Analytical Spectroscopy Library - Volume 7
Applications of Synchrotron Radiation to Materials Analysis
Analytical Spectroscopy Library
A Series of Books Devoted to the Application of Spectroscopic Techniques to Chemical Analysis Volume 1 NMR for Liquid Fossil Fuels, by L. Petrakis and D. Allen Volume 2 Advances in Standards and Methodology in Spectrophotometry, edited by C. Burgess and K.D. Mielenz Volume 3 lntroduction to Inductively Coupled Plasma Atomic Emission Spectrometry, by G.L. Moore Volume 4 Sample lntroduction in Atomic Spectroscopy, edited by J. Sneddon Volume 5 Atomic Absorption Spectrometry. Theory, Design and Applications, edited by S.J. Haswell Volume 6 Spectrophotometry, Luminescence and Colour; Science and Compliance, edited by C. Burgess and D.G. Jones Volume 7 Applications of Synchrotron Radiation to Materials Analysis, edited by H. Saisho and Y. Gohshi
Analytical Spectroscopy Library - Volume 7
Applications of Synchrotron Radiation to Materials Analysis
edited by
H. Saisho Liaison Office, Faculty of Science and Engineering, Ritsumeikan University, Noji-cho 7916, Kusatsu, Shiga 525-77, Japan
and
Y. Gohshi Department of Applied Chemistry, Faculty of Engineering, University of Tokyo, 7-3-1Hongo Bunkyo, Tokyo 113, Japan
ELSEVIER Amsterdam - Lausanne - Oxford - New York - Shannon -Tokyo
- 1996
ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 21 1,1000 AE Amsterdam, The Netherlands
ISBN 0 444 88857-8
O 1996 Elsevier Science B.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science B.V., Copyright & Permissions Department, P.O. Box 521, 1000 A M Amsterdam, The Netherlands. Special regulations for readers in the USA. This publication has been registered with the Copyright Clearance Center Inc. (CCC), 222 Rosewood Drive Danvers, M A 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the USA. All other copyright questions, including photocopying outside of the USA, should be referred to the copyright owner, Elsevier Science B.V., unless otherwise specified. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper Printed in The Netherlands
Preface This book is a part of the series "Analytical Spectroscopy Library" and is devoted to X-ray analysis by synchrotron radiation. X-ray analysis is one of the oldest and most well established techniques in analytical chemistry. The method, however, is still on the leading edge of instrumental analysis. X-rays are nonrestrictive and can be tuned to a very sharp bandwidth. Their very short wavelength enables high resolution imaging down to a molecular or atomic scale. These potential capabilities are not well realized in current X-ray analytical instruments. The largest drawback is the low intensity of an X-ray source. Recent developments in the synchronotron radiation (SR) X-ray source are the real breakthrough, and the photon deficiency problem is now being solved. The number of SR facilities, however, is limited and their distribution is not uniform in the world. Whereas research reports are circulated in the US and EU countries very well, reports from the orient are often inaccessible due to language barriers. This volume is intended to describe various facets of X-ray analytical methods by SR with some emphasis on Japanese activity. The editors hope that this volume will provide general perspectives of X-ray analysis by SR, with the added spice of Japanese activity in the Photon Factory and the future Spring-8 project. Chapter 1 (Mititaka TERASAWA and Motohiro KIHARA) is a concise introduction to synchrotron facilities and the related basic instrumentation. The chapter will be helpful to understand the discussion of machine operators and technicians and to communicate with design people in the planning of a new ring or an insertion device. In chapter 2 (Hideo SHAISHO and Hideki HASHIMOTO), X-ray fluorescence analysis is discussed. SR provides tunable excitation which is very effective in improving a detection limit (DL). The detailed discussion on DL is based on a theoretical estimation of background and the discussion can be extended to future research planning. Another important characteristic of SR is natural collimation, which leads to total reflection analysis. The theoretical treatment in this chapter will be especially useful for thin layer analysis by XRF. Chapter 3 (Shinjiro HAYAKAWA and Yohichi GOHSHI) covers microbeam and chemical state analysis. Instrumentation for microbeam optics is completely dependent on the fabrication technology of optical components which are state of the art. Xray analysts, however, can expect better performance of a microbeam optical system with the knowledge of optical systems described in this chapter. In addition to its imaging capability, an energy tunable microprobe can speciate analytes. Several sophisticated applications are discussed. Chapter 4 (Hiroyuki OYANAGI) covers X-ray absorption fine structure, which is the most widely used technique in the application of SR to material analysis. Instrumentation
in the detection system is included together with a general description of XAFS and a variety of applications.
Chapter 5 (Toshiaki OHTA, Kiyotaka ASAKURA and Toshihiko
YOKOYAMA) covers surface structural analyses.
Surface sensitive signal detection is
described, followed by SEXAFS and NEXAFS applications to surface phenomena. Other important techniques such as surface diffraction, standing wave, and angle resolved photoemission fine structure are also explained together with several applications. Chapter 6 (Kanji KAJIWARA and Yuzuru HIRAGI), Structure Analysis by Small-angle X-ray Scattering, describes the powerful traditional method for organic molecules, especially polymers. Chapter 7 (Fujio IZUMI) explains details of the Rietveld method and its applications to synchrotron X-ray powder diffraction data. The author has developed much software for the Rietveld method and his experience contributes to the comprehensive description given in this chapter. Chapter 8 (Katsuhisa USAMI) and Tatsumi HIRANO), Xray Microtomography deals with large scale structural analysis. Energy tunability of the SR X-ray source adds even more elemental and chemical state information to the tomographic image of information. Though there is still a gap between atomic scale structural analysis and mm or pm analysis, this will be bridged by the future SR X-ray method. The editors H.S. and Y.G. hope that this volume will be of help to research scientists and students who are interested in materials analysis by X-ray methods. H.S. and Y.G. finally appreciate the effort and patience of all the authors and the staff at Elsevier Science. February 1996 Y. GOHSHI H. SAISHO
CONTENTS Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Basic characteristics of synchrotron radiation and its related facilities and instrumentation by M. Terasawa (Hyogo 671-22, Japan) and M. Kihara (Ibaraki 305, Japan) . . . . . . . . . 2. X-ray fluorescence analysis by H. Saisho and H. Hashimoto (Shiga 520, Japan) . . . . . . . . . . . . . . . . . . . . . 3. Microbeam and chemical state analysis by Shinjiro Hayakawa and Yohichi Gohshi (Tokyo 113, Japan) . . . . . . . . . . .
4. X-ray absorption fine structure by H. Oyanagi (Ibaraki 305, Japan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Application to surface structure analyses by T. Ohta, K. Asakura and T. Yokoyama (Tokyo 113, Japan) . . . . . . . . . . . . . . . . . . 6. Structure analysis by small-angle X-ray scattering by K. Kajiwara (Kyoto 606, Japan) and Y. Hiragi (Kyoto-fu 61 1, Japan) . . . . . . . . . . . 7. The Rietveld method and its applications to synchrotron X-ray powder data by F. Izumi (Ibaraki 305, Japan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. X-ray microtomography by K. Usami and T. Hirano (Ibaraki 3 19-12, Japan) . . . . . . . . . . . . . . . . . . . . .
Subject index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Applications of Synchrotron Radiation to Materials Analysis H. Saisho and Y. Gohshi (Editors) 9 1996 Elsevier Science B.V. All rights reserved.
CHAPTER 1
BASIC C H A R A C T E R I S T I C S OF S Y N C H R O T R O N R A D I A T I O N AND ITS RELATED FACILITIES AND INSTRUMENTATION Mititaka T E R A S A W A Himeji Institute of T e c h n o l o g y 2167 Shosha, Himeji, H y o g o 671-22, Japan
Motohiro KIHARA National L a b o r a t o r y for H i g h E n e r g y Physics ( K E K ) Oho, T s u k u b a , Ibaraki 305, Japan
1.1. I N T R O D U C T I O N Electromagnetic radiation emitted by charged particles when they move at a highly relativistic velocity on a circular orbit is called synchrotron radiation, or SR in s h o r t - which was named after the first observation of the radiation in an electron circular accelerator, a synchrotron. Although the synchrotron radiation was, in its early history, considered simply to be a useless and harmful by-product generated in circular accelerators, it has become of great interest and is utilized in materials science as an intense and versatile photon source. At the beginning, the radiation was used as a powerful photon source for spectroscopy in the wavelengths ranging from soft X-rays to the vacuum ultra-violet (VUV). From the early 1960s many of the existing electron synchrotrons in the several hundred MeV class have been modified and turned over to synchrotron radiation facilities. In the mid 1970s, an electron storage ring became a normal photon source facility, in place of the electron synchrotron. The storage rings were originally developed for collision experiments in high energy physics. In the electron synchrotron, electrons are accelerated and, after the maximum energy is achieved, are extracted, usually at a rate of several tens per second. In
contrast, in the storage ring the previously accelerated electrons are injected into and stored in the ring orbit. As a consequence this is greatly superior to the former synchrotron as an intense and stable radiation source. Recently large storage rings in the multi-GeV class have been developed and the wavelengths of the radiation have been extended to the hard X-ray range. The use of such high energy X-rays is now having a great impact in studies on X-ray scattering, diffraction, inner shell ionization, and even nuclear excitation. Nowadays the usefulness of synchrotron radiation is well recognized in various research fields, and many storage tings dedicated to synchrotron radiation research have been constructed or are currently under construction. Moreover, insertion devices, which allow radiation of much higher brilliance to be obtained, are well developed and available as practical facilities. The synchrotron radiation is usually emitted by electrons moving in a circular orbit in the bending magnet. The radiation from insertion devices such as the undulator and the multipole wiggler, which are positioned on the straight section of the storage ring, is of extremely high brilliance - - and also has a shorter wavelength with multipole wigglers than with bending magnets. The storage tings of recent advanced synchrotron radiation facilities are designed to have many long straight sections so as to accommodate such insertion devices as low emittance and high brilliance photon sources. Consequently, the tings always tend to become larger.
1.1.1. Properties of Synchrotron Radiation Electrons submitted to an acceleration field emit electromagnetic waves. This phenomenon is well recognized in a classic antenna where the electrons are travelling in conducting wires, the acceleration is very weak, and the emission takes place in the radiofrequency range. With vacuum electron tubes, such as the klystron, it is possible to increase the acceleration on electrons which are travelling in the vacuum, and to push the emission into the ultra-high frequency (UHF) range. In these tubes the electron energy is fairly weak. In circular high energy electron accelerators,~with electron energies up to several giga electron volts (GeV), the magnetic field in the bending magnets induces a very strong centripetal acceleration in the highly relativistic electrons, with an energy E >> m c 2 where m is the rest mass of electron. The acceleration induces an electromagnetic wave emission up to several hundreds keV in the most energetic machines. This radiation is called synchrotron radiation. Relativistic positrons also emit synchrotron radiation in the magnetic field by the same principle. Thus, circular electron- and positron-accelerators and storage rings, in which the particles move with relativistic energies, have become man-made sources of synchrotron radiation. The radiation source is basically composed of three component facilities: a linear accelerator, the synchrotron and the storage ring. Electrons are injected into an electron synchrotron at a relatively low energy, say several hundred MeV, from the linear accelerator. They are then accelerated on a fixed circular orbit. The synchrotron consists of an array of magnets for focusing and bending the electron beam, and straight linear sections for accelerating the
particles. The magnetic field in the deflecting magnets is increased during the acceleration in order to keep the electrons on the same circular path as their energy is gradually increased. Finally, when they reach the appropriate energy, the electrons are transferred to the storage ring. Here the magnetic field remains constant. There are acceleration sections within the storage ring which compensate for the energy losses due to synchrotron radiation. Around the storage ring the radiation from the stored electrons can be used for various experiments. This photon production is of extreme interest for spectroscopy. Synchrotron radiation has a number of outstanding properties: (1) A continuous spectrum from the infrared to the X-ray region. (2) High intensity, owing to the high current electrons accumulated in the storage ring. (3) Collimation of the emitted radiation in the instantaneous direction of flight of the emitting particles (the angular spread is of the order of 1 mrad). (4) Linear polarization, with the electric vector parallel to the plane of the orbit. (5) Circular polarization above and below the plane of the orbit. (6) High brilliance of the source, because of the small cross section of the electron beam and the high degree of collimation of the radiation. (7) A time structure with pulse lengths down to 100 ps. (8) Absolute calculability of all the properties of the source. (9) Cleanliness of the source, since the light emission takes place in an ultra-high vacuum, in contrast to the situation in gas discharge or spark lamps. 1.1.2. Historical Remarks Although synchrotron radiation was first observed directly by Elder and his co-workers [1] at the General Electric 70 MeV synchrotron in 1947, the theoretical consideration of radiation by charged particles in circular motion goes back to the work of Lienard in 1898 [2]. Further theoretical work was made by Schott [3], Jassinsky [4], Kerst [5], Ivanenko and Pomeranchuk [6], Arzimovitch and Pomeranchuk [7] and others through to 1946. Blewett [8] was one of the first to be concerned with the effects of the radiation on the operation of electron accelerators and he observed the effects on the electron orbit in 1945, although he did not detect the radiation itself. After the observation of the radiation by Elder and his co-workers in 1947 the interest in synchrotron radiation was renewed. Comprehensive theoretical treatments were presented by Sokolov and his co-workers [9] and by Schwinger [10] in the late 1940s and later. With these works the theory was fully developed so that accurate predictions could be made regarding the intensity, spectral and angular distributions, polarization, and so on. Following the first observation Elder and his co-workers carded out experimental investigations in the late 1940s on the properties of the radiation, using the General Electric 70 MeV synchrotron. In the 1950s studies of synchrotron radiation were extended by several groups [11], using the 250 MeV synchrotron at the Lebedev Institute in Moskow, by Corson [12] and Tomboulian and
his co-workers [13] using the Comell 300 MeV synchrotron, and by Codling and Madden [14] using the 180 MeV synchrotron at the National Bureau of Standards (NBS) in Washington D.C. In the mid-1960s Haensel and his co-workers [ 15] were the first to utilize radiation from a multi-GeV accelerator, the 6 GeV synchrotron in Hamburg. The investigations mentioned above confirmed the basic theoretical predictions and provided much useful data and experience in the use of the radiation.
1.2. PRINCIPLE OF SYNCHROTRON RADIATION EMISSION 1. 2. 1. Principle of Synchrotron Radiation Emission The properties of synchrotron radiation (SR) can be derived by applying the methods of classical electrodynamics to the motion of relativistic electrons and positrons in circular orbits. Hereafter we confine the following discussion to the radiation emission only by electrons, because the principle of the emission by positrons is the same. First of all, we consider the circular motion of an electron with momentum p in a magnetic field B. The radius of the electron orbit, p, is given by the following equation. pc = Bep
(1-1)
where e is the electron charge and c is the speed of light. For relativistic electrons, E 2 = p 2 c 2 + m 2 c 4 ~ p2c2
(1-2)
where E is electron total energy and m is the electron rest-mass. From Eqns. (1-1) and (1-2), E = Bep
(1-3)
This relation is described using practical units, by Eqn. (1-4). B[tesla] p[m] = 3.336 E[GeV]
(1-4)
Angular distribution of SR When an electron is non-relativistic (fl = v/c << 1) and the velocity is significantly low compared to the speed of light, the electron on a circular orbit is accelerated toward the center of the orbit, and emits a dipole radiation whose axis is in the direction of the acceleration. The radiation power per unit solid angle, which is the energy of the electromagnetic wave emitted in unit time from the electron with the acceleration of dv/dt, is given by
d P ( z ) = e 2 (~t)2sin2z dO 4~rc 2
(1-5)
where Z is the angle between the directions of the acceleration and electromagnetic wave transmission. Therefore the angular distribution of the radiation power based on a dipole radiation shows the typical pattern of sin2z. As shown in Fig. 1-1, the radiation emission is at maximum in a direction normal to the acceleration (2' = 7r/2), and there is no emission at the two acceleration directions of Z = 0 and 27 = 7r.
dv
-Orbit- - - . . . . ~
Acceleration /--~~)~'~Ujt
)
at
'J
To
v
spectrograph
(a)
(b)
Fig. 1-1. Radiation patterns of electrons in circular motion at low velocity.
K
K' P
,-x-(x, y, z, t) (x', y', z', t')
vt
Otl
Z "JI
Z'
I
Fig. 1-2. Coordinate systems showing Lorentz transformation. Electrons moving at the velocity v are on the system K', and the radiations are observed on the system K. When an electron moves near the light speed, the properties of SR from the electron are described by relativistic electrodynamics. If an observer is in a fixed coordinate system located on the electron, he observes the same angular distribution of sin2z as for the non-relativistic
case mentioned above, even if the electron moves relativistically. However, if an observer is in the stationary coordinate system, the angular distribution becomes distorted due to the relativistic effect. Let us take a stationary coordinate system K(x,y,z,t), and another coordinate system K' (x ',y ",z ',t ") which is moving at speed v along the direction of the z axis. The relationship between the two systems is given by the Lorentz transformation,
z = 7(z' + vt') X ' - X'
y=y' t = 7 (t' + flz'/c)
(1-6)
where fl = v/c and, 1
r 41_/32 _
(1-7)
E
(1-8)
mc 2
= 1957 E[GeV]
(1-9)
Here, ?'is the energy of the electron represented in units of the electron's rest mass energy, that is mc 2 = 511 keV. For example, fl = 0.999999868 and ?' = 1957 for E = 1.0 GeV. The electron mass becomes 1957 times the rest mass, which means that the acceleration of relativistic particles increases the mass, while changing the velocity little, because it is almost the speed of light. As shown in Fig. 1-2, if the SR, which is emitted at angle 0 ' from the z' coordinate from an electron fixed at the K' system is observed at angle 0 from the z coordinate, the following relationship is obtained, using the Lorentz transformation for velocity.
cos0'+ fl
COS0 =
1 + fl c o s O '
(1-10)
sinO'
sin0 =
y(1 + fl cosO')
(1-11)
then, tanO=l
sinO'
? ' c o s O ' + fl
(1-12)
The angle 0 ' = + nr/2 in the K' system, at which no radiation is emitted, is proved to correspond to 0 = + 1/Tin the K system, according to Eqn. (1-12). The sin2z angular distribution of radiation emission is extremely distorted and a forward focusing, in which the radiation is concentrated within a sharp cone with its axis tangential to the electron circular orbit, is observed, as shown in Fig. 1-3.
....
Orbit . . . . .
Acceleration
dv dt
,
To"
V
(a)
spectrograph
(b)
Fig. 1-3. Radiation patterns of electrons in circular motion at velocity approaching that of light. The root mean square (r.m.s.) of the radiation emission angle, which corresponds to half the cone angle, is expressed as follows for the radiations near the emission spectrum center; (i//2)1/2 = 1 7
(1-13)
For example, 1/7 = 0.064 mrad for E = 8.0 GeV, which means only 6.4 mm spread of the radiation 100 m from the source point. It is understood that the SR is an ideal collimated beam. Actually, the intensity of the radiation drops rapidly with increase in the angle ~ i n the direction vertical to the electron orbit plane, while the plane is filled with the radiation because electrons are emitting their radiation all the way round.
Spectra In the case when an electron moves non-relativistically on a circular orbit with radius p, an angular frequency of radiation is equal to an electron revolution frequency, coo = v/p. When an electron moves relativistically, the main frequency components of the radiation are significantly
higher harmonics of the revolution frequency too. Let us consider an observer standing on a line tangential to the electron orbit, as shown in Fig. 1-4. The observer will be subjected to the radiation which is emitted by the electrons moving between A and B on the circular arc, the length of the arc being 2p/7' because the radiation emission angle is about 2/7,. The time interval of the radiation to which the observer is exposed will be the time that the radiation travels from A to B, subtracted from the time that the electron moves along the arc AB, that is, 2p At = v)'
2p sin~ 2p c = -U~-(c - 1)
(1-14)
Adopting
v/c = fl= 1 - 1/(2 7' 2) At=
p = C7' 3
(1-15)
1
(1-16)
too7' 3
The radiation is confined to an extremely short pulse, since this time interval is in inverse proportion to ),3. For example, At = 3.48 x 10-20 sec in the case of E = 8.0 GeV and p = 40.1 m.
observer
Fig. 1-4. Electrons in circular motion and synchrotron radiation emitted by the electrons in a tangential direction. An electric field of pulsed radiation as a function of frequency, E(to), is derived as the Fourier transformation of the time-dependent electric field, E(t). The power-spectrum of SR is determined by E(to) 2. In the present case, E(t) is a periodic pulse with the pulse width of
p/(c7"3), and the pulse interval 27rp/c, as shown schematically in Fig. 1-5a. The quantity E(to) 2 consists of a vast number of higher harmonics of the basic angular frequency too = c/p,
as shown in Fig. 1-5b. The frequency tOp of the major parts in the power-spectrum is determined from Eqn. (1-17). 2--~ = 2~ y3 too top = At
(1-17/
IJJ
q 2r~p
t
C
(a)
coo
Ill
._ v
~p
(b)
Fig. 1-5. Synchrotron radiation (SR) emitted by electrons in circular motion. (a) Time variation of the electric field induced by SR. (b) Spectrum of the radiation power. The spectrum is understood to include extremely high order Fourier components. While the whole spectrum consists of many spectral lines with the interval of too, it actually becomes a continuum because of the energy fluctuation of electrons caused by emitting radiations. Thus the wavelength spectrum extends widely from the infrared to the X-ray region. We use a critical (or characteristic) angular frequency toc as a parameter characterizing the power spectrum, that is,
toc = ~ 'Y 3 600
(1-18)
This parameter is defined to be an angular frequency at which the power spectrum is divided equally. A critical wavelength, ~,c, is then defined as Ze = 27rc = 4 ~
toc
3
P
~,3
(1-19)
Using practical units, A,c[A] = 5.59
p[m] _ 18.6 E 3 [GeV] E 2 [GeV] B [tesla] n
Also, a critical energy, e c, is defined by
(1-20)
I0 hc
(1-21) = 12.4 _ 2.22E3[ GeV] ee [keV] Ac[A] p[m]
(1-22)
For example, for E = 8.0 GeV and p = 40.1 m, the basic angular frequency is 090 = 7.48 x 106/sec = 7.48 MHz, and the critical angular frequency becomes o9c = 4.31 x 1019/sec. The critical wavelength Zc = 0.438 A, and the critical energy ec = 28.3 keV.
Polarization An electric field vector of an electric dipole radiation is represented by the sine-squared angular distribution as shown in Figs. 1-1 a and 1-1 b. The radiation emitted by an electron in the plane of its orbital is completely polarized, with the electric vector parallel to the orbital plane. Above and below this plane the radiation is elliptically polarized, to a degree determined by the viewing angle.
-
-
Orbit
~>0
'.-z.... ~
"
Accele "~--"O
-..
.~M
(a)
T
s S
SSS
~
~'~
~
e
v
(b)
Fig. 1-6. Polarization of synchrotron radiation. For emission in the orbit plane the electric vector lies in the plane, and for emission at some angle gt with respect to the orbit plane the vector also has a component normal to the parallel component. The polarization can be understood as a change of the acceleration vector being applied to an electron during its circular motion, as shown in Fig. 1-6. If an observer standing near a tangent line at the radiation source point traces the changing vector, the polarization is decided according to the locus of the vector. When the observation is made at 1g = 0 on the orbital plane, the
11 acceleration vector locus is always on the horizontal plane, and the radiation is linearly polarized. On the other hand, when the observation is made at gt > 0 and gt < 0, being off the plane, the acceleration vector locus is elliptical, and elongated in a horizontal direction, and the radiation has elliptical polarization. The rotation of the polarization is inverted for ~ > 0 and gt <0.
Coherency As we will discuss later (see Section 1.6. INSERTION DEVICES), if an undulator is employed in the storage ring, coherent radiation with an extraordinarily higher brightness is obtained. However, with a normal bending magnet, the radiation is incoherent, especially in the shorter wavelength range. Therefore the intensity is the total of the radiation emitted by each electron.
1. 2. 2. Basic Equations of Synchrotron Radiation
Radiation induced by electrons First of all we may take the Larmor formula for the power radiated by a single non-relativistic electron with mass m and charge e. The formula is obtained by integrating Eqn. (1-5) over solid angle, and is described as follows; e2/dv/2=2 e 2 /dp/2 W = 2--~ ~dt ! 3 n ~ 3 ~dt l
(1-23)
For a relativistic electron in circular motion, Eqn. (1-23) is rewritten by replacing the momentum p by an four dimensional momentum vector (P, iE/c), where P = TP. Adopting an intrinsic time ds =dt/?', e 2 //dP/2 1 /dE/2~ W = ~ m2c3 |~~s ] - ~ ~-~S] | _ 2 e___L_ 2
I(dPl2
- 3 m2c 3 ),2 I~ dt !
l/dE/2t
7~ ~-~! |
(1-24)
In a circular accelerator, the change of E with time is negligibly small compared to the change in p, and the second term in the bracket in Eqn. (1-24) is neglected.
d---PI2= r2 m2/0v dt !
~dt !
12
= ,y2 m 2 (evB~ (1-25)
= ~, 2 m 2 c4[p2
Eventually, W [erg/sec/electron] = ~_~_~__ ? ,2C e
4 (1-26)
Total radiation power is proportional to ~,4, i.e. E 4 , and inversely as p2. Eqn. (1-26) is the Larmor formula for a relativistic electron in circular motion. Radiation power and photon number
Schwinger [ 10] has provided the expression for the instantaneous radiation power (C.G.S. units) emitted per unit wavelength and per radian (in vertical angle gr by a monoenergetic electron in circular orbit: d2W Ierg/sec/rad(vertical)/cm/electron] d I/td~ -32tr3 p3
)'8{1+(?'1/~}2 K2/3(~)+ (~'l/t)2 K2/3(~) 1+()' V)2
(1-27)
where ~ = Ac { 1+(]~r and K1/3 and K2/3 are modified Bessel functions of the second kind, gt is the angle between the direction of photon emission and the instantaneous orbital plane, and A,c is the critical wavelength given by Eqn. (1-19). It is noted that the angular spread of the radiation is dependent on the wavelength. If ~, is in the vicinity of ~,c, the spread Agtis almost 2/7, as given in Eqn.(1-13). If ~ is far from A,c, the spreads are given by [I],11/2
Agt--- t2~3~1
AV =
' for ~, << Ac
, for ~ >> ~,c
(1-28)
(1-29)
The angular distribution becomes sharp as the wavelength decreases, in general. For ~, << ~,c, the distribution tends to a smaller angle than 2/7', while, for A, >> A,c, it spreads to a larger angle than 2/?,, and approaches a shallow dip at gt = 0 (see Fig. 1-7).
13 A representation of a photon number is sometimes useful instead of the powers expressed by Eqn. (1-27). The photon number is given as follows, by dividing the power by single photon energy, hc/L d3N [photons/sec/rad(vertical)/cm/electron] dtdvd,~ -87r 2 h /92 ~, 3{1+(~#')2}2
:o. 97
1+(~) 2
(1-30)
{
(1-31)
Assuming the electron beam current in storage ring is J, the number of electrons emitting the radiation is given by n
=
27rp j ec
(1-32)
= 1.31 • 1011 p [m] J[amp]
(1-33)
The photon number for the beam current of J [amp] is calculated by multiplying n to Eqn. (130). For the emission angle of 1 mrad in the electron orbital plane and 1 mrad in the direction vertical to the orbital plane, multiplying 10-6/27r by the Eqn. (1-30) gives the photon number. When 0.1% of band-width is adopted as a wavelength width, multiplying AA = 0.001~ = 0.001 x 47r/3 x 102 p[m]/y 3 x A/~c gives the corresponding photon number. Eventually, the number of photons per sec in 1 mrad 2 of solid angle, 0.1% band width, and a beam current of 1 mA is given by d3N [photons/sec/mrad2/0.1% b.w./mA] dtdl2dA,/& = 3.46 x 103 'J,'2
K22/3(~)+
1+('Bg)2
K2/3(~) (1-34)
This photon number is called a "brightness". Figure 1-7 shows an example of the angular distribution of photon number for various wavelengths [16]. As mentioned above, the shorter the wavelength, the smaller is the angular spread. In the orbital plane V = 0, Eqn. (1-34) is reduced to
14
d3N
]
dt d,QdMA, lqr=o
= 3 . 4 6 x 103 yg(Xc]2~"2 [~,c
-~--~] ~ 2/3~-~]
(1-35)
~d' 1
=
1oooA
.
--
I
r
.
.
.
.
.
.
l 1A
|
Azimuthal angle
.
.
.
.
.
~ (mrad)
Fig. 1-7. Angular distribution of SR. Photon number as a function of perpendicular angle V, calculated for the wavelength of radiations at E = 2 GeV and p = 5.55 m [16]. 1 014 <
E 013
d
E ~1
~
012
01 ~
,,_..
n'-
101~ 1 00.01
0.1
1
10
1 O0
E n e r g y , KeV 1 1000
I
100
I
10
I
1
I
0.1
1000 9
I
,,
0.01
Wavelength , A
Fig. 1-8. Typical spectral distribution of SR. Brilliance as a function of photon energy for (a) PF (E = 2.5 GeV, p = 8.66 m) and (b) SPring-8 (E = 8.0 GeV, p = 40.1 m). Examples of spectral distribution at V = 0 are shown for the cases of E = 2.5 GeV, p = 8.66 m (PF ring ), and for E = 8.0 GeV, p = 40.1 m (SPring-8) in Fig. 1-8. When Z =/2c, K2/3(!/2) = 1.206 and the photon number is given by
15 d3N
)
-
03
dtd-~MA ~o,Z=Z~- 5.04 x 1
y
2 (1-36)
Spectral distribution The spectral distribution of the radiation power is reduced by integrating Eqn. (1-27) over all vertical angles. dW[erg/sec/cm/electron] = 35/2 e2c r 7 aa(y) d~, 16n:2 p3
(1-37)
dW [erg/sec/~Jelectron] = 6.83 x 10 -24 7 7 G3(y) d~, p3Em]
(1-38)
y=~,c = e _ ec
(1-39)
or
where,
Here, G3(y) is the formula for n = 3 of Gn(y), which has been provided by Green [ 17].
Gn(y) = ynGo(y), G0(y) =
K5/ 3(r/ )d r/ (1-40)
where/(5/3(r/) is a modified Bessel function of the second kind. The values of various Bessel functions and integrals as a function of V are given in the Appendix of this chapter. Figure 1-9 shows the curves of Gn(y) for n - 1, 2, and 3 as a function of y. The radiation power spectrum is decided by G3(y), which has a maximum value of 1.24 and a fwhm of 0.84 ~,c, when y - 2.35, i.e.,/1, = 0.425 Ac. The radiation power for the electron beam current of 1 mA is represented by dW [erg/sec//~/mA] = 8.94 x 10-16 y 7 G3(y) dA p2[m]
(1-41)
Moreover, the power for the emission angle of 1 mrad in a horizontal direction is represented by dW [erg/sec/mrad(horiz.)/]k/mA] = 1.42 x 10 -19 ~t 7 G3(y) d0dZ p/Em]
(1-42)
16 where 0 is an angle in the horizontal direction. By dividing Eqn. (1-37) by hc/&, the power is converted to the photon number. d2N [photons/sec]cm/electron] = 33/2e 2 it 4 G2(y) dt dA, 47rh /92
(1-43)
d2N [photons/sec//~/electron] = 1.44 x 10- 5 ~,4 G2(y) dt dA, /921111]
(1-44)
or
101
I -
'
.......
I
'
.......
I
1 0~
........
I
'
' '''"1
:
1 0 -1
1 0 -2
1 0 -3
0.1
I
Fig. 1-9. Curves of Gn(y) for n
10
1/y = 1, 2 a n d
3.
100
1000
The radiation power spectrum is decided
by G3(Y). The photon number spectra is decided by G2(y), which is shown in Fig. 1-9. G2(y) has a maximum value 0.683 at y = 1.32, i.e., ~ = 0.76 2c, and has a fwhm 2.3 ~ . The photon number for an electron beam current of 1 mA is given by d2N [photons/sec/,2k/rnA] = 1.88 x 103 ~' 4 Gz(y) dt d,~, p2[m]
(1-45)
and the number for the emission angle of 1 mrad in a horizontal direction is given by d2N [photons/sec//~/mrad(horiz.)/mA] = 3.00 x 10-1 y4 G2(y) dtd0d~, p2[m]
(1-46)
17 Moreover, the photon number for a band width of AZ = 0.001 g is given by d3N dtdOdZi-----------~[photons/sec/mrad(horiz.)/O.l% b.w./mA]
=
1.26 x 107 ?'GI(y)
(1-47)
The photon number represented by Eqn. (1-47) is called "flux".
2 "~
1 02
..................... ~.............................. ~..................... p..................... .=..................... .
, oo
...........1 0 .
.
.
.
.
.
.
p=,i6m
10-2
10.4
~.
10s
10 -e 0.001
0.01
0.1
;~
1
10
(A)
1 O0
1000
Fig. 1-10. Spectral distributions of synchrotron radiation represented by the radiation power for various electron energies at an electron orbiting radius of 40 m. 1 0e
......................... ; ........................ ! ........................................................................ i i ~
o~
1 05
103 0.01
0.1
1
~(A)
10
100
1000
Fig. 1-11. Spectral distributions of synchrotron radiation represented by the photon numbers for various electron energies at an electron orbiting radius of 40 m.
18 The flux spectra depend on G l(y), shown in Fig. 1-9. G l(y) has a maximum value 0.92 at y = 0.29, i.e., ~, - 3.4 ~,c. The photon number at peak in the flux spectra is calculated by - d3N -
dtdOdM~,
[photons/sec/mrad(horiz.)/0.1% b.w./mA] = 1.16 x 107 ~'
(1-48)
Figures 1-10 and 1-11 are the spectral distributions of the radiation power and the photon number, respectively, showing the wavelength dependencies for various electron energies at a definite electron orbiting radius. The power and the photon number have remarkably wide spectral ranges of wavelength, and broad peaks. With an electron energy increase the peak shows an increase in the amount of power and the photon number, and a shift to shorter wavelength. The spectra have sharp cut-offs in the shorter wavelength range where both the power and photon number decrease steeply with increasing electron energy. In long wavelength range both spectra show gradually decreasing characteristics which is not likely to change much with electron energy.
Polarization The radiation is predominantly polarized with the electric vector parallel to the acceleration vector. The two terms in the last bracket of Eqn. (1-25) are associated with the intensities in the two directions of the polarization, Ip and I N, having the electric vector parallel and normal to the orbital plane, respectively. Figure 1-12 shows the angular distribution of the two components [ 18]. As is evident in Fig. 1-12a, the parallel component Ip has a sharp peak at ~r = 0 on the plane, while the normal component I N is zero at V = 0 and has small peaks above and below the plane (V > 0 and gr < 0). A degree of linear polarization Pl is defined as follows, /p - IN Ip + IN
Pl = ~
(1-49)
K~/3 (~) _ (]i i11)2 K2/3 (~)
l+(rv)
(1-50) Fig. 1-12b shows P l as a function of ~. At gr = 0, the radiation is of complete linear polarization and Pl = 1 (100% polarization). As I~ increases, PI decreases. Since the two components of the electric vector have a well defined phase relationship with respect to each other, namely +n/2 or-~/2 above or below the orbital plane, there is a degree of
19 circular polarization, Pc, with the decomposition of the elliptically polarized wave into rightand left-hand waves with intensities I R and I L, respectively. p c = IR - IL = +_ 1lip IN
IR + IL
Ip + IN
(1-51)
The positive and negative signs correspond to gt > 0 and gt < 0, respectively. As is evident in Fig. 1-12c, the circular polarization, Pc, is 0 at gt- 0. As I~ increases, the radiation tends to the circularly polarized condition, Pc - 1, although the total intensity decreases rapidly. e = 10 eV
"1 ~
e = 1 O0 e V
~1
+
e = 10000
'
eV
'
.__
m
(a)
g
.s 0 " " .....
0--
~...
"~
I
--2 0
I
-2
2
.9
'~ .+
-o "iI__ 0
t'-
-1
_!
1.
.
.
.
2 0
.~
.;
.I
.2
.9
(c)
.I..J
" "
-~.,
.s
0
I
V
(mrad)
2 o
u
•
I
1
(mrad)
2 0
V
(mrad)
Fig. 1-12. Angular distribution of intensity components with electric vector parallel (Ip) and normal (IN) to the plane of the synchrotron, linear polarization and circular polarization (from decomposition into left (IL) and fight (IR) hand circularly polarized components) for a storage ring with p = 12.12 m and an energy of E = 3.5 GeV calculated for three photon energies, gtis the electron angle perpendicular to the orbital plane [17]. By integrating Eqn. (1-27) over all wavelengths, the angular distribution of radiation power is obtained.
20 dW [erg/sec/rad(vertical)/electron] d~
=e2Cp275{1+(71//)2}-5/2 {1-7-6 - + 165 1+(~/i}2_ (Tgt)~_ }
(1-52) In this formula, the first and second terms in the last bracket correspond to parallel and vertical component of the polarization, respectively. Integration of Eqn. (1-27) over all angles and all wavelengths yields about 75% polarization in the orbital plane. Total radiation power and total photon number Total number is obtained by integrating Eqn. (1-37) over all wavelengths and the result is in agreement with Eqn. (1-26). The energy which a single electron loses by emitting radiation during one turn is given by AE [erg/electron/tuml = c2ZP W= 4z ep]t42
(1-53)
If we adopt practical units, AE [keV/electron/tum] =
88.5 E 4 [GeV]
p[m]
(1-54)
When the electron beam current is J, the total radiation power is given by
W_4ze2 _ __~ __p__,ff 4 j
(1-55)
In practical units, W[kW] =
88.5 E 4 [GeW] J [anlp]
p[m]
= 26.6 E 3 [GeV] B [tesla] J[amp]
(1-56)
For example, in the case of E = 2.5 GeV and p = 8.66 m, AE = 399 keV/electron/tum and W = 120 kW for J = 0.3 amp. In the case o f E = 8.0 GeV a n d p =40.1 m, AE = 9.04 MeV/electron/tum and W = 892 kW for J = 0.1 amp. The radiation power per 1 mrad on the horizontal plane is calculated for a beam current of 1 mA, as follows:
21 dW [watts/mrad]mA] = 0.0141 E 4 [GeV] dO p[m]
(1-57)
Then, the total number of photons emitted per unit time per electron, by integrating Eqn. (1-43) over all wavelengths, is dNdt[photons/sec/electron]
_5 ~
ff'e2~z hp
(1-58)
Moreover, the photon number per electron per turn is given by AN [photons/electron/turn] = 2 z p dN = 5 z ot 7 c dt
(1-59)
where a = 2 ~ 2 / ( h c ) = 1/137 is the fine structure constant. Brightness and brilliance
Previously we discussed the radiation emission from an electron in ideal circular motion. In practice, since the electron in the storage ring moves in an equilibrium orbit with some amount of fluctuation in space and angle, some corrections are necessary for the exact calculation of the spectral and angular distributions and the degree of polarization of the synchrotron radiation. Let us take x- and y-axes in the plane perpendicular to the tangent direction (z-axis) of a circular orbit, as shown in Fig. 1-13. Assuming that the electron beam has a Gaussian distribution in the xy-plane, and the standard deviations of the spatial distribution are represented by crx and Cry in the x- and y-directions, respectively, then the cross section of the electron beam, i.e., the size of the radiation source, is 2.352 Crx Cry (because the full width at half maximum is 2.35 times the standard deviation). The extents of angular distribution are also similarly represented by the standard deviations of Crx' and Cry'. The values of crx, Crx', Cry and Cry' are variable, depending on the location along the electron orbit. However, there is an interrelation between Crx and Crx', and also Cry and Cry'. We define an emittance ex and ey as follows, Ex = axax'
(1-60)
ey = Cryay"
(1-61)
The emittance is independent of electron motion, according to Liouville's theorem. When the spatial distribution of the electron beam is small, the angular divergence becomes large, and vice versa. Moreover, there is a correlation between ex and ey, depending on the storage ring structure and the oscillatory motion of electron beam in orbit.
22
r
X
/
-Orbit ~
12.35 y
y -<
2.35o'x
950'x' L..J'
z
..._1
2.35~y'
Fig. 1-13. Schematic diagram showing the size and angular divergence of synchrotron radiation source. As mentioned previously, the photon flux, which is the photon number emitted per unit time, per unit angle in the horizontal plane, and unit band width, is defined by d3N
dtd OdA/~
[photons/sec/mrad(horiz.)/O.1% b.w./mA]
The flux is decided only by the electron energy, and is not related to the size and angular spread of the electron beam. Brightness, which is the flux represented for unit solid angle, i.e., the photon number per unit time, unit solid angle and unit band width, is defined by d3N [photons/sec/mrad2/0.1% b.w./mA] dtdl-2dA/~,. According to this definition, the angular divergence is given by convolution of the angular width of the electron beam with the intrinsic angular width of the radiation. High brightness would be required in many synchrotron radiation experiments, especially when one uses an aberration-free optical system. "Brilliance" is the brightness divided by the size of the radiation source, i.e., the photon number per unit time, unit solid angle, unit area,.and unit band width. The brilliance is defined by
23
d~ dtdl'2dSdA/A
[photons/sec/mrad2/mm2/0.1% b.w./mA]
where S represents the area of the radiation source, being related to the size and angular spread of the electron beam. When one needs a focused sharp radiation, high brilliance is necessary. The low electron beam emittance gives high brightness and high brilliance. 1.3. TYPICAL SYNCHROTRON RADIATION SOURCES The synchrotron radiation facility is typically composed of a linear accelerator, synchrotron, and storage ring, although several variations are possible. The major part of the facility is of course the storage ring. High energy electrons accelerated by the linear accelerator and synchrotron are injected and stored in the storage ring. In this section we discuss the characteristics of the source facility in general and describe typical facilities in operation and under construction.
1.3.1. General System of SR Source In the storage ring, as a radiation source, high energy electrons moving in a definite circular orbit are stored for more than several hours. The electrons are injected into the storage ring after being accelerated up to a final storage energy by the linear accelerator, and by the synchrotron in the case of full energy injection. Sometimes, especially in small or intermediate size SR facilities, the injection system is designed to be operated at a lower energy than the final storage energy, and there must be one or more radiofrequency cavities in the storage ring in order to accelerate electrons to full energy. When the electron energy is several 100 MeV, the SR spectra have a peak around soft X-ray to VUV range. And when the energy is several GeV, the spectral peak is in the X-ray range. Dipole magnets are arranged along the electron orbit to bend the electron beam. The magnetic field of the magnets is kept constant in the storage ring because the electron beam energy has to be kept constant. There are photon beam ports at the location of the dipole magnets so that the synchrotron radiation may be utilized. Between the dipole magnets on the straight beam line section there are quadrupole magnets, which provide magnetic fields with gradients to act as lenses for the electron optics system. The electron beam is maintained in a stable oscillation (betatron oscillation) around an equilibrium orbit by the lens action with the magnets. The electron circulating in the ring loses energy by emitting synchrotron radiation. This energy loss is again supplied by radiofrequency cavities, which are set up on a straight section of the ring, and an associated power supply. The radiofrequency may be selected so as to be an integer multiple of the electron orbital revolution frequency. The integer is called the harmonic number of the ring. With this scheme of radiofrequency acceleration, electrons in the ring can be grouped in bunches of the harmonic number. The radiation is, therefore, pulsed at definite
24 intervals. A fairly high total electron current is available since many discrete bunches of electrons are in the ring, although there is a limitation to the number of electrons stored in one bunch. If time-analysis experiments are intended, the electrons should be stored in a small number of bunches, with the interval of the radiation pulse being spread. When the electrons are assembled in a single bunch, the photon pulse interval is the maximum and equal to the time when electrons turn over the ring. The radiation is pulsed to the bunch width, which is typically about 10% of the radiofrequency period. The electron beam should be of small emittance so that we obtain a radiation source having high brilliance. The chamber in which the electrons circulate has to be maintained at an ultra-high vacuum. For the stored electron beam's decay time to be of the order of many hours, the average pressure must be in the 10-9 to 10-10 Torr range to minimize scattering of the stored electrons by the residual gas atoms. In the current SR facility design, insertion devices such as wigglers and undulators are widely employed in the storage ring. The insertion devices are other magnetic structures, which for many experiments are more effective sources of synchrotron radiation than the dipole bending magnets. The principles of the radiation emission from the devices (wiggler and undulator) are shown in Fig. 1-14a and 1-14b. Both wiggler and undulator magnets are inserted into the periodically deflected, although there is no net deflection or displacement of the beam from the equilibrium orbit.
2
(a)
2
(b) Fig. 1-14. Radiation from insertion devices; (a) Multi-pole wiggler and (b) undulator. Let us define a parameter K in order to characterize the synchrotron radiation emitted from the insertion devices, as follows.
25
eBoAu
K =
2nmc 2 = 0.934 Bdtesla] Au[Cm]
(1-62) (1-63)
Here, B o is the peak field and )~u the magnet period in cm. K is called "deflection parameter", being independent of the electron energy. A maximum angle with which an electron oscillates around the axis (z-axis) is given by K/),. On the other hand, instantaneously emitted photons are focused in a very small cone with angular width 1/?'. Thus, in the case of K >> 1 (Fig. 114a), the radiation can be observed in the z-axis only when electrons move near the top of their oscillating trajectories. This photon emission is evident via the wiggler (multi-pole wiggler). The radiation shows an increase in the critical energy, and a shift of the overall spectrum to higher energies compared to the dipole bending magnet radiation, although the spectral profiles are similar in both two magnets. If the wiggler is composed of N poles, the electrons oscillate N times within the wiggler and the radiation intensity (brightness) is increased to N times. In the case of K ~ 1 or K < 1 (Fig. 1-14b), the radiation is observed without intermission in the z-axis. This emission of radiation is evident in the undulator. The electric vector of the undulator radiation changes sinusoidally. Therefore, the radiation spectrum, which is described by its Fourier transformation, is quasi-monochromatic and a primary wave is prominent. The radiation, which is emitted at an angle 0 with respect to the z-axis, has its k-th order higher harmonics of the wavelength 2 k, given by ,~k
=
K2 2 k ~ ( 1 + - ~ + ~'202)
(1-64)
In the case of K << 1, the major component of the radiation is the primary wave, with wavelength/1,1 = Au/(27'2), on the z-axis (0= 0). As we will see later, an electron emits waves which add coherently. The radiation intensity varies with the square of N, and the spectral width A&/A is of the order of I/N, where N is the number of periods of the undulator. Electrons travel incoherently with respect to each other, and the photon intensity is proportional to the number of electrons. Undulators are extremely intense tunable radiation sources. 1.3.2. Typical Synchrotron Radiation Source Facilities
In response to a rapidly increasing demand a number of synchrotron radiation facilities have been built or are under construction or planning throughout the world. All the existing or planned SR facilities fall into three general classes: (i) small scale facilities with the electron energy less than 1 GeV and ec < 1 keV; (ii) intermediate scale facilities with the 1 GeV to 3 GeV electrons and 1 < ec < 10 keV; and (iii) large scale facilities with the electron energy higher than 3 GeV and e c > 10 keV. Generally speaking, the larger scale facilities can produce higher
26 energy photons or X-rays. However, of course, what decides the photon energies is the orbital radius as well as energy of the electrons. Even if the electron energy is low, higher energy photons are emitted with a high dipole-magnet field, which makes the electron orbital radius small. The first class facilities, which mainly produce radiation with wavelengths from soft X-rays to vacuum ultraviolet (VUV) radiations or longer, are useful for research in photo-electron spectroscopy (XPS), photochemistry, and photoabsorption in the soft X-ray to VUV range, and lithography and photo-etching. On the other hand, using the small scale facilities is not always beneficial for X-ray spectrochemical analysis, because X-rays with energy higher than 1 keV are usually employed as primary X-rays in the analysis. Intermediate and large scale facilities may be preferable for X-ray spectrochemical analysis. The large scale ones, especially, can produce X-rays with energy higher than 100 keV with high intensity, which make possible the excitation of K X-rays of even the heaviest element, uranium. In this Section, we describe, as examples, three of the typical facilities dedicated for SR studies, that is, the National Synchrotron Light Source (NSLS) at Brookhaven National Laboratory, USA, the Photon Factory (PF) at National Laboratory for High Energy Physics (KEK) at Tsukuba, Japan and the Super Photon Ring (SPring-8) at Harima Science Garden City, Japan (under construction).
National Synchrotron Light Source (NSLS) The NSLS was designed and built as a combined light source for VUV and X-rays. The overall layout of the facility is shown in Fig. 1-15. Electrons are accelerated in a linear accelerator to 100 MeV, and injected into a booster synchrotron and accelerated up to 700 MeV. Then the electron beam is transferred either to the 700 MeV VUV ring or to the 2.5 GeV X-ray storage ring. Table 1.1 Parameters of the VUV- and X-ray- storage tings in the National Synchrotron Light Source. Parameters
VUV Storage Ring
Normal operating energy Maximum operating current Lifetime Circumference Number of beam ports of dipoles Number of insertion devices Maximum length of insertion devices ~c (ec)
745 MeV 1.0 amp
B (p)
1.23 T (1.91 m)
51.0 m 17 2 2.5 m 25.3 A (486 eV)
X-ray Storage Ring 2.5 GeV 0.25 amp --20 hr 170.1 m 30 5 <4.5m 2.48/~ (5.0 keV) at 1.22 T 0.60 A (20.8 keV) at 5.0 T 1.22 T (6.875 m)
27 Table 1.1 (continued) X-ray Storage Ring
VUV Storage Ring
Parameters Damping times, lrx = Vy
17 msec
6 msec
~e Electron orbital period Lattice structure (Chasman-Green)
9 msec
3 msec
170.2 nsec separated function quad., doublets 8 bending 3.14, 1.20 0.023 52.88 MHz 4.5 x 10-4
567.7 nsec separated function quad., triplets 16 bending 9.15, 6.20 0.0065 52.88 MHz 8.2 x 10-4
Horiz. dumped emittance (e x )
1.5 • 10-7 m-rad
1 x 10-7 m-rad
Vert. dumped emittance (ey)
22.8 x 10-10 m-rad
1 x 10-9 m-rad
Source size, trh, trv
0.5 mm, > 0.006 mm
0.35 mm,0.15mm
Number of superperiods Nominal tunes, v x , Vy Momentum compaction RF frequency Natural energy spread (ixE / E)
The details of the facility design have been described by van Steenbergen and the NSLS staff [19]. Recent operating parameters of the VUV and X-ray storage ring are listed in Table 1.1 [20]. The basic ring structure consists of 8 bending dipole magnets in the VUV ring and of 16 in the X-ray ring, delivering continuum synchrotron radiation with e c = 486 eV (~.c = 25.3/~), and e c = 5.0 keV (~.c = 2.48 !k), respectively. There are several insertion devices; i.e., four undulators (U5U, U13TOK, X1 and X13) and three wigglers (X17, X21 and X25). (see Table 1.2) With the incorporation of the high field superconducting wigglers in the X-ray ring, the wavelengths can be extended to hard X-rays exceeding 100 keV. Table 1.2 Energy parameters for the insertion devices in NSLS (January 1991). Parameters Magnetic field - Bmax [T] Number of poles Periodic length
U5U
U13TOK
X1
X13
X17
0.46-0.01 0.72-0.05 0.31-0.03 0.31-0.03
[email protected] 2poles @2.6 55 44 69 20 7 7.5 10 8 8 17.4
X21/X25 1.1 27 12
- ~, [ c m ]
Wiggler characteristic energy- ec [keV] Energy range [eV] 11.3-70 - Undulator fund. Deflection parameter 3.2-0.08 range - K Gap range [cm] 3.4-12.0
0.265 2.2-50
22.2 for 5.2T poles 0.19-0.73 0.19-0.73 10-100
4.57 4-30
6.7-0.4
2.3-0.2
2.3-0.2
84
12.3
3.35-12.0
3.3-9.8
3.3-9.8
3.2
2.4-12
tO OO
X19
X-RAY
% U3 VUV X5 •
u7 -~=
Insertion Beamline
Device
Control
R o o m ~
Fig. 1-15. Overall layout of the National Synchrotron Light Source (NSLS) at Brookhaven National Laboratory (by the courtesy of N. F. Gmuer).
29
10le E
..... VUV, 1:800mA " --XRAY, 1:250mA
.U~U
-=-_-.0.._.. . . . . . .-...
I0Is
~
X21,X25
'~ 1014
~I013
-
10~
u,ou
102
,,,
Ill!
\
.
.
.
.
.
.
.
105 104 Photon Energy [eV]
\ \
\
\
105
Fig. 1-16. Radiation spectra for NSLS bending magnet and insertion device (undulator and multipole wiggler) sources. The undulator curves are the envelopes of the first, third and fifth harmonics, 0.5 < K < 3. Figure 1-16 shows the radiation spectra (photon energy vs. flux curves) for the NSLS bending magnet and insertion device sources. Although the horizontal angular dependence of the bending magnet radiation is constant, output of the wiggler peaks on the axis, and falls to zero at ~ / y , where K is a deflection parameter given in Table 1.2 and y= Ering/mc 2 ; Ering = 745 MeV (VUV ring) and Ering = 2.5 GeV (X-ray ring). For the wigglers, the flux within one mrad (horizontal) centered on the axis is shown, even though most beam lines can collect more than one mrad horizontally. For the undulators, the central cone flux is plotted: the angular size of the cone depends on both the undulator and the electron beam parameters. Most beam lines can accept this entire cone. The undulator curves in Fig. 1-16 are the envelopes of the first, third and fifth harmonics, 0.5 < K < 3. In order to maintain precise control of the source locations and exit angles of the synchrotron radiation an elaborate electron beam deflection and correction system is incorporated in both storage tings. The VUV ring has 16 ports, and the X-ray ring 29 ports, of which some ports are split into two or more branches.
Photon Factory (PF) The PF is an X-ray storage ring fully dedicated to synchrotron radiation research. Figure 117 shows the layout of the facility [21 ]. A linear accelerator can accelerate both electrons and positrons up to 2.5 GeV. These are injected and stored in the storage ring. Since the full energy injection system is employed, there are two advantages in that the locations of the
30 sources of synchrotron radiation in the ring are controlled precisely and with good reproducibility, and the beam injection can be made in a shorter time. Table 1.3 Major parameters of the Photon Factory ring. Normal operating energy Maximum operating current Lifetime Circumference Bending radius Betatron tunes, horiz./vert. Emittance, horiz. vert. RF frequency
2.5 GeV (max., 3.0 GeV) 360 mA (e-), 350 mA (e+) 15 hr (300 mA), 30 hr (150 mA) 187 m 8.66 m 8.38 / 3.18 1.3 • 10 -7 m-rad 2.0 x 10-9 m-rad 500 MHz
0 ,
,
i
IA ~........,
2Bz
16C
~\11B
\
13m~"4 138z
Fig. 1-17. Layout of the Photon Factory at Tsukuba.
i
'
25m
31 The storage ring has an elliptical shape with diameters of 68 m and 50 m. The structure of the ring consists of 28 bending magnets, 58 quadrupole magnets and many small magnets for orbit corrections. Table 1.3 lists the main operation parameters of the PF facility. The radiation emitted from the electrons deflected by the bending magnets has a continuous energy spectrum with the critical energy of 4.11 keV (3.0 A). An electron loses energy of 400 keV per turn of the 187 m circumference, which means that a 300 mA electron beam loses 120 kW by emitting synchrotron radiation. The radiations from the bending magnets and the insertion devices are available at 24 ports, among which 15 ports are for the experiments and 3 ports are used for beam diagnosis. Table 1.4 Parameters for the insertion devices in PF. Line BL02
Type U
BL13
MPW/U
BL14
WS
BL16
MPW/U
2 u [cm]
N
Gmin [cm] Bmax [T] Kmax
hv [keV]
U0 [keV] 1.37 2.44
6.0
60
2.8
0.3 0.4
1.7 2.3
0.4 / 5.0 0.25 / 5.0
18.0
14
2.7
1.5
25.0
0.03 / 70
22.4
5.0
6.0
< 100
50.0 27.8
12.0
26
1.8
1.5
16.8
0.03/70 0.01 / 1.1
0.71 1.5 2.6 3.5
0.01 / 20
7.2
BL19
U
5.0 7.2 10.0 16.4
46 32 23 14
3.0 3.0 3.0 3.0
0.3 0.4 0.5 0.6
1.3 2.7 5.0 9.5
BL28
C
16.0
12
3.0 11.0
1.0 0.2
15.0 3.0
U, undulator: MPW/U, multipole wiggler and undulator: WS, wavelength shifter: C, circular polarization source: Au , magnet period: N, number of periods: Gmi n , minimum gap: B, magnetic field: K, K value: U0, radiation loss. The storage ring has two long straight sections (5 m long) and eight intermediate ones (3.5 3.75 m long). There are an electron injection apparatus, two sets of radiofrequency cavities and several insertion devices in the straight sections. Table 1.4 shows the major parameters of the insertion devices. An undulator made of SmCo 5 permanent magnets is installed in the long straight section (BL02). The undulator has 60 periods (6 cm each period) and the field is comparatively low (0.31 tesla maximum), producing high brilliance radiations with energy 0.4 to 5.0 keV. In BL16 is set a multi-pole wiggler (MPW), which has 26 periods with 12 cm length per period, producing the radiations between 0.03 and 70 keV with very high brilliance (almost 100 times that of the bending magnet radiations). The MPW is a complex of NdFe
32 permanent magnets (NEOMAX30H) and Permendur with the magnetic field variable from 1.5 to 0.05 tesla, by which one can select alternatively its operational mode,-- either the wiggler mode (fixed at 1.5 tesla) or the undulator mode (variable from 0.05 to 0.5 tesla). In BL14 a wiggler with super-conducting magnets of 6 tesla (maximum) is fitted to deflect electrons vertically to the orbital plane, which is useful for hard X-ray experiments. Other kinds of insertion devices are also available, which have been developed and installed recently.
Super Photon Ring (SPring-8) Up to the end of the 1980s more than twenty synchrotron radiation facilities have come into operation throughout the world, and studies using them have been made in many research fields. In due course, the need for high brilliance SR facilities was envisaged and new projects to construct the next generation SR facilities have started in Europe, the United States and Japan. The SPring-8, which is an 8 GeV SR source facility, is one of them. Construction started in 1990 and will be finished in 1997. Several requirements have been considered in designing the SPring-8 facility [22]. They are for: (i) a high brilliance source for both soft and hard X-ray radiations, (ii) a widely and easily tunable photon energy, (iii) a superior time structure of the photon beam (selective use of single- and multi-bunch beam), and (iv) a photon beam which has both a stable position and intensity. Insertion devices are to be the principal radiation sources for the next generation of SR sources. In particular, undulators with different spectral characteristics and polarization will be the most important radiation sources. In order to accommodate as many insertion devices as possible, the SPring-8 has a sufficient number of long, dispersion-free straight sections. A full energy injection method is adopted to make the beam position very stable. The injection system consists of a 1GeV linear accelerator as preinjector and a synchrotron to accelerate the electrons up to 8 GeV. Table 1.5 Major parameters of the SPring-8 storage ring. Energy Current (multi-bunch) Current (single bunch) Circumference Dipole magnet field Bending radius Number of cells
8 GeV 100 mA 5 mA 1435.95 m 0.665 T 40.098 m 48 (44 + 4)
Straight section length
6.4 m
Natural emittance Critical photon energy Tune, v x
6.89n nm-rad ze 28.32 keV Harmonic number 51.22 RF voltage 16.16 RF frequency
Vy
Synchrotron tune Momentum compaction Natural chromaticity Energy loss in the arc Energy spread Damping time, zx zz
0.0102 1.49 x 10 -4 - 13.17 - 43.32 9.04 MeV/rev 0.108% 8.473 msec 8.481 msec 4.242 msec 2436 17 MV 508.58 MHz
.~t I I 3
8
Experimental Hall
\,/ Linac
Storage Ring
-~
/~I~,~
"~~,.~~-~chro iron
Storage Ring Tunnel Long Beam Line
(30Ore)
r-------1
Long Beam Line
(Ikin)
"--"-'-'-'---~::z:~
Fig. 1-18. Plan view of the Super Photon Ring (SPring-8) at Harima science Garden City (by the courtesy of H. Kamitsubo).
34 The major parameters of the storage ring are summarized in Table 1.5 and a plan view is shown in Fig. 1-18. The circumference of the ring is 1436 m and is divided into 48 cells. Among them 44 are normal cells, each of which contains two 2.8 m bending magnets, ten quadrupoles, seven sextupoles and a 6.5 m dispersion-free straight section, whereas four are straight cells, which contain two 6.5 m dispersion-free straight sections, two 3-m long straight sections (in places of the bending magnets in the normal cells), and the same number of quadrupoles and sextupoles as in the normal cell. In addition to these, there are 12 steering magnets in each cell for closed orbital distortion (COD) correction and for dynamic feedback of the beam position movement. The chromaticity is corrected by three harmonic sextupoles which are located in the dispersion section between two bending magnets. The radiofrequency system (505.58 MHz) consists of four 1 MW klystrons and 32 cavities which are installed in 4 low-fi x straight sections each 6.5 m.long. The energy loss per turn due to SR emission in the bending magnets is 9.0 MeV, whereas the energy loss due to the insertion devices is estimated to be 3.2 MeV, and others due to the cavities and vacuum chambers are evaluated to be 0.5 MeV. In addition to this, an overvoltage is required for obtaining a long quantum lifetime. Calculations give the maximum peak voltage needed in the cavity as up to 17 MV. Table 1.6 Typical values brightness, flux and size of photon beam from SPring-8 Devices
Brightness Flux* [photons/s/ [photon/s/ mrad/0.1% b.w.] 0.1% b.w.]
Undulator 1
2 x 1019
1 x 1014
Undulator 2
2 x 1020
7 x 1014
Wiggler
2 • 1018
3 x 1014
Bending
2 x 1013
1 x 1013
* Flux;
Size [mm]
4x2
200 x 9
Au [cm]
L [m]
K
E0 [keV]
3.3
5
1
12.3
3.3
30
1
12.3
18
2
25
63.9
Undulator, photons through a hole ( ~ 0 - 64 mrad). Wiggler and Bending, photons / mrad.
Table 1.6 lists typical values of brightness and flux for the radiations from the undulators, wigglers and bending magnets, calculated assuming a stored electron current of 100 mA. From the bending magnets and insertion devices 23 and 38 beam lines are available, respectively. The average lengths of the beam lines are 80 m from the exit of the insertion devices or bending magnets to the photon beam dump at the end of the beam line. For studies requiring longer beam lines, 8 beam lines can be extended up to 300 m and three can be lengthened to 1000 m.
35 1.4. I N J E C T O R S
A synchrotron radiation source is composed of a storage ring, which accumulates and stores electrons (or positrons) over many hours, and attached beam lines for extracting the radiation to the experimental hall. The source of particles to a storage ring is called the "injector". One can classify injectors into two groups, depending on their energy relative to the operating energy of the storage ring. In many existing storage tings, as can be seen in Table 1.7, the injection energy is lower than the operating energy of the storage ring. In these cases, electrons must be accelerated to the operating energy by ramping the magnetic fields of the storage ring. In modern storage tings, however, the injector has the same energy as the operating energy. This scheme is called the full-energy injection. Table 1.7 List of typical synchrotron radiation facilities classified by their injector system Facility name
Storage ring energy [GeV]
(a)Synchrotron NSLS VUV (BNL) UVSOR (IMS) SRS (Daresbury) NSLS X (BNL) SPEAR (SSRL) INDUS-I (Indore) SRRC (Taiwan) ALS (LBL) APS (ANL) SP-8 (RIKEN/J AERI) (b) Linear accelerator SUPER-ACO(LURE) HESYRL (Hefei) Photon Factory(KEK) BEPC (Beijing) ELET1RA(Tfieste) PLS (Pohang) (c) Microtron MAX (Lund) ALADDIN(Wisconsin) op: in operation
0.75 0.8 2.0 2 . 6 - 2.8 4.0 0.45 1.3 1.9
energy [GeV] 0.75 0.6 0.6 0.75 2.3 0.45 1.3 1.5
Injector
rep. rate [Hz]
Status
0.66 2.5 10 0.66 30 2 10
op op op op op c c
7.0 8.0
7.0 8.0
2 1
1
c
0.8 0.88 2 . 5 - 3.0 2.8 2.0 2 . 0 - 2.5
0.8 0.2 2.5 1.4 2.0 2.0
12.5 1 25 50 10 10
op op op op c c
0.55 1.0
0.1 0.108
10 1.25
op op
c c
c: in construction
Needless to say, an injector is much more expensive for higher energies. However, the beam is generally subject to instabilities if it is injected at low energies and, in particular, is very sensitive to the operating parameters in the "low-emittance" storage tings which have been quite
36 common for several years in high-brightness synchrotron radiation sources. Table 1.7 shows that full-energy injection will be widely adopted in forthcoming storage rings. The accelerators which have been used as injectors for storage tings are synchrotrons, linear accelerators, and microtrons. Short descriptions of these accelerators will now be given.
1.4.1. Synchrotron The synchrotron is a circular magnetic accelerator which uses time-variable magnetic fields to bend and focus electrons. Since the magnetic field strength varies with the energy of the electrons, the orbit radius is always kept constant. Acceleration is achieved with the aid of radiofrequency (rf) accelerating cavities. In general, the rf frequency of a synchrotron must be changed with time in order for particles and fields to synchronize all the way. For an electron synchrotron, however, the time of revolution of the electrons around the ring does not depend on energy because of the constancy of their velocity, even at relatively low energies. Hence, a fixed rf frequency is adequate for electron synchrotrons. This simplifies their construction and operation, and makes it easy to operate them with a high repetition rate. As shown in Fig. 1-19, the basic components of a synchrotron are magnets for bending and focusing electrons, radiofrequency (rf) cavities for accelerating electrons, and specially designed pulsed magnets for injecting and extracting electrons. The electron beam passes through a hollow metal or ceramic pipe having a lateral size typically 5 cm x 10 cm. The pipe, which is called the vacuum chamber, is evacuated to a pressure of the order of 10-6 Torr.
extr
j l__lj ......
w IflJt~ULUI-
~
~
rf cavity
cross section
of magnet
Fig. 1-19. Schematic of a synchrotron. A small linear accelerator or microtron is used as the injector for a synchrotron. Electrons are injected into the synchrotron when its magnetic field is low (e.g. 1/10 to 1/20 of the maximum field strength), and are accelerated as the field strength increases. At the crest of the magnetic
37 field, electrons are extracted by means of pulsed magnets. This cycle is repeated at regular intervals. Usually, electron synchrotrons operate at a repetition rate between 1 Hz and 50 Hz.
(b)
Ca)
~t
--~t
Fig. 1-20. Magnetic field waveforms of synchrotron; (a) slow-cycling and (b) rapid-cycling. The synchrotrons are generally considered in two classes; the slow-cycling synchrotrons and the fast-cycling synchrotrons. The slow-cycling synchrotron, which operates at around 1 Hz, uses rectifying circuits for generating magnetic fields, and the field waveform is triangular (Fig. 1-20a). On the other hand, the rapid-cycling synchrotron repeats at a frequency of 10 to 50 Hz, and utilizes resonant circuits for generating magnetic fields. Hence, the field waveform is sinusoidal (Fig.l-20b). Generally speaking, a shorter injection time is better from the viewpoint of increasing the time available for the synchrotron radiation research. The injection time depends on the repetition rate and the number of particles which the injector can provide every cycle. Since there is a limit on the intensity which can be accelerated in a cycle, the rapid-cycling synchrotron gives us a shorter injection time. For technical simplicity and construction cost, however, the slowcycling synchrotron may be superior to the rapid-cycling synchrotron, since most technical components of the former are simpler. In the rapid-cycling synchrotron the magnet system needs an unusual high-power resonant circuit technique, while the thyristor rectifier used for the slow-cycling synchrotron is a popular component in power engineering. In rapid-cycling synchrotrons one also needs complicated corrugated or ceramic chambers to reduce eddy currents produced by the rapidly-changing magnetic field, and much higher power rf cavities. 1.4.2. Linear Accelerator
Radiofrequency linear accelerators have been the most popular injectors for synchrotrons, and can also be the direct injectors for storage rings. Figure 1-21 shows a simplified block diagram of a travelling-wave linear accelerator. The linear accelerator comprises an electron gun (a source of electrons), sections of microwave waveguides as the accelerating structure, and solenoid coils or quadrupole magnets for focusing electrons. The phase velocity of the microwave travelling down waveguide coincides with the velocity of electrons. Therefore, electrons which maintain the proper phase relationship with the microwave can be accelerated
38 continuously by the electric field pointing in the direction of motion. The most common accelerating structure is the disk-loaded waveguide working at a microwave frequency of about 3,000 MHz. Linear accelerators with energy lower than 20 MeV are widely used for medical applications and non-destructive inspection, and are commercially available. Their accelerating gradient per unit length of accelerator guide is typically 10 MeV/m, and a single microtron or klystron is used as a microwave power source. Linear accelerators of higher energies, which have been built for research applications, need many sections of accelerator guides and klystrons. For example, in the 2.5-GeV linear accelerator at the Photon Factory of KEK at Tsukuba, Japan, there are 160 sections of accelerator guides, each 2m long, with 40 klystrons used to energize them. A microwave power of 30 MW amplified in a klystron is split and fed into four accelerator guides. The KEK linear accelerator operates at a repetition rate of 50 Hz. The peak current of the accelerated beam is 50 mA in a pulse of 2 Its, in normal operation. The accelerating gradient per unit length of accelerator guide is 10 MeV/m. Accelerating structures with higher gradients, in the range of 30-50 MeV/m or even 100 MeV/m, are under development. Klystrons with powers of 50 to 100 MW are also becoming available.
Fig. 1-21. Travelling-wave linear accelerator; (a) accelerator guide and (b) disk-loaded waveguide. As mentioned before, linear accelerators have been used as injectors for synchrotrons. An energy of 20-30 MeV is appropriate for a synchrotron of an energy less than several hundred MeV, while an energy of 500 MeV to 1000 MeV will be necessary for a synchrotron of multiGeV. Linear accelerators have also been used for the direct injection to a storage ring. Most high energy linear accelerators in the multi-GeV range were built as injectors for high-energy electron-positron colliders for elementary particle physics, and some storage tings for synchrotron radiation rese~ch take advantage of these accelerators in parasitic use. However,
39 since high-energy linear accelerators are more expensive, synchrotrons are preferred as injectors for the direct injection into the storage ring, unless the beam is shared for other purposes. When positrons are stored in the storage ring a high-energy, high-intensity linear accelerator will be inevitable. Positrons can be produced by high-energy electrons bombarding a heavy metal target, such as tantalum. In the Photon Factory of KEK, a high-intensity linear accelerator of 230 MeV is used to produce positrons. Low energy positrons (around 5 MeV) produced at various angles and with various energies are focused using high-field magnetic solenoids, and accelerated to an energy of 250 MeV by another linear accelerator. The positron beam so produced is transferred to the main accelerator, in which the positron is accelerated to the final energy. Generally, the yield of positrons is roughly proportional to the beam power (i.e. the energy times the current) of the incident electron beam. At the Photon Factory the number of positrons accelerated to the final energy is around 0.1% of the number of the incident electrons. 1.4.3. Microtron The microtron is a circular accelerator in which a static magnetic field is used, and so the radius of curvature of the circular orbit grows with increasing energy. Acceleration is achieved by an rf accelerating cavity. As shown in Fig. 1-22a, each time the electron passes through the cavity it gains energy, and the orbit radius grows in discrete steps. If a proper relation holds between the energy and rf voltage, the orbits overlap at the position of the cavity.
(a)
i
agnet \rf cavity
e
f
f
(b)
Fig. 1-22. Schematic of microtron; (a) circular and (b) race-track. Small microtrons in an energy range of 10 to 50 MeV are commercially available, and are used in medical applications. The race-track design shown in Fig.l-22b has been applied to higher-energy microtrons of around 50 MeV. By this means, a cavity of multiple gaps can be inserted in a straight section between magnets, and a high accelerating energy per pass can be achieved. Although the intensity obtainable from microtrons is usually lower than from linear accelerators there are advantages in the beam quality, such as the excellent energy-spread and, more importantly, the cost is lower than that of linear accelerators.
40 Some synchrotron radiation sources have used a microtron of energy higher than 100 MeV as the injector to the storage ring. In addition to the examples listed in Table 1.7, a compact storage ring developed for micro-lithography has used a microtron of around 100 MeV. However, high-energy microtrons are not yet popular. 1.5. STORAGE RING
A storage ring is a circular ring, whose structure is essentially the same as a synchrotron's, and which keeps a beam of electrons (or positrons) circulating at relativistic velocities over many hours. It comprises magnets for bending and focusing the circulating beam, rf accelerating cavities for making up for energy lost as synchrotron radiation, and vacuum chambers for keeping an ultra-high vacuum along the particle trajectory. Figure 1-23 illustrates the principal functions of a storage ring. The magnetic system consists of bending magnets and focusing quadrupole magnets. Along the circular path in the bending magnets, synchrotron radiation is emitted in a tangential direction. Many beam lines for extracting the synchrotron radiation can be attached to normal bending magnets. Between the magnets there are straight sections which are free for installation of auxiliary components including rf accelerating cavities, pulsed magnets for injecting the beam, small magnets for controlling the orbit, diagnostic instruments, and vacuum pumps. In addition, insertion devices, which will be described later, are placed in the straight sections of the storage ring.
devlce
'~
l~bendlng magnet
7
/ from
L~
"~quadrupolemagnet \^
_
rf cavlty
_ ,./
Injector~l_l[__7,r_] ~
Fig. 1-23. Schematic of storage ring.
<synchrotron
I r~di-atio n
41
1.5.1. Storage Ring Parameters The principal parameters which characterize the performance of a storage ring will now be described.
Energy The energy of the storage ring is related to the spectrum of the synchrotron radiation. As described in the previous section, the spectral distribution of the synchrotron radiation is a function of the electron energy and the radius of curvature of the circular path. For normal bending magnets the critical wavelength of the radiation, Ac[A] = 18.6 / B[tesla] E2[GeV], as given by Eqn. (1-20). For high energy storage tings, the field strength of the bending magnet is usually chosen to be between 0.5 and 1 tesla, since these values are optima for economical and technical reasons. However, for compact storage rings which are designed for X-ray lithography, superconducting magnets with a field strength around 4 tesla have often been used. The advantages of taking high magnetic fields are that the size of the storage ring is reduced because of the small orbit radius, and that low electron energies are sufficient to give the radiation spectrum required for lithography. As described before, recent synchrotron radiation sources are based on the comprehensive use of insertion devices, particularly undulators. In this case, the required energy of the storage ring is determined by considerations different from those above, as will be described later.
Emittance While travelling in the storage rings, an electron makes a quasi-sinusoidal oscillation in the lateral direction owing to the focusing action provided by quadrupole magnets. This oscillation is called the "betatron oscillation"(Fig. 1-24). The electron with the design energy oscillates around the central orbit (the design orbit).
Fig. 1-24. Betatron oscillation.
42 Let s be the azimuthal coordinate along the orbit. The betatron oscillation is expressed by the following
y(s) = (e / fl
(S))I/2cos(O (S)+qbO)
(1-65)
where fl (s), which is called the betatron function, is a function of the azimuthal position, s. Here r (s) is the phase function which is given by $ (s)= ~(1/fl (s))ds, and $0 is an arbitrary phase: e is an arbitrary constant relating to the amplitude of oscillation and is called the "emittance". Equation (1-65) represents a pseudo-harmonic oscillation, whose amplitude and phase are modulated during motion, as shown in Fig. 1-24. If an electron has a slightly different energy from the design value, the orbit around which it oscillates will differ from the central orbit. This property is called dispersion. The equilibrium orbit for the off-energy electron is called the off-energy orbit. The quantity relating the orbit deviation to the energy is the dispersion function, usually designated as r/(s). The orbit deviation, Ay(s), for an electron of energy deviation AE/E is given by
Ay(s) = I"1 (s)AE/E
(1-66)
The betatron function fl (s) and the dispersion function r/(s) are uniquely determined if the magnetic configuration of the storage ring is given. Electron storage rings usually contain a periodic structure of magnets, and hence both betatron and dispersion functions have the same periodicity as the magnetic configuration. The amplitude and phase of the oscillation varies for the particles within the stored beam, which causes it to have a finite lateral size and angular spread. According to custom, we can define the emittance of the stored beam as the largest emittance within the beam. Roughly speaking, the emittance of the beam, which is defined independently in the horizontal and vertical direction, is given by the product of a lateral beam size and an angular spread. It is a measure of the transverse coherence of the electron beam and is a crucial parameter for determining the brightness of the synchrotron radiation. It is a unique property of the electron storage ring that the emittance does not depend on the injection condition, but is determined by the magnetic configuration (usually called the "lattice") and the electron energy. Owing to the emission of synchrotron radiation (or more precisely, owing to the compensation by rf cavities of the lost energy), the amplitude of the transverse oscillation is damped in an exponential manner with time, with a damping time which is given by the magnetic configuration of the storage ring and the electron energy. This effect is called the "radiation damping". However, due to the quantum nature of radiation and the randomness in the betatron oscillation phase at which a photon is emitted, the oscillation behaves like a Brownian motion, and its amplitude tends to grow. This effect is called the "quantum fluctuation". As a consequence of the existence of both effects, an equilibrium state is eventually reached. The distribution of the oscillation amplitude in the stored beam becomes
43 Gaussian, reflecting the probabilistic nature of the phenomenon. Hence we can define the r.m.s, beam size for any electron storage ring and, correspondingly, the r.m.s, emittance. The r.m.s, horizontal emittance, which is called the "natural emittance", has a particular meaning in the storage ring design, because it is determined just by the lattice structure and the electron energy. On the other hand, the vertical emittance is determined in practice from the horizontal emittance, through coupling between the horizontal and vertical oscillations, and it is usually a few percent of the horizontal emittance.
Bunch number and bunch length A storage ring applies the method of radiofrequency acceleration to make up for the radiation loss. If an electron is at a phase ~s, the energy gain balances the energy loss, and the electron will remain at this phase (Fig. 1-25). The phase ~s is called the equilibrium phase. An electron whose energy and phase deviate from the specified value is still stable if the deviations are small. In the stable condition the phase and energy oscillate around the equilibrium values. This represents "phase stability", and the oscillation is called the "synchrotron oscillation". According to the theory of phase stability, the stable area of the synchrotron oscillation is given by a separatrix which separates the stable and unstable areas of oscillation in the phase space made of phase and energy (Fig. 1-26). The area in which the phase stability holds is called the rf bucket. In consequence, electrons in a storage ring are bunched around a specified phase of sinusoidal rf voltage. The frequency of the applied voltage must be a multiple of the revolution frequency with which electrons circulate around the storage ring, in order for the electron and the rf field to synchronize. This ratio is called the harmonic number, and gives the maximum number of electron bunches which can be stored in the storage ring.
0 >
/0
~
) = Vsin
2~ft
~s ~_~2r~
radlatlon]oss energy galn
v
Urad Vsin ~s
Fig. 1-25. Sinusoidal rf waveform. The equilibrium phase is determined by the balance of radiation loss and energy gain.
44
(a)
0
2~
(b)
t
v
Fig. 1-26. (a) Rf bucket and (b) electron bunch. For the same reason as in the lateral distribution, the longitudinal distribution (in the moving direction) of electrons in a bunch also becomes Gaussian. In this case the r.m.s, length of the bunch is called the bunch length (usually expressed in units of sec). The rf frequency has been chosen between 100 and 500 MHz for technical reasons; hence the number of bunches will be several hundreds, or even more than a thousand in large storage rings. A typical value of the r.m.s, bunch length is 100 ps at the rf frequency of 500 MHz. If every rf bucket is filled with electrons, the radiation is observed at intervals of the rf period, and the radiation is effectively continuous in time from the experimenter's viewpoint. This is called the multi-bunch mode of operation. However, if a single rf bucket is filled, a train of very short pulses of radiation is observed at intervals of the revolution period. This is called the single-bunch mode of operation. Since the revolution time is of the order of gs in a typical high-energy storage ring, the single bunch operation is useful in studies of relaxation phenomena. Beam
current
The circulating beam contains a large number of electrons. The number of electrons (or intensity) is sometimes expressed by the circulating current (in units of mA). The radiation intensity is proportional to the stored current, so that higher currents are always desirable. However, the stored current is limited by instabilities of various kinds. Among them, the interaction of the beam with the metallic surroundings, and the ion-trapping effect, are dominant effects which limit the stored current. When the beam passes through conducting enclosures
45 such as rf cavities and vacuum chambers, the induced electromagnetic fields influence the beam and result in its instability. The ion-trapping effect is a phenomenon in which positive ions, produced through scattering of electrons by the residual gases, are trapped in the electron beam by electric attraction between electrons and positive ions. This can bring about complicated instabilities when the beam current increases. The ion-trapping effect can usually be observed as a growth in the lateral beam size. This leads to a reduction in brightness and brilliance of the synchrotron radiation. If the stored particles are positrons instead of electrons, however, no instabilities due to ion-trapping have been observed. This is the main advantage of selecting positrons as the stored particles. The stored current in most radiation sources ranges from 100 mA to 500 mA in the normal operation of multi-bunch mode. In the single-bunch mode of operation, however, the stored current is limited to several tens of mA because of the existence of single-bunch instabilities.
Beam lifetime The number of electrons stored in the ring decays exponentially with time. The time in which it decays to half is called the beam lifetime. The main cause of decay is the loss of electrons through scattering by residual gases in the vacuum chamber. The dominant processes are a loss of energy through the emission of Bremsstrahlung radiation and the Mr scattering with atomic electrons. If the energy lost in collision is greater than the energy acceptance of the stable area (rf bucket) of the synchrotron oscillation, the electron leaves the stability region of the synchrotron oscillation, and is eventually lost. Therefore, an electron storage ring requires a vacuum of the order of 10-10 Torr or lower in order to keep the beam alive over several hours. It is not necessarily an easy task to maintain this pressure while storing the beam, since the synchrotron radiation illuminates the wall of the vacuum chamber and this is the main cause of desorption of molecules and atoms. Therefore, the beam lifetime depends on the stored current. When the aperture of the vacuum chamber is extremely small, the Rutherford scattering, the scattering by atomic nuclei, causes a loss of circulating electrons. If the deflection angle of an electron exceeds a critical value, which is determined by the betatron function and the gap of the vacuum chamber, the scattered electron hits the wall of the vacuum chamber at the narrowest gap as it circulates in the storage ring. Since the probability of the Rutherford scattering increases at small angles, the rate of loss due to this scattering becomes significant when part of the ring vacuum chamber is extremely narrow. This is true when an insertion device such as an undulator with a narrow gap is placed in the storage ring. In low-energy storage rings the lifetime of the electron beam is affected by the Touschek effect. When the stored beam current is high, electrons in the same bunch collide with each other, and gain or lose the energy in the direction of movement. Some of them exceed the energy of acceptance of the rf bucket and are lost. A cure for this effect is to increase the applied rf voltage.
46 1.5.2. Low-Emittance Lattice
The importance of the low-emittance electron beam can be understood easily by thinking of the brightness or brilliance of the radiation. For high resolution spectroscopy radiation with a high brilliance is essential. Brilliance is a quantity conventionally defined as the number of photons emitted per unit of four-dimensional phase space, per unit bandwidth, per unit time. The size of the point source is given by the size of the electron beam, while the angular divergence of radiation is given by convolution of the angular divergence of the electron beam and the intrinsic angular divergence of the synchrotron radiation. As described in the previous sections, the intrinsic angular divergence of synchrotron radiation is very small; i.e. of the order of 1/y. Therefore, a low-emittance electron beam is essential in order to obtain the highbrilliance synchrotron radiation, particularly using undulators. As stated before, the horizontal emittance of the electron beam is determined by the betatron and dispersion functions and the electron energy. Hence the horizontal emittance depends on the type of lattice. Roughly speaking, in order to reduce the horizontal emittance the dispersion function should be as small as possible at the position of the bending magnets. The reason is easily understood by thinking of the nature of quantum fluctuations (Fig. 1-27). If the dispersion function is non-zero at the position of the quantum emission by an electron, the equilibrium orbit for the electron will change suddenly, due to the change of energy, and so the amplitude of oscillation changes, because the instantaneous position of the electron is not changed. Therefore, the growth of the amplitude of oscillation due to quantum emission depends on the dispersion function at the position of the quantum emission. The growth rate is small when the dispersion function is small. Since the horizontal emittance is determined by the balance of radiation damping and quantum fluctuation, small dispersion functions are essential to make the horizontal emittance small.
equilibrium orbit \
orbit
x~
jump
AY= ~l AE/E
/
\\
/
electron t r a j e c t o r y
X . /
| L-emission of a quantum i
Fig. 1-27. Illustration of the amplitude growth of the betatron oscillation owing to the emission of synchrotron radiation.
47 The most basic type of lattice which provides low emittances is the achromatic arc configuration (Fig. 1-28). The structure of this lattice consists of straight sections, for the installation of insertion devices, separated by an achromatic arc. The simplest achromatic arc is the double focusing achromat, commonly known as the Chasman-Green lattice, which uses a combination of two bending magnets with a quadrupole magnet placed at the middle. In this case, the dispersion function can be zero outside the bending magnets on both sides, as shown in Fig. 1-28, if an appropriate focusing strength is chosen for the central quadrupole magnet. The quadrupole doublets on both sides tailor the betatron functions.
achromatic arc
AV VA
QF QD
, J
I--'i , B
center of straight section
,
dispersion
0 ,i.._,V AIAV QF
I
B
QD QF
center of symmetry
Fig. 1-28. Chasman-Green configuration.
(a) Expanded Chasman-Green dispersion .f/
VAV ,_.-.,/~fA AV'i.--_, VAV AVA'~ 'AV VA'~ 'AVA
(b) Triple-bend achromat(TBA)
/,-- .... A\~, IlsperA 8iOn
AV I_.-"i 'A I I IV VLV 'V' 'V' 'AV B
B
B
Fig. 1-29. Typical low-emittance configurations.
48 There are many variations of the Chasman-Green lattice (Fig. 1-29). One can expand it by replacing the central quadrupole with several quadruploles. One can also make an achromatic arc with triple bending magnets and two focusing quadrupoles, called the triple bend achromat structure. Quadrupole doublets on both sides of the achromatic arc can be replaced by quadrupole triplets. In all cases, flexibility in the design of the optics and in the operation of storage tings can be improved with increasing numbers of quadrupole magnets, at the expense of cost and space. It should be noted that the horizontal emittance is strongly dependent on the angle of a bending magnet, or equivalently the number of bending magnets in the storage ring. As may be imagined from the nature of the dispersion function, the smaller the bending angle, the smaller is the dispersion function. According to calculations, the horizontal emittance is written in a general way as e = F.O.M. x E 2 /N 3
(1-67)
where E is the electron energy and N is the number of bending magnets. F . O . M . represents the figure-of-merit of a lattice with respect to emittance, which depends on the type of the lattice.
1.6. INSERTION DEVICES Insertion devices are new, powerful radiation sources. They are placed in straight sections of a storage ring and generate radiation with spectral properties different from those obtained using normal bending magnets. An insertion device comprises periodic magnets, whose polarity alternates periodically in the direction of motion of the electron beam. Therefore, the trajectory of electrons inside the insertion device wiggles like a section of sinusoidal curve. The effect of the magnetic field en the electron orbit is localized in the position of the insertion device, and the orbit is unperturbed in the rest of the ring. Insertion devices wb.ich have been used in operating synchrotron radiation sources are wavelength shifters, multi-pole wigglers and undulators. The wavelength shifter is sometimes just called the wiggler. Free-electron lasers (FEL) have seen unique progress in the past two decades. It has been conceived that they might bring us new technologies of generating very coherent, powerful radiation through the interaction of radiation with electrons passing through an undulator. The spectral range which FELs cover will be from the microwave to the vacuum ultraviolet region, or hopefully to the X-ray region. For the long-wavelength FELs electron beams from linear accelerators are used, while for the short-wavelength FELs electron storage rings are used. In any case, many proof-of-principle experiments have been done so far, and extensive developments are continuing. In the following section the basic properties of the insertion devices will be described.
49
1.6.1. Wavelength Shifter and Multi-Pole Wiggler The simplest insertion device is the "wavelength shifter". Consider three dipole magnets of alternating polarities shown in Fig. 1-30. If the field strength of each magnet is properly chosen, the electron trajectory makes a bump and the synchrotron radiation comes out of the magnets. The continuous nature of the spectrum of the synchrotron radiation from wavelength shifters is similar to that of the bending magnet radiation. Since the critical wavelength varies inversely with the magnetic field strength, however, wavelength shifters can generate shorter wavelengths than those achievable in normal bending magnets. To show an example, we consider the Photon Factory, at which the electron energy is 2.5 GeV. The field strength is 5 tesla for a superconducting wiggler magnet, while it is 0.96 tesla for the normal bending magnet. The corresponding critical wavelengths are 0.6A and 3.1A, respectively. It is clear that wavelength shifters with high magnetic fields provide practical means to obtain X-rays or even hard X-rays at moderate electron energies. A simple extension of the wavelength shifter concept is the "multi-pole wiggler". This is an array of many dipole magnets (the number of the magnets greater than three but not so many), and the radiation emitted in the forward direction is intensified by the number of magnetic periods. These magnets can be electro-magnets, either superconducting or normal ones, but are more often permanent magnets.
+
Fig. 1-30. Schematic of wavelength shifter.
1.6.2. Undulator When the number of periods increases, the synchrotron radiation exhibits particular features which are quite different from those of ordinary synchrotron radiation. As electrons pass through a periodic magnetic field, a quasi-monochromatic radiation, with a higher brightness than can be obtained from bending magnets, is generated by the interference effect. The devices using this concept are called "undulators". Let us consider a plane undulator, as shown in Fig. 1-31, with a magnetic field of the form By(s) = Bo cos(2zrS/&u)
(1-68)
50 where Zu is the period length of the undulator. If the field strength is weak, the electron follows a sinusoidal trajectory as
x(s) = (Zu/2Zr)(K/ y) cos(2nrS/Zu)
(1-69)
where K is the deflection parameter, K = eBo ,~,u/2a'mc2, as given by Eqn. (1-62). One sees from Eqn. (1-69) that the maximum deflection angle is a = K/y. In weak undulators the relationship K<
(1-70)
e
__4
[ Fig. 1-31. The principle of the plane undulator. If the electron is observed in the frame which moves, on average, with the electron passing through the undulator, the motion of the electron becomes a simple harmonic oscillation. Then dipole-radiation with a wavelength equal to the Lorentz-contracted undulator period, Z*=Zu/?'* is emitted. In the laboratory frame, the radiation is Doppler-shifted to a shorter wavelength which is given by ~1 =(,~,u/2T2)[ 1 + K2/2 + (719)2]
(1-71)
where 0 is the observation angle. By Lorentz transformation the angular distribution in the laboratory frame is confined to the narrow forward cone 0 < 1/),, as shown in Fig. 1-32a. The wavelength of the undu!ator radiation varies with the observation angle, but if one observes the
51 radiation at a specified angle one sees monochromatic radiation of wavelength A1. As Eqn. (171) shows, for the weak-field undulator the wavelength of the radiation is determined primarily by the magnetic periodicity, Au, and the electron energy, 7.
~
)
x
-
~
s-oscillation
in moving frame
--IIu
21/,t
In lab frame
Fig. 1-32. Distribution of undulator radiation with the acceleration vector lies, (a) in the x-axis, and (b) in the s-axis.
Fig. 1-33. Figure-eight shaped trajectory of electron in the moving frame. For K > 1 the motion of the electron in the moving frame is not a simple harmonic oscillation, but shows a figure-eight shape (Fig. 1-33). In this case, the oscillation is decomposed to a good approximation into the x-oscillation (Fig. 1-32a) and the s-oscillation (Fig. 1-32b). Since the oscillation is not simple harmonic, the radiation includes higher harmonic components in the spectrum, with peaks at ,~,k= ,~,1[ k (k = integer)
(1-72)
52 As Fig. 1-33 suggests, the spectrum of the radiation from the x-oscillation includes only odd harmonics, but that from the s-oscillation contains both even and odd harmonics. In the plane undulator the radiation is linearly polarized along the undulator axis, and elliptically polarized at non-zero angles. So far, we have assumed that the number of periods is quite large (N -.- oo). With a finite number of periods, however, the spectral line width of each line grows. Calculations show that the line width of the radiation from an N-period undulator is given by A~, / Ak =1 / Nk.
(1-73)
This is called the natural line width. It can also be shown that the peak power at the central wavelength is proportional to N 2. Therefore, the central brightness of the undulator is extremely high, about 104 times that from the bending magnet. However, the total power is proportional to N, since the line width is proportional to N-1. Considering that the maximum deflection angle, a, is small in undulators, the total power they radiate is relatively small.
Fig. 1-34. Plane undulator using permanent magnets: (a) basic design with permanent magnets only, and (b) hybrid design with soft iron as magnetic poles. Arrow indicates the direction of magnetization of permanent magnet. If one observes the radiation over a cone of finite angle, 0o, to the forward direction, the line width is broadened, as expected from Eqn. (1-71). A~ / ~ k .~ ~2 0o2 / (1 + K 2)
(1-74)
53 Comparing Eqns. (1-73) and (1-74), one can see that the line width becomes appreciable compared with the natural line width, unless the viewing angle is small enough; (1-75)
Oo <- 1 / ~'(Nk) 1/2
Equation (1-75) also shows the requirement on the angular spread of the electron beam in order to obtain a monochromatic radiation from an undulator. As described in Section 1.5.1., the electron beam has finite lateral size and angular spread which are determined by the betatron functions at the position of the undulator and by the emittances. Unless the angular spread of the electron beam is smaller than given in Eqn. (1-75) the spectral bandwidth becomes larger than the natural line width. Therefore, at the position of the undulators, the electron beam should be parallel to the undulator axis as far as possible. A focusing or diverging beam at undulators is not appropriate.
(a)
(b)
(c) "'d
"-&l
! I I I I I I ! I I ] I
illslIL..
! I I I I I I I I I 1
rV %
e
Fig. 1-35. (a) Helical undulator using double helix coils, (b) helical undulator using permanent magnets and (c) crossed field undulators.
54 A plane undulator is made of permanent magnets such as SmCo5 and Nd-Fe-B. An advantage is that small magnets can produce strong fields, and a short magnet period is possible compared with electro-magnets. The simplest configuration for undulator magnets is the socalled Halbach configuration, as is shown in Fig. 1-34a. The so-called hybrid configuration shown in Fig. 1-34b uses pole pieces made of soft iron as well as permanent magnets. Figure 1-35a shows a helical undulator which uses a double helix configuration (a pair of helical coils carrying electric currents in opposite directions). The direction of the magnetic field vector rotates along the undulator axis, and the electron follows a spiral orbit. The circularly polarized radiation is obtained on the axis of the undulator. The helical field can also be obtained with permanent magnets.- Figure 1-35b shows the magnet configuration of a helical undulator, in which permanent magnet pairs are placed alternately in horizontal and vertical directions. An example in Fig. 1-35c is a more elegant configuration which generates radiation with any polarization. Two plane undulators, one in the horizontal direction and the other in the vertical direction, are placed separately with a distance so as to keep the proper phase relation between two undulator radiations. In conclusion, undulators are powerful radiation sources, with high brightness, tunability of wavelength, and with the availability of various polarization states. The spectrum can be tuned by varying the magnetic field strength while keeping the magnetic structure and the electron energy constant. With permanent magnets the magnetic field is changed by varying the magnet gap (stronger fields can be obtained with narrower gaps).
1.6.3. Free Electron Laser
The possibility of operating a free electron laser was first demonstrated by the experiments made at Stanford in 1975 and 1977 [23, 24]. Madey and co-workers observed the stimulated emission of infrared radiation (~ = 10.6 l.tm) by relativistic electrons in a spatially periodic transverse magnetic field. Figure 1-36 shows the experimental setup. The periodic magnetic field, of strength 2.4 kG, was generated by a superconducting double helix with a 3.2 cm period and a length of 5.2 m. The electron beam and the infrared radiation were steered to pass through the magnet on the axis. A gain of 7% per pass was obtained at a current of 70 mA of 24 MeV electrons. The first operation of a free electron laser (FEL) was demonstrated by the same group, using the experimental apparatus shown in Fig. 1-37. In the experiment the injector was pulsed to emit a single bunch every 84.6 ns, equal to the round-trip transit time of the radiation between mirrors. Using the 43 MeV electron beam with a peak current of 2.6 A gave a peak power of 7 kW, which is greater by a factor of 108 than the spontaneous radiation, at a wavelength of 3.4 l.tm. The spontaneous radiation represents the usual radiation from the undulator. Since then, many FEL facilities have been built aiming at the wavelength range from microwave to infrared by using low-energy, high-current accelerators such as rf linear accelerators, and induction linear accelerators.
55
HELICAL MAGNET ( 3 . 2 crn PERIOD] -4-
/'-M,RaoR ~
. . . . .
.,, " . - .
"7. ~
~
24 MeV BUNCHED ELECTRON BEAM
..,-'"II~-- - ' -
M,RROR.--
_
.,.,,.,.,.....,
MOOULATEO 10.6~ RADIATION TO Cu:Ge DETECTOR ANO MONOCHROMATOR
MOLECTRON T-250 CO z LASER
Fig. 1-36. FEL experimental setup. The electron beam is magnetically deflected around the optical components on the axis of the helical magnet.
........
HELICAL MAGNET ( 3 . 2 cm PERIOD)
43 MeV BUNCHED .. . . . . . ELECTRON
=
Fig. 1-37. Schematic diagram of the free electron laser oscillator. The principle of operation of an FEL is that the electron loses (or gains) energy by interacting with the electromagnetic radiation in the periodic magnetic field, and that the radiation gains (or loses) the same amount of energy as that lost (or gained) by the electron. This process is a stimulated emission, which means a net amplification of the radiation. When the optical cavity is placed on each side beyond the undttlator one gets an FEL oscillator, if the gain in the radiation is greater than losses of radiation power in the optical cavity.
I
let
Pt~
~
let
s
Fig. 1-38. Principle of the interaction of electrons with the radiation field in the undulator.
56 In the following, the classical model developed by Colson [25] will be applied to describe the basic principle of operation of an FEL. Let us consider a plane undulator as shown in Fig. 131. The static undulator field wiggles the electron orbit, and gives the electron a transverse velocity fit. As a result, the energy of the electron changes by interaction with the transverse electric field E t of the radiation, by an amount proportional to fltEt . Whether the radiation gains or loses energy depends on the relation between the electron energy, 7, the wavelength of radiation, X, the undulator period, Xu, and the phase of the radiation, ~, at the time when the electron passes through the undulator. Consider the transit times for the electron and the electromagnetic radiation to traverse one period of the undulator. One obtains Xu / fl*c for the electron and Xu / c
for the radiation
where t * is the velocity with which the electron moves, on average, in the forward direction (see Eqn. (1-70)). The electron keeps a constant phase relationship relative to the electromagnetic radiation over many undulator wavelengths, if the difference of these transit times equals one period of the radiation (see Fig. 1-38), that is, when the following relationship holds: I c = Zu ( 1 / f l * c - 1/c )
(1-76)
Using Eqn. (1-70) for the average velocity one obtains the "resonance" condition: 7r 2 = (Au12A)( 1 + K212)
(1-77)
One should note that Eqn. (1-77) gives the same relationship as Eqn. (1-71), the frequency of the spontaneous radiation emitted on axis by the electron of an energy 7r from a plane undulator of a period Xu and a magnet parameter K.
Fig. 1-39. Phase space ($, 7) describing the motion of electrons interacting with the radiation.
57 As calculations show, the energy and phase motion of the electron in the field of the electromagnetic radiation is described by equations similar to those describing the synchrotron oscillation of electrons in a storage ring, but without radiation loss. Hence, the electron trajectories are represented by the phase space (~,),), which is shown in Fig. 1-39. If the electron is at a fixed point, at which the energy equals the resonance value ~'r and the phase relative to the radiation lies on 7r, the electron stays at equilibrium all the time, and there is no net effect on the electromagnetic radiation. However, if the electron lies at any point in the phase space other than the fixed point, it will move in the phase space along a corresponding trajectory, and the electron will change in energy. This produces a gain or loss in intensity of the electromagnetic radiation. Consider a case of a more realistic energy-phase distribution function for the electron beam. In most FELs one can assume that the electron beam is continuous in the phase direction, and that the energy distribution is narrow. If the electron energy is at resonance, some of the electrons gain energy, some of them lose it, and the amounts of energy gain and loss are balanced, so there is no net effect on the electromagnetic radiation. Calculations show, however, that if the electron energy is above (or below) the resonance value, the electron beam as a whole loses (or gains) energy, and the electromagnetic radiation is intensified (or faded). The energy gain, G, in the electromagnetic field intensity during one traversal of the undulator is calculated [26] for a helical undulator as: G = -32 x 21/2 7r2 ro ~ u 1/2 ,~,3/2peN 3 {K2[(1 +K2) 3/2 }f ( x )
(1-78)
Here ro is the classical electron radius (2.8 x 10-13 cm), Pe the electron density, and it is assumed that the transverse cross section of the electron beam is equal to that of the electromagnetic radiation. The functionf(x) is called the gain function, which is given by f ( x ) = ( l [ x 3 ) { c o s x - 1 + (x/2)sinx}
(1-79)
x = 47rN(?'o
(1-80)
and -
~)1~
and is plotted in Fig. 1-40. When the electrons interact with the radiation field, both the energy distribution and the phase distribution change. When observed at the undulator exit, the phase distribution of the electrons is modulated at the radiation wavelength, and the energy distribution is broadened. If the energy spread is larger than 1/2N (the line width of the spontaneous radiation of an ideal FEL) during passage in the undulator, the energy conversion rate is strongly reduced, and the gain reaches saturation.
58
~
f(x)
0.08 O.06 0.04
0.02 -I0
t
-5
I
I0 -0.02 -0.04
-0.06 -0.08
Fig. 1-40. The gain functionf(x). These energy-phase behaviors have brought us an idea of the "optical klystron". The idea of prebunching the beam was proposed by Vinokurov and Skrinsky in 1977 as a way to increase the gain over the uniform undulator [27]. A merit of prebunching can be easily understood by remembering that, if N electrons are bunched within an extremely small time scale (equal to, or smaller than, one period of the electromagnetic radiation), the radiation intensity is proportional to N 2 instead of the ordinary factor N. The basic scheme of the optical klystron is shown in Fig. 1-41. It comprises two undulator magnets separated by a specially designed dispersion section (a simple drift-space is adequate in principle, but it takes a much longer space to obtain an adequate bunching).
k/V /k kjV Fig. 1-41. Principle of the optical klystron. The electron is energy-modulated in the first undulator and bunched during passage through the bunching section between two undulators.
59 In the first magnet system, the electrons are energy-modulated as well as bunched. During passage through the dispersion section, the electrons are bunched at intervals of the radiation wavelength. The second undulator is for transferring the energy to the radiation field. The gain enhancement by using an optical klystron has been demonstrated experimentally to be a factor of 5to 10. For shorter-wavelength FELs electron storage rings have been conceived superior to linear accelerators, since storage rings can provide a beam with a high peak current (higher than 100 A) and a good emittance, and hence a high electron density, and good energy resolution (less than 0.1%). In the storage-ring FELs, however, the energy-spread of the beam grows, due to the interaction with the electromagnetic radiation, because the same bunch of electrons traverses the FEL many times. The energy-spread of the beam reaches an equilibrium, by balance with the radiation damping caused by the synchrotron radiation emission all around the ring. When the energy spread becomes comparable to the natural line width, 1/2N, the effective gain decreases and the laser power saturates. The first operation of a storage-ring free-electron laser was demonstrated at A =6500/~ on the ACO storage ring at Orsay (France) in 1983 [28]. Since then, several experiments have been made, mainly in the visible region, and one experiment was made in the ultraviolet region at Novosibirsk in 1988 [29]. As seen in Eqn. (1-78), the gain varies as the 3/2-power of the wavelength of the electromagnetic radiation, so a beam of extremely good emittance and high current is needed for obtaining a sufficient gain at shorter wavelengths. New storage ring projects dedicated to the FEL are now in progress. Storage rings with a small emittance, of the order of several nm-rad, have been conceived. 1.7. BEAM LINE
The synchrotron radiations generated in the bending magnets or insertion devices are taken from the ports of the ring and delivered to the experimental station through a beam line. The beam line is a light-path connecting the storage ring and the experimental station, and performs the function to extract or cut off the photon beam. Various kinds of optical instruments, such as reflection mirrors and monochromators, are set up in the beam line according to the characteristics of the synchrotron radiation to be used and the purpose of the experiment. Since one beam line can accept sufficient radiation (about 30 mrad) to serve several simultaneous experiments, the "main beam line" usually has several branch beam lines downstream. 1.7.1. Main Beam Line
Figure 1-42 gives an example of the layout of the front end of the main beam line in the Photon Factory (KEK) [21]. In order to shield high energy ),-rays and neutrons from the storage ring, a concrete shield wall 40 - 90 cm thick separates the ring from the experimental
Z;
o
5b" "&.
UMATIC
9
.
ABSORBER
Fig. 1-42. Layout of the front end of main beam line.
61 area. The front end of main beam line functions to protect the ultra-high vacuum of the storage ring and to deliver synchrotron radiation to the branch beam lines. The vacuum upstream of the main beam line is to be kept as low as that of the storage ring (10--10 Torr). This is attained by use of four triode ion pumps and two titanium sublimation pumps. As shown in Fig. 1-42, the line is typically composed, moving downstream along the light path, of a manual valve, a water-cooled absorber, isolation pneumatic valves, fast-closing valves, mask slits, an acoustic delay line, beam shutters and a large pneumatic valve. The manual valve is used to cut off the vacuum between the ring and beam line in the event of maintenance of the line. The light absorber is made of oxygen-free copper plate about 20 mm thick, cooled with flowing water, and serves to protect the beam shutter and valves from overheating by the synchrotron radiation. The mask slit is used to define the size of the photon beam delivered. The fast-closing valve, which is driven by compressed air or springs, is to isolate the vacuum from the storage ring, and acts within about 15 msec so that any accidental venting in the branch beam line, caused by equipment failure or experimenter error will not have an adverse effect on the storage ring. The acoustic delay line (ADL) is also used to provide a 200 - 300 msec delay in the arrival time of a sonic wave that could be initiated by an abrupt vacuum failure. The ADL in the Photon Factory is a stainless steel tube 31 cm in diameter and 200 cm in length, loaded with an array of nine conical diaphragms each having a rectangular aperture. To ensure that the valves close in the event of an accident, and that they can only be opened under proper vacuum conditions, a control system is essential. This system permits the storage ring operators or the experimenters to open gate valves only when the pressure measured by a vacuum gauge is below a preset level. The beam shutter, made of a copper and stainless steel block is to ensure that hard X-rays or T-rays do not escape downstream.
1.7.2. Branch Beam Line The main beam lines are divided into several branch beam lines. The branch beam lines are classified into two categories; VUV beam lines for VUV radiations and soft X-rays, and X-ray beam lines for X-rays and hard X-rays. The VUV beam lines are directly connected to the storage ring through the main beam line, which means that the beam lines must be kept at ultrahigh vacuum with an oil-free specification and that optical instruments in the lines, such as reflection mirrors and diffraction gratings are maintained totally clean. In designing the optical system special care is necessary to minimize deterioration of the optical devices by photon beam exposure and to remove beam-induced heat by means of flowing water or liquid metal coolants. The vacuum system of X-ray beam lines is usually separated from the main beam lines by a beryllium window, through which X-rays can pass to the experimental station. Even in the Xray beam lines the vacuum should be maintained below 10-7 torr so that reflection mirrors and other optical devices are always clean. Some X-ray beam lines are directly connected to the storage ring without a beryllium window, just like VUV beam lines.
