm of 2.5 mm thickness presents the best properties for irradiation and is also acid stable within a certain level. Therefore, it is one of the most popular support materials in XRS. However, small impurities of P, Ca, Zn, and Sb should be considered before using this material. Several new materials were investigated in the last few years and some of them seem to be quite promising. Double-adhesive polyester tape, for example showed good properties for the determination of Pb in dusted samples. The powdered sample was spread onto the tape, attached to a disk, and a thin ﬁlm obtained (Renauld and Mckee, 1995).

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Index Aberrations, 644, 735 Absorber, 17, 19, 299, 944, 956 Absorption, coefficients, 17, 21, 26, 27, 31, 32, 78–90, 243, 288, 440, 589, 823, 827, 935 calculation, 19, 827 Absorption, x-ray, 17, 19, 34, 242, 270, 283, 298, 436, 567 edge, 8, 20, 21, 26, 27, 30, 34, 96, 202, 221, 242, 270, 275, 279, 300, 571, 591 ﬁne structure 21 jump ratio, 20 self, 849 Absorption corrections, x-ray, 5, 283, 284, 288, 415, 416, 420, 680, 823 factor, 893 Absorption-enhancements effects, 893 Accuracy, 167, 260, 278, 284, 350, 379, 413, 470, 747, 852 Accelerator, 503 Algorithm: Claisse-Quintin, 390 De Jongh, 382, 386 KMC (K-means clustering) 675 Lachance, 382, 391, 394 Lachance-Traill, 155, 382, 384 Leverberg-Marquardt, 312 Lucas-Tooth and Price, 402, 439 Marquardt, 279, 291, 293, 300, 314, 324 PCA (principal component analysis), 675 peak-clipping, 739 Rasberry-Heinrich, 391 Rousseau, 395 Sherman, 402 Tertian-Vie le Sage, 399

Amplitude noise, 240, 241 Anode, 108 dual, 106 exchangeable, 107 rotating, 107, 110 Applications in, 474–495, 532–542 art and archaeometry, 489, 704, 775, 778, 787 atmospheric aerosols, 489, 629, 700, 761, 765, 787, 965 biology, 534–538, 629, 701, 709, 747, 750, 754, 755, 756, 757, 759, 785 environment, 481, 489, 423, 538–541, 549, 700, 746, 755, 761, 771 geology, 491, 538–541, 545, 618, 629, 773, 787 industry, 625, 696, 697, 700, 779, 781 material sciences, 474–481, 494–495, 541–542, 876 medicine, 534–538, 750, 752, 753, 754, 785 microelectronics, 697, 938 waste materials, 698 water analysis, 948, 950 Attenuation, 96 (see also Absorption) mass attenuation, 368, 412, 419, 682, 742 Auger: effect, 12 electron, 12 yield, 14, 66–67 AXIL, 280, 288, 292, 293, 298, 300

Background: estimation, 151 intensity, 638, 724, 731 977

978 [Background] sources, 151 in WDS, 150–153 Backscatter loss, 849 Becquerel, 442 Bethe, equation, 33, 818 Binders 942 Box-Muller method, 302, 328 Bragg: law, 29, 100, 118, 122, 155 polarizer, 622, 626, 628 reﬂection, 635 Bremsstrahlung production, 3, 5, 7, 115, 242 Brunetto’s result, 115

Calibration, 271, 272, 273, 282, 299, 360, 361, 376, 469, 588, 626 Capillary: alignment, 655 optics, 636, 647 Cathode X-ray tube, Coolidge, 105 Characteristic line spectrum, 11, 115, 242, 244, 639 Charge collection, 229 Chelation methods, 951, 956 Chemical effects on x-ray energy=wavelength, 16 Chemical shift, 169 Classical electron radius, 8 Coherent=incoherent scatter, 21, 22, 24, 25, 27, 68, 69, 243, 244, 286, 294, 295, 369, 437, 441, 603, 606 Collimator, 116, 240, 523, 610 microdiffraction beam collimator, 107 Compensation methods, 367 Composition-composition histogram, 923–926 Compton (see also Coherent=incoherent scatter): collision cross section, 17, 23, 34, 642 radiation, 368, 415 scattering, 156 Computer implementation, 315, 522 Computer routine: AXIL, 298 CASINO, 886 DERFUNC, 324 ENHANC, 412 FILFIT, 321 FITFUNC, 324, 325, 326 GETSPEC, 322 HEX, 298 LIMINV, 330, 335 LINREG, 322, 323, 329 LOCPEAKS, 317 LOWSFIL, 316

