Advances in Electronics and Electron Physics, Volume 60

ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS VOLUME 60 CONTRIBUTORS TO THISVOLUME G. BOATO J . P. BOUTOT P. CANTINI...

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ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS

VOLUME 60

CONTRIBUTORS TO THISVOLUME

G. BOATO J . P. BOUTOT P. CANTINI 0. DRAGOUN B. R. HUNT J. NUSSLI M. P. S m w D. VALLAT N . YILDIRIM

Advances in

Electronics and Electron Physics EDITED BY PETER W. HAWKES Lahoratoire d’optique Electronique du Centre National de la Recherche Scientifiqide Toulorrse, France

VOLUME 60 I983

ACADEMIC PRESS A Subsidiary of Harcourt Brace Jovanovich, Publishers

New York London Paris San Diego San Francisco Sao Paulo Sydney Tokyo Toronto

COPYRIGHT @ 1983, BY ACADEMIC PRESS,INC. ALL RIGHTS RESERVED. NO PART O F THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED I N ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM T HE PUBLISHER.

ACADEMIC PRESS, INC.

1 1 1 Fifth Avenue, New York, N e w York 10003

United Kingdom Edition published b y ACADEMIC PRESS, INC. ( L O N D O N ) LTD. 24/28 Oval Road, London N W 1 7DX

LIBRARY OF

CONGRESS CATALOG CARD

NUMBER:49-7504

ISBN 0-12-014660-6 PRINTED IN THE UNITED STATES OF AMERICA

83848586

9 8 7 6 5 4 3 2 1

CONTENTS CONTRIBUTORS TO VOLUME 60 . . . . . . . . . . . . . . . . . . . . FOREWORD . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii ix

Internal Conversion-Electron Spectroscopy 0. DRAGOUN

I . Introduction and Historical Remarks . . . . . . . . . . . . . .

I1 . Theory of Internal Conversion . . . . . . . . . . . . . . . .

111. IV. V. VI . VII . VIII .

Experimental Methods . . . . . . . . . . . . . . . . . . . . Treatment of Experimental Data . . . . . . . . . . . . . . . Comparison of Theory with Experiment . . . . . . . . . . . . Role of Internal Conversion in Nuclear Spectroscopy . . . . . . Environmental Effects on Internal Conversion . . . . . . . . . Summary and Outlook . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

1

3 22 43 60 61 71 82 84

Diffraction of Neutral Atoms and Molecules from Crystalline Surfaces G . BOATOA N D €? CANTINI

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . I1. I11. IV. V. VI . VII .

General Outline . . . . . . . . . . . . . . . . . . . . . . . Quantum Theory of Atom-Surface Scattering . . . . . . . . . Structural Information from Elastic Diffraction . . . . . . . . . Information on the Surface Potential Well . . . . . . . . . . . Information on Surface Lattice Dynamics . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

95 98 111 129 138 146 155 156

Digital Image Processing B . R . HUNT

I. I1. 111. IV. V.

Introduction . . . . . . . . . . . . . . . . Image Restoration (Deblurring) . . . . . . . Image Data Compression . . . . . . . . . . Reconstruction from Projections . . . . . . Steps toward Image Analysis/Computer Vision References . . . . . . . . . . . . . . . . . V

. . . . . . . . . . . . . . . . . . . . .

. . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

161 163 192 207 215 219

vi

CONTENTS

Recent Trends in Photomultipliers for Nuclear Physics J . BOUTOT.J . NUSSLI.AND D . VALLAT

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . I1 . State of the Art of Photomultiplier Technology . . . . . . . . . 111. Present Situation on Main Photomultiplier Characteristics . . . . 1V. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

223 224 262 299 300

Thermal and Electrothermal Instabilities in Semiconductors M . P. SHAWAND N . YILDIRIM

I. 11. I11. 1V. V. VI . VII .

Introduction . . . . . . . . . . . . . The Thermistor . . . . . . . . . . . . Thermally Induced Negative Differential Thin Chalcogenide Films . . . . . . . Vanadium Dioxide . . . . . . . . . . . Second Breakdown in Transistors . . . Summary . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . References . . . . . . . . . . . . . .

. . . . . . . . . . . . . . Conductance . . . . . . . .

. . . .

. . . .

. . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

307 310 321 332 358 364 369 369 382

AUTHOR INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . SUBJECT INDEX . . . . . . . . . . . . . . . . . . . . . . . . . .

387 403

. . . . . . . . . . . . . . . . . . . . . . .

CONTRIBUTORS TO VOLUME 60 Numbers in parentheses indicate the pages on which the authors' contributions begin.

G. BOATO,Istituto di Scienze Fisiche and GNSM-CNR, Universita di Genova, 16132 Genoa, Italy (95)

J. P. BOUTOT,Photoelectronic Device Department, Laboratoires d' Electronique et de Physique Appliquee, 94450 Limeil-Brevannes, France (223) P. CANTINI, Istituto di Scienze Fisiche and GNSM-CNR, Universita di Genova, 16132 Genoa, Italy (95)

0. DRAGOUN, Nuclear Physics Institute, Czechoslovak Academy of Sciences, 25068 Re2 near Prague, Czechoslovakia (1) B. R. HUNT,Department of Electrical Engineering and of Optical Sciences, University of Arizona, Tucson, Arizona 85721, and Science Applications, Inc., Tucson, Arizona 8571 1 (161)

J. NUSSLI,Photomultiplier Tube Development Laboratory, Hyperelec S. A., Brive, France (223) Department of Electrical and Computer Engineering, Wayne M. P. SHAW, State University, Detroit, Michigan 48202 (307)

D. VALLAT,Electro-Optical Device Marketing Department, RTC-La Radiotechnique Compelec, 75540 Paris Cedex 11, France (223) N , YILDIRIM," Department of Electrical and Computer Engineering, Wayne State University, Detroit, Michigan 48202 (307)

*Present address: Middle East Technical University, Ankara, Turkey. vii

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FOREWORD Like its predecessor, the present volume contains a variety of articles all commissioned by Dr. or Mrs. Marton, and I thank very sincerely all the authors for the time and trouble they have devoted to preparing these thorough and up-to-date surveys. The customary list of reviews planned for future volumes is given below. Offers of articles, however tentative, are extremely welcome, especially in the field of image processing, which we hope to cover regularly in this serial publication. Critictrl Rc~~ic,ll~.s: Atomic Frequency Standards Electron Scattering and Nuclear Structure Large Molecules in Space The Impact of Integrated Electronics in Medicine Electron Storage Rings Radiation Damage in Semiconductors

Visualization of Single Heavy Atoms with the Electron Microscope Light Valve Technology Electrical Structure of the Middle Atmosphere Microwave Superconducting Electronics Diagnosis and Therapy Using Microwaves Computer Microscopy Image Analysis of Biological Tissues Seen in the Light Microscope Low-Energy Atomic Beam Spectroscopy History of Photoemission Power Switching Transistors Radiation Technology Infrared Detector Arrays Polarized Electrons in Solid-state Physics The Technical Development of the Shortwave Radio Chemical Trends of Deep Traps in Semiconductors CW Beam Annealing Process and Application for Superconducting Alloy Fabrication Polarized Ion Sources Ultrasensitive Detection The Interactions of Measurement Principles, Interfaces, and Microcomputers in Intelligent Instruments Fine-Line Pattern Definition and Etching for VLSI Waveguide and Coaxial Probes for Nondestructive Testing of Materials ix

C . Audouin G. A. Peterson M. and G. Winnewisser J . D. Meindl D. Trines N . D. Wilsey and J. W. Corbett

J . S. Wall J . Grinberg L. C. Hale R. Adde M. Gautherie and A. Priou E. M. Glaser E . M. Horl and E . Semerad W. E. Spicer l? L. Hower L. S . Birks D. Long and W. Scott H. C. Siegmann, M. Erbudak, M. Landolt, and F. Meier E. Sivowitch r? Vogl

J. F. Gibbons H. E Glavish K. H. Purser W. G. Wolber Roy A. Colclaser F. E. Gardiol

X

FOREWORD

The Measurement of Core Electron Energy Levels Millimeter Radar Recent Advances in the Theory of Surface Electronic Structure Long-Life High-Current-Density Cathodes Microwaves in Semiconductor Electronics Applications of Quadrupole Mass Spectrometers

R. N . Lee and C. Anderson Robert D. Hayes Henry Krakauer Robert T. Longo J. L. Allen I. Berecz, S. Bohatka, and G. Langer

Advances in Materials for Thick-Film Hybrid Microcircuits Guided-Wave Circuit Technology Fast-Wave Tube Devices Spin Effects in Electron-Atom Collision Processes Recent Advances in and Basic Studies of Photoemitters High-Resolution Spectroscopy of Interstellar Molecules Solid State Imaging Devices Structure of Intermetallic and Interstitial Compounds Smart Sensors Structure Calculations in Electron Microscopy

J. Sergent M. K. Barnoski J. M. Baird H. Keinpoppen H. Timan G. Winnewisser E. H. Snow A. C. Switendick W. G. Wolber D. van Dyck

Supplemenfury Vo1rrme.s: Microwave Field-Effect Transistors Magnetic Reconnection

J . Frey P. J . Baum and A. Bratenahl

Volume 61: The Wigner Distribution Matrix for the Electric Field in a Stochastic Dielectric with Computer Simulation

Quantitative Auger-Electron Spectroscopy Impurity and Defect Levels in Gallium Arsenide Potential Calculation in Hall Plates

D. S. Bugnolo and H. Bremmer M. Cailler, J . P. Ganachaud, and D. Roptin A. G. Milnes G . DeMey

PETERW. HAWKES

ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS

VOLUME 60

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.

ADVANCES I N ELECTRONICS A N D ELECTRON PHYSICS V O L . 60

Internal Conversion-Electron Spectroscopy

I . Introduction and Historical Remarks ..................................... I1 . Theory of Internal Conversion ........................................... A . Electromagnetic Transitions in Nuclei ................................. B . Physical Models of ICC Calculations .................................. C . Numerical Calculations of ICCs . . . . . . . . . ................... D . Improvements of the Physical Model .................................. I11. Experimental Methods .................................................. A . Electron Spectrometers .............................................. B . Electron Detectors .................................................. C . Radioactive Sources................................................. D . Measurements of ICCs .............................................. IV . Treatment of Experimental Data ......................................... A . Factors Limiting the Spectrum Quality ................................ B . Analysis of Measured Spectra ........................................ C Overall Uncertainty of Measured Quantities............................ V . Comparison of Theory with Experiment .................................. VI . Role of Internal Conversion in Nuclear Spectroscopy ...................... A . Transition Energies.................................................. B . Transition Multipolarities ............................................ C . Nuclear Structure Effects ............................................ D . Inverse Internal Electron Conversion .................................. VII. Environmental Effects on Internal Conversion ............................. A . Decav-Rate Variations ........................... ....... B . Variations in the Conversion-Electron Spectra............................. C . Calibration of the Mossbauer Isomer Shifts ............................ VIII . Summary and Outlook .................................................. References.............................................................

1 3 3 5 9 14 22 22 31 37 42 43 43 49 59 60 61 63 65 68 70 71 72 75 81 82 84

I . INTRODUCTION A N D HISTORICAL REMARKS The electromagnetic interaction of an excited nucleus with surrounding orbital electrons may result in ejection of the internal conversion rlrctron out of the atom . The first monoenergetic electrons emitted in 1 Copyright 0 1983 by Academic Press. Inc . All rights of reproduction in any form reserved . ISBN 0- 12-014660-6

2

0. DRAGOUN

radioactive decay were recognized by von Baeyer and Hahn (1910). Rutherford and Robinson (1913) developed the first permanent-magnet spectrograph with photographic recording, and y-ray internal conversion was discovered in 1924 (Hahn and Meitner, 1924; Meitner, 1924). Contrary to original belief, this is another means of nuclear deexcitation independent of and competing with y-ray emission. The ratio of emission probabilities h,/hy for the two processes is the internal con\3rr.yion co@cirnt (ICC). Although the first correct theoretical treatment of internal conversion had been already presented by Hulme (1932) and by Taylor and Mott (1932, 1933), the relativistic calculations turned out to be too involved to be performed on a large scale, and interpretation of the experimental data had to rely on empirical rules (Goldhaber and Sunyar, 1951; Mihelich, 1952). The pioneer work of Rose et ul. (1949, 1951), resulting in 680 theoretical ICC for the K shell, is sometimes considered to be the first peaceful application of a digital computer. (It was the relay calculator Mark I at Harvard University.) The extensive ICC tabulations of Sliv and Band 1956, 1958) and of Rose (1958) for the K and L subshells enabled experimenters to assign multipolarities to the transitions in hundreds of nuclei. This was an important contribution to the verification of nuclear models. Later, the calculations were extended to the M shell (Hager and Seltzer, 1968) and to the N , 0, P, and Q shells (Dragoun rt a / . , 1969a, 1971). In most cases, the ICCs are independent of nuclear structure, but for hindered transitions, the experimental values may deviate from the tabulated ones by a factor reaching 20. This is the nuclrar structure kenetrution) eflect in internal conversion (Church and Weneser, 1956). Although the probability of penetrating the nucleus is extremely small for outer shell electrons, the effect was observed for N and 0 electrons, too (Dragoun et ul., 1970). Slight changes of the 99mTchalf-life caused by chemical changes of the outer shell ICC were recognized by Bainbridge et ul. (1951). Bocquet rt al. (1966) recorded 30% change of the valence-shell conversion intensity when going from metallic to oxide sample. Hybridization of relatively deep molecular orbitals in UO, and UF, was proved by Grechukhin et a / . (1980) in their conversion electron study of 235mU. First theoretical attempts to explain chemical (Hartmann e f ul., 1979) and solid-state (Hartmann and Seifert, 1980) effects in internal conversion have also appeared. At first, the conversion electrons were measured in radioactive decays only. In the early 1950s, experiments were extended to Coulomb excitation with accelerated ions (see, e.g., the review by Alder er a / ., 1956). At present, the studies also cover nuclear reactions with neutrons and charged particles including heavy ions. Energy of registered conversion electrons ranges from 30 eV to 9 MeV and their intensity varies up to 5

-

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

3

orders of magnitude within one isotope. Various spectrometers have been developed exhibiting either high transmission (up to 20% of 4 4 or a high resolution. The best momentum resolution reached up to now is A p / p = 8 X lop5(Mampe et al., 1978). An interesting history of the first 50 years of the internal-conversion studies (the 1910-1960 period) was described by Hamilton (1966a) and Mladjenovic (1980). In addition, several excellent treatments of the field are available, and we shall refer to them extensively. Our aim is to summarize basic knowledge and review critically the latest achievements. We hope this review will be helpful not only to nuclear spectroscopists but also to specialists in other fields.

11. THEORYOF INTERNAL CONVERSION A . Electromagnetic Transitions in Nuclei

I . Deexcitation Modes The excited nuclear state, the energy of which is not high enough for an emission of nucleons, deexcites to a state with lower energy mainly by electromagnetic transitions via (1) emission of a single y-ray photon (2) ejection of an orbital atomic electron (the y-ray internal conversion) (3) emission of an electron-positron pair (for E,,,,, > 2m,,cc2= 1.02 MeV) (4) higher order processes (such as simultaneous emission of two photons or two electrons) In this review, we consider mainly the first two processes. Let Ae,f and Ay be the probabilities of emission of the conversion electron from the ith subshell and of the y quantum, respectively, in a particular nuclear transition. The ratio ai

=

Ae,i/X,

(1)

is the subshell internal-conversion coefficient (ICC). Since there are no electric monopole photons, the ICC is not defined for the EO transitions. For a nuclear transition of mixed multipolarity,

4

0 . DRAGOUN

where L specifies the multipolarity order (2L pole), T sets the transition type [EL (electric) or ML (magnetic)], and p Lis the admixture of the multipolarity TL.The mixing ratio 6$(L)defined as

W L ) = Ay(L)/Ay(Lmin)

(3)

is connected with the admixture p L: PL

= 6$(L)/

y

L' =L ",,"

(4)

S$(L')

For a particular transition, only one or two (and, very rarely, three) lowest multipolarities are experimentally recognizable. In certain cases, the conversion-line intensities are very sensitive to a$, but the sign of 6, cannot be derived from their measurements alone. The total ICC is the sum of the ICCs of all atomic subshells i for which the conversion is energetically possible:

a = E q i

The total probability of the transition between the two nuclear states can be expressed with good accuracy as A

=

Ay(l

+ a)

(6)

In Eq. (6) we have neglected the coefficient for internal production of an electron-positron pair, a, = According to the tabulation of Schluter and Soff (1979), cr, < 2.8 x for 0 IZ I100, 1100 IEy I 8000 keV and E l , E2, E3, M I , M2, M3 transition multipolarities. In experiment, the relative intensities of conversion electrons emitted from various subshells are advantageously measured instead of the absolute conversion coefficients, and they can be compared with the ratios of the theoretical ICCs. These are denoted, e.g., as K/L

=

(YK/(YL,

LJL,

= CYLJGCL~,

NOP/M

=

(a"+ a o

+ (YP)/(YM

(7)

2. Energy qf Conversion Electrons For the system under consideration (nucleus + electrons + photons), the conservation laws of energy, momentum, angular momentum, and parity determine the quantum characteristics of outgoing photons and conversion electrons. The transition energy E,,,, equal to the energy difference of the two nuclear states Ex and E, is Etrans =

Ex

-

Eu

=

Ey + (AEyIrec

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

5

where the subscript i specifies the bound-electron state, E y is the photon energy, Ee,i is the kinetic energy of the conversion electron, and Eb,i is the electron binding energy. The quantities (AEy)recand (AEe,i)rec are the recoil energies of the nucleus during y or e- emission, respectively, and (AEe,i)exc is the energy eventually spent to excite another bound electron in the same event. For a free atom, Ey < 8 MeV and A > 25, the recoil Ey and the values for atoms in solid samples energy (AEe,Jrec< 2 x are even smaller. In nuclear reactions and a decay, however, the products can even be knocked out from a thin target or (Y emitter, and conversion electrons are then emitted by nuclei in flight. The increase of their kinetic energy gives rise to high-energy tails of the conversion lines (Section IV,B,2). Equation (8) usually simplifies to the form

Etrans = Ey = Ee,i f Eb,i (9) Note that the electron energy measured by a spectrometer is usually smaller than Ee,idue to energy losses within the source material and some other effects (see Sections IV,A,2 and 111,C). Shifts of the order of 0.1 to 10 eV in the binding energy Eb,i of nonvalence shells can result from changes of the chemical state of the atom (see, e.g., Sevier, 1972, 1979). The atomic vacancy created in the internal conversion is filled within the time interval of sec which yields the natural width of the conversion lines in the region of 0.07-70 eV. B . Physical Models of ICC Calculations I . Point Nucleus Approximation

The present theory of y-ray internal conversion is a well developed part of quantum theory. In order to calculate the ICC according to Eq. (l), both nuclear wave functions JIN and electron wave functions JIe are necessary since, even in the lowest nonvanishing orders of the perturbation theory (i.e., first order for y and second order for e- emission),

(Higher order contributions to both A, and Ay are discussed in Section II,D,2.) The simplest approximation in ICC calculations is that for the pure Coulomb field of a point nucleus. For the conversion process, the nucleus then serves only as a source of virtual photons with given energy, angular

6

0. DRAGOUN

momentum, and parity. In this case, the wave functions JIN and JI, can be separated in Eq. (lo), and the nuclear part is equal to that for A,: UJIN7

+el

=

(12)

4Jl,lh[JINI

Equation (12) holds for every subshell and transition multipolarity. In this approximation, no structural details of the nucleus, but only its gross properties, enter the expression for the ICC, CY

= a ( Z , 7,L , E,, n ,

(13)

K)

Here, n and K are the principal and relativistic angular momentum quantum numbers of the orbital electron, respectively, and Z is the atomic number. To specify the atomic subshell, nlj w i t h j = IKI - 1/2, spectroscopic notation is often used, e.g., IS,,^, 2p3,2, 5d5,2, . . . , o r K, L,, 0,, . . . . The qualitative dependence of the ICC on the variables listed in Eq. (13) was elucidated by Listengarten (1961) and Pauli et al. (1975). It is worth remembering that, in general, the ICC is an increasing function of Z and L , and a decreasing function of E, and n . This property is retained in better approximations, too (see Fig. 1). 104

102

100

10.;

10’’ I

I

50

100

I

I

500 1000

I 1

5000

transition energy (keV)

FIG. 1. Theoretical ICCs for the K shell, Z = 60, and multipolarities E l , . . . , E4, M1, . . . , M4, versus transition energy (Rose1 et ul., 1978). Displayed ICCs correspond to the finite nucleus with atomic screening. The electron binding energy Eb = 43.6 keV determines the energy threshold for the K-shell conversion in Nd ( Z = 60). Points on the curve for the E4 mutipolarity denote the transition energies for which the ICCs were tabulated.

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

7

The point nucleus approximation is the only one in which the analytical formulas for the ICC can be derived (Rose et al., 1951; O’Connell and Carroll, 1966). Yet, the numerical calculations are rather involved, since a considerable amount of slowly converging hypergeometric series of complex variables needs to be evaluated for each ICC. A general program is available (Dragounova et al., 1969; Dragoun and Heuser, 1971). 2. Effects of the Finite Nuclear Size

Discrepancies between the experimental ICCs and those calculated for a point nucleus were explained in the classic paper of Sliv (1951) as being caused by nonphysical singularities in electron wave functions and transition potentials. Assuming the nucleus to be finite, the singularities were removed and the ICCs turned out to be rather insensitive to assumptions about nuclear size and charge distribution. The described change of the electron wave functions is the static effect of the finite nuclear size in internal conversion. As a rule, it is incorporated assuming the nucleus to be a sphere with either homogeneous or Fermi charge distribution; Eq. (12) is still applicable. In addition, there exists the dynamic effect (called also the penetration or nuclear structure effect) connected with the appearance of new internal conversion matrix elements (Church and Weneser, 1956) which were absent in the point nucleus approximation. These matrix elements, also called penetration elements, correspond to the intranuclear contribution to the conversion probability. Owing to the transverse nature of the electromagnetic field, these are no EO photons and no EO interaction with atomic electrons outside the nucleus. Thus, the internal conversion of EO transitions is entirely due to the penetration effect. For other multipolarities, the effect is usually small. Therefore, in all published tables, the so-called static ICCs are presented which treat the intranuclear conversion in one of the following two ways: (1) The no-penetration model of Rose (1958)where the dynamic conversion matrix elements are put to zero (2) The surface-current model of Sliv (1951) where these matrix elements are calculated assuming the nuclear transition currents to be confined on the nuclear surface Typically, the ICCs of these two models agree within a few percent (see, e.g., Band et a/., 1981a), and the values are in accord with the experiment. For hindered nuclear transitions, however, the measured ICCs can deviate up to factor of 20 from their tabulated values. The necessary condition for this is that the selection rules hindering both y emission and

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0. DRAGOUN

conversion outside the nucleus have little or no influence on intranuclear conversion. We note that the term no-penetration model is rather confusing since the electrons always do penetrate the finite nucleus, although the probability for this is very small. There are different formulas for the ICC in the regions 0 5 rN Ire IRA and 0 5 re IrN IRN (re and rNare the electron and nuclear radial coordinates, respectively; RN and RA are the nuclear and atomic radius, respectively). It is the contribution of the latter region only which is put to zero in the no-penetration model. This is equivalent to the assumption re 2 rN. The region 0 IrN Ire 5 RN is always included, but its contribution to the ICC is small compared to that from the RN 5 re 5 RA region. The contribution of the re IrN 5 RN region is also usually small, but for hindered nuclear transitions, that may not be the case. From several expressions published for the ICCs influenced by the nuclear structure, ai(7L), we present those suggested by Pauli (1967): a i ( ~ L=) CX~~’(TL)A~(TL)

+ b,i(L)h + b,i(L)h2

A,(ML)

=

1

&(EL)

=

1 + ali(L)q

(14)

(15)

+ azi(L)q2 + aJL)q(

+ a,i(L)t + a,i(L)(’

(16) The a f ( d )term is the static ICC calculated in the no-penetration model. The penetration effects are included in the anomaly factor Ai(7L) in which the nuclear and electron variables were separated. The nuclear structure parameters, A, q , and 5 (ratios of penetration and y r a y matrix elements) depend on I/JN, whereas the dynamic correction factors a&), and bil(L), also called the electron parameters, as well as a i O ’ ( ~are L ) functions of I/Je. Due to different penetration of the nucleus of orbital electrons with different values of angular momentum, the Ai(7L) values differ substantially for various subshells of the same shell.For the d and f electrons this penetration is extremely small and their ICCs are insensitive to the nuclear structure. When dealing with highly accurate experimental data or those for hindered nuclear transitions, attention should be paid to possible deviations of measured ICCs from the tabulated ICCs (see Sections II,C,3 and V1,C). 3 . Screening o f the Nuclear Charge by Atomic Electrons

The ICCs computed for the pure Coulomb field disagree with experiment by a factor of 2-3 and -100 for the M and P shells, respectively. Surprisingly good results were obtained with the effective Coulomb field

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

9

which enabled Dragoun et ul. (1969b, 1971) to calculate the first ICCs for the outer shells, 0, P, and Q. This is connected with the formation region of the ICC (the region of the atom where -80-90% of the ICC is formed). Band et ul. (1970) proved numerically that for all atomic shells, the formation region lies deep inside the atom in a sphere of the radius re,,

5

2 min(r,,

9

r,)

(17)

where r,, , is the distance from center of the atom to the first density maximum of the bound-electron wave function I ) ~ , ~and , r, is the analogous distance for the free-electron state contributing most to the particular ICC. Near the nucleus, the shape of a realistic electron wave function is essentially Coulomb, only the amplitude is changed due to another field behavior at larger distances. These facts elucidate the approximate proportionality relation a'io,

- I)&(O)

(18)

which has been qualitatively known for a long time (see, e.g., Learner and Hinman , 1954). Sliv and Band (1956, 1958) and Rose (1958) used the atomic screening of the Thomas-Fermi-Dirac statistical model in their K- and L-shell ICC calculations. The newer tabulations were based on various types of the Hartree-Fock-Slater self-consistent field (see, e.g., Herman and Skillman, 1963; Lu ot a / . , 1971). Dragoun et ml. (1977) utilized the relativistic Hartree-Fock model where the effects of electron exchange are treated rigorously. Analytic formulas for the inner shell ICCs of neutral atoms or ions in the case of screened Coulomb potentials were derived by Bunaciu et a / . (1981). Current problems of the atomic screening in ICC calculations are discussed in Section II,D, 1. Detailed treatment of the present internal conversion theory and extensive quantitative discussion of all topics outlined in Section II,B of this review can be found in works of Pauli rt ul. (1975) and Band et ul. (1976a). The theoretical study of Borisoglebskii rt ul. (1972) is devoted mainly to the M-shell ICCs. Formulas for the ICC of partially occupied subshells were derived by Anderson et a / . (1980).

C. Numericul Crrlcirlutions of ICCs

I. Numc~ricalAccuracy In order to incorporate the effects of finite nuclear size and atomic term of Eq. (141, screening (Section II,B) into calculations of the aio'(~L) the following two procedures have to be carried out numerically:

10

0 . DRAGOUN

(1) Evaluate both bound- and free-electron wave functions for a spherically symmetrical isolated atom or ion (2) Obtain the static conversion matrix elements by integration of certain combinations of a large or small component of the bound- and free-electron wave functions, multiplied by the transition potential which is described by the Hankel spherical functions of the first kind For some energies and multipolarities, the ICC values are extremely sensitive to fine computation details due to the mutual cancellation of large positive and negative contributions to the imaginary part of the leading internal conversion radial integral (see, e.g., Dingus and Rud, 1968). Band et al. (1976/1977) and Rosel et u / . (1978) stated the overall of numerical accuracy 50.1% for their ICC. The latter authors checked their program by comparison with that of Dragounova et a / . (1969) for the ICCs of point nuclei with no screening. (This code is based on the analytical formulas and was extensively tested by Dragoun and Heuser, 1971.) A similar check was used by RySavy et a / . (1977) and by Hinneburg et al. (1981), who stated the overall numerical accuracy of their codes to be 0.5% (up to -1% for outermost shells) and 1% for all shells, respectively. As pointed out by M. VinduSka (private communication, 1981), such comparison may not be a decisive test for the ICCs outermost subshells since in the pure Coulomb field, the size of the atom is much smaller and the electron binding energies much higher than in a realistic field.

2. Tables of ICCs Recent tabulations of the theoretical ICCs which cover almost all cases of the nuclear spectroscopist’s interest, are listed in Table I. Detailed comparison with all previous ICC tabulations can be found in the introduction to the tables of Rosel et a / . (1978). As observed by Bhalla et al. (1966), imperfections in the interpolation procedures applied to get the ICCs for every Z value were responsible for a great part of the mysterious discrepancies between the former ICC tabulations. For this reason, the ICCs presented in newer tables have been calculated directly for every Z value. The physical models and numerical procedures of recent tabulations are not identical (see Sections II,C,3 and II,D,l), and there still remain discrepancies of a few percent among the resulting ICC. An example of the graphical comparison of Ewbank (1980) is displayed in Fig. 2. The differences among the ICCs calculated in the no-penetration model and surface current model (Section 11,B,2), however, should not be understood as discrepancies. With the help of Eq. (14) the ICC can easily be transferred from one model to another.

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

11

TABLE I TABULATIONS OF THEORETICAL ICCS" Transition energy Multipolarity order L (both EL, ML)

Atomic number

1-4

30-104 30- 104 30- 104 30-104 30-104 30- 104 3, 6, 10, 14-30 3, 6, 10, 14-30 10, 14-30 16-98h 50-100 50- 100 10-29 10-29 30-104 30- 104 30-104 30-98 30-98

5

1-5

1, 2

1-5

1-4

Zh

Atomic shell K Li-Qi" Total K

Lowest (keV)

L1-L3

26 26 2d -26 -2d

M1-M5

-2d

K Ll

15 I5 15 1 MeV 1 MeV 1 MeV 15 15 -2d -2d -2d

L,L3 K LI

LZ, L3 K L1-L3 K L1-L3 M1-M5

K-L, M,-M,'

Id

Id

Highest (MeV) 5 1.5 1.5 6 2 0.2 6 2 2 9 4.5 3.5 6 2 6 2 0.45 1.5 0.5

Reference Rose1 et a/. (1978)

Band et a / . (1978) Band et a/. (1976b)

Trusov (1972) Band and Trzhaskovskaya ( 1978)

Hager and Seltzer (1%8)

a Useful graphs of the K-shell ICCs (10 5 Z 5 100) and L-subshell ICCs (30 5 Z 5 100) for every tenth Z value and E l , . . . , E4, M1, . . . , M4 multipolarities are presented in the table of isotopes in Lederer and Shirley (1978). The comprehensive table of the L, /LZ and L / L 3 conversion-coefficient ratios for Ey 5 500 keV was prepared by Ewbank (1973). The ICCs are tabulated for all Z values within indicated intervals except the highenergy K-shell ICCs of Trusov (1972), which are given for every second Z value. For the outer shells, the highest transition energy is lower than indicated in several cases. This is the kinetic energy of the conversion electron E, .

3 . Overall Accuracy of Theoretical ICCs

Owing to the importance of internal conversion for nuclear physics (Section VI) and other branches (Section VII), it is natural to ask the following three questions: What is the overall accuracy of the present values of the theoretical ICCs? (2) Can this accuracy be improved within the framework of present theory? (1)

0 . DRAGOUN

12

FIG.2. Deviations between two sets of theoretical ICCs calculated by Rose1 et UI. (1978) and Hager and Seltzer (1968) for the L, subshell and E2 multipolarity. Both tables correspond to the finite nucleus with screening: the nuclear structure effect is treated identically [A,(T,L ) = 1 in Eq. (14)]. The graph is reproduced from the work of Ewbank (1980), where extensive comparison is also performed with the ICCs of Band and Trzhaskovskaya (1978). Symbol

Z

Symbol

Z

I

(3) Is such improvement necessary and worth the effort from an experimental point of view? The first question is considered here, the second in Section II,D, and the reasons for a positive answer to the third question follow from Sections VI and VII. The quality of various physical approximations was examined, e .g. , by Church and Weneser (1960), Listengarten (1961), Pauli et al. (1975), and Band et al. (1976a). There was a general belief that for unhindered transitions, an overall accuracy of 1-2% can be reached for the theoretical ICCs of the inner shells. Nevertheless, the scatter of actually computed values is sometimes much larger (see, e.g., Ewbank, 1980, and Fig. 2). Band et al. (1981b) reexamined these deviations and demonstrated that except for a limited number of cases, the difference among the ICCs from

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

13

the tables of Hager and Seltzer (1968), Rose1 et a / . (1978), and Band and Trzhaskovskaya (1978) is totally due to the difference in the physical assumptions underlying the ICC calculations (see also a review by Listengarten and Sergeev, 1981a). Listengarten and Sergeev (1981b) recommended the following uncertainty estimate of the theoretical ICCs for the subshells, K, L, , L,, and L, : [datheor)/%heorI

x 100%

max(A19 A31

(19)

where = $)

-

13 x 100%

(20)

[a(hole)/a(no hole)

-

13 x 100%

(21)

A1

= [a(C =

A,

=

l)/a(C

The ICCs in Eqs. (20) and (21) correspond to various treatments of the Slater exchange term and of the hole left in the atomic shell by a conversion electron, respectively (see Section II,D,I). Values of A, and A, are tabulated for 30 I2 I90, 10 keV 5 E, I400 keV, E l , . . . , M4 multipolarities (Band r t a/., 1981a). The max(A,, A3) range from 0.3 to 5.9% for the K shell and from 0.7 to 5.0% for the L subshells. Our guess for the overall accuracy of the outer shell ICCs is 5-20% where the second number refers to the atomic shell just below the valence shell. Generally, accuracy problems can be expected for (1) transitions energies near threshold; (2) the regions where a partial cancellation occurs in the leading conversion radial integrals; (3) internal conversion connected with a large transfer of angular momentum (high multipolarities and/or the d and f subshells); (4) the conversion of M1 transitions in the s subhells of heavy elements. The above estimates are meant for the absolute ICCs. For the ratios of the theoretical ICCs, especially those within one atomic shell, higher accuracy is expected (see, e.g., Dragoun and Jahn, 1967a; Dragoun et al., 1972). In nuclear spectroscopic practice, the measured ICCs or the relative conversion-line intensities are compared with theoretical values interpolated from the tables (see Table I). These tables involve up to lo4 values for various 2, Ey , rL, and atomic subshells. The ICC values cover the interval larger than lo4 is 1O1Ofrom which the region of about accessible with today’s experimental techniques. The roundoff error of the tabulated values does not exceed 0.5%. The energy scale is rather fine, and careful numerical interpolation in the log a versus log Ey coordi-

0.DRAGOUN

14

nates should not increase the uncertainty more than -1%. Typically, the ICCs are needed for an isotope other than for which it was tabulated. The ICC calculations for various nuclear radii (Sliv, 1951, for the K shell; Dragoun rt ul., 1972, for the N subshells) indicate that the isotopic effect should be <0.7% for Z z 30 and < 0.4% for Z 2 60. These limits hold for the M1 multipolarity and s-subshells; in other cases the ICC changes are even smaller. The ICCs were tabulated for the free neutral atoms, whereas the measurements were carried out for atoms in a chemical or solid-state environment. The calculations for isolated ions (Section II,D,3) indicate that ICC changes of order of lo-’ and lo’% can be expected for the innermost and outermost shells, respectively. On the experimental side there is no immediate need to calculate new tables of ICCs. Nevertheless, the computer codes should be further developed to enable the interpretation of unique experimental results exhibiting the accuracy of 5 1 % or even 5 0 . 5 % (see Sections III,D and IV,A, 1).

D . Improvements of the Physical Model I , Atomic Scrrening The authors of the 1978 ICC tables investigated the problem of what value of the parameter C should be used in the Slater exchange term Vex of the relativistic Hartree-Fock- Slater atomic potential,

vex= - C(Y3[(3/7r)pe(r)]”3

(22)

where pe(r)is the total spherically averaged electron density and (Y (not to be confused with the ICC) is the fine-structure constant. The value C = 1 is known to yield the energy eigenvalues closer to the experimental binding energies, whereas C = 3 supplied the electron wave functions closer to those of Hartree-Fock. The values 0.703 2 C 2 0.693 were derived for 42 5 Z 5 86 by Schwarz (1974), which either reproduce best the Hartree-Fock value of the atom total energy or satisfy the virial theorem. These values determined for free atoms are extensively used in molecular and solid-state calculations. Lindgren and Rosen (1968) introduced three free parameters into the local exchange potential of Slater type which are determined by minimizing the total energy of the atom. Band et ul. (1981a) tabulated relative changes of the theoretical ICCs caused by the following effects:

(1) Surface-current or no-penetration model (Section II,B,2)

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

1s

(2) C = 1 or C = 3 in the Slater exchange term (Eq. 22) (3) Hole or no hole in the atomic shell left by the conversion electron (see below)

As for the value of C , the ICCs (C = 1) were found to exceed ICCs (C = Q ) , except for the region of cancellation in leading conversion radial integrals (Section II,C,l). For the L, subshell, Z = 30 and E, = 10 keV, the difference amounts to 5% and it decreases with increasing Z , Ey , and decreasing L. Listengarten and Sergeev (1981b) pointed out that the problem of C = 1 or C = Q could be solved experimentally since, e.g., for Z = 30 and E, = 10 keV, the difference among the corresponding ICCs amounts to -lS% for the M,, M, subshells and -4-5% for the M,, M,, M, subshells. We remark that one should pay attention to interfering environmental effects since the M,, M, subshells are just below the valence shell for Z = 30. Yet, the influence of the C value should dominate as seen in Table I1 for a similar case for the N, , N, subshells in Te (Z = 52). Campbell and Martin (1975) investigated discrepancies among the theoretical ICCs near the energy threshold (see, e.g., Mai-tin and Schule, 1973; Listengarten and Band, 1974) and concluded that they cannot be explained by variation of the exchange term in the Hartree-Fock-Slater (HFS) potential. Dragoun er al. (1977) calculated the first ICC, CYHF, using the boundelectron wave functions which are self-consistent solutions of the relativistic Hartree-Fock equations with the magnetic interaction terms omitted (Grant, 1961). This model treats the exchange effect rigorously with the result that there is no longer an overall atomic potential. The authors found for MI and E2 transitions in 'i;Hg that (YHFS (C = 1) > (YHF, the deviations increasing from 0.0-0.9% for the K shell up to 27-31% for the PI subshell. Later, Dragoun er al. (1979, 1981) extended these calculations for eight transitions (50 keV IEy I412 keV, 47 IZ 5 80; E l , E2, E3, M4). The bound-electron wave functions were calculated with the program of Coulthard (1966, 1967). The a HF were found to be in substantially better agreement with 68 experimental data than are a H F s (C = 1). I. M. Band (private communication, 1980) carried out independent calculations for the investigated transitions in ls9Hg using the program complex RAINE (Band er al., 1976/1977). Their values for the subshells, 0,-05, P I , (YHF, and aHFS (C = 1) agreed with those of Dragoun et al. (1977) to within 4 and 2%, respectively. In addition, Band proved that aHF # (YHFS (C = 6), but the deviations did not exceed 10% for investigated transitions. Zilitis er al. (1981) presented a method of calculating the relativistic wave functions of a free electron with exact treatment of the exchange ef-

16

0. DRAGOUN

fect. This enabled the authors to investigate the role of the electron exchange in both initial and final states of the internal conversion process. The ICCs were calculated for the 3d,,, and 3d,,, subshells in Fe (E, = 1 keV; E l , E2, M1, M2). As expected, the ICC changes caused by various approximations of the exchange effect in the bound-electron state were found to be much larger (tens of a percent) than those for the continuum state (1-2%). The problem of considering the hole left by the converted electron, i.e., what atomic potential should be used in the final state, was studied extensively by the authors of the 1978 ICC tables. According to Band et al. (1981a), ICC(ho1e) > ICC(no hole). The deviation amounts to 6% (K shell, M4, E, = 10 keV); it depends weakly on Z and strongly decreases with increasing E, and decreasing L . Hinneburg et al. (1981) examined various “hole” models. The authors suggested yet another method that assumes that the converted electron moves in the field of the unrelaxed passive electrons without exchange interaction with them. The largest deviation of 36% (E4, E, = 1 keV) was found between the Rose (1958) ICC(ho1e) and the Hinneburg et al. (1981) ICC(hole), whereas the ICC the (no hole) result was in between the two. Listengarten and Sergeev (1981b) examined 18 (2+ + 0+) transitions for which the experimental K-shell ICCs were compiled by Mladjenovit et al. (1978a). These values, measured with 0.4- 10% accuracy and corresponding to 19.6 keV 5 E, 5 329 keV, are suitable for testing the theory since (1) the 2+ + Of transitions do not exhibit any multipolarity mixture or nuclear structure effect; (2) the difference between ICC(ho1e) and ICC(no hole) is largest for the K shell, amounting to 0.5-3.3% for the cases under investigation. The authors concluded that the ICCs, aFD(hole),are apparently in best agreement with the experiment. Here, F D denotes the Fock-Dirac atomic model (sometimes also called the relativistic Hartree-Fock model) in which the exchange effect is treated without any approximation. Some of the K-shell ICCs should be measured with higher accuracy to allow a more significant test of the theory (Section V).

2. Contributions 0.f the Higher Orders of Perturbation Theory So far, we have discussed ICCs calculated in the lowest nonvanishing order of the perturbation theory. Theoretical considerations as well as agreement among these ICCs and the measured values indicate that the contributions of higher orders will generally be small. Nevertheless, further development of the internal conversion theory together with increas-

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

17

ing acuracy of the experiment call for quantitative treatment of these effects. Let us consider first the influence of the nuclear static moments on the ICCs. The effect of the quadrupole deformation was scrutinized by Borisoglebskii and Tesevich (1980) who took into account (1) the appearance of the electronic satellite states as well as (2) the influence of the nuclear deformation on the basic bound-electron state. The effect on the K-, L-, M-shell ICCs did not exceed 0.7% even for nuclei with the Nilsson deformation parameter p 0.3. Schulyakovskii (1975) derived formulas for the ICC, (Y(~O)(TL), including effect of the nuclear magnetic dipole moment. His calculations for the M, , M,, M, subshells, L = 1, 2; 50 keV 5 E 5 500 keV, led to the ICC changes <0.7%. In particular, the M,/M, and M,/M, ICC ratios for the 316-keV-pure E2 transition in lg2Ptchanged by 0.5 and 0.3%, respectively, thus improving agreement with experiment. The vacuum polarization (the interaction with zero oscillations of the electromagnetic field) is the only quantum electrodynamical correction that can be included by means of an additional potential. Raff and Pauli (1975) studied the effect on the K-, L,-shell ICCs ( L = 1, 2; 1 keV 5 E, 5 1000 keV) and found that it is -0.1 and < 1.0% for Z = 26 and 80, respectively. Hager and Seltzer (1970) calculated the contribution to the conversion electron rate from the two-electron process which is one of the fourthorder contributions of perturbation theory. Detailed calculations were made for the 100.09-keV pure E2 transition in lSzW.Amplitudes involving EO, E l , M1, and M2 multipole interactions between two orbital electrons were considered. The resulting correction of +5.6% to the L1/L2 ICC ratio was in agreement with the average experimental anomaly observed at that time for the low-energy 2+ + O+ transitions. As noted by the authors, the above value is uncertain to -50% since contributions from the shells higher than M were neglected. In addition, some measurements confirm this anomaly, whereas others do not (Section V). A general theory of some higher order processes leading to the emission of one photon or conversion electron was developed (see Krutov and Fomenko, 1968; Krutov and Knyazkov, 1970). These processes form an additional means of electromagnetic deexcitation via intermediate electronic or electron-nucleus states (so-called electronic or electronnucleus bridges). In higher orders of the perturbation theory, several additional deexcitation modes appear:

-

(1) Since the energy does not need to be conserved in the intermediate state, the internal conversion of low-energy transitions may virtually proceed through the atomic shells for which it is energetically forbidden [Eq. (%I

18

0 . DRAGOUN

(2) In the EO transition, a single real photon may be emitted due to the multipolarity exchange which is enabled by the electronic bridge in the third order of the perturbation theory (3) The internal conversion of a particular transition is influenced by other possible nuclear transitions through the electron-nucleus bridge regardless of the point or finite nuclear size. Thus Eq. (12) holds only in the lowest nonvanishing order of the perturbation theory (4) The electron-nucleus bridge makes possible the one-electron internal conversion of the magnetic monopole transitions (0’ + O?), which is impossible in the lowest order approximation However, verification of these effects is beyond present experimental possibilities. For example, Hinneburg et [ I / . (1979) and Hinneburg (1981) obtained the following results for the 76.8-eV E3 transition in 235U: hk4) = hi2)with high accuracy and A?) = 1.5 x 105hk1).(The superscripts in parentheses denote orders of the perturbation theory up to which the contributions were included.) Thus the total ICC should not be 3.6 X lozo as calculated in the usual approximation (Table IV), but 2.4 X Unfortunately how to verify this tremendous effect experimentally is not yet clear. For the 2.17-keV E3 transition in 99Tc,the 1% effect in hb3)was calculated (Hinneburg et ( I / . , 1979). As noted by Hinneburg (1981) the correction terms of higher than third order will not contribute appreciably to the y- and internal conversion rates, since the qualitative change introduced by the inclusion of the electrons is accounted for already in A$3). Amus’ya et a / . (1968) estimated qualitatively the role of many-electron correlations in internal conversion by treating the orbital electrons of medium and heavy atoms as a high-density electron gas. The calculations for the MI subshell, E l multipolarity, and Z = 35 and 63 revealed large ICC changes near threshold (up to factor of 2 for E, 5 1 keV and a few percent for E, > 2 keV). Also, these estimates were not tested experimentally. Data for four transitions with 0.3 keV 5 E, 5 1.5 keV were discussed by Band et rrl. (1976a). Vatai and Szabo (1981) and Vatai (1981) examined the exchange correction in internal conversion which takes into account the monopole, first-order rerrrrangernent in the inner shell atomic processes (Vatai, 1970). For the electron capture and K,/K, X-ray intensity ratios, the effect was experimentally proved. The authors noted that the K / L conversion-line-intensity ratios (23 IZ 5 43; E, > 100 keV) deviated from the theoretical predictions by 6-996, similar to the K / L electroncapture ratios. The exchange correction for the latter improved agreement with experiment even in the case of the ICC ratios. A similar effect of -5% was observed by Makariunas (1981) who examined the K/L, and Ll/L3 conversion-line-intensity ratios for five M4 transitions of

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

19

80- 110-keV energy in Te isotopes. The author found much better agreement of measured intensity ratios with the ICC theory when the exchange correction for the electron-capture ratios (Bambynek et al., 1977) was included. It is obvious that a consistent and detailed treatment of the higher order effects in internal conversion is still needed. At the same time, it is important to know where their contribution to present theoretical values exceeds -0.5%. 3 . Valence Atomic Shells

The usual ICC calculations correspond to free, neutral, spherically symmetrical atoms. Thus the effects of the atomic environment and of the asymmetry arising from partially occupied subshells are neglected. The experiments on valence-shell conversion intensities and on decay-rate variations (Section VII) stimulated further progress on the theoretical side. In principle, the ICC variations can be expected not only for the valence shell but also for the inner shells due to changes in atomic screening. Andersonet ul. (1970) were the first to investigate the influence of ionization on the valence-shell ICCs. Besides the neutral Sn atom, calculations were also performed for the singly ionized free Sn ion having the valence-shell configurations 5sT./,5pt,, and 5sq,,5~:~,, respectively. The one-electron wave functions corresponding to the self-consistent fields of the atom and of the ion were utilized. The exchange effects were approximated by the exchange potential of Cowan (1968) which yields better electron wave functions of peripheral electrons than that of Slater (Eq. 22). This potential involves two parameters which were adjusted to meet the experimental binding energies of outer shell electrons. When one 5p electron was removed from the O2 valence shell, the 0,-subshell ICC for the (El, E2, M1, M2; 15 keV IEy I150 keV) cases increased by 12.3- 14.1%. The corresponding change of the bound-electron density, A$&(0)/$Es(O), was 12.7% demonstrating again the approximate proportionality between the two quantities (Section II,B,3). In environmental studies, this proportionality is utilized in two forms (see, e.g., Band et al., 1976a; Pauli ef al., 1975; Pleiter and de Waard, 1978): A a n , K / a n , K

A $ ~ K ( O ) / $ ; , K ( O )

(23)

anl,K/anz,K

$Cl%l,K(O)/$L(O)

(24)

Extensive numerical tests (see, e.g., Band et al., 1976a) revealed that these relations are often fulfilled to within 1%. It is generally not the case for low-energy transitions where the influence of atomic screening on the

-

20

0. DRAGOUN

free-electron wave functions can no longer be neglected. Salvat et al. (1980) proved that for the 14.4-keV M1 transition in 57Fe,Eq. (23) holds to within 1% even for the spin-polarized s-electrons. Using the relations (23) and (24), one can either derive the $%(O) changes from internal conversion experiments or predict the Aai/ai and AA/X variations from atomic structure calculations (see Section VII). However, as pointed out by Band and Trzhaskovskaya (1981), this method cannot yield reliable estimates for the chemical variations of the inner shell ICCs which are of the order of lo-'% and therefore below accuracy limit of the relations (23) and (24). To obtain theoretical values of Ah/X for the transitions proceeding also through the inner shells, direct ICC calculations with an accuracy of 5 0.1% are necessary. As an example of further studies of the ICC variations caused by ionization, we present in Table I1 results of RySavy et a / . (1978) calculated using the relativistic Hartree- Fock model. Eight configurations of the free Te atom and ions were considered. The ICC variations did not exceed 0.5% (the program accuracy) for the subshells K, LIP,, MI-,, and 1.0% for the N,, subshells. The 0.06% change of the total ICC is of the same order of magnitude as typical half-life changes found experimentally (see, e.g., Dostal et a / . , 1977). Hartmann et al. (1979) studied theoretically the chemical variations AX/h of the 99mTcdecay rate. This isomeric state decays through the TABLE I1 THEOUTERSHELLICCs IN

Valence-shell configuration Ion Te6+ Te5+ Te4+ Te3+ Te2+ Tel+ Te TelTe2+

0 1

0 1 2 2 2 2

2 2 1

0, 0 0 0

1 2 2 2 2 1

x lo5 FOR THE 35.892-keV M1 TRANSITION 125TeFREEI O N S ~

Atomic subshell

N4

N5

0 1

0 2

0 3

0 3

4d312

%2

5~112

5P,,*

5P312

0 0 0 0 0

4.30 4.25 4.20 4.16 4.13 4.12 4.12 4.12 4.17

3.03 2.99 2.% 2.94 2.93 2.92 2.91 2.91 2.93

0 459 844 782 725 672 626 592 369

0 0 0 25.5 44.9 39.5 34.4 29.8 22.8

0 0 0 0 0 2.40 4.09 5.09 5.51

1

2 3 2

a RySavq ef a / . (1978). These theoretical ICCs were corrected for the nuclear-structure effect [Eq. (IS)] with A = 1.3, as well as for the proper occupation number of the partially filled subshells.

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

21

highly converted 2.17-keV E3 transition. The authors assumed the applicability of Eq. (23) for this low-energy transition, and the assumption was later verified numerically by Hinneburg et [ I / . (1981). To calculate the nonrelativistic one-electron wave functions of several Tc compounds, Hartmann rt [ I / . (1979) utilized the X,-SW method (Johnson, 1972). In the X, method, the scaling parameter [denoted by C in Eq. (22)] is chosen to optimize the wave functions of separated atoms (Schwarz, 1971, 1974). Then the local potential of molecules or clusters is expressed approximately in the muffin-tin form by partitioning the space into three contiguous regions: (I) atomic spheres of central and ligand atoms, (11) the interatomic region, and (111) the extramolecular region. In regions I and I11 the spherically averaged potentials are used, whereas in region I1 the volume-averaged constant potential is employed. Boundary conditions are met by the method of scattered waves (SW). The compounds treated were the tetrahedral oxy-complex ion TcO; and the octahedral halogen-complex ions TcXi- (X = I, Br, C1, F). The calculated Aa/a variations of (0.2-3.3) x compared to the neutral isolated Tc atom were found to agree well with the experimental values of Nagel et u / . (1978). The calculations elucidated the role of the Tc bound states 3p, 3d, 4p, pval,and dvalin the total Aa/a (or Ah/h) change. In particular, the proportionality between the contribution Aadval/a, which is a measure of d-charge delocalization, and the total effect A a / a was confirmed for the compounds studied. The theoretical studies of the 2.17-keV E3 transition in 99Tc by the X,-SW method were extended by Hartmann and Seifert (1980) who simulated the Tc metal by a cluster consisting of 13 atoms. Thus the central Tc atom was supposed to experience surroundings adequate to the metallic state. The relative ICC variations of Tc metal versus Tc free atom were found to vary from -0.6 to + 0.4% for the states dval,pval,3p, 3d, and 4p, of the central-cluster atom, the total ICC variation being -0.7%. Using the previous result for the TcO; ion (Hartmann et ul., 1979), the authors derived the total ICC change of 0.40% between the Tc metal and TcO;. This is in fairly good agreement with the experimental value of 0.37% (Mazaki or id., 1980). The internal conversion theory was developed for fully occupied atomic subshells with j - j coupling among orbital electrons. Anderson et ul. (1980) generalized the formulas for the case of partially occupied subshells, which is important for the valence electrons. Atomic states with definite energy and angular momentum were considered. In a typical case when the ionization energies are much smaller than the nuclear transition energy, the ICCs of partially filled subshells were expressed as a weighted mean of the ICCs for fully occupied subshells. The weighting factors de-

22

0 . DRAGOUN

pend on generalogic characteristics of the atom. The authors also carried out numerical calculations for the 3d,,, + 3d,,, electrons in Fe and 1-keV transitions of E l , E2, M1, M2 multipolarity. The intermediate coupling of these d electrons was taken into account in the multiconfigurational approximation. The one-electron radial wave functions were obtained using the Hartree-Fock-Dirac method of the self-consistent field.

111. EXPERIMENTAL METHODS A . Electron Sprctromrters

I . Busic Chcircic,tc.ristics In order to investigate conversion electrons emitted from radioactive samples or nuclear reaction targets, a great variety of spectrometers have been developed. Usually, the differential momentum or energy spectra are measured. In some experiments, e-e or e-y coincidences and angular correlations are studied. Integral measurements combined with electronic differentiation (utilized extensively in Auger electron spectroscopy) are not employed here, since precise intensities are needed. The conversion-electron spectrometers make use of (1) semiconductor detectors, ( 2 ) magnetic fields, and (3) electrostatic fields, or their suitable combination. The Si(Li) detectors allow simultaneous measurements in a broad energy range but they do not exhibit superior resolution. This can be achieved with the other two types, mainly at the expense of tedious point-by-point measurements. This drawback can be somewhat compensated for by a position-sensitive detector (Section II1,B) placed in a focal plane of the spectrometer. When necessary, e.g., in coincidence experiments, the magnetic and electrostatic spectrometers can be constructed to have large transmission at moderate resolution. The electrostatic instruments are not used for E, > 50 keV since unconveniently high electric fields are needed. In addition, the focusing equation of present electrostatic spectrometers is not relativistically invariant, so that the parameters of the focus change with electron energy. These spectrometers, however, are suitable for the keV region as they are more easily shielded against the Earth and stray magnetic fields than are the iron-free magnetic spectrometers. In iron-core instruments, a remanent magnetism and inhomogenities of iron components cause difficulties at low energies. When comparing various spectrometers and evaluating the resolution needed to resolve close conversion lines, the following relativistic for-

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

23

mula should be kept in mind:

Here, moc2is the electron rest-mass energy, AEe and A p are full width at half-maximum (FWHM) of the line in the energy and momentum spectrum, respectively. In the contemporary conversion-electron spectrometers covering the energy range from 10' to lo6 eV, the instrumental resolution A p / p varies from 8 X to 1 x lo-'. The solid angle R is in the interval of 10-4-20% of 47r. The radioactive sources or beam targets of -0.5-mm2 to 200-cm2 area are applied and yield the spectrometer lumiosity of 10-7-10-1 cm2. Linear dimensions of the spectrometers vary from -10 cm to a few meters, the weight is between -1 kg and several tons. Torr are utilized in spectrometer chambers. For Vacuums of lop4some detectors, an oil-free vacuum is necessary. In magnetic spectrometers, a field of about 1-1000 G is needed, which is produced by currents of 10-3-102 A. Development of the magnetic @ray spectrometers over the almost 70 years of their existence was described in a monograph of Mladjenovic (1976). In 1979, the same author published an extensive review on inbeam use of magnetic spectrometers. Here, magnetic guides transporting electrons from the target to the Si(Li) detectors were also considered. Both works can be recommended especially to those who are searching for a spectrometer type suitable for their particular experiment. Modern electron spectroscopy with Si(Li) semiconductors was reviewed by Hamilton (1975) and Vylov et al. (1978). Vylov et uf. (1980) published an extensive atlas of the a-, p- and y-ray spectra of radioactive nuclei recorded with semiconductor detectors. The nonrelativistic theory of electrostatic spectrometers, including treatment of nonaxial electrons, was scrutinized in the monograph of Afanas'ev and Yavor (1978). For the relativistic calculations, see, e.g., papers of Sar-El (1968), KeskiRahkonen and Krause (1978), and Keski-Rahkonen (1978). The book of Abdurazakov r t a / . (1970) is devoted to magnetic spectrographs. Various aspects of the prismatic spectrometers (also called optical analogy spectrometers) were discussed in proceedings of specialized conferences (Kalinauskas, 1971, 1974, 1979). 2. Performance and Culculations We mention first several of the new spectrometers not covered in previous reviews. (1) Tret'yakov (1975) constructed a toroidal iron-free spectrometer

24

0. DRAGOUN

that exhibited very high luminosity of 1.5 x lop2 cm2 at the momentum resolution of 2.8 x lop4,fourfold path of electrons yielded the instrument dispersion of 3.7 m. In recent measurements of the neutrino rest-mass with this spectrometer (Lyubimov et a/., 1980, 1981), three independent proportional counters, placed in the focal plane at 7.5 mm distance between each other, were utilized. (2) Mampe rt a / . (1978) constructed a high-resolution iron-core magnetic spectrometer for measurement of conversion electrons in (n, e) reactions. The target situated in the reactor is irradiated in a thermal neutron flux of 3 x 1014 neutrons cm-2 sec-'. The 14-m distance between the target and the spectrometer resulted in a maximum solid angle of of 47r, which was compensated for by a target size reaching 40 3.4 x cm2. Electrons are detected with a set of proportional counters in the focal plane. The energy range of the instrument extends from 15 keV to 10 MeV. For the L, conversion line of the 334-keV transition in lSoSm,the which is the best reauthors achieved the resolution A p / p = 8 x ported until now for a p spectrometer. The target of natural Sm was 14 pg cmP2 thick and had an area of 0.2 x 10 cm2. It was supported by a 200-pg-~m-~ A1 foil. A demagnetization procedure developed by Jeuch and Mampe (1977) guaranteed the reproducibility of the line position < 2 X lop5,even at E , = 33 keV after having driven the field to 800 G (corresponding to E , = 10 MeV). (3) Backe et d.(1978) introduced the recoil shadow method for the in-beam spectroscopy of conversion electrons emitted by the reaction products in flight. The spectrometer consisted of an electron transport system with normal conducting solenoidal coils and a Si(Li) detector. Detection of prompt electrons was avoided by a special baffle between target and detector. The delay time was adjustable by shifting the target within the shadow region. From conversion-line intensities recorded at different target positions, the authors determined half-lives of 0.1-1 nsec for sevxn) 16*-"Yb reaction at a beam eneral levels populated in the 152Sm(160, ergy of 90 MeV. An example of the spectrum is shown in Fig. 3. (4) Arvay et al. (1980) developed a Si(Li) electron spectrometer with superconducting magnet transporters (Fig. 4). The equipment was installed on a beam of a 5-MeV Van de Graaf accelerator. The magnets providing a field up to 32 kG were made of a multifilament Nb-Ti cable. The spectrometer transmission is 76% of 47r for two detectors, and it is independent of E , up to -1600 keV at the 32-kG field. The ratio of the effective solid angle for electron detection to the geometrical one is 370. ( 5 ) Brianson et ul. (1979) constructed an electrostatic instrument for spectroscopy of conversion electrons (E, < 50 keV) emitted in the radioactive decay and heavy-ion reactions. The spectrometer consists of a

FIG.3 . Conversion-electron spectrum from the 154Sm(lfi0,n ) 170-xYbreaction with 85-MeV IfiOions measured by Backe rt al. (1978). The projected distance in beam direction between target and semicylindrical baffle was -0.8 mm. The spectrum was taken for 40 min with a 1 6 0 particle current of 2.9 nA. Mainly conversion-electron lines from excited 164.165J6fiYb levels with half-lives between 53 psec and some nsec were observable, as well as Coulomb excitation lines (CE) from the 154Smtarget. The K102 line was interpreted as the 2+ + Of transition from 162Erproduced with a (lfiO, (Y 4n) reaction.

26

0 . DRAGOUN ,9

10 11

I

3

~

12

L

5 13

, 14

0 (

20

10 .

'

.

crn

'

30 .

I

FIG.4. A cross-sectional view of the on-line Si(Li) electron spectrometer with superconducting magnet transporters (Arvay ('/ c i l . . 1980). (1) Preamplifier, (2) current lead, (3) He-gas return line, (4) liquid nitrogen, (5) liquid He, (6) Si(Li) detectors, (7) bombarding beam, (8) superconducting coils, (9) thermometer feed-throughs, (10) He filling tube, (1 1) nitrogen filling tube, (12) thermal screen, (13) superconducting switch, (14) Faraday cup, (15) target, (16) thermal anchor.

spherical retardator and a double-pass cylindrical-mirror analyzer. This combination enables the resolution A E / E 0.1% to be reached at large transmission of -3% of 4 ~ During . the spectrum measurement, the voltage on the retarder is varied (up to -50 kV with a stability of -+1 V), whereas the analyzer is kept at a constant voltage of -200 V. To obtain a

-

27

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

maximum of detection probability, the conversion electrons are accelerated in front of the channeltron to -3 keV. Using radioactive 233Paprepared by evaporation on A1 backing as source, the authors obtained resolution AEe = 40 eV at E, 7 keV and AE, = 30 eV at E, 23 keV. These values include contributions of the natural line widths in uranium of 12 and 5 eV, respectively. A similar spectrometer with linear dimensions increased by a factor of 2 was recently put into operation at the Joint Institute for Nuclear Research in Dubna (Brudanin el al., 1982). (6) Varga rt al. (1982) developed an electrostatic spectrometer in which the conversion electrons nf 0.1-20 keV energy can be analyzed with the resolution A E / E = 1 X low3to 1 x lov2.The sources of areas up to 1.5 cmz are applicable since the ring-shaped input slit serves as a virtual electron-optical object instead of the source itself. The permalloy shielding suppressed the Earth’s magnetic field inside the spectrometer to -0.3 mG. The analyzer operates at the oil-free vacuum of 3 X Torr. Electrons are detected with an electron channel multiplier having a background of 1.5 counts/min. Although the resolution AE/E < 1 X lop3was attained during test runs with an electron gun, the quality of radioactive sources has not yet allowed the authors to utilize fully the instrumental resolution in the keV region (see Fig. 5 and Section III,A,3).

-

-

800 ~

L YI

8

600-

L

0 L

4003

Z

200-

O‘

i

I

’ I

I

Electron energy (keV)

1

4

I

FIG.5. The K-shell conversion lines of the 63.1-keV transition in 169Tmmeasured with the electrostatic cylindrical-mirror spectrometer (Varga ct a / ., 1982). The 169Ybsources were prepared with a mass separator at implantation energies of 30 keV (dashed line) and -3 keV (solid line). The surface of the latter source was purified by reactive ion etching in the apparatus of Jech (1981). The spectra were taken at the instrumental resolution AEe < 30 eV. The natural width of the K-shell conversion line in Trn is -32 eV. Deterioration of the line shapes is due to energy losses within the sources (Section IV,A,2). The mass separation at 30 keV is known to provide excellent sources for electron spectroscopy at E, 2 100 keV.