TMP
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.
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.
.
.
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.
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TO CONSTANT DEVIATION 9.
LARGE /,MONITORMIRRORAI~U~F..R vAF~L,UI~TIC / ,,..B~..ANCHBEAM LINE LIGHT ABSORBER .... / /\ M!RRORADJUSTERS FOR G R A S ~ R \ / / \ ~ A___NOCONSTANT DEVIATION
I
SUPPORT STRUCTURE
BRANCH BEAM LINE LIGHT ABSORBER /~ TO SEYA-NAMIOI'r ~ ~ MIRROR M AD,,IUSTERSFOR SE_YA,NAMIOKA/ PNEUMATIC VALVE
\ X,Y,Z
ADJUSTER
Fig. 1-43. Layout of the beam-splitting section of VUV beam line (BL11 of PF).
63 Figure 1-43 shows the beam-splitting section of the VUV beam line (BL11 of the Photon Factory), located just outside the shielding wall. The front end is split into four beam lines. These are equipped with a Seya-Namioka monochromator, a grasshopper monochromator, a toroidal grating monochromator (TGM) and a double-crystal monochromator. Chemicalvapor-deposited silicon carbide (CVD-SiC) plane mirrors are installed in the mirror chambers. In the downstream part of the beam-splitting section are situated the beam transport tube, vacuum valves, optical system (monochromator, mirror, and so on), slits, pumps, vacuum gauges and the beam position-monitoring system. At the back end of the X-ray beam lines is set a beryllium window for extracting a beam to the atmosphere. The window and the experimental apparatus are accommodated in the shielded room which is protected from hard Xrays and ),-rays. At the back end of the VUV beam line are located a specimen chamber and other experimental apparatus, directly connected to the storage ring. BL-10 (X-RAY)
DOUBLE CRYSTAL MONOCHROMATOR (Si 111!
SL,r, I
/
EXPERIMENTAL HUTCH
TOROIDAL MIRROR
c
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,,,P"~' I ' J EXAFS ~ ' / / z ~ A SLIT \ / ~.A - - DOUBLE CRYSTAL FLAT SINGLE " ~ MONOCHROMATOR MONOCHROMATOR ~ (Si 311) (graphite / Si 111) " FOUR CIRCLE DIFFRACTOMETOR
5
BL-I I (VUV)
~
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B -,o,,~_.,,, MASK",..,,IVMI~RROR
10m
GRASSHOPPER D. 2m CONSTANT DEVIATION
r
-"
~"
~/A///////~,//~y,7`~/////////////////////////////////////////////////////////////////////////////`//. ~)
5
l~)m
Fig. 1-44. Layout of typical X-ray beam line and VUV beam line. The beam lines: BL10 (X-ray) and BL11 (VUV) of PF are shown. Figure 1-44 gives examples of the layout of an X-ray beam line (BL10) and VUV beam line (BL11), showing typical arrangements of the optical system and experimental station. The characteristics for these branch beam lines are described in Table 1.8 [21]. For BL10A a pyrolytic graphite is used as a dispersive crystal monochromator for obtaining a large integrated
64 intensity. For BL10C, a tuneable double-crystal monochromator is housed upstream, and a mirror downstream (See Chapter 6, Section 6.2 for the detail of BL10C). In order to have a fixed exit beam position in this monochromator either the first or the second crystal is translated along the incoming beam direction while rotating both crystals. The two crystals are driven independently by a microcomputer controlled unit. Table 1.8a Optics of the branch beam line BL10 in PF. Branch beam line
Horizontal Monochromator acceptance [mrad]
Mirror
Energy resolution (AE / E)
Energy range [keV]
BL10A
1
graphite germanium
none
5 x 10-3 5 x 10-4
6 - 24
BL10B
2
channel cut crystal, Si 311
none
1 x 10 -4
BL10C
4
double crystal Si 1:1 focusing 2 - 3 x 10-4 fixed beam height toroidal Pt-coated
4 - 10
4 - 10
Table 1.8b Optics of the branch beam line BL11 in PF. Branch beam line
Horizontal Pre-mirror acceptance [mrad]
Monochromator
Energy resolution (~,/A~,)
Energy range
[AI
BLllA
1.3
M0:88 ~ plane MI: 88~ spherical
Grasshopper Mark VII 2400 lhnm
(A~, =0.02/~)
10 - 145
BLllB
4.0
89 ~ bent cylinder
double crystal fixed beam height
> 2000
2 . 8 - 15
BLllC
4.8
77.5 ~ plane 42.5 ~ concave
1 m Seya-Namioka
20003000
4003500
BLllD
2.6
86 ~ cylinder 86 ~ plane
2 m constant deviation monochromator
10002000
60 - 600
65 BL-2 (SOFT X-RAY UNDULATOR)
LINEARTRANSLATOR
10-m GRAZING INCIDENCE
CONDENSERFRESNEL).--1"'"~ p,N~G BEAM BEAMLINE FAST CLOSING / ~ ~ SHUTTER DIAPHRAGM VALVE " r~ UR y i
P
A 9
i,
~
/===~
9
DEFLECTION MIRROR
9
=
a
~
9 '.
~ ~
- ~
9
9
i I 'l 't ='--
MONOCHROM.
~ ~ : : ] ~
. . . . . / . . . . .~-.... - ~ \ ~ ~~U~/~\GRATING
.~~ EXIT SLIT ~X!T~,S..I'-I
Iv nTrnllTlv v i' iJ,iJiJiiJAi
SECON6CRYSTAL I
HUTCH | ROTATABLE DOUBLECRYSTAL 7-RAY "PHOTODIODE MONOCHROMATOR STOPPER
SLE
STRAIGHT BRANCH
Fig. 1-45. Schematic diagram showing the beam lines for soft X-rays from undulator (BL2 of PF). Figure 1-45 gives an example of the beam lines for soft X-rays from an undulator (BL2), schematically showing an arrangement of beam-line components [30]. Two branch beam lines are installed, one utilizing the radiation deflected by a mirror, and the other being set on the line along the undulator. The two beam lines are used in a time-sharing mode, because the beam from the undulator is of extremely high directivity and small size, and consequently one cannot use both lines simultaneously. 1.7.3.
Beryllium
Window
A beryllium window is used to separate X-ray experimental areas from the ring vacuum. The windows must be thin, to ensure high X-ray transmittance, but thick enough to ensure mechanical strength and thermal conductivity. Figure 1-46 shows the window assembly for an X-ray beam employed in the Photon Factory. The beryllium foils 0.2 mm thick are set up at both ends of the assembly. In front of the upstream window, A graphite foil 20 l.tm thick is fixed to a copper frame in front of the upstream beryllium window in order to decrease the thermal load imposed by synchrotron radiation. This upstream window faces an ultra-high vacuum environment. The cross-shaped pipe is evacuated to 10-6 Tort with an ion pump. The beam transport ducts from the downstream window to the experimental apparatus are f'flled with purified helium at 760 Torr, because air causes too much X-ray beam attenuation and scattering. The downstream window, therefore, has to bear the helium pressure, but is only exposed to small thermal loads because of the absorption of low-energy radiation components by the upstream window. A major problem in utilizing thin foils lies in joining the beryllium to other materials so that it can be made part of the vacuum system. Several techniques are being employed, such as brazing or soldering the beryllium to copper or aluminum in an inert atmosphere, diffusion bonding to copper or Monel, glass-frit bonding to Monel, electron beam bonding to aluminum, or mechanical sealing with indium wire as a gasket.
66 0 i
5 i
. . . .
. . . .
loom i
HELIUM GAS
i
?'-~--L_
t
LL
i I
.J-"" ~ 1 [
LI
il S~'NCH~Or~ON
v=~uuM~L*~E
i
_
I
ION PUMP
Fig. 1-46. Schematic diagram of double beryllium window assembly.
1.8. OPTICAL SYSTEM Much effort is being devoted to the development of experimental equipment for synchrotron radiation research. Instrumentations that have been developed to meet the specific requirements of particular researches are described in the following chapters of this book. In this section we give an introduction and a general survey of the instn~mentation. The principal instruments are mirrors, monochromators and detectors. They are usually designed as components of vacuum or ultra-high vacuum systems, because they are accommodated in the vacuum chamber. 1.8.1. Mirrors Mirrors have the functions of deflection, focussing and filtration for synchrotron radiations. With beam splitting mirrors part of a single radiation beam can be deflected so that several experimental stations can share a single beam with space between them for equipment. Curved (cylindrical or toroidal) mirrors are used to obtain an image of the source point at some distant location. Filtration by mirrors is a function which makes a sharp cut-off in reflectivity above a certain photon energy. They can also act as low-pass filters, absorbing unwanted X-rays in VUV beam lines and achieving control over harmonics in VUV and X-ray lines. The reflectivity of a mirror depends on the photon energy, the angle of incidence, the mirror's surface material, and its smoothness. Figure 1-47 shows an example of the reflectivity as a function of the grazing angle of incidence for various photon energies [31 ]. The cut-off energy varies inversely with the grazing angle. Consequently harmonics can be controlled by varying the grazing angle of beam incidence.
67 1.0
------T~ "X " " ~
~
~+
"
"5--,
~w 05"'~
i|
---'---"
'
A
I
,
tt.
, 0.1
0
/. 8 12 GRAZING ANGLE e (DEGREES)
Fig. 1-47. The measured reflectivity of Au and C as a function of the grazing angle of incidence for various photon energies. Various materials are used as mirror substrates, including Be, Cu, SiC, fused SiO 2, zerodur, electroless nickel phosphorus (Kanigen), and float glass. The most frequently used coatings are Au and Pt, although uncoated mirrors are also used. The surface smoothness achievable varies for the various materials. In the case of fused SiO 2 and SiC, a micro-roughness lower than 5/~ r.m.s, has been attained by recent polishing techniques [32]. It becomes harder to achieve the surface smoothness in larger and more complex curved mirrors. It is reported that the autocovariance distribution of the surface micro-roughness becomes more important in X-ray ranges than does the micro-roughness height [33]. For mirrors in rather low energy storage ring beam lines, where thermal loading is not serious, fused SiO 2 is being employed. Even in high energy storage tings fused SiO 2 can sometimes be used for grazing incidence mirrors with an angle higher than about 88 ~, because much of the beam power is reflected and the absorbed power is distributed over a large area. However, surface deterioration is inevitable after long term use. When thermal loading is severe, as in the direct X-ray beam lines of multi-GeV storage tings, SiC or Kanigen-coated copper or aluminum, can be used. The former is the superior material, with a low thermal expansion coefficient and high thermal conductivity. The latter can be polished easily but, owing to its high thermal expansion coefficient, the surface will deform by thermal loading. The mirror must be cooled by flowing water or liquid metal in these applications. Segmented
68 mirrors are sometimes used where length of the mirror is especially required, as with in the case of an extreme grazing angle of incidence. Curved mirrors can be used to image the electron- or photon-beam source point at VUV and X-ray wavelengths. Magnification by the mirror system depends on the curvatures of the mirror surface, the grazing angle of incidence, and the mirror's location in the beam line. Demagnifying optics are used to obtain a high flux-density on samples as small as 100 l.tm in diameter. Magnifying optics are used to illuminate several square centimeter areas, as is required for pattern replication by soft X-ray lithography. 1.8.2.
Monochromators
Since most synchrotron radiation experiments are carded out by choosing a particular wavelength from the continuum and often scan over a range of wavelengths, a tuneable monochromator is indispensable. Many specialized monochromator systems have been developed for use in the VUV, soft X-ray, and hard X-ray regions of the spectrum. At synchrotron radiation facilities the light source, the beam line with the relevant optical components, the monochromator, and the experimental set-up have to be considered as a whole in order to maximize the flux of monochromatized photons onto the sample.
VUV
monochromators
VUV monoehromators fall into two classes, namely normal incidence monochromators (NIM) for the spectral range from about 6 eV to 50 eV photon energies, and grazing incidence monochromators (GIM) for the range about 30 eV to 600 eV. Three types of NIM are commonly used with synchrotron radiation sources, i.e., the standard (McPherson) NIM, the Seya-Namioka monochromator, and the modified Wadsworth monochromator, which are shown schematically in Figs. 1-48a, 1-48b, and 1-48c. They are all conventional mounts of concave spherical gratings with focusing properties. The McPherson NIM is a monochromator operating in a mode close to the Rowland, with one optical c o m p o n e n t , - a grating. In order to vary the wavelength the grating can be rotated and, to maintain focusing, can be translated at the same time. This system has the capability of high resolution with its ultimate values of the grating's constant and the radius of curvature. The Seya-Namioka monochromator has the simplest scanning mechanism, i.e., a single rotation of the grating about an axis through its center. Because of the fairly large angle of incidence this monochromator has abominable astigmatism, if a point light source is used near the entrance slit. However this drawback can be corrected by using, for example, an additional focusing mirror. The Wadsworth monochromator exploits one of the particular properties of synchrotron r a d i a t i o n - small d i v e r g e n c e - because it requires the light source to be at infinity for optimum performance. Wavelength scanning is achieved by simple rotation of the grating.
69
----,,
FG
I
FM i
(a) Mc Pherson Normal Incidence
(b) Seya-Namioka
(c) ModifiedWadsworth Fig. 1-48. Schematic diagrams of various normal incidence monochromators for use with synchrotron radiation. FG, focusing grating: FM, focusing mirror: PM, plane mirror. Various types of grazing incidence monochromators have been used in conjunction with synchrotron radiation. As typical examples of GIM, the plane grating monochromator and Rowland-type monochromator are shown in Figs. 1-49 and 1-50, respectively. Figure 1-49a shows the simplest instrument, with only one optical surface, a plane grating. Wavelength change is achieved by means of rotating a Soller slit system around an axis parallel to the grating grooves. Only moderate resolution is attained with this mounting. The instrument shown in Fig. 1-49b uses a plane grating and a spherical mirror. The grating is rotated by small angles so that the sum of the entrance and exit angles is fixed. The exit beam is spatially fixed, which allows its use in complex experimental facilities. Suppression of higher orders is made in only a limited wavelength range, by suitable choice of incident angle at the grating and the following mirror. Sometimes a plane mirror is used as a pre-mirror, so as always to illuminate the center of the grating in the arrangement shown in Fig. 1-49b. Figure 1-50 schematically shows the two typical grazing incidence monochromators based on the Rowland circle mounting. The entrance slit, grating, and exit slit are all located on the Rowland circle to minimize aberrations. This mounting can provide the best resolution for grazing incident monochromators. Figure 1-50a shows the simplest arrangement of a Rowland mounting. For absorption type experiments with the samples in front of the spectrometer, the
70 Rowland spectrograph is extremely useful, because the photographic plate or position-sensitive detectors allow the whole spectrum to be taken at a single exposure. In the instrument shown in Fig. 1-50b the entrance and exit slits are fixed while the grating slides along the Rowland circle. The directions of incoming and outgoing beams can be changed by a rotating mirror-slit combination. There are also several other variations of the Rowland mountings, with different arrangement of mirrors, gratings and slits.
PG
(a) PG
(b)
FM
Fig. 1-49. Two types of plane grating monochromators.
FM
(a)
.. FM
G
(b) Fig. 1-50. Rowland type monochromators. G is a concave grating.
71
X.ray monochromators Crystal diffraction can be used to monochromatize synchrotron radiation (SR). If d is the distance between atomic planes, Amax = 2d is the maximum usable wavelength. Table 1.9 lists values of d for several useful materials for this purpose. The inorganic crystals are generally quite resistive to the primary synchrotron radiation (SR) beam. High resolution can be achieved easily if the angular divergence of the incoming radiation is small. Table 1.9 The values of lattice constants in X.U. (~ 10-3/~). The geometrical distance between atomic planes, d, is given. Crystal
d [X.U.]
Crystal
silicon (220) germanium (220) quartz (1120) NaC1 (100) calcite (100) silicon (111)
1920 1997 2451 2814 3029 3135
germanium (111) quartz (1011) quartz (1010) gypsum (010) lead stearate cerotic acid
m
d [X.U.] 3262 3336 4246 7585 50300 72500
w
*~Si
r 0.6
9 , P.G.
o
LiF
9, o , m
O
>
'~
o.r
9m
9
9 #m
mm
m
[]
m
II
r'r" 0.2 mm mm
-20
-15
-10
-5
0
5 "
Angle (min.)
10
1'5
20
Fig. 1-51. Rocking curves measured with double crystal (+,-) arrangement, for Si(111) - Si(111), Si(111) - PG(0002) and LiF(200) - LiF(200). PG, a mosaic crystal of pyrolytic graphite. A perfect crystal of Si (111) is the typical crystal, showing high resolution and high reflectivity, as shown in Fig. 1-51 [34]. In some cases the very high resolution of a perfect
72 crystal is not necessary because, in general, it costs a considerable intensity loss. In some cases, crystals with a mosaic structure can serve to pass a wider wavelength band. Crystals of pyrolytic graphite (PG) show a wide mosaic spread of about 10 mrad, and low reflectivity, while the integrated intensity of reflected X-rays is usually large, which makes the mosaic crystals useful (see Fig. 1-51). For long wavelengths the so-called soap crystals can be used, although they have not yet found widespread application with SR. Reflectivities are of the order of 1-5 % for typically 100 layers in the 2 0 - 130/~ region. The evaporated multilayer is also another candidate for long wavelength X-ray diffraction. Double crystal monochromators (Fig. 1-52) are most widely used for continuous scanning of SR spectra with high resolution. The monochromators in the parallel (+,-) mode keep the direction of SR fixed while generating only a parallel displacement of the beam. In the (+,+) arrangement, because the angular acceptance in the dispersive direction is of the order of the width of the single crystal reflection curve, extremely high dispersion can be achieved without losing the X-ray intensity. The SR, with its high brightness, can make this arrangement useful, with very high resolution.
Crystal
Crystal I X Rays
(a)
Crystal
X Rays
(b) Crystal X Ray.~.s_s
(c)
Crystal
Crystal
Fig. 1-52. Double crystal monochromators (a) in the parallel (+,-) mode, (b) in the highly dispersive antiparallel (+,+) mode, and (c) in the combined mode of
(+,-) and (+,+).
73 The instrument combined with the (+,+) and (+,-) double crystal monochromator, shown in Fig. 1-52c, combines the highest resolution with simple scanning and independence of fluctuations in the position of tb.e source. Such multiple reflections serve to suppress the tails in the single crystal reflection curves, and the resolution function can be improved. Several other combinations of double crystal monochromators are also used. If a crystal is bent in one direction, it can focus radiation emerging from a source point on the Rowland circle back to an image lying again on the Rowland circle. The crystal is cut with the reflecting lattice planes parallel to the surface. Typical bent crystal monochromators, the Johann type and the Johansson type, are shown in Figs. 1-53a and 1-53b. 2
_ 0B
~_
2
0B
.~.--:.
..- .... sa
,
,
(a)
/
)
(b)
(c)
Fig. 1-53. Three types of bent crystal monochromators. (a) Johann type, (b) Johansson type (symmetric reflection), and (c) non-symmetric Johansson type. In the Johann type, a plane crystal is bent so as to be a cylinder with curvature of 2R and is set up contacting the Rowland circle with radius R. In the Johansson type, a crystal is cut so that the reflecting surface forms the curved surface of the column with radius 2R and is then bent with the radius R. For both types of bent crystals, let I be the distance between a source point and the center of crystal: the following relationship is given as 1 = 2R cos ( z r / 2 - 013) = 2R / d
(1-81)
where 013 is the Bragg angle. A focusing monochromator with asymmetric distances from the crystal to the source ll and to the focus 12 is shown in Fig. 1-53c. The crystal surface is cut at an angle ct with respect to the lattice planes. The source and the image lie on the Rowland circle, and the distances l 1 and l2 are given by l 1 = 2R sin (013- a)
and l2 = 2R sin (0 B + a)
(1-82)
74 For example, for ~ = 1.5 A and 013 = 13 ~ with quartz (10]-1) reflecting planes, when a = 7 ~ is selected, then this would give a focus at the position of l 1 = 11.3 m with 12 = 37 m and a bending radius R = 54 m. This one-dimensionally bent crystals can focus the X-rays emerging from a point source on to a line which is perpendicular to the scattering plane. In order to obtain a point focus a combination with one more bent crystal or reflection mirror is useful. A double bend crystal, which is bent at two directions perpendicular to each other, is also employed. 1.9. P H O T O N D E T E C T O R S In this section we survey several different types of photon detectors which are currently in use in the VUV and X-ray regions. Photographic film scintillation counters, proportional counters, channeltrons, microchannel plates, solid-state detectors, ionization chambers and positionsensitive detectors will be described.
1.9.1. VUV and Soft X-Ray Detectors Photographic detection for VUV is performed using nuclear emulsions such as Eastman Kodak 101 and the slower, but finer-grain emulsion 104. Photon counters and ionization chambers can be utilized for absolute intensity measurements of VUV radiations. For wavelengths shorter than about 100 A proportional counters are usually used as standard detectors, provided corrections are made to take account of absorption in the window materials of the moderately pressurized counting chambers. Although, in principle, proportional counters can be used to wavelengths about 1000 A, there are problems in finding a suitable radiation transmitting window material to withstand the differential pressure in the spectral region of 10(~ A, to 200 A. In addition, the energy resolution and sensitivity are reduced. For wavelengths above 250 A to 1000 A rare-gas ionization chambers are commonly used. This method is based on the fact that one photon absorbed in a rare gas produces just one electron-ion pair which can be measured by means of a collector electrode and a high sensitivity electrometer amplifier. At wavelengths shorter than 250 A, since 250 A (49.6 eV) corresponds to twice the ionization potential of He (2 x 24.56 eV), it is necessary to consider the effect of secondary ionization by emitted photo-electrons and of multiple photoionization. For photons below 50 eV the ionization chamber is used in the so-called two-chamber mode and above 50 eV the two chambers are connected for the measurement at low gas pressure. Behind the ionization chamber photocathodes are mounted and the total photoelectric current leaving the cathodes is measured. A variety of photoelectron multiplier systems is in use. The first of these were built with mechanically formed dynodes (mostly CuBe) mounted separately. Recently strip dynodes, which consist of semiconducting thin films on glass plates, are mostly used. An electric field is
75 generated across the continuous strip dynodes by an applied potential, causing an electron avalanche along the dynodes. The channeltrons are also of this type of detectors, with continuous dynodes. They consist of a semiconducting glass channel having an internal diameter of a few mm and a ratio of length to diameter of the order of 50 : 1. The microchannel plates (MCP), composed of many microchannels with diameters as small as 10 l.tm and lengths about 1 mm are manufactured in such a way as to form an area up to 10 cm in diameter. This device can detect photons and electrons with an extremely fast time-response (of the order of 50 ps). By using a resistant anode encoder readout behind the MCP one can use it as a position sensitive detector.
1.9.2. X-Ray Detectors The most interesting and frequently used detectors for X-ray measurement are the lithium drifted silicon and germanium semiconductor detectors [Si(Li) and Ge(Li)]. They have the highest energy resolution because only 3 - 4 eV is required to produce one electron - hole pair. The detectors can be used at count rates up to about 50,000 cps with no deterioration of the energy resolution. At too a high count rate the energy resolution decreases as a result of incomplete charge collection. Scintillation counters usually have very poor energy resolution. This is because about 400 eV is required for NaI(TI) and 2,000 eV for plastic scintillators to produce one photoelectron at the multiplier cathode. However, the decay times are very short (1 ns or less) which is an important property for experiments with very high count rates. Proportional counters are useful detectors in the soft X-ray region. Using very thin windows a high overall detection efficiency can be achieved and the energy resolution is just sufficient to suppress higher harmonics. For experiments utilizing a very high intensity primary beam, such as absorption measurements, digital electronics can not be used. One can use inert-gas or airfilled ionization chambers. The signal current is collected and amplified by a d.c. amplifier which usually has a sensitivity limit of 10- 1 4 A. The efficiency of all the detector systems depends on matching the size of the sensitive part of the detector to the mean free path of the detected photons. With higher X-ray energies the mean free path becomes large and the size or the pressure of the detector has to be increased. Position-sensitive detectors have been developed for the registration of diffraction patterns and for experiments such as topography. Photographic film (with a nuclear emulsion), a linear semiconductor detector, linear proportional counter, multiwire proportional counter or X-ray TV camera can be used. Photographic films are useful for experiments which need high spatial resolution, down to 0.3 ~tm. With photographic recording real time experimentation can not be carded out. Linear solid state detectors and proportional counters are used in such a way that they allow for the localization of individual events. Multiwire proportional counters are used for two dimensional detection, being equipped with crossed wires, and making use of the positive
76 signal from the ion current. The separation of the wires can be as low as 1 mm, and a positional accuracy of 0.15 mm can be achieved. Readout of individual wires into the computer allows very high count rates. A TV camera using a fluorescent screen and an image intensifier can be used with high and low intensity signals. Extensive use of these systems is expected in the future since they make use of highly developed TV techniques. REFERENCES 1. F. R. Elder, A. M. Gurewitsch, R. V. Langmuir and H. C. Pollock, Phys. Rev., 71, 829 (1947); J. Appl. Phys., 18, 810 (1947); F. R. Elder, R. V. Langmuir and H. C. Pollock, Phys. Rev., 74, 52 (1948). 2. A. Lienard, L'Eclairage Elect.,16, 5 (1898). 3. G. A. Schott, Ann. Phys. (Leipzig), 24, 635 (1907); G. A. Schott, "Electromagnetic Radiation", Cambridge Univ. Press, Cambridge, 1912, Ch. 7, 8. 4. W. W. Jassinsky, J. Exp. Theor. Phys. (USSR), 5 983 (1935); Arch. Elektrotech. (Berlin), 30 590 (1936). 5. D. Kerst, Phys. Rev., 60 47 (1941). 6. D. Ivanenko and I. Pomeranchuk, Dokl. Akad. Nauk SSSR, 44 315 (1944); Phys. Rev., 65 343 (1944). 7. L. Arzimovitch and I. Pomeranchuk, J. Phys. (Moskow), 9 267(1945); J. Exp. Theor. Phys. (USSR) 16 379 (1946). 8. J. P. Blewett, Phys. Rev., 69 87 (1946); Nucl. Instrum. Methods, A266 1 (1988). 9. D. Ivanenko and A. A. Sokolov, Dokl. Akad. Nauk SSSR, 59 1551 (1948); A. A. Sokolov and I. M. Ternov, Dokl. Akad. Nauk SSSR, 89 665 (1953); A. A. Sokolov and I. M. Temov, Sov. Phys. JETP (Engl. Transl.), 1 227 (1955); Sov. Phys. JETP (Engl. Transl.), 4 396 (1957); Sov. Phys. Dokl (Engl. Transl.), 8 1203 (1964). 10. J. Schwinger, Phys. Rev., 70 798 (1946); Phys. Rev., 75 1912 (1949). 11. I. M. Ado and P. A. Korolev, V. S. Markov, E. M. Akimov and O. F. Kulikov, Sov. Phys. Dokl. (Engl. Transl.), 1 568 (1956); F. A. Korokov and O. F. Kulikov, Sov. Phys. Dokl.(Engl. Transl.), 5 1011 (1961); F. A. Korokov, O. F. Kulikov and A. S. Yarov, Sov. Phys. JETP (Engl. Transl.), 43 1653 (1962). 12. D. A. Corson, Phys. Rev., 86 1052 (1952); Phys. Rev., 90 748 (1953). 13. P. L. Hartman and D. H. Tomboulian, Phys. Rev., 87 233 (1952); D. H. Tomboulian, U. S. A. E. C. NP-5803 (1955); D. H. Tomboulian and P. L. Hartman, Phys. Rev., 102 1423 (1956); D. E. Bedo and D. H. Tomboulian, J. Appl. Phys., 29 804 (1958); P.L. Hartman, Nucl. lnstrum. Methods, 195 1 (1982). 14. K. Codling and R. P. Madden, Phys. Rev. Lett., 10 516 (1963); Phys. Rev. Lett. 12 106 (1964); J. Opt. Soc. Am., 54 268 (1964); J. Appl. Phys., 36 830 (1965). 15. G. Bathow, E. Freytag and R. Haensel, J. Appl. Phys., 37 3449 (1966). 16. K. R. Lea, Physics Reports, Section C of Physics Letters, 43 337 (1978).
77 17. G. K. Green, Brookhaven Nat. Lab. Rep. BNL 50522, 90 (1977); Brookhaven Nat. Lab. Rep. BNL 50595, Voi.ll (1977). 18. E. E. Koch, C. Kunz and E. W. Weiner, Optik, 45 395 (1976). 19. A. van Steenbergen, Nucl. Instrum. Methods, 172 25 (1980); Nucl. Instrum. Methods, 177 53 (1980). 20. N. F. Gmuer, Brookhaven Nat. Lab. Rep. BNL 45764, NSLS User's Manual: Guide to The VUV and X-Ray Beam Lines, 4th Ed., February 1991. 21. K. Kohra and T. Sasaki, Nucl. Instrum. Methods, 208 23 (1983); S. Sato, Koide, Y. Morioka, T. Ishii, H. Sugawara and I. Nagakura, Nucl. Instrum. Methods, 208 31 (1983); A. Asami, H. Kobayashi and M. Ando, Houshakoh, J. Jpn. Soc.for Synchrotron Radiation Research, 1 47 (1988). (in Japanese) 22. H. Kamitsubo, Nucl. Instrum. Methods, A303 421 (1991). 23. L. R. Elias, W. M. Fairbank, J. M. J. Madey, H. A. Schwettman and T. I. Smith, Phys. Rev. Lett., 36 717 (1976). 24. D. A. G. Deacon, L. R. Elias, J. M. J. Madey, G. J. Ramian, H. A. Schwettman and T. I. Smith, Phys. Rev. Lett., 38 892 (1977). 25. W. B. Colson, Phys. Quantum Electron, 5 157 (1978). 26. C. Pellegrini, IEEE Trans. Nucl. Sci., NS-26 3791 (1979). 27. N. A. Vinokurov and A. N. Skrinsky, Institute of Nuclear Physics, Novosibirsk, USSR, Preprint 77-59, (1977) 28. M. Billardon, P. Elleaume, J. M. Ortega, C. Bazin, M. Bergher, M. Velghe, Y. Petroff, D. A. G. Deacon, K. E. Robinson and J. M. J. Madey, Phys. Rev. Lett., 51 1652 (1983). 29. G. N. Kulipanov, V. N. Litvinenko, I. V. Pinaev, V. M. Popik, A. N. Skrinsky, A. S. Sokolov and N. A. Vinokurov, Nucl. Instrum. Methods, A296 1 (1990). 30. H. Maezawa, M. Ando, T. Ishikawa, M. Nomura, H. Kitamura, A. Mikuni and T. Sasaki, Proc. SPIE - Int. Soc. Opt. Eng., Vol.733, Soft X-Ray Optics and Technology, 1986, p.96. 31. A. P. Lukirskii, E. P. Savinov, O. A. Ershov and Yu. F. Shepelev, Opt. Spectrosc. (Engl. Transl.), 16 168 (1964). 32. V. Rehn and W. J. Choyke, Nucl. Instrum. Methods, 177 173 (1980). 33. V. Rehn, V. O. Jones, J. M. Elson and J. M. Bennett, Nucl. Instrum. Methods, 172 307 (1980). 34. J. Harada, K. Ooshima and T. Sakabe, Nippon Kessho Gakkaishi, 24 256 (1982).
78
Appendix:
Table of Various Bessel Functions and Integrals
y
KI/3 (y)
1.000e-04 1.000e-03 2.000e-03 4.000e-03 6.000e-03 8.000e-03 1.000e-02 2.000e-02 3.000e-02 4.000e-02 5.000e-02 6.000e-02 7.000e-02 8.000e-02 9.000e-02 1.000e-01 1.500e-01 2.000e-01 2.500e-01 3.000e-01 3.500e-01 4.000e-01 4.500e-01 5.000e-01 5.500e-01 6.000e-01 6.500e-01 7.000e-01 7.500e-01 8.000e-01 8.500e-01 9.000e-01 1.000e+00 1.250e+00 1.500e+00 1.750e+00 2.000e+00 2.250e+00 2.500e+00 2.750e+00 3.000e+00 3.250e+00 3.500e+00 3.750e+00 4.000e+00 4.250e+00 4.500e+00 4.750e+00 5.000e+00 5.500e+00 6.000e+00 6.500e+00 7.000e+00 7.500e+00 8.000e+00 8.500e+00 9.000e+00 9.500e+00 1.000e+01
3.628e+01 1.672e+01 1.319e+01 1.038e+01 8.995e+00 8.116e+00 7.486e+00 5.781e+00 4.932e+00 4.386e+00 3.991e+00 3.685e+00 3.437e+00 3.231e+00 3.054e+00 2.900e+00 2.343e+00 1.979e+00 1.714e+00 1.509e+00 1.343e+00 1.206e+00 1.089e+00 9.890e-01 9.018e-01 8.251e-01 7.571e-01 6.965e-01 6.422e-01 5.932e-01 5.489e-01 5.086e-01 4.384e-01 3.079e-01 2.202e-01 1.594e-01 1.165e-01 8.581e-02 6.354e-02 4.727e-02 3.531e-02 2.645e-02 1.988e-02 1.497e-02 1.130e-02 8.545e-03 6.472e-03 4.904e-03 3.729e-03 2.159e-03 1.255e-03 7.317e-04 4.280e-04 2.509e-04 1.474e-04 8.679e-05 5.118e-05 3.023e-05 1.787e-05
K2/3 (y) 4.989e+02 1.075e+02 6.769e+01 4.262e+01 3.251e+01 2.682e+01 2.310e+01 1.450e+01 1.102e+01 9.052e+00 7.762e+00 6.837e+00 6.136e+00 5.581e+00 5.130e+00 4.753e+00 3.513e+00 2.802e+00 2.329e+00 1.987e+00 1.725e+00 1.517e+00 1.347e+00 1.206e+00 1.086e+00 9.828e-01 8.933e-01 8.148e-01 7.455e-01 6.839e-01 6.288e-01 5.794e-01 4.945e-01 3.406e-01 2.402e-01 1.722e-01 1.248e-01 9.132e-02 6.726e-02 4.981e-02 3.706e-02 2.767e-02 2.073e-02 1.558e-02 1.173e-02 8.853e-03 6.693e-03 5.069e-03 3.844e-03 2.220e-03 1.287e-03 7.495e-04 4.376e-04 2.562e-04 1.504e-04 8.842e-05 5.209e-05 3.073e-05 1.816e-05
K5/3 (y) 6.652e+06 1.433e+05 4.514e+04 1.422e+04 7.233e+03 4.478e+03 3.087e+03 9.723e+02 4.946e+02 3.061e+02 2.110e+02 1.556e+02 1.203e+02 9.625e+01 7.905e+01 6.627e§ 3.357e+01 2.066e+01 1.414e+01 1.034e+01 7.915e+00 6.263e+00 5.082e+00 4.205e+00 3.534e+00 3.009e+00 2.589e+00 2.249e+00 1.967e+00 1.733e+00 1.535e+00 1.367e+00 1.098e+00 6.712e-01 4.337e-01 2.906e-01 1.998e-01 1.399e-01 9.941e-02 7.142e-02 5.178e-02 3.781e-02 2.778e-02 2.051e-02 1.521e-02 1.132e-02 8.455e-03 6.332e-03 4.754e-03 2.697e-03 1.541e-03 8.855e-04 5.113e-04 2.965e-04 1.725e-04 1.007e-04 5.890e-05 3.454e-05 2.030e-05
GO (y) 9.960e+02 2.131e+02 1.336e+02 8.349e+01 6.329e+01 5.193e+01 4.450e+01 2.736e+01 2.045e+01 1.657e+01 1.403e+01 1.222e+01 1.085e+01 9.777e+00 8.905e§ 8.182e+00 5.832e+00 4.517e+00 3.663e+00 3.059e+00 2.607e+00 2.254e+00 1.973e+00 1.742e+00 1.549e+00 1.386e+00 1.246e+00 1.126e+00 1.020e+00 9.280e-01 8.464e-01 7.740e-01 6.514e-01 4.359e-01 3.004e-01 2.113e-01 1.508e-01 1.089e-01 7.926e-02 5.811e-02 4.286e-02 3.175e-02 2.362e-02 1.764e-02 1.321e-02 9.915e-03 7.461e-03 5.626e-03 4.250e-03 2.436e-03 1.404e-03 8.131e-04 4.725e-04 2.755e-04 1.611e-04 9.439e-05 5.543e-05 3.262e-05 1.922e-05
G3 (y) 9.960e-10 2.131e-07 1.069e-06 5.343e-06 1.367e-05 2.659e-05 4.450e-05 2.189e-04 5.522e-04 1.060e-03 1.754e-03 2.640e-03 3.722e-03 5.006e-03 6.492e-03 8.182e-03 1.968e-02 3.614e-02 5.723e-02 8.259e-02 1.118e-01 1.443e-01 1.798e-01 2.177e-01 2.577e-01 2.994e-01 3.422e-01 3.862e-01 4.303e-01 4.751e-01 5.198e-01 5.642e-01 6.514e-01 8.514e-01 1.014e+00 1.132e+00 1.206e+00 1.240e§ 1.238e+00 1.209e+00 1.157e+00 1.090e+00 1.013e+00 9.302e-01 8.454e-01 7.611e-01 6.799e-01 6.029e-01 5.312e-01 4.053e-01 3.033e-01 2.233e-01 1.621e-01 1.162e-01 8.248e-02 5.797e-02 4.041e-02 2.797e-02 1.922e-02
Applications of Synchrotron Radiation to Materials Analysis H. Saisho and Y. Gohshi (Editors) 9 1996 Elsevier Science B.V. All rights reserved.
79
CHAPTER 2
X-RAY F L U O R E S C E N C E ANALYSIS Hideo SAISHO and Hideki HASHIMOTO Inorganic Analysis Laboratory, Toray Research Center, Inc. 1-1, Sonoyama 1-Chome, Otsu, Shiga 520, Japan
2.1. I N T R O D U C T I O N Recent trace element analysis requires highly sensitive, simultaneous multi-element methods. Such methods include activation analysis, atomic emission spectrometry, mass spectrometry, and X-ray fluorescence analysis (XRF). XRF facilitates rapid, nondestructive analysis, and has a wide range of applications, including production processes and quality control. Although XRF is not necessarily a high-sensitivity technique (normally, it can only detect ~tg levels), researchers have recently found that synchrotron radiation (SR) greatly improves the sensitivity of XRF, enabling analysis over l.tg-pg ranges [1-3]. Synchrotron radiation is a continuum radiation with a wide wavelength range. Two to three GeV class storage tings currently available supply radiation ranging from microwaves to hard X-rays (20-30 keV). Furthermore, these storage tings provide an X-ray intensity 100-10 000 times greater than conventional X-ray sources. If a wiggler is used, hard X-rays of up to 50 keV can be supplied. In addition, when large scale SR facilities under construction are completed, the available intensity will increase by a few more orders of magnitude. Other advantages of SR include its almost perfect linear polarization and excellent collimation. More sensitive XRF can be attained by increasing the intensity of the signal and reducing the background. Greater signal intensity can be obtained by increasing the intensity of the excitation X-rays. The background is reduced by two characteristics of SR: continuum radiation and polarization. The excellent collimation of SR makes this method suitable for measurements using the total reflection method, which is useful for surface analysis, as well as for microarea analysis using a microbeam. The major characteristics of SR - - high intensity, continuum radiation, polarization, and c o l l i m a t i o n - greatly enhance the capabilities of XRF. Table 2-1 lists the characteristics of synchrotron radiation and the advantages when it is applied to XRF.
80 Table 2-1 Outstanding characteristics of SR for XRF SR characteristics
Advantages
High intensity Linear polarization Continuous spectrum
Signal enhancement Background reduction Selective excitation Total reflection
High collimation
{
Micro-trace analysis
SR-Excited XRF (SRXRF) was first developed in 1972 by Horowitz and Howell [4] for use in microbeam analysis. Working with the Cambridge Electron Accelerator, they used an ellipsoidal condensing mirror and a pinhole to produce a 2 I.tm focused X-ray beam. XRF using this beam attained a detection limit of 10-6 to 10- 9 g cm -2 at a resolution of 2 lxm. Sparks et al. [5, 6] conducted their well-known 1977 experiment at SPEAR of SSRL in order to find primordial superheavy elements using a curved pyrolytic graphite crystal and a Si(Li) detector. They demonstrated that it was possible to detect superheavy elements if at least 5 x 108 atoms were present in the sample. The announcement of this result, as well as the establishment of SR facilities at many locations, led to an increasing number of investigations of XRF analysis in the 1980s. Gilfrich et al.[7], at SSRL in 1983, systematically performed a series of experiments concerning detection limits. Similar research was carded out by Hanson et al. [8] at CHESS of Cornell, Kn6chel et al. [9] at DORIS of HASYLAB, and Bos et al. [10] at the SRS of Daresbury. Iida et al. [ 11] conducted a close investigation into the dependence of the detection limit on the excitation mode at the Photon Factory (PF) of the National Laboratory for High Energy Physics (KEK). Also, Gordon used beam intensity parameters to calculate theoretical values for detection limits at the NSLS of Brookhaven [ 12]. X-Ray total reflection produces a shallow X-ray penetration depth and very little scattering (background), and therefore total reflection XRF (TXRF) is useful for surface analysis and ultratrace analysis [13, 14]. While improvements have been made by Yoneda and Horiuchi [ 15] in the original technique, which used a conventional X-ray tube, the use of SR further enhances the effectiveness of TXRF because of SR's excellent collimation and strongly monochromatic beam [ 16]. Researchers often apply this method to analyze depth profiles [ 17] and layered structures [18]. In these analyses, the reflection curve and fluorescent X-ray profile are measured as changes are made in the incident angle near the critical angle of total reflection. In this chapter, we will discuss equipment used for bulk analysis and surface analysis using SRXRF. These techniques do not include analysis using a microbeam or chemical state analysis for obtaining information on chemical bonding by high energy-resolution measurements. These will be described in the next chapter.
81 2.2. E Q U I P M E N T SRXRF detects X-ray fluorescence caused by a white (non-monochromatic) or monochromatic incident SR beam. The radiation of hard and soft X-rays from the storage ring requires quite different equipment. With hard X-rays, beryllium windows can be used to form a barrier between the ultrahigh vacuum in the storage ring and the low vacuum or air in the sample chamber. Soft X-ray SRXRF requires ultrahigh vacuum through to the sample chamber. This means that completely different equipment is required for the analysis of light elements (atomic number less than 12) and of elements with higher atomic numbers. Hereafter, we will confine our discussion to analytical techniques using hard X-rays. Hard X-ray equipment is built in much the same way as equipment using X-ray tubes or rotating anode Xray generators. The major difference is that in SR equipment, the distance from the light source to the sample is longer (10-30 m). Despite this long distance, high collimation keeps the reduction of SR intensity to very low levels. Equipment known as beamlines introduces the X-rays into the detection system. A beamline is composed of a few beryllium windows, a vacuum system, shutters, and optical elements. Since SR obtained from a bending magnet is linearly polarized in the orbital plane of the accelerated electron, the optical elements usually have an axis of rotation horizontal to the electron's orbital plane. To protect the operator from radiation, the detection system is placed in an iron (or lead) hutch and is remote-controlled. Section 2.2.1. describes the optical elements used in SRXRF. Section 2.2.2. deals with the detectors, and Section 2.2.3. discusses the beamline for X-ray fluorescence incorporating these two components.
2.2.1. Optical elements Total reflection m i r r o r
The X-ray mirror is extensively used in SR experiments as a low pass filter, a high pass filter, or an X-ray focusing device. Since the refractive index for X-rays is very slightly less than unity (a difference of 10-5), an X-ray with a glancing angle smaller than a certain value (critical angle: a few mrad) is totally reflected. The critical angle depends on the kind of substance and the wavelength of the incident X-rays. The complex index of refraction n for an X-ray with a wavelength A is given as follows: (2-1)
n = l - 5 - i fl S = (rJ2~)(Nop/A)(Z
fl = ~/.t/4r~
+ Af' )Z 2
(2-2) (2-3)
82
(a)
(b)
1.0-
ad
1.0-
~: 1.5/~
0.8-
0.8-
ov,,~
~0.6-
Si " 0 = 4mrad
~,c" 1.6/~
0c " 10.5 mrad
Z,c" 1.7/t~ ;>0.6=!,~(
Si
t,,.,)
~0.4-
r
0.4-
ad 0.2-
0.2-
0.0- i 0
4
i
8
'
t
12
'
Glancing angle / mrad
o'o-t'- t ' "
1.2
1.4
i
1.6
t
1.8
Wavelength /
t
2.0
t
2.2
Fig. 2-1. The calculated reflectivities for silicon and platinum: (a) is expressed as a function of glancing angles at a fixed wavelength (1.5 A) of the incident X-rays; (b) as a function of wavelengths at a fixed glancing angle (4 mrad for Si, 12 mrad for Pt).
where re is the classical electron radius, No is Avogadro's number, p is the density, and A is the atomic weight. Thus, (Nop/A) represents the number of atoms found in a unit volume. The quantity (Z+Af) is the real part of the atomic scattering factor, where Z is the number of electrons per atom (atomic number) and zlf' represents the dispersion term. Far away from the absorption edge, Af' is very small, and therefore 6 is proportional to the electron density. The quantity/.t is the linear absorption coefficient. An incidence of an X-ray at a glancing angle less than the critical angle, 0c, which is a grazing angle, will result in total reflection. If we ignore absorption, the critical angle can be given as follows, according to Snelrs law: 0c = ~
(2-4)
Calculated reflection curves for silicon and platinum are shown in Fig. 2-1. Figure 2-1a shows the reflection curves as a function of the angle (glancing angle), and (b) shows them as a function of the incident X-ray wavelength. The critical angle (0c) and critical wavelength (20 can be seen. For a given substance and angle, X-rays of wavelengths less than a certain value (greater energy than a certain value) are not reflected: thus, the mirror works as a low pass filter. As can be seen in Fig. 2-1, for a given wavelength, the more electrons per unit volume, the larger the critical angle. However, as the amount of X-rays absorbed increases, the change in reflectivity near the critical angle is more gradual, which implies a lower
83 efficiency as a low pass filter. Since the critical angle for X-ray wavelengths is a few mrad, the use of an X-ray 2 mm high requires a mirror a few tens of centimeters long. Also, the mirror must be sufficiently smooth if its calculated efficiency is to be attained. Calculated reflection curves for mirrors of different roughnesses are shown in Fig. 2-2. Surface roughness is described as a group of flat planes distributed in a Gaussian manner. Figure 2-2 shows that less than 10/~ roughness is necessary. Surface roughness reduces the reflectivity. When using a mirror and employing a strong SR, it is necessary to ensure thermal stability. Therefore, it is important to place the optical system in a high vacuum or to cool the system. Another way is to position the mirror behind the monochromator, but then the monochromator would have to be protected from the heat. The mirror material is another point for careful consideration. A plane plate of thickness d with a difference in temperature, AT, between the surfaces, bends at a radius of curvature, R, which can be described as follows: R = d/(aAT)
= (kd2)/(otAq)
(2-5)
where k is the thermal conductivity, ct the coefficient of thermal expansion of the material, and Aq the amount of heat transferred through a plate of thickness d. Equation (2-5) indicates that the mirror material should have a high thermal conductivity and a small coefficient of thermal 1.0
t~=0 ~," 1.24 .~
0.8
~06
~
9
a=20
r
~ 0 4-
0.2
0.0 1
2
3
4
5
6
G l a n c i n g angle / m r a d Fig. 2-2. The calculated reflection curves for samples of silicon of roughnesses (t~ = 0, 10 ]~ and 20 ]~).
different surface
84 expansion, i.e., a large k/a. In addition, the mirror material must be able to be ground and polished to a sufficient degree. Fused quartz is widely used since large polishable mirrors can be fabricated. However, despite its small coefficient of thermal expansion, the thermal conductivity of this material is low and therefore it cannot be used where a substantial heat load is expected. Recently, SiC has received attention. It has a high thermal conductivity and a k]o~ value more than 20 times that of quartz. Since it is now possible to make large SiC mirrors, it will be used in most beamlines using an insertion device and in the next generation of large scale rings. The mirror chamber must be kept under ultrahigh vacuum to prevent carbon from adhering to the mirror surface, since this would greatly lower the mirror's efficiency. Total reflection mirrors are generally used for eliminating harmonics. They are also used, in combination with absorbers or transmission mirrors, to emit X-rays with a wide energy band or to focus X-rays. If the absorption characteristics of a very thin mirror are low, the unreflected X-ray comes out of the back of the mirror. Such mirrors are known as transmission mirrors. If a mirror is thin enough (less than 1 lxm thick) and is made of material with a low atomic number, it can be used as a transmission mirror. The calculated transmissivities of thin organic films are shown in Fig. 2-3. Transmission mirrors can be used as highly efficient high pass filters. Soap film is a smooth, stable transmission mirror. It is 300-10 000/~ thick, and very smooth, with a surface roughness less than a few/~. Lairson and Bilderback [19] used a solution of glycerin (35%), Ivory Liquid (2%), and distilled water (63%) to make soap films. They used a 20 Jam wire stretched inside a 10 cm x 30 cm frame to make 8 cm x 20 cm films. The films kept a few days in wet helium.
Figure 2-4 shows the transmitted spectra of the
1.0--
1.0" ~0.8
f
-
thickness: d = 0.5 ilm
9 t,,=q
r thickness: d = 0.1 I.tm
;~0.8 -
~,' 1.24 /~
~, 91.24 A
~0.6-
o~ 0 . 6 -
9 v,,,,l
9 t,,,,q
~04-
~0.4-
0.2-
0.2-
0.00
2
I
I
I
I
4
6
8
10
Glancing angle / mrad
0"0-1 0
'
'
2
I
I
I
I
4
6
8
10
Glancing angle / mrad
Fig. 2-3. The calculated glancing angle dependence of transmissivities for polystyrene films using incident X-rays with a wavelength of 1.24/~.
85
10
.--" .._-_ -
jJ
_
.;
9~
"~
0
-
~
-
-
I
_
I
t
0
i
I
i
t
I
12.5 Energy / keV
I
I
25
Fig. 2-4. Transmitted spectra of soap films at a thickness of > 10 000 ]k (broken line) and of 3 200 A (solid line). Both films are set to the same mirror angle (2.4 mrad). From Ref.[19], reprinted by permission of Elsevier Science Publishers B.V., Amsterdam.
films. It indicates the dependence of transmissivity on film thickness. Iida et al. [20] used 0.5 ~tm Mylar film as a transmission mirror. Compared with soap films, the Mylar film is more stable and durable, yet it is less smooth and more difficult to make flat. Mirrors with an ellipsoidal, cylindrical, or hyperboloidal cross section also can be used to condense X-rays. In microprobe analysis, a combination of these mirrors is used to create a micro-sized beam (microbeam). Reflection and transmission mirrors can be combined to select X-rays with appropriate energy bands. This will be discussed later in this section. Crystal monochromator In order to use SR as a monochromatic X-ray excitation beam it must be monochromatized using a monochromator. Crystals are used for this purpose. The diffraction of an X-ray by a crystal is described by Bragg's law 2d sin0 B= nA
(2-6)
where n is the diffraction order, d is the lattice spacing in the crystal lattice, ~, is the X-ray wavelength and 0]3 (the Bragg angle) is the angle which the incident X-ray makes to the
86 reflecting plane. Since sin0 is less than unity, X-rays with wavelengths longer than 2d cannot be reflected. Bragg angles of less than a few degrees or more than 70 degrees cannot be achieved because of the structure of the monochromator. Therefore, the lattice spacing of the crystal lattice plane must match the wavelength used. For example, for Si (111), where d is 3.136/~, the wavelengths that can be obtained range from 0.4 to 6 ]k. Equation (2-6) is differentiated to: AA/A = AE/E = A0 cot0a
(2-7)
From Eq. (2-7) it is evident that energy resolution is determined by the angular width A0 and Bragg angle 0B. The angular width is determined by the angular spread of the incident beam and the reflection width of the crystal monochromator. The angular spread of the incident beam depends on the size and angular divergence of the light source and the geometry of the experiment (distance, slit, etc.). The reflection width is greatly affected by the crystal'sstate of perfection. For a perfect crystal, the reflection width to is given as follows, according to the dynamical theory of diffraction [21 ]: to =
2 9re ~ 2 .C. [/7111.e -M sin20B uV
(2-8)
where re is the classical electron radius, V the unit-cell volume, Fh the crystal structure factor, e -M the temperature factor, and C the polarization factor. The integral reflecting power, I, for a weakly absorbing crystal is given approximately by 2
I = 1__. 8 .rex .C. [F~ .e -M ~ 3sin20B ~V
(2-9)
where b is a quantity called the asymmetry factor. For a mosaic crystal, kinematical diffraction theory can be applied to rewrite I as follows: \ I=
r2~3
(2-10) .C 2. ~t~2.e-2M
2/aVEsin2 0B The reflection width, energy resolution, and integral reflecting power of perfect crystals which are frequently used are shown in Table 2-2 [22]. The crystal used in a monochromator must not be seriously damaged by a strong SR. Generally speaking, it is impossible to use organic crystals in monochromators. Usually, silicon crystals are used, since large perfect crystals can be made easily and provide the lattice
87 Table 2-2 Parameters of crystal monochromators a Crystal
hkl
als
(second of arc)
b
c
AE/E
(•
5)
I
(xl 0 6 )
Silicon
111 220 311 333
7.395 5.459 3.192 1.989
14.1 6.04 2.90 0.88
39.9 29.7 16.5 9.9
Germanium
111 220
16.338 12.449
32.64 14.46
85.9 67.4
a Bragg reflection width, b Energy resolution, c Integral reflecting power. These parameters are used to determine experimental conditions such as incident X-ray energy and resolution. From Ref. [22], reprinted by permission of Elsevier Science Publishers B. V., Amsterdam.
spacing required in the hard X-ray region. Since SR is polarized in the horizontal direction, a non-horizontal monochromator rotational axis will jeopardize the efficiency. When the monochromator with a horizontal rotational axis is used, the diffracted beam goes up or down: therefore, changing the Bragg angle to change the wavelength varies the direction of the diffracted beam. This means that all units positioned behind the monochromator (slit, sample, detector, etc.) must be moved every time the Bragg angle is changed. A monochromator with a parallel double-crystal (or a channel-cut crystal) always reflects the beam in the same direction. In this case, the above units can simply be moved up and down, en bloc. It is also possible to keep the height of the outgoing beam constant by attaching a mechanism to translate the crystal while rotating the doub!e-crystal monochromator [23]. This makes the equipment even easier to use. Special care must be taken to eliminate harmonics when using a monochromator. It is clear from Bragg's law that for any X-ray, with a wavelength A, there will be an X-ray with a wavelength 2/n which will be reflected at the same angle. The relative intensity of harmonics depends on the spectn~m of the light source, the integral reflecting power of the crystal, etc. If absorption is ignored, the integral reflecting power is proportional to (2/n)2lF~e-M, a quantity that monotonically decreases with increasing order n. Considering the SR spectrum, therefore, the second and third harmonics are the problems in the X-ray region. In some cases, the second harmonics disappear due to the symmetry of the crystal. One such example is Si(111). In this case, harmonics pose very little problems if the storage ring can supply a relatively high incident energy (more than 15 keV for a 2 GeV class storage ring). Basically there are two methods of eliminating harmonics. One uses a total reflection mirror. As
88 discussed previously in this section, the mirror totally reflects X-rays of wavelengths longer than a certain value (the critical wavelength) and does not reflect those with shorter wavelengths. The other method employs a beamline without a mirror. Harmonics can be eliminated by slightly detuning two crystals used in a double-crystal monochromator. As shown in Table 2-2, the reflection width for Si(333) is much smaller than that for Si(111). This makes it possible to reduce the intensity of harmonics by detuning. Also used is an optical system that increases the intensity of the beam by focusing the incident X-ray using a curved second crystal (sagittal focusing) [24-26]. SR has a high degree of collimation in the vertical direction (angular spread: approx. 0.1 mrad), but it is also spread in the horizontal direction. An ordinary beamline can take in a few mrad of a beam in the horizontal direction, but given the long distance (10-30 m) from the light source to the hutch, the beam will be spread more than a few cm on the sample. Therefore, if the incident beam is not focused, a fiat-crystal monochromator can only supply less than 1 mrad of the beam. A focusing optical system collects much of the beam and improves the detection limit by producing more than 10 times the intensity. Another focusing optical system employs a fiat-crystal monochromator and a curved mirror. When using this system, however, the position of the mirror should be carefully considered. If the mirror is positioned in front of the monochromator its adjustment can be made at the beginning, and no further adjustment is needed when tuning the monochromator to change the energy. However, in this position SR could damage the mirror surface. When the mirror is positioned behind the monochromator damage is negligible, but then it requires more complicated adjustment. Protecting the monochromator from the heat of SR must also be considered when using an insertion device or a larger scale ring. Resolution may be sacrificed if higher intensity is desired. The properties of a mosaic crystal monochromator are between the wide band-pass (see the following section) and perfect crystal monochromators [ 11]. As shown in Eqs. (2-9) and (2-10) the mosaic crystal monochromator provides greater reflection intensity. The pyrolytic graphite mosaic crystal produces greater reflection intensity, but the band width is spread to about 170 eV, yielding an X-ray of 8 keV. Silicon with a rough ground surface also displays mosaic crystal properties. It is important to choose a monochromator that best matches the size and composition of the sample.
Wide band-pass monochromator In some cases, greater intensity is required and the high resolution of a crystal monochromator is not. A wide band-pass monochromator is defined as a monochromator with an energy resolution, AE/E, of more than 0.1. There are two types of such monochromators. One combines a total reflection mirror with an absorber or a transmission mirror. The other type uses a synthetic multilayer as a monochromator element. Iida et al. [20] studied monochromators that combined a total reflection mirror and an aluminum absorber (or a transmission mirror such as a soap film or Mylar film). Their calculations are shown in Figs. 2-5 and 2-6, and the results of their experiments in Figs. 2-7
89 1.0
_
P~ ~
p,.
%.I
~
0.5 -
-
//
/
e/
//
/ /
-
0.0
_/
200_/'-'11~\ // ~'k~ / 409.. "'.'~k,.x~
/
j"-
-.-'r"~-"l
5
m
'
m
,
m
10 Energy / keV
n
~
~
m
_
m
15
i
-
T - -
Fig. 2-5. Calculated fused quartz reflection mirror responses in combination with A1 absorbers (broken lines) and without an absorber (solid line). The glancing angle of the reflection mirror is 2.5 mrad. The absorber thicknesses are shown in the figure in ~tm. The cutoff on the low energy side is not sharp. From Ref. [20], reprinted by permission of Elsevier Science Publishers B. V., Amsterdam.
I
1.0 p~ p,.
-
~
. l
l
.I
/ i
/
_
I
I I
_1
I
I
0.0 5
3//
/
-
~~ 0.5
I
i
a
i
i
n
n
i
m
I
/
/
/
/,
a
t
2")[ ~ / 2.3 I / I I I I 2 I
I i
t
I I
I
I I
m
10 Energy / keV
m
15
Fig. 2-6. Calculated fused quartz reflection mirror responses in combination with (broken lines) and without (solid line) a soap film transmission mirror. The glancing angles of the transmission mirror are shown in the figure in mrad. The soap film thickness is 1 000 A. The cutoff on the low energy side is sharp compared with the cutoff of the reflection mirror / absorber combination. The glancing angle of the reflection mirror is 2.5 mrad. From Ref. [20], reprinted by permission of Elsevier Science Publishers B. V., Amsterdam.
90 ,=
1.0
(a)
~
aW~tshO%ter -
1.0
(b)
-
. ~ 4 - - - . . without f ,I,IiN absorbeq / ~ ~ 70/.tm
o N
70 m ~0.5
140 m
0.0:
0.0 I
4
0.5
i
i
I
6
8
.1
i_
10
i
i
12
i
i
i_
i
14 16
!
i
6
4
i
8
!
i
10
i
i
12
I
I
I
]
14 16
Energy / keV
Energy / keV
Fig. 2-7. Measured responses of the Pt coated (a), and non-coated (b), fused quartz mirror in conjunction with the A1 absorbers. A wide energy band-pass greater than 20% was achieved. From Ref. [20], reprinted by permission of Elsevier Science Publishers B. V., Amsterdam.
1.0-
1.0~ '(1))' ' ' A ~ - ~ [with p m '~,~
9
i
ttrans
No.o
,,, 4
6
8
'withou; ~ tr.ans. ]
l
o.o 10 12 14 16 18 20 Energy / keV
4
6
8 10 12 14 16 Energy / keV
18
Fig. 2-8. Measured responses of the fused quartz reflection mirror using soap film (a) and Mylar (b) transmission mirrors. For the use of the transmission mirror, each curve shows the response of a different glancing angle to the transmission mirror. A sharper cutoff at the low energy tail is achieved than with the reflection / absorber combination. From Ref. [20], reprinted by permission of Elsevier Science Publishers B. V., Amsterdam.
91 and 2-8. They obtained a wide band pass of more than 0.2. That using a transmission mirror has a sharper cutoff in the low energy region (see Fig.2-8a). Figure 2-9 shows the spectrum of X-ray fluorescence measured by the energy dispersive method, using chelated metal-resin beads as a sample. When using an absorber, the signal-to-background (S/B) ratio is large in the low energy region, but the X-ray fluorescence of Zn overlaps the scattered X-ray. On the other hand, the combination of a total reflection mirror and a transmission mirror provides an excellent S/B ratio for the X-ray fluorescence emitted by Zn. Synthetic multilayers have recently been receiving attention as excellent wide band-pass monochromators. These synthetic lattices stack two substances, with different atomic numbers (e.g., silicon and tungsten), one on the other at intervals between a few and a few tens of A. Early in their development there were problems resulting from the two substances diffusing at the interface. Recently, however, it has been possible to produce excellent multilayers in the desired combinations and intervals [27]. Because of their controllable lattice spacing they are useful not only as wide band-pass monochromators but also as monochromators for the soft X-ray region m a region that has lacked an appropriate monochromator. 2.2.2.
Detectors
It is often pointed out that a stable SR permits a correct calculation of its intensity, wavelength distribution, and angular distribution. In fact, however, the incident X-ray has to be monitored to ensure accurate measurements and experiments. This is because, even if the electron or positron beam put out by the storage ring is very stable, it cannot prevent the SR intensity from fluctuating because of the thermal instability of the optical elements. Fluctuations in position and direction of the electron or positron beam also contribute to fluctuations in the intensity of the X-rays. This is why the current in the storage ring beam cannot be used to monitor the incident X-ray. Monitoring part of the X-ray flux is not desirable either, since the flux does not necessarily have a uniform wavelength and intensity distribution. The ionization chamber is generally used as a detector for monitoring in the Xray region. When an X-ray irradiates a gas, the gas absorbs the radiation and is ionized. In the ionization chamber method, the electric charge caused by this ionization is measured to obtain the intensity of the incident X-ray. The height of the emitted pulse signals is too small to be measured. Therefore, normally the direct current is measured instead. The current produced in the ionization chamber is linearly related to the number of the incident photons: I = neEe/W
(2-11)
where I is the output current of the chamber (A), n the number of the incident photons (s-l), e the detector efficiency, E the energy of an incident photon (eV), e the electronic charge (1.6 x 10 -19 C) and W the energy to produce an electron-ion pair (eV). Despite the complicated process of gas ionization, W is known to be characteristic of a gas molecule, regardless of the
92
(a) |
1.0
l
(b) !
l
i
i
"2.5 GeV 92.3 mA
!
.0 2.5 GeV 75.9 mA
Zn MAX. CTS. 3628.5
Mn
Zn MAX. CTS. 4764.8
Ar
.
~05
Scattered
'.5
Mn
o~,,4
Ar Ca
.0
0.0 I
i
2
i
i
4
i
i
i
i
a
i
I
I
6 8 l 0 12 Energy / keV
i
.
I
14 16
2
4
6
8
10 12 14
16
Energy / keV
Fig. 2-9. Fluorescence spectra from a Ca, Mn and Zn adsorbed chelate resin with 20 ppm concentrations of each element, using the reflection mirror and A1 absorber combination (a), and the reflection / transmission mirror combination (b). Counting time, 100 s; MAX. CTS., maximum counts. From Ref. [20], reprinted by permission of Elsevier Science Publishers B. V., Amsterdam. kind of ionizing particle and the operating conditions of the ionization chamber. When n = 109 photons s-1, e = 0.2, E = 8 000 eV, and W = 30 eV, I will be 8.5 x 10-9 A. There are many kinds of ionization chambers. The structure of the one used at the Photon Factory is shown in Fig. 2-10. The parallel-plate type is employed because (1)the electric field between the
H.V I'1'
Electrode
X-Rays
Electrode
Amplifier
Fig. 2-10. Schematic diagram of a typical ionization chamber (parallel-plate type) used at the Photon Factory.
93 electrodes is uniform, (2) it is easy to set up, and (3) it is easy to use. A voltage is applied across the electrodes to keep ions and electrons apart. The ionization chamber has a wide dynamic range of more than 5 decades, high linearity, and is free of dead-time losses. The Xray absorption can be controlled by changing the pressure and kind of the gas contained in the ionization chamber, or the electrode length therein. The X-ray absorption by nitrogen and argon contained in a chamber with a 17 cm electrode is shown in Fig. 2-11. It is desirable to use 5-20% of the incident X-ray for intensity measurements, transmitting the rest to the sample. Another method of monitoring incident X-rays uses a scintillation counter to measure the radiation scattered by a foil made of a low atomic weight element such as aluminum. The method of detecting X-ray fluorescence is basically the same as that of laboratory X-ray devices. When using SR, the energy dispersive method is often employed because it is capable of rapid, simultaneous multi-element analysis. A solid state detector is saturated at a few thousand counts s-1. When XRF is used for trace element analysis, scattered radiation makes up the greatest percentage of X-rays reaching the detector. Fortunately, however, since SR is linearly polarized, scattered radiation can be dramatically reduced by positioning the detector perpendicular to the beam. The calculations for this are discussed in Section 2.3.2. A high counting rate system is also considered. Furthermore, multi-element detectors have been developed. They incorporate up to a few tens of detectors. Reduction of X-rays to individual detectors eliminates dead-time losses due to saturation, and the large solid angle which can be covered by the multiple detectors increases the overall detection efficiency. A large solid angle is of great importance when measuring by the wavelength dispersive method. 2.2.3.
Beamlines
The first thing to do when fabricating a beamline for XRF is to determine the energy region for measurement. Different beamlines are required for analysis or measurement of different energy regions: i.e., the analysis of light elements in the soft X-ray region of less than 1-2 keV, the measurement of X-rays in the region of 2-30 keV, or of the K-series X-rays of heavy elements. For measurement of heavy elements, the beamline must be fashioned in such a manner that it can accommodate a large scale SR facility or a wiggler. Strict shielding of radiation is also required. When using soft X-rays, the beamline must be made under high vacuum, since beryllium windows cannot be used in this case. The absorption of X-rays by beryllium is shown in Fig. 2-12. In a beamline for measuring the X-ray fluorescence in a 230 keV region, 2 to 4 beryllium windows are normally used. The total thickness is 300 ~tm1 mm. The design of the optical system depends on whether a microbeam is used or not. It must also be decided whether a crystal or a wide band-pass monochromator should be used. If desired, the optical system can be designed to be compatible with both types of monochromator. Different equipment in the same beamline should be substituted in the hutch for the energy dispersive and the wavelength dispersive methods. As an example, Beamline 4A installed at the Photon Factory is shown in Fig. 2-13. (In this chapter we will not discuss the focusing optical mirror system which is inserted between the
94
100-
1.080-
N2
6
0.8-
owit
'~
60-
.-,06-
o
9 1,,,~
~4oo
<
~0.4-
20
0.2-
0
' 0
I 10
'
I 20
'
I 30
Incident X-ray energy /keV Fig. 2-11. Absorption curves of X-rays passing through 17 cm lengths
0.0-
,
0
I
4
'
I
8
'
I
'
12
Incident X-ray energy /keV Fig. 2-12. The calculated transmissivities of X-rays through a 300 ~tm
of gases (N2 and air).
thick beryllium window.
monochromator and the hutch for analysis using a microbeam.) It is about 9 m from the source of light to the outer wall of the ring. A double-crystal monochromator is positioned about 1 m from the wall. Three beryllium windows are used, with a total thickness of 650 ~tm. To maintain the outgoing beam at a constant height, regardless of its energy, this monochromator is equipped with a mechanism to translate the first crystal parallel to the incoming beam whilst allowing it to be rotated [28]. It also can put out a strong beam by focusing rays in the horizontal direction using a curved second crystal. When Si(111) is used
Branch beam shutter i
R I1
Double-crystal monochromator Crystal Slit
P,
Fig. 2-13. Schematic drawing of Beamline 4A at the Photon Factory.
Hutch
95 as the monochromator crystal, the energy width for an 8 keV X-ray is 3.3 eV. The photon flux in this energy band is of the order of 108 photons s-1 for a beam 1 mm square. The equipment in the hutch must be remote-controlled, since it is impossible to enter during use of the X-ray beam. The equipment used for the energy dispersive method and built into a hutch includes a slit, a sample chamber, an incident X-ray monitor, and a solid state detector. A small TV camera is very useful for monitoring the position of the sample. A laser beam is sometimes used to help in positioning the equipment and the sample. The sample chamber should be able to hold a vacuum and be equipped with a remote control device for inserting several samples. The solid state detector must be positioned perpendicular to the beam. This minimizes scattering, since the SR is linearly polarized in the orbital plane of the storage ring [29]. Figure 2-14 shows the vertical divergence distribution of the SR (11 keV) from one electron [ 11]. The abscissa shows the angle from the electron orbit, and the vertical distance is measured at a position 14 m from the light source. The vertical polarization component vanishes on the orbital plane of the electron beam. This means that most X-rays are horizontally polarized when the beamline is horizontal to the orbital plane of the electron. Therefore, if the sample and the detector are positioned within 1 mm of the beam center, the
1.0
el
0.5
% 0.0
0
\
.1 .2 Angle / mrad
Displacement / mm Fig. 2-14. Calculated vertical divergence distribution of the SR from one electron at E = 11 keV. Parallel and perpendicular polarization components and their sums are shown by the solid, broken and dotted lines, respectively. The abscissa indicates the angle from the electron orbit and also the vertical displacement at 14 m from the source point. From Ref. [11], reprinted by permission of Elsevier Science Publishers B. V., Amsterdam.
96 vertical polarization component is very small. Iida et al. [11] measured the intensity distribution, in the vertical direction, of incident and scattered X-rays and X-ray fluorescence, as well as the distribution of the ratio of the intensity, in this direction, of the X-ray fluorescence and the scattered X-ray. The results are shown in Fig. 2-15. The monochromator has a single Si crystal, lapped with #600 SiC. The solid state detector is positioned perpendicular to the beam. While the intensity of X-ray fluorescence is proportional to the intensity of the incident X-ray, that of the scattered radiation is at a minimum at the beam center. Non-monochromatic excitation has the best detection limit in absolute (weight) amount because such excitation uses a very strong incident X-ray. However, when the scattering is strong or when the sample contains matrix elements with high atomic numbers, the emitted Xray fluorescence saturates the detector, resulting in a high detection limit in relative concentration. Monochromatic excitation coupled with a crystal monochromator allows the incident X-ray energy to be adjusted to the most efficient value for the element to be measured. For this reason, this method provides a lower detection limit in relative concentration. However, it yields a higher detection limit in absolute amount than excitation by a nonmonochromatic beam, because the total X-ray intensity for excitation is lower. The wide
[
I
I
I
I
I
I
.-.mr 30 I--
"~,
I
-1.0 ,d
\
-0.5~
20
e,
10-
I ' 0 1 2 Vertical d i s p l a c e m e n t / m m
,.$
0
Fig. 2-15. The variation of the incident ( 9 ), and scattered ( i ) , Zn K(t fluorescence (A) radiations, and the ratio of the Zn signal to the scattered radiation (o) as a function of the vertical displacement. The sample is a 20 ppm Zn adsorbed chelate resin. The monochromator used is silicon lapped with SiC. The excitation energy was 11 keV. Null vertical displacement indicates the center of the beam. The vertical resolution was 250gm. l~rom Ref. [11], reprinted by permission of Elsevier Science Publishers B. V., Amsterdam.
97 band-pass monochromator has features which lie somewhere between these two. The most efficient method should be chosen according to the composition of the sample and the elements to be measured. SR has a very small angular divergence in the vertical direction (approx. 0.1 mrad). Therefore, the maximum size of the beam's vertical component arriving at the sample is limited to a few millimeters. When a large beam is required, and it is necessary to enlarge it, the asymmetric reflection (asymmetric Bragg magnifier) of a crystal can be used (see Chapter 8, Section 8.3.5). Use of the asymmetric reflection easily permits the diffracted beam to have a cross section about 10 times larger than that of the original beam. From Liouville's theorem, a geometry which increases the cross section of a beam also improves its collimation. On the other hand, the beam's cross section can be reduced by using this method in the opposite way. Unlike the method using a slit, it maintains the intensity of the beam. Other components of a beamline include a vacuum system (vacuum pumps, vacuum indicators, valves), shutters, and an interlock system. In addition, a beam position monitor is useful for experiments using a microbeam. The control system for X-ray fluorescence detection is basically the same as the one used in the laboratory. 2.3. XRF USING M O N O C H R O M A T I C EXCITATION 2.3.1.
Characteristics
Despite its rapid, nondestructive analysis, XRF is not quite sensitive enough. To improve detection limits, the incident intensity must be increased to increase the intensity of X-ray fluorescence, and the background must be reduced. SR meets these requirements with its wide continuous spectrum and high intensity. The intensity of SR is 100-10 000 times that of conventional X-ray sources. It also is linearly polarized. This property can be Used to 'reduce the background: a detector positioned perpendicular to the incident X-ray beam reduces scattering. A monochromatic X-ray of a specified energy can also be separated from SR using a monochromator. The characteristics of monochromatic excitation are as follows: (1) The excitation energy can be designated to be slightly above the absorption edge of a specified element. This increases the sensitivity of measurement for that element. The absence of the unnecessary spectrum reduces the background and improves the S/B ratio. (2) Overlapping peaks can be experimentally separated. Choosing an excitation energy to be between the absorption edges of the elements involved can eliminate the effect of an extraneous element. (3) The strong signals emitted by predominant matrix elements quickly saturates the solid state detector, affecting the detection of trace elements. Setting the excitation energy below the absorption edge of the predominant matrix elements reduces the strong signals. (4) The known spectral characteristics of the exciting X-ray facilitate fundamental calculations and thus help improve the precision.
98 The difference between continuum- and monochromatic excitation is indicated in Fig. 2-16 [ 11]. Twenty ppm of calcium, manganese, and zinc each adsorbed in a chelated resin are used as the sample. Continuum SR, with and without an A1 absorber, and SR monochromatized by a crystal monochromator, are used as the excitation sources. The surface of the silicon crystal used in the monochromator is lapped using SiC. The results clearly show that the use of SR greatly improves the S/B ratio. Of particular note is that much less scattering occurs in the case of monochromatic excitation compared with other methods. Figure 2-17 shows the changes in the spectrum of X-ray fluorescence in relation to changes in the excitation energy using a NIST SRM 612 Glass Wafer (major constituents: SiO2 (72%), CaO (12%), Na20 (14%), and A1203 (2%), plus about 50 lxg g-1 of many other elements) [30]. Figure 2-17a is for excitation by a 19.5 keV X-ray, and Fig.2-17b for a 10.5 keV X-ray. It shows that the latter improves the (a) . . . . . .
":'. 1.0 .1,,,4
=
2.5GeV 105.0mA MAX.CTS_ Ar 1437 ,,.,) C ~ I'Mn ~ Z n
.0
0.5
).5
Ar
0.0 2
4 ' 6 ' 8;' 1'0"1~2' 14 Energy / keV
1.0
05
- MAX.CTS. "_ 2436.0 - ir
4
6
8
10
12
14
Energy / keV
l ZB fi il
~ "~ il Nil ]
0.0 2
4
6
8
10
12
Energy / keV Fig. 2-16. Comparison of fluorescence spectra from a 20 ppm metal adsorbed chelate resin excited using: (a) continuum SR; (b) continuum SR with a 280 ~tm thick A1 absorber; and (c) monochromatized SR. Counting time is 100 s for (a) and (b), and 200 s for (c). From Ref. [11], reprinted by permission of Elsevier Science Publishers B. V., Amsterdam.
99 (a) .
1.0
.
-
.
t
.
Ca
.
.
.
E~," 19.5 keV t: 500s
SRM (Glass612fer)
I[
Max.CTS.: 6323
![
Sr Zr Nb
scatl
d
0.5 -
l] [[
Si
Rare earth and transition
~Y~ Rbl[ /1 I] l[
Al[I
]l]ll
! "11
0 0
5
10 Energy/keV
15
20
(b) 1.0
Ca
-
_ _
E~," 10.5 keV t" 500 s
SRM 612 (Glass Wafer) Max.CTS.: 31251
m
0.5
Scat.
-
m
[" [--
.
Rare earth and 11 transition /I" [Vi elements I!
o 0
5
10 Energy/keV
15
20
Fig. 2-17. The change in SRXRF spectra from a NIST Standard Reference Material 612 caused by the excitation energy [E?'is 19.5 keV for (a), 10.5 keV for (b)]: t = counting time; Scat. = scattered radiation. From Ref. [30], reprinted by permission of Kodansha, Tokyo.
S/B ratio significantly for elements with absorption edges in the 5-10 keV range, such as rare earth and transition elements. This is because the excitation efficiencies of these elements are increased, and because unnecessary spectrum components which contribute to the background are eliminated.
100 ooo
(a)
800
d
(b)
3000 -
E r " 16.00 keV t :200s
As + Pb
Scat.
Ey" 12.50 keV
2500 -
t
9200 s
AsKs
Scat.
2000 -
600
1500-
. v,.=l r~
400-t
II / PbL~I
AI
A1
1000500 -
0
i
i
i
i
I
0
5
10
15
20
E n e r g y / keV
.it...
O0
"~ 5
_
i
10
i
15
i 20
Energy / keV
Fig. 2-18. Advantage of selective excitation. SRXRF spectra of a sample containing Pb and As caused by excitation energy above the Pb Lin and As K edges (a), and between them (b). Max. counts: 1 296 (a) and 2 330 (b). From Ref. [31], reprinted by permission of Elsevier Science Publishers B. V., Amsterdam.
Now we will discuss the effect of selective excitation. Figure 2-18 shows the spectra of an A1203 sample containing a 0.1% mixture of As and Pb in a 1:1 molar ratio [31]. The excitation energy is 16.0 keV for Fig. 2-18a, and 12.5 keV for Fig. 2-18b. The K absorption edge of As is at 11.862 keV and the LIII absorption edge of Pb is at 13.038 keV: therefore, if conditions (a) are used the peaks of As Ka (10.53 keV) and Pb La (10.55 keV) overlap. If conditions (b) are used, the excitation energy is set between As K and Pb LIII: hence, no peak appears for Pb La, and the peak of As makes its quantitative analysis possible. The detection limit and accurate quantitative analysis are the most important factors of XRF used for elemental composition analysis (percent) and trace element analysis (ppm, ppb). We will consider these factors below. 2.3.2. Detection limit
Determinants of the detection limit (a) Excitation efficiency and fluorescence yield. When an X-ray irradiates a substance, the X-ray photons excite the atoms of the substance, emitting inner-shell electrons as photoelectrons and creating orbital vacancies. This process of excitation is due to the photoelectric effect. Therefore, the probability of photoelectric absorption (photoelectric effect) is linearly related to the excitation efficiency. This depends on the element and the energy of the incident X-ray. The contribution of photoelectric absorption to total absorption for incident X-ray energies of 10, 30, and 100 keV is shown in Fig. 2-19, which is plotted
101 100
..-
...
80
J ~-60/I /I
I
/I'
:""
/
x : photoelectric absorption. Oc:Comptonscatte ng.
/
f
./" ......
OT:Th~176 scatte~ng" ~: : electron-pair creation.
20
0~
I
20
I
I
40 60 Atomic n u m b e r
I
80
Fig. 2-19. Ratio of total absorption (~t) to photoelectric absorption (x) [32].
using the tables reported by McMaster et al. [32]. This shows that, except for the 100 keV radiation, the total absorption coefficient can be regarded as the excitation efficiency, since most of the absorption coefficient is due to photoelectric absorption. The absorption coefficient's dependence on incident X-ray energy is shown in Fig. 2-20, based on Sasaki's table [33]. The absorption coefficient (excitation efficiency) is highest when the energy is slightly above the absorption edge. The vacancies caused by this excitation are filled by electrons from the outer shells. As shown in Fig. 2-21, the process of filling the vacancies is accompanied by a release of energy, which occurs by one of the following two processes. In the first case, an electron in an outer shell drops to the energy level of a vacancy, emitting an X-ray photon with energy equal to the difference between the two energy levels involved. In the other process, the released energy is transferred to another electron to eject an Auger electron. The ratio of the number of emitted X-ray photons to the number of primary vacancies created is called the fluorescence yield, which depends on the element and the absorption edge. Figure 2-22, plotted using the values tabulated by Bambynek et al. [34], shows the fluorescence yield for the K-series X-ray fluorescence. Elements with small atomic numbers have a low probability of releasing X-ray fluorescence, most of them emitting Auger electrons. In the detection of X-ray fluorescence using the K absorption edge, Ka and K~ lines are emitted. As shown in Table 2-3, the intensity ratios of the Ka to KI~ lines are also element dependent.
102 500 -
7 r
I " absorption edge
~400O C
o300-
91=,,4
O
0
ZrK
o200 0
~lO0r~
d~
<
0-
I
I
0
5
Fe K I
10
Pt LIII
I_ I
15
~ I
20
I
25
Energy / keV Fig. 2-20. Energy dependence of absorption coefficients for Fe, Pt and Zr, plotted as a function of incident X-ray energy [33].
X-Ray fluorescence emission
Auger electron emission (the Auger effect) ctron
2P3/2 2Pl/2 ..... [ O-'---~
2S Ko~1 X-r
ls
i
--O
Fig. 2-21. Energy release mechanism after the X-ray absorption process.
103 1.00.8
Table 2-3 Relative Ko~ line intensities in the K-series
""
Atomic number
9
o0.6-
~
0
.
4
20 24 28 32 36 40 44 48 52 56
-
02
0.0-
I 0
'
I 40
'
I 80
Relative intensity (Kct)
'
0.887 0.883 0.881 0.871 0.856 0.844 0.833 0.824 0.816 0.809
Atomic number Fig. 2-22. Fluorescence yields for K absorption edges [34].
The product of the excitation efficiency, the fluorescence yield, and the ratios of the Ka line represents the intensity of the Ka X-ray fluorescence detected. Calculated values for this product (the fluorescence intensity) are indicated in Fig. 2-23. Since the most suitable excitation energy for the desired measurement can be selected, SRXRF using monochromatic excitation improves the detection limit.
lOO~ 6"
4: . 2-
0 0 0 0r ~ @
_= 1~ ........ 15
I
I
I
I
I
20
25
30
35
40
Atomic number Fig. 2-23. Variation in fluorescence intensity (mass absorption coefficient x fluorescence yield x relative intensity of Kt~ line) with atomic number at 15 keV of the incident X-ray energy. The 15 keV incident X-ray energy is smaller than the binding energies of the electrons in the K shells for atomic numbers above 37.
104
(b) Scattered X-ray and bremsstrahlung. The detection limit is affected to a great extent by the background. The X-ray scattered by the sample contributes greatly to the background. Environmental and biological samples, both the subjects of trace element analysis, are composed mainly of light elements such as carbon and oxygen, and contain very small amount of metals. Much of the radiation intensity from these samples, measured by the energy dispersive method, is the result of scattering caused by light elements. Baryshev et al. [2] and Hanson [35] made thorough studies of scattering by polarized X-rays. In the following we will describe the theory explaining why polarized X-rays reduce the background. There are two classifications of X-ray scattering: coherent scattering and incoherent scattering. The differential cross section of coherent scattering (Thomson scattering) for a free electron is as follows: dtYT = dO
rg.le. ol =
(2-12)
where e 0 is the polarization vector (in the direction of the electric field) of the incident X-ray, e* is the conjugate complex polarization vector of the scattered radiation, and r0 is the classical electron radius. For linear polarization: (d~)
= rZ.sin2O
(2-13)
pol
where ~ is the angle between the direction of polarization of the incident X-rays and the direction of observation. Therefore, the differential cross section of Thomson scattering becomes zero when the direction of observation coincides with the direction of polarization of the incident X-rays. Since the SR obtained using a bending magnet is polarized in the horizontal direction, Thomson scattering can be minimized by positioning the detector perpendicular to the SR beam. On the other hand, for a fully unpolarized X-ray: (d-~o)
= 1 .r~.(l+cos20) unpol 2
(2-14)
where 0 is the angle between the direction of propagation and the direction of observation. Therefore, observation from the perpendicular direction minimizes Thomson scattering also. However, in this case most of the scattering results from electrons bound to atoms rather than from free electrons. This form of Thomson scattering is called Rayleigh scattering. The differential cross section of this scattering in a plane horizontal to a detector is as follows:
105 dO'R)
-d-~/pol
= r~-{f(q, Z)}2.(1 - sin20)
(2-15)
wherefis the atomic scattering factor, q = sin (0/2)/~ is the momentum transfer, and Z is the atomic number. Observation from the perpendicular direction minimizes the scattering. Incoherent scattering is called Compton scattering. The differential cross section of Compton scattering for a polarized X-ray is expressed by Klein-Nishina's formula: dO'c)
= ur 2 .(KK0)2 " (__~+ K
4cos2~ - 2)
(2-16)
where ~ is the angle between incident and scattered photons; K0 = 2rr/2o and K = 2n/~, are the wave numbers of the incident and scattered X-rays, respectively. Using the Compton formula,
K/Ko =
[1 +
a .(1 -
cos0 )]-1
(2-17)
a = Eo/mc 2
(2-18)
where E0 is the incident photon energy. When 0 = ~ = 90 ~ the intensity of the scattered radiation is minimized. For an unpolarized X-ray,
d--~/unpol - -~_
~-0 -- sin20 )
(2-19)
The dependence of the intensity of Compton scattering upon the polarization is indicated in Fig. 2-24 [ 10]. We have previously confirmed that the linear polarization of SR greatly reduces the scattering. However, even though SR is perfectly linearly polarized on an electron's or positron's orbital plane, it contains another polarization component away from the orbital plane. The beam and the window of the detector both have finite size : therefore, Compton scattering cannot be completely eliminated (see Fig. 2-15), although scattering can be minimized by positioning the detector perpendicular to the beam. Compton scattering can be reduced by increasing the distance between the sample and the detector: however, this reduces the solid angle and weakens the signals reaching the detector. Another cause of a background is bremsstrahlung. In the energy region normally used for XRF the sample absorbs most X-rays through the photoelectric effect. The photoelectrons created in this process decay in the sample while emitting continuous X-rays, which also contribute to the background. The energy of a photoelectron ejected from the K shell of an atom by the photoelectric effect is as follows:
106
1.0xl0
-3 0~~89"
// /
0.5•
-3
/
/
/
J
/
/
0=~=90"
5 10 15 E n e r g y / keV
20
Fig. 2-24. Ratio of theoretical difference cross sections for Compton scattering of polarized to unpolarized radiation under scattering angles 0 = ~ = 90 ~ and 0 = ~ = 89 ~ From Ref. [10], reprinted by permission of Amsterdam.
Elsevier Science Publishers
Ex = E I - EK(Z)
B.V.,
(2-20)
where EI is the energy of the incident X-ray and EK(Z) is the energy (work function) required to excite an electron from the K shell up to the continuous energy level. For a sample with a matrix of an element of atomic number Z, if Nx photons with energy between Ex and Ex + dEx are emitted [36], Nx = 2.5•
10-6.Z.Ni.cYip.t ~ X "(Ez9
Ex)
(2-21)
where NI is the number of incident photons, t is the thickness of the sample (g cm-2), and crw is the photoelectric absorption cross section of the incident X-ray. The distribution of the bremsstrahlung for a 10 keV X-ray with a carbon matrix is shown in Fig. 2-25, which is calculated from Eq. (2-21).
107 1.0 0.8 .~..~ r,r
~
I=
"
"~0.4 r~
o.2 0.0 0
2
4
6
Energy / keV
8
10
Fig. 2-25. Calculated bremsstrahlung intensity versus its energy in a carbon matrix using a 10 keV incident X-ray energy [36].
Calculated values of the detection limit It is important in SRXRF to estimate the detection limit correctly. The detection limit is commonly defined as the minimum concentration or absolute amount, of an element in a sample, which can be detected with a confidence level of above 90%. Gordon [12] calculated this theoretically. His calculations are based on the parameters of the NSLS, a 2.5 GeV ring. He considered two kinds of samples. One contained a carbon matrix, as an example of a bioorganic substance, and the other a mineral sample containing nine kinds of elements. Both samples were assumed to be attached to a carbon-matrix substrate of 1 mg cm -2 (e.g., a Kapton film). Both the energy dispersive and wavelength dispersive methods were considered. In the energy dispersive method a Si(Li) detector with a 30 mm 2 crystal area was assumed to be used, and in the wavelength dispersive method a multi-channel crystal spectrometer system. To calculate the detection limit, Gordon used the following formula: CD(ppb) = N x 109 / (I (7 T G A)
(2-22)
where: N is the detectable signal, I is the integrated beam intensity, cr is the cross section in cm 2 g-l, T is the sample thickness in g c m -2, G is the fraction of the solid angle subtended by the detector, and A is the self-absorption correction.
108 From the criteria of Currie [37], N is equal to 3.29(nNbg)a/2 (Nbg is the background beneath the signal; 1 -< n --<2). In this calculation, Gordon took the size of the detector and the distance from the sample into consideration. He also calculated the detection limit for the photon fluxes, further increased at higher energies by using a 6-pole wiggler. Figure 2-26 shows Gordon's calculations of the determination (quantitation) limits for the carbon matrix sample measured by the energy dispersive method. The determination limit (CQ) was obtained from the detection limit (CD) using CQ/CD=I0f/3.29, wherefis a correction coefficient for the amount of background. The distance from the sample to the detector is 10 cm. The value (CQ) is for a one-minute measurement. The detection limit (Co) is from a third to a fifth of this value. 3
10
i
i
i
i
Ktx
9
l
i
LO~
~- ld 30 keV
t\
0 eV,
"~ 101
,
15 keV v
~o 10 0 .~ "~
0keV
!~~,~~3 keV \ -11 10 10
N \20keV
10keY 15 keY
1
I
I
I
I
20
30
40
50
t
60
t
70
80
Atomic number Fig. 2-26. Determination limits for 2 mg cm -2 carbon matrix sample using a Si (Li) detector and a one-minute measurement. Sensitivity curves are shown for five excitation energies ranging from 5 to 30 keV. The sensitivity expressed is a determination limit (concentration measurements) with a 10% standard deviation attributed to counting statistical errors. The detection limits are a factor of 3 to 5 lower than the determination limits shown here. From Ref. [12], reprinted by permission of Elsevier Science Publishers B. V., Amsterdam.
109 Experimental values of the detection limit
Starting in the 1980s, XRF experiments were conducted at various SR facilities, and the detection limit studied experimentally. Gilfrich et al. [7] made their experiments at SPEAR of SSRL. The beamline they used was not equipped with a mirror or a monochromator. It produced a non-monochromatic beam with 1 mrad divergence in the horizontal direction. The X-ray energy ranged from 2 to 60 keV. The experiments were carried out using the energy dispersive method and the wavelength dispersive method employing a fiat crystal. A vacuum deposit of metal on a Mylar film, and a solution dropped on a Millipore filter were used as samples. The X-rays were obtained using a 3 GeV ring with a storage ring beam current of 40-80 mA. The energy dispersive method allowed a 100-s measurement using a 2.4 • 10-4cm 2 incident beam size. The wavelength dispersive method employed a 0.4 cm 2 beam (100-s measurements). It used LiF and PET as analyzer crystals and a proportional counter as the detector. All the samples measured were very thin. The detection limit was assumed to be three times the standard deviation of the background (Fig. 2-27). The energy dispersive method gave a detection limit of the order of 10-12 g in absolute amount.
::i. ..
100
!
|
!
(a) Energy -1 dispersion 10
o ,~ o o
-2 10
m
10.3
i
/ /)/
K-lines I
20
I
I
40
i
i
_
10~ (b) . . . . Wavelength/ 10 -1 - d i s p e r s i i ~
_
10 -2
l
- - - - On filter On Mylar
-
L-lines I
I
60
Atomic number
I
I
80
10-3
~K .i
20
--~'--O'n filter On Mylar j ~ A 1
lines ~ n e s n
n
40
n
i
60
.
,
80
Atomic number
Fig. 2-27. Detection limits as a function of atomic number as measured by (a) energy dispersion and (b) wavelength dispersion. From Ref. [7], reprinted by permission of the American Chemical Society, Washington, D. C.
110 Also at SPEAR, Giauque et al. [38] studied the dependence of the detection limit on the excitation energy, taking advantage of the fact that monochromatic excitation can be set to any level. They reported that, under optimum conditions, they obtained a detection limit of 20 ppb for many elements. Bos et al. [ 10] compared SRXRF detection limits with those of conventional XRF and an analytical technique using proton excitation (proton induced X-ray emission spectrometry: PIXE). The conventional X-ray source used a Mo anode and Zr, Mo, and Ti filters. This source produced a 17.5 keV X-ray at 26 kV and 12 mA. For proton excitation, they used 3 MeV protons produced using a cyclotron. In the SR experiment, they used X-rays of 16.5 keV and 9.1 keV in beamline 7 at the SRS of Daresbury incorporating a pyrolytic graphite crystal monochromator. Measurements were conducted using NIST Orchard Leaves (SRM 1571) and Human Hair (IAEA-HH1). The detection limits obtained for each sample are shown in Fig. 2-28. The calculation was based on the criteria of Currie [37]. Measurements were made for 1 000 s at 2 GeV with a beam current of 200 mA. Compared with conventional XRF, SRXRF substantially improved the detection limits. It was found that SRXRF is nearly equivalent to PIXE up to Z = 30, and has an advantage for heavier elements. Specifically, the detection limit can be improved further because this method allows the user to set the excitation energy to any desired value (see the results using 9.1 keV in Fig. 2-28). They also studied how the V K~ X-ray arising from the 9.1 keV X-ray changed, depending on the sample thickness (Fig. 2-29). The results indicated that the detection limit is very sensitive to sample thickness when the sample is thin. Hanson et al. [8] performed their experiments at CHESS. This ring has an electron energy of 5 GeV and generates strong SR in the hard X-ray region. They used a channel-cut silicon monochromator, and chose the same sample that Bos et al. [10] used (SRM 1571). X-Rays of 13.0, 16.25 and 24.9 keV were used as incident beams. The detection limits were calculated using the criteria of Currie [37] (Fig. 2-30). The present authors performed a similar experiment at the Photon Factory. Using a 2.5 GeV ring with a storage ring beam current of about 200 mA, we measured several types of NIST standard reference materials. Powder samples were applied thinly to adhesive tape. The measured spectra of Bituminous Coal (SRM 1632a), Oyster Tissue (SRM 1566), Orchard Leaves (SRM 1571), Citrus Leaves (SRM 1572), and Aluminum-Silicon Alloy (SRM 87a) are shown in Fig. 2-31. A Si (111) double-crystal monochromator was used, with a 15 keV exciting X-ray. The X-ray fluorescence measuring device was developed by Iida et al., and the measurement time was 500 s. The sample and the Si(Li) detector were 45 mm apart. The size of the beam's cross section is shown in the figure. The cross section was changed so that the detector would not be saturated under the same geometrical conditions. The measurement was made under vacuum. The detection limits calculated based on the criteria of Currie [37] are given in Table 2-4 and in Fig. 2-32. Organic and alloy samples were found to have quite different detection limits. This is due to the different matrices of the samples. Even among organic samples, different compositions give different detection limits. Measurements of Citrus Leaves and of Aluminum-Silicon Alloy using an X-ray tube are shown in Fig. 2-33.
111 3-
10
7
e~o
(a)
102
o 1
10
l
o
10
10
0
7
e~
SRXRF (E?= 9.1
I
-1
3
102
-
I
20
10
10
i [IET = 16.5 keV)
,[
30
40
(b)
-SRXRF
(ET = 16.5
O
o~
O o
~
10
10
0 -
-1
10
PIXE
-
SE,RXRF1 k ,V " ~ ~ ( T = 9.1 keV) ~
.,
, -. _~
I
!I
i
I
20
I ~"~-
30
,
40
A t o m i c number Fig. 2-28. Detection limits of several elements in, (a) NIST Orchard Leaves, and (b) IAEA Human Hair, for different excitation modes. From Ref. [10], reprinted by permission of Elsevier Science Publishers B. V., Amsterdam.
112
A
-
3000 ~
/
~
X
X
2 GeV, 130 mA
/
o 2000
-0.03
1000 s
/
E~,= 9.1 keV -
1000 I - d
1,7 1,1
0
~:It:x_____
.
.
.
.
.
.
b) I
I
10
i
20
-
30
Thickness (mg c m
i
40
0.04
I
Z
0.02 k~ 0.01
"~
50
-2)
Fig. 2-29. Number of V K~ counts of samples with well defined thickness. Curve b shows the root of the number in the background beneath the peak normalized to the .peak content (proportional to the detection limit). From Ref. [10], reprinted by permission of Elsevier Science Publishers B. V., Amsterdam.
3
I0
I
I
I
I
+ 13.0 keV x 16.25 k e V -
It
§
~.~
1
' 24.9 keV
§
9~ lO -
s 9
0
§247
It e
N
m NIlI
9
++
~
x
Ig Ig
...b
lo -~-2
10
I
10
16
I
I
22
28
I
34
40
Atomic number
Fig. 2-30. Detection limits of several elements in NIST Orchard Leaves when fluoresced with 13.0, 16.25 and 24.9 keV X-rays [8]. (Reprinted with permission from A.L. Hanson, H.W. Kraner, K.W. Jones, B.M. Gordon, R.E. Mills and J.R. Chen, "Trace Element Measurements with Synchrotron Radiation", IEEE Trans. Nucl. Sci., NS-30, 1339 (1983). 9 IEEE.)
113 5
10 t 04
(a) SRM 1632a (Bituminous Coal) Fe " Ti+V ^ ,,. K+Ca I /~Fe
1
~"
"
[(vkI ~~.'s ls
o~
o' 1 o
I 0
5
10 10 r~
0
/~h"
.JVi
Br
~
1
1
AY
Zn
,,,I,I /I
B.S.: 2.2 x 3.0 mm I 5
I 10
Energy / keV
I 15
.dJ._
_ll ......
I
20
(b) SRM 1566 (Oyster Tissue) Zn
4 -
K p , SCI ~ ~Ca
1 03 -
Scat. As+Pb
Fe
~ L
Br
10 2 I ..I]
1
10 o
MAX.CTS:7337 B.S.: 2.2x 3.0 mm2
10 -
I 5
0 5
10 -
(c)
1 02 -
K A Ca pS
1
10 -i 0
LLL_JtU,"ll=l...
I
20
Scat.
fV~Ca
Fe
~MA
10 o
I 15
SRM 1571 (Orchard Leaves)
10 1 0z -
I 10
Energy / keV
4
r~
I
.CTS..
10 10
EL~
Scat. /~
I~ll
Mn
Pb+As Fe
Zn
Br
X. B.S.: 1.5X2.9 mm2
/l_d.I 1 ~
I
I
I
I
5
10
15
20
Energy / keV
Fig. 2-31. SRXRF spectra from NIST Standard Reference Materials under vacuum. Excitation energy, 15 keV; counting time, 500 s; MAX. CTS, maximum counts; B.S., beam size; Scat., scattered radiation. Except for SRM 87a, thin samples were attached to adhesive tape: SRM 87a was used as it was.
114
10 10 :"4. r.~
1
5
-
4
SRM 1572 (Citrus Leaves)
B.S.: 0.7 X 0.4 mm
p~ICa
-
As+Pb
S
2
MAX.CTS" 4108
Ca
-
03
10
(d)
-
Mn
Fe
Cu
t~
2
Scat.
13r
Zn
Pb
1
10 10
0
0
I
I
10
15
..........
t
|
I
20
Energy / keV 5
10 "] 1 04 i...i 9 r,r
3
(e) SRM 87a (A1-Si Alloy)
1
10"]
Zn Fe Ni Cu Zn Ga/1 Mn,~e ~ \ ~ ~ 77APb
A1,Si
02
Pb
Scat. ~
9
101 10
o
I"
I
0
I
I
I
5
10
15
.....
~
I
20
Energy / keV Fig. 2-31. SRXRF spectra from NIST Standard Reference Materials under vacuum. continued Table 2-4 Detection limits in ppm (l.tg g - l ) obtained by the measurements of NIST SRMs at the Photon Factory on Beamline 4A Element Ti Cr Mn Fe Co Ni Cu Zn
SRM 1632a SRM 1566 SRM 1571 SRM 1572 SRM 87a (Bituminous Coal) (Oyster Tissue) (Orchard Leaves) (Citrus Leaves) (Al-Si Alloy) 60 m ~ 30 . 9 5 4
m -10 5 .
m 4 5 3 .
~ 4 3
.
2 1 1
.
m ~ 7 6
800 350 260 140
3 3 2
70 60 60
Measuring conditions are described in Fig. 2-31. Detection limits are calculated on the basis of the Currie criteria [37].
115
1000 _
[] Bituminous 9 Coal zx Oyster 9 Tissue 9 "Orchard Leaves O "Citrus Leaves x 9AI-Si Alloy
~......,~
o
~
~
~:~ 1 0 0 ..r
@
10-
0.1
I
20
I
22
I
24
!
26
I
28
I
30
I
32
Atomic number Fig. 2-32. Detection limits for NIST SRMs, derived from the spectra shown in Fig. 2-31 on the basis of Currie's criteria [37].
The measurements were made under vacuum, using a Mo anode of 40 kV and 20 mA, and a Mo secondary target. Measurement times were 1 000 s for Citrus Leaves and 500 s for the Alloy. Compared with measurements employing SRXRF, there is a clear difference in the background level. The detection limits of Orchard Leaves obtained from the experiments of Hanson et al. [8], Bos et al. [10] and ourselves are listed in Table 2-5. Each used a different ring, optical system, excitation energy, and sample thickness; yet the results all agree within 1 ppm. Iida et al. [ 11, 20], at the Photon Factory, studied the variation in the detection limits with the excitation mode. The samples used were 0-100 ppm of Zn, Mn, and Ca, adsorbed in a chelate resin, attached to adhesive tape. X-Ray fluorescence was measured by the energy dispersive method to allow calculation of the detection limits, using a white (nonmonochromatic) beam; an aluminum absorber only; a single-crystal monochromator using a silicon crystal surface lapped with SiC; a wide band-pass monochromator using a total reflection mirror and an aluminum absorber; and a wide band-pass monochromator using a total reflection mirror and a transmission mirror. The results are shown in Table 2-6. The strong intensity of white (non-monochromatic) excitation gives a low detection limit, in absolute amount. Using an absorber improves the detection limit for Zn, in relative
116 concentration. This is because scattering in the high energy region is reduced. The crystal monochromator gives the best detection limit in relative concentration since it reduces the background caused by scattering. The wide band-pass monochromator exhibits excellent performance in both relative concentration and absolute amount.
sRM1572
U' [ ~,
, t
(Citrus Leaves)
] 103~-
I , Scat. I A I
t~a Ca
10
'1
TM
.
1
.
.
I
5
. . . .
I
. . . .
10
I
.
.
.
.
15
1
sRMZTa
04
_/ 103~102
I
(Al-Si Alloy)
, Scat. I /~ 'l
Ni b tb
l,Si TiCr~
10
ii
20
1
5
10
15
20
Energy / keV
Energy / keV
Fig. 2-33. Spectra of NIST SRMs by XRF using a conventional secondary target (Mo) energy dispersive system, measured with a Mo tube, 40 kV, 20 mA, 1 000 s. The samples are those used in Fig. 2-32.
Table 2-5 Comparison of detection limits (ppm) of four elements in NIST Orchard Leaves (SRM 1571) by three different workers Element Mn Fe Cu Zn
Hanson et al. [8] a 4 2 0.6 0.5
a CHESS: 5.144 GeV, b SRS" 2 GeV, c PF: 2.5 GeV,
Bos et al. [10] b 2 2 0.6 0.5
16.25 keV, 16.5 keV, 15 keV,
300 s. 1000 s. 500 s.
Authors c 5 3 1 1
117 Table 2-6 Comparison of detection limitsa(DLs) for different excitation modes [11, 20] Excitation mode
DL in relative concentration
Irradiation area/ mm 2
DL in absolute amount Zn/pg
Zn(ppb) Mn(ppb) Ca(ppb) Continuum
550
410
440
3.5 x 10-3
0.13
Continuum 170 with absorber b Crystal 60 monochromator Mirror/absorber b 430
240
750
2.8 x 10- 2
0.34
70
200
1.1
4.7
180
410
1.1 x 10-2
0.33
Reflection/ 100 transmission(soap)
140
470
4.2 x 10-3
0.03
a The definition of the DL was a signal at least three times the square root of the background criterion. The counting time was 100 s. b A1 of 280 mm in thickness was used.
2.3.3.
Calibration
Calibration is performed in basically the same way as when calibrating conventional XRF. However, monochromatic excitation and collimation further facilitate calibration calculations. The results of an experiment on thin samples of several elements, performed by Kntchel et al. [39] using DORIS at HASYLAB are shown in Fig. 2-34. The results show excellent linearity. When using monochromatic excitation, the quantity of an element can be determined from the quantitative analysis value of another element using an X-ray fluorescence cross section.
Table 2-7 Calculated contents of several analyte elements in NIST Orchard Leaves (SRM 1571) using the known content of Fe (300 ppm) from the data shown in Fig. 2-23 Element K Ca Cr Mn Ni Cu Zn
Calculated values
Certified values
0.5% 0.9% 2.6 ppm 46 ppm 0.8 ppm 12 ppm 24 ppm
1.47% 2.1% 2.6 ppm 91 ppm 1.3 ppm 12 ppm 25 ppm
118
An example of determining the concentration of elements, based on the quantitative analysis value for iron, is shown in Table 2-7. SRXRF is also useful when using the fundamental parameters method, since it produces monochromatic incident X-rays and high collimation.
lO3 102
G 101
n
m
10~ 10-1 ,.,
10~ ~
..... 10I ~
1(~2
1~)3
Concentration (Bg g-1) Fig. 2-34. Calibration plots : (A) chromium; (m) arsenic (• 1.50); (0) yttrium; ( , ) molybdenum (• 0.75); (O) cesium (x 0.50): Co is the ratio of the counting rate to the standard. From Ref. [39], reprinted by permission of Elsevier Science Publishers B. V., Amsterdam.