Index [Computer routine] MARQFIT, 324, 325 NLRFIT 324 NRAND, 328 OPOLBAC, 320 ORTPOL, 320, 321, 330 PIXAN, 298 PIXASE, 298 PRAND, 328 PROZA, 836 SAMPO, 299 SGSMITH, 315, 319 SNIPBG, 319 TOPHAT, 317, 321, 322 URAND, 327 Contamination peak, 225 Continuous spectrum, 115, 242, 243, 255, 260, 267, 270, 280, 281, 286, 293, 295, 523 polarization, 5, 7 spatial distribution, 5 spectral distribution, 4, 5 Continuum (see Continuous spectrum) Convolution, 245, 247, 251, 252, 255, 269, 287 coefﬁcient, 252 Cooling systems, 217 (Co)precipitation, 951 Coprex method, 952 Coster-Kronig, transition, 13, 14, 60–65 Count rate, 139, 231–233, 469 determination 515 Counting: capacity, 201 statistics, 240, 246, 248, 261, 267, 270, 285, 287, 288 Critical absorption wavelength, 8, 36–39 Critical angle, 559, 564, 587, 598 Critical depth, 165 Critical excitation energy, 8 Cross sections, 8, 22, 24, 26, 27, 292, 399 Crystals, 118–124 curved, 122, 871 Debye-Wallner factor, 127–128 integral reﬂection coefﬁcient, 118 Johann, 122 Johannson, 122 material, 125, 215 mosaicity, 118 multilayer, 124–136 reﬂectivity, 118, 127 resolution, 118 smoothness, 127 stability, 121 thermal expansion coefﬁcient, 122 thickness, 127

Index Curved crystal optics, 139 Dead layer, 228 Dead time, 148, 235, 357 Deconvolution, 245 Delbruck scattering, 18 Depth: proﬁling, 588 resolution, 857 Detection limit, 164, 270, 297, 354, 434, 448, 459, 473, 511, 514, 523, 525, 559, 575, 577, 580, 581, 604, 610, 615, 634, 639, 660, 720, 733, 744, 794, 859, 870, 949, 953 Detector, 139, 239, 278, 281, 288, 298, 442, 445, 573, 574, 618 background, 229 CCD camera (charge-coupled device), 657, 660 dead time, 147 duplex, 175 efﬁciency, 31, 227, 449, 850 escape peak, 143 fabrication, 214, 454 gas-ﬁlled, 140 gas-scintillation proportional, 146 HP Ge, 658 microcalorimeter, 135, 147, 190, 871 noise, 217, 230 p-i.n. diode, 659 proportional, 141–143, 454, 459 resolution, 219, 232, 445, 448 (see also Resolution) response function, 242, 304 scaled proportional, 141 scintillation, 144, 461 semiconductor, SiLi, solid state, 214, 230, 454, 456 sensitive area, 449 Si(Li), 242, 658, 730 solid state drift-chamber, 659 surface barrier detector (SBD), 788 windowless, 729 windows, 242, 281, 283, 299 Diameter, radiometric, 424 Diffraction, 28 orders, 28 peaks, 225 Dispersion, 30 DIXE (deuteron-induced x-ray emission), 787 Drift-correction, 360, 362 Dry ashing, 960, 961 Dual-channel instrument, 139 Dual-target x-ray tube, 106, 210 Duane-Hunt law, 3, 203 Dynode, 144