28

0. DRAGOUN

(7) Bergkvist (1972) substantially improved the focusing properties of the iron-yoke 7 r d magnetic spectrometer at E, 18 keV utilizing (i) four trimming coils around the inner and outer tank wall of the spectrometer and (ii) the electrostatic corrector placed intermediately between the source and detector, which reduced the aberrations for a point source by a factor of 15 (see Fig. 6). (8) Toriyama et a / . (1974) utilized the magnetic spectrometer to measure conversion electrons emitted after Mossbauer excitation. The method introduced by Bonchev rr a / . (1969) enables one to investigate surface properties of solids. It is based on the fact that conversion electrons emerging from different depths of the absorber suffer different energy losses. In the experiment of Toriyama et a / . (1974), the iron-free 7 r d spectrometer of p = 75 cm was set to a momentum resolution of 1%. The solid angle was -1% of 47r. The investigated sample (the 91% i.e., -400-A thickness) was put at the enriched 57Felayer of 30 pg source position of the p spectrometer. The 100-mCi 57C0source of 2.5 X 2.5-mm2 area was placed outside the spectrometer vacuum chamber at 5-mm distance from the sample. The energy loss of 0-300 eV suffered by -7-keV electrons corresponded to the resonance absorption in the depth of 0-250 A. From the Mossbauer spectra, the dependence of the Fe,O,/Fe concentration ratio on the absorber depth was determined. For more recent studies of the l19Sn and 57Feconversion-electron Mossbauer spectra, see the papers by Deeney and McCarthy (1979) and Sawicka and Sawicky (1981), respectively.

-

FIG.6. Aberration patterns in the iron-yoke ~4 magnetic spectrometer at a focused energy of 18 keV (Bergkvist, 1972). The patterns correspond to the point source realized by an electron gun and they were recorded on a fluorescent screen. The radial direction in the spectrometer coincides with horizontal direction in the figure. An initial deficiency in the shape of the magnetic field, apparent in (a) (magnet alone), prevented a satisfactory performance of the electrostatic corrector as indicated by (b) (magnet + corrector). After realization of the proper magnetic field by means of trimming coils (c) magnet + coils), the desired functioning of the corrector was achieved (d) (magnet + coils + corrector). The actual radial image width in (d) is -1 mm.

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

29

(9) By means of computer calculations, Baverstam et u / . (1978) designed a cylindrical-mirror analyzer for the conversion-electron Mossbauer spectroscopy, which reached the highest luminosity at a requested energy resolution of 2.2-3.2%. The optimalization was carried out for a constant spectrometer “size,” in particular, for a constant volume of the outer cylinder. The instrument built exhibited a transmission of -6% of 47~.For the source of 10-mm diam., the luminosity was roughly 5 mm2 at AEIE = 2.3%. (10) Parellada et a / . (1981) constructed a spectrometer which allowed the simultaneous recording of three groups of conversion electrons corresponding to the resonance absorption at different depth of the sample. The calculated resolution and solid angle were AEIE = 0.47% and s1 = 2.8% of 4 ~ . Several authors examined theoretically more complicated focusing fields to exclude abberations of higher orders. For instance, Bergkvist and Sessler (1967) and Schmutzler and Daniel (1970) obtained promising results for azimuth-dependent transverse magnetic fields. Two new types of the axial-mirror analyzers were suggested: (1) a cascade analyzer consisting of the serially set external and internal mirrors with stepwise-changing radius (Kaczmarczyk, 1979), and (2) a conical exponential analyzer (Kaczmarczyk et a / ., 1979). The authors calculated that such constructions would compensate for the spherical aberration coefficients of higher than third order. The influence of the following imperfections on the resolution of cylindrical-mirror analyzers was calculated, which is of practical importance for designers of spectrometers: (1) Nonaxial setting of external cylinder (Kaczmarczyk and Wierzbowski, 1980) (2) Elliptical deformation of cylinders (Kaczmarczyk, 1980) (3) Conical deformation of cylinders (Kaczmarczyk and Pyt.towski, 1980) Dube et N / . (1981) proposed utilizing such potential distributions which both deflect and retard the electrons during their passage through the electrostatic spectrometer. These potentials could be produced by properly shaped o r coated electrodes. For a retardation factor of 10 applied in the 127” cylindrical electrostatic analyzer, the authors calculated a gain of about 100% in the transmitted intensity at constant energy resolution. The instrumental (or response) function, i.e., the spectrometer response to monoenergetic electrons, was derived theoretically (see, e.g., Siegbahn, 1965; Draper and Lee, 1977; Parellada et a / . , 1981) and measured using radioactive sources (e.g., Cretu et d.,1977) or electron guns (e.g., Baverstam Ct ul., 1978). Generally, a reasonable agreement was ob-

30

0. DRAGOUN

tained between the experimental and computer-simulated line profiles. Present knowledge of the response function of real spectrometers, however, is not good enough to allow deconvolution of measured spectra (see Section IV,B,2).

3. Improvrmrnts of Dcitrr-Collrc.tion Efjciency In conventional p spectrometers, both the source size and the baffle opening have to be reduced in order to improve the resolution. This yields an inconvenient reduction of the luminosity. Bergkvist (1964) developed a method allowing sources of extended size to be used in high-resolution work. The source surface is no longer equipotential, but a variable voltage is applied to correct for the variation of emission point with respect to the central orbit. These voltages depend on the energy of focused electrons. Ground potential in the source vicinity is achieved by means of suitable electrods. Usually, multistrip sources with a stepwise change of potential are applied. For example, Daniel et ul. (1970) utilized an extended source combined from 25 single strips of 0.5 X 20-mm2 area and obtained A p / p = 6.8 x lop4,R = 0.67% of 4rr and luminosity of 1.7 mm2 with the ( ~ / 2 )x -spectrometer. In their study of the internal conversion in the valence shell of tellurium, Davidonis rt ul. (1976) employed an extended source composed of 14 independent strips of 0.6 x 20-mm2 area. The measurements were carried out with a prismatic spectrometer. Using the same instrument and the extended source made of 13 strips (0.5 X 20 mm2 each), Kalinauskas rt a / . (1971) rezched overall resolution of 3.1 x lop4for the L-shell conversion lines at E, 104 keV. The geometrical limit to the resolution arising from the width of a single strip and that of the exit slit was 2.6 X lop4. The additional line broadening by 5 X lop5 was not far from that measured for a real single-strip source, thus demonstrating high performance of the extended source. Apparently the largest source (20 X 10 cm2) ever used in electron spectroscopy is that of Bergkvist (1972) for measurement of the tritium p spectrum in the region of E, 18 keV with regard to a neutrino mass. The source was cut into strips on which the compensating voltage was applied. The spectrometer luminosity increased by three orders of magnitude compared to usual single-source performance. For a similar experiment, Rode and Daniel (1972) utilized an extended source which was electrostatically corrected by a continuous potential change along the source. This approach eliminates (at least in principle) the aberration arising from the finite source width. Although many of the spectrometers could be used for stimultaneous

-

-

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

31

focusing of electrons of different momenta (see, e.g., Lee-Whiting and Taylor, 1957), they had to be operated as single-channel instruments. The reason was the lack of appropriate detectors. The only exception was a magnetic spectrograph, however, where the photographic recording made the precise determination of conversion-line intensities extremely difficult. Microchannel plates and special proportional counters have subsequently appeared (see Section 111,B,2) which are suitable for the detection of electron positions. Some of the commercial photoelectron spectrometers have already been equipped with microchannel plates. Fujioka e f al. (1981b) calculated the expected performance of the iron-free 7 ~ f i spectrometer with p = 75 cm as a high-resolution multichannel instrument: (1) For the line source, a normal incidence of the analyzed electrons onto a position detector was assumed, and the opening of the entrance baffle was reduced to preserve the desired resolutions. For A p / p = 1 x lop4 and 1 x lop3, the largest momentum acceptance of 2 2 . 5 and ? 5%, respectively, was found. Neglecting the finite resolution of the position detector, the authors estimated a gain in the data-collection efficiency to be 140 and 40 for the two resolution settings, respectively (2) For the extended source with voltage compensation, an additional gain by a factor of 2 2 0 with respect to the line source can be expected. The actual performance will strongly depend on the position resolution of the detector for electrons of unavoidable nonnormal incidence.

B. Electron Detectors I . Con vent iona 1 Detectors The following properties of electron detectors are important for their use in magnetic and electrostatic spectrometers: (1) The electron detection efficiency should be high in the requested energy region, known with sufficient accuracy, independent of counting rate, and stable in time (2) The background counting rate should be as low as possible. In particular, low efficiency to y rays is needed. Adjustable energy threshold enables further reduction of background (3) The detectors should have suitable shape and size allowing the full utilization of the spectrometer luminosity.

Widely used are the Geiger-Muller counters. For E, > 4Ecutoff, their detection efficiency approaches 100% and is practically independent of

32

0. DRAGOUN

energy. The background is typically 1 count min-l cmW3of the counter volume, and it can be further reduced by coincidence or anticoincidence arrangements. Shestopalova (1 962) reached background < 1 coincidence 40 hr-' using three Geiger-Muller counters in the twofold focusing 2 x .rr& spectrometer. Due to scattering of electrons in the counter placed at the first focus, the setup was applicable for E, 2 500 keV. Proportional counters allow operation at higher counting rates and reduction of background by impulse discrimination. The counter windows are advantageously made of thin organic films [for preparation techniques at thicknesses 2 2 pg c m 2 (i.e., -200 A) see, e.g., Vobecky and Dragoun, 19571. In measuring 10-keV conversion electrons with the .rr-\/2 spectrometer, Fujioka er al. (1981a) applied the gas-flow-type proportional counter of 1.4 cm3 sensitive volume (6-mm diam., 50-mm length). The window of area 5 X 35 mm2 was made of -15 pm cm+-thick film supported by a 90% transmission photoetched nickel mesh 10 pm thick. The cutoff energy of the window was measured to be 1.35 +- 0.25 keV. The Si(Li) detectors enable multichannel measurements of conversion-electron spectra at moderate resolution of -0.9 and 1.8 keV at E, = 100 and 1000 keV, respectively. These values correspond to the resolution of magnetic spectrometers, A p / p = 0.5 and 0.13%. The conversion lines are superimposed on a high background (especially at low energies) caused by Compton scattered y rays and backscattered electrons. The former can be reduced by magnetic transport systems (see Fig. 3 and Section 111,A72).Merits and drawbacks of the solid-state detectors were discussed in detail by Hamilton (1975) and Vylov er al. (1978). The windowless electron-channel multipliers (the channeltrons) are well suited for detection of low-energy electrons. They are used in the electrostatic spectrometers mentioned in Section II17A,2(Brianson et al., 1979; Varga ef al., 1982), as well as in magnetic spectrometers, e.g., by Hansen et al. (1973). The channeltron sensitivity to y rays is rather low (see, e.g., Macau et al., 1976), but the electron detection efficiency varies with energy (Fig. 7). As found by Manalio et al. (1981), this efficiency can be improved by more than a factor of 3 when the detector cone is coated with MgO. It is preferable to operate the channeltrons in an oil-free vacuum. Conversion electrons emitted by sources of activity of s 0.1 pCi are sometimes measured by proportional counters in 4.rr geometry. Kitahara et al. (1977) developed a multiwire proportional counter operating at pressures up to 30 atm. For conversion electrons of 624-keV energy, the resolution of 8.5% was achieved. Isozumi er al. (1981) built a proportional counter for conversion-

-

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

33

Electron energy CkeVl

FIG.7. Absolute detection efficiency of the windowless channel-electron multiplier (Hansen rt a/.,1972). The measurements were carried out with the magnetic spectrometer Torr. The channeltron was surrounded by a double-foil operated at oil-free vacuum, magnetic shielding serving simultaneously as electrostatic protection. The calibration was performed with an electron gun (0-6 keV) and a radioactive source of 241Am(20-60 keV). Uncertainty of displayed results is -10% for measurements with 241Am(+) and between 10% at 6 keV and 20% at 50 eV for measurements with the electron gun (0).

electron Mossbauer spectroscopy which operates at temperatures up to 800°C. Massenet (1978) described the apparatus for carrying out this spectroscopy at a temperature of 4.2 K. 2. Position-Sensitive Proportionul Counters

Development of position-sensitive proportional counters applicable to high-resolution spectroscopy of conversion electrons was reported by Yoshida et ul. (1978) who constructed a three-wire proportional counter having a sensitive volume of 120-mm length x 20-mm height x 2.5-mm thickness and filled with propane of 100-400 Torr. The resistive anodes were made of carbon-coated quartz fibers of 25-pm diam. and of 8 kR cm-' resistance. This detector, placed in a focal plane of the 75cm-radius iron-free .rr.\/z- spectrometer, covered the momentum range of f2%. Adjusting the spectrometer baffles to 0.01% momentum resolution, the authors obtained the overall position resolution of 1.34 mm [corresponding to (Ap/p)detector = 0.046%] for the K-shell conversion line of the 1064-keV transition in zo7Pb. Very recently, Yoshida et ul. (1981) constructed an improved version of this counter, the anode of which was formed by a nickel-chromium wire of 10-pm diam. and of 12 R mm-l resistance. For electrons of energy between 100 and 1000 keV, the spatial resolution of 0.5 mm was achieved, The momentum resolutions which corresponded to A p / p = 1.6 x

34

0. DRAGOUN

were obtained for E, = 45 and E, > 10 keV, of 4.7 X and < 1 x respectively. Using two counters of this type simultaneously, the authors decreased the natural background counting rate to < 1 count hr-I FWHM-' of a peak. Fujita rt a / . (198 1) developed another position-sensitive proportional counter for the above-mentioned P-ray spectrometer. This counter exhibited the position resolutions (FWHM) of 0.3 and 1 mm for electrons of energy E, > 200 and E, = 40 keV, respectively. The integral position linearity of 0.2% was reported. 3. Micuoc han n rl-Plii t r Dr t r ct ous

Although microchannel plates (MCP) are well suited for positionsensitive detection of low-energy conversion electrons, they were apparently applied only in studies of the 77-eV transition in 235U(Zhudov et a / . , 1979; Grechukhin rt a / . , 1980) and of the 2.2-keV transition in 99Tc (Gerasimov rt a l . , 1981, 1982). These authors employed a commercial photoelectron spectrometer (Kelly and Tyler, 1973) with the MCP in its focal plane. In consideration of the vital importance of position-sensitive detectors for further development of high-resolution conversion-electron spectroscopy, we summarize here some properties of the MCP and demonstrate their use in other branches of electron spectroscopy. The MCP is an array of millions of miniature electron multipliers operating independently. The electron multiplication factor is lo4 for single MCP and lo7 for two MCP in cascade (the chevron configuration). Due to the small diameter and center-to-center spacing of individual channels (8-30 pm), the MCP exhibit high position resolution. At higher energies, the incident electron can induce an avalanche in neighboring channels. This improves the detection efficiency but deteriorates the spatial resolution. The MCP are produced in disks form with diameters up to 13 cm or in rectangular configurations up to 8 X 10 cm2. The thickness of single MCP varies from 0.4 to 2 mm. Performance characteristics and numerous applications of the MCP were reviewed by Wiza (1979) and Dmitriev et a / . (1982). Detection efficiencies r ) for UV, X, and y radiation and for electrons and protons of various energy and angles of incidence were compiled by Macau et al. (1976). As for the electrons, qmax= 80-90% for E , 300 eV. The efficiency falls off rapidly at lower energies, but the accelerating grid can be applied immediately in front of the MCP. Wijnaendts van Resandt (1980) measured r) = 50 and 30% for E, = 8 and 28 keV, respectively, and estimated r ) 5 25 and 20% at 50 and 100 keV energy, respectively. At the

-

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

35

same time, the y-ray efficiency is below 2% for E, > 6 keV, which is advantageous when measuring conversion electrons in the presence of y rays. The MCP have background counting rates of the order of 1 count cm-2 sec-l even in chevron configurations at room temperature. However, this background begins to increase at pressures of >lop6 Torr because of positive-ion feedback (Wiza, 1979). An oil-free system extends the MCP lifetime, but good results were also obtained with the system cooled by liquid nitrogen. Important for the conversion-electron spectroscopy are the following MCP properties: low sensitivity to external magnetic fields, no entrance window, no necessity of cooling, large dynamic range, and superior time characteristics (an effective dead time of sec, pulse width < 1 nsec with risetime < 500 psec). Galanti r t a / . (1971) utilized MCP for an image intensifier in the electrostatic parallel-plate spectrograph for 1- 10-keV electrons. The position-sensitive detector, placed in a spectrometer focal plane, consisted of the 1.6-mm-thick single MCP, a 3-pm-thick plastic scintillator, and a 3-mm-thick piece of fiber optics followed by a photographic plate. Altogether, lo5 photons per incident electron hit the photographic plate. For electrons of 3-5-keV energy, the exposure time was reduced by a factor of 4 x lo3, whereas the resolution was deteriorated only slightly. Karlsson et a / . (1976) constructed a multichannel detector for the electrostatic spectrometer analyzing photoelectrons excited by the 21.2- and 40.8-eV UV radiation. The spectrometer had a resolution of < 10 meV and operated at vacuum of < 6 x Torr. The detector consisted of a chevron MCP, screen, and TV camera, which was connected to the PDP 15 computer via direct-memory access. The 5 x 30-mm2effective area of the MCP covered a +1.5% range around the observed mean energy. Since the efficiency of the detector varied over its area, it was not suitable to record parts of the electron spectra at fixed analyzer voltage. Instead, this voltage was incrementally increased, and every channel of the complete spectrum received contributions from each of the 256 sampling channels in the camera picture. The computer simultaneously controlled voltage increments and proper addressing of memory cells in the complete spectrum. The described multichannel detector increased the spectrometer information speed by a factor of 100 compared to the previous single-channel detector. A similar position-sensitive detector using the microchannel plates and a vidicon TV camera was developed by Kudo er a / . (1978) for the computer-controlled electron spectroscopy for chemical analysis (ESCA) spectrometer with hemispherical electrostatic analyzer. An event position on the MCP can also be derived without trans-

-

36

0. DRAGOUN

forming the electron avalanche into a lightspot. For instance, Wiza (1979) fabricated the one-dimensional resistive anode encoder shown in Fig. 8. The event location was determined by using the risetime method with commercial electronics. Integral nonlinearity of 0.3% over the 25-mm diam. format and spatial resolution of about 50 pm were found during the test with a 100-pm-wide beam of IO-eV UV radiation. The MCP can also be employed in two-dimensional position-sensitive detectors. Lampston and Carlson (1979) combined a cascaded pair (Chevron) of two 25-mm-diam. MCPs with a circular-arc-terminated resistive anode. The electrical connections at the anode corners were connected to four field-effect transistor (FET)-input preamplifiers. A test with 63Nip-rav source (EPmax = 67 keV) revealed very good linearity of the assembly. Weeks et a / . (1979) employed the MCP-vidicon camera detection syc.tem for two-dimensional angle-resolved electron spectroscopy. A 70" X 70" collection geometry with 21.5" angular resolution was achieved. Again, the information speed increased by two orders of magnitude when compared with a conventional movable single-detector

FIG.8 . Two microchannel plates in chevron configuration (usable diameter of 25 mm). together with the one-dimensional resistive-anode encoder. (Reproduced from work of Wiza, 1979.)

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

37

system. Two-dimensional position-sensitive detectors are important for conversion-electron spectrometers with curved exit slits because better combination of resolution and transmission can be achieved due to the elimination of higher order aberration terms. C. Radiouctiiv Soirvces

In order to make the best use of the spectrometers and to gain undistorted conversion-electron spectra, the radioactive sources should fulfill several requirements. (1) The source thickness should be small compared to the inelastic electron mean free path (Section IV,A,2) to avoid energy losses within the source material (2) The source backing should be thin enough and made of a low-Z material to suppress electron backscattering ( 3 ) Radioactive atoms should not diffuse into the backing or escape into the spectrometer vacuum chamber, even during lengthy measurements (4) The source strength should be sufficiently high to allow measurements of weak conversion lines; ( 5 ) The source should not involve radioactive atoms other than those required (usually one isotope, or a mixture of the isotope investigated and the energy or intensity calibration standard) (6) The source should be of appropriate shape and size to utilize fully the spectrometer luminosity (7) The source should have the necessary potential distribution on the surface to ensure proper focusing. Usually, the sources are grounded or given an accelerating or retarding potential of several kilovolts. For extended sources, more complicated potential distribution is needed. It is absolutely necessary to avoid charging the source during measurement (8) When investigating environmental effects in internal conversion, the radioactive atoms should, in addition, be in a stable and known chemical or physical state.

Obviously, some of these requirements are contradictory and their realization depends on the conversion-electron energy, half-life, preparation and chemical properties of the radioisotope, and on the aim of the experiment. To demonstrate the problem more quantitatively, we mention a few examples. First we note that the discussed source requirements are similar to those in P r a y spectroscopy. Albert and Wu (1948) succeeded in preparation of the 35Ssource of thickness < 1 pg cmP2deposited on 3 pg cm-* collodion film. Using this source the authors demonstrated

38

0. DRAGOUN I

I

I

I

I

I

I

I .o

0.8 > k u)

W z

I-

5

0.6

W 0

N A

a I

B

0.4

0.2

0

0.4452 VOLTS ON

0.2

0.4456

n

0.4460

0.4464

STANDARD R E S I S T O R

FIG.9. The L, conversion lines at 223 keV of the 238.6-keV transition in 212Pb(the I line of ThB) recorded by Graham rt crl. (1965). Three different sources were measured with identical setting of the iron-free T& spectrometer. Curve A is from a ThB source collected for 20 hr on 0.4-mm commerical A1 sheet; curve B from a source collected for 4 hr on 0.4-mm commercial Al sheet, and curve C from a source collected for 3 hr on 800-pg c m P (i.e., 3 p m ) Al foil. (Source, 1.5 mm X 20 m m ; counter, 1.5 mm X 50 mIn; baffle, 0.03%.)

that the 35S source of 5 p g cmP2 thickness only deteriorated the p-spectrum shape under 70 keV. Douglas (1949) prepared the 177Lupsources on 500-pg cm-2 A1 and 20-pg cm-2 nylon backings and observed that conversion lines corresponding to the source on the insulator surface were shifted up to 19 keV with respect to the grounded source. Practically, monoatomic layers of 212Pb(ThB)should be obtainable by collecting decay products of 22sTh(RdTh) on a backing having negative potential. The method has been widely used for calibration purposes. Yet, the shape of the L, conversion line of the 238.6-keV transition recorded by Graham r t a / . (1965) and reproduced in Fig. 9 showed some dependence on the collecting time and the backing surface. The effect was apparently connected with the atomic recoil after a-decay in the complicated decay chain of z2sTh, as discussed in monograph of Sevier (1972). The a-ray spectroscopy meets similar, though not identical, problems.

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

39

Doubtless, the a source must be extremely thin, but owing to very low backscattering of a particles, the sources can be deposited on rather thick backings of heavy elements. Thus, e.g., platinum, which allows treatment in acids and at high temperatures, is applicable. In electron spectroscopy, Pt backing leads to electron backs,cattering and, therefore, A1 or C backings are preferred if allowed from a chemical point of view. Often utilized are the aluminized organic foils. Self-supporting carbon foils of thickness down to 3 pg cmP2are commercially available. When a source of short half-life is to be prepared by vacuum deposition of a stable isotope and subsequent irradiation in a nuclear reactor, the -50-pg cmP2 carbon foils can advantageously be used since they are almost not activated and can be handled without frames (see, e.g., Dragoun et al., 1970). Auger and photoelectron spectroscopies deal with low-energy electrons of very small mean free path. Best results were obtained with vapor targets (see, e.g., reviews by Berenyi, 1976; Siegbahn, 1977; Holloway, 1980). The solid surfaces are excited from the outside and only those electrons that escape without any energy loss are examined (e.g., see the reviews by Nefedov, 1982, and Wild, 1981). When investigating noncon-ducting materials, thin metallic or carbon deposits are sometimes used as calibrants for the charge correction. Uwamino et al. (1981) measured the influence of gold deposits on the half-widths and apparent energies of photoelectron lines. The authors determined the optimum thickness of this deposit to be -6 A, i.e., 1.2 pg cm-2 (which corresponded to -9 A path length in direction to the spectrometer entrance baffle). Note that the continuous phase is formed when the thickness of the deposited gold film exceeds 90 A, i.e., -17 pg cm-2 (Ebel and Ebel, 1972), which represents absorber too thick for the photoelectrons emerging from the investigated surface layer of the sample. Bergkvist (1972) examined eventual accumulation of an electric charge on tritium sources prepared by ion implantation at 400-800 eV on A1 foils. The beam of 2 keV electrons focused by the magnetic spectrometer was allowed to pass through a deflector, one plate of which was formed by the tritiated foil and the second by the brass plate of the spectrometer material. A search for the beam displacement revealed that (1) the work functions of the two deflector plates equaled within 0.1 V and (2) no charge resulting in potential larger than 0.1 V was built up on the tritium source. Methods of source preparation are a subject of special reviews (see, e.g., Yaffe, 1962; Parker and Slatis, 1965; Dobrilovic and Simovic, 1973; van der Eijh, 1977). Extensively used are electroplating and vacuum deposition. Unfortunately, impurities are deposited simultaneously with the radioactive atoms thus increasing the source thickness. Even the commercial “carrier free” activities are often not good enough for preparing

-

40

0. DRAGOUN

the electron sources of requested quality. When only a small amount of the investigated nuclei produced in a scarce nuclear reaction, the highest yield of the source preparation method may be of primary importance. One of the best methods providing sources of outstanding purity is the electromagnetic mass separation. Although an efficiency approaching 80% was reached for several elements (Beyer et al., 1971; Latuszynski et al., 1974), the typical efficiency is of the order of percents and sometimes even lower. Schmeing et al. (1981) reported overall efficiency of the isotope separator as a function of atomic number (8 5 Z I36) of the observed activities and their half-lives (<1 sec, 1 sec- 1 min, 1 min- 1 hr, >1 hr). Contamination from the isotope of adjacent mass number is of or even better. the order of The proceedings of the tenth international conference on electromagnetic isotope separators have been published (Ravn et al., 1981). The applicability of the method to conversion-electron spectroscopy was throughoutly examined by Bergstrom et al. (1963). The projected range R, of ions of kinetic energy E into various targets can be calculated by means of general expressions derived by Lindhard et al. (1963):

+ Zi/3)-1A,(A, + A,)-, = 32.5EA,(A1 + Az)-lZ;lZ;l , + Z2/3)-1/2

p, = 166Rp(2!’3 E

(22’3

2

(26) (27)

Here, ppis the reduced projected range; E is the reduced energy; Z, and A, are the atomic and mass numbers of the ions, respectively; 2, and A, are the same quantities for the target atoms; E is expressed in keV and R, in pg cm-,. The dependence of the reduced projected range on the reduced energy was measured for amorphous materials by Kalbitzer and Oetzmann (1978) and is reproduced in Fig. 10. In conversion-electron spectroscopy, the polycrystalline aluminum foils are ofted utilized as the source backings. The ion penetration into these foils is larger than calculated from Eqs. (26) and (27) using the pp = f ( ~dependence ) for amorphous materials due to the ion channelling in microcrystals. On the surface of the A1 foils there is a natural A1,03 layer of -45 A (i.e., 1.2 pg crn-,) thickness which may be considered as amorphous. Much thicker Al,O, layers can be prepared by anodic oxidation. It should be kept in mind, however, that the A1,0, layers are nonconducting. After the ion implantation into the A1 target, a surface layer of definite thickness can be removed by electrolytic peeling (Davies et ul., 1960). Unfortunately, a disturbing tail of the concentration profile in the direction of the material is not removed. To achieve good focusing and efficiency the mass separators operate with the extracting voltage 2 3 0 kV. However, the ions can be retarded to the kinetic energy of -0.5 keV (or

-

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

41

" 2 ol

c

4

REDUCED ENERGY 6

FIG.10. Dependence of the reduced projected range ppon the reduced-energy E of ions implanted into amorphous materials [see Eqs. (26) and (27)]. (Reproduced from work of Kalbitzer and Oetzmann, 1978.)

Symbol

0

v

0

0

A

n

Ion

+ target

Bi + Si Bi + Ge Sb + Si As -+ Si As + C Ge + Si Ge + C

Symbol

Ion + target

V

P+C Au + Si Au + Al Cs + Si Eu -+ Si Gd -+ Si Tb + Si

X

+ 0

0 0

even < 25 eV) just before hitting the collector foil. This yields much lower ion penetration and significant improvement of the conversion line shape (see, e.g., Fig. 5 ) . Conversion lines of 7.3-keV energy measured by Porter and Freedman (1971) with sources implanted at 500 and < 25 eV are shown in Fig. 1 1 . A similar 57C0source was utilized by Porter et al. (1971) in their study of the shake-off accompanying K-shell internal conversion. Ions of <25-eV energy were deposited onto the cleaved and etched surface of a natural graphite crystal. At this energy, the ions could not penetrate into the graphite lattice. The crystal surface was believed to be smooth on a scale of a few atomic layers. From the collected charge of the 57-mass ions, the mean source thickness was estimated t o be <1/10 monolayer. Nevertheless, the K-shell conversion line at 7.3 keV exhibited a pronounced low-energy tail, the intensity of which exceeded 70% of the main line (see Fig. 11). One can speculate about a thin absorber layer on the

42

0 . DRAGOUN

38.20

38.24

38.28

38.32

38.36

POTENTIOMETER (momentum) UNITS

FIG. 11. The K-shell conversion-electron lines at 7.3 keV of the 14.4-keV transition in 57Femeasured by Porter and Freedman (1971). The iron-free toroidal magnetic spectrometer was adjusted to the resolution A p / p = 0.05%. Two sources of 57C0were prepared by using the electromagnetic mass separator. The source backings were the cleaved surfaces of natural-graphite crystals. The T o ions of 0-25-eV energy could not penetrate the graphite lattice and oxidized on the surface. The 500-eV ions penetrated the lattice (at least one but no more than - 5 atomic planes of graphite) and remained in a metal-like environment protected from oxidation. The K-shell binding energy of the “oxide” state is larger by 3.3 eV than that of the “metallic” state. A more pronounced low-energy tail of the line corresponding to implantantion at 500 eV was caused by greater energy losses of conversion electrons emerging from inside the backing.

source surface, but the origin of the low-energy tail in this solid radioactive source is not fully understood. As an example of special methods, we mention here 235mU samples prepared by using collection of recoiled nuclei after 239Pua! decay (Section VII). Draper and McDonald (1981) described the centrifuge method of target fabrication for in-beam electron spectroscopy. The method provides uniform targets of thickness from 0.5 to -5 mg cm-2. It is suitable for cases where vacuum deposition and mechanical rolling cannot be applied. The retrieval efficiency is 250% of material used, which is advantageous for enriched isotopes. D . MPasurements of ICCs

The relative intensities of conversion electrons corresponding to various nuclear transitions and different atomic shells or subshells are measured with the spectrometers described in Section II1,A. In conventional spectroscopy, the accuracy of these intensities ranges between 5 and 20% in dependence on the line intensity, half-life, and spectrum complexity. In special cases, accuracy of 1-2% or even 0.4-0.7% (Gelletly et ul., 1981) was reported. Numerous methods for measurement of the absolute ICCs were developed, and their applicability and accuracy were extensively discussed in

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

43

the literature (see, e.g., Subba Rao, 1966; Hamilton, 1975; Mladjenovic et al., 1978b; Hansen, 1981). For illustration, we mention here only few of the more general methods: (1)

Absolute

p- and y-ray counting, which yields the total ICC, a

= (Ip/Iy)(Ey/Ep) -

1

(28)

(2) Absolute conversion-electron and y-ray counting, which usually provides the K-shell ICCs, a K = (zeK/ly)(Ey/%)

(29)

(3) Absolute X- and y-ray counting, a K =

(4)

(ZxK /

z y ) ( ~ y / ~ x K ~

(30)

Normalized conversion-electron and y-ray measurement,

In Eqs. (28)-(31), I is the counting rate of the indicated radiation after correction for background and dead-time losses; E is the detector efficiency including solid angle; wK is the fluorescence yield of the K shell; and a, is the reference ICC, the value of which is known either from experiment or theory. In nuclear spectroscopic practice, the fourth method is widely utilized. In general, the absolute ICCs are measured with 10- 15% accuracy. In favorable cases, a few percent accuracy is attainable. In the newest edition of the Table of Isotopes (Lederer and Shirley, 1978), there are 10 calibration standards of the K-shell ICCs, accurate to 0.6-1.8 and 3 . 3 4 2 % for transition energies 88-662 and 81 1 - 1 1 16 keV, respectively. I v . TREATMENT OF EXPERIMENTAL DATA A . Factors Limiting the Spectrum Quality

I . General Considerations The usual task of conversion-electron spectroscopy is to measure energies and intensities of electrons emitted during electromagnetic deexcitation of the nucleus. These data are either used directly or they are combined with the p-, X-, or y-ray intensities to provide the absolute ICCs (see Section 111,D). For nuclear structure studies, additional information about time and angular correlation with other radiation is of importance.

44

0. DRAGOUN

Channel number

FIG.12. Conversion-electron spectrum of the '$:Lu + '$%Ybdecay (T,,, = 4.1 m) measured by Adam e r c i l . (1980) with a toroidal iron-free magnetic spectrometer. Radioactive ions of the mass A = 163 produced by irradiation of Ta target with 660-MeV protons were implanted into 5-pm Ta foil heated to 14OO0C,which resulted in enrichment of the 1 6 3 L ~ . Measurements started 10 m after the end of irradiation. A second series of measurement which started 10 m after the beginning of the first series is shown in the lower part. A group of lines at the beginning of the spectrum corresponds to the L-Auger electrons.

Special experiments supply data useful in atomic and solid-state physics and chemistry (Section VII). The main emphasis in this review is to determine precisely relative conversion-line intensities. Conversion-electron spectra are often rather complex (see Figs. 3 and 12) because tens or even hundreds of nuclear transitions can proceed in a particular isotope and many atomic subshells are involved. Even in the simplest cases, precise and detailed analysis of the spectrum is not easy because of the following reasons. (1) Quantum mechanical phenomena, such as stalistical scattering of measured counting rates, the natural width of conversion-electron lines, and many-body effects, that result in the excitation energy not being transferred to only one electron (2) Electron energy lossrs by scattering in radioactive sources and their backings or in nuclear reaction targets (Section IV,A,l)

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

45

(3) A Jinite instriimentcrl resolution resulting from the finite width of entrance and exit slits and from higher order aberrations (Section IV,B,2) (4) Impeyfections o f u real spectrometer, e.g., a departure from idealized geometry for which the focusing has been computed; mechanical inaccuracies; electron scattering on the spectrometer baffles and walls and on molecules of a residual gas; inliomogeneities in pole pieces of iron instruments; disturbing electric and magnetic fields; a departure from the predicted detector response to electrons of various energies and counting rates (5) Time instabilities of any kind, i.e., changes of conditions under which different parts of the spectrum are taken (Section IV,B,l). In the energy region above 100 keV, conventional measurements of conversion-line intensities are carried out to 10% accuracy. To achieve accuracy of a few percent one has to look carefully into possible systematic errors. The most precise measurement of the absolute ICCs seems to be that of Schonfeld and Brust (1977) who determined the atotof the 166-keV transition in 139Lawith a +0.25% uncertainty. As for the relative conversion-line intensities, the most precise results are apparently those of Gelletly et ul. (1981) who reported 0.4-0.7% accuracy for the L-subshell ratios of the 79.8- and 184-keV E2 transitions in ls8Er. For conversion-electron spectroscopy at energies below 100 keV, high-quality radioactive sources (Section II1,C) and thin reaction targets are necessary. Although measurements at E, 5 10 keV are difficult at the present state of the art, remarkable results were achieved by Gerasimov et ul. (1981). For the E3 transition in 99Tc, these authors determined the transition energy to be 2172.6 + 0.4 eV and the conversion-line intensity ratios, M, :M, : M4,5:N2,3:N,,,O,, with 3- 11% accuracy.

-

2. Electron Energy Losses Although elaborate methods have been developed to provide highquality radioactive sources for the conversion-electron spectroscopy (see Section III,C), the thickness of normal radioactive layers ranges from 1 to lo2 p g c m P , whereas those of the supporting backings and nuclear reaction targets vary typically from -50 p g cm-, to a few mg For 2 keV 5 E, 5 500 keV, the cross section for the electron interaction with a solid is a decreasing function of E, (see Fig. 13). In the energy region 102-103 keV it is possible to utilize fully the resolving power of the present spectrometers. More specifically, the momentum resolution A p / p = 3 x lou4can often be reached for E, 2 150 keV. On the other for E, < 50 keV, although hand, it is not easy to obtain A p / p 5 5 x such values were achieved even for E, 5 10 keV (see, e.g., Fujiokaet al., 1981a).

-

46

0. DRAGOUN

Electron energy (keV)

FIG. 13. Total electron stopping power for various materials. The values for energy ranges of 1- 100 keV and 0.1-6 MeV are those calculated by Brown (1974) and Akkerrnan c't uI. (1972), respectively. Thickness ( t ) of I pg cm-* layer Element

cu

Element

29

92

5.2

Because of its practical importance, the penetration of electron beams through layers of solid absorbers is well understood at higher energies. For instance, results of detailed calculations for 0.4 MeV IE, I6 MeV, 3 < 2 I82, angle of incidence between 0" and 81", and the absorber thickness from 0.01 to 3 g cmP2were published by Akkerman et a / . (1972). During the last decade, the low-energy region was studied both experimentally and theoretically to facilitate quantitative interpretation of the Mossbauer conversion-electron spectra at 5 keV 5 E, 5 20 keV (see, e.g., Liljequist et a / . , 1978; F'roykova, 1979, 1980), as well as the Auger electron and photoelectron spectra (E, < 1.5 keV). In the theoretical studies, the Monte Carlo method of randomly sampling the individual scattering events was extensively employed. For example, Spalek (1982) investigated the energy and angular distributions of conversion electrons under conditions similar to measurements with p r a y spectrometers: (1) A monolayer of radioactive atoms was supposed to be on the surface of a Au backing the thickness of which exceeded the range of 30-keV electrons (4.4 mg cmP2, i.e., 2.3 pm). Some of the isotropically emitted 30-keV electrons were backscattered and reached the spectrometer en-

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

47

trance slit. These electrons created a low-energy tail of the conversion line representing 90% of the intensity in the zero-energy-loss peak. In similar calculations for the 3.7-keV electrons and A1 backing, the corresponding tail intensity amounted to 45%. Here, the electron range in A1 was 100 pg cm-2 (0.4 pm). The examples illustrated the negative role of the source backings. (2) In connection with in-beam electron spectroscopy, the author considered 15-keV electrons emerging from the 135-pg ern+ (0.5 pm) A1 target. The uniform volume distribution of deexciting nuclei and isotropic emission of conversion electrons was assumed. The calculations revealed that the number of electrons leaving the target into 5% of 47r solid angle around the normal to the target surface was 1.6 times greater than the value corresponding to an ideal thin source. One of important quantities in the above calculations is the inelastic scattering mean free path Xine,(Ee).Szajman and Leckey (1981) have derived an analytical expression for Xinel in solids on the basis of a dielectric formalism (see Penn, 1976; Sevier, 1972). As seen in Fig. 14, their results for A1 and 200 eV 5 E, 5 100 keV are in very good agreement with the experiment. The authors distinguished between nonlocalized valence/conduction-band electrons and core electrons of a solid. In the case of Al, the three conduction-band electrons contributed by 97 and 87% to the total inelastic scattering of the 0.5- and 100-keV electrons, respectively. In this energy region, the elastic and inelastic mean free paths in A1 are roughly comparable. A compilation of about 350 measured

102

103 10' lo5 ELECTRON ENERGY (eV)

FIG. 14. Inelastic-scattering mean free path for electrons in Al. The figure is reproduced from work of Szajman and Leckey (1981) whose theoretical results are shown by the solid line. Open circles denote experimental data of Tracy (1974), triangles those of Swanson and Powell (1966), and solid circles those of Blackstock et a / . (1955).

48

0. DRAGOUN

values of Ainel for various materials and electron energies is available (Seah and Dench, 1979). These values are grouped around the “universal curve” which exhibits a minimum for E, = 35 eV. Szajman et a / . (1981) concluded that for a wide range of materials, Aine, = AE0,.75,where A is a sample-dependent constant. The empirical mean free path curves for the electron scattering in solids were also discussed by Ballard (1982). Undoubtedly, the above-mentioned calculations of electron energy losses within radioactive samples and nuclear reaction targets will be helpful in choosing optimum conditions for conversion-electron experiments. In order to utilize these calculations in the quantitative analysis of measured spectra the following are necessary: (1) A more elaborate model of electron interaction with a solid should be employed, especially in the low-energy region (2) Detailed information about distribution of radioactive atoms in the source is necessary (3) Exact knowledge of the actual instrumental function is needed These data are generally not yet available, and therefore the shape of the conversion line can only be expressed phenomenologically (Section IV, B ,2). The work of Bergkvist (1972) on the end-point behavior of the tritium p-spectrum may serve as an example of an excellent electron spectroscopic study in the 20-keV region, including a search for possible systematic errors. The work was devoted to the measurement and analysis of the continuous p r a y spectrum with regard to the neutrino rest mass. Nevertheless, the energy calibration, as well as the determination of the overall response function, expressing both instrumental effects and those caused by energy losses within the radioactive source, were determined by means of the 22.9-keV conversion electrons from the I7OTm decay. The problem of the overall response function for tritium sources of various thickness (<2.4 pg cmP) measured with the magnetic spectrometer (Lyubimov et al., 1981) has been examined by Soloshchenko (1981). Another correction of the measured spectrum for electron energy losses was applied by Shirley (1972). Using a high-resolution spectrometer and monochromatized A1 K, X radiation, the Shirley investigated photoelectrons emitted from the valence bands of gold. The apparent resolution was about 0.6-0.8 eV. The smoothed and background-subtracted spectrum Z(N)was corrected for the inelastic scattering by using the procedure which yielded the counting rate

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

49

Here, it was assumed (1) that Aine,(N)is constant over the narrow spectrum interval and (2) that the number of inelastically scattered electrons recorded in the Nth channel is proportional to the spectrum area at higher energies N ' > N . Equation (32) should be solved by iterations until Z'(0) = 0. The resulting spectrum depends critically on the correct choice of the channel at which the counting rate should be put at zero. The correction (32) apparently requires that the width of the instrumental function be negligibly small in comparison with details of the true spectrum.

B. Analysis of Measured Spectra I . Test of Measurement Stability High-resolution measurements of conversion-electron spectra with magnetic or electrostatic spectrometers often require exposure times of several days. When scanning a spectrum point by point with a counting period of 10- 1000 sec/point, the spectrum may be deformed due to time changes of the apparatus and its environment. The rapid-scan counting technique, when the measured region is scanned many times with an exposure of 0.1- 1 sec/point in each cycle, enables one to average the fluctuations quite uniformly over all points. For instance, Gelletly et ul. (1968) utilized this method to minimize the effect of background variations on the observed spectrum. Dragoun rt al. (1974) suggested a simple numerical procedure which compares partial spectra taken at different time intervals and examines whether the observed deviations are compatible with counting statistics of if they should be assigned to changing conditions of the experiment. The method requires that intermediate outputs, taken in successive time intervals during the cycle measurement, be stored, e.g., in a computer memory. The authors supposed that the ratio of any two partial spectra, j t h and mth, can be expressed by a constant kjm which is to be determined by the least squares method:

where m = 1, 2, . . . , M ;j = 2, 3, . . . , M ; and j # m. Here, n is the number of points in the spectrum; Nj is the background-corrected counting rate in the ith channel of the j t h of the M partial spectra; wjm is the statistical weight of the ratio N { / N y , and xt is the x2 quantity per 1 degree of freedom. An example displayed in Fig. 15 corresponds to the conversion-electron spectrum consisting of 428 points, which was taken in 172 cycles during about 40 hr. The intermediate outputs punched every

50

0. DRAGOUN I

I

: I

COMPARISON OF TWO PARTIAL SPECTRA

199 31d 199 79AU80 Hg

10,000~

208keV

-

1

20001

J

N.0

M

L3

0

I

....................

step1

step2 9700

step2

9800

9900

step5

step2

cop00

step1

IOJOO

step2

14200 CUWRENT SETTINGS

FIG. 15. Conversion-electron lines of the 208-keV transition in lg9Hg measured by Dragoun ef rrl. (1974) with the iron-yoke magnetic spectrometer. In lower part, counts accumulated during 172 successive forward-backward cycles are depicted. Comparison of the second and seventh partial spectra [see Eq. (29) and Fig. 161 yielded residuals shown in upper part. These residuals together with ~ 3 7 2), = 0.86 indicated negligible small variations of the measurement conditions between the two series.

6 hr enabled the authors to divide the whole spectrum into 7 independent series. The comparison of these partial spectra with each other according to Eq. (33) yielded the x: values depicted in Fig. 16. It is seen that the first series was measured under different conditions from the others. In fact, the spectrum of residuals revealed that the first spectrum was shifted by

FIG.16. Comparison of seven partial conversion-electron spectra measured in successive time intervals under supposedly identical conditions. Displayed are the x2y(j, m) values calculated according to Eq. (29) for the partial spectra, the sum of which is shown in lower part of Fig. 15. Systematic shift of the first series is discussed in Section IV,B,l. (Ig8Au-+ Ig9Hg208-eV transition; total time of measurement, 40.8 hr.)

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

51

1 x with regard to the rest. No such deviations are seen, e.g., in upper part of Fig. 15 where the residuals (in units of standard deviations) resulting from the comparison of the second and seventh series [x2y(2, 7) = 0.861 are exhibited. Therefore, the first series was subtracted from the total spectrum, thus improving the quality of the experimental result. This method represents a sensitive test of the measurement stability and can be also applied to the spectra taken in a multichannel mode.

2 . Shape of the Conversion-Electron Line In most cases, the single conversion line is a simple peak-shaped curve characterized by a certain half-width and a more or less pronounced low-energy tail. On occasion, more complicated structures appear on one or the other side of the peak. The line shape results from several processes, the relative importance of which varies from case to case:

(1) The spread around the central energy Ee,odue to the natural width

r of an excited atomic state. This spread is expressed by the Lorentzian distribution Ne

- [(Ee - Ee,$

-

(r/2)'1-'

(34)

Values of r range from 0.07 to 70 eV, depending on Z and the atomic subshell (Sevier, 1972; Keski-Rahkonnen and Krause, 1974). (2) The shaking process (see review by Freedman, 1974) also called the double ionization accompanying the internal conversion. This process leads to satellites on the low-energy side of the conversion line which, however, were recognized only in special experiments on the K-shell conversion (see, e.g., Porter et a/., 1971; Artamonova et a/., 1975). (3) The Doppler component on the high-energy side of the line corresponding to conversion electrons emitted by recoiled atoms in flight. This component, which is wider than the undistorted one, may be observed in spectra of thin a emitters (e.g., Ewan, 1963) or in nuclear reactions (e.g., Daniel et al., 1970). (4) When the atom is recoiled into vacuum and its nucleus deexcites via a cascade of converted transitions, the rearrangement of atomic electrons after the first conversion may result in the production of multiple vacancies. The next conversion then proceeds in a highly ionized atom in which the electron binding energies are higher than in the neutral one. This results in the satellite band on the low-energy side of the line (see the review by Walen et al., 1973). ( 5 ) When a conversion-electron line of small natural width is measured with the resolution AE of a few electron volts and internal conversion takes place in atoms which are not e x p s e d to the same chemical

52

0 . DRAGOUN

environment, the shifts in the electron binding energy may lead to broadening or even splitting of the conversion line (Gerasimov et a / . , 1981). (6) The electron energy losses within the radioactive source or nuclear reaction target and their backings (Section IV,A,2). The effect leads to a low-energy tail of the conversion line, the shape of which varies with electron energy. If both the natural width of the conversion line and the width of the instrumental line are sufficiently small (typically, < l o eV), the electrons emitted without any energy loss can be distinguished from those which suffered discrete energy loss. Measurements of the zeroenergy-loss peaks forms a basis of the X-ray photoelectron spectroscopy (see, e.g., Carlson, 1975). These peaks were also resolved in a few conversion-electron spectra and enabled precise determination of the transition energy (e.g., Geiger et ul., 1963; Johansson et ul., 1967; Gerasimov et a / . , 1981). First application of the zero-energy-loss peaks to the determination of the relative conversion intensities is described in Section IV,B,3. Perhaps not all the nuclear spectroscopists know that investigation of discrete electron-energy-losses in thin foils is a well-developed part of solid-state physics. For example, Blackstock et a / . (1955) measured the energy loss of 45- and 100-keV electrons in 15-pg cm-* (i.e., -550 A) A1 foil with the instrumental resolution of 2 eV. The loss peaks recorded at multiples of 14.9 eV corresponded to plasmon excitations. Boersch et a / . (1969) examined penetration of 30-keV electrons through the 30-pg c m P (i.e.,' -2000 &-thick NH,Cl film by using the resolution I10 meV. Strong satellites on both sides of the primary line were observed which corresponded to the energy losses and gains of about 20 meV. These energy transfers were interpreted as excitations of the translational lattice vibrations. The energy losses of the electrons caused by scattering in solids were also treated in a monograph by Sevier (1972). For more recent experimental studies, see, e.g., papers of Ballu et [ I / . (1976) and Schubert and Wolf (1979). (7) In addition to the above-mentioned atomic and environmental effects, the shape of the conversion line is also determined by the spectrometer itself. The function f ( E e ) ,which describes the energy distribution of electrons emerging from the source, is convoluted with the instrumental (or response) function K ( x , Ee):

Here, g(x) describes the recorded spectrum and El,E2determine the interval wheref(E,) # 0. Generally, this procedure leads to line broadening and a shift of the line peak toward lower energies; details of the original distribution are also hidden.

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

53

In conventional conversion-electron spectroscopy, the line shape is determined mainly by the last two of the discussed effects. The instrumental function K ( x , E,) is the spectrometer response to monoenergetic radiation of unit intensity,f(EL) = 6(E, - EL). For spectrometers of several types, the theoretical instrumental function was derived (see, e.g., Siegbahn, 1965; Draper and Lee, 1977). In principle, Monte Carlo calculations enable one to obtain K ( x , E,) for any spectrometer, assuming that its actual dimensions and field shape are known precisely enough. The main reason that Eq. (35) is not utilized to derive the true conversion-electron spectrum f ( E e ) is due to insufficient knowledge of K ( x , E,) for a particular spectrometer. Nevertheless, the deconvolution represents an uneasy task even if K(x, E,) is precisely known. The solution of Eq. (35) belongs to incorrect mathematical problems in the sense that the resultingf(E,) is not stable with respect to small changes of the input function g(x). Obviously, one loses information about details of the true spectrum when scanning it with the “wide window” of a real spectrometer. In addition, the measured spectrum g(x) suffers from experimental errors, which further complicates the task. The application of deconvolution methods in electron spectroscopy was reviewed by Carley and Joyner (1979). Regularization methods (e.g., Tikhonov and Arsenin, 1974) can be used to find the correct solutionf(E,) of Eq. (35). To achieve this, information about the accuracy of g(x) is necessary. Stolyarova et a / . (1979) utilized the regularization method in y-ray spectroscopy. An another approach is to convolute successively the theoretical predictions f,(E,), fi(Ee), . . . , with the experimental response function K ( x , E,) and compare the resulting g,(x), g,(x), . . . , with the measured spectrum. For example, Ewan and Graham (1965) determined in this way differences in the natural width of several inner atomic subshells in Pu. The experimental conversion line of small natural width served as K ( x , Ee). It was successively convoluted with Lorenzian curves f l ( E e ) , f,(E,), . . . , of various widths, providing gl(x), g,(x), . . . . The procedure was repeated until the shape of another measured line, g(x), was met. The mentioned approach was also applied in the analysis of the tritium p spectrum, assuming various values of the neutrino rest mass (Bergkvist, 1972; Lyubimov et a/., 1980). 3. Decomposition of the Spectnrm into Components

Formerly, the conversion-electron spectra were treated graphically. In the 1960s, computer analyses started aiming to determine relative intensities in the multiplets of partially resolved conversion lines. These

54

0 . DRAGOUN

values are of great importance for the spin-panty determinations of excited nuclear states (Section V1,B). Dragoun and Jahn (1967b) and Hennecke et a / . (1967) expressed the line shape in a tabular form derived from a single line of the same spectrum. Von Egidy et a / . (1966) utilized an analytical function of the following type, which is often used today:

N i= a,,{exp(-a,(i + [a3 exp(a,(i - a,)) + - a,))] x [1 - exp(- a 4 i - a,>z)l>

+ a, + a,(i

- io)

for i

5 a,

(36)

for i > a, Ni = alO{exp(-a6(i - a,)'),) + a, + a,(i - io) Here, N iis the counting rate at the point i and a, the fitted parameters. Sometimes, only one asymmetric term is utilized on the low-energy side of the line; at others, a similar term is added to the high-energy side. These functions can approximate a variety of experimental conversion lines in a broad energy region (see, e.g., Mampe et [ I / . , 1978). According to our latest experience, however, they fail to express accurately the low-energy tail at E, s 30 keV. A common drawback of all these phenomenological descriptions (not considering the physical processes outlined in Section IV,B,2) is the uncertainty as to whether the true shape of the conversion line can be reproduced with sufficient accuracy by fitting parameters of the chosen function. The best values of these parameters (including the line areas and their uncertainties that follow from the complete error matrix) are determined by the least squares fit. It should not be forgotten, however, that the statistical reliability of the spectrum fit does not mean its correctness. It only means that this particular solution could not be rejected as statistically unreliable. It is therefore advisible to include in the fit as much input information as possible. For instance, data about the number of conversion lines and their relative positions (following from the binding-energy differences) greatly improve results of the decomposition of the multiplets. In the narrow energy interval, the conversion lines are often assumed to be of similar or identical shape (the fact that the instrumental resolution AEIE or A p / p is constant instead of the width AE or A p , can easily be taken into account). Of course, this is only an approximation, since the natural width, shake-up effects, electron energy losses, etc., are not the same for conversion lines of various origin and energy. When determining the intensities of conversion lines described by different profiles, the rule of constant tail fraction (Fujioka and Shinohara, 1974) should be followed. According to this rule, the tail fractions (defined as the intensity of the tail

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

55

part divided by the total line intensity) have to be equalized at every step of the profile change. As a rule, smoothing of measured data is avoided so as not to lose the spectrum details. Only when the high-resolution spectrum is taken in the multichannel mode with very fine step width, can the smoothing be appropriate (e.g., Gerasimov et af., 1981). It is advantageous to fit all the components including background simultaneously in order not to increase the statistical error of measured counting rates by successive subtractions. Even the complicated background expressed in tabular form can be taken into account only when the background intensity is fitted while its shape is kept fixed. For example, Fujioka ef al. (1981a) fitted the LMX Auger spectrum of 207P0as one component into the complex spectrum of zOsPoin order to separate overlapping Auger and conversion lines in the region of 10 keV. During the measurement with the Heidelberg (.rr/2) magnetic spectrometer, Dragoun ef al. (1970) also recorded the time-dependent background from the neighboring tandem accelerator operating with deuterons. A large-volume Geiger-Muller counter was utilized to determine that effect with sufficient statistical accuracy. The variable background counting rate was then fitted into the measured electron spectrum as one of its components. Surprising improvement of the quality of fit was achieved in this simple way. An elaborate computer program for the analysis of conversion spectra was briefly reported by Zhirgulyavichyus (1979). The line is described by a linear combination of several functions of the type exp(-x2/a2), b'/(x' + b'), x2ex'c,xz/(x2 + d z ), and a step function accounting for the far part of the low-energy tail. Here, x is a distance from the line peak and the other letters denote the fitted or fixed parameters. As an alternative approach, the author divided the conversion line into certain number (e.g., eight) of parts of various widths and described each part by a third-order polynomial. Requirement for smoothness of the overall function and its derivatives resulted in additional equations for the polynomial coefficients. Thus the number of fitted parameters was reduced to the number of the line parts. For application to the M, N, 0 conversion lines of the 191.6-keV E4 transition in lI4In, see the paper by Davidonis ef af. (1978). Fujioka and Takashima (1979) developed a method of spectra analysis that uses empirical line profiles expressed numerically. Two different profiles are applied for the core and valence-shell conversion lines, respectively. During the iterative procedure, these profiles are optimized by means of 5-6 incremental quantities that express changes of linewidth

56

0.DRAGOUN

and of tail height and gradient on both sides of the line. Details can be found in the application to the 10.84-keV transition in ,06Bi done by Fujioka et al., 1981a. According to our latest experience, the initial profiles have to be close to the actual profiles otherwise the iterative process may result in distorted line shapes. Gelletly et al. (1981) and K. Schrechenbach (private communication, 1981) reported the high-resolution measurement and analysis of the conversion-electron spectra which yielded the L-subshell intensity ratios of the 79.804-keV E2 transition in lssEr with 0.4-0.7% accuracy. The measurements were carried out at the resolution A p / p = 2.5 x (FWHM = 33 eV), so the L-subshell lines were completely resolved (Eb,LI - Eb,L2 - 487 eV, &,L2 - Eb,L3 = 907 eV). The only problem was the precise determinatioh of a slight low-energy tail of the L, line under the weak L, line. This tail resulted from a finite thickness of the target (15 pg cm+ of I6'Er metal) and that of the backing (200-pg cmV2A1 foil). The authors scanned the L group twice and recorded three spectra simultaneously by means of proportional counters placed in a focal plane of the spectrometer. Each scan took 5 hr. The different scans were evaluated separately as well as after summing them up. The authors analyzed the spectra by using two methods: (1) The conventional computer fit with a standard line shape (Mampe e f al., 1978) fitted to all three lines simultaneously (2) The direct subtraction of the line having the measured shape of the L, line from different lines in the spectrum. The intensity-reduction factor and the position were varied until the line of interest disappeared from the spectrum in an optimum way with respect to the remaining intensity (see Figs. 1 and 2 in the paper of Gelletly et al., 1981). Systematic errors arise in this substraction method from the definition of background and from differences in the line shapes due to different natural widths as well as electron energy losses. The subtraction method enabled the authors to eliminate effectively the L, line background under the L, line. The reported uncertainties included an estimate of the systematic errors; random errors amounted to only 0.2%. Gerasimov et al. (1981) utilized the zero-energy-loss peaks in the conversion-electron spectrum of the 2.17-keV E3 transition in 99Tcto determine the relative intensities of the M, , M, , M4,5,N2.3, and N4,5O, lines. The Tc atoms of the radioactive source formed a layer of thickness x > 100 A on Pt backing. There was a contamination layer of thickness d = 7.5 k 2.5 A on the Tc surface composed mainly of C, 0, and H atoms. Using X-ray photoelectron spectroscopy (Powell and Larson, 19781, the authors expressed the experimental ICC ratio for the subshells i

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

57

and k as

(37) where

Here, Zi is the intensity (area) of the zero-loss-energy peak of the ith conversion line, A(E) and A’(@ are the electron mean free paths (Penn, 1976) in the source and contamination layer, respectively. The coefficient Ri takes into account the influence of many-electron transitions, and S ( E ) describes the detection efficiency. In the cases under study, 1.7 keV 5 E, 5 2.2 keV, 0.842 5 X(EM,)/A(Ei)I1.008, [Aik- 11 < 0.01 for x > 100 A, 0.93 5 BiM3I1.00 for d = 5-10 A, R,/R, = 1, and S(E)= const. Error of the z i / z k intensity ratios followed from the uncertainty in subtracting the background of inelastically scattered electrons. For the resulting @-i/@-kratios, Gerasimov et a/.(1981) derived 3- 11% uncertainties. The possibility of estimating the thickness or composition of a covering layer on a solid by means of photoelectron spectroscopy using neither a series of standard samples nor argon-ion etching was examined by Hirokawa ef a / . (1981). Dragoun ef a/.(1972) analyzed two groups of conversion lines at E, -300 and -400 keV measured in two laboratories with the overall resolution A p / p 5 5 x lop4. Four different computer programs operating with either numerical or analytical line shapes were applied. The resulting relative intensities were found to be in mutual agreement. It happens, however, that even the successive measurements taken with same instrument and analyzed with the same code do not yield results consistent within computed statistical uncertainties. This is undoubtedly caused by neglecting other random and systematic errors of the measurement and analysis. In order to gain insight into what can be expected in a particular case from purely statistical scatter of measured data, we propose here the following simple method based on the numerical experiment: (1) The artificial spectrum is constructed in such a way that the number, shape, distances, and intensities of conversion lines, as well as the counting rate and the background level, are very near to those of the real measurement.

58

0. DRAGOUN

t

I

I

3700

I

3800

I

3900

Electron momentum (arbitrary units)

FIG. 17. Analysis of an artificial conversion-electron spectrum yielding information about contribution of statistical errors into overall uncertainty of an actual measurement. The fitting procedure was the same as for the measured data: the common parameters of the line shape [a3 through a8 in Eq. (36)] were allowed to change as well as the line intensities and their positions. The line distances were kept constant. As expected, the fit was statistically reliable and residuals did not reveal any systematic deviations. From the fit, the line uncertainty of the shape parameposition was derived with relative accuracy of 2 9 X ters, a 3 , . . . , a,, ranged from 3 to 11%. Results concerning the line intensities are displayed in the table. “True” values are those postulated prior to randomization. The fitted (i,e., 0 . 3 ~toward ) higher energies, the fitted parameters spectrum was shifted by 2.6 x ~ the original ones. (Numerical experiment: 35.5-keV M1 deviated by - 0 . 3 ~to + 1 . 8 from transition in lz5Te, x: = 1.06.)

Shell M,

MZ M3

(%I

Shell

(%I

0.3 2 2.1 5.4 f 3.7 -10 f 12

N,

0.0 f 2.6 38 2 13 -2.8 k 6.5

N*.3 0 1

(2) The calculated counting rates N iare randomized according to the Poisson distribution with the standard deviations (Ni)liZ. (3) This man-made spectrum is analyzed in the same manner as the actual one. In this artificial case, however, there are no doubts about the stability of the “radioactive source” and “spectrometer,” correctness of the line shape, number of lines, etc. The computed errors provide an estimate of the statistical contribution

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

59

to the overall uncertainty of the actual measurement. Accuracy of this estimate can be improved by repeating the procedure with another set of random numbers and calculating the averages. An example of such a numerical experiment is depicted in Fig. 17. In addition, the method makes it possible to verify whether measurements with a finer step or higher counting rate are worth the effort at the resolution available. C. Overall Uncertainty o j Measured Quantities

Since the actual measurement suffers from both systematic and random errors, the determination of the overall uncertainty represents a difficult and ambiguous task. In the literature, various contradictory points of view and recommendations can be found. For instance, Williams et d. (1973) argued that the final statement should give the random and systematic uncertainties separately. Muller (1979) questioned a clear-cut distinction between the two errors. Grinberg et al. (1973) attempted to establish rules for determining one best value and its uncertainty at a 68% confidence level for every parameter of a nuclear decay scheme. The authors suggested the following formula for the overall uncertainty A of the final result:

A

=t,s/fi

t 6/3

(40)

Here, t, is the Student t value for v degrees of freedom and a 68% confidence level; s is the standard deviation, rz the number of measurements, and 6 the overall systematic uncertainty. Usually, it is the sum of all residual systematic uncertainties ai that remain after correction of the result for the systematic effects. For application of Eq. (40), see, e.g., the works of Hansen et al. (1979), Nylandsted Larsen et al. (1980), and Mouchel et al. (1981). Methods different from Eq. (40) were used, e.g., in works of Morii (1980) on y-ray energy measurements with A E I10 eV and those of Geidel’man et al. (1979, 1980) on half-Iife determinations. In these papers, a detailed treatment of systematic uncertainties can also be found. The Particle Data Group issues every even year a review of particle properties and applies rules and methods (Trippe et al., 1976) that proved very successful in averaging results of independent measurements and assigning realistic uncertainties to these averages. Both Gaussian distribution with scale factor [ x 2 / ( n - 1)]1’2and empirical Student’s distribution (Roos et id., 1975) is used. The inconsistent data are displayed by means of illustrative ideograms. For an application of the scale factor see Section V.