2.3.4. A d v a n t a g e s
and d r a w b a c k s
We would add two more to the many advantages of SRXRF we have already described. First, because of its high sensitivity, this method can be used in the analysis of lower concentrations, using smaller samples, than conventional XRF. The X-ray fluorescence spectrum of a strand of hair is shown in Fig. 2-35. Even a small amount of sample can provide a spectrum with a satisfactory S/B ratio. This suggests that SRXRF is suitable for microbeam analysis. Another advantage is that SRXRF causes far less radiation damage to samples than methods such as PIXE which use charged particles as the excitation source. It is reported that the beam does more than 102 of the damage to blood cells caused by photon excitation [40]. We will now discuss the drawbacks of the present SRXRF technique. The SRXRF analysis of two kinds of aluminum alloys was compared with chemical analysis and glow discharge mass spectrometry of the same samples. We attempted to determine sample B, based on the quantitative analysis value (chemical analysis value) of sample A. The results are shown in Table 2-8 [31]. The detection limits for trace element analysis of metals show that SRXRF has no advantage over the other methods. Even with SR excitation, it is difficult to reduce the
119 spectral background below the lag g-1 level for many industrial materials containing elements with high atomic numbers. Attention should be paid to the effect of diffraction caused by the sample. Sutton et al. [41] discussed the effect of diffraction in continuum radiation. We will describe this same problem due to diffraction that we experienced when using monochromatic excitation [31]. We used the highly pure aluminum employed in VLSIs as a sample. The sample was cut from a block of material, using a superhard steel cutter, while dripping alcohol, and degreased. It is known that this sample contained about 80 ppb of U and Th. A preliminary XRF
(a) 1000
MAX.CTS: 2379 Cu
Scat.
.s..
~o 100
N
A /~
lO i .
1 -! o
I
I
5
.1..... I__
I
10
I
15
20
Energy / keV ,=
(b)
,=
ooo
s
-
MAX.CTS: 1068
.
.I==4 r~
Scat.
Br
Zn
,=
~
100-!..
N
-
Ar Ca
Fe
Br
Cu Zn Pb
lO. ,=
1
-
0
I
5
I
10
I
15
I
20
Energy / keV Fig. 2-35. SRXRF spectra of a strand of human hair (male)" (a), white part; (b), black part. Excitation energy, 16 keV; counting time, 1 000 s.
120 Table 2-8 Analytical results for two aluminum alloys obtained by three different methods Element
Cr Cu Fe Mn Zn Th(ppm) U(ppm)
Chemical analysis a (%)
GDMS b (%)
Sample A
Sample B
Sample A
Sample B
0.059 0.0073 0.11 0.060 0.018
0.0005 0.0009 0.11 0.0030 0.0050
0.062 0.007 0.12 0.068 0.018
<0.001 0.001 0.14 0.003 0.005
0.06 0.38
0.06 0.18
0.07 0.33
0.07 0.15
SRXRF (%) Sample B c
0.01 0.001 0.10 0.002 0.004 Not detected Not detected
a U was analyzed by fluorophotometry, Th by colorimetry and other elements by inductively coupled plasma atomic emission spectrometry. b GDMS is glow discharge mass spectrometry. c SRXRF analysis values of sample B are calculated on the basis of the chemical analysis values of sample A and the SRXRF measurement values of both samples. From Ref. [31], reprinted by permission of Elsevier Science Publishers B. V., Amsterdam.
measurement gave the amounts of U and Th shown in Fig. 2-36a. However, neither element was detected in the final measurement (Fig. 2-36b). Since we thought that the peaks that appeared in the preliminary test were caused by X-rays diffracted by the sample, we changed the incident angle slightly. This changed the spectrum in the 12-15 keV region, confirming that the results obtained in the preliminary test were incorrect. When we measured silicon containing a small amount of U, a diffraction peak appeared near the U Lo~ peak, as shown in Fig. 2-36c. Transition metals were also detected in this sample. These could have contaminated the sample when it was being prepared. After washing the sample with acid, only a small amount of iron was detected, as shown in Fig. 2-36d. Special attention must be paid to these problems, since the peaks from diffraction and contaminants on the sample surface might overlap the data used for analysis. Another problem with SRXRF is the increase in background due to resonant Raman scattering, as pointed out by Jaklevic et al. [42]. We have repeatedly maintained that when monochromatic excitation is employed, the energy of the incident X-ray can be set to any appropriate value to maximize efficiency. In addition, for an element with a large atomic number, the background can be reduced by setting the energy of incident radiation just below the absorption edge of that element. However, Jaklevic et al. [42] suggested that this was not always true. They suggested that setting the excitation energy just below the absorption edge of a matrix material caused resonant Raman scattering, and increased the background. Raman
121
scattering is a continuous spectrum with a cutoff in the high energy region. At SSRL, they studied Cu in GaAs. The X-ray fluorescence was measured with incidences of 9.2, 9.8, 10.0 keV. The absorption edge of Ga is at 10.37 keV. The intensity of resonant Raman scattering, which changes with the sum of Rayleigh and Compton scattering, was found to increase as the
1.0,
"~ ~" (a) ,~d [
,
I
'
I
,
I
,
9
L
9
9
"1
(?)
, E~ 22.70keV
OI i,,
11
(b)
?"~
|
Th Ltx ET. 20.50 keV (9) U L o ~ t:2000s
,
I
,
13 15 Energy / keV . . . . ET: 19.60 keV t : 2000s Fe
=
17
Ey. 22.70 keV
~
t:2,00s
0
~ !f,~#
Fe
.,./v
It
Th Ltx I J IU Lo~.f"-
~k%...TiCr~Fe ^ / ] , Ni
,, ]~,,.[
1~,!
/J t: 200 s
~
5
,
'i~| "1
,
10 15 20 Energy / keV
25
1.(l
~(d
E7:. 19.60 keY t: 1000 s
Th La
I k T Cr/! Fe
,
[ /
I U L a t'1 ~ 0
Fe
,
~ ,~ "~
Ey. 19.60 keV 3
5
t'2000s 7 9
0 11
13
15
2
4
6
8 10 12 14 16 18 20 Energy / keV
Energy / keV Fig. 2-36. SRXRF spectra of high-purity aluminum (a, b and d) and silicon (c). (a) Preliminary; (b) at two different tilt angles using the same sample as in (a); (c) at two different tilt angles of silicon containing a trace amount of U; and (d) after surface cleaning of the sample, following the measurement of (b). From Ref. [31 ], reprinted by permission of Elsevier Science Publishers B. V., Amsterdam.
122 incident energy gets closer to the absorption edge. The ratio of the peak areas of resonant Raman scattering and Rayleigh-Compton scattering was found to be 2.3 at an incidence of 10 keV, and 0.64 at an incidence of 9.2 keV. The detection limit for Cu was 1 ppm at an incidence of 10 keV, and 0.6 ppm at an incidence of 9.2 keV. The same workers compared this GaAs sample with a cellulose matrix sample and found that the background for Cu was 12.9 times larger than for cellulose at 10 keV, and 4 times larger at 9.2 keV. 2.4. E L E M E N T A L ANALYSIS BY TOTAL R E F L E C T I O N XRF 2.4.1. Total reflection method
The detection limit in XRF is improved by a greater signal intensity and lower background. To reduce the background in the energy dispersive method, it is necessary to reduce the scattering caused by the sample and the sample holder, and to keep down bremsstrahlung. We saw in Section 2.3.2. that polarized SR greatly reduces the background. In ordinary XRF, the sample supports should be as thin as possible. Various micrometer-thick materials, such as Mylar films and Kapton films, are used depending on the purpose of the analysis. To further reduce the background, XRF can use the total reflection of X-rays. As will be described in Section 2.4.2. if an X-ray is totally reflected by a surface, it hardly penetrates the substance. Therefore, attaching the sample to an optically fiat surface (reflector) keeps the scattering caused by the sample holder to a minimum, thereby improving the detection limit. Yoneda and Horiuchi [ 15] were the first to demonstrate the effectiveness of this method, using an X-ray tube (W target, 35 kV, 15 mA) and the energy dispersive method. A glancing angle was adjusted to be much lower than the critical angle, to achieve nearly 100% reflectivity. Detection limits of the order of ng were achieved for Cr, Fe, Ni and Zn. They also reported that this method eliminates the matrix effect and facilitates calibration. Experiments employing the total reflection method were made later using an X-ray tube or a rotating anode, and have become very popular, particularly in recent years [ 13, 14]. Analysis in the pg range is now performed routinely. Today, total reflection XRF spectrometers for silicon wafers take advantage of the nondestructive, highly sensitive, simultaneous multielement analysis method. These spectrometers, equipped with an automatic measurement device, are used to measure trace quantities of metal contaminants on the surface of silicon wafers, and make it possible to map them with a detection limit of 108-1010 atoms cm -a. An advantage of these units is that they can be installed "in-line" at semiconductor plants. In the laboratory, total reflection X-ray fluorescence (TXRF) spectrometry is mainly used for highly sensitive detection of impurities in solution, which may be dropped on a reflector and dried, or for the analysis of contaminants on silicon surfaces. However, most problems concerning calibration have yet to be solved. Iida et al. [16] thoroughly investigated the SR method at the Photon Factory. The advantages of using SR for TXRF are as follows: (1) Monochromatic excitation can be used to improve the S/B ratio.
123 (2) The excellent collimation of SR is suitable for total reflection experiments using grazing incident beams applied at small glancing angles. (3) The linearly-polarized beam further reduces the background. (4) Selective excitation improves the detection limit. Thus, SR is the most suitable excitation source for TXRF. 2.4.2. The principles of the total reflection method
As discussed in Section 2.2.1., the refractive index of X-rays is very slightly smaller than unity. Therefore, a highly collimated X-ray beam applied to a fiat surface at an angle smaller than the critical angle is totally reflected. Such a beam penetrates the sample only slightly, and the reflectivity of the X-ray is independent of its polarization. Under these grazing incidence conditions, the reflectivity R can be expressed as follows, using Fresnel's formula [43]: Ar ]2=
R=lAi
01- 02 '12 01+02
(2-23)
I
where Ai and Ar are the electric field amplitudes of the incident and reflected X-rays, respectively, and 01 and 02 are the respective glancing angles, as shown in Fig. 2-37. Generally, 02 is a complex number, described as follows using the complex index of refraction. 02 = ( 0 7 -
2~-
(2-24)
i 2fl2) 1/2 = P2 + i q2
p2 = 1 {[(01 - 262) 2 + 4/~] 1/2 + ( 0 2 _ 262)}
(2-25)
q22 = 1 {[(01 -- 2 ~ ) 2 + 4]322]1/2 - (0 2 - 2~)}
(2-26)
The values $2 and t2 are those indicated in Eqs.(2-2) and (2-3) for the reflector. The amplitude of the transmitted (refracted) X-ray is attenuated in the z direction. The depth at which the X-ray intensity attenuates to 1/e is given by Zp(O1) = 1
2k l.q2
=
~,
.[{ (0 2 _ 262)2+ 4132} 1/2 _ (0 2 _ 262)1-
1/2
(2-27)
2"~
where ~ is the wavelength of the incident X-ray, and kl=2n/A. The results of the calculation of the reflectivity and penetration depth (Zp) for silicon near the critical angle are given in Fig. 2-38. The wavelength of the X-ray is 1.24 A. For glancing angles smaller than the critical angle, the penetration depth of the X-ray is very small (a few tens of A). There is a standing wave on the surface of the sample due to the interference of the incident and reflected X-rays. The intensity of the X-ray on the surface of the sample is given by
124 Medium 1 (n l )
Medium 2 (n 2 ) Z
Fig. 2-37. Schematic drawing of X-rays when incident X-rays from air or vacuum (n1-1) irradiate the surface of a material having refractive index n2.
1.0-
.:-10
0.8-
-10
5
4
"~-06.~ 9
l,io
-10 ~0.4-
3
O
=
-lO
0.2-
2
>o
=,
0.0 0
I
2
I
4
I
6
Glancing angle / mrad
I
8
lO
1
10
Fig. 2-38. The calculated glancing angle dependence of reflectivity and penetration depth for a silicon wafer using an incident X-ray with a 1.24/~ wavelength.
125 M(O1) = 4 O ? / { ( O1 +p2)2 + q2}
(2-28)
Calculation of this value for silicon gives the results shown in Fig. 2-39 9 The product of the intensity on the surface and the depth of penetration of the X-ray gives the intensity distribution of the X-ray fluorescence emitted by the substrate shown in Fig. 2-40. At a glancing angle smaller than the critical angle the intensity of X-ray fluorescence is very close to zero. The scattered X-rays are caused by X-rays that go into the substrate without being reflected. Therefore, the closer the reflectivity is to unity, the less is the scattering 9 The condition of the surface of the substrate affects its reflectivity. A rough surface reduces the reflectivity, and thus increases the intensity of the emitted scattering and X-ray fluorescence. Surface roughness can be described using a group of fiat planes distributed in a Gaussian manner. Taking the standard deviation of the Gaussian distribution as tr, the reflectivity is given by R = Ro-exp[ - (4 11;010"/~)2]
(2-29)
where Ro is the reflectivity for a surface with no roughness. The reflection curve for a (r = 10 /~ is shown in Fig. 2-41 a. The corresponding X-ray fluorescence intensity is shown in Fig. 24lb. The angular dependence of the intensity of X-ray fluorescence when a sample is spread very thinly on a substrate was calculated by Iida et al. [16] and is shown in Fig. 2-42a (tr is hereafter assumed to be zero, unless otherwise noted). The intensity is about twice as large above the critical angle as below it. The intensity of the X-ray fluorescence emitted from the 4 m
1.0--"
=0.8"~0.6-
~2-
9~,.,.,,I
r~ = 0.4-
r~
~_~ 0.2-
00
I
2
I
4
I
6
I
8
G l a n c i n g angle / m r a d
I
10
Fig. 2-39. The calculated X-ray intensities on the silicon surface (perfect surface; tr= 0) as a function of glancing angles, using a 10 keV incident X-ray.
0.0-
0
I
2
I
4
I
6
I
8
I
10
G l a n c i n g angle / m r a d Fig. 2-40. The calculated X-ray fluorescence intensities from a silicon substrate ( a = 0) as a function of glancing angles, using a 10 keV incident X-ray.
126
"-:'.1.00.8-
(b)
, - , 0 8. -
.1,.=1
t~
"-06-
9- 0 . 6 -
0 0
O
0 L)
00.4-
~0.4O
0.2-
r
0.2-
c~
0.0-
I
I
0
2
~I~ -
I
4
I
6
~0.0
I
8
10
!
0
Glancing angle / mrad
J
I
2
I
I
4
I
6
I
8
10
Glancing angle / mrad
Fig. 2-41. The glancing angle dependence of reflection from a silicon surface (a); and the X-ray fluorescence intensities from a silicon substrate (b). Surface roughness, 10/~; incident X-ray energy, 10 keV.
(a) .1,.-I
-
(b) Sample
/
.
o
O
f
/ f J
9i,..,.I
O
f
/
-
/
/
/
.
/
-
!
-
-
I. . . .
~'I
I Reflector )I 1
B
I
!
0
,
I
I I
2
Normalized glancing angle (0/0c)
s
1
~
I
2
3
=
I
4
=
I
,
5
Glancing angle / mrad
Fig. 2-42. (a) Calculated intensities of the signals from the sample (solid line) and from the reflector (broken line) as a function of glancing angles. (b) Angular dependence of the Zn ( 9 ) and Si (o) fluorescence signals, which are from the sample and the reflector, respectively. From Ref. [16], reprinted by permission of the American Chemical Society, Washington, D. C.
127
(a) 1.0
(b)
-2"5MGeA~lgr
:lA50
2.5 GeV 133.9 mA | MAX. CTS. :651 ~Zn
1.0
I Zn
0.5
!i Sca'"
0.5 si
i
II
0.0
0.0 I
2
I
I
4
I
I
6
I
I
I
8
I
10
Energy / keV
I
I
12
2
4
6
8
10
12
Energy / keV
Fig. 2-43. Spectra (a) and (b) obtained at different glancing angles, corresponding to A and B in Fig. 2-42, respectively. From Ref. [16], reprinted by permission of the American Chemical Society, Washington, D. C.
substrate shows an S/B ratio which greatly improves below the critical angle. The results of the experiment by Iida et al. [16] are shown in Fig. 2-42b. A sample of 2 lal of a Zn solution was used, dried on an optically flat reflector. The intensity of X-ray fluorescence for Zn and Si is indicated in Fig. 2-42b. The measured values agree well with the calculated values. When the spectrum was measured near the critical angle shown in Fig. 2-43 [ 16], there was very little scattering below the critical angle. The beam width was a few tens of ILtm.
2.4.3. Liquid sample analysis Figure 2-44a [31 ] shows the spectrum of a solution containing 1 ~tg m1-1 each of V, Fe, Ni, Zn, and Pb dried on a fused quartz reflector, measured by the total reflection method at the Photon Factory. The XRF spectrum of the same solution dried on filter paper is shown in Fig. 2-44b [31 ]. The experiment was conducted using Beamline 4A and equipment developed by Iida et al. The horizontal axis of rotation of the sample holder makes this device suitable for SR experiments. Measurements are made in the air using a 14 keV monochromatic X-ray obtained from a Si(ll 1) double-crystal monochromator. It is evident that using the total reflection method produces very little scattering and improves the S/B ratio. Similar experiments on Ti, Mn, Cu, and Ge were carded out. The detection limits obtained are indicated in Table 2-9 [30]. In the total reflection method, any absorption and secondary excitation in the sample is negligible because the sample is very thin. We applied TXRF to a synthetic sample from 100 ~tl of a solution containing 50 ng g-1 of V, Cr, Mn, Fe, Ni, Cu and Zn, dried on a silicon
128 wafer (Fig. 2-45). The intensity corrected for air absorption, excitation efficiency, relative Ko~ line intensity and fluorescence yield is shown in Fig. 2-46. This shows that a fixed amount of sample gives a constant X-ray fluorescence intensity, within the experimental error. Therefore, we can ignore the matrix effect and secondary excitation.
1.0
~=s" tET' "20014. : lilLl 0s it _a) Total Reflection Zn
~, ~ /,,' ~,."
keVFe
~U.3" k
,!1, il ,
i!t
I 0
"
0
Scat,
i,~!t ~'-" '
2
4
6
"
8
Analysis
tE"T'5~'0keY
Scat!
I ,i
i~t! " "
I .~~
(b)
o
i,/ :
Ni '
10 12 14
0
.*~~.
1
3
~"~"'.-"'~
5
7
9
.
11
'
13 15
Energy / keV
Energy / keV
Fig. 2-44. Comparison of SRXRF spectra: (a) 100 ng of each element on the fused quartz, by total reflection; and (b) 100 ng of each element on the filter paper, by ordinary bulk analysis. From Ref. [31], reprinted by permission of Elsevier Science Publishers B. V., Amsterdam. Table 2-9 Comparison of the detection limitsa between two different methods by SR excitation Element V Fe Ni Zn Pb Ti Mn Cu Ge
Total reflection (ng) 0.03 0.02 0.01 0.01 0.02 0.04 0.02 0.01 0.005
Bulk analysis (ng) 0.07 0.07 0.05 0.04 0.04 0.07 0.06 0.06 0.03
a The detection limit is defined as the quantity which gives a signal equal to three times the square root of the background. From Ref. [30], reprinted by permission of Kodansha, Tokyo.
129
Zn
1 400-
12 0 0 -
Cu Ni
"-:"
=. I O 0 0 ;~
800
~o
600
~9
-
E~" 12 keV t 91000 s MAX.CTS 91361
Fe
Mn Cr
-
Scat.
400200 0
I
I
i
I
I
0
5
10
15
20
Energy / keV Fig. 2-45. SR-Excited TXRF spectrum from the sample prepared by drying 100 lxl of a solution containing 50 ng of each element (V, Cr, Mn, Fe, Ni, Cu, and Zn) on a silicon wafer.
10,,-:,. ::::186-
v
..
--
^
v . X
.
.
.
v
X
4-
~20 i
I 22
I 24
I 26
I 28
I 30
I 32
Atomic number Fig. 2-46. The X-ray fluorescence intensity values of each element, corrected for excitation efficiencies, fluorescence yields, relative Ktx line intensities, and air absorption between the detector and the sample shown in Fig. 2-45.
This method, however, has many problems when it comes to quantitative analysis. One of these involves sample preparation. It is virtually impossible to make a quantitative estimate because of questions concerning where the sample solution is dropped on the reflector, how large an area it covers, and whether it is spread uniformly. It is also difficult to determine where on the sample the X-ray falls. One solution to this problem is to use an internal
130 standard. Pella and Dobbyn [44] used this method to measure the ppb concentration level of Se in human blood serum, using germanium as an internal standard. For trace analysis in the total reflection geometry, using the internal standard, monochromatic excitation with SR allows us to quantify elements reliably and easily, and to improve detection limits compared to X-ray tube excitation. As an example, we will describe an experiment on a NIST reference material (Trace Elements in Water, SRM 1643b). The spectrum of the total reflection X-ray fluorescence for this sample is shown in Fig. 2-47a. A 15 keV X-ray obtained using a Si(111) double-crystal monochromator was used for excitiation. Next, Ge, which was absent from the original sample, was added to bring the concentration up to 70 ng g-1. Then, 100 l.tl of this solution was dried on the reflector. Its spectrum is shown in Fig. 2-47b. The values for the other elements, determined from the amount of Ge, after corrections for excitation efficiency, fluorescence yield, relative Ko~ line intensity and air absorption between the sample and the detector, are indicated in Table 2-10. The detector efficiency is assumed to be constant for the given energy region and the values are in good agreement with the NIST certified values. However, this method has limited applications, since an element not originally present in the sample has to be added. The following procedure can deal with any detectable element as an additional element and is expected to be applied to trace element analysis in liquid samples [45]. For measurements employing the energy dispersive method using monochromatic excitation, the following relationship holds between the concentration of element i contained in the sample and the intensity of X-ray fluorescence [46]: Tabel 2-10 Quantitative determination of several elements in NIST SRM 1643b (Trace Elements in Water) by SR-excited TXRF using Ge a as an internal standard Element
Determined (ppb)
Certified (ppb)
V Cr Mn
49 23 27
45.2+0.4 18.6+0.4 28 +2
Fe Co
120 42
99 +8 26 +1
Ni
49
49 +3
Cu
30
21.9+0.4
Zn
71
66 +2
a Ge was added to SRM 1643b so as to contain 70 ng g-1 in the aqueous solution for measurement.
131 Ci_
li4xR 2(]2s, o +/Zs,isin ~/sin ~)[Ps PoOi[(lIK/P)oOJKfK]i(1 - exp{-[(/Zs,o +/2s,isin tF/sin ~)/ps]psT/sin tF} )
(2-30)
where: C i is the concentration of analyte element i. li is the fluorescence intensity for the analyte line of element i. Po is the intensity of the exciting X-ray beam. Di is the detector efficiency. R is the distance from the sample to the detector. l,ts,o is the linear absorption coefficient of the sample for the exciting X-ray beam energy. l.ts,i is the linear absorption coefficient of the sample for the fluorescence energy of element i. is the incident angle of the exciting X-ray beam. is the angle between the sample surface and the detector, the takeoff angle.
Ps is the density of the sample. T is the thickness of the sample. (,UK/P)o is the mass absorption coefficient of analyte element i for the exciting X-ray beam. tOKis the fluorescence yield of the K-series line.
fK is the intensity ratio of Ks to the total K-series lines. When the sample is very thin, this can be approximated by Ci =
li 4 x ' R 2
(2-31)
PoDi[(l.tK/P)oCOKfK]iPsT / sin tp The term [(l.tK/P)otOKfK]i is a quantity peculiar to an element, that is the constant of its element parameters; the remaining terms, excluding li, 4r~R2/(PoDiPsT/sin tF), are a constant parameter determined by experiment (assuming the detector's efficiency to be independent of X-ray energy). If the term [(PK/P)oOJKfK]i denotes ei and the remaining terms, excluding li, are represented by K, the above equation will be reduced to
Ci =
K
li
(2-32)
9e i
which is validated by the data shown in Fig. 2-46. However, when a solution with a high salt concentration is used, the salt will remain on the reflector after drying, and affect the measurement by absorbing some X-rays. Using a first approximation, to correct this absorption by the matrix we add the correcting term exp(tz~i3). This is based on the assumption that the absorption coefficient of an X-ray is approximately proportional to the cube of its wavelength: 2~"is the wavelength of the fluorescent X-ray of the analyte element i, and a is a quantity which depends on the absorption, and will be determined experimentally.
132 5
10
4
10
-
/
\
(a)
Ca Ca
Fe
Ni
4"
"]
E r" 15keV t 9500 s
MnCO Cu Cu
-
tCr+lCo! +As ~ +Fe JIN~//I/Z l ~ S e
10 3-
i rMn~ I
n
Scat.
]As
0
1 02 1
10
-
MAX.CTS 918468 B.S. 90.11 x 2.8 mm 2
10 ~ 0
i
!
i
5
i
10
15
20
Energy / keV 10 10 "
.1-,~
5
4
10 1
(b)
-
-
Cu Mn Ni ,7+ + [, ~n -Fe F e l N i l
F_7 915 keV t 9200 s
+
iv
-
o~
Co
Ca ~Ca
As ~ ~ Ge
Scat. /~
C
-
1
10 10
:0.15 x 3.0 mm
B.S.
0
0
~[
I
I
I
I
5
10
15
20
Energy / keV Fig. 2-47. SR-Excited TXRF spectra from the samples obtained by drying 100 ~tl of NIST SRM 1643b, (a); and the solution prepared so as to contain 70 ng g-1 of Ge in SRM1643b, (b).
Let us consider the case of three elements in unknown amounts contained in a sample. The intensity of the X-ray from each element is related to its concentration as follows:
C1 -
ll.exp(aZ13)
K.el
(2-33)
133
C2 =
C3 =
12.exp(aA 3)
K.e2 13.exp(aA 3)
K.e3
(2-34)
(2-35)
If two of these elements are added at specific concentrations (Ci', i = 2, 3), then
C1 =
C2+ C2' =
C3+ C3' =
ll'.exp(aA13)
K'.el 12'.exp(a~,23)
K'.e2 13'.exp(o~A33)
K'.e3
(2-36)
(2-37)
(2-38)
In these six equations, (2-33) to (2-38), li and li' are measured values and ,~,i and ei are peculiar to these elements - - v a l u e s that can be determined using constants. Thus, there remain six unknown quantities, K, K', Ci (i = 1, 2, 3), and a, which can be determined by solving simultaneous equations. Quantitative analysis will be possible for elements other than these three, using the values of K and a determined using the equation
Ci = li . exp(aA/3) / K.ei
(i>4 )
We made measurements on two kinds of samples. One was the synthetic standard used in the experiment shown in Fig. 2-46, and the other the NIST standard reference material used for the experiment shown in Fig. 2-47. For the calculation of quantitative values, the synthetic sample and NIST SRM were each divided into subsamples, A and B. Vanadium and nickel were added to subsamples A until the concentration of each element reached 50 ng g-1. Likewise, manganese and zinc were added to subsamples B until the same concentrations of each element were reached. Then, 100 lxl of each sample was dropped onto and dried on a silicon wafer in the same manner as the experiments shown in Figs. 2-45 and 2-47. These preparations were performed in a Class 1 000 clean booth. A 12 keV monochromatic X-ray created using a Si (111) double-crystal monochromator, was applied to each sample at a glancing angle of about 2 mrad. The counting time was 1 000 s. The results of the quantitative analyses listed in Tables 2-11 and 2-12 suggest that this is a very satisfactory method for trace element analysis. It requires no consideration of the spot size or the area irradiated, and provides rapid, simultaneous multi-element analysis of ppb concentration levels using a simple sample treatment. It has a wide range of applications since unknown amounts of the elements to be added can be selected. However, the expedient correction used above for absorption by the matrix may be further improved by a more precise approximation.
134 Table 2-11 Results for the synthetic sample Found, ng g-1 Element
Added a, ng g-1
V Cr Mn Fe Ni Cu Zn
50 50 50 50 50 50 50
Addition of V and Ni for calculation b
Addition of Mn and Zn for calculation c
42 48 49 49 55 49 56
42 47 47 47 51 46 52
a Added values are given for comparison. b For the calculation of K and ct, the fluorescence intensities of V, Ni and Cr were used. c For the calculation of K and o~, the fluorescence intensities of Mn, Zn and Ni were used. From Ref. [45], reprinted by permission of the Japan Society for Analytical Chemistry, Tokyo. Table 2-12 Results for NIST SRM 1643b Found, ng g-1 Element
Certified a, ng g-1
V Cr Mn
45.2+0.4 18.6+0.4 28 +2
Fe
99
Co Ni Cu Zn
Addition of V and Ni for calculation b
Addition of Mn and Zn for calculation c
38 14 22
43 15 21
+8
120
114
26 +1 49 +3
28 53
24 47
21.9+0.4
23
20
66
99
86
+2
a Certified values are given for comparison. b For the calculation of K and ct, the fluorescence intensities of V, Ni and Mn were used. c For the calculation of K and ct, the fluorescence intensities of Mn, Zn and Ni were used. From Ref. [45], reprinted by permission of the Japan Society for Analytical Chemistry, Tokyo.
135
2.4.4. Near surface analysis When the incident X-ray is totally reflected it penetrates only a few tens of A into the sample. This property of the TXRF method makes it possible to carry out the elemental analysis of the area near the sample surface. Conventional XRF only performs bulk analysis, and cannot be used to analyze the sample surface. A significant feature of TXRF is that while it maintains the rapid, nondestructive, and simultaneous multi-element analysis of conventional XRF, it can also be applied to surface analysis. This method has been extensively used in the laboratory for surface analysis of silicon wafers. However, there have not been many cases where SR has been used for surface analysis where the samples must be handled carefully. It is very difficult not to contaminate samples from the time they are prepared in the laboratory, through transportation, until they are measured at a SR facility. In this section, we will discuss precautions for using SR-excited TXRF for surface analysis. Any element on the surface will cause X-ray fluorescence. However, the intensity of the fluorescence varies, depending on whether the elements are on the sample surface or slightly below it. This difference is illustrated in Fig. 2-48, which shows the calculated dependence of the X-ray fluorescence intensity on the glancing angle, for the same quantity (1012 atoms cm -2) of iron distributed inside silicon in the following ways: (1) close to the surface, (2) homogeneously from the surface to a depth of 10 A, (3) homogeneously from the surface to a depth of 100 A, and (4) homogeneously from the surface to a depth of 1 000 A. It should be noted for quantitative analysis that the X-ray fluorescence intensity caused by the same amount
,--:,.4A
: top surface ..... : surface to 10 A - - -: surface to 100 A
~39~ 2 o o r,o
1s,
0-
s"
........ ...I
2
I
4
I
6
I
8
I
10
G l a n c i n g angle / m r a d Fig. 2-48. The calculated glancing angle dependence of Fe Ko~ intensities on the following distributions of Fe (1012 atoms cm -2) in the silicon substrate: (1) only on the top surface; (2) on the homogeneous distribution from the surface to 10 A in depth; (3) on the homogeneous distribution from the surface to 100 A in depth; (4) on the homogeneous distribution from the surface to 1 000 A in depth.
136 of element varies greatly depending on the distribution and the glancing angle. At the Photon Factory, Iida et al. [47] measured the dependence of X-ray fluorescence on the glancing angle, using a 3.5-l.tm-thick Gal_xAlxAs thin film (x - 0.298) epitaxially grown on GaAs. In comparison with GaAs, there were fewer As atoms near the surface of this thin film. This is mostly due to sublimation that occurs during sample preparation. To estimate the depth of the reduced As concentration they calculated the exhausted length (the distance from the surface to where the concentration is 1/2 of the bulk concentration value) to be about 100 /~, on the assumption that the As concentration increases from the surface value of zero to the bulk value in the manner of a complementary error function. Changes in the chemical composition of compound semiconductors can also be measured by destructive methods, such as ion channeling and SIMS; however, the present method does not destroy the sample. Bloch et al. [48] performed an experiment at SSRL on the interface between a polymer solution and the air. The method they employed allows measurements in any atmosphere, and therefore is suitable for analyzing the surface of liquids. They positioned an optically flat reflector in front of the sample, to control the glancing angle of the incident X-ray. The sample was a slightly sulfonated polystyrene dissolved in dimethyl sulfoxide. The polymer had a molecular weight of 115 000, with a chain containing about 10 mole% manganese sulfonate. By measuring the dependence of the S Ka from the solvent and the Mn Ka from the polystyrene on the glancing angle, they found that the polymer concentrations increased on the interface, since the Mn intensity was higher at smaller angles. With uniformly soluble MnC12, the intensity ratio was found to be constant. They also compared this method with the optical method. The minimum penetration depth is a few tens of A for X-rays and hundreds of A for the optical method. Also, in the optical method, polymer solutions are generally transparent to a depth of a few thousand/~, allowing penetration to infinite depths at angles above the critical angle. This means that the X-ray method provides a far more sensitive surface analysis for this kind of sample.
2.4.5. Depth profile analysis As shown in Fig. 2-38, the penetration depth increases from a few tens of/~ to a few l.tm as the glancing angle of the incident X-ray increases beyond the critical angle. This means that the depth profile of the element being studied can be analyzed by changing the glancing angle in small increments. Iida et al. [17] used a silicon wafer ion-implanted with As to measure the dependence of the intensity of As K a X-ray fluorescence on the glancing angle (1-5 mrad). They also made the same measurements on a sample heat-treated at 1 000~ for 20 min. As shown in Fig. 2-49, the increase in As concentration on the sample surface brought about by the heat treatment gives a greater intensity of X-ray fluorescence below the critical angle. The decrease in the intensity at glancing angles above the critical angle corresponds to a reduction of the total amount of As. The demonstrated intensity of X-ray fluorescence has been found to agree well with the value calculated using the profile analyzed by the SIMS method.
137 _1
'
I/.~
/ -
/
/
/
/
\
'
I
'
I_
\ '
A
-
o~==l
tD - / / o1-,~
\
I
1
""
B
/
-
2
\
i
I
-!
i
3
4
Glancing angle / mrad Fig. 2-49. Reflected X-ray intensity (broken line) and As K intensifies (solid lines) for Si wafers, before (A), and after (B), annealing, as a function of glancing angles. From Ref. [17], reprinted by permission of Elsevier Science Publishers B. V., Amsterdam.
Figure 2-50 (dotted line) shows a depth profile analysis of a silicon sample ion-implanted with iron (40 kV, 1 x 1015 atoms cm -2) [31]. Iron ion-implanted in this manner shows a Gaussian distribution, with a 123 A standard deviation around the peak ion concentration at a depth of 302 ,~. The dependence of the X-ray fluorescence on the glancing angle, calculated from this profile, is shown by the solid line in the figure. The slight difference from the measured values is probably due to the spread in energy and angle of the beam. We made the following calculations to ascertain how sensitive the curve of the X-ray fluorescence plotted against the glancing angle is to the depth profile. Using Fresnel's formula [43], the amplitude, At, of the transmitted X-ray is given by
A t _ 201 Ai 01+02
(2-39)
where Ai, 01 and 02 have the values indicated in Eq. (2-23). The intensity of the transmitted X-ray just beneath the surface is given by
~_~ 2
=
4 012 (01 +p2) 2 + q22
(2-40)
138 [In the actual calculation, the values of 01, P2 and q2 are used 9See Eqs. (2-24)-(2-26).] The transmitted X-ray, expressed by Eq. (2-40), is transmitted in the medium while being absorbed. The X-ray fluorescence intensity for an element distributed at a given depth in the medium is calculated from the transmitted X-ray intensity and the concentration of the element at that depth. Figure 2-51 shows the calculations for the ion-implanted iron described in Fig. 2-50, with different standard deviations for their Gaussian functions 9 The calculated Fe Ka intensity vs. the glancing angles, at different depths of peak concentration, are shown in Fig. 2-52. The X-ray fluorescence profile is found to be quite sensitive to these parameters. Application of this characteristic to thin film samples will be discussed in detail in the following section. Measuring the depth profile of impurities at the surface during the analysis solves the problems mentioned in the preceding section, i.e., the difficulty in making a quantitative determination of an element caused by the dependence of the X-ray fluorescence intensity on the elemental concentration profile. Various applications of the above method are expected because of the nondestructive depth profile analysis in air that it provides.
1E22
_ _ _ Calc. 302.~
"
Exp.
d
"IE16 0
O
Depth (/~)
1000
O
0
1
2
3
I
4
I
5
6
Glancing angle / mrad Fig. 2-50. Depth profile analysis of implanted Fe in a silicon wafer. From Ref. [31 ], reprinted by permission of Elsevier Science Publishers B. V., Amsterdam.
139 2.5-
.
,~
:300A
i
:c=100
~~ .2~ . 0 -
.......
: o =
5 0 ko
~15-
~
~
9v.,,l
~l.Or~
os-
0
"
0.0 . . . .
0
I
2
I
I
4 6 Glancing angle / mrad
I
8
I
10
Fig. 2-51. The glancing angle variation in Fe Kcz intensifies with different Fe concentration distributions in a silicon wafer, changing standard deviations (cr) at a fixed peak depth (Zp) of 300 ,~ below the surface in a Gaussian distribution. (o" is the halfwidth at half of the maximum concentration)
3 . 0 - -
c~" 100 .,~
,.~,,~2 . 5 -
:
9
'i ~~,,,
~t=2 0 - "
zp = 3oo A
....... : Zp = 200 .A Zp 400 A
o
"-" 1 . 5 o
i
o10-.
=._.
. . . . . . . . . .
=_.
r~
o0.5-
se
#
0.0-
0
2
4
6
8
10
Glancing angle / mrad Fig. 2-52. The glancing angle variation in Fe Ko~ intensifies with different Fe concentration distributions in a silicon wafer, changing peak concentration depth (Zp) below the surface at a fixed standard deviation (o') of 100 A in a Gaussian distribution.
140 2.4.6.
Application
to thin film s a m p l e s
TXRF is effective not only for the elemental analysis of a sample surface, but is also useful for the analysis of thin films. As explained above, the depth of penetration of the incident Xray can be controlled by changing its glancing angle to the sample. This makes it possible to match the measurement conditions to the sample thickness. We will discuss an experiment using a 1.2-gm-thick photoresist polymer coated on a silicon wafer [ 18]. Small amounts of impurities in photoresist polymers are normally measured by chemical analysis. In this case, TXRF was chosen because the impurities were in a thin film spread on a wafer. The experiment was conducted using Beamline 4A at the Photon Factory. A 9.6 keV monochromatic X-ray with a Si(111) double-crystal monochromator was utilized. The intensities of the incident and reflected X-rays were measured using an ionization chamber, and the intensity of the X-ray fluorescence measured using a Si(Li) detector. The sample and the detector were 5 mm apart. The measured reflection curve is shown in Fig. 2-53. The curve shows two critical angles. That in the low glancing angle region is the critical angle for the resist polymer, and the other, in the high glancing angle region, the critical angle for the silicon wafer. This means that, if the incident angle is gradually increased from a low value, the behavior of the incident X-ray will be changed in the following manner. First, it is totally reflected on the resist surface. Then, as the glancing angle becomes greater than the critical angle of the resist (2.15 mrad),
1.0 0.8 ~. 0 0
0.6 0.4
0
0.2 I 2
I
I 3
I
i--r------4
Glancing angle / m r a d Fig. 2-53. The observed angular dependence of reflectivities from a silicon wafer covered with a resist polymer. The critical angle of the resist is around 2 mrad, and that of silicon around 3 mrad [ 18].
141 the X-ray penetrates the polymer, and is totally reflected on the silicon-resist interface until the glancing angle reaches the critical angle of the silicon wafer (3.20 mrad). The reflection curve rises until the glancing angle reaches the critical angle, because of the change in the effective cross section of the incident X-ray. We measured the X-ray fluorescence using glancing angles of 2 and 3 mrad. The measurements were made in air and lasted for 2 000 s. At 2 mrad, the maximum penetration of the X-ray into the resist is only 100 J., and most X-ray fluorescence is emitted from the surface. At 3 mrad, the measured X-ray fluorescence comes from all parts of the resist film. The spectra of the X-ray fluorescence for these two cases are shown in Fig. 2-54. Both show X-ray fluorescence from the silicon substrate, argon in the
0=2 mrad 0 =3 mrad
~ ~
es'st
~, ~
t
Silicon (a)" Schematic drawing of total reflection with changing glancing angles (2 mrad and 3 mrad).
10
105 /L
0= 2 mrad
0= 3 mrad
/
~~
4|
10[ Ar 10 3
8
Scat.
Si
Scat.
l~
]
/
,o31 - tt
102
0
5
Energy / keV
10
0
5
10
Energy / keV
(b): SR-Excited TRXF spectra obtained. On the left side (2 mrad) total reflection occurs on the resist surface; on the right side (3 mrad) on the interface between the resist and silicon wafer. Counting time is 2000 s. Fig. 2-54. Impurity (Fe) analysis in a thin resist film on a silicon wafer [18].
142 air, and iron which is regarded as an impurity in the sample. The problem with measuring the quantity or distribution of the profile of iron is the dependence of the X-ray fluorescence intensity on the depth profile, as described in the previous discussion on surface analysis. Assuming the total quantity of iron to be constant, we calculated the intensity using the two glancing angles (2 mrad, 3 mrad) for the following four kinds of distributions: (1) homogeneous from the surface to 10 A, (2) homogeneous from the surface to 100 A, (3) homogeneous from the surface to the interface, (4) decreasing linearly from the surface to the interface. The calculated values for 2 mrad are: (1) 0.74, (2) 0.49, (3) 0.007, and (4) 0.014; and for 3 mrad: (1) 0.31, (2) 0.28, (3) 0.24, and (4) 0.25. Note that the unit for these values is arbitrary. The peak area of the X-ray fluorescence of iron measured in this experiment has been found to be four times as large at 3 mrad as at 2 mrad. We attempted a quantitative analysis, based on the assumption that the iron concentration was homogeneously distributed on the thin resist film and that it also contaminated the surface of the film. Comparing the measured and calculated values of the X-ray fluorescence intensity using 2 and 3 mrad we found that 92% of the iron detected was homogeneously distributed in the film, with 8% contaminating the surface. Our estimate of the amount of iron showed that 1.7 x 1012 a t o m s cm -2 were contained inside the film with 1.5 x 1011 atoms cm -2 attached on the surface. The amount of iron contained inside is converted to approx. 1.3 ppm in relative weight concentration- a value in good agreement with that obtained by chemical analysis (1.0 ppm). However, the depth concentration profile must be measured if one needs a more precise quantitative determination. 2.5. C H A R A C T E R I Z A T I O N OF LAYERED STRUCTURES BY GRAZING INCIDENCE
2.5.1. X-Ray spectroscopy using grazing incidence As discussed in the preceding section, the penetration depth changes as the incident angle of X-rays changes around the critical angle of total reflection. Also, under these conditions the incident X-rays interfere with reflected X-rays. Interference occurs inside the film in a sample with a layered structure, which affects the reflection curve and the X-ray fluorescence intensity profile for the constituent elements. The interference effects depend on factors which include the number and thickness of layers, the concentration profile, the roughness of the surface or interfaces, and the existence of transition layers on interfaces. This makes it possible to determine important parameters of the film structure, by analyzing the reflection curve and the X-ray fluorescence intensity profile measured as functions of the glancing angles. For multilayered materials, the following parameters can be determined by this method: (1) The thickness and density of individual layers. (2) The concentration profile of constituent elements. (3) The roughness of the surface or interfaces. (4) The existence of transition layers between the surface and the thin film, and between a
143 thin layer and the substrate, as well as the thickness, density and composition of the transition layers. (5) The depth profile of impurities. Interference shows up as oscillations on the reflection curve. Back in the 1930s, Kiessig [49] analyzed oscillations in the curve exhibited by reflection from a monolayer thin film, and described a method of determining film thickness. He regarded the oscillations as fringes of equal inclination, and determined the film thickness from a relationship between it and the optical path difference. Parratt [50] formulated a rigorous method for calculating the reflection curve for multilayered materials, and attempted to analyze these materials from the changes in the reflection curve, especially around the critical angle. He made systematic calculations on oxidized layers of copper, and compared them with experimental results. N6vot and Croce [51] gave a more rigorous definition for the conventional method of incorporating interface roughness as a Debye-Waller factor [a scalar treatment, see Eq. (2-29)] and obtained an equation that enabled a vectorial analysis. They used this equation in a detailed analysis of surface layers of polished glass. They measured the reflectivity at glancing angles varying from the critical angle to much larger values. This is important in determining very sensitive roughness values, since roughness has a great effect on the reflection curve, particularly when the glancing angle is large. Generally speaking, reflectivity is very small at large glancing angles. Therefore, careful experimentation is required to determine oscillations in reflectivity with high precision. Vidal and Vincent [52] developed a matrix calculation method to systematically determine reflectivity for multilayered materials using a computer. The roughness term is also incorporated in the matrix elements. They used this method to evaluate layered synthetic materials. Kr61 et al. [53] established a method of calculating the angular dependence of X-ray fluorescence intensity. Using the matrix treatment of Vidal and Vincent, this method enabled a rigorous determination of X-ray fluorescence emitted from individual layers. The experiments were carded out on the thin layers of semiconductor wafers. In the above experiments, only Kr61 has used SR. This is not necessary for measuring reflection curves; careful measurement using specially designed equipment produces satisfactory results. In fact, as well as N6vot and Croce, Huang and Parrish [54] performed precise experiments using equipment they had developed incorporating a channel-cut monochromator. However, monochromatized X-rays obtained from SR are useful for the measurement and analysis of X-ray fluorescence intensity profiles. The high intensity, tunability, and high collimation of SR are indispensable for measurements of minute compositions and applications to very thin films and transition layers. Sakurai and Iida [55] have proposed a method for determining each layer thickness of mutilayered thin films by a Fourier transformation of oscillations known as the Kiessig structure [49]. Other research using the grazing incidence method includes a laboratory study, in combination with ellipsometry [56], of a thin film on a compound semiconductor, and an effort to examine heterojunction roughness using soft X-rays (SR) [57, 58]. Heald et al. [59] used the grazing incidence method to measure both reflection curves and EXAFS, in order to analyze metallic multilayers.
144
2.5.2. Determination of the thickness of monolayer thin films (Kiessig's method [49]) Kiessig established a simple method for determining the thickness of monolayer thin films. He assumed the oscillations that appear in the reflection curve to be fringes of equal inclination. When the X-rays are incident at an angle slightly larger than the critical angle of the sample, part of the X-ray beam goes into the thin film. As shown in Fig. 2-55, X-rays reflected by the surface and by the interface interfere with one other. The reflectivity from the interface is so small that X-rays emitted after repeated reflection within the film can be neglected. The X-rays can be assumed to be parallel beams (coming from a light source infinitely far away), and the two beams (AB and A'B') which interfere are parallel and come from the same incident X-rays. We write N for the foot of a perpendicular line, A'N, dropped from A' to AB. Since there is no optical path difference between the two beams beyond A'N, the difference in the optical paths is expressed by A = n2.(AC + CA' ) - nl.AN
(2-41)
where nl is the refractive index of air or a vacuum (=1), and n2 is the refractive index of the film, which takes a complex number if there is any absorption. Let the film thickness be dE, then, since 01, 02<<1, the optical path difference is expressed by
Air (vacuum): n = n 1 = 1
B ~ / B
N
d2
~
~monolayer:
Substrate: n = n 3 Fig. 2-55. Schematic representation of fringes of equal inclination.
n =n
145
A
=
2d2 02 = 2d2( 0 7 -
02 ]1/2
(2-42)
lc/
See Eqs. (2-4) and (2-23) for the meanings of 01, 02 and 01c. The intensity of reflected Xrays is a maximum or minimum value when the optical path difference is an integer (m) or a (m + 1/2) multiple of the wavelength. Therefore, changes in the intensity of reflected X-rays depend on 01. Let the wavelength of the incident (reflected) X-rays be ~; then the intensity becomes a maximum value when:
mA = 2 d2(O2 - 021e!]1/2
for&<&,
(m + 1/2) A, = 2 d2(07 - 02) 1,2
for & > &
and (2-43)
The term (m + 1/2) is required because a phase shift occurs during reflection when d;2 > 63, i.e., the electron density is higher in the film than the substrate. See Eq. (2-2) for definitions of 62 and t~3. From Eq. (2-43), the values of 01 corresponding to the m-th maximum is related to m as follows:
021(m) = 02 + $2m2
for $2 < 63,
4d~ and
O2(m) = 02c + A 2(m + 1/2) 2
44
for ~ > ~
(2-44)
Therefore, 012(m) plotted against m 2 or (m + 1/2) 2 forms a straight line, and d2 can be determined from the slope. Therefore, highly accurate determinations of the film thickness can be obtained without contact with or destruction of the sample. Further, since 01c [= (262) 1/2] can be determined from the intercept of the line, the density of the thin film can also be obtained. We carded out an experiment on a Ni thin film deposited on a silicon wafer using a sputtering method. Measurements were made using 1.2 A monochromatized X-rays at Beamline 4A of the Photon Factory. The reflection curve measured is shown in Fig. 2-56. Oscillations can be seen in the curve. Since nickel has a larger t~than silicon (62 > 53), 012(m) corresponding to the maximum oscillation was plotted against (m + 1/2)2 (Fig. 2-57). The result is a straight line. From the slope of the line, the thickness of the Ni thin film was determined to be 454 + 3/~. The critical angle (01c) was found to be 5.6 + 0.2 mrad. As described above, it is easy to determine the film thickness and the critical angle, i.e., the sample density and composition of a monolayer thin film. However, this method cannot be applied to samples having more than one layer or having transition layers.
146
10
;~' 10-
o
1
;;>
-
-
2
lo-~
1
-
10
-4
-m
I
0.0
0.2
I
I
0.4
I
0.6
I
0.8
I
1.0
Glancing angle / degree
1.2
Fig. 2-56. The reflection curve of a Ni thin film deposited by sputtering. The incident X-ray wavelength is 1.2 A. 1.0 0.8 ~~
0.6
~
0.4 0.2 0.0
I
I
I
I
o
20
4o
60
I
a0 (m+l/2) 2
I
I
I
loo
12o
14o
Fig. 2-57. A plot of maximum oscillation values for the Ni thin film on a silicon substrate shown in Fig. 2-56. The straight line was obtained by least-squares.
2.5.3.
Calculation
of reflection
curves-1
(Parratt's
method
[50])
Kiessig's method uses only the period of the oscillations that appear in the reflection curve. More information would be available if it were possible to analyze the entire reflection curve, including the amplitude of the oscillations. It is also important to have a method for analyzing multilayered thin films. One way to analyze the reflection curve is to determine the parameters such as the film thickness and composition, and to construct a theoretical curve; the parameters are then adjusted to make an experimental curve fit the theoretical curve. It is therefore necessary to have a theoretical method for calculating the reflection curve. Parratt formulated a method for determining the
147 reflectivity of grazing incident X-rays for multilayered samples, using the electromagnetic theory (continuity of the tangential components of electric and magnetic fields on interfaces) and Fresners optics formulae. When monochromatized X-rays irradiate a sample, (without multilayers), the electric field vectors of the incident [El(Zl)], reflected [E~(zl)], and refracted [E2(z2)] X-rays are expressed as follows:
El(z1) - El(0)exp{i[(.ot -(kl,xX1 + kl,zZl)]) E~(z1) = E[(0)exp{i[~ot -(kl,xXl - kl,zZl)]} E2(z2) = E2(0)exp{i[~ot -(k2.~x2 + k2.zZ2)]}
(2-45) (2-46) (2-47)
where z denotes the perpendicular distance from the surface (which has a positive value in the sample), and k i (= 2~/Ai, i=1, 2) is the propagation vector of X-rays. Suffix 1 represents the outside of the surface, and suffix 2 the inside. The plane of incidence is an xz plane. The continuity of the tangential component of the electric field vector on the interface requires that, if there is a grazing incidence, the following holds true: k2.x = kl, x = kl, and k2.z = k102 = kl (012- 2 6 2 - 2ifl2) 1/2. From Fresners formulae, the reflectivity can be expressed as a function of the glancing angle by
01 + 02
(01 + p2)2+ q2
(2-48)
where P2, q2, 01, and 02 are defined in the same way as in Section 2.4.2.: see Eqs. (2-24)-(226). Now consider a sample having N layers (air, 1; layers, 2 to N - l ; substrate, N). Assume also that the interfaces are perfectly smooth. Denote the thicknesses of individual layers as dn. The thickness of the air, dl (vacuum), will not be considered. From the boundary condition of the continuity of the electric vectors on the interface, there is the following relationship between layers n-1 and n (n = 2, 3,..., N):
an-lEn-1 + an-l_l Er_l = a-nllEn_ (an-lE,-1-an-l_l E r _ l ) f , - l k l = (
+ an E~
a~lxE.-a. Er)Lkl
fn = Pn - iqn an = exp( - iktfndn / 2 )
(2-49) (2-50) (2-51) (2-52)
[En is the value on the n-th interface of the electric field that propagates downward (to the substrate) through the interface; En r is the value on the n-th interface of the electric field
reflected upward from the interface (to the surface)]. From these equations, the reflection
148 coefficients of the electric fields' amplitudes are expressed by the following recursion formulae:
rn-l,n =
a4_l( rn,n+l + Fn-l,n)
rn,n+lFn-l,n + 1
(2-53)
rn,n+1 = a2n (E r / En)
(2-54)
fn-1 - f n Fn-l,n =fn-1 + fn
(2-55)
The calculation starts with n = N, and proceeds until the reflectivity R = I rl,212 is determined. Note that rN,N+I = 0, since there is no upward electric field, EN r, in the substrate. The parameters required for these calculations are dn (film thickness), Sn (composition and density), and fin (composition and density). A comparison of the calculated results with the experimental values makes it possible to establish an optimum model of the multilayered structure. Parratt measured the reflection curve of a copper film (thickness approx. 2 000 A) deposited on glass, and compared the results with the values calculated by the above method. First he used a two-layered structure of copper and glass. The reflection curves did not agree very well, and the value of ~, which is proportional to density, was found to be about 10 percent smaller than the value for bulk copper. Then he considered a three-layered model with an additional oxide (Cu20) layer, and searched for an optimum model by changing the thickness of each layer. He went as far as a five-layered structure: Cu20 (90% density of the bulk value) + Cu20 + Cu (90% density of the bulk value) + Cu + glass. Although the calculated results did not coincide perfectly with the experimental values, they provided quite a good approximation. These results suggest that, in order to examine the structure of thin films, even monolayer materials must be regarded as multilayered. This means a satisfactory analysis cannot be based only on the period of the oscillations; the entire reflection curve must be considered. This is why it is very important for thin film analysis to have a method, such as the one shown in this section, for calculating the reflectivity of multilayered structures. Unfortunately, the calculations are tedious when the sample has many layers.
2.5.4. Incorporation of roughness and transition layers Real samples have rough surfaces and interfaces. They can also contain transition layers. Omitting these factors from the reflectivity calculation prevents precise comparisons with experimental results. A rough interface and transition layers reduce reflectivity, which can be observed in the reflection curve. We will describe the method for incorporating these factors which was established by N6vot and Croce [51 ].
149 Consider an interface where homogeneous substances meet. The refractive indices can be represented by n l (upper layer) and n2 (lower layer). The roughness of the interface, as shown in Fig. 2-58, is expressed as the root-mean-square of deviations of the depth coordinate ZD.
0.2 = ( z 2 )
(2-56)
If the roughness value is not very large, it will be smaller than kln -1, k2n-1 (kin is the normal component of wave vectors for each layer, i). A real rough interface can be considered as many smooth planes distributed in a Gaussian manner. The peak of the Gaussian distribution is the average plane (P0) of the deviations of the rough interface. Therefore, using this assumption, if the reflection coefficient of a perfect interface is represented by ri, that for a rough interface, re, can be described by rR = rlexp(
-
81r,2klnk2n0.2)
(2-57)
The reflectivity is given by the square of the absolute value of the refection coefficient (R = I rR 12). The above equation also gives the reflectivity of a transition layer whose refractive index changes in the manner of an error function. If we assume that
9
m
n(z)= nl + (n2- nl).F(z)
(2-58)
F(z)=l~fZ___ooex[~ -v2x0-2
(2-59)
20.2 ,u2]du
,
+++
+
+++P0
/I
I
++
Fig. 2-58. Schematic representation of a rough interface: Po is the average plane surface (perfect interface); P is the actual interface.
150 when nl is unity (the refractive index of air or a vacuum), we obtain the same reflectivity in a similar manner to the method for the rough interface above. Namely, consideration is made by substituting a transition layer for a rough interface, and the refractive index for the parameter representing the roughness. Conversely, a transition layer as shown in Eqs. (2-58) and (2-59) can be used to analyze interface roughness. Now consider a multilayered material, as shown in Fig. 2-59. The amplitudes of the electric fields are represented by a, b, a, and ft. Since only transmitted X-rays exist in the substrate, if aj_ 1 is known, Eq. (2-57) can be used to determine bj-1 o n the Oj-1 interface. The following relations hold on the Dj_ 2 interface.
OtJ-1 + rR'j-2~J-I 1 + rR, j_ 2
(2-60)
bj_2 = otj_l + flj-1- aj-2
(2-61)
aj-2 -
where rR, j-2 is the refection coefficient on the Dj_2 interface, defined by taking into consideration its roughness and transition layer. Consecutive calculations to the first layer, D1, determine the reflectivity of the multilayered material, I bl[al I2. N6vot and Croce [51 ] used the above method to analyze a thin layer formed on a surface by polishing, oxidation, or contamination. We will describe their analysis in the following. The a M
bl
1
D1
M
M i-2
~
~
b1_2
DF2
M j-1 u.
DF1 i
aj Substrate
Fig. 2-59. Sketch of electric fields for a multilayered material.
r Z
151 surface layer in question is assumed to be sufficiently thin (less than a few tens of A). Its refractive index and thickness are also assumed to be sufficiently uniform. If the refractive index and the average thickness of the surface layer are denoted by no and 1:o, respectively, the refractive index for a given depth (z) between the surface layer and the substrate (refractive index = n2) is expressed by n2(z) = n 2 + (n 2 - n2)F(z-'ro, or)
fz
(2-62)
0u
F ( z - v o , o') = V2xcr2 J-oo
(2-63)
And, when the layer composition and density continuously change from the surface to depth z2, the refractive index is expressed by n2(z) = n~ + (n~ - n~)[ 1 - exp( - z/z2)]
(2-64)
Using Eqs. (2-58) and (2-59), the surface roughness can also be included in n(z). Ntvot and Croce measured the reflectivity of grazing incident X-rays to examine how various methods of polishing change the surfaces of various kinds of glass. An X-ray tube was used in their experiments. Reflection curves were observed using glancing angles between 0 and 3 ~ For pure silica glass, polished with pine resin mixed with cerium oxide, the refractive index had a maximum value at 51 A, below the surface. This value is obtained by adding Eqs. (2-58), (2-62), and (2-64) and by comparing the reflectivity determined from the experimental results with that from the calculated ones. This means a high-density layer was formed at that depth. It was also found that it could be eliminated by heat treatment. The value for z2 obtained in this case is 225 A, and the refractive index decreases continuously according to Eq. (2-64). It was also found that the surface was not dense, having a roughness of 7.75 A. They also discovered that when aluminosilicate glass (density, 2.63; SiO2, 60%; A1203, 20%; CaO, 20%) was polished using pine resin mixed with iron oxide, a very thin low-density layer (density, 2.04; Zo, 21 A) was formed, with a high-density layer below (maximum density, 2.71 at 48 A; z2, 200 A). The surface roughness was found to be 9 A.
2.5.5. Calculation of reflection curves-2 (Matrix treatment) The methods for calculating reflectivity indicated in Sections 2.5.3 and 2.5.4 require complicated data handling when there are many layers. Vidal and Vincent [52] reported a very convenient method for calculating the reflectivity of multilayered samples incorporating roughnesses. In this method, one matrix corresponds to each layer. TO handle a multilayered film, the matrices representing the layers can simply be multiplied.
152 First, we will discuss the case where roughness is omitted, as shown in Fig. 2-60. The electric fields above and below the interface (y > Yo, Y < Yo) can be expressed by
e(x,y) = [E2exp( - i(2y) + e~exp(+i(2y)] exp(itrx)
(for y > yo )
E(x,y) = [El'exp( - i(ly) + El'rexp(+i(ly)] exp(iax) (for y < yo )
(2-65a) (2-65b)
ko=2~/~,
(2-66)
o[ = kocos01
(2-67)
(i = {(koni) 2 - a2} 1/2 (i=1, 2)
(2-68)
where ko is the wave number of the incident X-rays, 01 their glancing angle and ni (i = 1, 2) the refractive index. In the case of S polarization (the electric field vector is perpendicular to the incident plane), the electric fields can be expressed as follows, using the boundary condition of the continuity of electric field on the interface.
/p11
E~
P21
P22
, 69,
E1 'r
Pll -- alext~-i((1 - (2)Y]
(2-70)
P12 = a2ex~+ i ( ( l + (2~]
(2-71)
P21 = a2ext~-i((1 -!- (2)Y]
(2-72)
P22 = a l e x ' + i ( ( 1 - ~2~]
(2-73)
where al = 1+~1/~'2, a2 = 1-~1/(2.
The reflectivity, R, is given by
Medium 2 ( = n 2 ) E2
yo E1 r !
E 1'
Medium 1 ( = n 1 ) r
x
Fig. 2-60. Sketch of electric fields of a perfect interface between two media of refractive indices n l and n2.
153
R = [P21/Pll
I2
(2-74)
The case of P polarization (where the electric field vector is parallel to the incident plane) can also be handled if the above (1/~2 is replaced by (~l/ff2} X (E2/E1)
(2-75)
where ei = ni 2 is the dielectric constant. In the case of a multilayered material, the product of the matrix corresponding to each interface can be simply calculated. This method is very useful because it permits the use of the same procedure to calculate the reflectivity for samples having any number of layers. If cr is the roughness, then for S polarization the matrix will become
(E2) = (p 11exp[4 ~"1-~'2)20"2/2] E:~ p2lexp[4 r + ~'2}20-'2/2]
(2-76) P22ex~4 ~1_~2)2o-2/2]
E1 'r
The roughness of each interface can be determined by comparing the experimental results with the calculations. Vidal and Vincent [52] used the reflection curves for grazing incident X-rays to evaluate a multilayered X-ray mirror. We will employ this method to compare the effects of the surface roughness and interface roughness of a sample with a monolayer. Figure 2-61 shows the reflection curve of a nickel thin film (300 A) on a silicon substrate. It indicates that surface roughness mainly reduces the reflectivity, and interface roughness decreases the contrast of oscillations.
2.5.6. Calculation of X-ray fluorescence intensity The discussions above deal with the angular dependence of reflectivity, which is used for analyzing the structure of thin films. This structure can also be elucidated using the dependence on the glancing angle of the X-ray fuorescence emitted from constituent elements. Since this can be observed for each constituent element, it provides more information than the reflection curve alone. However, X-ray fluorescence from thin films is so weak at grazing incident angles that it was rarely used experimentally until SR became available. In this section, we will describe a method of calculating the intensity of X-ray fluorescence. The structure of a film can then be analyzed by comparing the experimental profile with the calculated results. Kr61 et al. [53] developed a method for calculating the intensity of X-ray fluorescence from multilayered samples, using Vidal and Vincent's treatment of matrices for calculating the reflectivity. Consider a multilayered material shown in Fig. 2-62. Layer j is located above the
154
10
-1
10 ;>.
......
~ , .
o"= 20/~ "'":
-
on a
Ni surface
( ~ = 0 on an interface between Ni and Si )
-2
9v,,,4 .v-,I
0
10
-
9
10-3
;,.
\
_
,,"_'~ :'',
,..
-4 10
-
-5 10
-
I
0
5
I
10
I
I 20
15
Glancing angle / mrad 0 10
-
. . . . . .
_
. . . . . .
_
_
_
_
~ = 20]k on an interface between Ni and Si -1 10
;> r O O
-
-2 10
-
10-3
" "J
_
-4 10
-
-5 10
m,
I 5
0
i 10
I 15
~i I 20
Glancing angle / mrad Fig. 2-61. The calculated reflection curves of a Ni thin film (thickness = 300 A) on a silicon substrate. Dotted lines indicate the case of neither interface nor surface roughness. Solid lines indicate the case of either surface or interface roughness: the upper figure shows values for a Ni surface where (r = 20 ~, and the lower figure shows values for an interface between Ni and Si where o" = 20/~.
j-th interface. If S polarization is assumed, then transmitted X-rays
Ej § and reflected X-rays
Ej- are expressed by
Ef (x,z) = Aj exp(ipjz) exp(ikoxx) E](x,z) = Bj exp(-ipjz)exp(ikoxx)
(2-77) (2-78)
155
kjx = kj cosOj = kj+lCOSOj+1 = kj+l, x = kox pj = kj sinOj + iaj = (kZn 2 - gn2cos20)l/2
(2-79) (2-80)
where Aj, Bj, and pj are complex quantities, and kj and aj are the magnitudes of the real and imaginary parts, respectively, of the complex wave vector. Refer to Fig. 2-62 for Oj. The complex wave vector for layer j is expressed by k02n~, using the complex refractive index, where k0 is the wave number (2~/~) of the incident X-rays and 0 their glancing angle. If the matrix connecting the electric fields above and below the j-th interface is lj, then
l e;(zj) l(1
~j rj l
ei( j)
Ej++l(ZJ)t Ej++l(Zj)=lj t Ei+l(Zj) E;+l(Zj)
where rj and tj are the Fresnel coefficients of reflection and refraction, respectively.
9Air or vacuum
E-~ (Zo)
E+ (Zo)
nj
"
o.
o.
Ej+I(Zj ) -
+ j) Ej+I(Z
E+ (ZN)
j-th interface nj+l
EN (ZN)
nN ns
Substrate
z Fig. 2-62 9Schematic representation of electric fields of a multilayered material.
(2-81)
156
zj,
The matrix, Tj+I(Z-Zj) , that connects the z coordinate, at the j-th interface with a point z inside the layer, (/+1) Zj+l), is determined as follows:
(zj
I
Ej++I(ZJ)~exp[-ipj+,(z-zj)] 0 exp[ipj+ l(Z-Zj)] Ej+1(zj)
E j+ 1(Z)
Tj+,(z-zj) E;+I(Z)
I
E;+l(Z)
(2-82)
Matrix Sj corresponds to the roughness of the j-th interface:
e;-r}e; rj(e;-e;)) SJ = l-~r~ rj(ef-ef ) ef-r~ef e+=exp[-(Pj+l+pj)24[2] ef =ext~-(Pj+l-pj)24]2 ]
(2-83) (2-84) (2-85)
where crj denotes the roughness parameter of the j-th interface. From Eqs. (2-81)-(2-83), the following relation is obtained:
( Ei(zj~ )=
Ei+,(z)
(2-86)
If the matrix that connects the electric field at the substrate with the electric field at the interface between the surface and the air or vacuum is PON,then
pl~
PoN=( pl0N p201N P22 ON
(2-87)
If this matrix is calculated, we can obtain
E~(zo)) Eo(zo)
= PON
where
Es+(ZN)is
leo(Zo)/e~(zo~
E+(ZN) 0
(2-88)
the value of the electric field inside the substrate.
can be obtained by calculating [p2~ ' / pl~
I~
The reflectivity
157 The electric field at below. If
z,
(Zj_1 < Z < Zj), Can be determined using matrix PiN, defined as shown
PjN = IjSj
T j + I ( Z j + I - Z j ) - " 9INSN
(2-89)
ET(z) =Tj(zTz)Pju(e+s(ZU)o) E;(z)
(2-90)
then
Es+(ZN) can be expressed as follows using the electric field of the incident X-rays, Ei. (2-91)
E+(zN) = E~(zo____.._~)= E__i_i p l ON
p l ON
Equations (2-89)-(2-91) make it possible to determine the electric fields in individual layers and in the substrate. Also, using Maxwell's equations, and taking the permeability/lj - 1, the magnetic field can be expressed by -
Hjy(Z)
-
/ko
-
(2-92) (2-93)
=0 +
(2-94)
In order to calculate the X-ray fluorescence intensity, the time average of the density of the energy flow, i.e., the Poynting vector P(z), must be determined. P(z) = C(Re(ExH*) )
(2-95)
where Re is the real part; (), the time average; H*, the complex conjugate of H; and C, the multiplicative constant. Substituting the electric and magnetic fields for the corresponding variables provides the components of the Poynting vector. We now describe the calculation of the intensity of X-ray fluorescence from a given layer. The excitation efficiency, fluorescence yield, and absorption of X-ray fluorescence are ignored. With this simplification, all we have to know is the intensity of the incident X-rays required to excite X-ray fluorescence in layer j. It is necessary also to subtract the X-rays that penetrate the sample without being reflected by the surface, those absorbed by layers up to j - l , as well as those going into layer j+ 1. The Xray fluorescence intensity, for z from zero to zj_l, is given by
158
Yj=C
[
1-R-I=I k
tan ~j = Pjz/Pjx
sin [~z/c)]Pi0(z) J
(2-96) (2-97)
where Pi0 represents the incident X-ray flux, R the reflectivity, ~j the direction of Poynting vector Pj(z), and Pjx and Pjz the x and z coordinate components, respectively, of Pj(z). The quantity ~)j is different from Oj representing the direction of propagation of the electric field [see Eq. (2-80) and Fig. 2-62]. Then, the X-ray fluorescence intensity from layer j can be expressed by
Yj+I- Yj (2-98) In the next section, we will discuss examples of thin film analysis using both the angular dependence of reflectivity and of X-ray fluorescence intensity. The analysis was performed by comparing the results of a SR experiment with the reflection curve and X-ray fluorescence profile obtained by the method of Kr61 et al [53]. 2.5.7. Characterization of titanium wafers
and
carbon-titanium thin films on
silicon
The angular dependence of reflectivity and X-ray fluorescence intensity, measured and analyzed using the grazing incidence method, is described in the following [60, 61 ]. The two samples used were a titanium thin film deposited on a silicon wafer by a sputtering method, and a carbon-titanium thin film also sputtered on a silicon wafer. Parts of these samples were cut off and heated in argon. The experiment was conducted using Beamline 4A at the Photon Factory. A Si (111) double-crystal monochromator was used to monochromatize 10 keV X-rays, which were shaped into a beam of less than 0.1 mm in height and 2 mm in width by passing them through a slit. The intensity of the incident and reflected X-rays was measured with an ionization chamber, and the intensity of X-ray fluorescence with a Si(Li) detector. Figure 2-63 shows the reflection curve measured for the titanium thin film. It reveals that the period of the oscillations is reduced after heat treatment. This suggests that high temperatures thicken titanium layers. However, since the amount of titanium remains the same, this effect is believed to be caused by titanium silicide formation during heat treatment. The observed X-ray fluorescence profiles are shown in Fig. 2-64. An attempt was made to create an optimum model by comparing the results with the calculations done by Kr61 et al. [53]. Curves calculated for a metallic titanium thin film, which was not heat-treated, shown in Fig. 2-65, did not agree with the experimental results. Therefore, the model was modified to fit the reflection curve and X-ray fluorescence profile, with the assumption that additional layers were formed on the surface and interface. Figure 2
159
Heat treatment temperature
/'/
\
"1-x
50~
V
"
'
"
"r.
8 Without heat treatment f
4
f
5
f
6
Glancing angle / mrad Fig. 2-63. Experimental reflection curves for Ti on Si wafer samples (with and without heat treatment). Heat treatment temperatures 9250~ 500~ and 750~ From Ref. [60], reprinted by permission of Plenum Publishing Corp., New York.
I
(d)
~i
(c)
*w~
8
3
4 5 6 Glancing angle / mrad
7
Fig. 2-64. Ti Ktx fluorescence intensity profiles for Ti on Si wafers. Without heat treatment (a); and with heat treatment at 250~ (b), at 500~ (c), and at 750~ (d). From Ref. [60], reprinted by permission of Plenum Publishing Corp., New York.
160
9 I,,,,,I
j~176
f
9 ~,,,I 9 1,,,,,I
i
ov,,~
~
o..,. "
t
i
e~
3
4
5
6
""
7
I
4
G l a n c i n g angle / m r a d
I
5
6
7
Glancing angle / m r a d
Fig. 2-65. Comparison of the calculated curves (broken lines), based on the model, with the experimental reflection curve and Ti Ktx fluorescence profile.
......
-...... Calc.
.=.
9 ~,,,I
. . . j
2
4
5
6
Glancing angle / m r a d
7
3
Exp.
I
4
I
5
I
6
7
Glancing angle / m r a d
Fig. 2-66. Comparison between the experimental results for the sample without heat treatment and the calculations based on the model for a Ti0.42Oo.58 (11 n m ) / T i (43 nm)/ TiSi2 (6 nm)/ silicon substrate. For the oxide layer tS= 9.16x10 -6, fl = 3.55x10-7; for the titanium t~ = 8.56x10 -6, fl = 4.84x10-7; for the silicide t~= 8.08x10 -6, fl = 2.67x10-7; and for the silicon substrate d; - 4.75xl 0 -6, fl = 7.42xl 0-8. From Ref. [60], reprinted by permission of Plenum Publishing Corp., New York.
161 -66 shows that the calculated curves based on this modified model are in good agreement with the experimental ones. The thickness, concentration, and density of each layer were determined. From these results, the total amount of titanium was calculated to be 24 l.tg cm -2, a value that agrees well with that determined by chemical analysis. Analysis of heat-treated samples revealed that transition layers on the surface and interface had been eliminated; a thick, homogeneous silicide layer had been formed instead. The experimental results for the thin film sputtered with carbon-titanium, before heat treatment, are shown in Fig. 2-67. The reflection curve shows two critical angles. The small one is that of carbon and the large one that of titanium. The X-ray fluorescence profile detected the Ka line of iron. Although iron is an impurity, analysis must take its existence into consideration. Assuming a carbon-titanium two layered structure, it was found that the sample had a surface roughness of 50/~, a carbon layer density of 1.7 g cm -3, and a titanium silicide (65% Ti and 35% Si) layer with a density of 4.4 g cm-3. The density of the carbon layer was considerably smaller than that of graphite (2.26 g cm-3). However, as shown in Fig. 2-68, there is a significant difference between the experimental and calculated reflection curves in the vicinity of the critical angles. Therefore, it was necessary to once again incorporate transition layers into the model. The curve from the experimental results and the curve which resulted when the transition layers were incorporated in the model are shown in Fig. 2-69. This modification suggests that there are two high-density transition layers (carbon, p = 2.0; carbon, p = 1.7) near the surface. The total concentration of iron contained in each of the three
~,
TiKtx
FeK~ m~ = (t.24
= .
;> O
!
I
I
I
2
3
4
5
I
6
Glancing angle / mrad
I
7
2
I
3
I
4
I
5
I
6
I
7
Glancing angle / mrad
Fig. 2-67. Experimental results for the carbon and titanium sputtered sample without heat treatment. From Ref. [60], reprinted by permission of Plenum Publishing Corp., New York.
162 1.0
.~n-I
0.5
1
2
3
4
5
6
Glancing angle / m r a d Fig. 2-68. Comparison of the experimental reflection curve (bold line) with the calculated curve; the sample is assumed to be carbon ( 2 8 0 n m ) / Ti0.65Si0.35/ silicon substrate [61 ]. The carbon layer contains iron as an impurity. The values of t~and fl of the carbon layer are affected by iron. For the carbon layer t5= 3.45x10 -6, fl = 2.57x10 -8 and for the silicide layer t~= 8.46x10 -6, t = 3.57x10-7.
carbon layers was 2%, which is in good agreement with the value determined from the X-ray fluorescence intensity. The results of the 250~ heat treatment sample, shown in Fig. 2-70, indicate much steeper curves near the critical angles, in the reflection curve and in the Fe Ks fluorescence profile. This suggests that the two high-density transition layers previously mentioned have been eliminated during heat treatment. In the 750~ heat treatment sample shown in Fig. 2-71, the titanium fluorescence profile is similar in pattern to the iron fluorescence profile. These similar patterns point to the formation of a homogeneous monolayer on the silicon substrate surface. We have shown that measurements of the reflection curve and X-ray fluorescence profile at grazing incident angles provide an excellent method for analyzing thin films. The benefits of this method are that it is nondestructive, and that sample composition, thickness, density and elemental concentration can be determined. Although still used in very few studies of this kind, SR is expected to be widely employed in the near future.
163 Carbon
20nm
S = 4 . 1 4 x 10 ~ , f l = 4 . 1 0 x 10 -8 , p - 2 . 0 Carbon S = 3 . 4 7 x 1 0 -4, f l = 3 . 4 4 x l O 4, 0 = 1 . 7
20 nm
Carbon 260 nm
S = 3.13 x 10-6 , f l = 2 . 5 7 x 10 4 , p = 1.5
Titanium silicide t5 = 8.46 x 10 -6, fl = 3.57 x 10-7,/9 - 4.35
23 nm
Silicon substrate
1.0
o~,~
0.5
1
2
3
4
5
6
Glancing angle / mrad
Fig. 2-69. Comparison of the experimental reflection curve (bold line) with the calculated curve [61].
164
Ti Ko~ fluorescence profile ]
~ v...~
Reflection curve
I
"7.
I
I
Fe Ko~ fluorescence profile m
I
2
I
3
4
5
2
Glancing angle /mrad
3 4 6 7 Glancing angle / mrad
Fig. 2-70. Experimental results for the samples without heat treatment (bold line) and with heat treatment at 250~ From Ref. [60], reprinted by permission of Plenum Publishing Corp., New York.
~
TiKo~
.~,,I
e~
r~
O
3
I
I
4
I
I
5
i
i
6
I
Glancing angle / mrad Fig. 2-71. Comparison of the Ti Kcx fluorescence profile with the Fe Ko~ fluorescence profile using the 750~ heat-treated C / Ti / Si substrate sample. From Ref. [60], reprinted by permission of Plenum Publishing Corp., New York.
165
2.6. PROSPECTS FOR FUTURE DEVELOPMENT With the advent of a third generation high-brilliance synchrotron source, and further extensive use of insertion devices, we can expect further advances in X-ray fluorescence analysis. We will discuss two such prospects: bulk analysis and surface analysis. The former mainly uses monochromatic excitation XRF, and the latter is based on the total reflection of Xrays.
2.6.1. Bulk analysis
Heavy element analysis Since conventional X-ray methods using ordinary tube excitation cannot provide sufficiently high excitation energy, researchers have been forced to use L- or M-series X-rays to analyze lanthanoids and actinoids. Compared with K-series X-rays, these characteristic X-rays have an intensity at least one order of magnitude lower and suffer from considerable interference because of closely arranged spectral lines, which result because the X-rays have small energy level differences. Furthermore, the energy region where these spectral lines are observed includes the characteristic X-rays of common, smaller atomic number elements (the third and fourth period elements). This results in a complex spectrum and makes accurate analysis even more difficult. Generally speaking, higher energy analyte lines reduce interference because their excellent penetrating powers decrease absorption by a matrix. Also, since no naturally occurring elements emit L-series lines of more than 20 keV, use of K-series lines has great advantages for the analysis of elements heavier than rhodium, both in precision and in sensitivity. Uranium is a good example. The K absorption edge of this element is approximately 115.6 keV. Therefore, if it were possible to excite uranium using an X-ray source with a higher energy level than 115.6 keV, analysis of uranium with very little interference would be possible. Consider another example. The K absorption edge of neodymium, used in materials such as laser glass, is at about 43.6 keV. If it were possible to excite it with X-rays of more than 50 keV, sufficient excitation efficiency would be achieved to conduct analysis of its trace quantities. In fact, an analysis of rare earth elements with K-lines, using VEPP-4 at Novosibirsk, equipped with a high-energy ring (5 GeV), detected amounts lower than ppm [62].
XRF using soft X-ray excitation X-ray fluorescence of second period elements (superlight elements) causes the release of a very small number of photons, compared with the number of photoelectrons and Auger
166 electrons observed. Also, the energy required for them to fluoresce is very low. These characteristics pose difficult problems for fluorescence measurement which uses an ordinary X-ray optical system. However, if the optical system is improved, and radiation from a large scale ring and an undulator combination is used as an excitation source, it will make the system sensitive enough for bulk analysis of these elements. This was tried at the Photon Factory, where an experiment has been reported using the undulator as the excitation source and a synthetic multilayer as the optical element [63]. Absolute (standardless) method The energy tunability and excellent collimation of SR are indispensable for accurate elucidation of the excitation and emission mechanisms of X-ray fluorescence, and for clarifying the interactions between analyte lines and any spectral lines emitted by elements other than analytes. Development of these analysis techniques will open up the possibility of an absolute method: i.e., a quantitative technique that requires no reference standard. When it is freed from reference standards, XRF will not be affected by the precision and accuracy of reference samples which have to be standardized by other methods such as chemical analysis. This will permit not only highly reliable ultratrace analysis, but also highly accurate measurements of the stoichiometric relationships of unknown substances. An example of analysis without standard references, based on the characteristics of SR, is a study by Bowen et al. [64]. They determined the concentration of As contained in a silicon sample ion-implanted with As, simply by measuring As and Si X-ray fluorescence intensities of the sample. They assumed a Gaussian distribution for the As concentration, and based the quantitative analysis on fluorescence yields and absorption coefficients available in the literature.
2.6.2. Surface analysis Excitation sources with higher brilliance, and improved measuring equipment and optical systems, will certainly combine to yield determination limits in the order of ppt (10-12 g g-l) and femtogram quantities (10 --15 g) using TXRF. As the use of SR increases, the structural analysis of thin films as described in Section 2.5 will also be employed to evaluate the surfaces and interfaces of various advanced materials. Recently, the X-ray standing wave method has received considerable attention as a total reflection surface analysis technique. It has been found to be useful for determining the structures of ultrathin organic films [65] and diffuse layers on liquid-solid interfaces [66]. This method is based on the phenomenon of diffraction and the accompanying oscillations in X-ray fluorescence intensity. In experiments, the angular dependence of the fluorescence intensity is measured and analyzed. Thus, depth profile analysis using X-ray standing waves has provided an experimental means of researching interfaces, an area that was previously only investigated theoretically.
167 As discussed above, the use of SR as an excitation source is creating a new range of possibilities. XRF, even itself, has made analysis more sensitive and has helped in the development of structural analysis techniques. In the future, SRXRF will provide more useful information for the characterization of a wide range of advanced materials.
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168 18. H. Hashimoto, H. Nishioji, H. Saisho and A. Iida, Adv. X-Ray Chem. Anal. Jpn., 20, 143 (1989). 19. B. M. Lairson and D. H Bilderback, Nucl. Instrum. Methods Phys. Res., 195, 79 (1982). 20. A. Iida, T. Matsushita and Y. Gohshi, Nucl. Instrum. Methods Phys. Res., Sect. A, A235, 597 (1985). 21. B. W. Batterman and H. Cole, Rev. Mod. Phys., 36, 681 (1964). 22. T. Matsushita and H. Hashizume, "Handbook on Synchrotron Radiation ", ed. E.-E. Koch, Vol. la, Chap. 4, p. 273, North-Holland, Amsterdam, 1983. 23. H. Oyanagi, "Photon Factory Activity Report 1989", KEK, Tsukuba, 1990, p. I-4. 24. C. J. Sparks, Jr., B. S. Bode and J. B. Hastings, Nucl. Instrum. Methods, 172, 237 (1980). 25. C. J. Sparks, Jr., G. E. Ice, J. Wong and B. W. Batterman, Nucl. Instrum. Methods Phys. Res., 194, 73 (1982). 26. B. W. Batterman and L. Berman, Nucl. Instrum. Methods Phys. Res., 208, 327 (1983). 27. See, for example, H. Takenaka, Y. Muramatsu, H. Takaoka, Y. Ishii and H. Hashizume, Rev. Sci. Instrum., 60, 2018 (1989). 28. T. Matsushita, T. Ishikawa and H. Oyanagi, Nucl. Instrum. Methods Phys. Res., Sect. A, A246, 377 (1986). 29. H. Winick, "Synchrotron Radiation Research", ed. H. Winick and S. Doniach, Chap. 2, p. 11, Plenum Press, New York, 1980. 30. H. Saisho and H. Hashimoto, "Trace Analysis (in Japanese) ", ed. K. Kurosaki and F. Konishi, Chap. 9, p. 245, Kodansha, Tokyo, 1988. 31. H. Saisho, Trends Anal. Chem., 8, 209 (1989). 32. W. H. McMaster, N. Kerr Del Grande, J. H. Mallett and J. H. Hubbell, "Compilation of X-Ray Cross Sections", UCRL-50174 Sec. II, Rev. 1, University of California, Livermore, CA, 1969. 33. S. Sasaki, "X-Ray Absorption Coefficients of the Elements (Li to Bi, U)", KEK Report 90-16, November 1990, M/D, KEK, Tsukuba, 1990. 34. W. Bambynek, B. Crasemann, R. W. Fink, H.-U. Freund, H. Mark, C. D. Swift, R. E. Price and P. V. Rao, Rev. Mod. Phys., 44, 716 (1972). 35. A. L. Hanson, Nucl. Instrum. Methods Phys. Res., Sect. A, A243, 583 (1986). 36. F. S. Goulding and J. M. Jaklevic, Nucl. Instrum. Methods, 142, 323 (1977). 37. L. A. Cunie, Anal. Chem., 40, 586 (1968). 38. R. D. Giauque, J. M. Jaklevic and A. C. Thompson, Anal. Chem., 58, 940 (1986). 39. A. Kn6chel, W. Petersen, G. Tolkiehen, Anal. Chim. Acta, 173, 105 (1985). 40. K. W. Jones, B. M. Gordon, A.L. Hanson, J.B. Hastings, M.R. Howells, H.W. Kraner and J. R. Chen, Nucl. Instrum. Methods Phys. Res., Sect. B, B3, 225 (1984). 41. S.R. Sutton, M. L. Rivers and J. V. Smith, Anal. Chem., 58, 2167 (1986). 42. J. M. Jaklevic, R. D. Giauque and A. C. Thompson, Anal. Chem., 60, 482 (1988). 43. M. Born and E. Wolf, "Principles of Optics", Pergamon Press, New York, 1980.
169 44. P. A. Pella and R. C. Dobbyn, Anal. Chem., 60, 684 (1988). 45. H. Hashimoto, Y. Iida, H. Saisho and H. Nishioji, Anal. Sci. [Supplement], 7, 577 (1991). 46. C. J. Sparks, Jr., Adv. X-Ray Anal., 19, 19 (1976). 47. A. Iida, K. Sakurai and Y. Gohshi, Adv. X-Ray Anal., 31, 487 (1988). 48. J. M. Bloch, M. Sansone, F.Rondelez, D. G. Peiffer, P. Pincus, M. W. Kim and P.M. Eisenberger, Phys. Rev. Lett., 54, 1039 (1985). 49. H. Kiessig, Ann. Phys. (N.Y.), 10, 715 (1931). 50. L. G. Parratt, Phys. Rev., 95, 359 (1954). 51. L. N6vot and P. Croce, Rev. Phys. Appl., 15, 761 (1980). 52. B. Vidal and P. Vincent, Appl. Opt., 23, 1794 (1984). 53. A. Kr61, C. J. Sher and Y. H. Kao, Phys. Rev. B, 38: Condens. Matter, 8579 (1988). 54. T. C. Huang and W. Parrish, Adv. X-Ray Anal., 35, 137 (1992). 55. K. Sakurai and A. Iida, Jpn. J. Appl. Phys., 31, L113 (1992). 56. M. Gentry, J. Durand, M. Erman, J. B. Theeten, L. N6vot and B. Pardo, Appl. Surf. Sci., 44, 309 (1990). 57. A. Kr61, C. J. Sher, D. R. Storch, S. C. Woronick, Y. H. Kao, L. L. Chang and H. Munekata, Surf. Sci., 228, 108 (1990). 58. A. Kr61, H. Resat, C.J. Sher, S. C. Woronick, W. Ng, Y. H. Kao, T. L. Cole, A. K. Green, C. K. Lowe-Ma, T.-W. Nee and V. Rehn, J. Appl. Phys., 69, 949 (1991). 59. S. M. Heald, H. Chen and J. M. Tranquada, Phys. Rev. B, 38: Condens. Matter, 1016 (1988). 60. H. Hashimoto, H. Nishioji and H. Saisho, Adv. X-Ray Anal., 35, 807(1992). 61. H. Hashimoto, H. Nishioji and H. Saisho, "Proceedings of the International Conference on Materials and Process Characterization for VLSI, 1991(ICMPC "91)", ed. X.F. Zong, Y. Y. Wang and X. Y. Gu, p. 592, National Microanalysis Center for Microelectronic Materials & Devices, Shanghai, October 1991. 62. V. B. Baryshev, A.E. Gilbert, O.A. Kozmenko, G.N. Kulipanov and K.V. Zolotarev, Nucl. Instrum. Methods Phys. Res., Sect. A, A261, 272 (1987). 63. A. Iida, Y. Gohshi and H. Maezawa, Adv. X-Ray Anal., 29, 427 (1986). 64. D. K. Bowen, S. T. Davies and T. Ambridge, J. Appl. Phys., 58, 260 (1985). 65. M. J. Bedzyk, G. M. Bommarito and J.S. Schildkraut, Phys. Rev. Lett., 62, 1376 (1989). 66. M.J. Bedzyk, G. M. Bommarito, M. Caffrey and T. L. Penner, Science (Washington, D.C.), 248, 52 (1990).
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Applications of Synchrotron Radiation to Materials Analysis H. Saisho and Y. Gohshi (Editors) 9 1996 Elsevier Science B.V. All rights reserved.
171
CHAPTER 3
M I C R O B E A M AND C H E M I C A L STATE ANALYSIS Shinjiro HAYAKAWA and Yohichi GOHSHI D e p a r t m e n t of A p p l i e d Chemistry, Faculty of Engineering, University of T o k y o , 7-3-1, H o n g o , B u n k y o , T o k y o 113, J a p a n 3.1. I N T R O D U C T I O N Since the discovery of X-rays, X-ray microscopy has received great attentions because of its potential in realizing the highest spatial resolution of all the types of light microscopes. A number of types of X-ray microscopes have been designed since the 1950s [1,2]. However, there have been few attempts to make them, and the performance of these microprobes has been poor because the requirements for the optical elements were far beyond the technologies available at the time. Most of the X-ray microscopes which use a conventional X-ray source have been of the imaging type ( the projection type ) because the acceptance, or aperture, of the grazing-incidence optics is too small to realize the small intense X-ray beam which is needed as a probe beam for a scanning X-ray microscope or an X-ray microprobe. The advent of synchrotron radiation (SR) as a light source for use in materials science has changed the situation. Horowitz and Howell were the first to fabricate a scanning type SR Xray microscope with analytical capabilities [3,4]. Figure 3-1 shows a schematic layout of their microscope. Because of the small divergence of the SR, X-ray focusing optics could be used effectively to achieve the photon flux density. Fabrication of an ellipsoidal or a toroidal figure was difficult, so a bent cylindrical mirror was used instead [5]. The size of the resultant beam spot was 1 mm x 2mm in the vertical and horizontal directions. To realize sub-mm spatial resolution, a small pinhole of 2 ~tm diameter was placed in front of the sample. The advantage of the scanning X-ray microscope is its ability to detect various types of signals from a sample, and X-ray images were demonstrated with transmitted X-rays and fluorescence X-rays. Although the idea of the analytical X-ray microprobe was excellent, its analytical capability was limited to elemental analysis because the microprobe used "white" SR. Furthermore, there were difficulties in improving its spatial resolution of the X-ray microprobe. There have been great improvements in the spatial resolutions of electron and ion microscopes as a
172 result of progress in the focusing of charged particles. Because electrons and ions can be used as primary beams for X-ray emission analysis, an electron probe microanalyzer with an X-ray detection system can be regarded as an X-ray microanalyzer. The importance of the SR X-ray microprobe has been recognized widely since Sparks demonstrated the feasibility of using monochromatized SR X-rays for X-ray fluorescence analysis, as described in the preceding Chapter. Sparks has calculated the expected performance of the SR X-ray microprobe in terms of the trace sensitivity. The heat load on a sample was also compared for SR X-ray-, electron-, and proton microprobes [6]. Recently, several types of SR X-ray microprobes have been realized. They are used in materials science, and the advantages of SR X-ray microscopy are well recognized. These advantages are not limited to their trace-sensitivity but also apply in
chemical state analysis. Moreover, the
importance of the X-ray microprobe is widely understood as being a fundamental excitation technique for various X-ray analyses. In this Chapter, the design considerations and reported performance of the analytical SR X-ray microprobes are reviewed. In addition to the analytical X-ray microprobe, which mainly employs hard X-rays, there is great progress with
soft-X-
ray microscopes for the observation of biological sections. For information on the soft-X-ray microscope the reader is referred to recent conference proceedings[7-9].
ctron synchrotron
X,Y,Z mechanical stage l,,,,,,Sampleholder
\
~~
Vacuum
Air
" / ~ l"/ . ~ " ~
Gas-filled proportionalcounter
Synchrotron ~ . ~ ~ ~ Beam stop radiation ~ 4 ~]~~.__ X-ray condensing mirror 10 #m | beryllium | window " Pulse amplifier Pinhole Energy discriminator in lOOmmgold
X,Y transducer Fig. 3-1.
0
Oscilloscope display
A prototype of a scanning X-ray microscope by Horowitz and Howell [3].
173
3.2.
X-RAY FOCUSING OPTICS
In this Section, the background for the design of SR X-ray focusing optics will be described. Since the 1950s, there have been several important papers about X-ray imageforming optics. Their most important characteristic is the beam size which is obtainable. However, in the analytical sense, the obtainable photon flux or the gain in photon flux density also have great importance.
3.2.1.
X-ray optical elements
The X-ray optical elements and their applicable energy ranges are shown in Table 3-1. For the hard X-ray region, the optical elements are limited to crystals, multilayers and totalreflection mirrors. Crystals and multilayers are used with glancing angles which satisfy the Bragg condition. Total-reflection mirrors are used with glancing angles less than 1 degree for hard X-rays. Although the difference in the glancing angles required for the total-reflection mirror and the crystals and multilayers is large, the optical systems composed of these optical elements use the grazing incidence geometry. Table 3-1 X-ray optical elements and applicable energy ranges which are commonly used. Optical element
Function
Applicable energy range
Total-reflection mirror
Reflective
~ 20 keV
Multilayer
Diffractive
~ 20 keV
Crystals
Diffractive
~ 60 keV
Diffraction grating
Diffractive
~ 1 keV
Zone plate
Diffractive
~ 1 keV a
a Recently, several attempts have been made to fabricate hard-X-ray zone plates: fairly good performance has been reported [ 10,11 ].
174
3.2.2. Grazing incidence optics. The parameters for grazing incidence optics are shown in Fig. 3-2. The optical axis lies between the source S and the image I. The typical reflecting point is represented as P. The plane including S, P and I is a meridional plane and the plane normal to this is a sagittal plane. The radius of curvature of the reflector should be different for the meridional plane and the sagittal plane when focusing the divergence from a point source with grazing-incidence optics. With the distance between SP and PI, the famous mirror formula describes the radii of curvatures in the meridional and the sagittal planes.
1/FI +I/F2= I/f
(3-1)
f=Rmsin O/2=Rs/(2sin 0)
(3-2)
The magnification of the optics is defined as
M=F2/Fl =sina/sina' =a/a'
(3-3)
By considering the nature of an ellipsoid, one can see that rays from one of the foci are reflected and converged into another Iocus. It is evident that the ellipsoidal figure satisfies the mirror formula at all the points on the surface. Moreover, the sum of distances from the foci is constant at all the points on the ellipsoidal surface. Therefore the ellipsoidal mirror is ideal for a point source, for both geometrical and wave optics. When one is designing practical X-ray-focusing optics, one should consider a real source having finite dimensions. One of the practical ways to evaluate the performance of the optics is to use the ray-tracing method. The focused beam spot is expressed in terms of the sum of intersections between rays and the screen at the image plane. Another way is to
use the
theoretical descriptions which are classical but are suited to comparisons of optical systems. In this method it is helpful to consider the point-spreading functions for point sources at on-axis and off-axis positions. In the next Section, the performance of the optics is discussed, along with the theoretical descriptions.
175
$ -1
$ FI
F1
2 ~..~1
e
I
I p
m
Fig. 3-2.
L s
Parameters for a grazing incidence optical system: S(xs,Ys), I(xi, Yi) refer to coordinates in the source plane and image plane, respectively; Lm, Ls are mirror lengths in the meridional plane and sagittal plane; Rm, Rs are the radii of curvature in the meridional plane and sagittal plane; 0 is the glancing angle; FI=S 1/cosa,
F2=S2/cosa'
3.2.3. Image formation
of a point source
An ellipsoidal surface is ideal for use with a point source. However, the aspherical figure is difficult to fabricate. Therefore a substitutional configuration is often employed. One methtxt uses a toroidally shaped mirror which satisfies the mirror formula in its central part. Another uses a Kirkpatrick-Baez mirror [12] which consists of two concave (or elliptical ) mirrors lk~r
176 vertical and horizontal divergence, respectively (Fig. 3-3). With the concave mirror the beam size is determined by the spherical aberration [ 12]. Figure 3-4 shows a cross section of a concave mirror in the meridional plane. Points S and I are on the so-called Rowland circle. A ray SOI shows a principal ray. As other rays from a point source S are not converging into point I, blurring of the image occurs in the image plane. The size of the image in the meridional plane, Sm,
is represented by the following
expressions.
Fig. 3-3.
Kirkpatrick and Baez mirror [12]
jt
jO
i s
i
/ i
/
P
i J J~
H
!
I
Fig. 3-4.
i
ss ss
Js
,I',
9 9 9
I
I"
i J i
._-.-.--- "--.--re "," L,""
,,."F ~ / /
91 j S
s S
O
(Left) Reflection of X-rays from a spherical mirror at small glancing angles. R is the radius of a circular reflector. (Right) Details of ray intersections at the image of a point source formed by a spherical mirror illuminated at a small glancing angle [12].
177
Sm=-3/2Ry2g(M)
(3-4)
where
(3-5)
g(M)={( l+2a)M 2-1 }/(M+ 1)
a=~,/0
M=F2/F1 In focusing optics whose magnification is far less than unity,
Sm
can be regarded as
Sm=_3/2R.~2 If we consider a point source and a concave mirror of magnification M=I/10
(3-6) with a
glancing angle of 8.4 mrad placed at a distance of 10 m from the source, the radius of the curvature R must be 216 m. When the dimension of the mirror is 24 cm in the meridional plane the focused beam has a finite size of 0.4 mm in the meridional plane even with the point source. To reduce the spherical aberration while maintaining the acceptance, the glancing angle must be larger to reduce the effective mirror size in the meridional plane.
3.2.4. I m a g e f o r m a t i o n
of a real
source
In considering the performance of the optical system with a real source of finite dimensions, it is most convenient to think about the image of a point source placed at an off-axis position. When the effects of an off-axis point can be neglected, the optical system should obey Abbe's famous criterion, which is sin a=sin a'= con st For grazing incidence optics,
(3-7)
Abbe's criterion should be satisfied independently, both in the
meridional and sagittal planes. Figure 3-5 demonstrates that the Abbe's criterion cannot be satisfied with single reflector and that it is easier to realize the criterion when double reflectors are employed. Abbe's criterion suggests that the ellipsoidal mirror has limitations in covering a large field of the view. The Wolter mirror was proposed in 1952 to overcome this limitation [13,141.
178
S
$
I
Fig. 3-5.
I
Abbe's sine theorem. A comparison between single (Left) and double (Right) reflectors.
3.2.5. Ellipsoidal and W o l t e r m i r r o r s The schematics of ellipsoidal and Wolter (type I) mirrors are shown in Fig. 3-6. For an optical system having axial symmetry rays from a point source are equivalent wherever the source is on the optical axis. To compare the performance of the two types of mirrors, traces are shown for two rays from the displaced point source in the meridional plane. The meridional ray crosses the image plane at the point Im and the sagittal ray crosses at the point Is. For the ellipsoidal mirror, Im and Is come on opposite sides of the optical axis. With the first order approximation about 0, the distance between two points can be expressed by
Imls=2Md
(3-8)
where M is the magnification of the ellipsoidal mirror and d is the displacement of the point source S 'from the optical axis. For the Wolter mirror, Im and Is fall in the same position. The resultant images of the point source with rays of various directions are summarized in Fi g. 3-7. When the point source is located on the optical axis the resultant image also becomes a point shape for both cases. However, when the point source is displaced the resultant images are quite different for the ellipsoidal and the Wolter mirrors. With the ellipsoidal mirror the resultant image becomes an arc of radius Aid. The length of the arc is determined by the portion of the annular aperture as indicated in Fig. 3-2. Using a Wolter mirror of ideal shape the resultant image is a point shape. Considering the real source of finite dimensions, the image obtained with the ellipsoidal mirror becomes circular when more than half of the annular aperture is used. The radius of the image is determined by the maximum displacement of the source from the optical axis. The
179 Wolter mirror provides a uniform transformation, neglecting the higher order aberrations. The results show that the Wolter mirror is not as sensitive to misalignment of the optical axis as is the ellipsoidal mirror.
80 v
~
~v
Is
z-~ SOURCE ~
ELLIPSOIDAL
~IMAGEr.....
80
S
z
j
Fig. 3-6.
~
I ~~ SOURCE
I~
ELLIPSOIDAL
a'
MAGE HYPERBOLOIDAL
A schematic representation of, (a) ellipsoidal and, (b) Wolter mirrors. To show the characteristics of these mirrors, meridional ( in plane) and sagittal (out of plane) rays from a point source at the focus (S) and those from a displaced source (S') are
shown in the Figures.
180
SOURCE
IMAGE Ellipsoidal Wolter
y x
k Y
Y
' ' "
x
Fig. 3-7.
•
I
l
l
"=
x
Comparisons of images between ellipsoidal and Wolter mirrors. From top to bottom, a point source on the axis, a point source off the axis, and a real source of finite dimensions. ~brepresents the portion of the annular aperture as indicated in Fig. 3-2.
3.2.6. Expected photon flux with an X-ray focusing system For an analytical X-ray microprobe, both the spatial resolution and the sensitivity are important. In order to estimate the photon flux expected with an X-ray focusing system the brilliance of the source, B, the acceptance of the optics, and the reflectivity (or efficiency) of
181
the optics should be considered. The acceptance can be defined as the divergence from the source which can be received with the optics. If we consider the grazing-incidence mirror shown in Figure 3-2, the vertical acceptance qJv can be expressed in terms of the mirror length in the meridional plane (Lm) and 0, as
~Pv=LmsinO/F1
(3-9)
For the ellipsoidal mirror, the horizontal acceptance ~Ph at the center of the mirror can be expressed in terms of Rs and the half-width of the annular aperture, AR, as ~Ph=2V'((Rs+AR)2-Rs2)/F1
(3-10)
where
AR=Lmsin O/2
(3-11)
By considering the monochromatic X-rays of energy E from a synchrotron radiation source of brilliance B, the expected incident photon flux to the optical system in a small divergence from a source of unit dimensions can be expressed as
AP(E) =Ba ~v k Wh
(3-12)
The photon flux after the optical system can be given in terms of the reflectivity at the mirror, R(E, 0), and the absorption through the optical path, A(E). Therefore, the photon flux at the focus can be expressed as P(E)=A(E) ~ ~ ~ S BR(E,O) d~Pvd~Phdxdy
(3-~3)
If we neglect the angular distribution of the brilliance, the photon flux can be shown to be
P(E) =A (E)B ~ crxcryR (E, O)~Pv~Ph
(3-14)
where Ox and cry are the source size in the x and y directions. With a distance L between the source and the focus, the approximate beam size without the focusing system is ~pvL and ~PhL in the vertical and horizontal directions, respectively. Therefore, the intensity gain with the optical system will be defined by
182
G(E)=L2R(E, O)~Pv~h/(S x' Sy ') where
Sx'
and
(3-15)
Sy' represent the beam size at the focus. In general, the intensity gain is largely
affected by the acceptance of the optical system. Therefore, a larger glancing angle is advantageous not only for the spherical aberration but also for the photon flux.
3.3. 3.3.1.
EXAMPLES
OF X - R A Y F O C U S I N G
SYSTEMS
Introduction
Characteristics of most of the X-ray focusing systems currently employed for X-ray fluorescence microprobes are shown in Table 3-2. For the soft X-ray region there are several types of optical systems which are successfully used for photoelectron microprobes.
These
are mentioned in the following Sections. Most of optical systems are classified into two categories; reflective optics and diffractive optics. A combination of reflective focusing optics and a monochromator can produce an energy tunable X-ray microbeam while maintaining the beam position. However, diffractive focusing optics can usually provide a higher photon flux than can be obtained with the reflective optics because they can perform both as monochromators and focusing devices of larger acceptance. 3.3.2.
Reflective
optics
Figure 3-8 shows examples of reflective optics. By using a Wolter mirror a micron-sized hard X-ray beam was first realized. However, effects of the surface roughness were also indicated by the difference in beam size in the sagittal plane and meridional plane. In the Kirkpatrick-Baez (K-B) mirror systems, crossed elliptical mirrors are used instead of the concave mirrors, to eliminate spherical aberration. Although a spatial resolution of several microns has been obtained with the Wolter and K-B mirrors,
the fabrication of large
aspherical mirrors is still limited by technical difficulties. Therefore, the photon flux with these optical systems is not sufficient for various types of analytical applications. In order to optimize both the spatial resolution and the trace sensitivity a combination of the focusing optics with the pinhole is employed, as was first demonstrated by Horowitz and Howell [3]. Figure 3-8(c) shows a microprobe system with an ellipsoidal mirror. In this case, the surface-
183 finishing of the mirror was carried out by skilled technicians to achieve sufficient reflectivity but at the sacrifice of the accurate contour. Table 3-2 X-ray focusing systems for X-ray fluorescence microprobes in the hard X-ray region. DCM, double crystal monochromator; CCM, channel cut monochromator; SMM, synthetic multilayer monochromator Source Optics
Beam size
Photon energy
Refs.
PF Wolter mirror
1.6 ~tm x 34 ~tm
K-B mirror
3.8 ~tm x 1.7 ~tm
K-B mirror
3.0
gm x 4.2
~tm
5.5 gm x 6.1 gm Ellipsoidal mirror Zone plate
10-200 ~tm with a pinhole 6.5 ~tm x 7.7 ~tm
8keV tunable(DCM) 5.4 keV tunable(CCM) 10.5 keV tunable(DCM) 12.6keV tunable(SMM) 4-20 keV tunable(DCM) 8keV
15, 16 17, 18 19
20,21 10
NSLS
Ellipsoidal mirror Multilayer coated K-B mirror
5 ~tm x 5 ~tm
10keV
22 23,24
Curved Si crystal
10 ~tm x 20 ~xm
15keV
25
20 keV
26
8.3keV
27 28
SRS
VEPP-3 Pyrolytic graphite LURE Bragg-Fresnel (K-B or Elliptical)
3.3.3.
20 gm
Diffractive optics
Typical examples of diffractive optics are shown in Fig. 3-9. Following the invention of the sagittal focusing monochromator [29] most proposed microprobe optics have included bent crystals of variable radius. However, it is difficult in practice to realize a spatial resolution of less than several tens of microns with these optics, and the bent crystal of fixed radius is
184 employed instead. Owing to the fixed radius, the incident X-ray energy is not tunable. However, a 15 keV microprobe shown in Fig 3-9(a) is successfully used for XRF trace element analysis.
M=1/13
a
M=1/20
SR
2.45m
F2
~:: SLtT . I CONDENSER MIRRO~~~
FOCUSING
MIRROR
PINHOLE Ta 60~m~
"(i
11.2m/~' g
MONOCHROMATOR M~
F
Mz
b
C
I
Sample
Pinhole +Beam monitor ~ ~ J ~ Ionization chamber Ellipsoidal mirror
Si(Li)
Monochromator SR Fig. 3-8.