979 EDXRF (energy-dispersive X-ray fluorescence), 239, 255, 264, 267, 304, 933 comparison with WDS, 103 Electron backscatter, 5, 34 factor, 820 Electron beam, 33, 504 Electron excitation, 33 Electron hole-pair, 219 Electronic noise, 230 Electron-probe-X-ray microanalysis, 811–814 Energy loss, 3 Enhancement, effect, 31, 32, 347, 366, 398, 421, 582, 727 Emission-transmission method, 413 EPXMA (electron-probe x-ray microanalysis), 5, 239, 272, 637, 719, 794, 796, 934 specimen preparation, 964 ERD (elastic recoil detection), 784 Escape fraction, 243 Escape peaks, 143, 221–224, 243, 279, 289, 290, 294, 298, 299, 300, 453, 739 ETAAAS (electrothermal atomization atomic absorption spectrometry), 795 EXAFS (extended x-ray absorption fine structure spectroscopy), 21, 545, 634 Excitation: by charged particles, 34 by continuous spectrum, 31 direct, 201, 465 electrons, 33 equations, 31–34 ﬁltered, 201 by monochromatic x-rays, 523 principles, 31–34, 201 by radioisotopes, 433 by secondary ﬂuorescence, 207

Factor: efﬁciency, 349 enhancement, 422 fano, 282 heterogeneity, 424 Filament, 811 Filters, x-ray, 463, 950, 956, 965, 967, 968 balanced, 463 primary-beam, 204 Filter technique, 465, 950, 951 Flow-proportional counters, 141 Fluorescence: analysis, 439 correction, 893 emission, 344 secondary ﬂuorescence emission, 346

980 [Fluorescence] tertiary, 381 yield, 13, 14, 61–65, 208, 243, 848, 862 Forbidden transition, 10 Fourier: deconvolution, 248 transformation, 21, 245, 246, 248 Freeze-drying, 960 Fundamental parameter, method, 348, 375, 378, 379, 382, 690, 742 Future prospects, 189, 551, 870 FWHM (full width at half maximum), 118, 140, 143, 219, 244, 249, 258, 260, 262, 267, 270, 282, 284, 288, 292, 298, 445, 665 Gamma rays, 239, 456 Gas-scintillation proportional, 146 Gaussian, 279, 283, 284, 299, 351 correlator function, 258 peak, 240, 244, 246, 281, 282, 286, 287, 293, 300 response function, 242, 299 shape, 242, 286, 287, 288, 298 Geometry factor, 19, 32 Glancing angle, 29 Goniometer, 136–139 Grinding, 939, 941 Half-life, 442 Heterogeneity, 674, 680, 753, 936, 939 High-voltage generator, 114 Homogeneity, 31, 685, 699, 938 IBA (ion beam analysis), 783, 793 IBT (ion beam thermography), 789 ICP-MS (inductively coupled plasma mass spectrometry), 795 Incoherent scattering (see Coherent=incoherent scatter) Infinitive thickness, 356 Inner shell vacancy creation, 214 Instruments, intercomparison, 179 Intensity, 96, 437 characteristic spectra, 34 continuous spectra (see Continuous spectrum) distribution, 242 equations, 31–34, 437 Ion-exchange methods, 951, 954 Ionization cross section, 721, 743, 848 charged particles, 8 electrons, 8 inner atomic shell, 8 photons, 33 IUPAC notation, 34