60

0. DRAGOUN

V. COMPARISON OF THEORYWITH EXPERIMENT The discrepancies among calculated and measured ICCs, found during the course of internal conversion studies, stimulated ( 1 ) improvements of the physical approximations and computational methods as well as (2) development of experimental techniques including treatment of measured data. Not all precise experimental data are suitable to test the ICC theory, since necessary nuclear parameters may not be known with sufficient accuracy. For example, the total ICC of the 166-keV (5/2+ -+ 7/2+) M1 transition in 139La,measured with 0.25% accuracy (Schonfeld and Brust, 1977), could not be directly compared with the theoretical value due to possible nuclear structure effect and E2 admixture. In fact, the analysis yielded A = 3.6 0.2, <0.2% E2 (Dragoun et al., 1981). From this point of view, the enhanced pure E2 transitions are suited for testing the theory. Mladjenovic et a / . (1978a) listed measured aKfor 39 (2+ + Of) transitions of energy 73 keV 5 Ey 5 1133 keV in nuclei with 26 5 Z 5 86. In most cases, results of several independent measurements were averaged as WXai/(AaJ2 k W1I2, W = l/X(AaJ2. Agreement of these averages with the values calculated according to the program of Bander al. (1976/1977) was not satisfactory. We notice that the aKvalues measured for a particular transition are often inconsistent within stated errors. Nevertheless, increasing the errors of the means by the scale factors [ x z / ( n (see, e.g., Trippe et ul., 1976), we reach for the whole set of the experimental ICC, a K agreement , with theory on a 1% confidence level. To test the ICC calculations, the L-subshell conversion-line intensity ratios of the pure-multipolarity transitions were extensively measured. In general, the results agreed very well with the theory (for more recent references, see, e.g., Gelletly er al., 1981). The only exception occured for pure E2 transitions of -80-keV energy in rare-earth nuclei. In this region, the calculations of the af9,'(E2)are complicated by a partial cancellation in leading conversion matrix elements and, in addition, weak L, lines are superimposed on low-energy tails of stronger L, and L, lines. The L1/L, and L,/L3 intensity ratios were found to exceed theoretical values by 4-7%, whereas agreement within -1% was observed for the LJL, ratios. Hager and Seltzer (1970) suggested an explanation of the anomaly by taking into account some of the higher order effects in the ICC calculations (Section II,D,2). Another explanation was offered by Bulgakov et al. (1978, 1979) who assigned the higher L, line intensity to the contribution of the double ionization accompaning the internal conversion (Section IV,B,2). In the rare-earth region, the electrons shaken off the N1-, subshells during the internal conversion in the L, subshell have kinetic en-

*

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

61

ergy approximately equal to that of the L, conversion electrons. Thus the authors argued that no higher order effects are necessary to explain the measured results. Gelletly et al. (1981) reported the following results for the 79.8-keV (2+ + Of) transition in 168Er:the L,/L, ratio by 5.7 ? 0.7% higher and the M,/M, ratio by 7.0 t 1.1% lower than those predicted by theory, and perfect agreement for the L,/L, and MJM, ratios. These authors ascribed the deviations to the mentioned cancellation in the leading conversion-matrix elements which made the precise calculations of the afP1 difficult. Undoubtedly, more effort on both experimental and theoretical side is still needed to solve the problem. No serious discrepancies were reported among the intensity ratios, including the higher shells and newer ICC calculations listed in Table I. The precise experimental data for E2 transitions are scarce (e.g., Chu and Perlman, 1971; Bulgakov et NI., 1980). Precise experimental data for the transitions of mixed multipolarity and those influenced by the nuclear structure enable one to test another aspect of the ICC theory. In particular, it is possible to examine whether the various quantities measured for the same transition yield the consistent value of 8; and A (or v ,6). No discrepancies of this kind were reported during the last decade. When the nuclear parameters 8; and h (or 7, 6 ) were determined from the data for inner atomic shells, yet another test of the theory could be performed. ICC measurements for outer shells can be compared with theoretical predictions. Especially the atomic model (Section II,D, 1) can be examined in this way (see, e.g., Dragoun e f NI., 1979, 1981). The authors also recommended measuring the conversion-line intensity ratios of the outer to inner shells, e.g., O/L, N/K, with an accuracy of better than 5%. These quantities would be more sensitive to the atomic models employed in the internal conversion calculations than are the absolute ICCs for inner shells or the ICC subshell ratios. When two different sets of theoretical predictions are to be compared with a set of experimental values, the Ax2 statistics developed recently by Betvaf (see Dragoun et al., 1979, 1981) can be advantageously utilized since they are more powerful than the usual x2 test. VI. ROLE OF INTERNAL CONVERSION I N NUCLEAR SPECTROSCOPY Measurement of conversion electrons started soon after the discovery of radioactivity. Over the years, it has turned into a powerful method of examining excited nuclear states. Electron and y-ray spectroscopies have always complemented each other, but their achievements did not often

62

0. DRAGOUN

I

I

100

200 CHANNEL NUMBER

FIG. 18. The conversion-electron spectrum of the 123mC~ decay = 1.7 sec) measured by Marguieret c i l . (1981) with the Si(Li) detector of l .6-keV resolution at 624 keV. The 123"'Cs was prepared by La(p, 3p14n) reaction at ED = 600 MeV, followed by on-line mass separation. (a) The single spectrum (collection time 0.1 sec, counting time 4 sec, number of cycles 7000). (b) The coincidence electron spectrum gated by the y rays of 94.6 keV, obtained in the same experiment as in (a). (Copyright @ 1981, The Institute of Physics; reproduced by permission.)

coincide. In the late 1950s, resolution A p / p of magnetic spectrometers apin energy proached the value of lop4 and the relative accuracy of determinations became attainable; y- spectroscopy, however, suffered from the poor resolution of scintillation detectors. (For this reason, magnetic spectrometers were also used for analysis of photoelectrons ejected by y rays from thin targets of heavy elements.) The conversion-electron spectra of hundreds of radioactive nuclei were recorded up to the mid-l960s, which provided valuable information about energies, intensities, and multipolarities of electromagnetic transitions. The invention of solid-state detectors revolutionized y-ray spectroscopy. When highresolution data on y-ray energies and intensities became attainable, inter-

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

63

est in the conversion-electron spectroscopy was renewed because of its ability to determine transition multipolarities even in the case of very complicated decay schemes (see, e.g., Alexeev rt al., 1978; Abdurazakov et al., 1980). Using the methods outlined in Section 111, conversion-electron measurements were extended to radioactive nuclei with half-lives of a few minutes (Fig. 12) or even seconds (Fig. 18). Conversion-electron spectra are also measured in nuclear reactions with neutrons (see the review of Schreckenbach, 1980) as well as heavy particles including heavy ions (e.g., Gono rt al., 1974; Nagai et al., 1977; Backe et al., 1978). A . Transition Energies

For almost half a century, measurements of conversion electrons provided the energies of electromagnetic transitions populated in radioactive decay (see, e.g., Groshev and Shapiro, 1952; Siegbahn, 1955, 1965). The method based on knowledge of electron binding energies [Eq. (9)] not only covered the energy range from 1 keV up to several MeV, but also enabled one to determine the atomic number Z of the nucleus in which the transition occured (see, e.g., Toriyama et al., 1970). The conversion-electron spectra taken with high-resolution magnetic spectrometers revealed the existence of doublets, which led to substantial improvements of the nuclear decay schemes (see Figs. 19 and 20 for example). Measurement of conversion electrons is the only direct way of proving the existence and determining the energies and intensities of the electric monopole transitions. It is also indispensable in investigating the transitions of energy E y s 10 keV where competing y rays are hardly detectable. Siegbahn (1941, 1944, see also 1965) proposed a method for deterrelative accuracy by meamining the transition energies with -5 x suring the ratio of the B p values of two conversion lines corresponding to the same transition but different atomic subshells. The method is less suitable for the iron spectrometers where the profile of the magnetic field may depend on the field intensity. Nevertheless, Dragoun et al. (1976) applied the Siegbahn method to the cycle measurements with the 50-cm n-d iron-yoke spectrometer. The computer analysis of 27 independent spectra of conversion lines of the 88-keV transition in loSAgyielded the ratios of the positions of the K , L,, M2,, and N,,, lines [the parameter a, in Eq. (36)] with 4 x relative accuracy. The agreement of the resulting transition energy E y = 88.023 ? 0.008 keV with the adopted value of 88.032 5 0.002 keV (Bertand, 1971) served as an argument for the spectrometer linearity in the measured energy interval. In order to calibrate the momentum scale of the 2 X n - 4 iron mag-

-

64

0. DRAGOUN

FIG.19. The conversion-electron spectrum of lo5Agdecay in the regions of (a) 443 and (b) 644 keV measured by Kawakami and Hisatake (1970) using the iron-free P-ray spectrometer. The measurement proved the existence of the 442.25 + 443.44-keV and 644.50 + 645.95-keV doublets which enabled the authors to improve substantially the lo5Pd decay scheme (see Fig. 20).

netic spectrometer, Medvedev et al. (1978) applied 31 B p values of conversion lines in the interval of 246-3264 keV and expressed the calibration curve in the form Y = a + b / B p , with a = 1 + (8.6 k 1.5) x and b = -0.44 0.07 for 2000 G cm IB p 5 12500 G cm.

*

keV

1087.93 - r ' 0 8 8

727. I 7 650.69 644.50 489.1 I 442.23

- -7- 599 489 443 U Q

- 0

-0 (a)

(b)

FIG.20. The improvement of the losPd decay scheme resulting from the existence of two close transition doublets revealed by high-resolution conversion-electron spectroscopy (Fig. 19). Reproduced from work of Kawakami and Hisatake (1970) who constructed the right-hand decay scheme (b). The former scheme by Pierson and Rengan (1967) is shown on the left-hand side (a).

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

65

Measurements of conversion electrons in the zero-loss-energy peak and photoelectrons excited by A1 K, X radiation from the same radioactive source enabled Zhudov et al. (1979) and Gerasimov rt al. (1981) to determine the energy of the 77 eV and 2.17 keV transitions with ? 0.5 eV uncertainty. The present measurements of the transition energies are mostly carried out with y-ray multichannel spectrometers using Ge(Li) detectors (see, e.g., review of Vylov rt ul., 1978). Problems of the energy and intensity standards in the y-ray spectroscopy were scrutinized by Dzhelepov and Shestopalova (1980). This monograph can be recommended not only for detailed and consistent treatment of the methods and their uncertainties, but also for listing present values of nuclear spectroscopic standards. For instance, there is a list of 148 transitions of 9.1 keV s Ey I1.38 MeV, for which the transition energy was determined with an uncertainty AE < 10 eV, compared to the primary wavelength standards, X6057.8 A 8sKr, XWKa,, and Xy411.8 ls8Hg. Renewed application of the p r a y spectrometer to the measurement of photoelectrons ejected from a 2.5-mg-~m-~-thick 238U02convertor by y rays from the 48Ti(n, y)4sTi reaction was reported by Schreckenbach (1980). The magnetic spectrometer operated at the instrumental linewidth of 1.3 keV at 6.5 MeV and the FWHM of the photoelectron line corresponding to Ey = 6.762 MeV (and including the spread in the uranium convertor) was 2.6 keV, which is superior to that of a Ge(Li) detector. This measurement provided precise energies of primary transitions after neutron capture.

B . Transit ion M u 1tipolu rit ies Comparison of measured absolute ICCs with the theoretical ICCs (see, e.g., Fig. 1) often enabled experimenters to determine the multipolarity of the nuclear electromagnetic transitions. Sometimes, the ratios of the ICCs for various atomic subshells were even more sensitive than the absolute ICCs. Of course, the circumstances were not always as convenient as in Fig. 21, but the method was applied to thousands of transitions. Although these determinations were always not unambiguous, they contributed substantially to the spin-parity assignments of the excited nuclear states. As an excellent example of current y-ray and conversionelectron studies, we quote here the work of Davidson rt ul. (1981) on the ls7Er(n, y)ls8Er reaction in which the decay scheme consisting of 79 levels grouped into 20 low-K rotational bands was established. The experimental values of the ICCs and their subshell ratios for nuclei with Z 5 60 can be found in the extensive compilation of Hansen (1981). In general, the ratios of the nuclear matrix elements, i.e., the mixing

66

0. DRAGOUN

Transit ion

Energy

FIG. 21. Comparison of the L-subshell conversion-line intensity ratios for the 49.63-keV transition in 156Tbwith theoretical predictions for various multipolarities (Hager and Seltzer, 1968). Reproduced from the work of Toriyamaet NI. (1970) who carried out the measurement and determined the transition multipolarity to be El + <0.6% M2.

ratios 6?(L/Lmi,)defined by Eq. (3) and the nuclear structure parameters

A, 77, and 5 for hindered transitions [Eqs. (15) and (16)] can also be deter-

mined in the conversion-electron experiments. For example, Gavrilyuk er ul. (1977) investigated the E l + M2 E3 transitions of 536.7- and 920.9-keV energy in lE4W.Comparing the measured and calculated data on a K ,L,/L, , y- y angular correlations and angular correlations of y rays emitted from polarized nuclei, the authors obtained the values of 8,(M2/El) and 6,(E3/E1) and two possible solutions for the nuclear current parameters ?(El). To determine the sign of 6,, angular correlation measurements are necessary. For an example of the apparatus for y-ray angular correlation studies of radioactive nuclei polarized at low temperatures, see the paper by Gromova rt a / . (1979). Surveys of the E2/M1 multipole mixing ratios 8, were published by Krane (1975, 1976, 1977). The merits and drawbacks of conversion-electron studies for the determination of the quantum characteristics of nuclear transitions were

+

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

67

scrutinized by previous authors (see, e.g., Listengarten, 1961; Graham, 1966; Hamilton, 1966b; Dzhelepov, 1974), so we restrict ourselves to several comments only. (1) The importance of the described method justifies both experimental (Sections III,A and IV,B) and theoretical (Section II,D) efforts to provide more accurate values for comparison (2) When several quantities that depend on mixing ratios and nuclear structure parameters are measured, it is advantageous to combine the graphical analysis (e.g., Dragoun rt al., 1970; Hansen, 1979) with the least-squares fit. The former illustrates the sensitivity and number of solutions, the latter is more rigorous in weighting and error estimates. The computer code of RySavy and Dragoun (1980) can be used for this purpose. (3) In order to derive reasonable uncertainties of 8: and A, q , 6 , the analysis of precise experimental data should include the uncertainty of theoretical ICC (Section II,C,3). This leads to the problem of combining the random and systematic errors (Section IV,C). Dzhelepov (1974) recommended adding both uncertainties quadratically, i.e., using the weighting factors 1/(u&, &) in the least squares fit and the x2 test. Following the proposal of Listengarten and Sergeev (1981), the u t h values for the K, L,, L,, and L, subshells could be calculated from Eq. (19). It should be kept in mind, however, that differences in the theoretical assumptions often result in the systematic shifts of the ICC values which makes the procedure questionable. As an alternative approach, one could carry the analysis twice, using either the ICCs calculated under different physical assumptions or the values interpolated, e.g., from the tables of Rose1 et al. (1978) and of Band and Trzhaskovskaya (1978), and try to estimate the overall uncertainty of the fitted parameters from the spread of fitted values and their computed uncertainties. (4) If possible, the relative conversion-line intensities should be presented in the form ( I , ? u,):(1, ? c,): . . : ( I , f c,),rather than in the form 1 :(Z, t u,) : * :(Z, ? un), since the former presentation enables the use of any pair of the intensities without increasing the uncertainty due to error propagation. ( 5 ) In some cases, even the inaccurate intensities of conversion lines corresponding to atomic subshells with higher angular momentum (e.g. , the M4,5 lines in the work of Plajner et al., 1967) can help to assign the transition multipolarity . (6) If the group of conversion lines (e.g., the L group) cannot be decomposed into components, one can try to fit the group as a whole with one composed line, constructed from several single lines whose intensities correspond to various multipolarities or multipolarity mixtures.

+

- -

68

0 . DRAGOUN

Sometimes, this method exhibits surprising sensitivity (see, e.g., Dragoun et ul., 1969~). The numerical experiment outlined in Section IV,B,3 easily shows what can be expected in a particular case.

C . N u d e u r Structure Efiects

As pointed out in Section II,B,2, there is a nonzero probability for any atomic electron to penetrate the nucleus and interact there with the transition currents and charges. Therefore, the ICCs of all electromagnetic transitions have to depend, at least in principle, on the structure of the nucleus. In most cases, however, the effect is too small to be detected with the present experimental accuracy. According to Listengarten (1978), the contribution of the penetration-matrix elements into the ICCs varies, in the case of nonhindered transitions, from (low Z, all multipolarities except probably M1 and M2) up to 10% (highZ, M1 multipolarity). It was therefore reasonable to calculate extensive tables of the ICCs in which the nuclear structure effect was either neglected or included very approximately (Section II,B,2). The importance of such calculations for the spin-panty determination of the excited nuclear states (Section VI,B) is not doubted. There exist about 70 hindered nuclear transitions, the experimental ICCs of which are known to deviate from the tabulated ICCs. Systematic analysis of these anomalies was presented by Listengarten and Feresin in the monograph of Band et ul. (1976a). The authors examined the experimental data for 29 MI transitions in spherical nuclei and nuclei of the transitional region, 12 M1 transitions in deformed nuclei, 5 MI components in the EO + M1 + E2 (2' + 2+) transitions in even-even nuclei, 25 E l transitions in deformed nuclei, and 1 M3 transition of 73.4-keV energy in lS1Os measured by Loeweneck and Martin (1973). More recently, the nuclear structure effect was also reported for the 391.7-keV M4 transition in l131n (Dragoun et ul., 1976) and for the M2 component of the K-forbidden M2 + E3 transition of 500.6-keV energy in lsoHf (Sergeenkov, 1977). The investigated transitions are 10'- 10l6times slower than the Weisskopf single-particle estimate predicts. The magnitude of the ICC anomaly [expressed by the quantities Ai(7, L) in Eq. (14)] varies from a few percent up to a factor of 20. There is no universal correlation between the anomaly factor Ai(7, L) and the hindrance factor Fw.Some of the transitions with Fw = lo* have normal ICCs, whereas others, with Fw= 102, exhibit anomalous ICCs. For transitions of a certain type, e.g., hindered transitions in deformed nuclei, the ICC anomaly increases with increasing Fw. A summary of the present knowledge of the nuclear structure effect in internal conversion has been published by Listengarten (1978). Feresin

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

69

(see Band et al., 1976a) scrutinized the formulas and selection rules for the anomalous conversion. The author also presented theoretical values of the penetration-matrix elements for a number of transitions. The cases for which the effect was expected, but experimental data not yet available, are included, too. For the MI transitions in odd-A nuclei, the y-ray and penetration-matrix elements were calculated with the nuclear wave functions of the Nilsson model by Browne and Femenia (1971) and by Krpic et al. (1973), respectively. Predictions of the nuclear structure parameter h for 35 I-forbidden M1 transitions were presented in a graphical form by Subba Rao (1975). In order to determine the experimental values of the nuclear structure parameters [Eqs. (15) and (16)] for the investigated transition, three sets of data are needed: (1) Measured absolute ICCs, the ICC subshell ratios and angular correlation coefficients L ) that can be either interpolated from ex( 2 ) The static ICCs, a(i0)(7, isting tabulations listed in Table I or calculated directly for the transition under study (Section I1,C) (3) The dynamic correction factors a&) and b,(L), calculated with the code of Pauli and Raff (1975). Similar quantities can also be interpolated from the tables of Listengarten and Feresin (1968), reproduced in the book by Band ef ul. (1976a), or from the tables of Hager and Seltzer (1969).

The analysis is often complicated by the simultaneous presence of the penetration effect and the multipolarity mixture. The comments concerning determination of the mixing ratios (Section VI,B) are relevant to this analysis as well. In many cases, the present experimental data are not complete and accurate enough to allow unique and precise determination of the nuclear structure parameters (see, e.g., systematics in the monograph by Band et al., 1976a). The resulting nuclear structure parameters multiplied by the y-ray matrix elements (derived from measured y-ray emission rates) provide the experimental penetration-matrix elements. Comparison of these quantities with the calculated ones supplies the effective spin-gyromagnetic ratios. For the time being, these values provide unique information about the magnitude of the spin-dipole interaction. If the anomalous conversion of the hindered E2 transitions were observed and the corresponding effective g factor determined, the very first information about the constant of the spin-quadrupole interaction could be gained. For detailed discussions of these topics, see the paper by Listengarten ef al. (1976) and the monograph by Band et al. (1976a).

70

0. DRAGOUN

Listengarten er (11. (1981) have proved that the matrix elements of the nuclear toroidal moments (Dubovik and Cheshkov, 1974) are proportional to the well-known internal-conversion penetration-matrix elements of the electric multipole transitions. From experimental data on anomalous conversion, the authors determined the matrix elements of toroidal moments for seven E l transitions. The values were shown to be proportional to the contribution of the spin-transition currents into the E l y-ray emission probability. When comparing experimental and theoretical values of the nuclear structure parameters, it should be kept in mind that, for electric transitions, these quantities were defined differently by Voikhanskii and Listengarten (1959) (see also Voikhanskii er al., 1966; Band et al., 1976a), Pauli (1967), Hager and Seltzer (1969), and others. Formulas allowing one to recalculate the values from one notation to another were summarized, e.g., by Pauli et (11. (1975) and Band et al. (1976a). For magnetic transitions, conventional notation is used. In a few cases, the measurement of conversion-line intensities of the outermost atomic shells enabled one to calibrate the Mossbauer isomer shift (Section VI1,C). This calibration yielded the relative change ARN/RN of the nuclear-charge radius between the excited and the ground states.

D. Inverse Internal Electron Conversion An investigation of this scarce phenomenon was motivated by a problem of great practical importance, namely, the search for more effective methods of 235U enrichment. It was pointed out by Morita (1973) that a vacancy in the atomic shell could lead not only to the usual X-ray or Auger electron emission. The energy of the excited atom could, at least in principle, be transferred via electromagnetic interaction to the nucleus, thus picking it up into the excited state. The decay of the 235mU produced in this way could result in ionization of the atom or breaking of the chemical bond thus allowing the effective separation of 235U from the other uranium isotopes. Morita (1973) and Okamoto (1977, 1980) developed a theory for nuclear excitation by electron transition between two bound atomic states. Gol’danskii and Namiot (1976) considered the process exactly inverse to the y-ray internal conversion, i.e., the capture of electrons from a continuum state into the bound state. In particular, the authors examined the possibility of exciting the 235U ground state into the isomeric state at 76.8 eV by the capture of electrons from a laser-produced uranium plasma with temperature of 50-100 eV. The production of the 235mUin this plasma was reported by Izawa and Yamanaka (1979) who detected con-

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

71

version electrons of the isomeric transition. Gol’danskii and Namiot (1981) continued their theoretical studies and concluded that the capture of electrons from a continuum state into the 6~312vacancy was responsible production since it was by three orders of magnitude more for the 235mU probable than the process involving two bound atomic states. Otozai et a l . (1973, 1978) bombarded the 0 s target by a 60-100-pA beam of 70- 100-keV electrons from the electron microscope and re~ In accordance corded the conversion electrons from the l a s m Odecay. with the theory of Morita (1973) and Okamoto (1977, 1980), the following explanation of the effect was possible: (1) The electron bombardment produced a vacancy in the lsl/2shell of the 0 s atom. (2) When the electron from the 3d3/, subshell filled the vacancy, the virtual photon of M1 or E2 multipolarity became available. (3) This photon was absorbed by the lasOs nucleus in the 3/2- ground state thus producing the 5/2- excited state at 69.6 keV which deexcited to the 18smO~ level. The 3ds/2 + atomic transition also contributed to this process. Gol’danskii and Namiot (1981) predicted that with the increasing power of future laser beams, an excitation of the isomeric levels in 57Fe and lEITacan be expected. VII. ENVIRONMENTAL EFFECTSON INTERNAL CONVERSION Internal conversion is one of the nuclear decay modes that involve the atomic electrons directly. The probability for conversion-electron emission [Eq. (lo)] should therefore depend, at least in principle, on the chemical and physical environment of the atom. The first successful experiment of this kind is due to Bainbridge et a / . (1951) who observed the rela~ between the KTcO, tive change of (2.7 f 0.1) x lop3in the 9 s m Thalf-life and Tc2S7samples. The ssmTcisomeric state decays via the highly converted 2.17-keV E3 transition. Variations of nuclear decay rates were later reported in a number of cases, and now they are a prosperous field of interdisciplinary research (see, e.g., Johannsen et a / . , 1981). Environmental effects were the subject of several excellent reviews. Emery (1972) summarized the theoretical background and discussed in detail the experimental results for 10 transitions in nuclei with 4 5 2 I92. Freedman (1974) treated the shaking process as well as the energetics of atomic electron interactions in nuclear events. Crasemann (1973) and Dostal et a / . (1977) reviewed the higher order processes influencing the nuclear decay-rate variations. The latter authors also provided a detailed list with commentary of the experimental results concerning electron capture, p- decay, and internal conversion. The paper of Pleiter and

72

0. DRAGOUN

de Waard (1978) was devoted to the interrelationship of the isomer shift and internal conversion, whereas the review of Daniel (1979) dealt with the influence of chemical environment on lifetimes in nuclear physics, including muonic X-ray studies. As in previous sections, we concentrate here on experimental results not covered previously in review articles; theoretical studies were summarized in Section II,D,3. A . Decay-Rate Variations

In accordance with Eq. (6), the probability of electromagnetic transition depends on the total ICC. The largest change of the ICCs caused by the environmental effects can be expected for the outermost atomic subshells. These subshells, however, contribute to the total ICCs very little unless the conversion on the inner shells was energetically impossible [see Eq. (9)]. To illustrate the magnitude of the effect, we present in Table I11 the ratios avaJaof the valence-shell and total ICCs for several transitions of various energies. With regard to the treatment of the effect throughout previous reviews, we shall consider here only two studies devoted to solid-state and chemical variations of the half-life of the 76.8-eV E3 transition in 235U. This transition provides a unique opportunity to investigate the environmental effects in internal conversion. Owing to exceptionally low transition energy, the conversion is forgidden on 21 atomic subshells, K through 0, , with binding energies ranging from 115.61 keV to 96 eV. Thus the transition proceeds almost entirely via internal conversion in the outermost subshells with a tremendously low probability for competing y-ray emission (see Table IV). TABLE 111 RELATIVE CONTRIBUTION OF VALENCE-SHELL ELECTRONS TO TOTALICCs FOR ISOLATED NEUTRAL ATOMS

THE

~~

Transition Nucleus energy (keV) '8:Ba ':gTe P83 OGBi ZTC 2 92 35U

661.7 35.5 10.8 2.17 0.077

Multipolarity M4 M1 M1 E3 E3

Internal conversion possible in shells

K , L, M, N , 0 , P K, L, M, N , 0 M, N , 0 , P M, N , 0 P, Q

Assumed valence subshells 6s sssp 6s6p 4d5s 6d5f 7s

ava,/ a

(%)

0.01 0.05

0.5 3.4 2.5

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

73

TABLE IV THEORETICAL ICCs"

Atomic subshell 6s112 PI 6Pll2 p2 6P312 p3 5f512 0, p4

7~112

Q1

FOR THE

Energy eigenvalue (eV) 52.0 33.9 24.4 8.5

4.6 5.7

76.8-eV E3 TRANSITION IN

Occupation number 2 2 4

3 1 2

235U

ICC 2.7(17)* 1.8(20) 1.7(20) 4.0( 17) 8.5(18) 3.0(16) 3.6(20)

The ICCs correspond to the 5f36d17s2.configuration of the free uranium atom and were obtained from the results of Grechukhin and Soldatov (1976a,b). The values presented refer to the relativistic Hartree-Fock-Slater atomic model; the ICCs corresponding to the Thomas-Fermi-Dirac model are higher by 20-40%. 2.7(17) means 2.7 x lo".

Neve de Mevergnies (1972) measured the half-lives of the 235mU nuclei implanted into 20 various transition metals. Since the 76.8-eV isomeric level in 235U is fed from the a decay of 239Pu,the radioactive sources were recoils from a thin 239Pu advantageously prepared by collecting the 235mU layer into the foils. The whole procedure including measurements with the channel-electron multiplier (Section III,B, 1) was carried out without breaking the vacuum of about 5 x lop7Torr. Under these conditions, the 235mU atoms kept their recoil energy ( 5 9 0 keV) and penetrated into the collector up to the depth of about 100-250 A, depending on the metal used. Several checks of the etching and cleaning procedures of the collector surface were performed to verify that implanted nuclei, and not those lying eventually on the surface, were detected. Low concentration of the collected 235mU recoils excluded any interaction between the implanted uranium atoms themselves. Decay-rate variations up to 5% were observed with experimental accuracy of about 0.2%. The authors explained these solid-state effects qualitatively in terms of the interaction between the outer atomic electrons of the 235mU impurity and the free electrons of the metallic environment. Quantitative interpretation requires ICC calculations with respect to the environment of the investigated U atom in a real sample. The only ICCs for this transition available until now, however, are those of Grechukhin and Soldatov (1976a,b) for a free natural atom.

74

0 . DRAGOUN U5f

Binding energy Eb (eV)

FIG.22. The X-ray photoelectron spectra of various uranium oxides measured by Teterin rt d.(1980); VZ denotes the peak of photoelectrons ejected from the valence zone.

Neve de Mevergnies and Del Marmol (1974) succeeded in measuring chemical effects on the 235mU decay rate. The compounds studied were the uranium oxides, U 0 2 , U,O,, and UO,. The oxides were prepared from 50- 150 pg of Pu in 2 M HNO, solution to which 50 pg of natural U carrier was added. Although the stoichiometric oxides of U are not easy to obtain, the authors were able to carry out the necessary chemical procedures within 45 min (235mU half-life is about 26 min). Recorded X-ray photoelectron spectroscopy (XPS) spectra of the 4f photoelectrons proved that the radioactive samples should have been close to the desired oxidation states. [See the paper by Teterin et al. (1980) and Fig. 22 for an example of the 4f and 5f XPS spectra of six various uranium oxides taken by using monochromatized A1 K, X radiation.] Results of the half-life measurements of Neve de Mevergnies and Del Marmol (1974) carried out with the channel-electron multiplier are shown by the solid circles in Fig. 2 3 . The half-life of 235mU in the 6+ oxidation state was found to be larger by 5 . 6 k 1.1% than that in the 4+ oxidation state. In addition, the half-life measurements were carried out for 235mU implanted in the UOz, U,O,, and UO, layers of a few mg cmP2thickness. The results shown by open circles in Fig. 23 are different from those obtained with the chemically separated sources. According to the above authors, there are two possible reasons for this difference:

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

75

26-

*

25 -

1

24

uo2

4’ 5’ OXIDATION STATE

6’

-

FIG.23. Effect of the oxidation state of uranium on the half-life of 235mU measured by Neve de Mevergnies and Del Marmol (1974): ( 0 )chemically prepared sources; (0)sources recoils into layers of UO,, U,08, and UO,. The authors prepared by implantation of 235mU noted that a horizontal error bar on the nominal oxidation state of the samples studied should probably be added.

recoils and the U host (1) “Incomplete exchange between the 235mU atoms, leaving a fraction of interstitially located 235mUnuclei with a half-life of about 27 min or longer” (2) “Perturbation of the original lattice configuration around the 235mU recoils, due to a ‘thermal spike’ effect” In Section VII,B, we present a qualitative argument against the first possibility. The 235mU half-lives measured for 235mU02 (Neve de Mevergnies and Del Marmol, 1974) and 235mUimplanted into Ag metal (Neve de Mevergnies, 1972) differ by 9.8 % 1.1%. This is the largest variation of the nuclear decay rate reported until now.

B . Variations in the Convrrsion-Electron Spectra The chemical and solid-state environment of the atom influences mostly the bound-electron wave functions. With respect to internal conversion, the important effects are the following: (1) Changes of the electron densities near the nucleus (2) Appearance of new bound-electron states of the whole system which were absent in an isolated atom (3) Shifts in the electron binding energies

Changes in the conversion-electron spectra due to the above effects are

76

0. DRAGOUN

usually too small to be detected by conventional measurements. Under suitable circumstances, several interesting features were recognized. The environmental shifts of the electron binding energies range typically from tenths of eV to a few eV and are approximately equal for all atomic subshells. In photoelectron spectroscopy for chemical analysis (ESCA method; see, e.g., Carlson, 1975), these shifts are derived from the zero-energy-loss peaks. The necessary condition for observing the environmental shifts of binding energy in the conversion-electron spectra is that the shift should not be a negligibly small part of the full width of the conversion line. A most illustrative example can be seen in Fig. 11, demonstrating the 3.3-eV shift between K-shell conversion lines of 57C0 samples in “oxide” and “metallic” states (Porter and Freedmann, 1971). The first variation of the conversion-line intensity with the chemical state of radioactive atoms was reported by Bocquet et ~ 1 (1966) . for the 23.87-keV M1 transition in l19Sn. Using the magnetic spectrometer, the authors found the 0-line intensity to be 30 ? 5% smaller in SnO, than it was in white tin. The relative intensities of the other conversion lines were constant within experimental accuracy of about 3-4%. Later, the internal conversion of valence-shell electrons was examined by several investigators (see, e.g., review of Dostal rt al., 1977). The main problem in these studies is the identification of the chemical form of the radioactive source. This source has to be very thin to allow highresolution electron spectroscopy (Sections II1,C and IV,A,2). Therefore, the total amount of radioactive material studied is often below the detection limit of conventional analytical methods. In a few cases, the chemical form of the radioactive atoms was verified directly by means of the Mossbauer absorption spectroscopy (Section VI1,C). In other cases, one had to assume that the radioactive atoms and those of the carrier are in identical chemical states. This is the usual assumption in all extensive applications of the chemical compounds labeled with radioactive isotopes. The chemical states of the uranium and technetium carriers in the surface layer of the 235mU and 99mTcsources were determined by Zhudov et a / . (1979), Grechukhin et a / . (1980), and Gerasimov et a / . (1981, 1982), respectively, case is discussed in more detail using the ESCA method. The 235mU below. When radioactive atoms are implanted into a suitable backing, their environment may be well defined from a physical point of view (e.g., a position within the crystal lattice), however, as pointed out by Porter and Freedman (1971), it may not be easy to assign a definite chemical state to such an environment. In measurements of decay-rate variations by means of y-ray spectroscopy, the necessary amount of stable carrier can be added to the radioac-

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

77

tive atoms to allow investigation of the chemical state by conventional spectroscopic methods. This carrier does not deteriorate the y-ray spectrum, which is not the case in conversion-electron spectroscopy. This spectroscopy, however, provides information on environmental effects in individual atomic subshells, whereas measurements of the half-life variations yield only the sum of these effects over the whole atom. The experimental ICC ratios involving the valence-shell electrons provide an additional test of the internal conversion calculations. For example, Fujioka et ul. (1981a) measured the O,,,P/O,-line intensity ratio for the 10.84 keV M1 transition in 20sBi.In Fig. 24, the result is compared with the theoretical predictions corresponding to various configurations of the Bi valence shell involving the 6s,/,, 6p,/, , and 6p3/2 electrons. The calculations were based on the bound-electron wave functions of the relativistic Hartree-Fock model, corresponding to various configurations of isolated bismuth atoms and ions. The best agreement was reached for the 6s,/, configuration (Bi3+), which was consistent with probable chemical form Bi203of bismuth atoms in the radioactive source. The configuration 6~?/,6ppkwith any q2 I0.3 was also compatible with the measured value within one standard deviation (see Fig. 24). We describe two recent environmental studies in the internal conversion of the 76.8-eV E3 transition in 235U(Zhudov er al. 1979; Grechukhin er al., 1980). (The decay-rate variations of this 26-min isomeric transition were discussed in Section VI1,A.) The measurements were carried out with the commercial electrostatic spectrometer HP 5950 A (Kelly and Tyler, 1973), the energy range of which was extended from 300- 1500 to 0-2500 eV. The instrumental resolution was 5 0 . 5 eV. The apparatus enabled the authors to measure the conversion-electron as well as X-ray photoelectron spectra from the same sample. To prepare the 235mUsources of 1.5 x 5-mm2 area and of -3-pCi activity, Zhudov et al. (1979) applied the 239Pu layer of 25-pg recoils were first, slowed down thickness and of 100-cm2area. The 235mU in high-purity He of 100 Torr pressure, and then they were collected on the surface of the polycrystallic UF, layer. The electrostatic field of 100-V cm-l intensity was utilized, and the corona-discharge current was limited to 1 p A in order not to destroy the UF, layer. The X-ray photoelectron spectra did not reveal any uranium oxides on the surface of the UF, layer. The authors expected that a drift into this layer followed by the isotope exchange aided formation of the 235mUF4 compound. Comparing the XPS and internal-conversion spectra reproduced in Fig. 25, Zhudov et al. (1979)

-

(1) concluded that the internal conversion of the 1/2+ + 7/2- iso-

78

0 . DRAGOUN

t heor 2 00 210 20 I 220 21 I 221 212

I10

121

0 01

~~

02 I 010 000

0 00 1

FIG.24. Comparison of the theoretical ICC ratios O,,,P/O, of the 10.84-keV MI transition in zOsBiwith the experimental result. The theoretical values correspond to various configurations of the valence shell of the free atom and ions of bismuth. The triads of digits on the right-hand side represent the occupation numbers of the 6sll,, 6p1,, , and 6p,/, subshells, respectively. [From Fujioka C t a / . (1981a).]

meric transition in 235Utakes place in the 6p1/2 and 6p3/2 subshells of the U atom, in the “valent zone” of the UF, compound and, apparently, in the 2s subshell of the F atom (2) determined the transition energy to be 76.8 2 0.5 eV in a way independent of the electron binding energies and work functions of the particular arrangement. The theoretical ICCs of this transition presented in Table IV are in accord with the experimental result that the 6p1/2 and 6p3/, lines dominate, whereas the 5f line is absent in the conversion-electron spectrum. The strong 5f line in the XPS spectrum in the upper part of Fig. 25 demonstrates that at least some of the uranium 5f electrons are not taking part in the chemical bond of UF, . Note that this line is missing in the XPS spectrum of UO, (Fig. 22). Grechukhin et al. (1980) continued these studies using the same appa-

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

... -.:-

. v:.

I

I

40

50

79

,$, I

60

*> :.

>*W?

7u

Kinetic energy ( e V )

FIG.25. The internal conversion-electron spectrum (shown by points) of the 76.8-eV E3 transition in z35U corresponding to the 235mU atoms deposited on the polycrystallic layer of UF,. The X-ray photoelectron spectrum of the same layer is shown by the solid line. The numbers on the horizontal axes indicate the kinetic energy of electrons in electron volts. [From Zhudov ef ul. (1979).]

ratus but collecting the 235mU nuclei on the y-UO, layer instead of the UF, layer. There appeared to be a great difference between the conversionelectron spectra of the two experiments, since they differ in the number of recorded conversion lines (see Figs. 25 and 26). Therefore, not only the intensities and energies of conversion lines but also their number in the spectrum can be influenced by the environment of the atom undergoing the internal-conversion process. , ~ caused by the elecThe electrostatic splitting of the core U 6 ~ , level, tric field of ligands at the place of the uranium atom, is seen in both XPS and internal-conversion spectra of the y-UO, sample (region I1 in Fig. 26). Using the XPS method, Veal et al. (1975) investigated this splitting systematically as a function of the primary uranium-oxygen separation for a number of hexavalent uranium compounds. The most interesting feature of the y-UO, conversion-electron spectrum (Fig. 26) is the line in region I11 where the oxygen 2s line is seen in the XPS counterpart. Since internal conversion proceeds in the near vi-

80

0 . DRAGOUN

a

b

-40

E P m I 1 J I 1 -30 -20 -10

I

Binding energy Eb(eV)

corFIG.26. The conversion-electron spectrum (b) of the 76.8-eV E3 transition in 235U responding to 235mU atoms deposited on the y-UO, layer. The X-ray photoelectron spectrum (a) of the same layer is shown for the comparison. [From Grechukhin et al. (1980).]

cinity of the excited nucleus (see Section 11,B,3), the said conversion line cannot be explained as a result of the electromagnetic interaction between the 235mU nucleus and 2s electrons of the surrounding 0 atoms. In particular, Grechukhin et al. (1980) calculated the probability of this process to be 10+ of that for a usual conversion of uranium electrons. They also proved that, within 1% accuracy, the interaction proceeds in the sphere of 0.05-A radius around the U nucleus. The authors therefore consider the above conversion line as experimental evidence for substantial hybridization of rather deep 0 2s and U 6p bound-electron states in the y-UO, compound. Similarly, the doublet in the conversion-electron spectrum of UF, (Fig. 25) was then interpreted as being caused by the hybridization of the F 2s and U 6p states, which are of similar energy. Thus the internal-conversion studies of Grechukhin et al. (1980) revealed molecular orbitals with evident p-state admixtures and binding energies up to -40 eV. The authors conclude that conversion-electron studies of low-energy transitions (Ey < 10 keV) will contribute effectively to clarifying the nature of the chemical bond. The similarity between the XPS and conversion-electron spectra of the uranium compounds (Figs. 25 and 26) supports the assumption on forming 235mUF4 and 235mU03 within the investigated UF, and y-UO, layers, respectively. In Section VII,A, the variations of the 235mU half-life measured by Neve de Mevergnies and Del Marmol(l974) were discussed. In view of the above similarity in the electron spectra, we consider the

INTERNAL CONVERSION-ELECTRON SPECTROSCOPY

81

argument on incomplete isotope exchange less probable than that on the lattice perturbation.

C. Calibration of the Miisshauer Isomer Shifts Mossbauer spectroscopy (see, e.g., Shenoy and Wagner, 1978) provides shifts of transition energies caused by different chemical environments of deexciting nuclei: S = const. Ape(0) A(vB)

(41)

Here, S is the isomer shift, Ap,(O) is the difference of the electron densities at the nucleus in two environments, and A(rE) is the difference between the mean square nuclear charge radii of'the two nuclear states involved. Although both Ap,(O) and A( r ; ) are of great interest in quantum chemistry or solid-state physics and in nuclear physics, respectively, they cannot yet be determined separately by direct experimental methods. In general, none of these quantities can be calculated with sufficient accuracy at this time. The only exception seems to be the calculation of Ap,(O) for several iodine compounds (Hartmann and Eifrig, 1981). Band and Fomichev (1979) tabulated the electron charge densities at the nuclear center and surface for each of the ns1,2and np,,, subshells for 129 atoms and ions relevant to Mossbauer spectroscopy. These densities were obtained by solving the relativistic Dirac- Fock equations. Self-consistent calculations of this type for free atoms and ions are known to provide realistic results for the inner shells. To obtain Ap,(O), however, the contribution from the outermost shells of atoms in solids and chemical compounds is needed, too. In some cases, this contribution can be obtained from the internal conversion experiments. The method makes use of the approximate proportionality between the ICC, a { , and the subshell electron density peJO) (Eq. 24). It was introduced by Bocquet et a / . (1966) who measured the O/N, conversion-line intensity ratio for the 23.9-keV transition in l19Sn. To check the chemical composition of the conversion-electron sources, the authors recorded their Mossbauer spectra and compared them with the standard ones. The psS(O)densities in the Sn and SnO, sources were derived from measured O/N, intensity ratios and the theoretical value of the 4s electron density. Owing to difficulties with preparing the radioactive sources suitable for both conversion-electron and Mossbauer spectroscopies, further measurements were accomplished only for the 14.4-keV transition in 57Fe(Fujioka and Hisatake, 1972; Shinohara and Fujioka, 1973; Shinohara et al., 1976); the 27.8-keV transition in lZ9I(Spijkervet and Pleiter, 1979); and the

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9.4-keV transition in =Kr (Spijkervet et a / . , 1981). The experimental techniques and results were scrutinized by Pleiter and de Waard (1978), who also discussed the change of the inner shell electron density pe,co,,(0) induced by the environmental effects. Dautov et al. (1975, 1981) examined the pe,core(0) variations in detail and calculated that they may not be a negligible part of the overall change Ape(0). In particular, the authors demonstrated the following: (1) The conversion-line intensity ratio anmax ,s/~nmax-l,s (where nmaxis the principal quantum number of the outermost bound electrons), holds for the determination of Ape(0)only when d and f electrons do not participate in the chemical bond (2) For the transition elements, it is necessary to derive the ICC changes with good accuracy at least in two s subshells, n,,, and nmax - 1 (3) For the rare earths and actinides, such ICC changes have to be determined at least for three s subshells, nmax,nmax- 1, and nmax- 2.

Dautov et a / . (1981) also analyzed all the methods for calibration of the isomer shifts and provided the best current values of the A( rk) for 54 Mossbauer transitions. A N D OUTLOOK VIII. SUMMARY

We have summarized the basic knowledge of the y-ray internal conversion and reviewed critically the theoretical and experimental results obtained mainly during the last decade. At present, extensive tables of the ICCs are available for all subshells of neutral isolated atoms. These calculations, based on the lowest nonvanishing order of the perturbation theory, are generally in accord with newer measurements. Nevertheless, consistent treatment of the higher order contributions is still needed. The regions where these effects should be large are in most cases not yet experimentally accessible. Transitions of very low energy are of special importance due to increased sensitivity of the ICC values to the physical assumptions on which the calculations are based. When analyzing precise experimental data, (1) the ICCs calculated directly for the investigated cases should be preferred to those interpolated from the tables, and (2) simultaneous searches for possible multipolarity mixture and the nuclear structure effect should be carried out. In the 1970s, the resolution of magnetic spectrometers was further improved and electrostatic instruments were applied for conversion-electron investigations at very low energies. For measurements with moderate res-

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olution, multichannel solid-state detectors are utilized. Recent development of the microchannel plate detectors and of the position-sensitive proportional counters is a great promise for multichannel measurement of conversion electrons at high resolution. The number of nuclear reaction studies with conversion electrons has increased substantially. The spin-parity determination of electromagnetic transitions remains a domain of conversion-electron spectroscopy. This spectroscopy also provides valuable information about the nuclear matrix elements of hindered and electric monopole transitions. The results of both experimental and theoretical studies of the environmental effects on internal conversion demonstrate that it is a promising field of interdisciplinary research. As for applications, conversion-electron Mossbauer spectroscopy has begun to provide valuable information about the surface layers of solids. During the next decade, we can expect the following: (1) The theoretical conversion coefficients for inner shells, including all effects larger than 1% (2) The ICC calculations for chemical compounds and solids, together with corresponding experiments (3) The multichannel measurements of conversion electrons at the momentum resolution of better than 0.05% (4) The multipolarity determination for many transitions (including high-energy ones) which are populated in nuclear reactions and in the radioactive decay with half-life smaller than 1 min ( 5 ) Some unpredicted results (as was the case in previous decades) which make the research so interesting.

ACKNOWLEDGMENTS It is pleasure to acknowledge Dr. V. Brabec and Dr. M. RySavy for critical reading of the manuscript and valuable comments. I am grateful to cited authors for permission to include figures from their papers in this review. I am obliged to Prof. M. Fujioka, Dr. K. Schreckenbach, and Dr. J. L. Wiza for sending me details on their published or current investigations. My thanks are also due to the following publishers for granting me permission to reproduce previously published figures: The American Physical Society, New York (Fig. 1l ) , Central Research Institute for Physics, Budapest (Fig. lo), Elsevier Scientific Publishing Company, Amsterdam (Fig. 14), The Institute of Physics, London (Fig. 18), Joint Institute for Nuclear Research, Dubna (Fig. 12), North-Holland Publishing Company, Amsterdam (Figs. 4, 5 , 6, 8, 15, 16, 19, 20, 23), Oak Ridge National Laboratory, operated by Union Carbide Corporation for the U.S. Department of Energy (Fig. 2), Physical Society of Japan, Tokyo (Fig. 21), Publishing House “Nauka,” Moscow (Figs. 22, 25, 26), Research Journals, National Research Council of Canada, Ottawa (Fig. 9), Springer-Verlag, Berlin and New York (Figs. 3, 24), and The Technical Information Center of the U.S. Atomic Energy Commission, Oak Ridge (Fig. 7).

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Wijnaendts van Resandt, R. W. (1980). J . Phys. E 13, 1162. Wild, R. K. (1981). Vtrcrrrim 31, 183. Williams, A,, Campion, P. J., and Bums, J. E. (1973). N i d . Instrum. Methods 112, 373. Wiza, J. L. (1979). Nrrcl. 1n.strrrm. Methods 162, 587. Yaffe, L. (1962). Annrr. Re\.. Nrrcl. Sci. 12, 153. Yoshida, Y., Tsuji, K., Marubayashi, K., and Matsumoto, Y. (1978). Nricl. Instrum. Mrthods 154, 261. Yoshida, Y., Tsuji, K., Umesaki, S . , and Marubayashi, K. (1981). Nrrcl. Instr. M e t h . 189, 423. Zhirgulyavichyus, R. K. (1979). In “Methods for Research of Radioactive Transformations in Chemical Compounds” (K. V. Makariunas, ed.), p. 92. Phys. Inst. Acad. Sci. Lith. SSR, Vil’nius (in Russian). Zhudov, V. I., Zelenkov, A. G . , Kulakov, V. M., Mostovoi, V. I., and Odinov, B. V. (1979). Pis’nzrr Zh. Eksp. Teor. Fiz. 30, 549. Zilitis, V. A,, Trusov, V. F . , and Eglais, M. 0. (1981). l z v . Akad. Norrh S S S R , S e r . Fiz.45, 690.

ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS. VOL . 60

Diffraction of Neutral Atoms and Molecules from Crystalline Surfaces G . BOATO

AND

P. CANTINI

Istituto di Scienze Fisiche and GNSM-CNR Universita di Genova Genoa. Italy

I . Introduction ............................................................ I1. General Outline ......................................................... A . Brief Review of Experimental Techniques .............................. B . Elastic Diffraction ................................................... C . Inelastic Diffraction .................................................. D . Rotational Diffraction ................................................ 111. Quantum Theory of Atom-Surface Scattering .............................. A . Static Potentials and Elastic Diffraction Theories ........................ B. Vibrating Surfaces and Inelastic Scattering ............................. IV . Structural Information from Elastic Diffraction ............................. A . Simply Corrugated Surfaces........................................... B. Reconstructed and Periodically Deformed Surfaces ...................... C . Adsorbate-Covered Surfaces .......................................... V . Information on the Surface Potential Well ................................. A . General ............................................................. B . The He-Graphite System............................................. VI . Information on Surface Lattice Dynamics.................................. A . Debye-Waller Factor ................................................ B. Structures in the Angular Distribution .................................. C . Energy-Loss Measurements ........................................... VII . Conclusions ............................................................ References .............................................................

95 98 98 100 106 109 111 112 121 129 130 132 135 138 138 139 146 146 148 152 155 156

I . INTRODUCTION The scattering of neutral atomic beams. particularly helium. at thermal energies from crystalline surfaces has become a very practical and useful tool for studying both the surface structure and lattice vibrations. The diffraction of thermal atoms differs from low-energy electron diffraction (LEED) and other surface-scattering techniques essentially by the fact that in the first case the very top layer of the surface is explored. The 95 Copyright @ 1983 by Academic Press. Inc. All rights of rcproduction.in any form reserved. ISBN 0.12-01466&6

96

G. BOAT0 AND P. CANTINI

Fig. 7. Die gestrichelten Kurvenstiicke zeigen den EinfluB einer kleinen Verdrehung (- 20) des Kristalls in seiner Ebene.

\, ,, , \\

..I,'

,,'

_,

FIG.1 . Reproduction of some of Frisch and Stem (1933) illustrations of selective adsorption experiments: (a) diffraction figure; (b) part of the apparatus; (c) experimental geometry.

reason for this is the impenetrability of atoms, or, in more physical terms, the repulsion originating from the Pauli exclusion principle. The first observation of atom diffraction from surfaces is due to Estermann and Stem (1930), who had the particular objective of demonstrating the wave nature of atomic particles. However, Stem and collaborators

ATOM DIFFRACTION FROM SURFACES

97

carried out in Hamburg a series of very ingenious experiments that were the prelude to the present atom-surface diffraction method. They studied different ionic crystal surfaces, built a double-crystal diffractomer (Estermann et al., 1931), and observed the phenomenon of selective adsorption, later explained by Lennard-Jones and Devonshire (1937). For historical record and respectful memory, some of the results obtained by Frisch and Stem (1933) on selective adsorption are shown in Fig. l a (a photostatic reproduction of one of Stem’s figures). Similarly, Figs. l b and l c show the apparatus and the geometry used in this outstanding set of experiments. Unfortunately in 1933 Stem was forced to leave Germany and never resumed work in this field. About 35 years elapsed before significant progress was accomplished. This happened at the end of the 1960s because of two main reasons (1) the development of the supersonic beams and (2) the availability of controlled surfaces by the use of ultrahigh vacuum. Further motivatim was given by the theoretical paper of Cabrera et al. (1970), who first used a nonperturbative approach to the theory of atom diffraction. Interest in the gas-surface interaction was also aroused by the growth of rarefied-gas dynamics, in connection with space aeronautical problems. Important experiments were carried out by Smith et al. (1969), Fisher et al. (1969), and Subbarao and Miller (1969). At the beginning of the 1970s, the first detailed diffraction experiments were carried out with the use of nearly monochromatic supersonic beams; among these the experiments of Williams (1971a,b) are worth mentioning, since they first showed fine structures in the diffraction pattern and, from the technical point of view, made use of cryogenic vacuum. The atom diffraction technique was then ripe for a gradual development. The first meeting where the subject was thoroughly discussed and a new look taken at the problem was the LVIII Course of the E. Fermi International School (Varenna) held in 1973 (proceedings published by Editrice Compositori, Bologna, Italy, 1974). Subsequently, rapid progress involving an increasing number of workers was made. The purpose of the present review is to give an overview of the subject as it stands now, to discuss the main points in both theory and experiments, and to indicate possible future trends. This review does not pretend to be complete: we regret that some important work may not be sufficiently discussed or even mentioned. In return, we hope to give the reader a critical discussion of the principal information one can obtain from atom diffraction experiments. Review articles on the subject that appeared and the topics covered by these articles are mentioned below; these topics are not given wide space in this review. Cole and Frank1 (1978) and Armand and Lapujoulade (1979) wrote brief general reviews, where details on techniques and

98

G. BOAT0 AND P. CANTINI

theories may be found. Hoinkes (1980) treated thoroughly the gassurface interaction potential as determined by atom diffraction experiments. Engel and Rieder (1982) wrote a very detailed review on structural studies of surfaces, where much information and a complete bibliography can be found. Finally, Cardillo (1981a) gave a general survey of recent experimental work on all types of gas-surface interactions studied by means of molecular beam techniques. The present article starts with a qualitative treatment of the experimental methodology and a simplified explaination of the physical phenomena involved in atom diffraction (Section 11). A review of the theoretical methods useful for understanding and interpreting the experimental data is then given (Section 111). In Section IV a concise account of the structural information obtained by elastic diffraction is given, and in Section V the problem of determining the atom-surface potential from experiment is treated. The information obtained on surface lattice dynamics is discussed in Section VI, and finally, the conclusions and perspectives are reported in Section VII.

11. GENERAL OUTLINE

A . Brief Review of Experimental Techniques The apparatuses presently used in different laboratories for the study of molecular beam diffraction do not differ much in their main constituents. The production of neutral beams is based on the supersonic expansion from a high-pressure region (5-100 atm) through a small nozzle (50-5 pm) into a vacuum, the source being held at temperatures ranging from 10 to about lo00 K. The central part of the beam is skimmed and angularly defined by means of suitable collimators. As a result, one disposes of very narrow beams of neutral atoms and molecules, which are intense (- 5 x 1019 particles/steradians sec) and nearly monochromatic (velocity spread, Av/v = 1-5%). Details of molecular beam techniques and of scattering apparatuses for surface studies can be found by the reader in review articles by Cole and Frank1 (1978) and Engel and Rieder (1982), and they are very briefly discussed in this article. Special sources must be used for the production of beams of radicals or excited molecules. An important source of this kind is that of atomic hydrogen, which was extensively used for diffraction studies by the Erlangen group (Hoinkes el al., 1972a) and further developed in Genoa. The effusive beam is produced by a radiofrequency discharge and velocity selected by a magnetic hexapole (Barcellona et al.,

ATOM DIFFRACTION FROM SURFACES

99

1973) or by a rotating slotted disk (Finzel er al., 1975). With the source at room temperature, the velocity spread can reach 10% and the intensity can be 10l6molecules/steradians sec. The detection of molecular beams has also reached a high degree of sophistication. Although stagnation detectors (like that used by Stern) are still in use, electron bombardment followed by mass analysis of the produced ions is the most widespread method of detecting neutrals. A sensitivity of times the incident beam intensity has been reached. This technique is easily combined with time-of-flight measurements. Another detection method which proved to be effective and equally sensitive is based on the measurement of the energy delivered to a bolometric device-usually a low-temperature doped-semiconductor crystal-during the accomodation of the neutral particles on the sensitive surface. Special detectors must be used for radicals or metastable particles. In order to improve the signal-to-noise ratio, chopped beams and lock-in techniques are widely used. The scattering region is enclosed in a ultrahigh vacuum chamber under dynamical conditions. The vacuum is mantained with conventional techniques. Of great advantage has proved the use of cryogenic vacuum, particularly for the study of surfaces in the low-temperature region. As shown below, the use of cold surfaces may be determinant both for structural studies (the elastic peaks are attenuated by the presence of a temperature-dependent Debye- Waller factor) and for inelastic phenomena, when one-phonon processes are the main object of investigation. The crystal surface is cleaned or prepared and subsequently characterized in situ by the standard techniques used in modern surface studies [cleavage, heating, ion bombardment, annealing, etc., for surface preparation; LEED, Auger electron spectroscopy (AES), secondary-ion mass spectrometry (SIMS), etc., for surface cleanliness, analysis, and characterization]. The intensity and sharpness of diffraction peaks are very sensitive to the quality of the surface, so that atomic scattering itself can be advantageously used to obtain a check of the surface perfection. The surface is mounted on more or less elaborate crystal holders, possibly rotatable in all spatial directions. In fact, precision measurements need accurate determination of both the incident angles (polar and azimuthal) of the beam with the surface net and the outward scattering angle. Collimation problems are therefore important and usually delicate. For these and other experimental problems the reader is again referred to the above-cited review articles. We may conclude by saying that even if atom-surface scattering techniques are sophisticated and only partially commercialized, they are not particularly difficult to be installed and utilized.

100

G . BOAT0 AND P. CANTINI

B . Elastic Diffraction When a beam of particles having a mean wave vector k,, (and an energy E, = h2ki/2m) impinges on a stationary periodic surface (lattice vibrations are for the moment discarded), a discrete set of diffraction peaks is backscattered from the surface under the condition that the wavelength h = 2rr/ko is comparable with the surface lattice parameter a. Under these circumstances, there is no energy exchange between the particles and the surface. Because of the surface periodicity, the particle-surface potential can be expanded in a two-dimensional ( 2 D ) Fourier series V&) exp(iG R)

V(r) =

(1)

C

where r(R, z) defines the position of the particle, R is the projection of r on the surface, and the z direction is assumed to be the outward normal to the surface; G(m, n) represents a surface reciprocal lattice vector defined by G = ma* + nb* (2) where m and n are integers, and a* and b* are related to the surface unit vectors a and b by a . a* = b . b* = 2 r (3) a . b* = a * . b = 0 (4) The condition for having non-zero-order diffraction peaks is that some of the V , terms are different from zero. Otherwise, only the specular peak (G = 0) survives. The kinematic conditions for having elastic diffraction are

+

(5)

&=K,+G

(6)

ki = kg = k& (conservation of energy) and

(conservation of parallel momentum). Here kC(& + G, kGz) is the final wave vector of the particle, labeled by a reciprocal vector G. Expression (6) is the Bragg condition for 2 D diffraction. What we have stated until now is equally valid for neutral particles and low-energy electrons. Both techniques are in fact able to give quite similar information on the surface-lattice symmetry. However, low-energy electrons (Eo = 2-500 eV) explore a few atomic surface layers, whereas neutral atoms (E, = 4-200 meV) probe only the uppermost Dart of the surface and are nondestructive in character. The two techniques are therefore complementary for structural information.

ATOM DIFFRACTION FROM SURFACES 00

. o.p . .

ii i o

ii

i 2

'0,

0:

02.

1 ;

1p

:1

2 1.

2T.

2p

2 ;

2.2

101

' L i F (001)

22

20

I

FIG.2. Diffraction pattern of He scattered from LiF(001).

A typical experimental diffraction pattern is shown in Fig. 2 for the system He-LiF(001). In this example the incident beam (E,, = 63 meV, A = 0.58 A) was normal to the surface and the crystal was at 10 K. Only the diffraction peaks in an octant are shown; the others can be derived by the square symmetry of the surface unit cell. The peak widths are determined by the angular resolution and the velocity distribution of the beam. The diffraction-peak intensities are related to the atom-surface potential V(r). It is then important to give a qualitative description of this interaction potential. The discussion may remain specialized to systems showing effective diffraction, namely, to the case of light atoms (A large) impinging on chemically inactive periodic surfaces. The appropriate physisorption potentials are not exactly known, but they exhibit some common features, which are of great importance in understanding what kind of information on the surface structure a diffraction experiment can give. As an example, we consider the potential of the system He-graphite, one of the most accurately known. The methods used to derive it are extensively discussed in Section V,B,3. The potential as a function of z for a few positions R in the surface unit cell is shown in Fig. 3. It is seen that at short distances, very strong repulsion is present, originating from the mutual overlap of outer electron wave functions of the incident particle with those of surface atoms. The z dependence is nearly exponential and has very short range. On the other hand, at large

102

G. BOAT0 AND P. CANTINI

-201 1.0

I

I

I

2.0

1

3.0

I

1

I

4.0

I

5.0

z 181

FIG.3. Gas-surface potential of He-graphite.

distances, attraction due to London dispersion forces is prominent. According to the general treatment of Lifshitz (1956), the asymptotic form of the potential for large z is given by

V(z) = -cz-3

(7)

with C related to the dynamical polarizability of the atom and to the dielectric function of the solid (see Hoinkes, 1980). At medium distances, around the equilibrium position, there is the well region, where the potential is usually less accurately known. Figure 3 shows that for z values just above the equilibrium distance the incident atom sees an attractive potential which does not depend on R and gives only a minor contribution to the diffraction figure. In contrast, the potential at short distances is laterally dependent and periodic; of a similar nature is the region near the minimum. The potential region at short distances is the most important for diffraction. The effects of the periodically lateral dependence can also be represented by the Fourier components V,(z) of the interaction potential [see Eq. (l)]. In the long-range

ATOM DIFFRACTION FROM SURFACES

103

attractive region and for G # 0, the Vc's are in fact completely negligible, but they strongly increase when z becomes lower than the equilibrium distance (for example, see Fig. 22 later). A very interesting quantity is the zero Fourier component, given by

V(R, z) d2R

V,(z) = d - l / U.C.

where d is the area of unit cell (u.c.); Vo represents the laterally averaged gas-surface potential and can be used as a first approximation to V(r) when the other Fourier components are small. The description of V(r) through its Fourier components is fully exploited in the next sections. The above-illustrated trend of the gas-surface potential is common to all systems of interest for atom diffraction. The modulation of the repulsive potential may be more or less strong, and in a parallel manner the well depth may change with R (the two effects are described by the number and strength of sizable Fourier components), but two features remain: the repulsive potential is very steep and modulated, whereas the attractive region is long range and nonperiodic. The average well depth D , namely that associated with Vo(z),can range from 5 to 20 meV for He and rise to about 100 meV for H and H,, but its effect can be viewed, at least in a first approximation, as an acceleration of the particle before a hard collision against the steep repulsive periodic potential takes place. From the above analysis, we can understand the physical reasons for the commonly adopted approximation of representing the diffractive potential with a corrugated hard-wall model (see Section III,A,2). In all cases the observed diffraction pattern, namely, the intensity distribution among diffracted beams, essentially reflects the periodic shape (or corrugation) of the repulsive wall. The corrugation can be visualized by constructing the equipotential surfaces at the incident energies (25- 100 meV); in other words, the periodic repulsive wall is the locus of classical turning points of the particle scattered by the potential V(r). As a first example of the above discussion, we return to Fig. 2, showing the case of He-LiF. The diffracted peak intensities display a peculiar trend: their envelope as a function of the scattering angle Of has a maximum around the second-order peaks, i.e., at Of = 35-40'. This effect is called quantum surface rainbow, by analogy with the classical surface rainbow observed when rays are scattered by a periodic reflecting surface. The classical surface rainbow for atom scattering was discussed by McClure (1972); the theory of quantum rainbow is expounded in Section 111,A,2. A second example is shown in Fig. 4. Here the diffraction pattern is more complicated, due to the complex structure of layered compounds

G . BOAT0 AND P. CANTINI

104 He/2 H -Ta Se 2

............ ........... 0

0

.