X-ray focusing systems wit h reflective optics with:
(a), tandem Wolter
mirrors[ 16]" (b) Kirkpatrick-Baez mirrors [18]" (c) an ellipsoidal mirror [20].
185
a Synchrotron source (white radiation)
M1
Multilayer coated mirrors Solid state Si(Li)detector M2 Sampl ~ ~ spot ~ ~ Focal ,.,
Fluorescent x-rays
Scanning stage
b
Ion
Storage Sourceslit ring
chamber Si(Li) Anti-scat. i Object detector slit ~1 ~[ slit [~ ~~~,J~] ~ ~
t
image plate
Toroid Si(111) S.C
C
lit S"
F"~""~ [~rPinhole
, ~ ' ~
Zone plate
"~'*~~ !~'~~ Monochromator X-ray d
Slit 7
10 1. Inputslit 2. BFMFE 1 3. BFMFE 1 stage 4. Axisof rotation 5. Goniometer 6.7. Slits 8. BFMFE2 9. BFMFE2stage 10. Recording plate
Fig. 3-9.
X-ray focusing systems with diffractive optics with: (a) multilayer coated K-B mirrors[23]" (b) a bent Si(111) crystal[25]" (c) a zone plate[10]" (d) Bragg-Fresnel lens (K-B) [27]
186 Figure 3-9(a) shows a multilayer coated K-B mirror system. The coating of multilayers on the mirror surface limits the applicable energy range. However, it can make the glancing angle much higher than achieved with the total-reflection mirror when X-rays of the same energy are focused. As mentioned previously, the larger glancing angle is advantageous both for the spherical aberration and the acceptance of the optics. Figure 3-9(c) shows an optical system with a zone plate. Despite the successful fabrication of zone plates for the soft-X-ray region, using lithographic techniques, it is difficult to fabricate zone plates for hard X-rays because of the rays' large penetration. In order to obtain a zone plate of sufficient stopping power, a high aspect ratio is required. Therefore a novel technique was invented to replace the conventional lithographic techniques. Saitoh et al. first fabricated a hard X-ray zone plate using the sputtering method [10]. Layers of C and WSi2 were alternately sputtered around a fine Au wire of 18 ~tm diameter. The performance of the zone plate was evaluated with 8 keV X-rays, and a focused beam spot of 6.5 ~tna by 7.7 ~tm in the vertical and horizontal directions was realized. Recently, another hard-X-ray zone plate (phase zone plate) has been reported, which can produce a focused beam spot of 3 ~tm by 8 ~tm in the vertical and horizontal directions, respectively [ 11]. A recent advance in X-ray optics is the Bragg-Fresnel lens (BFL) which uses a combination of the Bragg reflection and Fresnel zone. Since the first report of a linear Bragg-Fresnel lens [30], two types of focusing system have been proposed. One is the K-B configuration with a crossed linear BFL, and another uses an elliptical BFL. An X-ray fluorescence microprobe with the elliptical BFL is now in operation at LURE [28].
3.4. C H E M I C A L S P E C I A T I O N USING SR 3.4.1.
Introduction
Several kinds of X-ray analyses can provide chemical state information; X-ray photoelectron spectroscopy
(XPS),
Auger
electron
spectroscopy
(AES),
X-ray
fluorescence
spectroscopy(XFS) and X-ray absorption spectroscopy(XAS). Of these, XPS, AES and XFS are the popular analytical methods with conventional X-ray sources. Therefore, it is not necessary to describe their principles in this Section. The use of XAS has become popular since the advent of SR as an X-ray source having a continuum energy distribution. Spectral structures which appear around the absorption edge energy in X-ray absorption spectra are called X-ray absorption fine structure (XAFS), and both chemical state information and local structural information can be obtained from XAFS.
187 Various combinations of SR X-ray microprobe and spectrochemical techniques have the promise of becoming powerful methods for the characterization of advanced materials whose function is strongly related to the chemical state of a local area in a material. Recently, several reports have appeared of X-ray spectrochemical analysis with spatial resolution. In this Section,
X-ray spectroscopies for chemical state analyses are
briefly reviewed.
The
advantages of energy tunability will also be discussed in order to point out the potential of Xray spectrochemical analysis with an energy-tunable X-ray microprobe.
3.4.2. Chemical speciation utilizing X-ray absorption fine structure The chemical states of X-ray absorbing atoms are reflected in the fine structures of the Xray absorption spectra. A detailed analysis of X-ray absorption near-edge structure (XANES) reveals the electronic structures of an element of interest, and the extended X-ray absorption fine structure (EXAFS) is used to obtain local structural information on the element of interest -i.e., the nearest-neighbor distance and coordination numbers. XAFS spectra can be used for detailed analysis for revealing the chemical state of unknown materials and as finger prints for the identification of chemical species. In order to obtain XAFS spectra one may use several signals from a sample: transmitted Xrays, fluorescent X-rays, and Auger or secondary electrons. The XRF detection becomes advantageous in the following two cases. The first case is when the element of interest is a minor component or when the sample is too thin for measurement of XAFS spectra of sufficient S/N by the transmission method. The second case is when an XAFS spectrum is measured from a small region of interest in a heterogeneous sample and the absorption of the sample is not always in the appropriate range for transmission measurements. Figure 3-10 shows XANES spectra of various oxides [30].
Incident SR X-rays were
monochromatized with a Si(111) double crystal monochromator, and fluorescence detection was employed using a Si(Li) detector. The first inflection point of the spectrum from the pure metal is set to be relative zero. Figure 3-11 shows the relationship between the absorption edge energies measured using the transmission method and the XRF detection method. A fairly good correlation was obtained.
188
23
CuK,-'/ /! ~.','~Cu2C
;s
FeO~;~ Fe304
03
I!
," "',,
Ni
MnO--~,!].-~-Mn203 -./Mn02
x
i
-20
0 20 40 X-ray Energy / eV
-20
0
20
40
X-rag Energy ! eV
Fig. 3-10 XANES spectra of various oxides with the XRF detection method [31 ]. .....
~
o f
l~0I
Fe203j~U' L Fe304D" K4Fe(CN)6 FeO /'O'FeCI3 I o,/ I FeSO4 P" I FeFsS,,.,O,'O FeCI2
1 F
ul
U
I
0" LII
i
i
I
i
I
0 Fig. 3-11.
I
,.-Y
f
i i
I
i
I
i
!
i
[
i
i
5
!
i
i
i
i
i
l
i
i
i
i
10
i
i
i
l
!
l
i
Edge Energy(F)/eV
i
!
l
i
I
!
15
Comparisons of the absorption edges of various iron compounds: (F) and (T) refer to the fluorescence and transmission methods, respectively.
189 Although XRF detection becomes advantageous with the X-ray microprobe,
spectral
distortions caused by self-absorption effects should be considered when the element of interest is a major component in a sample. Figure 3-12 shows Fe K-edge XANES spectra from a 400 nm evaporated thin film on Mylar and from an infinitely thick iron section with different takeoff angles [32]. The incident beam size was restricted to 19 ~tm and 12 l~m in the horizontal and vertical directions, respectively. The XANES spectrum from the iron section with a takeoff angle of 40 ~ clearly shows spectral distortion resulting from the self-absorption which is described by the theoretical predictions (dotted lines). The results show that the effects of selfabsorption can be overcome with a detection-geometry having a small takeoff angle. The small-takeoff-angle technique is also effective with self-absorption effects in EXAFS spectra [33].
Z21 v
2=-
I--
03 Z I,.1.1
IZ
iii (.3 Z iii r.D 03 iii r'v" O Zl .,,.J Lt_
/i ,
I,
] !
I
7.10 X-RAY ENERGY/keV Fig. 3-12.
I
7.15
Micro XANES spectra of Fe obtained using the fluorescence detection method: (a), from a 400 nm evaporated thin film on Mylar and from an infinitely thick iron section with takeoff angle of, (b) 5 ~ and (c) 40 ~ Solid lines show the data obtained and the broken lines show the calculated XANES spectra [32].
190 3.4.3.
Chemical speciation with
XRF spectroscopy
Chemical speciation using X-ray fluorescence spectroscopy has had a long history since the 1920s. The XRF, caused by electronic transitions between the inner shells of an atom, shows chemical shifts which depend on the environment of an X-ray emitting atom. Band spectra caused by the electronic transition from the outer shell to the inner shell reflect the partial density of states of the valence band. The K-L(Ka) spectra of second row elements and KM(K[3) spectra of the third row elements are typical examples of valence band spectra. Figure 3-13 shows B K-V XRF spectra and boron K edge XANES spectra of B, BN and B203 [34]. As is widely known, the fluorescence cross section is small for light elements. Therefore these measurements were difficult without the brilliant X-rays from an undulator.
.! o ~
B
emission I
o
2?
h-B.
_
_
f'-J"
r
B203
!
x
.,..9":" o
150 Fig. 3-13.
ption
/
160
:
t :
o. !-
180 190 170 Photon energy / eV
P-200
210
X-ray fluorescence spectra and boron K-edge XANES spectra of B, BN and B203. To obtain the XRF spectra, monochromated first harmonics undulator radiation (212 eV) was employed.
191 For the observation of chemical effects in the XRF spectra from heavier elements, conventional XRF spectrometers have insufficient energy resolution. To overcome the limitations, a double crystal spectrometer which achieves adequate energy resolution for a wide angular range is preferable [35]. Although the spectrometer has been used successfully with the conventional X-ray sources, there has been no report of its use with the SR, owing to its low efficiency. Recently, another type of optical geometry has been reported which can realize high energy resolution utilizing SR. Figure 3-14 shows the experimental scheme [36]. A dispersed XRF spectrum is detected with a position-sensitive proportional counter(PSPC) and the energy resolution of the spectrometer can be adjusted via the beam-irradiated area and the sampledetector distance. In order to realize an energy resolution sufficient for chemical state analysis the beam size was restricted by narrow slits to 20 ~tm in the horizontal plane and the sampledetector distance was set to be 1010 mm. Figure 3-15 shows an XRF spectrum of Cu metal obtained with the (444) reflection of a mirror polished Si crystal. The FWHM of the Cu KL3(Kal) peak was 2.6 eV, which is almost identical to the natural width of the spectrum. Therefore the energy resolution, dE/E, of the spectrometer is supposed to be less than 3 x 10-5.
l-
1st slit
Di UI
I.C.
2nd slit
Sample
( Analyzer crystal
Fig. 3-14.
C
A wavelength-dispersive (WD) spectrometer with a position sensitive proportional counter (PSPC) [36]: IC is a He-filled ionization chamber.
192
| m
. . . . . . .
Cu K-L3 A
....
!
c
m
Cu N-L2
l
LL
Fig. 3-15.
Wavelength
Energy---~-
XRF spectra of Cu metal from a high-energy resolution WD spectrometer equipped with a Si(444) crystal [36].
3.4.4. Advantages of energy tunability As shown in this Chapter, the tunability of the energy of SR can be used for the selective excitation of an element of interest. In the XRF elemental analysis the selective excitation is applicable to trace element analysis in a matrix of heavier elements. When the incident X-ray energy is tuned to be just above the absorption edge energy of an element of interest, interferences of the XRF lines from matrix elements which are heavier than the analyte can be avoided. The sensitivity is also optimized for the element of interest. Moreover, when the incident energy is selected carefully, selective excitation of the element in a specific chemical state can be realized. Selective excitation also has importance in X-ray spectroscopies because it provides a well defined excitation condition. In conventional XRF spectroscopy, primary X-ray photons usually have sufficient energy to excite a core electron to the continuum level (or out of the atom). This means that the polarization is not conserved between the primary X-ray and the fluorescent X-ray and that there are chances of multiple ionization in an atom. When selective excitation is employed and the incident X-ray energy is tuned around the absorption edge of the element of interest, a core electron can be selectively excited to an unoccupied level in an atom, and the transition should satisfy the selection rule. Spectroscopic techniques which use
193 this selective excitation around the absorption edge
are sometimes called threshold
spectroscopies and can provide a variety of chemical information. One typical example of threshold spectroscopy is the elimination of multiple ionization satellites in XRF spectra. In X-ray fluorescence spectra there often appear satellite lines whose origin is not tabulated as diagram lines. Some of these satellites are attributed to multiple ionization of the X-ray-absorbing atom. Beside the well separated satellite there exists a linewidth satellite (or parasite) which is observed as a slight profile change or asymmetric smearing of the spectrum line. Kawai et al. have investigated the profile changes in chlorine KL2,3(K~l,2) XRF spectra, and the spectral changes have been explained in terms of the chemical state of the C1 atom [37]. The assignments of these lines were derived from theoretical calculations. However, the threshold technique provides spectral profiles without the multiple ionization satellite. The difference between with and without multiple ionization data will be useful for chemical state analysis. Figure 3-16 shows argon K-M(K[3) spectra obtained with several incident X-ray energies around the K absorption edge of Ar [38]. Fig 3-16(a) shows a typical spectrum obtained at high excitation energies. Satellite peaks are observed to the higher energy side of the main line. With incident X-rays of smaller excitation energy the satellite lines become smaller and finally there only appears the main line. Satellite peaks are therefore attributed to multiple ionization. By using the threshold techniques, gas molecules of similar orientation can be selectively excited and the anisotropy of X-ray fluorescence has been demonstrated [39]. The threshold technique is also important for studies of X-ray Raman scattering where resonant effects play an important role [40]. As threshold techniques require a narrow energy bandwidth, both for incident X-rays and fluorescent or scattered X-rays, most of the experimental results have been obtained with the high brightness X-rays from the undulator. In the present SR facilities the undulator source is limited to the soft X-ray region. However, with the next generation of SR facilities, threshold spectroscopy in the hard X-ray region will become a promising field of Xray spectroscopies.
194
~
a
.i--'
,:,,. "-"
23
x:i L. v
o3
b
o3
v
9
l
03 Z w
.
U.! "
; .~ 3175
Z
":-
9
Z
03
.&
9
3190
"~
. . . . z. . . .
3205
-~ .$,
-
I--
J
3175
-
-
,
3190
ENERGY/eV
X.__ .--
c
i_
j
r ":z5
L.
d
k.. o3
,.,._..
v
:=--
-1
F-
03 Z
9
9
I--
Z
UJ
I-
i
3175
B
0 3
"
Z
,3 ]90
32-05
3-i--7S
ENERGY/eV Fig. 3-16.
'
3205
ENERGY/eV .~.a
,-.
9
l--
F-
"
:" 9
A
".'- -
9
"
3190
3205
ENERGY/eV
Argon K-M(KB) XRF spectra obtained with incident photon energy at: (a), 3281.4 eV" (b), 3245.9 eV" (c), 3213.1 eV" and (d), 3199.2 eV[37].
3.5. A N A L Y T I C A L 3.5.1.
X-RAY MICROPROBES
Introduction
A scanning X-ray microprobe can provide most X-ray analyses with spatial resolution 9 Since the paper by Horowitz and Howell [3], various types of analytical microprobes and their applications have been reported. In this Section, the analytical feasibility of the SR X-ray microscope (microprobe) is shown 9 3.5.2. Trace element analysis with micro X R F
One of the strong advantages of the SR X-ray microprobe is its small heat load to a sample 9 Figure 3-17 shows a photograph and XRF images of a tree ring from an 80 year old Japanese
195 cedar. The trace element localized in the tree ring gives information about the environmental changes that have occurred over the last few decades. Owing to the low heat conductivity of the tree sections it is usually difficult to avoid severe heat damage. However, by utilizing the X-ray microprobe system shown in Fig 3-8(c) trace element localization in the sample was visualized without damage being observed.
Fig. 3-17.
(a) a photograph of a tree ring after irradiation by 10 keV monochromatic SR for XRF imaging and (b) Ca, (c)Cu XRF images of a tree ring. A region of 3mm • 3mm was analyzed with a data acquisition time of 10 s for each pixel. Spatially averaged
concentrations of Fe, Cu and Zn were less than lppm.
Figure 3-18 shows a SR XRF spectrum obtained from a chelate resin bead, which was prepared to contain 100 ppm of Ca, Mn and Zn. The experimental system shown in Fig. 3-8 was employed, with a spatial resolution of 2001am. For a data acquisition time of 100s the minimum detection limit (MDL) of Zn is less than 1 ppm.
196
4000 3000 0') I-, Z 2OO0 O 0
1000 C __/L 2 3 4
5
L A 6
7
8
9
10 11 12
X-RAY ENERGY/keV
Fig. 3-18.
A SR-XRF spectrum from a chelate resin bead, prepared so as to contain 100 ppm of Ca, Mn and Zn, excited with 10 keV monochromatic X-rays.
3.5.3.
Micro X A F S m e a s u r e m e n t and chemical state i m a g i n g
The advantage of the energy-tunable X-ray microprobe appears in
micro XAFS
measurements. By employing an experimental system as shown in Fig. 3-8 the Cu K-edge XAFS of an evaporated Cu film 15 nm thick was measured with a spatial resolution of 100 gm (Fig. 3-19)[21]. A Si(Li) detector was used for detecting Cu K-L fluorescence with a sample-detector distance of 15 mm. The obtained XAFS spectrum corresponds to that obtained from a Cu foil of appropriate thickness with transmission measurement. Consideration of the beam size shows that the absolute amount of measured Cu was less than 1 ng with a data acquisition time of approximately 2h.
_J UJ >m
klJ O tu
8.8 ~9
LL
Fig. 3-19.
9.0
9.2
9.4
9.6
9.8
X-RAY ENERGY/keV
Copper K-edge XAFS spectrum obtained from a Cu-deposited thin film of 15 nm thickness.
197 As described previously,
XANES spectra can be used to provide "finger prints" of
materials. However, when the sample is in a mixture of several chemical states it is usually difficult to obtain the chemical state information because the measured XAFS spectra are affected by the overlap. Sakurai et al. have employed a novel technique to separate the chemical state in a binary system [41]. Figure 3-20 shows evaporated test patterns with Cr. The character "PF" is patterned with Cr203 and the rest of the area is patterned with Cr metal. The surface density of Cr is uniform within the sample. It is usually difficult to distinguish differences in chemical states using the XRF imaging technique. However, when the excitation energy is tuned to that in the absorption-edge region the difference in an excitation cross section can be emphasized. To separate the XANES spectra of Cr and Cr203, two excitation energies were chosen. One is the excitation energy that gives equal XRF signals for the same Cr surface density of the different compounds. The other is an energy that gives a large contrast between the two compounds. After the XRF imaging with two energies the XRF images are numerically processed to obtain separate images of each component. Figure 3-20(b) shows the extracted Cr image that clearly shows the characters. In this experiment the images were obtained using the image reconstruction technique, the principle is applicable to all kinds of XRF imaging techniques.
Fig. 3-20.
(a) Optical micrograph of the sample. The region measured is shown in the circle. The letters "PF" were Cr, and Cr203 was deposited around them (b) Cr image processed by the image reconstruction technique; the Cr203 image obtained was a reversal of the Cr image [41].
198 In the soft X-ray region, micro XANES measurements have been applied to polymer blends of polypropylene and a 50:50
weight percent random styrene-acrylonitrile copolymer [42].
Figure 3-21 shows carbon K-shell XANES spectra of polymers, obtained using transmission measurements. The incident X-rays were focused using a zone plate and a spatial resolution of 55 nm was realized. As the focus of the zone plate depends on the incident X-ray energy, focusing adjustments were carried out at each incident X-ray energy while the XANES spectra were measured. By choosing the incident X-ray energy for the resonance peaks of each polymer, X-ray images with chemical contrasts were obtained (Fig. 3-22). As a peak at 285.5 eV corresponds to an antibonding ~-orbital resonance of styrene an X-ray image from transmitted X-rays gives a high contrast to the styrene-acrylonitrile copolymer in the polymer blends. Micro-XANES measurements in the soft X-ray region are receiving great attention, especially in the field of biology. A trial to visualize the DNA distribution in a chromosome has also been reported, utilizing a technique similar to that of Sakurai et al. [41 ]. ,
i
,
I
. . . .
i
. . . .
. . . .
i
. . . .
A
,L
B
4 ~. ....
-.. ~ . . . _ L
c
e3 < o
2
u
2
i
r
In o u
1
0
. . . .
280
i
. . . .
285
i
. . . .
i
290
photon
295
energy
. . . .
i 300
. . . .
0 305
(eV)
280
,
,
,
,
i 285
. . . .
,
. . . .
290 pholon
i 295
energy
. . . .
,
,
i
,
,
30fl
(eV)
4
c
~2
._ u
| i e
i o
280
285
290
photon
Fig. 3-21.
295
energy
300
305
(eV)
Carbon K-shell XANES spectra of: (A), polyacrylonitrile; (B), polystyrene" and (C), polypropylene (dashed line) and a 50:50 percent by weight random copolymer of polystyrene and acrylonitrile (solid line) [42]
199
Fig. 3-22.
Images of a 0.5 ~tm-thick section of a blend consisting of polypropylene and a 50:50 percent by weight random styrene-acrylonitrile copolymer at the following photon energies: (A), 285.5 eV; (B), 286.2 eV; (C), 286.8 eV; and (D), 287.9 eV. The contrast arises from differences in the near edge absorption cross section of the different domains as shown in Fig. 3-21(C) [42].
3.5.4. Trace element characterization with micro X R F and micro X A F S
A combination of micro XRF and micro XAFS measurements becomes a powerful tool for the characterization of trace elements in a sample. Large synthetic diamond crystals are grown under high pressure and temperature with some metallic solvents. For the industrial use of the diamonds the characterization of impurities originating from the metallic solvent receives great attention. However, very few elements concentration.
have been characterized because of their
low
Figure 3-23 shows X-ray images of a diamond crystal grown with Ni-Fe
solvents, measured using the experimental system shown
in Fig. 3-8(c) with a spatial
resolution of 200 ~tm [43]. To maintain the spatial resolution the diamond crystal was polished into a plate 277 ~m thick, parallel to the { 110} plane. The sample was fixed onto a plastic sheet to reduce inelastic scattering which overlaps with the XRF signal. The Ni XRF image clearly shows that Ni exists selectively in the { 111 } growth sector. Figure 3-24 shows the SR XRF spectra obtained from { 111 } and { 100} sectors in the sample. Iron was not detected in either of these, twithin the sensitivity available. The X-ray images and XRF spectra confirmed selective dissolution of Ni into the { 111 } growth sector.
200
Fig. 3-23.
X-ray images of a sliced single crystal of synthetic diamond under high pressure and high temperature. (Upper left) Ni image, (Upper right) Fe image, and (Lower left) transmitted X-ray image.
1.2
1.6
Scattered x-rays
,,--., 1.0
1.4
0.8
0
1.2
Ni K~
,, 0.6
1.0
0.4
m
0.8
03 Z 0.2 LLi
{111}
-0.2
{100}
IZ
0.6
o.o
0.4
-0.4 5
0.2 0.0
6
7
8
9
10
X-RAY ENERGY/keV Fig. 3-24.
SR-XRF spectra of the synthetic diamond crystal taken at two different positions on the sample.
201 To evaluate the Ni concentration in the { 111} growth sector an absorption correction method was employed which measures the transmittance at each small area of interest and corrects for the self absorption effects inside the sample [44]. The resultant concentration value ik~r Ni was supposed to be 30 ppm for the { 111} growth sector and less than 0.3 ppm for the { 100} sector. Micro XAFS measurements have also been applied to the trace Ni in the diamond [45]. Figure 3-25 shows the Ni K-edge XANES measured at a { 111 } growth sector in a diamond. Its profile is quite different from that of metallic Ni. Therefore, it is confirmed that the Ni atoms inside the diamond crystal are not part of the small inclusions but are dissolved in the diamond lattice. From empirical knowledge about the pre-edge peak in the XANES profile it is suggested that the Ni is in a tetrahedral site. 1.0 0.8 0.6 0.4
z
0.2
l
0.0
8300
1
8320
I
8340
|
8360
I
8380
8400
X-RAY ENERGY/eV Fig. 3-25.
Nickel K-edge XANES of Ni dissolved into { 111 } growth sector of the synthetic diamond crystal.
3.5.5.
Other analytical m e t h o d s with the m i c r o b e a m
With the use of an X-ray microprobe various kinds of X-ray analyses can be perIormed with spatial resolution. In the field of micro X-ray diffraction (XRD) there have been several reports [46--49] following the initial preliminary report [49]. The local structure of the zig-zag defect boundaries in surface stabilized ferroelectric liquid crystal cells has been investigated using an X-ray microprobe equipped with Kirkpatrick-Baez mirrors. Rocking curves measured as a function of the sample position have determined the local layer structure [48]. Another use of the X-ray microbeam is in high angular resolution measurements of signals from a local position on the sample. One successful application is in grazing-exit detection for
202 surface sensitive measurements. As has been theoretically and experimentally demonstrated by several groups using conventional X-ray sources [50], the reciprocity of incident and exiting X-rays suggests the existence of a critical takeoff angle which is similar to the critical angle of total reflection. By detecting XRF with takeoff angles around the critical angle, layers near to the surface down to several tens of angstroms, can be selectively investigated. After the surface-sensitive XAFS application [51], a combination of the X-ray microprobe and the grazing-exit detection has been reported [52]. For chemical-state analysis one of the promising spectrochemical microscopes is the photoelectron microscope. Experiments can be carried out using a zone plate [53], ellipsoidal min-or [54], multilayer coated Schwarzschild optics [55] scanning photoelectron microscopes,
many
or Wolter mirror [56]. As well as the
electron-imaging microscopes have been
reported[56]. Incident SR X-rays are used to illuminate the sample in the field of view. The spatial resolution of the microscopes is determined by the electron-image-forming optics and, particularly when one considers the difficulties in realizing a focused microbeam in the hard Xray region, this method is of promise in realizing a spectroscopic microscope.
3.6.
FUTURE
EXPECTATIONS
WITH
THE
THIRD
GENERATION
SR
SOURCE
In the preceding Sections, several types of microprobes and their applications Ior chemical state analysis are described. Although the potential of an energy tunable X-ray microprobe has been recognized with the second generation SR sources the practical applications in the hard Xray region are still limited owing to the difficulties in realizing an intense X-ray microbeam. As is successfully demonstrated in the soft-X-ray region the characteristics of the undulator radiation, such as small emittance and high brilliance, are suited to an X-ray microprobe, and the X-ray microprobe with the hard-X-ray undulator has received great attention. The recent innovation of an in-vacuum undulator has made it possible to produce hard-X-ray undulator radiation. The in-vacuum undulator was installed in the 6.5 GeV TRISTAN Accumulation Ring (AR) at the Photon Factory. By changing the magnet gap from 1 to 5 cm, the photon energy of the first harmonics can be tuned from 5 to 25 keV. Brilliant 14.4 keV X-rays were used successfully for M0ssbauer experiments [58]. It is clear that the high brilliance
of undulator radiation is suited for a variety of
spectroscopic techniques. Fig. 3-26 shows an example of an experimental arrangement for high-energy-resolution X-ray fluorescence spectroscopy [59]. To realize chemical state analysis using the threshold technique, an energy resolution (AE/E) of better than 10. 4 is
203 required for both the incident X-ray monochromator and XRF spectrometer. Owing to the poor efficiency of the XRF spectrometer, the threshold spectroscopy is difficult to achieve with SR from the bending magnets. However, threshold spectroscopy in the hard X-ray region is one of the promising applications to use the third generation SR sources. The high brilliance source helps not only high-energy-resolution spectroscopies but also measurements with high angular resolutions and with high spatial resolutions. The combination of high resolution spectroscopies has promise in characterizing heterogeneous systems. An X-ray microprobe equipped with the grazing exit XRF detection system can provide information on three dimensional trace-element-distilbution. In addition to the expansion in spectroscopic applications the low emittance source will widen the choice of optical elements. A smaller source directly implies a reduction in the focused beam size. Moreover, the small divergence makes the practical use of focusing optics of small acceptance. The limitation of the total-reflection mirror owing to the small sagittal acceptance becomes less important and a focusing mirror which covers more than 20 keV can be used. The small divergence is also suited to capillary tubes [60, 61] whose acceptance is extremely small. By using the third generation SR source, an adequate photon flux may be obtained with the capillary tubes.
SR_
Excitatior
XRF/
U
hv
Detector
Sample Fig. 3-26.
Precise c.hemical state analysis[59]
204 REFERENCES
1.
V.E. Cosslett and W.C. Nixon (Editors), X-ray Microscopy, Cambridge University
2.
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Applications of SynchrotronRadiation to Materials Analysis H. Saisho and Y. Gohshi (Editors) 1996 Elsevier Science B.V.
207
CHAPTER 4
X-RAY A B S O R P T I O N FINE S T R U C T U R E Hiroyuki OYANAGI Electrotechnical Laboratory 1-1-4 U m e z o n o , T s u k u b a , Ibaraki 305, Japan
4.1. INTRODUCTION 4.1.1. Interactions of photons with atoms and molecules By varying the energy and polarization of incident photons, one can study the microscopic structure and electronic states through the various interactions of the photons with a variety of materials ranging from clusters to macromolecules as schematically illustrated in Fig. 4-1. Reflected and transmitted photons conventionally provide information on the electronic states of a system. Elastically and inelastically scattered photons are used to study the microscopic structure. One can obtain information on the electronic states of occupied states by analyzing the energy of photoelectrons. On the other hand, X-ray absorption spectroscopy (XAS) provides a means of studying the unoccupied states and the local structure around an excited species of atom [1]. The advantage of XAS is its selectivity for atomic species and sensitivity to the atomic arrangements [2,3]. The relation between the fine structures and crystal structures has been recognized from an early stage. Historical developments which established the fundamental aspects of the fine structures based on a short-range order theory are reviewed in detail in the literature [4]. Even if the system has many elements, a particular species of atom can easily be selected by tuning the photon energy to one of the inner shell absorption edge energies. Moreover, since the interaction is strongly polarized, the information on the geometrical arrangements of atoms, or the radial distribution in a particular direction, can be obtained. These characteristics make synchrotron radiation an ideal light source, in that it provides an intense polarized light over a wide range in energy. XAS has advanced rapidly since the application of synchrotron radiation in the early 1970s [5]. The advances in experimental techniques have been inspired by a simple interpretation of fine structures, based on a single scattering theory [6] and a Fourier transform analysis [7]. Reviews on modem XAS
208 Incident
photons
Surface r Interface [
Scattered photons
Fluorescence
/
/
//
,/Photoelectron
Aucjerelectron
Bulk
Fig. 4-1. Schematic representations of the various interactions of photons with atoms and molecules. are available [8,9] and the theories and analytical methods are described in detail. In this Chapter, the recent experimental advances in XAS as a structural probe in a hard X-ray region (> 4 keV) are reviewed. The capabilities of high-brilliance photon sources, i.e., insertion devices, for the studies of dilute systems are demonstrated, and the future prospects of a tunable X-ray undulator for the third-generation storage rings [ 10] are discussed.
4.1.2. X-ray absorption fine structure (XAFS) X-ray absorption fine structure (XAFS) is a general terminology for extended X-ray absorption fine structure (EXAFS) and X-ray absorption near edge structure (XANES) techniques. In general, EXAFS refers to the sinusoidal oscillations observed over a wide energy range extending ~1000 eV above the edge, while XANES refers to fine structures around an absorption edge extending typically -50 eV. In Fig. 4-2, the Ge K-edge absorption spectra are shown as a function of photon energy for glassy and crystalline GeO2 [ 11 ]. In the trigonal crystal structure, each Ge atom is coordinated by four oxygen atoms forming a tetrahedral unit. These tetrahedra are linked by two-fold coordinated oxygen atoms which occupy the corner positions of each tetrahedron [ 12]. Clearly, the absence of long-range order has little effect on the XANES and EXAFS regions; the spectral features in a glass are broadened but the essential features are preserved. This fact naively suggests that the glass has a short-range order as the fine structures arise from interference between the outgoing and scattered photoelectrons. The fine structure in X-ray absorption spectra and its relationship to
209 I
'
I
I
'
I
'
I
'
I
'
I
"
'
I
GeO2 80K
Ge K- edge
z D
~...,..,_..,,~, f'\,_. . . . . . .
I-IIl
trigona I glass
A .m
I !
0 0,~
I C
I
10.8
,
I
11.0
,
I
11.2
,
I
1t.4
,
I
,
11.6
I
11.8
,
I
12.0
,
I
12.2
ENERGY (keY) Fig. 4-2. Ge K-edge absorption spectra for glassy and trigonal GeO2" taken from ref. 11. Above the K-shell excitation energy (11.11 keV), near-edge and extended fine structures are observed. the crystal structure have been recognized from an early stage. Until recently, however, the quantitative understanding of XAS has been hindered because of poor quality. The experimental techniques of XAS have been improved dramatically by the use of synchrotron radiation. Synchrotron radiation has outstanding advantages as a light source for XAS. These are, (1) a wide spectral range covering the VUV and soft X-ray regions, where the light elements such as C and O have absorption edges (250 eV-4 keV), to a hard X-ray region (4-30 keV) where elements heavier than Ca (Z = 20) have absorption edges: (2) an intense quasiparallel beam which allows us to measure XAFS with a high energy resolution [13] or energy dispersive geometry [14]: and, (3) a variety of polarization characteristics with high purity and controllability. In general, a linear polarization from a bending magnet is used for polarization dependent experiments, while one can obtain linear [ 15], circular [ 16], and elliptical [ 17] polarization using special insertion devices. The improved quality of XAS obtained with synchrotron radiation made possible the quantitative analysis of experimental data. Partly driven by a motivation to utilize the technique as a unique local structural probe, the theoretical understanding has been accelerated [ 18-20]. In the 1970s, various new experimental techniques were evolved, such as an electron detection technique [21 ] as a means of surface-sensitive measurement, a fluorescence detection technique
210 [22] to enhance sensitivity for dilute systems, and an energy-dispersive technique [ 14] for timeresolved experiments. These new techniques have spread the capabilities of XAFS to cover a wide area from physics to biology. Dedicated synchrotron light sources, often referred to as the "second-generation" storage ring, are characterized by a low emittance and high stability. Such facilities made possible a high-energy-resolution near-edge spectroscopy called XANES [23] or NEXAFS [24] (near-edge X-ray absorption fine structure) which can probe elaborate changes in coordination geometry. It should be noted that the capability of XAFS is strongly dependent on the characteristics of the light source. In this sense, the XAFS technique is indeed still developing with the evolution of the light source. 4.1.3. Uniqueness as a structural probe
In contrast to diffraction techniques, which give the atomic coordinates as a macroscopic average, XAFS provides the information on a local structure, i.e., the radial distribution and electronic states around a particular species of atom. The uniqueness of XAFS is attributed to its physical origin, the interference of outgoing and scattered photoelectron waves from the central atom, which modulates the matrix element of a dipole transition. Such an interference does not require a long-range order and reflects a "partial" radial distribution around an excited atom, extending to several A because of the rather short mean free path of ejected photoelectrons. Schaich [20] has shown that the short-range [1] and long-range order [25] theories are equivalent if the inelastic energy loss of photoelectrons is taken into account. A great advantage is the tunability for atomic species; the energies of various inner shells are distributed over a wide range in energy and one can tune to a particular species of atom and inner shell by choosing the energy of incident photons. The superior capabilities of XAFS as a local structural probe are fully exhibited in studies of dilute, multi-element systems, in particular, those without long-range order. XAFS is complementary to crystallography or scattering techniques, which provide the average or macroscopic structure with a long-range order. On decreasing the quantity of the atom species of interest, or increasing the number of constituent atoms, the uniqueness of XAFS as a local structure probe becomes particularly evident. For a structurally disordered system, the correlation between atoms is described by a radial distribution function for elements. For a disordered system consisting of more than two elements, it is necessary to separate partial radial distribution functions from either an isomer effect in neutron diffraction or an anomalous scattering effect in X-ray diffraction. This problem is difficult for disordered systems consisting of more than two elements and it is practically impossible to handle three-component systems. A general problem of crystallography is the lack of accuracy in the local structure. Biological systems usually contain only a few atoms in a macromolecule consisting of several tens of thousands of atoms, and the analysis of local structure is not reliable even if single crystals are prepared [26]. For biological systems, the local structure is strongly related with biophysical functions and it is important to study the coordination in enzymes and heme proteins in vivo [27].
211 4.1.4. Evolution of XAFS
The recent advances in the experimental techniques of XAFS are demonstrated in Fig. 4-3 where the Ge K-edge absorption spectra for trigonal GeO2 powder taken on various spectrometers and synchrotron radiation facilities are compared [11 ]. The top spectrum, (a), was obtained using an in-house EXAFS apparatus using Bremsstrahlung, by Lytle et al. [28] while the middle spectrum, (b), indicates the data obtained by using the first-generation storage ring. Although it appears that the spectral quality is much the same between (a) and (b), the statistics and data collection time are improved in (b) by more than two orders of magnitude [29]. Since a typical magnitude of modulation is of the order of 10-2 and the statistical error must be less than 10-3 , one needs to collect at least 106 photons for the incident and transmitted beam intensities. Since the photon flux is usually reduced by several powers of ten after the absorption, the XAFS experiments require a high photon flux. Synchrotron radiation has not only reduced the data collection time by several orders but has also improved the accuracy as a structural probe. In Fig. 4-3, new near-edge features are resolved in (c), taken on
GeOz
(13)
in-laboratory system Lytle (1965) .m t.-
(b) <:1:
DCI storage ring of LURE (Orsay, FRANCE) Lapeyre et al. (1983)
~=~
(c)
Photon Factory (present work)
l
11.0
I
I
11.5 12.0 Energy (keV)
Fig. 4-3. Comparison of Ge K-edge absorption spectra for trigonal GeO2 taken on various spectrometers 9from ref. 11.
212 a spectrometer at the second generation storage ring. A high-energy resolution is achieved by a low emittance and a narrow width of monochromator-crystal rocking-curve. As a result, the near-edge features have been revealed and the theoretical understanding of XANES has been advanced. The evolution of the XAFS technique is evidenced by the surface-sensitivity in a hard X-ray region, which improved by several orders of magnitude in the late 1980s, as will be discussed. It is interesting to note that the recent progress in XAFS is certainly due to the correlated developments in experimental techniques and theories which strongly inspired and accelerated each other's progress. 4.2. P R I N C I P L E S
4.2.1. Extended X-ray absorption fine structure (EXAFS) EXAFS is an interference phenomenon of photoelectrons, which occurs between the outgoing waves and the scattered waves and modifies the matrix element of core-excitation or X-ray absorption. Since the dominant interference involves the path lengths between the absorbing (central) atom and a scattering atom 2r, the 2kr oscillations are observed in k-space, in contrast to kr oscillations in a well-known formula for electron scattering. The propagation of photoelectrons is described in terms of the exact spherical waves [30] or curved waves [31 ]. Since the interference of photoelectrons is detected on an excited atom, the information is intrinsically local. Within a single scattering formalism [6], the interference among the three components, i.e., the outgoing wave, the single-scattering path, and the double-scattering path of the lowest order, is dominant. The photoelectrons emitted from an excited atom as spherical waves rapidly damp because of inelastic effects. This fact limits the probed spacial range but it also ensures that contributions of multiple-scattering beyond double-scattering are neglected. The fact that multiple-scattering effects can be ignored for the nearest-neighbor scattering allows us to analyze the data by a simple Fourier transform [7]. In Fig. 4-4, a scheme of the photoemission process is shown. In X-ray absorption, the bound electrons are ejected when the incident X-ray with an energy hv exceeds the binding energy E0. Accordingly, the absorption coefficient increases sharply at the threshold energy E0, above which fine structures are observed over 1 keV or more. Figure 4-2 shows a steep rise of absorption coefficient at ~11.11 keV, the onset of the Ge K-edge or the s-state absorption. The fluorescence X-rays [22] and Auger electrons [21], and, in some cases, the luminescence yields [32] are proportional to the absorption coefficient, and these quantifies provide alternative means to monitor the absorption coefficient. Photoelectrons propagate as a spherical wave having a kinetic energy Ekin, which is given as follows.
Ekin -- h v - EO
(4-1)
213
Photoemission
Process
Ekin hv - F'o =
T
Auger electron
Valence bond
~ ~
Core Levels
i
=
2pz/ 2 2p~/z
2L_,,- Fluorescent ~--~---- ls
v
X-roy Absorption
X-roy Emission
Fig. 4-4. Schematic representation for the photoemission process. Photoelectrons are emitted when the incident photon energy exceeds the binding energy. Radiative and nonradiative decays of excited states are associated with the characteristic X-rays or Auger electrons. where h v is the incident X-ray energy and E0 is the binding energy. The photoelectron wave vector k is then defined as
k = 2n h-1 [2m ( h v - E 0 ) ] 1/2
(4-2)
The excited states return to the ground states either radiatively or nonradiatively. In the former case, characteristic X-rays are emitted as a fluorescence with an energy difference between the excited shell and outer shell. In the latter case, Auger electrons with the same amount of kinetic energy are ejected. In Fig. 4-5, a scheme for the interference of photoelectrons with the wave vector k is shown, where 0 denotes the angle between k and radial vector r. Dashed arrows indicate the outgoing photoelectron and solid lines indicate the scattered photoelectrons. The EXAFS modulations in an absorption coefficient result from the interference between the direct wave going from an excited atom (0) and waves scattered by surrounding atoms (1, 2). Among various pathways for photoelectrons, the interferences between the direct wave and singly scattered paths (0-1, 0-2) and the lowest order doubly scattered paths (0-1-0, 0-2-0) are important. The probability of emitting a photoelectron in the direction k is given by
214
EXAFS
XANES
o
o-z-~
@ o_,_o',x,, 0 / / o-2-6 \~-/,r
o0-
~
,.
o-2-o ' , \ / /
o-I-z-o
Central Atom ,oz Scatterer Atom
\, ,
o 9
I
o-l-2 \ \ \
" -
r~
-@2
Fig. 4-5. Principle of EXAFS. Photoelectrons ejected from an excited atom (0) are scattered through the potentials of near-neighbor atoms (1, 2). Outgoing spherical waves interfere with scattered waves causing a modulated transition probability. 0 denotes the angle between k and radial vector r.
P(k) = D[e.k + f(O)/r exp(ikr(1-cos 0)) e.r] 2
(4-3)
where e is the electrical field vector, D is a constant and f(/9) is the scattering amplitude. Implicit assumptions in a single-scattering formalism are the following; a plane wave approximation is assumed, a single-scattering pathway is considered as the dominant interference; a muffin-tin type of potential is assumed for excited and scatterer atoms; and a sudden approximation for the core hole potential. The first term in Eqn. (4-3) expresses the outgoing wave and the second term is the scattered wave, where r(1-cos0) denotes the interference path length. The total absorption coefficient/1 is obtained by spherical integration of Eqn. (4-3) and if the normalized modulation of the absorption coefficient is defined as A/1//1, EXAFS Z as a function of k is obtained from the cross-term of the fight hand side in Eqn. (4-3), since other terms are smooth functions of photon energy or vanish, canceling with a part of the cross-term. The interference term is given by
~dk/4n e.k f(O)/r exp(ikr(1-cosO)) e.r
(4-4)
Using a spherical wave expansion and integration angle over 0, Eqn. (4-4) is given by
215
z(k) = -3 [(e.k) 2 / kr2]f(k,n) exp(2ikr) = -3 [(ok) 2 [ kr 2] If(k,n)l exp(2ikr+~k))
(4-5)
The forward scattering term in Eqn. (4-4) cancels with the second term of the square in Eqn. (4-3). For single-scattering paths (0-1, 0-2), the central atom phase shift is canceled and only the back-scattering phase shift ~k) is left in the equation. If we take only the double scattering paths with the shortest path lengths (0-1-0, 0-2-0) into account, and generalizing the formula for Ni scatterer atoms at ri, with a mean-square relative displacement (MSRD) oi, then z(k) is given by
z(k) = -3E [Nj (e.ki) 2 / krj 2] I~(k,n)l sin(2ikr+~k)) x exp(-2oj2k 2) exp(-2rj/~,j) = - 3 E [Nj cos20j/krj 21 I~(k,n)l sin(2ikr+gt(k)) • exp(-2oj2k 2) exp(-2rj/~,j)
(4-6)
where Aj is a mean free path of photoelectron and ~k) is a total phase shift function which is a sum of the central atom phase shift 2~0(k) and scatterer atom phase shift 0(k).
~ k ) = 2~0(k) + ~(k)
(4-7)
Since the total phase shift is approximately a linear function of k, the magnitude of the complex Fourier transform gives a peak maximum shifted to smaller r. Empirical data analysis procedures are based on the fact that this phase shift is transferable, i.e., once the total phase shift is established for a given pair of absorbing atom and scatterer atom, it is independent of the nature of the chemical bond, allowing one to extract structural parameters from one compound and apply them to other systems. On the other hand, Teo and Lee published tabulated values of theoretical amplitude and phase shift functions using a plane wave approximation for outgoing photoelectrons[33]. Such theoretical calculations are quite useful, in particular when an appropriate model compound with a known structure is not available. Later, McKale et al. have calculated the scattering amplitude and phase shift on the basis of a
216 curved wave approximation [34], which improved tile calculated phase shifts in the low kregion for high-Z elements. Figure 4-6 shows the back-scattering amplitude and phase shifts calculated for In (Z=49), Ge (Z = 32), Si (Z = 14) and O (Z=8) [34]. For a low-Z atom such as Si or O, the back-scattering amplitude has a large value in a small k-region which sharply reduces intensity with an increase in k. For higher-Z elements, the amplitude extends to a high k-region: the back-scattering amplitude has a maximum at k -- 6 A-1 for Ge while two maxima at k ~ 3.5 A -1 and 10 A-1. A characteristic k-dependence of a back-scattering amplitude is often used to distinguish atom types. Equation (4-6) shows that the modulation has a fundamental frequency 2rj while the envelope reflects the k-dependence of the back-scattering amplitude I~(k,~)l. Thus the analysis of EXAFS is essentially a process to obtain structural parameters, Nj, rj and crj from EXAFS oscillations using known I~(k,~)l and qt(k) as a function of k. Conventionally, the experimental phase shift is determined by fitting Eqn. (4-6) to the data of an appropriate model compound with a known structure.
20
1.2 1.0
i
... I0'
0.8
In
In
~0.6
Eo o
0.4
Ge
0.2 -I0 10 .-.
o
8
"-"
6
e-
0
In
10
k (~-i)
20
Ge
4 "
20
Si
!
0
I
10
!
20
k (~'~)
Fig. 4-6. Calculated back-scattering amplitude I~(k,~z)l as a function of k (A-1) and phase-shift functions for various scatterer atoms 9taken from ref. 34.
217 The details of data analysis are described in Section 4.4. Here, some important aspects of Eqn. (4-6) are summarized. First, the effect of multiple scattering is large in the low-k region where photoelectrons are strongly scattered. Figure 4-7 illustrates the scattering pathways for the three atom system which originates from the excited atom (0), scattered by the nearestneighbor atom (1) and the second-nearest neighbor atom (2). fl denotes the angle between the pathways (0-1) and (1-2). The strong forward scattering by a low-Z nearest-neighbor atom affects the magnitude of scattering by the second-nearest neighbor atom when the central atom, nearest neighbor and second-nearest neighbor atoms form a triangle. In general, for high energy electrons (kr >> 1), the magnitude of back-scattering is much smaller than that of forward scattering. This is the reason why the multiple scattering is not important in EXAFS at least for the nearest-neighbor, in contrast to low energy electron diffraction (LEED) where multiple scattering must be taken into account. In LEED, the electrons are irradiated outside the surface and the interference between forward-scattering photoelectrons is monitored by a detector placed away from the scatterer, while in EXAFS, the absorbing atom emits the photoelectrons and simultaneously detects the interference of back-scattered waves coming back to the origin. Secondly, in EXAFS, the interference pathlength of multiple scattering is long and the high frequency sin(2kr) oscillations due to multiple scattering cancel out. Actually, possible combinations of multiple-scattering pathways increase rapidly on going to higher shells beyond the nearest-neighbor atoms, and such long pathways would reduce the photoelectrons because of intrinsic and extrinsic inelastic scattering. These facts make the simple single-scattering EXAFS formalism valid for the nearest-neighbor atoms where the multiple-scattering pathways are separated in r. For multiple-scattering calculations, the spherical wave expansion [35] or small-atom approximation [36] can be used. However, there are several special cases where multiple scattering cannot be neglected [37,38]. In the low-k region, particularly for low-Z elements, the photoelectrons are strongly scattered and the multiple scattering becomes important. Thus, if a light-element atom is placed between the direct scattering pathways, its large scattering amplitude in the low-k region would enhance the multiple-scattering effect. As shown in Fig. 4-7, the strong forward scattering of
'
I
o
,2
92
(a)
--"" ,,/
0-2-0
o
2 (b)
0-I-2-0
. ....-" 1
(c) 0-t-2-1-0
o
Fig. 4-7. Schematic representations for scattering pathways (dashed line) for the three atom system which originates from the excited atom (0), scattered by the nearest-neighbor atom (1) and the second-nearest neighbor atom (2). fl denotes the angle between the pathways (0-1) and (1-2).
218 the intermediate atom (1) gives rise to a substantial multiple-scattering contribution along a triangle pathway (0-1-2). The effect of the multiple scattering is clearly seen in the Fourier transform of the Ga K-EXAFS oscillations measured for GaP, GaAs and GaSh (See Fig. 4-8). All three compounds have a zinc blende-type structure where each cation (Ga) atom is tetrahedrally coordinated by anion (P, As, Sb) atoms. In Fig. 4-8, although the second-nearest neighbors are twelve Ga atoms for all cases, the second-nearest peak observed around 4 ,/k is strongly dependent on the species of the first-nearest atom located at the apex of a triangle. A large second-nearest neighbor peak observed for GaP is due to the strong forward scattering power of the P atom. Thus it is clear that the analysis of the higher shells must take the multiple-scattering effect into account. With the decrease of t , the scattering by the secondnearest neighbor atom sharply increases due to the multiple scattering involving the nearestneighbor atom. Thus, when three atoms separated by an equal distance are linearly aligned, the second-nearest neighbor peak is observed with a magnified intensity, which is known as a focusing effect. The effect of multiple scattering can be corrected by a theoretical formula taking three pathways shown in Fig.4-7 into account. i
0.08 0.04 0 -0.04 -0.08
i
I
i
i
i
i
Go K- edge
1
V
0.04 0.02 0 -0.02 "" -0.04 >~ -0.06 0.08
0.04 0 -0.04 -0.08 2
fluorescence -t transmission z~ ~ ~ 10 1'2 14 16 18 k (~-1)
Fig. 4-8. Ga K-edge EXAFS oscillations as a function of photoelectron wave number k for GaP, GaAs and GaSb.
219 Atoms are vibrating around equilibrium positions which are usually displaced from ideal ones. The effect of dynamic and static disorder is described by the Debye-Waller-like term exp(-2o'j2k 2) for a harmonic oscillator with a Gaussian distribution. The displacement measured in EXAFS is the mean-square relative displacement between a central atom and a scatterer atom, in contrast to the Debye-Waller factor in the diffraction formalism, which measures the mean-square displacement. Physically, the mean-square relative displacement oj 2 includes the correlation term which vanishes for independent motions [39].
aj 2 =<[rj.(uj - u0)]a>
(4-8)
where uj and u0 denote the displacement vector for scattering and absorbing atoms originating from the thermal equilibrium positions. Based on various lattice dynamical models, o] 2 can be calculated [40,41 ]. If the phonon spectrum obtained by neutron diffraction measurements is available, trj 2 can be obtained [39]. Simple approximations based on a Debye model [39] or Einstein model [40] are often used for convenience. In order to describe the disorder by the exp(-2o'j2k 2) term, it is assumed that atoms are harmonic oscillators with a Gaussian distribution. However, for highly disordered systems the Gaussian distribution must be replaced with a pair-distribution function gij(r). It should be noted that the effect of disorder gives rise to the additional term in the total phase [42]. Even a symmetric gij(r) results in an additional term in phase. Since the integral over r, the pairdistribution, cannot be obtained in a straightforward manner such as a simple Fourier transform, a cumulant expansion of the pair-distribution function is used. One of the most difficult problems in EXAFS theory is the inelastic effect, i.e., energy loss. The many-body effects were discussed by Lu and Rehr [43]. In general, a simple simulation based on Eqn. (4-6) has a smaller amplitude by a factor of 0.1-0.4. This reduction of amplitude arises as a result of inelastic effects, (1) around the excited central atom via a multielectron process, or (2) associated with the scatterer atom, and (3) the inelastic effects of photoelectrons which are taken into account by an exponential decay term exp(-2rj/2,j) in Eqn. (4-6) using 2.j, a mean free path of a photoelectron. Since the proper theoretical estimation of (1) and (2) is difficult at present, in order to correct the amplitude reduction a simple constant is used in a conventional data analysis. Stern et al. introduced a damping factor S02 to account for an overlap of the initial and final wave functions of passive electrons which decrease the oneelectron matrix element [44]. The calculated S02 factors lie in the range 0.6-0.9 [45]. Since this value is dependent on both k and chemical bonds, it is reasonable to treat it as an empirical parameter. The contribution to the inelastic scattering within the back-scattering atom is included in the scattering amplitude using an optical potential whereas the inelastic effects at the central atom are accounted for by the S0 2 factor. Thus, adding an S0 2 factor should be associated with replacing the exp(-2rj/Zj) term with exp(-2(rj - A)/2,j) where A accounts for the
220 energy loss already accounted for in the S02 term. It should be noted that the mean free path between the first and second shells is not modified because the energy loss suffered by photoelectrons between the first and second shells is described by the mean free path term. Readers should refer to theories on intrinsic [46] and extrinsic [47] energy losses and optical potential [48] for a detailed discussion on many-body effects.
4.2.2. X-ray absorption near edge structure (XANES) The near edge structures (XANES) observed up to about 50 eV above the absorption edge have oscillations with much higher frequencies than EXAFS. They are caused by multiple-scattering resonances of photoelectrons within a cluster around an excited atom. Because of the lifetime broadening and solid-state effect, the energy resolution of 1-2 eV is sufficient for continuum features in most cases. Figure 4-9 shows typical XANES spectra on the Ge K-edge (11.11 keV) for crystalline and glassy GeO2, where a transition from the ls to 4p* states is observed as a sharp "white line" at the threshold. Several characteristic features are observed for both the crystalline and glassy phases although they are broadened in the latter. In general, the magnitude of XANES modulation is larger than that of EXAFS since the scattering factor is larger in a low energy region. As will be shown in Section 4.5, the analysis of EXAFS data shows that the short-range order within a GeO4 tetrahedral unit is strictly maintained in the glass although the long-range order is degraded. This indicates that the near-edge features are dependent on atomic arrangements beyond the first-nearest neighbors. As "slow" photoelectrons are strongly scattered, the multiple scattering is dominant in the near-edge region. Since the multiple scattering is sensitive to the bond angle, this means that the information on the bond angle as well as bond length is contained in the near-edge features. I
'
I
/.~'\
I
"i
'
i
'
I
i
Ge 02
,.....
Ge K-edge
--
f,,\\
o:>-
I\
trigonol ( z = 4 )
...... glass
\
(z 61
rren
I
-I0
l,.,
I
0
t
I
,
I
10 20 ENERGY leVI
i
I
30
'
9
40
Fig. 4-9. High resolution Ge K-edge XANES spectra for trigonal and glassy GeO2, taken on a spectrometer using a Si(311) channelcut crystal monochromator 9from ref. 11.
221 The scattering of low-energy electrons is sensitive to the shape of potential. This gives rise to sensitivity to chemical bonds or valency. However, these make it difficult to analyze the XANES by a simple method as in the case of EXAFS, although it has rich information on site symmetry and electronic states. In order to calculate XANES, multiple-scattering pathways must be taken into account. In fact, the state density of conduction bands for the 3d transition metals reproduces the experimental absorption curves for both XANES and EXAFS regions. The band calculations are equivalent to calculating all possible multiple-scattering pathways based on the long-range order or Bloch theorem [49]. The method of calculating multiplescattering paths is essentially the same as that for dynamic LEED intensity calculations. The number of possible pathways increases rapidly on taking outer shells into account and various approaches for full multiple-scattering calculations have been proposed [50]. For molecules, excited photoelectrons sometimes form a strong "shape resonance" state, giving rise to sharp transitions in the near-edge spectra [51], while insulators such as alkali halides show an exciton-like bound state transition [52]. Calculations of XANES using a molecular orbital approach have also been proposed [53]. Furthermore, features related to bound-state transitions are often observed below and above the absorption edge. In Fig. 4-10, polarized XANES spectra are shown for single-crystal La2CuO4 (above) and for a powder sample (bottom) [54]. The strong, polarization-dependent features observed for the single-crystal data arise from the anisotropic coordination geometry of square-planar Cu atoms in the CuO2 plane. In La2CuO4, the Cu atoms are four-fold coordinated by oxygen atoms at ~1.9 A in the basal plane. They are coordinated by two other oxygen atoms at ~2.4 ,~, forming a CuO6 octahedron distorted along the c-axis. The in-plane features are due to the bound-state transition from the ls to 4p*(cr) states while the out-of-plane features are the Is to 4p*(zr) transitions. These features correspond to the fine structures in XANES for powder La2CuO4, where anisotropic features are averaged out. The polarization dependence thus provides not only the information along a particular direction but is also helpful for the assignment of the near-edge features. It was pointed out by Natoli that the energy positions of these fine structures are related to the bond length [55]. As shown in this figure, the polarization-dependent out-of-plane features are sensitive to the ligand along the caxis. One can thus obtain the information on the apical-oxygen position from the out-of-plane polarized XANES. In high Tc superconductors, it is recognized that the apical-oxygen affects the superconducting properties and their role in the pairing mechanism is important. The recent development of the full multiple-scattering approach made it possible to analyze XANES and to obtain information on subtle changes in coordination, bond length, effective charge, or valency.
222
Cu K-edge 41~{r) O ? Is-41)'Io'I k[--~ d
Is-
~
A rE') I---
e
z >.cIc cIc
Lo2Cu04
Singlecrystal
~k'/O:9O~
g
k~
n...._~ ix..
! I I I I 8980 8990 9000 9010 9020 9030 9040 ENERGY (eV) Fig.4-10. Polarized Cu K-edge XANES spectra of La2CuO4 taken for a single crystal (above 9 from ref. 54) and unpolarized data for a powder specimen (bottom).
4.3. E X P E R I M E N T A L 4.3.1. Transmission
TECHNIQUES
XAFS
Monochromator
Classical X-ray absorption spectroscopy uses the transmission experiment, where a monochromatized X-ray is passed through a sample and from the incident and transmitted beam intensities the absorption coefficient p(E) is obtained as a function of the photon energy, E. In order to obtain the spectrum, the Bragg angle 013 of the monochromator is scanned. For Bragg reflections with dhkl, the energy of the incident beam E is given as
Ehkl = 12.4/~,hkl = 6.2/[dhkl sin0B] (keV)
(4-9)
223 The energy resolution AE/E is a convolution of the Darwin width ~50w and geometrical resolution of the incident beam ~50gwhich are determined by collimating optical components, a source size and an angular divergence. Since the polarization for a bending magnet radiation is horizontal, the vertical emittance is a dominat factor. AE/E is expressed by the following equation.
AE/E = [(~50w)2 + (~50g)2]1/2 cot 013
(4-10)
The vertical angular divergence of synchrotron radiation is an order of magnitude larger than the width of the rocking curve. For Si(111), 80w is ~8 sec or 1 x 10-5 rad for 8 keV Xrays while the vertical divergence is ~ 3 x 10-4 rad for the 2.5 GeV storage ring at the Photon Factory. Since the angular divergence is larger than the Darwin width roughly by an order of magnitude, it is necessary to collimate the beam either by a slit or a mirror. A typical energy resolution for Si(111) is ~2 eV at 9 keV when a slit with an aperture of 1 mm is placed before the monochromator located ~25 m from the source point. From Eqn. (4-10), the energy resolution increases with the increase of photon energy, and for high-photon-energy experiments, reflections with a smaller dhkl value such as Si(220) or Si(311) are used. Figure 4-11 shows a schematic representation of a variable-beam-height double-crystal
reflected beam stepping motor i
Oa 72
Y-translation
stepping L motor
\ ~
rotatingtable
Fig.4-11. Schematic diagram of a variable beam-height double crystal monochromator. The first crystal is placed on a computer-controlled XY stage. When the XY stage is controlled according to simple functions of the Bragg angle (see text), the output beam height is kept constant on rotating the axis.
224 monochromator. On rotating such a monochromator, the output-beam-height varies as a function of the Bragg angle. Various schemes have been proposed for achieving a constant output-beam-height [56]. The dominant source of experimental error in XAFS experiments is the systematic error which is caused by the fluctuations of beam positions due to monochromator scanning and the light source instability. In most cases, the mechanical stability in the parallel setting of the two crystals is the most important factor. In Fig. 4-11, the incident beam irradiates the center of the first crystal while the second crystal is placed on an XY stage. In the original design, a mechanical linkage was used for controlling the XY stage [57]. By replacing a mechanical linkage with computer control, the vertical position of an output beam can be controlled easily [58]. It should be noted that the dominant source of error in the parallel setup of the two crystals is the wiggles of X stage since the magnitude of the X-stage motion is larger than that of the Y-stage motion by an order. Figure 4-12 shows the positions of an XY stage as a function of the Bragg angle for Si(111) which achieve a constant output-beam height. E (keY)
Si(111) 100
Si (220)
151311109 8 7 6 t~,ll,l,~,l,l, l, ~ , z~181614151211 10 9
/
5 '
~}
"~
4 ~
6
3.5 ,
'51
3 J
Position of XY stage H=25mm
90 80 .-. 70E E "-" 60'~ 50-
>X
o 40r o
= 30-
o
20-
Y=H/2cos8
100
I
5
I
10
I
15
I
I
20 25 8 (deg.)
I
:30
I
35
410
Fig.4-12. Positions of the XY stage as a function of the Bragg angle for Si(11 l) which achieve a constant output-beam height.
225 HUTCH
Be WINDOW SLIT I SOURCE UHV POINT
MONOCHROMATOR
He
AIR
IF~I
SAMPLE
MASK
I,ON IONON , l CHAMBERII ll IICHAMBERIICHAMBER
li X"
SHUTTER
i STAGE ITO PZT
1
FEEDBACK ELECTRONICS
IV AMP i
VFC II~COUNTER I
POSITION
VFC I COUNTER I
I
MONOCHROMATOR
IV AMP
COMPUTER
STAGE
POSITION
Fig.4-13. Experimental scheme for XAFS measurement in a transmission mode. Intensities of the incident and transmitted beam are measured by the two ionization chambers as the photon energy is scanned. In this system, the vertical positions of the two ionization chambers and the sample are controlled by a lifting stage so that the beam irradiates the same position. Absorption spectra are obtained by dividing i0 by i and taking a logarithm according to Eqn. (411). The second term in Eqn. (4-11), which varies smoothly with photon energy, derives from the absorption of the two ionization chambers. Although this does not affect the normalization of EXAFS, it can be estimated from a simple measurement for a blank sample, if necessary. Higher harmonics and sample inhomogenity severely degrade absorption spectra [59]. Higher harmonics arising from higher-order reflections can be minimized by either a mirror or detuning the two monochromator crystals. For a typical X-ray mirror, varying the grazing angle can select the cut-off energy, above which the reflectivity rapidly decreases. Detuning the two crystals is a simple method for elimination of higher harmonics but it also loses beam intensity, and the random error increases.
4.3.2. Special techniques Fluorescence XAFS
Fluorescence detection [22] is a technique for increasing sensitivity for dilute samples. In a transmission experiment for a dilute sample, the thickness of sample is adjusted so that the edge jump is, but the transmitted beam intensity is exponentially attenuated while the ratio for the element of interest to the total absorption increases linearly. Thus the signal-to-noise ratio
226 rapidly decreases on increasing the total absorption since the signal (absorption) is proportional to the number of absorbed photon, which is a linear function of thickness, while the transmitted beam intensity decreases exponentially. As fluorescence and Auger yields are proportional to the absorption coefficient, these quantities provide alternative means to monitor absorption spectra. The distinction between these two methods arises from a difference in escape-depth. The fact that Auger yield is surface-sensitive is utilized in surface XAFS, while the escapedepth for the fluorescence X-rays is of the same order as the penetration depth of the incident beam. Above the absorption edge energy, the fluorescence X-rays are ejected and the fluorescence intensity I(E) accepted by a detector with a solid angle 12/4n is expressed by the following formula as a function of photon energy E,
I(E) = #(E) e ~ [I0 exp(-#T(E)t)) exp (-~tT(Ef) (sin0/sinr t ] dt dl2/4rc = I0 bt(E) e cosec0 j" [ 1- exp (- (,uT(E) cosec0 +/~T(Ef) cosec~) a)] dl2/4n; / [ktT(E) cosec0 + btT(Ef) cosecr ]
(4-12)
where I0 is the incident beam intensity, e is the fluorescence yield, p(E) and pT(E) are the absorption coefficient for an excited species and the total absorption coefficient, respectively; tx is the sample thickness, Ef is the energy for the fluorescence X-ray, 0 is the angle between the sample surface and incident beam, and r is the angle between the detector and sample surface. For bulk dilute systems where (pT(E) cosec0 + pT(Ef) cosec~) t~ << 1 (dilute limit), Eqn. (4-12) reduces to the following equation [60] :
I(E) .... S I0 #(E) e cosec0 dl-2/4n: / [pT(E) cosec0 + pT(Ef) cosec~ ]
(4-13)
Thus the fluorescence yield is proportional to the absorption coefficient and Eqn. (4-13) shows that the intensity is maximized as sin0/sin~ is minimized and 12 is maximized. However, the signal-to-scattering ratio degrades if s is simply increased. On the other hand, for a ~ 0 (thin-film limit), Eqn. (4-12) is approximately given by
227 I(E) ... ~ I 0 t.t(E) e cosec0 dl-2/4~
(4-14)
It should be noted that Eqn. (4-14) shows that the intensity is proportional to 1/sin0 but is not dependent on r suggesting that the intensity is gained by a grazing incidence angle. In both the dilute and thin-film limit, the incidence angle should be minimized. This is one of the reasons why a high brilliance X-ray beam is important in fluorescence detection. Figure 4-14 shows a typical energy spectra for fluorescence X-ray excitation. In general, the elastic scattering peak shifts to higher energy when the monochromator is scanned, and the characteristic X-ray lines with a fixed energy appear as the primary beam energy exceeds the absorption edge. In a fluorescence detection, the dilute limit is determined by the energy resolution of a detector and, conventionally defined as the signal-to-background ratio, is nearly unity. Other characteristic lines in a system, and the elastic and inelastic scattering, are sources of background. One can reduce the intensity of scattering by an order of magnitude by inserting an X-ray filter which has the absorption edge between the characteristic line and the elastic peak. Conventionally, a thin metal foil or a homogeneous powder containing the (Z-l) element is used as a filter [61 ]. An ionization chamber [62] or a scintillation counter array [63] are used in combination with a filter. Figure 4-14 shows the effect of an X-ray filter in reducing the elastic scattering background. This method, however, cannot remove the other characteristic lines because of an insufficient energy resolution. A solid-state detector (SSD) can remove the
ELASTIC
Go Si
A
0 0 0 ...-.
x .......
>.1
without Zn filter
t--
c-
GoKa ZnK..AKe/.,~wi,h Zn filler Channel number
2047
Fig.4-14. Fluorescence X-ray spectrum for Ga impurities in Si. The elastically and inelastically scattered photons and characteristic X-rays from other elements which have lower absorption edges are the source of background. Using an X-ray filter, the intensity of scattered photons can be reduced.
228 background although the count rate is limited by detector electronics. A multi-element SSD can enhance the count rate by an order of magnitude [64]. Although defined by signal-tobackground ratio, the dilute limit is practically determined by a detector count-rate and incidentbeam-intensity. Figure 4-15 shows the Zn Kt~ fluorescence yield for thermolysin + L-valyl-Lleucine measured using a scintillation counter array. It should be noted that a high signal-tobackground ratio is achieved. In some case, the self absorption cannot be neglected, and several procedures for correction have been proposed [65,66].
Zn K-edge
Thermolysin + L-volyI-L-leucine A If) t-:3 ..c:i t_
__.o LL.
9.7
9.8
I
I
I
9.9 10.0 10.1 Photon Energy (keY)
I
10.2
10.3
Fig.4-15. Zn K-edge fluorescence yield spectrum for +L-valyl-L-leucine measured with a NaI scintillation counter array.
Dispersive XAFS In an energy-dispersive mode [ 14,67-70], X-ray absorption spectra are measured as a spatial distribution of dispersed X-ray beam intensity. A bent crystal monochromator is used to disperse the X-ray beam and a position-sensitive detector is used to record the incident or transmitted beam simultaneously. The arrangement of an energy-dispersive spectrometer [14] is schematically illustrated in Fig. 4-16. A linear photodiode array with 1024 sensor elements separated by 25 ~tm is used as a position-sensitive detector. Various phosphor materials, such as YVO4:Eu and Gd202:Tb are used to convert hard X-rays into visible photons [68,71]. The most sensitive method is to irradiate the sensor with X-rays peeling off the optical window, although the direct exposure causes some radiation damage to the sensor element [69]. Figure 4-17 shows the Fe K absorption spectrum for iron foil measured by a linear photodiode array (RETICON RL1024SF) in an energy-dispersive geometry. The extracted EXAFS oscillations are shown in Fig. 4-18 for various exposure times. In this Figure, the results of a conventional step-by-step scan are also shown for comparison. Except for slight differences in magnitude
229 CURVEDCRYSTAL ,
SLITS
FROMSYNCHROTRON RADIATION SOURCE
/~
~j;.......~J~'~^ /Y~ " SAMPLE / ~
/
/
MIRROR
i I/
/
/
\
X-RAY FILM OR POSITIONSENSITIVE DETECTOR
_ \
\
ROWLANDCIRCLE~f\
Fig.4-16. Schematic diagram for energy-dispersive X-ray absorption spectroscopy (ref. 14). Incident and transmitted photons with various energies are focused at a sample position and simultaneously detected by a position detector.
due to a nonlinearity of the photodiode array, the results for the two modes agree well. It is noted that the one shot experiment with a 35 msec exposure gives essentially the same EXAFS features with those of a conventional method, indicating a small systematic error, which is the advantage of this method. As noted already, mechanical instability is the dominant source of systematic error. In an energy-dispersive mode, the primary source of error is a statistical one.
Fe EXAFS 55msec x 100
=< I--
z
I
I
I
I
I
I
I
I
7200 7500 7400 7500 7600 7700 7800 7900 ENERGY (eV) Fig.4-17. Fe K-edge absorption spectrum for iron foil taken in an energy-dispersive geometry. A self-scanning photodiode array is used as a linear detector with 50 lxm spatial
resolution.
230 l
J
i
,
i
i
"'
Fe K-edge
0.1 0.05 0 -0.05 -0.1
0.1 z ......
0.05 0 -0.05 l
0.05 0 -0.05 -0.1 I
4.0
I
6.0
I
8.0
!
I0.0
I
12.0
I
14.0
16.0
Fig.4-18. Fe K-EXAFS oscillations measured by an energy-dispersive method for iron foil, with various exposure times. The spatial resolution of a linear detector is mainly influenced by the cross-talks and broadening in a phosphor (--50 l.tm). The total energy resolution is a convolution of the source size, divergence and spatial resolution of the detector. Although the geometrical energy resolution is 1.73 eV for a Si(111) crystal with R = 2460 mm, the total energy resolution is --5.6 eV at 7.1 keV. A better energy resolution (2 eV at 9 keV) is achieved by either a direct exposure of the photodiode or a Si(311) bent crystal. The energy-dispersive geometry is particularly important for small samples, such as a diamond anvil apparatus for high pressure. Recently, it was shown that a small focus size (~100 ~tm) is obtained using an elliptically-bent crystal. In a time-resolved experiment, the time-resolution is dependent on photon statistics, and is of the order of a second for phosphor-coated photodiodes, while it can be decreased by an order of magnitude for direct exposure. Further, for an experiment which can be repeated, the time-resolution is reduced by several orders of magnitude. A fast chemical reaction in solution was recently studied by a stopped-flow experiment with a 25 milli-second time
231 resolution. Figure 4-19 shows the time-dependent change of Fe K-XANES associated with a chemical reaction in solution with a time scale of 10 msec [72]. The time-resolution limited by photon statistics will be improved by more brilliant photon sources, such as a multipole-wiggler or an undulator if the response of a position detector is sufficient enough.
i
!
,
,
|
|
i
i
Fe K EXAFS e
. ~ /
71oo
I
I
i
O.SM C6H4(OH)2 d e
/// /// J//
71 o
i
O.SM Fe(N03)3 +
oo
%
i
b: 100 C: 250 d: 550 e:1150 I
71:4o
I
71' o
I
71 o
PHOTON ENERGY (eV)
I
msec msec msec msec
72'00 7220 1
Fig.4-19. Time-resolved Fe K-edge XANES spectra after mixing 0.3 M ferric nitrate and 0.3 M hydroquinone. Curves a, b, c, d and e are the spectra taken by integrating the signal during the periods of 0-25, 75-100, 225-250, 525-550 and 1125-1150 ms following the mixing: from ref. 72.
Surface-sensitive XAFS In a hard X-ray region ( > 4 keV), because of the large penetration depth (of the order of a micron) absorption experiments are not surface-sensitive. In order to apply the XAFS technique to surfaces and buffed interfaces, a surface-sensitive detection scheme is required. As discussed already, fluorescence detection [22] is a highly-sensitive technique but it is not surface-sensitive in a hard X-ray region because the fluorescence escape-depth and the penetration depth of incident photons are of the order of one micron. However, surfaceselective excitation is made possible by a grazing-incidence geometry [73] illustrated in Fig. 420. Below a critical angle for the total reflection (0 < ~ ) , X-rays are totally reflected as they cross the interface between the two media, reducing the extinction length by several orders of
232 8>>8c Fluorescence
Photons Reflected
X-rays~
\ photons P~otoelectrons \
-
X-ray penetration depth
.
,.,
() \
x..,
" ,.~ l ~Electron I ~escapedepth /~-2 I
V/ " \~,( (~_===~' ....
8<~c
L)
~
~
z
~
_~.~~;~.~" X-roy " penetration depth
~-"~ I
L
Ix- raY
lescapedepth
Fluorescence
X-rays
~__.~eEIsecCa~re~p,h IY .... I~ "~do,~.,,,, i ~,--w - ~ v , -
Fig.4-20. Schematic representations of the normal (above) and surface-selective (bottom) excitations. The grazing-incidence geometry can reduce the probing depth by orders of magnitude, achieving a surface-selective excitation. magnitude, which enhances the surface sensitivity by the same amount [74]. Although the capability of surface-sensitive XAFS using a fluorescence detection and a grazing-incidence geometry has been demonstrated, a conventional method using a large-area fluorescence detector such as an ionization chamber has limited the application to non-crystalline or polycrystalline specimens because of a diffraction problem. Recently, it was shown that a combination of grazing-incidence geometry with an energy analysis using a solid state detector (SSD) which can discriminate a fluorescence signal from a background scattering can dramatically improve the surface-sensitivity, allowing us to apply this technique to studies of epitaxial layers on single crystal substrates [75]. Figure 4-21 shows a schematic geometry of surface-sensitive XAFS experiments [76]. Because of polarization dependence, the two orientations of specimen, i.e., a vertical mount and a horizontal mount, provide the information on the radial distribution around an excited atom, parallel and perpendicular to the surface normal, respectively. Since a typical critical angle for the total reflection of Si at 12.3 keV is --3.5 mrad, a high resolution goniometer with translational adjustment capability is used to control the incidence angle. The vertical divergence of incident photons is limited by the slits (--50 ~tm) placed in front of a beam monitor so that
233 only a central region of a specimen (15 mm x 15 mm) is irradiated. In Fig. 4-22, the incidentangle-dependence of a fluorescence spectrum is shown schematically. On decreasing the angle of incidence, 0, the fluorescence signal increases. Below the critical angle 0c, the inelastic scattering peak drops sharply in intensity. A single-element Si(Li) SSD subtends a solid angle
Reflected beam monitor
z
_
z
,--.~/
tO
9~ S p e c i m e n
beam
Z X~ . . . . ~ Y
Vertical mount
~X Horizontal mount
Fig.4-21. Experimental arrangement for surface-sensitive XAFS by fluorescence detection. The grazing-incidence angle is monitored by the intensities of the fluorescence signal, elastic peak, and reflected beam" from ref. 76.
Or) Z
~.2.0
1313 13::: >-
.-'~~
Si/Get/Si (001) 12.50 keV
X
~1.5'
(,/") U,J I---
,,,1.0 UJ
"'0.5 0 __1 U_ I
-1
:
I
0
i
r ~ - - ~ - ~- -" ~-i--- ;
1 2 ANGLE (mred)
~.:e
Fig.4-22. Variation of fluorescence signal and reflected beam intensities for Si23/Gel/Si(001) as a function of angle between the incident beam and the surface 9taken from ref. 75.
234 of 1% while a multi-element SSD can improve the acceptance of detector by an order of magnitude. In this setup, the white X-ray beam is monochromatized by a fixed-exit doublecrystal monochromator [57] which horizontally focuses the output beam by sagittal bending of the second crystal. A typical photon flux N1011/photons/sec/mm2 with an energy resolution AE/E --2 x 10-4 at 9 keV is obtained at a normal-bending-magnet beamline at the Photon Factory. 4.3.3. Insertion devices in XAFS research
Beyond the limitations Synchrotron radiation from dedicated low-emittance storage rings are characterized with high quality of beam characteristics, i.e., long lifetime, stability and capability to provide high brilliance photons using various insertion devices. The sensitivity and time-resolution in fluorescence detection are determined by the incident photon flux and efficiency of the detector. One of the advantages of high-brilliance photon sources is the sensitive and rapid XAFS, i.e., data can be collected for more dilute samples using a smaller quantity with a better timeresolution. Insertion devices such as wigglers and undulators can increase the brilliance; a multipole wiggler inserted in a low-emittance storage ring provides more brilliant X-rays than bending-magnet radiation, by more than an order of magnitude over a wide energy range. The vertical angular divergence of a wiggler is, however, still larger than the monochromator acceptance by an order of magnitude. The mismatch becomes more serious in a glancing angle geometry for surface-sensitive experiments. Either a collimating X-ray optics or an ultra-low emittance storage ring would improve the mismatch. The advantage of an ultra-low emittance storage ring is obvious and its potential in acceptance matching will be discussed in Section 4.6. Here we focus our attention to the application of a multipole wiggler to XAFS researches at a low-emittance storage ring.
Multipole wiggler A multipole wiggler consists of an array of magnetic poles which locally bend electron/positron beam and highly directional radiation from each pole is added. Figure 4-23 compares the brilliance of bending-magnet radiation with that of 27-pole wiggler magnet inserted at the Photon Factory. This device can be used either as an undulator with a weak magnetic field (B0) or as a multipole wiggler with strong magnetic field. The magnetic field can be varied by changing a gap of magnet array. In a wiggler mode, the maximum B0 is 1.5 T [77], for which the total power is 5.44 kW. From the figure, an increase of brilliance by more than an order of magnitude is expected in a hard X-ray region (4-30 keV). Such a high power causes heat load problems for optical components which are irradiated with white X-rays. The thermal distortion of the monochromator crystal due to heat load deteriorates its throughput and
235
10,6 undulator mode 1st 1# 5
BL13 MPW wiggler mode 1.5T
o~
.~ 1014 C3
bending magnet
E E
.e 10~3 e,O or
~" 1012 Z _J _J
N 1011
1
1 ~ 10z 103 104 105 PHOTON ENERGY (eV) Fig.4-23. Calculated brilliance of a 27-pole wiggler (1.5 T) installed at BL 13 of the Photon Factory. For comparison, the calculated spectra for undulator radiation and normalbending-magnet radiation are also shown. The smooth function indicates the envelope of the fundamental peak as the undulator gap is varied. 10
0
angular divergence or energy resolution. Figure 4-24 shows a schematic representation of the plan view of the optics at 27-pole wiggler beamline at the Photon Factory. White X-rays pass through the water-cooled graphite absorbers and Be windows, absorbing 1/3~1/2 of the heat load, before irradiating a double crystal monochromator. The horizontal acceptance of the first crystal (3:1 focusing) is 4 mrad, which is sagittally focused to a 4 mm x 1 mm spot at ~36 m from the source point. For vertical focusing and rejection of higher harmonics, a bent Pt-coated fused quartz mirror is placed behind the monochromator. The total Fe Ket signal intensity from 500 t.tM myoglobin is 5 x 104 cps for B0 = 0.75 T, Is = 337 mA without focusing where Is denotes the stored current. The total count rate increased further by about 2.2 times with B0 = 1.5 T. By use of horizontal focusing, the flux would further increase by an order of magnitude, suggesting that maximum 106 counts per second is obtained for 500 l.tM biological samples. It is expected that the fluorescence detection
236 BL-13 branch beam line water-cooled monochromator be windows ~
\\
2oo~mx?
~
~
/
/
branch beam line shutters
r
A
Ir
_ . . . . . ._. . .
2 7- p o l e - - -
--..-7"/-/-- . . . . . . . .
_
JI
~1
eom width of magnetic pole : 12cm deflection parameter K : 0.M,,,25 magnetic field 0.02,-,1.5T 9 total power : 5.44kW (I.5T) _ power density: 2.3 kW/mradz (~ I I 5 10
monochromotor branch beam line shutters and slit assemblies I
15
I
20
I
I
25 30 Distance from the source point (m)
I
35
I
40
4=5
Fig.4-24. Optics of a 27-pole wiggler beamline. A sagittaUy-bent second crystal focuses the central beam horizontally while a bent fiat mirror focuses the beam vertically. can be increased by about three orders of magnitude by combining a multi-element solid state detector, a multipole wiggler, and focusing optics [78]. With this count rate, the main source of error is the systematic noise caused by various sources of instability. A continuous scanning monochromator [79] or energy-dispersive geometry [14] can provide a stable incident beamintensity, removing the mechanical instability of step-wise motion of the monochromator. It is also important to reduce higher harmonic contributions using mirrors which are enhanced for high magnetic fields. Thermal distortion of the first crystal which deteriorates the throughput by broadening the rocking curve is critically dependent on power and power density. The temperature gradient causes bending and local bump of diffracting plane as well as gradient in lattice spacing along the surface normal. More importantly for XANES, the deformed crystal results in the beampower dependence of energy resolution. The effect of heat load on the first crystal has been studied for various types of grooved Si crystals [80,81]. The recent studies indicate that the cooling efficiency of grooved silicon crystals can be improved by replacing a conventional semicircular cooling channel [80] with a fiat one with optimized dimensions [81 ]. The distance between the surface and the fiat water channel is 1 mm and the width of water channels and cooling fins is 0.6 mm. Using such an optimized grooved crystal, the energy resolution required for XAFS experiments (AE/E--2 x 10-4) was obtained within a limited power range (B0 < 1.25 T). In order to dynamically correct the deformation of a grooved crystal caused by
237 heat load and internal water pressure, several "adaptive" approaches are proposed for silicon crystal monochromators. The curvature of the diffracting plane for the first crystal is controlled by pushing the lower block parallel to the grooves, using a piezoelectric translator or air dumper, so that the rocking curve profile is independent of the beam power. Sagittal bending [82] is widely used to focus the multipole wiggler radiation, which extends over 4 mrad. The conventional sagittal bending used for horizontal focusing has a serious problem; the horizontal focus size depends on the radius of the bent crystal. In XAFS experiments, a constant focus size over a wide energy range (~1 keV) is required. In order to meet this purpose, some approaches to achieve an energy-independent focus-size have been reported [83]. Dynamic sagittal bending is a technique which keeps the focus size constant. There are several proposals, such as a translational movement of a monochromator [83] or a crystal bender using an inchworm motor [81 ].
Surface sensitivity Surface-sensitive XAFS experiments using a grazing-incidence geometry and a fluorescence detection requires high-brilliance beam. A combination of multipole wiggler and a multi-element SSD achieved sub-monolayer surface sensitivity [78]. Surface-sensitive XAFS with submonolayer sensitivity has allowed us to probe the atomic rearrangements on the surface upon layer-by-layer growth such as organometallic vapor phase epitaxy (OMVPE) or molecular beam epitaxy (MBE). In OMVPE, one of the epitaxial growth techniques for III-V materials, the exchange reaction on the first layer has been an important issue. For example, the surface exchange reaction of a group V element (As, P) during the epitaxial growth of InAs or InAsP alloys on InP substrates has been a serious problem from viewpoints of the atomic layer epitaxy (ALE) since such an exchange would degrade interface sharpness in chemical composition. In order to clarify whether the substrate P atoms exchange with As atoms when exposed to ASH3, surface-sensitive XAFS on the As K-edge was measured for an InP(100) substrate exposed to AsH3 for a very short period (0.5 sec). This experiment can be a critical test for surface sensitivity as the coverage of As is expected to be well below 1 monolayer (ML). In Fig. 4-25, the As Ktx fluorescence-yield spectrum for AsH3-exposed InP(100) is shown. The As coverage determined by comparing the fluorescence intensity with those for strained InAs layers grown epitaxially on InP(100) was ~0.1 ML. The k-dependence of EXAFS oscillations indicates that the As atoms are bonded with In atoms, which evidences that ~1/10 of surface P atoms are replaced by As atoms within a short interval of 0.5 sec [84].
238
As/InP(lO0) (,,,1014/cm2) 0.1 ML 300K
As K- edge 0.10 0.08 ._o U_
0.06 0.04
I
I
0'11.6
I
I
I
I
I
I
I
11.8 1 2 . 0 12.2 12.4
I
I
12.6
PHOTON ENERGY(keY)
I
I
12.8
I
I
13.0
Fig.4-25. As K-edge fluorescence yield spectrum measured in a surface-sensitive geometry using a 27-pole wiggler and a multi-solid-state detector (SSD) for 0.1 monolayer (ML) of As on InP(100). The As atoms substitute P atoms when the InP surface is exposed to AsH3 gas flow : from ref. 78. 4.4. DATA ANALYSIS 4.4.1. Initial data treatment
In this section, the fundamental data analysis procedures to obtain structural information, i.e., extraction of the EXAFS oscillations, Fourier transform and curve fit are described. Figure 4-26 illustrates the schematic absorption spectrum and extracted EXAFS oscillations. The pre-edge region of absorption spectra contains the absorption due to: (1), ionization chambers as beam intensity monitors; (2), higher shells and; (3), other elements in a system. For fluorescence yield spectra, the pre-edge region is a featureless background due to the scattered photons and fluorescence lines from higher shells or other elements which smoothly varies as the excitation energy is scanned. The signal-to-background ratio reflects the energy-resolution of a fluorescence detector. The pre-edge region for spectra measured with an SSD is usually negligibly small. The first step in data analysis is the elimination of the smooth background functions which can be extrapolated from the pre-edge region. The mass absorption coefficients vary as a function of photon energy given by
p/p = Z (Ci/13 + Di/1,4)
(4-15)
239
0, ....
~
~
~
_
k
#o
"M
/~
/VV
re-o~
absorption
E Fig.4-26. Schematic representations of an X-ray absorption spectrum measured by a transmission mode. EXAFS is defined as modulations of absorption normalized with a free-atom absorption for a particular inner shell. where ~, is the X-ray wavelength, p is the density and Ci, Di are coefficients of Victoreen's function given as a function of photon energy for a particular element i [85]. Conventionally, the background absorption is approximated by a simpler function such as aEfl where a and fl are empirical parameters determined to fit the pre-edge. Using tabulated values of coefficients Ci and Di and normalization of Eqn. (4-15) using the edge jump, p/p above the absorption edge can be generated and subtracted which varies according to Eqn. (4-15). This evaluation of background absorption is useful since it is valid for both transmission and fluorescence modes, which allows us to compare the fluorescence data with transmission data. If the composition is known, the only parameter is a thickness which is determined by normalizing Eqn. (4-15) with the net absorption defined as a difference between the pre-edge absorption and a smooth fleeatom absorption of interest. A free-atom absorption P0 can be evaluated by fitting a smooth function, such as a cubic spline, to the EXAFS oscillations. The normalized EXAFS oscillations are then obtained by normalizing the modulations of absorption coefficient which is obtained by subtraction of a flee-atom absorption.
z(k)
= AI~
I laO
(4-16)
240 where A# and/-tO are the magnitudes of modulation in # and atomic absorption, respectively. Figure 4-27 shows the normalized Ge K-EXAFS oscillations as a function of wave number of photon energy k for crystalline (solid line) and glassy (dashed line) GeO2 measured at 80 K (a), and 300 K (b). On reducing the temperature, the magnitude of oscillation increases as a result of temperature-dependent term in Eqn. (4-6) which describes thermal disorder. It is concluded that the glassy and crystalline GeO2 have a similar short-range order, since the fundamental EXAFS oscillations arising from the interference between the nearest neighbors are essentially the same for the two forms.
0.04
(a) 80K
GeO2
I
0.03 0.02 0.01 0.00 -0.01 -0.02 -0.03 Trigonol GeO2 Crystol
--- -0.04
,,_... X
(b) 300 K
GeO2 Gloss
0.03
0.02
0.01 0.00 -0.01 -0.O2 -0.03 -0.04 0
I
2
I
4
I
6
I
I
I
I
I
8 10 12 14 16 18
R(~I
Fig.4-27. Normalized Ge K-EXAFS oscillations for crystalline (solid line) and glassy (dashed line) GeO2 measured at room temperature and at liquid nitrogen temperature" taken from ref. 11.
241 4.4.2. Fourier transform
For a quantitative analysis in real space, or to filter out the contribution of a particular shell, the normalized EXAFS oscillations are Fourier-transformed according to the following equations.
F(r) = (1/2n) ~k z(k) w(k) exp(-2ikr) dk = (1/2n)
~k z(k) w(k) cos(2kr) dk - i (1/2n) ~ k z(k)
= Re
(4-17)
F(r) + i Im F(r)
w(k) sin(2kr) dk (4-18)
where w(k) is a window function in order to minimize the cut-off effects in the Fourier transform. Figure 4-28 shows the magnitude of Fourier transform IF(r)l of the Ge K-EXAFS oscillations for crystalline (trigonal) and glassy GeO2[11 ]. Several sharp peaks correspond to positions of near neighbor atoms of an excited atom, although its position is shifted to small R because the total phase shift g1(k) in Eqn. (4-6) is neglected. The first peak located around 1.7 A is due to the tetrahedrally coordinated nearest oxygen atoms while the second peak at ~ 3 ./k is due to the second-nearest Ge peak. In trigonal GeO2, Ge atom is located at the center of a tetrahedron in which four oxygen atoms occupy each corner. The GeO4 tetrahedra are connected each other sharing oxygen atoms. On going from a crystalline phase to a glassy one, the first peak does not change while the magnitude of the second-nearest neighbor peak decreases. This indicates that the short-range order within the GeO4 units is strictly maintained but connectivity is disordered in a glassy state. These results are consistent with the near-edge structure data. Figure 4-9 shows that the near-edge structures observed for a crystalline phase broaden in a glassy state but essential features remain unchanged indicating the conservation of the short-range structure. Following Eqns. (4-17) and (4-18), the region of interest in real space can be Fourierfilterd into k-space. Contribution of each shell is separated and a non-linear curve fit analysis is used to yield structural parameters. In order to determine the structural parameters such as coordination number, bond length and mean-square relative displacement, the total phase-shift function for each pair and the back-scattering amplitude functions must be known. There are several approaches to fit the experimental data with Eqn. (4-6). A total phase shift and a backscattering amplitude can be extracted empirically from a model compound data with a known crystal structure, on the basis of chemical transferability of these functions. If a model compound is not available, a theoretically-calculated phase shift and amplitude are used with a
242
(a) 80K
GeOz
(b) 30OK
Trigonol GeOz Crystol GeOz Gloss
..-...
..ci
0
1
4
5
0
1
4
5
Fig.4-28. Magnitude of Fourier transform of the Ge K-EXAFS oscillations for trigonal crystal and glassy GeO2 measured at room temperature and at liquid nitrogen temperature : from ref. 11.
slight adjustment of the E0 value to match the zero of muffin-tin potential. An alternative approach is the intermediate method which uses the experimental phase-shift function and theoretical amplitude as will be described in the next subsection.
4.4.3. Curve fit analysis The least-squares curve fit analysis is a procedure to fit the experimental data with Eqn. (4-6) with structural parameters as variables. The step is to filter out the first-shell contribution by a back-Fourier transform into k-space to yield the simple sinusoidal oscillations as a function of k. The total phase shift and back-scattering amplitude I~(k,Tt)] calculated using a curved wave method are plotted in Fig. 4-6. Clearly, ~j(k) and I~(k,,t)] depend on the atomic species and show characteristic k-dependence. The total phase shift ~(k)is approximated by a polynomial ~j(k). In order to extract the total phase shift from the reference data, it is parametrized in a simple formula such as,
tDi(k) = a0 + al k + a2 k2 + a3 k-3
(4-19)
243 where coefficients a0, al, a2 and a3 are adjusted so that the calculated EXAFS oscillation according to Eqn. (4-6) best fits the reference data using a least-squares method. In a curve fit to the experimental data based on a single-scattering formalism using a theoretical I~(k,n)l, two other parameters are introduced: the damping factor in order to account for the inelastic energy loss of photoelectrons, which apparently reduces the amplitude, and the disorder parameter which accounts for the relative displacement. Although the former parameter S02 is k-dependent, it is often approximated as a constant. Unlike diffraction experiments, oi in EXAFS is the mean-square displacement, which refers to a "relative" displacement between an excited atom and a scatterer atom. For a single shell, there are six parameters to be determined by a least-squares curve fit. If the reference material has a coordination geometry similar to that of unknown material, it is reasonable to assume that the empirical phase shift is transferable [86], although there is uncertainty in the amplitude, depending on the electronic states [87]. Figure 4-29 shows the Ga K-EXAFS oscillations for GaAs powder, plotted as a function of the photoelectron wave number k, in ,~-1 (above). The Fourier-filtered first-shell contribution (solid line) is compared with the results of a least-squares curve fit (dashed line) below.
/
0.06 0.04
Ge K-edge
..-. 0.02 "-"
0 -0.02 -0.04 -0.06 I
I
I
I
I
I
I
First shell t--
A
I
4
I
6
I
8
I
10 k (All
I
12
I
14
I
16
Fig.4-29. Ga K-EXAFS oscillations for GaAs powder plotted as a function of photoelectron wave number, k, in/~-1 (above). The Fourier-filtered first shell contribution (solid line) is compared with the results of a least-squares curve fit (dashed line) below.
244 4.4.4.
Disorder
A well-known single scattering expression of EXAFS, Eqn. (4-6) is justified for a case where the disorder is small enough to be approximated as a harmonic oscillator. Then, the distribution of atoms around an excited atom can be expressed by a Gaussian function. The mean-square relative displacement (MSRD), oij in EXAFS is the ensemble average, and is smaller than the sum of mean-square displacement (MSD), o52 + o] 2, by 2crio'j which expresses the correlation between an excited atom and a scatterer atom where o-12 and crj2 are MSD's for the i-th (excited) atom and the j-th (scatterer) atom, as expressed by the following formulae.
oij 2 = < [ R j . (uj - u 0 ) ] 2 > = < ( u 0 . Rj)2> + < ( u j . R j ) 2 > - 2 <(u0. Rj) ( u j . Rj)>
(4-20)
Here, Rj is the vector along the i-j bond and u0 and uj are displacement vectors for the excited atom and the scatterer atom, respectively, originating from the equilibrium positions. One can calculate the MSRD from a proper lattice-dynamic force model which describes the phonon distribution. A simple model such as the Debye model or the Einstein model, is used to evaluate the effect of correlation or a temperature-dependent dynamic disorder. In highly disordered systems, one must replace the Gaussian distribution with a pair correlation function gij(r) given by z(k) = -1/k ~ I~(k,rt)l [ gij(r)/r 2 ] sin(2kr + ~j(k)) exp (-2r/Aj) dr
(4-21)
It should be noted that even if gij(r) is symmetric with respect to r, the integration over r introduces a higher order term in the total phase shift function. Moreover, in general, gij(r) is asymmetric, reflecting the potential shape. Thus, neglect of the disorder-induced term in the phase would result in a smaller bond length. This effect can be corrected by a cumulant expansion [88]. Eqn. (4-21) can be rewritten as z(k) = -1/k I~(k,n)l Im [ exp i (2kr + ~ij(k)) exp {E ((2ik)n/n!) Cn}]
(4-22)
In most cases, it is sufficient to introduce C2 and C3 in order to correct the higher-order terms in amplitude and phase, respectively, where C2 and C3 are coefficients of cumulant expansion, determined by fitting the amplitude and phase part of Eqn. (4-6) to the data. The physical meaning of the second-order term is the thermal expansion. The relation between the thermal expansion coefficient, a, and the cumulants of the vibrational amplitude, has been elucidated by Frenkel and Rehr [89].
245
4.4.5. Multiple scattering The advantage of EXAFS is that the multiple scattering effect is negligibly small for the nearest neighbor, in sharp contrast to LEED which must take into account the multiple scattering. The difference arises from the fact that in EXAFS, the electrons are emitted as a spherical wave and the interference is detected, at the origin between the outgoing wave and back-scattered waves which are strongly damped because of inelastic energy loss. In contrast, in LEED the electrons come in and are reflected as a plane wave in a forward-scattering geometry. One of the reasons why the multiple scattering cannot be neglected in LEED is that the forward-scattering amplitude is large in low energy region. Since the total path length increases as a result of multiple scattering, the effect of multiple scattering is often observed in a low-k region as higher frequency oscillations than those for the nearest-neighbor spacings, and these oscillations rapidly damp, canceling out each other in EXAFS. In Fig. 4-7, the multiplescattering pathways are illustrated for the three atom (0-1-2) system. Photoelectron waves originate and are terminated at the excited atom, 0, after being scattered at other atoms (1, 2). In addition to the single scattering between 0 and 2 (a), a double-scattering path (0-1-2-0, b) and a triple-scattering path (0-1-2-1-0, c) are indicated. The multiple-scattering pathways would modify the amplitude and phase of EXAFS oscillations as a strong function of the geometry, and it is well known that the triple scattering is magnified for the case of a linear array (lens effect) in Fig. 4-7. It should be noted that the double-scattering is important for covalent materials with an sp 3 configuration, where the second-nearest neighbor-peak contribution depends strongly on the atomic species at the second-nearest sites. Figure 4-30 shows the Fourier transform of Ga K-EXAFS oscillations for; (a), GaP; (b), GaAs and (c), GaSb. All these materials have a zinc blende type of structure where each cation is coordinated by four anions as nearest-neighbor atoms and by 12 cations as the second-nearest neighbor atoms. Although the species of atoms at the second-nearest sites are the same for the three compounds, the peak profile at the secondand third-nearest neighbor atoms is quite different. This demonstrates that in this geometry, i.e., fl -- 71 o, the double-scattering pathways are strongly affected by the forward scattering at the second-nearest neighbor atom, and in particular for atoms with large scattering amplitude in the low-k region, such as P. In such a case, the multiple-scattering paths appear as shoulders with larger r, which makes the peak in the Fourier transform broader as shown in Fig. 4-30. 4.5. A P P L I C A T I O N S
4.5.1. Dopants in oriented polyacetylene Polyacetylene, (CH)x, has been extensively studied because of its unique properties as a conducting polymer when it is doped with electron donors or acceptors [90,91]. The electrical conductivity of pristine (CH)x increases by several orders of magnitude when it is doped with electrons as in AsF5, Br2, or 12 [92-94]. Transport, optical, and magnetic properties of these
246
K-edge GaP
A
e-
GaAs
r'r LI_
GaSb
012545678910
RC I
Fig.4-30. Magnitude of Fourier transform of the Ga K-EXAFS oscillations for GaP, OaAs and GaSb. On going from a light element scatterer (P) to a heavy one (Sb), the peak profile for the second-nearest neighbor (Ga) changes as a result of multiple scattering pathways via the first-nearest neighbor group VI atoms. systems were discussed in terms of solitons [95] or the spinless defect states resulting from the bond-alternation domain wall in a conjugated carbon chain. XAFS studies of doped (CH)x have been aimed at investigation of the local structure of dopants, in order to understand the mechanism of doping and the role of charged solitons. The structure of pristine trans-(CH)x was first determined by Baughman et al. [96] from the packing analysis of X-ray diffraction data, which was later refined by Fincher et al. [97] to provide direct evidence of bond alternation. For the case of iodine doping, a structural model similar to the first-and third-stage graphite intercalation compound has been reported [96], in which dopant molecules are contained in the planes separated by one or three close-packed (CH)x chains forming a linear column. This structure model accounts for a long X-ray diffraction spacing in heavily-doped trans-(CH)x and the existence of I3- or I5- polyiodine ions which are evidenced by Raman [98] and M6ssbauer [99] experiments. EXAFS has recently been applied to doped polymers
247 such as AsF5-doped (CH)x [100], FeC13-doped (CH)x [101], Br2-doped (CH)x [102], and Br2-doped (SN)x [103]. In case of bromine doping, the room temperature conductivity increases by many orders of magnitude on increase of bromine concentration, to the metallic region (y ~ 0.1-0.2) [92,93], followed by a saturation and finally a gradual decrease. With further doping (y > 0.6), a sharp drop of conductivity occurs and (CHBry)x becomes an insulator [104]. Early EXAFS studies on unoriented (CHBry)x [102] showed that the rather short Br-C spacing (1.96 ,~) dominates the radial distribution of bromine in the concentration range 0.05 < y < 0.55, which suggests that either a substitution or addition reaction takes place over a wide range of dopant concentration. Because of the highly polarized nature of synchrotron radiation, one can study the orientation of a linear molecule by analyzing the polarization dependence of X-ray absorption spectra. The geometrical arrangement of bromine species within fibrils can be obtained by measuring the absorption spectra as a function of an angle between the electrical field vector E and the fibril axis of oriented (CHBry)x. The polarization factor of EXAFS is given by the following formula.
A(0j) = 3 cos20j
(4-23)
where 0j denotes the angle between the electrical field vector and the radial vector for the i-th scatterer atom. The integral of the right hand side of Eqn. (4-23) over 0 is unity, and for higher symmetry than three-fold, or randomly oriented powder samples,
z(k) becomes a normal
formula given by Eqn. (4-6). The polarization dependence provides a useful means to investigate anisotropy in both structural and electronic states, for single crystals and surfaces. The X-radiation of synchrotron radiation is more than 95 % polarized in the central part of the horizontal plane which is further enhanced by the Lorentz polarization effect in the monochromatization process. Polarized Br K-EXAFS oscillations for trans-(CHBry)x with y = 0.02 are shown in Fig. 4-31, where (a) and (b) denote the spectra taken with E parallel (E//c), or with E perpendicular (E_Lc) to the fibril axis, respectively [ 105]. The fibril axis is parallel to the c-axis of the unit cell with P21/n symmetry in trans-(CH)x [96]. Also shown in this figure, denoted as (c), are the data for Br2 gas. The EXAFS oscillations for (CHBry)x with E parallel to the caxis (E//c) extend to the high-k region, whereas those with E perpendicular to the c-axis (Elc) rapidly damp in magnitude. This shows that doped bromines exist as polybromine ions, highly oriented along the c-axis for the following reasons; I~(k,n)l of bromine has a maximum at k ~ 6-7/~-1 and extends to k ~ 15/~-1. In contrast, Ifj(k,r01 of low-Z elements such as carbon has a peak at a small k ~ 2-3 ,~-1 and falls off sharply with increase of k. Therefore, we attribute the oscillations extending more than ~1 keV above the edge to those caused by the scattering by bromine atoms, while the rapidly damping oscillations observed only in the low-k region result
248 I
.....
I
I
I
I
l
I
I
Br K-edge
(CHBrylx y -0.020 0.02 0.01 0-0.01 -0.02~
x" . - -
o.oz-I-
-0.01 ~Z
0.02 0.01 0-0.01 -0.02
L L i 2
t
4
~
6
i
8
I
I
10 12 k (/1-~ )
~ t4
16
9 ]8
Fig.4-31. Polarized Br K-EXAFS oscillations for trans-(CHBry)x with y =0.020, together with those for Br2 powder : taken from ref. 105. from scattering by carbon atoms. The Br-Br scattering dominates the E//c EXAFS oscillations whereas the Br-C scattering dominates the E_Lc EXAFS oscillations. The fact that a small amount of Br-C scattering is present in the E//c z(k) oscillation of (CHBry)x, in contrast to the Br-Br scattering only observed in the E//c EXAFS suggests a high degree of orientation of polybromine ions. Figure 4-32 shows the results of Fourier transform of k times z(k) for the data shown in Fig. 4-31 over the range of k from 2/~-1 to 16.8/~-1. Peaks in this figure are shifted toward a smaller radial distance, as a result of the phase-shift effects. A striking difference between the results for (CHBry)x with E//c and Elc data is the magnitude of the prominent peak at 2.21 A corresponding to the first-nearest Br-Br distance. The peak at 4.78 /~ corresponds to the second-nearest Br-Br distance. The first-nearest Br-Br spacing in (CHBry)x is 2.551 /~ +_ 0.10/~, which is longer than that of Br2 gas by 0.27 A. This bond-length expansion is interpreted as a result of the charge transfer from carbon chains. Such a bondlength expansion is expected to bring about the reduction of a bond strength, which is
249 |
i
,
(CHBr~lx 80K
Y=0.015
Or) z >-. ,,~ rlr" i-.
E//c ............ E l c
rlr"
!
o
1
2
5
4
5
6
7
RADIAL DISTANCE (A)
8
9
10
Fig.4-32. Results of Fourier transform of the Br K-EXAFS oscillations for trans-(CHBry)x with y = 0.015. Solid and dashed lines indicate the results for the electrical field vector E parallel and perpendicular to the fibril axis" taken from ref. 105. evidenced by the softening of Raman frequency of the stretching mode [106]. The secondnearest Br-Br distance is 5.12,~ + 0.20 A, which is close to the sum of the first-nearest Br-Br distances, indicating that a symmetrical structure is maintained with an average inter-atomic spacing of 2.55 A. These results rule out the possibilities of Br- and Br2 molecules. Therefore, the dominant bromine species are either Br3- or a longer Br2n+ 1- chain such as Br5-. It is difficult, however, to determine the average size of polybromine ions from the second-nearest neighbor coordination number, which is expected to vary between 1.33 and 2 depending on the size of a linear chain. The Br-Br spacing of 2.55/~ matches the unit dimension (2.46 A) of (CH)x along the c-axis, making the periodicity of polybromine ions commensurate with that of the polymer backbone in the columnar direction. The effective coordination number of bromine for (CHBry)x (y = 0.015) with E parallel to the c-axis is 1.1 + 0.6 from the curve-fit analysis. The real coordination number of bromine is estimated to be 0.367 + 0.2. Since the oriented Br3- polyion would give the coordination number of 1.33, and a longer linear chain has a larger value, this implies that less than-27 % of bromine species are in the form of polybromine ions. This suggests that the unpolarized EXAFS of (CHBry)x would be dominated by the Br-C oscillation. Indeed, previous EXAFS results [102] for unoriented (CHBry)x showed that most of bromine atoms are covalently bonded to the polymer chain in the higher concentration range (y > 0.05), which is consistent with these results. The number of anion species in (CHBry)x is of the order of 1020 in the concentration range studied (0.015 < y < 0.036). Since the coordination number of bromine is hardly dependent on the dopant concentration in this concentration range, the fraction of bromine in the state of polybromine ion is constant. This indicates that the number of anion
250
species acting as acceptors is proportional to the dopant concentration in the lightly doped region. By employing the Fourier transform using the low-k region (2 < k < 8.5 A -1) which emphasizes the carbon scattering, a small peak at 1.29 A and two more distant peaks at 2.68 A and 3.08 A are assigned to the Br-C correlations for E//c (CHBry)x. The shortest Br-C spacing (2.0 A) is close to the Br-C bond length between bromine and aromatic carbon, which is reported to vary from 1.82 A to 1.96 A [107]. If trans-(CH)x is perfectly oriented and bromines are substituted for hydrogens of the polymer chain, there would be no contribution of Br-C scattering in the E//c EXAFS oscillations. These results indicate that the brominesubstitution reaction takes place at sites with imperfect orientation or induces disorder. More distant Br-C spacings located at 3.4 A and 3.8 A are less dependent on the polarization direction and are close to the sum of the van der Waals radii of bromine and carbon atoms. Since these spacings are only observed in the specific concentration range where Br-Br scattering is prominent for E//c, these indicate the correlation between polybromine ions and a carbon chain. As these spacings are close to the half of the length of the b-axis in the unit cell for trans-(CH)x (7.32 A) [96] polybromine ions seem to take ordered sites intercalated in closepacked(CH)x chains. Figure 4-33 shows the Br K-XANES for: (a), Br2 gas; (b), (CHBry)x for E//c ; (c), (CHBry)x for E.Lc and, (d), bromobenzene. Characteristic features in XANES are denoted as A, B and C. Although the features B and C are weak in the E//c XANES for (CHBry)x and
I
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I
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l
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I
'
I
'
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A
Br2
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l--n-1 s
I/,, I
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-20
,
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0
;
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;
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,
PHOTON ENERGY (eV)
I
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80
Fig.4-33. Polarized Br K-XANES spectra for (a)Br2 gas, (b) (CHBry)x for E//c, (c) (CHBry)x for E_Lc and (d) bromobenzene. A strong anisotropy indicates the highly oriented polybromine ions" taken from ref. 105.