Index Jump ratio, 20, 243 Klein-Nishina cross section, 22, 24 Kossel structure, 21, 27 Kramer’s equation, 4, 5, 115, 242, 281 Kronig structure, 21, 27 Kulenkampff equation, 4 Lateral resolution, 685 Least-squares fit, 27, 239, 244, 249, 250, 251, 260, 262, 268, 270, 273, 278, 281, 288, 290, 292, 294, 295, 296, 329 orthogonal polynomials, 309 Limits of XRF: application limits, 165 high-Z limit, 168 precision and accuracy limit, 167 Linear least squares fitting, 260, 268, 269, 279, 280, 282, 298, 299, 306, 315 Lorentz distribution, 242 Low statistics digital filter, 253–255, 262, 316 Mapping (x-y), 909–923 Matrix, 96, 273 correction factor, 299 effects, 32, 275, 278, 357, 365, 383, 678 Mass-absorption coefficient (see Absorption coefficient) Methods: backscattered fundamental parameters, 417 dilution, 374 emission-transmission, 408 gradient, 378 linear interpolation, 377 particle size correction, 426 standard addition, 373 Micro: beam, 631, 633, 637, 651 beam instruments, 634, 646, 666 PIXE, 668, 732, 736, 737, 797 X-ray source, 632, 634, 647 XRF, 660 Microwave digestion, 962, 963 MDL (see Detection limit) Monochromatic excitation, 100 Monochromatization, 118 Monochromators, 100 Monte-Carlo simulation, 882 Moseley law, 96, 155 Moving average smoothing technique, 249 Multichannel spectrometer, 139 Natural decay, 442 Neural networks, 162

Index Nondiagram lines, 12 Nonimaging optics, 636 Nonlinear least squares fitting, 272, 279, 280, 282, 290, 293, 294, 298, 299, 300, 306, 310, 311, 315, 324 NRA (nuclear reaction analysis), 785, 787, 793 Optics: capillary, 636 nonimaging, 636 Orthogonal, polynomials, 262, 263, 264, 265, 320, 330 Overvoltage, 115–116, 858 Particle: analysis, 690, 693, 700 size effects, 423, 425, 894, 938, 939, 940, 966 Pattern recognition, 158–160 Peak: broadening, 258, 270 detectability, 258 distortion, 249, 250 model, 297 overlap, 153, 240, 255, 270, 342 proﬁles, 242, 287 searching, 252, 255, 267, 317 shape, 286 shape correction, 288 shift, 270 stripping, 261, 319 Peak-to-background ratio, 115, 169, 206, 230, 260, 511, 559, 618, 860, 870, 906 Pelletizing of samples, 942 Penetration depth, 564, 588, 638 PESA (particle elastic scattering analysis), 784 f(rz), 831–840, 863, 877–882 Photoelectric effect, 8, 638 Photoelectrons, 381 Photon emission, 504 PIGE (particle-induced gamma-ray emission analysis (PIGE), 785 Pile up, 243, 298, 299 Pile up element, 290, 299 PIXE (particle-induced x-ray emission analysis), 239, 242, 244, 253, 261, 262, 280, 283, 289, 290, 298, 299, 599, 719, 727, 737, 792, 934 Poisson distributed random variables, 240, 300, 302 Polarization, 603–630 Polarized, beam XRF, 603–630 Polishing, 934 Polychromatic excitation, 598 Precision, 167, 270, 284, 747 Preconcentration of trace analytes, 950, 951

981 pRNA (photon-tagged NRA), 788 Pulse: pair resolution time, 243, 244 pile-up, 233 Pulse-height, 261, 263 discriminator, 140, 144 distribution, 143 shift, 147 Qualitative analysis, 150, 153, 249, 268, 300 Quality control, 359, 360 Quantitative analysis, 3, 42, 150, 343, 524, 676, 676, 740, 742, 816, 840, 863, 876, 889, 893 standardless, 847–857 Radioisotope excitation, 433 Radioisotope sources, 442, 456 Raman: effect, 245 scattering, 31, 245 Random errors, 351 Rayleigh, scattering, 369, 416 RBS (Rutherford backscattering spectrometry), 719, 780, 783, 784, 793 Reflection of x-rays, 29 Reflectivity, 559, 571, 621, 622, 624, 643 Refractive index, 29, 30, 125 RMF (Regenerative Monochromatizing Filter), 205 Resolution, 219, 232, 239, 258, 270, 281, 287 calibration function, 292 depth, lateral, 857 Rowland circle, 871 Rutherford backscattering, 886 Sample (see also Specimen): heterogeneity, 472–473 liquid, 949 nuclear fuel, 950 preparation, 933 solid, 934 Satellite lines, 12, 359 Saturation thickness, 33 Savitsky-Golay, 250, 262, 315 polynomial ﬁlters, 250, 261 Scatter, 530, 608 Secondary excitation, 96, 207 Secondary fluorescer, 34, 275, 346, 883 SEM (scanning electron microscope) (see EPXMA) Sensitivity, 364, 434, 525, 559, 566, 568, 582, 590, 743 curve, 529 factor, 258, 889