.

o

$0.

o

* *

io!I

00’

2c

01

5c

.-A0

017

Reciorocal lattice

I ..,....: ...

.......’ -5OO -40°

-30° -

I

30O

FIG.4. Diffraction pattern of 2H-TaSeZ. A superstructure caused by charge-density waves is present.

such as 2H-TaSe2. The intense integer-order peaks are the result of the corrugation of the top layer of selenium atoms (having trigonal symmetry owing to the metal-atom layer below). The small fractional order peaks are due to a superstructure brought about by the presence of chargedensity waves (see Section IV,C), which slightly deform the top layer with a periodicity three times larger than the principal one. In other words, a (3 x 3) commensurate superstructure, stable below 90 K, is detected (P. Cantini and R. Colella, unpublished results). The previous discussion is a first qualitative approach to understand the diffracted intensities. There is, however, another effect of the attractive well that is relevant and, under special circumstances, can become dominant and eventually upset the above picture. This effect was observed by Frisch and Stern (1933) in the first experiments on atom diffraction and was explained by Lennard-Jones and Devonshire (1937) as the result of resonant transitions into the bound states of the attractive well. It was named “selective adsorption” by the latter authors, and this evocative name is still widely used today. In brief, under given incident conditions, the particle can make a resonant transition into a state bound to the surface insofar as the z direction is concerned, while traveling with in-

ATOM DIFFRACTION FROM SURFACES

105

creased kinetic energy parallel to it. The kinematic condition controlling a bound-state resonance (BSR)is k2Gz - k2 - [K,+ G(m, n)I2 = 2mej/h2 (9) where ej(< 0) is one of the energy level of the average potential V,(z) and (h2/2m)(& + G)2represents the parallel kinetic energy. From Eq. (9) it is seen that a BSR is labeled by three indices: ( m ,n), which define G, and j . The kinematics of a BSR process is illustrated graphically in Fig. 5 , using the K plane. The total energy hzki/2m is positive, so the particle lives temporarily in this metastable state; it ultimately leaves the surface via an elastic transition in one of the diffraction channels or it will suffer an inelastic process. As a result, the distribution of intensity among diffraction peaks can be, at resonance, significantly different than out of resonance. Various aspects of the selective adsorption and its importance for the experimental determination of the shape of the potential in the region of the well are discussed at length in later sections. We want only to mention here how the selective adsorption structures are experimentally observed. One usually works at constant energy E, = h2kg/2m,whereas one of the angles (0, , 4,) that the incident wave vector ko makes with the surface net is continuously changed. When the condiction (9) is met, an anomaly may be observed in both the specular and diffracted beams; the resonant structure has the form of a minimum (more often), a maximum, or a

Example:

j =1

.' FIG.5. Kinematics of a BSR in the K plane.

106

0.2

L Of 2

I 10

01

a (0)

(b)

FIG.6 . Bound-state resonance structures in the specular intensity plotted (a) as a function of +o at constant Ba, and (b) as a function of Bo at constant + o . [From Frankl et a / . (1978).]

more complex shape. This is shown for the system He-LiF(001) in Fig. 6a and b where either the polar O0 or the azimuthal angle 4owas changed. The angular location of minima and maxima satisfies Eq. (9) with excellent agreement. Rewriting Eq. (9) in the f7rm

[KO

+ G(m, n)]'

=

(2m/h2)(Eo-

~ j )

(10)

one deduces that the values of & (the projection of ko on the surface) leading to each bound state ej (
Owing to thermal motions, during a scattering process surface-atom displacements take place so that a static gas-surface potential is unrealis-

107

ATOM DIFFRACTION FROM SURFACES

Ky

(i-') ' ..

...

'\.* '\

\

--

*

.

\ \ \\-

\

\ \\

-

.€0

\

01-threshold

-A;,

-.El '\

..

:

- €2.. -::..

..

... ..

. .. .. .. .. €3: .:. . - .. . < :. .. .. K,

(i-' )

I

I

I

4

5

6

FIG.7. Location of observed BSRs in the K plane: k,, NaF(001) [From Meyers and Frank1 (1975).]

=

6.10 k

l ; He scattered from

tic. In fact, lattice vibrations are present not only at finite temperature but even at 0 K by virtue of the so-called zero-point motions. The result is that a true potential must be time dependent and that in real scattering experiments, exchange of energy between particle and surface is always possible, in the form of created and annihilated phonons. At surface temperatures T, lower than a characteristic temperature, the inelastic efusually identified as the surface Debye temperature 8,, fects are sufficiently weak to leave almost unaffected the picture given previously in Section I1,B. Some intensity is subtracted from the elastic peaks and scattered in a wide solid angle, but no change of the intensity ratios among diffraction peak intensities is observed. This fact is operatively described by the introduction of a temperature-dependent corrective factor, smaller than unity, which multiplies the intensity of diffraction peaks. The experimentally measured diffraction probabilities can then written as ( PG)expt = ( PG)el exp( - W G ) (1 1) where (PdeXpt is the ratio of the observed peak intensity to the incident

108

G . BOAT0 AND P. CANTINI

beam intensity, whereas (P&, is the same quantity to be expected for a purely elastic process (static potential). The quantity exp(- 2W,) is called the Debye-Waller factor by analogy with what is found in X-ray, neutron, or electron diffraction; however, in these cases, Eq. (11) is easily justified and W, is found to be the thermal average W,

=

:((Ak

*

u)~)

(12)

- k, and u is the thermal atom diswhere Ak is the scattering vector placement. In molecular beam scattering, Eq. (12) is only approximately valid and even unapplicable, as we discuss in detail in Sections III,B,l and VI,A. The existing experiments essentially show that the thermal attenuation of the diffracted beams is approximately exponential in character at temperi.e., P , exp(A/T,). atures around 8,, Under the condition that 0 < T, << 8,,the intensity subtracted to elastic peaks appears as tails to these peaks and as a weak background between them (see Figs. 2 and 4). This inelastically scattered intensity contains in a concealed way a wealth of information on the elementary inelastic processes occurring during scattering. For low-mass particles and low surface temperatures, exp( - 2 W,) can be of the order of 0.1 and only a few phonons are involved at each inelastic collision. More precisely, it can be shown that when W, is of the order of unity, one-phonon processes are dominant: in this case, for a coherent process, the conservations of energy and of parallel momentum require that

Ef - E,

=

(hZ/2m)(kf2- kg) = k f w ,

AK = Kf - & = G

k

Q

(13)

(14)

where hugand hQ are the energy and parallel momentum of the annihilated (+) or created (-) phonon, and the subscript f refers to final states. Among all possible involved phonons, surface phonons are making a large contribution, as is discussed later. Assuming that the conditions for the occurrence of one-phonon processes are met, the experimental method for detecting them is based on a combination of both angular distribution and energy measurements. As pointed out by Williams (1971b), it must be realized that creation and annihilation processes can be spatially separated at sufficiently large incident angles. For in-plane scattering this is illustrated in Fig. 8. Different attempts to detect surface phonons and their dispersion relation by purely angular distribution gave satisfactory results, as is discussed in Section V1,B. However, in order to obtain sufficiently accurate information, an energy analysis must be carried out: this question is treated in Section V1,C.

ATOM DIFFRACTION FROM SURFACES

109

Reciprocal S u r f a c e Array

FIG.8. In-plane inelastic events involving phonons of parallel momentum hQ (Ewald representation).

D . Rotational Diffraction

When molecules are used as impinging particles another kind of surface diffraction process is possible, namely, that involving both conservation of parallel momentum and energy exchange between translational and internal degrees of freedom of .he incident molecule. In the real world, the only detectable processes of this kind are rotational transitions of very light molecules in the act of interzcting with a surface. In fact, no case is presently known of surface diffraction coupled with rotational transitions other than that occurring with H, and its isotopic species; on the other hand, vibrational transitions of hydrogen molecules involve too high energies in comparison with those of thermal beams used in practical experiments. We give here a short account of this interesting phenomenon but the subject is not further treated in this article. The reason for this choice is that “rotational diffraction” has not yet been studied in depth, and, what matters most, no satisfactory theory of the diffracted intensities and resonance phenomena is yet available. For the interested reader, an extended bibliography is reported in this section. Logan (1969) was the first to predict on theoretical grounds the existence of rotationally inelastic diffraction for H, scattering. The phenomenon was experimentally observed by Boato et al. (1974) and independently by Grant Rowe and Ehrlich (1975a), using LiF(001) and Mg0(001), respectively. The phenomenon was further studied in different systems by various authors (Grant Rowe et al., 1975; Grant Rowe and Ehrlich,

110

G. BOAT0 AND P. CANTINI

00 -450

'i,,UL

t

t m z W

c

- 30°

Z T Oool0

H*-LIF(001)

00

30"

60"

30'

60'

01

W

a I

3

It 1

1

O

2

Jo

w I-

c

a

0

m

-

- 30°

0f

FIG.9. Elastic and rotationally inelastic diffraction in H,-LiF(001). [From Boato et al. (1976b).]

1975b; Boato et af ., 1976b,c; 1977; Cantini et a / ., 1977; Lapujoulade et al., 1981). The kinematic conditions for rotationally inelastic diffraction are given by the conservation of parallel momentum

Kf=K,+G

(154

and by the conservation of energy

A2kf/2m = Azk;/2m

+ AE,,,

(13))

where AE,,, = kBT,,,(Jo - Jf)(Jo+ Jf + l), Trotbeing the rotational temperature and J the rotational quantum number. An example of rotational diffraction is given in Fig. 9, where the scattered intensity is shown for the system H,-LiF(001). Here Jf - Jo = 2 2 and only the rotational transitions 0 + 2 and 2 + 0 forpara-H, and 3 + 1 for ortho-H, are detected as additional diffraction peaks. Conditions (15a,b) are sufficient to determine the angular location of rotational peaks for all isotopic species of Hz. They lead also to a focusing, resulting from extremal effects in Of as a function of k,, (Grant Rowe et al., 1975). It is due to this focusing that the latter authors were able to detect rotational peaks, even using an effusive source with large velocity spread. In connection with the experimental results we would like to mention

ATOM DIFFRACTION FROM SURFACES

111

two main features. On the one hand, in systems like H,-LiF, H, , D,, HD-MgO, and H,-NiO, the total diffracted intensity associated with rotational peaks is comparable to the elastic intensity, which gives a considerable importance to rotational transitions in gas-surface interactions. On the other hand, when metal surfaces are considered, the ratio of the rotationally inelastic intensity to the elastic one is much smaller, i.e., of the order of 10+ (HD is an exception, owing to mass asymmetry). The effect may be attributed to a presence of a “softer” repulsive potential in metal surfaces. The diffraction probabilities observed in H, scattering for both elastic and rotational peaks are only partially understood and a reliable theory of all phenomena involved is still missing, as mentioned above. There are inherent difficulties in this theoretical problem; however several qualitative and significant aspects of the phenomenon have been clarified using different approaches (Wolken, 1975; Goodman and Liu, 1975; Garibaldi et al., 1976; Gerber et al., 1980; Adams, 1980).

111. QUANTUM THEORYOF ATOM-SURFACESCATTERING

The rigorous quantum scattering theory of a nonpenetrating particle interacting with a crystalline surface is a complex problem featuring all aspects of many-body theories. From the beginning, two main points should be kept in mind: (1) The interaction of a light projectile (such as H or He) with the surface does not affect appreciably the surface statics (equilibrium position of the atoms) and dynamics (lattice vibrations at the impact location) during the collision (2) Owing to simmetry breaking, the surface layer finds itself in static and dynamic conditions which may be appreciably different from the bulk

These facts allow separation of the whole theoretical problem into two parts, i.e., the determination of surface structure and lattice dynamics, and the solution of the scattering problem. However, a problem still remains, namely, the choice of a proper atom-surface potential V = V(r; r l ,

...

, rN)

(16)

where r and rl , which are functions of time, denote the coordinates of the gas atom and of the lth lattice atom respectively. Attempts were made to assume V to be well represented by the sum of pair interactions V(r)

=

C vl(r 1

112

G. BOAT0 AND P. CANTINI

This choice is approximately valid for a few systems, such as He on heavy noble-gas surfaces, where the three-body contributions are relatively small and vl can be substituted by the well-known atom-atom potential. Until now, however, a detailed check of the validity of pairwise summation was obtained only for static potentials used to describe elastic scattering; an example is given by Carlos and Cole (1979), who studied the He -graphite system. The most significant attempts to formulate applicable scattering theories were carried out by using “model potentials,” which were chosen ad hoc in order to theoretically describe processes of experimental relevance. The model potentials may be separated into two classes: (1) static potentials used for elastic diffraction, and (2) dynamic potentials used for inelastic scattering.

A . Static Potentials and Elastic Diffraction Theories

Many static “model potentials” have been proposed or used to discuss a great many experimental results. For an extensive and critical review we refer to the paper by Hoinkes (1980), who also gives semiempirical rules to derive them. We concentrate here on three different families of model potentials to outline three different and productive theoretical approaches to elastic scattering. 1 . Pairwise Summation and Coupled Differential-Equations Method

As discussed in Section II,B the interaction potential is composed of a short-range repulsive and long-range attractive part which both originate from pair interactions. For several solids, the potential is commonly approximated by a sum of static pair potentials vl(r - rl) between the gas atom and each atom of the solid in its equilibrium position. Several forms for 2rl have been proposed, as well as different ways to practically perform the summation (see Steele, 1974). In any case, the gas-solid potential must take into account the surface periodicity and can be expressed by the Fourier series (1). The Fourier components VG(z)=

d-l

V ( R , z ) exp( - iG R ) d 2 R

(18)

with G # 0, are generally much smaller than the laterally averaged zero component Vo given by Eq. (8). For many observed systems only few Fourier components are relevant; generally, they are of repulsive character.

ATOM DIFFRACTION FROM SURFACES

113

The Schrodinger equation for a particle of mass m and wave vector k,, in a potential V(r) is

By the Bloch theorem in two dimensions, the wave function +(r) can be expressed by =

C +G(z)exp[i(&

+ GI Rl

(20)

G

Substituting in Eq. (19), one obtains a set of coupled differential equations - (2m/h2) C VG--dz)+G’(z)= 0 [(d2/dz2) + k&zl+G(~)

(21)

G’

where k&, is given by Eq. (5). Far from the surface, the potential is zero, hence (d2/dz2 + k&)+G(Z) = 0

(22)

At sufficiently large distances, the solutions of Eq. (21) are therefore plane waves for all G’s giving k:, > 0 (open diffraction channels, identified hereafter by F vectors), while they are evanescent waves for all G’s giving k2,, < 0 (closed channels, identified either by N vectors, if leading to possible BSRs, or by E vectors, if k2,, < -2mD/h2, D being the well depth). The asymptotic form of the solutions for z -+ may then be written +F(z) = exp(- ikozz)6F,0 + AF exP(ikF,z)

(23)

for open channels, and

+dz)

= ANexp(- K ~ Z )

(244

for closed channels, K being (-k;)l’*. If the AF are known, from Eqs. (23) the diffraction probabilities PF(peak intensities) can be derived as PF =

(kFz/kOZ)IAFJ2

(25)

The sum of diffraction probabilities over all open channels must obey the condition 2, PF = 1 (unitarity condition for elastic scattering). The problem is now to solve the infinite set of coupled differential equations (21)-with the boundary conditions given by Eqs. (23) and (24)-in order to find the amplitudes AF . Approximate solutions were ob-

114

G . BOAT0 AND P. CANTINI

tained by Wolken (1973) and Tsuchida (1%9, 1974,1975),and in more detail by Chow and Thompson (1976b) and Chow (1977a,b), using different model potentials. The work of Chow and Thompson (1976b) is of particular importance because it showed for the first time several of the bound-state effects that were experimentally studied afterward. These authors obtained a satisfactory approximate solution of Eqs. (21) by including a finite basis set of G vectors, namely, the lower order open channels F and some of the closed channels N leading to possible BSRs. The unitarity of the approximate solution guaranteed the satisfactory choice of the basis set. The numerical results for He-LiF(Wl), obtained by using a Yukawa-6 pairwise potential, showed the appearance of several sharp resonant structures having the shape of both maxima and minima in the specular and other diffraction-peak intensities as a function of the azimuthal angle $I~.These results cleared the field from the prejudice that only experimental minima could be observed, owing to inelastic processes occurring during the residence time of the atom in the metastable state. A second important result by Chow and Thompson concerns the validity of the free-atom approximation, expressed by k;,

=

g

-

(K,,

+ N)2 = 2mei/h2

(26)

[compare with Eq. (9)]. This approximation is acceptable when Eo >> ej , which is usually the case for thermal atom collisions. The free-atom approximation is no longer valid for really adsorbed atoms 0 > Eo > - D , or-a more interesting case for diffraction-when two different resonances (ej ; N) and (ej, ; N’) happen to occur at the same incident conditions. Around this “crossing region” a better approximation is

(ki,

-

2 m ~ ~ / h ~ ) (-k 2rnq/h2) i,~ = I(xjJVN-,Ixj,)1’

(27)

(see also Chow, 1977b),where is the eigenfunction corresponding to ej, associated with the laterally averaged potential V,, . Equation (27) shows that the observed splitting of two near BSR anomalies is proportional to the (N - N’) Fourier component of the total potential (this is a bandstructure effect). Returning to the general problem, the solution of the infinite set of coupled equations (21) is often a formidable task, even with modern computers. On the other hand, a “brute force” solution of this type may cause lack of insight into the physical problem. For these reasons, and in trying to interpret the experimental data, other approximate theoretical methods were used, examples of which are the unitarized distorted-wave Born treatment by Cabrera et al. (1970) and the semiclassical methods. To this last class belong the theories given by Doll (1974), Berry (1975), Miller (1975), and McCann and Celli (1976). A more successful method is that based on a hard-wall corrugated model, which is discussed in the next.

115

ATOM DIFFRACTION FROM SURFACES

Close-coupling calculations, with their accurate results, still remain a possible check, together with experimental data, of the validity of approximate theories in the simplest cases.

2. The Corrugated Hard- Wall Potential The corrugated hard-wall model, also known as hard corrugated surface (HCS) model, has been widely used to interpret experimental data. A HCS model corresponds to a potential r

where {(R) is known as corrugation function; {(R) has 2D periodicity, i.e., it may be expressed as =

c, { c exP(iG

R)

*

(29)

G

The HCS potential is a first but realistic approximation for He interacting with a surface, for the reasons qualitatively explained in Section I1,B. A quantum theory using this potential was first used in an early paper by Boato et al. (1973) in order to interpret the diffraction of He and Ne scattered from LiF(001). The theory was fully developed by Garibaldi et al. (1975). It was found very useful to understand the peculiar shape of the diffraction pattern, known as quantum surface rainbow. Historically, the quantum theory of surface rainbow is akin to the problem of diffraction of sound waves having wave length A from a undulated reflecting mirror of period a A , studied by Lord Rayleigh (1896). In the HCS model the asymptotic form of the wave function for z + co is written

-

+(r) = exp(- iko r)

+

AF exp(ik,

r)

(30)

F

valid for z > max {(R). Rayleigh assumed that the far-field solution expressed by Eq. (30) is strictly valid all the way to the surface. With this Rayleigh hypothesis and imposing the boundary condition $(r) = 0 at the surface we get -exp[ - iko,<(R)] =

C AF exp[iF

*

R

(3 1)

i- &,{(R)]

F

Multiplying both sides of this identity by exp{-i[G and integrating over the unit cell, one obtains

A8 =

BG,FAF F

*

R

+ kG,{(R)]} (32)

G . BOAT0 AND P. CANTINI

116

where the coefficients A$ = BG,F

-&?-I

= &?-’ ]u,c,

exp[-i(kol exp[+

i(kFz

-

+ k,,){(R) - iG kG,)((R)

R] d2 R

+ i(F - G)

*

R] d 2 R

(334 (33b)

can be evaluated from the assumed ((R). The infinite system of linear equations (32) may be approximately solved, yielding the scattering amplitudes A F [and the diffraction probabilities PFfrom Eq. (25)]. It follows from Eq. (33b) that B G , G = 1 for all G’s; if, in addition, for all relevant channels (kFz - kGz){(R)is very small and can be neglected, the approximation A F = A: applies. This is commonly known as the eikonal approximation first used in atom scattering by Garibaldi et al. (1975) to explain the quantum rainbow. The eikonal approximation can be quite satisfactory for a qualitative interpretation of experimental data, particularly when many F vectors are involved. However, Chow and Thompson (1976a) and other authors warned against the danger that the nondiagonal elements B G , F may not be negligible under physically relevant circumstances. Their approach was to truncate the sum in Eq. (32) and retain enough terms to obtain convergent results. Another way to obtain the scattering amplitudes from Eq. (3 l), probably the most useful one, is that proposed by Garcia (1976) and known as the GR method. Briefly, it simplifies to writing Eq. (31) in the form

with

MG,R= exp[iG R + i(kGz + k,,)((R)] (35) Equation (34) must be satisfied at every point R of the unit cell. The GR method assumes a finite number n of points Rj, withj = 1, . . . , n, distributed in the unit cell, whereas the summation is truncated after n terms. Equation (34) can now be regarded as a system of n linear equations, which is solved for the A G ’ s . The computational applicability of the GR method was shown by Garcia (1977). The Rayleigh hypothesis gives a convergent solution of Eqs. (32) and (34) when the surface corrugation does not exceed a certain amplitude. A more general solution of the quantum mechanical HCS problem was consequently searched for (Masel et al., 1975; Goodman, 1977; Toigo et al., 1977; Garcia and Cabrera, 1978; Armand and Manson, 1978). The starting point is the Lippman-Schwinger equation which gives for a HCS

117

ATOM DIFFRACTION FROM SURFACES

$(r) = exp(-zk,,

- r) +

d2R’f(R’)

{exp[i(&

+ G)

(R - R‘)

G

+ ikGzlZ

-

((Rf)l}/k3z

(36)

The wave function $(r), with the asymptotic behavior expressed by Eq. (30), yields the following scattering amplitudes:

The density of sources f(R) necessary to calculate AF from Eq. (37) is obtained by the appropriate boundary condition $[R, z = ((R)] = 0, giving - exp[-

ik~,S(R)l =

[exp(iG * R)/k,l G

LC.

f(R’) exp[-i(&

+ G) *

+ ik,lS(R) - C(R’>lId’R’

R’ (38)

Garcia and Cabrera (1978) proposed a numerical method to obtain f(R) from Eq. (38). Model calculations based on this procedure for 1D corrugations of different shape and amplitude seem to prove that the method is applicable to any kind of cumgation, no restriction concerning the corrugation amplitude or discontinuities in ( ( x ) or d((x)/dx being necessary (Armand et al., 1979). The solution of the problem, however, requires handling of very large matrices. We can conclude by saying that the HCS is a model potential suited to describe diffraction patterns obtained with He. In this description the real missing part is the existence of BSRs; the introduction of a potential well in front of the hard wall improves the calculations and explains the presence of BSRs, as is shown next. Other details on the use of the HCS model can be found in the review article by Engel and Rieder (1982).

3 . Corrugated Surface with a Well As discussed in Section III,A, 1, the close-coupling calculations of Chow and Thompson (1976b) proved that the observed BSR features could be well reproduced by using a purely elastic theory, except for a Debye- Waller correction. Several calculations carried out by using simple but realistic atom-surface potentials showed remarkable agreement with the experiments (Harvie and Weare, 1978; Wolfe et al., 1978; Garcia et al., 1979b). An elastic theory capable of reproducing in detail BSR effects without large computational efforts and giving physical in-

118

G. BOAT0 AND P. CANTINI

sight into the problem is that proposed by Celli et al. (1979). For brevity, we limit ourselves to the discussion of this theory. The calculation is based on the use of a model potential of the type V(r)

=

VR[z - C(R)1 + V,(Z)

(39)

where V, represents a short-range repulsion, [(R)being the effective corrugation function, and V, a long-range attraction. More precisely, V, and V, are separated by two planes, at z = zo - 6 and z = zo, with 6 > 0, such that in the region between them vR[z -

5(R>1 =

for z > zo - 6

(404

VA(z) = - D

for z < zo

(40b)

where D is the well depth and 6 can be arbitrarily small. In the calculations reported on p. 121, VR is often taken as a hard corrugated wall, whereas V, can be chosen as a zr3 term [see Eq. (7)]; in this case, the resulting well has a flat bottom. This model potential is shown in Fig. 10, with the observation that the indicated AD is set equal to zero. Returning to potential (39), the wave function can be found separately for VR - D and V, and matched at z = zo. In the region zo - 6 < z < zo , the wave function must have the form

{B&exp[i(&

+(r) =

+ G)

R + kh,~]

G

+ BE exp[i(K,, + G) R - k&,z]} (41) where k&%= ki - (K,, + G)2 + 2mD/h2. Far from the surface, +(r + m) = exp(zKo R - kozz) + C, AG exp[i(& + G ) R + kGg] _G_

(42) If it is now assumed that the scattering problem for V, - D has been solved (e.g., with the methods of Section 111,A,2), then for z < zo, the amplitudes B& of the diffracted waves are related to the incident amplitudes BE by B& = S(G, G’)B,, (43) G’

where S(G, G’) are known scattering-amplitude coefficients. Assuming that the scattering problem has been solved also for V,(Z > z,)-a simple problem since V, shows no periodicity-one then obtains for G # 0, AG = TGB&,

BG

=

RGB&

(44)

where RG and TG are the reflection and transmission coefficients, respectively, for incidence from the left (toward positive z), with kGzlTG12 + kbzlRGIZ= kbz

(45)

ATOM DIFFRACTION FROM SURFACES

I

rn, k o

I

T

I I

' '

!

"R

119

I I

4

Al I+ I I

2,-

s

20

FIG.10. HCS potential with a flat-bottom well; V , and/or D are modulated by the corrugation function. The corrugation amplitudes are represented by A( and AD, respectiveli.

For G = 0 , one has

where R; and T; are coefficients for incidence from the right, where [TAP = kOz/k&in agreement with Eq. (45). Equations (46) include the incoming wave. Combining Eqs. (44)and (46), one obtains

B&

= S(G, 0)T;

+ 2 S(G, G')R&B&t

(47)

G'

According to Celli et al. (1979), G' in Eq. (47) can be restricted in most cases to the set of the N reciprocal lattice vectors for which 2rnD/A2 > kk: > 0 (diffraction into the well, see Section III,A, 1). As a consequence, Eq. (47) becomes a matrix equation of small rank for the unknown quantities B&. Finally, the diffraction intensities PFcan be obtained in the form of PF

=

(k,,/koz)JA2,1 = (kkZ/khz)

/C S(F, N)B&RN/G+ S(F, O ) l z

(48)

N

as a combination of the elastic amplitudes S(F, N) and S(F, 0) for diffrac-

120

G . BOAT0 AND P. CANTINI

tion from the potential V , - D and of the reflection coefficients R, from the attractive well. A simple way to characterize the reflection from the well uses the phase shift BN as a function of the perpendicular energy EN = h2kir/2m;BN is related to the reflection coefficient RN by RN = exp(isN).Following Celli et al. (1979) and, in more detail, Hutchison (1980), the phase shift BN = 277j(EN) can be obtained by interpolation with the use of the relationship EN(j)

=

-D[1 - c(j + 8)y

(49)

which is a simplified version of the equation for the bound-state energy levels ej obtained by Matteraet al. (1980a);C and a are parameters related to the width, steepness, and asymmetry of the potential well. The above theory was used by Celli et af. (1979) to discuss in physical terms the selective adsorption process: a simple result is obtained when only one N vector is involved (isolated BSR). In this case Eq. (48) can be written

which contains a typically resonant term with a width

r = 2(ds/dE)-yi

-

p ( ~N)I) ,

(51)

In Eq. (50), S(F, 0) is the nonresonant amplitude corresponding to AF in Eq. (31). Equation (51), together with Eq. (49), shows that the resonance width goes to zero as the bound-state level approaches the threshold ( E ~+ 0). For intermediate levels the width increases, passes through a maximum, and reduces again for the deepest levels (for which 1 - IS1 decreases). The line shape of an isolated resonance can be written as a function of x =2 ( ~ E j ) / r in the form x + i

= 1 +

by

+ bz + 2b1 - 2bZx 1

+ x2

(52)

Forb, = 0, Eq. (52) becomes 1 + b,(b, + 2)/(1 + x2),which represents a Lorentzian line shape and is a maximum for b, > 0 or b, < -2, and a minimum for 0 > b, > -2. On the other hand, following this theory, the line shape is a maximum-minimum (i.e., non-Lorentzian) when b, # 0. The formulation given here is valid for any surface symmetry. In addition, when two N vectors are involved (BSR crossing region, see Section III,A,l), Celli and co-workers are able to give a simple explanation of resonance splitting and symmetry effects. In conclusion, the

ATOM DIFFRACTION FROM SURFACES

121

theory of Celli et al. (1979) gives a satisfactory physical picture of the intensities, widths, and line shapes observed in BSR structures. Whereas this and other elastic theories (see beginning of this section) give a good qualitative explanation of resonance effects, a detailed agreement with experiment was obtained only for He scattered from LiF(OO1) (Harvie and Weare, 1978; Garcia et al., 1979a). Similar calculations carried out for He-graphite (Garciaet al., 1980) showed only a partial agreement with experimental data; for example, the resonant structures are predicted to be narrower and more intense. Quantitative discrepancies in the BSR line shapes with respect to the prediction of elastic theories are observed in several experiments. These discrepancies have to be ascribed to inelastic effect, that are certainly effective in resonant transitions, as initially suggested by Lennard-Jones and Devonshire (1937). The use of an optical potential (Chow and Thompson, 1979; Garcia et al., 1980; Wolfe and Weare, 1980) can phenomenologically account for some of these deviations. Alternatively, Hutchison (1980) and Greiner et al. (1980) proposed a variant of the elastic theory by Celli et al. (1979), by multiplying each scattering coefficient S(G, G ’ ) by exp[- W(G, G ’ ) ] , 2W(G, G ’ ) being the appropriate Debye- Waller coefficient. This prescription works surprising well for explaining the experimental data at low incident energy or near-grazing incidence. The importance of the connection between BSRs and inelastic scattering is further discussed in Section III,B,3.

B . Vibrating Surfaces and Inelastic Scattering If one wants to go beyond the static potential approximation, only very simplified model potentials can be adopted to solve the theorical quantum problem. In quantum scattering theories, the relevant information is contained in the transition T matrix, which has to be evaluated between states compounded of particle states labeled by the initial wave vector k, and final wave vector k (hereafter written as subscripts), and of crystal states labeled by phonon occupation numbers a, p. A T-matrix element is then where Tk+ko is an operator in the space of crystal states. Following Manson and Celli (1971), the differential scattering probability for an atom to go in the solid angle dfl, losing energy A E = E, - E , is

122

G. BOAT0 AND P. CANTINI

where L is a quantization length. To obtain the probability p a , Gibbs' thermal equilibrium distribution may be used. As suggested by Levi (1979), the transformation introduced by Van Hove (1954) can be applied by using a Fourier representation of the 6 function, to obtain the correlation function expression

where the time evolution of the T matrix is driven by the free Hamiltonian of the solid. Equations (54) and (55) are exact and are the starting point for several approximate inelastic scattering theories. We now discuss three different theoretical problems. The first is the form of the Debye-Waller factor; the second is the derivation of the inelastic scattering probability; the last is a theoretical treatment of inelastic BSRs. 1 . Debye-Wuller (D-W) Factor

In order to derive the elastic diffraction probability, we must specialize the general correlation function in Eq. (55) to the case AE = 0. The exponential factor vanishes and the integral is dominated by long times. Asymptotically, a factorization of the correlation function can be applied (Levi and Suhl, 1979), thus

The next step is a direct computation of the average (Tkcko),a problem which presumes the knowledge of both the lattice dynamics and the time-dependent atom-surface potential. Levi and Suhl (1979) proved that for the interaction of fast light particles with heavy atom substrates, a D-W factor completely identical to that used in X-ray, electron, and neutron scattering is obtained, namely, Eqs. (1 1) and (12) are valid. The result was found to be equally valid for both completely correlated and completely uncorrelated (Einstein oscillators) surface-atom displacements. For slow atoms, however (i.e., when the collision time is comparable with the characteristic time of lattice vibrations), the application of the standard theory is not straightforward. Several corrections to the stand) proposed, depending on the mass, size, and ard exponent ((Ak * u ) ~were velocity of the incoming particle and on the crystal structure and dynamics (Beeby, 1971; Hoinkes et al., 1972b; Armand et ul., 1977). All these corrections can be justified by a more general formulation based on Eq. (56) and/or making use of semiclassical approaches (Levi and Suhl, 1979; Meyer, 1981). A detailed description of the semiclassical approximation is

ATOM DIFFRACTION FROM SURFACES

123

beyond the scope of the present work. We only give some of the conclusions which are relevant for interpreting the experiments, as derived from Levi and Suhl(l979). These authors find that a D-W factor exp(- 2W) is still valid, with the exponent given by 2W(k, k,,) = h-’

2 [[ F,(t)F,(t’):

(um(t)u,(t’))dt dt’

(57)

m,n

where F,(t) is the force by which the mth crystal atom in its equilibrium position acts upon the incident particle, whereas u,(t) is the corresponding displacement. The forces F(t) in the integral are evaluated along the trajectory of the particle in the field of the static lattice (u, = u, = 0) from the incoming k,, to the outgoing k state. Using the stationary properties of lattice vibrations and introducing the tensors A,,,(T) = h-’ Bm,n(T)

I

F,(t)F,(t

+ 7 ) dt

= (um(O)un(~))

(58b)

the formally simple and general formula for the D-W exponent 2W(k, 16) =

IAmn(T):Brnn(T) d7

(59)

m,n

is obtained. This formula is able to qualitatively describe the size and time-of-interaction effects, observed and/or proposed by different authors. Leaving the discussion of experimental results to Section VI,A, we limit ourselves to a few important consequences contained in Eq. (59)

(I) Ordinary D-W factor. As expected, for fast collisions and in case the well is neglected, Eq. (59) reduces to the elementary formula ((Ak * u)’) . (2) Beeby effect. In the presence of a nonnegligible well and fast collision, the formula given above (1) can only be used for the repulsive part of A,,,. For the part of A,,, involving attractive forces, B,,, in Eq. (59) has to be evaluated not at T = 0, as in (1) above, but at considerably longer times, when all displacement correlations ultimately vanish. The result is that in the elementary formula Ak ought to be replaced by Ak‘, with the vertical energy effectively increased by the well depth so that K2 = k”, + 2mD/h2, i.e., the particle is accelerated in the well before the “hard” collision. This correction was first proposed by Beeby (1971) in a somewhat different context. ( 3 ) Levi effect. When the overall collision time is large compared to the period of typical surface-lattice vibrations, the D-W factor is substantially increased. This effect was proposed by Levi (1975) and extensively treated in the paper by Levi and Suhl (1979).

124

G. BOAT0 AND P. CANTINI

(4) Armand effect. In atom-surface collisions, the assumption of single atom-atom interactions is not justified, owing to the finite size of the incident atom. A rough correction was first suggested by Hoinkes et al. (1972b). A more rigorous theory was proposed by Armandet al. (1977), who took into account correlations between displacements of neighboring surface atoms. The correction is important at short phonon wavelengths and leads to a reduction of Wand to an increase of the diffracted intensity. Equation (59) is inclusive of the Armand effect, as shown by Levi and Suhl (1979). 2. Inelastic Scattering Probability Quantum theories of inelastic scattering have been proposed in considerable number during the last 10 years. All theories contain some approximation. The close-coupling formalism proposed by Wolken (1974) was used by Lin and Wolken (1976) to calculate energy transfers in the He-Ag( 111) scattering. The renormalized distorted-wave Born approximation was initially employed by different authors (Manson and Celli, 1971; Goodman and Tan, 1973; Benedek and Seriani, 1974; Lagos and Birnstein, 1975,a,b). The approximate method more often applied in recent years has been the impulsive approximation (Beeby, 1972a,b, 1973; Weare, 1974; Adams and Miller, 1979), particularly when connected with a time-dependent HCS (Benedek and Garcia, 1979; Armand and Manson, 1979; Marvin and Toigo, 1979; Levi, 1979). To give a summary description of the inelastic formalism proposed by Levi (1979) for the vibrating HCS, we introduce the eikonal approximation in the 7'-matrix elements of Eq. ( 5 3 , i.e., Tktko

-ih

m

I

exp[iq(R, t ) ] d2R

where q(R, t) is the semiclassical phase, the integral being extended over the impact parameters R. The phase can be split into three parts q(R, t ) = qi(R) + r/z(R) + qAR, t )

(61)

where ql(R) = A K . R is the phase difference for a reflecting plane; q2(R) = (kO2+ k,){(R), where 5 is the corrugation function of the static surface; and q3(R, t) is the fluctuating phase, related to the surface lattice vibrations. By assuming a hard vibrating surface, one finds d R , t ) = (koz + kA @XR*,t ) (62) where 65 is the lifting of the surface at (R*, t), R* being given by R <(R)K,Jk,, . In turn, 6[(R*, t) may be written in terms of local displace-

+

ATOM DIFFRACTION FROM SURFACES

125

ment as 6<(R*, t ) = 6(R*) * u(R, t )

(63) The inelastic scattering probability for a vibrating HCS then becomes, if the Armand effect is neglected,

d3P/(dEd2a) klkozl 81S12exp( - 2W) (64) where S is an amplitude akin to that occurring in elastic scattering [see Eqs. (33a) and (SO)]. More explicitly,

where AK is the parallel momentum transfer, no more restricted to be a reciprocal lattice vector G. A first result obtained from Eq. (64) is then that all effects occurring in elastic scattering (like the rainbow), also occur in the inelastic scattering; in particular, in the neighborhood of intense diffraction peaks, strong inelastic scattering is expected. Similar results were obtained by Armand and Manson (1979) and by Benedek and Garcia (1979). The remaining factor 5,containing the effects of lattice dynamics, may be usefully expanded as

where 8,describes, in the approximation of an harmonic solid, the contribution to the scattering event brought about by the simultaneous exchange of 1 phonons. At this point, we can limit ourselves to one-phonon scattering, occurring when 2W [Eq. (57)] is small; in these circumstances the kinematic conditions (13) and (14) apply. Following Levi (1979) the final expression for the inelastic scattering probability is given by Eq. (64) with E substituted by the first term of the sum (66)

where M is the lattice-atom mass and A E is the energy loss Eo - E. Equation (67) contains on the one hand the Bose-Einstein population factor and on the other the surface-projected phonon density

where e,(:) are the surface components of the polarization vector of the phonons belonging to branch v and having parallel momentum Q (see Chen et al., 1972, 1977). The surface-projected phonon density contains

G . BOAT0 AND P. CANTINI

126

contributions from both the continuum spectrum associated with bulk bands and, with a larger weight, the discrete surface modes. In conclusion, the eikonal approximation applied to the vibrating HCS model, as expressed by Eq. (64),contains the most relevant features of the inelastic scattering probability. The dependence on the static potential is contained in ISP, whereas Eqs. (67) and (68) show that atom scattering is sensitive to low-frequency modes and in particular to Rayleigh phonons. More refined theories include multiple-phonon processes, size effects, interaction time of incident particles, etc. Particular mention should be made of the paper by Marvin and Toigo (1979), who took into account the effect of the attractive well. However, they did not consider explicitly the presence of BSRs. A description of how this last phenomenon can be treated in inelastic scattering is given next.

3 . Inelastic Scattering and Bound-State Resonances Structures in the inelastic background of the scattered intensity as a function of er were repeatedly observed in different atom- surface experiments (Williams, 1971a; Boato et al., 1976a; Frank et al., 1977). An interpretation of these structures as inelastic BSRs was early given by Cantini et al. (1976). A deeper study and a theoretical description of the processes involved in inelastic BSRs were presented by Cantini and Tatarek (1981). The proposed formalism can be considered an extension of the paper by Celli et al. (1979) [see Section III,A,3]. The starting potential is a vibrating repulsive wall plus a time-independent attractive well, that is, V(r,

1) = VR[Z -

s(G

k

m, 111 + V*(4

(69) The formal extension of Eq. (47) to the inelastic one-phonon processes gives b&

=

Q , 0)T;

+ 2 [s(G k Q , G’)&,b& G‘

+ S(G * Q , G’ * Q)&,*Q~&,Q]

*

(70)

where S(G k Q , G’ Q ) and s(G k Q , G ’ ) are the elastic and onephonon inelastic scattering amplitudes, respectively; they refer to the interaction of the particle with the time-dependent repulsive wall described by V , - D . Kinematic conditions (13) and (14) must be fulfilled for each intermediate state. For any final scattered state, represented by f(Ko + F k Q , Eo k hw,), Eq. (70) gives a matrix equation for b+ which can be solved in complete analogy with the elastic problem, and from it the inelastic probability can

ATOM DIFFRACTION FROM SURFACES

127

be derived. As in Section III,A,3, a simple result is obtained when only one N vector is involved, whereas two different energy levels ej and ejt are supposed to take part in the process. In this case the inelastic scattering probability to go in a final state is given by

d3P/dEd 2 0 = (d3P0/dE d 2 0 ) ) 1+ A ,

+ A, + A,[*

(71)

where d3Po/dEdzO is the probability for a single interaction with V , - D , given by Eq. (64). Equation (71) contains three resonant contributions A,, A, , and A,, whose kinematics is illustrated in Fig. 1 1 ;A, corresponds to an elastic res, followed by a phonon stimulated desorption. It onant transition in ( E ~ N), is given by

akin to the second term in parenthesis of Eq. (50). In Eq. (72), eNis the energy associated with the motion perpendicular to the surface for an elastic transition into the N channel; i.e., 2mEN/fi2 = kiz = kg - (I?, + N)2. The resonant condition = ej selects the incident wave vector lC,,(ko,Bo , +o). The process corresponding to A, is schematically illustrated with process (2) in Fig. 11. The term A, corresponds to an inelastic resonant transition in ( E ~ , ,N), say a transition in a bound state assisted by one phonon, followed by an “elastic emission” of the particle. It is given by A , = - - iC - iS(f, n)R,s(n, O)/s(f, 0)(1 - IS@, n)l) 2(€, - Ej,)/rn +i x’ + i

(73)

Now E , is the perpendicular energy after the inelastic transition in the N channel;i.e.,2rne,/fi2 = (k2, ? 2rno,/h) - (& + N k Q),. Theresonance , at every incident lC,, a different family of phonons condition E , = E ~ selects wN,j,(Q). Such an inelastic resonant process is indicated by process (3) in Fig. 11. The last term A3 has the form of a double resonance A, = -B’/(X+ i)(x’

+ i)

(74)

where X and x’ are defined by Eqs. (72) and (73), and

Of course, A, is different from zero only when both the direct elastic resonant transitions are allowed by the initial lC,, ; A, gives then, at selected incident conditions, an (ej, N) and the inelastic, from (ej, N) to ( q ,N),

128

G. BOAT0 AND P. CANTINI

FIG. 11. Perpendicular energy changes for a direct (1) and three resonant (2), (3), and (4) inelastic processes associated with phonon creation: (1)

K,=&+F-Q

(2)

Kf = (&

(3)

Kf = (&

+ N) + (F - N - Q) + N - Q) + (F - N)

(4)

Kf = (&

+ N) - Q + (F - N)

extra resonant contribution for the same selected phonon family wNj,(Q) as in process (3) in Fig. 11. The resonant effect described by A, is shown as process (4) in Fig. 11. Since the process indicated by process (4) occurs at the same “selected” incident ko as process (2), it can manifest itself as a spectacular change in the resonance line shape, as it is shown by experimental data reported in Section V1,B. We point out finally that the resonance amplitudes B, C , and B‘ do not contain effects due to statistics and to the phonon spectrum. This can be shown immediately by using the eikonal approximation for the vibrating HCS, with the result that

Idf,0)l2 lS(f, 0)l’El o(

(76)

where El is given by Eq. (67). As a consequence the ratio of two s (always contained in Al ,A 2 , and A3) is equal to the ratio of the corresponding S, given in turn by Eq. (65).

ATOM DIFFRACTION FROM SURFACES

129

IV. STRUCTURAL INFORMATION FROM ELASTICDIFFRACTION The best available atom diffraction probe for structural investigations is helium. The reasons for this are manifold. Not only He is light, but it has small polarizability, thus giving rise to a shallow potential well (and few bound states) with most surfaces. The weak He-He interaction prevents condensation in the gas phase and therefore allows the production of very intense beams by supersonic expansion at quite low energies. The size of He atoms is not negligible, but is small enough to enable one to explore the profile of outermost electron density of most surfaces. The accuracy with which the profile is reproduced depends on the He hard-sphere radius and on the corrugation function. On the other hand, the reliability of the profile determination by He scattering has received a strong theoretical support from the recent work by Esbjerg and Ngirskov (1980). They approximate the potential V(r), through which He interacts with the surface-electron-density profile n(r), by replacing the nonuniform electron distribution in which the atom is embedded with an homogeneous electron gas having density ff = n(ro), r,, being the location of the center of He atom. Moreoever, they prove that the energy needed to embed He in a homogeneous electron medium is well approximated by

v,

=

pii

(77)

with p = lo5 meV A3 in the relevant energy range. Of course, the resulting V(r) is always repulsive in this approximation. For all mentioned reasons, whenever structural investigations are carried out, He is largely preferable to H and H2 (having larger well depth and often being reactive) and to Ne (having larger atomic mass and higher polarizability). The entirely nondestructive character of He must also be emphasized, in contrast to electron, ion, and X-ray probes. In this section we give a short account of the work accomplished to arrive at structure determination. A much more detailed survey can be found by the reader in the review article by Engel and Rieder (1982). We divide structural investigations into three main classes depending on the surface complexity; (1) simply corrugated surfaces, (2) reconstructed and periodically deformed surfaces, and (3) adsorbate-covered surfaces. A few typical examples for each of these classes are given. In most cases, the HCS model has been employed to derive the static corrugation function [(R)from the diffraction-peak probabilities. Equations (25) and (33a) and the eikonal approximation (AF = A$) have been used for this purpose. A choice of the experimental scattering conditions

130

G . BOAT0 AND P. CANTINI

may be useful in order to minimize the effect of both BSRs and multiple scattering. This aim has been reached by using sufficiently large incoming energies and small incident angles. Some times corrections were applied for the D-W factors by making use of expressions based on Eq. (11) and for the potential well by replacing k,, in Eq. (33a) with k& [see Eq. (41)]. A . Simply Corrugated Surfaces

The first example is the (001) face of LiF, mentioned previously in Section I1,B. From a detailed study of the diffraction pattern under different incident conditions (Boato et al., 1976a), the corrugation function can be found by using the HCS model with the eikonal approximation (Garibaldi et al., 1975). Neglecting all but the first-order terms of the Fourier expansion (29), which now reads 2TX

+ cos-

+ 251, [cos--(X277 2T + cos-((x a

-

a

+ Y) Y)

1+

*

.

*

one finds = 0.0752 f 0.0009 A, corresponding to a peak-to-peak corrugation of 0.602 A. With a more detailed analysis and using the GR method, Garcia (1976) found that the best-fit values for the first two = 0.0767 0.0007 A and = Fourier components of [(X, Y) were 0.0042 2 0.0007 A. The introduction of a well with 5-meV depth causes to decrease to about 0.072 A. The peak-to-peak corrugation of about 0.6 A is in qualitative agreement with the difference in ionic radii of Fand Li+. Further measurements, a more complete analysis of the data, and an improved theoretical description of the ionic and electronic distribution at the surface, should eventually allow information to be obtained on the relaxation of this and other ionic surfaces. The second example of a simply corrugated surface is the basal plane of graphite, a lamellar solid. No relaxation effects are expected at the surface, owing to the weak Van der Waals forces binding the layers. A diffraction pattern by He scattering is shown in Fig. 12 together with a sketch of the graphite surface geometry. From the whole set of measurements (Boato et a f . , 1979b) of this hexagonal surface one derives = 0.023 f 0.002 A, corresponding to a peak-to-peak corrugation of 0.21 A. The higher order Fourier components are negligible. The graphite surface appears to be less corrugated than that of ionic crystals.

*

cl1

ATOM DIFFRACTION FROM SURFACES 0.04

,0.03

r

I

Surface

131

unit cell

-

> k 20.02-

(I)

Y

I-

f

SCATTERING A N G L E

ef

FIG. 12. Diffraction pattern of He scattered from the basal plane (OOO1) of graphite:

9,

=

00;

eo = 100.

A third example is given by simple metal surfaces. A close-packed metal surface is supposed to appear very flat to an incoming He atom. In fact, as first pointed out by Smolukowski, conduction electrons have the tendency of filling out the space between ion cores and therefore to smooth out the surface corrugation. The extent to which this smoothing occurs can be inferred from the diffraction experiments. These have shown that for hexagonal compact surfaces such as Ag( 111) (Boato et al., 1976c; Home and Miller, 1977),the corrugation is indeed very small, as it appears from the almost complete absence of diffraction (see Fig. 13). From the weak intensity of the first-order diffraction peaks-the only equal to (2.5 ? 0.2) x A can be inones observed-a value of fered, corresponding to a peak-to-peak corrugation of -0.022 A, a factor of 10 less than graphite. For less packed surfaces a larger corrugation is expected. For instance, in Ni(ll0) close-packed rows of atoms are present, separate by throughs a d wide, d being the Ni-Ni distance. Perpendicularly to the rows, a peak-to-peak corrugation of 0.05 & 0.01 A was found by Rieder and Engel (1979), whereas the corrugation is almost absent along the rows. In order to find corrugations comparable to graphite or ionic crystals, one must choose such loosely packed metal surfaces as W(112) (Tendulkar and Stickney, 1971) or the stepped-surface Cu(ll7) (Lapujoulade and Lejay, 1977) (see Fig. 14), which are strongly corrugated only in one direction. In all these cases the parallel troughs are so

132

G. BOAT0 AND P. CANTINI Surface

0.15

unit

cell

z

,

0

>

0.10

(I]

z

w

I-

z

a

; a05

a = 2.88

w

Il-

'.,.\.

!!

a

0 v)

I

00

I

I

..'

I

100 SCATTERING

.....'.

...

i

10 :.

.. '....

I 200

ANGLE

I

I

I

30' 8f

FIG.13. Diffraction pattern of He scattered from Ag(lll), do= 0".

wide that conduction electrons are unable to fill them out completely. It is interesting to note that the profile across the steps reported in Fig. 14 was derived on the basis of Eq. (37), since the eikonal approximation is too rough for treating such a case (see Garcia et al., 1979b). B . Reconstructed and Periodically Deformed Surfaces

The atom diffraction technique has proved to be of great advantage for understanding the structural features of complex surfaces, i.e., surfaces with large unit cell or complicated corrugation or both. By giving informa-

r

0.03

FIG. 14. Diffraction pattern of He scattered from Cu(117). [From Lapujoulade and Lejay (1977).]

ATOM DIFFRACTION FROM SURFACES

133

tion on the topmost surface layer, He diffraction is able to complement the results obtained by LEED, ion scattering, and surface-extended X-ray adsorption fine-structure (SEXAFS) techniques. The principal problem area for which rapid progress has been made is that of the reconstruction of semiconductor surfaces, through work carried out at Bell Telephone Laboratories by Cardillo and co-workers. It is well known that the geometrical configuration of these reconstructed surfaces is very complicated and in several cases not yet understood. Surface reconstruction takes place to saturate the strongly directional dangling bonds produced when the surface is created. Three surfaces were thoroughly studied, namely Si(OOl), GaAs(1 lo), and Si( 111). Apart from the experiment on Si(OO1) 2 x 1 (Cardillo and Becker, 1980) which confirmed this surface to be reconstructed in the form of dimer arrays but showed the presence of a lack of ordering, very interesting results were obtained for GaAs( 110) 1 x 1 and Si( 111) 7 X 7. The reconstructed surface of GaAs(ll0) is considered to be experimentally well understood. The unreconstructed surface is composed of parallel ridges and troughs of gallium arsenic bonded chains; the surface reconstructs by simply tilting the Ga-As bond by about 27". Thus the resulting surface is strongly corrugated across the troughs, but only slightly along them. The already established structure was fully confirmed by He scattering (Cardillo er al., 1981). Diffraction patterns obtained at different incident angles O0 and in the two opposite directions across the throughs (indicated by +o = 180" and +o = 0") are shown in Fig. 15. The asymmetry in the two azimuths is due to the tilting of Ga-As bonds during surface reconstruction. Simple evaluations of the two main corrugations by means of both the HCS model (eikonal approximation) and the locations of the classical rainbow angles (as deduced from the diffraction peak envelopes) give average values for cl, = 0.54 and tO1= 0.14 A, respectively. The corrugation parameters are in satisfactory agreement with close-coupling calculations carried out by Laughlin (1982) using a realistic soft-wall model potential. Moreover, sophisticated theoretical techniques were used by Hamann (1981) in order to calculate the electron charge density at the surface down to very low values. By using Eq. (77), Hamann derives that the classical turning points of He at 20 meV occur along the a.u. The two principal corrugations electron-density contour at 3 x are thus evaluated to be about 0.44 and 0.15 A, respectively, in remarkable agreement with experiment. In conclusion, the data by Cardillo and co-workers on GaAs(ll0) have given greater confidence both in the experiments and in the simple potentials (such as the HCS) used to interpret them. Contrary to GaAs( 110) 1 x 1, the Si(ll1) 7 x 7 face is a well-known

134

G . BOAT0 AND P. CANTINI 12 ~ 1 0 - ~

(a)

FIG. 15. Diffraction pattern of He scattered from reconstructed GaAs(ll0): (a) = 0.98 A; (b) I$,, = o", A = 0.98 A. [From Cardillo et al. (19811.1

b0 = 18o", A

puzzle in surface reconstruction. The surface was studied by Cardillo and Becker (1979) who measured the He diffraction pattern at different incident angle and energies. The 7 x 7 reconstruction was confirmed to be present up to the top layer. A multitude of diffraction peaks over a wide range of scattering angles was observed; their intensities are of comparable magnitude and change rapidly with the incident angle. The whole trend is difficult to disentangle; the only clear result is that the surface is strongly corrugated. However, Cardillo (1981b) was able to overcome this difficulty by making accurate measurements of the specular intensity as a function of the incident angle. The results are reported in Fig. 16; they show an oscillatory structure with at least one definite period. The observed trend may be interpreted in terms of interference of waves reflected from terraces displaced 3.2 A along the surface normal. This finding, together with other knowledge on the Si(ll1) 7 X 7 surface, enabled Cardillo to propose a simple model for the 7 x 7 reconstruction: the unit cell is essentially made of two triangular terraced regions displaced one from the other by two silicon layers (see lower part of Fig. 16). Although Cardillo's results cannot be considered conclusive, they are likely to lead soon to the resolution of the 7 x 7 puzzle. To conclude this section, we report briefly on some work carried out on another kind of complex surface, namely, that of a layered material de-

ATOM DIFFRACTION FROM SURFACES

135

12 10 0

-; v)

8-

64-

2-

20'

I

30'

I

1

40°

50°

I

80

I

€Do 70°

80°

90°

FIG.16. The specular intensity as a function of Bo for He-Si(ll1) 7 x 7. The proposed model for reconstruction is sketched ( h = 0.57 A, [01]* azimuth; terrace spacing: .1 3.28 A, j 2.95 A). [From Cardillo (1981b).]

formed by the presence of charge-density waves. The 3 x 3 commensurate superstructure of 2H-TaSe2 is an example and it has already been shown by the diffraction pattern in Fig. 4.Another example is that studied by Cantini er al. (1980) who measured the perpendicular deformation due to charge-density waves in the x superstructure of 1T-TaS,. Large superstructure peaks were observed. By using the HCS model, the overall peak-to-peak deformation was estimated to be quite large at T = 80 K, namely, about 0.4 A.

m

C . Adsorbate-Covered Surfaces

The study of the properties of surfaces covered by an ordered monolayer of adsorbed atoms or molecules has become a subject of wide interest because of fundamental aspects (2D phase transitions, dynamical

136

G . BOAT0 AND P. CANTINI

( i1 -r 5)

2.6

1 5 (*-I-)

2 6

4 = f 1.3'

,100

00

100

200

30° 40°

50°

60°

Qf

FIG. 17. Diffraction pattern of He scattered from Ni(ll0) Bo = 25", Ts = 105 K. [From Rieder and Engel (1980).]

+ H (2 x

6): h

=

0.63 A,

properties of monolayers, structural investigations, etc.) and interest in applications (physical and chemical adsorption, surface catalysis, etc.). The use of atom diffraction techniques is also very promising in this field, since it gives direct information not only on the monolayer structure, but also on the outer electron distribution of adatoms at the surface. A number of structural studies have been carried out, among which we like to mention the papers by Rieder and Engel (1979, 1980) on H-covered Ni(llO), by Lapujoulade et al. (1980b) on 0-covered Cu(llO), and by Ellis et al. (1981) on Xe-covered graphite (this last system was studied by H-atom diffraction). For some of these studies, details can again be found in the review article by Engel and Rieder (1982). We concentrate on one selected structure, i.e., the 2 x 6 phase of H adsorbed on Ni(llO), occurring for a coverage of 0.8. We first emphasize the He-atom diffraction is superior to LEED when investigating H-

ATOM DIFFRACTION FROM SURFACES

FIG. 18. Surface corrugation (a) and proposed structure (b) for Ni(l10) [From Rieder and Engel (1980).]

137

+ H (2 x

6).

covered metal surfaces, since electrons are only weakly scattered by H atoms. Among the various phases found by Rieder and Engel (1979, 1980), some of which were previously detected by LEED, the 2 x 6 phase is particularly interesting, owing to its structure and complexity. The diffraction patterns corresponding to both in-plane and out-of-plane detection for one particular incident angle are shown in Fig. 17. Rainbow scattering is present in-plane, and alternate-order superstructure peaks are observed both in-plane and out-of-plane. In order to interpret these results, Rieder and Engel used the composite corrugation function

<(x,y ) =

-+<0,1,3

-f
+ 1&?3/6

cos 2dY/3az) - !i
<1/2,5/6

sin 2.rr(X/2aI)sin 2.rr(5Y/6aZ)

sin 27r(X/2a1) sin 27r(3Y/6a2)

(79)

where a, = 2.49 and az = 3.52 A are the lattice constants of the surface unit cell of Ni( 110). A best-fit analysis led the authors to derive the values of the 5 coefficients. The fit with the experimental data is shown in Fig. 17 by the dashed line and appears to be quite good. The resulting corrugated

138

G . BOAT0 AND P. CANTINI

surface is displayed in 3D in Fig. 18a and shows paired zig-zag hydrogen chains parallel to the bare-surface troughs. The proposed structure is sketched in Fig. 18b. The lower height of 0.18 A for the twofold sites (dashed circles) compared with the height of 0.25 A for the three fold sites (solid circles) seems to be in agreement with the expected H-Ni bond lengths. This final study is surely a most effective example to show the capability of the atom diffraction technique for understanding surface structures.

V . INFORMATION ON

THE

SURFACE POTENTIAL WELL

A. General

As briefly described in Section II,B, the study of BSR structures in the elastically diffracted intensity is very effective for obtaining, through the levels ej ,information on Vo(z), Eq. (8). In addition, from the measurement of energy splittings occurring at the crossings of different BSRs the matrix are determined, as pointed out in Section III,A,l in elements (~jlVN-Nt1~) the discussion of Chow and Thompson’s theoretical findings. Therefore, the measured splittings can give information on the Fourier components V,(z) of the potential in the region of the well. The power of this experimental method, when applicable, lies in its simplicity, since the overall information on the potential is obtained by the use of kinematic relations, i.e., by measuring the angles at which resonant structures appear for any given k,, . Suitable systems and favorable experimental conditions for accurate measurements of the levels ej and especially for the determination of the matrix elements should be sought. First, V , must be much larger than any other V , , otherwise the free-atom approximation [Eq. (26)] is no longer valid over wide regions of the K space. Second, only a few Fourier components (one or two) and a limited number of levels ej must be present, otherwise a large number of crossings occur; under these circumstances, splitting make the measurements confused and intricate. On the other hand, the relevant Fourier components must not be too weak, otherwise BSRs may not be observed, as in the case of He-metal systems. A last requirement is to work at low incident energies and large incident angles, where the resonant structures are usually best resolved. Systems carefully investigated for the BSR levels were H and D on (001) alkali halide surfaces (LiF, NaF, KCl) (Finzel et al., 1975; Franker al., 1977); H and D on graphite (Ghio et al., 1980); 3He and 4He on LiF and NaF (Derry er al., 1978); 4He on NiO(001) (Cantini et al., 1979); 3He

ATOM DIFFRACTION FROM SURFACES

139

and 4He on graphite (0001) (Derry et al., 1979, 1980; Boato et al., 1979b); studies of energy levels were also carried out for H2 and D2diffracted from LiF(OOl), Ni0(001), and graphite. Several of these studies allowed more or less accurate determination of the form of Vo(z),but only a few of them gave information on the Fourier components. For a more complete review of the experimental results we refer to the previously mentioned article by Hoinkes (1980). We should like to add a promising piece of information: from BSRs associated to rotational peaks in the scattering of HD from Pt(11l), it has been possible for the first time to measure the energy levels ej of the interaction potential between a simple molecule and a close-packed metal surface (Cowin et a / . , 1981). In Section V,B we discuss in detail only the results relative to the He-graphite system, which is the most accurately studied until now.

B . The He-Graphite System 1 . Determination of Energy Levels

The selective adsorption in the elastic scattering of He from the basal plane of graphite was systematically studied by Derry et al. (1979, 1980) and by Boato et al. (1978, 1979a,b). The study of Derry et a / . (1979) concentrated on a very accurate determination of the energy levels for both 3He and 4He, by absolute measurement of the incident angles. The specular intensity was explored with an incident energy E, = 17.3 meV (corresponding to ko = 5.76 A-l) in a region of the K space where the freeatom approximation is clearly valid. In Fig. 19 a typical polar scan is presented. Precise measurements with 4He were also carried out in similar conditions by Boato et al. (1979b), with some uncertainty in the absolute measurements of O0; on account of this, e4 was set to coincide with the value by Deny et al. (1979). The final results obtained by the two experimental groups are shown in Table I. A remarkable agreement between the two sets of measurements is present: the statistical error for the deepest level E, is less than 1%, which shows the capability of the experimental method. 2. Determination of Energy Splittings One reason for the high accuracy secured for the energy values ej in He-graphite is the presence of only one sizable nonzero Fourier component in the interaction potential. This was predicted by simple pairwise evaluation of the potential and was confirmed by the measurements of Boato et al. (1979a,b), who carefully studied the crossing of possible reso-

140

G. BOAT0 AND P. CANTINI

I 40 '

1

5L

300

I

I

,

500

I

60'

I

Oo

FIG. 19. Accurate absolute measurements of angle of incidence Oo at which BSR minima appear in 4He-graphite: +o = 17", Eo = 17.3 meV. [From Derry et al. (1979).]

nances. The specular intensity was measured at an incident energy Eo = 22 meV (ko = 6.49 k l ) by exploring azimuthal angles between +o = 0" [(lo) azimuth] and +o = 30" [(ll) azimuth], and polar angles O0 up to about 80". As an example, typical polar scans for 30" < O0 < 46", obtained at different +o's are reported in Fig. 20, where several crossing of the ej (10) and ej (01) BSRs are observed. The resonances are split TABLE I ENERGYLEVELS~j A N D MATRIXELEMENTS (xjlvl~lxp)OF 4HE-GRAPHITE

~

0 1

-12.06 -6.36

2 3

-2.85 -1.01 -0.17

4

~~

-11.98 -6.33 -2.85 -0.99 -0.17

0.28 0.19 0.12 0.09 0.03

0.18 0.16 0.10 -

0.12 0.11 -

0.08 -

Uncertainties are *O. 1 meV. Uncertainties are kO.06 meV for cj, and from kO.01 to k0.02 for matrix elements; E* is set to be equal to that given in the first column. Boato et al. (1979b). See footnote b for errors.