251 Br2, feature C merges into a shoulder of B in the E_l_c XANES of (CHBry)x and bromobenzene. A sharp peak is observed in all spectra 9-10 eV below the threshold, or the continuum limit (13.474 keV). This resonance peak A is the transition from the 1s core level to unfilled bound p states. In the Br2 molecule, this state is the 4p*(cr) molecular orbital, since 4p(tr), 4p(n:) and 4p*(n:) orbitals are all occupied. For a linear molecule such as Br2, the molecular orientation can be studied from the peak intensity which varies as cos20j [21 ]. If a charge transfer occurs, an extra charge is expected to partially fill 4p*(n:) states, and therefore the magnitude of this peak can be a measure of charge transfer. For a closed-shell configuration of halogen atom, 4s24p 6, this peak would not be observed. Thus the results in Fig. 4-33 shows that bromines exist in the form of polybromine ions with partially filled 4p* states, which is consistent with the EXAFS results. Peak A shifts to higher energy in E//c (CHBry)x compared to Br2, by .-2 eV, indicating the decrease of covalency in the Br-Br bonds, consistent with the bond-length increase observed in the E//c EXAFS data. This sharp peak shifts slightly further to higher energy, by --2 eV, with a decrease in intensity in the Ed_c orientation. Remarkable similarity in XANES features observed between E.l_c (CHBry)x and bromobenzene suggests that bromines are bonded to carbon atoms with the sp2 configuration. This implies that a substitution reaction takes place in the concentration range 0.015 < y < 0.036. In summary, it was shown that bromine atoms in (CHBry)x exist as linear polybromine ions with an average inter-atomic spacing of 2.55 A, highly oriented along the fibrillar axis in the concentration range 0.015 < y < 0.036. The observed coordination number of polybromine ions suggests that only a small portion of bromine atoms form polybromine ions. The number of polybromine ions intercalated in close-packed (CH)x chains is, however, roughly proportional to the dopant concentration. In the direction perpendicular to the fibril axis, the radial distribution of bromine is dominated by the Br-C bond, indicating that polymer chains are brominated by either substitution or addition. A substitution reaction is likely to be the case, since the bromines are covalently bonded with carbon atoms with the sp 2 configuration, as suggested by the near-edge data. The structure model of trans-(CHBry)x emerging from these results is characterized by the coexistence of highly oriented polybromine ions and (CH)x chains brominated by a substitution reaction.
4.5.2. Oxide high Tc superconductors After the discovery of high Tc superconductivity (HTSC), the mechanism of pairing has been a subject of intensive studies from the theoretical and experimental viewpoints. One of the striking features of HTSC materials is the fact that superconducting properties are highly sensitive to the microscopic structures; in particular, the role of oxygen defects as interstitials [108] and vacancies [109] has been recognized as one of the key factors for HTSC. In
252 Nd2-xCexCuO4-t~ (NCCO), for instance, superconductivity is observed over a narrow Ce concentration range (0.14 < x < 0.18), only after heat treatment under reducing conditions. The role of reduction in superconductivity has been a puzzling problem since the additional carder concentration originating from an observed small change of oxygen deficiency, t~, cannot account for a large change of Hall coefficients [110]. Although one may expect that the excess oxygen atoms provide additional holes, the effect of oxygen interstitials is sometimes not so simple. X-ray absorption near-edge structure studies for Nd2-xCexCuO4-t5 indicated that Cedoping may induce oxygen interstitials [ 111 ]. EXAFS studies confirmed that such interstitials are formed at apical sites by reduction which suppresses electron doping [ 112]. It is well known that the ordering of the oxygen vacancy in the Cu 1-O 1 chain strongly affects the transport properties of YBa2Cu3Oy (YBCO), which varies from an insulator to a superconductor depending on y. There are two classes of superconducting phases, i.e., low Tc (50 ~ 60 K, 6.3 < y < 6.7) and high Tc (90 K, y > 6.7) phases, while the oxygen deficient phase (y < 6.3) is a tetragonal insulator [113-115]. X-ray absorption near-edge structure studies [116, 117] showed that ordering of oxygen vacancies in the Cul-O1 chain for y < 6.7 dopes extra carders into the conducting plane via a self-doping mechanism [ 118]. Although the physical reason for vacancy ordering at specific y values is not established yet, we show that the phase diagram of HTSC as a function of y for YBCO can be qualitatively explained. Vacancy ordering is consistent with the phase diagram for CaLaBaCu3Oy (CLBCO) where Tc drops sharply at y -- 6.7 because of disordered oxygen vacancies. Since the parent compounds are antiferromagnetic insulators, the exchange effect which produces intermediate range spin correlations is expected to be important in the doped superconductors also. Antiferromagnetic correlation exists in the superconducting regime. Theories on the basis of magnetic interactions have been proposed as a possible mechanism of HTSC [119]. On the other hand, the importance of the lattice has been indicated experimentally. Lattice anomalies have been observed by EXAFS [120], neutron scattering [121,122], and ion channeling [ 123,124], The observation of dynamic fluctuation between the two structures has been related to polaron or bipolaron formation [120,121]. Although previous studies focused on the anomalous lattice distortions at Tc, recent experiments showed that lattice anomaly is also present at much higher temperature than Tc [125,126]. In this subsection, the temperature-dependent EXAFS anomalies at high temperatures in highly oriented YBCO is discussed in relation to elastic properties and spin excitation. The detailed knowledge on the lattice anomaly became available from polarization-dependent EXAFS experiments using untwinned single crystals and a fluorescence detection technique [54]. This technique can probe the local and dynamical nature of atomic displacements of particular species from the temperature-dependence of EXAFS. The use of fluorescence detection has made possible to observe the polarized XAFS for HTSC single crystals which are free from spectral distortions due to the strong absorption. This technique can provide reliable data for asgrown HTSC thin films prepared by reactive co-evaporation or with a typical thickness of 1000
A.
253 Electronic states of high Tc superconductors
Since the discovery of HTSCs, XAFS has been used to characterize the local structure and electronic states, particularly, those of the CuO2 plane, a common structural feature in these materials. L a 2 - x M x C u O 4 - y (M=Ba, Sr) is a class of materials having an octahedral CuO6 unit and Tc .'. 30--40 K [127,128]. XANES studies on these materials have shown that the fundamental Cu valence is 2+ or the d 9 configuration. Heald et al. measured the polarization dependence of the near-edge structure on the oriented polycrystalline ceramic samples and found a strong anisotropy in the near-edge structure[ 129]. Although transmission XAFS experiments were difficult because of a strong absorption for HTSCs grown in small flakes, Oyanagi et al. showed that polarized XAFS spectra can be obtained by a fluorescence detection technique [54]. Polarized Cu K-XANES for single crystal La2CuO4 demonstrate capabilities of XAFS technique as a means to investigate the electronic states and symmetry of unoccupied states. The polarized Cu K XANES spectra for single crystal La2CuO4 (5 x 5 x 1 mm) with 0 = 90 ~ and 0 = 10 ~ are shown in Fig. 4-10, which were grown by a top-seeded solution growth (TSSG) method [130]. Figure 4-34 shows the crystal structure of La2CuO4 where lattice constants (a = 3.78 /~, b = 3.82 A, c = 13.13/~) and the position parameters (ULa = 0.3607, uO = 0.1824) are taken from the neutron data by Jorgensen et al.[131]. Since the electrical field vector E is
768A
(
Lo2Cu04 OLo
OCu
9
Fig.4-34. Crystal structure of La2CuO4 in which the copper atom is octahedrally coordinated by oxygen atoms :from ref. 54.
254 parallel with the ab-plane for 0 = 90 ~ while E is nearly parallel to the c-axis for 0 = 10 ~ it is clear that near-edge features a, b and e are polarized along the c-axis, while c and d are polarized within the ab-plane. The polarization dependence of thes,e structures strongly suggests that a and b are due to the ls-4p*(Tr) transition, while c and d are assigned to the ls4p*(tr) transition, though the origin of peak e is less clear. The split of the ls-4p* transition commonly observed in the near-edge spectra of square-planar Cu 2+ compounds was interpreted in terms of a ligand-to-metal charge transfer. Higher and lower energy peaks observed at the threshold (a,b) have been explained in terms of the ls-lj_s3d94p * and the ls1...~.s3dl0L4p* transitions where the underline and L denote the hole state and a ligand, respectively [132,133]. Since the linear Cu + compounds or flattened Td Cu 2+ compounds show only the single ls-4p* transition peak, polarized in the direction perpendicular to the Culigand bond [132,134], the presence of a split ls-4p* transition confirmed the 2+ valence of Cu atoms in undoped La2CuO4. Similar conclusions have been reached by chemical shifts of absorption edge [135-138]. It should be noted that the polarization-dependence of Cu KXANES rules out a possible assignment of peak e to the contribution of Cu 3+ proposed by Alp et al.[137]. It is also unlikely that the Cu(La) antisite disorder [138] contributes to in-planepolarized peak e since the Cu and La atoms form a bcc-like sublattice which does not have a polarization dependence. Strong anisotropy is observed also in the EXAFS region in the polarized Cu K-edge absorption spectra for La2CuO4 single crystal as shown in Fig. 4-35 (above). These EXAFS features are directly related to anisotropic radial distribution between the c- and a(b)- axis directions which are indicated in the magnitude of Fourier transform shown in Fig. 4-35 (bottom). The first peak at ca.l.4 ~ in IF(r)l is due to the fourfold Cu-O1 bonds (1.90 ,/k) within the CuO2 plane, while the second broad peak located at ca. 3.4/~ consists of the Cu-La distance (3.25 A), the Cu-Cu distance (3.80/~) and the Cu-O1 distance (4.25 ,/k). Although the adjacent peaks are not resolved due to the limited range of k-space used in the Fourier transform, the overall anisotropy in the Fourier transform is well reproduced by taking the effective coordination number [ 139] into account. It should be noted that the second-nearest Cu-Cu distances, along the c-axis (5.38/~) and within the CuO2 plane (3.80/~), are observed for 0 = 10~ and for 0 = 90 ~ respectively.
Doping-induced oxygen defects Figure 4-36 shows the Cu K near-edge spectra for NCCO, annealed under reducing conditions with various pO2 (upper curves). For NCCO, samples synthesized from a mixture of CeO2, Nd203 and CuO were annealed at 1050 ~ for 20 hours in an argon-oxygen gas mixture with various oxygen partial pressures (pO2 = 1-10 -3 atm). On decreasing pO2, a systematic change is observed in the absorption edge features; an increase at -8990 eV and decrease a t - 9 0 0 8 eV which are correlated with one another. The shoulder-like ls-4p*(Tr)
255 I
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Cu K-edge
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LezCu04-y single crystel 8-10"
if) e-
8=90 * ..,.,..
.m O
u..
l
c hv@ J I
I
p
I
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9.0
8.8
I
I
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9.2 9.4 9.6 Energy (keY)
'
Cu K- edge
~
I
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La2Cu04-y single crystel
/~
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8:90 ~ O= 10 0
D
25
i!
;" 1
A/'A 2
5
4
R
5
(A)
6
7
8
9
10
Fig.4-35. Polarized Cu K-edge absorption spectra for single crystal La2CuO4 measured in a fluorescence detection mode (above)and the magnitudes of Fourier transform of these data (bottom). X-Ray absorption spectra are measured in a fluorescence mode for various 0, the angle between the incident X-rays and the ab-plane (ref. 54).
256 I
i
Cu K-
i
I
edge
i
Z
p02 (Oxygenpressure) 1 9 1 arm 2" 0.1 aim 3 0.001 arm
5
>.Q::
i
I 2
J
l--
I
'
ls - 4p*((7)
O::
Nd1.s5Ceo.15CuO4-y
cl
Id
p02 =O'O01atm'~ ls-4p*(Tr)//
~ J
__~ I
I
8990
"Nd~.4Ceo.2Sro.4CuO4-y
(T*phase)
I
I
I
I
9000 9010 ENERGY (eV)
I
9020
Fig.4-36. Cu K near-edge structures for Ndl.85Ce0.15CuO4 and Ndl.4Ce0.2Sr0.4CuO4. Oxygen-partial-pressure dependence for Ndl.85Ce0.15CuO4 is shown above, and the data for the two structures with and without apical oxygens (T' and T*) are compared below. features, a and b, increase the intensity while the ls-4p*(G)features, c and d, decrease in intensity as pO2 is lowered. This spectral change indicates the decreased number of apical ligands since full multiple scattering calculation of XANES clarified the effect of ligands by a comparison between the T*- and T'-type structures which have a five-fold pyramidal geometry and a square planar one, respectively [ 140]. In the lower column, the data for NCCO with pO2 = 10-3 atm are compared with those for Ndl.4Ce0.2Sr0.4CuO4-6 (NCSCO, T*-type phase). NCCO with the lowest pO2 was chosen in order to obtain the ideally square-planar copper with no apical oxygen atoms. For a fair comparison between the two structures, with and without apical oxygens, the Ce composition of NCSCO was matched to that of NCCO. The observed systematic spectral change for NCCO on decreasing the pO2, although small, is essentially the
257
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,
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Fig.4-37. Variation of the characteristic near-edge features in a difference spectrum between Ndl.85Ce0.15CUO4 and pure Nd2CuO4 as a function of oxygen-partial-pressure. Without Ce, the oxygen pressure dependence is weak, in sharp contrast to Nd1.85Ce0.15CUO4. same as the difference between the data for NCCO and NCSCO, i.e., pO2 dependence indicates that apical oxygen interstitials are removed by heat treatment with under reducing conditions. This is more clearly demonstrated by taking the difference spectra, and is confirmed by the EXAFS experiment discussed below. In Fig. 4-37, the magnitude of the characteristic features of difference spectra is plotted as a function ofpO2 together with the diamagnetic fraction Z measured at low temperature and weak magnetic field. A strong correlation between the near-edge structure and Z is revealed [111 ]. This indicates that the subtle change in the coordination of copper strongly affects the HTSC fraction. The amazing fact that the oxygen deficiency d; in quenched NCCO (t~ < 0.01), obtained from the thermogravimetric measurement, is smaller than the observed spectral
258 change, by an order of magnitude suggests the existence of a special mechanism of electron doping or mobility change, suggested by Hall experiments [110]. Since the Ce L3-edge results indicate that the Ce valence (4+) is not affected by reduction and Ce concentration, a simple picture of electron doping originating from the oxgen deficiency cannot explain the magnitude of spectral change. This can be explained if oxygen interstitials induced by doping at apical sites balance in number with oxygen vacancies in the Nd-O plane. In fact, quenched Nd2CuO4-t~ has an oxygen deficiency (S ~ 0.08) which roughly coincides with the Ce concentration in superconductive NCCO. An apparent oxygen stoichiometry can be achieved if the number of excess interstitial oxygen atoms at apical sites matches that of vacancies. Weakly-bound excess oxygen interstitials would be formed or removed with ease, depending on the annealing conditions. In NCCO, excess oxygen would give rise to five-fold oxygen-coordinated Cu atoms and nine-fold oxygen-coordinated Ce 4+ ions. Apical oxygen atoms are unfavorable for electron doping to the CuO 2 plane because of the repulsive Coulomb interaction, as shown by the Madelung energy calculation by Kondo et al. [118]. Thus both the electron density and mobility in the CuO2 plane would be strongly affected by oxygen interstitials at apical sites. Figure 4-37 shows that quenched Nd2CuO4-t~ is insensitive to pO2, indicating that the oxygen defects in the nondoped sample is not dependent on reducing conditions but on a cooling rate, suggesting that these vacancies are in balance with an approximately equal number of interstitials in NCCO. The existence of oxygen interstitials is further confirmed by the Cu K-EXAFS measurements for NCCO and Nd2CuO4. Figure 4-38 shows the Fourier transform of EXAFS oscillations as a function of the photoelectron wave number, k. A sharp peak located at ~ 1.7/1, is due to the oxygen atoms while the peaks at around 3.2 .~ are the contribution of Nd and Ce atoms as the second-nearest neighbors. The increased intensity of the first peak in the Ce-doped sample indicates that oxygen interstitials are indeed induced by doping. A schematic diagram of the generalized oxygen interstitial model is shown in Fig. 4-39. In T*- and T-type compounds, oxygen vacancies are formed at apical sites near Sr sites which are removed by heat-treatment under high oxygen partial pressure [ 112]. However, doping induced defects might be locally disordered. It would be difficult to detect them by structural techniques based on long-range order, such as a powder neutron diffraction technique which reported no evidence for oxygen interstitials. The X-ray near edge study showed that the doped Sr atom is nine-fold coordinated by oxygen atoms, including one interstitial oxygen atom [141] consistent with the present results.
Oxygen defect ordering In Fig. 4-40, the Cu K near-edge spectra are shown for YBCO with various oxygen contents y. In this experiment, the oxygen content y was systematically varied by changing the quench-temperature in the final heat treatment. A systematic spectral change is observed in the
259 i
z
i
I
Cu
I
I
K-edge
Nd2Cu04
0
|
A
L.. o
c:[
Nd1.a5Ceo.15Cu04
LL
0
2
4 6 8 10 RADIAL DISTANCE {A) Fig.4-38. Fourier transform of the Cu K-EXAFS oscillations for Ndl.85Ceo.15CuO4 (solid line) and pure Nd2CuO4 (dotted line). The nearest-neighbor oxygen peak at around 1.8 A increases on doping Ce, indicating that oxygen interstitials are formed at apical sites.
ls-4p*(lr) (a, b) and the ls-4p*(cr)(c, d) regions, as a function of y. The essential feature, i.e., the increase in a lower energy region ( 8 9 8 2 - 8989 eV) and the decrease in a higher energy region (8997 - 9004 eV), is the same as in Fig. 4-36. Figure 4-41 shows the Cu K difference XANES spectra for YBa2Cu3Oy obtained by normalization and subtraction of the reference spectrum (YBa2Cu306.96). The results indicate that: (1), the "apical" oxygen atoms at O1 sites in the Cul-O1 chain decrease in number with decrease in y, and (2), the two-fold coordinated monovalent Cul is formed above a critical y value (y ~ 6.7). Figure 4-42 (top) shows the magnitude of characteristic near-edge features in a difference spectrum and measured Tc values as a function of y. Clearly, a sharp increase in spectral change at y ~ 6.7, which indicates the formation of two-fold coordinated copper, is strongly correlated with Tc, indicating that oxygen-vacancy ordering gives rise to a self-doping [118]. For y > 6.7, the monovalent copper sites increase in number almost linearly with a decrease in y, although Tc sharply drops at y = 6.3 where the crystal symmetry changes into a tetragonal one, as indicated in the lattice constants shown in Fig. 4-42 (bottom). A possible model for oxygen vacancy ordering is schematically illustrated in Fig. 4-43. In orthorhombic YBCO with y ~ 7, oxygens occupy O1 sites forming a square-planar CuO4 unit extending along the b-axis, sharing oxygens at O1 sites. With a decrease of y, oxygen
260 T
I
OCe ~(MOD) 0 Ce. Lowp02 , o ~ )e Ce Anneol doping e
Nd2Cu04 Slowcool
Quenched
eo
D~(NMOD) Quenched B~o.o8
e DopedCu02plone
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T*,T
LozCu04 Slowcool
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D3*(NMOD)
p
DopedCu02plane
Lot.eSro.zCu04
~o.o
Quenched Fig.4-39. Schematic diagram of oxygen interstitials at apical sites in a square-planar (T'), a pyramidal (T*) and octahedral (T) coordination induced by doping. For the T'-type structure, interstitials suppress electron doping while in the T- and T*-type structures, vacancies suppress hole-doping.
vacancies are formed but they are randomly distributed within the Cul-O1 chain. The disordered vacancies keep most of the copper ions three-fold coordinated. A sharp increase of the two-fold coordinated copper atoms at y -~ 6.7 indicates that a d i s o r d e r - o r d e r transition occurs at a specific oxygen content, i.e. y ~ 6.7. As the experiment shows, the number of twofold-coordinated coppers increases almost linearly with y, indicating that ordered oxygen
261 I
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A O1 e--
=)
I..=. e--.m
'
i
'
I
Y= 6.22 Y = 6.52 Y = 6.70 Y = 6.71
'
'
I
Y= 6.91 Y = 6.96 A
B
I
8982
C
/
/'
Is-4p*(r)~ ,
Ol
J.---"~
i
I
'
'
i
i
I
I
j
i
~"
,,// ' 32
1
i
I
'
,d
i
D
I
I
YB(]eCu30y
is-4p~ (o')
c, /~~6,7
Y =6.88
8976
'
Cu K- e d g e
i
I
'
,
,
E
I
8988 8 9 9 4 9000 ENERGY (eV)
,
,
I
9006
,_
Fig.4-40. Oxygen-composition dependence of the Cu K near-edge structures for YBa2Cu3Oy. The characteristic features change systematically on changing oxygen composition, indicating that two-fold coordinated copper sites are formed. I A
'
'
I
'
'
I
'
i
I
'
'
I
Cu K-edge
u') I---
'
i
I
'
YB(]zCu30y
i-.-.,
Z :Z)
A
C '
B
A
1" Y = 6.22 2" Y = 6.52
1-7
3y-6.70
3-7 4-7
I--
4" Y =6.71 5" Y = 6.88 6" Y = 6.91 7"Y =6.96
,,=, ,,=, ,HI
8976
,
,
I
8982
,
i
I
i
i
I
,
,
ID I
,
8988 . 8 9 9 4 9 0 0 0 ENERGY (eV)
,
I
,
9006
Fig.4-41. Cu K-XANES spectra for YBa2Cu3Oy obtained by normalization and subtraction of the reference spectrum (YBa2Cu306.96).
262 50
ioo
.YBo2Cu30y
40 -
9
~
80
o~ so ~=
60 ---
20
40 h'~
I0
20
0 6.(
'
--
I
I
6.2
6.4
Oxygen Content y
_o.o,f f z0.0t oosr.L YBo,2C u3. ~y ~ v
6.8
~ ~
o~
O~--L-~--~ 60
6.2
/
'
6.4
0 7.0
I
6.6
~3_J
t10.,o o~
<1
'
,
6.6
'
Oxygen Content y
'
6.8
~==P-JO 7.0
Fig.4-42. Oxygen composition-dependence of the intensities of characteristic features in a difference XANES spectrum (closed circles and squares) and Tc values (closed triangles) for YBa2Cu3Oy (above). Variations of lattice constants of YBa2Cu3Oy as a function of oxygen content y (below) where b - a and Ac values are indicated by open circles and squares, respectively. For Ac values, the minimum c value is taken as a standard.
vacancies are formed for y > 6.7. Oxygen vacancies are then disordered at y --6.3, inducing the symmetry change from orthorhombic to tetragonal in a macroscopic scale. CLBCO is a HTSC material (Tc ~ 80 K, y > 6.87) which has a similar structure to that of YBCO, although the crystal symmetry is tetragonal throughout the possible y range [142]. In CLBCO, Tc gradually decreases with the decrease in y, and a sample with y -- 6.69 is an insulator. The distinct difference between CLBCO and YBCO in HTSC phase diagram has been a puzzling problem. Here, we interpret the difference as arising from the disordered oxygen vacancies in CLBCO. Kuwahara et al. showed that in CLBCO, at least for y > 6.7, two-fold coordinated Cul sites are not formed [143]. This indicates that oxygen vacancies are disordered, possibly because of a random distribution of cations. Kondo et. al. showed that oxygen vacancies are spontaneously ordered, from Madelung-energy calculations [ 118]. When vacancies take adjacent sites and the two-fold copper sites are formed, holes are provided through a valency change from Cu 2+ to Cu + which is called a "self-doping". The charge compensation via this mechanism is dependent on the vacancy ordering. For YBCO with y >
263 6.7, since ~0.3 oxygen vacancies per unit cell can pair with ~0.3 additional vacancies, this mechanism can compensate for a loss of carders, at least by 1/2 for y > 6.4. Interestingly, YBCO has a low Tc phase over a wide range for 6.3 < y < 6.7, while CLBCO only has a HTSC phase for y > 6.7. These results indicate that a HTSC phase diagram is strongly
dependent on the charge compensation via a self doping mechanism. order
-
..~-~>~,4-c.~=~
y =7
01 defect disorder disorder or~ler order
-
1
or o +2 ~
-
o .0-,L o
+2
+2 ~ - ~ + 2 +2 9 0--0.4
+2 +2 +2 o-O-o-O-o
-
o
order
;--
+1 9
y=6.7
+2 +2 oOo
y =6.5 Ay =0.2
f
oOo
order I disorder
y=6.8
01 defect
9
9
oOo
oOo
9 +1
oOo
9 oOo
9 +1
9
9 9 +1 +1
9 ~XO
y=6.3
oOo O
O
+1+1
y :6.0
Fig.4-43. Schematic model for oxygen ordering for YBa2Cu3Oy. At oxygen content y ~ 6.7 decreased, the oxygen vacancies are ordered so that monovalent copper species are formed which partly compensate the loss of carders. At y --- 6.3, residual oxygens are disordered, to give rise to a macroscopic structural change from orthorhombic to tetragonal symmetry.
Lattice anomalies The interplay of charge, lattice and magnetic fluctuations has been recognized as an important issue in understanding the mechanism of HTSC. Recently, lattice anomalies have been observed near T c , using various local structural probes such as EXAFS [120,144], pulsed neutron scattering [ 121,122] and ion channeling [ 123,124]. The anomalous temperature dependence of the mean-square relative displacement in EXAFS data for YBCO has been interpreted as an indication of dynamic fluctuations or tunneling between the two closely separated oxygen positions [120]. Pulsed-neutron experiments also indicated the dynamic oscillations between the two positions for T12Ba2CaCuO8 [ 121]. Most of the observed lattice
264 anomalies are characterized by temperature-dependent anomalies at Tc involving displacement of oxygen atoms along the c-axis. However, they are not consistent with a simple picture of phonon softening indicating the importance of local distortion. Temperature dependence of phonon frequency is strongly dependent on phonon modes. For YBCO, the Ag Raman mode shows a small hardening which is interpreted in terms of phonon self-energy change associated with superconductivity [145] while infrared absorption in the 584 cm-1 mode shows a slight softening on decreasing the temperature across Tc [146,147]. The discrepancy between the local structural probes and vibrational techniques is a subject of intense discussion [147,148]. Recently, in-plane lattice anomalies well above Tc were reported from polarized EXAFS experiments for T12Ba2CuOy [149,150], Bi2Sr2CaCu208 [125] and YBCO [151] which suggest an interplay between lattice and spin fluctuations. A highly oriented YBCO thin film (Tc = 87 K) has been synthesized by a reactive coevaporation technique [152]. Polarized Cu K-EXAFS data taken with the electrical vector parallel (e//ab) and perpendicular (e//c) to the ab plane were analyzed in order to obtain the temperature dependence of the mean-square relative displacement, or, and bond length R. In Fig. 4-44, the local structure of YBCO is illustrated. The filtered Cu-O EXAFS oscillations were fitted by a single oxygen position model in order to minimize errors arising from correlations among parameters, although Mustre de Leon et al. claimed that the split-apical-
Cu2
Correl!tion 04 Cu2 YBa2Cu30T Fig.4-44. Schematic local structure of YBa2Cu3Oy (y -- 7) observed by polarized EXAFS. At Tc, the relative displacement between copper and oxygen for all Cu-O bonds increases. The Cu2 atom shifts toward apical oxygen, associated with a charge transfer between the two apical oxygens coupled to their motions along the c-axis.
265 oxygen-position model fits better the experimental data [ 120,153,154], which became the basis for a tunneling picture within a double well potential [ 11 ]. As the apical oxygen, 04, bridges Cu2 atoms in YBCO, the contributions of Cu 1-O4 and Cu2-O4 pairs were determined from e//c EXAFS data whereas the in-plane Cu-O pairs were averaged for el/ab data. Figure 4-45 shows the variation of R as a function of the normalized temperature T/Tc. The Cu2-O4 bond length (open circle) clearly decreases by 0.01 /~ while the Cul-O4 distance (closed circle) is unchanged below Tc, which indicates that the Cu2 atom slightly shifts toward apical oxygens
below Tc. Figure 4-46 shows the T-dependence of o'for the Cul-O4 bond (closed circle), and for the Cu2-O4 bond (open circle). A sharp increase in trfor the Cul-O4 bond is observed at Tc, which is essentially the same as previously reported anomalies [120,144,153,154]. Reflecting the fact that the Cu2-O4 bond is weaker than the Cu 1-O4 bond, tr for the Cu2-O4 bond has a larger temperature coefficient; tr decreases gradually on lowering T until it sharply increases at
Tc, which is followed by a gradual decrease below Tc [144,154]. In contrast, o'for the inplane Cu-O bond shows a deviation from a normal T-dependence at a much higher temperature than Tc, 120 K ~1.5 Tc. The observation of a sharp increase in relative displacement for the out-of-plane Cu-O is striking considering the fact that both Raman and IR phonon softening is of the order of 1%. Thomsen et al. attributed the Raman frequency shift (softening) below Tc to a resonant electron-phonon
'
2.31
'
'
interaction
'
I
'
'
associated with
an HTSC state [145].
In
'
o YBo2Cu307/MgO
Cu2-04
9YBo2Cu3OT/MgO
Cul - 0 4
t
2.30 -
o,,~ ..=..
2.29
1.96
1.9
tt ,
0.,5
,
q
t |
n
1.0 T I TC
.5
Fig.4-45. Normalized temperature dependence of the Cu2-O4 distance (above) and the Cul-O4 distance (below) for highly oriented YBa2Cu3Oy (y ~ 7).
266
|
'
'
'
'
I
'
'
'
'
o YB0zCu3Or-a/MgO
Cu 1 - 0 4
9YB0zCu3OT-a/MgO
Cu - Oeq
|
1
,_.., ro '0 x
,...,.
t II
0
0
_.N b n
-1
I-oa v
b 0
-0.5
t ttt t t |
O5
,
|
,
,
I
1.0
TI Tc
,
,
t
i
115
Fig.4-46. Normalized temperature dependence of the mean square relative displacement trin YBa2Cu3Oy ( y - 7) for the Cul-O4 and Cu2-Oeq bonds. The in-plane bond length Cu-Oeq is obtained by averaging the values parallel with the a and b directions. Bi2Sr2CaCu208, much greater phonon softening is observed at Tc for copper vibrational modes within the CuO2 plane although the copper-phonon-density is normal across Tc for the c-axis [155], in sharp contrast to the softening oxygen-phonon-density of states for YBCO [ 156]. Although further efforts are necessary to relate observed lattice anomalies to particular phonon modes, it seems that the out-of-plane anomaly arises from the dynamic fluctuation of apical oxygens. A sharp increase in the magnitude of anharmonic motion for apical oxygens at Tc, along the c-axis, may indicate a strong coupling between the two CuO2 planes via a charge transfer between the two apical oxygens, assisted by oxygen motions, as illustrated in Fig. 444. The fact that Cu2 shifts toward 04 indicates that some charge is indeed transferred into the O4-Cu 1-O4 region. The coupling of oxygen atom and the charge transfer is quite similar to a tunneling picture although the recent X-ray diffraction study on untwinned YBCO found no evidence for the split apical oxygen positions [157]. The fact that tr values for the in-plane CuO increase sharply at Tc indicates the development of local phonon modes. The most stable Jahn-Teller-active vibrational mode for square-pyramidal CuO5 clusters, asymmetric stretching, is one of candidates [158]. A recent neutron-scattering study supported a phonon softening for the asymmetric bending mode below Tc [155]. We note that these asymmetric bending/stretching modes are important for commensurate charge modulation or coherent bipolaron conduction [159]. Bianconi et al. attributed the in-plane lattice anomaly at ~l.5Tc in Bi2CaSr2Cu208 to the development of polarons [125].
267 Lattice anomalies observed in EXAFS are strongly related to elastic anomalies and spin fluctuations. Ultrasound velocity shows an anomaly at Tc and well above Tc, Ts ~ 120-130 K in YBCO [160], the characteristic temperature at which the magnetic susceptibility changes. Moreover,Ts is close to the onset temperature of the lattice constant anomaly in X-ray diffraction profile ( ~125 K) initially interpreted as an indication of Gaussian fluctuation [ 161,162]. Below Tc, hardening of ultrasound velocity [ 160] was observed, which is related to the shear-modulus anomaly, indicating a strong coupling between the HTSC order parameter and shear-distortion [163]. A bipolaron formation [164] is one of the candidate mechanisms compatible with a strong lattice distortion. The fact that the anomaly exists at, and well above, Tc, indicates an interplay between lattice effects and spin excitations. (T1 T)-1 in NMR at Cu2 sites [165], where T1 is the nuclear spin-lattice relaxation time, and neutron inelastic scattering experiments [166] are interpreted as evidence for the spin excitation at much higher temperatures than Tc. More recently, from careful transport measurements of YBCO, Ito et al. showed that deviation of resistivity from a T-linear behavior arises from the formation of a "spin gap" [ 167]. The critical temperature TO, defined as an inflection point of dp/dT in ref. 167, corresponds to the elastic anomaly at Ts and the lattice anomaly in EXAFS well above Tc. Within the framework of resonating valence bond (RVB) theory, the "spin gap" corresponds to the formation of a spin singlet [119] and Tanamoto et al. recently explained the anomalies in (TIT) -1 by an extended t-J model [ 168]. In summary, the effects of oxygen defects (interstitials and vacancy) and their ordering, are revealed. Oxygen interstitials are formed at apical sites associated with Ce doping. A simple oxygen interstitial model can explain the effect of heat treatment under reducing conditions; surplus oxygen interstitials which suppress electron doping are removed by heat treatment under reducing conditions. The dependence on oxygen content of the Cu K-XAFS for ceramic YBCO shows that the ordering of oxygen vacancies seriously affects HTSC via a self-doping mechanism. A puzzling difference in the HTSC phase-diagram between YBCO and CLBCO is attributed to the order/disorder of oxygen vacancies. Lattice anomalies in YBCO, studied by polarized EXAFS on a highly oriented YBCO thin film, are related to anomalies in elastic and magnetic properties. Anomalies in the out-of-plane Cu-O bond along the c-axis are observed at Tc ~ 87 K while the magnitude of the in-plane oxygen motion reduces its magnitude well above Tc ~ 120 K; the mean square relative displacement increases sharply at Tc, for in-plane and outof-plane Cu-O pairs, while in-plane anomaly is observed at 1.5Tc, which coincides with anomalies in elastic properties and spin excitations. The lattice anomalies observed at Tc seem to be related to low-energy excitations, such as polarons or spinons in a RVB picture. Lattice anomalies well above Tc seem to be related to spin-gap and elastic anomalies, suggesting a connection between the lattice and spin fluctuation.
268
4.5.3. Biological systems Local structure of heme-irons
Fluorescence XAFS is used as a local structural probe of dilute metals in biological systems, such as metalloproteins and enzymes. In this subsection, the capability of XANES to detect subtle changes of coordination is demonstrated. Some ferric hemoproteins such as hydroxide complexes of ferric hemoglobin, and myoglobin show values of spin susceptibilities intermediate between those characteristic of 5- and 1-unpaired electrons. The spin-states of these complexes have been interpreted in terms of thermal spin equilibrium between two magnetic isomers, one in a high-spin and the other in a low-spin state. These results have been obtained from the analysis of the temperature dependence of the magnetic susceptibilities and light-absorption spectra [169,170]. Spin states are strongly related to the coordination geometry and species of ligand. While fluoride and cyanide complexes were found respectively to be in purely-high, and purely-low spin states, hydroxide, azide, imidazole and cyanate complexes exhibited intermediate magnetic susceptibilities and the optical spectra characteristic of an intermediate spin state. Although the thermal spin-equilibrium has been studied extensively for a variety of hemoproteins and their model heme-compounds, from a thermodynamic view-point [169171], the relationship between the local structure of heme-iron and the spin states has not yet been established. Heme-iron is expected to be out of the heme plane in high spin states, which is favored from the view-point of metal-d- ligand-p interaction. The magnitude of the displacement of heme-iron from the heme plane in deoxyhemoglobin (Hb(II)) has been of considerable interest in relation to the affinity of oxygen binding [ 172-174]. The movement of heme-iron into the heme plane in an oxygenated form has been proposed as the mechanism for transfer of the information on oxygen-binding from one subunit to the other, causing the transition from the low-affinity T state to the high-affinity R state [172]. The local structure of heme-iron has been studied for a variety of hemoproteins by means of extended X-ray absorption fine structure (EXAFS) [173-176], which determined the distance between heme-iron (Fe) and the nitrogen atoms of porphyrin (Np) and that between heme-iron and the nitrogen atoms of proximal histidine (Ne), together with the distance between heme-iron and the sixth ligand, such as oxygen, within 0.01 A accuracy [174,175]. However, the magnitude of displacement of heme-iron from the heme plane is difficult to determine by EXAFS because it is sensitive to bond-distance but insensitive to the threedimensional arrangement of the atoms, i.e. the bond angle [174,177]. On the other hand, XANES is suitable for probing the coordination geometry in hemoproteins because of its sensitivity to the atomic arrangement, which affects the interference of photoelectron waves through multiple scattering [178].
269
Spin-state equilibrium in myoglobin Figure 4-47 shows the temperature dependence of Fe K-XANES for dilute (2 mM) Mb(III)OH- measured by a Si(Li) solid state detector [179]. At 80 K, where Mb(III)OH- is purely in the low-spin state, several characteristic features are observed which were originally discussed by Bianconi et al [177]. On going from the predominantly high-spin state (300 K) to purely low-spin state (80 K), the spectrum changes systematically. Feature A indicates the position of a shoulder structure, observed for low-spin Mb(III)OH- at 7125 eV, which appears as an inflection point in the first derivative. This shoulder gradually disappears with increase of the high-spin content; at 300 K no shoulder structure is observed. A broad bump structure C2 observed at 7148 eV for low-spin MbOH- reduces in its intensity at higher temperature and is not observed at 300 K. Peak P, observed at 7112 eV, is a quadrupole-allowed transition from Fe l s to empty 3d states. This is observed as a weak but rather sharp peak at 80 K, which broadens at 300 K. It is found that there are three features, P, A and C2 within 40 eV from the edge, which are spin-state sensitive. In particular, feature A is strongly correlated with a decrease in broad peak intensity at C2 in the high-spin state. If a spectrometer and beam are stable, a difference between a sample and a reference can provide a sensitive means for detecting and analyzing quantitatively an elaborate change in XANES. Difference spectra are obtained by subtracting 80 K data from spectra at higher temperatures. The inset of Fig. 4-48 indicates the difference spectra around feature A, as a |
!
!
8OK(LOW SPIN) C~ 300K (HIGH SPIN) I-
.-.
/
2D
|
D -"
|
Fe K-edge MbOH
C2
h"
I
I
>...
rr ,~
I
i ,
CD
oc.
#//
-
ta_
e
"
1
7.10
.11
#
,., f i ,..I N
'L I
I
I
7.12 7.1.3 7.14 PHOTON ENERGY (keV)
I
7.15
7.16
Fig.4-47. Temperature-dependence of the Fe K-XANES spectra and their first derivatives for 2 mM Mb(III)OH- in low-spin (solid line) and high-spin (dashed line) states measured by a fluorescence mode : taken from ref. 179.
270
Mb(EI)OH-
I-z
=.,...,
..~
>... n,.I-13o n..."
a = - - - - - A H o AS* 1+exp - - - + - -
nS0=-5 col/mol.deg
w
~I
/
/,//
z w rrw LJ_ LJ_
/
0
7110
I
I
I
50
100
150
I
200 Y/K
I
250
7120
7130
E,~ERGYU (eV)
300
Fig.4-48. Intensity variation of the spin-state-dependent feature in XANES for 2 m M Mb(III)OH-(open circles). The inset shows the difference spectra obtained by subtracting the reference measured at the lowest temperature from data taken at higher temperatures. The high-spin content, ct, calculated according to the formula in the figure, is also shown. function of temperature. The intensity of this peak, which decreases with decrease in temperature, is plotted as a function of temperature and compared with the high-spin content calculated from thermodynamic data [169,170]. The high spin content is normalized at 300 K to the relative intensity of this difference peak. The overall behavior is well predicted by thermodynamic data, which suggests that the difference is proportional to the high-spin content. The magnitudes of C2 and A in a difference spectrum were found to be inversely correlated with one another. Bianconi et al. interpreted the change of structure "A" as a result of increased multiple-scattering of photoelectrons within the heme plane [ 177], which is sensitive to the motion of heme-iron out of the heme plane. Their calculations showed that near-edge features 40-50 eV above the edge are sensitive to the heme-iron displacement from the heme plane. However, the spectral change for features A and C2 is not completely reproduced by their calculation for the photoelectron scattering within the heme plane. Feature B observed by Bianconi et al. [177] halfway between features A and C1 for deoxy-Hb(Hb(III)), can be observed for deoxy-Mb(Mb(II)) but feature B is absent for other Mb derivatives. On the other hand, recent polarized XANES studies [ 180,181 ] on square-planar Cu complexes have shown that sharp features ascribed to a ls-4pz transition aopear 4-5 eV above the Is-continuum transition. The former transition is polarized normal to the plane while the
271 latter is polarized along the plane. The interaction between Fe 4pz and ligand 4p*~(]r) states are sensitive to the distance between metal and axial ligand [ 181]. Bianconi et al. have reported that the features C 1 and C2 observed for single-crystalline Mb(II)CO are strongly polarized in the direction normal to the heme plane [182]. The interaction between the 4pz and ligand n'* orbital reflects the symmetry around a metal atom. In either case, the axial movement of heme-iron can affect the absorption threshold regions. If the metal-ligand distance is not changed appreciably between the high- and low-spin states, the observed change in spin-state sensitive structures arises from the change in the metal-ligand interaction which is caused by the displacement of heme-iron out of the heme plane. The heme-iron of Mb(III)OH- is coordinated with four nitrogen atoms of pyrrole rings and with another nitrogen atom of proximal histidine, as indicated in Fig. 4-49. The sixth ligand of Mb(III)OH-, denoted by X, is a hydroxyl ion. In Mb(II), heme-iron has no sixth ligand and is displaced from the heme plane by ~0.40 ,~ and from the nitrogen plane by -0.27 .~, according to the structural analysis by Takano [183]. Because of a doming of the pyrrole ring, the iron-nitrogen distance is slightly shorter than the iron-heme plane distance. Such a displacement of heme-iron from the mean heme plane has been reported for various high-spin porphyrin compounds [184]. On the other hand, heme-iron in low-spin myoglobin derivatives such as Mb(III)CN-, Mb(II)CO, and Mb(II)O2 is expected to be within the heme plane or only slightly displaced from it. The displacement of heme-iron out of the heme plane is expected to weaken the octahedral ligand field (Oh) by lowering the symmetry. High-spin states are stabilized by this distortion of the square-planar geometry because the weaker ligand field reduces the eg-t2g splitting energy, which is favorable for high-spin states. In hemoprotein, HIGH SPIN S=5/2 eg t,l
t2g
t
LOW SPIN S=1/2
tt
fl' i i
~.N..-~(5th) N M
I
X
(6Ih)
~.,~ .-~{5th)
V
~N~
, i
N
\
3M
V
- ' " Fe - - . ~ , , _ _ L L
..... ~N.--~'-,,,M X (61h)
Fig.4-49. Schematic local structures of Mb(III)OH- associated with a temperature-dependent distortion around a heme-ion in relation to spin states.
272 the spin states of heme-iron are strongly affected by the chemical character of the sixth ligand. Low-spin states are stable for strong ligands such as CN- while high-spin states are commonly found in hemoproteins with a weak ligand, such as H20. The heme plane-normal component of the ligand field due to the axial distortion also allows p-d mixing and lowers the dz 2 orbital energy, which contributes to the stabilization of high-spin states. The change in distance between heme-iron (Fe) and the center of the heme plane (Ct) on going from a low-spin to high-spin state is not associated with an appreciable change in Fe-Np proximal, Fe-Ne or Fe-O(OH2) bond lengths, since the difference in the nearest-neighbor distance would appear as an energy shift. These results are consistent with recent EXAFS studies [173,174] on oxy- and deoxy-hemoglobin which found that the Fe-Np distance is 2.05 A for both forms. If the movement of heme-iron is not associated with the change in Fe-Np distance, the four nitrogen atoms (Np) are pulled toward the center of a square. The strain energy caused by the Fe-Np bond-bending can be partly released if the porphyrin ring is further deformed so that the doming is enhanced. The displacement of heme-iron may involve deformation of the entire porphyrin ring which can stabilize the high-spin states. In this respect, the XANES results are consistent with the recent interpretation of EXAFS data for oxy- and deoxy-hemoglobin by Perutz et al. [ 174]. Bianconi et al. estimated that the Fe-Ct and Fe-Np distances in carp azide hemoglobin, associated with the T-R transition are less than 0.1/~ and 0.01/~, respectively [185]. They also observed that specific spectral regions change during the T-R transition, which is essentially the same trend as for the spin-state-sensitive features discussed above. Their data show that features A and C2 change in such a way that T and R states correspond to low-spin and highspin states, respectively, indicating that the T-R transition involves the movement of heme-iron out of the heme plane. Chance et al. have shown that variations of the Fe-Np distance in carp hemoglobin associated with the T-R transition are less than the detectable limit (0.01 A), directly from EXAFS experiments [ 186]. Further, they reported that the ratio of the two bump structures observed at 7171 eV and 7190 eV in the low-energy EXAFS region is also spinstate-sensitive. It has been demonstrated that there are particular spectral regions in the near-edge spectra for Mb(III)OH-within 40 eV from the threshold which are spin-state-sensitive. The spin-state-sensitive near-edge features are primarily due to the change of the heme-iron and ligand-orbital interaction as a result of the axial movement of heme-iron and following deformation of the porphyrin ring. It is also shown that these features are generally observed for other myoglobin derivatives and therefore can be used as spin-state markers. It is expected that the variation in the Fe-Ne distance for Mb(III)OH- should not be large on going from a low-spin to high-spin state, suggesting that the porphyrin ring should be deformed to enhance the doming.
273
4.5.4. Alloys and impurities Isoelectric impurities in semiconductors
The local structure around impurities in semiconductors is important for understanding the nature of doping-induced electronic states. For example, midgap states, known as DX centers are formed by n-type doping in ternary semiconductor alloys such as GaAsxPl-x [ 187] and AlxGal-xAs [188]. A large lattice relaxation, arising from strong electron-phonon coupling, has been considered to lower the impurity-levels deep into the bandgap [188]. Impurities also influence crystal growth. Dislocations are reduced in GaAs by impurity doping [189] or in InP by co-doping of Ga and As [190]. The effect has been ascribed to solutionhardening or a simple elastic interaction between impurity (solute) and dislocations [191]. In both cases, however, quantitative discussions require direct information on the local structure around impurities. The radial distribution of atoms in alloys or around impurities can be obtained by EXAFS [192]. Fong et al. theoretically, predicted the bimodal distribution of bond lengths or a deviation from Vegard's law in pseudobinary alloys [193]. Mikkelsen et al. found that Ga-As and In-As distances in InxGal-xAs are similar to those in pure binary compounds, i.e., GaAs and InAs, deviating from the average interatomic distance or virtual crystal approximation (VCA) [192]. In this subsection, the local structures around Ga and As impurities in liquid encapsulated Czochralski (LEC) grown InP are discussed. Impurities (1018-1019/cm3) can be studied by use of a fluorescence detection technique [22]. Ga/As-doped bulk InP crystals with a low dislocation density (102-103/cm2), prepared by a magnetic field-applied LEC method with a small temperature gradient, were studied [ 194]. The concentration of Ga and As dopants was determined by inductively-coupled plasma (ICP) emission spectrometry as 1.16 x 1019/cm 3 and 7.32 x 1019/cm 3, respectively. A latticemismatch between the Ga/As-doped InP and pure InP is less than 1 x 10-4. Dilute (In, Ga) (As, P) alloys lattice-matched to InP grown by the liquid-phase epitaxy (LPE) technique at 650~ were also studied. For dilute alloys or impurities, the background is primarily caused by elastic scattering and fluorescence lines of other components. These must be eliminated by an X-ray filter and high energy resolution detector or a focusing crystal monochromator. Thin Zn and GeO2 X-ray filters are used for Ga and As edges, respectively, while residual elastic scattering is eliminated by a pulse height analysis of a Si(Li) detector output. Normalized As K- and Ga K-edge EXAFS oscillations z(k) are plotted as a function of photoelectron wave number k in Fig. 4-50. Near-neighbor species are identified from the kdependence of the EXAFS envelope, which reflects I~(k,Tt)l of the scatterer atom. Analysis of the As K-EXAFS envelope indicates that doped As atoms are coordinated by In atoms, i.e., As atoms substitute P atoms. This directly rules out a possible clustering or precipitation in Ga/Asdoped InP. In a similar manner, it is found that Ga atoms also substitute for the In atom. In Fig. 4-51, the Fourier transform of kz(k) are indicated for the As K-edge data. The left and fight curves indicate the results for Ga/As-doped InP and pure InAs, respectively. The Ga-P
274 I
I
I
I
0.04
I
I
As K-edge
I
I
GaAs :InP
. -0.04~
o.of2" 0-
-0.02-
II
-0.04 2
I
4
V ~
~ I
6
ea(1.2xlO19/cm3): InP I
8
I
I
10 12 k (A-')
I
I
14 16 18
Fig.4-50. As K- and Ga K-EXAFS oscillations for Ga/As doped InP measured by a fluorescence detection technique: taken from ref. 194. and In-As distances are determined by a curve fit analysis using the Ga-P and As-In phase shift functions extracted from the pure GaP and InAs data and theoretical I~(k,r01 functions by Teo and Lee [33]. The Ga-P distance (2.41/~) in InP is shorter than the interatomic distance of the host lattice (2.541 ,/k) by 0.131 A and is rather close to that of pure GaP (2.360/~). This implies that.tetrahedrally coordinated P atoms are displaced toward the Ga atom by AuGa-P = 0.131 /~. If we assume the isotropic local distortion, this gives rise to a compression of the GaP4 tetrahedron along the [ 111 ] direction. In Fig.4-52, the In-As bond length is close to that in pure InAs (2.623/~), indicating the local expansion of the Asln4 tetrahedron. If the local distortion is isotropic, the displacement along the [111] direction Auln-As is 0.06/~. Information on the symmetry and extent of local distortion can be evaluated by analysis of the second-nearest neighbors or XANES. The results show that the host lattice is either expanded or compressed around isoelectronic impurities depending on the covalent radii difference. The resulting rearrangement of atomic positions is schematically shown in Fig. 4-52. The Coulomb interaction between the adjacent sp 3 orbitals will influence the second-nearest neighbors in addition to a Jahn-Tellertype distortion of tetrahedra sharing an impurity atom at an apex. Isoelectronic impurities thus induce a structural disorder, as evidenced by an increased width of the X-ray diffraction peak profile. The preservation of covalent radii predicted by Fong etal. [193], and confirmed by
275
As K-edge
As ( 7.5 x IOm/cm s) "lnP
InAs
..--,..
t--
..ci
L.I_
0
1
2
5
4
5
0
I
2
5
4
5
6
Fig.4-51. Magnitude of Fourier transform of the As K- EXAFS oscillations for Ga/As-doped InP measured by a fluorescence detection technique 9taken from ref. 194.
Fig.4-52. Local structures of Ga and As impurities in InP: taken from ref. 194.
276 EXAFS [192] causes the local distortion around impurities. The "bond length mismatch" is relaxed by: (1), the bond-stretching/shortening of the nearest neighbors and, (2), bond-bending of the second-nearest neighbors. In the dilute limit, the displacement is isotropic and the interaction is within the second-nearest neighbors. Isoelectronic impurities often induce a structural disorder as evidenced by an increased width of the X-ray diffraction peak profile. The impurity-induced structural disorder modifies both electronic states [195] and crystal growth [ 190,191 ]. Dislocations are preferentially accommodated near internally-expanded or -compressed regions, blocking the propagation in a manner similar to the dislocation pinning mechanism well known as solution hardening [189,191]. The motion of the dislocation may be prevented by the presence of structural disorder which increases the activation energy of cooperative dislocation motion. Semiconductor alloys
Let us compare the bond length relaxations in III-V alloys with the above dilute-limit case. Figure 4-53 shows the Ga-P and In-As bond lengths as a function of the relative number of bond pairs, normalized to pure binary compounds [196]. For Ga-P and In-As pairs in Ga/As-doped LEC-grown InP, the bond length variations from binary-compound values, AuGa-P and Auln-As, are ca. 1/4 (28 %) of the difference between the interatomic spacing of the host lattice and the bond length in pure binary compounds, Auln-P. This ratio coincides with those for LPE-grown (Ga, In) (As, P) quaternary alloys lattice-matched to InP. In this system, the Ga-P and In-As distances are almost constant over a wide range of concentration for which the lattice spacing is kept constant [ 196]. For InxGal-xAs ternary alloys, Mikkelsen et al. have reported that the bond length deviation in the dilute limit (2 mol %) amounts to roughly 1/4 of the difference between the values in pure binary compounds [ 192]. Recently, Shih et al. have shown that this bond length relaxation in the dilute limit can be ascribed to the 2.70 ,.--,,,
0
*~ 2.60 -Il-(.9 z
In -As
0
--InP
Go-As
,,, 2.50 .._1 z o
Go-P
,-,-, 2.40
2.30
--InAs
0-r162
--GoAs
r o ---o--
,-GoP
0-4
I
I
,
10-3
,
I
t
i
,
i
10-2 ]
t
,
I
I
10-s ]
I
I
'
|
RELATIVE NUMBEROF PAIRS
Fig.4-53. Variation of Ga-P and In-As bond lengths in InGaAsP quaternary alloys and Ga/As impurities in InP as a dilute limit: taken from ref. 197.
277 bond-stretching term of Keating's potential [197]. On the other hand, bond lengths of III-V alloys in the concentrated region deviate from the interatomic distance of virtual crystal by the same amount [ 192]. These results imply that in the concentrated region, a host lattice for the dilute-limit case can be replaced by a virtual crystal, and bond lengths in the concentrated region are also dominated by bond-stretching interactions. It is found that the bond lengths of isoelectronic impurities (Ga, As) doped in LECgrown InP relax from those in pure binary compounds (GAP, InAs) by ~28 % of the difference between the interatomic spacing of the host lattice and bond lengths in pure binary compounds. The displacement of second-nearest atoms along the [ 111 ] direction gives rise to structural disorder, such as internal expansion or compression, localized at impurities. A reduction in dislocations can be explained in terms of dislocation pinning where internal expansion (compression) accommodates dislocations and preventing their spread.
4.5.5. Superlattices and interfaces
GaAs/Si(O01) Despite growing interests in heteroepitaxy on Si substrates such as GaAs/Si, the role of the heterointerface in subsequent epitaxial growth is not understood from the microscopic viewpoint, although it is widely recognized that the initial growth conditions strongly influence the subsequent epitaxial growth [ 198-200]. In heteroepitaxy of GaAs on Si substrates, there are several intrinsic problems related to polar on nonpolar, such as an antiphase domain due to the surface step, the effect of tetragonal strain arising from a lattice mismatch, and an electrostatic field at the heterointerface. These problems have been dealt with, mostly on an empirical basis such as growth on a vicinal surface [201] or two-step growth [200], and the microscopic mechanism of epitaxial growth still remains to be solved. In particular, a large electrical field produced at a heterointerface is expected to cause atomic rearrangements of the interface [202, 203]. It is well known that the growth mode of GaAs on Si(001) depends critically on the growth temperature of the As prelayer [200, 204]. A defect-free twodimensional GaAs epitaxial growth is achieved when the As prelayer is grown at low temperature (300~ while three-dimensional growth is observed for high temperature ( > 600~ growth of the As prelayer. It is also puzzling that the As-covered Si(001) surface at high temperature gives a (1 x 2) reconstruction which is rotated by 90 ~ degrees from the (2 x 1) super structure observed for clean Si(001) and As-covered Si(001) at low temperature [205]. Since the surface reconstruction of GaAs grown on As-covered Si(001) is determined by the reconstruction of an As prelayer, the growth mode should be related to the microscopic structure of the As-Si heterointerface. The GaAs/Si(001) samples were prepared by molecular beam epitaxy (MBE) on a slightly (2 ~ misoriented Si(001) surface toward [011], with various growth temperatures for the As prelayer deposition [ 197]. Prior to the growth of the 1ML GaAs, the As prelayer was
278 grown at high temperature (> 600 ~ HT), and low temperature (300 ~ LT). In addition to As/Ga/As(HT)/Si and As/Ga/As(LT)/Si, a sample without the As prelayer, As/Ga/Si was prepared. After depositing the As layer, a thin aluminum layer was grown for protection from oxidation. Figure 4-54 shows the extracted Ga K-EXAFS oscillations z ( k ) f o r As/Ga/As(HT)/Si, As/Ga/As(LT)/Si and As/Ga/Si [206]. The k-dependence of the EXAFS profile indicates that the Ga atoms in As/Ga/As(HT)/Si are coordinated by As atoms, but in As/Ga/As(LT)/Si and As/Ga/Si are coordinated by Si atoms. The results indicate that the As prelayer grown at low temperature does not form chemical bonding with the first Ga layer. Figure 4-55 shows the Fourier transform of the EXAFS data shown in Fig. 4-54. In case of As/Ga/As(HT)/Si, the first-nearest neighbor appears as a sharp peak at -2.1 ]k, which is assigned to As atoms from the k-dependence of the EXAFS profile. In the Fourier transform of the data for As/Ga/As(LT)/Si and As/Ga/Si, the main peak shifts to larger R and reduces in intensity. The Ga-As bond length determined in As/Ga/As(HT)/Si is 2.45/~, which coincides I
I
I
I
I
I
I
Go K-edge As/Ga/AslHT)/Si(O01) 0.02 0.0~) -0.01
0.02 ,~
z
As/Go/As(LT)/Si(001)
0.10
._.. 0.05
:_o.o ~ -0.10
-0.15 -0.20 0.10~ 0"0~)I
-o.o51o.lolo. 5t-
O.20P
l
I
4
I
6
I
8
I
10
K (A-I }
I
12
I
14
I
16
18
Fig.4-54. Ga K-EXAFS oscillations for As/Ga/As(HT)/Si, As/Ga/As(LT)/Si and As/Ga/Si where HT and LT denote the high (>600~ and low (300~ substrate temperatures, respectively.
279
~
I or) l-l,.-w Z
I
I
I
I
I
I
Ga K- edge
>- . 13:: l,.-w -rrl
13C ,,::I:
As/Ga/As(HT)/Si (001) HT > 600 ~ As / Ga/As (LT)/Si (001) LT =SO0*C
13C i, ..,,.. _
2 4 6 o RADIAL DISTANCE (A)
8
Fig.4-55. Magnitude of Fourier transform of the Ga As/Ga/As(HT)/Si, As/Ga/As(LT)/Si and As/Ga/Si.
K-EXAFS oscillations
for
with that in bulk GaAs powder. This clearly indicates that the bond length fully relaxes to the bulk equilibrium value, even for the first layer, when the As prelayer is grown at high temperature. This is evidence for a three-dimensional growth mode with misfit dislocations at boundaries, because the bond-length relaxation should be observed in a two-dimensional growth under the tetragonal strain. The position and magnitude of the main peak for As/Ga/As(LT)/Si and As/Ga/Si agree well with each other, indicating that the local structure of the Ga atoms is essentially the same in two samples, i.e., the Ga atoms are bonded with Si atoms rather than As atoms. This suggests that the As prelayer grown at low temperature is not involved in the heterointerface. On Si(001) at high temperature, symmetric As dimers are formed giving rise to the (2 x 1) reconstruction [207]. Subsequent GaAs growth on As-terminated Si(001) would be dominated by the electrostatic effect and tetragonal strain. The fact that no Ga-As bonds were observed in the Fourier transform for As/Ga/As(LT)/Si and As/Ga/Si indicates that the Ga atoms are not bonded with As layers--not only in the prelayer but also in the overlayers. The determined Ga-Si bond length (2.54/~) is unusually long, which is similar to that observed for Ga adsorbed on a Si(111) surface [208]. As observed in the bond length, if the interaction between the first Ga layer and Si is weak or the charge transfer as a result of polar covalent bonding is small, the electrostatic effect would be small enough for a two-dimensional growth. Thus the absence of chemical bonding between the Ga overlayer and substrate Si qualitatively explains the subsequent growth mode, although it is not consistent with the fact that the As prelayer grown at low temperature gives axis rotation by 90 ~ However, this picture can explain why good crystal-quality is obtained when the As prelayer is grown at low temperature and the substrate should not be exposed to As4 beam at high temperature [209].
280 For low temperature As prelayer growth, the observed local structures around Ga atoms in As/Ga/As(LT)/Si and As/Ga/Si suggest that the As prelayer is on top of Ga atoms rather than the interstitials. Since the Ga overlayers on Si(001) deposited at low temperature form (2 x 1) reconstruction [210], the mechanism of axis rotation might be due to the reconstruction of the top As layer as adsorbates. This growth mechanism is consistent with the fact that the Auger electron intensity increases when the As prelayer is grown at low temperature [211] although As atoms would occupy Si sites, according to the electrostatically-driven intermixing model [203]. Secondly, the absence of Ga-As bonds in As/Ga/As(LT)/Si and As/Ga/Si suggests that large structural disorder exists in the As overlayers. The results demonstrate that the growth temperature of the As prelayer dominates the chemical bonding of the heterointerface. When the As prelayer is deposited at low temperature, or no As prelayer is grown, only the weak Ga-Si interaction, and no Ga-As bonds are observed. When the As prelayer is grown at high temperature, the Ga-As bond length is fully relaxed, even for 1 ML GaAs overlayer. These experimental results suggest that the As atoms are located on top of the Ga overlayers, if deposited at low temperature, and that this rotates the axis of reconstruction by 90 ~. Such disordered Ga overlayers may not cause the electrostatic instability, achieving a subsequent two-dimensional growth mode.
GeSi Superlattices The role of the heterointerface in strained-layer superlattices (SLSs) with a very short period attracts much attention towards understanding their unique optical properties which are not explained by a zone-folding scheme and a simple SLS structure with an ideal interface and strain confinement. For example, the strong optical transition (0.80 eV) observed for Ge4Si4 SLS [212] has been interpreted as an indirect transition, from previous band calculations [213215] based on an ordered SLS structure with a sharp interface and strain confinement in the Ge layers. However, the calculated oscillator strengths were weaker than the observed intensity, by several orders of magnitude. This can be interpreted from two different viewpoints: there might exist some enhancement factor for an indirect transition, or the transition might be direct, if a realistic model is used. Structural studies using XAFS on Si/Gen/Si(001) (n < 12) heterostructures did indeed indicate direct evidence for interface mixing [75]. Such a chemical disorder in the interface would greatly affect the transition matrix, possibly enhancing the oscillator strength due to a relaxed k-conservation rule. On the other hand, it is known that the distribution of strain affects the lowest energy in the A-direction, which essentially determines the direct/indirect nature of the transition. The energy levels for Ge4Si4 SLS grown on Si, GeSi and Ge substrates are indicated in Fig. 4-56 [215]. The minimum energy of the conduction band in the A-direction falls on going from Ge to Si substrates, as a result of compressive strain in the Ge layers. This shows that GeSi SLSs on Si substrates would lead to an indirect transition, as predicted by band calculations. A possible superlattice structure with a relaxed bulk-like Ge-Ge distance
281 Ge4Si4
substrate
e-
/Xz
W
r2,3
F Wave vector k Fig.4-56. Schematic representations of energy levels for Ge4Si4 strained supeflattice with an ideal Ge/Si interface : taken from ref. 213. predicted a direct transition [216], although its structural basis has been ruled out by the recent XAFS study [217]. The SLS consisting of the strained Ge layers and unstrained Si layers would result in an indirect transition, whereas the reversed strain-distribution is expected to result in a direct transition [215]. Recently, strong photoluminescence has been observed at low temperature for Ge4Si6 [218] and Ge5Si5 [219] SLSs grown on the GeSi alloys. These experiments renewed interest and attempts to realize a direct transition, based on the idea of strain-control by reducing bond-length mismatch, although the details of the microscopic mechanism for a strong light emission are still unclear. GenSi SLSs were prepared on a well-oriented Si(001) surface by molecular beam epitaxy (MBE) [220]. The oscillatory intensities of reflection high energy electron diffraction (RHEED), taken from the [010] azimuth during growth, were used to control the number of Ge layers: one period of oscillation corresponds to the growth of one monolayer. After the deposition of the Ge layers, 20-22 MLs of Si were grown as a cap layer. In Fig. 4-57, the Fourier-filtered first-shell contribution of the Ge K-EXAFS oscillations are shown for GenSi SLSs (Tg = 400~ n = 4, 8) together with the data for Ge0.05Si0.95, Ge0.5Si0.5 on Si(001) and pure Ge. The observed systematic change of EXAFS profile indicates that the relative ratio of the Ge-Ge pair to the Ge-Si pair increases with the increase in n. The number of Ge-Ge and Ge-Si pairs around the Ge atom was determined by a least-squares curve fit of these data. If the Ge-Si interface is ideally sharp, the average local Ge composition, x, defined as the fraction of Ge-Ge bonds around each Ge atom, would be 0.75 and 0.88 for n = 4 and n = 8, respectively. The EXAFS profile for Ge4Si indicates, however, that the Ge-Si bonds dominate for n = 4, and the observed x is only --0.5 for n = 8. This gives evidence for Ge/Si interface mixing, which would relax the k-conservation rule and consequently enhance the oscillator strength, so explaining the strong intensity for the observed transition at 0.80 eV. Moreover, such disorder would modify the electronic states, thus leading to a direct transition.
282 I
I
f~
I
I
I
I
Ge K-edge Si/Gen/Si (001)
A
I-z 22)
q~
L
~
/ n : 8,Tg :400~
rr Fro
9 Sio.5
I
4
I
6
I
8
I
I
10 12 k (~-1)
I
14
Fig.4-57. Ge K-EXAFS oscillations of GenSi(001) (Tg =400~ n =4, 8) and those for Ge0.05Si0.95/Si(001), Ge0.5Si0.5/Si(001) and pure Ge plotted as afunctionof photoelectron momentum k : taken from ref. 217. The change of oscillation profile is due to the difference in backscattering amplitude between Si and Ge. Figure 4-58 shows the Fourier transform of the Ge K-EXAFS oscillations for the Ge4Si and Ge8Si SLSs (Tg = 400~ together with data for the Ge0.05Si0.95 and Ge0.5Si0.5 alloys. The prominent peak at --1.8/~ consists of the nearest neighbor Ge and Si atoms which are not resolved as separate peaks because of the limited range in k. The results indicate that the EXAFS oscillations for Ge4Si and Ge8Si SLSs are remarkably similar to those for Ge0.05Si0.95 and Ge0.5Si0.5, respectively. The presence of the second- and third-nearest neighbor peaks for Ge4Si shows that the SLSs for n < 4 have long-range order within --6/~. The determined Ge-Ge distance in Ge4Si4, (2.42/~) is shorter than that in bulk Ge, (2.45/~), indicating that the Ge-Ge distance is relaxed as a result of tetragonal strain. This result clearly rules out a possible structure model proposed by Wong et al. [216], in which the Ge-Ge distance in bulk Ge is retained in the Ge4Si4 SLS.
283 I
I
I
I
I
Ge K - e d g e
I--.
:5oo ~ Geo.5Sio.5
/~
.,_..... Q:: i,
Geo.05 Si0.95 n=4
f,#
['~
,1 ~
0
t.
I
Si/Ge./Si(O01)
A-
rr"
I
I
2
~
I
I
~, ~ / ~
4 6 RADIAL DISTANCE (A)
~.~_~r~,~ ""
8
Fig.4-58. Magnitude of Fourier transform of the Ge K-EXAFS oscillations for Ge4Si and Ge8Si on Si(001) grown at Tg = 400~ together with those for Ge0.05Si0.95 and Ge0.5Si0.5 alloys on Si(001) 9taken from ref. 217. The local Ge/Si ratio around Ge atoms in GeSi SLSs is determined as a function of n from the curve fit analysis. For GeSi SLSs grown at 400~ the Ge-Si site exchange amounts to c a . 1 ML. The total Ge-Si interchange is almost independent of n, but is strongly related to Tg. The Ge-Ge pairs are only formed for n > 4. Figure 4-59 indicates a model structure for Ge4Si4 SLS in which an ordered double-layer Ge0.5Si0.5 interface, similar to the reported ordered GeSi alloy [221], is indicated although there is no evidence for an ordered interface so far. The interatomic spacings in this figure are taken from the Ge-Ge and Ge-Si bond lengths. In this model, --1/2 of the Ge sites are replaced with the nearest neighbor Si sites. The average local Ge/Si ratio, --0.4 for the Ge4Si4 SLS with the smallest interface mixing (Tg = RT) is still smaller than this model. This suggests that the Ge/Si interchange during the Si growth on Ge overlayers, or surface segregation, is much larger than 1/2 ML. This gives rise to an asymmetric Ge/Si interface which intervenes between the Ge and Si layers. Such a chemical disorder is also associated with a structural disorder as a result of bond length mismatch between the Ge-Ge and Ge-Si pairs. The important feature in the distorted Ge/Si interface is the
284
Ge4 Si4
"~_/
~
{'~
x., .J~
Ge---~.47(4) I(+o.I18)
Ge/Si-~.40 (8)
St/Ge-~ +0"051) Si ___~ .35 161
Fig.4-59. A schematic representation of the double layer Ge-Si interface 9taken from ref. 217. Half of the Ge atoms at the Ge/Si interface exchange their sites with the nearestneighbor Si atoms. The ordered double-layer Ge0.5Si0.5 interface is formed. fact that there exist strain components other than a uniaxial compression. This would strongly affect the selection rules of optical excitation so that weak indirect transitions can be enhanced or, on the contrary, normally inhibited direct transitions are allowed. In conclusion, the origin of the 0.80 eV transition may not involve the zone-folded states but can be ascribed to a disorder-enhanced indirect transition or quantum confinement. It should be noted that a "band picture" based on a long-range order might not be meaningful for a very short-period SLS. Figure 4-60 shows the Ge K near-edge spectra for GenSi(001) (n - 0 . 5 - 8 ) and (SiGe)4/Si(001) grown at Tg = 400~ together with those for Ge0.5Si0.5 and Ge0.05Si0.95 alloys. Because of a dipole selection rule, near-edge spectra primarily reflect the p-like state density of conduction bands. For GenSi(001) SLSs (Tg - 400~ a sharp peak is newly observed ~4 eV above the conduction band edge which is not observed for samples with Tg RT and gradually smears out with increase in n. This suggests new electronic states are formed on the strained Ge atoms which replace Si sites. Since the GeSi SLSs grown at room temperature show typical near-edge features found for GeSi alloys, such as Ge0.sSi0.5, the local Ge/Si ratio cannot explain the spectral change. This new feature could be related to an ordered heterointerface as a result of preferential site exchange, or the nature of strain on the Ge atoms.
285 I
I
I
Ge K-edge
I
I
I
I
I
~._.n~o~ Si/Gen/Si(001)
ni 7 w-.n"
.,.-... O'3 Z E3
.
~
i
~_~
Si/6en/Si 1001)
n=2
r n..." I.t-n n,., >-
Si/lSiGe)4/Si 1001) ~""x
o
Geo.05Si0.95 on.
u_
6eo.5Sio.5 on Si 1001)
11100
I
I
I
11120 11140 ENER6Y leVi
I
I
11160
Fig.4-60. Ge K-XANES spectra for Gen/Si ( n = 0.5, 2, 4, 6, 8) and (SiGe)4/Si. The growth was made on Si(001) at Tg = 400~ Each sample has a c a . 20 ML Si cap layer for protection of oxidation. For comparison, the data for Ge0.5Si0.5 and Ge0.05Si0.95 alloys are also shown" taken from ref. 217. 4.5.6.
Surfaces
m
Ge overlayers
on S i ( 0 0 1 )
A Si(001) surface shows reconstruction with a (2 x 1) ordered structure. The formation of dimer stabilizes the unreconstructed Si(001) surface [222], reducing the number of dangling bonds at the sacrifice of surface strain, which affects at least three sub-surface layers by elastic distortion. The nature of surface strain [223] is intrinsically layer- and siteselective.Such strains would open up specific pathways or channels which preferentially migrate atoms so that the rearrangement reduces the strains [224]. Secondly, the surface reconstruction with the presence of additional strain due to bond length mismatch or the atomicsize effect is an interesting problem itself. In the present study, we focus on the structure of Ge overlayers on Si(001) in relation to the number of Ge layers. Here, we show that the Ge overlayer structure is strongly connected with layer- and site-specific surface strains having various origins. As the XAFS technique can provide the atom-selective local structure, the observed bond lengths can be a good measure of strain on a probed species of atom. By tuning
286 the photon energy to the Ge K-absorption edge, XAFS can probe the bond lengths and coordination number for the Ge-Ge and Ge-Si pairs around the Ge atoms [75,225]. The strains in the Ge overlayer and the interface Si atoms are directly estimated from the observed bond lengths. It is found that the Ge overlayers show a unique surface rearrangement depending on the number of Ge layers. This indicates that the surface strains arising from different origins, i.e., surface reconstruction [222,226] and the atomic-size effect [223], greatly affect the surface rearrangement. Here, we use the word "rearrangement" in contrast to "reconstruction", meaning that an interchange of an atom is involved. The structure of Ge epitaxial overlayers on Si(001) has been studied by the surfacesensitive XAFS technique, in situ, after the growth by molecular beam epitaxy (MBE). Gen overlayers (n < 7) were prepared on a well-oriented undoped Si(001) surface at 400~ and the XAFS spectra were measured as a fluorescence yield for each sample immediately after the growth. The Ge K-EXAFS spectra for Ge overlayers on Si(001) have been obtained by detecting the fluorescence signal, using the 27-pole wiggler-magnet radiation described in I
I
I
I
I
I
I
I
I
I
Ge K- edge
9
~
v X
N*= 1.T5Ge + 2Si
"
6
N*=3.13Ge + 0 7 5 S i
8
10
1
1
k (~-~) Fig.4-61. Fourier-filtered first-shell Ge K-EXAFS oscillations for the Ge overlayers on Si(001) (solid line) together with the calculated curves for model structures (ref. 221). N* indicates the effective coordination of Ge atoms, taking the polarization factor into account. The first-shell EXAFS oscillations were generated for various model structures, and best fit was obtained for 1 ML and 2 ML Ge overlayers, assuming Ge-Si site exchange of 20% and 50%, respectively.
287 Section 4.3.3. Surface-sensitivity of the order of ~0.1 ML is obtained by combining a grazing incidence geometry with an energy analysis of the fluorescence spectrum [77]. In Fig. 4-61, the Fourier-filtered Ge K-EXAFS oscillations are shown for Gen/Si(001) (n = 1-3). Since the backscattering amplitude I~(k,~)l for Si and Ge atoms are quite different, the ratio of the Ge-Ge pair to the Ge-Si pair (NGe/NSi) is determined from the k-dependence of the EXAFS profile. The NGe/NSi value increases with increase in the number of the Ge layer, n. The number of Ge-Ge and Ge-Si pairs, as well as the bond lengths, were determined by a least-squares curvefit analysis for various structure models, taking the polarization factor into account. The total phase shift for the Ge-Ge and Ge-Si pairs were determined experimentally from the data for crystalline Ge powder and Ge0.05Si0.95 grown on Si(001) by MBE, while theoretical amplitude functions by curved wave calculations [34] were normalized to fit the experimental data for Ge and GeSi alloys. The fitting procedure was repeated to obtain model-independent bond lengths and mean square relative displacement. Figure 4-62 compares the dimer geometry for 1 ML Ge on Si(001) with Si[227] and As[228], where the inset arrows and values indicate the directions and magnitude of atom displacement as a result of surface reconstruction. The results of curve-fit analysis indicate that the 1 ML Ge on Si(001) forms an elongated dimer structure with the average adatom-adatom distance (Ra) of 2.51 + 0.01 A and the adatom-substrate distance (Rs) of 2.40 + 0.01 /~, respectively. Interestingly, the observed Ra is much longer than the reported value (2.46/~) for
Si dimeron S i ~ - ~ , Si(OOl) C~,o--~ ~
sps
Si- ~
dayaramel el.
Ge dimeron G e ~
sePxPl
Si(O0') C~o-~ ~ ~ Si- ~
Oyanagiel al.
As~c-,~__~
As dimeron(~_~ Si (001) si
.~..~'~:,
sep3
KrOcjeel r ol.
Fig.4-62. Schematic representations of dimer structures on Si(001) (ref. 224). The value for 1 ML Ge is determined experimentally in this work. The Si-Si distance is taken from experiments in ref. 227, and the As-As value is obtained by the energy minimization by Kruger etal. (ref. 228). For Si(001), the asymmetric dimer geometry is observed, whereas for As dimers on Si(001), the symmetric configuration is predicted from a total energy calculation.
288 (2 x 1) Ge(001) by surface X-ray diffraction [229] or the interatomic distance in bulk (R0, 2.45/~), despite the uniaxial strain due to lattice mismatch, which would shorten the bond length within a simple elastic-deformation model. Here, we define the relative relaxation factor, r as Ra/RO and Aa as a-1. For GeSi SLSs on Si(001), the Ge-Ge distance is 2.42/~ (Aa = -
0.012), which is almost independent of the choice of superlattice period if the epilayer is
coherently grown [217]. Although Aa usually takes negative values for epitaxial layers under uniaxial compressive strain, Aa for a 1 ML Ge overlayer is positive (0.025). This unusual expansion of atomic size is not observed for pure semiconductor surfaces. According to the total energy calculation for (2 x 1) Si(001) [228], the adatom-adatom bond length is 2.25/~ (Act = - 0.043). It is well known that the partially ionic bonding character or charge transfer between adatoms opens up a semiconductor gap in the surface band [222]. The lower Si atom of an asymmetric dimer takes an sp2-1ike geometry while the upper Si atom is in an s2p3-1ike geometry. Recent XAFS experiments using photoemission yield for (2 x 1) Si(001) reported that Ao~ increases from -0.064 to 0.04 upon surface doping by 1 ML Na adsorption [230]. This indicates that an electron transfer to a dimer increases the adatomadatom distance as a result of repulsive lone pair interaction for s2p3 bonds. From inspection of Ra values, such an intra-dimer charge transfer between adatoms through the formation of an asymmetric dimer, is unlikely for 1 ML Ge on Si(001) although asymmetric dimers have been reported for the (2 x 1) Ge(001) surface [231,232]. The local structure of 1 ML Ge on Si(001) is quite similar to that of As dimers on Si(001). Their common features are an elongated adatom-adatom bond and an s2p3-like bonding geometry. For a p3 configuration, adatom-adatom bonds tend to elongate in order to achieve orthogonal bond angles which are favorable for p-orbital overlapping. Secondly, a repulsive Coulomb interaction between lone pairs also favors the elongated dimer geometry. In the case of Ge on Si, elongated adatom-adatom bonds are favorable in terms of surface strain because of the atomic-size effect. If the adatom bond angles were distorted beyond a certain limit, to match the interatomic spacing of substrate as a result of coherent growth, the total energy would sharply increase as a result of repulsive interaction between the adjacent bond charges. It should be noted that the mismatch between adatom and substrate is much larger than a common value deduced from the lattice constants of pure elements (4%). Partly assisted by the fact that the sp mixing decreases on going from Si to Ge, as a chemical trend, this enhanced mismatch is likely to induce a charge transfer between substrate and adatom which would stabilize the structure by relieving strain. The observed RGeGe and RGeSi values for 1 ML Ge deviate significantly from the sum of covalent radii, which means that the mismatch strain effectively increases its magnitude. This gives rise to an important feature; the mismatch strain amounts to more than 10 % for 1 ML Ge, which is much larger than the conventional lattice mismatch (4 %). Such a large mismatch strain works to sharply increase the elastic strain in the second layer when another Ge layer is deposited on 1 ML Ge. Careful examination of the Ge K-EXAFS profile for 2 ML Ge did indeed indicate Ge/Si exchange [224]. The curve-fit analysis based on several model
289 structures, assuming various exchange sites, indicated that-1/2 of the second-layer Ge atoms are replaced with Si atoms in the third layer. To the author's knowledge, this is the first direct observation of Ge/Si site exchange in Ge overlayers on Si(001). Kelires and Tersoff have shown that the composition of GeSi alloys at a reconstructed Si(001) surface varies in an oscillatory way as a result of surface stress [233]. Calculated EXAFS oscillations for various model structures were fitted to the experimental data for 2 ML Ge, taking the polarization factor into account. Starting from a simple model where 1/2 of Si atoms in the third layer exchange their sites with the Ge atoms in the second layer, various exchange schemes between the second layer and the third layer were tested for the same amount of interchange, 1/2 ML. The N*Ge/N*si value is sensitive to the choice of exchanged sites and the degree of exchange, where N* expresses the effective coordination number, taking a polarization factor into account as expressed by Eqn. (4-23). One of the candidate model structures consistent with the experiment is shown in Fig. 4-63 together with the structure for 1 ML Ge. The value of N*Ge/N*si for this model (1.1) agrees well with the experimental value (1.14 + 0.2). Although the model cannot be uniquely determined and no evidence has been obtained for the presence of ordering, this model structure provide some insights on the mechanism of site exchange, as will be discussed. For a (2 x 1) (001) surface, the second-layer sites and 1/2 of the third layer sites are compressed by surface reconstruction while 1/2 of the third-layer sites has a tensile stress. For 1 ML Ge, the surface stresses due to dimer bonds and mismatch have the opposite signs in the
1ML Ge on Si(O01) Ge dimer i
N*,-, 1.5Ge + 2Si 2ML Ge
N*'-' 1.75Ge + 2Si Fig.4-63. Model structures for Ge overlayers on Si(001). : taken from ref. 224.
290 first two layers canceling one another. For 2 ML Ge, however, the compressive stress at the second layer sites sharply increases because of the surface reconstruction and atomic size effects which have the same stress components. The second layer sites are therefore unfavorable for elements with larger atomic size than Si, such as Ge. According to Tersoff [226], the total energy per added Ge atom does indeed increase at n = 2 if the surface stress is large enough. Interchange through special channels connecting these sites would stabilize the structure, relieving the interface strain. Ordered interchanges through these channels would further lower the elastic strain along a dimer row by a similar mechanism, with a missing dimer [234]. The idea of strain-induced site exchange can be applied to a surface segregation problem [235-237]. Upon the growth of 1 ML Si onto the Ge layers (n > 2), the Ge atoms in the second layer under surface stress would interchange with Si atoms in the first layer through special channels discussed above. Enhanced migration through these channels can be a driving force for the surface segregation of Ge atoms associated with a subsequent growth of Si. Surfactant atoms such as Bi and Sb inhibit the interface mixing, and are often related to the suppression of diffusion length in the growth plane [237]. However, surfactants can also inhibit the strain-induced migration channels since they can remove the stress due to surface reconstruction, and the larger atoms do not need to occupy the unstable sites under compressive strain. It is likely that surface strain is a dominant driving force of atomic migration during the epitaxial growth.
4.6. F U T U R E P R O S P E C T S
4.6.1. Third generation storage rings In this section, some future prospects of XAFS research using high-brilliance photon sources are described. The third-generation storage rings, such as ESRF, APS and SPring-8, are planned to provide high-brilliance photons over a wide range in energy. At such facilities with ultra-low (<10 nm rad) emittance, undulator radiation would be routinely used in a hard X-ray region as a well-collimated, quasi-monochromatic, coherent light source. The feasibility of the undulator as a circulaly polarized light source for magnetic XAFS experiments has been discussed [238]. Let us focus our attention on the potential of the undulator as a high-brilliance photon source, from the viewpoint a structural probe. A "tunable" X-ray undulator is a promising insertion device which is designed to cover a hard X-ray region (4-30 keV) using the fundamental and the third-higher harmonic radiation by varying a gap. The use of the undulator in XAFS researches would dramatically improve sensitivity and resolutions in time, space and energy. Because of the small source size and high degree of collimation, undulator radiation is ideal for a beamline with high spatial and energy resolutions. For studies of dilute systems using a fluorescence detection technique, the angle between the incident beam and a sample should be as small as possible because of the geometrical factor in the fluorescence
291 yield as discussed in Section 4.3. Moreover, in surface-sensitive geometries, such as a grazing incidence [73] or grazing exit [239], a well collimated beam is required. In biological studies, high-brilliance photons would increase a sample-damage [240]. High-power insertion devices have the disadvantage of inducing serious heat-load problems, which degrade not only the throughput but also the quality of beam, such as an energy resolution. A high-power density (> 100 W/mm 2) gives rise to a serious heat problem for optical elements [241,242]. A clear advantage of a high-brilliance beam for studies of dilute systems is its capability of sensitive and rapid XAFS measurements. The feasibility of a variable gap/band-width undulator and new fast- scanning techniques for applications in timeresolved studies will be described in the following subsections. For a fluorescence yield above a dilute limit, the increase in the incident-beam flux and detector solid angle 12 would further reduce a volume-fraction, time-fraction and concentration. The ultra-low-emittance storage ring is also favorable for compatibility with contradictory limitations to achieve high resolutions in space and energy.
4.6.2. Tunable X-ray undulator Figure 4-64 shows the calculated brilliance for an X-ray undulator designed for SPring8 [243]. One can shift the energy of the fundamental and harmonics radiation, as indicated by the envelope-functions (dashed line), by varying the undulator gap. The hard X-ray region (430 keV) where the K and L-edges of all elements heavier than Ca (Z = 20) are included, can be
10 20
x5
~ t0
18
i
i
i
i
i
i
i
Si(lll)
---
st
. . . .
i
i
Si12201
-
,,,-.:
0 ~
~
10
16
r ~
-
-9 ~
--1n14 _
m~
-
1012_ e--.
ca-
lO 10 3
I
I
I
i
I
i
[
I
10 4
Photon Energy [eV]
I
I
10 5
Fig.4-64. Calculated brilliance for tunable X-ray undulator for SPring-8 9taken from ref. 243.
292 covered by combining Si(111) and Si(220) crystals and switching from the fundamental and third-higher harmonic radiations. Note that the energy for exchanging the Si crystal coincides with the switching-point from the fundamental to higher-harmonic radiation. Thus, this broad energy range is practically split into two regions from the viewpoints of both the light source and monochromator. In general, tunability in XAFS experiments can be treated in two categories: spectroscopic tuning (AE < 1 keV) and edge-to-edge tuning (1 keV < AE < 26 keV). The former tuning is routinely used to scan a spectral energy range ~-1 keV, while the latter tuning is necessary when one wishes to move the monochromator in order to change the absorption edge. Ideally, the undulator gap should be varied so that the fundamental peak coincides with the energy of the monochromator. Alternatively, if a band-width of undulator radiation is wide enough to cover a spectroscopic energy range, the undulator gap can be varied only when one wishes to change the absorption edge. In Fig. 4-65, the concepts of various quick scanning techniques are shown schematically. The first approach is a combination of full independent tuning and in-phase I Ky = 2.5
ii ii=~.-----Non-tapered iI Undulator spectram
I I, ~Tapered //i ~
,/
A r,y :o.5
//
~ Edge-to-edgetuni~ ,~ /]/ keY /I . K. _ ........ jr",, I.-.~AE .,.1keV ~ --E Monochrorhator i l acceptance I | Spectroscopic ~E-,-2eV_~L_~ tumng JiUi~ IQ scon']J ----E
"-~AE"I
Polychromotor I ' - - (bent crystal)
Movobleslit-----,.-~ ~-----Somple Fluorescence / \ detector I// '\ I
,
I Q scan]TJ
,
Fig.4-65. Schematic diagram for novel quick scanning techniques using a variable gap/band width undulator 9taken from ref. 78.
293 scanning of the monochromator. The undulator gap should be varied, in the worst case, to cover about 26 keV, while a spectroscopic tuning requires only 1 keV variation. As shown in Figure 4-65, for quick scanning of the monochromator, a tapered undulator with a broader distribution of the fundamental peak can be used alternatively in a partial independent tuning mode, sacrificing the brilliance. In this case, the undulator gap is fixed during a scan. The concept of quick scanning and use of a tapered undulator is called Q-scan I. In order to optimize the undulator band width, a new undulator having variable band width capability and tunability of gap is proposed (a variable gap/band-width undulator) [78].
4.6.3. Time-resolved studies in dilute system For rapid measurements, one can use the energy-dispersive geometry in order to fully utilize the polychromatic beam in a transmission mode. In an energy-dispersive geometry [ 14,67,68], a cylindrically-bent crystal irradiated with quasi-parallel white X-ray beam reflects X-rays which are focused at the sample so that the transmission spectra are recorded as a function of the position behind the focus. At present, a typical time-resolution in the order of 100 msec is obtained for concentrated samples. The feasibility of time-resolved experiments using energy-dispersive geometry in a transmission mode, using a highly brilliant beam, is discussed elsewhere [244]. For concentrated samples, an ultimate time-resolution in the order of msec is expected. The intrinsic problem in energy-dispersive XAFS is that it cannot be applied to dilute systems, since the fluorescence-detection technique requires a point-by-point data collection. In order to solve this problem, a new technique called Q-scan II was proposed. In this approach, a narrow slit, which limits the energy spread, oscillates on a linear motor drive. The fluorescence signal from a dilute sample is collected by a high-density multi-element solid state detector, as a function of the linear position of the oscillating slit. Time-resolved XAFS spectra are obtained for dilute systems by sequential data collection, while the timeresolution can be varied by changing the frequency of slit oscillation. The intensity-variation for the incident beam, for normalizing fluorescence yield spectra, can be measured simultaneously by an ionization chamber in front of the sample.
4.6.4. Microprobe XAFS with high energy resolution Resolution in space and energy is strongly related to the angular divergence of the incident beam. The present storage tings have a large acceptance-mismatch between the incident beam and optical elements, in both the vertical and horizontal directions. A vertical mismatch for the monochromator degrades the throughput, and a high energy-resolution spectrometer sacrifices intensity. A horizontal mismatch between the divergence and acceptance functions of the focusing optics degrades the throughput and the focus size. Undulator radiation allows us to use a four crystal monochromator and microprobe optics, without serious intensity-loss. Figure 4-66 shows an example of X-ray optics designed to achieve high resolutions in space and energy. The proposed beamline design consists of two branches; a high flux medium-focus
294 branch or for high-energy/space-resolution branch. An eniptically-bent mirror is placed in front of the monochromator in order to increase vertical collimation. The main feature of the proposed beamline is a fixed exit (2 + 2) crystal monochromator. This monochromator can be operated in either the four-crystal mode or double-crystal mode for the high-flux medium focus branch and a high-energy/space-resolution branch, respectively. The upper beam is focused by an ellipsoidal mirror while the lower beam is focused by crossed elliptical mirrors. Two sets of Si(111) and Si(220) crystals are mounted on each axis and can slide to accept the fundamentaland third-higher-harmonic undulator radiations, respectively. Thus, by switching from one crystal to the other without breaking the vacuum, the full energy range (4-30 keV) can be covered. For a high-flux medium-focus branch, a typical focus size less than 6 x 3 microns is achievable. For micro-probe XAFS in a fluorescence mode, a ring-shaped multi-element solid state detector can be used to accept the fluorescence signal over a large solid angle around a point [77].
Fig.4-66. Schematic diagram of the X-ray optics for microprobe with a high energy resolution, or a high-flux XAFS experiment : taken from ref. 78.
4.6.5. Concluding remarks It is hard to predict the future prospects of XAFS research since this field is still developing rapidly. However, the evolution of XAFS associated with the advances in the light source gives us a hint. The use of the quasi-monochromatic well-collimated beam available from an ultra-low-emittance storage ring will improve experimental limits and open up new fields of XAFS research. A "tunable" X-ray undulator provides a highly collimated quasimonochromatic beam with controllable polarization characteristics. Undulator radiation can enhance the sensitivity and resolutions in space, time and energy which would change the present XAFS measurement quantitatively. A well-collimated beam from an undulator can be used for high energy resolution XAFS spectroscopy. Mapping of not only the distribution but also the chemical states would be practiced using a microprobe XAFS beamline. Because of the acceptance-mismatch between the optical elements and the angular divergence, resolutions in space and energy have been incompatible. The variable-gap/band-width undulator can provide a highly brilliant X-ray beam in a hard X-ray region, using the fundamental and third higher
295 harmonics. The variable band-width can optimize the spectral width of quasi-monochromatic undulator radiation which is essential for partial independent tuning. The author certainly wishes that these experimental developments will lead to the third generation XAFS research in which progresses in theories and analytical methods of XAFS would be accelerated as well.
ACKNOWLEDGMENTS
The preparation of this chapter was made possible by the cooperation of many colleagues and friends. The author would like to express his sincere thanks to the following collaborators: T. Matsushita, T. Iizuka, M. Okuno, H. Hashimoto, Y. Kuwahara, H. Yamaguchi, H. Kimura, K. Haga and R. Shioda. He thanks T. Sakamoto, K. Sakamoto, Y. Yokoyama, H. Ihara, K. Oka, H. Unoki, H. Kawanami, T. Terashima and Y. Bando for sample preparation. He also appreciates valuable discussions with A. Bianconi, B. Chance and J. Goulon and encouragement by T. Ishiguro, D. Sayers, E. Stem, T. Sasaki and K. Kohra.
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Applications of Synchrotron Radiation to Materials Analysis H. Saisho and Y. Gohshi (Editors) 9 1996 Elsevier Science B.V. All rights reserved.
307
CHAPTER 5
APPLICATION TO SURFACE STRUCTURE ANALYSES Toshiaki OHTA, Kiyotaka ASAKURA and Toshihiko YOKOYAMA D e p a r t m e n t of C h e m i s t r y , School of Science, The U n i v e r s i t y of T o k y o 7-3-1, H o n g o , B u n k y o - k u , Tokyo, 113, Japan
5.1. INTRODUCTION The characterization of solid surfaces is a very important subject in analytical chemistry. So far, a number of surface analytical techniques have been developed, as summarized in Table 5-1. For surface analysis, high sensitivity towards the surface is essential, as the number of surface atoms is roughly ten orders of magnitude smaller than that of bulk atoms. Electrons and ions have strong interactions with materials and as a result they have short mean free paths and are very surface sensitive probes. Thus electron (ion) spectroscopy and scattering have been adopted as the major methods for surface analysis. In contrast, photons, especially those with high energy, interact with materials very weakly. In other words, photons themselves are not surface sensitive and when we use photons as the probe of surface analysis, it is necessary to adopt techniques to enhance surface sensitivity. For example, we combine the photon probe with surface-sensitive phenomena such as photoemission and photodesorption, or apply an experimental technique such as total reflection. The advantage of the methods using photons is that the analysis is straightforward compared with those using electrons or ions, where we have to deal with collision phenomena and cannot neglect the multiple scattering effects. In fact, there are several techniques using laboratory photon sources, but the advent of synchrotron radiation as a light source has brought about the revolution in the analytical methods using photons. Synchrotron radiation has several unique features: high intensity, continuous spectral distribution from infrared to X-ray photons, high degree of polarization and pulse structure. Synchrotron radiation has made it possible to apply the conventional techniques to more difficult and sophisticated subjects and also to exploit new methods of surface structure analysis. In this
308 chapter we will review the most prominent advances in surface structure analysis using synchrotron radiation, especially addressing their potential as well as some selected examples of research. Table 5-1. Surface analysis techniques. Incoming Outgoing electron
Method
Phenomenon
Information
LEED RHEED EELS EXELFS AES IPES XMA ESD
diffraction diffraction energy loss energy loss core hole decay electronic transition core hole decay ion desorption
structure structure vibration structure elemental analysis electronic states elemental analysis structure
electron photon
RBS MEIS INS IEXR
scattering scattering electron transfer X-ray emission
chemical analysis structure electronic states elemental analysis
photon
IRAS
reflection and absorption scattering and diffraction absorption diffraction photoemission (core) photoemission (valence) photoemission photoemission and diffraction
vibration
electron
photon ion ion
photon
ion
XRD XAFS XSW XPS UPS ARUPS ARPEFS
structure structure and electronic states structure electronic states and chemical analysis electronic states electronic states structure
LEED = Low Energy Electron Diffraction; RHEED = Reflection High Energy Electron Diffraction; EELS = Electron Energy Loss Spcetroscopy; EXELFS = EXtended Energy Loss Fine Structure; AES = Auger Electron Spectroscopy; IPES = Inverse PhotoElectron Spectroscopy; XMA = X-ray MicroAnalysis; ESD = Electron Stimulated Desorption; RBS = Rutherford BackScattering; MEIS = Medium Energy Ion Scattering" INS = Ion Neutralization Spectroscopy; IEXR = Ion Excited X-ray Radiation; IRAS = Infrared Reflection Absorption Spectroscopy; XRD = X-Ray Diffraction; XSW = X-ray Standing Wave; XPS = X-ray Photoelectron Spectroscopy; UPS = Ultraviolet Photoelectron Spectroscopy; ARUPS = Angle Resolved Ultraviolet Photoelectron Spectroscopy; ARPEFS = Angle Resolved Photoelectron Extended Fine Structure
309 5.2. SURFACE EXAFS AND NEXAFS
5.2.1. Introduction The Extended X-ray Absorption Fine Structure (EXAFS) technique has been widely used to determine local structures of crystalline and non-crystalline materials [1]. The EXAFS method applied to surface systems is usually named SEXAFS (Surface EXAFS), and has also been developed extensively owing to the use of synchrotron radiation. SEXAFS is now a powerful technique for investigating the local adsorption geometry of adsorbate-substrate systems quantitatively since SEXAFS can extract local structural information only around Xray absorbing atoms. Local structures of clean surfaces are, in contrast, difficult to examine because of the same elements are in the surface and bulk atoms. In this subchapter, we will mainly focus our attention on the surface structure analysis of adsorbate-substrate systems. We will first discuss a simple application of SEXAFS, the determination of adsorption sites of atomic adsorbates, and subsequently present further examples of SEXAFS studies on surface reconstruction, molecular adsorbate systems, and the use of the temperature dependence of SEXAFS to investigate surface dynamic properties. Near-Edge X-ray Absorption Fine Structure (NEXAFS) [2] appears in the vicinity of Xray absorption edges, and is caused by the electron transition from a core orbital to discrete or quasi-discrete unoccupied levels. NEXAFS contains information not only on electronic structures but also on local atomic structures. Although NEXAFS includes a higher order of information on local structures than EXAFS, it is not always easy to analyze quantitatively for atomic adsorbate systems because of the complicated contributions of multiple scattering. The spectra of molecular adsorbate systems are much easier to understand since the intramolecular resonance dominates the whole spectrum. NEXAFS is now mainly applied in order to determine the orientation angles of molecules on surfaces and the intramolecular bond distances. These X-ray Absorption Fine Structure (XAFS) spectroscopies, SEXAFS and NEXAFS, have several advantages compared to diffraction techniques. First of all, XAFS spectroscopy requires no long-range order which is needed for LEED (Low Energy Electron Diffraction) and surface X-ray diffraction. Adsorbate-substrate systems often provide no LEED patterns except for fundamental spots from substrates, even when the adsorbates have uniform shortrange order. Secondly, electron-probed techniques such as LEED happen to induce molecular dissociation by the impact of high energy electrons, whose probability would be several orders of magnitude greater than in photon-probed techniques. Thirdly, the XAFS technique is available for most elements in the periodic table; there have been published SEXAFS and NEXAFS from carbon (Z=6), while it is difficult to investigate such light elements by means of surface X-ray diffraction. In contrast, XAFS spectroscopy has several disadvantages. SEXAFS contains information only about the vicinity of the X-ray absorbing atoms (less than ~5 A). Furthermore, information for the higher than first nearest neighbours is often
310 difficult to extract reliably due to the existence of closely-lying other neighbours and multiple-scattering paths. Even for the first nearest neighbours, if they contain more than two different kinds of X-ray absorbing atoms with different environments, reliable SEXAFS analysis is not straightforward because the EXAFS spectra give only averaged structural information which is difficult to disentangle. Thus SEXAFS would not be suitable to investigate systems with large unit cells. Another problem in SEXAFS is that it requires sophisticated, excellent beamlines which covers a large energy range with a high photon flux. On the other hand, NEXAFS is not always easy to understand theoretically, although its experiment is much easier. 5.2.2. Experimental
techniques
In order to obtain reliable SEXAFS and NEXAFS spectra from a small amount of surface adsorbate, a sophisticated beamline with a high photon flux is strongly desired. SEXAFS does not always require high energy resolution of the monochromator (a few eV is sufficient), while NEXAFS needs high resolution for detailed discussion. For soft X-ray regions below 1500 eV, a grazing-incidence grating monochromator is available, while for higher energies a double-crystal monochromator has been developed. Details of the beamline optics are described in the other chapters. A measurement chamber for SEXAFS and NEXAFS requires standard surface analysis systems such as LEED and Auger optics, and a sample-cleaning apparatus with an ion sputtering gun, a heater and others. The base pressure should be in the ultra-high vacuum range of better than lx10 .8 Pa. Because of low concentration of surface atoms, the usual transmission measurements cannot be applied to SEXAFS or NEXAFS. Indirect detection modes have therefore been examined. Fluorescence yield and electron yield measurements have been shown to provide reliable spectra equivalent to the absorption spectrum, based on the simple concept that the absorption coefficient is proportional to the number of created core holes, which should also be proportional to the emitted fluorescence or (Auger) electron yields. Figure 5-1 shows these decay processes schematically. The incident X-rays come deeply into the bulk material, and fluorescence X-rays are emitted from both the surface and bulk, while electrons come out only from the vicinity of the surface (~50 ,~ for secondary electrons and ~10 .& for Auger electrons). The electron yield mode is thus surface-sensitive, while the fluorescence yield cannot be applied if the same element as the adsorbate is present in the bulk. The other decay process depicted in Fig. 5-1 is the desorption of adsorbate atoms. The desorption probability is also proportional to the absorption cross section in some cases of ionic bondings. Although this detection method is highly surface-sensitive, it is not always applicable because of complex nature of the desorption process. We will not discuss this method further, here. We have to consider two different factors before determining which mode is a superior one. The signal-to-background (S/B) ratio is usually much higher in the fluorescence yield mode than in the electron yield mode, because in the electron yield
311
fluorescentX Adsorbed at Electro 9
bstrate
th
Fig. 5-1. Several decay processes after X-ray absorption. The incident X-rays come deeply into the bulk material, and both surface and bulk atoms are excited. The escape depth of Auger electrons is typically ~ 10 A, and the electrons come out only from a few surface layers. Low energy secondary electrons have a larger escape depth of -50 .A, implying that they are less surface-sensitive. Fluorescence X-rays are not at all surface sensitive, the escape depth being at least thousands of ,~. mode there exist significant amounts of secondary electrons which have the same kinetic energy as the Auger electrons of interest and are derived from substrates. In the fluorescence yield mode, background contribution can be neglected when the atomic numbers of the adsorbate and substrate are quite different. The signal-to-noise (S/N) ratio is also very important for one to analyze spectra. Usually more than five hundred thousand counts of signals are required. For soft X-ray regions such as the C, N, O and F K-edge (at less than 1000 eV), the probability of the fluorescent decay process is much smaller (0.5 % at the C K-edge) than that of the Auger decay. Furthermore, detection of very low energy fluorescence is not easy because of the presence of electronic noise. This means that the electron yield detection is superior to the fluorescence yield. The Auger electron yield mode, using an electron energy analyzer such as CMA (Cylindrical Mirror Analyzer) is, however, not applicable to SEXAFS. This is because the elastic photoelectron signals enter the analyzer at a certain photon energy, resulting in a huge increase in the yield spectrum. Instead of the Auger electron yield mode, a partial electron yield technique has been proposed.
312 Metal grid
sample
Retarding field
Io monitor (chaneltron)
Total or partial electron yield (chaneltron)
Fig. 5-2. Experimental system for the partial electron yield mode. A fine Cu grid is placed upstream of the sample, which gives the intensity I0 by measuring emitted electrons with a channeltron. Electrons from the sample are measured by a channeltron under the sample. The upper grid is grounded, and the lower grid is biased with a retarding voltage to prevent low energy secondary electrons from entering the detector. It is important to place the detector underneath the sample in order to avoid a change of background during rotation of the sample to measure polarization dependence. Figure 5-2 shows a schematic view of the partial electron yield measurements. In this method, a channeltron or a multichannel plate is used, and a retarding voltage is applied before the detector to eliminate low-energy secondary electrons. The grid upstream of the sample is used to monitor the intensity of the incident X-rays (I0). Although the S/B ratio is better in the Auger electron yield mode, the partial electron yield mode can avoid a photoelectron contribution and is thus suitable for a long energy scan as in the case of SEXAFS. The lowest limit of the adsorbate coverage in these energy range would be ~0.1 ML because of a low S/B ratio. Further improvements of the S/B ratio are required especially when large molecules are studied which contain only a small number of atoms of interest. In order to perform experiments for such systems, a development of a high-count-rate fluorescence detector is nevertheless desirable from the viewpoint of the S/B ratio. For bulk materials, it is already shown that the fluorescence yield mode, using a Si(Li) detector, gives more reliable spectra than the electron yield mode, and thus we hope to obtain a sophisticated high-count-rate, low-noise fluorescence detector in these energy range for SEXAFS measurements. For the higher-energy soft X-ray regions of the K absorption edges of the second row elements in the periodic table, both Auger electron yield and fluorescence yield modes are available. Figure 5-3 shows the experimental design for the Auger electron yield mode by uisng a CMA and for the fluorescence yield mode, uisng a proportional counter, together
313 CMA(electron energy analyser) "~'~,.'~h,~ " ~
.