982 Sequential spectrometers, 174 Shirawai-Fujino, 156 Signal-to-noise ratio, 610, 258 Simplex method, 5 Simulation, Monte Carlo, 278, 304, 327, 612, 614, 615, 624, 641, 687 Smoothing, 239, 248, 251, 261, 315 effect, 270 ﬁlter, 250, 251 Source radiation, 105 Spatial resolution, 514, 523, 814 Specimen: biological, 958, 959 briquet, 938, 940, 943 collection, 748, 763 fused, 944, 946 heterogeneous, 680 homogeneity, 365 inﬁnitely thick, 341, 742, 743, 744 intermediate thick, 407 layered, 876 liquid, 948, 950 metallic, 936 orientation, 935 powder, 938 preparation, 748, 933 thick, 33, 343, 345 thin, 740, 744 Spectra: continuous spectra, 348 overlap, 342, 358 PIXE, 729 Spectral: background, 603 distribution, 3, 660 lines, 96 overlap, 459 resolution, 118, 131, 134, 168 Spectrometers, 270 energy dispersive, 199 milliprobes, 178 semiconductor, 629 sequential, 174, 179 simultaneous, 177, 187 wavelength, 104, 870 Spectrum: artifacts, 221 evaluation, 31, 239, 240, 241, 249, 267, 268, 269, 272 information content, 240 processing, 241, 245, 248, 668 Standard deviation, 297, 747 Standards, 359, 360, 362, 371, 661, 692, 741, 948 Stopping power, 34, 352, 720, 742, 818, 848

Index Sum peaks, 224, 294, 300 Synchrotron: facilities, 506 microbeam, 633, 635 properties, 503 radiation, 503 SRXRF (synchrotron radiation X-ray fluorescence), 501, 560, 571, 590, 596, 597, 605, 934 Takeoff angle, 108, 166 Target, 7 thin, 213 Thick sample, 33 Thin-film, 940, 949, 950, 951, 969 Thin specimen, 889 Thomson scatter, 607 Tomography, 542 Transition-edge sensor, 147 TXRF (total-reflection x-ray fluorescence), 559, 573 Valence effects, 17 Variable-width smoothing filters, 252 Voight profile, 242, 281, 287, 288, 294 Walter’s equations, 19 WDXRF (wavelength dispersive X-ray fluorescence), 95, 239, 240, 267, 377, 933 comparison with energy dispersive, 103 instrument and method, 104, 173 Wet digestion, 960 White radiation, 243 XANES (x-ray absorption near-edge spectroscopy), 21, 545, 634 X-ray: aberrations, 644 capillary, 648 focusing, 633 intensity distribution, 240, 654 lenses, 190 microtomography, 692 optics, 647 production, 639 reﬂectivity, 643, 645 transmittance, 650, 654 tubes, 209, 275, 278, 615, 622, 624 X-ray microscopy: apparatus, 507 collimated, 509 focused, 515 X-ray preferential absorption technique, 440, 467

Index X-ray scattering analysis, 441 X-ray scattering technique, 469 X-ray source, 105 compact ﬂash X-ray source, 113, 190 X-ray tubes: end window, 105, 108, 110, 169, 174, 212 side window, 105, 108, 110, 169, 174, 210

983 [X-ray tubes] transmission target, 212 windowless, 106 ZAF (atomic number Z, absorption, fluorescence), 817 correction, 817, 838, 898