141

I D

I

I

1

I

I

1

I

00 FIG.20. Polar scans of specular intensity at different azimuths I # I ~in He-graphite. Several BSR crossings are observed. [From Boato et al. (1979b).] 40'

by the first Fourier component V,, of the periodic potential (in the case of Fig. 20, N,, - No, = G,i and GI, = Glo). Figure 21 shows how the splitting associated with the crossing of two selected resonances appears in the O 0 , plane. The reported experimental points correspond to the minima of the observed resonant structures. The solid lines refer to the free-atom approximation, whereas the dashed lines represent the best fit obtained by using Eq. (27), withj = 1, N = (lo), and N' = (01). The matrix elements (>(JIVlolG)obtained with this method by Boato et ul. (1979b) are reported in Table I together with their evaluated errors. A few matrix elements were determined also by Derry et af. (1980) at very low incident energy (E, = 4.7 meV); they agree with the values of Table I.

+,

142

G. BOAT0 AND P. CANTINI

I

I

I

43"

I

I

44O

I

1

45O

00

FIG.21. A detailed view of the splitting of two BSRs. [From Boato et al. (1979b).]

3. The He -Graphite Potential

The very accurate values of energy levels ej and matrix elements (xi[Vlolxjr) for He-graphite were used to propose reliable interaction potentials. This can be done in different ways, either by employing atomsurface model potentials or by looking for sums of appropriate pairwise potentials. A rough model potential, but one still well suited for describing the experimental results, is a modified HCS potential with a well (see Section III,A,3). The hard wall is assumed to have the shape function

((R) = 2{,,[cos(2r/a)X

+ cos(2r/a) Y + cos(2r/a)( Y

-

X)]

(80)

with = - 0.023 A as experimentally determined by Boato et al. (1978). The attractive well is chosen in a shape convenient for scattering calculations as proposed by Garcia et af. (1979a,b), namely, V*(z)

=

-D,

z < P

(814

V*(Z)

=

- m a + P)Aa + z)?,

z

P

(8 1b)

2

ATOM DIFFRACTION FROM SURFACES

143

With the parameters D = 14.4 meV, (Y = 1.33 A, and p = 0.93 A, the experimental ej are well reproduced (Garciaet al., 1980); moreover the C, coefficient for the zP3dependence at large distances [see Eq. (7)] turns out to be 166 Hi3 meV, close to the theoretical values 173 A3meV of Bruch and Watanabe (1977) and 186 A3 meV of Vidali et al. (1979). However, the matrix elements are not in agreement with the experiments. Hutchison and Celli (1980) suggested that this disagreement depends on the assumption of a R-independent flat bottom. By assuming a periodic variation of the well depth AD(R) = 2D,[~0~(2rr/a)X + cos(2r/a) Y

+ COS(~T/U)( Y - X)] (82)

in the range 0 < z < p (see Fig. lo), Hutchison and Celli showed that experimental matrix elements can be well reproduced if D, is chosen equal to -0.25 meV. The relevance of the HCS with a modulated well depth rests on separating the role of the different potential parameters: represents the hard-wall corrugation function; the well depth D and width p are related to the BSR levels, whereas the bottom variation amplitude D, determines the splitting of the deepest levels. A different model potential well suited to explain the experimental results is the Tommasini potential

V~(Z) = D{[1

+ A(z

- ~ , ) / p ] --~2[1 ~

+ A(z

- ~,)/p]-’}

(83)

discussed extensively by Mattera et al. (1980a). In Eq. (83), D is the well depth, A a reciprocal range parameter, p a variable exponent such that - 1 Il/p I1; A essentially describes the width of the potential, while p is related to asymmetry; z, denotes the position of the minimum and is a parameter not sensitive to scattering. The three-parameter Tommasini potential has the advantage of containing as special cases such widely used potentials as the ( 2 4 n) Lennard-Jones potential (X = p/z,), and the Morse potential Cp + 03). Its energy levels have approximately the form given by Eq. (49). The best-fit parameters needed to describe the ej levels by Derry et al. (1979) are D = 16.1 meV, A = 1.485 A-l, and p = 4.4. The flexibility of the Tommasini potential was proved by using it in a large number of systems; it does not give the correct zP3 asymptotic behavior, but the observable BSR levels approaching the continuum are likely to be nonsensitive to the long-range zP3 term. Starting from model potential (83), a parametrization of the first (and only relevant) Fourier component for He-graphite was proposed by Boato et al. (1979b) in the form V d z ) = -PloD[1

+ X(Z - z e > l / ~ ) - ~ ~

(84)

144

G. BOAT0 AND P. CANTINI

t FIG.22. Zero-order (V,) and first-order (Vl,,) Fourier components of the He-graphite potential.

The measured matrix elements are well represented by using Eq. (84) with a = 3 and plo = 0.019; the choice a = 3 gives a much better fit to the experimental results than the more conventional choice a! = 2, used by several theoreticians in conjunction with a Morse potential having a modulated repulsive term. However both a! = 2 and a = 3 give too large a corrugation-compared to the HCS model-within the experimental energy range from 20 to 65 meV. Figure 22 shows V,(z) and VlO(z) determined by Boato et al. (1979b). The whole potential V(R, z) is that represented in Fig. 3. A third approach to the He-graphite potential is based on the assumption that V(r) can be expressed as the sum of He-C pair interactions. A potential of this type was first proposed by Steele (1973, 1974) and further critically discussed by Carlos and Cole (1978, 1979, 1980a). The calculation assumes that the potential is a sum of contributions arising from different graphite layers, each labeled by the integer n , in the form

ATOM DIFFRACTION FROM SURFACES

145

In Eqs. (85a,b), z, = z + nd ( d = 3.37 A) and 4 is a lattice vector in the basal plane. Further, j = 1, 2, 6, and 6, being the positions of the two carbon atoms in the unit mesh. By Fourier transforming Eq. (85b) one obtains the contribution to Vc(z) originating from the nth plane as

V,, = Pcd-' /u,c, exp( - iG * R)u(z,, R) d2R with

PC = exp(-

iG 6,)

+ exp( - iG

6,)

(87)

Starting from different He-C pair potentials u(r), Carlos and Cole (1978, 1979, 1980a) calculated the He-graphite potential and, from it, the energy levels ej and the matrix elements (xjlVlo(xj,).A first attempt to fit the experimental data was made by assuming a variety of central potentials u(lr - rll). With these potentials the energy levels were reproduced by an appropriate choice of parameters. In each case, however, the matrix elements computed with isotropic forms of u were smaller by 20-70% than the experimental values. Only by using anisotropic pair potentials was good agreement obtained with the experimental matrix elements (Carlos and Cole, 1979, 1980a). The necessity of such a choice was explained in terms of the anisotropy of dielectric function and charge density in graphite. The resulting Fourier components Vo(z) and Vlo(z), are very close to those shown in Fig. 22 in the region of the well. High-order Fourier components are again found to be negligible. However, the potential determined by Carlos and Cole also shows a high-energy corrugation amplitude which is larger than that experimentally determined. Two concluding remarks should be made. First, there is no unique potential that explains the experimental data. However, the physical arguments put forward by Carlos and Cole render the potential of Figs. 3 and 22 highly reliable. Second, excellent agreement is found between atomic beam scattering data and accurate measurements of thermodynamic properties of He submonolayers physisorbed on grafoil (Elgin and Goodstein, 1974; Elgin et al., 1978). The binding energy and the low-coverage specific heat deduced from these measurements (see also Silva-Moreira et al., 1980) are in excellent agreement with those evaluated from bandstructure calculations making use of the matrix elements derived from atom diffraction (Carlos and Cole, 1980b). We should like to emphasize that the He-graphite potential used prior to atom-scattering experiments was significantly in error; it overestimated the well depth by about 20%, also giving it a somewhat different R dependence. [For a critical review of this subject, see Cole et al. (1981).]

146

G. BOAT0 AND P. CANTINI

VI. INFORMATION

ON

SURFACE LATTICE DYNAMICS

Whereas bulk-phonon disperson relations are currently determined from inelastic neutron diffraction, little is known experimentally about the dispersion relations of surface phonons. Although such knowledge is an important prerequisite for a complete understanding of solid surfaces and their dynamical properties, standard techniques such as EELS or Raman and Brillouin scattering are not very effective in studying surface phonons. Because of their strong interaction with the surface and their thermal energy, atomic beams provide a unique tool for investigating lowfrequency surface phonons. The required angular and energy resolutions have become available only in the last few years; very recent experiments have finally shown that the success of atomic beams to perform highresolution surface-phonon spectroscopy is now guaranteed. The older efforts to obtain information on surface lattice dynamics were devoted either to “integral” studies of the inelastic atom scattering, through the D-W factor, or to “differential” studies, where structures in the inelastic angular distribution were uniquely related to the energy ha, and to the parallel wave vector Q of the exchanged phonons. Both old and quite recent experimental results will be discussed in the following sections. A . Debye- Wuller Factor

As discussed in Sections II,C and III,B,l the D-W factor exp(- 2W), operatively described by Eq. (ll), contains information on the mean square displacement (2) of the surface atoms, which in turn depends on the phonon spectrum. It was shown that due to the potential well, to the finite size of the atom probe, and to the nonnegligible interaction time, appropriate corrections must be introduced in the ordinary expression of 2 W . Unfortunately, no experimental evidence exists to demonstrate without doubt the validity limits of each proposed correction. Hereafter, we shall give a short description of the results and their possible explanation. In many experiments the D- W factor was measured from the thermal attenuation of the specular beam, made of He, H, or D. In order to apply the conventional formula, now expressed by 2W

=

((Ak * u)’)

=

2k&(u2,)

(88)

at least two corrections should be considered for reasons discussed in Section III,B,l, namely, the Beeby correction (kgz is substituted by kh2, =

I47

ATOM DIFFRACTION FROM SURFACES

k;, + 2rnD/h2, D being the well depth) and the size-effect correction. The validity of the Beeby correction is now generally accepted. It was first verified by Hoinkes et a/. (1972b, 1973) for the system H- LiF. The results are shown in Fig. 23, where the effect of taking into account a realistic well depth is clearly proved, The same authors applied also a size-effect correction, but did not use the better justified expression later proposed by Armand. By using the Debye model, (u:) can be expressed by

where M is the mass of surface atoms, eSis the surface Debye temperais the Debye function. Hoinkes and co-workers applied to ture, and their data the high-temperature limit of Eq. (89); that is,

+,,

(uz) = 3hTS/MkB@

(90)

A value for eSof 415 ? 42 K was found, compared with the bulk value 8 = 730 K . The ratio eS/0agrees well with the theoretical predictions (Allen et a/., 1969; Chen et al., 1972), based upon the weaker forces experienced by the surface atoms of the solid. An extensively studied surface was NaF(001) (Wilsch et al., 1974; Krishnaswamy et a/., 1978) using H, D, and He probes. The surface

TS . ( c o s 2 0 0

+&J

-

0

400K 8M)K

1200K 1600K

FIG.23. The D-W attenuation in scattering of H from LiF(001). Only the introduction of a well depth D = 18 meV makes the experimental points align on the same straight line: D = 17.8 meV (a), 0 meV (b), 71 meV (c); Ts = 725 K ( O ) , 475 K ( x ) , 225 K (0). [From Hoinkes et a / . (1973).1

148

G . BOAT0 AND P. CANTINI

Debye temperatures obtained in the three different experiments (370,416, and 425 K, respectively, without the size-effect correction), compare rather well with each other if account is taken of the experimental errors; eS/8is again in agreement with theoretical predictions. Another accurate study of the D-W factor was carried out by Lapujoulade et al. (1980a) for the system He-Cu(001); in this case also the conventional theory appears to work quite well when both well-depth and size-effect corrections (Armand et al., 1977) are taken into account. In spite of this apparent success, several doubts were thrown upon the D-W elementary expression given by Eq. (88). The cited paper by Krishnaswamy et al. (1978) showed that with or without the Beeby correction, Eq. (88) is not able to explain the angular dependence over the entire range 0" < O0 < 90". Much stronger doubts were earlier aroused by the large elastic diffraction probabilities found for Ne scattered by LiF(001) (Boato et al., 1976a). Other discrepancies were observed in the thermal attenuation measured in Cu(OO1) by using beams of both He (Mason and Williams, 1978) and Ne (Lapujoulade et al., 1980a). Some of these anomalies are qualitatively explained by Levi and Suhl (1979) and by Meyer (1981), but several features are not well understood. As expected, it seems certain that the more one departs from the fast-collision approximation, the stronger are the anomalies. But, as a matter of fact, a complete systematic experimental study of the D- W factor in atom scattering is yet to be accomplished. B . Structures in the Angular Distribution

In a number of cases, structures in the angular distribution of the scattered intensity are observed. In particularly favorable cases and under the condition that one-phonon events are dominant, it is possible to relate univocally the kinematic parameters ( O o , +o; Of, +f) of the structures to the energy fiwq and parallel momentum hQ of the exchanged phonons (see Section 1I.C). One kind of such experiments is based on the existence of a spatially forbidden region for inelastically scattered particles. At the boundary of this region particles associated with Rayleigh phonons should concentrate. The method was suggested by Williams (1971b) and an interpretation of the results for both LiF (Williams, 1971b) and NaF (Mason and Williams, 1974) in terms of surface-phonon dispersion relations was given. However, Williams' analysis was incorrect since he assumed Q to be perpendicular to the plane of incidence, and, in fact, the results for NaF were unsatisfactory. A reinterpretation of Williams' results was recently given by Avila and Lagos (1981),who observed, following a remark

ATOM DIFFRACTION FROM SURFACES Q,

[k’]

149

\ (a)

\

\

\ \

I

F

’1

0.5 L)

I

1.0

[1013rod sec-l] / W W R

NaF(001)

b

< 100 > < 110 >

. /’

FIG.24. Data of Mason and Williams (1974), as analyzed by Avilaand Lagos (1981) (a). The Rayleigh-phonondispersionrelation (b)is obtained: G = A,(00); 0, (1i)annihilation; 0 , (11)creation; +, (01) annihilation; 0,(Of) creation.

by Benedek (1979, that the kinematic relations (13) and (14) applied to Rayleigh waves identify for each elastic peak F a cone in k space having its vertex in kF. This cone corresponds to the allowed region for onephonon inelastic events in the real space. A “frontier of allowed zone” thus exists for out-of-plane scattering, which is experimentally observable as a sudden cutoff in the inelastic intensity (this fact was already noted by Williams). Avila and Lagos pointed out that for Rayleigh phonons, the intensity should have a singularity at the frontier of allowed zone. The angles corresponding to such singularity determine both kw, and Q of the involved phonons. The data of Mason and Williams (1974) on NaF are reported in Fig. 24 together with the proposed interpretation. The Q values now obtained from the experiment tend to align along the ( 110) direction (dashed line in Fig. 24a), whereas the w , Q plot (Fig. 24b) gives the surface-phonon dispersion relation. This compares well with the expected linear part of the Rayleigh-phonon dispersion relation (dashed straight line). A similar analysis can be made for LiF results.

G . BOAT0 AND P. CANTINI

150

A different kind of experiment is based on the study of inelastic BSR structures (see Section 111,B73). An inelastic BSR, characterized by (ej, N), selects at each incident angle and energy a well-identified family of phonons. Their frequencies oNJare given by the kinematic condition

k$

?

2rn~,,~(Q)/h - (K,,+ N

f Q)2 =

2rnej/A2

(91)

On the other hand, when a scanning of the angular distribution is taken, each final scattering angle 0, corresponds to inelastic processes which select a family of phonons of(Q) given for in-plane scattering by the parabolic equation.

kg

f 2rnof(Q)/A -

(I?,

+ F f Q)”sin2

0, = 0

(92)

Since the family of phonons yielding a resonant contribution [Eq. (91)] and that selected by 8, [Eq. (92)] in general do not coincide, the location (do, 0,) of a sharp resonant structure in the tail of the diffraction peak fixes through conditions (91) and (92) both the energy Awq and the parallel momentum AQ of the particular phonon involved. Thus, this procedure is also capable of yielding a dispersion curve without energy analysis of the scattered particle. The existence of inelastic BSR structures was first demonstrated by Cantini et al. (1976, 1977), who also showed that a rough dispersion relation compatible with Rayleigh phonons could be determined for He- LiF. More recently, the inelastic BSRs were observed and carefully studied in the system He-graphite (Cantini and Tatarek, 1981). In Fig. 25, typical examples of inelastic BSRs measured with high sensitivity are shown. They were observed in the tail of the specular peak near the scattering angle e, = 57”. These structures correspond to the transition into the BSR labeled eO(10-01) after a phonon has been exchanged with the surface. As the incident angle is increased the line-shape changes from a minimum to a maximum (around O0 = 51”) and again to a minimum. This is due to the intervention of the elastic BSR labeled el( 10-01). A double resonance takes place, as described in Section III,B,3 [process (4) of Fig. 111. Further, Cantini and Tatarek (1981) and, in more detail, Boato et al. (1982) showed how a systematic study of inelastic BSRs at different incident angles do and final angles 0, is able to furnish a dispersion relation for the graphite surface phonons. The dispersion relation is indistinguishable, within sizable experimental errors, from the TA, branch of the bulk graphite, relative to phonons traveling along the hexagonal plane. Figure 26 shows a different type of inelastic BSR structure which appears in the scattered intensity, namely, that corresponding to the transition in a bound state after the creation of one phonon, without a change of G. The process, called “specular inelastic” selective adsorption, was

151

ATOM DIFFRACTION FROM SURFACES

. . . .

00

52.45O

. . ..

52.20°

. .

. .

52.03'

.. ..

51.86' 51.70°

. . .. . .

51.53' 51.36' 51.20' 51.03O 50.86O 50.70"

I

55"

I

Of

I

60'

FIG.25. Inelastic BSR structures observed in the tail of specular peak in He-graphite: k, = 9.07 A, & = 30". [From Cantini and Tatarek (1981).]

experimentally studied by Cantini and Tatarek (1982). The resonant process appears as a sequence of maxima, each associated to a level ej of Vo(z),which become particularly strong at low surface temperatures Ts . The intensity of each inelastic structure of Fig. 26 can be understood from Eq. (73). Applying the approximation given by Eq. (76), one obtains C == Si&nSoo/Sio(l -

I ~ o o l )2: Soo/(1 -

ISOol)

(93)

where the S terms are elastic amplitudes. As the corrugation of graphite is small, lSoolapproaches unity and ICI >> 1 (strong maximum). In addition, the temperature dependence is weakly influenced by Bose statistics (phonon creation is in fact involved), but strongly affected by the D-W factor; the result is that as the temperature increases, the inelastic maxima reduce their height in the same way as the elastic peaks. Once again, the angular position of the observed maxima can be used to obtain the phonon dispersion relation.

152

G . BOAT0 AND P. CANTINI r

-

10

€2

€1

€0

FIG. 26. Specularly inelastic BSR structures in He-graphite: ko = 11.05

kl,

00 = 74", l#Jo = 0".

Finally, Brusdeylins et al. (1981b) studied inelastic BSRs in the systems He-LiF and de-NaF, in connection with measurements of energy losses by the time-of-flight (TOF) method (see Section V1,C). They observed a large number of structures in the inelastic background between diffraction peaks, most of them maxima, but some minima, which they attributed to resonance transitions followed by the exchange of one phonon [selective desorption, or process (2) in the scheme presented in Fig. 113. Brusdeylins and co-workers prove that one-phonon exchange takes place, by directly measuring the energy of the outgoing particles. The large number of structures seen by them is due both to the high monocromaticity of the incident beam and to the presence in He-alkali halides of a larger corrugation amplitude. The observed resonant events and line shapes can be understood in the framework of the theory by Cantini and Tatarek (1981). C . Energy-Loss Measurements

Energy analysis of the scattered beam is a prerequisite for a valuable surface-phonon spectroscopy by atom scattering. We limit our attention to one-phonon inelastic scattering of He from LiF and mention an early attempt to reach this goal made by Fisher and Bledsoe (1972), who observed some structure in the TOF spectrum. Their energy resolution (AEIE = 20%) was largely insufficient to give a clear interpretation of the data. Later, Horne and Miller (1978) used better resolution to obtain evidence for a single Rayleigh mode, similar results were reported by Feuerbacher et al. (1980).

153

ATOM DIFFRACTION FROM SURFACES

It was the experiment by Brusdeylins et al. (1980, 1981a) which showed that the crucial point was the angular and energy resolution of the apparatus. Using a supersonic expansion from 200 atm at T = 80 K through a 5-pm-diam. nozzle, these authors were able to get a beam having an angular spread [full-width at half-maximum (FWHM)] of 0.30" and an energy spread (FWHM) of less than 2%. Long flight distances were used, and very sharp TOF spectra were obtained, with an energy resolution of about 0.6 meV. A typical TOF spectrum obtained by Brusdeylins et al. (1981a) for in-plane scattering is shown in Fig. 27, at Ts = 300 K. Six peaks are well resolved. Peak 3 is the remnant of elastic scattering [the (fi) diffraction peak is near in angle]; peak 2 is due to a weak but widespread tail in the incident-velocity distribution. The other peaks are related to one-phonon events. Three of them, namely, 1,4, and 6 are due to Rayleigh phonons, whereas peak 5 is very likely a structure in the bulk-phonon contribution at low frequencies. This can be understood more easily by looking at Fig. 28, which is a plot of the measured frequency w versus parallel momentum transfer AK in the extended Brillouin-zone representation. The points correspond to events detected in a large number of TOF spectra taken at different values of Oo . The solid lines show the theoretically expected Rayleigh-phonon dispersion curve (Chen et al., 1977; Benedek, 1976), whereas the dashed line is the parabola 0

= (A2k;/2m){[(AK/k, cos 0,)

+ tan OOl2 -

1)

(94)

derived from the kinematic conditions (13) and (14), if account is taken of the fact that in the experimental apparatus of Brusdeylins and coworkers, the difference Of - Oo was fixed and equal to 90".Equation (94) is then the locus of possible in-plane phonons events (w. A m , at the incidence condition ( k o , 6,). The location of the six peaks detected at ko = 6.06 k1 and Oo = 64.2" in the TOF spectrum of Fig. 27 is indicated in Fig. 28. The results can be summarized by saying that for the first time, the full dispersion curve for Rayleigh phonons was measured, and a high accuracy was attained. The curve agrees with the theoretical expectations, with an exception made near the Brillouin-zone boundary; the lower frequency observed is likely due to changes in interatomic forces (polarizability, relaxation, etc.) not taken into account in the surface-dynamics calculations. Bulk-phonon events (as peak 5 in Fig. 27) are shown to participate in the inelastic scattering, but the dominance of Rayleigh phonons, particularly at low frequencies [see Eqs. (67) and (68)], is clearly demonstrated.

154

G. BOAT0 AND P. CANTINI

2.0

1.5 Time

of

Flight

2.5 [msec]

FIG.27. A high-resolution TOF spectrum for the system He-LiF(Wl):( 100) azimuth. k, = 6.06 k*, Oo = 64.2". [From Brusdeylins et al. (1981a).]

The measurements by Brusdeylins and co-workers fulfill the longawaited expectation of the specialists in atom scattering and, more important, open a fruitful field for future developments. A second important result in energy analysis is given by the experi-

FIG. 28. A plot of measured inelastic TOF peaks in the o,AK diagram. The whole Rayleigh-phonon dispersion curve appears. [From Brusdeylins et al. (1981al.l

ATOM DIFFRACTION FROM SURFACES

155

ments of Mason and Williams (1981), who used a LiF(001) surface as the energy analyzer (this technique is similar to the triple-axis spectrometer used in inelastic neutron diffraction). They applied their method to the Cu(OO1) face and to an ordered monolayer of Xe on the same surface at low temperatures. They observed both surface and bulk phonons and were able to measure a part of the Rayleigh-phonon dispersion curve for Cu(001). Of greater interest are the measurements carried out on the Xe overlayer: Mason and Williams find that the layer is ordered, as expected from previous LEED measurements. More important, crystal analyzer scans at fixed B0 were made; the data correspond to energy losses or gains of 2.5 and 5.0 meV, showing that a single frequency is involved. This frequency should correspond to the vibration of Xe atoms, perpendicular to the surface; the dispersion relation appears to be flat, resembling that of an Einstein oscillator. To end this section we should like to mention that interesting energychange measurements by TOF analysis were made in the scattering of Ne from LiF(001) (Semerad and Horl, 1982; Mattera et al., 1980b). Here the time spectra show only broad structures, which appear to be related not only to multiphonon effects, as expected from qualitative arguments (large incident mass), but also to one-phonon events, thus confirming the large D-W factor previously reported (Boato et al., 1976a).

VII. CONCLUSIONS The present review shows how the atom-surface scattering method is rapidly advancing along different pathways, with respect to both experimental and theoretical techniques. We may distinguish between two main lines of development, namely, elastic diffraction and inelastic scattering. The first line has already reached a highly satisfactory level from the experimental point of view. Elastic He diffraction has become a commonly used technique for determining structures of the topmost surface layer, thus giving independent or complementary information compared to other standard techniques such as LEED or ion scattering. In connection with this, the examples reported on semiconductor reconstructed surfaces and H-covered metal surfaces should be convincing about the capabilities of the method. A systematic study of physisorbed monolayers and other surfaces largely affected by destructive techniques such as LEED might be a subject for future investigation. Further, we have repeatedly emphasized that the study of elastic BSR structures is a subject of great interest. Since the experimental technique is quite simple and accurate, this method is a powerful one for the deter-

156

G. BOAT0 AND P. CANTINI

mination of atom-surface potentials in the region of the well. The extension of the method to state-selected beams of H, , D, , and HD might clarify the dependence of the gas-surface potential on the rotational state of the molecule. Along with experiments, we have described how simple elastic theories, capable of explaining the experimental results, were developed. Although the HCS model was proved to be sufficiently flexible for describing most diffraction patterns, further theoretical effort should be made toward the study of more realistic “soft” potentials. The second line of research, i.e., inelastic scattering, is just now in full development, particularly since the experimentally needed energy resolution has only recently been reached. A first problem in inelastic scattering is the proper use of the D-W factor exp(-2W). In a few cases the conventional formula 2W = ((Ak * u ) ~ )modified , by appropriate corrections, was proved to be able to furnish reliable information on the mean square surface displacement ( u z ) (or on the surface Debye temperature @). However, the behavior of 2W becomes more intricate when multiple scattering is involved or heavier (and slower) incident atoms are used. A deeper study of the D- W attenuation, both by systematic experiments and by more refined theory, is still necessary. On the other hand, recent measurements have definitely shown the usefulness of atomic beam energy spectroscopy in giving direct information on the spectrum of surface phonons. The TOF technique seems now capable of determining the phonon-dispersion relation for several bare and adsorbate-covered surfaces. The theory of inelastic scattering needs further refinement, particularly for metal surfaces, where the energy transfer from incident atoms to lattice ions is still a problem owing to the presence of free electrons. Anyway, the theoretical approach should go beyond the vibrating HCS model; moreover, the effect of the surface-projected phonon density [Eq. (68)] on the inelastic scattering probabilities should be further clarified. With these refinements, the He atom ought to become a very useful probe for the investigation of surface dynamics. ACKNOWLEDGMENT The authors are greatly indebted to all members of the Genoa group on gas-surface scattering for useful discussions and constant cooperation.

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ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS, VOL. 60

Digital Image Processing B. R. HUNT Department of Elecirical Engineering and of Optical Sciences University of Arizona Tucson, Arizona and Science Applications, Inc. Tucson, Arizona

I. Introduction ............................................................. 11. Image Restoration (Deblumng) ............................................ A. Matrix Forms of Convolutions ......................................... B. Matrix Derivation of Image-Restoration Algorithms. ...................... C. Blind Deconvolution .................................................. 111. Image Data Compression ................................................. A. Correlation/Decorrelation.............................................. B. DPCM Compression .................................................. C. Transform Domain Compression.. ...................................... D. Hybrid Compression .................................................. E. Color and Temporal Compression ...................................... IV. Reconstruction from Projections. .......................................... V. Steps toward Image Analysis/Computer Vision ............................. References ..............................................................

161 163 164 174 189 192 192 193 197 203 204 207 215 219

I. INTRODUCTION We begin by defining digital image processing. Digital image processing is the employment of a digital computer to carry out some sequence or set of processing actions upon an image. Obviously, within this definition, the exact nature of the processing actions which are chosen to be applied to the image determines the objectives which are meant to be fulfilled by the processing. This gives the impression that digital image processing has a distinctly application-oriented flavor. The impression is a correct one. Digital image processing arose from very specific problems in space science and physics, and the solution of those problems was sought by applying digital computing to the imagery in question. Success in the initial endeavors provoked still other applications. 161 Copyright @ 1983 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-014660-6

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Digital image processing is a subset of the total area of research in image processing. There are obvious methods for image processing other than digital. Analog circuitry can be used to process images. Analog image processing has been applied in a variety of ways in the commercial television industry for a number of years. The “special effects” seen in television programs are a consequence of such analog image-processing techniques. Another technology for image processing is optics. The invention of coherent light sources by laser, and the dissemination of high-quality lenses at reasonable cost, has made optical image processing a technology that can be usefully employed in real systems and applications, e.g., synthetic aperture radar. In this article, however, it is digital image processing which chiefly interests us. This is an appropriate focus for this publication because the revolution in semiconductor digital electronics has been profound in the consequent impact upon digital image processing. First has been the continuing surge in digital computer performance. Today digital imageprocessing operations are routinely undertaken in general-purpose computers that would have been considered unthinkable 10 or 15 years ago. But as dramatic as the growth has been in general-purpose computers, the growth in special-purpose computers, of a variety suited for image processing, has been even more dramatic. It is possible to purchase auxilliary pipe-lined computers which, when interfaced with a modest minicomputer, can provide more image-processing calculations (e.g., convolutions, correlations) than the largest general-purpose mainframes, but a fraction (e.g., one-twentieth) of the mainframe cost. A final impact in image processing has been the economical design and construction of systems which provide real-time display and processing of digital imagery. The dramatic decline in the cost of random-access semiconductor memory has made it feasible to directly store the digital values of an image in a large semiconductor memory which can be employed as the direct source of refresh data that can be used to generate a directly viewable image on the face of a cathode-ray tube. Likewise, the speed of semiconductor circuits has increased, while cost was decreasing, to allow direct modification or computation on the image values at the refresh rate of the display process. The increasing capability in hardware has been associated with a parallel growth in theoretical and algorithmic knowledge. For example, the discovery of the fast Fourier transform algorithm spurred the study of a variety of fast orthogonal transforms and their application to image-data compression. The increasing computer power led to consideration of processes which were more computationally intensive, as researchers acted on the faith (a faith which is still proven justified) that processing

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algorithms which were of great utility were worthy of investigation, regardless of the computational load. In this article we wish to give an overview of the progress in digital image processing over the past decade. We will examine progress in both the theoretical and algorithmic development of image processing. Our time frame for the overview is roughly the year 1970 to the present. For comparison, the reader may wish to consult a source which summarizes the state of the art circa 1970. The most readily available source of such a summary is probably the July 1972 issue of the Proceedings of the ZEEE. We will, in this article, report developments which have reached their full bloom since this special issue of the Proceedings. Finally, we wish to emphasize that the overview we give in this article is not complete. The field of digital image processing is growing so rapidly that it would be impossible to make a complete survey in anything less than thousands of pages. (The reader should consult Pratt’s book, which is only partially complete as of 1978, if he doubts the truth of this assertion.) Instead, we will try to concentrate upon those developments which have been of greatest importance. As always, the criterion of importance is a subjective judgement of the author. To those who disagree with the judgement of this author, or who feel that important work was neglected, this author offers apologies and extends an invitation for the offended parties to develop their own summary and overview for the benefit of the readership.

11. IMAGE RESTORATION (DEBLURRING) The problem of image restoration can be stated succinctly. Under the assumption of space-invariant image formation, the image go(x,y) of an object f(x, y) is determined by the convolution integral (Goodman, 1%8),

where h(x, y) is the point-spread function of the image formation system. The quantity g o , the image, must be sensed and recorded by a suitable mechanism, e.g., photographic film. Since image sensors and recorders have response characteristics, and since the process of sensing any physical quantity introduces noise, the actual image which is available for processing is described mathematically by g(x, Y) = sko(x, Y)I

+ n(x, Y)

(2)

where s[g,(x,y)] is the sensor characteristic function, and n(x, y) is the

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sensing/recording noise process. For example, in the case of photographic film, s is the D log E curve, and n(x, y) is a film-grain-noise process (Mees, 1954). The image restoration problem is to infer, as best possible, the original object f from the recorded image g. This problem arises in a variety of practical applications. For example, any relative motion between object and image plane during the exposure of the image leads to a blur of the object, and it is the goal of image restoration to estimate the object in the absence of motion. Image resotration was one of the first problems subjected to analysis for digital image processing. A number of algorithms for image restoration were developed and applied by the early 1970s (see the article by Sondhi, 1972, for example). The early methods utilized Fourier concepts in the restoration of the imagery (Andrews and Hunt, 1977). However, a concept which proved equally useful, and mathematically equivalent to the Fourier approach, was the formulation of restoration in terms of linear algebra. We shall turn our attention first to this methodology. A . Matrix Forms of Convolutions

We begin by assuming that the sensitometry associated with x in Eq. (2) is linear. Without loss of generality we can let s be the identity transformation. (We shall relax this linearity assumption in Section II,C below.) The resulting equation becomes fm

Fa

+ 4x9 Y )

(3) Equation (3) is in continuous variables. It is intended to perform the image restoration in a digital computer, however, so the quantities in Eq. (3) must be sampled into a finite number of picture elements (pixels), and each pixel must be quantized into a finite number of bits of numeric representation. The effects of quantization can be described in terms of an additional noise process (Oppenheim and Schafer, 1975). If the quantization scheme is well designed, then quantization noise should be less than the sensing/recording noise, and we can neglect it in comparison to the term n(x, y). We assume that the image-sampling process is ideal, i.e., the aperture through which the image is examined for sampling is infinitesimally narrow, so that no degradation of the image spectrum occurs due to sampling (Hunt and Breedlove, 1975). (This assumption can be relaxed

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and the effects upon the sampled image can be determined and compensated; see Hunt and Breedlove, 1975, for details.) The mathematical notation for sampling the image then becomes

+ n ( j A x , k Ay)

(4)

The problem remaining in Eq. (4) is the sampling of the convolution integral, since the A x , Ay sampling only constitutes specific evaluations of g and the associated values of the kernel function h. It is clear that the evaluation of the integral in sampled form will not, in general, be exact. The approximation of an integral by samples of the integrand is the province of numerical analysis, and is referred to as the problem of numerical quadrature (Ralston, 1965). Quadrature can be carried out with either equally spaced samples or unequally spaced samples, the latter leading to greater accuracy in the approximation. However, unequally spaced samples are inconsistent with the usual standards and procedures for image-sampling devices and, for the most exotic of quadrature algorithms, can result in sample spacings dependent upon the actual data being sampled. Equally spaced samples are readily extracted by conventional image-sampling equipment and are not dependent upon the data, except for the specification that the sampling frequency satisfy the Nyquist criterion (Oppenheim and Schafer, 1975). Consequently, we will assume the approximation of the integral in Eq. (4) will be in terms of equally spaced samples. Numerical quadrature with equally spaced samples is known by the class of Newton-Cotes quadrature formulas, of which a variety are known and familiar, e.g., the trapezoidal rule, the parabolic rule, and Simpson’s rule. The higher order quadrature approximations, that is, all quadrature rules of order higher than the trapezoidal rule, require the multiplication of the integrand samples by a periodic repetitive pattern of quadrature weights. It can be shown that the periodic weight pattern induces a second-order frequency modulation in the spectra of the functions it has been multiplied by, leading to a higher sampling frequency to suppress aliasing (Hunt, 1971b). Consequently, it is desirable to use only a trapezoidal approximation for the evaluation of the integral in Eq. (4) in sampled form. Since the trapezoidal approximation applies unity weights to all samples, except the first and last samples which receive a weight of 1/2, the approximation formula is trivial. We neglect the end-point weights (at a calculable and small increase in integration error) and use unity weights for all samples. The resulting sampled convolution expres-

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sion becomes gci Ax,

k Ay) = C, C h ( j Ax m

- m Axl,

k AY

- P Ay,lf(m Axl , P

AYJ

p

+ n(j k

(5)

j AY)

There are a number of simplifications to be made to this expression. First, we recognize it is an approximation, but we shall delete the approximation symbol in future discussion. It is a lack in current research that no real analysis of the accuracy of the approximation has been carried out. Second, we note that there is a completely general set of sampling intervals on both the left and right sides of Eq. (5). There is, however, no utility to allowing Axl and Ax or Ay, and Ay to be different. The opposite is true. If Ax, # Ax, then a physical sample distance in the object is mapped into a different sample distance in the image. Since there is one sample distance which is suitable for information recovery, i.e., the Nyquist sample distance (Oppenheim and Schafer, 1975), it is most desirable to let both object and image be sampled at the Nyquist distance of the object, since the linear operation of convolution will not change the Nyquist frequency. Without loss of generality, we set Ax = Axl = Ay = Ayl = 1 and have the simplified convolution summation:

dj,4 =

h ( j - m, k - p l f ( m , P) m

+ n(j, 4

(6)

p

We have yet to specify the range of indices m a n d p in Eq. (6). Two important cases must be distinguished. The first case we refer to is that in which the object and image are smaller than the image frame. We mean the following. In imaging an extended object, we recognize that the field-of-view of our imaging system is not of indefinite extent. The action of convolving the point-spread function with the object creates an image whose physical dimensions, as imaged in the focal plane of the imageformation system, are greater than the object. A trivial example of this is the point-spread function itself, which is the finite size image of an infinitesimally narrow object, the ideal point source of light. If the resulting image can be encompassed within the field of view, then no information is lost. A simple example is the imaging of an object on black background, e.g., the moon on a night sky, by a telescope whose field of view is large enough to encompass the moon’s image. A second case immediately comes to mind, of course. If the focal length of the telescope were increased until the moon’s image filled and exceeded the available field of view, there would be an information loss. Most important, there is information in the image which is affected by the “cropping” action of the field of view. Specifically, the point-spread function of the imageformation system, being of finite extent, will cause parts of the image near

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the edge of the field of view to be composed of object information which is outside the field of view. Figure 1 indicates the effects of these two cases in graphic form. It is the necessity of describing both these cases that governs the limits of indicates m and p in Eq. (6). The book by Pratt (1978) devotes a chapter to the analysis of these two cases. We summarize the two cases with the context of a more general description, the matrix-vector convolution. Examination of Eq. (6) reveals a sum of products, linear in the transformation off into g. A linear transformation can always be described by a matrix operation. To cast Eq. (6) in this form we define a lexicographic ordering on the image samples. Let

f=

where f j k =f(j, k)

(7)

which corresponds to a stacking of the columns of the sampled image matrix f into a single vector. Likewise, the matrix of samples g can be stacked into a vector. This having been done, Eq. (6) directly implies that g = [Hlf

(8)

where the elements of the matrix [HI are samples of h (j , k) placed into a matrix format in such a way that the product of [HI with f calculates only the terms indicated in Eq. (6).

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168

7 m+j-

-P+k-l-

1

...

I

*

...

"A'"""

A-

i-1

k-

.. ................ Field of view of size L x L (d)

FIG. 1. (a) Point-spread function. (b) Object. (c) Case I. Field of view larger than image; image extent is sum of object and point-spread function dimensions. (d) Case 11. Field of view smaller than object being imaged, i.e., extended object "cropped" by the field of view: A, region of object space that can contribute to the field of view; B, region of image space which can be contributed to by objects outside the physical field of view; C, region of image space which is contributed to solely by objects within the field of view.

Equation (8) and the discussion leading up to it is important, because key realizations in much of image restoration/deblurring during the past 10 years came about through analysis of the matrix nature of Eq. (8). From Eq. (8) it is obvious that the problem of image restoration is equivalent to the solution of a set of simultaneous equations. What makes image restoration a unique (and trying!) problem is the magnitude of the problem. For example, if the image is sampled at 500 x 500, then on the order of 250,000 simultaneous equations must be solved in the process of image restoration. This is clearly prohibitive unless there is special structure in [HI which can be exploited. Fortunately, such special structure exists. The nature of the special structure can be demonstrated with a simple one-dimensional example. Take the case of five samples of the object, three samples of the point-spread function, and assume that Case I dis-

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cussed above governs the indices of Eq. (6). It is direct to show that the object-image relation can be written

fl

fi f3

f4

(9)

f5

The matrix [HI has interesting structure, with every element on the diagonal being the same. The matrix is nonsquare, however, because of the relation between object and image size governed by Case I outlined above. We can extend the size of this matrix to be square, without affecting the left-hand side, if we append zeros to the vector f. Thus,

hl h2 hl h3 h2 hl h3 h2 h3 hl The matrix [HI in Eq. (10) has a special form, being square and with every element on a diagonal being the same. This is a Toeplitz matrix. Furthermore, we see that the appending of zeros to fallows us to place anything in the upper right corner of this matrix without affecting the left-hand side. We choose to insert elements to produce the following:

fl

fi f3

f4 f5

0 0-

In Eq. (1 1) the matrix is still Toeplitz, but with an even more particular structure. We note that in this matrix, each row is a circular right shift of the row above it, the top row being such a shift of the bottom row. We call

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this shift circular because an element shifted out of the row of the matrix at the right boundary reappears at the left of the row. The matrix in Eq. (1 1) has a special designation as a subclass of Toeplitz matrices: it is known as a circulant matrix. We see that in terms of the left-hand side of Eq. (9), the left-hand side of Eq. (11) is the same. Thus, we can replace the initial description of convolution with the description in Eq. ( l l ) , which we symbolize as ge = [Hclfe

(12)

where [H,] is the circulant, f, has been extended by zeros, and g, may also be extended, depending upon the number of zeros appended to f. Equations (1 1) and (12) are important because of a unique property of circulant matrices, and the relevance of this property to digital signal processing in general was made clear during the period of the 1970s. We define the discrete Fourier transform (DFT) of a sequence as

From the discussions above, it should be obvious that we can write this transform as a matrix-vector product, i.e., F = [W]f

(14)

where the matrix [W] has the form {[Wlljm = ~ x P [ -(i27rlWmI

(15)

and { Ijm is a notation for thejmth element of the matrix enclosed by the brackets. The remarkable property of [Hc]can now be stated in the context of eigenvalues. We know from linear algebra that under quite general conditions, it is possible to determine the eigenvalues of a matrix [A] by a transformation [A1 = [TI[AI[TI-'.

(16)

It can be shown (Hunt, 1971a) that for any circulant matrix, the matrix which casts it into diagonal form is the matrix [W], the DFT operator. Thus, [Ah1 = [ w l [ ~ c l [ W l - l

(18)

where [Ah]is a diagonal matrix of the eigenvalues of [H,]. Furthermore, the eigenvalues of [Hc]are determined as the DFT of the typical row:

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where there may be zeros within h ( j ) , depending on the relative dimensions of g, h, andfin Eq. (11). Rearranging Eq. (18) and substituting into Eq. (12), we have which gives

[Wke = [M[Wlfe We recognize that the product of [W] times a vector gives the DFT of that vector, so

Ge =

[&We

(21)

and since [Ah] is diagonal we readily recognize this expression as a matrix description of the discrete Fourier convolution theorem: the product of Fourier transforms of object and point-spread function yields the Fourier transform of the image. There are a number of important facts revealed by this result. First, we see that for a linear shift-invariant system, eigenanalysis of the matrix structure of that system is equivalent to Fourier analysis of the corresponding sequences. Second, we see that what is often proved by Fourier transforms of sequences, the discrete convolution theorem, is actually a property of deeper importance, i.e., the discrete convolution theorem is governed by the eigenanalysis of a corresponding set of linear equations. From the viewpoint of image restoration and image deblurring, Eq. (21) is important because of what it states about the problem of dimensionality. We have been able to convert a set of simultaneous linear equations [Eq. (12)] into a set of equations in which all cross-terms are zero and each variable on the right-hand side of the equation is related to a variable of the left-hand side by only one coefficient. Because of the diagonal nature of [A,] in Eq. (21), we can write

Ge(m) = H(m)Fe(m)

(22)

as a product in Fourier transforms. Equation (22) states that the inversion of Eq. (12) can be computed by division in the Fourier domain, followed by transformation back to the original image space. Soution of deblurring by Fourier methods was, of course, the first method for realistic solution of these problems. However, the establishment of a linkage between Fourier methods and the solution of simultaneous equations greatly expanded the realm of conceptual methods in restoration and deblurring. It is obvious that the full range of the tools of linear algebra become applicable to the solution of image-restoration problems, and that even though a conceptual solution to a problem may

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require matrix equations of incredible dimensionality, the equivalence between Fourier analysis and linear equations can be used to reduce the solution to something of reasonable size. Furthermore, the Fourier analysis can be implemented by a fast algorithm (the FFT) for extremely efficient computation. Thus, a number of image-restoration methods can be developed and analyzed in terms of matrix calculations, but implemented by Fourier computations. The discussion above was inaugurated in the context of a onedimensional example, but it can be readily extended to two dimensions, as we now summarize.

(1) The “stacking” of the image-sample matrix into a single vector [see Eq. (7)] leads to the imposition of a structure for [HI in Eq. (8). The structure has the property of describing convolution between a matrix of point-spread-function samples and a matrix of image samples, and is analogous to the one-dimensional case where the matrix described the convolution of sequences. (2) The structure of the matrix in Eq. (8) can, by extension of the individual column components in f of Eq. (7), be made into a square matrix which is a block Toeplitz matrix. This is not, in general, a Toeplitz matrix but has the following structure. The matrix is composed of partitions. The partitions are arranged in Toeplitz form; that is, along diagonals ordered according to partitions all partitions on a diagonal are the same. Each of the partitions (which are arranged in Toeplitz fashion) is itself a Toeplitz matrix, of the form of Eq. (10). (3) The block Toeplitz matrix can be converted into a block circulant matrix. This is done by placing partitions into the upper right corner, analogous to the process in constructing Eq. (1l), and each partition is itself made circulant by the same process. (4) The resulting block circulant matrix is diagonalized by the twodimensional discrete Fourier transform, so that the simultaneous equations in Eq. (8) are converted into a product of two-dimensional Fourier transforms. We shall now briefly touch upon the solution to the system of linear equations represented by Eq. (8). As discussed above, we actually replace Eq. (8) with a two-dimensional block circulant version:

Ostensibly, we would calculate [H,]-’ to determine f. However, the inverse of our circulant approximation in Eq. (23) is not the same as inverting the original Toeplitz matrix which it approximates. However, we can work out the difference explicitly. We represent [Hc]as the sum of the

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Toeplitz matrix and a matrix [D], which is highly sparse, that circularizes the Toeplitz. Thus, [He1 = [HI + [Dl

which implies

[HI = [He1

-

[Dl = ([I1 - [DI[HcI-')[HcI

(24)

and thus,

[HI-' = [Hc]-'([Z]

-

(25)

[D][He]-')-'

we now expand into a series: ([I1 - [DIIHcl-')-' = ([I1 + [DI[Hel-' + ([D1[HcI-')2

+ ([D][He]-')3 +

*

*)

(26)

and it is possible to show that the series converges if

ll[~l[~cl-'Il < 1 Equations (25) and (26) have the following significance. To compute [HI-', we first compute [He]-' and then multiply that by the terms of the series in Eq. (26). As many terms need be retained as necessary to adequately approximate the inverse, i.e., as the power of the terms in the series increases, the norm of the terms decreases, and as the norm of the next term in the series falls below a certain magnitude, the series can be terminated with allowable error. [See Ekstrom (1972) for complete details of the method.] The above formulation indicates the method by which only Fourier calculations need be used to compute the inverse of a Toeplitz matrix of two-dimensional convolution. Matrix [He]-' can be found from Fourier calculations; as seen Eq. (18), one derives [He]-' by calculating the eigenvalues of [Hi] by Fourier transform, calculating their reciprocal and inverse transforming. The multiplication represented by [Hc]-'[C] requires little computation because matrix [D] is so sparse. Thus, Ekstrom was able to invert Toeplitz matrices as large as lo6 x lo6 using this method! (Ekstrom, 1972). Of course, one quick approximation to the inverse of [HI-' is to use [HJ'. This can be faulted as neglecting the higher order terms in the expansion of Eq. (26). However, it is this author's experience that such errors as represented by the higher terms of the series are of no appreciable magnitude. Indeed, it is usually of greater concern that [He]-' does not exist, so that the problem of greatest concern is dealing with the singularity. That [Hc]can be singular should be obvious from the eigenanalysis.

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Since the eigenvalues of [H,] are the Fourier transform of the sequence which generates the Toeplitz form, the occurrence of a zero in the Fourier transform is sufficient to cause [H,] to be singular. Dealing with the singularity can be done in several ways. First, it is possible to show by eigenanalysis that filter structures (such as the Wiener filter) can prevent singularities in the inversion of [H,]. Second, there are matrix techniques, such as the pseudoinverse, which are defined for solution of linear equations even when the matrix of the equations is singular. This latter class of techniques has also been applied to digital image restoration (Pratt, 1978). The majority of image-restoration techniques actually implemented and used on real data have been of filter types which prevent singularities in the inversion process. B . Matrix Derivation of Image-Restoration Algorithms

We employ the notation and formulations of the previous section in the description of several image-restoration algorithms. Our discussion herein will be focused on some of the image-restoration techniques which have been developed in the past ten years, in keeping with the stated intent at the beginning of this article to review developments during the past decade. Thus, we shall not review techniques, such as the Wiener filter, which were known during prior years. A major difficulty with image restoration is sensitivity to noise. This is seen explicitly in the matrix formulation of the problem. Since randomness is inherent in all physical data and measurements, we must use a complete model for image deblurring of the form g = [H]f

+n

where n represents noise sources of all types, e.g., thermal noise in electronic components that sample the image, random fluctuations in the device which originally sensed and made a permanent record of the image (such as film), and photon fluctuations in the initial light which created the image. Equation (27) explicitly states that we must solve a linear system of equations in the presence of uncertainties in the data. Such a solution is made difficult, even when [HI is nonsingular, if [HI possesses small eignevalues. A matrix with small eignevalues is said to represent an illconditioned set of equations. The term ill-conditioned refers to the fact that only minor perturbations in the data can appear as very major fluctuations in the solution. There is a simple Fourier domain interpretation of this ill-conditioned behavior. Since eigenanalysis of the Toeplitz form of [HI is approximately equivalent to Fourier analysis of the corresponding

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sequence h(x), the existence of small values of the Fourier transform corresponds to ill-conditioned behavior. Expressing Eq. (22) in the case of noise, we have G e h ) = H(m)Fe(m)+ N(m)

(28)

and inversion of the circulant in Eq. (22) proceeds by the Fourier domain computation: Frest(m)

=

Ge(m)/H(m)= Fe(m) + [N(m)/H(m)I

(29)

The small eignevalues in [HI correspond to an amplification of whatever noise power exists at frequencies associated with small values of H(rn). Various mechanisms for dealing with the ill-conditioned behavior have been proposed and investigated, and the Wiener filter is simply a frequency domain description of one means to accomplish this control. Other means of achieving such control can be developed in the space domain. Ill-conditioned behavior, as noted immediately above, is associated with amplification of any noise that is present. When this occurs, the resulting solution is dominated by high-frequency noise, i.e., the restored image consists of little correlated structure, being mostly uncorrelated noise. Such behavior is associated with the derivative of the solution being very large; that is, the noise which was amplified has many oscillations in a small image region. A real image has a number of smooth regions, as a rule, and even edges have a bound placed upon the derivative at the edge by the bandwidth of the imaging system (Papoulis, 1965). Consequently, one criterion for controlling an ill-conditioned solution is to minimize or constrain some measure of the derivative in the solution. The above discussion suggests the following as an image-restoration procedure : minimize f t[C]t[C$ f

subject to (g - [H]fIt(g - [Hlf)

=

e

(30)

the interpretation of these equations is as follows. Since ill-conditioned behavior leads to large derivatives in the solution, from amplified noise, we will compute a measure sensitive to this behavior and minimize it. The matrix [C] is a matrix which represents constraints on the behavior of the solution, constraints which we will assess for smoothness of the solution to suppress the noise behavior. If, for example, we wish to have our solution smooth in the sense of first derivatives, the resulting first difference

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176

smoothness is invoked from a matrix of the form: -1

1 -1

1

0

1

-1 -1

[CI =

1

-1

0

1

-1

1

We have obviously written [C] in a circulant form, the purpose of which will be immediately evident below. However, minimizing derivative energy and maximizing smoothness is not sufficient by itself, for the minimum of ft[CIt[Clf can be achievied by any constant. So we must place a side constraint which will guarantee that useful picture structure is also present in the restoration. Suppose that we are given a vector fo and wish to test how well it serves as a solution. One measure is to substitute it into the imaging equation and compare it to the original data, i.e., calculate the residual:

(32) Because of noise and the fact (as discussed in the previous section) that the matrix [HI is associated with either an overdetermined or underdetermined system of equations, the residual r will not be identically zero. Indeed, suppose that the actual object f were to be made known, then Eq. (27) shows that = (g - [Hlfo)

r

=

g - [H]fo = n

(33)

for the case where fo = f the actual object. The value of n is random, however, and we cannot hope to know anything about it except on a statistical basis. Since r'r = n'n =

N

2 n:

(34)

i=l

we see that r'r has the property r'r = ( N - 1)(+2,

(35)

where N is the number of points in the vector r, and cr; is the variance of

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the noise process, assuming that the noise is zero mean. Thus, we set

e = (N

- 1)a;

(36)

in Eq. (30). Our intuitive justification is that even in the best of cases, where the actual object f is known, the residual must be related to the variance of the noise; thus, we impose such as a constraint on our minimization problem. Equations (30) are a minimization problem with an equality side constraint. Thus, the Lagrange theorem applies and we can construct the equivalent problem: minimize f

f'[C]'[Clf

+ y(g

-

[H]fIt(g - [ H Y )

(37)

where y must be varied until the residual takes the magnitude e in Eq. (35). The expression in Eq. (37) can now be used to derive the optimum solution. Differentiating with respect to f and setting the result equal to zero, we can derive the result f =

([HITHI + y[ClTCI)-"Hltg

(38)

This expression depends upon the parameter y . It is possible to show that the quantity rtr = (g - [H]f)l(g - [Hlf) varies montonically with y . Thus, it is possible through multiple iterations to establish the value of y which results in satisfying the constraint-the second equation of Eqs. (30). The algorithm is the following: pick a value of y , solve for the corresponding value off, using Eq. (38), and compute the value of the residual rtr, which corresponds to the value of y. If the residual is larger than rtr = (N - l)(r;, decrease y . Otherwise y is increased. The actual magnitude of the increase or decrease can be determined similar to the root-finding procedure for polynomials (Newton's method). Convergence to the proper value of y within five to seven iterations is common (see Hunt, 1973, for details). Expression (38) is intimidating if viewed as a matrix equation. As discussed in Section II,A of this article, the two-dimensional nature of imagery makes the size of the matrices beyond realistic computations. However, it is here that the circulant approximation can become so useful. It is direct, using the results of Section II,A to convert Eq. (38) directly into a Fourier transform expression (Hunt, 1973). The result has the form:

where the (m, n) quantities are the Fourier transforms of all the corresponding quantities in Eq. (38). Thus, the procedure for computing the so-

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178

24 34

40

FIG.2. Original test image.

lution takes place with iterations as discussed above, but with all calculations in the Fourier domain, as per Eq. (39). Figure 2 is an original image sampled as a matrix of size 450 x 450; Fig. 3 is a blurred version of it. Noise has been added, the noise being from a uniform distribution, the ratio of peak signal to RMS noise (SNR) being approximately 14: 1. Figure 4 is the restoration produced by y = 0 in Eq. (39). The image was blurred by a radially symmetric Gaussianshaped point-spread function, with the standard deviation of the pointspread function being 24 samples, so that the Fourier transform H in Eq. (39) also possessed Gaussian behavior. Figure 5 is the result of iterations on y to satisfy the constraint on the noise (the variance of the noise was known, of course, through the SNR). The above discussion illustrates the power and utility of matrix analysis in image restoration. A matrix description of the problem was directly

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FIG.

3.

179

Blurred test image.

converted into a Fourier domain calculation, using the circulant approximation. This was, of course, the motivation for expressing the difference operator [C] as a circulant in Eq. (31). We now demonstrate an even more complex process for image restoration derivable through matrix analysis. The method expressed in Eq. (38) could be categorized as a deterministic algorithm; that is, all quantities were treated as though fixed, but known or unknown, quantities. On the other hand, an equally valid viewpoint, which is the basis of the Weiner filter theory, is to treat the quantities as random. The maximum

LNnH X ‘8

08 I

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This nonlinear model is a suitable description of the process of capturing a blurred image on photographic film (Hunt, 1977).

FIG.5. Restored image, optimum y .

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182

We now assert that there exist probability density functions to describe the variation of the quantities g, f, and n in Eq. (40). Furthermore, we adopt the philosophy of using these probability densities to determine f. As can be learned from a standard text in estimation theory, a variety of estimates are possible (Van Trees, 1968). For example, if we let p(.) symbolize the probability density function of a random variable, then J

is a minimum-mean-square-error (MMSE) estimator, maximum p(f lg) is the maximum a posteriori (MAP) estimator, and maximum p(g If) f

is the maximum likelihood (ML) estimator. The MMSE estimator is extremely difficult to derive for a general nonlinearity, as in Eq. (40). The ML estimator reduces to an ill-conditioned inverse filter in most cases (van Trees, 1968). The estimator of both utility and computable efficiency is the MAP estimator, which we shall derive and show an implementation algorithm. Conceptually, we observe that the MAP estimate is the one which finds the most probable object data f, given the recorded image data. From Bayes' law we know that p(flg> = [P(P If)p(f)I/p(g)

(42) and since the logarithm is monotonic, an equivalent maximization problem is f maximize

In p(f)g) = In p(g)f) + In p(f) - In p ( g )

(43)

Since the maximization is on f, the last term involving p ( g ) can be discarded without affecting the optimization. The problem posed by Eq. (43) cannot be solved without specifying a form for the probability density functions. Since the quantities f and g are vectors, a multivariate density model is required. The most useful and well-known multivariate probability density is the Gaussian. We adopt it for modeling for two reasons: first, a set of tractable equations results, which is pragmatic, second, it is possible to show that images can be made to satisfy a probability model of the form: p(f) = [ ( 2 7 ~ det([Cf])ll'z )~ exp[-i(f

-

f)-'[C,]-'(f

-

TI1

(44)

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where det( ) is the determinant of a matrix. This model assumes Gaussian fluctuations about a mean f , the fluctuations possessing a multivariate covariance matrix [C,]. The utility of the model results from assuming that the meanf is nonstationary and is basically a low-pass version of the image f, whereas the Gaussian fluctuations are a spatially stationary random process. The stationary assumption makes [C,] a Toeplitz matrix, and it can be approximated by a circulant. Surprisingly, virtually any image can be made to fit into a model such as this (Hunt and Cannon, 1976). The power of the assumption is thatf is a nonstationary mean, so that the major portion of the structure information is carried i n f . The random fluctuations about this nonstatinary mean represent individual realizations of the ensemble of the random vector process which is generated by Eq. (44),and these fluctuations are stationary. The meanf is basically a low-pass filtered version of the original image f, and the quantity (f - f ) represents assumed high-frequency stationary Gaussian fluctuations about 1. Using this model, we can directly show (Hunt, 1977) that the MAP estimator is equivalent to maximize -$(g - s([ff]f))TICnl-'(g - s([fflf)) f

-&(f - T)'[C,]-'(f

-T )

(45) where the log transformation has been used to simplify, and constants that do not affect the optimum have been discarded. Two approaches to the optimization in Eq. (45) can be adopted. One method is to differentiate Eq. (45) with respect to f, set the resulting equations to zero, and solve for the f which is the necessary condition. The other approach is to treat Eq. (45) by a direct computational optimization. Since the former approach leads to nonlinear equations, the latter approach has proven more feasible computationally. A number of iteration procedures exist for optimization. If we let +(f) be the function to be optimized, then fk+l

=

fk

- a k v+(fk)

is the conventional gradient or hill-climbing method, where

(46)

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B. R. HUNT

We define +(f) as equal to the expressions in Eq. (45). Then we can directly write fkfl

= fk - ak([HIT[SbI[CnI-'(g - [cfI-'(fk

s([H1fk))

- 7))

where [Sb] is a diagonal matrix of partial derivatives

0 (49)

of the sensor function s(bk),and where bk is defined as bki

=

hijfkj

(50)

j

and is the blurred image associated with the kth estimate of the object. The derivatives are obviously present to correct for the nonlinear sensor distortions. The factor (Yk in Eq. (46) is, in essence, the step size at each step of the hill-climbing iteration. It is shown as a scalar quantity in Eq. (46), but more complex behavior (a matrix [a]) can lead to even more rapid convergence, and optimum algorithms can be constructed that will lead to the optimization of Eq. (45) in only five to ten iterations (Trussell and Hunt, 1979). Figure 6 is an original image, sampled as a matrix of size 128 x 128. Figure 7 shows the same image after undergoing a linear motion blur, in the horizontal direction, and having been recorded through a nonlinear characteristic. Gaussian noise is added, as in Eq. (40), and the ratio of signal- to-noise variance is 200: 1. The nonlinearity of the characteristic is readily visible as the distortion in intensities between comparable regions in Figs. 6 and 7. Figure 8 shows the image created after 60 iterations of the algorithm in Eq. (48). Note that the blur has been removed simultaneously

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FIG.6 . Original image.

with the correction of gray tones to yield the original intensity rendering. Again, we have seen how the matrix formulation of the imaging model leads to a set of equations for the restoration process which are simple to derive and which can be implemented by Fourier calculations when the circulant approximation is applied. A final image-restoration mechanism which we wish to discuss is the random-grain or maximum-entropy method. We assume that is a fixed number, say P , of random units (say, quanta of radiant energy) that can be arranged into an object f. That is, we place quanta into the components of the vector f such that i= 1

Likewise, we assume there will be noise, and noise will be represented as a

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B. R. HUNT

FIG.7. Blurred image (nonlinear sensor).

total of No quanta which must be partitioned over the vector n, i.e., N

C n, = No i=l

From simple combinatorial laws, we know that the number of ways we can assign a fixed number P of identical particles into (fi, fi, f3, . . . ,fN), the components o f f is given by

(53) i=l

Likewise, for assignment of No noise quanta we have (54) i=l

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FIG.8. Restored image, 60 iterations.

Since we assume that the processes between object and noise are statistically independent, we compute the joint set of combinations for both as the product

w = WfW,

(55)

Given Eq. (55) we can pursue a desire to maximize the combinations off and n, acting under the assumption that the resulting object is, in some sense, the most probable one. This will not yield anything meaningful without other constraints, however, because the randomness inherent in the combinatorial laws would only generate random objects. Therefore, we impose side conditions that reflect a recorded image and the associated image-formation process. Thus, we formulate the optimization problem: (56) maximize W, W, fa

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188

subject to g = [H]f

+ n,

l'f = P,

1% = No

(57)

where 1 is a vector of unity at each position. Since Eq. (56) presents no obvious treatment, we now simplify WfWn. The maximum remains the same regardless of a monotonic transformation, so In WfWn = In P! + In No! -

N

N

i=l

i=l

C In&! - C In n i !