M etal grid
h'v -
v
. . , , ~ . , , l ~
F ~
A
A
X-ray detector (Proportional counter) Io monitor
Total electron detector(channeltron)
Fig. 5-3. Experimental design of the Auger electron yield method using CMA, the fluorescence yield method using a proportional counter and the total electron yield method using a channeltron. with a channeltron which measures the total electron yield from the sample. In these energy ranges, photoelectron sweeping in the Auger window does not occur very often, and we can obtain Auger-yield SEXAFS spectra in a sufficiently wide energy range for most systems without disturbance by photoelectrons. Nevertheless, the fluorescence yield mode usually provides a much higher S/B ratio and a comparable S/N ratio. For example, in the S K-edge SEXAFS of c(2x2)S/Ni(100), the S/B ratio is more than ten, using the fluorescence yield mode with a proportional counter, while the S/B ratio is only 0.1 in the Auger electron yield mode using CMA. Although the fluorescence yield mode is not surface-sensitive, the fluorescence yield detection is usually a better solution, as long as there is little fluorescence emission from the bulk with similar photon energies. Inhard X-ray regions (higher than 4000 eV), although the probability of fluorescence decay is further enhanced, we will encounter another problem. Because of the use of crystalline substrate, some Bragg reflections come into the detector and give intense spikes in the yield spectrum. In order to avoid Bragg reflections, a solid state detector which separates the adsorbate fluorescence completely from elastic diffraction is required. To improve the S/N ratio, a multielement solid state detector has been exploited. It is essential to be able to measure the intensity of the incident X-rays (I0) with an extremely good quality. The resultant SEXAFS spectrum will be obtained by dividing the yield spectrum (/) by I0. Usually it is not easy to divide out because the transmission function (I0) has many spikeS and edges caused by absorption (or diffraction) by the mirrors, monochromators and other optics. If the S/B ratio is not good, especially in the electron yield mode, a simple I0 monitor such as that shown in Figs. 5-2 and 5-3 does not always provide
314 suitable data. There are several established techniques to allow one to divide out successfully. First, an identical detection method should be used for I0 and I measurements. For the electron yield mode, the yield spectrum of clean surfaces is first measured for I0 and that of adsorbate-deposited surfaces is subsequently taken for I. By use of this method, simultaneous measurements of I0 and I cannot be performed, and stability of the beam is therefore required. For the fluorescence yield mode, if the substrate fluorescence (I0) can be completely isolated from the adsorbate signal, simultaneous measurements are possible. If the above identical detection is not available, one has to take care to measure the same beam between I0 and I. The intensity I0 can be measured by use of a fine grid upstream of the sample, which gives an electric current proportional to I0. Normal incidence- are much easier than grazing incidence measurements. In order to avoid these difficulties in SEXAFS measurements, it is natural that one should choose optical materials before construction of the beamline. Absorption edges of the elements of the mirrors and monochromators should be avoided. For the measurements of soft X-ray SEXAFS, great care must be taken to avoid carbon and oxygen contaminations. Although we have so far discussed SEXAFS measurements, NEXAFS measurements can basically be carded out using the same experimental system and one does not have to take such great care in I/Io normalization as in SEXAFS. Improvements in NEXAFS measurements would be provided by a higher energy resolution of the monochromator. For soft X-ray regions, highly monochromatic X-rays are available, which give vibrational fine structures. 5.2.3. Determination of molecular orientation by NEXAFS NEXAFS corresponds to the electronic transition from a core orbital to unoccupied discrete and/or quasi-discrete levels. Let us consider a simple molecule such as N2. The unoccupied valence orbitals of N2 are lng* and 3Ou*, which consist of N 2p orbitals. The lng* orbital is located below the ionization threshold, while 3Ou* is strongly antibonding and lies beyond the threshold, exhibiting a quasi-discrete level. Since the l~g* (3Ou*) orbital is perpendicular (parallel) to the molecular axis, the transition probability I from the N ls core orbital is different from each other, depending on the direction of the electric field vector E of the incident X-rays [3]: I o~
I(ileE. rlf)l2
(5-1)
where i is the initial-state one-electron orbital, Nls, andfis the final-state one, lng* or 3Ou*. If E is parallel to the molecular axis, the Nls-to-3Ou* transition is allowed and the Nls-tol~;g* one is forbidden, while in the case of E perpendicular to the molecular axis, only the l~;g* transition is expected. Taking account of the linear polarization factor P, the transition probability I is given as a function of a polar angle 0 of the electric field vector of X-rays, in
315
the case of higher than threefold symmetric substrate. It depends on the type of orbitals" a vector-type orbital (3Ou* in the present case) or a plane-type orbital (l~g*), and these transition probabilities, Iv and Ip, are respectively given as [3]"
1 E1
I v = -~AP 1 + ~ ( 3 c o s 2 0-1)(3cos 2 w - l )
]1+ ~ A ( 1 - P ) s i n 2 w
(5-2)
and
Ip =
3BP1 11--~-(3cos 2 0-1)(3cos 2 a ) - l ) ]1+ B (1- P) (l + cos 2a))
(5-3)
where m is the polar angle of the bond direction for Iv or the normal vector of corresponding planes for Ip. These formulae are generally available for the transitions from ns (n=l,2..) core orbitals to p-type orbitals. Figure 5-4 shows N K-edge NEXAFS spectra of gaseous N2 and adsorbed N2 on Ni(110) [3,4]. In the gas-phase spectrum, peaks A and B are attributed to the lng* and 3Ou* transitions, respectively, and other features are also observed around 405-410 eV (Rydbergtype transitions) and -414 eV (multi-electron transitions). In the adsorbed-phase spectra, the n* and o* resonances are observed, while the Rydberg transitions disappear. The n* and o* resonances exhibit a noticeable polarization dependence; the o* resonance is completely i "~
I
A
1.0
i
I
il
Nz Gas B
o
3 0.5 >z w
0
z
400
410
420
ENERGY LOSS (eV)
1.25 1.00 0.75 rid
O.5O 0.25
z w o
4
5
3
2 ! 400
410
420
PHOTON ENERGY (eV}
Fig. 5-4. N K-shell excitation spectra of molecular nitrogen [3,4]. (a) The electron energy loss spectrum (EELS) of gaseous N 2 which gives a spectrum essentially equivalent to NEXAFS. Peaks A and B, respectively, correspond to Jt* and o* transitions. The N ls binding energy relative to the vacuum level is denoted as "XPS." (b) The N Kedge NEXAFS spectra of N2 chemisorbed on Ni(110). The pronounced polarization dependence of the ~t* and o* resonances is caused by the vertical orientation of N2 on the surface. The N ls binding energy for a lower energy peak (well-screened state) is given with respect to the Fermi level.
316 quenched in the normal incidence spectrum, while the intensity of the x* resonance is significantly reduced in the grazing incidence one, clearly demonstrating that the nitrogen molecule is standing up on the Ni(ll0) surface. After detailed analysis, it is concluded that the polar angle of the molecular axis is 0+5*. Although the accuracy of the determined orientation angle strongly depends on the quality of the spectra, it is tentatively estimated to be +8* for typical cases [3]. Polarization and coverage dependent N K-edge NEXAFS spectra of pyridine/Ag(111) were measured [5]. Figure 5-5 shows the intensity of the n* resonance at normal incidence (190) and the intensity ratio between 20* and 90* incidence (I2o/I9o), as a function of the pyridine dosage. The 12o/19o value remains constant up to ca. 3.5 L (1 L = l x l 0 -6 (Torr.s)), subsequently decreases abruptly until ca. 5.5 L, and finally increases again, approaching unity, which signifies random orientation of multilayers. The 190 curve shows the opposite trend, and both results indicate that the orientation of pyridine molecules is suddenly changed around 4.5 L like a phase transition, and that the molecule which has adsorbed relatively flat on the surface (~o=45", where a~ is the angle between the surface and molecular planes) is beginning to stand up ((0=70*). These results imply that at a lower coverage pyridine favours a flat orientation, to increase the strength of its interaction with the substrate Ag through the pyridine n electrons, while at a higher coverage pyridine is likely to stand up so that a larger
2.2
--
K
2.0-
_
x
[
~,
181.6
I
x x
~
-36
2-'-4_"
-3L -
i
01
0
-28o"
9
--~:) I 0
32,.~"
-30~-
l./, .~ o12 i
-26g
~
,-~ 0.8
condensed multilayer
OG O.L, 9 9
0.2
9
9
-20
J I
9
0
-22 -IB -16
-- I monolayer I
I
I
I
2
I
3
i
~
i
!
I
i
5 G 7 Dose IL ( u n c o r r e c t e d )
i
B
I
9
//
I
15
Fig. 5-5. The intensity of the N Is-to-Jr* resonance at normal incidence (/90) and the intensity ratio (120/190) between normal and grazing (20*) as a function of pyridine dosage to a A g ( l l l ) surface [5]. At ca. 4.5 L, a phase transition occurs from the parallel to perpendicular orientations of pyridine. Multilayer pyridine shows random orientation.
317 amount of pyridine can directly interact with the substrate through its N lone-pair electrons. It is noted that, even at a low coverage, pyridine is not lying completely flat on the surface, indicating the greater importance of the interaction of the N 2p o lone pair than the n electrons.
5.2.4. Estimation of intramolecular bond distance ,by NEXAFS The other useful application of NEXAFS is in estimating the intramolecular bond distances from the energy position of u* resonances. The o * orbital is of a highly antibonding character, and its resonance energy is strongly dependent on the corresponding bond distance. The shorter the bond distance is, the more the antibonding character is enhanced and the higher the o* resonance energy becomes. The o* resonance energies of many kinds of atom pairs relative to the ionization threshold were plotted as a function of the bond distance [6]. Figure 5-6 gives the results, indicating that the resonance energy decreases linearly as the bond distance becomes longer, and the line is identical for the same Z, where Z
2O i
15i-
t
Z
9 ol
oo .
U
CH3HCO'~
< z o oq
CF4
~.~
(CH3)
]
H2
H6
ILl n."
-I0. '
_--
. I
I
t.zo
!
!
!
I
!
1.2o
NF~ ~ I
I
!
t
!
1.3o
l
!
!
~
t !
t.4o
t
t
~
~
!
Lso
~
t
t
BONO LENGTH R(,~)
Fig. 5-6. The correlation between the o* resonance energy, relative to the ionization threshold, and the distance of the corresponding bond [6]. Z denotes the sum of the atomic numbers of atom pairs. By use of this plot, one can estimate the bond distance of structurally unknown molecular systems.
318 denotes the sum of the atomic numbers. Although the linearity should be a crude approximation, from the theoretical point of view, and the estimated error bar (+0.03 ,~ or more) is larger than from SEXAFS results, it provides valuable information since SEXAFS signals for light element scatterers are difficult to detect. Figure 5-7 shows the polarization dependent C- and O K-edge NEXAFS spectra of CO/Pt(lll) and CO/Na(0.2 ML)/Pt(lll) [7]. Both C and O K-edge spectra reveal that the o* resonance appears at a lower energy in the presence of Na than when Na is absent, implying an increase of the C-O distance owing to the presence of Na. By use of Fig. 5-6, the C-O distance is estimated to be 1.15+0.03 A in CO/Pt(lll) and 1.27+0.06 A in CO/Na/Pt(lll). These remarks indicate that the electron back donation from the substrate to the CO 2~* antibonding orbital is enhanced in Na coadsorption, resulting in the weakening and elongation of the C-O bond. Correspondingly, the ionization threshold measured by XPS is lowered and the 2Jr* intensity is reduced because of a greater amount of charge transfer from the substrate to the 2Jr* orbitals. The O-O bond distance of chemisorbed 02 molecules on several metal surfaces such as P t ( l l l ) [8,9], Ag(ll0) [9] and Cu(100) [10] has been determined by O K-edge NEXAFS. Although the O-O distance is not perturbed in the physisorbed states compared to the gasphase distance, the chemisorbed oxygen exhibits significant elongation of the O-O bond:
o,0
fl/
"(~
t
,
o o::
9
! , 280
i.lJ / ILl .J
-
I-
,
t
, I , 290
' ( h- '.l '
'
'
t
'
T t
I t 300
I . . . .
t
! t 310
t
I . . . .
,
, 320
I"''
t h, i
~I,
o 0.9
t
I
n.n
I.I
I
W'[~ I r , I 530
A ,
,tt
B
I , ~ , , 540 PHOTON
I
,t
t
550 ENERGY
,
i 560
(eV)
,t
-
Fig. 5-7. C- and O K-edge NEXAFS spectra of C O / P t ( l l l ) and CO/Na/ P t ( l l l ) [7]. Peaks A and B are respectively assigned to the n* and o* resonances. The o* resonance energy is shifted to the lower energy side because of the Na coadsorption, implying the elongation of the C-O bond.
319 1.37+0.03 A, in Pt(111), 1.47+0.03 A for Ag(110) and 1.52+0.03 A, for Cu(100), compared to the gaseous, solid and physisorbed length of 1.21 A,. These results indicate that the amount of charge transfer from the substrate to oxygen ~* orbitals is dependent on the substrate; Cu interacts with 02 more strongly than Ag and Pt. 5.2.5. Determination of adsorption geometry by SEXAFS The principles of EXAFS are given in the preceding chapter. The polarization dependent EXAFS function z(k) ~ is the wave number of emitted photoelectrons) can be written as
z(k) = Z Ni *Fi (k) exp [ -2C9 ik2 - ~ ' i l lsin 2kRi + q~i(k) - 4-~C 3 i k 3 1 ( 5 - 4 ) i
k1~'2
"'
where Fi(k) and r~i(k) are respectively the back-scattering amplitude and total phase shift of the i-th shell with the averaged distance of R i and the effective coordination number Ni*, and Ai is the mean free path of the photoelectron. C2,i and C3,i are the second (EXAFS DebyeWaller factor) and third cumulant moments, which describe respectively the width and asymmetry of the distribution function of the i-th shell. Higher cumulant moments are neglected in this formula and the intrinsic damping factor is implicitly included in El(k). The effective coordination number Ni* is given as a function of angle 6 between the electric field vector of the incident X-rays and the bond direction. In the case of K-shell (S)EXAFS, Ni* is expressed as Ni
Ni* = 3~] COS200 9
(5-5)
i=1
where ~3- is the angle for thej-th bond in the i-th shell, and Ni is the true coordination number of the i-th shell. For the sample with higher than threefold symmetry, the polar angle 09 of the bond can be given by a formula similar to Eqn.(5-2):
1 2 ~-~1)(3cos 2 0 - 1 ) ]3+ ~ N i ( 1 - P ) s i n 2 o9 N i, = NiP [1+~(3cos
(5-6)
Let us consider a simple application. The polarization dependent S K-edge SEXAFS spectra of c(2x2)S/Ni(100) were measured [11], and N*(O) and R (0), -where 0 is the polar angle of the electric field vector of the incident X-rays, for the first nearest neighbour S-Ni coordination were determined experimentally in a similar manner to the bulk EXAFS analysis. The Fourier transforms of the SEXAFS spectra are shown in Fig. 5-8. The value of R(O) is found to be independent of 0, implying the presence of the single S-Ni shell for the first nearest neighbour. The numerical results of N*(O) and its ratios are given in Table 5-2. Next, one can calculate N*(O) for typical adsorption geometries such as hollow, bridge and atop sites, as shown in Fig. 5-9, by use of the experimentally-determined S-Ni distance R of 2.19+0.03 ,&. Comparing the calculated and experimental values, one can immediately
320 conclude that the adsorption site of S atoms is fourfold hollow. It should be noted that the polarization dependent measurements are necessary for determining the adsorption geometry precisely because the error bars of the N* ratio are much smaller than those of the absolute N* values. Table 5-2. Experimental and calculated effective coordination numbers N* and their ratios for the first nearest neighbor S-Ni pair of c(2x2)S/Ni(100). 0[ ~
Experimental
10/90 10/45 10 45 90
Calculated* hollow
bridge
atop
1.09 1.04 4.22 4.06 3.88
4.08 1.58 3.96 2.51 0.97
1.94 2.91 1.50 0.00
1.16__+0.15 1.15-t-0.15 4.42+1.04 3.77+0.79 3.94__+0.75
oo
* The S-Ni distance is assumed to be 2.19 A. Although the original Ref. [11] gives R=2.23 and hence slightly different N* and its ratios, the present authors have recalculated these values because recently corrected S-Ni phase shift is more reliable and the S-Ni distance of 2.19 ,~ is in complete agreement with that given by LEED. .;
i I ~ I , ~ , ~ I I I t I I J ~ I 1_1200
:
~-/l', ~(~x~)soo N,(,oo) :1
,~oIO0
Ill'
o,~o"
"
I
I
.:-
I
l
--t--l--r-;--
i
-'1,~o
it
e-45"
i
'00
-
50 i
50
_
200
:/TI,
"
II Ill t
t50
tOO
0
o:,o"
t II~
0
.
"li
t
I
2
I I! NiS Powder Somple I/I t ( lOlOI yield )
'/l
t 50
I i
tOO
4
6
8
0
DISTANCE
0
i t
2
4
6
(~)
Fig. 5-8. Fourier transforms of polarization-dependent S K-edge SEXAFS spectra of c(2x2)S/Ni(100) and also that of nickel sulphide as a standard [11]. The dominant peaks for all the spectra are ascribed to the first nearest neighbour S-Ni shell.
321
(
(
hollow
bridge
atop
Fig. 5-9. Typical adsorption geometries such as fourfold hollow, bridge and atop sites of S on Ni(100). The adsorption site gives the different polarization dependence of an effective coordination number N*. 5.2.6. Surface reconstruction by SEXAFS Surface reconstructions have been reported for many kinds of adsorbate-substrate systems. The adsorption of S on Ni(111) provides a stable and clear LEED pattern of so-called (5~/3x2), while simple states such as p(2x2) (1/4 ML) and (~/3x~/3)R30* (1/3 ML) are unstable, even at room temperature. The polarization dependent S K-edge SEXAFS of (5~/3x2) S/Ni (111) was measured [ 12], and drastic reconstruction of the Ni (111) substrate surface was elucidated. By use of the polarization dependence of N*(O) for the first nearest neighbour S-Ni shell, the averaged bond angle co (see Eqn. (5-6)) is determined to be 57+_5*, which is much larger than the values for simple adsorption geometries such as threefold hollow (co--40.4"), bridge (34.1") and atop (0.0") sites. As a structure model which satisfies the SEXAFS results and also the LEED pattern, a pseudo-c(2x2)S/Ni(100) surface is proposed, which is depicted in Fig. 5-10.
[171
2.492 A L.
J
2.697A= ~
x 2.492
Fig. 5-10. A surface reconstruction model of so called (5~/3x2)S/Ni(lll) [12]. The surface loses the threefold symmetry of the original Ni(111) plane and is reconstructed into the pseudo-c(2x2)S/Ni(100) with a nearly fourfold symmetry.
322 The calculated co for this model is 55.8 ~, which is in excellent agreement with the experimental value of 57 ~ Although the (5~/3x2) unit mesh is quite large, all S atoms adsorb on the equivalent hollow sites of the drastically reconstructed surface, which loses the threefold axis of the original Ni(111) surface and exhibits nearly fourfold symmetry. The p4g(2x2)N/Ni(100) system shows rotational reconstruction of the Ni (100) top layer [13]. The N atom is found to adsorb on the fourfold hollow site with a bond distance between N and the surface-layer Ni of 1.88+_0.03 ,~, also interacting with the second-layer Ni with a distance of 1.85+_0.03 .~. This result requires a squeezed square of the four Ni atoms at the top layer, as shown in Fig. 5-11(b). The polarization dependence of the N* ratio also agrees with this, and gives a lateral displacement (see Fig. 5-11) of 0.68+_0.10 ,~. This result is quite interesting as a comparison with the c(2x2)O/Ni(100) which exhibits no surface reconstruction. The O atoms are located on the fourfold hollow site on the unreconstructed surface with an O-Ni distance of 1.93+_0.02 ,~ and have no direct bonding with the second layer Ni. The difference in the chemical bonding between N and O with Ni can be linked to the occupation of 2p orbitals. Other surface reconstructions have been characterized by SEXAFS for (~/7x~/7)R19*S/Cu(lll)[14], (~/17x~/17)R14~ [15], (2xl)O/Ni(ll0) [16], (2xl)O/Cu(ll0) [17], (~/2x2~/2)R45~ [18] and so forth.
9 O
9
e
oQe
9 <.im
---tl00)
(b) t o p view
Fig. 5-11. A surface reconstruction model of p4g(2x2)N/Ni(100) (lower panel) compared to the non-reconstructed c(2x2)O/Ni(100) surface (upper) [13]. The surface Ni atom is squeezed maintaining a fourfold hollow adsorption geometry for nitrogen. The N atom is also bonded with the second layer Ni atom.
323 5.2.7. Adsorption of molecules studied by NEXAFS and SEXAFS As mentioned in Section 5.2.1, applications of SEXAFS to molecular adsorption are quite important. Thiophenol (C6HsSH) adsorption on Ni(100) has been investigated by means of S K-edge SEXAFS and NEXAFS [19]. Multilayer thiophenol is deposited on the surface below 180 K, and most molecules desorb at that temperature. A chemisorbed species (0.17 ML) remains on the surface around 200 K, which was structurally studied in detail. Figure 512 shows polarization dependent S K-edge NEXAFS spectra of this stage. Peak a corresponds to the S ls-to-o*(S-C) transition, which is emphasized at grazing incidence and suppressed at normal incidence. This implies that the S-C bond is nearly vertical to the surface plane, and its polar angle to is calculated to be 19_+8~ Peak b is associated with the S-
~
e -----
r/'/-r''~'~
-~
90"
75"
60"
o o o o o
t...
o
3o"
LL
15
!
I
2/.60 2/.70 2480 2/.90 2500 2510 Photon Energy(eV)
Fig. 5-12. Polarization dependent S Kedge NEXAFS spectra of thiophenol (C6HsSH) adsorbed on Ni(100) annealed at 200 K [19]. The adsorbed species is identified as C6H5S-. Peaks a and b correspond to the transitions of the o*(S-C) resonance and the S-Ni bondings, respectively. The polarization dependence of the o*(S-C) resonance indicates the nearly vertical orientation of the S-C bond (to=19~
324
E
kO E c0 L.
k_
z
O lJ _
I
0
1
t
2 3 Distonce
90
9
5
9
!
!
4 (~,)
5
6
Fig. 5-13. Fourier transforms of the polarization dependent S K-edge SEXAFS spectra of thiophenol adsorbed on Ni(100) annealed at 200 K [19]. The dominant peak is assigned to the first nearest neighbour S-Ni shell.
2.21
2.49/k Fig. 5-14. Schematic surface structure of thiophenolate (C6H5S-) adsorbed on Ni(100) determined by S K-edge NEXAFS and SEXAFS [19].
325 Ni bonding, which also appears in the spectrum of atomic S on Ni(100). Taking account of the configuration of the S-C bond, the adsorbed species is identified as thiophenolate (C6H5S--). The Fourier transforms of the S K-edge SEXAFS are depicted in Fig. 5-13. The dominant contribution to all the incident angles is attributed to the first nearest neighbour SNi shell, and the shoulder on the shorter distance side (~ 1.4/k) which is enhanced in grazing incidence corresponds to the S-C coordination. A similar analysis to that of c(2x2)S/Ni(100) shows that S locates at the fourfold hollow site with an S-Ni distance of 2.21_+0.03 ,~, which is nearly the same, or a little longer than, that of c(2x2)S/Ni(100), and that the S-C distance is 1.84+0.05 A. A schematic view of the thiophenolate adsorption is depicted in Fig. 5-14. SEXAFS studies on molecular adsorption have also been reported for CH3S/Cu(111) [20], CS~/Ni(100) [21], O2/Cu(100) [10] and other systems. 5.2.8. Temperature dependence of SEXAFS As we have discussed in this section, SEXAFS is regarded as a tool of surface structure analysis. SEXAFS contains, however, further information on dynamic properties. The EXAFS formula given in Eqn. (5-4) includes C2 and C3, which are apparently temperature dependent. By use of a simple anharmonic pair potential for the probed shell as: 1 V(r) = ~a(r-r0 )2 -fi(r-ro )3
(5-7)
C2=<(r-r0)2> and Ca=<(r--r0)3>, where < > denotes the thermal average, can be given through a quantum mechanical calculation: lch2/1/2 c o t h ( 2 ~ ) c2=2t, ,)
(5-8)
and
c3=
,L23coth2( )
11
s9,
When one measures temperature-dependent SEXAFS, the differences of C2 and C3 are evaluated with high accuracy since other parameters such as N*, AE0, (correction of edge energy), F(k), ~(k) and A(k) are temperature independent and the only parameters to be fitted are R, C2 and C3. Once the differences in C2 and C3 are known, the cubic potential V(r) is determined through Eqns. (5-7)-(5-9), and important dynamical parameters such as (Einstein) characteristic temperature OE and thermal expansion coefficient at are immediately obtained. Systematic temperature-dependent studies have been carried out for p4g(2x2)N/Ni(100) [22,18,24], c(2x2)N/Cu(100) [23], c(2x2)O/Ni(100) [24], pseudo-c(2x2)O/Cu(100) [18], (~2x2~/2)R45*O/Cu(100) [18] and c(2x2)S/Ni(100) [25]. Table 5-3 summarizes the values of OE and at(300 K). One can find in this Table much information on the strength and
326 anharmonicity of the chemical bonds for many kinds of atom pairs, including their anisotropy. Although one might think that vertical motion would be larger and more anharmonic at the surface, the results given for p4g(2x2)N/Ni(100) and c(2x2)S/Ni(ll0) contradict the simple concept, and it can be concluded that the shorter the bond distance becomes the higher the characteristic temperature is, and the more harmonic is the potential. It is interesting to note that the potential is more harmonic in the reconstructed surfaces than in the unreconstructed ones, indicating that more stable chemical bonds between adsorbate and substrate require the reconstruction of the surfaces. Table 5-3. The equilibrium distance r0, the Einstein characteristic temperature @E and the thermal expansion coefficient at(300 K) for several adsorbate-substrate systems determined by temperature dependent SEXAFS analysis.
System
Reconst.
Bond
ro[,~]
OE[K]
oil [K -1]
Ref.
p4g(2x2)N/Ni(100)
yes no no no yes
1.85 1.88 1.85 1.93 1.86 1.84 2.05 2.19 3.16 2.19 2.28
690 557 414 414 364 667 260 328 203 386 330
0.9x10 -5 1.4x10 -5 2.2x10 -5 2.1x10 -5 2.5x10 -5 0.9x10 -5 4.2x10 -5 2.7x10 -5 3.0x10 -5 0.6x10 -5 2.6x10 -5
22,18,24
c(2x2)N/Cu(100) c(2x2)O/Ni(100) "c(2x2)"O/Cu(100) (~2x2~/2)R45*O/ /Cu(100) c(2x2)S/Ni(100)
N-Ni(2) N-Ni(1) N-Cu O-Ni O-Cu O-Cu(1) O-Cu(2) S-Ni(1) S-Ni(2) S-Ni(2) S-Ni(1)
c(2x2)S/Ni(110)
no no
23 22,18,24 22,18,24 18 18 25 25
For "bond" notation, "N-Ni(2)" denotes the bond between adsorbate N and the second layer Ni, and for "reconst.", "yes" means reconstructed surface while "no" means unreconstructed surfaces. "c(2x2)"O/Cu(100) provides only diffuse c(2x2) LEED patterns. Some values were calculated by the present authors.
327 5.3. S U R F A C E X-RAY DIFFRACTION
5.3.1. Introduction X-ray diffraction has been used for many years for the structural analysis of crystalline materials. One of the advantages of this technique is that a kinematical (single scattering) approach can be used for data analysis. In an other sense, it means the weak interaction of Xrays with materials, and the detection of surface diffraction is very difficult because the number of atoms on surfaces is extremely small. Accordingly, an intense X-ray source such as synchrotron radiation is indispensable for performing the X-ray diffraction experiments on solid st~rfaces. In fact, the number of surface X-ray diffraction experiments rapidly increased with the increasing availability of synchrotron radiation. For details one should refer to the recent review paper by Feidenhans'l [26]. 5.3.2. Principle of the surface diffraction The principle of the surface diffraction is quite similar to that of LEED. The electric vector Ee of the scattered X-ray observed at a distance R from the sample can be written as, Ee = E0
e2 p1/2~_,fj (q)eiq.R mc2R
(5-10)
J
where j~.(q) is the form factor directly related to the electron density, q the momentum transfer, e the charge of electrons, m the mass of electrons, and c the light velocity. P is the polarization factor. The value of P=I if E0 is normal to the scattering plane, and P=cos220B when E0 is in the scattering plane. If atoms are arranged in a two-dimensional lattice with vectors a l and a2, the position of each atom is given as Rj = jlal + j2a 2 + rj
(5-11)
where rj describes the atomic position measured from the origin of the unit cell. The amplitude of the scattered electric vector is given by,
e2
Ee = E0
N1-1N 2-1
mc2R p1/2F(q ) ~ E eiq'(jlal +j2a2) Jl J2
(5-12)
where N1 and N2 are extensions of two dimensional crystal lattice and F(q) is the structure factor summed over the unit cell: F(q)
= ~ , f j (q)e -Bj (q/4Z)eiq'rj J
(5-13)
328 where Bj represents the thermal vibration. Since the intensity l(q) of the scattered wave is equal to the square of Ee, l(q) is given as e4
sin2/2Nlq "al)sin2/2N2q "a2 )
I(q) =l~ m2c4R21~F(q)[2 sin2(2 q.al)sin211q.a2 )
(5-14)
l(q) has peaks when two-dimensional Laue conditions are fulfilled: q . a I = 2Jrh
and
q .a 2 = 2xk
(5-15)
where h and k are the integers. This means that the component parallel to the surface q//is the point of two-dimensional reciprocal lattice. Then the peak intensity at hk is
iPhkeak(q) = 10 m2c4R2 e4 PlFhk(q)I2N12N22
(5-16)
where
Fhk (qz ) = ~ f j (q) e2xi(hxj+kyj)+iq~zj J
(5-17)
Note Fhk assumes an intensity continuously along the z direction called a reciprocal rod like LEED pattern. Accordingly, the diffraction from the surface lattice gives the rod profile normal to the surface. It is important to separate this from the normal spot-like diffraction from the bulk. The diffraction along the Bragg rod arises from the surface lattice point and, if it is large enough compared with the thermal diffraction from the bulk, one can determine the positions of surface atoms from the profiles of the Bragg rod. Moreover, when the surface is reconstructed, fractional order Bragg rods would appear, which do not involve any strong bulk Bragg peaks. In order to determine the surface structure, we perform a surface-projected Patterson analysis, which gives the relative position of each atom on the surface. The important point for obtaining good data suitable for the analysis is to reduce the thermal diffuse scattering from the bulk to a level where the surface signal can be observed with reasonable accuracy. This can be achieved by adopting the glancing incidence geometry [27,28], as shown schematically in Fig. 5-15. When the incident angle oci is smaller than the critical angle ac, the penetration depth of the X-rays is reduced to a few tens of A and total external reflection occurs. Typical values for the critical angle are in the range of 0.2*-0.6* for X-ray wavelengths around 1.5 ,~.. Consequently, background scattering such as thermal diffuse scattering from the bulk is greatly suppressed. Fig. 5-16 shows the transmission coefficient Ti vs. the incident angle [26]. The value of Ti is defined by the ratio of the amplitude of the reflected or evanescent waves and that of the X-ray incident at the surface. The maximum transmission coefficient is at ai=ac. This means that the amplitude of the
329 evanescent wave at the surface is twice as large as that of the incident X-rays at the critical angle because of the constructive interference between the incident and the reflected X-rays Therefore, performing a surface diffraction experiment with the incident angle equal to the critical angle greatly enhances the surface signal. The situation is similar to the emitted Xrays. If the exit angle ae is less than ac, the X-rays scattered from the bulk cannot come out. Only the X-ray from the area with the penetration depth A from the surface can be detected.
q
*': ~
"
: " "" "i -. , ; , ~
9. . . .
'~
Fig. 5-15. Glancing angle geometry. The X-ray impinges on the surface with the wave vector kin at the angle ai. Diffracted X-rays exit with kout at the angle ae. 0 is the diffraction angle and q is the momentum transfer.
z~-O
~
i
~
I
I
3.0
m
_
~
2.0
m
o
1.0 0.0
,
0.0
l
,
1.0
I
2.0
o= ,
1
3.0
cti/Ctc Fig. 5-16. Transmission coefficient IT/I2 as a function of the incident angle ai for InSb(111) with the wavevector of 1.2 ,~ and the critical angle ac of 0.25 ~ [26].
330 When the incident angle ai and exit angle ae are both set at the critical angle, the surface sensitivity is doubly enhanced. The structure information normal to the surface can be obtained by measuring F(q) with various exit angles changing q along the Bragg rods.
5.3.3. Experimental System In order to analyze the surface structure it is necessary to construct an UHV environment for preparing and maintaining clean and well-defined surfaces. Moreover, it is important to carry out other surface characterizations in addition to the X-ray diffraction. Several types of apparatus for surface X-ray diffraction experiments have been developed [29,30]. A typical example [31,32] is shown in Fig.5-17. Many required tools for sample preparations and characterizations are assembled in the diffractometer. A sample manipulator is surrounded by a 360~ Be window, through which the incident X-rays can enter and the scattered Xrays exit to the detector. Four kinds of rotation are possible in this apparatus: 0 (incident angle), 20 (scattering angle), a (the incident angle by tilting the whole system), and af (detector moves out of plane to detect diffractions of different qz). Because the X-ray is polarized horizontally, the vertical scattering geometry is better. In this case the polarization factor P is unity. There are two types of scanning method. One is the m-scan and the other is rod-scan. The co scan involves the scanning of crystal around the axis normal to the scattering plane spanned by kin and kou t and the detector is fixed. The diffraction occurs when the momentum transfer q(= kin-kout) is intersecting the Ewald sphere. In order to get the three dimensional information of the surface one has to carry out detecto~
20 d r i v e ~ 0 drive ,,
"
I
J
Fig. 5-17. A sketch of an X-ray diffractometer with the UHV chamber [31]. The sample is located inside the chamber. The whole chamber is rotated around three axes (0, 20, a ) . The detector moves out of the surface plane. The chamber is evacuated by ion pump which is connected to the chamber through the diffractometer. It also contains the LEED, ion sputtering gun and metal evaporator.
331 out-of-plane measurements called rod scan to get the Fhk(q) with various oz. 5.3.4. Application to InSb(111) surface structure The III-V semiconductors, such as InSb have a zinc-blende type crystal structure. There are two possible planes; one consists entirely of Group III atoms (In) and the other of Group V atoms (Sb). The Sb plane shows a 3x3 surface structure and the In plane shows a 2x2 surface structure. An X-ray diffraction measurement was carried out for this system and the
0 Unreconstructed
9 Reconstructed a
.1
A/J w
/
f 32
3
C
Fig. 5-18. (a) Contour map of the Patterson function for (2x2)InSb(111). (b) Interatomic vectors as derived from the vectors 1 to 4 in (c). (c) Undistorted and the distorted hexagonal arrangement of the IriSh(ill) surface. The open and closed circles indicate the unreconstruced and reconstructed atoms, respectively [33].
_I
.C}
/
I
( I
J
/
,$
I
i
t. /
/I
I
/
'
I
I I
; / J
Insb(lll)
Fig 5-19. Projected atom positions of unreconstructed surface (a) and (2x2)InSb(lll) surface (b) determined by surface X-ray diffraction. The open and filled circles respectively represent Sb and In atoms [33].
332 half order reconstructed spots were recorded [33]. The Patterson function did not agree with that expected from the undistorted structure as shown in Figs. 5-18 (a) and (b). The surface was reconstructed as shown in Fig. 5-18 (c). Further refinements of the model structure were made to reproduce the observed IFhk(Obs)l2 and the reconstructed structure shown in Fig. 5-19 (b) was proposed, where the one element moves inward by 0.24 ,~ and the other outward by 0.48 A, keeping the 3m symmetry. It is impossible to discriminate between In and Sb because the scattering powers of In (Z--49) and Sb (Z-51) are almost the same. The studies of similar systems, GaSh and GaAs made possible to determine that the inward displaced atoms were In and the outward displaced atoms were Sb [34]. The driving force of the above reconstruction is orbital rehybridization. The Group III atom (In) at the surface prefers a planar sp 2 geometry with a bond angle of 60 ~ while the Group V atom (Sb) favors an s2p 3 configuration with the p-type bonding which prefers the bond angle of 90 ~. As can be seen in Fig. 5-19, the angle around In is nearly 60 ~, though that around Sb is more obtuse. The rod scans were performed and showed that the surface structure of In and Sb was almost flat within --0.2 ,~. 5.3.5. Wavelength scanning X-ray diffraction As with the I-V curves of LEED, the wavelength scanning with X-rays gives information about qz. Takahashi et al. applied this method to the structure analysis of (~3x~/3)Ag/Si(111) [35-37]. The measurement was carried out with the geometry of equal incidence and exit angles, as shown schematically in Fig. 5-20. In this arrangement, the scattering vector q is perpendicular to the surface i.e., parallel to the qz which involves the out-of-plane structure. etector
,,'\
Surface" normal
,,',
/
hv
.i' "- ./"_. .i' Fig. 5-20. Experimental geometry of the wavelength scanning mode.
\
333 Fig. 5-21 shows the intensity change of the (00) spot as a function of the wavelength of Xrays. The huge peaks correspond to the bulk Bragg diffractions. It should be noted that the regions between the strong Bragg diffraction peaks are not symmetrical. This originates from the interference between diffracted waves of the surface Ag layer and bulk Si substrate. By comparison with the calculated intensity profiles, it was concluded that the Ag atoms were located at 2.9 A above the unreconstructed Si first layer. A similar wavelength-scanning measurement was also carried out in an asymmetrical arrangement with unequal incidence and exit angles. This provides information on the in-plane structure. By combining these results, the authors proposed a possible model for Ag adsorbed structure on Si(111) surface as shown in Fig. 5-22.
l
/,
>-
l o3 3 3
3
~o Z LU I-Z
O0
_
2 I 0 1.0
1.2
1.~
1.6
1.a
WAVELENGTH
?.o 2.? (A)
2.~
?.~
Fig. 5-21. The intensity profile of the (00) spot as a function of the wavelength of X-rays. Large peaks correspond to the bulk diffractions [36]. 4g
o
0 lsl Si o 2nd Si
o
oo
t 112] ---,--
00
o 9 o 9
OoQOdoqOGoqO Oo 9 9 9 9169 cO
o
o
o--~0--.. .. < D
~.x~' "
o
o
t .
o
I
$_6~(-ho x ox,_6
Ca)
o (-ho o"-6
o .
%-Q--.o, '
.
9
o o
'
o(~." ~(~ o
x"Jo Xo
~"1o
o o o
(b)
Fig. 5-22. The adsorption site of the Ag atoms with the reconstructed Si crystal [35]. The large open circle is Ag. The middle and small circles indicate the 1st and 2nd Si atoms, respectively. Good fits were obtained for sites (a) and (b).
334 5.4. X-RAY STANDING WAVE METHOD 5.4.1. Introduction The X-ray standing wave technique gives the height of the adsorbate atoms relative to the X-ray diffracted plane. This provides information complementary to SEXAFS, which gives the local atomic structure around an absorbing atom. Until the early 1980s it was applicable only to a perfect single crystal with a very low mosaicity like Si and Ge [38-42]. However, since the X-ray standing wave was found in the soft X-ray regions, it became one of the most powerful surface structure analysis techniques. In this section we will describe principles and applications of the X-ray standing wave method, mainly focusing on the works in the soft Xray region. 5.4.2. Principle of X-ray standing wave The dynamical theory of X-ray diffractions predicts that a well-defined X-ray standing wave is generated when an X-ray plane wave is Bragg diffracted from a thick and perfect single crystal in the Bragg case geometry [38]. In this geometry, the interference between incident and scattered X-rays creates a standing wave normal to the (hkl) diffracted plane with the same periodicity, as illustrated in Fig. 5-23.
incident X-ray
diffracted X-ray ~lalding wave
9<--~176176 ~~176 ~ ~ o ~o~ o--..9~o~o o~o~ o ~ ~ o ~ o ~ o ~ o
(! p ( (
o~--o~o-"o-...9~ o ~o-'-m~o 0% o o ~ o 0"6
amplitude intensity
( l
) )
P
)
Fig. 5-23. Schematic picture of the X-ray standing wave pattern at a Bragg diffraction condition by interference between the incident and diffracted beams.
335 The amplitude of the X-ray electric field in the crystal, E(r), is expressed by: E(r) - E 0 exp(-ik 0 9r) + n n exp(-ik n 9r)
(5-18)
where E0 and EH are the incident and diffracted X-ray fields, and k0 and kH are the wave vectors of the incident and diffracted X-ray beams. The intensity I of the X-ray field at the distance ziz from the crystal plane whose spacing is dH is expressed as:
I
l=lE(r)l 2 = l+(EH/exp(
IE01
2xiAz
,n0)
dH
/I2
(5-19)
Eqn. (5-19) shows that the X-ray field varies in a sinusoidal way with the periodicity of dH. The EH/E0 is expressed by the X-ray structure factors FH and F~ for the H and H reflections and the displacement parameter W: EH
_(FHI1/2[W++_(W2_I)]
(5-20)
The intensity of the standing wave field changes with the incident photon angle and energy. For a Bragg angle OB far from 90 ~ W is linearly dependent on the deviation AO from the Bragg angle:
W = AOsin(2OB) + FF~
(5-21)
[P~(FHF~)1/2 where P is the polarization factor just as in Section 5.3 and F is the function of the X-ray wavelength A, as expressed in Eqn. (5-22): F =
4r
2 JrV
(5-22)
with V the volume on the unit cell. By advancing the incident angle through the Bragg reflection, the equal-intensity plane of this standing wave moves in a continuous fashion in the - q (=kH-k0) direction by one-half of a d spacing. On the lower-angle side, the nodal planes of the standing wave field become close to the diffraction planes and at the higherangle side the antinodal planes become close to the diffracted planes. A characteristic profile can be observed in the angular dependence of the X-ray absorption cross section of atoms because of the drastic change in the interaction between the atoms and the standing wave field. The variation in the X-ray absorption cross section can be monitored using the X-ray fluorescence or Auger electron yield [39-41]. Since Eqn. (5-19) contains the exp(-2xiAz/ds) term, the profile of the intensity depends on ziz as shown in Fig. 5-24.
336 4.0 1.0~,
cO
:~ 3.0 EL k-
0 vl r'l
o 2.0 r > 0
-~ 1.0
0
--------___.__
I
-2
I
!
t
-1 0 1 Retative photon energies (eV)
I
2
Fig. 5-24. Schematic view of the X-ray standing wave profiles with various z3z. [57]. By analyzing the profile of the standing wave using Eqns. (5-19) and (5-20), zlz, the height of an atom of interest, can be determined inside the crystal. Since the X-ray standing wave extends over the perfect crystal region, the position of adsorbates on the surface of a perfect crystal can be determined as for Br on Si(111) and on Ge(111) [43-46]. These experiments in hard X-ray regions have been carried out mostly on perfect single crystals using a wellcollimated beam with a high-precision goniometer. On the other hand, in the soft X-ray region (1-4 keV), the Bragg conditions for most single-crystal metals are fulfilled at 0B=90~ At 0//=90 ~ the Bragg reflection width (Darwin width) is enormously broadened [47], relaxing the demanding requirement of a high degree of crystalline perfection of the sample as a prerequisite to the application of the X-ray standing wave method. As shown in Eqn. (5-20), the region of total reflections is limited to the range of W between-1 and 1, which corresponds to a Darwin-angle width co. co is approximately 1 minute of arc for Cu(111) with a 45 ~ incidence. Thus, the structural imperfections such as mosaic structure in the crystal directly affect the standing wave amplitude. On the other hand, in the normal-incidence case, where the Bragg angle is nearly 90 ~ co is enormously broadened. For example the Cu(111) reflection has co of as large as 2.9 ~ at 0B=90 *. As an usual mosaic spread of the single crystal is 0.2-0.5 ~ the standing wave can be observed on the metal surface and it is applicable to the structure analysis of adsorbates on metal surfaces [48-50]. If OB is close to 90 ~ W is no longer linear in A0. Instead, W is a linear function of the energy deviation z~E from the Bragg condition: W = -2(A~/E)sin2 0B + rFo
II~(FrlF~)vz
(5-23)
337 In the soft X-ray region the measurement of the standing wave is usually carded out with the energy scan mode. Another advantage is that a high precision goniometer is not necessary. In summary, the use of the normal incidence standing wave in the soft X-ray region offers a much more convenient and powerful mehod for surface analysis. 5.4.3. Analysis of standing wave In the practical analysis we have to take into account the coherence factor Fco which reflects thermal and static fluctuations in z3z and Eqn. (5-19) should be rewritten as l=]E(r)] 2
IEol =
I
l+Fco.EH.exp. t, E0) t,
ii
2 EH " + ( 1 - F co)
dH
E0
(5-24)
In addition, the standing wave profile is modified by the mosaic spread of substrate and the energy resolution of incident photons. Thus the observed profile can be determined by the four parameters; the energy width of the incident X-ray bE, the mosaicity of the substrate crystal 60, the coherence factor, Fco, and Ziz. The first two parameters can be estimated from the reflectivity measurements at various incident energies and angles since the observed reflectivity is expressed via convolution of the Gaussian distribution functions G of incident angle 0 and E [50]:
>.. I--
0.O4 A
,,.-.,
t./3
;-
o.o8
~
0.00 ~,
Zl.2
W I,-Z ,,--.,
O W N .g
Fig. 5-25. Simulation of standing wave profiles for the Ni(200) Bragg reflection, z~z is changed from 0.00 ,~ to 0.08 ,& [52].
,,..--,
<1.0
o z 3510
i
!
1
3530 PHOTON ENERGY /eV
3550
338
R(O,E) = ~~ R(O',E') . G((O'-O)/60) . G ( ( E - E')/6E) d O'dE'
(5-25)
A similar convolution is carried out on the intensity of the standing wave expressed by Eqn. (5-24). Therefore the curve fitting analysis is carried out to obtain the best fit values of ziz and Fco. In the case of adsorption on a metal single crystal, Fco is usually 100 %. As an example, simulated results for the Ni(200) standing wave from c(2x2)C1/Ni(100) is shown in Fig. 5-25, which demonstrates the sensitivity to Zlz. 5.4.4. Experimental The normal-incidence standing wave experiment is usually carried out in the energy scan mode with the same configuration as the conventional surface EXAFS experiment. As mentioned above, the energy spread reduces the intensity of the standing wave. Thus a high energy resolution for the beamline is preferred with a well-collimated beam and a high energy resolution monochromator. A standing-wave signal is monitored either by the Auger electron yield or by the fluorescence yield emitted from the adsorbate. In general, Auger electron signal has a large background of photoelectrons, as in surface EXAFS experiments. Especially for the cases of S/Ni and C1/Ni, the Auger electrons from S or C1 appear on a huge Ni 2p photoelectron peaks and the S/B ratio is less than 0.1. It is essential to remove the background for the quantitative analysis of the standing wave profiles because the background signal has its own standing wave profile. An adequate S/t3 ratio can be obtained by the fluorescence yield method [51-53]. The fluorescence can be monitored by a gas-flow proportional counter. However, its energy resolution is not high enough (z1E/E is about 10 %) and it is necessary to remove the background arising from the elastic scattering X-rays by the independent measurement of the background X-rays if their energy appears near of the fluorescence as in the cases of C1/Ni and S/Ni. Further improvements to remove the scattering X-rays is possible by using an SSD (Solid State Detector), which provides spectra with an S/B ratio one order of magnitude higher than obtained using gas-flow proportional counters [54]. In order to determine the mosaicity of crystals and the energy resolution of the monochromator, it is necessary to measure reflectivity curves with different angles and energies. In the soft X-ray standing wave method, normal incidence and normal reflection occur. Thus the reflectivity (Rm) can be determined by measuring the electric current on a Cu mesh in front of the sample using the following equation: I m = I o (1 + RmT)
(5-26)
Here, Im and I0 are the measured electric currents of the Cu mesh, with and without the Bragg diffraction from the sample, and T is the transmission factor of the Cu mesh. Fig. 5-26
339 40 ,.-.
(a)
/~
~ o~. ......... calc.
s >
90"
/ / /
~ 20 U W _J U.. W
0
L -
~..~
......
."
(b) ~
obs.
i
80 ~
......... calc.
v I--,,_..
U ILl _J la.. I,.IJ or"
20
-6 0 6 RELATIVE PHOTON ENERGY /eV
Fig. 5-26. Reflectivity curves of the Ni (200) Bragg reflection from Ni(100). The solid and dotted lines represent the observed and calculated data, respectively [52].
shows the reflectivity of the Ni(200) diffraction, together with the calculated data. In this case the mosaic spread was 0.3 ~ and the energy resolution was 2.0 eV [52]. 5.4.5. Application of soft X-ray standing wave
(1) c(2x2)Cl/Ni(l O0) The soft X-ray standing wave profile of c(2x2)C1/Ni(100) was investigated [52,55]. Fig. 5-27 shows the C1 Kcx fluorescence-yield and total current spectra of c(2x2)C1/Ni(100). The standing wave profile from the C1 fluorescence yield provides information on the C1 position with respect to the (200) diffraction plane. The total electron yield reflects the standing wave field profile of the bulk Ni substrate. The fact that these two standing wave profiles resemble each other indicates that the C1 atom is located near the virtual lattice plane of Ni(100). Accurate analysis revealed that the vertical displacement of the C1 atom from the bulk substrate is (0.04_+0.03) .~. The layer spacing between C1 and Ni is 1.80 ,~ since the Ni layer spacing in the Ni(100) plane is 1.76 ,~. The surface EXAFS was also taken and it was found that a C1 atom is located at the fourfold-hollow site with the C1--Ni bond distance of 2.38 A, corresponding to a vertical layer spacing between C1 and Ni of 1.60 ,A. These two results indicate that outward relaxation of the first Ni layer occurs and the first and second layer spacing expands by 0.20 A as shown in Fig. 5-28. It was explained that the difference arises from the expansion between the first and the second Ni layers from 1.76 .~ to 1.96 A. Similar expansion of the first-to-second Ni layer spacing has been also reported by Sette et al. [56],
340
1.30
>-
z w
CI-Kc~ obs.
......calc.
~-I15 Z " 0 W U
,..-,
_a 1.00
Q~ 0Z
0.85 >-
1.30
z ILl I.-Z1.15
_
o LU N .a 1.00
I
TEY --
obs.
::
...... calc.
0 Z
0.85
I
-10
RELATIVE
(a)
/
0
PHOTON
I
10
ENERGY /eV
Fig. 5-27. X-ray standing wave profiles of c(2x2)C1/Ni(100) [55]. (a) C1 Kct fluorescence yield and (b) Total electron yield. The latter corresponds to the profile from the bulk Ni. The solid and dotted lines are observed and calculated ones, respectively.
(b)
0.04 A o
.2.38 A
1.76A ~ ~ ( ~ _ ' ~ . ~ . _ _
9
i~9~ ,&,
Fig. 5-28. The surface structure of c(2x2)C1/Ni(100) obtained both from the SEXAFS and soft X-ray standing wave methods [55]. (a) Plan view (from SEXAFS), (b) Side view (left side from X-ray standing wave and right side from SEXAFS).
341 from the analysis of higher coordination shells in the surface EXAFS. The standing wave measurement can be performed using the same system as for surface EXAFS measurement. The two techniques provide complementary information. Standingwave technique provides the vertical displacement of adsorbates relative to the substrate lattice. Surface EXAFS gives the bond distance and adsorption site. The relaxation of the surface layer has been determined by standing wave analysis for c(2x2)C1/Cu(100) [53] and (~/3x~/3)R30*C1/Cu(111) combined with surface EXAFS techniques [57]. (2)
(V/3x~/3)R30~
l l)
The C1 atom adsorbed on N i ( l l l ) forms a ({3x~/3)R30 ~ superstructure and the EXAFS analysis has shown that C1 is located at a threefold hollow site with a C1-Ni distance of 2.33+0.02 ~, [58]. The standing wave analysis of C1 Ka fluorescence showed 3z=1.84 ~,, corresponding well to that derived from surface EXAFS (1.83 ~,), indicating that little surface relaxation occurs on (~/3x~/3)R30~ There are two adsorption sites on Ni(111). One is thefcc site, which would be occupied by the next layer of Ni atoms in building the bulk crystal, and the other is the hcp site where the C1 atoms sit directly above the second-layer Ni atoms as shown in Fig. 5-29.
(Top
(Si
V i cw]
de
View)
dlti
dlli 1.97A
'1.29 A
Cl Ni (1) dill dill
#
t
O fC:C 0
hcp
Ni (2) Ni (3)
Fig. 5-29. The surface structure of (~/3• obtained from SEXAFS [58]. There are two possible adsorption sites; fcc and hcp for the threefold hollow site.
342
1.4
(]1i)
b
I~ I
....... f . c . c . .... h.c.p. -.
!
.'. I
I
." "
"
"
/ ~: I "
7
;
obs.
"
iI
09
o
i I I ~J
|
9. . ~
I
"
I
1.0
~ s
I
3042
i
,i,
1
[
!
I
!
3047
1
3052
Photon energy (eV)
Fig. 5-30. Comparison of the observed profile with model calculations for the (11 1) standing wave profile of C1/Ni(111). The solid curve is the observed profile, while the dashed and dotted curves are simulated curves for the hcp and)%c sites, respectively [59]. Although both sites have the same zlz with respect to the (111) diffraction, the zlz values of these two sites are different with respect to the (11 1) diffraction as shown in Fig. 5-30. The former site is located 1.97 .& above the (11 T) plane, while the latter site is 1.29 ,~. The standing wave experiment using Ni (11 ]-) diffraction was carried out [59]. Fig. 5-30. shows the standing wave profile of the (11 1) diffraction of C1/Ni(111). The calculated standing wave profiles based on the fcc and hcp models had opposite energy dependencies. Thus a definite site assignment is possible. As a result, the observed data are in good agreement with the calculated ones based on thefcc model, indicating the C1 atom is located at thefcc site.
5.5 ARPEFS
5.5.1. Introduction Photoemission is a phenomenon in which a bound electron is excited to a free electron state by a photon with an energy higher than the ionization threshold. The photoemisson spectra of valence and core levels reflect on the electronic strucutures of materials, and this method has been widely used for surface analysis. In this section, we focus on another aspect of the photoemission; its application to surface structure analysis. In core level photo-
343 emission, the initial state is localized on a particular atom, and the final state consists of one component emitted directly toward the detector and another component scattered off the nearby atoms before travelling to the detector. If the system is an adsorbate-deposited single crystal, we can expect interference between these components. This phenomenon, named photoelecton diffraction, is regarded as a kind of LEED, in which the source of electrons is a localized species. In fact, we can use the same algorithms developed in treating LEED and obtain information about the local atomic structure around the photon-absorbing atom. Photoelectron diffraction is carried out using two experimental modes. One is to observe the diffraction pattern by varying the azimuthal and polar angle of a detector with fixed photon energy. Liebsch [60] suggested the theoretical possibility of deriving the structural information from the photoelectron diffraction. Farrel et al. [61] studied the adsorption structure of (~/3x~/3)R30~ by the azimuthal photoelectron diffraction. They concluded that the iodine atom sits on the threeholdfcc sites using comparison with theoretical calculations. This technique can be conducted in a laboratory system and we shall not describe it further. In the other experimental mode, one detects the photoelectron as a function of the photon energy with a fixed detector position. In the early 1980s, normal photoelectron diffraction (NPD) was used to determine the vertical distance between the adsorbate and substrate layers by measuring the modulation in intensity of the photoelectron in the normal direction. Rosenblat et al. [62] studied the structure of c(2x2)O(ls)/Ni(100) and c(2x~2)S(2p)/Ni(100) using this method. This technique was developed to ARPEFS (Angle Resolved Photoemission Extended Fine Structure) with the improvement of a theoretical treatment. 5.5.2. Principle of ARPEFS The angle-resolved photoemission intensity varies with the photon energy, the interatomic distance between the photon-absorbing atom and its neighbouring atom, j , and three angles ctj, flj, 7 as given in Fig. 5-34 [63]. aj is the angle between the bond vector rj and the direction to the detector; ~. is the angle between the electric vector, E, and t).; and 7 is the angle between E and the detector direction. In the region of the electron kinetic energies less than a few hundred eV, one must treat the modulation using the multiple-scattering theory. However, when the kinetic energy of the electron is large enough, the single-scattering theory can reproduce the ARPEFS modulation quite well [64,65]. Although the final quantitative analysis requires spherical-wave multiple-scattering theory, a single-scattering plane wave analysis is useful to analyze the data before starting a more accurate but much more laborious multiple scattering calculation. In the single scattering plane wave theory, a modulation in the angle-resolved photoelectron intensity, z(k)(=(l(k)-lo)/lo), is given by,
344 electron energy analyser e-
e-
primary X-ray
.._
scattered P?OvtOelectr0n
____~,y" E " \\
0
0
0
0
-'---
0
0
0
0
0
primary photoelectron wave Fig. 5-31. Experimental geometry of ARPEFS. The angle resolved electron energy analyzer is along the vector labeled e-. See text for the angles, aj, flj and 7.
J
cos
r
~)
exp[rj(1-c~ A,
.l_cos.j .
~(k, aj
a2 (1- c~
where k is the photoelectron wave number as an EXAFS. The scattering factor expressed as Eqn.(5-28):
~ (~,~J) -l~ (~,~J ~I~'*~~''%
(5-27)
F(k,aj) is (5-28)
Here, A. is the mean free path of a photoelectron and o)2 is the mean square relative displacement between the photoemitter and the scattering atom (Debye Waller like factor) owing to thermal and static disorder [65]. The Fourier transform of the cosinusoidal modulation of X(k) produces peaks in the radial distribution function, whose strengths are proportional to IFj(k,a~.)l and whose positions are approximately equal to 1).(1-cosaj). Thus the information given by ARPEFS is similar to that from the surface EXAFS, but the Fourier trasnforms vary with the detector direction because of the cosa] term. Accordingly, if we analyze the ARPEFS spectra at several detector positions, a complete surface structure should be obtained.
345
5.5.3. Applications of ARPEFS (1) (~/3x~/3)R30"CI/Ni(l l l) C1 Is ARPEFS spectra of (~/3x~/3)R30~ in two geometries [111] and [110] were recorded, which are shown in Fig. 5-32 [67]. In the [111] geometry, photoelectrons were collected along the surface normal with the electric vector 35 ~ from the surface normal to the [112] direction. In the [110] geometry, the detector direction and the electric vector were collinear along the [110] direction. The thin curve with dots is x(k) measured at 300 K and the thicker curve is at 120 K. Enhancement of the oscillation is caused by the reduction in thermal fluctuation at low temperature. In the Fourier transform of the data from the [110] geometry, the strong peak at 4.6 .~ arose from the first-nearest Ni atom, as shown in Fig. 5.33. This peak only appears when C1 is located at the threefoldfcc site of N i ( l l l ) faces, as shown in Fig. 5-29, which is in agreement with the standing wave experiment [59]. The CINi-Ni bonding lies nearly collinearly along the [110] direction. In this case the double -scattering amplitude has a similar amplitude to that of the single scattering and these two waves constructively interfere. The consequent wave is nearly as large as that expected from a simple single scattering theory and produces a strong peak in the Fourier transform. Such an enhancement is called a focusing effect. The peak appearing at 7.6 A and 9.1 ,i, in the [110] direction could be attributed mainly to the scattering from the atoms in the second Ni(ll0) plane and the peak at 2.5 ,i, is attributed to the other nearest neighbour atoms. The Fourier peaks in the [111] geometry appear at different positions from those in the [110] geometry. This is because the peak position depends not only on the interatomic distance but also on the scattering angle aj. In the [111] geometry, the first peak arises from the three nearest-neighbour Ni atoms and the second peak corresponds to the scattering from three third-nearest neighbour atoms in the second Ni layer. Thus, by varying the detector direction, the bondings with similar bond distances can be separated. Further quantitative analysis, to obtain the accurate bond length, requires the multiple-scattering spherical-wave calculation (MSSW) [60]. As a result, the separation between C1 and the first Ni layer and between C1 and the second Ni layer are 1.837 and 3.763 ~k, respectively and the C1-Ni distance is 2.332 ,~. Consequently, the separation between the first and second Ni layers is 1.926 ,~ (5 % contraction). The C1-Ni distance is in good agreement with the surface EXAFS data. However, the above result for the relaxation between the first and second Ni layers contradicts with the standing wave observation, which indicates no remarkable contraction, as described in the previous section. m
(2) c (2x2) S/Ni (l O0) Fig. 5-34 shows the Fourier transform of x(k) of c(2• [68]. The detector was located along normal to the surface and the polar angle of the electric vector of the light, E, was 30*. The main peak at 6.2 A comes from the back-scattering from the second-layer Ni atoms and the peak at 10 .A arises from the third-layer atoms [69]. In this geometry,
346 ,-
.
.
~
0.,5
IJ..I
0.0
-o.5
0~5 I,j v
;,< o.0
•-t1.5
FiB. 5-32. ARPEF~q modulatior~s of (~/3x'~13)R30°Cl/Ni(l 11 ) in two geometries 1111] and [[10] (Ref. [67]). Thick solid t[ne was obtained at 120 K, ~nd tMn solid line with dots at 300 K.
z(k)
I
O0
i
r
r
i
ZOO
~00
400
500
Kinetic
Energy
E(eV)
qJ "O
,Cr,,
[11t)
L_ O 1.4_
n~
2
C
O~ O
Fig. 5-33. Fourier ~pectra of AI
O
o
5
path
length
10
di"fv.runce
(~)
347 the first peak should appear at 3.5 ,~, corresponding to the back-scattering from the firstneighbour. However, positions of two peaks appearing below 5 ,~ does not correspond to those of the expected structure. Barton et al. [69] showed that this abnormal behaviour in the Fourier transform is caused by the generalized Ramsauer-Townsend effect. At a certain scattering angle, aj and kinetic energy, the scattering amplitude is weak and the phase changes rapidly, almost by ~ tad. These abrupt changes in the amplitude function modify the shape of the Fourier transforms. The S-Ni nearest neighbour distance could be derived from the %(k) data, in the presence of the generalized Ramsauer-Townsend effect, using curvedwave multiple-scattering calculations. The S-Ni distance was found to be 2.20 ,~ and the SNi interplanar distance 1.32 ,~.
c(2X2)S/Ni(00 i)-[ 001 ] |
-
Auto-reg.
~
l I
f
.,-4
k~
O
"
0 path
Y
length
L
5
l
difference
10
(~)
Fig. 5-34. Fourier transforms versus scattering path length differences of ARPEFS of c(2x2)S/Ni(100): (a) The conventional Fourier transform and (b) The auto-regressive linear prediction method [68]. (3) c (2 x2) S /N i (l l O) S.W.Robey et al. investigated S/Ni(ll0) by means of ARPEFS [70]. They carried out experiments with different geometries and determined the S-Ni(first layer) distance of 2.31 and the S-Ni(second layer) one of 2.18 ,~, which corresponds well to the SEXAFS results [71,72].
348 5.6. CONCLUSION Synchrotron radiaton offers new and powerful techniques to surface science, with its high intensity, linear polarization and energy tunability. One can get considerable information about surface structures, such as the orientation of a molecule, adsorption sites, surface reconstruction, and surface relaxation. The information on the surface strucuture had been less easily and less accurately obtained before the advent of synchrotron radiation. Although experiments are limited by the sharing of beam time at present, the experiments using synchrotron radiation have increased year by year. We now have the third-generation synchrotron source which will enable us to carry out new surface science experiments, such as time- and space-resolved experiments and spin-polarized experiments. In the next ten years, our knowledge on the surface of materials will be renewed by these new techniques. REFERENCES
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350 35. T. Takahashi, S. Nakatani, T. Ishikawa and S. Kikuta, Sulf ScL, 191, L827 (1987). 36. T. Takahashi, S. Nakatani, N. Okamoto, T. Ishikawa and S. Kikuta, Jpn. J. Appl. Plus., 27, L753 (1988). 37. T. Takahashi, S. Nakatani, N. Okamoto, T. Ishikawa and S. Kikuta, Rev. Sci. Instrum., 60, 2365 (1989). 38. B.W. Batterman and H. Cole, Rev. Mod. Phys., 36, 681 (1964). 39. B. W. Batterman, Pl~s. Rev., 133, A759 (1964). 40. R. Patel and J. A. Golovehenko, PlUs. Rev. Lett., 50, 1858 (1983). 41. T. Takahashi and S. Kikuta, J. Phys. Soc. Jpn., 47, 620 (1979). 42. P.L. Cowan, J. A. Golovchenko and M. F. Robbins, Plus. Rev. Lett., 44, 1680 (1980). 43. J. A. Golovehenko, J. R. Patel, D. R. Kaplan, P. L. Cowan and M. J. Bedzyk, P/~s. Rev. Lett., 49, 560 (1982). 44. G. Materlik and J. Zegenhagen, Plus. Lett., 104A, 47 (1984). 45. G. Materlik, A. Frahm and M. H. Bedzyk, Pl~s. Rev. Lett., 52, 441 (1984). 46. M. Bedzyk and G. Materlik, Su~S Sci., 152/153, 10 (1985). 47. K. Kohra and T. Matsushita, Z. Natulforsch., 27a, 484 (1972). 48. T. Ohta, Y. Kitajima, H. Kuroda, T. Takahashi and S. Kikuta, Nuc[. bzstrum. Methods, A246, 760 (1986). 49 D.P. Woodruff, D. L. Seymour, C. F. McConville, C. E. Riley, M. D. Crapper, N. P. Prince and R. G. Jones, Pto,s. Rev. Lett., 58, 1460 (1987). 50. H. Hashizume and T. Nakahata, Jpn. J. Appl. Phys., 27, L1568 (1988). 51. T. Yokoyama, M. Funabashi, Y. Kitajima, T. Ohta and H. Kuroda, Physica B, 158,237 (1988). 52. T. Yokoyama, Y. Takata, M. Yoshiki, T. Ohta, M. Funabashi, Y. Kitajima and H.Kuroda, Jpn. J. Appl. Phys., 28, L1637 (1989). 53. J. R. Patel, D. W. Berreman, F. Sette, P. H. Citrin, J. E. Rowe, P. L. Cowan, T. Jach and B. Karlin, Phys. Rev., B40, 1330 (1989). 54. Y. Takata, T. Yokoyama, Y. Kitajima, M. Funabashi, H. Kuroda and T. Ohta, X-ray Absorption Fine Structure ed. by S.Hasnain, Ellis Horwood, p.254, 1991. 55. T. Yokoyama, Y. Takata, T. Ohta, M. Funabashi, Y. Kitajima and H. Kuroda, Phys. Rev. B42, 7000 (1990). 56. F. Sette, T. Hashizume, F. Comin, A. A. MacDowell and P. H. Citrin, Plo~s. Rev. Lett., 61, 1384 (1988) 57. D. P. Woodruff, D. L. Seymour, C. F. McConville, C. E. Riley, M. D. Crapper and N. P. Prince, Surf Sci., 195, 237 (1988). 58. M. Funabashi, T. Yokoyama, Y. Takata, T. Ohta, Y. Kitajima and H. Kuroda, Surf. Sci., 242, 59 (1991). 59. Y. Takata, H. Sato, S. Yagi, T. Yokoyama, T. Ohta and Y. Kitajima, Surf Sci., 265, 111 (1992). 60. A. Liebsch, Phys. Rev. Lett., 32, 1203 (1974).
351 61. H. H. Farrell, M. M. Traum, N. V. Smith, W. A. Royer, D. P. Woodruff and P. Johnson, SulJ~ Sci., 102, 527(1981). 62. D. H. Rosenblatt, J. G. Tobin, M. G. Mason, R. F. Davis, S. D. Kevan, D. A. Shirley, C. H. Li and S. Y. Tong, Plg's. Rev., B23, 3828 (1981). 63. J.J. Barton, C. C. Bahr, Z. Hussain, S. W. Robey, J. G. Tobin, L. E. Klebanoff and D. A. Shirley, Phys. Rev. Lett., 51,272 (1983). 64. P.J. Orders and C. S. Fadley, Phys. Rev., B27, 781 (1983). 65. J.J. Barton, S. W. Robey and D. A. Shirley, Plo,s. Rev., B34, 778 (1986). 66. J.J. Barton and D. A. Shirley, Plo2s. Rev., B32, 1892 (1985). 67. Li-Qiong Wang, Z. Hussain, Z. Q. Huang, A. E. Schach von Wittenau, D. W. Lindle and D. A. Shirley, Pl~s. Rev., B44, 13711 (1991). 68 J.J. Barton, Z. Hussain and D. A. Shirley, Phys. Rev., B35, 933 (1987). 69 J.J. Barton, C. C. Bahr, S. W. Robey, Z. Hussain, E. Umbach and D. A. Shirley, Plo,s. Rev., B34, 3807 (1986). 70. S. W. Robey, J. J. Barton, C. C. Bahr, G. Liu and D. A. Shirley, Plo,s. Rev.. B35, l 108 (1987). 71. T. Ohta, Y. Kitajima, P. M. Stefan, M. L. Stefan, N. Kosugi and H. Kuroda, J. Phys. (Paris) Colloq., 47, C8-503 (1986). 72. D. R. Warburton, G. Thornton, D. Norman, C. H. Richardson, R. McGrath and F. Sette, SluS Sci., 189/190,495 (1987).
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Applications of Synchrotron Radiation to Materials Analysis H. Saisho and Y. Gohshi (Editors) 9 1996 Elsevier Science B.V. All fights reserved.
353
CHAPTER 6
STRUCTURE ANALYSIS BY SMALL-ANGLE X-RAY SCATTERING Kanji KAJIWARA Faculty of Engineering & Design, Kyoto Insitute of Technology, Matsugasaki, Sakyo-ku, Kyoto 606, Japan
Yuzuru HIRAGI Institute for Chemical Research, Kyoto University, Uji, Kyoto-fu 611, Japan 6.1. G E N E R A L P R I N C I P L E S OF S M A L L - A N G L E S C A T T E R I N G
6.1.1. Characteristics of small-angle scattering Small-angle X-ray scattering is characterized by its small scattering angle. A scattering process is described in terms of a reciprocal law. Imagine that a particle of diameter D is immersed in a beam of X-rays. The X-rays are scattered by all the electrons of the particle and interfere with each other, giving a scattered intensity which decreases with increasing scattering angle. The scattered intensity has its maximum in the direction of the incident beam (i.e., at zero scattering angle), where the scattered rays are all in phase. This intensity at the zero scattering angle is proportional to the number of electrons in the particle. The phase difference between scattered rays of wavelength A becomes more prominent as the scattering angle increases, and eventually diminishes at a scattering angle of the order of A/D. For example, the limiting scattering angle to be observed is equal to 0.45~ when D = 100 A, or to 0.045* when D = 1000 .~, for the X-ray wavelength of 1.54 A from the CuK~-line. Thus the scattering pattern provides a method for evaluating the particle size. However, the particle size should lie in a certain range for the scattered X-ray to be observed with reasonable accuracy. The scattered X-rays are coherent, and the scattered intensity is given by the
354 square of the amplitude resulting from the summation of all the amplitudes of scattered X-rays. The amplitudes are of the same magnitude but differ in their phase, depending on the position of electrons in space, so that the total amplitude A(q) of the scattered rays at a point q is given by [1] A(q) =
p(r). e-iq'~dr
(6-1)
where p(r) is the electron density of the particle at a point r, and the scattering vector q is defined as q = (27r/A)(s - So) (6-2) Here s and so denote the unit vectors in the direction of the scattered and incident X-rays, respectively. As illustrated in Fig. 6-1, the scattering vector q has the same direction as ( s - so), and the magnitude is given by q = (47r/A) sin 8
(6-3)
Scattered
s ,, \S_s~q
v
Incident
rays
rays v
Figure 6-1" Diffraction by a single particle Eqn. (6-1) shows that the amplitude A(q) of scattering in the direction of q is the Fourier transform of the electron density distribution p(r). The scattered intensity is then given by the product of the amplitude A and its complex conjugate A* as
]'(q) =
fo f0
/9(1"1)/9(1"2)"e-iq'(r'-r2)drldr2
(6-4)
355 which involves the separation (rl - - r 2 ) for every pair of scattering points. Constraining (rl - r2) = r, eqn.(6-4) is modified as X(q) =
e-iq"dr
(6-5)
where ~2(r) is regarded as the density of electron pairs with separation r. The value t52(r) is determined by the structure of the particle, and is given by the inverse Fourier transform of the scattered intensity as /52(r) = (1/27r) a
I(q). eiq'rdr
(6-6)
, Assuming the sytem is statistically isotropic, the phase factor e - i q ' r is replaced by its space average according to Debye [2]: sin qr qr
(6-7)
4rr2/52(r) sin qr dr
(6-8)
< e -iq'r >--
and eqn.(6-5) reduces to I(q) =
fO ~
The distance distribution function
qr
p(r), defined from
I(q) = 47r
p(r) sin qr dr qr
fO ~
(6-9a)
or its Fourier transform: p(r) = ~ 1 fo~ qrI(q) . sin(qr)dq,
(6-9b)
is related to the correlation function -/(r) of Debye and Bueche [3] as p(r) = Vr 2. 7(r)
(6-10)
where the correlation function 7(r) represents the correlation of the electron density fluctuation at a distance r, and is introduced by subtracting the constant term at a long distance (which should be equal to Vp 2 with ~ being a mean value) from the density of electron pairs (/~2(r)). In the special case of r = 0, the mean square fluctuation of electron density ,~(0) is obtained from eqns. (6-9) and (6-10) as VT(O) =
q2I(q)dq if0~176
(6-11)
356 which is invariant with respect to the particle structure. Conventionally the invariant Q is defined [4] as
O =
q~I(q)dq
(6-12)
6.1.2. Particle scattering In this section we consider briefly the scattering from a single particle in solvent. The particle and solvent are assumed to be homogeneous and have constant electron densities p and po, respectively. The electron density difference Ap = p - p o causes the excess scattering from the particle, which contains the information on the shape and size of the particle. The excess scattering amplitude at the zero scattering angle should be equal to the number of excess electrons An, [5,6], i.e., A(0) = Anr = Ap. V
(6-13)
irrespective of the size and shape of the particle. Eqn.(6-1) implies that the normalized scattering amplitude Ao(q) is given generally for a homogeneous particle with or without a hollow as [7] ao(q) = f D P ( r ) ' e - i q ' ~ d r / f D P ( r ) d r fD(2) p ( r ) " e - i q ' r d r - f D ( 1 ) p ( r ) , e - i q ' r d r
=
fD(2) p ( r ) d r - fD(a)p(r)dr
(6-14)
where D denotes the domain specified by the number 2 (the whole body) or 1 (a hollow domain). Assuming the homogeneous electron density, p ( r ) in eqn.(6-14) is replaced with Ap, as argued above, and the denominator is given by eqn.(6-13). Thus eqn.(6-14) reduces to: Ao(q) = (F~ - F2)/(V~ - V2)
(6-15)
with Fk = fo(k) e-iq'rdr and Vk = lock) dr (k = 1 or 2). Apparently, V~ corresponds to the volume of the domain specified by i. When the term e - i q ' r is replaced by r 2, eqn.(6-15) yields the radius of gyration Ra as shown in the next section. The scattering amplitude from a triaxial body of orientation specified by 0 (zenith angle) and r (azimuth angle) is a function of the semi-axes A, B and C as designated as Fi(A,B, C, ri, O, r where rl = r sin 0 (6-16) r2 -- r cos
357 and is calculated as: (a) Isosceles triangular prism (6-17)
V~ = 4 A B C 8 A B C e x p ( - 4 i A r l cos r
F/=
(B 2 sin 2 r - 4A 2 cos 2 r
• [exp(2iAri cos r
cot r sin(Br, sin r
- cos(Brl sin
r
+ 1]~c (6-18)
(b) Rectangular prism
(6-19)
Vi = 8 A B C
Fi
8ABC
c~ r [t sin(Arl Arl cosr
[ sin(Bra sin r ]~I/c Brlsinr
(6-20)
(c) Regular polygonal prism of n sides (6-21)
Vi = 2nA2C tan(r/n)
---~--k)} ~c,
(6-22)
-n-k) sin(rlA sin(r - 2~r k ) ) - tan - cos(rlAtan - sin r
+ tan ~ (6-23)
Fi = 4A2C
k=l
fk(rl, r
27r k ) e x p { i A r l c o s ( r n
where fk =
x
1 AZr~ {tan 2 ~ sinZ(Tr - ~-~r) - cos 2 r
i cot(r
n
Tt
Tt
(d) Elliptical cylinder Vi = 2 A B C
(6-24)
Fi = 4 7 r J l ( r 1 K ) ~ e
(6-25)
rlK
where K = x/A 2 cos 2 r + B 2 sin 2 r and J1 denotes the first order Bessel function. (e) Ellipsoid Vi = (4/3)TrABC
Fi = 4~r sin(rL) - rL cos(rL) (rL) 3
(6-26) (6-27)
358 where L = x/A 2 sin2'0 cos2 r + B 2 sin 2 0 sin 2 r + C 2 cos2 0. As the normalized scattering amplitude is defined by eqn.(6-15), the particle scattering factor of an oriented particle P(q, O, r is given by the product of A0(q) and its complex conjugate A~(q) as P(q, O, r = A0(q)A~(q). If a particle in solution prefers no specific orientation, the observable particle scattering factor is calculated by integration of P(q, O, r with respect to 0 and r as
p(q) = fo~=of4,~o P(q, O, r
0d0dr
fo f02'~sin 0d0dr
(6-28)
which is written by specifying the size of a hollow by aA, fiB, and 7C as
p(q) = fo f2o'~ IF~(A, B, C, r, O, r - F2(aA, 13B, TC, r, tg, r 2 sin0d0dr 47rIVI(A, B, C) - V2(aA, fiB, 7C)I 2
(6-29)
The particle scattering factor is calculated for simple homogeneous triaxial bodies as summarized below: (a) Sphere (radius R) P(q) = r (6-30) where (6-31)
X3
(b) Hollow sphere (outer and inner radii Ra and R2, respectively)
P(q)
[R~ f(qRi ).~I_ R]r
]2
(6-32)
(c) Ellipsoid of revolution (semi-axes a, a, c~a)
P(q) =
[
~r/2
+2(qajcos2 8 + a2 sin 2 0/cos 0d0
J0
(6-33)
(d) Cylinder (radius R and height 2H)
P(q) = which reduces to
cos0) [ ,~/2 sin2(qH q2H2 d0
C082 ~
4J~(qRsin O) sin 0d0 q2R2 sin2 0
(6-34)
p(q) = Si(2qH) _ sin2(qH) qH q2H2
(6-35)
Si(x) = fo ~ sintt dt
(6-36)
where
359
when R ~ 0 (in the case of a rod of length 2H and infinitesimal radius), or to
P(q)=q22R2 [ 1 - --~J~(2qR)] 1
(6-37)
when H ---, 0 (in the case of a disk of radius R and infinitesimal thickness). The Gaussian chain consisting of N units jointed by a link of length b yields also a simple particle scattering factor [8] P(q) = ( 2 N 2 / x 2 ) [ e x p ( - x ) - 1 + x]
where x =
(6-38)
q2Nb/6.
6.1.3. Structural Parameters Although the scattering from a single particle in solvent was considered in the preceding section, in practice the scattering factors derived above are valid for a dilute system consisting of N scattering particles, which are sufficiently isolated so as not to interact with each other. The sine expansion of the particle scattering factor yields, at very small angles
P(q) =
lq fv
1- ~
p(r)-
9dV/
fv
p(r)dV
(6-39)
where the second term in eqn.(6-39) represents the radius of gyration Ra. The Guinier approximation [9] replaces the sine expansion with the exponential function as
P(q) '~ exp(-q2R~/3)
(6-40)
Conventionally, the Guinier approximation is also applied to evaluate the radius of gyration corresponding to the cross-section of a rod-like molecule or the thickness of a flat particle. For example, the scattering function from a cylinder, eqn.(6-34), is composed of two factors which are regarded as nearly independent, and is approximated as 71"
Pcylinder(q) "~ " ~ q " P~(q)
(6-41)
where P~(q) denotes the scattering function from the cross-section. The factor 1/q is characteristic of a rod-like molecule, and eqn.(6-41) is valid only when the condition Hq < 1 is satisfied. Then P~(q) is given in terms of the cross-sectional radius of gyration R~ as P~(q) = exp(-q2R2~/2) (6-42)
360 In the case of a circle of r in radius, the cross-sectional radius of gyration is given as R~ =~/vS. Similarly, the scattering function from a fiat particle is approximated as 271" PIt,t v,~tid,(q) ~ -7-~_~ ~q- " Pt(q)
(6-43)
with A being the cross-sectional area. Here Pt(q) denotes the scattering function from the thickness, and is given by the Guinier approximation as Pt(q) = exp(-q2Rt2)
(6-44)
The thickness radius of gyration Rt is related to the thickness T as Rt = T / x / ~ . Thus, a ln qPcylinder vs. q2-plot or a lnq2pllat particle vs.q2-plot yields the crosssectional radius of gyration (as R~/2) or the thickness radius of gyration (as Rt2) from the respective slopes. Examples are shown in Figs. 6-2 (a), (b), (c) and (d) for the model trixial bodies of Re = 50 /l., where the lines are drawn with the slopes specified by the corresponding radii of gyration (see eqns. (6-40), (6-42) and (6-44)). Radii of gyration RG and cross-sectional radii of gyration R~ are summarized below for simple triaxial bodies: Table 6-1: Radius of gyration R e of simple triaxial body Triaxial body Sphere of radius R Hollow sphere of radii R1 (inner), R2 (outer) Ellipsoid of semi-axes a, b, c Prism with edge lengths A, B, C Elliptic cylinder of semi-axes a, b and height h Hollow cylinder of radii R1, R2 and height h
R~ 3R2/5 3(R~ - nl~)/5(n~ - n~)
(a 2 + b 2 +c2)/5
(A2 + B 2 + C2)/12 {(a 2 + b2)/4} + (h2/12) {(R1~ + R~)/2} + (h~/12)
Table 6-2: Cross-sectional radius of gyration R~ Cross sectional shape Circle of radius R Ring of inner radius R1 and outer radius R2 Ellipse of semi-axes a, b Rectangle with edge lengths A, B
R~ R2/2
(R~ + R~)/2 (~ + b~)/4
(A 2 + B2)/12
361
1.0
10~
o o
0
(a)
0.8-
0
0
(b)
0 0
10-1
0 0
0.6-
o
10.2 -
c~ o o o
0.4-
10.3 -
0.2
10.4 _
0.0 0.00
~ o
o o
0o ~
i
0.05
0.10 q
0.15
1.1u"5
O.2O
i
2
1
i 1 iii1
4
68
0.001 5OO
0
2
0.01
4
q
6 8
oOO
2
68
(d) 400 i-
~~
-5
o
o
~300
o
o
4
O .1
(c)
o
~-10
9
-15
100
, o
-20
o ~o
I
0
10
30
2~
1.2
0
40x10 -3
o
0.00
I
0.05
o
ooo
oo
o
I Oo:ooo
0.10 q
00
?Oo~o
0.15
q (e)
1.0--
~
0.8-
/
~0.6 "K 0.4-
g o gg
~
o
o
/
o
oe
0.2
o
g
0.0
I
o
,o
I
I
150
200
Figure 6-2: Scattering from a sphere of radius 64.55 A (Re - 50 A); (a) the normal plots, (b) the double logarithmic plots, (c) the Guinier plots where the line represents the initial slope specified by the value R~/3 (eqn. (6-40)), (d) the Kratky plots (q2I(q) vs. q) and (e) the distance distribution function (eqn. (6-9)).
....
0.20
362
1.0
o
(a)
0
o
o
o
o
o
1 0 -1 ___--
o
(b)
o o
_ _ _ _ _
o
0.6-
~
_
o
0.8-
0
10 ~ _~
o
o o o o
10 -2 ==_
o
-
o
_
0.4-
89 o o o
1 0 "3 ___
o
_
o
0.2-
~oo
_
10 -4 _
o
~o
_
~
I
0.0 0.00
_
~
~ .....
0.05
l .........
l ..........
0.10 q
0.15
0.20
i
1 0 -5 2
4
6 8
0.001
2
0.01
2
k
i i ii 4
6 8
0.1
ooo
(c)
-2
6 8
q
500
0
4
(d)
400 l
-4
~ 300 ~-
Oo
~-6
0
0
-8
0
0
~200 0
0
0
0
0
0
0
-
o o
0
100 t-o
-10
o
-12 0
10
I
I
20 2 q
30
40x10 -3
0~ ~ 0.00
i
0.05
oOOOoo
~176176176 0.10 q
~
ot ......... 0.15
1.2 1.0
(e)
--
\
o g o o o
0.8-
o o o o
~0.6-
s
g o
0.40.20.0 / 0
o g o o
I
50
~(~
150
200
Figure 6-2: (continued) Scattering from an ellipsoid of three semi-axes 91.29 A, 45.64 A and 45.64 A (RG = 50 A); (a) the normal plots, (b) the double logarithmic plots, (c) the Guinier plots where the line represents the initial slope specified by the value R~/3 (eqn. (6-40)), (d) the Kratky plots (q2I(q) vs. q) and (e) the distance distribution function (eqn. (6-9)).
0.20
363 1.0'
10~
o o
0.8-
o o o~
(a)
(b)
o 1 0 "1 o o
0.6-
1 0 -2
o o o
o
o
0.4-
10 .3
_
1 0 -a
_
o ~ o
0.2-
o
~176176
I
0.0 0.00
o
,
0.05
I
oo .....
r .........
0.10
0
10-5 0.20
0.15
q
i
I
2
i
i i ii|l
i
4
6 8
2
o.ool 1.0
1
0 .Ol
i
i iiill
4
68
i
i
2
i
I i ill
4
68
0.1
q
oOOOOo
-1
(c)
o
0.8
o
~ o
(d) o
o
o
o
-2
o
o Oooo
~-3
0.6
~
o
o o o o
o
~
-5
o
0.2
o
o --
o
0.8
o
o ~
2
o
0.4
~
-4
1.0
o
o ~
0
o
o
~
-6
o
o
l
I
l
10
2 20 3 q xlO
30
o o
o
40
~
0.0 o 0.00
I 0.05
I 0.10 q
I 0.15
o
o
o
o
0.20
(f)
(e)
~ ~N~
o
o
.0.6 :
o
o
~
~ o
-7I 0
,
,
50
100
_
150
_
I
-8
200
0
10
I
z 20 q xl03
I
30
r(A)
Figure 6-2; (continued) Scattering from a cylinder of cross-sectional radius 16.82 A and height 168.23 ,~ (Ra = 50 .~); (a) the normal plots, (b) the double logarithmic plots, (c) the Guinier plots where the line represents the initial slope specified by the value R~/3 (eqn. (6-40)), (d) the Kratky plots (q2l(q) vs. q), (e) the distance distribution function (eqn. (6-9)), and (f) the Guinier plots for cross-section where the line represents the slope specified by the value R~/2 (R~ = 11.90 A in eqn. (6-42)).
40
364 1.0
o
(a)
o
0.8-
8
10~
~
0 0 0
o~
o
~o
10-]
\
o
0.6-
o
o~
10.2 o
0.4-
10.3
o
o
o
~
0.05
0.10 q
0.15
0.20
O
i
2
i i i iiiI 4
6
0.01
0
0
4
2
68
0.1
i
i~
i i1|
4
68
1
0
_
~
o
o
40xlO 3
o
--o o
I
30
q
i
o O
100
2~ q
2
i I i ii1[
(d)
o
-10
10
i
0
0 0 0 0 00 O
i
0
~176176176 ~
0
8
0.001 500 400-
oo
-12
i
10-5
(c)
-2-4 '
o
10-4 ~176
0.0 0.00
\
%
o
0.2-
(b)
Oo\
0 0.00
~176176176176176176176
I
I
1
0.05
0.10 q
0.15
~
*o
0.20
1.0 (0
(e) 0.8
_8 ~
g 0.6
"~10~ 0.4
o
-12
0.2 0.0
1
0
50
1O0
150
-14 200
I
0
10
I
2~ q
I
30
r(A)
Figure 6-2: (continued) Scattering from a disk of radius 69.79 A and thickness 27.91 .~ (Ra = 50 .~); (a) the normal plots, (b) the double logarithmic plots, (c) the Guinier plots where the line represents the initial slope specified by the value R~/3 (eqn. (6-40)), (d) the Kratky plots (q2I(q) vs. q), (e) the distance distribution function (eqn. (6-9)), and (f) the Guinier plots for thickness where the line represents the slope specified by the value Rt2 (Rt = 8.06 ]~ eqn. (6-44)).
40xlO 3
365 1.0
10 ~
o
"
o
.
o ~
~
(a)
0.6-
o
~
0 . 8 -- o
10-1
(b)
-
o~
10.2 F-
o
0.4-
o
10"3 ~ _
0.2-
o
00 "0.00
~. . . . . . . . . +. . . . . . . . . 1. . . . . . . . . 10-s 0.05 -0.10 . . . . . . 0.15 . . . . . . . 0.20 0 001 '2
0V--------
"-~ ~
-~b\
-4t~176176176176
"~
~_6~
~
t
10
oo\
o
\
o
I
o
\
,o
1.4 [
I
_
1~V /
~o
I
, ....... '
4'6'i'.101
i~
""
I
2
400V . . . . . .
/
o
0 l~
I %1
o,o :,2~ 2
4--6
o
I ~ ~
o.o~
/
o.1o
o
/
o.,~
q (e)
':
/
o ~o
0.2
~ 50
100 r(h)
150
.... 81
,.)
/
4ox,o-~ o.oo
4 6
q
I ~00~
~
0.4
0
'~)
0000%0
o oo ~
OoOOl,ko
~
~ 0.8
00
Oo~
I
~o 2
~
1.2~-
~0.6~
o oo.
o
o
_-121
o
10"4 ~-
200
Figure 6-2: (continued) Scattering from a hollow sphere of radius 62.36 h. with a spherical hollow of radius 31.18 in the center A (Re = 50 A); (a) the normal plots, (b) the double logarithmic plots, (c) the Guinier plots where the line represents the initial slope specified by the value R2/3 (eqn. (6-40)), (d) the Kratky plots (q2I(q) vs. q) and (e) the distance distribution function (eqn. (6-9)).
o.~o
366 1.0'
10 ~
0 o
0.8-
~
~ o
(a)
o
10 -1
~
(b)
o o o o
10 -2
0.6-
o o o o
10 "3
0.4o o
0.2 1 0.0 0.00
~176176 I ~
.....
0.05
J, . . . . . . . . . 0.10
_1. . . . . . . . . . 0.15 0.20
q
0
o o
10 -4 10-5 0.001
I
I
2
I
I III1[
t
4
68
2
4
(c) 400 -
0 0
-4 0
0
I ~_ I I l l 4
68
0 0
0 o o ~
o
0
o
o
o
o
e"C: T,
200
-
100
-
o
-10
o
I
20 q2
I
30
40x10 3
0 0.00
o o
o
10
i
(d)
0
-8
0
2
O.1
300 -
o o
-12
I
6 8
q
500
-2
I IIII1[
!
0.01
o
oo
o
o
o
oOO
o
o o
o o
I
I
0.05
0.10 q
~ o
I ~176 ......
0.15
1.2 (e)
1.0 0.8
0
50
100 r(h)
150
200
Figure 6-2: (continued) Scattering from a hollow ellipsoid of three semiaxes 8.97 A, 4.48 A and 4.48 ~ with an ellipsoidal hollow of semi-axes 4.48/~, 2.24 A and 2.24/~ in the center (Re = 50 A); (a) the normal plots, (b) the double 19garithmic plots, (c) the Guinier plots where the line represents the initial slope specified by the value R~/3 (eqn. (6-40)), (d) the Kratky plots (q2I(q) vs. q) and (e) the distance distribution function (eqn. (6-9)).
0.20
367
10 ~
1.0
o
(a) 0.8
o
0.6-
o
1 0 .2
?
o o
0.4-
(b)
o o OoooO~oo
10 -1
-
10 .3
o
0.2-
~176176176
0.0 0.00
0.05
?
10 .4
~
0.10 q
r
10 .5 0.20 0.001
0.15
0
,
2
,
,
,,,,,n
4
6 8
,
,
, ,,,,,I
2
4
0 .01
,
6 8
,
2
,
, , , , ,
4
68
O.1
q
1
1.0[ (c) 0.8-
-2
o
~ 0.6 ~
~---~4
o
~ ~ ~ o o
o
o
o
oOOo
(d) ~
o
o
o
o
o
o
o 0
1~0.4 ~ -6
o
10
1.4
1.0
I
I
2~
30
oo oo o o o o o o o o
_
o ~176
o o
.30. O~
o
40x10
o
0.00
I
I
I
0.05
0.10 q
0.15
q
1.2
o
o
o
0
0
o
0.2-
o o
-8
0
~ Oooo+
0.20
(e)
(f)
-5 -6
~ 0 . 8 -- ~ x 0.6o
o
o
~-7 "d
~
g
o
o o
-8
0.4 - go
o
o
o o
0.2 -~ o
-9
-
O O
o.o~' 0
50
100 r(A)
150
200
-10
0
I
I
I
10
20 2 q
30
40x 10 .3
Figure 6-2" (continued) Scattering from a hollow cylinder of cross-sectional radius 16.71 .& and height 167.05 .& with a cylindrical hollow of cross-sectional radius 8.35 A and height 167.05 A (Re - 50 A); (a) the normal plots, (b) the double logarithmic plots, (c) the Guinier plots where the line represents the initial slope specified by the value R~/3 (eqn. (6-40)), (d) the Kratky plots (q2I(q) vs. q), (e) the distance distribution function (eqn. (6-9)), and (f) the Guinier plots for cross-section where the'line represents the slope specified by the value R2~/2 (Re - 13.20 ~ in eqn. (6-42)).
368 1.0
10 ~
~o o
~176
0.8-
(a)
~
(b)
~o
10 -1
\
o
0.6-
1 0 "2
0.4-
10 .3 _
%
o
o
0.2-
10 -4 _
o o
~176176176176176
~~ . . . . . . .
i
0.0 0.00
0.05
~ .........
0.10 q
I
10 .5
0.15
0.20
0
i
2
I
lilili
4
68
I
i
2
0.01
0.001
I
I lili[
4
68
I
i
2
I
I llil
4
68
0.1
q
1
800
-2
(c)
-4
~ 1 7 6
o
(d)
000
600-
0
0
0
o
o
o
o
~-6
o
o
O
o
400 -
O
o o o,
%
0
o
0
o
o
o
ooo o
-8
200-
Oooo
o
o
o
-10
o o
o
o
o
o ~
o
1
-12 0
10
I 30
20
40x10 3
0 o 0.00
I
I
0.05
0.10
I 0.15
~
0.20
q 1.2 1.0
(e)
m
0.8 e~ o
-9
~0.6
s
0.4
_
-10 o o
0.2
.
o o
0.0 0
50
i 100 r(A,)
-11 -
g ~,_
_.1 . . . . 150
200
-12
0
I 10
I 20 q
I 30
Figure 6-2: (continued) Scattering from a hollow disk of radius 62.58 and thickness 25.03 A with a hollow of radius 31.29 A and thickness 25.03 ,~ (Re = 50 h); (a) the normal plots, (b) the double logarithmic plots, (c) the Guinier plots where the line represents the initial slope specified by the value R2a/3 (eqn. (6-40)), (d) the Kratky plots (q2I(q) vs. q), (e) the distance distribution function (eqn. (6-9)), and (f) the Guinier plots for thickness where the line represents the slope specified by the value Rt2 (R~ - 7.23 ~ in eqn. (6-44)).
40xlO -3
369 6.1.4. Molecular Mass Determination Since the absolute scattering intensity I . ( q ) at the scattering angle 20 is given in terms of the molecular mass M of colloidal particles as [10] In(q) =
L " A z 2" M . d . N A " c a2 9~(q)
(6-45)
where I, = 7.9 x 10 -26 [cm 2] is the Thomson factor representing a scattering intensity from a single electron, q the reduced scattering variable (see eqn. (6-3)), d [cm] the sample thickness, N A the Avogadro number, c [g/cm 3] the concentration, a [cm] the distance between the sample and detector, and ~(q) the normalized form factor. Az denotes the number of effective mole-electrons given by Az = z - ~'Zo
(6-46)
with z and zo being the the number of mole-electrons per gram of particles and per cubic centimetre of solvent, respectively, and ~ the specific partial volume of a particle. By definition ~(0) = 1, so that the molecular weight is determined from the scattered intensity at zero angle as M =
I,(0). a 2 A z 2. d. c" L " NA
(6-47)
Eqn. (6-47) can be rewritten in terms of the scattering amplitude b [cm/g] of a particle (given by the sum of coherent scattering lengths) and the scattering density P0 [cm-2] of solvent as M= NA a2In(O) (6-48) c(Ab) 2 d where the excess scattering amplitude Ab [cm/g] is defined as Ab = b - ~po,
(6-49)
and is related to the number of effective mole-electrons as I~Az 2 = (Ab/NA) 2
(6-50)
Eqn. (6-48) requires the absolute scattered intensity and the excess scattering density to be measured with areasonable accuracy in order to evaluate the molecular mass. Several techniques are proposed for the absolute measurement in X-ray scattering, although none can be applied satisfactorily to the synchrotron radiation
370
X-ray source of extremely high intensity. A large error in evaluating the excess scattering density is involved through the partial specific volume ~, which should be measured with less than a 1% error to yield the molecular mass with a reasonable accuracy. Thus, the molecular mass determination by small-angle X-ray scattering is less practical than other methods such as light scattering, osmotic pressure and sedimentation. The mean square fluctuation of electron density (Ap)2 is given by (Ap)2 = 27r 21 rio~ a ~In(q)d
" q2 . dq
(6-51)
which reduces, in the case of a two-phase system characterized by the scattering density po and/91, to (Ap)2 = Vo. V~. (pl - po) 2 (6-52) Here V0 and V~ represent the volume fractions of the two phases, so that V1 = 1 - V0. Considering the solution of homogeneous particles as a special case of a two-phase system, we may assume that eqn. (6-52) is valid to a good approximation. Since the volume fraction is given in terms of concentration and partial specific volume as El = c v ,
(6-53)
Eqns. (6-49) and (6-52) yield (Ap)2 c
1
= - . (Ab) 2 - c(Ab) ~ v
(6-54)
where the scattering density Pl of a particle is substituted by b/~. Thus the initial slope of the (Ap)2/c vs. c curve is identified with the squared excess scattering amplitude as d (Ap)~ _ _(Ab)2
dc
c
(6-55)
In general, the scattered intensity is considered to be composed of two terms due to single particles and interparticular interferences, P(q) and Q(q), respectively.
I.(q) ~ cP(q) + c2Q(q)
(6-56)
The interference term in eqn. (6-56) can be neglected when c << 1 and/or q is sufficiently high. That is, no interference effect will appear in the scattering profile when q exceeds a certain finite value qm which may depend on concentration.
371 To evaluate the mean square fluctuation of electron density, eqn. (6-51) requires the value of the scattered intensity I.(q) to be measured over a whole q range from 0 to cr Since no concentration dependence is expected in the observed I.(q) when q > qm, eqn. (6-51) can be converted as [11] d (Ap)2 _-- i
dc
c
d fo q"~ a2I.(q).q2, dq
27r2 dc
cd
(6-57)
where the integration limit qm denotes the maximum q value above which no concentration dependence will be observed in the scattering profile. Combining eqs. (6.55) and (6.57), the excess scattering density is given by
(Ab) 2 =
27r 21 dcd fo am a2In(q).cd q2. dq,
(6-58)
which is inserted in eqn. (6-48) to yield the molecular weight as
M =--27r2NA
a2In(0) d fo qr" a2In(q), q2. dq cd /-dcc cd
(6-59)
Here the term In(q) appears in the numerator and denominator, so that the term I,(q) in eqn. (6-59) can be replaced by an scattered intensity I(q) in arbitrary units. This procedure for molecular mass determination is demonstrated for the synchrotron SAXS from lysozyme [12] (taken by the SAXES system installed in the BL-10C of the Photon Factory described in the next section), which yielded I(O)/c = 232,000 in a.u./g.cm -3 and dZ/dc = -10.09 • 1024 in a.u./g 2 from the data shown in Fig. 6-3. Here Z is defined as
Z - -~1 2 fo am I(q) c
q2.dq,
(6-60)
and the molecular mass is calculated from
M~
= --NA
I(O)/c Z
(6-61)
as 13,800, which agrees with the value of 14,000 calculated from its primary structure.
372 1200 o
o
.qr~
"6-"
b,O
=i6oo "7
o
, ~,,,,~_
0
400
L ,2~0 0 1 ~ / ~r' 0~ 0.00
0 /
~
A 0.05996g~ cm"s [] 0.12407g.cm "3 I 0.10 q (A -1 )
I 0.05
:3.~
0.02981g. 0.04161g 9cm"3
~ ~ qm
I 0.15
0.20
6.2 6.0
" o 5.8 5.6 =i 5 . 4 d 5.2 N 5.04.8 0.00
0.02
0.04
0.06
0.08
_30.10
Concentration (g.cm )
0.12
0.14
Figure 6-3: SAXS from lysozyme solutions. (a) Kratky plots (q2I(O)/c vs. q) of scattered intensities. Concentrations are as indicated in the figure, and qm denotes the upper integration limit, see eqn. (6-59). (b) Z - (27r2)-1 foqm[I(q)/c] 9q2. dq as a function of concentration, see eqn. (6-60).
6.2. S A X S Instrumentation at the Photon Factory 6.2.1. Basic design of beamlines Since the general instrumentation of synchrotron radiation has been given already, the present section outlines the special features of two beamlines, BL-10C and BL-15A, at the Photon Factory, Tsukuba, Japan, which are allocated to smallangle X-ray scattering.
373 All X-ray beams are extracted from the storage ring through beryllium windows which separate the ring vacuum from the low vacuum or helium gas atmosphere in the down-stream beamline components. The layouts of Beamline 10 and Beamline 15 are schematically shown in Fig. 6-4 [13]. These two beamlines have a similar construction, where the beamline is branched to three sources with horizontal divergences 4.5 mrad, 2 mrad and 6 mrad, respectively, extracted for A, B and C. The beam pipes and chambers for optical elements of these beamlines are evacuated by a turbomolecular pump to 10-~ Torr.
Figure 6-4: Layout of the beamlines at the Photon Factory, Tsukuba, Japan. (a) BL-10, and (b) BL-15. Each beamline is branched to three sub-beamlines A, B and C.
374
BL-IOC
The optics are composed of a double crystal monochromator (located 14 rn from the source) and a doubly-focusing mirror (located 16 m from the source) as schematically shown in Fig. 6-5. Silicon 111 crystals of approximately 10 cm in diameter are used for the monochromator placed in the upstream of the focusing mirror. Three modes of motion of the monochromators are synchronously controlled to rotate the first and second crystals and shift the second crystal horizontally, using three independent computer-controlled stepping motors, so that the height of the successively Bragg-reflected beam is always constant. This optical system yields X-rays with energies varying from 4 to 10 KeV. The focusing mirror has a platimum-coated surface polished cylindrically with a radius of curvature of 12.4 cm. The mirror is bent in the incident beam direction to a radius of curvature of 2000 m. The glancing angle of the X-ray beam to the mirror is set to 8 mrad, which gives a cut-off energy of approximately 10 KeV. An X-ray beam has a horizontal divergence of approximately 4 mrad, and the mirror yields one-to-one focusing at a distance of 32 m from the X-ray source. The beam size at the focus is approximately the same as at the source, i.e., 6 mm (H) x 1 m m (v).
The resolution of this optical system is better than 2 eV at 9 KeV, as estimated by observing the pre-edge structure from a copper foil. The intensity is estimated t o b e 12 -,~ 16 x 1011 photons/sec at 9 KeV with a 2 eV, when the storage ring is operated at 2.5 GeV and 300 mA. The small-angle X-ray scattering equipment for solutions (SAXES) [13] was designed for scattering experiments on non-crystalline materials including polymer solutions, metals and alloys, and was constructed to fit to BL-10C. The SAXES is composed of a slit assembly made of a tapered 3 mm thick tantalum plate, a specimen holder, vacuum pipes and a detector holder assembled on 2.5 m double optical rails as schematically illustrated in Fig. 6-6. Basic requirements in the SAXES design were specified by the wavelength from 2.4 to 1.24 A (4 to 10 KeV), the small-angle resolution better than dmax "~ 1 , 5 0 0 ~k (2sin0/)~ ,,~ 7 x 10 -4 with ~ = 1.5 ~) [13], the angular resolution being 1 mrad, the beam size at the specimen not exceeding 10 mm, and the range of scattering measurement up to 5 A. The SAXES is placed at the very end of BL-10C which is situated 29.5 m from the X-ray source, and is all installed in a safety hutch. Three slits are employed to achieve a high scattering resolution, separated from the X-ray source by 11 m to limit incident X-rays, at 15 m between the monochromator assembly and the
375 .
.
.
.
.
16m
.
.
.
.
.
.
~
02 -~~._..
=
16m
I
~ _ . ,
-
-/
~ ~
....... ~..~\ ~ ~ ~ ~ _ _ - - / / / / . . ~ focal point
~_ 16mrad
SR ,~ ~
. ~.c,. ,- ~
X-RAY
"JOt doubly focusing mirror double-crystal ( bent cylinder ) monochromator
OPTICS ( SAXES )
. . . . . . . . .
Figure 6-5: Optics of BL-10C, which consists of a double crystal monochromator (ups-tream) and a bent-cylindrical mirror (down-stream).
i~bc~:._.:~::.:i1_:ll-.-. i J.-._.i e 9
t
"
~]~
. . . . .
-i
I!i!:!!!::!iiii.iiill iiiii1 i .....
30m
L..
t
31m
l
32m
Figure 6-6: Small-angle X-ray scattering equipment for solutions (SAXES) installed on BL-10C: a, the exit window of X-ray beam; b, the third slit assembly; c, specimen holder; d, vacuum pipe; e, detector (PSPC); and f, double optical rails.
SOURCE
BENT CRYSTAL
MMRAD.
h'~SCM
10MM I./~
~_...~ ~.~.
SAMPLE
~.~. ~ , . ~ . .
Figure 6-7: Optics for the muscle diffmctometer.
"~
~~
376 thickness, respectively) from the helium pass and the which is evacuated by a rotary pump. The exit of the employs a Kapton window of 0.025 mm in thickness. specimen to the detector has two exit windows made of is also evacuated to minimize unwanted scattering.
monochromator chamber monochromator chamber The beam pipe from the Kapton at both ends, and
6.2.2. Data acquisition systems Two types of data acquisition systems are currently employed at BL-10C and BL-15A in the Photon Factory. One dimensional position sensitive detector (PSPC) of size 200 x 20 x 6 mm 3 (an effective length x a window height x a chamber depth) is mostly used at BL-10C, coupled with a CAMAC data acquisition system. The window is made of beryllium 1 mm thick, and xenon + CO2 or argon + CH4 is used as a counter gas. The PSPC adopts the position read-out of delay line type, with a fast 400 nsec delay line along the effective length of the probe. The probe has a standard structure except that if is constituted of six anode wires of 40 ~m in diameter to minimize the space charge effect of the proportional chamber. Its spatial resolution is approximately 0.3 ram, and the uniformity of sesitivity is within 4 % along the effective length of the probe.
Figure 6-8: Data acquisition system in BL-10C.
377 mirror, and at 29.5 m prior to the specimen holder. The beam pipe between the mirror box and the exit window is evacuated to avoid scattering by helium or air. The SAXES is coupled with a detector (at present a one-dimensional PSPC) and a CAMAC data acquisition system.
BL-15A BL-15A is allocated to the Small Angle Scattering (Muscle) working group. A small-angle resolution of approximately 1000/1, is achieved in the vertical direction at a wavelength of 1.5 /~ with a demagnifying focusing mirror-monochromator optical system [14] (see Fig. 6-7). The intensity is 4.5 ~ 6 x 101~ photons/sec at 1.5/~, with the storage ring operating at 2.5 GeV and 300 mA. The mirror-monochromator optical system is composed of two focusing assemblies in vertical and horizontal directions. For focusing in a vertical direction, seven fused silica mirrors of 20 cm x 6 cm x 1.5 cm in length, width and thickness, respectively, constitute a 1.4 m long reflective surface, where the mirrors are fixed tightly by mirror clamps onto the highly polished bottom surface of a 1.5 m long steel beam with an H-shaped cross-section. Thus the reflective surface of the mirrors is expected to bend, following curvature of the bottom surface of the H-beam which is bent elastically to a radius of a few kilometers. The whole mirror assembly can be positionaUy adjusted with respect to its inclination and height by two independent linear motors at each end of a bench on which the H-beam and the bending mechanism are mounted. Focusing in a horizontal direction is achieved by using a triangular curved crystal (50 mm base x 170 mm in height x 1 mm thick)of silicon with its l 11plane inclined at 7.8 ~ with respect to the surface. Here the tip of the triangular crystal is pressed by an eccentric cam, while the base is tightly fixed. Three sets of slits are used in the diffractometer. The first set is placed just upstream of the mirror, and limits the size of X-ray beam to suit the specimen size. The second set is located at the exit of the crystal monochromator to reduce undesirable scattering from the monochromator. The third set is placed immediately in front of the specimen. Two viewing ports are provided upstream, just prior to the mirror and the monochromator, respectively, in order to monitor the shape and position of X-rays. A TV camera monitors the X-ray beam at the specimen. The diffractometer is evacuated independently of the storage ring, and a short helium pass separates two parts. The mirror chamber is evacuated to 10-5 Torr by a turbomolecular pump in order to prevent contaminaton and damage to the mirror. The mirror chamber is isolated by beryllium windows (0.2 mm and 0.07 mm in
378 Data acquisition is performed as shown in the block diagram (Fig. 6-8). The main amplifiers and timing circuits are the standard NIM modules. The position determining system consists of a Le Croy 4202 time-to-digital converter (TDC) followed by a Le Croy 3588 histogram memory, which cuts the dead time to 1.2 #m at most. A moderately high counting rate can be measured with this short dead time (probably up to 3 x 105 counts/sec/chamber). The 4202 TDC has a function for routing, and the time-resolved measurements are easily performed with a time interval controlled by a Kinetic 3655 timing pulse generator. Since the 3588 histogram memory has a memory size of 16 K words with a depth of 24 bits in its double width CAMAC module, 64 time frames can be recorded with 512 channels/frame. These CAMAC modules are controlled by a Kinetic 3920 crate controller which is connected to a DEC MNIC11/23 computer with a Kinetic 2920 Bus adopter for the LSI 11 Bus. A photo-stimulable phosphor screen (imaging p/ate) is employed at B L-15A as an area detector for X-ray diffraction and scattering experiments. An imaging plate (IP) [15-17] has advantages as an area detector over conventional integrating detectors such as X-ray film or an X-ray TV detector with respect to the dynamic range and the detection quantum efficiency. For example, the area detector made of a 250 x 200 mm 2 IP has more than 80 % detection quantum efficiency for 8 ~ 17 KeV X-rays, a dynamic range of 1 9105, a spatial resolution better than 0.2 mm (in terms of FWHM; a full width at half maximum) in two orthogonal directions and no counting limitation. An IP is made of a flexible plastic plate coated with a 150 #m thick layer of suspended polycrystals (crystal size 4-5 #m in diameter) of photo-stimulable phosphor (BaFBr:Eu 2+) in an organic binder. A phosphor layer surface is coated with a 10 #m thick polyethylene terephthalate sheet. A typical IP has dimensions of 250 mm x 200 mm x 0.5 mm. Here the phosphor layer stores a fraction of absorbed X-ray energy in the form of quasistable color centers, when exposed to X-rays. These color centers are stimulated by visible light, and emit photo-stimulated luminescence (PSL) with an intensity proportional to the number of absorbed X-ray photons. The phosphor can be stimulated by light of wavelength approximately 400 nm to 800 nm with two PLS intensity peaks at 540 nm and 600 nm, and the response time of PLS is 0.8 #s. The PLS-radiation spectrum falls within the range of 300-500 nm with Am~ = 390 rim, which is sufficiently different from the wavelength of stimulating light (for example, a He-Ne laser beam). The image stored on the IP lasts for several hours without substantial fading. An IP with an active area of 251 x 200 mm 2 has 2510 x 2000 pixels, where
379 the pixel size is 0.1 x 0.1 mm 2. The full width at half maximum (FWHM) was found to be less than 2 pixels (equivalent to 0.2 mm), by exposing to a 20 #m wide line-shaped monochromatic X-ray beam. The linearity and dynamic range of PSL were examined by CuKa (8.9 KeV) and MoK~, (17.4 KeV) X-rays, and the relative intensity of PSL was found linear with the incident X-ray intensity over a range of 8 to 4 x 104 X-ray photons/pixel, with a relative error less than 0.05. The absorption efficiency is 96 % for 17 KeV X-rays with a 150 #m thick phosphor. The absorption edge is observed at 37.4 KeV due to barium. Defining an ideal detector as having a relative uncertainty equal to that in the incident X-ray intensity, the IP was found nearly ideal in the middle exposure range between 101 and 103 photons/pixel, whereas a highly sensitive Kodak DEF-5 film requires 15 times and 25 times more exposure for CuKa and MoK,~, respectively, than would be needed by an ideal detector to obtain a 10 % relative accuracy.
6.3. A P P L I C A T I O N S Small-angle X-ray scattering is suitable for observing particles of dimensions 10-500 A, which covers most colloidal particles including biological or synthetic polymers. Since the high flux of synchrotron radiation enables one to record scattering data within a matter of seconds, when coupled with a suitable detector system, synchrotron radiation X-ray scattering is thus best applied where a quick measurement is required. The wavelength continuum and low divergence are other prominent features of synchrotron radiation X-rays, which, however, are not yet fully utilized. This section introduces some work performed at BL-10C and BL-15A in the Photon Factory, Tsukuba, Japan. BL-10C is devoted mainly to small-angle scattering from solutions, and BL-15A to that from muscle and alloys in the bulk state.
6.3.1. Wide-angle and small-angle X-ray scattering from films and membranes Time-resolved small-angle X-ray scattering from polyethylene tereph~halate film The time-resolved small-angle X-ray scattering was observed from poly(ethylene terephthalate) film during isothermal crystallization [18-20], using a double focusing camera for synchrotron radiation at the storage ring DORIS. Here, a linear
380 position-sensitive detector was used for the data acquisition system composed of an IN90 (Intertechnique) programmable multichannel analyzer. Amorphous polyethylene terephthalate films were oriented by stretching at 92~ and crystallized at temperatures ranging from 90~ to ll0~ The smallangle X-ray scattering was monitored by a vidicon-system during crystallization, and the long period and azimuthal half-width of the diffraction maximum were evaluated from the obtained patterns. Here the long period was found to decrease with time and increasing crystallization temperature, while the azimuthal half-width remained constant, irrespective of crystallization temperature, after a certain time as shown in Fig. 6-9. The result implies that the orientation of the crystal lamellar surfaces improves with time but not the chain orientation itself. When a polymer is annealed above its crystallization temperature, the crystal lamellae thicken, as observed by an increase of the long period. Amorphous polyethylene terephthalate films were crystallized at 120~ Then the films were heated with a rate of 100~ to the temperatures of 230~ 240~ 245~ and 250~ in sequence (below the melting poin0 by cooling down once to 120oC between two successive heating processes with the same rate. The scattering power Q = f I(q)q2dq was observed to increase and decrease simultaneously with heating and cooling, respectively, due to the independent thermal expansion of crystalline and noncrystalline regions which changes the density difference between the regions. An additional partial melting and recrystallization was observed at 240~ and 2500C, as shown in Fig. 6-10 which shows the result of the last two heating/cooling cycles.
Time-resolved small-angle and wide-angle X-ray scattering from crystallizing polyethylene The time-resolved small-angle and wide-angle X-ray scattering was observed from polyethylene film under stretching [21]. The two-dimensional pattem of SAXS and WAXS was recorded on imaging plates (126 x 126 mm 2) coupled with an IP rapid exchanger, as a function of time. The film of low density polyethylene LLDPE (Sumitomo Chemical FA101-0) was stretched at a constant rate of 204 mm/min. (for present specimens, 14.75 %/s), while the scattering patterns were taken at 0 (0 %), 6 (88 %), 12 (177 %) and 18 (265 %) seconds (stretching ratio) after stretching started, with an exposure time of 1.0 s for SAXS and 0.1 s for WAXS. Since the wavelength of the incident beam was 0.1507 nm, the q range covers 0.1 to 1.3 nm -1 for SAXS with a camera of ~ 2000 mm in length, and from 2.0 to 22 nm -1 for WAXS with a camera of ~ 100 mm in length.
381 200
i
\
180
"<
_J
160 ILo
-
~o~'c 9 ~91'c~\~oc
120 I00
a
108~
8(3 120"
I
I
'~,~
100'
I
1
\.~96"c 0~o~'.~...o ~ ' ~ ~ ~ _
80*
I
I
I
~
60*
L0* b
0
I
2
I
L
I
1
6 8 t k {mln ) - - - - - - ~
I
10
I
12
1L
Figure 6-9" Long period L (a) and azimuthal half width A~ (b) of oriented polyethylene terephthalate film as a function of time during isothermal crystallization (the crystallization temperature is indicated in the figure).
Figure 6-10" Change of the small-angle X-ray scattering during stepwise heating/cooling of unoriented polyethylene terephthalate, where Q denotes the scattering power Q = f I(q)q2dq.
382 Fig. 6-11 shows the SAXS and WAXS patterns from stretching polyethylene film, where the stretching direction is vertical in respective patterns. Polyethylene crystals are seen from the WAXS pattern change to align along the c-axis during stretching, while the rearrangement processes of lamellae are observed from the SAXS patterns.
Figure 6-11: WAXS and SAXS patterns of polyethylene film during stretching. The stretching ratios (and the times after starting stretching) are (a) 0 % (0 sec.), (b) 88 % (6 sec.), (c) 177 % (12 sec.) and (d) 265 % (18 sec.). Time-resolved small-angle X-ray scattering from systems undergoing phase transition
Many biological membranes are known to undergo phase transitions,which can be observed in real time by synchrotron radiation small-angle X-ray scattering. Fully hydrated phosphatidylethanolamine, for instance, exhibits a phase transition from gel (L~) to liquid crystalline (L~), while fully hydrated phosphatidylcholine has a ripple gel phase (Pz,) between gel (Lz,) and liquid crystalline (L~) phases.
383
DPPE (1,2-dipalmitoyl-sn-glycero-3-phosphorylethanolamine) was dispersed in doubly distilled deionized water to the concentration of 20 wt%. Small-angle X-ray scattering was recorded every 0.5 min. through PSPC at BL-15A in the Photon Factory, while the temperature was reduced at a rate of 0.1~ [22] The scattering profile changes as the temperature decreases from 64~ to 61.5~ where the La ~ L~ transition takes place around 63~ (see Fig. 6-12, where the sample was incubated for 15 min. in the L~ phase prior to the cooling run). The scattering profile has revealed two lamellar spacings corresponding to the L~ and L~ phases, which coexist in the temperature range from 63.5~ to 62.5~ No other phase appears in this transition range, although the high-sensitivity differential scanning calorimetry (HSDSC) exhibits a multi-peak thermogram depending on the incubation time in the L~ phase, and suggests that the cooling La ~ L~ transition is not a simple two-state transition. Multiple inflection points observed in the temperature dependence of scattering intensities also confirm the result of HSDSC. Thus the equilibrium state of the L~ phase of hydrated DPPE is a mixture of domains which differ in thermal behavior, but their structural difference is too small to be distinguished by S AXS. 3000
65
.--.60
AA~'A j~AAAA
a 9
z 55
2000
a
m
z
z
i.-..i r'l
50 a
a
~A~ ~a~ a
A
AA
1000
9
O Z ..-_4 (.f3
aa ,
62
<"
~ 6 s
o
I
63
i
I
64
TEMPERATURE ('C)
65
5 c"
62 5 ~~-'~ 62 65
60 55 SPACING (A)
50
...<
.$~
61.5
Figure 6-12: L~ --, L~ transition of DPPE in excess water with a cooling rate of 0.1~ observed by time-resolved SAXS. x denotes the spacing, and A, A the intensity of L~ and L~ phases, respectively.
384 Dipalmitoylphosphatidylcholine (DPPC) is known to possess three phases in its aqueous dispersion; the non-rippled gel phase La, below Tp (the pre-transition temperature ~ a4~ the rippled gel phase Pa, between Tp and Tm (the main transition temperature .,~ 41.5~ and the liquid crystalline phase L, above Tin. The system exhibits a temperature hysteresis in the Pa, temperature region, and takes over 24 hours to attain the stable Pa, phase from the Pa,(ms0 phase (i.e., the metastable rippled gel phase) when cooled from L, to Pa,. Time-resolved small-angle X-ray scattering was observed from the system quenched from the stable Pa, phase to the La, phase temperature region [23] with mirror-monochromatic optics coupled with a PSPC at BL-15A in the Photon Factory, where the wavelength of the incident beam was set to 1.5 A and the specimento-detector was 1.113 m. The ordering process below Tp is quite fast, and will be completed within several tens of seconds. Synthetic L-a-dipalmitoylphosphatidylcholine was dispersed in water to the concentration of 30 wt%. Fig. 6-13 presents the time-resolved SAXS profiles from the DPPC-water system quenched from 37.7~ (the stable Pa, phase region) to 23.3~ (the La, phase region), where each profile was measured for 7 seconds and the temperature was jumped from 37.7~ to 23.3~ after the initial four profiles were taken. The Pa, rippled structure is characterized by three Bragg peaks at q =~ 0.05, -.~ 0.11 and ~ 0.15 A-~, and the Pa, repeat distance of the lamella is represented by the Bragg peaks at q =~ 0.09, ,.~ 0.18 and -.~ 0.27 h-1. When the temperature was jumped down to 23.2~ these peaks were soon replaced by the Bragg peaks corresponding to the La, repeat distance of the lamellar structure at q =~ 0.095, ,,~ 0.19 and ,.~ 0.285 .h,-~, which grow gradually. The relaxation process from the Pa, phase to the La, phase was found to be much quicker than that from the Pa,(mst) phase to the Pa,.
Structural analysis of biomembranes Most biomembranes disintegrate within a couple of hours of dissection, and thus require a quick measurement before the disintegration takes place. The structure of invertebrate rhabdomes was analysed from the X-ray diffraction pattem from unfixed retina, taken by the use of synchrotron radiation and the imaging plate (a storage phosphor screen) [24]. Here the retina was dissected in dim red light from squid rhabdomes (Watasenia scintillans), and consists of hexagonally arranged photoreceptor microvilli accomodating visual pigments as schematically shown in Fig. 6-14. A 1 mm thick retina was kept in an artificial seawater chamber with Mylar windows at 4~ during the X-ray diffraction measurements. The X-ray diffraction patterns were recorded on imaging plates attached to mirror-
385
', , '
,I
-- ~- - lnlensity of 1st Bragg Peak
=
ll"ittl,~ d'
//'/:'-~"
~
~
+--22884
5 .....11 ]I[! i]~,'I'~"!:''~ '.',"-'''llIilll~ " ~'~'~ ~ '' .... ~ ~------~' "--=--~-~-..6 T 172 tlme '"0~0 . . . . . .
02
.,
% ~ -1. . . .
Q (A)
0'6 . . . . . . .
0'8
~
700
~>" 600 ._. 1500 /
.
:'
'
232~
400 0 . . . . . . . .I00 . . . . . . . . . . .200 . . . . . . . . . 300
time (sec)
1 400
500
600
Figure 6-13: Time-resolved SAXS profile and evolution of the first Bragg peak intensity from DPPC/water system, quenched from 37.7~ (the stable P#, phase) to 23.2~ (the L#, phase).
Figure 6-14: Schematic diagram of a vertical slice of squid retina. The retina consists almost entirely of photoreceptor cells sectioned verticaUy into inner and outer segments. The photo-receptive outer segments are seen in the upper layer and consist of microvilli, which are cylindrical extensions of the cell membrane, packed hexagonally in the rhabdomes. The microvilli are 600 A in diameter and 1 #m in length.
monochromator optics (Muscle Diffractometer) at BL-15A of the Photon Factory. The wavelength of the incident beam was 1.5 A , and the specimen-to-detector distance was set to 2276 mm.
386 Fig. 6-15 shows the X-ray diffraction patterns from outer segments of a live squid retina in the dark (left) and in light (right). The low angle X-ray diffraction spots are due to the oblique lattice of microvilli. One observes that the lattice constants and the X-ray intensity distribution change immediately upon illumination by the light (~ 500 nm in wavelength) of a 100-W halogen lamp through an interference filter and a heat filter. The results suggest the increase of the microvilli diameter and the inter-microvilli distance in response to light stimulation. The Patterson function was calculated from the observed two-dimensional Xray diffraction intensity (Fig. 6-15 left) as Fig. 6-16a. The model was constructed as shown in Fig. 6-16b, where the microviUar membrane was assumed to have a bilayer electron density distribution. The membrane junction is represented by inter-microvillus materials. The Patterson map (B) calculated from the model A (Fig. 6-16b) reproduces reasonably that calculated from the observed X-ray diffraction intensity, except for the regions around the lattice comer.
Figure 6-15: The X-ray diffraction patterns from a live squid retina in dark (left) and light (right). The X-ray diffraction spots are due to the oblique lattice of microvilli. The lattice constants are a = 60.0 nm, b = 59.0 nm and 7 = 118~ in the dark, and a = 65.5 nm, b = 64.0 nm and 7 = 118~ in the light. The exposure time was 4 minutes for each pattern with a beam current of 300 mA. The recording of both patterns was finished within 40 minutes after decapitation.
387
b
0
Figure 6-16a: The Patterson map calculated from the observed X-ray diffraction intensity (see Fig. 6-15). The lattice constants are a - b = 60 nm and 7 = 120~ Negative contours are indicated by broken lines.
A
b
0
B
~(~
~ ""(
,, " " ' O I
Figure 6-16b: A model microvillar membrane (A) and the Patterson map 03) calculated from the model A. Each large circle in the model structure represents the cross-section of photoreceptor microvilli, which contain a cytoskelton core (filled circle) at the center. The discrepancy factor is 16 % for this model.
388 Bacteriorhodopsin (bR) is the sole protein found in the purple membrane of The bR is composed of 248 amino acid residues and the chromophore retinal, and is folded into seven a-helices spanning the lipid and additional segments in aqueous regions. The bR forms trimers arranged in a hexagonal lattice in the purple membrane. The bR undergoes a photo-reaction cycle on absorbing light with the M-intermediate considered to play a key role in its proton transport process. ttalobacterium halobium.
The arginine treatment was found to stabilize the M-intermediate for a considerable period without changing the bR and membrane structure or disturbing the photo-reaction cycle of the bR [25]. The purple membrane was isolated from a R1M1 strain of Halobacterium halobium. Its solution was dialysed against ArgHC1 solution (pH 10), and then dried on a sheet of Mylar. The X-ray diffraction profiles were observed from this purple membrane while irradiating yellow light (A > 530 nm; the M-intermediate state) or purple light (A = 410 nm; the trans state), with the Muscle Diffractometer coupled with a PSPC detector system at BL-15A of the Photon Factory. As the results show in Fig. 6-17a, the X-ray diffraction patterns exhibit small but significant differences between the t r a n s state and the Mintermediate. The lattice constants of the purple membrane were found to be 62.7 A and 62.8 A, respectively, in the trans state and M-intermediate, indicating that the chromophore-chromophore interaction between adjacent trimers was weakened in the M-intermediate. The calculated difference electron density map (Fig. 6-17b) reveals a small structural change in two helices B and G which will tilt toward the inside of the trimer during formation of the M-intermediate, in accordance with the fact that the helix G contains amino acid residues responsible for proton-pumping. The time-resolved measurement was also performed during the conversion of the M-intermediate to the trans state, and no disordering was confirmed during the conversion. Muscle contracts rapidly when subjected to a low load: muscle shortening is complete within 200 msec. The sliding mechanism has been proposed for this muscle shortening [26], and experimental confirmation is required by following the structural change of muscles during shortening. A two-dimensional X-ray diffraction pattern from frog skeletal muscle was recorded on imaging plates during shortening [27]. Frog sartorius muscle was stimulated electrically for 1 s (20 Hz stimulation) at 4~ The sarcomere length was adjusted to 2.6 #m in a resting state, and the tendon end of the specimen was connected to a solenoid using a stainlesssteel thread on which a small LED was fixed. The solenoid was activated to make
389 2800
.,... m
I
~- II ,'=~
..,...
~/
mJ
mJ
t.-
c ,,...
0.02
0.085 S
(
O.
= 2:in8/~.)
(A-~]
Figure 6-17a: X-ray diffraction profiles from the purple membrane in the t r a n s state (solid line) and M-intermediate (dotted line). The exposure time was 300 sec. for each measurement.
-0. ..L
50
O. O 0 .
.
.
.
.
I
O. f O ,
.
,
Figure 6-17b" Difference electron density map. The region where the electron density increases in the M-intermediate is designated by thick lines.
the thread slack at 280 ms after a first stimulus, and then the muscle was allowed to contract (by about 16 % of its length) until the thread was taut. The length of the muscle was monitored by the LED. The load including the weight of the thread and LED, was small (about 1 g) in comparison with the tension developed in the specimen (300 ~ 500 g), and the specimen was supposed to contract virtually with no load with a velocity of about 6 ~m/sarcomere/s. The contraction lasted for about
390 7 ms. The X-ray diffraction was taken at 5 different phases with 300 ms intervals; (1) in a resting state before contraction, (2) during isometric contraction before shortening, (3) during shortening, (4) during isometric contraction 330 ms after shortening, and (5) during isometric contraction 630 ms after shortening. A shutter was opened for 30 ms in each phase to take diffraction patterns. The diffraction patterns were accumulated from 20 contraction experiments for each specimen, so that the exposure time amounted to 600 ms on the respective imaging plates. The measurements were repeated four times with flesh muscle specimens, and the diffraction patterns were summed to produce an X-ray diffraction pattern shown in Fig. 6-18.
Figure 6-18: An X-ray diffraction pattern from frog sartorius muscle during unloaded shortening. A total exposure time was 2.4 sec. The intensity and spacing changed during shortening as follows. (1) The intensity of the (1,1) equatorial reflection decreased during shortening, and that of the (1,0) reflection also showed a slight decrease, making the intensity ratio slightly increase. (2) The 14.3 nm and 7.2 nm meridional reflections decreased in intensity. The reflections moved away from the origin by about 5 % during shortening. (3) The off-meridional part of the myosis layer-lines exhibited no change in intensity. (4) The meridional reflection at 1/21.4 nm -~ increased in intensity during shortening. (5) The intensities of the actin layer-lines at 1/5.9 and 1/5.1 nm -1 decreased during shortening. The results support the sliding mechanism for muscles.
391 6.3.2. Time-resolved small-angle
X-ray scattering from
solutions
Synchrotron radiation affords a powerful source of X-rays for solution smallangle scattering, and its high intensity has opened up the possibility of timedependent scattering measurement of solutions for the investigation of the kinetics of biologically important transient phenomena [28,29]. This method can be applied to studies of enzymatic reactions, assembly/disassembly of biological systems, denaturation/renaturation, and phase transitions, which has been studied spectroscopicaUy so far by techniques such as stopped-flow rapid mixing, temperature-jump, and flash photolysis. The advantage of small-angle X-ray scattering (SAXS) over spectroscopic methods may be that the SAXS reflects directly the structure of a macromolecule itself. A combination of SAXES and the CAMAC data acquisition system described in 6.2.1 and 6.2.2. is well suited for the purpose of realizing time-dependent measurements of SAXS. The data acquisition scheme for stopped-flow and temperaturejump methods is shown in Fig. 6-19. All scattering data were corrected for variation in intensity of the primary beam which is monitored by an ionization chamber in front of the specimen. mixing
mixing time
1' _
time
'
;J /li~
i
Eu~
g' .1time frame =1
2
3
n-|
n
(nn~x= 64)
Figure 6-19: Data acquisition scheme in SAXES at BL-10C. The SAXS profiles of each time frame are recorded during the course of reaction in 512 channels of histogram memories which carry a total of 32 K words with a depth of 24 bits. A total of 64 time frames can be recorded in a single experiment. The shortest interval of reaction time recorded is less than 1 ms.
392 Stopped-flow study of denaturation and aggregation of bovine serum albumin [30]
Bovine serum albumin (BSA) is a protein of molecular weight 69,000 daltons. It is easily soluble in water and transports fatty acid, bile pigments, drugs, etc. in the blood. Each BSA molecule has 17 disulfide bonds. Cleavage of the disulfide bond by dithiothreitol(DTI') induces denaturation (unfolding) of the protein and aggregation of unfolded BSA. The BSA-DTI" reaction was performed by rapid mixing with a stopped flow apparatus [31]. The scheme for the stopped flow measurement is shown in Fig. 6-20. The SAXS intensities were recorded in each time frame immediately after mixing of the BSA and DTF solutions with intervals of 50 s. The concentration of BSA in the reaction mixture was either 0.5 % (w/v) or 1% (w/v), i.e. either 1% or 2 % BSA solution was mixed with an equal volume of buffer solution containing DTr. . J Data Aquisilion
l' s2 II
-
L_l
II ~
-- N2 gas
Figure 6-20: Outline of the stopped flow rapid mixing apparatus used with SAXES at the Photon Factory, National Laboratory for High Energy Physics, Tsukuba. R1 and R2 are reservoirs for reactant solutions; S 1 and $2, a pair of syringes; M, the mixing chamber. The chamber between the optics and PSPC (position-sensitive proportional counter) is the specimen chamber. A trigger pulse immediately after mixing initiates the data acquisition system. Actions of valves and syringes are controlled by pressure of N2 gas.
The time-course of the change in the radius of gyration is plotted in Figs. 6-21 and 6-22 for 0.5 and 1% BSA solutions respectively: the initial stage has an Ra value of 29 .~ (the same value as the native BSA), the final equilibrium having an Ra value of about 60 A. As seen in Figs. 6-21 and 6-22, I(0) values increase 4 to 5 times in both cases, where I(0) is proportional to the molecular weight (see eqn. (6.47)). Thus, an increase in the radius of gyration is attributable to the aggregation of partially or fully reduced BSA monomers, probably as a result of unfolding of
393 tnll(o]
Rg (1)
I(OI Rg(~) 8 70
tn
60
?
50
z,O
,o - - d l 6
:':I/t/
9 30 ..' .
~,n,,,o~
~ tnll(0)l
20 .~t i |
|
250 time
500 (sec)
i
0
750
250 time
500 (sec)
750
Figure 6-21" Plots of radius of gyration and ln[I(0)] for 0.5% BSA solution, obtained from Guinier plots. The time delays in the initial rise are unclear Figure 6-22: Plots of radius of gyration and In[I(0)] for 1.0% BSA solution, obtained from the Guinier plots of time resolved measurement of small-angle scattering profiles. The time delay in the initial rise is very obvious in these plots. the compact form by cleavage of disulfide bonds in the monomers. This reaction can be represented by a number of equilibria as follows: A~
Ak + lA' ~
A'
Ak+t (k, l = 1, 2, 3, ...)
(6-62)
where A is a monomeric BSA in the native state and A' the monomer with partiaUy unfolded structure. The second equation denotes a number of reactions: this equation indicates that a k-meric A particle reacts with l moles of A' resulting in a (k +/)-meric A particle, for example. Certainly, k-meric and/-meric particles react to produce (k +/)-meric A particles. We do not have available parameters such as the equilibrium constants of such reactions in the final stage of the reaction mixture, except for the behavior of the I(0) values. Therefore, the numbers of monomers involved in aggregation are estimated to be at least four for the particles of the 0.5 % BSA-DTI' mixture and five for the 1% BSA-DTI' mixture, assuming that the equilibrium is completely to the right and that the degree of hydration does not change upon aggregation of monomers.
394 A delay in the initial increase in the radius of gyration and ln[I(0)] for 0.5 % BSA is not clear. On the other hand delay is evident for the 1 % BSA reaction mixture, and is about 150 s. This delay is attributable to the molar ratio of [DTr]/[BSA] in the solutions. The disulfide bonds in proteins are classified into three groups, depending on their relative activity: fully exposed (reactive with 0.5 mM dithioerythritol), partly buried (reactive with 10 mM dithioerythritol), and buried (unreactive in the native state without denaturing reagents). For complete reduction, a 50-fold molar exess of D T r over the number of moles of disulfide bonds in the protein molecule is necessary, in addition to a denaturing reagent such as urea or guanidine hydrochloride to unfold the polypeptide chain and expose all the disulfide bonds. Even for limited reduction, about a 60-fold molar excess of D T r is required over the number of moles of 7a-immunoglobulin, and this amount is dependent on the nature of proteins. In the present reaction mixture, a 218- or 110-fold molar excess of D T r was added with respect to the number of moles of disulfide bonds in 0.5 and 1.0 % BSA, without a denaturating reagent. Such an amount of D T r seems to be about the critical concentration, resulting in a definite time being necessary in the first reaction of eqn. (6-62) and a delay time occurs in the 1.0 % BSA solution as compared with that of the 0.5 % BSA solution. Therefore, the SAXS results obtained from 0.5 and 1 % BSA-DTr reaction mixtures differ from one other in the delay time of the reaction, leaving the rest of the timecourses of the reactions approximately parallel. A 218-fold molar exess of D T r is certainly sufficient to unfold the polypeptide chain of 0.5 % BSA by reducing fully exposed or partly buffed disulfide bonds, without an appreciable reaction time, and aggregation starts immediately after mixing of the solutions. On the other hand, a l l0-fold molar excess of D T r in the 1 % BSA solution is probably insufficient to reduce the minimum number of disulfide bonds to unfold the polypeptide chain and cause aggregation. About 150 s after mixing, D T r reduces a sufficient number of disulfide bonds to induce aggregation of BSA monomers. After the initial rise in the reaction, the concentration of BSA(I%) accelerates the aggregation process so that it tends to proceed a little faster than that in the 0.5% BSA reaction mixture. Temperature-jump study of the association process of tobacco mosaic virus protein
[32] Tobacco mosaic virus is a rod shaped virus composed of RNA and coat proteins. The coat protein (TMVP) self-assembles to form a variety of aggregates of virus-like rods without RNA [33], which depends on the pH, ionic strength, temperature and protein concentration. Fig. 6-23 shows the scheme for the association
395 K§ K- 2 <
~%~.
>
,
>
<--
/~
oee
O0
9
~C I ~/~-JJ/JJJJJJJ
A
B
C
D
I
Figure 6-23" Scheme for temperature-induced self organization of TMVE The reaction was not sequential but a random association-dissociation mechanism. The following assumptions are made for the reaction. (1) Formation of the 20 S disk from A-protein is very fast, to set up an equilibrium between them; (2) further association to form the stacked disks (short rods) proceeds by stacking of the disk or rods with the single velocity constant of k+2, and the velocity of dissociation of the rod to shorter rods or the disk is k_2. Formation and dissociation of the disk or rods is interconvertible, i.e. 1 double-layer disk and 4-layered rods form a 6-layered rod; one 10-layered rod disassembles into 1 double-layered disk and an 8-layered rod or 4-layered rod and a 6-layered rod, etc. process of TMVE At low temperature TMVP forms a small aggregate (A-protein, Fig. 6-23 (a)) comprising 4 to 6 monomer subunits. As the association force of TMVP is entropic (hydrophobic) [34], raising the temperature forms tobacco mosaic virus-like rods (Fig. 6-23 (b) to (i)) via stacking of intermediate double layer disks whose radius of gyration, Ra, is 66.5 /1 [35] (20S disk, Fig. 6-23(b)). A kinetic study of the association process was performed using a temperature-jump apparatus [36]. The scheme for the temperature-jump measurement is shown in Fig. 6-24. The present purpose is to determine the kinetic parameters of the equilibrium constant K~, the association rate constant of k+2 and dissociation constant k_2 (see Fig. 6-23). Fig. 6-25 shows a result of the time resolved temperature-jump (5~ to 25~ SAXS experiment of TMVP. Fig. 6-25 (a) shows the Guinier plots with time interval of 0.8 s for each frame, with 7.5 seconds of accumulated measuring time.
396
....
I~ ~[~,_
:~WAT~a Ic'~cU~T'~ ~J~il
NZ GAS--"
--r--
~ ~ EXCHANGER
c'~w~!~
DATA ACvQUISITIONJ"
#~
l c~176 ~ I
Figure 6-24" A schematic diagram for time-resolved temperature-jump measurements used with SAXES at the Photon Factory. Sample solution S of temperature T1 is injected by the pressure of N2 gas into the heat exchanger kept at the temperature T2. A trigger pulse opens the electromagnetic valve and initi.~tes the data-acquisition system simultaneously.
!
110
E o
86
I ;l
I
I
Rg,z(Z) 9O
a % u ~
o
6.2
80
-~ 3.8 f--
70
1.4
-~.o
0
I
0.8
I
I
1.6 2.4 Q2 x 103 (s
I
3.2
4.0
60
0
!
8
I
16
Time(sec)
Figure 6-25" Result of temperature-jump study of TMVP (jump from 5~ to 25~ (a) Guinier plots of the time course of temperaturejump. (b) Least squares fit of the radii of gyration. Solid line: estimated value analysed by the method written in the text. :experimental value obtained from Figure 6-25 (a).
!
24
397 Fig. 6-25 (b) shows the time dependence of the z-average radius of gyration. A kinetic study combined with SAXS cannot be analysed simply by the conventional spectroscopic methods: There, the characteristic absorbance of intermediates or final products measured by spectroscopic methods is proportional to their concentrations. In contrast, the scattered intensity of each time frame is the sum of the constituent components, because the system is not monodisperse but polydisperse. The observed intensity I(q) from a transient solution composed of L components is the sum of each component in the solution, L
I(q) = ~ Z
L
niJi(q) = a Z
i+1
niJi(O)exp(-R2aiq2/3)
(6-63)
i=1
where ni is the number of each component particle, and Ra~ is the radius of gyration of the ith particle of L components. Thus, the radius of gyration obtained from the Guinier plots corresponds to the z-average and Y(O)/c is proportional to the weight average molecular weight Mw. For analysis of the kinetic SAXS experiment, let us describe the association process of TMVP rod as an example. To obtain the kinetic parameters, K~, k+a and k_2, combine the following equations. 1) Conservation of total monomer concentration. Total monomer concentration CT should be conserved. Thus, CT = hA" a(t) + nDi y ~ di(t)
(6-64)
i=1
where nA is the number of components of A-protein, no~ the number of components of i stacked rods of 20 S disk: a(t) is the concentration of A-protein at time t, and di(t) the concentration of the i stacked disk with maximum stacked number m. 2) Equilibrium between A-protein and 20 S disk. The transition between A-protein and 20 S disk is very rapid; this process is expressed in terms of an equilibrium of A-protein and 20 S disk, specified by an equilibrium constant Ka: d l / a " = K1
(6-65)
Here d 1 and a denote the molarities of the 20 S disk and A-protein, respectively. 3) Time dependence of the average radius of gyration. The radius of gyration at time t obtained from experiment is the z-average of the proteins existing in the solution: R ~ = a(t) . na . R~A + Eim=l di(t) " rtbi . R2GDi (6-66) a(t) " n2A "~- Eim=l di(t) . nDi
398
RGA, RGDi denote the radius of gyration of A-protein and/-stacks of the 20 S disk, respectively. 4) Kinetic equations of association and dissociation of rods. A change in the concentration of the/-stacks of the 20 S disk, di(t) where
m
d(di)/dt = k+2
m
m--1
di.dj_l~-k_2 ~ di .(i-2 1).k-2.di-k+2.di ~ di (6-67) j+i+l j+l l<_i~_j/2
By use of least-squares fitting, optimized values of K1, k+2 and k_2 were determined by solving eqn. (6-67), with the constraint of eqn. (6-64), to yield the best fit of the mean-square radius of gyration to the observed value for each timeframe. Fig. 6-25 (b) shows an example of the calculated radius of gyration with estimated rate constants from eqn. (6-67) and the experimentally estimated value as a function of time. The kinetic parameters evaluated for the TMVP assembly process are shown in Table 6-3. In the case of Fig. 6-25 (b), where the buffer concentration of 100mM at pH 7.2 jumped from 5~ to 25~ estimated values of
Table 6-3" Kinetic parameters for TMVP assembly process obtained by temperature-jump SAXS measurements at pH 7.2 Protein Buffer conc conc (mg/ml) (mM)
At (s)
log K1
k+2M-ls -1
k_2sq
(xl0 4)
(xl0 -1)
20013
5.0
1'
12.0
50
12.0
360
28.32
0.0857 0.1457
5.0
100
5.0
150
32.03
0.4984
5~ 250C
T 5"I;
50 40.0
Final time (s) 1200
0.1035
12.0
100
5.0
150
28.12
1.2719 0.3914
5.0
50
4.0
120
31.22
1.4894 0.3156
12.0
50
2.5
75
29.09
0.8423 0.3530
5.0
100
1.2
36
32.89
8.8512 0.5361
12.0
100
0.8
24
31.82
3.5760 0.5203
At denotes the measuring time in each time frame, and the final time the total measuring time for 30 time frames.
399
kinetic parameters a r e K1 -- 663 x 1028, k+2 = 3.58 x 10 - 4 M - 1 s - 1 , k _ 2 - 0.52 x 10-1s -1. The k+2 value of around 104M-l-S-1 in the present system is lower than that expected for normal diffusion-controlled association reactions [37], and falls in the lowest range of protein-protein interaction systems. This result could suggest that rod elongation of TMVP is due to disk stacking, which would not proceed as rapidly as formation of oligomers comprising only a small number of subunits. The analytical procedure proposed here is applicable not only to the temperature -jump method but also to other kinetic small-angle scattering experiments, for example those using the stopped-flow technique.
Time-resolved smM1-angle X-ray scattering from gelling system Polymer gel is specified by the three-dimensional network structure formed by a small amount of interchain (either chemical or physical) links. This network structure can be characterized by the spacial correlation of a scale from 10 to 100,000 .~. The SAXS is expected to reveal the spacial characteristics of gel in some extent [38], and thus is applied to the systems of various gels [39]. An SR X-ray source of extensive intensity has an advantage in measuring the excess scattered intensity of small-angle X-rays from solutions within a minute. An attempt has been made to observe the network formation in real time by SR SAXS in the system undergoing gelation by end-crosslinking of two-functional and four-functional monomers [40]. The stoichimetric mixture of the 2-functional monomer [ 1,4 bis(1,1,3,3-tetramethyl- 1,3-disilya-2-oxa-4-pentenyl) benzene (VTM)] and the 4-functional monomer (crosslinker) [tetrakis (dimethyl siloxy) silane (F4-C)] in toluene (20 wt%) yields randomly branched polymers with a broad molecular weight distribution and eventually forms gel by hydrosilylation according to the reaction scheme H CH3-S~i-CH 3 CH 3 C H - ~ I I "'J~ CH2 = C H - S i - O - S i - " ( \ /)"I I ~k~.__.~ CH 3 CH 3
VT-M,_3
CH 3 CH 3 I I Si- O-S -CH=CH,,+ I I ': CH 3 CH 3
cH~ o cH~ .-si-o-si-o-si-, ' CH 3
d
'
CH 3
H2PtCI66H20
Gel
Toluene
CH3-Si-CH 3 H F4-C
where the catalyst H2PtCI6.6H20 is added. The time-resolved SAXS was observed from the system of VT-M/F4-C toluene solution encapsulated in a glass capillary of r = 2 mm with the SAXES optics (see Fig. 6-6). The scattering from gel is considered to be caused by the constituent polymer chains and/or the spatial inhomogeneity due to the statistical fluctuation of
400 density constrained in the three-dimensional architecture of the gel. The scattering from the constituent chains can be formulated in terms of the Debye equation for non-interacting particles dispersed in the medium: n
71
I(q) = Z ~ fifj sin(qrij_______~) i=1 j=l
(6-68)
qrij
with f~ being the scattering length of the ith unit and rij the distance between the ith and jth units. The sum is extended over all pairs of units, and is calculated in the simplest case of f-functional polycondensation as oo
I(q) c< 1 + ~ f(f - 1)n-lolnq~ n
(6-69)
n=l
r = exp(-q2b 2/6)
(6-70)
where a is the fraction of reacted functionalities corresponds to the mean-square distance between weight-average degree of polymerization diverges I(q) remains finite as far as ( f - 1)ar < 1. That is,
1.0 r
10000
1000 i"
sol 9
O
(b) o
9
%
gel
sol
9 gel
100
oo
r
0
"
(a)
0.8
0.6
on a monomeric unit, and b2 adjacent units. Although the at a gel point c~ = 1 / ( f - 1), the scattering will be observed
0.4 10 r 0.2
0.0 0.00
1 0.05
0.10
q (A-1)
0.15
0.20
0.001
........ 2
4
I 68
........ 2
4
I 68
0.01
........ 2
0.1
q (Al)
Figure 6-26: The scattered intensity profile of the 4-functional (f = 4 in eqn. (6-68)) randomly branched polymer system before and after gelation (a = 0.332 and 0.334, respectively, where b is fixed at 5 ~). (a) Kratky plots; (b) double logarithmic plots.
4
68
1
401 over a whole range of q before gelation, while the scattering intensity diverges to infinity at q satisfying ( f - 1)aq~ = 1 when gel is formed. The scattering behavior from the gelling system is demonstrated in Fig. 6-26, where the sharp upturn of the scattering intensity at q ~ 0 is characteristic of gel formation. The system of VT-M/F4-C exhibited the upturn of q2I(q) as q ~ 0 due to the gel formation by end-crosslinking. Fig. 6-27 shows the time-course of the SAXS profile in Kratky plots. The gel point lies around 7900 s at 37~ although the exact gel point should be less than 7900 s considering the present resolution of the SAXES equipment. A similar tendency of the divergence in the Kratky plots was observed in other reaction temperatures, where the gel point was found to become shorter in terms of the gel time as increasing the reaction temperature. The result suggest that the network formation seems to follow the classic scheme of Flory-Stockmayer [41].
Figure 6-27: 3-dimensional Kratky plots of time-resolved SAXS during polysiloxane network formation at 37~
402
REFERENCES
0
3. 4. 5. 0
o
8. 9. 10. 11. 12. 13. 14.
15. 16.
17. 18.
19.
L. D. Landau and E. M. Lifshits, Quantum Mechanics, Pergamon, Oxford, 1985. P. Debye, Ann. Phy. (Leipzig), 28, 809 (1915). P. Debye and A. M. J. Bueche, J. Appl. Phys., 20, 518 (1949). G. Porod, Kolloid Z., 125, 51 (1951). A. Guinier and G. Fournet, Small-Angle Scattering of X-Rays, Wiley, New York,1955. L. A. Feigin and D. I. Svergun, Structure Analysis by SmMI-Angle X-ray and Neutron Scattering, Plenum, New York, 1987. Y. Hiragi and S. Ihara, Acta Cryst., A37, 378 (1981). P. Debye, J. Phys. Colloid Chem.,51, 18 (1947). A. Guinier, Ann. phys. (France), 12, 161 (1939); see also Ref. 5 p26. O. Kratky, G. Porod and L. Kahovek, Z. E1ectrochem., B55, 53 (1951). J. Plegtil, Makromol. Chem. Macromol. Syrup., 15, 185 (1988); J. Ple~til, H. Posp~gil, Yu. M. Ost~evich and G. Degovics, J. Appl. Crys., 24, 659 (1991). M. Mimura, S. Yamanaka, H. Urakawa and K. Kajiwara, Bull. Inst. Chem. Res. Kyoto Univ., 69, 199 (1991). T. Ueki, Y. Hiragi, Y. Izumi, H. Tagawa, M. Kataoka, Y. Muroga, T. Matsushita and Y. Amemiya, Photon Activity Report, 1, V7, V29, V170 (1983). H. Hashizume, K. Wakabayashi, Y. Amemiya, T. Hamanaka, T. Wakabayashi, T. Matsushita, T. Ueki, Y. Hiragi, Y. Izumi and H. Tagawa, Design of Diffractometer for Small-Angle Scattering (The Muscle Diffractometer) at Photon Factory, KEK Internal 81-11, National Laboratory for High Energy Physics, Tsukuba, 1982. M. Sonoda, M. Takano, J. Miyahara and H. Kato, Radiology (Easton, Pa.), 148, 833 (1983). Y. Amemiya, N. Kamiya, Y. Satow, T. Matsushita, K. Wakabayashi, H. Tanaka and J. Miyahara, in A. Bianconi and A. C. Castellano (Editors), Biophysics and Synchrotron Radiation, p.61, Springer, Berlin, 1987. Y. Amemiya, Y. Satow, T. Matsushita, J. Wakabayashi and J. Miyahara, Topics in Current Chemistry, 147, 121 (1988). G. Eisner, H. G. Zachmann and J. R. Milch, Makromol. Chem., 182, 657 (1981); G. Eisner, M. H. J. Koch, J. Bordas and H. G. Zachmann, Makromol. Chem., 182, 1263 (1981). G. Eisner, Ch. Riekel and H. G. Zachmann, Adv. Polymer Sci., 67, 1 (1985).
403 20. S. R0ber, P. BiSsecke and H. G. Zachmann, Makromol. Chem. Macromol. Syrup., 15, 295 (1988). 21. T. Kojima, A. Uemura, K. Chikaishi and Y. Amemiya, Photon Factory Activity Report, 8, 310 (1990). 22. H. Yao, I. Hatta, R. Koynova and B. Tenchov, Photon Factory Activity Report, 9, 251 (1991). 23. S. Ueno, T. Takeda, S. Komura, H. Seto and Y. Amemiya, Photon Factory Activity Report, 9, 254 (1991). 24. T. Hamanaka, Y. Kito, M. Seidou, K. Wakabayashi and Y. Amemiya, Photon Factory Activity Report, 6, 124 (1988); T. Hamanaka, Y. Kito, M. Seidou, K. Wakabayashi and Y. Amemiya, Photon Factory Activity Report, 8, 296 (1990); T. Hamanaka, Y. Kito, M. Seidou, K. Wakabayashi and Y. Amemiya, Photon Factory Activity Report, 9, 252 (1991). 25. M. Nakasako, E Tokunaga, M. Kataoka and Y. Amemiya, Photon Factory Activity Report, 7, 122 (1989); M. Nakasako, M. Kataoka and E Tokunaga, FEBS Left., 254, 211 (1989); M. Nakasako, M. Kataoka, Y. Amemiya and E Tokunaga, FEBS Left., 292, 73 (1991); F. Tokunaga, M. Nakasako, M. Kataoka and Y. Amemiya, Photon Factory Activity Report, 9, 258 (1991). 26. Y. Harada, K. Sakurada, T. Aoki, D. D. Thomas and T. Yanagida, J. Mol. Biol., 216, 49 (1990). 27. N. Yagi, S. Takemori, M. Watanabe and Y. Amemiya, Photon Factory Activity Report, 9, 265 (1991). 28. R. S. Goody and K. C. Holmes, in H. B. Stuhrmann (Editor), Use of Synchrotron Radiation in Biology, p.203, Academic Press, London, 1982. 29. M. H. J. Koch, H. B. Stuhrmann, A. Tardieu and P. Vachette, in H. B. Stuhrmann (Editor), Use of Synchrotron Radiation in Biology, p.223, Academic Press, London, 1982. 30. T. Ueki, Y. Hiragi, M. Kataoka, Y. Inoko, Y. Amemiya, Y. Izumi, H. Tagawa and Y. Muroga, Biophys. Chem., 23,115-124 (1985). 31. T. Nagamura, T. Kurita., E. Tokikura and H. Kihara, d. Biochem. Biophys. Methods, 11, 277-286 (1985). 32. Y. Hiragi, H. Inoue., Y. Sano, K. Kajiwara, T. Ueki and H. Nakatani, 3. Mol. Biol., 213, 495 (1990) 33. P. J. G. Butler and A. C. H. Durham Adv. Protein Chem. 31, 187 (1977). 34. M. A. Laufer, Entropy Driven Processes in Biology, Springer, New York, 1975. 35. Y. Hiragi, H. Inoue, Y. Sano, K. Kajiwara, T. Ueki, M. Kataoka, H. Tagawa,
404 Y. Izumi, Y. Muroga and Y. Amemiya, J. Mol. Biol., 204, 129 (1989). 36. Y. Hiragi, H. Nakatani, K. Kajiwara, H. Inoue, Y. Sano and M. Kataoka Rev. Sci. Instrum., 59, 64 (1989). 37. R. Koren and G. G. Hammes, Biochemistry, 15, 1165 (1976). 38. K. Kajiwara, S. Kohjiya, M. Shibayama and H. Urakawa, in D. DeRossi, K. Kajiwara, Y. Osada and A. Yamauchi (Editors), Polymer Gels; Fundamentals and Biomedica/Applications, p.3, Plenum, New York, 1991. 39. See, for example, J.-M. Guenet, Thermoreversible Gelation of Polymers and Biopolymers, Academic Press, London, 1992. 40. T. Ando, S. Yamanaka, S. Kohjiya and K. Kajiwara, Poly. Gel. Network., 1, 45 (1993). 41. See, for example, P. J. Flory, Principles of Polymer Chemistry, Comell U.P., Ithaca, 1953.
Applications of Synchrotron Radiation to Materials Analysis H. Saisho and Y. Gohshi (Editors) 9 1996 Elsevier Science B.V. All rights reserved.
405
CHAPTER 7
THE RIETVELD METHOD AND ITS APPLICATIONS TO SYNCHROTRON X-RAY POWDER DATA Fujio IZUMI National Institute for R e s e a r c h in Inorganic Materials 1-1 N a m i k i , T s u k u b a , Ibaraki 305, Japan
7.1. INTRODUCTION Since the discovery of superconductivity in the KaNiF4-type oxide (Lal_xBax)2CuO4 in 1986 [ 1], numerous papers concerning Rietveld refinements of high-To superconductors and related compounds have been published, chiefly in the field of physics [2,3]. Information about the average and defect structures of the superconductors reported in them was indispensable to searches for new copper oxides and understanding the mechanism of carder doping. Therefore, they have been noted and cited repeatedly by researchers of superconducting copper oxides. The Rietveld method [4] had been well known, prior to 1986, to some crystallographers, physicists, chemists, mineralogists, and materials scientists studying metals and inorganic compounds. However, it became much more popular after superconductivity "fever" spread round the world, and has now been recognized as a fundamental technique of characterization in solid-state science. 7.1.1. Raison d'etre of the Rietveld method
The Rietveld method is a technique for refining structure and lattice parameters directly from whole X-ray or neutron powder diffraction patterns without separating peaks contained in them [4-6]. In contrast with single-crystal diffraction, the projection of the three-dimensional reciprocal lattice onto the single dimension of a powder-diffraction pattern leads to a serious loss of structural information. Solving phase problems in compounds with unknown structures are, therefore, very difficult with powder-diffraction data unless the compounds have fairly high symmetry and contain few atoms in their asymmetric units. However, once structural
406 models can be constructed by some means, the Rietveld method is available as a most powerful procedure for structure refinements. Many scientists, including even crystallographers, still have a preconceived idea that the Rietveld method should be applied only when single crystals cannot be grown or when twins are inevitably formed during crystallization processes and phase transitions. Such an idea is judged to be too superficial from the standpoint of materials science. Most metal and inorganic materials, e.g., intermetallic compounds, metal hydrides, solid-state ionics, superconductors, zeolites, catalysts, inorganic ion exchangers, and ceramics, are polycrystals. The crystal (defect) structures of single crystals may differ to some extent from those of polycrystalline materials. As described above, the powder method always suffers from the disadvantage that an appreciable amount of structural information is lost owing to the overlaps of diffraction peaks. However, it possesses several advantages over the single-crystal method: (i) easy preparation of polycrystalline samples, (ii) simple procedures for measurements, (iii) the ease of in situ diffraction experiments in special sample environments (highfiow temperature, high pressure, etc.), and (iv) negligible secondary-extinction effects.
7.1.2. Amazing ability of the Rietveid method The Rietveld method substantially contains the following data-processing procedures: (i) separation of overlapping peaks in diffraction patterns, (ii) separation of K~I and K~2 peaks when using characteristic X-rays, (iii) background subtraction, (iv) refinement of lattice parameters, (v) refinement of structure parameters (fractional coordinates, occupation factors, and thermal-displacement parameters), (vi) correction of preferred orientation, (vii) determination of mixing ratios, (viii) identification of impurity peaks, (ix) indexing of reflections, and (x) determination of integrated intensities, full-widths at half-maximum intensities (FWHM), and peak positions. The Rietveld method, in which these complex calculations are executed simultaneously, is an exquisite technique worthy of being the ultimate method for the analysis of powder-diffraction data. It owes its dramatic development to the spread and improvement of computers in recent years. The Rietveld method is widely applicable to metals, inorganic compounds, and organic compounds of low molecular weights only if they are crystalline. Conventional X-ray powder diffractometers using characteristic X-rays are changed into high-performance machines with which both structure and lattice parameters can be refined accurately by the Rietveld method. The combination of X-ray powder diffraction and Rietveld refinement can provide us with much more reliable information about average structures than the direct observation of crystal structures by high-resolution transmission electron microscopy (HRTEM) and analysis of extended X-ray absorption fine structures (EXAFS). When high-resolution powder-diffraction data are measured using synchrotron or neutron sources, structure parameters can be obtained which are comparable in accuracy to those determined by the single-crystal method using four-circle goniometers.
407
7.1.3. Applications in analytical science The starting point of all studies on crystalline materials is to learn their crystal structures and chemical compositions accurately, and also the relationships between them. Synchrotron X-ray powder diffraction [7-10] is particularly useful for the analysis of complex structures because of its extremely high resolution. Rietveld analysis can be regarded as the method for a kind of state analysis by which the positions, thermal displacements, and occupancies of atomic sites in crystalline materials are quantitatively determined. The principle of the Rietveld method can be applied not only to elastic powder diffraction but to other spectroscopic techniques that produce complex spectra containing overlapping peaks. Furthermore, precise determination of compositional ratios in mixtures is possible with this method (cf. Section 7.13) [11,12]. The introduction of basic information on the Rietveld method is, therefore, important for researchers and students of analytical science. In this chapter, emphasis is placed upon the various calculations involved in Rietveld analysis. An excellent review article on the Rietveld method was written by Albinati and Willis [6] in 1982. In addition, a book dealing with all the aspects of this method [13] was written by invited lecturers at the international workshop on the Rietveld method, held at Petten where it was invented in 1967. Those who wish for more detailed information about Rietveld analysis are strongly recommended to read these articles in addition to this chapter. 7.2. ELEMENTARY INFORMATION ON THE RIETVELD METHOD
7.2.1. Principle Before the Rietveld method was introduced, structure parameters were refined from diffraction data using integrated intensities of respective reflections obtained by the curve fitting of overlapping peaks. This procedure is effective when dealing with those compounds of high symmetry and simple structures which display relatively few diffraction peaks. However, it is no longer available when peaks overlap so heavily that they cannot be separated by curve fitting [ 14]. In the Rietveld method [4], a set of variable parameters, x (Xl, x2, x3...... ), that represent powder-diffraction patterns are refined by fitting the calculated powder pattern to the observed one by a nonlinear least-squares method (cf. Section 7.9). In other words, the sum of weighted squares of residuals
S(x) : E w i [yi- fi(x)] 2 i
(7-1)
is minimized by adjusting x. In this equation, i is the step number, Wi (= 1/yi) is the weighting based on counting statistics, yi is the observed intensity, and fi(x) - f(Qi; Xl, x2, x3 ...... ) is the calculated intensity, Q is the magnitude of the scattering vector Q (Q = I Q I = 2x/d = 4xsin0/;L,
408 14400
.
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.
.
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20/~ Fig. 7-1. Observed, calculated, and difference patterns of fluorapatite (Cu Ks). The solid line is calculated intensities, and small crosses superimposed on it are observed intensities. The difference between the observed and calculated intensities are shown by points appearing at the bottom. Tick marks below the profile indicate the positions of all allowed Ko~I and Ko~2peaks.
d is the lattice-plane spacing, 20 the diffraction angle, and A the X-ray or neutron wavelength). Either 20 (angle-dispersive method) or energy E (energy-dispersive method) may be the experimental variable. The sum is over all data points in the powder diffraction data. Figure 7-1 exemplifies the result of a Rietveld refinement for fluorapatite using angle-dispersive X-ray diffraction data. The model function f/(x) contains structure parameters as a part of the parameter vector x; details of fi(x) will be described in Section 7.3. Because structure parameters are refined using the whole diffraction pattern in the least-squares calculation, maximum structural information can be directly extracted from the powder pattern without any pre-processing.
7.2.2. Development and spread of the Rietveld method The above idea is very simple but was a creative achievement that seemed impossible until it was actually tried and confirmed to be effective by Rietveld [ 14,15] in 1967. He originally devised it for the analysis of constant-wavelength neutron-powder data (he was a researcher at
409 a reactor center at Petten in the Netherlands). Since then, many neutron-diffraction data have been measured using reactor neutron sources and analyzed with a FORTRAN program developed by Rietveld [4], and its extended version. Up to 1977 the number of crystal and magnetic structures analyzed by this method reached as high as 172 [5]. In that year, Malmros and Thomas [ 16] first applied the Rietveld method to the analysis of X-ray diffraction data. Many researchers made this pioneer work an occasion to start studying Rietveld analysis with X-ray powder data [6]. The structural information extractable from powder-diffraction data increases markedly as the resolution is enhanced because the overlapping of peaks is diminished. In any experimental system with a high intrinsic resolution, it is always possible to trade intensity for resolution. Synchrotron radiation (SR) facilities [10,17] and pulsed-spallation-neutron sources [18] built in several countries made it possible to collect high-resolution powder-diffraction data over relatively short periods, and opened paths to structure refinements as good as those by the single-crystal method. Structure refinement according to the procedure that Rietveld developed had been referred to by several different terms such as profile refinement, profile fitting, and pattern-fitting structure refinement (PFSR). In 1982, the International Union of Crystallography (IUCr) adopted Rietveld analysis/method/refinement as the formal technical terms [ 19]. These names should be used when reporting the results obtained by this method. 7.2.3. Estimated standard deviations
The estimated standard deviation, r
for the jth parameter, xj, is usually evaluated as (7-2)
where Mj~1 is the diagonal element of the inverse coefficient matrix in the normal equation (cf. Section 7.9), N is the number of independent observations, P is the number of refinable parameters, and C is the number of constraints applied. In recent years, much discussion has appeared in the literature about the reliability of crj in Rietveld refinements [20]. The r calculated in the above manner are correct, provided there are no unaccounted systematic errors. However, in most Rietveld refinements, systematic errors result from the preferred orientation, inadequate profile shape and structure models, poor background fit, inclusion of unknown impurities, etc., which cause serial correlation among neighboring residuals. If there are no systematic errors unaccounted for, the calculated r is no longer a valid measure of uncertainty. Under these conditions, o'j's calculated by Rietveld refinement may be significantly smaller than those obtained by the integrated-intensity refinement of the same data set. Scott [21] proposed a method of adjusting crj's in Rietveld analysis to provide comparability with integrated-intensity refinement:
410
~j ~
- { S(x)-N+Pp}II/2 Mjj 1 1 + NB - Pc
(7-3)
where Pp is the number of parameters describing the profile (all those parameters in the model that do not directly affect the integrated intensities), Na is the number of Bragg peaks, and Pc is the number of structure parameters. Because this equation is based on the assumption that only the integrated intensities are subject to model errors, it should be applied only to structure parameters. The adjustments calculated above are only approximations because they assume that Bragg peaks are completely resolved and that the model for the peak shape is good, which are both seldom true. Nevertheless, this procedure imposes reasonable restraints on the uncritical use of trj's generated by Rietveld refinements as measures of the accuracy of the refined structure parameters.
7.2.4. Agreement indices Table 7-1 summarizes numerical criteria used to measure the agreement between the observed and calculated intensities and the progress of Rietveld refinement. RF closely resembles the R factor (= E I IFo]- [Fell / E IFol; Fo: observed structure factor, Fc: calculated structure factor) widely used in single-crystal structure analysis. In the definition of RB and RF, the notation "o" has been enclosed in quotation marks because the Bragg integrated intensities, Ik("o") are not actually observed but are derived from a parceling out of the observed net intensity, in a given 20 range, in proportion to the calculated Bragg intensities l~(c) [4]. Consequently, Ik("o") is heavily biased by the structural model, and these two R factors tend to be too optimistic [20]. Nevertheless, they are valuable indicators because they depend more strongly on the fit of the structure parameters than do the other agreement indices. The most significant R factor is Rwp because the numerator of Rwp is the quantity that is actually minimized in the least-squares refinement procedure. Both Rwp and Rp are influenced mostly by the intensity of the diffraction line as well as the background because the sum of the observed intensities is used in the denominator of these R factors. To judge the quality of the fit, the final Rwp value should be compared to the expected Rwp, Re, which is derived from the statistical error associated with the measured intensities:
Re
,, wiy ll'2
710,
l
Thus, the goodness-of-fit indicator, S, equals Rwp/Re. An S value of 1.0 indicates that the refinement is complete; it can get no better statistically. An S value of 1.3 or less is usually considered to be quite satisfactory. S includes the number of variables, P, undergoing refinement. Therefore, it may help in determining whether or not a change in P significantly decreases the residual error.
411 Table 7-1. Indices for evaluating the results of Rietveld analysis: cri is the standard deviation of the observed intensity for the ith data point (= q-~), I~("o") and Ik(c) are respectively the estimated and calculated integrated intensities for reflection k, N is the number of data points, and P is the number of refinable parameters.
E wi{Yi--ft" (X)}2 1/2
__Rwp= 9i
i
wiy?
R-weighted pattern
(7-4)
R-pattern
(7-5)
R-Bragg factor
(7-6)
R-structure factor
(7-7)
Goodness-of-fit indicator
(7-8)
Durbin-Watson d-statistic
(7-9)
~_~lYi - f i (X) l
Rp= i
EYi i II~ ("o") -Ik(c)l
RB =
k
E I~: ("o")
k
E ]Vlk("o") - VI~(c) ]
RF =
t
E Vlk ("o")
k
~iwi{Yi_fiN_p (x)}2 .]1/2 S
_..
~2 [Yi -f i(x) Yi-1 Gi--fi-ll(X)]2" d= i= •i i~lN [Yi --f i (x) ]
The Durbin-Watson d-statistic, d, is very useful for assessing the reliability of estimated standard deviations in Rietveld analysis, by providing quantitative information about serial correlation in the residuals [22,23]. Moreover, the d statistic is a sensitive measure of the progress of a refinement, and is still discriminating even when other indices fail.
412 In addition to the above agreement indices, the reliability of the refinement must be checked on the basis of final structure and lattice parameters. Occupation factors and thermaldisplacement parameters need to be physically meaningful values; interatomic distances and bond angles should be reasonable from a crystal-chemical point of view. 7.3. M O D E L F U N C T I O N The observed intensity, Y i, at a particular step, i, is modeled by the calculated intensity, fi(x), which is the sum of contributions from Bragg reflections plus background correction
Yb(Qi): fi(x) = ~(Qi)A(Qi)s E mklFkl2EkPkL(Qk)~Qi -Qk) + Yb(Qi) k
(7-11)
where t J b ( a i ) = incident intensity (corrected for the efficiency of counters if necessary), A(Qi) = absorption factor, s = scale factor for the particular phase, k = reflection number, mk = multiplicity, Fk = structure factor, E~ = correction factor for extinction (needed in time-of-flight neutron diffraction), P~ = correction factor for preferred orientation, L(Qk.) = Lorentz and polarization factors (the polarization factor is unnecessary in neutron diffraction), Qk = peak position of reflection k, q)(Qi - Qk) = profile-shape function to approximate the profile of each peak. The sum in Eq. (7-11) must be carried out over all reflections contributing to the net intensity at the ith step. 7.3.1. Structure factor
The structure factor consists of the crystal-stracture factor, Fk(cryst.), and the magnetic-structure factor, Fk(magn.):
IFk i2 = IFk(cryst.)] 2 + IFk(magn.)l 2
(7-12)
The second term is required in neutron diffraction only when compounds containing magnetic atoms such as Fe, Co, and Ni exhibit magnetic scattering in addition to nuclear scattering. Refer to Refs. 4, 24, and 25 for the parameters contained in Fk(magn.). The crystal-structure factor for reflection hkl is represented as: Fk(cryst.) = Y' gjf jTjexp[2~i(hxj + kyj + lzj)]
(7-13)
J with
Tj = exp (-Bj/4d 2) = exp [-Bj(sinO~/A, )2]
(7-14)
413 for isotropic thermal motion and
Tj = exp [-(h2fll lj + k 2f122j + t2f133j + 2hkfll2j + 2hlfll3j + 2klf123j)]
(7-15)
for anisotropic thermal motion. In Eqs. (7-13)-(7-15), j = site number, gj = occupation factor (occupation probability), j~ = atomic scattering factor, 7) = temperature factor, xj, yj, zj = fractional coordinates, B j= isotropic thermal-displacement parameter, d~ = lattice-plane spacing, 0h = Bragg angle, fll lj, f122j, f133j, fll2j, fll3j, f123j - anisotropic thermal-displacement parameters. In neutron diffraction,j~ must be replaced with the bound coherent scattering length, bcj. 7.3.2. Unit-cell dimensions Q~ can be calculated from reciprocal lattice parameters, a*, b*, c*, a*, fl*, and 7":
Q~ = 2rr,/dk = 4~sin0~/A = 2~(h2a .2 + k2b .2 + 12c'2 + 2klb*c*cosa* + 2lhc*a*cosfl* + 2hka*b*cosy*) 1/2
(7-16)
Thus, it is not (direct) lattice parameters (a, b, c, a, fl, and y) but a *a, b .2, c .2, b*c*cosa*, c*a*cosfl*, and a'b'cosy* that are actually refined in Rietveld analysis [4]. The metric tensor for the direct lattice equals the inverse matrix of the metric tensor for the reciprocal lattice:
a2 bacosy
abcosy accosfl I b2
cacosfl cbcosa
bccosa~ = c2
]
a .2
a'b'cosy* a*c*cosfl*
b'a'cosy*
b .2
c*a*cosfl* c*b*cosa*
b*c*cosa*
-1
(7-17)
C.2
The elements of the metric tensor for the direct lattice can be easily converted into the direct lattice parameters. In the Rietveld method, Qt is calculated from lattice parameters, and IF&l2 from structure parameters. Therefore, the diffraction pattern does not need to be decomposed into the respective peaks. Because the whole powder pattern is fitted by the least-squares method, lattice parameters can be refined more accurately by Rietveld analysis than by conventional least-squares refinement with only peak positions. 7.3.3. Classification of refinable parameters The calculated intensity, j~(x), contains the following five kinds of refinable parameters, x (functions in brackets are those containing each parameter): 1. Parameters to correct for integrated intensities
Table 7-2. Comparison of the Rietveld method with the two pattern-decomposition methods
Individual profileRietveld method
fitting method
Pawley method
Refine structure parameters
Refine peak position and
Refine lattice parameters
and lattice parameters
integrated intensity
and integrated intensity
Range of refinement
Whole powder pattern
Parts of whole powder pattern
Whole powder pattern
Peak position
Function of lattice parameters
Assigned to each reflection
Function of lattice parameters
Integrated intensity
Function of structure parameters
Assigned to each reflection
Assigned to each reflection
Profile parameters
Global and Q-dependent
Assigned to each reflection
Global and Q-dependent
Object of analysis
415 s, powder extinction coefficient [Ex], and preferred-orientation parameters [Pk]. 2. Parameters related to peak positions Lattice parameters [Qt] and zero-point shift of the counter [Qk]. 3. Profile-shape parameters Parameters to approximate FWHM, rate of decay, peak asymmetry, etc. [O(Qi - Qk)]. 4. Crystal-structure parameters
Xj, yj, Zj, gj, Bj, fll lj, f122j, f133j, fll2j, fll3j, and f123j [Ft(cryst.)]. 5. Magnetic-structure parameters Parameters to describe the magnetic moments of magnetic atoms [Fk(magn.)]. 6. Background parameters Parameters to describe the background
[Yb(Qi)].
The zero-point shift and background parameters are global parameters independent of phases contained in a sample, and all the other parameters are phase-dependent. 7.4. COMPARISON WITH PATTERN DECOMPOSITION Pattern decomposition is a technique to decompose overlapping peaks by a method of nonlinear least squares without refining any structure parameters [19]. This feature makes it widely applicable to powder data analysis. It is complementary to Rietveld refinement because no structural model is required. Pattern decomposition is classified into two methods: (i) individual profile-fitting [26,27] in which overlapping reflections within a relatively narrow Q range are separated and (ii) Pawley refinement [27,28] where the whole powder pattern is decomposed in one step. Table 7-2 summarizes the major characteristics of the two methods of pattern decomposition and the Rietveld method. Individual profile-fitting is position-unconstrained pattern decomposition. It can extract information about the integrated intensities, peak positions (i.e., lattice parameters), and profileshape parameters of individual reflections. Since these parameters are independently assigned to each reflection, it is difficult to separate a group of peaks overlapping very heavily with each other. This method has been carried out in a variety of fashions by many researchers. Pawley refinement is position-constrained pattern decomposition. In this method, only integrated intensities are assigned to respective reflections; the lattice, profile-shape, and background parameters are "global" ones common to the whole Q range and refined in a similar manner to the Rietveld method. The main difference between the two methods is that the integrated intensities of reflections are refined in the Pawley method whereas structure parameters contained in structure factors are directly refined in the Rietveld method. Pawley refinement has the disadvantage that peaks can no longer be separated when they come closer to each other and pass a certain limit. Because lattice parameters are refined using the whole powder pattern in the Pawley method, it has a much higher ability for pattern
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50"
60'
20/~ Fig. 7-2. Pawley analysis of synchrotron X-ray data for cimetidine [29] (by permission of Oxford University Press). The observed pattern is shown as dashes, and the calculated pattern as a smooth curve; a difference pattern is also given below. decomposition than the individual profile-fitting method. Nevertheless, overlapping peaks often have to be grouped together in high-Q regions where the density of peaks becomes very high. Information about the structure is then partly lost when structure parameters are refined using integrated intensities determined by the Pawley method. If a structural model is available for a compound, the Rietveld method utilizing that structural information can provide more accurate structure parameters than the Pawley method. Therefore, integrated intensities determined by the Pawley method are seldom used for structure refinement. Pawley refinement is often used (i) as a process of ab initio structure analysis [29] in which unknown crystal structures are solved only from powder-diffraction data (Fig. 7-2), (ii) for making d, 1/11, and hkl tables similar to those in the Powder Diffraction File, and (iii) as a convenient software tool to refine lattice parameters. 7.5. COLLECTION OF INTENSITY DATA The precision of the intensity measurement can be improved by increasing the intensities and/or the number of the steps across the peak. This is, however, only effective up to the point where the counting variance becomes negligible in comparison with other sources of
417 error. Hill and Madsen [22,30] indicated that excessively large step intensities/numbers reduce the derived estimated standard deviations of parameters to physically meaningless values. They recommend a step width 1/4-1/5 as wide as the minimum FWHM and a maximum step intensity of 5000-10000. When extracting information about the crystal structure from powder diffraction data, it is desirable to reduce FWHM's and overlapping of peaks as much as possible. If peak overlapping is too serious, the weighted sum-of-squares, S(x), does not decrease sharply near the minimum and correlations between parameters become high, which increases the possibility that refinements converge to one of the local (false) minima rather than the global minimum (true solution). The methods of measuring powder-diffraction data for Rietveld analysis fall into four classes according to the radiation (X-ray or neutron) and the abscissa (20 or energy E) specifying experimental conditions [6]. Diffraction techniques in which intensities are collected as 20 and E are varied are referred to as angle- and energy-dispersive methods, respectively. In what follows, the difference between X-ray and neutron diffraction, and the principles and features of the four diffraction techniques will be explained successively.
7.5.1. Scattering of X-rays and neutrons by atoms X-Rays are scattered by electrons, and neutrons virtually by atomic nuclei. Such a difference in the mechanism of elastic scattering leads to the following difference in the scattering ability for X-rays (atomic scattering factor f) and that for neutrons (coherent scattering length bc): 1. In X-ray diffraction, f decreases monotonously with increasing Q, whereas in neutron diffraction bc remains constant, regardless of Q. 2. As the number of electrons for an atom is increased, f increases monotonously. On the other hand, bc changes irregularly and takes positive or negative values, depending on the atomic nucleus [31 ]. The intensities of reflections with large Q are relatively higher in neutron diffraction than in X-ray diffraction because of the constancy of bc. This is favorable for collecting information about those atoms with small atomic numbers because they display marked thermal motion and theirf values decrease considerably in high-Q regions. The second characteristic of neutron diffraction is utilized for the analysis of compounds with a combination of constituent atoms which is not suitable for X-ray diffraction. Figure 7-3 shows relative scattering amplitudes of nine elements. O (atomic No.: 8, bc = 5.803 fm) has abc value 70% as large as that of Bi (atomic No.: 83, bc = 8.5307 fm); the bc values of Ti (be = - 3 . 3 0 fm) and Mn (bc =-3.73 fm) are negative; bc changes irregularly with increasing atomic weight. Evidently, neutron diffraction is very useful for refinement of the structure parameters for light elements (e.g., D, N, and O) in compounds containing heavy elements as principal constituents and for distinguishing elements (e.g., Mn and Fe, Ba and La) with comparable atomic weights. It is also probable that two atoms with bc values close to each other have considerably differentfvalues, e.g., 0 and Ba.
418 In neutron diffraction, not only nuclear scattering but also magnetic scattering is often observed in compounds containing magnetic atoms such as Cr, Mn, and Fe. This is caused by interaction between the magnetic moment of atoms with unpaired electrons in 3d or 4f orbitals and that of the neutron. Intensities of peaks due to magnetic scattering decay rapidly with increasing Q because of the extreme decrease in magnetic form factor (corresponding to bc in nuclear scattering) with Q. Reference 32 gives the coefficients in an analytical approximation to the magnetic form factors for the 3d and 4d transition series, the 4f electrons of lanthanoid ions, and the 5f electrons of some actinoid ions.
Fig. 7-3. Bound coherent scattering lengths of nine elements. An atomic number is attached to each atom. The radii of the circles are proportional to Ibl, and atoms with negative b are shaded.
7.5.2. Angle-dispersive X-ray powder diffraction In this traditional diffraction method, incident beams of characteristic or monochromatic synchrotron X-rays are scattered by samples, and diffraction intensities are measured by scanning 20 with a constant step width of 20 (e.g., A20 = 0.03~ The angle-dispersive method is used almost exclusively for collection of high-resolution intensity data for Rietveld analysis by means of synchrotron X-ray diffraction, which is one of the major subjects of this chapter. Therefore, it will be described in detail in Section 7.6. Measurement techniques using conventional X-ray sources, i.e., characteristic X-rays, are classified into: (i) the Bragg-Brentano parafocusing method [33], (ii) the position-sensitivedetector (PSD) method [34], and (iii) the Guinier method (using counters or cameras) [ 16,35,36]. The intensity distribution of incident beams, @(Qi) = ~(20i), is constant over the whole range of 20. The parafocusing geometry has a merit that A (20i) is 1/2/~ (/1, linear absorption coefficient), regardless of 20 because plate samples are used which are at the same angle to the incident and diffracted beams. Placing a crystal monochromator between the sample and counter is desirable to cut fluorescence and Compton-modified X-rays as much as possible. Even in this popular and simple diffraction method, the resolution can be enhanced by using fine-focus X-ray tubes, enlarging the radius of the focusing circle, or reducing the width
419 of the receiving slit sufficiently. However, peak broadening due to the deviation from ideal parafocusing conditions (e.g., flat-specimen, specimen-transparency, and focal-size effects) [33] and overlapping of Koc2 peaks is inevitable. Another serious problem is the presence of the K~ doublet in the incident radiation. LouSr and Langford [37] recommend a high-resolution goniometer which uses Ko~I incident radiation monochromatized perfectly by using a Johanssontype curved-crystal monochromator. Because intensity data over the whole range of 20 can be collected simultaneously and promptly by the PSD method, it is suited for studying non-equilibrium systems undergoing physical and/or chemical changes, i.e., kinetic and structural features of transformations in the solid state or the structural response of a sample to external perturbations (temperature, pressure, atmosphere, etc.). It opens up the field of real-time crystallography when combined with the Rietveld method. The Guinier method has, in principle, high resolution, and an advantage that K~I radiation is used which is strictly monochromatized by a Johansson-type crystal monochromator for incident beams. Both transmission and reflection focusing arrangements can be adopted which are suitable for collection of intensity data in low and high 20 regions, respectively. Goniometers fitted with scintillation counters having high linearities of intensity, and wide dynamic ranges, are now being used in place of cameras. The Lorentz-polarization factor for X-ray diffraction using characteristic X-rays is a function of the Bragg angle for the kth reflection, Ok, and the diffraction angle of the monochromator, 20M:
L(Qk) = L(Ok) = (1 + COS220MCOS220k)/sin20ksin Ok
(7-18)
7.5.3. Angle-dispersive neutron-powder diffraction This diffraction method [5,38-40] is very similar to the angle-dispersive X-ray diffraction method, except for the much larger scale of the diffractometer. White neutron beams from a nuclear reactor are monochromatized with a crystal monochromator. Since there is no polarization factor in neutron diffraction, only the Lorentz factor
L(Qi) = L(Ok) = 1/sin2OksinOk
(7-19)
is needed. Samples, usually in cylindrical cans made of vanadium, are irradiated with the resulting neutrons with a constant wavelength, and diffracted beams are counted with a constant step width of 20. A high-resolution powder diffractometer, D1A, utilizing the high-flux reactor at the Institut Laue-Langevin (ILL) is a representative machine [39]. Application examples are described in detail in Refs. 5 and 40. A high-resolution powder diffractometer, HRPD, was installed in 1990 at the 1G beam hole of the new JRR-3M reactor at the Japan Atomic Energy Research Institute (JAERI). In
420 this machine, a monochromator is used in which eleven pieces of Cu or Ge crystals are united, and neutron beams are collimated through three collimators (6'/12'-20'/40'-6'). To collect intensities efficiently over a 20 range of 160 ~ a set of 64 3He counters arranged at an interval of 2.5 ~ is rotated within a 20 range of 2.5 ~ The highest resolution, A d / d (Ad: FWHM), attained with this machine is as high as 2.2x10 -3. The principal advantage of this method is that correction of physical quantities dependent on wavelength is unnecessary because samples are irradiated with neutron beams of a constant wavelength. Furthermore, it is favorable for the analysis of magnetic structures [4,5,24,25] and refinement of the occupation factor that the intensities of low-Q reflections can be measured with satisfactory counting rates. However, if the wavelength is set a high value, the Q range in which intensity data can be collected becomes narrow; the resolution must counterbalance the Q range appropriately. A preferred orientation mainly occurs near the surface of a sample. Neutron beams can penetrate in most cases, because the neutrons are scarcely absorbed. Particles of the sample hardly orient along the wall of the cylindrical container. Therefore, preferred orientation is much less marked in neutron diffraction than in X-ray diffraction using flat samples. 7.5.4. TOF neutron-powder diffraction The angle-dispersive diffraction methods described in Sub-Sections 7.5.2 and 7.5.3 use X-ray and neutron beams incident upon samples successively without any intervals. On the other hand, energy-dispersive-type neutron diffraction utilizes pulsed-neutron beams generated at definite time intervals [ 18,41-44]. Injection of pulsed proton beams activated by an accelerator into a target made of W, U, etc. brings about spallation of nuclei. The resulting fast neutrons are slowed down by passage through a moderator containing a hydrocarbon such as methane, or water or hydrogen, and converted into pulsed white beams of neutrons, which are used as incident beams for neutronpowder diffraction. Counters are fixed at constant 20 positions, and neutrons diffracted by the sample are counted with multi-channel time analyzers as a function of the elapsed time (time-of-flight: TOF), t, after the target has been hit by the pulsed proton beams. The time required to travel from the target to the counter is proportional to the wavelength of the neutron; thus, diffraction patterns in a wide Q range can be recorded simultaneously. Since intensities are collected using the time analyzers, the abscissa of the diffraction pattern is plotted with t or Q not & (Fig. 7-4). ~(ti) has to be included in fi(x) to take into account the intensity distribution. The TOF, t, can be converted into d by combining the Bragg condition, & = 2d/sin 0, and the equation of de Broglie, ;L = h/mv = ht/mL (h, Planck constant; m, neutron mass; v, neutron velocity; L, flight path from the target to the counter): d = ht/2mLsinO
(7-20)
421
8 1 0 0
,
9
,
9
,
9
,
9
,
9
,
'
,
'
,
9
.
.
.
.
.
7200 6 3 0 0
5400 4500 0 u
3600 2700 1800 9 0 0
l
It
2'o
II I
"
I
Ill II I IIIII
3'o
'
I I
IIII I IIHIIIII
'
s'o
IIIII
'
Illl
tl IIIlalliHIIll
|HIHIIII
IlgllliHilllll|llililillmllH
6'o ' ,'o ' .'o ' 9'o 'i;o'iio'i Q / n m -I
-o
Fig. 7-4. Rietveld refinement patterns for TOF neutron diffraction data of superconducting Lal.82Cal.~8Cu206+8 [45]. The scattering vector is plotted as abscissa and the net intensity as ordinate. Background was fitted as part of the refinement but has been subtracted before plotting.
For TOF neutron diffraction, the Lorentz factor is L(Q~) = L(tk)
= d4sin0/c
(7-21)
Most high resolution is obtained in back-scattering configurations using long flight paths, in particular in a "slowing-down" high-Q region. The TOF neutron-powder diffractometer, HRPD, installed at the Rutherford Appleton Laboratory [44], achieves an ultra-high resolution as high as A d / d ~ 4xl 0 -4 in back scattering (170 ~ < 2 0 < 178 ~ where geometrical contributions are negligible. Structures of many organic compounds displaying fairly complex diffraction patterns are now being refined routinely with HRPD. Reflections with dk = 0.03 nm can be observed by TOF neutron diffraction because the dimensions of Ewald's sphere are no longer limited by the ~ value and the resolution is surprisingly high in a high-Q region. The ability to probe such low lattice-plane spacings offers significant advantages; properties with different )~ dependences such as occupation factors, thermal-displacement parameters, crystallite size, and strain may be readily de-correlated from one another. However, the wide wavelength range requires a sophisticated treatment of the formalization of profile shape and wavelength-dependent phenomena such as absorption, ex-
422 tinction, thermal-diffuse scattering (TDS), and multiple scattering. At the present stage, much of this work is under development.
7.5.5. Energy-dispersive synchrotron X-ray powder diffraction This is a white-beam technique. Diffraction patterns are recorded at fixed scattering angles by measuring scattered X-rays with the combined use of solid-state detectors (SSD) and multi-channel analyzers. If the reciprocal relationship between energy, E, and wavelength is noted, Bragg's law can be recast as Ed = hc/2sinO
(7-22)
where h is the Planck constant, and c is the velocity of light. This equation means that a complete range of the d space can be scanned at once if we have a powder sample and a detector capable of energy discrimination using the SR source containing a broad spectrum of energies. In contrast with TOF neutron-powder diffraction, this method has seldom been applied routinely to Rietveld analysis [46--48] because of the limited energy resolution of the SSD. The best apparatus has an FWHM of the order of 140 eV (1 eV = 8065.541 cm-1) at a photon energy of 7 keV. The width increases as E 1/2, and reaches 300 eV at 30 keV. For a scattering angle of 12~ the resolution, Ad/d, is as large as 10-2, which is nearly two orders of magnitude worse than an angle-dispersive diffractometer on the same beam line [17]. However, the fixed-20 geometry makes it possible to record intensity data in a wide Q range simultaneously and rapidly. This method is, therefore, very suitable for in situ diffraction experiments at high temperature and/or high pressure. The improvement of the SSD is urgently needed. Bourdillon et al. [49] and Parrish and Hart [8,50] carried out energy-dispersive analysis using scanning incident-beam monochromators. In the high-resolution energy-dispersive method developed by Parrish and Hart [8,50], the wavelength is varied by step-scanning two parallel monochromators for incident beams and measuring scattered X-rays with a scintillation counter at a fixed 20 position. Extra measuring times are required because intensities for the whole Q range cannot be measured by this method. The resolution is, however, two orders of magnitude better than conventional energy-dispersive diffraction (Fig. 7-5), and applications to Rietveld analysis may be possible with this method. The energy-dispersive method has been successfully applied to high-pressure studies. A multi-anvil type X-ray system (MAX 80) employing a cubic anvil was installed at the Photon Factory of the National Laboratory for High Energy Physics (KEK) [51,52]. Remarkable advances in in situ observation of crystal structures and phase transformations under high pressure and high temperature were achieved using this apparatus. Major experiments carried out with MAX 80 include (i) precise determination of phase boundaries, (ii) precise determination of compressibilities, (iii) dynamic observation of phase transformations, (iv) EXAFS, (v)
423 d range 1 I. 7--*3. I A
10"
m
d range
d range 2.7--*0.71A
~
45 ~
d range I . ! - + 0 . 3 A
.4~0.4A r
i
.
.
.
.
,
.
.
.
.
.
.
.
.
*
.
.
.
.
i 2.04
.........
().55 2.04. . . . . . . .
" .......
0.55
z/h Fig. 7-5. High-resolution energy-dispersive diffraction patterns of quartz using 0.55-2.04 A incident X-rays [8]. The d range recorded depends on the 20 setting (upper left comer) of the detector.
measurement of viscosities, and (vi) determination of thermal-displacement parameters. 7.6. ANGLE-DISPERSIVE SYNCHROTRON X-RAY POWDER DIFFRACTION
7.6.1. Advantages of synchrotron X-ray diffraction Synchrotron X-ray powder diffraction [7-10,17] is rapidly evolving as a powerful technique for structural studies. The use of SR sources that provide extremely parallel and intense X-ray beams makes it possible to measure diffraction data of much higher resolution within practical times. The profile shape is very sharp and nearly Gaussian (decaying rapidly at tails), which is very favorable for high-precision Rietveld analysis. The high resolution and peak-to-background intensity ratios, and the high accuracy with
424 which the low-angle peak positions can be determined, are very suitable for the application of automatic indexing methods, to subsequent solution of the phase problem by ab initio techniques [29], and finally to the refinement of structure parameters by the Rietveld method. As the wavelength, ~, approaches the absorption edge, A,e,of an atom, the atomic scattering factor, f, changes with ~ because of the interaction between the incident beam and the atom. This phenomenon is referred to as anomalous dispersion, and coherent scattering under conditions where anomalous dispersion takes place is called anomalous scattering. The atomic scattering factor, f, in this case is f = f0 + f,(A) + f,,(A)
(7-23)
Near the absorption edge, the real term, f', changes like a very steep valley, and the imaginary term, f'; stepwise. Thus, atomic scattering factors can be considerably changed by the anomalous dispersion effect if the wavelength is set near an absorption edge in contrast with conventional sources.
7.6.2. Apparatus The most pronounced difference between X-ray powder diffraction experiments in the laboratory and at a SR facility is that, at the latter, the source is situated 10-20 m from the sample, and the beam divergence is 2-3 orders of magnitude less [ 10]. Incident beams are strictly monochromatized by two fiat crystal monochromators parallel to each other. For example, a channel-cut, double-crystal Si(111) monochromator scattering in the vertical plane is used in an instrument optimized for high-resolution powder data installed at the NSLS (National Synchrotron Light Source) beam line X7A [17]. The instrument resolution is substantially increased for a modest loss in intensity owing to a reduction in the energy bandwidth of the primary beam at the sample. Since the synchrotron beam is fully polarized in the orbital (horizontal) plane, no polarization factor is need; the Lorentz factor has the same form as that in angle-dispersive neutron diffraction. The intensity of the incident X-rays, ~(Qi) decreases gradually in diffraction experiments using SR; therefore, it must be always monitored by measuring diffracted and fluorescent X-rays from some materials. To enhance the resolution as much as possible, (i) a receiving slit [53], (ii) Soller slits [8], or (iii) a perfect crystal analyzer [7,9] are placed between a sample and a counter. The diffracted beam is usually measured with a counter, but the measurement time can be dramatically reduced by using an Imaging Plate [54].
Diffractometers with parallel-beam optics Figure 7-6 schematically illustrates four different types of sample-detector geometry used for synchrotron X-ray powder diffraction in the vertical scattering plane [ 10].
425
D
(a)
(c)
D
IS
IS I I o s---~ D
i ~
(b)
I CA
(d)
IS I Io S
I
Ib
S- ' ~
I Io !
Fig. 7-6. Four different types of sample-detector geometry used for synchrotron X-ray powder diffraction experiments in the vertical scattering plane [ 10]. I0 denotes a monochromatic beam from the source (typically 20 m distant), IS the incident beam slits, S the sample, and D the detector. (a) Flat-plate geometry with single receiving slit RS. (b) Capillary geometry with single receiving slit RS. (c) Flatplate geometry with Soller slits SS. (d) Hat-plate geometry with crystal analyzer CA.
Figure 7-6(a) shows the flat-plate geometry with a narrow receiving slit. The powder diffractometer (PFPD) at the Photon Factory consists of an incident slit, a monochromator, a Soller slit, a vertical-type goniometer, a monitor, a receiving slit, a Soller slit, and a scintillation counter [53]. The monochromator consists of a special type of monolithic Si crystals for a fixed-exit beam position with the (111) plane. The angle resolution of diffraction lines is comparable to that obtained by Parrish et al. [8] at the Stanford Synchrotron Radiation Laboratory (SSRL). In the capillary (Debye-Scherrer) geometry [55,56] illustrated in Fig. 7-6(b), samples are contained in capillary glass tubes turning on the extension axis. For a 1 mm capillary, there would be no loss of resolution compared with the fiat-plate situation. However, this arrangement is unsuitable for strongly absorbing samples. Parrish and coworkers [8] adopted the flat-plate geometry with Soller collimator for diffracted beam, as shown Fig. 7-6(c). This type of arrangement is used extensively in angle-dispersive neutron-powder diffraction [38], making it possible to use a much wider incident beam and, therefore, a much larger sample area, with a corresponding gain in the integrated intensity of diffracted peaks [10]. In a powder diffractometer at the SSRL [8], the diffracted beam is defined by vertical parallel slits (VPS) to limit the axial divergence to 1.8 ~ and 365 mm-long
426 horizontal parallel slits (HPS) with 0.05 ~ aperture. The former greatly reduces the asymmetry in peak shape caused by axial divergence. The HPS determine the resolution and shape of the profiles. The long distance between specimen and detector (59.1 cm) reduces the fluorescence background without the loss of intensity caused by a diffracted-beam monochromator. The geometry permits uncoupling the 0-20 sample-detector relationship without changing the profile shape, and makes possible new applications such as grazing-angle incidence depth analysis of thin films. The final approach is the crystal-analyzer geometry given in Fig. 7-6(d). A perfect Si or Ge crystal is mounted in the diffracted beam to obtain high resolution [7,9,10]. This can be regarded as a very narrow "angular" receiving slit capable of providing very high resolution (0.01~176 over a wide range of 20. The bandpass of the beam diffracted by the perfect crystal is a few eV, which serves to reject any unwanted fluorescence radiation and attains very high peak-to-background counting ratios. In common with the Soller-slit geometry, displacement-type effects are eliminated with a crystal analyzer. Guinier Diffractometer
Arnord [57] proposed a new type of powder diffractometer that can be operated in a Guinier mode, as well as the Debye-Scherrer and Bragg-Brentano modes, without changing the equipment. The counter arm is rotated around the center of the sample by the angle 20, and the entrance slit of the counter is shifted on the counter arm so that it coincides with the focal circle. Weissenberg camera
Honda et al. [54] collected synchrotron X-ray data of 5-aminovaleric acid with a largeradius (28.65 cm) Weissenberg camera equipped with an Imaging Plate, which is an area detector for X-rays using a photostimulate phosphor screen. The sample was sealed in a glass capillary, and it took only 6 min to measure its powder pattern by the Debye-Scherrer method. The angle resolution was fairly high (FWHM = 0.06 ~ at 20 = 19.13 ~ owing to the use of the large-radius camera, and ambiguous index assignments could be achieved with these wellresolved data. The structure of this compound was successfully solved from these data by combination of Patterson synthesis and a trial and error method, followed by Rietveld analysis.
7.6.3. Sample preparation The low divergence and intrinsic collimation of synchrotron radiation causes two major problems. First, the effect of preferred orientation on the integrated intensity becomes more and more marked. Second, crystallites situated at positions needed to satisfy Bragg conditions are fewer than those in conventional sources. The intensities of some reflections may become extraordinarily high because of the diffraction from coarse grains. These effects lead to a poor
427 reproducibility and an inferior precision of the diffraction pattern. Spinning (rotation) of samples is essential in synchrotron X-ray powder diffraction to reduce the second effect. If the crystallite size is below 3 l.tm, fiat sample holders may be used because preferred orientation is negligible in such a fine powder. However, the preferred-orientation effects are usually encountered for fiat-plate X-ray samples in synchrotron powder diffraction. The capillary geometry shown in Fig. 7-6(b) should be preferred to the flat-plate geometry when preferred orientation is serious. When the sample is very absorbing, the effective absorption coefficient can be reduced by diluting the sample with a non-absorbing powder (e.g., submicron diamond) that has a simple, highly symmetric structure [17]. In addition to reducing the absorption, the diluent helps to reduce preferred orientation in the mount. 7.7. PROFILE-SHAPE FUNCTIONS
7.7.1. Angle-dispersive powder diffraction
Symmetric profile-shape functions The line shapes of individual Bragg peaks need to be approximated by appropriate profileshape functions [58,59]. Table 7-3 lists profile-shape functions which have been used to approximate the profile shape of reflections observed in angle-dispersive powder diffraction. All the functions are normalized in such a way that integrals from -oo to oo are equal to 1. Simple Lorentz (7-24) or Gauss (7-25) functions do not satisfactorily fit to peak shapes, even though the peaks are symmetric except at low scattering angles. Poor fitting of profiles strongly affects occupation factors and thermal parameters, but fractional coordinates are not significantly influenced by the choice of profile-shape functions [59]. Profile-shape functions which are implemented in most Rietveld-refinement programs for angle-dispersive X-ray and neutron diffraction as well as energy-dispersive X-ray diffraction are the pseudo-Voigt function (7-28) [55,60] and the Pearson VII function [61]. The Gauss and Lorentz functions are the two extremes of these two profile-shape functions as regards the degree of decay from peak tops to tails. In X-ray powder diffraction, the Gauss function is usually too broad near the peak and too narrow at the tails, whereas the Lorentz function is unsatisfactory in the opposite way. Both the pseudo-Voigt and Pearson VII functions can be varied from Gaussian (7/= 0 or m = ~o) to Lorentzian (77 = 1 or m = 1) by changing the mixing parameter, 1/(Fig. 7-7), and the exponent m. Diffraction peaks in angle-dispersive neutron diffraction can be approximated fairly satisfactorily by the Gauss function [4]. The dependences of 77, m, and Hk on 20, which are required in Rietveld analysis, are investigated in detail experimentally and theoretically. The Voigt function (7-30), which is the convolution of the Gauss and Lorentz functions, is also implemented in some computer programs [62,63]. Thompson et al. [55] gave a series expansion relating 7/in the pseudo-Voigt function to Hk and the FWHM's of the Lorentzian component, HkL, for the Voigt function as:
428 Table 7-3. Symmetric profile-shape functions for angle-dispersive powder diffraction. A20 = 2 0 i - 20k, 20i is the diffraction angle at step i, Oh is the Bragg angle of reflection k, Hk is the FWHM of the profile-shape function, eL is Lorentzian, ~6 is Gaussian, F is the F function, fig and flL are respectively the integral-breadths of Gaussian and Lorentzian components, Re is the real part of the function, and to is the complex error function.
1 + 4 (A20/2] -1 2 = _-77-, Ilk I] ~t-/k
Lorentz
r
Gauss
O(A20) = q-x-Hk 2r exp [- 41n2 IA20/21 ~--~! j
(7-25/
Modified Lorentz
/A2012] -2 O(A20) = 4~]2 l/z- 1 1 + 4(~t2 - 1)~--~ / J
(7-26)
Intermediate Lorentz
O(A20) =
Pseudo-Voigt
O(A20) = r/q~L(A20) + (1 - r/)~G(A20)
(7-28)
Pearson VII
$(A20) = :l-'(m)~/:l/m- 1 Vi1 + 4(21/m _ 1)/A20//-m[/2-] " ~ F ( m - 0.5)Hk / ~ Hk ] J
(7-29)
Voigt
r
(7-24)
zn~
~ 2 2/3- 1
H,
= ~-Re
1 + 4(22/3- 111A201_|-1.5, ~2]
[o3/q
-~-A20 + i ilL(:;)1 ~ fiG ~/-~-fl
(7-27)
(7-30)
(7-31)
rI = 1.36603(HkL/Hk) - 0.47719(H~L/Hk) 2 + O. 11116(HkL/Hk) 3
They also used a set of numerically convoluted profiles to obtain the series approximation for the Hk of the pseudo-Voigt profile: 3
2
2
3
Hk = (HSG + 2.69269H2aHkL + 2.42843HkGHkL + 4.47163HkGHkL +
0.07842HkGH4L + H5L) 0"2
(7-32)
where the Gaussian FWHM, HkG, for the Voigt function is related to the variance of the
429
i
i
i
i
a
i
a: 77=0.0 b: 77=0.5 c: r/-- 1.0
r,t]
0
0
a
20
Fig. 7-7. Pseudo-Voigt-type profiles with an equal integrated intensity (area surrounded by the profile and the abscissa) and an equal FWHM. (a) 77 = 0 (Gaussian), (b) r / - 0.5, and (c) r/= 1 (Lorentzian).
Gaussian component cr by the relation H~G = ~ 8cr21n2
(7-33)
Dependence of the FWHM on the diffraction angle Separate refinement of H ~ and HkL is preferred when they are used to account for the strain and crystallite-size effects, respectively, on peak broadening. Strain broadening is Gaussian in shape whereas size broadening is Lorentzian. Convolution of these two with the profile-shape functions of well-crystallized samples of adequate crystallite sizes affords profiles for actual samples displaying strain and size broadening. Thermal-displacement parameters and occupation factors are strongly influenced by crystallite size and strain [59], which must therefore be included in the profile-shape function. Here, expressions adopted by Larson and Von Dreele [64] in their GSAS program will be introduced instead of simpler ones given by Cox and coworkers [10,55] because they enable one to model anisotropic broadening. The variance of the Gaussian component, ~2, varies with 0h as o"2 = HZG/81n2 = Utan20k + Vtan0~ + W + Psec20~
(7-34)
The angular dependence of cr is thus a function of the three parameters U, V, and W and the Scherrer coefficient, P, for Gaussian broadening. HkL varies with Ok as
430
HkL (X + XeCOStpk)sec0k + -
-
(Y + Yecostp~)tan0~
(7-35)
where Xe and Ye are anisotropy coefficients, and tp~ is the angle between the scattering vector and a broadening axis. The first term is the Lorentzian Scherrer broadening, and the second term is the strain broadening. Thus, there is a direct relationship between the parameters and a physical model in the Larson-Von Dreele formulation, which is therefore appealing in terms of its soundness based on physics motivation.
Peak asymmetry A variety of instrumental and sample effects, such as axial divergence of the X-ray beam and sample transparency, cause marked asymmetry in the observed profile shape, especially at low diffraction angles. The symmetric profile-shape functions listed in Table 7-3 are usually modified for peak asymmetry by multiplying O(A20)and an asymmetric function containing the asymmetry parameter, A [4]: a(A20) = 1 -
AcotO~A201A201
(7-36)
Peak asymmetry can be alternatively corrected [64] by employing the multi-term Simpson's rule integration described by Howard [65]:
1
~ gidp(A20")
~p'(A20) = 3 ( n - 1)i=
1
(7-37)
The 2 0 difference modified for peak asymmetry, As, and specimen shift, Ss, is A 2 0 ' = A20 +fiAscot20~ + Sscos0k
(7-38)
The sums in Eq. (7-37) have 3, 5, or 7 terms, depending on the size of As. The corresponding Simpson's coefficients,
gi and f ,
are:
n = 3: gl =g3 = 1,g2 = 4 n = 5: gl = g5 = 1,g2 =g4 = 4, g3 = 2 n = 7: gl = g7 = 1, g2 = g4 = g6 = 4, g3 = g5 = 2 f / = [(i - 1)/(n - 1)]2
(7-39)
431 Although this method of describing asymmetric peaks gives a better fit to X-ray and neutron profiles than that using the asymmetric function (7-36), it fails to fit strongly asymmetric peaks at very low scattering angles [64]. In fact, the Simpson's rule integration can break up into multiple peaks for very strong asymmetry. The profile-shape function in which the pseudo-Voigt function is combined with the Simpson's rule integration contains ten refinable parameters: U, V, W, P, X, Xe, Y, Ye, As, and Ss. Interpretations of these coefficients are described in Ref. 64. Asymmetric peaks can also be approximated by a split profile-shape function where two independent sets of profile-shape parameters are assigned to the left and right sides of peaks [66]. The main problem with the split function is the difficulty in assigning a physical meaning to the functional form. Profile-shape functions for synchrotron X-ray diffraction
The extremely parallel beam and the use of incident-beam monochromators in synchrotron X-ray diffraction eliminate many of the geometrical aberrations inevitable in conventional sources, giving nearly Gaussian and more symmetric line shapes. The pseudo-Voigt function was applied to synchrotron X-ray powder data taken using the capillary geometry [55] and the flat-plate geometry with crystal analyzer (Fig. 7-8) [10]. Will et al. [67] tested the Gauss, Lorentz, Pearson VII, and pseudo-Voigt functions as profile-shape functions and, in agreement with Thompson et al. [55], found that the pseudo-Voigt function yielded the best fit. Lehmann et al. [68] successfully used the Voigt function. Will et al. [69] also reported that the profile shape could be approximated by superimposing two Gaussian functions with a common peak position and a ratio of 2:1 for the FWHM and the peak height. Will et al. [70] later reported another better profile-shape function in which 20k of the Lorentz function was shifted 0.03 ~ to the low 20 side to match the asymmetry of the experimental profiles and added to the Gauss function. The profiles measured by the Weissenberg camera equipped with the Imaging Plate (cf. Sub-Section 7.6.2) could be satisfactorily approximated by the Gauss function; surprisingly, no peak asymmetry needed to be corrected [54]. 7.7.2. TOF neutron-powder diffraction Neutron production at a pulsed-spallation-neutron source involves two complex physical processes, "slowing-down" and thermalization which dominates at epithermal (A, < 0.1 nm) and thermal (A, < 0.1 nm) energies, respectively [18]. Consequently, the characteristic peak shape and incident neutron flux are rather complicated, depending on wavelength (cf. Fig. 7-4). Reflections in the epithermal region are very sharp and nearly symmetric, while those in the thermal region are broad with very long tails on low-Q sides. Because profile shapes depends on the materials, temperatures, and shapes of moderators and instrument features, they have to be optimized for different diffractometers. Examples are the convolution of arising and falling exponential with a Gaussian for the GPPD and SEPD diffractometers at the
432 1000 -
................... S~ ( 2 2 0 ) 1.5216A
300 ...................... Si ($:51) 1.5216A
"G 240 E o o
.... 8OO
(z)
o
600
~ ~BO
w 400
8 120
)
l-
e,-
-~ 200
~
60
0
'
'
-
.
.
.
.
.
.
" -
A
A
I
,oo F ...................... -100
.........
46.5
:"
........
46.7
t 46.9
20/~
f .....
t 20/~
Fig. 7-8. Least-squares fits to observed 220 and 331 peak profiles from Si with an Si(111) channel-cut monochromator and an Si(400) crystal analyzer [10].
The data
points from a step-scan at 0.0025 ~ intervals are shown by the dots, the solid line is a fit to a pseudo-Voigt function, and the difference plot is indicated below.
IPNS [44], and the summation of two functions based on a Gaussian leading edge and a second Gaussian trailing edge with an exponential tail 172] for the HRP diffractometer at the KENS [73]. Because these two profile-shape functions are not convoluted with the Lorentz function in the present forms, fits between observed and calculated patterns are often not very satisfactory for samples with small crystallite sizes. David [44] developed another TOF profile-shape function adopting the moderator pulse shape of Ikeda and Carpenter [74]. The Ikeda-Carpenter function is a convolution of the slowing-down spectrum from the moderator S(~') = ct3"r2exp ( - a ' r ) / 2
(7-40)
and a mixing of a &-function and an exponential decay R(~:) = (1 -R)6('r)
+
Rexp(-/3~:)
(7-41)
These two describe the leakage of fast and slow neutrons from the moderator and respectively contains 'fast' and 'slow' decay constants, ~x and j3, which are related to the material and
433 dimensions of the moderator. The mixing coefficient, R, is related to the moderator temperature. The Ikeda-Carpenter function must further be convoluted with the Gauss and Lorentz functions to give the full profile shape function. This function is most complex mathematically but excels at including physically meaningful parameters. 7.8. VARIOUS CORRECTION FACTORS
7.8.1. Absorption No absorption factor is required in Bragg-Brentano-type X-ray powder diffraction using fiat-plate samples; it is constant regardless of 20. On the other hand, absorption correction is necessary for the Debye-Scherrer geometry. Rouse et al. [75] gave analytical approximations of the absorption factor, A(Qi), for cylindrical and spherical samples with the linear absorption coefficient,/z, and radius, r:
A(Qi) : exp[-(a 1 + a2sin20)lzr - (a 3 + a4sin20)(gr) 2]
(7-42)
with
al
a2
a3
a4
Cylinder
1.7133
-0.0368
-0.0927
-0.3750
Sphere
1.5108
-0.0315
-0.0951
-0.2898
(note that there is a printing error for a4 in the original paper [75]). In the case of TOF neutron diffraction,/.t is dependent on wavelength because the absorption cross section increases with increasing wavelength.
7.8.2. Extinction The extinction in powder diffraction is a primary extinction effect within each perfect crystal block. Extinction is dependent on both wavelength and scattering angle. It may be neglected in X-ray diffraction but must be included in Rietveld analysis of TOF neutronpowder-diffraction data because of the wide range of wavelengths used in TOF experiments. The extinction correction, Ek, can be calculated according to a formalism developed by Sabine [76] and Sabine et al. [77]. E~ has Bragg and Laue components
Ek = Essin20k + ELCOSZ0k
(7-43)
434 where EB = (1 + x) -1/2
(7-44)
and EL = 1 - x / 2 + x 2 / 4 - 5x3/48 + 7x4/192
for x < 1
(7-45)
EL = (2/~c)1/2(1 - 1/8x- 3/128x 2 - 15/1024x 3)
for x > 1
(7-46)
with
X = Ex(l],Fk ]V) 2
(7-47)
Ex is the powder extinction coefficient which is a direct measure of the block size in a powder sample, Fk is the structure factor, and V is the unit-cell volume.