We recognize that the terms involving P and No have no effect on the optimization over f and n. We can now use Sterling's approximation In a! = a In a which is most accurate when a is large. (This is likely to be the case for photons used as the quanta and an image taken in all but extremelylow-light conditions.) We can replace the criterion of Eq. (56) by the expression maximize W = -fT In f - nTIn n f,n

(58)

subject to the equality constraints in Eqs. (57). From the meaning of (a In a) in thermodynamics and communication theory, we can see why the problem of Eqs. (57) and (58) is referred to as maximum entropy. As is usual, we can form an unconstrained optimization from the problem of Eqs. (58) and (57). The result is maximize -fT In f - nt In n f,n

+ AT([H]f + n - g)

+ Ml(lTf - P) + M2(lTn- No)

(59)

We define this function o f f and n as W(f, n), equal to Eq. (59). One method of optimization is to differentiate with respect to f, n, the Lagrange multiplier vector X and the two additional multipliers M I and M2,set the resulting equations equal to zero, and try to derive a closed-form expression. This was originally the approach taken by Frieden (1972). However, the resulting equations are highly nonlinear. Frieden developed a Newton-Raphson procedure for solving the equations, but the calculation of the resulting solution is still extremely difficult on images of many samples, e.g., greater than 64 x 64 pixels. An alternative approach is to solve Eq. (59) directly as a maximization problem. For example, the hill-climbing solution would be constructed in

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the following way. Define a vector x that has the components f n x =

A

MI M2

then, clearly, W(f, n) = W(x). We can now solve simultaneously for all variables by Xkfl =

xk

+

(Y

vw(xk)

(60)

where the values of the Lagrangian variables must be solved in a manner such as discussed in conjunction with Eq. (38). In practice, there are too many variables. Consequently, one can adopt fixed values for the Lagrangian parameters, which converts the optimization in Eq. (59) from a Lagrangian problem into one with a multicomponent criterion function. The magnitudes chosen for the various Lagrangian parameters determine the weight given each component with respect to the others. For example, if we let all Lagrangian parameters be unity weight, thereby weighting equally all components of the optimization criterion, we have W(f, n) = - f T In f

-

nTIn n - lT([H]f

+ n - g)

+ (lTf - P ) + (lTn - No) and the optimization described in Eq. (60) takes place by hill-climbing on the vector x=

fn

I

An even more sophisticated technique than hill-combining can be used, of course. Cheng (1981) was able to compute maximum-entropy restorations using a conjugate gradient maximum-entropy procedure. The combination of optimization and matrix descriptions can be extremely useful in the treatment of image-restoration problems, as we believe the above discussions indicate. C. Blind Deconvolution

In all of the discussions above concerning image deblurring, the basic assumption has been that the point-spread function is known, i.e., h(x, y )

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is available. Of course, this assumption is not always valid. For example, consider the case of an individual using an amateur or hobbyist camera to photograph a Presidential motorcade. Something unexpected happens: shots, a flurry of motion, and the photographer keeps taking pictures through it all. Of course, in the excitement and confusion the camera is not precisely focused, and in the motion is not held sufficiently on target to prevent blurring of objects. The result is a set of pictures that are out of focus and motion-blurred. Is this a far-fetched case? No, it is precisely the situation that resulted from the assassination of President John F. Kennedy in Dallas, Texas, November 22, 1963. A number of photographers took pictures of the event, and the images suffered from a variety of degrading phenomena. The nature of the phenomena was not known, needless to say. It is a very important problem to be able to take an image and, using only the image, identify the specific properties of the point-spread function which caused the degradation. This has been called the “blind deconvolution” problem by Stockham (see Stockham et al., 1975). It has not been completely successfully solved in general, but two very important special cases of it have been solved, as we shall discuss below. We assume the same notation as before, i.e., the image, point-spread function, and object are g , h, and f, respectively. We now divide the image into a set of overlapping segments. The size of each segment is not important, as long as each segment is appreciably greater, i.e., an order of magnitude greater, than the physical extent of the point-spread function. (This is not a problem, since the physical extent of the point-spread function can usually be crudely estimated by inspection of the image.) Convolution will be preserved on these segments (except near the edges, which is the motivation for making the segments large with respect to the point-spread function.) Thus, for each image segment, we have h(x, Y ) * A b , Y )

(62) We now take the Fourier transform of these segments and average the squared magnitude of them: gi(x, Y )

We see that the right-hand side of Eq. (63) contains the transform squared magnitude multiplied of the point-spread function times the averaged squared magnitude of the segments. If the transform of the point-spread function contains any characteristic features, then observation of Z IGilz may make clear the nature of H. Stockham et al. (1975) showed this to be the case for the following two important situations.

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(1) Focus blur. Simple geometrical optics arguments indicate that when a lens is out of focus, the image of a point-source object is transformed into a disk of light, the actual shape of the disk being the shape of the limiting pupil in the optical system. For example, if the limiting pupil is circular, the point-spread function is defined as

h(x, y ) = ho

for (xz + y2)”2 5 ro

0 elsewhere (64) where r is the radius of the blur disk and h, is the strength. The transform of this function can be shown to have the following form (Goodman, =

1WS):

H(m, n) = Jl(ror)/ror

for r = (m2+ n2)1’2

(65)

This function has “rings” of zeros surrounding the origin of coordinates, each zero ring corresponding to the position in radial frequency space where Jl(ror)= 0. Therefore, inspection of the average squaredmagnitude transform can determine the existence of a focus blur and its actual extent (Stockham et al., 1975). (2) Motion blur. Again, simple geometry is sufficient to show that if the object and camera focal plane are in relative motion with respect to each other during the time of exposure of the image, the image of a point-source object is stretched into a “streak” in the direction of motion. This can be described by a point-spread function which has a Fourier transform given by the expression

H ( m , n) = sin(dow)/dow

(66)

where w = m cos 8 + n sin 8; 8 is the direction of the motion relative to the horizontal axis of the imagery; and dois the physical extent of the motion blur in the image plane. We see that this function, if observed in the Fourier plane, has “rows” of zeros which lie perpendicular to a*line that passes through the origin at an angle of 8. The rows are all spaced (.rr/do) apart. Again, we see that direct inspection of the Fourier transform averaged-squared magnitude will yield the direction and extent of the point-spread function. The Fourier spectral features of these two point-spread functions are so unique that it is possible to develop a computer program with sufficient “intelligence” to recognize these two cases and determine the appropriate parameters of the blurring (Stockham et al., 1975). More general degradations represent an unsolved problem. It is possible, for example, to make an assumption about the average of X i and use this assumption to infer the behavior of IH$. However, this can determine only the magni-

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tude of H ( m , n) and not the phase. The phase characteristics of a blur are of equal, if not greater, significance than the magnitude characteristics when maximum quality in the restored image is required. At least one method has been investigated for estimation of the phase in the absence of a priori information (Morton and Andrews, 1980); however, it was not a sufficiently unqualified success to warrant extended discussion here. 111. IMAGEDATACOMPRESSION

In terms of qualitative results, image data compression is the opposite of image restoration. That is, when an operation of image restoration is most successful, we see a dramatic result in the deblurred image. However, image data compression has the goal of most accurately approximating the original image in the fewest number of bits of information. A properly working image data-compression system, therefore, will produce an image of little or no visible degradation while significantly reducing the number of bits required to represent the image. Many of the relevant concepts for image data compression were known at the beginning of the decade of the 1970s. The past decade has seen the development and implementation in hardware of these concepts. Therefore, in this section we briefly review the relevant concepts and then indicate what systems have proven to be of greatest interest. A . Correlation /Decorrelation

The successful application of an image data compression system is a demonstration of the existence of significant amounts of correlation in the image data. That images are strongly correlated in the space of the image plane should not be surprising. For example, reflection upon the nature of physical objects indicates that structural integrity of an object is usually associated with spatial correlation in the object image. An automobile, a face, a mountainside, a house, etc., are all objects whose very structural nature ensures that any point in the image of the object will very closely resemble points in a surrounding neighborhood. This could be made untrue only by (literally) painting the object with an uncorrelated twodimensional random field. Since, in cases except camouflage, we do not encounter such, we can expect to use and exploit the neighborhood correlation properties of imagery in the development of image data-compression systems. Indeed, the only thing which will distinguish different kinds of image data-compression systems is the mechanism by which image correlation is exploited.

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If correlation between image points is associated with uncompressed imagery, then it is logical to conclude that image-compression processes will be associated with the decorrelation of the image. There are three principal methods that have been successfully used for image decorrelation: spatial domain prediction, transform domain bit allocation, and hybrid space-transform domain processing. We review the basic workings of each of these methods and then indicate the refinements which have occupied much of the attention in creating working compression systems. B . DPCM Compression The basic architecture is that seen in Fig. 9. The system accepts successive pixels and processes them through the feedback-prediction loop, with the quantizer located in the feed-forward path. In the following we use the z-transform notation to indicate the frequency response properties of this architecture. Since the z transform is associated with temporal sampling and time delays, we are making a tacit assumption that the image is scanned and sampled along a single line and that the sample

?-=-I

Quantizer

FIG.9. (a) Compression loop; (b) reconstruction loop; (c) quantizer noise method.

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number also relates the temporal sampling as well as the spatial position along the scan line of the sample. At the node marked 1 in Fig. 9a we can write the relation

where hj are the weights of a linear predictor whose order is N. Now at node 2 in Fig. 8a we must make a substitutution for the quantizer. We use a conventional model, as seen in Fig. 9c; i.e., we let the quantizer be an additive noise source. Thus at node 2 we can write gk =

dk

+ nk

= fk

-fk

+ nk

(68)

We now recognize that the z transform of Eqs. (67) and (68) is

F(z) = H(z)[G(z)+ F(z)] G(z) = F(z) - &)

+ N(z)

Solving Eq. (69) for F we have

Rz)= [H(z)G(z)/[l- H ( z ) ]

(71)

Substituting this result into Eq. (70) and then solving for G(z), we have G(z) = [ 1 - H(z)l[F(z)+ N(z)l

(72)

The meaning of this can be interpreted directly as a frequency response if we wish to use the conventional equivalence between the discrete Fourier transform and the z transform (Oppenheim and Schafer, 1975). A similar analysis for the reconstruction loop leads to the z transform of the architecture in Fig. 8b as follows:

P(z) = G(z)/[l - h(z)]

(73)

and we see the factor [ l - H(z)]-' in Eq. (72), the output of the compression loop. The result is that the reconstructed image contains only the quantization noise. However, if the quantizer were placed outside the feedback loop, e.g., to the right of node 2, the result would be a term N ( z ) / [ l - H(z)],and this would represent an amplification and integration throughout the image of the quantization noise. The operation of DPCM is easy to understand from the equations and models above. The best prediction that we can make of the current pixel using a linear combination of previous predictions is fed back and subtracted from the current pixel. The difference is quantized and then coded for transmission. This is basically a differentiation process, so the reconstruction is, in essence, an integration in which values are fed back

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through a prediction loop, which is identical to the prediction loop in the compression step, and added to the incoming differences. Obviously, a total of N initial pixels must be sent unquantized to start the reconstruction loop running in synchronism with the compression loop. These N initial pixels are analogous to the initial values for solution of an Nth-order difference equation. The heart of DPCM lies in the predicting and differencing. If a prediction is accurate, i.e., if the image is strongly correlated in the direction along the scan line, then the predicted pixel value will be quite close to the actual pixel value, and the difference dk will be small. Thus, dk can be quantized with fewer bits than fk , but with the same absolute error. This reduction in dynamic range, and the quantization of same, constitutes the data compression that occurs. The prediction and differencing eliminates the low-frequency information in the scan-line direction, and lowfrequency information is strongly correlated. The remaining highfrequency differences typically have a smaller dynamic range and can be quantized with fewer bits. Notice that the method allows the low frequencies to be uniquely reconstructed, which is the function of the prediction loop at the reconstructor. Given this description and discussion of basic DPCM, it is obvious what is required to make an optimum DPCM system. Whatever mechanism we use to improve the prediction process will contribute to making the differences dk small, and thereby allow the degree of data compression to be increased. The primary interest during the past decade, therefore, has been in developing ways to improve the prediction process and/or concurrently improve the quantization of the dynamic range of the differences dk. Principal developments include those listed below. 1 . CoefJicient Optimization /Predictor Order Optimization The simplest possible scheme for the predictor and hj coefficients is to assume that the previous pixel is the best possible predictor of the current pixel, i.e., hj = 1, N = 1. Remarkably good results can be obtained for this choice. However, optimal choices can improve the results. First, the choice of coefficients can be such as to minimize the mean-square prediction error: minimize E h

[

N

(fk

- C hjfk-)'] j= 1

Differentiating with respect to h j , we have Eq. (75):

(74)

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which requires that hi satisfy the set of equations:

(76)

where (77) commonly referred to (by electrical engineers) as the autocorrelation of the sequence fj . Equations (76) give the optimum set of weights, but the order of the predictor N governs the number of weights. The optimum value for N depends on the data. For example, if the data were generated by an Nthorder autoregressive process, then the choice of an Nth-order predictor would be optimum Jenkins and Watts, 1968). Real image data cannot be precisely modeled as an Nth-order autoregressive process, however. The best recourse is to study, for a given class of images being subjected to data compression (such as aerial photography or patient X rays), the mean square prediction error as a function of predictor order N . Surprisingly, the prediction error shows little decrease after only two or three terms, and a third-order predictor (N = 3) is adequate for a wide set of circumstances and images (Pratt, 1978). 2. Coefjcient Adaptation The derivation of the above equations for optimum hi is based on the assumption of stationarity for the image datafj . It is seldom that an image having truly stationary spatial behavior ever presents itself, however, so one means of achieving better prediction is to let the coefficients hj be adaptive to specific regions of the image. This can be done by assuming a correlation function for the imagery (such as first-order Markov) and reestimating the parameters of the correlation function, which are then used in Eq. (76), to resolve for h, in different image regions. An alternative is to simply save the pixel values fj and estimate the Rj coefficients by

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In either case, adaptation cannot be responsive to abrupt changes in the image data because of the necessity to have sufficiently many values of& to reliably estimate the changes in the correlation structure which governs the hj terms.

3 . Difference Quantization The differences dk must be quantized for coding and transmission or storage. Extensive studies have shown that on the average, the probability density of the differences dk can be characterized as a Laplacian function (Pratt, 1978). However, within an image region the width of the Laplacian may change substantially. The optimum quantizer of the differences can be constructed by known principles (Pratt, 1978), but changes in the width of the underlying difference density function mean that a fixed quantizer can introduce errors. The obvious solution is to let the quantizer vary with the image regions, so that the width of the reconstruction levels changes with the image structure. In this way one can attempt to minimize the two most troublesome errors in DPCM: idlingnoise, which is a patterned oscillation at the level of the least significant bit in a smooth or constant image region; and slope overload, which is a blurring of sharp edges caused when the maximum quantizer level is insufficient to integrate the slope at an abrupt edge in the picture DPCM data compression is simple, direct to implement in hardware, and easy to analyze theoretically. Unfortunately, it does not produce outstanding results. A DPCM system can reliably produce a 2 :1 reduction in data, e.g., from 8 bits/pixel to 3 or 4 bits/pixel. But compression below this level is usually associated with unacceptable losses in image quality. A variety of attempts at optimizing and increasing the compression performance of DPCM systems have shown that 2 bits/pixel is the best that can be expected (Pratt, 1978). However, modest compression requirements lead to the choice of DPCM, because of simplicity and ease of implementation. C . Transform Domain Compression

DPCM acts in the spatial domain of the image itself. In transform domain compression the compression of the data takes place in the domain produced by an orthogonal transform of the image. There are two rationales behind the employment of an orthogonal transform domain, both rationales having been verified by extensive experimentation with a variety of images and transforms. The first rationale is that of decorrelation,

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as mentioned at the beginning of this section. Image data is highly correlated in the spatial domain. The natural description of the spatial correlation is the two-dimensional spatial covariance matrix (Andrews and Hunt, 1977). It is a well-known fact that transformation of a data set by the eigenvectors of its covariance matrix accomplishes a “whitening” or decorrelation of the data. This transformation, known as the Karhunen-Loeve transformation, provides a minimum mean square representation of the data for compression purposes (Wintz, 1972). However, the Karhunen- Loeve transformation requires calculation of the covariance matrix and its eigenvectors, and there is no fast or rapid algorithm to implement it. An image of N 2 points requires calculation time proportional to N 4 , and for large image data sets, the computation is prohibitive. The important property of the Karhunen-Loeve transformation, aside from its whitening effects, is that it is an orthogonal transform. A reasonable question, therefore, is: Do there exist orthogonal transforms which posses$ a fast algorithm and which can approximately decorrelate the image data? The answer, fortunately enough, is yes; there are a number of such transforms. Besides the decorrelation behavior of transform domains, the second rationale for transform compression is energy compaction. In the original image domain the wide variety of pixel intensities makes it impossible to economize in the assignment of quantization bits. For example, any pixel may be presumed to be capable of taking on the entire dynamic range of the image, and quantization bits must be allowed accordingly. In the transform domain, however, images exhibit a great deal of regularity, virtually independent of the image source or the orthogonal transform used. It is usually the case that image transforms possess the largest magnitude transform coefficients in a particular region of the transform domain. The greatest number of quantization bits can be assigned to this region, and smaller numbers of bits to regions of lesser magnitude. This, in effect, distributes the bits to the transform regions where they are most efficiently utilized; since the transform uniquely represents the image, this efficient distribution of bits proves to be an efficient means of representing the image itself. In the following paragraphs we review the features which have become commonplace in the most efficient of transform domain compression systems. First, however, we wish to introduce a basic set of notations and terminology to describe transform compression systems. Let the image to be compressed be given asf(j, k). We form the orthogonal transform off(j, k ) by the transform kernel A ( j , k , m , n): F(m, 4 =

k

C A( j,k , m, n ) f ( j , k ) j

(79)

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The quantity F is the transform domain version off. The transform must possess a unique inverse kernel so that

f d i , k) = 2 m

B ( j , k, m , n)F(m, n)

(80)

n

For example, two-dimensional Fourier transforms operate with the kernels i2rr [ N ( j m + kn)] [ + ( j m + kn)1

A(j, k, m, n)

=

B(j, k , m , n )

=-

exp N

-

(81)

Within the transform domain it is necessary to assign or allocate quantization bits to each of the transform coefjicients F(m, n). That is, we assume the existence of a function N,(m, n ) which assigns a unique number of bits to each ( m , n) coefficient of the transform domain. The output of this bit-allocation rule determines how many bits are used to design quantizers for the coefficients. That is, the coefficients are initially computed with far more significant bits than will be retained in the compressed form. The reduction of bit requirements is done by treating the coefficients as random variables, with an underlying probability density function. Quantization of the computed coefficients with the smallest error requires constructing the quantizer levels and breakpoints in accordance with the probability density that governs the coefficients. Given this discussion of the basic steps in a transform domain datacompression process, we now wish to review the state of practice in same. 1 . Transform Selection A great deal of activity went into the study of different orthogonal transforms during the past decade. The activity focused on answering two different questions: which orthogonal transforms could be computed most quickly, and which transform produced the best results in data compression? The former question has been made somewhat moot by the semiconductor revolution. First, the prospects of very large scale integration (VLSI) and pipelined computations offer the opportunity to implement almost all fast transform algorithms at the same data rate. This is so because all fast transforms require computation steps whose total number is dependent upon the number of multiples and adds. Some algorithms, such as the Hadamard transform, require no multiplications (Pratt, 1978), but as VLSI makes less intimidating the prospect of circuits with a number of

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multipliers and adders in parallel, the advantage of the Hadamard diminishes on computational grounds. Finally, pipe lining of the computations makes the data rate of transform hardware more dependent upon the physical circuit technology (e.g., MOS, ECL) than the number of multiplies and adds. The question of which transform to choose for the best performance in compression efficiency has been distilled to a small number of options. At highest compression ratios, e.g., 8 : 1 reduction in bits, the Hadamard transform causes undesirable loss in image quality, because of the discontinuities in the basis functions of the transforms. Transforms possessing “smooth” basis functions, such as the Fourier or slant transform (Pratt, 1978), will yield the best image quality at the lowest bit rates. Thus it is that the consequence of the research in transform-domain selection has been to indicate the overall utility of the first algorithms investigated: the fast Fourier transform and its symmetric version, the cosine transform (Ahmed et al., 1974).

2. Transform Size Ostensibly, the calculation of the orthogonal transform should take place on the entire image, requiring an N X N transform for an N x N image. However, experimentation has shown this to be unnecessary. The usual convention for determining the image quality of an imagecompression processs is to compare the mean square error between the original image f(j,k) and the image f(j, k ) reconstructed from the compressed data: e2 = ZjEk(f(j, k ) - f(j, k))2.The variation of e2with transform size has been studied extensively for a number of different transforms, and the conclusions are uniform for all transforms: above a certain size the mean square error shows little decrease as the transform size decreases. This has led to transform compression by blocks. The standard practice is to break up the image into a set of contiguous, nonoverlapping blocks, say of size P x P where P << N, the image size. The transform of each block is computed, and the compression steps are executed on the transform domain of the blocks. Studies of mean square error versus block size show that errors escalate for blocks smaller than 10 x 10 pixels in compression of “natural” imagery, i.e., images of the sort associated with television or photography. A common choice of block size is 16 x 16 or 32 x 32 pixels. 3. CoefJicient Bit Allocation

Once the transform domain is computed it is necessary to determine how many bits should be assigned to each coefficient. The usual proce-

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dure is to first fix the bit rate desired for the compressed imagery and then use that to calculate the number of bits which must be allocated over the transform block. For example, if a bit rate of B, bits per pixel is desired over the P x P block, then a total of B,P2 bits must be assigned to the block and then distributed among the transform coefficients in optimum fashion. The bit allocations can be made by either a fixed or an adaptive rule. The adaptive case is one in which the highest quality image reconstruction occurs; but it is obviously more complex. Fixed-allocation rules can be constructed in one of two ways. One method is to study the power spectra of a large number of images which are assumed to be typical of the images being subjected to data compression. If a general model of the power spectra of these images suggests itself, for example, first-order Markov, then this model is denoted (Df(m, n). Since it is possible to show that the variance of a transform coefficient is related to the power spectrum of the coefficient, i.e., Var(F(m, nN

=

W W m , n))

(82)

+

where is a function that can be determined (Jenkins and Watts, 1968), then the power spectral model makes it possible to estimate the variance, on the average, of a given transform coefficient. This is important, because for a fixed number of bits of quantization, the mean square quantization error increases directly as the variance of the random variable being quantized increases. Thus, the largest number of bits should be assigned to those coefficients associated with the largest variance, e.g., the largest power spectral values. A power spectral model is a direct route to bit-allocation decisions. The alternative is to compute, from a “typical” set of images, the variance of the transform coefficients and then use these variance values directly in the allocation of bits. Among the bit-allocation rules which have been suggested, one of the simplest is

NB(m, n) = ( N , / P 2 ) + 2 logio[Var(F(m, n>>I ? ” ” n=l m = l

where

N B = B,P2

(84)

is the total number of bits allocated for the P x P block [9]. Experience with this algorithm shows that it will usually be slightly suboptimal, i.e., a few bits more than N B may be allocated or a few bits less than NB allocated. It is best to use this rule in conjunction with a direct inspection of the resulting bit allocations and engage in “fine-tuning” the allocations manually until desired allocations result. This is usually possible for small

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i

FIG. 10. Bit-allocationmatrix.

blocks (16 X 16) when a fixed rule is employed for all the blocks of an image. Figure 10 shows a bit-allocation matrix on a 16 x 16 block, with an average bit rate of 1.5 bits/pixel, using the rule of Eq. (83), a first-order Markov power spectrum, and manual adjustment to achieve the final rate. Note that there are zeros in this bit-allocation matrix. They obviously represent a loss of information since the corresponding transform coefficients are not retained or transmitted. This is typical behavior when bit rates approach 1 bit/pixel in a transform domain compression scheme. The fact that reconstructed image quality is acceptable even with such an information loss indicates the virtue of concentrating bits in the places where the human visual system is less aware of the effects. This is the basis of a very powerful image data-compression scheme developed by Hall (1978), which uses the spatial frequency sensitivity of the human eye to allocate compression bits in such a way as to produce errors in the reconstructed image which are the least visible (and, hence, least objectionable) to the viewer.

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4 . Coefficient Quantization

Once the number of bits has been assigned to each coefficient, the coefficients must be quantized. The probability distribution of each coefficient can be assumed to be a Gaussian distribution, or a complex Gaussian in the case of Fourier transforms. Quantization with minimum mean square error leads to the Max quantizer (Pratt, 1978). Tables exist for this quantizer, and they can be applied to make simple the actual specification of quantizer breakpoints and reconstruction levels (Pratt, 1978). The above discussion outlines the principle steps in transform domain data-compression techniques. We now wish to indicate what the specifics of an optimum compression scheme would be. One of the most successful compression schemes, in terms of compression efficiency and quality of reconstructed image, is the adaptive cosine transform scheme (Chen and Smith, 1977). In this scheme the cosine transform of the image is computed in 16 x 16 blocks using a fast cosine transform algorithm. The bit allocation is done with a rule similar to Eq. (83) above, except that it is computed across all blocks, as well as within a block, and multiple iterations are carried out until the total desired bit rate is achieved. The performance of the algorithm is very good, having been demonstrated to achieve 1 bit/pixel (from 8 bits/pixel source) with little if any degradation of image quality perceived in the reconstructed images.

D . Hybrid Compression The one disadvantage of a transform domain compression scheme is the necessity for a buffer to contain data during the transform process. For example, if the transforms are computed in P x P blocks on an N x N image, then two buffers of size P X N are required, one to receive incoming image data, the other to contain working memory for the computation of the P X P transform blocks. The buffers are “ping-ponged” back and forth, one filling while the other is scratch-memory being emptied. A compression scheme which Combines some of the simplicity of DPCM with the performance of transform is the hybrid transform/DPCM compression. The basic concept is to perform one-dimensional transforms of length N on each image line. Since there is two-dimensional correlation in the image, successively transformed lines will have transform coefficients which are very similar. Rather than directly quantizing these lines, therefore, the remaining line-to-line correlation is removed by subjecting the transform coefficients to DPCM. That is, if we examine thejth transform coefficient in lines k and k + 1, the correlation between these two coefficients is removed by DPCM. Thus, the quantization of the coef-

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ficients takes place in the Laplacian quantizer associated with a DPCM processor, and the number of bits to be used for each quantizer is governed by a bit-allocation rule such as discussed above. Architecturally, one can achieve this by one DPCM processor which is multiplexed among the N transform coefficients, by N DPCM processors acting in parallel, or by a combination thereof. The performance of a hybrid system can be judged as having slightly more image degradation than a wholly transform system, but at superior bit-rate reduction than a wholly DPCM system. Hybrid systems at 1 bit/pixel have shown acceptable image degradation, although higher levels of quality loss than a wholly transform scheme (Habibi, 1974). E . Color and Temporal Compression

Any of the methods for image data compression discussed above would be considered spatial compression techniques in that they eliminate data redundancy in the spatial two-dimensions of the image. They are usually referred to as intraframe techniques, since the computations to accomplish them take place within a single frame. Image data may contain redundancy in dimensions other than the spatial, however. Inspection of the individual red, green, and blue channel signals of a color television indicates the great degree of redundancy that exists between primary colors in a color image. Remote sensing in narrowband wavelengthdiverse windows also indicates the great redundancy between different spectral observations of an object. Another source of redundancy is temporal variation. Except for objects in very rapid motion relative to the frame time, successive images acquired by a motion-picture or television camera show very great redundancy. The thing that links spectral and temporal variations in imagery is the existence of a third dimension in which redundancy exists, a dimension separate and (in some sense) orthogonal to the two space dimensions of the usual redundancy source. Any of the techniques discussed above are applicable to the reduction of this additional redundancy, and we briefly review some of the experience accumulated by experiment. Color-Image Compression

As noted above, color primary components of a color image are strongly correlated, as are the spectral components of images from observation through a set of parallel spectral narrow-wavelength bandpass filters. Since the basic direction of redundancy reduction is one of decor-

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relating the data, the first step is to reduce the correlation between color or multispectral components of the image. We designate the image to be fi(j,k), for i = 1, 2, . . . , N , where N is the number of spectral components. We are seeking a matrix [A] which has the properties:

and where the transformed components Fi(j, k ) have no correlation in color or spectral content. Note they may still possess spatial similarity. But the transformation should, on the average, result in a particular pixel ( j , k ) in image m,F&, k ) , being statistically unrelated to the same pixel ( j , k) in the Ith image, Fl(j, k). Following the construction of the transformation [ A ] and its action upon the image components, we can treat the transformed images F i ( j , k ) as though they were independent images, with only two-dimensional spatial redundancy within each image being subjected to compression by transform, DPCM, or hybrid techniques (Pratt, 1978). The only difficult step, therefore, is the construction of the transformation [A]. For conventional color imagery, particularly imagery obtained from color television, the trnasformation [A] is already known. In the development of color television the necessity to conserve color bandwith and ensure compatibility with black-and-white television led to the development of color-signal decomposition in luminance, chrominance, and hue signals. These are nearly independent statistically, and the resulting Y , Z, Q signals in U.S. television standards are highly suited for the decorrelation of color images. In such a case we have

I);:;[

Q ( j ,k )

=

[

I[ 1

0.299 0.587 0.114 FR(j,k) 0.596 -0.274 -0.322 FG(j,k ) 0.211 -0.523 0.312 F&, k)

where the subscripts R, G, B refer to red, green, and blue primaries. The Y , Z, Q images are then processed in parallel and independently with cosine transform algorithms similar to that discussed above. It has been found experimentally that Y requires the most bits allocated, with Z next, and Q least among bit allocations (Pratt, 1971). Compression of multispectral images is no different, except the matrix [A] must be constructed. For a set of N multispectral channels, we first

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construct the spectral covariance matrix, whose (m,n) element is CA(m, n) =

(fm(j9

k,

-J‘m>
k,

- J‘n)

(87)

.i,k

for m ,n = 1, 2, . . . , N. If we now determine the eigenvectors of [CAI according to the transformation [A] = [@~-‘[CA~[@I

(88)

where [A] is a diagonal matrix of the eigenvalues of [CAIand [@Iis the matrix of eigenvalues, then the matrix [A] we seek in Eq. (85) is just [A] =

[a]-’

(89)

This is often referred to as the “whitening” transformation and is a consequence of the optimal Karhunen-Loeve representation of a stochastic process (Fukunaga, 1972). Following the decorrelation and transform coding, the imagery is transmitted or stored. The imagery is reconstructed in the obvious way, using the inverse of the transform (e.g., inverse cosine) followed by the inverse of the decorrelation transform (i.e., [A]-’). The compression of temporal image sequences could be carried out in a similar way, except there is no transformation [A] that is universal in achieving temporal decorrelation of a set of image framesf,(j, k ) , h ( j ,k), h(j,k), . . . . Consequently, it is most useful for interframe temporal coding to extend the transform processing in two dimensions to a transform process in three dimensions. A 16 x 16 block transform would thus become a 16 x 16 x 16 transform, the third dimension being 16 successive frames in the temporal sequence. Quantization and bit-allocation processes can be pursued as before. Pratt and his colleagues report that a three-dimensional cosine compression can achieve a bit rate as low as 0.25 bits/pixel with acceptable degradation in image quality (Roese et a f . , 1977). The three-dimensional transform approach has a great drawback-the great amounts of memory required to store imagery for a set of P x P x P transforms ( N 2 P total storage locations). More easily implemented is a spatial transform/temporal DPCM technique. Two-dimensional spatial transforms are computed as discussed above. The storage of only one previous frame of transform coefficients allows the computation of DPCM in the temporal dimension, analogous to the hybrid computations within an image frame discussed in Section II1,C above (Lei et al., 1977). However, the lowest bit rate attainable with a DPCM system is 1 bit per transform coefficient, so that bit rates lower than 1 bit/pixel result from some transform coefficients being discarded (allocated zero bits of quantization). Another option is to compute interframe DPCM with another compres-

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sion scheme applied to redundancy within the frames, e.g., frame replenishment (for example, see the paper by Haskell et al., 1977, concerning interframe techniques). In summary, interframe data-compression techniques represent more research that needs to be done than research successfully completed. As the cost of semiconductor memory continues to decrease, the system investment necessary to achieve the storage of frames, which is an integral part of interframe compression, will decrease and we may expect more results of significance. IV. RECONSTRUCTION FROM PROJECTIONS As notable as have been the visible successes of image bandwidth compression and image restoration, there is one area of image processing which could reasonably be said to be the most successful in terms of dollar value of products, human lives affected, and recognition by the general public in the form of the Nobel prize. This is the area of reconstruction from projections, which is the basis for an entirely new method of imaging-computer-aided tomography (CAT) scanning, which is the basis of a substantial market in medical electronics. The basic mechanisms of computer tomography are encompassed in Fig. 11. An object is illuminated by a planar “sheet” of X rays, and the sheet slices the object to produce a projection of all interior twodimensional structure into one-dimension. The sheet or planar source and a suitable detector are now rotated around the object, collecting a set of such one-dimensional projections. The explicit geometry is seen in Fig. 12, which can be thought of as looking down from the top of Fig. 11 directly onto the axis of rotation. We see that the projections make an angle 13 with an arbitrary reference, such as the horizontal. The projection angle 8 is stepped around the object in small increments, e.g., 1” increments, collecting a large number of one-dimensional shadowgram views from the two-dimensional slice of the object. The problem of interest is to take the collection of one-dimensional projections and accurately recon-

FIG. 11. Planar projection.

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B. R. HUNT direction of rotation

I source

FIG. 12. Rotation/projection geometry.

struct the two-dimensional interior which is sliced by the planar source. For notational convenience, we assume that the projections always occur in the direction of the x axis. The basic projection of the slice we define as

where f ( x , y) is the two-dimensional density distribution of the object at the point of the planar slice.' It is obvious that we can determine a projection on any angle 8 with respect to the horizontal by a rotation of the object or the coordinate system. For example, p ( y 18) =

f ( x cos 8 - y sin 8, x sin 8

+ y cos 8) dx

(91)

is the projection obtained from rotating the object counterclockwise through an angle 8 to construct the projection along the direction 8. It is a complete set of projections collected over increments in 8, i.e., p ( y l j A8) f o r j = 1 , 2, 3, . . . , N which must be used to recoverf(x, y). A clue as to why the collection of projections is sufficient to recover f ( x , y) can be obtained from the projection-slice theorem. Consider again the projection at 8 = 0, p ( y ( 0 ) .If we calculate the Fourier transform of this quantity:

We neglect exponential attenuation characteristicsof X-ray transmissionthrough objects, since they do not affect the mathematics of slice reconstruction.

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then clearly we have that PO(^) =

[/I

f ( x , y ) exp[-ia.rr(ux

+ u y ) ] dx dy

(93)

--cc

where the notation in Eq. (93) is to indicate that the two-dimensional transform is calculated for the spatial frequency u = 0. Therefore, the Fourier transform of a projection is equal to the two-dimensional Fourier transform of the original data evaluated along the direction of the projection. It is, in other words, a “slice” through the original two-dimensional Fourier transform, the slice being in the direction 8 of the projection. The theorem is obviously true, regardless of the angle 8, given the description in Eqs. (90) and (91) for projection in any direction in terms of 8. The importance of the theorem is that it demonstrates that a projection preserves information, namely, the information on the slice at angle 8 through the Fourier domain. It should be obvious that if the angular increments A8 are closely enough spaced, then any point in the entire Fourier-transform plane will be either intersected by a slice or lie only a very small distance from a slice. The angular increments A8 could be thought of, in fact, as a sampling of the Fourier domain on a Polar coordinate raster, rather than a conventional x - y coordinate raster. The reconstruction of an image is obviously calculable from the polar Fourier raster, by interpolating from the slices in the Fourier domain to an x - y rectangular grid and using the fast-Fourier-transform algorithm to calculate the inverse that is the function f(x, y ) (Mersereau and Oppenheim, 1974). The preferred reconstruction method, however, is not a Fourier reconstruction but the “filtered back-projection’’ method. The basis of the filtered back-projection method can be seen in Fig. 13. We “backproject” the individual projections, an action which corresponds to converting the one-dimensional projection into a two-dimensional function by “smearing” it in the direction 8. Heuristically, the smearing is the same as convolving the one-dimensional projection with a “sheet” of Dirac impulses which is defined on a line at an angle of 8 with the horizontal, as we see illustrated in Fig. 14. We can describe this mathematically in the following way. The projection p ( y 18) is parallel to the y axis, being a projection along x. The angle between the projection and the x axis horizontal is 8, or measured from the projection to horizontal with the algebraic sign, -8. The impulse sheet is defined as existing on the line y

=

x tan(-@

(94)

which passes through the origin. Using the convolution property of im-

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Region where back projections sum

FIG.13. Back-projectiongeometry.

pulses, which evaluates the function at the point where the argument of the impulse is zero, we have the back-projection fB(X,

Y ) = P(Y - x tan(- 8))

for all values of 8. We must now sum these over 8. For the case where the sum is over an infinite number of infinitesimal increments in angle

which we can simplify by recalling that the argument of the projection is zero, so y - x tan(-8)

=

y

-

x

sin( - 8) = y cos 8 cos( - 0)

+ x sin 8 = 0

FIG.14. Creation of back-projectionsfrom a single projection.

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21 1

and thus

where f is our estimate of the interior of the slice. We can now be motivated to ask: How well doesfrepresent the original slicef(x, y)? This can be determined by letting the original object be a point source, i.e., a Dirac impulse centered at the origin of coordinates. In this case, each projection is also a Dirac impulse, and the estimate f comes from summing in 8 a set of lines or “rays” which are back projected. Thus we have the estimate

with the integration in 8; this is simpler to evaluate in polar coordinates. Using the conventional transformations, x = r cos 8 and y = r sin 8, we have

A well-known property of the Dirac impulse function (Papoulis, 1961, p. 274), i.e., (98)

6(at) = la(-’ 6 ( t ) allows the integral to be written as

The Dirac function is defined and has a value only at the point where argument is zero, and for such we have J?,S(x) dx = 1, which has a nonzero contribution to the integral only in the neighborhood of x = 0, i.e., Jrm6(x) dx = J?: 6(x) dx = 1 for E as arbitrarily small as desired. Examining Eq. (99),we see that the argument of the Dirac is equal to , Since the projection zero at three different values of 8, i.e., 8 = 0 , ~ / 2T. at T contains exactly the same information as the projection at 8 = 0, we can change the upper limit of integration to 7 ~ - .There are then two zeros and we can evaluate the integral in terms of the impulse contributions at those two zeros. Thus

1;-

a(sin(28)) de

=

2

(loo)

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The important behavior is the variation as since the Dirac-function integral evaluation only constitutes a scale-factor change in this variation. Equation (101) gives the point-spread function in polar coordinates of the process of back-projection and summation of back-projections. The point source is obviously reconstructed in a blurred way, since the function 1rI-l has appreciable width. Recognition of this blurring, which is inherent in the summation of back-projections is the basis for the use of the jiftered back-projections. The concept is direct: before doing the back-projection, filter each projection by a transfer function whose frequency response is the inverse to the Fourier response of the point-spread function 1rI-l. To derive this, we recognize that the polar coordinate form shown in Eq. (101) has no variation with 8, i.e., it possesses rotational symmetry. The Hankel transform is the appropriate transform relation, and we use the Hankel transform pair (Bracewell, 1978):

lrl-l&bll

(102)

where p is the radial frequency variable corresponding to r. It is now obvious what processing of back-projections is required, namely, the multiplication of the Fourier transform of each back-projection by a frequency characteristic of lpl, i.e., a differentiator. The filtering caused by this characteristic corrects for the point-spread function of the backprojection process. Realistic algorithms implementing the above processes are common in the hardware which is manufactured to reconstruct the planar cross section of objects that have been X rayed in projection. Important features of such algorithms typically include the following. (1) A selection of projections from 0 to 7r is taken, the projections from 7i- to 27r being redundant. The projections are taken in sampled form, for direct processing in a high-speed array computer within the tomography machine. As many as 500 samples per projection can be reoutinely processed. The number of projections has a direct bearing on image quality (as can be realized from the projection-slice theorem). Increments of 1" (180 projections) are not uncommon. (2) The projections are filtered prior to back-projection and summation. The filtering can be done either by calculating the discrete Fourier transform (DFT) of each projection, using the fast Fourier transform (FFT) algorithm and multiplying by [ P I , or by implementing a direct convolution with a filter whose space-domain impulse response is the inverse

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DFT of the filter characteristic IpI. The choice of transform or convolution is made on the basis of efficiency. If no special convolution hardware is available, the FFT algorithm yields greater efficiency. The availability of special-purpose convolution hardware can make the direct convolution filtering just as rapid as Fourier methods. (3) After the filtering, the back-projections are calculated by evaluating the back-projection p ( y cos I3 + x sin 0) at the x - y locations on a rectangular grid, for each value of 8. Finally, the back-projections at each grid point are summed to create the final image. The number of grid points is a variable, but obvious constraints apply to the resolution and spacing and the x - y grid and the resolution and spacing of the original projections. In particular, the collection of samples on each projection at the spacing required by the Nyquist relation to prevent aliasing, constrains the sample spacing in the x - y reconstruction grid (Mersereau and Oppenheim, 1974). The above discussion has been made in the context of parallel projections. The creation of a planar source with parallel beams is not simple. A much simpler process is to use planar collimation on a point source, which leads to a fan-beam projection, as in Fig. 15. It should be obvious that every fan-beam projection ray is equivalent to a projection from a parallel beam system located at a different projection angle. The recon-

FIG. 15. Fan-beam geometry.

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FIG. 16. Computer tomography image through a human head.

struction of image cross sections from fan-beam data is equivalent, therefore, to sorting appropriately the projection samples until all the samples in a single projection appear as though they were obtained from a parallel beam system at constant angle of projection (Peters and Lewitt, 1977; Wang, 1977). As intuition would suggest, from inspection of Fig. 15, the re-ordering is an approximation, and is most valid when the fan angle is small.2 For large fan angles, the filtered back-projection method can be used without any reordering. However, a new filter transfer function is required, and the projections must be modified for the projection fan (Kak, 1979). These modifications to the parallel-beam case do not represent any significant additional computations. The majority of computer tomography systems now being built use fan-beam geometry, as a consequence, because of the lower cost associated with generation and collimation of the X-ray source. In the limit, as the fan angle approaches zero, i.e., point source at infinity, no reordering is necessary at all.

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The wealth of literature in computed tomography is rich, as can be discerned most dramatically from the papers referenced above. Figure 16 is a computed tomography image, representing a “slice” through the head of a human, the slice being taken at eye level so that the eyes and sinus cavities in the head are visible. Finally, we note that even though our interest has been digital image processing in this article, tomography can be achieved by analog computations (Banett and Swindell, 1977). V. STEPSTOWARD IMAGE ANALYSIS/~OMPUTER VISION

As noteworthy as the successes discussed above have been, there has been one area where feverish effort has not brought forth results at a rate which would be desired by the research workers involved. It is the area which we wish to refer to as image analysis, and which has also been characterized by others as computer vision or image understanding (Barrow and Tenenbaum, 1981). The stated objective of efforts in this research area can be posed in terms of human capabilities. Image analysis/computer vision would be successful if a given human task in the exploitation or utilization of visual imagery could be carried out by a machine, the machine being totally competent to replace the human being. This is obviously a very broad objective, and it encompasses a variety of topics known by other research labels, e.g., pattern recognition or artificial intelligence. In pondering an approach to this broad problem, one is always confronted by what might be referred to as the “two-legged existence theorem,” i.e., we know that such is possible because people do it. Should we, therefore, seek to develop machine vision systems on the basis of human information processing, a natural response to the “twolegged existence theorem”? This is not necessarily a rewarding approach. First, we clearly know very little about the basics of human information processing at the detailed level which would be necessary in computing system design. Second, even if we did know the detailed level, there is no guarantee that we could implement it in a design, given the great differences between the biological vision computer (the brain) and the computer we can build from nonbiological hardware. Undaunted by this dilemma, work has been going on in trying to implement systems that analyze imagery with some measure of human-like skill and autonomy. In general, we can state the following. If the domain of the problem is restrictively confined and can be precisely defined, it is encouraging to believe that a computer-based system can be constructed to accomplish the desired task. As the problem domain expands, or the

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problem definition grows imprecise, the computer image analysis becomes more difficult. The failing of computer image analysis is associated with robustness. An analysis of an image by a human being can automatically draw upon the human ability to function in imperfectly defined environments, environments where the human virtues of logic, intuition, and recollection of past experience prove to be of greatest importance. An example of the success for an appropriately constrained and defined problem can be seen in the analysis of certain medical X rays. The X ray of a human being can be described in terms of great amounts of a priori information, and individual human beings, therefore, represent specific instances and variations on the a priori information. Medical pathology, as imaged in an X ray, is a specific variation which must be recognized and classified into a diagnostic category. A successful example of this is the diagnosis of rheumatic heart disease from chest X rays of children (Kruger et al., 1972) and the recognition of black-lung disease from chest X rays of coal miners (Kruger et al., 1974). In both of these cases the problem of interest was specifically well defined, so that variations in the structure of each image were sufficiently small to make success possible. Indeed, by the measure of the “two-legged existence theorem,” the system reported in Kruger er al. (1974) was accurate enough to pass the United States standards for certification as a reader of X rays for purposes of compensating coal miners under the U.S. Occupational Safety and Health Act. (This raises the interesting question as to whether or not passing the certification standards is a status reserved for human beings or can be granted to machines!) The success of these two X-ray image-analysis systems should not lead to false optimism, however. Within the biomedical image-processing arena, a variety of problems have been tackled with varying amounts of success. For example, cell cytology represents an area where tremendous amounts of expense in manpower could be reduced or eliminated by systems capable of analyzing microscope slides of biological tissue, typically arising from medical procedures such as Pap smears, biopsies of expected tumors, blood analysis, and culturing of bacteria for identification of infections. In spite of quite substantial investments and voluminous computer programs, success in this area is fragmentary and restricted to certain problems, which, again, are appropriately defined and constrained. Industrial quality control represents an area where automatic image analysis is currently thought to offer some of the greatest potential. A variety of products go through a final quality-control stage which is visual. That is, a human being views the product for defects which will be detectable by visual means. Typical examples are products such as

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integrated-circuit chips; fabrics with imprinted patterns; the filament wires in light bulbs; the finish on a painted automobile fender; and the integrity of a welded joint. What makes the industrial quality-control problem so attractive is that such situations can be characterized by the existence of a reference image, i.e., an image of how an object from the production line should appear if there are no defects of the sort detected by visual means. The image of the production-line object can be compared with the reference and any differences flagged and interpreted for their significance as product defects. The only truly complex parts of the problem are the description of the number of defects and how they arise, and the development of the process for carrying out the comparison between the image of the production-line object and the reference image. Often, a straightforwad registration and differencing is sufficient. The application of such image-analysis techniques to a variety of integratedcircuit defects has been demonstrated, and appears capable, with suitable extension and development, of carrying out virtually any defect identification currently implemented by human means (Fu, 1981). A major point for all consideration of automated image-analysis techniques is the question of costs. In most areas of digital image processing, the technique under consideration (such as image debluning, bandwidth compression, or computed tomography), represents a unique capability that cannot be achieved by a human being alone. However, image analysis is currently done by human beings, and the cost of the “human system” can be directly compared to the cost of the “machine system” in a total life-cycle cost sense. Clearly, replacing a human being in the analysis of an image makes no sense economically if the computer life-cycle cost, when reduced to the cost per image, is greater than the per image cost of the human viewer. Or even if the machine cost is less on a per image basis, what if the human viewer becomes permanently unemployed by the machine? How are the economic costs of unemployment (wage compensation, lost revenue in taxes and wages), not to mention the social costs, factored into the cost calculations and comparisons? As formidable as the implementation of general-purpose computer vision systems might seem, it is the opinion of this author that dealing with the social and economic consequences of same could turn out to be even more formidable. There is a variety of research activity in image analysis/computer vision. It is not possible to summarize even briefly the activity in this field; the Additional References should be consulted for reference. As stated above, true successes in the general problem are not at hand. A recent review of the area by a panel of research workers has indicated where the short-fall lies between current capabilities and what is desired. The imageanaIysis/computer-vision problem can be broken down into a number of

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Collections of Names, Functions, etc.

FIG. 17. The image-analysis/computer-visionproblem: KS, knowledge sources.

specific steps. Figure 17 represents a diagrammatic rendering of these steps. The specific functions which are present in the paradigm are as follows : ( 1 ) Restoration. This is the process of employing models of the sensor and image-formation process so as to invert or remove any deleterious effects of image formation and provide an image that corresponds as closely as possible to the original object radiance distribution. ( 2 ) Primitive segmentation. Understanding of the image requires that different entities-segments-be recognized as existing in the scene. This problem has received much attention in research. The basic methods popular in research rely upon the computation of primitive properties of the image, e.g., edges, gradients, and color clusters, and to use similarity measures (or dissimilarity measures) to group sets of pixels together into contiguous segments. ( 3 ) Texture grouping. Conceptually a part of the wider class of functions we referred to as segmentation, texture is by itself such a rich and varied key to image segments as to merit a separate function block. It is also one of the most difficult problems; textures that are subtly different, yet easily distinguished by the human viewer, are not easily distinguished by machine. (4) Three dimensional shape, form, reflectance. As noted previously, the image is a mapping from the three-dimensional world to the two-dimensional scene. One of the most important tasks, therefore, is to recover intrinsic scene characteristics. The first level of this recovery problem is the three-dimensional shape and form of an object. Reflec-

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tance and reflectance discontinuities are also an important part of the shape and form recovery process. ( 5 ) Completion of full recovery. When completed, not only shape, form, and reflectance are understood, but also the host of other cues which a vision system can use: stereo depth/distance, object orientation, relative location of objects with respect to each other, etc. (6) Names, functions. Once object recovery is complete, the labeling of objects can take place (assuming that recovery not only distinguishes objects, but furnishes the information for unique names or functions to be associated with the distinguished objects). (7) Collections of names, functions. Recognizing that vision can fruitfully organize objects into hierarchies, the names and functions can also be so organized. The paradigm shown in Fig. 17 represents the product of an interdisciplinary team concerned with developing the best new tools for automating image analysis (Hunt, 1981). As such, it represents a new look at the information processing or computational processes of image analysis. Actual system inplementation based on this paradigm must be carried out before its real value will be known. The ‘‘two-legged existence theorem” continues to inspire researchers in image analysis, and the difficulty of achieving comparable behavior in a general-purpose machine continues to plague them, as well. REFERENCES Ahmed, N., Natarazan, T., and Rao, K. R. (1974). On image processing and a discrete cosine transform. IEEE Trans. Comput. C-23, 90-93. Andrews, H., and Hunt, B. (1977). “Digital Image Restoration.” Prentice-Hall, Englewood Cliffs, New Jersey. Barrett, H., and Swindell, W. Analog reconstruction methods for transaxial tomography. Proc. IEEE 65, 89-107. Barrow, H., and Tenenbaum M., (1981). Computational vision. Proc. IEEE 69, 572-5%. Bracewell, R. (1978). “The Fourier Transform and its Applications,” 2nd ed. McGraw-Hill, New York. Chen, W., and Smith, C. (1977). Adaptive coding of color and monochrome images. IEEE Trans. Commun. COM-25, 1285- 1292. Cheng, Y. (1981). “Optimization of Maximum Entropy Equations for Image Restoration,” Rep. No. SIE-DIAL 81-008. Digital Image Anal. Lab., University of Arizona, Tucson. Ekstrom, M. (1972). Numerical restoration of random images. Ph.D. Dissertation, Dept. of Electrical Engineering, University of California, Davis. Frieden, B. R. (1972). Restoring with maximum likelihood and maximum entropy. J . Opt. SOC.Am. 62, 511-518. Fu, K . S. (1981). Dept. of Electrical Engineering Purdue University, Lafayette, Indiana @rivate communication).

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Fukunaga, K. (1972). “Introduction to Statistical Pattern Recognition.” Academic Press, New York. Goodman, J. W. (1%8). “Introduction to Fourier Optics.” McGraw-Hill, New York. Habibi, A. (1974). Hybrid coding of pictorial data. IEEE Trans. Commun. COM-22, 6 14-624. Hall, C. (1968). “Digital Color Image Compression in a Perceptual Space,” Rep. No. 790. USC Image Process. Inst., Los Angeles, California. Haskell, B. G. (1977). A survey of interframe compression techniques. IEEE Trans. Commun., Spec. Issue Image Bandwidth Compression, 1977. Hunt, B. (1971a). A matrix theory proof of the discrete convolution theorem. IEEE Trans. Audio, Speech, Signal Process. AU-19, 285-289. Hunt, B. (1971b). Spectral effects in the use of Newton-Cotes approximations for computing discrete Fourier transforms. IEEE Trans. Comput. C-20, 942-943. Hunt, B. R. (1973). The application of constrained least-squares estimation to image restoration by digital computer IEEE Trans. Comput. C-22, 805-812. Hunt, B. R. (1977). Bayesian methods in nonlinear digital image restoration. IEEE Trans. Comput. C26, 219-229. Hunt, B. R. (1981). Univ. of Arizona Report on NSF Grant: “Automation of image processing.” Hunt, B., and Breedlove, J. (1975). Scan and display considerations in processing images by digital computer. IEEE Trans Comput. C-24, 848-853. Hunt, B. R., and Cannon, T. M. (1976). Nonstationary assumptions for Gaussian models of images. IEEE Trans. S y s t . , Mun, Cybernet. SMC-6, 876-881. Jenkins, G., and Watts, D. (1%8). “Spectral Analysis and Its Applications.” Holden-Day, San Francisco, California. Kak, A. (1979). Computerized tomography with x-ray, emission, and ultrasound sources. Proc. IEEE 67, 1245-1272. Kruger, R. P., Townes, J. R., Hall, D., Dwyer, S., and Lodwick, G. (1972). Automated radiographic diagnosis via feature extraction of cardic size and shape descriptors. IEEE Trans. Bio-Med. Eng. BME-19, 174-186. Kruger, R. P., Thompson, W., and Turner, A. F. (1974). Computer diagnosis of pneumoconiosis. IEEE Trans. Sysf., Man, Cybernet. SMC-10, 40-49. Lei, T., Scheinberg, N., and Schilling, D. (1977). Adaptive delta modulation systems for video encoding. IEEE Trans. Commun. COM-25, 1302- 1314. Mees, Q . (1954). “Theory of the Photographic Process.” Macmillan, New York. Mersereau, R., and Oppenheim, A. (1974). Digital reconstruction of multidimensional signals from their projections. Proc. IEEE 62, 1319-1338. Morton, J., and Andrews, H. C. (1980). An a-posteriori technique of image restoration. J . Opt. Soc. A m . 70,981-987. Oppenheim, A., and Schafer, R. (1975). “Digital Signal Processing.” Prentice-Hall, Englewood Cliffs, New Jersey. Papoulis, A. (l%l). “The Fourier Integral and its Applications.” McGraw-Hill, New York. Papoulis, A. (1%5). “Probability, Random Variables, and Stochastic Processes.” McGraw-Hill, New York. Peters, T., and Lewitt, R. (1977). Computed tomography with fan-beam geometry. J. Comput. Assist. Tomogr. 1, 429-439. Pratt, W. (1971). Spatial transform coding of color images. IEEE Trans. Cornmun. COM-19,980-982. Pratt, W. (1978). “Digital Image Processing.” Wiley, New York. Ralston, A. (1%5). “Introduction to Numerical Analysis.” McGraw-Hill, New York.

DIGITAL IMAGE PROCESSING

22 1

Roese, J., Pratt, W., and Robinson, G. (1977). Interframe cosine transform image coding. IEEE Trans. Commun. COM-25,592-599. Sondhi, M. (1972). Image restoration: The removal of spatially invariant degradations. Proc. IEEE 60, 842-852. Stockham, T. G., Cannon, T. M., and Ingebretsen, R. B. (1975). Blind deconvolution through digital signal processing. Proc. ZEEE 63, 678-692. Trussell, H. J., and Hunt, B. R. (1979). Improved method of maximum a-posteriori image restoration. IEEE Trans. Compui. C-28, 57-62. Tsui, E., and Budinger, T. (1979). A stochastic filter for transverse section reconstruction. IEEE Trans. Nucl. Sci. NS-26,2687-2690. Van Trees, H. L. (1%8). “Detection, Estimation, and Modulation Theory,” Vol. 1 . Wiley, New York. Wang, L. (1977). Cross-section reconstruction with a fan-beam scanning geometry. IEEE Trans. Comput. C-26, 264-268. Wintz, P. (1972). Transform picture coding. Proc. IEEE 60, 809-820.

Additional References In addition to the references cited within the body of this article, the following is a set of books, papers, and journals which the reader is recommended to consult for further detail. Books

Castleman, K. (1980). “Digital Image Processing.” Prentice-Hall, Englewood Cliffs, New Jersey. Duda, R., and Hart, P. (1973). “Pattern Classification and Scene Analysis.” Wiley, New York. Gonzalez, R., and Wintz, P. (1978). “Digital Image Processing.” Addison-Wesley, Reading, Massachusetts. Hall, E. (1979). “Computer Image Processing and Recognition.” Academic Press, New York. Rosenfeld, A., and Kak, A. (1981). “Digital Picture Processing,” 2nd ed. Academic Press, New York. Journals Compuier Graphics and Image Processing. Academic Press, New York. IEEE Transactions on Paiiern Analysis and Machine Intelligence.

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ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS, VOL. 60

Recent Trends in Photomultipliers for Nuclear Physics J. P. BOUTOT Photoelectronic Devicc. Deportment Lahorutoires d'Electroniqlte et de Physique AppliqrnPe Limeil-Brlvannes. France

J. NUSSLI Photomultiplier Tube Development Lnhoratory Hyperelec S . A . Britv, France

D. VALLAT Electro-Optical Device Marketing Depurtment RTC -La Rudiotechnique Compelec Paris, Frunce

1. Introduction. . . . .

........................................

11. State of the Art o

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A. Photocathodes.. . . . . . . . . . . . . . . . . . .

111. Present Situation on Main Photomultiplier Characterist' A. PMT Timing Performance.. .....................

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. . . . . . . . . . 262 ....................... 277 ........................... 287

References

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I. INTRODUCTION Photomultipliers have been used for more than 40 years in nuclear physics. During that period, their performance has been continually im223 Copyright 0 1983 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0- 12-014660-6

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proved. The introduction of solid-state detectors in the early 1960s did not bring serious competition except in those fields of nuclear physics where the highest level of energy resolution was needed. The reason why the photomultiplier was not superseded by the solid-state detector can be attributed to its three main characteristics: large sensitive area, good timing characteristics, high gain combined with low noise factor in the multiplier. The second and third characteristics signify that photomultipliers have an exceptionally high-gain- bandwidth product. It should also be noted that most inorganic scintillators have a higher stopping power for gamma rays than has silicon. This also brings cost advantages. The aim of this article is to describe the present-day state of the art of photomultiplier technology, particularly concerning photocathodes, secondary emission, new structures. In addition, information is also provided on important characteristics of these tubes, such as timing, pulse linearity, stability, dynamic range, reliability. Previous articles published in Advances in Electronics and Electron Physics by Poultney (1972) entitled “Single Photon Detection and Timing” and by Seib and Auckerman (1973), “Photodetectors for the 0.1-1 pm Spectral Region” were not entirely devoted to photomultipliers, although there were large sections dealing with them. The present article has been limited to the use of photomultipliers in nuclear physics, covering mainly the high-energy physics and the medical markets. The first requires wide dynamic range and good timing as the main characteristics, whereas the second requires energy resolution. Both markets have benefited during the past ten years by significant price reductions in photomultipliers. This article does not cover photomultipliers used in industrial markets, particularly tubes having photocathodes sensitive in the red region of the spectrum.

11. STATEOF

THE

ART OF PHOTOMULTIPLIER TECHNOLOGY

In nuclear physics, the use of photomultipliers (PMTs) is almost exclusively restricted to applications where ionizing radiations produce light pulses during their partial or total absorption in an optically transpar-

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225

ent medium called a scintillator. The fundamentals of scintillation counting and the properties and applications of scintillators have been described by Birks (1967) in an excellent book which is still the basic document on the subject. In a scintillator, the absorption length generally exceeds centimeters or even meters except for very low energy applications. The scintillating volume can therefore be cubic centimeters, cubic decimeters, or even more. The light emitted in this volume is collected through one or more output faces by one or more photomultipliers. This light reaches the output face within a solid angle of up to 27r steradians. Therefore the maximum quantity of light will only be captured if the total photocathode area is at least equivalent to the area of one face, and if the light can reach the photomultiplier cathode without an intervening air gap (or vacuum gap). The only practical solution is to have a semitransparent cathode deposited on the inner surface of the entrance window which forms the front face of the tube, this face being coupled (with grease, for example) to the scintillator. In some applications, it is not possible to place the tube close to the scintillator; for example, if there is a high magnetic field environment. It is then necessary to transmit the light through a light guide (generally made of plastic). The light at the output of the light guide has characteristics similar to those at the input. The required sensitive area is usually still large (square centimeters or square decimeters) because a reduction of area in the light pipe would result in reduction of the light level. Scintillator- PMT assemblies, or scintillation counters, are particularly suitable for measurements on the ionizing radiation such as: occurrence frequency (counting), energy (energy spectrometry), time of occurrence (timing), and location in space. The performance of a scintillator-PMT assembly is defined by the characteristics of each component, both of which have seen developments and innovations during the past ten years. Although it is not strictly the subject of this article, a short survey of recent trends in scintillating materials is given in Section II,A. The developments carried out during that period on PMTs of conventional structure were centered on improving the detection efficiency. These include the photocathode sensitivity over the operating wavelength range (see Section II,A), the electron collection efficiency in the input optics system and the first multiplication stages (see Section II,C), and the secondary emission power of the dynodes (see Section 11,B). It is related to the growing demand for PMTs to meet a high-energy resolution requirement in spectrometry measurements in nuclear medicine instrumentation. Research and development (R and D) studies on PMTs have also been

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aimed at improving other important characteristics such as the timing properties and at introducing new detection capabilities such as high immunity to magnetic fields and a position sensitivity. The use of conventional PMT structures in these applications is reviewed in Section II,C, while the innovations in PMT design are described and evaluated in Section I1,D. A . Photocathodes

The alkali antimonides are still the only photocathode materials which present a high quantum yield over a wide spectral range and which can also be produced economically in large area. Their spectral range extends from the cutoff wavelength (150-350 nm) of the material used for the window o r the scintillator itself to 650 or 800 nm depending on the type of cathode. For many years, photocathodes using 111-V compounds have been extensively studied and may eventually become a practical possibility.

I . Scintillators Scintillating materials have been developed to take advantage of the high blue sensitivity of alkali antimonide photocathodes, and so it is not a coincidence that most of scintillators have a blue emission (similarly, new scintillators are being developed to cover the spectral range of silicon solid-state photodiodes). A comprehensive survey of scintillators is beyond the scope of this article, but the main families are summarized here. a . Inorganic Scintillators. These are mostly crystals but also include some glasses. The most common is NaI(T1) with an emission ranging from 300 to 500 nm (Fig. 1). Heath et ul. (1979) have given a comprehensive and up-to-date account of the techniques used in the production and operation of NaI(T1) detectors and have reviewed their applications particularly in high-energy physics. There is a recent evaluation of some inorganic scintillation detectors for gamma-ray logging applications by Stromswold (1981). Information useful in selecting a scintillation detector for gammaor X-ray, neutron or charged-particle measurements can be found in the scintillator manufacturer's catalog (e.g., Harshaw Chemical Company,' Bicron Corp.,2 Quartz et Slice3). Radiation Detector Catalog, The Harshaw Chemical Company, Crystal and Electronics Products, 6801 Cochran Road, Solon, Ohio 44139. * Bicron Corporation, Scintillation Detector Catalog, 12345 Kinsman Road, Newbury, Ohio 44065. Quartz et Silice, Les Miroirs, Cedex 27, 92096 Paris-La Defense, France.

227

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0

300

FIG.1. shifter.