7.8.3. Preferred orientation
Dollase [78] tested several preferred-orientation functions listed in Table 7-4 and selected a special case from the more general description by March [79] as the best preferred-orientation correction: mk
ek = E (r2c~ j=l
+ r-lsin2otJ )-3/2/mk
(7-48)
where aj is the angle between the preferred-orientation direction and the jth member of the symmetry-equivalent set of mt diffraction planes. The sum is over all the equivalent reflections. The refinable parameter, r, represents the effective sample compression or extension due to preferred orientation. The March-Dollase formulation is applicable to both plate- and needleshaped crystals and independent of the diffraction geometry. Ahtee et al. [63] proposed another very effective preferred-orientation function, in which the preferred-orientation effect is modeled by expanding the orientation distribution in spherical harmonics. They implemented the model in their Rietveld-refinement program where the Voigt function was used as the profile-shape function. In tests using samples with textures known from pole-figure measurements, they found that the corrections obtained from the refinement agreed very closely with the measured values. If satisfactory results could not be obtained with the March-Dollase approach, this more complex but, in principle, more powerful approach is worth trying.
435 Table 7-4. Functions to correct for preferred orientation. Notes: a is the angle between the direction of preferred orientation and the scattering vector, Qk; G, b, and r are variable parameters which are related to the degree of preferred orientation. Two functions which are maximum at a = 0 ~ and minimum at a = 0 ~ should be selected according to the method of packing samples (plate or cylindrical sample) and crystal habit (plate- or needle-like).
No.
Maximum at a = 0 ~
Minimum at a = 0 ~
1
exp (-G a 2)
exp[-G (~/2 - a ) 2]
2
exp[G (71;/2 - or)2]
exp(Ga 2)
3
exp(-G sin2a)
exp (-G cos2a)
4
exp[-G (1 - cosaa)]
exp[-G(1- sin3a)]
5
b+(1-
b+(1-b)exp[-G(rc/2-ot) 2]
b) exp ( - G a 2)
r2cos2a + r_lsin2a)-3/2
7.8.4. Background The background, Yb(Qi), results from several factors, such as fluorescence from the sample, detector noise, TDS from the sample, disordered or amorphous phases in the sample, incoherent scattering, air scattering of X-rays or neutrons, diffractometer slits, and sample holder. In the original Rietveld refinement program [4], the background was subtracted from the observed intensity data. However, background parametel:s need to be included as variables for complex diffraction patterns. In a Rietveld-refinement program RIETAN [80], a background function which is linear in six refinable background parameters bo-b5 is used for the angledispersive diffraction method: 5
Yb(Qi) = yb(20i) = E bj[(20i- Omax- Omin)/(Omax- Omin)]j j=O
(7-49)
where 0max and 0min are respectively maximum and minimum Oi's. Therefore, 20i is normalized between-1 and 1 to reduce the correlations between bo-b5. Alternatively, the GSAS program [64] provides a cosine Fourier series with up to 12
436 refinable parameters including a leading constant term: 12
Yb(Qi) = bl + E bjcosPj_l
(7-50)
j=2
P is 20i in degrees in the case of constant-wavelength data. For TOF data, the tfs are scaled by 180//max, where/max is the maximum TOF allowed by the incident spectrum. 7.9. NONLINEAR LEAST-SQUARES METHODS Almost all computer programs for Rietveld refinement employ some form of the GaussNewton algorithm to find parameters which minimize S(x) apart from XRS-84 [81] and MINREF [82] adopting a variable metric method. However, when applied to Rietveld analysis, the Gauss-Newton method suffers a disadvantage that the range of convergence is rather narrow [59], and the refinements often converge to local minima rather than the global minimum [83]. Since none of algorithms has proved to be so superior that it can be classified as a panacea for nonlinear least-squares solutions, it is advantageous to have more than one method available on call. In RIETAN [73,80], three different techniques for nonlinear least-squares fitting are adopted: the Gauss-Newton method [84], a modified Marquardt method [85], and the conjugate-direction method [86]. All of them are designed to give stable convergence. RIETAN also has the very convenient features of incremental and combined refinements. The algorithms implemented in RIETAN will be introduced shortly. 7.9.1. Gauss-Newton method
In this algorithm, changes in n variable parameters at each iterative step, Ax, are calculated by setting up a normal equation:
MAx = N
(7-51)
where M is the coefficient matrix with n rows and n columns, and both Ax and N are nxl column matrixes. Although Ax is evaluated from M-1N in most structure-refinement programs, there is little to recommend such an old-fashioned technique because of the long computation time and low precision. In RIETAN, only a lower triangle of the positive-definite symmetric matrix M is kept in a one-dimensional array to save storage, and the Choleski decomposition of M and forward- and back-substitutions for the solution of consistent sets of linear equations are carried out [87]. A new set of x and x', is readily obtained by
437
x" = x + dAx
(7-52)
with d =
2-m
(m = 0, 1, 2, 3, 4)
(7-53)
The variable damping factor, d, is initially set at 1 (m = 0). If S(x') > S(x), d is decreased, and x' is calculated again with Eq. (7-52). The value of d is adjusted appropriately, according to the rule adopted in SALS [84].
7.9.2. Modified Marquardt method This method also calculates M and N but adds &.diag(M) (A,, Marquardt parameter; diag, diagonal matrix) to M to stabilize the convergence to the minimum: M + A.diag(M)Ax = N
(7-54)
Then, Ax tends towards the steepest descent direction as ~ becomes larger, while the GaussNewton solution is obtained when A, becomes negligible. Even if M is not positive definite, it can be made computationally positive definite by choosing ~ to be large enough. The value of A,is automatically adjusted during a series of iterations using a most efficient method developed by Fletcher [85]. The motivation for his strategy is that if the ratio of (actual reduction in S(x))/(predicted reduction in S(x)) is near 1, then A, ought to be reduced, and if the ratio is near to or less than 0, then A, ought to be increased. Fletcher's algorithm improves the performance of the Marquardt method in certain circumstances, yet requires negligible extra computer time and storage. The modified Marquardt method is very effective for dealing with highly nonlinear model functions, fi(x), or problems in which starting values for refinable parameters differ markedly from the true ones.
7.9.3. Conjugate-direction method The conjugate-direction method [86] is one of the most efficient algorithms for minimizing objective functions without calculating derivatives. The minimum of S(x) of a quadratic function with H > 0 (H: hessian matrix of S(x)) is located by successive unidimensional searches from an initial point along a set of conjugate directions generated by effective algorithms. In RIETAN, a combination of Davies-Swann-Campey and Powell algorithms [88] is adopted as a method of unidimensional minimization. Estimated standard deviations of refinable parameters are obtained by calculating M and inverting it (cf. Sub-Section 7.2.3) after convergence to the solution. Since the directions for minimization are determined solely from successive evaluations of
438 the objective function, S(x), this procedure is much slower than the two least-squares methods with derivatives It is, however, capable of solving ill-conditioned problems in which very high correlations exist between parameters. Because the conjugate-direction method is very fast in any nearly quadratic region near a minimum, it is mainly used in the late stages of refinement to test the prospect of a local minimum being the global minimum or to escape from a local minimum by using sufficiently large step sizes in line searches. On the other hand, someone using the Gauss-Newton and Marquardt algorithms can check the convergence to the global minimum simply by using different starting vectors.
7.9.4. Auxiliary techniques for stable convergence We usually proceeds in steps in Rietveld analysis, first refining only one or two parameters and then gradually letting more and more of the parameters be adjusted in the successive least-squares refinement cycles [20]. RIETAN requires only a single input to refine parameters incrementally; that is, variable parameters in each cycle can be pre-designated by the user or selected appropriately by the program when using the Gauss-Newton and modified Marquardt methods (incremental refinements). Repetition of batch jobs is, therefore, unnecessary in most Rietveld refinements. For example, linear parameters (background parameters and a scale factor) are refined in the first cycle, lattice parameters in the second cycle, profile-shape parameters in the third cycle, and subsequently all the parameters simultaneously. Even if initial parameters are far from the true solution, incremental refinements coupled with the appropriate adjustment of d (Gauss-Newton method) or A,(modified Marquardt method), enable very stable convergence to an optimum solution in most cases. Combined refinements are also possible in which the parameters obtained by the incremental refinements described above are further adjusted by the conjugate-direction method to ensure that there are no lower minima in the vicinity of the one found by the initial refinement. 7.10. INTRODUCING ADDITIONAL INFORMATION TO RIETVELD ANALYSIS Although high-resolution synchrotron X-ray and neutron powder data can be measured almost routinely at present, the amount of information in these data is still limited in comparison with that in corresponding single-crystal data. Rietveld refinements generally converge more slowly, and it is not possible to refine all parameters together from the start. It is essential to have a good initial structural model and to proceed with Rietveld refinements slowly and carefully. The chances of finding false minima increase particularly if the lattice parameters are not initially well known. Parameters cannot be refined with small estimated standard deviations, particularly when dealing with compounds showing complex diffraction patterns or severe line broadening. In such cases, peaks overlap heavily with each other, the sum-of squares S(x) does not decrease sharply near the minimum ("flat" minimum), and there can be quite a number of false minima around the global minimum [83]. When the positions of sites for light elements are poorly
439 defined because of the coexistence of heavy and light elements, the calculated interatomic distances and bond angles often deviate from crystal-chemically reasonable values. Occupation factors are strongly correlated with thermal-displacement parameters, and their simultaneous refinement leads to extremely large crj's.
7.10.1. Restraints Introduction of a priori geometric and chemical relationships into Rietveld analysis is often very effective for overcoming the above problems, and needs only the addition of the relationships and their estimated uncertainties to the observed intensity data [89]. These "pseudo-observations", referred to as restraints (soft/slack constraints), include expected structural features such as interatomic distances [80,90], bond angles, relationships between thermaldisplacement parameters, and those between occupation factors. Mathematically, there is no difference between the pseudo-observations and the X-ray or neutron diffraction data. The weighted sum-of-squares, S(x), can be calculated in a similar fashion:
S(x) = E wi[Yi- j~(x)]2 + E IzJ - gj(x)]2/tTfj i j
(7-55)
where zj and gj(x) are respectively observed and calculated relationships between parameters, and trj is the estimated error for zj. Thus, the restraints supplement the diffraction data, increasing the substantial number of observations significantly. Rietveld analysis under restraints leads to a prompt and sure convergence, makes it possible to refine more structural parameters than conventional analysis, and reduces the possibility of trapping into a false minimum.
7.10.2. Hard constraints Hard constraints are used to reduce the number of independent parameters by defining geometric and chemical relationships which have to be satisfied by variable parameters in Rietveld analysis. For example, if we regard an atomic group as a rigid body (group refinement), the number of fractional coordinates can be limited to 6: the coordinates for the center of the rigid body (x0, y0, and z0) and the rotation angle with its center as an origin ((/91, (/92, and tp3) [91 ]. Such an approximation does not hold strictly in the actual compound; the actual interatomic distances, bond angles, etc., more or less violate the assumed constraints. Hard constraints are therefore not so flexible or versatile as restraints.
7.10.3. Use of information from other experimental methods As described above, powder diffraction is often not powerful enough to show a definite conclusion as to structural details. Initial structural models should be constructed by taking into account not only crystallo-chemical information described in the literature but also results
440
I Selected-area ~
__~Powderdiffraction~
diffractionpatternsJ
patterns J
~r
Ir
Determinationof I Refinementof lattice F parameters I I~HRTEMimages1 [Composition3 ~Structuremodel
-I Simulation I
C stal c,emistry) Fourier/DI ._1 Rietveld L synthesisL ik(,,o,,)Irefinement I~1_ 1 ] Constraint Calculationof I distances& angles Fig. 7-9. A flow chart of Rietveld analysis. Squares with shadows are data analysis or calculations, and frames with rounded comers are some kinds of data.
441 obtained by some experimental means (e.g., HRTEM, chemical composition determined by quantitative analysis, spectroscopic measurements) other than powder diffraction. For example, when the structure parameters of Ba2YCu3OT_8 (0 < ~ < 1) [92] are refined, determinations of ~ values by chemical analysis such as iodometry make it possible to impose a linear equality constraint between the occupation factors (g) of two oxygen sites, O(1) and O(5), in a z = 0 plane: g(O(1)) + g(O(5)) = 1 - ~. In particular, contributions of oxygen atoms to diffraction intensities are so subtle that the ~ values are essential in the X-ray Rietveld analysis of this copper oxide. In the case of aluminosilicates, where (Si,AI)-O bond lengths can be estimated fairly reliably from A1/(AI+Si) ratios [93], restraints can be imposed on the (Si,AI)-O bond lengths. If the oxidation state of a metal is determined by X-ray photoelectron spectroscopy (XPS) [94] or X-ray absorption near edge structure (XANES), partition of the metal between two different sites can often be estimated without ambiguity. It is very dangerous to rely upon powder diffraction data alone when estimating the space group and lattice parameters of an unknown structure; selected-area electron diffraction should also be used [95]. Determination of the space group by convergent-beam electron diffraction is also very helpful when some possible space groups afford comparable R factors [96]. 7.11. REFINEMENT STRATEGIES When applying the Rietveld method to actual samples, one usually proceeds in the following way (Fig. 7-9): 1. Index peaks in powder-diffraction patterns and/or reflection spots in selected-area electron diffraction patterns, and determine possible space groups on the basis of conditions limiting possible reflections. 2. After determining peak positions of reflections, refine lattice parameters by a linear leastsquares method. For this purpose, use a refinement program such as that developed by Appleman and Evans [97]. Rietveld analysis often does not converge to the global minimum unless the initial values of the lattice parameters are fairly close to the true values. Therefore, it is safe to refine the lattice parameters prior to Rietveld analysis. 3. Roughly infer atomic configurations through structural data described in the literature, a search for an isomorphous compound or a compound with a similar structure, or direct observation of the crystal-structure image by HRTEM. 4. Simulate a powder diffraction pattern on the basis of the structural model. If the calculated diffraction pattern is not similar to the powder pattern actually measured, return to step 3 and assemble another structure model. 5. Perform Rietveld refinement. The lattice parameters determined in step 2 are used as initial values. Use profile-shape parameters for standard samples such as Si (e.g., NIST Standard Reference Material 640b) as initial ones unless broadening of diffraction lines due to strain and particle size is not very marked.
442 6. Modify the structural model and return to step 5 if R factors are not decreased to sufficiently low values. Fourier or D synthesis based on lk("o") is often useful in this process [5,20]. 7. If the Bj value of a site is extraordinarily large or small, return to step 5 after checking the validity of the Wyckoff-position assignment and occupation model for the site. 8. Calculate interatomic distances and bond angles from structure and lattice parameters obtained by the Rietveld analysis. Some values of them may be unreasonable in view of crystal data reported in the literature [98], effective ionic radii [99], bond-valence sums [ 100], etc. In such a case, return to step 5 after correcting the structure model or imposing appropriate constraints on the interatomic distances and/or bond angles. 9. Check the coordination numbers of atoms and/or calculate electrostatic (Madelung) energies from the lattice and structure parameters. If unreasonable results are obtained, modify the structure model and return to step 5.
X-ray
12
1
I
In1
SR
12
1
I
In2
Reactor neutron
TOF neutron
3
2
I Structure model I
Rietveld refinement
I
1 Structureand lattice1 parameters
Fig. 7-10. Rietveld refinement with combined neutron and X-ray diffraction data.
443 100
Neutron
80
4.1
60
m
O H
40
20
'i 9
.
.
I
.
.
~o 100
I
;o
I
.
'
I
'
I
.
~'o
i
'
'
i
I
i
'
I
6 "0
I
"
I
"
"
i
SR 80
4-) ,,.4 m
H
60
40
20
Ia
30
9
"
I
--
40
.
I
.
.
.
I
l,,
50
i l l
li
i
l
6O
I i
i
I
i
I
20/~ Fig. 7-11. Simulated neutron and synchrotron X-ray powder diffraction patterns for the spinel-type oxide MgTi204 for ~ = 0.15418 nm radiation.
7.12. SIMULTANEOUS RIETVELD REFINEMENT OF X-RAY AND NEUTRON DIFRACTION DATA
A new technique of Rietveld analysis, where two or more sets of intensity data measured by different diffraction methods (or under different experimental conditions) using the same sample are combined and refined simultaneously [101,102], are now being used more and more widely (Fig. 7-10). In such refinements, the profile and background parameters are refined separately for the respective sets of intensity data, and the structure and lattice parameters are refined as parameters common to the all the data sets. As exemplified in Fig. 7-11, X-ray and neutron diffraction are complementary in the sense
444 that they afford different structure factors, i.e., different diffraction patterns, for the same sample. Simultaneous refinement of X-ray and neutron diffraction data, therefore, leads to a pronounced increase in the amount of structural information. This technique is particularly useful for determining the distribution of elements between two or more sites and reducing correlations between refinable parameters. Even if X-ray and neutron diffraction data are not combined, separate Rietveld refinements using them may be useful for reinforcing one of the two intensity data sets. Kanke et al. [ 103] refined the crystal structure of NaFe3V9019 by Rietveld refinements of both neutron and X-ray powder diffraction data. Some structure parameters for V(Fe) sites were fixed at those refined using X-ray data in the Rietveld analysis of the neutron data because the coherent scattering length of V is negligibly small: bc = -0.3824 fm [31 ]. Complementary use of single-crystal and powder-diffraction data is also very effective. Iyi et al. [ 104] proposed a new model of defect structures in Lil_sxNbl+xO 3 using the results of single-crystal X-ray analysis. According to their Li-site vacancy model, the chemical formula of the nonstoichiometric oxide can be represented as (Lil_sxNbxI--14x)NbO3. Further, they obtained conclusive evidence for their model by Rietveld analysis of TOF neutronpowder-diffraction data. It is also very effective to combine diffraction data taken using two or more X-ray (neutron) diffraction methods or the same diffraction method under different conditions of measurement. For example, a technique is useful in which two or more sets of synchrotron X-ray diffraction data taken by changing X-ray wavelengths appropriately are combined and refined simultaneously. In particular, the tunability of SR to wavelengths near absorption edges can dramatically vary the X-ray scattering factor and enables the distinction of two elements with similar atomic numbers (cf. Sub-Section 7.6.1). In TOF neutron diffraction, measuring two sets of diffraction data with counters placed at different 20 positions makes it possible to cover a wide range of Q. Various diffraction methods show different dependences of the FWHM and diffraction intensity on Q. For example, peaks in a high Q region, which are grouped together in angle-dispersive neutron diffraction, can often be separated by using TOF neutron-powder diffraction, although there is the drawback that peaks in this region are weak. Their combined use evidently increases the amount of structural information. Combination of synchrotron and characteristic X-ray and neutron diffraction data not only increases the amount of information extracted from powder data but also reduces correlations between structure parameters. Accordingly, the problems of flat minima in sums-of-squares, and of false minima are at least partially solved, which increases the possibility of obtaining a more reliable solution. The structure parameters of orthorhombic Ba2YCu3OT_S [105] and BaPbO3 [106] were refined by combining X-ray and neutron diffraction data. The structure of LaSrCu0.sCo0.504-~ was also refined by the combined analysis of angle-dispersive synchrotron, X-ray Guinier, and neutron diffraction data [ 107].
445 Table 7-5. Quantitative X-ray determination for mixtures of Cr203 and A1203 [12]: C, Cr203; A, A1203.
Weight percentage
Observed percentage*
C
A
C
A
C
A
20.0 40.0 60.0 80.0
80.0 60.0 40.0 20.0
20.5 39.3 59.2 79.6
79.5 60.7 40.8 20.4
21.2 40.3 57.9 78.0
82.3 58.2 37.4 20.4
System
CA28 CA46 CA64 CA82
Observed percentaget
* Values determined by Rietveld refinement without an Si internal standard, totals constrained to 100%. ~- Values determined by Rietveld refinement using an Si internal intensity standard, totals not constrained to 100%.
7.13. APPLICATION TO QUANTITATIVE ANALYSIS Most Rietveld-refinement programs have a feature for dealing with mixtures of two or more phases. Weight fractions of phases j can be easily calculated from scale factors, sj, obtained by using this multi-phase capability [ 11,12]:
I~ = sj(Zmg)j / E si (ZMV)i i
(7-56)
where the summation is carried out over all the phases contained in the mixture, Z is the number of a formula unit contained in the unit cell, M is the mass for the formula unit, and V is the unit-cell volume. Table 7-5 lists an example of quantitative analysis by Rietveld refinements of X-ray diffraction data with and without an Si internal intensity standard. The March-Dollase function, Eq. (7-48), for correction of preferred orientation displays the best overall performance for structural studies. In addition, this function has the advantage that it conserves scattering matter, allowing its use in quantitative phase determination. This method does not require any working curves and affords more reliable results than the conventional method using only a limited number of reflections. Because structure and lattice parameters are refined at the same time, it is useful as a versatile data-processing method for powder diffraction. In addition, the content of an amorphous substance can be determined by adding an internal standard [ 12].
446
7.14. RIETVELD ANALYSIS OF MODULATED STRUCTURES The Rietveld method may be applied to sophisticated analysis which has hitherto been regarded as almost impossible. Yamamoto [108] developed a computer program PREMOS which can refine incommensurate structures as well as superstructures by the Rietveld method. It also makes possible joint refinement of X-ray and neutron diffraction data under nonlinear constraints. The algorithms adopted in REMOS for the analysis of single-crystal intensity data [108] have been combined with the Rietveld method. Elsenhans [82] independently developed a MINREF program for Rietveld analysis of incommensurate nuclear and magnetic structures with neutron-powder data, but neither fractional coordinates nor occupation factors are refinable in the present version. Peak positions and structure factors for (one-dimensional) incommensurate structures are calculated in more complicated ways than for commensurate ones. Four integers, hklm, are needed to index main and satellite peaks systematically. The reciprocal-lattice vector, q, can be written in vector notation q = ha* + k b * +lc* + m k
(7-57)
with k = k l a * + k2b* + k3c*,
(7-58)
where k is the wave vector of the modulation wave, and a*, b*, and c* are reciprocal unit-cell vectors for the subcell. Then the lattice-plane spacing, d, can be obtained by d=lql -~.
(7-59)
For example, d is expressed simply as d =[(h + mkl)2a .2 + (k + mk2)2b .2 + (l + mk3)2c'2] -1/2
(7-60)
in cubic, tetragonal, and orthorhombic forms. Additional parameters are necessary to calculate structure factors for incommensurate structures. For example, the atomic position, r, is calculated by adding cosine and sine waves to the average position, ~: r = ~ + uccos(27~t) + ussin(2~t),
(7-61)
where uc and Us are respectively the amplitudes of the cosine and sine waves, and t (= k-~) is the phase of wave. The occupation factor and isotropic thermal-displacement parameter can
447 be expressed by similar equations. Superconducting oxides with the ideal compositions Bi2Sr2Can_lCu,,O2n+6 (n = 1-3) have modular layer structure containing CuO2 sheets typical of all the high-Tc superconductors and double BiO sheets [2,3]. Yamamoto et al. [109] refined the complex incommensurate structure of Biz(Srl_xCax)3Cu208+ 8 (n = 2) using PREMOS and proposed a possible model for configurations of Bi and O atoms, including interstitial oxygen, on BiO sheets. This technique has not yet been applied to SR data but must be very effective because of the excellent resolution and high S/N ratio in synchrotron X-ray diffraction. Walker and Que [ 110] indicated that the modulated crystals of Bi-containing superconducting oxides can be described as composite crystals which consist of two substructures with average periods incommensurate to each other. On the basis of this idea, Yamamoto et al. [111] analyzed the crystal structure of the superconductor Bie+xSrz_xCuO6+S (n = 1) by the Rietveld method with the combined use of X-ray and TOF neutron diffraction data. This compound consists of two interpenetrating one-dimensionally modulated substructures: O atoms on BiO sheets and all the other atoms. The analysis revealed large atomic displacements from average positions in BiO, SrO, and CuO2 sheets. Oxygen arrangements in BiO sheets proved to be similar to those in Biz(Srl_xCax)3Cu208+ 8 [ 109]. 7.15. CONCLUDING REMARKS Refinable parameters and their estimated standard deviations will have considerable errors if the model function, fi(x), is not calculated strictly. In recent years, attempts have been made to take into account preferred orientation [63,78], primary extinction [76,77], multiple scattering, TDS, peak broadening due to the strain and crystallite size effects [112-114], etc. A Fourier-filter method was developed which removes the contribution of an amorphous substance from the diffraction pattern [115,116]. Because the model function is being refined more extensively, "problems under the carpet" that remain unsolved will disappear gradually. The technique of ab initio structure determination from powder diffraction is now being developed actively [29]. Rietveld analysis is used as the last step of this method of solving unknown structures. The HRTEM images correspond to Fourier maps in single-crystal X-ray analysis and help one to construct initial structural models easily without complex data processing to solve the phase problem. Thus, HRTEM is an excellent complement to Rietveld analysis. Crystal analysis with the combined use of HRTEM and Rietveld analysis will hereafter be performed more frequently. Third-generation synchrotron sources are now being built in the U.S.A., Europe, and Japan, i.e., the Advanced Photon Source (APS) at Argonne National Laboratory, the European Synchrotron Radiation Facility (ESRF), and SPring-8 (Super Photon ring-8 GeV) at Japan Synchrotron Radiation Research Institute. High-resolution powder diffractometers are planned to be installed at these synchrotron sources. The Rietveld method will be applied more widely to synchrotron X-ray diffraction data and will contribute greatly to advances in structural studies of various metals, inorganic and organic compounds.
448 REFERENCES
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Applications of Synchrotron Radiation to Materials Analysis H. Saisho and Y. Gohshi (Editors) 9 1996 Elsevier Science B.V. All rights reserved.
453
CHAPTER 8
X-RAY M I C R O T O M O G R A P H Y Katsuhisa US AMI and Tatsumi HIRANO Hitachi Research L a b o r a t o r y , Hitachi Ltd. O m i k a - c h o 7-1-1, Hitachi-shi, Ibaraki 319-12, Japan
8.1. I N T R O D U C T I O N X-Ray computerized tomography (CT) is a well known method for imaging the internal structure of an object. 1he CT image is reconstructed by a computer from a large number of Xray projection images of the object and is obtained as the two-dimensional distribution of Xray attenuation coefficients. Since Hounsfield presented a practical CT imaging scanner [1 ], much development work has been done for medical diagnostic devices. As internal structures are obtained nondestructively, many CT scanners for nondestructive testing and evaluation of industrial materials have also been developed [2-4]. In most of these CT scanners, a continuous or broad-band spectrum X-ray beam from high voltage X-ray tubes is used as a source and a xenon gas-chamber or a scintillator is often used as an array detector (spatial resolution < 0.2 mm). Therefore, elemental analysis is difficult and the spatial resolution is inadequate for nondestructive inspection for small defects in industrial materials. Although a quasi-monochromatic X-ray CT scanner, which uses a conventional CT scanner and an image reconstruction method which takes account of the incident energy spectra have been proposed [5,6], the quantitative treatment of the attenuation coefficients is not simple. Some attempts have been made to improve the spatial resolution [7-9] using conventional Xray generators. For example, 10 pm resolution was achieved using microfocus X-ray tubes, by improving the effective spatial resolution of the array detectors with geometrical magnification of projection images. Synchrotron radiation (SR) is a better X-ray source for CT scanners used to characterize industrial materials because it provides a highly collimated, high intensity, and tunable monochromatic X-ray beam. In the SR-CT (CT using SR as a source), an energy tunable monochromatic X-ray CT can be realized, having the following expected advantages. 1) Imaging of a specific element in a material on the basis of the difference between two CT images just above and below the absorption edge.
454 2) Improved quantitative treatment of X-ray attenuation coefficients due to absence of beam hardening effects. 3) High spatial resolution because of high SR collimation. 4) CT measurements with optimum absorption contrast by selecting the proper X-ray energy. From the above it can be expected that SR-CT will be developed into a spectroscopic CT by means of which elemental distribution analysis and chemical state imaging with high spatial resolution will be possible. The application of SR to X-ray CT has been studied theoretically [10,11], and the dependence of its spatial resolution/sensitivity on the photon energy was reported, as well as the estimated sensitivities for several samples. The experimental feasibility of SR-CT has been demonstrated [12] in a system using an Si(Li) detector of 30 elements and a combined scanning method of sample translation/rotation to produce CT images of a pig heart. The subtracted CT image of the iodine distribution from the difference just above and below the iodine K absorption edge was also obtained. Another study has described the development of an SR-CT scanner using a pencil beam and a photo-multiplier with an NaI scintillator as a detector [13]. However, all these studies were intended for medical diagnostic applications and the spatial resolution, 0.1 m m - 1 mm, was not so high. An SR-CT developed for high spatial resolution and elemental analysis of industrial materials, i.e. microtomography, has been reported by Flannery et al. [14,15]. This was followed by similar CT scanners [16-23], so that at present an SR-CT of 10 ~tm spatial resolution has been realized. In the following sections, the basic principle of CT is outlined, an experimental set-up and results of SR-CT are shown, and future prospects are described. 8.2. BACKGROUND OF C O M P U T E R I Z E D T O M O G R A P H Y
8.2.1. Principle of CT CT is a method of reconstructing a two-dimensional (2-D) image from a large number of one-dimensional (I-D) projections. In general, it is demonstrated by Radon transform that an n-dimensional image is reconstructed from an infinite set of (n-1)-dimensional projections. Presently, several CT scanners such as X-ray CT, NMR-CT, and positron CT have been put to practical use in medical diagnostic fields. X-Ray CT provides a distribution image of the X-ray attenuation coefficient/z in a material. In a sample X-ray shadow such as that shown in Fig. 8-1, the X-ray intensity It(x',O) transmitted through the sample is given by
It(x',O) = I S(E) e~ u(x,y,e)dy' dE
(8-1)
455
J ~'p(x,y) /IX-ray s !
X
c.
P(x;O) ~'- X'
Fig. 8-1. The principle of computerized tomography.
where S(E) is the energy spectrum of incident X-rays and #(x,y,E) is the energy dependent attenuation coefficient. Coordinates (x,y) are fixed at the sample, coordinates (x',y') are fixed in space, and 0 is the angle between these two coordinate pairs. If the incident X-rays are monochromatic with photon energy EO, S(E) can be approximated by S(E)=Io(Eo)~E-Eo) and Eq. (8-1) is represented as
It(x',O) =
Io e-I U(x'y'E~
dy'
(8-2)
where I0 is the X-ray intensity incident on the sample. When the incident X-rays are monochromatic, the projection data P, defined as ln(Io/It), can be regarded as the line integral of lU(x,y,Eo)and is simply expressed as
P(x',O)=ln(~t)= f l.t(x,y,Eo) dy'
(8-3)
A C T image is obtained by numerically solving Eq. (8-3) for/1, using many projections
P(x',O) measured as a function of sample rotation 0. 8.2.2. Reconstruction algorithm
There are several approaches to the reconstruction algorithm for obtaining a CT image [24]. Among them, analytical methods are generally used. In this section, we review two analytical methods briefly, i.e., two-dimensional Fourier transform, and filtered back projection.
456
Two-dimensional Fourier transform method (2D-FT) Two-dimensional FT is the simplest method for obtaining a strict solution of Eq. (8-3) in order to reconstruct a CT image from many projections. In order to solve Eq. (8-3), the Fourier transform P(~o,O) of P(x',O) is taken with respect to x'.
A
P(r
=
P(x',O) exp(-icox') dx'
(8-4)
Since the coordinate transform between (x',y') and (x,y) is
X
!
y'
/cos0 sin~ )Cx -sin 0
cos 0
(8-5)
Y
A
P(to, O) is rewritten by substituting Eqs. (8-3) and (8-5) for Eq. (8-4) as
P( oo,O) =
la(x,y) exp(-ioaxcosO--io~sinO) dxdy
(8-6)
where E0 in 12(x,y,Eo) is omitted for simplicity. On the other hand, the two-dimensional A
Fourier transform 12(r
a~) of l.t(x,y) is expressed as
12(Ogx,COy)=
bt(x,y) exp(-i COxX-iogyy) dxdy
(8-7)
Since COxand COyare written as
(8-8)
COx= 09 cos 0, axe = co sin 0 Eqs. (8-6) and (8-7) are equivalent. A
A
/.t(COx,COy)= P(w,0)
(8-9) A
Consequently, la(x,y) can be obtained from the inverse Fourier transform of/2(r
a~y) as
457 1
oo
.,,.
la(x,y) =
P( to, O) exp(it_OxX+i~yy) dtOxda~
(8-10)
.....oo
A
Since the coordinate (to, 0) expressing P differs from the coordinate (Obc,a~y), it is necessary to interpolate the transform between two coordinates in calculating la(x,y) from Eq. (8-10). Also, a FFT (Fast Fourier Transform) algorithm is useful for quick calculation of the Fourier transform.
Filtered back projection method (FBP) FBP gives a strict solution of Eq. (8-3) similar to that obtained by 2D-FT. Expressing l.t(x,y) of Eq. (8-10) in polar coordinates instead of rectangular coordinates gives
la(x,y)
--~2j 0
P(to, o) exp(imx') 1r daxi0
(8-11)
where I~ is derived from the Jacobian of the transformation into polar coordinates. Equation (8-11) can be rewritten as =
bt(x'Y)
1
f2~
2-~Jo P'(x',O) dO
(8-12)
where
P'(x',O) = l~-~I._~ P(to,0) exp(ioax') g(to) dto
and
g(r
(8-13)
--I~1
Eqs. (8-12) and (8-13) show that la(x,y) can be obtained by the back projection of P'(x',O) with respect to 0, after calculating P', which is the inverse Fourier transform of P multiplied by Ir as a filter function in the frequency space. Although the filter function g(w) is theoretically equal to la~, la~ is not suitable for the numerical calculation of Eq. (8-13) for the following reasons: 1) noise in the projection data; 2) discontinuous sampling of the projection data for x; and 3) a finite number of projection data for 0. Therefore, the following filter functions are used in general,
458
g(o)) = ~a sin a2-~
I~/sinacO/2(ac~
(Shepp-Logan Type)
(8-14)
where a is a sampling interval of P(x',O) and the region in which the filter functions are applied is I~1 < ~/a. The Shepp-Logan type filter function [25] decreases high frequency components and improves the signal/noise ratio in calculating/.t. Since the coordinates used in the left- and right-hand sides of Eq. (8-12) differ, interpolation between them is necessary. On the other hand, two-dimensional interpolation with respect to COx and COy is necessary for 2D-FT. In general, the higher the order of interpolation, the lower the accuracy in calculating the interpolation, so FBP is superior to 2DFT. 8.2.3. CT measurement methods
The scanning time during CT measurements depends on both the sample scanning method and the detector which measures the projection data. A position-sensitive detector which has many channels makes it possible to collect many data simultaneously, so scanning methods using multi-channel detectors can shorten the scanning time. Some scanning methods for CT measurements are described next and position-sensitive detectors are described in Section 8.3.4. Figure 8-2 shows three scanning methods to measure a CT image which consists of M x M pixels. Scanning method (a) is a first generation X-ray CT and is the basis of later X-ray CT scanners. An X-ray detector with one channel is located on the opposite side of an X-ray source which generates the collimated X-ray beam. Projection data P(x',O) composed of M data points are measured by translating a sample point by point. Next, the sample is rotated by a certain small angle 60, and P(x',O+60) is measured again. These measurements are repeated
X-rays
X-rays I
X-rays
I
.~_..1 ~ a m p l e !~ Detector (a) 1st generation
Detector (b) 2nd generation
Detector liiii':':iiiiiiiiiii':i':':i':l (c) 3rd generation
Fig. 8-2. Scanning methods for X-ray CT imaging. (a) uses a detector with 1 channel. (b) and (c) use a position-sensitive detector with a few tens up to several hundreds channels, respectively.
459 by a number of sample rotations, N. Position resolution of a CT image in this scanning method is determined by the width of the collimated X-ray beam. The measurement time for this scanning method is M • N times the exposure time to measure one point. Scanning method (b) uses a position-sensitive detector with a few tens of channels ( a channels). Since projection data corresponding to the a channels of the detector are measured at the same time, the translation number of the sample is M/a. Therefore, the total measurement time in this scanning method is a times shorter than that in method (a). The third generation method, (c), uses an M channel position-sensitive detector and no translation of the sample is needed because the projection data which consist of M data points are simultaneously measured at a certain angle 0. Therefore, the scanning time in this method is much shorter than in scanning methods (a) or (b). This is one reason why method (c) is used frequently. Moreover, the method needs only sample rotation so that its mechanical reliability is very high, and the efficiency of use of the incident X-rays is very good.
8.2.4. Quality of CT images The quality of CT images is evaluated by sensitivity, 5p/p, and spatial resolution, 5x. High sensitivity is required to distinguish between tissues in which X-ray attenuation coefficients are close to each other. Spatial resolution indicates the power to resolve fine tissues in a sample. When the total number of X-rays incident on the sample is constant, the relation between the sensitivity and the spatial resolution is a trade-off; improvement of one degrades the other. In this section, the sensitivity and spatial resolution are described.
Numbers for sampling and projection A CT image is discretely reconstructed from a large number of projection data P(x',O) because the measured P(x',O) consists of finite sampling data with respect to x' and 0. So, there is an optimum relationship between the sampling points M of P(x',O) for the x' direction, and projection number N for the 0 direction, in order to reduce noise in the arithmetic reconstruction of the CT image. When two intervals in the radial and angular directions at maximum frequency fmax in the frequency space are equal, the optimum value Nop of N is given as Nov = 2/rRfmax
(8-15)
where R is the maximum outward extent of the sample from the center of rotation. On the other hand, the sampling interval Ax is equal to 1/(2-fmax) according to the Nyquist theorem, so that M is written as
460 M = 2__R__R = 4Rfmax Ax
(8-16)
Substituting Eq. (8-16) for Eq. (8-15), we get Nop as Nop -- ~ M
(8-17)
Noise generated in the calculation to reconstruct the CT image depends on N, if M is given. When N > Nop, the noise decreases because the projection data are averaged, while the condition of N < Nop increases the calculated error. In particular, the latter condition causes the generation of a tangent-like artifact from the spot at which p changes suddenly.
Sensitivity/resolution The relationship between sensitivity dJ#//.t and point resolution t~x, which was studied by Grodzins [10,11], is described below on the basis of his results. We consider the first generation scanning method of CT, shown in Fig. 8-3a, for simplicity in the following discussion. The section to be reconstructed is a disc-shaped slice of diameter D and thickness h from a long homogeneous cylinder. When the X-ray beam width is Ax, the sampling number is M(=D/Ax), and the projection number is also M, the CT image of the section is composed of ffM2/4 pixels, each of the size dx x ix. The spatial resolution of the CT image is nearly equal to At. The transmitted X-ray intensity lout along the diameter is written as lou t = I 0
e-UD = I0 e-2;m ax,
(8-18)
where I0 is the incident X-ray intensity and 12i is the X-ray attenuation coefficient in pixel i. When I' is the total number of photons traversing one pixel during the scan of one section, then Q-", the statistical uncertainty of I', gives a measure of the minimum observable change in bt. Since the change of the number of photons absorbed in the pixel due to the change in #t is given as zlx6/z/'
4-iv= AX tSl.tl' =I' (l.tAr,)(~-~)
(8-19)
From Eq. (8-19),/tot, the total number of incident X-ray photons to the section for the scan, is expressed approximately as
/tot-- D__D__ ePD Ax (#z~)2(~) 2
(8-20)
461
Fig. 8-3. (a) Schematic drawing of first generation scanning method of measuring a discshaped section of diameter D and thickness h in a long cylinder. The collimated beam area is At h. The cylindrical sample is translated and rotated. (b) Schematic drawing showing the section containing ~r pixels. The raster scan of the sample consists of M rotational increments for each translational increment [10].
where epD is an absorption correction. The factor
D/zix comes from the fact that each of the
M 2 independent transmission measurements involves the traversal of M pixels. When/tot is constant, Eq. (8-20) means that the higher the spatial resolution is, the lower the sensitivity. Moreover, if the spatial resolution is constant then an increase of/tot is necessary for better sensitivity. Next, the optimum condition is obtained to minimize/tot. This condition can be met by the proper choice of the incident X-ray energy. The value of/.t depends on the composition of material and the X-ray energy E. In the low energy region (E < 40 keV) where the photoelectric absorption is dominant, /.t varies approximately as E -3. When a tunable, essentially monochromatic X-ray beam can be used,/1 is a variable in Eq. (8-20). If dItot/dl.t--O and 5/~//l=constant, then u = 2
(8-21)
This is the optimum condition to obtain a CT image at a given sensitivity from the minimum number of photons. In other words, when the optical depth "r(=/zD) of the sample is equal to 2 through proper choice of the X-ray energy, the number of photons sufficient to provide the CT image at a given sensitivity is optimized. Substituting Eq. (8-21) for Eq. (8-20), this minimum /tot to take one CT image is expressed as /tot =
2M31 (t~/,//,/./)2
(8-22)
462 From this equation,/tot needed to obtain a spatial resolution of 1% of the diameter and a sensitivity of 1% of the X-ray attenuation coefficient in a scan of one disc-like section is 2 x 1010 photons. The X-ray brilliance, B0, from the normal bending magnet at the Photon Factory in the National Laboratory for High Energy Physics is B0 -- 1011 photons/(s mm 2 mrad 2 0.01% b.w. 250mA) at an energy, E, of 20 keV. This means that an exposure time of 300 s is needed to measure the CT image of a disc-like section with diameter of 5 mm and thickness of 0.5 mm at 26 m from the source, when the source size is 1 mm • 1 mm, the storage ring current is 250 mA and the monochromator band width is 0.01%.
Beam hardening effects When an incident X-ray beam has a continuous spectrum, the CT value for the inside of a homogeneous material is less than that at the outside. This is caused by the spectral change in transmitted X-rays. This spectral change is known as a beam hardening effect. Since the X-ray attenuation coefficient in the low energy region is larger than that in the high energy region, the spectrum of the X-rays reaching the inside of the specimen is harder than that of the incident X-rays and depends on the specimen thickness. When a continuous X-ray beam is used, the projection data P(x',O) cannot be simply expressed as in Eq. (8-3). In this case, the mean attenuation coefficient It(E), averaging It(E) with respect to the transmitted X-ray spectrum, is expressed as
It(E) S(E) e-u(E)x dE
I
(8-23)
S(E) e-U(e)x dE
From this it is clear that It(E) depends on the specimen thickness x. On the other hand, if the incident X-ray beam is monochromatic with photon energy E0 then S(E), the energy spectrum of incident X-rays, can be approximated by a delta function and/z=it(E0). Therefore,/z is m
independent of x. Then the calibration of the CT value to/~ is easy and very reliable.
8.2.5. Elemental mapping One advantage of SR-CT is that the incident X-ray beam is monochromatic and tunable. Therefore, elemental distributions in a material can be obtained nondestructively from a subtraction method using an absorption edge of a specific element. Furthermore, the CT value, the brightness intensity in a CT image, has a very good correlation with the X-ray attenuation coefficient It, if a monochromatic X-ray beam is used, so the CT value can be treated
463 quantitatively. In this section, we discuss a method of taking a specific elemental image using monochromatic X-rays. The X-ray attenuation coefficient, /,t(E), of a material which is composed of many elements, is written as
,L/,(E) ='O~"ci(~~---)i
i =~i{~)i pi.
(8-24)
where p is the density of the material; (t,t/P)i is the mass absorption coefficient of the element i at the energy E; Pi is the density of the element i in the material; and ci is the concentration of element i. Figure 8-4 shows the energy dependence of/t for water, silicon and iron [26]. For iron,/,t changes sharply at about a photon energy of 7 keV, due to the K-absorption edge. The absorption edge of an element is intrinsic so its energy differs for each element. The reconstructed CT image is obtained as a two-dimensional shading image and the CT value is proportional to/,t. Two methods of obtaining an elemental mapping will now be explained. In one, the reconstructed CT image can be approximated to an image of a specific element. This approximation can be used if the attenuation coefficient/,tA of a specific element A is much larger than that of other composite elements. In this case, Eq. (8-24) is approximated as
~. lO s
~9 ]0 2
8 ~
lo'
-
Fe
10 ~
10 -1
10
100
Photon Energy / keY
Fig. 8-4. Mass absorption coefficient as a function of energy for water, silicon and iron.
464
,/2=/9CA(p)A
(8-25)
so that # is proportional to CA. Since/z and (/z/p)A can be estimated from the CT value and a numerical table of mass absorption coefficients, respectively, the density distribution of the specific element can be obtained. The second method is a subtraction method using the intrinsic absorption edge of the element. The difference t~/.tbetween two attenuation coefficients at X-ray energies just above and below the element's absorption edge is expressed as (8-26)
where
~la/p)
is the difference in mass absorption coefficient between two energies. The
subscripts A and B denote the specific element A and matrix B, respectively. Since the change of the mass absorption coefficient of the matrix is minimal in the two energies, ~/,t/p)B is negligible. Therefore, t~/zcan be approximated as d/,t-- ~p)APA
(8-27)
The expression means that the subtracted CT image shows the specific elemental distribution. As a result, PA can be obtained from Sjt as it is estimated from the differential CT value in the subtracted CT image. The CT image of an arbitrary element can be obtained by this subtraction method because the absorption edge energy of each element is different.
8.3. EXPERIMENTAL In this section, the SR-CT is outlined based on the scanner we have developed. The functions of the main parts of the CT scanner are reported. A method of magnifying the projection image is also explained which compensates for the relatively poor resolution of the detector, and improves the overall resolution of the system.
8.3.1. SR-CT system Figure 8-5 is a schematic illustration of the SR-CT. The third generation scanning method is used to shorten the scanning time. The SR-CT is installed at an experimental station in a branch beam line from a storage ring. The SR passes through the beryllium windows and enters the scanner. It is collimated by slit 1 and monochromated through the crystal monochromator. Diffracted X-rays with a desired photon energy are selected by slit 2. The radiation passing through the specimen is measured with a one- or two-dimensional position-
465 Array Detector Sample Crystal Slit 2 I Monochromator ~ _~.....~
"-
Slit 1
i Photo-Diode Array Detect:or (POA)
Synchrolron Radiation
_ ....
l
_
Pickup Tube _j
itnterface j 1
Fig. 8-5. Schematic diagram of SR-CT.
sensitive detector. Detector types are discussed in Section 8.3.4. The amount of charge stored in each detector channel is proportional to the incident X-ray intensity. After measuring one set of projection data, the charges at each channel are read out and amplified. The output signals are stored in a computer memory, after being digitized through an analog/digital converter. Next, the specimen is rotated at a fixed small angle and projection data are measured again. These measurements are repeated until the rotation angle of the specimen is 180~ The output signal from the detector includes not only signals proportional to the X-ray intensity, but also background signals caused by dark current and preamplifier offset. Moreover, it must be normalized by the intensity of the incident X-ray beam, and the SR intensity decreases with time. Therefore, corrections for these effects are necessary. In our system, the dark current intensity, the offset value and the incident X-ray intensity are measured before and after a series of the projection data measurements. The correction values of incident beam intensity, dark current intensity and offset during projection data measurement are then calculated by linear interpolation. 8.3.2.
Monochromator
There are many methods of monochromating SR. Since a continuous energy scan is not necessary in CT measurements, a channel-cut crystal is widely used for the monochromator. Adjustment of the channel-cut monochromator is easy and it has long-time reliability. On the other hand, a narrow energy bandpass (rE/E
466
~
E E
1
E c o o
/ ~ % ~ ~
Si(111) Channel'cut ~"
10 _9.i(400). C.h,an.nel-
cD c r .
Si(111 ) Double Crystal "
m
10
3l 0
_,
I
10
.
,
I.
20
i
I
30
,
"It, 40
X-ray Energy /keV Fig. 8-6. Intensity of the monochromatic output beam measured by an ionization chamber. Typical ring current is 300 mA.
Si(511)-Si(333) combination of crystals in a double-crystal configuration, for example, is used for this purpose [ 17]. Furthermore, the heat load from SR requires cooling of the first crystal in order to maintain good crystal properties. The crystal monochromator with a (+,-) configuration allows easy adjustment of an optical system positioned downstream because the monochromated X-ray beam is parallel to the incident X-ray beam. Examples of measured monochromator output beam intensities which were obtained with an ionization chamber are shown in Fig. 8-6. The crystal monochromators of the Si(111) double crystal, Si(111), Si(220) or Si(400) channel-cut were used. An X-ray beam with a narrow energy bandpass (SE/E
8.3.3. Sample stage The sample stage consists of a horizontal translation stage (Y-stage), a horizontal rotation table (O-turntable) and a vertical translation stage (Z-stage). The Y-stage is used to adjust the
467 center of the 0-turntable to the midpoint of the position-sensitive detector with a high accuracy. The accuracy should be at least one-fifth of the width of one detector channel. The 0-turntable, mounted on the Y-stage, rotates the sample with high precision. The Z-stage on the 0-turntable is used to select the slice position of the sample. Sample mounting on the Z-stage is done through a suitable attachment. Each mechanical stage is driven by a computer controlled stepping-motor. 8.3.4.
Detectors
The spatial resolution of a CT image is mainly determined by both the detector resolution and the SR source size. The SR angular divergence (<10 -3 rad) and electron beam size in a storage ring (N1 mm) are not negligible so that the spatial resolution is degraded by penumbral blurting by several microns. Then the spatial resolution of the image up to several microns is limited by the detector's spatial resolution. Moreover, the image quality is affected by the detector's dynamic range and linearity. The following are required of the detector: 1) high spatial resolution; 2) wide dynamic range; 3) good linearity; and 4) low noise. At present, Xray photodiode arrays, X-ray pickup tubes or CCDs (Charge Coupled Devices) with a scintillator screen are usually used as the detector of the SR-CT scanner.
X-ray photodiode array detector (X-PDA) The X-PDA is a one-dimensional photodiode array with an optical-fiber-coupled X-ray scintillator layer. The scintillator layer converts incident X-rays into visible light which is detected with the photodiode array (PDA) through the optical fiber. The PDA consists of 1001000 P-I-N type silicon photodiodes in a one-dimensional array on a silicon substrate. The element width is a few tens of microns. The diode is usually loaded with an inverse bias voltage. Visible light converted from incident X-rays produces electron-hole pairs in the silicon active layer due to excitation of electrons in the valence band. The electrons and the holes are collected towards each electrode and are detected as an inverse current. During exposure, the PDA usually accumulates the inverse current as charges in the condenser found on every diode, and then the integrated charges are successively read out using a digital shift register. We used a Reticon RL-1024 SFX, 1024-elements self-scanning photodiode array. Each element is 25 l.tm wide and 2.5 mm high and its maximum charge is 1.4 x 10-11 coulombs. The PDA was cooled to-30~
to reduce the diode dark current, which at room temperature is
high, e.g. 2 pA. Cooling reduced the dark current to 1/100. The dynamic range and the energy dependence of the sensitivity are shown in Figs. 8-7 and 8-8, respectively. The dynamic range of the X-PDA has a magnitude of about 3, and the saturation photon numbers per element at energies of 20 and 30 keV are 3 x 105 and 2 x 105 photons, respectively. The spatial resolution of the X-PDA is about 125 ~tm, as estimated from a rectangular pattern X-ray test chart (see Section 8.4.1).
468
i
0 r .Q 9
IE
i
0
v
10
1
"i"
r o
0.8
0 t-'
0 0 i i
1,0
x
..Q
10
E 0
tO'J
0,4
o 0
r -1 :o 10 . i i
0,2
>
0 10
0,6
........ 3
I
........ 4
I
.... 5
10 10 10 -1 X-ray Intensity / photons s
Fig. 8-7. Dynamic range of X-PDA.
9
0,0
I 10
I 20
I 30
I 40
50
X-ray Energy / keY
Fig. 8-8. Dependence of X-PDA sensitivity on X-ray energy.
X-ray-sensing pickup tube (X-SPT) The X-SPT, which is a kind of X-ray TV camera, is similar to a light-sensing pickup tube. Its spatial resolution is high because X-rays are sensed directly. The X-SPT was developed to observe real-time topography [27,28]. The structures of the tube and faceplate of the X-SPT (HS 501X, Hitachi Denshi Ltd.) we used [29] are shown in Figs. 8-9 and 8-10, respectively. A narrow electron beam from the diode electron gun is focused onto the photoconductive layer by magnetic focusing and the beam is scanned on the layer twodimensionally by magnetic deflection. The faceplate consists of a 500 l-tm beryllium plate, a 5 l.tm epoxy layer, 25 lxm glass plate, 800/~ transparent electrode, 160 A cerium oxide layer, and a 20 ~tm photoconductive layer. The beryllium plate mainly supports the force from the atmospheric pressure. The glass plate on which the photoconductive layer is deposited is used o suppress white defects appearing in the image due to the ber3rllium surface roughness. An amorphous Se/As alloy is used as the photoconductive layer. X-Rays absorbed in the photoconductive layer produce electron-hole pairs. While the electron beam scans on the photoconductive layer two-dimensionally, the holes at a point irradiated by the electron beam move to the irradiated surface and are discharged by the electron beam. On the other hand, since the transparent electrode is kept at a positive potential, charges equivalent to the amount of the discharges become the output signal, passing through the electrode and charge-sensitive preamplifier. The spatial resolution of the X-SPT depends
469 Beryllium //,-ScanningElectronBeam Window/--target "x ~~~ ,,~[_ "~i~ I n
cusing a n d Deflection S e c l i o n i"
145mm '
,..
I
,q
Fig. 8-9. X-SPT structure.
Dm
X'RaY>
~V~-~~Am
Jtl I Photoconduc~~-
Fig. 8-10. Faceplate structure.
on the electron beam size. The spatial resolution is improved by decreasing the electron beam size, but this decreases sensitivity. Present technology gives the X-SPT a spatial resolution of 6-8 ~tm [29]. We use a standard scanning area of 12.7 x 9.5 mm 2. The electron beam scan is controlled by a TV camera and consisted of 1024 pixels x 960 scanning lines, which are non-interlaced. Scanning is done at 7.5 frames/s. Frame blanking (FBL), an intermittent beam scan, is used to accumulate weak signals. The dynamic ranges of the X-SPT at FBL of 1.07 and 0.27 s are shown in Fig. 8-11. The dynamic range is larger than 102. It is limited by the electron beam current and the signal/noise ratio of 57 db in the video amplifier. Thus, the dynamic range of the photoconductive layer alone is larger. The energy dependence of the sensitivity is shown in Fig. 8-12. The step-like variation around 13 keV is caused by the Se K-absorption edge (12.65 keV). The sensitivity is a maximum at a photon energy of 20 keV and a minimum at 40 keV. The limiting resolution of the X-SPT is about 8 ILtm,as estimated from a radial pattern Xray test chart (see Section 8.4.2).
470
o
.%./ (D x
v
~ . ]0 c:D O tO
'T
2
l0
I Si(111
x
to
8
I
Si(400) r
-ii.-,
O t-
1
.Q
I
Se
6
edge
E O
5 . i .
,-, ]0
O o
0
4 2
r~ . i
O
. i .
]0
r
-1 10
5
10
6
10
7
X-ray Intensity / photons s 4 mm -2 Fig. 8-11. Dynamic range of X-SPT.
0
0
I 10
I 20
I 30
I 40
50
X r a y Energy / keV
Fig. 8-12. Dependence of X-SPT sensitivity on X-ray energy.
Charge coupled device (CCD) SR-CT scanners with a CCD have been reported [14,16]. A schematic illustration of a CCD with a scintillator screen (X-CCD) is shown in Fig. 8-13. X-Rays pass through the sample and are converted into visible light on the high resolution scintillator screen. The image thus formed is projected onto the CCD chip with a bi-convex lens. The spatial resolution of CT images depends on the resolution of the scintillator screen and the element size of the CCD. A single crystal of CdWO4 [30] or segments of CsI crystals [31] of size of the order of microns are used for the scintillator screen. In order to reduce blurring on the screen due to fluorescence refraction, the former has an anti-reflective coating and the latter very small optically-isolated pieces. The CCD is an MOS (metal Oxide silicon) device which consists of many picture elements in a two-dimensional array. The light detection method for the CCD is similar to that for the PDA. However, stored charges are read out in digital format by transferring the charges from the elements to an analog shift-register. The element size is 1020 l.tm and the maximum accumulated charge per element is about 10-14 coulombs [ 16]. The dynamic range of the cooled CCD array is 103-104 [ 16]. The blurring of the image on the scintillator reduces X-CCD resolution. The factors in the blurring of the image are re-absorption of the fluorescent X-ray and diffusion of photoelectrons. These effects limit the resolution of the scintillator screen to about 5 ~tm [30]. The spatial resolution of a CT image measured by the SR-CT scanner with the X-CCD array is then a little better than 10 l.tm [30].
471
X-rays ~~~ .ScintillatorScreen .......;~.......-.'~[ Visi/bleLightI /
"\ .
'..~..
CCD1
Optical Mirror
--._2
Fig. 8-13. Schematic drawing of a two dimensional position-sensitive X-ray detector using a CCD. The scintillator screen converts X-rays to visible light. The image formed on the screen is projected onto the CCD with a biconvex lens.
8.3.5. Asymmetric Bragg magnifier Magnification of a transmitted X-ray beam by X-ray optical elements allows CT images of higher spatial resolution to be obtained, without the limit of detector resolution. Asymmetric Bragg diffraction is used for magnification of the transmitted X-ray beam in the hard X-ray region. This magnification method compensates for the relatively poor resolution of the detector, and improves the overall resolution of the system. The principle of asymmetric Bragg diffraction is illustrated in Fig. 8-14. A perfect single crystal surface is inclined to a lattice plane at angle a. Then, the incident X-ray beam of width din is expanded to the diffracted Xray beam of width dout because the reflection angle 0out to the crystal surface is 2 a larger than the incident angle 0in. The magnification factor Mf of the X-ray beam width is expressed as Mf = dout = sin 0out = sin (0B+a) din sin 0in sin (0B-a)
(8-28)
where 0B is the Bragg angle. The asymmetric Bragg magnifier is suitable for CT imaging, because it works only for one-dimensional expansion optics and positional linearity is retained. The Bragg angle 0t3 depends on the X-ray energy so that preparing several asymmetric crystals of different a is necessary to magnify the X-ray beam with the desired Mf. Figure 8-15 shows the energy dependence of Mf and/gin for two asymmetric crystals (Si(220)) with a=4.1 ~ and a=6.2 ~ Both angles give Mf of about 5 at X-ray energies of 20 and 30 keV, respectively. In order to avoid magnified image distortion, an optically flat surface of perfect Si or Ge crystals is used for image magnification. Although fluorescent X-rays generated by incident X-rays may degrade the contrast of the CT image, this effect can be suppressed by using the asymmetric magnifier.
472
an
dout 0out ":::--. a t ......
i.:i---}--}-{{{!{{{!:::!!:.:.:..::::::---. .--.'.; . . . . . . .
"_.. . . . . . .
.~
Bragg Plane
Fig. 8-14. The principle of asymmetric Bragg diffraction.
"-
,
I
/
i
I
~
// I
[ '~ 15--a
u_ 8
6
o-
25
io~
I
10
I
a=6.~ I '--..~, I
~ I
20 30 X-ray Energy / k e Y
I
40
o
Fig. 8-15. Dependence of the magnification factor Mfor incident angle 0in to the crystal surface on X-ray photon energy. The Si(220) crystal is used for the asymmetric Bragg magnifier in the calculation: a is the angle between the crystal surface and the lattice net plane.
8.3.6. Computer system
A high speed micro-computer with an image processor is used for both data acquisition and image reconstruction. The image processor is used for the Fourier transform calculations and makes high speed CT imaging possible. We used the HD-68000 with an image processor (Hitachi Medical Corporation), which is designed for CT imaging. High speed image
473 reconstruction is achieved by the special pipe-line processor. The CT image is reconstructed by a filtered back projection (FBP) method with a Shepp-Logan type filter. Under typical measurement conditions of a 1024 pixel one-dimensional projection with 180 projections in 1.0 degree steps, image reconstruction with the FBP takes only half a minute per slice. The reconstructed CT image is displayed as a gray-scale image on a 512 x 512 pixel video display.
8.4. E X P E R I M E N T A L RESULTS Results on the spatial resolution of detectors and CT images of biological samples are described next. Elemental analysis using the SR-CT scanner and other results are also presented.
8.4.1. Spatial resolution of detectors Evaluation of the spatial resolution of detectors is important in investigating the spatial resolution in measured CT images. The spatial resolution of detectors is estimated by a modulation transfer function (MTF) as shown in Fig. 8-16. The projection image of uniformly spaced lead slits (X-ray test chart) is measured with a position-sensitive detector and the MTF value is defined as the modulation amplitude divided by the average modulation. The modulation depends on the lead slit interval. The pair number of lines and spaces per mm
IX-raYl,,,
X-ray Test Chart
,.
/m YA oeteo, to , MTF Value - b- a
_c
(b)
Fig. 8-16. (a) Schematic drawing showing evaluation method for detector resolution using uniformly spaced lead slits (X-ray test chart). The projection image of the X-ray test chart is measured with a pos ition-sensitive detector. (b) Schematic diagram of output signal intensity of the detector as a function of position.
474 100
I
I
I
I
I
80
cO Q.
n-
60 40
------.e....._
20 0
0
10
20
I 30
I 40
I 50
60
Spatial F r e q u e n c y / I p m m -1
Fig. 8-17. Measured square-wave MTFs of the detectors. Filled circles are MTF of the X-PDA without magnification at E=15 keV. Squares are that of the X-PDA with magnification (M=4.9) at E =20 keV. Open circles are that of the X-SPT without magnification at E =20 keV.
(lp/mm) is used for the spatial frequency. For instance, 10 lp/mm corresponds to the 50 l.tm line-and-space width. The spatial frequency at MTF = 5% is usually adopted as the limiting resolution. The square-wave MTFs measured with detectors are shown in Fig. 8-17. The limiting resolution of the X-PDA is estimated to be about 125 ~tm, in spite of the fine array pitch (25 l.tm). This spatial resolution degradation is attributed to visible light diffusion in the scintillator layer. Then, the transmitted image of the X-ray test chart was magnified by asymmetric Bragg diffraction and measured with the X-PDA. When the magnification is 4.9 and the X-ray energy is 20 keV, MTF = 23% is obtained at the spatial frequency of 10 lp/mm. The limiting resolution is estimated to be about 30 ~m from extrapolation of the MTF curve. Therefore, magnification by asymmetric Bragg diffraction is very effective. Since the spatial resolution of the X-SPT is much higher than that of the X-PDA, an X-ray test chart with high spatial frequency is necessary for evaluation of the X-SPT resolution. It is, however, very difficult to fabricate fine lead slit patterns of under 10 l.tm. Therefore, the transmitted image of the original X-ray test chart was reduced using asymmetric Bragg diffraction and measured with the X-SPT. This corresponds to the situation in which the X-ray beam shown in Fig. 8-14 enters the asymmetric crystal from the opposite side. The MTF value of the X-SPT is 9% at 10 l.tm line-and-space width, and the limiting resolution is estimated to be about 8 l.tm by extrapolation of the MTF curve. Moreover, the transmitted image of a radial pattern X-ray test chart was also reduced to one fifth by asymmetric Bragg diffraction and measured with the X-SPT. The results are shown in Fig. 8-18. The finest pattern in the image is about a 6 ~m line-and-space width, and fine patterns up to about 8 l.tm are visible.
475
Fig. 8-18. Measured image of a radial pattern X-ray test chart, which is reduced to 20% by asymmetric Bragg diffraction. The finest pattern in the picture is about a 6 l.tm line and space width. X-ray energy is 20 keV. Exposure time is 4.3 s. Limiting resolution is about 8 lam.
8.4.2. Photon energy dependence of contrast It is important to select the proper photon energy for good contrast in CT images. The dependence of contrast on photon energy was examined using a meat sample. The sample was prepared as shown in Fig. 8-19a by filling a 13 mm diameter plastic vessel with muscle and fat. The slice width was 0.5 mm and the X-PDA was used as a detector. The contrast was examined in the photon energy region of 15 to 35 keV. Figures 8-19b and 8-19c are the CT images at photon energies of 25 and 15 keV. The bright area in the CT images corresponds to a high X-ray attenuation coefficient. The contrast of muscle to fat changes with photon energy. Figure 8-20 shows the dependence of contrast on photon energy quantitatively. The contrast was calculated from the mean CT values of 21 x 21 pixels for muscle and fat according to the definition shown in the Figure. The contrast becomes better for lower photon energy. Figure 8-21 shows the dependence of X-ray attenuation coefficients on photon energy for muscle and fat. These were calculated from the experimentally obtained CT values of water and acrylic resin for which X-ray attenuation coefficients can be calculated. A lower photon energy gives better contrast, and X-ray absorption coefficients can be obtained quantitatively.
476
Fig. 8-19. (a) Schematic of test sample. The plastic vessel was filled with muscle and fat. The CT image of the sample was measured with X-PDA. Slice width was 0.5 mm. (b) CT image at 25 keV. (c) CT image at 15 keV.
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8.4.3. Elemental analysis One of advantages of SR-CT is that elemental and density distributions in a material can be obtained nondestructively. The results of CT imaging of a specific element taken by the XPDA are shown in Fig. 8-22. The test sample consists of five teflon tubes. One was filled with a 10 wt% Mo solution, a second was filled with 4 wt% Cu solution, and the other three tubes were empty. Elemental mapping of Mo can be obtained by the subtraction method using the Mo K-absorption edge (20.0 keV). The high-energy image at 20.6 keV and the low-energy image at 19.6 keV are shown in Figs. 8-22b and 8-22c, respectively. The differential CT image obtained by simple subtraction of the low energy CT image from the high energy one is shown in Fig. 8-22d, where the pixel brightness corresponds to the difference in absorption
Fig. 8-22. Monochromatic X-ray CT images of a test sample at X-ray energies below and above the Mo K-absorption edge (20.0 keV). (a) Cross sectional drawing of a test sample. (b) CT image at 20.6 keV. (c) CT image at 19.6 keV. (d) Differential CT image obtained by subtraction, image (b) - image (c).
478 coefficients. Only the pixels corresponding to the tube containing Mo are brighter than their surroundings in Fig. 8-22d. This result means that the spatial distribution of specific elements can be observed nondestructively. Next, quantitative elemental analysis was investigated [32]. A schematic cross-section of the.test sample is illustrated in Fig. 8-23a. One of the seven teflon tubes was filled with a 4.5% Mo solution, which was prepared by diluting a concentrated Mo stock solution. This was prepared by mixing ammonium molybdate ((NH4)6Mo7024o4H20, 20 g), ammonia solution (25 wt%, 11 ml) and pure water (100 ml). Five tubes were filled with diluted Mo solutions of 2.8, 1.0, 0.5, 0.3, 0.1 wt%, prepared from the stock solution and the last was filled with pure water. Two CT images of the test sample, taken with the X-PDA at 20.1 and 19.9 keV, are shown in Figs. 8-23b and 8-23c, where a brighter pixel corresponds to a higher X-ray attenuation. Areas with higher Mo concentration are brighter than those of lower Mo concentration. The differential CT image obtained by subtracting the low energy CT image from the high energy one is shown in Fig. 8-23d. The brightness of the parts containing Mo is
Fig. 8-23. Monochromatic X-ray CT images of a test sample taken at X-ray energies below and above the Mo K-absorption edge (20.0 keV). (a) Cross-sectional drawing of a test sample. Seven Teflon tubes are filled with Mo solution and pure water. Numbers represent Mo concentrations in wt%. (b) CT image at 20.1 keV. (c) CT image at 19.9 keV. (d) Differential CT image obtained by subtraction, image ( b ) image (c).
479 related to their Mo concentrations. The subtraction method provides not only the spatial distribution of a specific element, but also its density distribution. In Fig. 8-24, the relation between the Mo concentration and the differential CT value in the image shown in Fig. 8-23d is plotted. The data have good linearity so the Mo concentration can easily be estimated from the differential CT value. Consequently, the density of a specific element can be obtained by the subtraction method. However, a weakness of this method is that the X-ray energy to be used is determined by the element. Thus, it is difficult to apply this method to a sample which absorbs almost all the X-ray photons at that energy. In Fig. 8-25, the average CT values for the six Mo solutions and pure water in the two CT images shown in Figs. 8-23b and 8-23c are plotted against the X-ray attenuation coefficient ~t. The value of ~t is calculated from the compositions of the Mo solutions and water at each energy. The linearity between the CT value and/.t is very good and shows an advantage of SRCT because monochromatic X-ray CT allows quantitative treatment of X-ray attenuation coefficients due to absence of the beam hardening effect. When the subtraction method is not applied to a sample due to high attenuation, the X-ray energy is set so as to obtain a sufficient transmitted X-ray intensity and the X-ray attenuation coefficient of the sample can be estimated from the CT value measured at that energy, using the calibration curve as shown in Fig. 8-25. 1000
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480 If composite elements or compounds in the sample are known, the estimated attenuation coefficients allow a relationship to be obtained between the elements/compounds and regions in the CT image.
8.4.4. High resolution CT images Because of the high SR collimation, a SR-CT scanner provides high spatial resolution CT images, which need a high spatial resolution detector. We confirmed that the asymmetric Bragg magnification method was effective in improving detector resolution [22]. A crosssection of a standard sample used is illustrated in Fig. 8-26a. The sample was a 4 mm diameter half cylinder made of optical glass (BK-7), with many grooves cut along the axis. The grooves were 50, 100, and 200 l.tm wide. The CT image measured by the X-PDA with a magnification factor of 4.9 (E = 20 keV) is shown in Fig. 8-26b. The 50 l.tm width grooves can be clearly discerned in the CT image, and the spatial resolution is higher than 50 l.tm. The magnification method is confirmed to be very effective in improving the spatial resolution of CT images. Next, a SiC fiber-reinforced Si3N4 ceramic sample was examined. Its structure is illustrated in Fig. 8-27a. The 140 lam-diameter SiC fiber consists of a 30 l.tm-diameter carbon core and a SiC layer. The SiC layer was deposited around the carbon core by chemical vapor
Fig. 8-26. (a) Cross-sectional drawing of a 4 mm diameter half cylinder made of optical glass (BK-7) with many grooves. (b) CT image of the sample obtained by the magnification method (4.9-fo!d); E =20.0 keV. Slice width was 1.0 mm. Exposure time per projection was 10 s.
481 deposition. The sample was cut into a prism with the slice plane perpendicular to the fiber. The CT image in Fig. 8-27b was taken unmagnified by X-PDA. Although individual SiC fibers are visible, the carbon core is not. The image in Fig. 8-27c was taken by X-PDA with a magnification factor of 9.0 ( E-24 keV). The carbon core is visible, but the fiber-matrix boundary and the core boundary are not clear. The last image, in Fig. 8-27d, was taken unmagnified by the X-SPT and the 30 I.tm-diameter carbon core can be clearly observed. In order to estimate the spatial resolution of the CT image measured with X-SPT we used the CT value, which numerically expresses the brightness in the CT image [23]. The CT values around the SiC fiber (denoted by an arrow in Fig. 8-27d) are plotted against the distance from the center of the carbon fiber core in Fig. 8-28. Since the boundaries between the carbon core, the SiC layer, and matrix are very sharp, the components can be clearly distinguished from each other. The slight spread of the boundaries is caused by both the reconstruction of the CT image and the spatial resolution of X-SPT. When the boundary
Fig. 8-27. (a) Illustration of the SiC fiber-reinforced Si3Ni4 ceramics. (b) CT image taken by X-PDA without magnification at 32.9 keV. Slice width was 0.8 mm. Exposure time per projection was 10 s. (c) CT image taken by X-PDA with magnification (9-fold) at 24 keV. Slice width was 1.0 mm. Exposure time per projection was 30 s. (d) CT image taken by X-SPT without magnification at 24 keV. Slice width was 110 l.tm. Exposure time per projection was 7.5 s.
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spread due to these effects is expressed by a Gaussian distribution, the distribution of the CT value around the boundary is represented by the convolution of a step function and the Gaussian distribution. We adopted twice the standard deviation (6) of the Gaussian distribution for the spatial resolution of the CT image. The distance between the transition levels from 14% to 86% in the boundaD, corresponds to 2~ of the Gaussian distribution. From the CT value in the boundary between the SiC layer and the matrix, the resolution of this CT image was estimated to be 10 ~tm. 8.5. A P P L I C A T I O N S
8.5.1. CT images of biological samples Some CT images of biological samples were taken. The main constituent elements of the sample were hydrogen, carbon, nitrogen and oxygen, and their X-ray mass absorption coefficients are very small. Therefore, it is better to use low energy X-rays for high contrast CT images. A photon energy of 15 keV was selected in consideration of the results in Section 8.4.2. The CT images of two vegetables, okra and green bean, are shown in Figs. 8-29a and 8-29b, while that of salami is given in 8-29c obtained by X-PDA. The complex cross sectional structure of okra is clearly observed. The outer pentagon is the okra pericarp and seeds are
483
Fig. 8-29. CT images of (a) okra, (b) green bean and (c) salami sausage.
located at five sites within the pod. The fine structures of the pericarp are also observed. The green bean is mostly watery tissues, but the seed and seed coat are observed distinctly since a low photon energy was used.
8.5.2. Observation of cracks and defects in composite materials
Ceramic composite In order to observe cracks in a ceramic composite, a cavity was formed by indenting with a Vickers hardness tester on the surface of the SiC fiber reinforced Si3N4 ceramic described in Section 8.4.4. Its CT image is shown again in Fig. 8-30. The induced flaw is indicated by arrow A. A black line (B) from one fiber to another, and some dark regions (C) between the matrix and SiC fiber, are observed in the CT image. The former can be considered as a crack produced by the external force and the latter as debonding produced by the external force or by some steps in the manufacturing process of the composite. Moreover, three very bright spots, which strongly attenuate X-rays, can be observed in the middle of the CT image. Their X-ray attenuation coefficients are about 20 cm -1. The matrix consists of Si3N4, A1203 and Y203 whose attenuation coefficients at 24 keV are 5.6, 5.6 and 171 cm -1, respectively. From these values, the concentration of Y203 at three bright.~ots is estimated at about 8.7 wt%. As mentioned above, cracks and defects in a ceramic can be clearly observed, but their spatial spread is not obvious. This indicates that multi slice observation is necessary.
484
Fig. 8-30. Monochromatic X-ray CT image of the SiC fiber-reinforced Si3N4 ceramic composite ta&en at 24.0 keV. Slice width was 110 lxm. Exposure time per projection was 7.5 s.
Metal matrix composites Metal matrix composites axe becoming very useful structural materials. They are fabricated by combining two or more different materials. In the development of these new materials, it is important to examine their fundamental properties. Nondestructive inspections seem to be especially necessary for detecting defects such as debonding, cracks and voids. Threedimensional microtomography is a very useful method for these purposes, and internal structures of aluminum based composites have been observed [33,34]. Figure 8-31 shows three slice CT images of a SiC fiber reinforced aluminum based composite. The 140 l.tm diameter SiC fiber consists of a 30 p.m diameter carbon core and a SiC layer. A photon energy of 21 keV was selected, based on X-ray attenuation through the sample. The slice width was 82 ~tm and the exposure time 13.5 s/projection. The X-ray attenuation coefficients of A1 and SiC are 8.02 and 8.71 cm -1, respectively. Both can be observed distinctly in spite of there being only a 9% difference in coefficients. The dark region around the fiber, indicated by arrow A, can be considered as showing debonding of the matrix and fiber produced in the manufacturing process: it ca.n be seen in all three CT images. This indicates that debonding spreads over a large region.
485
Fig. 8-31. Cross section and three consecutive CT images of SiC fiber/A1 composite. The slice planes were perpendicular to SiC fibers.
Fig. 8-32. Six consecutive CT images of SiC fiber/Al composite. The slice planes were nearly parallel to the SiC fiber.
486 Figure 8-32 has images showing fiber damage produced by indenting with a Vickers hardness tester on a surface of a sample similar to that described above. The slice planes are parallel to the longitudinal direction of the fibers. The slice width was 35 l.tm and the slice interval was 52 ~m. The dark striped regions are carbon cores and the brighter ones around them are SiC fibers. The black square in the middle of the first slice CT image is the cavity produced by indenting. A crack in the SiC fiber just below it is clearly observed. From these results, observations of the plane perpendicular to the fibers are suitable for determining debonding of the interfaces, while studies on damage to the fiber itself can be made using parallel plane observations. The final example showing observations of internal damage of a metal matrix composite is shown in Fig. 8-33. The sample was prepared by pressing aluminum with 60 l.tm mean diameter SiC particles dispersed in it. The test piece was fractured by repeated loading. The CT sample was cut from around the fractured region. In the CT images, white spots (A) are SiC particles and the dark regions (B) are defects caused by repeated loading. The defects are concentrated near the SiC particles. It seems that the fracture occurs at the interface between the matrix and the SiC particles in this sample.
Fig. 8-33. CT images of SiC particle dispersed AI matrix composite. The sample was cut from around the region fractured by repeated loading. The bright spots (A) are SiC particles and the dark spots (B) axe defects induced by the load.
487
8.5.3. Applications to other materials As mentioned above, three-dimensional SR-CT is useful for nondestructive inspection of industrial materials. Besides these, it is applicable to such samples as archaeological artifacts, ore minerals, and fossils. As an example of these objects, internal fine structures of the Allende meteorite were observed [35]. The spatial distributions of fine structures and chemical compositions in meteorites have usually been analyzed by cutting and polishing the sample repeatedly. This takes a long time in sample preparation, and defects and partial collapse sometimes occur in the process. In three dimensional SR-CT, these problem can be avoided. Figure 8-34 shows six consecutive CT images of Allende meteorites in 20 slice planes, taken at 30 keV photon energy. The slice thickness was 37 l.tm and the slice planes were spaced at 110 l.tm intervals. The exposure time was 28.1 s/projection. The metallic minerals (bright region), matrix (silicate including iron) and chondrules, which were confirmed by comparison of the CT image and elemental mapping with X-ray microanalysis, can be clearly
Fig. 8-34. Consecutive CT images of Allende meteorite measured at 30 keV with twenty slice planes. The slice thickness was 37 l.tm and the slice planes were spaced at 110 lxm intervals. The exposure time per projection and the total exposure time were 28.1 s and 5120 s, respectively.
488 observed. The 20 observed CT images indicate that the metallic minerals surround some chondrules, and the largest chondrule has two humps and well crystallized olivine in its center. These observations suggest that three dimensional SR-CT is a useful method for identification of the intemal structure of stony meteorites. 8.6. FUTURE PROSPECTS OF M I C R O T O M O G R A P H Y USING SR 8.6.1. Spatial resolution The present spatial resolution of the SR-CT is about 10 lxm, and efforts to improve the resolution are now underway. In this section the possibility of achieving micron resolution SR-CT is discussed [36]. The spatial resolution is determined by both the detector resolution and SR source size. The SR cannot be considered to be completely parallel for micro-resolution CT imaging. The SR angular divergency (<10-3 rad) and electron beam source size (~1 mm) are not negligible, and the spatial resolution is degraded by penumbral blurring. The distance between the sample and detector is relatively long in CT, because of the sample rotation mechanics. When the source to sample distance is 20 m, and sample to detector distance is 50 mm, a source size of 2 mm results in 5 lxm resolution. However, the problem of source size and parallelism can be overcome by placing an asymmetric diffraction collimator in front of the sample. The problem of detector resolution can be solved by using asymmetric Bragg diffraction for magnification. Although small angle scattering and fluorescent X-rays may reduce the contrast and spatial resolution, these effects can also be suppressed using asymmetric diffraction for collimation behind the sample. A SR-CT with nominal micron-resolution can be achieved by using these techniques. 8.6.2. Future directions Microtomography based on SR has experienced the greatest progress in improved spatial resolution. Therefore, objects for which the microtomographic scanners are applicable are limited to relatively small and high transmissive samples, with the main constituents being light elements. On the other hand, there is a strong need for nondestructive inspections of larger samples, including heavier elements, in the fields of industrial products, metal engineering, archaeology, and biology. From this requirement, the techniques of microtomography are expected to progress in two ways: the development of techniques for larger samples with the present spatial resolution; and the search for more functional microtomography, with better spatial resolution of sub-l.tm levels and chemical bonding state imaging. For the former, an X-ray source with higher photon energy and intensity is required. Recently, the use of a 50-100 keV photon energy using the wiggler line of SR has been reported [37]. Moreover, large scale SR facilities are now being constructed, and microtomography with monochromatic X-ray above 100 keV will
489 be developed in the near future. A new reconstruction algorithm for sparse data sets using the maximum entropy method has also been reported [38] and techniques for image reconstruction of only the region of interest are under development. CT measurements of 0.5 meter sized samples will become possible by combination of the above techniques. For the second requirement, the following techniques need to be developed: 1) To micronize the X-ray source. 2) To improve the energy resolution. 3) To improve the detector's spatial resolution. 4) To develop magnification techniques for projection images, with no reduction of spatial resolution. 5) To reduce scattering of X-rays by samples. The use of SR with high brilliance and low emittance is considered to be essential. As a result, it is expected that microtomography with subfftm resolution and chemical bonding state imaging can be realized. ACKNOWLEDGMENTS We would like to thank the staff of the KEK Photon Factory for their helpful advice and for use of the synchrotron facilities. We also thank Y. Suzuki for his help for a great part of this work, K. Hayakawa for his encouragement throughout the work, and K. Sakamoto, H. Kohno and H. Shiono for their help in the scanner development. This work has been performed under the approval of the Photon Factory Advisory Council (proposal No. 86-Y014, No. 86-Y018).
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ab initio structure analysis 416 ab initio structure determination ab initio technique 424
447
Abbe's sine theorem 178 absolute method 166 absorber 84, 88, 115 absorption 100, 101,421,427, 432 absorption coefficient 101, 131,214, 222, 427 absorption correction 433 absorption cross section 433 absorption edge 97, 101, 120, 424, 444 absorption factor 412, 433 acceptance 181 acoustic delay line (ADL) 61 actinoid 165 adsorption geometry 319 AES 308 Ag/Si(lll) (v~ x x/~)R30 ~ 332 agreement index 410 AI 484 A1203 483 AlxGal_xAs 274 angle-dispersive diffraction 420, 435 angle-disperswe method 408, 417, 418 angle-disperswe neutron diffraction 424, 427, 444 angle-dispersive neutron-powder diffraction 419, 425 angle-disperswe powder diffraction 427, 428 angle-dispersive synchrotron diffraction 444 angle-dispersive synchrotron X-ray powder diffraction 423 angle-dispersive X-ray diffraction 408, 419, 427 angle-dispersive X-ray powder diffraction 418 angular width 86 anisotropic broadening 429 anisotropic thermal motion 413 anomalous dispersion 424 anomalous scattering 210, 424 APS 291,447 ARPEFS 342 ARUPS 308 AsF5 246, 248 AsH3 238 asymmetric Bragg magnifier 97, 471 asymmetric function 430, 431 asymmetry parameter 430 atomic layer epitaxy (ALE) 238 atomic scattering factor 82, 105, 413,417, 424
Auger electron 101, 212, 213 Auger electron yield 310 Auger yield 227 axial divergence 426, 430
back scattering 421 back-scattering amplitude 216, 242, 243 background 97, 99, 104, 105, 120, 406, 409, 410, 412, 415, 421,435 background function 435 background parameter 415, 435, 438, 443 bacteriorhodopsin 388 beam hardening effect 462 beam lifetime 45 beamline 59, 81, 93 Beamline 4A (BL-4A) 93, 127, 140, 145 bending magnet, see also dipole magnet 29, 209 bent crystal monochromator 73 beryllium window 63, 65, 81, 93 betatron function 42 betatron oscillation 23, 41 binding energy 212 biomembrane 384 Bi2Sr2CaCu208 265, 267 BL-15A 372, 376, 383, 384, 388 BL- 10C 372, 374, 376 Bloch theorem 221 bond angle 439, 442 bond-valence sum 442 bovine serum albumin (BSA) 392, 394 Br2 246, 248, 249, 251,252 Br3 250 Br5 250 Bragg angle 85, 86, 214, 222, 413, 419, 428 Bragg reflection width 87 Bragg-Brentano mode 426 Bragg-Brentano parafocusing method 418 Bragg-Brentano-type X-ray powder diffraction 433 Bragg-Fresnel lens 183 Bragg's law 85 branch beamline 61 bremsstrahlung 45, 105, 122, 211 brightness 21 brilliance 21 bunch 23, 44 bunch length 44
494 CaLaBaCu30y 253 calibration 117 CAMAC 377, 378, 391 Cambridge Electron Accelerator 80 capillary 425, 426 capillary geometry 425, 427, 431 capillary tube 203 central orbit 41 ceramic composite 483 channel-cut crystal 87 channel-cut monochromator 432 channeltron 75 characteristic X-ray diffraction 444 characteristic X-rays 213,406, 418, 419 charge coupled device 470 Chasman-Green lattice 47 (CHBry)x 248, 249, 251,252 chemical relationship 439 chemical speciation 187 CHESS 80, 110, 116 chromaticity 34 chromophore retinal 388 circular polarization 19 C1/Ni(100) c(2x2) 339 CI/Ni(lll) (x/~ x x/~)R30 ~ 341,345 closed orbital distortion (COD) 34 CO/Na/Pt(111) 318 CO/Pt(111) 318 coarse grain 426 coefficient matrix 409, 436 coherent scattering 104 coherent scattering length 413, 417, 418, 444 collimation 79 collimator 420 combined refinement 436, 438 composite crystal 447 compositional ratio 407 Compton-modified X-ray 418 Compton scattering 105, 121 computerized tomography 454 concave mirror 175 conjugate-direction method 436-438 constraint 409, 440 continuum radiation 79, 119 convergent-beam electron diffraction 441 coordination number 442 core excitation 212 correlation function 355 cosine Fourier series 435 criteria of Currie 108, 110 critical angle 81, 82, 122, 143, 145 critical angular frequency 9 critical (characteristic) energy 9
critical wavelength 9, 82, 88 cross field undulator 53 cross-sectional radius of gyration 359, 360 crystal analyzer 424-426, 431 - geometry 426 crystal chemistry 440 crystal diffraction 71 crystal monochromator 85, 96, 116, 183, 187, 191, 418, 419, 424 (2 + 2) crystal monochromator 295 crystal-structure factor 412 crystal-structure parameter 415 crystallite size 421,429, 432, 447 crystallography 210 CT value 462 cubic anvil 422 cumulant expansion 245 curve fit analysis 243 curved wave 216, 243 cylinder 358, 363 cylindrical sample 435 cytoskeleton 387 D synthesis 440, 442 damping factor 437 Darwin width 223 Debye equation for non-interacting particles 400 Debye model 219, 245 Debye-Scherrer geometry 425, 433 Debye-Scherrer method 426 Debye-Scherrer mode 426 Debye-Waller 219 deflection (K) parameter 25, 50 demagnifying focusing mirror-monochromator 377 deoxy-Hb 271 deoxy-Mb 271 deoxyhemoglobin 269 depth profile 136, 143, 166 detection limit 97, 100, 107, 122, 127 determination limit 108 differential scanning calorimetry 383 diffraction grating 173 diffractive optics 173, 183 L-c~-dipalmitoylphosphatidylcholine 384 dipole magnet 23 dipole radiation 4 dipole transition 210 disk 364, 395 dispersion 42 dispersion-free straight section 34 dispersion function 42 dispersive XAFS 229 distance distribution function 355, 361-368
495 dithiothreitol (DTT) 392, 394 DORIS 80, 117, 379 double-crystal monochromator 72, 87, 94, 223, 235, 374 double logarithmic plot 361-368 doubly-focusing mirror 374 DPPC 384 DPPE 383 Durbin-Watson d-statistic 411 dynamical theory of diffraction 86 EELS 308 effective coordination number 319 effective ionic radius 442 Einstein model 219, 245 Einstein temperature 326 electrical field vector 214 electron density distribution 354 electrostatic energy 442 elemental mapping 462, 477 ellipsoid 357, 358, 362 ellipsoidal mirror 171, 174, 177-184, 202 elliptical cylinder 357 elliptical mirror 175, 182 emittance 21, 42, 43, 46 energy-dispersive diffraction 422, 423 energy-dispersive methods 93, 104, 107, 408, 417, 422 energy-dispersive synchrotron X-ray powder diffraction 422 energy-dispersive X-ray diffraction 427 energy resolution 86-88, 191,202 enzyme 269 ESD 308 ESRF 291,447 estimated standard deviation 409, 417, 437, 438, 447 EXAFS, s e e extended X-ray absorption fine structure excess scattering amplitude 369, 370 excess scattering density 369-371 excitation effeciency 99-101, 103 EXELFS 308 extended X-ray absorption fine structure (EXAFS) 189, 196, 208, 212, 220, 226, 239, 246, 250, 309, 406, 422 - temperature dependent 325 extinction 412, 421,433 f-functional polycondensation 400 false minimum 417, 438, 439, 444 FeCI3 248 filtered back projection method 457 flat minimum 438, 444 fiat-plate geometry 425, 427, 431
flat-plate sample 433 flat-plate X-ray sample 427 flat sample 420 flat-specimen effect 419 Fletcher's algorithm 437 fluorescence 435 fluorescence detection 226 fluorescence radiation 426 fluorescence X-ray 212, 227, 228, 418 fluorescence XAFS 226 fluorescence yield 101, 103, 227, 310 focal-size effect 419 Fourier filter method 447 Fourier synthesis 440, 442 Fourier transform 8, 207, 242, 251,255, 259, 280, 355, 456 fractional coordinate 406, 413,427, 439, 446 free electron laser (FEL) 48, 54 Fresnel's formula 123, 137, 147 fringes of equal inclination 143, 144 frog sartorius muscle 388, 390 full-energy injection 23, 35 fundamental parameters method 118 FWHM 406, 415, 417, 420, 422, 426-429, 431,444 GaAs 218, 244, 246, 274, 281 GaAs/Si(001) 278 GaAsxPl -x 274 GaP 218,246 GaSb 218, 246 Gauss function 427, 428, 431 Gauss-Newton method 436, 438 Gaussian 423, 429, 431 Gaussian broadening 429 Gaussian chain 359 Gaussian FWHM 428 gel 382, 384, 399, 400 gel point 400 GeO2 211,220, 241-243, 274 geometric relationship 439 Ge4Si4 283 GeSi superlattice 281 glancing angle 81, 82, 122, 135, 142, 147 global minimum 417, 438 global parameter 415 goodness-of-fit 410, 411 grazing exit detection 201,203 grazing incidence 142, 158, 173-175, 177 grazing-incidence geometry 233 group refinement 439 GSAS 429, 435 Guinier approximation 359, 360 Guinier diffraction data 444
496 Guinier diffractometer 426 Guinier method 418, 419 Guinier plot 361-368, 393, 396 - for cross-section 363, 367 for thickness 364, 368 -
388 hard constraint 439 hard X-ray 81 harmonic number 23, 43 harmonic oscillator 219 heat damage 195 helical undulator 53 heme-ion 269 high-pass filter 81, 84 high-Te superconductivity (HTSC) 252 high-Te superconductor 221,252, 405, 447 higher harmonics 226 hollow cylinder 367 hollow disk 368 hollow ellipsoid 366 hollow sphere 358, 365 HRTEM 406, 441,447 HRTEM image 440 hydrated phosphatidylcholine 382 hydrated phosphatidylethanolamine 382 hydrosilylation 399 Halobacterium halobium
12 246 13 247 15 247 IEXR 308 Ikeda-Carpenter function 432 ILL 419 imaging plate 378-380, 384, 424, 426, 431 impurity analysis 199 in situ diffraction experiment 406, 422 in vivo
210
InAs 238 InAsP alloy 238 incident-beam monochromator 431 incident intensity 412 incident X-rays 423 incoherent scattering 104, 105, 435 incommensurate structure 446 incremental refinement 436, 438 indexing 424, 440 individual profile-fitting 414, 415 Inductively-coupled plasma (ICP) 274 rejector 35 InP 274, 278 INS 308
InSb(lll) (2x2) 331 insertion device 48, 208, 235 integral reflecting power 86, 87 integrated intensity 406, 407, 409-411,413-416, 425, 426, 429 intensity gain 173, 181 mteratomic distance 439, 442 Intermediate Lorentz function 428 internal intensity standard 445 internal standard 129 mterparticular interference 370 mtramolecular bond distance 317 mvariant 356 InxGal_xAs 274, 277 ion-trapping effect 44 ionization chamber 74, 75, 91-93, 225, 226 IPES 308 IPNS 432 IRAS 308 isosceles triangular prism 357 isotropic thermal-displacement parameter 446 isotropic thermal motion 413 JAERI 419 Jahn-Teller-type distortion 275 Johann-type monochromator 73 Johansson-type monochromator 73, 419 joint refinement 446 K absorption edge 101,463 K parameter, see deflection parameter k-space 212, 242, 255 Keating's potential 278 KEK 422 KENS 432 Kiessig structure 143 Kiessig's method 144 kinematical diffraction theory 86 kinetic equation of association and dissociation Kirkpatrick and Baez mirror 176, 183, 201 Klein-Nishina's formula 105 klystron 38 Kratky plot 361-368, 401
398
La2CuO4 221,222, 254-256 Lamor formula 11 lanthanoid 165 Larson-Von Dreele formulation 430 lattice 42, 46 lattice anomaly 264, 268 lattice parameter 405, 406, 412-416, 438, 440-443, 445 lattice-plane spacing 408, 421,446
497 L a 2 _ x M x C u O 4 _ y (M=Ba, Sr)
254 layered structure 142 least-squares method 413 least-squares refinement 410 LEED, see low energy electron diffraction linear absorption coefficient 82, 418, 433 linear accelerator 23, 37 linear equality constraint 441 linear parameter 438 linear photodiode array 229 linear polarization 18, 79, 81, 93, 95, 97, 105, 123 liquid crystal 382, 384 liquid encapsulated Czochralski (LEC) 274 liquid-phase epitaxy (LPE) 274 lithium drifted silicon detector, see Si(Li) detector local minimum 417, 436, 438 long-range order 210 Lorentz factor 412, 419, 421,424 Lorentz function 427, 428, 431,432 Lorentz polarization 248 Lorentz transformation 6 Lorentzian 429 Lorentzian Scherrer broadening 430 low emittance 35, 46, 235 low energy electron diffraction (LEED) 217, 221, 308 low-pass filter 66, 81, 82 lysozyme 371 M-intermediate 388, 389 Madelung energy 259, 442 magnetic atoms 412, 415, 418 magnetic form factor 418 magnetic moment 415, 418 magnetic scattering 412, 418 magnetic structure 420, 446 magnetic-structure factor 412 magnetic-structure parameter 415 main beamline 59 many-body effect 220 March-Dollase formulation 434 March-Dollase function 445 Marquardt method 436-438 Marquardt parameter 437 mass absorption coefficeint 463 matrix effect 128 matrix treatment 151 Maxwell's equation 157 Mb(II)CO 272 Mb(II)O2 272 Mb(III)CN- 272 Mb(III)OH- 270 McPherson monochromator 68
MDL 195 mean free path 215 mean-square displacement (MSD) 219, 245 mean-square relative displacement (MSRD) 215, 219, 245 MEIS 308 meridional plane 174 metal matrix composite 484 metalloprotein 269 meteorite 487 method of nonlinear least squares 415 metric tensor 413 microchannel plate 75 microfocus X-ray tube 453 microtomography 454 microtron 36, 39 microvilli 386, 387 MINREF 436, 446 mirror 66, 83 mirror formula 174 Mo 477 model function 408, 412, 437, 447 moderator 420, 431-433 modified Bessel function of the second kind 12 modified Lorentz function 428 modified Wadsworth monochromator 68 modulated structure 446 modulation transfer function 473 molecular adsorption 323 molecular beam epitaxy (MBE) 238, 278, 287 molecular mass 369-371 molecular orientation 314 Moller scattering 45 monochromatic beam 81, 85 monochromatic excitation 96, 97, 103, 110, 122, 130, 165 monochromator 222, 225, 226, 374, 377, 385, 419, 420, 422, 424-426, 465 monolayer (ML) 238 mosaic crystal 86, 88 muffin-tin potential 214, 242 multilayer 173, 185 multiphase capability 445 multiple scattering 217, 221,246, 422, 447 multiplicity 412 multipole wiggler 31, 49, 235 multiwire proportional counter 75 Mylar film 85 myoglobin 236, 270 N/Cu(100) c(2x2) 326 N2/Ni (110) 315 N/Ni(100) p4g(2x2) 322
498 natural emittance 43 Ndl.4Ce0.2 Sr0.4CuO4 _ 6 257 Nd2_xCexCuO4_/~ 253, 258 near-edge X-ray absorption fine structure (NEXAFS) 210, 309 needle-shaped crystal 434 network 399 neutron diffraction 210, 219, 412, 413, 417, 418, 444 neutron powder data 438 neutron powder diffraction 420 non-monochromatic beam 81, 115 non-monochromatic excitation 96 nonlinear least-squares method 407, 436 normal equation 409, 436 NSLS (National Synchrotron Light Source) 26, 80, 107, 424 nuclear emulsion 74 nuclear scattering 412 nuclear structure 446 Nyquist theorem 459 O2/Ag (ll0) 318 O2/Cu(100) 318 O/Cu(100) "c(2 x 2)" 326 O/Cu(100) (x/2x2x/2)R45 ~ 326 O2/Pt(111) 318 occupation factor 406, 412, 413, 420, 427, 429, 439, 441,446 optical klystron 58 optical path difference 144, 145 optical potential 219 organometallic vapor phase epitaxy (OMVPE) 238 P polarization 153 pair correlation function 245 pair distribution 219 parafocusing geometry 418 parallel-beam optics 424 Parratt's method 146 partial electron yield 311 particle scattering 356 particle scattering factor 358 pattern decomposition 415 Patterson function 386 Patterson map 387 Pawley method 414 Pawley refinement 415 peak asymmetry 415, 430, 431 peak broadening 419, 429 Pearson VII function 427, 428, 431 penetration depth 123 penumbral blurring 488 phase factor 355
phase shift 215, 242 phase stability 43 phase transition 382 photo-stimulable phosphor screen s e e imaging plate photo-stimulated luminescence (PSL) 378 photoelectric absorption 100, 106 photoelectric effect 100, 105 photoelectron 105, 202, 212, 213, 215 photoelectron multiplier 74 photographic film 75 Photon Factory (PF) 29, 80, 92, 93, 110, 115, 116, 122, 127, 136, 145, 158, 166, 223, 235, 236, 372, 376, 383, 384, 388, 392, 396, 422, 425, 462 photon flux 181 photoreceptor 387 PIXE 110, 118 plane grating monochromator 69 plane undulator 52 plane wave approximation 214 plate sample 418, 435 plate-shaped crystal 434 polarization 10, 18, 207, 209 polarization factor 412, 419, 424 polyacetylene 246 polyethylene 380, 382 polyethylene terephthalate 379, 380 polymer blends 198 polymer gel 399 porphyrin 269 position-constrained pattern decomposition 415 position-sensitive detector (PSPC) 75, 376, 388, 392, 418 position-unconstrained pattern decomposition 415 positron 39 powder extinction coefficient 415, 434 Poynting vector 157 preferred orientation 406, 409, 412, 415, 420, 426, 427, 434, 435, 447 preferred-orientation function 434 PREMOS 446, 447 primary extinction 433, 447 profile parameter 414 profile-shape function 412, 427-429, 431-434 profile-shape parameter 415, 431,438, 441 proportional counter 74 proton excitation 110 proximal histidine 269 PSD 418, 419 pseudo-observation 439 pseudo-Voigt function 427-429, 431,432 PSPC 191 pulsed-neutron beam 420 pulsed-spallation-neutron source 409, 431
499 purple membrane 388 pyridine/Ag(111) 316 pyrolytic graphite 88 Q-scan I, II 294 quadrupole magnet 23 quantitative analysis 441,445 quantum lifetime 34 quaternary alloy 277 R factor 410, 441,442 radial distribution function 210 radiation damage 118 radiation damping 42 radiofrequency (r0 cavity 23, 36 radius of gyration 359, 393 Radon transform 454 Rayleigh scattering 104, 121 RBS 308 reactor neutron source 409 receiving slit 419, 424-426 reciprocal lattice parameter 413 reconstruction algorithm 455 rectangular prism 357 reflection coefficient 149, 150 reflection curve 80, 82, 140, 142-145 reflection high energy electron diffraction (RHEED) 282, 308 reflection width 86 reflective optics 173, 182 reflectivity 66, 123, 125, 147, 149, 150, 152, 156 refractive index 81, 123, 144, 149 regular polygonal prism 357 resolution 409, 420-423,426, 447 resonant Raman scattering 120, 121 resonating valence bond (RVB) 268 restraint 439, 441 rf bucket 43 RHEED, s e e reflection high energy electron diffraction RIETAN 435-438 Rietveld analysis 409, 411,413,417, 418, 422, 423, 427, 433,438, 439, 441-444, 446, 447 Rietveld method 405, 407-409, 413-416, 419, 424, 441,446, 447 Rietveld refinement 409, 410, 415, 438, 440-442, 444, 445 rigid body 439 rocking curve 212 rod-like molecule 359 roughness 83, 125, 142, 143, 149 Rowland circle 176 Rowland-type monochromator 69
S/B ratio 97, 118, 122, 127 S/Ni(lll) (5V~• 321 S/Ni(100) c(2• 319, 326, 345 S/Ni(110~ ~2• 326, 347 S polallzation 152, 154 sagittal bending 235, 238 sagittal focusing 88 sagittal plane 174 sample transparency 430 satellite 193 SAXES 374 SAXS, s e e small-angle X-ray scattering scale factor 412, 438, 445 scanning incident-beam monochromator 422 scattered intensity 354 scattered radiation 93, 96, 104 scattering amplitude 214 scattering power 380 scattering vector 354, 407, 421,430, 435 scintillation counter 75, 93 second-generation storage ring 210 secondary excitation 128 secondary extinction 406 selected-area electron diffraction 440, 441 selective excitation 100, 123 self absorption effects 189 self doping 263 separatrix 43 SEXAFS 309 Seya-Namioka monochromator 68 shape resonance 221 short-range order 207, 210 SiC fiber 480, 484 SiC mirror 84 Si(Li) detector 75, 107 Simpson's rule integration 430 simulation 440 simultaneous refinement 444 simultaneous Rietveld refinement 443 Si3N4 480, 483 single-crystal monochromator 115 single scattering 214, 217, 244 size broadening 429 slack constraint 439 small-angle X-ray scattering 353, 370 Snell's law 82 (SN)x 248 soap film 84 soft constraint 439 soft X-ray 81, 91, 93, 165 solid-state detector (SSD) 75, 93, 228, 233,422 Soller collimator 425 Soller slit 424, 425
500 Soller-slit geometry 426 space group 440, 441 SPEAR 109, 110 specimen-transparency effect 419 sphere 358, 361 spherical aberration 176 spherical wave expansion 214 spin-equilibrium 269 spin gap 268 split profile-shape function 431 SPring-8 (Super Photon Ring) 32, 291,447 squid rhabdomes 384 SRS 80, 110, 116 SSD, see solid-state detector SSRL 80, 109, 121, 136, 425 standing wave 123, 166 step intensity 417 step width 417 stopped-flow 231,392 storage ring 23, 41,208 strain 421,429, 441,447 strain broadening 429, 430 strained-layer superlattice (SLS) 281 strip dinode 74 structural model 410, 415, 416, 439, 441,442, 447 structural parameter 439 structure factor 412, 415, 434, 446 structure model 409, 440-442 structure parameter 405-408, 410, 412-4 17, 424, 441-445 subtraction method 464 sum-of-squares 438 sum of weighted squares 407 superstructure 446 surface reconstruction 321 surface relaxation 341 surface-sensitive XAFS 232, 233 surface X-ray diffraction 327 symmetric profile-shape function 427, 428, 430 synchrotron 23, 36 synchrotron oscillation 43 synchrotron powder diffraction 427 synchrotron radiation 1, 79, 207, 248 synchrotron X-ray diffraction 418, 423, 431,444, 447 synchrotron X-ray powder data 438 synchrotron X-ray powder diffraction 407, 423,424, 427 synthetic diamond 200 synthetic multilayer 91, 166 T-type compound 259 T~-type structure 261
T*-type structure 261 t - J model 268 T - R transition 273 takeoff angle 189 TDS 422, 447 temperature factor 413 temperature jump 394 thermal-diffuse scattering 422 thermal-displacement parameter 406, 412, 413, 421, 423, 429, 439 thermal expansion coefficient 326 thermal motion 407 thermal parameter 427 thermal stability 83 thickness radius of gyration 359, 360 thiophenol/Ni(100) 323 Thomson factor 369 Thomson scattering 104 threshold spectroscopy 193 time-resolved SAXS 380, 382 time-resolved WAXS 380 T12Ba2CaCuO8 264 tobacco mosaic virus coat protein (TMVP) 394, 395, 399 TOF (time-of-flight) 420 TOF (time-of-flight) neutron diffraction 412, 421, 433, 444, 447 TOF (time-of-flight) neutron-powder diffraction 420, 431,433, 444 data 444 top-seeded solution growth (TSSG) 254 total reflection 80, 82, 232 total reflection method 79, 122, 127 total reflection mirror 81, 84, 87, 88, 115 trace element analysis 79, 100, 133 trans-(CH)x 248, 251 transition layer 142, 148, 149, 161 transmission mirror 84, 88, 115 triaxial body 356, 358, 360 triple bend achromat 48 two-phase system 370 TXRF method 122, 135
-
undulator 24, 49, 166, 190, 193, 291,292 unpolarized X-ray 104, 105 UPS 308 variable metric method 436 VEPP-4 165 Victoreen's function 240 virtual crystal approximation (VCA) 274 Voigt function 427, 428, 431,434 voltage-to-frequency (VF) converter 225 VUV monochromator 68
501 wave vector 213 wavelength-dispersive method 93, 107 wavelength-dispersive spectrometer 191 wavelength scanning X-ray diffraction 332 wavelength shifter 49 weight-average degree of polymerization 400 weight fraction 445 weighted sum-of-squares 417 weighting 407 Weissenberg camera 426, 431 white line 220 wide band-pass monochromator 88, 91, 93, 96, 115, 116 wiggler 24, 79, 93, 108 Wolter mirror 177, 178, 202 X-ray absorption fine structure (XAFS) 186, 187, 189, 207, 210, 225, 226, 237, 253, 308 X-ray absorption near edge structure (XANES) 189, 197, 198, 201,207, 208, 210, 220, 221,237, 251, 252, 254, 255, 257 X-ray absorption spectroscopy (XAS) 207-209, 230 X-ray attenuation coefficient 454, 463 X-ray CT 454 X-ray diffraction 210, 417 X-ray filter 228
X-ray fluorescence analysis (XRF) 79 X-ray fluorescence profile 138, 142, 143, 158, 162 X-ray monochromator 71 X-ray optical elements 173 X-ray photodiode array 467 X-ray-sensing pickup tube 468 X-ray test chart 473 X-ray tube excitation 130 XAFS, s e e X-ray absorption fine structure XANES 441 XANES, s e e X-ray absorption near edge structure XAS, s e e X-ray absorption spectroscopy XMA 308 XPS 308, 441 XRD 201,308 X R F , s e e X-ray fluorescence analysis XRS-84 436 XSW 334 YBa2Cu3Oy 253 Y203 483 z-average radius of gyration 397 zero-point shift 415 zone plate 173, 183, 185
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