I

400

I

500 Wavelength (nrn)

I

600

700

Relative spectral emission characteristics of some scintillators and the BBQ

h . Orgcinic Scintillators. These are plastics o r liquids in a very large variety of forms with different compromises of timing, efficiency, and cost. The emission of these is also in the blue range. Development work and performance measurements carried out on this type of scintillator during the past 15 years have been reported by Brooks (1979). The scintillator catalog from Nuclear Enterprises Ltd4 (the most important manufacturer of such materials) gives valuable information on plastic scintillators and on light pipes. In addition, a team of physicists from C E N Saclay (France) have reported on developments of new low-cost acrylic scintillators (Aurouet et al., 1980; Bourdinaud and Thevenin, 1981). c. C?‘erenkov Radiators. These are not true scintillators, as the light is not created by absorption phenomena but by the shock wave produced when the speed of the charged particles exceeds the velocity of light in the medium. The important fact is that the emission spectrum of this light has the form which means that its maximum wavelength is just above the cutoff wavelength of the window or the medium. Depending on the material, this cutoff wavelength ranges from 350 to 200 nm (fused silica). Aerogel radiators are of this type (Carlson and Poulet, 1979). d . Recent Research. These follow two main directions.

( I ) For large-size organic scintillators, the BBQ shifter-bar collection technique makes use of a shift of the light wavelength to the 450-600 nm range in which light guides have a larger attenuation length (Barish et al., 1978; Berlman, 1971) (see Fig. 1). This technique permits a reduction of the number of PMTs because many scintillators can be read by each bar. Shifters to even longer wavelengths have also been investigated (Eckardt et a/., 1978; Franks e r al., 1978). “Plastic Scintillators, Light Guides, and cerenkov Detectors,” Catalog, Nuclear Enterprises Ltd, Slighthill, Edinburgh EHI 1 4EY, Scotland.

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(2) For medical and similar applications such as computer-assisted tomography (CAT) scanners, investigations to find scintillators with higher stopping power led Harshaw to develop BiGeO (bismuth germanate) commonly called BGO, with an emission spectrum from 400 to 600 nm, similar to the BBQ spectrum (Fig. 1) (Farukhi, 1978).

The search for scintillators with a stopping power higher than plastics, high efficiency, and faster than NaI(T1) (0.2 psec) has led to the study of new types of crystals such as CsF (Moszynski et al., 1981; Mullani et al., 1980) and pure NaI (Persyk et a/., 1980). In each case, the increased speed is associated with a shift to the UV range (down to 250 nm) of the emitted light spectrum. 2. Dejinitions and Units

A photocathode is characterized mainly by its spectral response as a function of wavelength. The radiant sensitivity Sk,h is related to the quantum efficiency p by the formula

Luminous sensitivity is defined as the current measured when 1 lm of light emitted by a tungsten-filament lamp operated at 2856 K color temperature reaches the cathode. The justification of the use of such a unit in nuclear physics applications seems laborious. The main advantage is that it is easily reproducible as an industrial measurement since the spectral response itself cannot be measured on all tubes for practical reasons. It is clear that such a source emits little in the blue range and so does not result in a reliable figure of merit. A much better figure of merit is obtained if a suitable filter is placed in the path of the same light, for example, a Corning CS 5.58 (half-stock thickness) filter, with a transmission curve peaking at 400 nm. The light transmitted has a spectral distribution similar to the spectrum of NaI(T1). If in this case, 1 Im (2856 K) impinges on the filter and the transmitted light reaches the cathode, a sensitivity can be defined, measured in pA/lm F (F for “filtered”) and called the “Corning blue sensitivity.” More generally, this is a way of characterizing the sensitivity integrated over a given range. It is possible to define as many “pA/lm F” as there are sources. This emphasized the irrationality of such a definition: for example, the BBQ spectrum can be simulated by a combination of filters (Corning CS 4.72 and Schott G G 455). With such filters, a S 1 1 cathode would exhibit a sensitivity of approximately 15 pA/lm F.

RECENT TRENDS IN PHOTOMULTIPLIERS

229

It has been proposed that the concept of quantum efficiency be extended to an integrated quantum efficiency (IQE) by the definition

where n,(A) is the distribution of photon density and p(A) the quantum efficiency versus wavelength. In practice, this IQE has to be measured by the method described above; a proportionality factor has to be determined experimentally by measuring the complete spectral response of a certain number of samples and calculating the IQE. The usual way of evaluating the performance of a PMT for spectrometry applications is the measurement of the energy resolution by using a reference scintillator and a given radionuclide. The radioactive sources are usually 137Cs(Ey = 662 keV), T o (Ey = 122 keV), and 55Fe (Ex= 5.9 keV). This test characterizes the whole PMT as it indicates the reduction in signal-to-noise ratio after conversion of photons into electrons and amplification in the tube. The energy resolution is related to the PMT characteristics by RE

=

2.36[(1

+ ~,,,)/E~p7]~’~

(3)

where EP is the mean number of photons coming from the scintillator collected by the PMT photocathode, p is the quantum efficiency at the wavelength corresponding to the maximum emission of the scintillator, 7)is the efficiency of collection of the photoelectrons by the first dynode, and v,,, is the relative variance of the multiplier gain (see Section 11,B). Nevertheless, experimental results on RE with NaI(T1) and bialkali photocathode PMTs are not as good as those estimated from Eq. (3). Experience shows also that a close relationship does not generally exist between R Eand the photocathode quantum efficiency, whereas it does if the photocathode sensitivity is evaluated by its Corning blue (CB) sensitivity or by its IQE. The energy-resolution measurement also gives an indication of the collection efficiency of the tube, but it is not a reliable indicator of the statistical characteristics of the multiplier. The measurement of the single-electron pulse-height distribution is a better test (see Section I1,B).

3 . Survey of the Various Blue-Sensitive Photocathodes The typical spectral response curves of the photocathodes suited for scintillation-counter applications are shown in Fig. 2. The SbCsJMnO photocathode, progressively improved during recent years, is still in use. It is well adapted to NaI(T1) (CB sensitivities of 10 pA/lm F are typical), and also to BBQ (with IQE typically above 15%),

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J . P. BOUTOT, J. NUSSLI, AND D. VALLAT

"

300

400

500

600

700

Wavelength (nrn)

FIG.2.

Spectral response of some blue-sensitive photocathodes.

but its high dark noise makes it unusable for high gain tubes; PMTs with this type of photocathode generally have wider stability spreads than bialkali tubes. The bialkali SbK,Cs photocathode is the best adapted to NaI(T1) (CB sensitivities can reach typical values of 12 pA/lm F). Extensive efforts have been made to reduce the spread in large-scale production. For example, by using computerized monitoring of the photocathode processing, tubes designed for gamma-camera applications exhibit CB sensitivities in the range 1 1 - 13 pA/lm F for 90% of the tubes. Its resistivity is a serious drawback in applications having high-peak cathode currents. The sensitivity to BBQ light is lower (IQE = 12%), but its noise and stability are quite attractive. The SbRb,Cs/MnO cathode has now appeared in the literature (Stapleton and Wright, 1979). Its blue sensitivity is equivalent to the sensitivity of SbK,Cs as the latter has also improved. The dark noise and stability also compare favorably. The resistivity and BBQ sensitivity are similar to the corresponding figures of SbCs3. This type of cathode should replace the other two in many cases. The SbNa,K cathode is restricted to high-temperature applications (Persyk et al., 1976). 4 . Windows

The response of a PMT is limited in the UV band by the transmission of the window. Table I summarizes the cutoff wavelengths (1 mm thickness, 10% transmittance) for various materials. Unfortunately, because the sealing techniques have become more difficult, the price has increased more rapidly than the spectral range. Note: Most cathodes have a quantum efficiency between 30 and 35% when incident light is perpendicular to the cathode plane. This is equivalent

RECENT TRENDS IN PHOTOMULTIPLIERS

Material Soda lime glass Pyrex “Kovar” glass (7056) 9741 (Corning), 8337 (Schott)

Material

Wavelength (nm)

Fused silica (Suprasil) Sapphire Magnesium fluoride

160 145 1I5

Wavelength (nm) 300

280 260 200 (UV glass)

23 1

1 mm thickness, 10% transmittance.

to approximately 60% if related to the light actually absorbed, and it is unlikely to reach significantly higher figures. Furthermore, a large part of the light reflected by the cathode layer is often reflected back to it and only a small part is really transmitted. This is due to the large solid angle of the light arriving on the cathode when coupled to a scintillator.

B . Secondary Emission Electrons emitted by the ca,thode are focused and accelerated on the first dynode d, (with a collection efficiency q). With a primary energy of 200-500 eV, they produce low-energy secondary electrons; these electrons are again focused and accelerated to the second dynode d2 (with a collection efficiency q,), etc. The number $ ( V ) of electrons emitted by dynode i is a function of the energy in eV. The curve of this function is a characteristic of the material used as the secondary emitter on the surface of a dynode. Allowing for the interstage collection efficiency, the gain per stage is given by gi = &qi (noted g for iterative stages). 1. Muterials

The secondary-emission curves for the materials commonly used at present in PMTs are shown in Fig. 3. Types AgMg (MgO + Cs), CuBe (Be0 + Cs), and AlMg (MgO + Cs) consist of an alloy base (AgMg, CuBe, AlMg) on which an adequate oxidation treatment produces a thin oxide layer. This material can be prepared prior to assembling the tube and will withstand some storage in air. After cathode activation, some of the alkali used for the cathode treatment remains on the surface of the oxide, increasing the emission slightly. Only CuBe is now used extensively because of its low price and good stability performance at high current. Nevertheless, it is suspected of pro-

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100

500

1000

Primary electron energy ( e v )

FIG.3. rials.

Secondary-emission coefficient versus primary-electron energy for some mate-

ducing hysteresis effects due to the insulating nature of the oxide, as such phenomena are clearly observed with a much higher thickness. The B e 0 secondary-emission coefficient has its maximum value of -8 at 600 eV energy of the primary electrons. Alkali antimonides are made exactly like photocathodes. They are prepared in the tube and cannot withstand presence of air, but they confer a much higher emission coefficient; SbCs, has been known for many years but its gain stability, even at low currents, is poor. Bialkali antimonides are better, provided the difficulties of obtaining correct composition on all dynodes can be overcome. SbK,Cs and SbN%K(Cs) have typical emission coefficients of 7 and 15 at 100 and 300 eV, respectively. Negative electron affinity (NEA) materials (111- V compounds) like GaP(Cs) or GaInAs(Cs) exhibit emission coefficients of about 20 at 300 eV and 35 at 600 eV. They are prepared prior to assembly by conventional epitaxy techniques on metal plates. These are mounted as dynodes in the tube and activated at the same time as the cathode by alkalis. These materials are still expensive.

2 . Dzferent Multipliers a . Multipliers Using Only One Type of D y n o d e . It is interesting to compare a multiplier having a number N of CuBe dynodes with a multiplier

RECENT TRENDS IN PHOTOMULTIPLIERS

0

1

23 3

2

Anode pulse height (Photoelectron equivalent number)

FIG. 4. Single-electron pulse-height distribution of several PMTs having a high-gain first dynode (curve 1 corresponds to dynode inhornogeneity).

having the same number N of bialkali antimonide dynodes for a given application and a given required gain. The voltage needed in the second case is only half the voltage required in the first: 500-750 V, compared to 1000- 1500 V, for example. In many cases, this is a distinct advantage, but the linearity (which varies at least as the square of the voltage) is drastically impaired and the timing properties are also seriously degraded. These drawbacks can be partly reduced by using a small number of stages; e.g., there is now a new group of tubes which only use eight stages. Note that much earlier, RCA introduced a five GaP-stage PMT (C 31024). b. Multipliers with High-Gain First Dynode. In this case, the first dynode is made of a high-gain material, the other dynodes being generally CuBe. The main features of such a combination are the following: ( 1 ) Much better multiplication statistics because the resolution of the single-electron pulse-height distribution (SEPHD) depends mainly on the first dynode gain (Morton et ul., 1968) (2) The initial velocities of the secondary electrons of the high-gain materials are much lower. This makes the collection of electrons from d , to d, much easier and facilitates the best use of these electrons. It also contributes to the reduction of transit-time fluctuations (see Section 111,A).

The features of typical SEPHD obtained with PMTs using materials with different secondary emissions are clearly visible in Fig. 4. The SEPHD can be characterized by its resolution (FWHM/peak amplitude). The definition does not apply if the valley between the peak and the vertical axis is not “deep” enough. In this case, it is possible to define a

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J . P. BOUTOT, J . NUSSLI, A N D D. VALLAT

“half-resolution’’ which is the half-width at half-maximum, measured on the upper side, divided by the peak amplitude (Brown rt d., 1981). When the ratio P/V (peak/valley) exceeds 2, an estimation of the resolution can be obtained by the formula 1)]“2

(4)

with gi = S i q i . In “good” cases, resolution figures of 40% for GaP and 70% for bialkali antimonides are obtained, but the spread of the P/V ratio is rather large, and the final values of P/V are always lower than expected. Theoretically, the left part of the curve must reach the origin of the axis. In fact, the P/V ratio never exceeds 5 for GaP and 4 for SbK,Cs. One possible explanation is inhomogeneity , but another cause suggested by Coates (1973) is that low-amplitude pulses are probably due to primary electrons backscattered after having lost only part of their energy. Experience clearly shows that the P/V ratio is poorly correlated to S1.

3 . Advantages of a High-Gain First Dynode a . Photon Counting. A well-defined peak for the SEPHD aids the adjustment of the gain of the PMT relative to the threshold of the associated electronic circuitry. With a P/V ratio more than 2.5 and a threshold at 0.15 photoelectron equivalent, the detection efficiency exceeds 90%. With such a low threshold, noise due to electrons leaving the first dynode could be counted with a low-gain dynode but is eliminated with a highgain one. Interest in tubes showing a well-defined peak for the SEPHD is also obvious for experiments where elimination of high-counting-rate background is necessary (Roberts, 1980; Hayakawa and Hayashi, 1980). b. Energy Resolution. In a scintillation detector (multielectron pulses), [see Eq. (3)] the energy resolution is proportional to the term ( 1 + with vm = vo + k/gi(g - 111,

g 1. = 6.1%

(5)

For focusing multipliers and CuBe, v, can reach a minimum of 0.3, which corresponds to vo = 0.1. Doubling 6, permits a value for v, of 0.2, as observed with GaP and bialkali antimonide dynodes. In that case, an improvement of energy resolution of 4% can be expected [see Eq. (3)] equivalent to an increase of cathode sensitivity of 8%. It must be pointed out that with a resolution on the SEPHD of 60%, but with a P/V ratio of only 2.5, a variance of 0.28 is obtained, almost equivalent to the one obtained with a conventional CuBe dynode multiplier.

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C . Conventional Photomultiplier Structures

The performance of a PMT depends considerably on its design, that is to say, as seen in the Sections II,A and II,B, on the emitter materials used for the photocathode and the dynodes but obviously also on the shape, the arrangement, and the number of electrodes that constitute the tube. The most common structures and some of their variants are reviewed in this section. Common or conventional structures mean those which have been used for many years by the majority of PMT manufacturers.

I . Review of the P M T Structures Before reviewing the improvements made recently to conventional structures, we recall the functions of the different parts of a PMT and the main electro-optical characteristics which have to be considered for optimum performance. The electron-optical features of photomultipliers have been described in detail in application books published by some of the PMT manufacturers (Philips, 1970, 1971; 1982; RCA, 1970, 1980; RTC, 1981). a . Input Optics System. The input optics system plays a prominent part in the overall performance of a PMT. It must ensure the complete collection of the photoelectrons by the useful area of the first dynode no matter where they are emitted from the photocathode or what their initial velocities may be. The input optics system is characterized by its collection efficiency 77, that is the ratio between the number of electrons impinging on the useful area of the first dynode and the total number of electrons emitted by the photocathode. As with the photocathode sensitivity, the collection efficiency may vary with the location of illumination and so is also characterized by its uniformity. In a PMT with an opaque photocathode, it is generally integrated into the input of the multiplier and consequently has a small useful area (0.1- 1 cm2). A simple configuration of the input optics system is thus adequate for focusing all the photoelectrons onto the first dynode. Such types of PMT are not generally used in nuclear physics applications because the location of the photocathode away from the window (tube glass envelope) makes good light coupling with a scintillator or light guide impossible. In a PMT with a semitransparent photocathode (photoemissive layer deposited directly onto the inside face of the input window), the useful area of the cathode may be made very large (a few cm2 up to over 100 cm2). This makes the collection of the photoelectrons by the first dynode more difficult because of the much smaller area of the dynode.

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For nuclear physics general-purpose PMTs, i.e., those used in experiments not requiring a very high timing performance, a collection efficiency q higher than 80% is generally obtained with a simple input optics system. In fast-response PMTs, the optics configuration must not only ensure a good collection efficiency but also keep the transit time of the photoelectrons in this space significantly independent of their initial velocities and emission locations, i.e., make the transit-time fluctuations as small as possible. These fluctuations have two causes, which are analyzed in Section 111: (1) Fluctuation of the initial velocities of electrons emitted from the same point of the photocathode, thus introducing a transit-time spread (chromatic contribution) ( 2 ) Differences in length of the paths of electrons emitted from different points on the photocathode and nonuniformity of the electric field between the different parts of the space between the photocathode and the first dynode, thus introducing transit-time differences between two principal electron trajectories (geometrical contribution)

As is explained in Section 111, reduction of the spread of electron transit times is obtained by making the electric field at the photocathode surface as high as possible and by optimizing the paths of the electron trajectories in the space between the photocathode and the first dynode. That is achieved in a fast-response PMT by using a curved photocathode and one or more additional focusing electrodes. b. Electron Multiplier. The electron multiplier consists of a cascade of reflection-emission secondary electrodes or dynodes. The shape of these electrodes and the distribution of applied voltages determine the geometry of the field distribution in the space between two consecutive dynodes. Each interdynode space must have the same electrooptical properties as those described for the input optics system. So, optimization of the configuration of the multiplier structure is obtained from the determination of the secondary electron trajectories. Computer simulation is currently used for the calculation of electron optics systems with axial symmetry. However, for the multiplier part, the three-dimensional problem cannot be reduced to a two-dimensional one as is done for the input optics system. Therefore optimization is often undertaken by determining electron trajectories in planes with specific orientations. Taking into account the spread of secondary-electron initial velocities, a simple analysis shows that convergence of the electron trajectories is necessary (in two dimensions) between two consecutive dynodes so that the

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diffused electron spot on each next dynode be confined to its useful area. An ideal collection efficiency of 100% cannot be achieved in practice because part of the secondary electrons have high initial velocities and are not properly collected (the percentage may amount to more than a few percent). The configuration of the first interdynode space d , , d, is rather critical as it constitutes the coupling space between the input optics system and the following part of the multiplier. Therefore, in most multiplier structures, the first (and often the second) dynode differs considerably in shape from the following iterative dynodes. For the same reason, these coupling spaces require higher potential differences. The most common structures have solely electrostatic focusing. They are generally referred to by their time performance. The so-called slow-multiplier structures, i.e. , those whose configuration does not provide a high electric field at the dynode surface, are the following: (1) The venetian-blind structure (Fig. 5a). The large area of the first dynode provides a high collection efficiency of photoelectrons even with a simple input optics system ( 2 ) The box-and-grid structure (Fig. 5b). This structure also provides a good collection efficiency

The so-called fast-multiplier structures are the following: (1)

The 1inearfi)cusing dynode structure (Fig. 5c) in which a progres-

(C)

(d)

FIG. 5. Configurations of conventional electron-multiplier structures: (a) venetianblind multiplier; (b) box-and-grid multiplier; (c) linear focusing multiplier; (d) circular-cage multiplier.

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sive focusing of the electron trajectories is achieved. This provides a much smaller spread in transit time compared with the two previous structures ( 2 ) The circular-cage structure (Fig. 5d). This is also a focusing structure with the advantage of compactness over the previous one. c . Collection Space. In the output part of the multiplier, the shape and arrangement of the electrodes differ from those of the iterative part because this stage is designed to

collect all the electrons emitted by the last dynode, provide a high pulse linearity (space-charge limitation), match the output stage with the external transmission line (very important for fast time-response transmission). The anode generally consists of a grid set a short distance (‘1 mm) from the last dynode so that the secondary electrons coming from the penultimate dynode go through it. Such a configuration provides a high electric field in the last collection stage and thus minimizes the space charge effect. The PMT structures most commonly used in nuclear physics applications may be simply described as follows:

(1) They belong to the “head-on” type; i.e., they have a semitransparent photocathode deposited onto a front window ( 2 ) The general-purpose PMTs have a plane photocathode, a simple input optics system design (generally diode type), and make use of one of the four multiplier structures mentioned above (3) the fast PMTs have a curved photocathode (generally with plano concave window), a more sophisticated input optics system design (several focusing electrodes), and an in-line focusing dynode structure. 2. 1mprovement.s in Conventional PMTs a . PMTs with New Input Structures. As outlined at the beginning of this article, research and development work in the past ten years on PMTs has been focused on the improvement of the energy resolution of general-purpose tubes. That is related to the increase in ionizing radiation applications in many areas of activity, particularly the medical field. Indeed, scintillation counters are now currently used in various detection systems in medical imaging instrumentation (Heath et a/., 1979). Among these systems, the gamma-ray scintillation camera and the X-ray computerized tomography (CT) scanner are the imaging systems bearing the most important research efforts since their invention.

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An X-ray CT scanning system provides a density image representing a cross section of the inspected organ of the patient, whereas conventional radiology provides a projection image. A tomographic image is reconstructed by a computer from several tens of thousands of attenuation measurements. Continuous efforts at improving CT scanner performance, particularly in the reduction of acquisition time, has resulted in several generations of systems. A survey of different scanning assemblies and their performance has been given by Kowalski er a / . (1979) and Zonneveld (1980). The first generations used scintillator-PMT assemblies as X-ray detectors (up to several hundreds per scanner). The detection characteristics required by such an application led the PMT manufacturers to develop small-diameter tubes (19- and 13-mm types) with moderate gain, wide dynamic operating range, low dark current, short time-constant recovery of anode current after illumination, good stability and reliability. Later generations also make use of other types of detectors such as multicell high-pressure gas (xenon) devices or even scintillator- silicon or vacuum photocell assemblies (Zonneveld, 1980). The gamma-ray scintillation camera provides a representation of the distribution in the body of radionuclides previously injected into the patient, giving either morphological, functional, or even metabolic information. The camera head includes a gamma-ray collimator, a thin but large diameter (30-50 cm) NaI(T1)-crystal scintillator, a light guide, and a large number of PMTs. According to the principle devised by Anger, each scintillation following the absorption of a gamma ray within the crystal is detected by a set of PMTs generally in hexagonal arrangement. The electrical signals delivered simultaneously by the PMTs are processed by analog circuits which analyze the coordinates of the scintillation and compute its amplitude. This information is used to discriminate between the absorbed gamma rays and those due to scatter phenomena which do not have to be visualized. Finally, the image is constructed from points representing the unscattered gamma rays. The camera performance usually depends on the intrinsic characteristics of the detection head, i.e., the spatial and spectral resolving powers, the distortion, and the response uniformity in the field of view. All these characteristics depend on the whole of the design parameters. The number, shape, and sensitivity characteristics of the PMTs have a large influence on detection-head performance (Jatteau er a / ., 1979). Improvements made during the past ten years are mainly in the increase of the spatial- and energy-resolving powers and the enlargement of the field of view (FOV) (Patton et a / . , 1980). The improvement in spatial resolution has resulted primarily from the use of a larger number of tubes with smaller diameter for a given FOV (e.g., 37 x 50-mm PMTs instead of

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19 x 75-mm PMTs for a 26-cm FOV and 61 x 50-mm PMTs instead of 37 x 75-PMTs for a 38-cm FOV, or even 61 x 60-mm PMTs for a 40-cm FOV); second from improvements in the energy resolution of the PMTs; third from an optimization study of the camera head; and finally from the introduction of on-line correction circuits (Jatteau, 1980; White, 1979). Among the main parameters affecting the energy resolution of the system other than those related to the energy-conversion factor of the scintillator block, are these related to the PMT itself: the collection efficiency of photons by the photocathode, the quantum efficiency of the photocathode, the photoelectron collection efficiency of the PMT input optics system, the gain fluctuation of the first stage gl (gl= &ql). Concerning the first point, it can be shown that the best shape suited for the input window section is hexagonal or square and that a few percent improvement in photon collection can be expected compared to a circular shape (assuming that each tube is in contact with the surrounding ones); PMTs with such shapes have recently been put on the market by all of the gamma-camera PMT manufacturers. The value of these tubes is only real if the photocathode sensitivity in the peripheral area is as high as in the central part and above all only if all the photoelectrons emitted by this area are effectively collected by the first dynode, but there is the additional advantage that the area between the tubes is distributed better than with round tubes (Brenner er al., 1982). As mentioned in Section II,B, a marked improvement in the energy resolution of the scintillation counters has resulted from an increase in sensitivity in the blue region of the alkali antimonide photocathodes. At the same time, it has become evident that the photoelectron collection efficiency, particularly the uniformity of collection in the input optics system, could be further improved. Development work is being carried out on this particular point by all the PMT manufacturers. Optimization of these two characteristics has improved as a result of better knowledge of the photoelectrons initial velocity distribution and of the application of computerized simulation techniques which are now available in most development laboratories. The response uniformity depends on geometrical parameters as well as optical and electrical ones. Indeed, for a simple input optics system, the lack of axial symmetry in the multiplier structure and particularly in the first stage leads to a spatial nonuniformity in the collection of the photoelectrons. As a result, the collection is only optimum on the photocathode diameter which is parallel to the dynode generating line (low

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variation of the electron incidence angle on the first dynode). On the other hand, a small fraction of the blue photons impinging on the photocathode is not absorbed and is thus transmitted. This light is more or less reflected by the input optics electrodes and is eventually sent back to the photocathode. The transmitted light can also, under particular incidence angles, reach the photoemissive area that may be present on the input optics electrodes. For certain input optics configurations, a large part of the photoelectrons emitted by these stray photoemissive areas is collected by the first dynode and contributes to the increase of the tube sensitivity. Both the coefficient of the light transmission through the photocathode and also the reflection coefficient of the electrode surfaces depend on the thickness of the layer, on the electrode material, and also on the wavelength and the incidence angle of the photons (Chabrier ef al., 1973). Consequently, the variation of anode sensitivity depends considerably on the angle of incidence of the light beam, its aperture angle, and its azimuthal position in the photocathode plane. Now each PMT in a gamma-camera head “views” the scintillations under these different illumination conditions. There appears to be no unique solution to the complex problem of optimizing the collection in the input optics system and in the input part of the multiplier of a PMT. Various manufacturers have solved this problem more or less successfully by quite different methods. RCA chose to design a new multiplier structure incorporating a large-area cup-shaped first dynode known descriptively as the “tea-cup” structure (Fig. 6a). A detailed description of this input structure has been given by Faulkner (1976) in his patent. Focusing of the photoelectrons is assisted by a very fine mesh (with a very high optical transmission). Secondary electrons from the first dynode are directed to the second large box-and-grid-type dynode through an aperture in one side of the “teacup” electrode. The third dynode, also of unconventional shape, has been specially designed to fit this new input configuration onto a conventional circular-cage-type multiplier structure. The particular advantage of the tea-cup design claimed by Engstrom (1977), against the previous RCA venetian-blind structure design, is an improvement in the spatial and off-axis collection uniformity. This results from the good collection efficiency for photoelectrons emitted from the front-end photocathode as well as for photoelectrons emitted from the photoemissive layer on the side walls (see also RCA, 1980). RCA has modified the “tea-cup” structure by introducing an additional focusing electrode which also plays the role of a mirror and thus improves the reflection of the photons toward the cathode. Other manufacturers have obtained similar improvements without significantly modifying the design of the conventional electron multiplier

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FIG.6 . Input section configurations of state-of-the-art PMT structures for gamma-ray cameras: (a) RCA tea-cup dynode structure (by courtesy of RCA, Lancaster). (b) SRC “gamma-target” dynode structure (Persyk and Moi, 1978; copyright @ 1978 IEEE). (c) HTV box-and-grid dynode structure (by courtesy of Hamamatsu TV Co., Ltd., Japan). (d) Philips venetian-blind dynode structure (by courtesy of RTC, Paris).

structures. SRC5 Laboratories chose to replace the first dynode of the conventional venetian-blind structure (parallel slats) by a dynode with a circular symmetry referred to the tube axis (Fig. 6b). The new dynode, referred to as the “gamma target,” is made of concentric slats in the shape of a truncated cone (Morales, 1975). The SRC tube fitted with the new first dynode is reported to have improved angular uniformity of response for off-axis photons. Hamamatsu and Philips have also improved the design of the input of their box-and-grid- and venetian-blind-structure PMT families, respectively. The Hamamatsu design includes a large-area (square section) box-and-grid first dynode following the same idea developed in the “tea-cup” design (Fig. 6c). Philips has replaced the diode-type input electron optics system by an asymmetrical triode configuration (Fig. 6d). As shown in this figure, such a configuration permits the efficient collection of photoelectrons emitted from the photoemissive layer on the side walls because of the smaller incidence angle of the electrons on the first dynode by reducing the part of the electrons going through the first dynode, i.e., nonimpinging on it. As a result, an improvement in energy resolution and in response uniformity has been obtained, particularly in tubes having large diameters of 60 and 75 mm. S.R.C. Laboratories, Inc., Fairfield, Connecticut 06430.

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The evaluation of the performance of PMTs for gamma-camera application is not easy because the test procedure conditions have to be very similar to the operating conditions of the PMTs in the camera head in order to obtain significant results. A typical test procedure for energyresolution measurements has been described by Persyk and Moi (1978). Most of the diagnostic applications in nuclear medicine make use of low-energy gamma-ray emitters (99mTcwith Ey = 140 keV being most common), and consequently the signals detected by the PMTs located far from the scintillation correspond to equivalent energies as low as a few keV. Thus energy-resolution measurements are carried out with 57C0 (Ey = 122 keV) and 55Fe(Ex= 5.9 keV) radionuclide sources and small NaI(T1) crystals. Such measurements indicate the values of photocathode sensitivity and of the collection efficiency of the input optics systems. The energy-resolution comparison study carried out by Persyk and Moi (1978) between the state-of-the-art PMTs for gamma cameras concludes that no one multiplier structure was at that time superior to the others. A simple PMT response uniformity examination can be made by scan-

Cornera-head axis

Collimated F r a y sourc

-

0 0

50

100

d (mm)

FIG.7. (a) Arrangement for measuring the response uniformity of PMTs for gamma-raj cameras; (b) “signal-distance” curve of a 70-mm PMT for a given scintillator-light-guide configuration (from Jatteau and Lelong, 1981).

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FIG.8. Relative deviation of the anode signal S(d,B)against the mean signal &) for 70-mm PMTs (scale, 5% per centimeter): (a) tube with good azimuth uniformity; (b) tube with bad azimuth uniformity; (1) dynode orientation with figure axis as a reference (from Jatteau and Lelong, 1981).

ning the photocathode with a collimated light beam. Such a method only detects large collection defects in the input optics system. A more meaningful method of estimating the response uniformity of a PMT for scintillation camera applications is to measure the variations of the mean value of the signal amplitude at the PMT output3 against the position (d, 6) of a scintillation source for a given scintillator-light-guide configuration, similar to that used in a camera head. As shown in Fig. 7a, d is the source-PMT axis distance and 6 the azimuthal position of the source. The variation of and d is the so-called signal-distance curve (Fig. 7b). The response symmetry of a tube may then be evaluated by plotting in a two-dimensional representation of either the source position (d, 6 ) for equal output signals S (isocline chart) or the deviations of the

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signals S(d, 6 ) against the mean values ?(d) (Hayashi, 1978). The difference between typical response charts featuring PMTs with good and bad azimuth uniformity are shown in Fig. 8. b. PMTs with Hemispherical Photocathodes. As pointed out previously, experiments in nuclear physics provide a very broad range of applications for PMTs. In high-energy-particle physics, the detector requirements become increasingly complex because of the large detection volumes involved. Most of the experiments in this field make use of large arrays of PMTs (a few hundreds up to several thousands per experiment such asWAl, R807, UAI, UA2, at CERN), putting much more stringent requirements upon the PMT characteristics, particularly upon stability (see Section 111,C). In large-volume gas (or liquid) Cerenkov counters, e.g., those designed for proton-decay measurement experiments (Weinberg, 1981) or for cosmic neutrino detection (Learned and Eichler, 1981), the Cerenkov light is not only feeble in intensity but also extended and diffuse. It is therefore important to use PMTs having a good detection efficiency and to maximize the collection of photons by the tubes. PMTs with large photocathode area (up to 200 mm useful diameter) are used to minimize the number of tubes per experiment. The improvement in light collection brought about by this type of tube is generally obtained at the expense of photoelectron collection efficiency and of time performance (particularly increased center-edge transit-time differences; see Section 111,A). Hearing and Wright (1979) have presented a new type of PMT specially designed for this application in which immersion of the tubes in the Cerenkov medium (clean water) is possible. A hemispherical photocathode is a feature of the so-called bulls-eye EM16 tube which offers isotropic photon collection over a large surface; versions with 200 and 500 cm2 currently exist (Fig. 9). Its input electron optics system, which includes two focusing electrodes, has been designed by computerized simulation in order to obtain the best compromise between electron collection efficiency, response uniformity, and timing. An extensive performance evaluation of the tube has been carried out by Cory et al. (1981a). For similar applications, HTV7 have developed and made hemispherical tubes, including one of 50-cm diam. (model HTV-R 1449 X). The results of a series of tests run on such a big tube have been reported also by Cory et al. (1981b). c . Position-Sensitive Photomultipliers. The location of light events distributed randomly in space and time is one of the fields of application of EM1 Industrial Electronics Ltd, Electron Tube Division, 243 Blyth Road, Hayes, Middlesex, UB3 1H5, England. ' HTV, Hamamatsu TV Co., Ltd., 1126 Ichino-Cho, Hamamatsu, Japan.

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\ocusing

electrodes

FIG.9. Cross section of the “bulls-eye’’ EM1 9870 photomultiplier having a large-area hemispherical photocathode (Cory et u / . , 1981; copyright @ 1981 IEEE).

PMTs. In many nuclear physics experiments, the determination of particle trajectories or of the shape and the intensity distribution of particle beams is essential. These are generally achieved with scintillation-counter hodoscopes, particularly if good timing is also important. Other detection devices using large quantities of PMTs are found in nuclear medicine instrumentation, such as in scintillation- Anger cameras, in X-ray CT scanners, and in positron imaging systems (Heath et al., 1979). In all these systems, the requirements of improved spatial resolution entail a larger number of PMTs, and thus smaller size tubes are used. Scaling down the conventional multiplier structures is limited to a 10- 15-mm outside diameter for many reasons but particularly because the light collection is decreased by the relative increase of wall thickness and tolerances. Ideas for making a photomultiplier position sensitive have often been suggested and tubes have been developed for special purposes. Among these ideas was the image-dissector tube developed from the original work by Farnworth (1934). It includes a high-resolution imaging input optics system (with magnetic or electrostatic focusing and deflection) and a multiplier section. It is in fact more of a type of image pickup tube than a PMT. The main applications of image dissectors are high-resolution TV systems, star-tracking systems, and scanning spectrometers (e.g., see ITT catalog, ITT, Electro-Optical Products Division, Tube and Sensor Laboratories, 3700 East Pontiac Street, Fort Wayne, Indiana 46803). Another type of PMT specially designed for electronic star-tracking systems (control of satellite attitude, autoguidance of telescope, etc.) is the quadrant-multiplier phototube (QMP). Two versions of this have been developed. One designed by EMRS and described by Rome et al. (1964) contains four independent and adjacent photocathode quadrants and one single-multiplier-anode structure; the second, designed by ITT (1964), has one single photocathode, an electrostatic imaging input electron optics system, and a four-multiplier-anode structure (ITT model F 4002). LocaEMR Photoelectric, Box 44, Princeton, New Jersey 08540.

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lization of a light spot (e.g., a star, the image of which is defocused on the photocathode plane) is achieved by means of zero balance between the four quadrants. In the first version, the four signals are delivered sequentially (by operating each photocathode quadrant one after the other), whereas they are delivered simultaneously in the second version. Following the same basic idea, Charpak (1967) has shown that localization of light events can be achieved by using only one conventional design PMT. His experiments show that a spatial and temporal modification of the photocathode potential of a PMT (and consequently a modulation of the photoelectron emission) can be induced by an electrical voltage (a few tens of volts) applied to electrodes (meshes or wires) located against the outside of the tube window. Localization is then performed by a gating sequential mode if the light events have sufficient duration. According to this principle, Charpak and Fillot (1968) have designed and built a hodoscope consisting of ten scintillators viewed by a single tube through light guides. Boutot and Pietri (1972) have designed an experimental photomultiplier based on the same principle. The originality of the tube lies in the use of a cross-bar control grid embedded in the glass of the window at a short distane from the photocathode (= 1 mm). Preliminary experiments have shown that a spatial resolution of 1 line/cm could be achieved, making possible a light-event localization on 8 x 8 positions. Although the realization of tubes having a larger number of picture elements (pixels) seems feasible, particularly by using a microchannel plate (MCP) electron multiplier (see Section II,D), the applications of such types of tubes are limited by the requirements that the duration of the light event must be long enough for a full scanning to be achieved. There has also been interest shown by nuclear medicine instrumentation designers in QMPs or PMTs with several electron-multiplier-anode structures. Muehllehner (1974) has described in his patent a QMP having a design similar to that of the ITT tube F 4002. It includes one single photocathode, an electrostatic lens input optics system (inverter type), and a four venetian-blind electron-multiplier-anode structure. The use of such a type of tube in scintillation Anger cameras improves the spatial resolution and allows cameras to be designed with enlarged FOV without increasing the PMT number. SRC has designed a QMP which differs in many ways from previous structures (Morales, 1975). The input of the tube has a square section and a venetian-blind structure separated into four discrete channels from the photocathode to the anode by means of adequate barrier walls in each stage. The tube is so divided into four independent sensitive sections. Collection uniformity in the first stage is achieved by the use of the proprietary “gamma-target’’ dynode previously described. An evaluation of such a tube has been reported by Persyk ef al. (1979). Its advantages as a

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multidetector device are lower cost for systems requiring a large number of PMTs plus a larger packing density. This applies when a resolution less than 1 cm is required, for example in multilayer positron tomographs (Derenzo et al., 1981). Following another approach, Kuroda et al. (1978) have proposed a new type of position-sensitive PMT by reviving the principle of the transmission secondary electron (TSE) image-intensifier tube in which focusing of the electron between two consecutive dynodes is achieved by means of an axial magnetic field; TSE dynodes are replaced here by a grid-type dynode derived from the venetian-blind one, the geometry of which has been studied by computerized simulations. A prototype tube including a 4 x 4 matrix of anodes has been realized by Hamamatsu and its performance evaluated by Kuroda et al. (1981). The intrinsic resolution in both directions (parallel and normal to the dynode wires) is around 3 mm full width at half-maximum (FWHM), i.e., equal to the anode pitch. It has been shown that better accuracy in light-event localization (say a fraction of a millimeter) can be achieved by adequate processing of the signals delivered simultaneously by the anodes. Applications of this type of tube are foreseen in the field of high-energy physics as well as in nuclear medicine. The main drawbacks of this tube are those previously met with cross-field PMTs (see Section II,D,2) related to the size and handling of the magnetic source. Other design proposals have been made for a PMT sensitive in one dimension. The proposal of Prydz (1973) was based on the variation in secondary-emission coefficient that can be obtained in the first stage depending upon the origin of the photoelectron on the photocathode, combined with the use of a suitable input optics system and of a high secondary-emission material such as GaP(Cs). Such an idea has not been really evaluated. A similar idea, based on the variation of photoelectron mean transit time with the emission location on the photocathode, has been realized in a tube developed by a team of Russian physicists (Vasil’chenko et al., 1980). Other poroposals making use of so-called nonconventional electron-multiplier structures are presented in Sections II,D,l and 2). In spite of the advantages the position-sensitive PMT may bring to system performance, it requires very special devices with a consequent low production rate. It also means tubes with more complex structures than conventional ones, resulting in a lower production yield and operating reliability. For these reasons, such tubes are likely to remain very costly. None of the tubes mentioned above is commercially available yet.

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D. Unconventional Photomultiplier Structure3

I . Channel-Electron-Multiplier PMTs a . Single-Channel Electron-Multiplier ( S C E M ) PMTs. The principle and characteristics of SCEMs have been the subject of many papers (Schmidt and Hendee, 1966; Adams and Manley, 1966; Eschard and Manley, 1971; Pook, 1971). This type of multiplier has found its full application potential as a windowless detector (Millar, 1971; Timothy and Timothy, 1971), whereas its use as a multiplier structure in a photoelectric tube arouses less interest. Indeed, its advantages over conventional structures -compactness, ruggedness, lightness, low power consumption, and operating simplicity-all properties well suited for space application, disappear when incorporated into a sealed tube for more general applications (small input area, large gain fluctuation when not operated in the saturated mode, low output current, and slow time performance). Some of its features such as high gain (up to lo8), saturation operating mode giving flat counting plateau, and low noise are of interest in designing photon-counting phototubes for low light-level detection applications. The ITT F 4125 PMT has been developed for such a purpose. This tube includes a SCEM with a conical entrance in combination with a restricted photocathode area. In a patent, Sharpe (1975) has described a version of such a tube in which the use of a conventional dynode as the first amplification stage permits the signal-to-noise ratio of the device to be improved. Such a version does not appear to have been developed. As reported by Petley and Pook (1971), the use of an SCEM in an imagedissector tube allows its overall package size and weight to be reduced. A few years ago, Duckett (1972) described a rugged miniature PMT (EMR model 521) that took advantage of the very small intrinsic section of an SCEM. The ceramic envelope of the tube is only 8 mm in diameter due to the use of a “sinewave”-shape multiplier. Its size makes it ideal for use in closely packed arrays in particle or radiation imaging systems where energy resolution is of less importance. b. Microchannel-Plate ( M C P ) PMTs. The MCP electron multiplier, which is basically a matrix of tightly packed microscopic channel-electron multipliers, has been developed primarily for use in the so-called second generation of image intensifiers for night-vision aids. In the past ten years, intensive research and development investigations on its technology and applications have been carried out in a number of laboratories, and a large amount of literature deals with this subject (Eschard and Woodhead, 1971; Washington et al., 1971; Graf and Polaert, 1973; Emberson and Holmshaw, 1973; Schagen, 1974; Ruggieri, 1972; Wiza, 1979).

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J . P. BOUTOT, J. NUSSLI, AND D. VALLAT Window,

,Photocathode Proximity focusing 'input optics system

MCP multiplier'

U

'Anode

(a 1

Electron multiplication in one channel

LZZZZZZ

Chevron ossembly

(b) Z assembly

MCPwith curved channels

FIG.10. (a) Schematic cross section of a proximity-focusing MCP-PMT; (b) some of the MCP multiplier configurations used in MCP-PMTs.

The MCP has the unique property of electron image amplification with very high spatial resolution (more than 30 line pairs mm-') when compared with conventional electron-multiplier structures, and its geometrical and electrical features also give it other advantages such as fast time response and relative immunity to stray magnetic fields. Many original PMTs have been designed by taking advantage of these unique properties (Pietri, 1975, 1977; Dhawan and Majka, 1977; Lecomte and Perez-Mendez, 1977; Leskovar, 1977; Wiza, 1979; Dhawan, 1981). An MCP-PMT is simply a phototube with a semitransparent photocathode in which an MCP multiplier (including one or more MCP units) replaces the discrete dynode structure (Fig. 10a). Its input electron optics system is either an electrostatic focusing structure similar to that used in conventional PMTs or a proximity focusing one similar to that used in some MCP imaging devices (Graf and Polaert, 1973). Before reviewing the advantages and performance limitations of such PMTs, it is useful to recall the gain properties of MCP multipliers: i. Gain properties qf MCP electron multipliers. The multiplication process in one microchannel is identical to that of an SCEM. The gain of the plate is governed by the geometrical and electrical parameters 1 , d , and V which are respectively the length and diameter of the channel and the voltage applied to its ends (Guest, 1971; Oba and Maeda, 1972). As in an SCEM, a saturation effect caused by transient space charge occurs in the electron avalanche with a high output charge. This limits the gain almost in proportion to the channel diameter (e.g., maximum gain of about

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lo5f o r d = 1 2 . 5 p m and of a few lo6 f o r d = 40 pm). Also, an unavoidable amount of ionic feedback restricts the operation of an MCP with straight channels to a gain of not higher than a few lo4, otherwise random run-out of some channels could appear. Ion suppression resulting in higher gains and charge-output saturation may be achieved by curving all the channels of the MCP (Boutot et al., 1976; Henkel et ul., 1978) o r by stacking two o r even three straight channel MCPs (known respectively as the chevron and Z assemblies), each operating at gains in the 103-104 range (Parkes and Gott, 1971; Colson et al., 1973; Cekowski and Eberhardt, 1976; Fig. lob). These types of MCP multiplier differ slightly in gain performance. An MCP with curved channels can be operated in the full charge-saturation mode (as an SCEM) but has a limited gain (a few lo5- lo6). A Z-plate configuration (three plates in cascade without spacing between each other) provides a slightly higher gain but at the expense of resolution of pulseheight distribution and count rate. Cascading two MCPs with space between them allows higher gains (up to lo8) even with small channel diameter to be achieved. But, at the same time, this results in worse gain fluctuation performance (Audier et ul., 1978). So-called nonion-feedback NIF) plates have been introduced by Philips (Eindhoven, The Netherlands). They consist of 3-4 thin MCPs fused together into one plate and have the nonion-feedback features of the MCP with curved channels at the expense of a slightly degraded pulse-height resolution. This technology enables combinations of different channel diameters in different parts of the multiplier and offers an easier way of making nonion-feedback plates of large dimensions. The operation of an MCP multiplier in the saturation mode has the same interesting features as an SCEM: a quasi-Gaussian single-electron pulse-height distribution gives a better signal-to-noise ratio (lower relative variance of the gain), a flat counting plateau, and an efficient discrimination of the multiplier noise, all of which improve the single-electron detection efficiency. The response linearity of the MCP as a whole is retained as long as the number of individual channels in the plate is large compared with the number of electrons contained in the input pulse. As mentioned above, the unique advantages of MCP-PMTs over conventional PMTs are better timing performance, the ability to operate in magnetic fields, and the possibility of imaging. ii. Timing performance of MCP-PMTs. As is explained in detail in Section III,A, the overall timing properties of a PMT are defined by the spreads of electron transit time in both the input electron optics system and in the multiplier-anode structure. The advantage of using an MCP multiplier in designing fast PMTs comes from the fact that each individual channel in a plate is very short (e.g., 0.5 mm to a few millimeters) and the

252

J . P. BOUTOT, J . NUSSLI, AND D . VALLAT

electric field applied across it is very strong (between 1 and 2 kV mm-'1. As a result, the electron transit time in the MCP is very small (a few 100 psec to about 1 nsec), and consequently the transit-time fluctuation is much smaller than in a conventional multiplier (Guest, 1971; Boutot, 1972; Oba and Maeda, 1976; Eberhardt, 1981). The shorter the multiplier structure the better the impulse response so that a gain-impulse response compromise has to be taken into account in designing an MCP-PMT. Furthermore, the plane-parallel structure of an MCP makes possible the use of a proximity-focusing input design more attractive than a conventional input optics system. Indeed, proximity focusing between the photocathode and the MCP input results in major improvements: the transit-time differences due to the origin of the photoelectrons are quite negligible compared to those found in conventional tubes with electrostatic focusing (see Section 11,A); in addition, the uniform and very high electric field (a few 100 V mm-' up to 1 kV mm-') near the cathode reduces the transit-time spread caused by the different photoelectron initial velocities to less than a few picoseconds. Various fast MCP- PMTs have been designed following these guidelines, but so far only on a prototype or developmental basis rather than on an industrial one. They differ from one another either in their multiplier structure (useful area, size and shape of the channels, number of MCP units), in their input optics, in their collector stage structure, o r in their body technology. Two main categories corresponding to types of application exist: (1) MCP-PMTs with ultrqfast response. Such tubes include only one MCP unit with typically d = 12.5 wm and l/d = 40, a proximityfocusing input optics system (distance of a few tenths of a millimeter to 1 mm between photocathode and MCP input) and a coaxial anode output (Boutot and Pietri, 1970; Boutot, 1972; Cekowski and Eberhardt, 1976; Cekowski et a / . , 1981; Pietri, 1977; Boutot and Delmotte, 1978-1979; Oba, 1979). Typical performance is as follows: gain in the lo3- lo4 range and impulse response between 200 and 300 psec (FWHM) (see Section III,A,2). They find applications in subnanosecond real-time detection experiments commonly used for single-shot light-shape analysis as in laser plasma physics in conjunction with a wide-band oscilloscope ( 5 GHz) (Hocker et a / . , 1979; Lyons et a / . , 1980). (2) MCP-PMTs for single-photon timing. They differ from the previous type in that the high-gain multiplier either includes a single MCP with curved channels (Boutot, 1976) or a cascade of two or three MCPs (Bateman and Apsimon, 1976; Cekowski and Eberhardt, 1976; Cekowski et al., 1981; Oba, 1979; Oba and Rehak, 1981; Boutot and Delmotte,

RECENT TRENDS IN PHOTOMULTIPLIERS

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1978-1979). Typical performance is as follows: gain more than lo6, impulse response between 700 psec and 1.2 nsec, and a single-electron transit-time fluctuation lower than 200 psec (FWHM) (see Section III,A,4). iii. Operation of MCP-PMTs in magnetic fields. In a conventional dynode structure, the distances traveled by electrons are relatively large (i.e., a few millimeters to 1 cm between consecutive dynodes and several centimeters in the input optics system). The electron trajectories are then very sensitive to stray magnetic fields, and as a result the gain may vary markedly under the influence of the terrestrial magnetic field. The anode sensitivity is more affected by a transverse magnetic field than by an axial one, and so the input optics system is consequently the most sensitive section of the tube. Thus conventional PMTs in high-energy physics experiments using strong magnetic fields have to be located far from the scintillators and very long light guides have to be used. Because the electron trajectories inside a channel are very short (a few tens of microns) and because the electron avalanche is confined, MCP multipliers are relatively immune to magnetic fields. Measurements carried out by Long (1974) have shown that the operating limits of an MCP are about 0.1 T in a transverse magnetic field and more than 1 T in an axial field. With a proximity focusing input and output design (with distances around 1 mm or even smaller) and applied voltages in these spaces of a few hundred volts, MCP-PMTs can be operated up to a few tens of mT in transverse magnetic fields and up to higher levels (a few tenths of a tesla) in an axial field (Bateman and Apsimon, 1976; Lo et ul., 1977; Lo and Leskovar, 1979, 1981b; Oba and Rehak, 1981). This is of course of great potential interest because the use of PMTs inside the large magnetic fields surrounding particle accelerators will greatly simplify the detector design by shortening or eliminating the light guides and will improve light collection, timing, and signal-to-noise ratio. iv. Position-sensitive MCP-PMTs. Large useful area, two-dimensional multiplication with high resolution, high gain, and fast time response are features that make MCP multipliers well suited for the design of high-performance position-sensitive detectors. A straightforward design for a two-dimensional read-out is the use of a multianode structure, that is to say, an array of discrete anodes in proximity focusing at the output of the MCP multiplier, each anode defining a picture element (pixel) (Fig. 11). A variety of multianode MCP-PMTs have been designed mainly to be used in new-concept hodoscopes and Cerenkov counters (Benot et nl., 1973, 1979; Majka, 1976). An experimental tube with proximity-focusing input including a chevron MCP multiplier and 100 anodes arranged in a 10 x 10 square array was first reported by Catchpole

254

J . P. BOUTOT, J . NUSSLI, AND D. VALLAT Photocathode

,MCP

multiplier

u

Multianode collecting electrode

FIG. 11. Schematic cross section of a proximity-focusing MCP-PMT with a multianode collector.

and Johnson (1972). Prototype tubes of a similar design including an MCP with curved channels and a 5 x 5 anode array arranged with 2.54-mm pitch have been made by LEPS (Boutot, 1976; PiCtri, 1977; Boutot and Delmotte, 1976). Evaluation of such a tube for focusing-Cerenkov-counter applications has been carried out at CERN by Meunier and Maurer (1978). Tubes with higher spatial resolution (higher anode density) have also been realized. Cekowski et al. (1981) have described a proximity-focused PMT (ITT model F 4149) using a 3 MCP multiplier (Z configuration) and a 10 x 10 matrix of collectors, each 1.7 mm square, with a pitch of 1.8 mm. The performance parameters of such a tube have been discussed by Sandie and Mende (1982). This category of tubes also includes the multianode microchannel array (MAMA) detector with 10 X 10 pixels, each 1.2 mm square, developed by Timothy et al. (1981) for ground-based and space-borne photon-counting imaging systems. The use of a multiple discrete anode structure has obvious practical limitations. Indeed, with an anode density greater than about 1 mrnp2,the wiring of the tube is quite impractical, and the number of sensing circuits (one per pixel) becomes prohibitive for a large useful area. The most ambitious multianode MCP-PMT intended for cerenkov ring imaging is probably the 400 anode prototype tube described by Oba (1979). It includes an invertertype input optics system, a chevron MCP multiplier, and a multianode base consisting of 400 pins of 0.3-mm diam. arranged in a matrix 12 x 16 mm (0.7-mm pitch). A full evaluation of this tube has also been reported by Oba et af. (1979). Another type of multianode structure that allows a large number of pixels to be identified by using a relatively restricted number of data circuits is the coincidence anode read-out system. This is called the crossbar system because it consists of two sets of orthogonal linear wires or strip anodes (rows and columns) insulated from each other. Such a mulLEP, Laboratories d’Electronique et de Physique Appliquee, 3 avenue Descartes, 94450 Limeil Brevannes. France.

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tianode structure has been used in some prototype tubes. Audier and Boutot (1975) have made an MCP tube of large useful area (20 cm2) having proximity focusing and a 4 x 5 cross-bar collector dividing the tube into 20 elementary 9 x 9 mm2detection areas. A preliminary evaluation of the tube when associated with a mosaic of miniscintillators has also been reported by the same authors. Using a similar design, Timothy et a / . (1981) have shown the feasibility of a 512 x 512 pixel coincidence anode array with pixel dimensions of 25 x 25 pmZ. Many other readout techniques have been explored for high-accuracy sensing of the position of the charge pulse at the output of an MCP multiplier. Reviews of possible techniques have been reported by Lampton (1976), Wijnaendts van Resandt et a / . (1976), and Knapp (1978). Some of these techniques have been used in experimental visible sensitive MCP detectors designed mainly for photon-counting image systems (Kellog et a f . , 1979; Cekowski et a/., 1981; Rees et a / . , 1981). v. Limitations in MCP-PMT performance. MCP-PMTs do not surpass conventional photomultipliers on all characteristics and they present some drawbacks in MCP operation, namely, small electrondetection efficiency, low dc and pulse linearity, and a relatively short lifetime. The electron-detection efficiency of an electron multiplier depends on its geometrical input open-area ratio (IOAR) and on the secondaryemission coefficient and collection efficiency of the first stages. The IOAR of a standard MCP is about 65%. It may be increased to 80-90% with funneled input channels (Pollehn et a / ., 1976; Asam, 1978), although the production of this has not yet been realized. The effective IOAR can in fact be higher than the geometrical IOAR for tubes with proximity input optics because some of the backscattered and secondary electrons emitted by the closed area (glass webbing) enter the channels (Polaert and Rodiere, 1974; Panitz and Foesch, 1976). Rather poor secondary-emission coefficient of the channel material (a,, = 2 ) means that the probability of zero signal in the first amplification “stages” cannot be neglected (Lombard and Martin, 1961). This would appreciably lower the effective detection efficiency to values not much higher than the geometrical IOAR (Pollehn et al., 1976). Concerning linearity, the behavior of an MCP as an electrical signal amplifier differs according to whether the dc mode or the pulse mode is considered (Loty, 1971; Baumgartner and Gilliard, 1976). Assuming a homogeneous current input, the amplification departs from linear when the mean output current reaches about 10% of the MCP strip current. Typically, the strip current is a few pA cm-2 (set by a few mW cm-2 heatdissipation limit for a plate), and consequently the maximum direct cur-

256

J . P. BOUTOT, J. NUSSLI, AND D . VALLAT

rent available is only some hundred nA cm-2. This is nearly 100 times smaller than the value currently available from a conventional multiplier structure. In the pulse-mode operation, parameters other than the strip current have to be considered (Boutot and Delmotte, 1976; Audier et al., 1978; Eberhardt, 1981). The high resistivity of the MCP material limits the number of electrons which can be supplied to the channel walls in a given time. The recovery time needed by the strip current to neutralize the positive surface charge is long compared with the signal pulse durations generally found in nuclear detection. A few milliseconds would be a fair order of magnitude for the recovery time when a channel has delivered a charge of lo6 electrons. During this time, this channel is nonoperational; other channels are hardly affected. This unusual behavior for a multiplier calls for an examination of the pulse arrival rate in the case of repetitive signals. With signals having a short duration compared with the recovery time and having a frequency low enough to allow a full charge replenishment of the dead area between two consecutive pulses, the charge linearity can be estimated by considering the MCP as a simple capacitor. For a standard MCP (i.e., d = 12.5 pm, l / d = 40), the maximum linear output charge is about lop9 C cm--2. As a result, the peak current linearity will depend directly on the pulse duration. For durations of 5-50 nsec, the response linearity of an MCP-PMT is lower than that of a conventional focusing structure. On the other hand, linear peak currents greater than 1 A cm-2 are available for subnanosecond pulses. Such a performance is only obtained by very special conventional structures. Multichannel plate behavior under high pulse-counting rates, i.e., with a mean time interval between pulses short compared with the charge recovery time of a channel, shows that the response linearity is retained as long as the mean output current delivered by the MCP does not exceed 10% of the strip current of the operating area of the MCP. The maximum counting rate for a single-electron pulse is about lo6 pulses sec-I cm-2 for a gain of lo6 and, of course, the rate is lowered by a factor of 100 for input pulses containing an average of 100 electrons. The lower the gain the higher the maximum available counting rate. This is a severe drawback which limits the use of such PMTs to rather low counting-rate applications (Nieschmidt P r al., 1982). Finally, the feature which has up to now severely hampered attempts to incorporate MCP-PMTs in nuclear physics detection experiments, is their very limited operational lifetime. The loss of anode sensitivity during operation is due to a degradation of both photocathode sensitivity and MCP gain. It was first observed in image wafer tubes, that to a certain ex-

RECENT TRENDS IN PHOTOMULTIPLIERS

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tent, this loss is due to photocathode damage by ion bombardment (Boutot, 1971; Peifer, 1976). Many aspects affect this feature, including input optics system design, production processes, and of course, tube operating conditions. Damage of the photocathode is more rapid in tubes having proximity-focusing input compared with more conventional input where ion trapping by some electrodes may take place. Whatever the tube design, the drop of the photocathode sensitivity depends directly on the total amount of output charge delivered by the MCP multiplier. In spite of the use of curved channels or of a stack of MCPs, plus a rigorous MCP scrubbing process, the lifetime of the photocathode of MCP-PMTs having proximity-focusing input is currently considered as being too short. Meunier and Maurer (1978) have observed, during the evaluation of a multianode MCP-PMT, a drop of the blue sensitivity by a factor of 2 for an accumulated output charge of a few lop2 C cm+. Lo and Leskovar (1981b) have carried out a preliminary life test on a PMT containing three MCPs with no better results. A solution for overcoming this problem could lie in the use of a very thin (5- 10-nm thickness) aluminum film acting as an ion barrier in front of the MCP multiplier (Pollehn et a / . , 1976; Woodhead and Ward, 1977). Such a solution proved very efficient in the third generation of wafer tubes (Csorba, 1979). At present this improvement is obtained at the expense of the detection efficiency of the MCP. Use of an ion barrier film in MCP-PMT, planned for future generations of devices, should offer the possibility of a reasonable photocathode operational lifetime (say, above 1 C cmP). As for the photocathode sensitivity, the MCP gain variation is affected by the total amount of accumulated output charge (change of the secondary-emission coefficient of the channel walls). Only a few results have been reported in literature about MCP lifetime. As observed with SCEM, the results are not always consistent because gain variation depends both on the processing history of the multiplier and on its operating conditions (Ruggieri, 1972; Sandel et al., 1977). Measurements of variation of multiplier gain carried on high-gain MCP-PMTs have been reported (Oba and Rehak, 1981; Lo and Leskovar, 1981b). No matter what the multiplier configuration, all the results show a monotonic curve of gain degradation when the accumulated charge reaches a few C cmP2. However, it has been shown that it has few practical consequences because the gain can be kept constant by progressively increasing the voltage on the MCP multiplier (within, of course, certain limits) (Timothy et al., 1981). Oba and Rehak (1981) have even observed on cascaded MCP multiplier PMTs that the MCP gain becomes steady when the accumulated output charge reaches about 1 C cmP.

258

J. P. BOUTOT, J . NUSSLI, A N D D. VALLAT

In spite of the current drawbacks, MCP-PMTs are very attractive devices for future experiments in the nuclear physics field. However, it must be remembered that the sophisticated technologies needed to make MCP multipliers and to process proximity-focusing tubes will result in such PMTs remaining more expensive to manufacture than PMTs of conventional structure, even in large-scale production because MCP technology is capital intensive. 2. Other PMT Structures

a . Static Crossed-Field Multiplier Structures. A static crossed-field multiplier structure uses both static electric and magnetic fields to control the electron trajectories between the dynodes. Such a structure was first reported by Zworykin et ai. (1936). The high-speed response features of this geometry were stated by Smith (1951), and a full evaluation of a crossed-field photomultiplier specially designed for the detection of wideband modulated light was carried out by Miller and Wittwer (1965). An experimental version having a design more suitable for use in nuclear physics applications (mainly making use of a large-area semitransparent photocathode and a larger number of multiplication stages) was described by Lodge et al. (1968), but further development of this tube was not undertaken. Tubes with the design suggested by Miller and Wittwer have been manufactured for some time by Sylvania Electronic Systems'O (Sylvania model 502). More recently, a crossed-field PMT family with improved electron optics system and microwave-collector design has been developed by Varian" (Wilcox et al., 1979). These tubes are particularly useful in applications demanding wide bandwidth, high modulation rates, or the detection of very short light pulses. Their timing performance is discussed in Section II1,A. The opaque photocathode having a small area of a few mm2, only accessible with collimated light, and the limited gain (lo3- lo5) make these tubes unsuitable for use in scintillation counters and single-photon timing experiments. The bulky structure (size and weight) and fixed gain also hamper their use in nuclear physics experiments. b. Hybrid PMTs. A hybrid photomultiplier tube is one which combines in one evacuated envelope a photocathode and a solid-state silicon-diode multiplier element, operating in the electron-bombarded semiconductor (EBS) mode. Focusing of the photoelectrons on the very small area of a silicon diode is obtained with either electrostatic or magnetic electron optics systems. The high tension of the photocathode relal o Sylvania Electronic Systems, Western Division, P.O. Box 188, Mountain View, California 94040. Varian LSE, 601 California Avenue, Palo Alto, California 94304.

RECENT TRENDS IN PHOTOMULTIPLIERS

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tive to the semiconductor is such that the photoelectrons strike the diode with an energy between 10 and 30keV. The amplification mechanism in a solid-state diode is based on the creation of multiple hole-electron pairs by each incident high-energy electron. Since the pair creation energy in silicon is approximately 3.6 eV, this results in electron gains in the target of a few lo3. The description and the principle of operation as well as the capabilities and limitations of EBS diode amplifiers have been described by Silzars er a / . (1974) and by Bates ef a / (1977) in review papers on EBS power devices. The first attempts to use an EBS diode in a PMT dates from the late 1960s. At that time, the use of such a multiplier device promised significant advances compared with the conventional multiplier structure in pulse-response linearity and statistical gain fluctuation (i.e., in signal-tonoise ratio). Kalibjian (1966) and later Fertin er al. (1968) demonstrated the feasibility of hybrid tubes having a peak output linearity of several amperes. A larger improvement of the resolution of the single-electron spectrum was obtained with the tube designed by Chevalier (1967) ( R = 30% compared with about 130%for a cascade of dynodes at that time). The potential applications of such tubes are in weapon and laser plasma diagnostics and in nuclear scintillation spectrometry , particularly for lowenergy radiation. The development of these tubes was dropped for many reasons. Early in the development, the combination of an alkali antimonide photoemissive layer and a solid-state diode within a vacuum envelope presented some compatibility problems due to damaging of the diode characteristics by alkali poisoning and baking. These were partly solved by using the transfer method for processing the photocathode (Pietri, 1968). However, a large development effort was needed on the diode technology. Moreover, although a hybrid design has advantages, it also has severe drawbacks such as low gain (needing very high performance associated electronics) and the requirement for a high voltage supply with consequent electrical limitations on the device. Finally, the unique advantage of low gain fluctuation lost ground when Simon er a / . (1968) and Krall er al. (1970) showed that an NEA material, with high secondary-emission power, could be used in a practical device. Improvements in technology and electron optics design later showed also that high-current output (well over 1 A) could be obtained by means of conventional technology, e.g., the Philips photomultiplier XP 1143 (Philips, 1971). At the beginning of the 1970s however, the unique property of single-electron detection of an EBS diode was combined with its very small size to produce a new class of multidetector device by including an array of PN diodes in one envelope. Such a tube having 38 diodes was

260

J . P. BOUTOT, J . NUSSLI, AND D. VALLAT

first reported by Beaver and McIllwain (1971). The name DIGICON (digital image tube) was given to such a category of multichannel photon-counting image tubes developed by Electronic Vision Company (EVC, Division of Science Applications, Inc., 11526 Sorrento Valley Road, San Diego, California 92121). Various types of such tubes have been manufactured during the past ten years. These can be divided into two general categories depending on the type of diode array used (Choisser, 1976a,b). The parallel-output DIGICONs have either a linear or a matrix array of diodes, each connected by an output lead to external electronics. DIGICON tubes have been produced in this configuration with up to 212 diodes for use in astronomical instrumentation. The development of tubes with a single large-area diode (23 mm2),intended for scintillation detection in some specific exploration techniques (borehole, airborne radiometric instrumentation), has also been reported (Orphan et a/., 1978). The advantages expected of such a solid-state PMT over conventional tubes are in improved photoelectron statistics, gain stability, ruggedness, and compactness. A tube having a 2 x 2 array of silicon diodes on a common substrate is known as a quadrant photosil detector (QPD) and is also marketed by EVC. An evaluation of such a device as a photoelectric sensor in telescope autoguiding systems has been made by Jelley (1978) and Fegan and Craven (1977). A nine-channel DIGICON tube was also investigated for a very special airborne scintillation detector in which it appeared suitable for replacing an array of photomultipliers (Ginaven et al., 1976). As a general feature, all these tubes, which can be considered as members of the PMT family, are operated at a relatively low voltage, 10-15 kV, compared with 35-50 kV used in the tube designed by Chevalier (1967), and have maximum gains around lo3. Their operation therefore requires the use of a high-gain preamplifier. The limited bandwidth of such electronic circuitry precludes the use of hybrid tubes in applications where high count rates (> lo5 counts sec-') or short time resolution are required. To overcome this drawback, EVC has been investigating the use of avalanche photodiodes (APDs) in DIGICONS (Choisser, 1976a,b). Preliminary tests carried out in a demountable vacuum system have shown that gains of lo5 or more can be achieved with APDs operated in the EBS mode. The successful development of single or multichannel widebandwidth APD-DIGICONS should offer distinct advantages for the detection of faint high-frequency optical signals. For applications requiring more than a few hundred detection elements (diodes), such as low-light-level imaging, EVC is also developing a serial-output DIGICON family by using either self-scanned diode linear arrays or two-dimensional charge-coupled devices arrays (Science Appli-

RECENT TRENDS IN PHOTOMULTIPLIERS

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cations Inc., 1980). These tubes are built around existing monolithic silicon photon detectors (marketed by Reticon, Fairchild, and some other firms) which are adequately modified to be operated in the EBS mode. Development work on similar imaging devices was also carried on in other laboratories (Cekowski, 1976; Brown ef al., 1976; Caldwell and Boyle, 1976). Johnson (1982) has described the various types of electron-bombarded selfscanned array photodetectors that have been made since the invention of CCDs in 1970. The operational characteristics, possible new designs, and applications of such detectors are also reviewed in this well-documented paper. To summarize, the main advantage of an EBS phototube is probably its position-sensitive potentiality. Other features such as stability, ruggedness, low power requirement only appear essential in some particular applications. Its realization needs very sophisticated technologies which at present make it very expensive because it is not yet in large-scale production. The problems met at an early development stage, such as radiation damage induced in the diode by electron bombardment and afterpulses, are now said to be solved. Two other original structures of hybrid phototubes which are being studied should be mentioned here. One, proposed by Varian (Wilcox et al., 1979), uses an EBS diode as the anode in conjunction with a conventional but optimized all-electrostatic multiplier structure. A response time of 225 psec (FWHM) has been reported for this PMT using only two dynodes with gains ranging from lo2 to lo4. The other is a twodimensional position-sensitive detector making use of a single large-area diode which acts as an amplifier and localizer (Roziere, 1977). Eventposition information is obtained by analogical processing of the signals collected on the ends of resistive electrode(s) acting as a signal divider. The design of such a tube is similar to that of a first-generation imageinverter tube in which the output screen is replaced by the positionsensitive silicon target operating in the EBS mode. It has been particularly investigated for the detection and localization of scintillations in image-intensifier gamma-camera systems by replacing the set of PMTs now used. The evaluation of such a detector in a large FOV imageintensifier gamma camera has been reported (Roziere er al., 1981). c . Transmission Secondary-Emission ( T S E ) Dynode Structures. A TSE dynode structure is simply a stack of flat, thin parallel foils of secondary-emitter materials operated in the transmission mode. A high interstage voltage (i.e., a few kV or even more) is required, depending on the film thickness and on the material density. The combination of a high interstage voltage with uniform close spacing of the dynodes (e.g., a few millimeters) makes, in principle, such a structure very suitable for the de-

262

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BOUTOT,

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NUSSLI,

AND

D.

VALLAT

sign of fast PMTs (reduction of transit-time differences and transit-time spreads-see Section 111,A). Low-density evaporated layers of alkali halides, such as KCI and CsI, have been thoroughly investigated as TSE materials (e.g., Smith et a / . , 1966; Hagino et a / . , 1972). They have been used mainly to manufacture high-gain lo3- lo4 image-intensifier tubes in which image focusing with high spatial resolution is achieved by a combination of uniform electrostatic and magnetic coaxial fields. The poor secondary-emission statistics of the materials used together with instability phenomena induced by charging effect have hampered the development of PMTs with a TSE dynode structure. During the early 1970s, a large research and development effort was dedicated to the evaluation of NEA materials for use as transmission-mode secondary emitters (Williams, 1972; Fisher and Martinelli, 1974; Howorth et al., 1976). Among the various materials investigated, the most promising was probably GaAs. High-transmission secondary-emission coefficients, around 200 at a primary energy of 10 keV, have been reported, and the realization of thin dynodes having a suitable useful diameter (e.g., 10-20 mm), albeit very fragile, has been considered achievable (Olsen et a / ., 1977). The impulse time response of a TSE GaAs dynode structure is not likely to be as fast as conventional dynodes because the time for diffusion of the secondary (thermalized) electrons to the surface is long (1 nsec or more) compared with that of the “hot” electrons in a conventional positive electronaffinity material (estimated as lower than sec). Theoretical calculations indicate that the response time of such dynodes should be in the range of a few hundred picoseconds (Bell, 1973). A full experimental evaluation of a photomultiplier with a TSE dynode structure using NEA material does not appear to have been carried out yet. The application of such a technology to photomultipliers appears to be very limited at present.

111. PRESENT SITUATION ON MAIN PHOTOMULTIPLIER CHARACTERISTICS A . PMT Timing Performance

Many investigation methods and measurement techniques in nuclear physics depend on the detection and timing of fast but generally lowlight-level phenomena by photomultipliers. Timing means either measurement of the precise time of occurrence of an event or the determination of the temporal profile of a light phenomenon. The timing performance of the detection systems used is particularly linked with that of the PMT. The timing capability of PMTs has been the subject of intensive theoretical

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and experimental investigations (Gatti and Svelto, 1966; PiCtri and Nussli, 1968; Krall and Persyk, 1972), and comprehensive surveys of the literature have been given over the past ten years by Poultney (1972), Seib and Aukerman (1973), Pietri (1973), Leskovar and Lo (1975), and Moszynski and Bengtson (1979). This section is devoted to the time-related properties of PMTs. The four main characteristics that relate to PMT time performance are the following: signal transit time, pulse response, transit-time differences, transit-time fluctuation or time resolution, and these are defined below. The influence of the PMT operating conditions on each of these characteristics is discussed and the state of the art in fast PMT structures, as well as further improvements, is reviewed. 1 , Signal Transit Time

The signal transit time within a PMT is the time interval between the illumination of the photocathode by a very short light pulse and the appearance at the anode of an arbitrarily determined characteristic point (e.g., the peak) on the corresponding anode current pulse. This time interval varies from one light pulse to another due to transit-time differences and transit-time fluctuation of the electrons in all the PMT stages. The signal transit time ft is the time averaged over a large number of pulses (Fig. 12); 7, varies as V i i / 2for a given tube where Vhtis the total applied voltage between photocathode and anode. From a practical point of view, the transit time introduces a delay within the experimental system. It is typically several tens of nanoseconds for fast high-gain tubes of conventional structure. That is not generally a limiting factor in applications. A much shorter transit time can only be achieved with unconventional structures: a typical value for a high-gain MCP-PMT with a proximity-focusing input is 1 nsec. 2. Pulse Response a. DrJinitions. The pulse response &(f) is the resulting anode current pulse when the photocathode is illuminated by a delta function light pulse 8(t) (Fig. 13). The pulse respoiise has a definite width (anode pulse duration). The broadening of the response is due to fluctuations in the electron transit time in the various stages (in the input optics system and between

264

J . P. BOUTOT, J . NUSSLI, A N D D . VALLAT Anode pulse RS(t)

n

_

_

-_- - __ - _- - - -

__ I 5 ///'I /

S -

-- - - -- --_ -- -

0

___

VIXI

Probability density R(t) of anode pulse arrival instants

I = P

A

I

I

I I

-1

I

-'

I I

1-

,-I ;-

//"

Mean transit time

'\\

Full width ot half maximum R t of the distribution R ( t )

-I

Tt

FIG.12. Photomultiplier time characteristics.

FIG. 13. Characteristics of photomultiplier pulse response.

RECENT TRENDS IN PHOTOMULTIPLIERS

265

the successive dynodes of the multiplier). The fluctuation in each of the stages can be characterized by a probability density function (PDF) with standard deviations U K , d , , ( T d l , d z , . . . , u ~ , \ - etc. ~ , ~Assuming +, the PDF at the anode is represented by a Gaussian distribution R,(t)

= (2mR)-1'z

exp[ - (t - tJ2/2uk]

(6)

the variance u i of the anode pulse response is then the sum of the variances of the responses of the stages

+ (J2di.dz + ( N

&l= d C , d l

- 1)u2d,d

(7)

The pulse response is generally characterized by its FWHM t , (anode pulse duration) related to uRby t,

=

2.36~~

(8)

or by its rise time t,,, defined as the time difference between the 10% and the 90% amplitude points of the leading edge (Fig. 13). The time response of an electronic device is generally characterized by the output pulse rise time t , , (10-90%) with a step-unit-shaped input signal E ( t ) (Fig. 13). The system bandwidth is then related to tr,, by B = 0.35/tr,, (9) Light sources with a step-unit-function emission are not currently available for recording photodetector step response. By fitting the PMT pulse response to a Gaussian distribution, it can be shown that t,, the FWHM of the pulse response, is roughly equal to the step rise time tr,, and consequently the PMT bandwith is related to t, by

B,,,

= 0.35/tW

(10)

The transit-time fluctuation in each of the stages of the PMT has two main causes (PiCtri and Nussli, 1968):

(1) Differences in transit time of electrons emitted from different points of the electrode (photocathode or dynode). It has been shown that the difference in the transit time At between two electrons emitted with zero initial velocity, normal to the electrode and describing two trajectories of length L differing by a small amount AL in a homogeneous electric field of strength E , is given by At = (m/2e)1'2E-1'2L-1 AL

(11)

where m and e are the mass and the charge of the electron, respectively. ( 2 ) Spread in initial velocity (energy and direction) of the emitted electrons. The transit-time difference between two electrons emitted with an initial velocity normal to the electrode and with an energy difference

266

J . P. BOUTOT, J . NUSSLI, A N D D . VALLAT

A W is given by At'

=

(2m/e)1/2 AW1'2E'-1

where E' is the electric field at the emitting-electrode surface and m and e are as previously defined. The tangential initial-velocity component shifts the point of impact of the electrons on the electrode by a certain distance, but generally the transit-time change introduced is negligible. Relations ( 1 1) and (12) show that the pulse-response width t , varies as V;ll"with 4 < n < 1 . As shown in Eq. (7),the input optics system plays only a small part in the pulse broadening between photocathode and anode. However, the illumination level of the photocathode and the surface of the excited area has an influence on t, . The shortest t , value for a given tube is obtained for a single-electron response ((T;,~~ = 0) or for illumination on a very small photocathode area (At = 0). Equation (12) also shows that the wavelength of excitation may have a small influence on t , as it determines AW. Other factors affecting the PMT pulse-response width are the spacecharge effect taking place in the final stages with high-level output currents, and the electrical matching of the anode to the output transmission line. h. Pulse-Response Measurement. Because a delta light pulse is not easy to produce experimentally, the measurement of the pulse response is generally carried out by exciting the photocathode of the tube in one of two ways: ( 1 ) By single photons: the so-called single-electron response (SER) which characterizes the multiplier structure only; SER measurement can lo7) since limitations be performed only with high-gain tubes (e.g., G are introduced by the sensitivity of large bandwidth oscilloscopes ( 2 ) By light pulses having a duration t,,, short compared to the expected pulse response of the tube. Various types of sources of subnanosecond-duration light pulses are available: spark sources, cerenkov sources, light-emitting diodes, solid-state laser diodes, Nd: YAG mode-locked lasers, etc. The main properties of these sources together with current oscilloscope-measurement techniques were reported by Krall and Persyk (1972), but it should be noted that more recently the performance of oscilloscopes has improved considerably. For repetitive signals, sampling oscilloscopes are superior in both bandwidth and sensitivity to the best traveling-wave oscilloscopes. An acquisition system with 10-GHz bandwidth and a few millivolts sensitivity has been described by Hocker et ul. (1979). However, in some cases, sampling techniques are not usable and real-time measurements have to be carried L-

RECENT TRENDS IN PHOTOMULTIPLIERS

267

out with a high-speed oscilloscope. The CRT performance has been much improved thanks to the use of an MCP (Clement and Loty, 1973). Oscilloscopes including such a CRT, having a 4-GHz bandwidth and 2-V cm-' sensitivity, are now commercially available (e.g., Thomson-CSF TSN-660 oscilloscope"). Other oscilloscope systems with higher sensitivity (140 mV cm-') have also been made (Sipp r t al., 1975; Kienlen el a / . , 1976). The experimental pulse response R,Yi,oscillogram is the combination of the lighting function L,,), the pulse response of the tube R6(,),and the system pulse response S(,) (delay lines, cables, oscilloscope): RT,, = Lo, *

R6,t)

* S,t,

(13)

and R,,,, can then be deduced. If L,,, has Gaussian distribution with a FWHM t,,, , the pulse response t , of the tube can be simply deduced from the experimental pulse-width response t z using the following relation: (14) t g = t$,, + t& + tf,, where tr,s is the rise time of the acquisition system step response. The shorter the light pulse and the higher the system bandwidth, the higher the accuracy of determination of the pulse response of the tube. c . PMT Pulse-Response Data. All the fast conventional PMTs use a large-area semitransparent photocathode and a linear-focusing multiplier structure having typically 10-12 stages. They are designed to obtain high-detection efficiency and time resolution in single-photon-condition measurements. These tubes have a typical pulse response t , of 3 nsec (FWHM). Early efforts to improve the pulse response of PMTs have been centered around the use of larger interdynode voltages, and as a result, tubes with t, = 1 nsec and an overall operating voltage of 5-6 kV have been produced (e.g., Amperex13 XP 1210, RCA C 7045). As indicated by Eq. (7), a reduction oft, can be obtained by reducing the number of stages. By using high secondary-emission-power material, high gains can be retained. With five GaP(Cs) dynodes, the RCA C 31024 has a t , = 1.3 nsec and a gain of lo6 for an interdynode voltage of 500-600 V. A t , value of about 800 psec was reported by Krall and Persyk (1972) for a developmental three-stage version (RCA C 3 1050). As previously mentioned, the circular-cage multiplier structure, because of its focused-dynode arrangement, has a short time response. With only a few dynodes operated with a

Thomson-CSF, Department Applications Speciales de I'Instrumentation, 23-27 rue Pierre Valette, 92240 Malakoff, France. l 3 Amperex Electronic Corp., 230 Duffy Ave., Hicksville, New York 11802.

268

J. P. BOUTOT, J. NUSSLI, AND D . VALLAT

high interdynode voltage (400-500 V) and with carefully designed circuitry, such a type of tube can provide t , values in the range 550-700 psec with gains = lo4 (Beck, 1976; Lyons et al., 1980). One of the potential applications for tubes with such characteristics is the observation of very fast processes in flash photolysis and pulse radiolysis. The optimum compromise for the pulse response and gain which can be obtained with a conventional dynode structure is probably that provided by the five-stage Varian VPM 152 ( t , = 400-500 psec with lo4 gain). The practical bandwidth limit for an all electrostatic conventional design PMT is certainly around 700 MHz. To obtain a larger bandwidth, one needs to use other designs of multiplier structure. The static crossed-field PMTs (see Section I1,D) developed by Varian to meet the requirement of ultralarge bandwidth optical receivers for high-speed laser communications may give a reasonably good gain-bandwidth factor (typically, lo5 x 1 GHz o r lo3 X 1.5-2 GHz). As previously mentioned, the small cathode area (a few mm2) and the extremely limited linearity (= 10 mA) of such a tube restrict its application range (Leskovar and Lo, 1978). Originally, Boutot and Pietri (1970) reported on the use of a channel plate multiplier in a structure very similar to that of an ultrafast vacuum photodiode to obtain a high-gain-bandwidth performance and a high linearity. These detector features are essential for many applications which have emerged recently, particularly those dealing with laser-matter interaction studies. They include laser-fusion research (plasma diagnostics), molecular physics (short fluorescence or radioluminescence decay determination), radiation-to-light converter decay-time characterization (faster plastic scintillator development), fiber-optic pulse dispersion studies, weapon diagnostics, etc. As explained in Section II,D, the pulse response, gain, and linearity of an MCP-PMT depend considerably on the geometrical characteristics of the MCP multiplier. The shortest response is obtained by using one single thin MCP (small d and l / d ratio) and by integrating the output into a coaxial structure matched to a 50-0 transmission line. This is obtained at the expense of gain. It has also been mentioned that the maximum charge available at the output of an MCP and the pulse duration determine the peak current linearity: the charge linearity is about C cmP2. Fast MCP- PMTs with various geometrical configurations have been built by ITT, Varian, ITL,14 HTV, and L E P (see Section 11,D). They have different gain- bandwidth linearity performances. Pulse-response measurements on MCP-PMTs have been reported in many papers. A full inl4 ITL, Instrument Technology Ltd, 29 Castleham Road, St. Leonard-on-Sea, East Sussex TN38 9NS, United Kingdom.

RECENT TRENDS IN PHOTOMULTIPLIERS

269

vestigation was carried out by Hocker rt a / . (1979). A minimum FWHM of 210 psec has been recorded with a Varian VPM 221 D photomultiplier having a small useful area (6-mm diam.) and consequently a linearity down to 1 A. Maximum peak linear currents up to 1 1 A have been reported with an 18-mm ITT tube ( F 4126 G) having a 330-psec FWHM pulse response. Values of 230-250 psec have been also reported by Boutot and Delmotte (1978- 1979) for an experimental tube made by LEP (type HR 305). The time performance of a fast scintillation detector including such a PMT has been documented by Gex et a / . (1978). As expected, the space-charge saturation occurring at very high gain (G = a few lo4)into the channels affects the pulse response (Boutot et a / . , 1977). Lyons et a / . (1980) have shown that MCP-PMTs can satisfy many specific needs for laser fusion diagnostics, particularly in conjunction with newly developed fast plastic scintillators that have decay times of only a few 100 psec (Lyons et a / . , 1977).

3 . Transit-Time Differences The mean transit time depends on the location and size of the illuminated area of the photocathode. Transit-time difference is the difference of the mean transit time between two spots (or very small areas) of illumination on the photocathode. The shortest mean transit time is generally obtained when the photocathode is illuminated at its center. It increases from the center to the edge, mainly because of the difference in the trajectories of electrons between consecutive electrodes in the tube. The largest transit-time differences appear to arise in the input optics system because all the photoelectrons, irrespective of their point of emission, are focused on a small area of the first dynode, whereas in the multiplier stages the electron trajectories are more closely bunched. So the center-edge transit-time difference AtCEis mainly a characteristic of the input optics system of a PMT. Transit-time differences are minimized by designing the input optics system with a curved window. As indicated by relation (11), the transit-time difference varies as and thus as V-ll2 where V is the voltage across the input optics. It is therefore beneficial to increase the voltage in the input optics system in order to reduce the transit-time difference. In fast PMTs, the input optics configuration is such that the electric field near the photocathode can exceed 100 V cm-l under normal operating conditions. Electrons leaving the photocathode are therefore quickly accelerated to a high speed so that their final energy on arrival at the first dynode is large (a few hundreds of electron volts) compared with their initial mean energy (a few tenths of an electron volt in the visible

270

J . P. BOUTOT, J . NUSSLI, A N D D. VALLAT 1

20

15

10

5

0

5

10

15

20

Distance from the photocathode center d (mm)

FIG.14. Mean transit-time differences as a function of the distance from the center of the photocathode when illuminated by a small spot for a 50-mm fast PMT (Philips XP 2020). Measurements along two perpendicular diameters X and Y (by Courtesy of RTC, Paris).

range). The mean transit time of electrons in the input optics system is then independent of their mean initial energy and, consequently, of the wavelength of the illumination light. It also can be shown that relation (1 1) is still valid whatever the mean initial energy of the photoelectrons. Thus, in most cases, the wavelength of the illumination has no significant influence on the transit-time differences. As the first dynode configuration generally introduces asymmetry in the input optics system, the evolution of the transit time along a diameter may depend slightly on its orientation. Results are generally given for two orientations of diameter, one parallel and the other perpendicular to the dynode axis (Fig. 14). For conventional fast photomultipliers, with 50-mm diam., the center-edge transit-time difference ATCEcan be as small as 0.25 nsec (de La Barre, 1973; Sipp and Miehe, 1974). As is seen in the next section, this characteristic contributes largely to the time resolution of the PMT. It should be noted that this contribution will be much reduced if the illumination source can be collimated, which is not the case in nuclear physics applications. The only way to make the transit-time differences negligible is to use a proximity-focusing input optics system. Such a design cannot be adapted to fast conventional multiplier structures but is very well suited to an MCP multiplier used to produce a PMT with better timeresolution performance (see next section). 4 . Transit-Time Fluctuation and Time Resolution

a. Definitions. For a given illuminated area on the photocathode, the transit time fluctuates around a mean value it from one light pulse to another. The transit-time fluctuation is characterized by a probability density function which can be fitted with a Gaussian function 92(o, the

RECENT TRENDS IN PHOTOMULTIPLIERS

27 1

FWHM of which defines the time resolution $?it of the detector (Fig. 12). The time resolution is an important feature because it limits the accuracy of the measurements of the time elapsed between two events. By convention, the time resolution $?it of a PMT is the FWHM of the distribution $?i(t, for single-electron pulses. The variance a;, of this distribution is controlled by certain tube parameters according to the relation:

where the terms (+;,dl and u& are the variances of the transit-time spreads of electrons in the input optics system and in the multiplier, respectively. This relation holds for a PMT with all the stages identical except the first one. So, (T%,,dZ and @$,d are, respectively, the variances of the transit-time spreads in the first stage and in the successive stages of the multiplier, g, and g are the mean gains of these stages, and v,, and v, their relative variances. The transit-time fluctuation of a PMT depends essentially on the fluctuation in transit time of electrons in the input optics system and in the first stages of the multiplier. The fluctuations in each of the stages have two main causes, as stated previously in Section III,A,2:

(1) The transit time differences (geometrical contribution) (2) The electron initial velocity spread (chromatic contribution) Each of these contributions can be characterized respectively by their variances u& and (+;". The variance u k , d , of the fluctuations in the input optics system is then given by u;,dl

= ufv

+ UEE

(17)

The two contributions can be theoretically calculated by relations (1 1) and (12), considering that uCEand uiv vary as Ar ( u C E = 0.3 At) and Art (uiv= 0.5 Ar'), respectively (de La Barre, 1973). As in the input optics system, the contribution of the first stage u&2 to the total fluctuations can be split into two components: a chromatic one depending on the spread in secondary-electron initial velocity and a geometrical one depending on the diffusion-zone area of the photoelectrons on the first dynode. Before investigating the various parameters having an effect upon the time resolution $?it of a PMT, it should be mentioned that the time resolution varies with the mean number FiK of emitted photoelectrons per pulse

272 as

J. P. BOUTOT, J . NUSSLI, AND D . VALLAT

that is,

So the higher n K , the smaller the transit-time spread and, consequently, the better the accuracy of time measurement. This number obviously depends on the pulse illumination level, on the collection of the photons by the photocathode, and on the matching factor between light-source spectral emission and photocathode spectral response. As shown by Eqs. (lo), ( l l ) , and (12), the main operating parameters affecting the single-electron response-time resolution 9it of a PMT are the following: (1) The interelectrode voltage. The chromatic and geometrical components in each stage vary as V-' and V-1/2, respectively, and the gain g per stage [in the denominator of Eq. ( l l ) ] varies as V" (with 0.7 < a < 0.8) as long as its maximum is not reached. Also, the gain variance vg generally varies as g-lI2.It is therefore always beneficial to operate a PMT with high interelectrode voltages, particularly in the input optics system and in the first stage of the multiplier. As shown by Eq. (1 l), the contribution of the multiplier to the time resolution of the PMT can be considerably reduced by making g, very high, i.e., by using a high secondaryemission-power material (e.g., RCA 8850, RCA C 31024 tubes). (2) The illuminated area of the photocathode. When only a small spot of the photocathode is illuminated, the time resolution may depend on the location of the illuminated point because the electrical field strength near the photocathode and the electron-collection efficiency in the input optics system may not be homogeneous. In fast conventional PMTs, the electrode configuration is optimized to ensure a high and homogeneous electric field strength at the surface of the emissive electrodes, thus the time resolution is almost independent of the location of the illumination point on the photocathode (except for spots very close to the edge); i.e., the contributions of mCEis negligible. When a large part of the photocathode is illuminated, the transit-time differences add to the transit-time fluctuation and impair the time resolution. However, in fast conventional PMTs, this additional contribution uCE is only significant when a rather large area of the photocathode is illuminated. (3) The wavelength of the illumination source. For a given photocathode spectral response, the mean energy and the energy dispersion of the photoelectrons depend on the incident light wavelength: they increase monotonically with increasing photon energy. As a conseqence, the time resolution is dependent on the relative spectral incident light emission [see Eq. (7)]. That has been shown experimentally by several authors (de La Barre, 1973; Wahl et al., 1974; Sipp et al., 1976; Moszynski and

RECENT TRENDS IN PHOTOMULTIPLIERS

273

Vacher, 1977; Calligaris et a l . , 1978). A change of wavelength has an influence not only on the chromatic contribution of the input optics system but also on the geometrical contribution uEof the first stage of the multiplier since the spreading area of the electron beam on the first dynode results from the spread of the photoelectron tangential initial velocity component (de La Barre, 1973). For fast PMTs with a bialkali (Sb, K, Cs) photocathode spectral response, the improvement in singleelectron time resolution may reach 30-40% from A = 400 nm (photocathode spectral response peak) to A = 560 nm (close to the photoelectric threshold) (de La Barre, 1973; Moszynski and Vacher, 1977). In spite of a lack of experimental data on the variation of the photoelectron initial velocity distribution with wavelength, the theoretical estimations of the influence of wavelength on time resolution carried out by de L a Barre (1973) and Moszynski and Vacher (1977) are in close agreement with the experimental results they have obtained. b. Time-Resolution Measurement. The time resolution 9$ of a photomultiplier is dependent on the incident light relative spectral emission but is theoretically independent of the time distribution of the photons on the PMT photocathode since the transit time is referred t o the instant of arrival of one single photon on the photocathode. However, in practice, the zero time reference is the instant of arrival of the light pulse on the photocathode. Thus the practical time resolution 92: also depends on the probability density function of the instants of emission of the photons. If L,,, is this illumination function, then the time distribution 92;) is given by

36,= Lo, *

%t,

(19)

and the FWHM of this distribution defines the experimental singleelectron time resolution %& of the PMT. If the illumination function &) has a duration short enough compared with the intrinsic PMT time resolution 92, (i.e., FWHM of I100 psec if fast conventional PMTs are considered), then the experimental time resolution %? is a good evaluation of %t (92; = 92,). However, in practice, the light-pulse sources such as spark sources and light-emitting diodes (LEDs) have a time duration (FWHM) which is not negligible compared with 92,. As L(,)and 92(t,can be generally described by Gaussian distributions with standard deviations of uLand ut, respectively, then 926)is also a Gaussian function with a standard deviation uz given by u;, =

[Ui

+u;y

(20)

So the experimental time resolution becomes 92r

=

2.36~:

(21)

In the case where EK photoelectrons are emitted per pulse, then by follow-

274

J. P. BOUTOT, J . NUSSLI, A N D D. VALLAT

ing the same mathematical process, the experimental time resolution is given by

92;

(;#,=

2.36[(ai

+ uf,)/iiK]”’

(22)

The principle of PMT transit-time fluctuation measurement has been described by many authors (Vallat, 1969; Poultney, 1972; de La Barre 1972a,b; Leskovar and Lo, 1975; Leskovar, 1975). The LED is the most commonly used light source for this measurement because it is inexpensive and easy to operate. It can provide pulses of very short duration (down to about 200 psec FWHM). The only fast LEDs available at present have their spectral emission centered in the green-yellow range, so intrinsic time-resolution data obtained on PMTs with a blue-sensitive photocathode (e.g., Sb, K, Cs type) may not reflect the effective contribution of the PMT in a scintillation counter because of the possible minimization of the chromatic contribution. Transit-time fluctuation measurement with a high resolution needs a suitable detection circuit (also called a time pick-off circuit or timing discriminator) to determine the arrival time of the current pulse at the anode of the tube. Different timing methods have been investigated in the past. They have been described and discussed by Gedcke and Williams (1968), Poultney (1972), Lo and Leskovar (1974), Wright (1977), and more recently by Moszynski and Bengtson (1979). Very useful information concerning timing techniques and instrumentation has been given by Ortec (1977) in an application note. The contribution of the time-derivation circuits to the experimental time resolution 92; is generally very small, say around 25 psec, which may be considered as negligible in single-electron timing measurements but not in multiphotoelectron time-resolution evaluations. c . Time-Resolution Data. In many nuclear physics timing experiments, such as particle-velocity measurement by time-of-flight (TOF) determination, measurement of the lifetimes of excited states, scintillator timing characteristics evaluation, the measurement accuracy depends, among other parameters, on the time resolution of PMTs. In the experiments using the single-photon method (Poultney , 1972), the limitation on accuracy is determined primarily by the PMT, whereas in scintillation counters there are several sources of time uncertainty many of which may originate in the scintillator itself such as finite decay time of the light-emitting states and in the light coupling between the scintillator and the PMT (Bengtson and Moszynski, 1970; Lynch, 1975; Moszynski and Bengtson, 1979; Brooks, 1979). All the time contributions may generally be added quadratically. Analysis of the possibilities of improving the accuracy of an experimental measurement requires first a determination of the part played by each of the contributions. As the determination of the timing

RECENT TRENDS IN PHOTOMULTIPLIERS

275

properties of the scintillators as well as the light dispersion in scintillators and light guides are obtained by using PMTs, it is obvious that any improvement in the time resolution of the PMT brings direct o r indirect improvements in the measurement accuracy and in the phenomena analysis. Although conventional fast structures had reached timing performance very near the limit which could be expected in the early 1970s, PMT manufacturers have improved these tubes and also developed completely new structures mainly using MCP multipliers during the past ten years. Progress has been made on the photocathode sensitivity of fast conventional PMTs, particularly in the blue range where quantum efficiency has been increased by nearly 20%, bringing a direct improvement on multielectron time resolution. RCA has introduced GaP(Cs) high-gain dynodes in a fast structure (RCA 8575) to reduce the multiplier contribution in the overall transit-time fluctuation (RCA 8850, RCA C 31024). Critical evaluation of the improvements brought about by this new material, as well as evaluation of the part played by the various contributions to the time resolution of the fast PMTs commercially available, have been carried out by de La Barre (1972a,b, 1973, 1974), Sipp and Miehe (1974), Sipp et al. (1976), Leskovar and Lo (1972, 1975, 1978), Leskovar (1979, Lo and Leskovar (1974, 1981a), and Moszynski and Vacher (1977). Analysis of these papers shows that it is difficult in some cases to draw objective comparisons between tubes because the results depend on many operating conditions which are often not specified or not comparable. Moszynski and Bengtson (1979) have collected the time-resolution data obtained by these experimenters in a table which proves this point (see Table 5 of their paper). However, a statement on the timing capabilities of the most commonly used 50-mm fast conventional tubes, such as RCA 8850 (with a high-gain GaP(Cs) first dynode) and Philips XP 2020, can be drawn: their single-electron time resolutions with full photocathode blue illumination, when optimized, are nearly the same and equal to about 450-500 psec (FWHM). That means that the tubes could theoretically be improved further by reducing the multiplier first-stage contribution by means of a high-gain dynode in the Philips tube and by reducing the transit-time fluctuations in the input optics system of the RCA tube. Manufacturers have also recently put some effort into the development of small-diameter tubes (19- and 29-mm diam.) with reduced transit-time spread to meet the requirements of new positron-emission tomography (PET) imaging systems. Classical PET cameras use a time-coincidence technique with a resolving time of 10-20 nsec for the determination of the direction of the two 511-keV anihilation photons (Heath et al., 1979; Ter Pogossian et al., 1980). As investigated by Allemand et al. (1980) and Mullani et al. (1981), the use of TOF information in conjunction with the conventional coincidence technique may improve the image contrast pro-

276

J . P. BOUTOT, J . NUSSLI, A N D D . VALLAT

vided the time resolution between two scintillation counters is small enough. Thus an image-contrast improvement by a factor of 2 is expected for a so-called TOF-PET system having a resolving time of 0.5 nsec (FWHM) between two PMTs. From the evaluation carried out on various potential scintillators, it appears that CsF best fulfills the requirements for this application (Allemand et a / ., 1980; Moszynski et a / ., 1981; Mullani et a / . , 1980). Alongside this work, 29-mm diam. PMTs having a singleelectron time resolution of about 550-600 psec (FWHM), i.e., values very near those presented by 50-mm fast conventional tubes, have been developed to obtain a resolving time of about 0.5 nsec (Moszynski and Vacher, 1981, private communication). A direct localization of positrons with a low enough spatial resolution (e.g., 1 cm) by only using the T O F technique would need a resolving time below 100 psec for which much faster detection systems need to be developed. It should be observed, however, that PET applications require a high detection efficiency which means long scintillators, and so the timing characteristics are mainly set by the scintillator dimensions. Prospects of improving scintillation-counter timing characteristics have been reviewed by Lyons (1977) and more recently by Moszynski and Bengtson (1979). Scintillation counters with organic (plastic) scintillators produce the best time resolution compared with inorganic scintillation counters, but their lower conversion efficiency and energetic radiation-detection efficiency make them unsuitable for this particular application at present. However, improvements are expected in the near future from the development of high-Z loaded plastic scintillators (Lyons, 1977). Regarding the PMT time resolution, only a small improvement may be expected with fast conventional structure PMTs. As explained in Section II,D, the use of an MCP electron multiplier in a proximity-focusing configuration is at present the most successful approach for obtaining a marked improvement in time performance. The study of timing capabilities of MCP-PMTs has been the subject of many papers in the past five years. Preliminary evaluations carried out by Chevalier et a / . (1970) on a prototype tube with a proximity-focusing input optics system showed that such a new design structure was very promising (an upper limit of 200 psec was indicated for the time resolution). A prototype tube with a conventional input optics system associated with a chevron MCP multiplier was also tested by Catchpole (1972). As was to be expected, the time resolution of this tube was defined by the input optics system. Thanks to the gaincharacteristics improvement obtained on the MCP multiplier, a more precise determination of the MCP-PMT time-resolution performance was made possible. Lo et a / . (1977), then Lo and Leskovar (1979), have carried out characterization studies on L E P experimental proximity-focusing tubes (models HR 400 and PM 137) using MCP with curved channels to

RECENT TRENDS IN PHOTOMULTIPLIERS

277

reduce ion feedback. With full photocathode illumination (i.e., 15- and 20-mm diam. for HR 400 and PM 137, respectively), the single-electron time resolution of these tubes was found to be the same with an upper limit of 200 psec (FWHM). The time resolution was not improved by illuminating only a small area, showing that the transit-time differences in the input optics system are effectively negligible. More recently, Lo and Leskovar (1981b) have reported an upper limit of approximately 125 psec (FWHM) for a high-gain proximity-focusing tube (ITT F 4129), including a 3-MCP multiplier and an 18-mm diam. photocathode. The same experimenters have also carried out multiphotoelectron time-resolution measurements on all these tubes with very similar results, i.e., about 30-50 psec for 6000 photoelectrons/pulse. Oba and Rehak (1981) did not obtain such good results on a developmental high-gain (3 MCP multiplier) tube manufactured by Hamamatsu (model R 1294 X). The differences may originate from the use in this tube of an electrostatic focusing input optics system. Because the intrinsic time-resolution performance of PMTs is very delicate to investigate and is not always significant, the time-resolution capabilities of MCP-PMTs could be better demonstrated on a scintillation-counter spectrometer. This has not yet been done, other than a preliminary evaluation made by Uyttenhove et ul. (1978) with an MCP-PMT included in one of the two counters in a subnanosecondlifetime spectrometer. In such systems, the advantages of the use of MCP-PMTs will only be obtained if the tubes have a photocathode sensitivity as high as that of a fast conventional PMT. There is no fundamental reason why this should not be achieved.

B . Linrurity

The proportionality between the number of incident photons and the number of electrons collected at the anode is called charge linearity. Taking time into account, the corresponding proportionality between incident flux and anode current is called current linearity. When the flux is itself a function of time + ( t ) , it is interesting to look for the conditions for which the anode current follows a similar law k+(t). The limitation on this is set by the time fluctuations existing in the photomultiplier. Limits on charge and current linearity are set by both internal and external factors.

I . External Factors Affecting Linearity a . Power Supply. Changes in interelectrode potentials can affect the gain by influencing the secondary-emission factors of the dynodes and the trajectories of the electrons.

278

J . P. BOUTOT, J . NUSSLI, A N D D . VALLAT

FIG. 15. Current sharing between electrodes of a photomultiplier.

(1) Divider cirrrrnt. When electrode potentials are derived from a resistive divider across the terminals of a stabilized power supply, the photocurrent tends to decrease the potential between the last dynode and the anode by an amount AV. Because it affects the current and hence voltage distribution throughout the divider, this causes an increase of gain comparable to what would be caused by increasing the high voltage by the same amount. The various currents flowing in resistors and to electrodes are described on Fig. 15. The gain G is the photomultiplier gain for the anode current I , . For an N-stage tube and a supposed iterative divider (R,= R1 = Rz = . . . = R N , and the voltage V between each dynode), G is given by

n N

G

=

KV*

i=l

where K is a constant and a an exponent of value between 0.65 and 0.75, depending on the type of secondary-emitting layer of dynodes. The ratio of G to the gain G , at zero anode current is given by the formula (RTC, 1981)

where IN is the current flowing in dynode N and Zpthe current given by the power supply for I, = 0. As the quantity under the summation is always less than unity, the quantity between the inner parentheses is positive and the gain G is an increasing function of the ratio ZN/Zp = Za/Zp. Throughout the range INIZ,, the coefficient (between the inner parentheses) of the ratio ZN/Zp does not depend very much on the gain of

RECENT TRENDS IN PHOTOMULTIPLIERS

279

each stage, so, by setting Zi/&

= gi = g

(25)

and noting that g N >> 1 , Eq. (24) can be simplified to G Gm Or, for large values of N , AG -G

G-Gm G

a-- N Za N + 1zp

21

Equation (27) expresses the relative variation of the gain as a function of the ratio of the anode current to the theoretical divider current, provided the voltage decrease across the terminals of the last stage does not impair collection efficiency. The ratio AG/G has the same sign as Za/Zp: any increase in Za results in an increase in gain (Fig. 16). consequently, this increase as a function of I , can be described as an overlinearity. It can be noticed that it is approximately independent of N . For example, assuming N = 10, a = 0.7, and Za/Zp = 0.1, the relative gain change AG/G amounts to about 7%. When the ratio Za/Zp approaches unity, Eq. (27) no longer applies. The voltage drop in the last stage, which increases with I,, becomes too great and the collection efficiency of this stage quickly deteriorates. Hence there is an abrupt reduction in the gain (Fig. 16). Other phenomena can occur at the same time to accentuate this effect. I

I 80

40

:I

11 II

60

I

to-3

10-2

10-1

I

t0 0

la -

IP

FIG. 16. Gain variation as a function of the ratio of the mean anode current to the divider current (overlinearity).

280

J . P. BOUTOT, J. NUSSLI, A N D D . VALLAT

The maximum value of the ratio AG/G mainly depends on the voltage distribution across the tube and on how the currents due to the voltage drop between the anode and the last dynode are distributed among the first stages. The dependence of gain on photocurrent can be modified by substituting a Zener diode for the resistance R , between the last dynode and the anode. It can be shown theoretically, by the same analysis as before, that the overlinearity is then eliminated. Moreover, the anode current at which the rapid drop in gain occurs becomes about ten times higher. This is only so if the multiplier is strictly iterative. If that is not the case, it may be necessary to fit Zener diodes to several of the last stages in order to obtain the same result and/or to increase the divider current, bearing in mind that the heat dissipation in the divider becomes a hindrance as soon as it reaches a few watts. In certain applications, a drawback of using Zener diodes is that they limit the freedom of gain adjustment. Altering the supply voltage to adjust the gain also alters the overall voltage distribution for it affects the voltages across the resistor stages but not the Zener-stabilized ones. As linearity is dependent on the overall voltage distribution, a divider with Zener diodes should be designed for a specific supply voltage and that voltage should be adhered to as closely as possible. Departure from it invites the risk of either overlinearity or premature saturation. The risk is considerably less if only the very last stage is Zener stabilized. A voltage divider composed solely of Zener diodes should never be used as it presents two big drawbacks: the impossibility of adjusting the gain by means of the high-voltage supply and the absence of current limiting in the event of accidental exposure to ambient light during switch-on. Whether Zener diodes are used in the higher or lower stages of the divider, they should be shunted by resistors to protect those stages from receiving the full supply voltage in the event of a diode going open circuit. The Values of the resistors should be two to three times what they would be in a purely resistive divider. The temperature coefficient of the Zener diodes is an important consideration. Variation of Zener voltage with temperature can cause variation of gain. (2) Reservoir capacitors. If the anode current can reach high values for only a small fraction of the time (short-pulse operation), it is preferable to connect reservoir (or decoupling) capacitors to the dynode. The charge stored by the capacitors must be large compared with the forseeable high charge supplied by each dynode when pulses pass through the tube, so that the dynode potentials will not vary by more than 1 or 2 V. The voltage-divider current has to be chosen so that the capacitors are recharged between each pulse. However, this gives the same result as calculating it as a function of the mean anode current. Calculation of the ca-

RECENT TRENDS IN PHOTOMULTIPLIERS

28 1

pacitance values is slightly different depending on whether the decoupling is parallel or series because, in the latter case, voltage variations are cumulative. When space (or insulation) considerations limit the size of the capacitors that can be used, their values must be calculated with especial care; defects due to inadequate decoupling are often fairly misleading because they limit not the current but the charge. (3) Damping resistors. When the photomultiplier is operating in the pulse mode, a very high frequency stray oscillation superimposed on the anode pulses can be observed, even with pulses as wide as a few hundred nanoseconds. This oscillation, which interferes with the linearity characteristic of the tube, usually producing an overlinearity, can appear abruptly when the anode current exceeds a certain level. One way of overcoming this effect is to connect a 5142 resistance in series with each of the last two or three dynodes. In fast photomultipliers with plastic bases such resistors are sometimes built-in. For other types, they must be wired into the socket, between the base and the decoupling capacitors. b. Anode Load. The anode load is usually composed of either a resistance (e.g., the iterative impedance of a coaxial cable) or a resistancecapacitance ( R C )network (pulse preamplifier). The voltage V , developed across this load is subtracted from the last dynode-to-anode voltage VdN,,. As photomultipliers do not usually have an ideal set of I, = F( Vd,,,) characteristics, this may result in a linearity error when V , is not negligible compared with VdN,a(e.g., V , > 10 V). In most applications, the voltage across the load rarely exceeds a few volts. 2. Internal Factors Affecting Linearity a . Space Charge. At very high currents, space-charge effects can modify some trajectories, causing collection losses. At even higher currents, the space charge can cause some electrons to return to the surface from which they originated. This condition resembles that of a diode with parallel-plane electrodes, the saturated current density of which is given by the Child-Langmuir equation:

Z, (A/cm2) = 2.2 x 10-6(V3’2/a2) (V3’2/cm2)

(28)

This relationship, in which V is the interelectrode voltage and a their distance apart, shows that the ability to obtain high-current depends on both area and electric field. In principle, the current is highest between the last dynode and the anode. This is why the last stage is always designed to ensure a higher field by the use of a grid-shaped anode through which the secondary elec-

282

J . P. BOUTOT, J . NUSSLI, A N D D . VALLAT

0

4

2

3

4

5

6

7

Light flux (0.u.)

FIG.17. Typical linearity diagram of a PMT operated with a progressive voltage distribution.

trons pass on their way from dNe1to dN.But then it is between dNPland the anode, where the electric field is three to five times lower, that the most inconvenient space charge arises which sets the limit for current linearity in most photomultipliers. This limit can be raised by using a progressive instead of an equal voltage distribution in the last stages, so as to raise the voltage between the last two stages to as much as 300 V or more. To maintain correct focusing between dynodes without unduly increasing the gain, the interelectrode voltages are successively decreased in the anode-to-cathode direction until the nominal value is obtained at the terminals of the first stages. If, for a given progressive voltage divider, a PMT exhibits overlinearity or premature saturation, it is possibly because the d,,,_,-stage space charge deviates the electron beam toward an area of the last dynode dN that provides gain change. An improvement can usually be obtained by slightly altering the voltage of the dynode d,-,, which finally leads to a compromise to be found between overlinearity and saturation. For tubes with focusing dynodes, using progressive distribution, the maximum current values in linear operation can be increased from the 10-50-mA range to the 100-300-mA range. Figure 17 gives a typical diagram of pulse linearity. For some special tubes, a linear operation can be obtained at currents exceeding 5 A. For tubes with venetian-blind or box-and-grid dynodes, the maximum currents for linear operation are smaller (a few tens of milliamperes) because of the very low electric field present at all dynodes other than the last. For most types of photomultipliers, the current linearity limit due to space charge varies as V & , where n is between 2 and 3. This is only an approximation, but in most cases it is sufficient to indicate a practical limit

RECENT TRENDS I N PHOTOMULTIPLIERS

283

at a specific operating voltage when the linearity limit at some other voltage is known from the published data of the tube. As an approximation, it is the more useful as the current saturation only comes into effect progressively. If current linearity is not required, the maximum anode current that can be obtained before saturation is several times greater than the maximum specified for linear operation. The space-charge effects that cause current nonlinearity occupy times comparable to the transit times between dynode, that is, 1-2 nsec. Even when linearity errors are severe, the electric fields do not cancel out at any place in the tube. Therefore, there is no charge accumulation and the errors are strictly related to the electron current passing between the last dynodes. Current linearity is an important parameter of photomultipliers operating in the wide pulse mode, i.e., with pulses much wider than their own pulse response. For operation with short light pulses such that the current pulses at the anode have a width comparable to the pulse response, it is no longer possible to define a current linearity but only a charge linearity. Higher peak values of anode current can be obtained while still maintaining a good charge linearity, but these values depend mainly on the shape and width of the anode pulse. Only the currentlinearity (not the charge linearity) limitation is quoted in the data sheets; it is given for a worst-case situation with relatively wide anode pulses (about 100 nsec). h. Cuthodr Resistivity. The cathode is a semiconducting surface forming the first element of an electron-optical input system which is designed on the assumption that the cathode is an equipotential surface. If that is not the case, the electron trajectories are modified and the collection efficiency of the first dynode may vary. This is what happens if the cathode current is too large with respect to the surface resistivity, at least in the case of semitransparent cathodes having no underlying conductive layer. As an example, consider a circular cathode of uniform sensitivity, illuminated by a uniform incident flux and emitting a total current (Fig. 18). Let R o b e its surface resistance (Rm is the ratio of the resistivity of the semiconducing layer to its thickness); then the potential difference between the center and the edge is AVk

= (&Zk)/'h

(29)

If it exceeds a few volts, this potential difference increases the convergence of the input system and results in the loss of electrons emitted from the edge. More complicated effects occur in the case of localized illumination of small areas; these lead to a dynamic variation of gain as a function of the

284

J . P. BOUTOT, J . NUSSLI, AND D. VALLAT

-r

0

+r

Dlstance from center of photocathode

FIG. 18. Potential distribution along a photocathode diameter for large mean emitted photocurrent.

cathode current or, in other words, to linearity errors. However, for the cathode currents normally encountered, these phenomena assume practical significance only in the case of bialkali SbKCs cathodes. When tubes are operated at very high count rates and low gain, excessive cathode currents can be reached even with cathodes such as S 11 and SbRbCs which have resistivity about 100 times lower than SbKCs. Fortunately, the distributed capacitance of the cathodes (about 1 pF) is sufficient to store a charge of 10-l2 C. At a gain of lo5, this corresponds to an anode pulse of 100-mA amplitude and 1-psec duration. Therefore, instances in which the resistivity of the cathode is actually a hindrance are fairly rare. c . Gain D r i f t . Gain may undergo more or less reversible variations when the mean anode current varies. Strictly speaking, these constitute a linearity error but, by convention, are treated as an instability (see S e e tion 111,C). /

3 . Measurement of Linearity Only two measurement methods, one applicable to nonrepetitive light pulses and the other a more general method, are mentioned here.

RECENT TRENDS IN PHOTOMULTIPLIERS

285

attenuotor

FIG. 19. Pulse linearity test: block diagram of the X Y method.

a . X Y M e t h o d . This method makes use of an oscilloscope having identical X and Y deflection factors (Fig. 19). The anode pulse of the photomultiplier under test deflects the beam parallel to one axis, and the anode pulse of a reference photomultiplier operating in its known linear region deflects it parallel to the other. It is therefore a method of measuring instantaneous current linearity. Both tubes are excited simultaneously by light pulses lasting a few hundred nanoseconds, for example. At each level of pulse current corresponding to setting of an optical attenuator, an oscillogram is obtained which has a linear part and a curved part. The linearity error is the maximum percentage difference between the curved part of the oscillogram curve and the straight line corresponding to its slope at the origin. This is a measurement of integral linearity. b. Method Using Bursts of Three Pulses. A special pulse generator, giving bursts of three calibrated pulses of increasing amplitude is used to drive one LED. The light reaches the cathode, passing through a neutral filter of attenuation five times. The anode pulses feed a multichannel analyzer (MCA) (Fig. 20a). The registered channel numbers p l , p 2 , and p 3 become p i , p ; , and p i when the filter is withdrawn and an electrical five-times attenuator is placed before the MCA. An on-line calculator can be used to make the ratios a = p 2 / p l and a' = p i / p i and to calculate the linearity deviation which is given by (a' - a ) / a . In the same way, the ratios b = p 3 / p 1and b' = p i / p i give the linearity deviation (b' - b)/b. A practical example of test conditions can be the following: burst of three pulses of 50 nsec giving, when the filter is ON, three anode pulses of 6-, 20- and 30-mA peak value at the operating gain; time between each pulse is 100 psec; burst frequency is a few kilohertz. When the filter is OFF, the anode pulses should be 30, 100, and 150 mA.

286

J . P. BOUTOT, J . NUSSLI, A N D D . VALLAT Optical

&A~ultichann;l Attenuator

attenuator

x5

onalyzer

W Z J j

Pulse generator

LED (0)

W

6

6

P3

PZ

(b)

O6 2O3O4O

60 80 100 120 140 Peak anode pulse current ( m A )

FIG.20. (a) Pulse linearity test: block diagram of the “three-pulses” method; (b) example of an experimental linearity curve obtained with the “three-pulses” method.

The results of the test are plotted on a graph as shown on Fig. 20b. The linearity deviation is then calculated between the channel numbers corresponding to the ideal values 100-30 mA on the one hand, and 150-30 mA on the other hand, with respect to the tube linearity at very low level. One advantage of this method (which measures the differential linearity) is that the result is not affected by the possible gain shift, due to the change of mean anode current which occurs when the filter is withdrawn. Other methods have been described in the literature. Fenster et al. (1973) have measured differential linearity by using a small calibrated light pulse superimposed on an adjustable larger pulse. A charge-linearity measurement of a scintillation counter, using a multienergy gamma-ray source is described in the RTC photomultiplier manual (1981). Note. Pulselinearity deviation can of course also be caused by the electronic modules associated with PMTs. As an example, overlinearity can occur in highenergy physics when a PMT is associated with an analog-to-digital converter (ADC). The input impedance of an ADC can decrease slightly when the amplitude of the applied pulses increases. When the ADC is directly driven by the PMT (high-impedance source), the conversion is always linear. But if a 5 0 4 attenuator is placed before the ADC, the latter

RECENT TRENDS IN PHOTOMULTIPLIERS

287

is driven by a low-impedance source and so overlinearity appears. The recommendation that ADCs be driven by a current source is given in the equipment-operating manuals, particularly for the “qVt module” of LeCroy Research Systems Corp., Spring Valley, New York 10977.

C . Stability

“Stability” (the accepted euphemism for instability) covers any variation of anode sensitivity with time, temperature, mean current, aging, etc. Except for changes with temperature, most such variations are traceable to the electron multiplier as shown by Breuze (1978) who investigated the PMT sensitivity temperature dependence upon over 200 various tubes from four manufacturers. The main factors which determine the stability of alkali photocathodes have been discussed by Sommer (1973). Special notice applies to S1 photocathodes, as mentioned at the end of this section. The types of instability with which we shall be chiefly concerned are the following:

(1) Long-term drift, which is a time-dependent variation of gain under conditions of constant illumination (2) Short-term drift, which is a time-dependent variation of gain following a change in mean current 1. Long-Term Drift

Two modes can be distinguished, according to whether the mean anode current is high or low. a . High-Current Drijt; Operating Life. Certain more or less irreversible effects are observable at anode currents of more than 10 PA. After long storage (e.g., a few months), a photomultiplier exhibits a large drift in gain for the first few dozen hours of operation. After that, the gain does not vary much for some thousands of hours. Then it slowly decreases, as a function of the total electric charge handled. The rate of these variations varies roughly as the current. Operating life, defined as the time required for anode sensitivity to be halved, appears to be a function of the total charge delivered. Values of 300- 1000 C are typical. If the incident flux is reduced (by a factor of, say, 10) or cut off completely, or if the supply voltage is switched off for several days, the following sequence can be observed when the original operating conditions are restored: first, a certain recovery of sensitivity accompanied by a renewed initial drift; then, a tendency to catch up, fairly quickly, with the slow decline of sensitivity at the point at which it was interrupted. Figure 21 illustratres the relative

288

J . P . BOUTOT, J . NUSSLI, AND D . VALLAT

3 40

t

i

Interruption during a few tens of hours

&

2

-

%

FIG.21. Relative gain variation of a PMT operating at medium to high anode current (mean anode current, I , = 30 PA).

gain variation of a photomultiplier operating at a mean anode current of 30 PA. The initial drift, which can be considered an aging period, is greatest at the beginning-about 20-40%. The length of the aging period depends on the anode current; at 10 PA, it is about 24 hr. As long as the mean current does not fall below some hundreds nanoamperes, the aging effect is still observable though very slow. In most applications, the cathode current is low (<1 nA), and the variations of anode sensitivity reflect variations of gain due to changes in the surface state of the dynodes. This is commonly attributed to the mobility under electron bombardment of the alkali metal molecules (mainly potassium and cesium) with which they are coated, although the exact mechanism is probably more complicated than the literature has suggested (Youngbluth, 1970; Coates, 1975; Yamashita, 1978). When the mean anode current is only a few microamperes, the total charge delivered is no longer the decisive factor for operating life. Other phenomena, such as helium migration through the glass or the establishment of internal migration and diffusion balances, determine the end of useful life, which is then measured in years and is independent of the mode of operation. The experience of many users would even seem to indicate that continuous, uninterrupted operation results in better long-term stability of performance characteristics than storage. Photomultipliers with S 1 cathodes deserve separate mention. Even at anode currents of only a few microamperes, they exhibit large short-term drift which is independent of the gain adjustment. That this drift can to a large extent be reversed by heating the tube for a few hours at the maximum permissible temperature, lends support to the hypothesis that there is some exchange of molecules (such as cesium) between the dynodes and surfaces not subject to electron bombardment. b. Low-Current Drift. For the first few hours or days after it is

RECENT TRENDS IN PHOTOMULTIPLIERS

0

I

I

12

24

289

I

Time (hrl

FIG.22. Relative gain variation of a PMT operating at low anode current.

switched on, a photomultiplier subject to constant illumination undergoes change of gain. The amount of change varies considerably from type to type and even betwen specimens of the same type. In most cases, though, the rate of change quickly decreases to a few percent per month (Fig. 22), and the higher the current the quicker the gain stabilizes. It is sometimes worthwhile speeding this aging process by operating the tube initially at a current ten times higher than that expected in the intended application. It is also advisable to leave the tube switched on even when it is idle. If the tube is stored for a time comparable with its former operating period, the change in gain is reversible and is repeated when the tube is again put into service. The American National Standards Institute (ANSI) test (1972), which is used to characterize this type of drift, employs a scintillator and a source of 13Tspositioned so as to produce a fixed count rate between lo3 and lo4 counts/sec. After a stabilization period of 30 min to 1 hr, the height of the 13'Cs peak (662 keV) is measured every hour, and the anode sensitivity drift over 16 hr is determined on the basis of the mean gain deviation (MGD) calculated with the formula

I

where p is the mean height of the peak and pithe height corresponding to the ith measurement. This type of drift is not related to the high-current long-term drift described in the previous section. Its major cause is also to be sought in the change in the structure of the emissive surfaces although other factors, such as the charge distribution at insulator surfaces (e.g., dynode spacers), also sometimes play an important part. The drift is much less (MGD < 1%, typically) in tubes with bialkali cathodes and CuBe dynodes than in those with S 1 1 cathodes.

290

J . P. BOUTOT, J . NUSSLI, AND D. VALLAT

0

v

2

4

6

Stabilization period before test

8 10 12 Time (hr)

14

16

FIG.23. Examples of variation curves of anode sensitivity having the same maximum deviation over 16 hr but different MGD values; p is the mean amplitude over 17 readings. (MGD values: curve 1, 1.6%; curve 2, 1.1%; curve 3, 0.75%.

The ANSI test specification does not mention the anode sensitivity at which the test is to be performed. However, when a figure for long-term stability is given, the mean anode current during the test must be specified. Values of about 1 p A are generally used because they are broadly representative of most applications. For convenience, the scintillator and the source used in the ANSI test may be replaced by a LED. Figure 23 gives some examples of anode-sensitivity-variation curves having the same maximum deviation but different MGD values: it can be seen that the same MGD figure can mean a low-level continuous drift during 16 hr, or a higher level sudden drift followed by a fairly constant gain. For some applications, such as nuclear medicine, it is desirable to know the stability over a long period (for example, a month). Very little information exists on monthly stability of photomultipliers. Some nonpublished results from tests performed in our application laboratory show that standard PMTs used in gamma cameras have typical MGD values Long-term drift of the goin with mean anode current T

\

02

Gain with mein anode current i a 2

I

Time Shik against operating level

L&ng-term drift of the gain with mean anode current To,

FIG. 24. Example of relative variation of photomultiplier gain after a change of the operating level.

RECENT TRENDS IN PHOTOMULTIPLIERS

29 1

not higher than 1% over 1 month, the better results being obtained with venetian-blind-structure photomultipliers (probably thanks to their less focused electron beam). It should be noted that some tubes do not stabilize at all and consequently exceed 10% drift after some months. Equipment designers should take such facts into consideration. However, the impact of these instabilities is often minimized because most of the tubes of a given type (but not all of them) drift in the same direction: this is important when many tubes have to be used in the same equipment. Measuring MGD values over such long periods is difficult because of the possibility of drift in the measuring system itself. In this case, a radioactive source in combination with a scintillator is preferable to a LED because its long-term stability is much better. 2 . Short-Tc.rrn Drift

When the average flux to which a photomultiplier is exposed gives rise to an anode current of less than 10 PA, the gain is usually surfficiently stabilized after 10- 15 min for its long-term drift to be disregarded. If the flux is then changed abruptly, the anode current, instead of assuming the new value that it should reach immediately, starts a new drift phase before stabilizing to another level (Fig. 24). Thus the gain becomes a function (often an increasing one) of the mean value of the anode current reckoned over an interval of 1 sec or longer. For most photomultipliers, the time required for stabilizing the gain after changing the average flux is very low, of the order of 1 sec. But in some cases, and especially for tubes with S11 cathodes, a drift with a much longer time constant (about 1 hr) is added to this fast shift. Figure 25

I

-8 -

0

alu

,

I

heon anode current ( P A )

2-

1-

Hysteresis-

4 I

292

J . P. BOUTOT, J . NUSSLI, AND D . VALLAT

I'

I

I

I

I

I

I

Mean anode current ( F A )

Time (min)

FIG.26. Relative gain variation with change of operating level for a PMT with a long stabilization time constant.

gives an example of short-term drift with a single short-time constant and Fig. 26 an example with two time constants, one short and one long. Two methods are used to characterize the short-term gain drift due to change of average flux. The ANSI test uses a 13'Cs radioactive source and NaI(T1) scintillator. After a stabilization time of at least 15 min, the position of the 137Csabsorption peak is recorded at a count rate of lo4 counts/sec. The source is then moved to reduced the rate to lo3 counts/sec and the new position of the peak is recorded. The drift is characterized by the relative shift of the peak, A p / p . As in the test for long-term drift, the ANSI test specification does not mention the anode sensitivity at which the test is to be performed. For the stability figures to have meaning, the extreme values of the anode mean current must be quoted. Taking account of typical photomultiplier applications, the test is performed between 300 nA (at lo4 counts/sec) and 30 nA (at lo3 counts/sec), or between 1 pA and 100 nA. Another method, easier to set up, uses two independent LEDs as proposed by Kerns (1975). One emits pulses of adjustable intensity and frequency, or simply a continuous flux of adjustable intensity, for setting the mean anode current to any desired level throughout the applicable range. The other emits light pulses of fixed intensity and frequency. The mean height of the anode pulses due to these is a measure of the relative gain at the set level. One cause of short-term drift (shift) may be that charges on the insulators (dynode spacers) are affected by the passage of electrons (scatter, for example) and that this in turn modifies the focusing between stages. In present-day photomultipliers, particularly those with venetian-blind

RECENT TRENDS IN PHOTOMULTIPLIERS

293

dynodes, careful design of the electrode configuration practically eliminates this effect. But, even so, variations in gain due to changes in secondary emission can still be observed, which suggests the existence of phenomena at the level of the emissive layer itself. The supposition of an effect similar to the Malter effect has been put forward but so far without experimental proof (Cantarell and Almodovar, 1963; Yamashita, 1978). Tubes with bialkali cathodes and CuBe venetian-blind dynodes are usually considered the most stable in respect of short-term drift, gain variations of less than 1% being commonly obtained at anode current variations of ten to one (from 100 nA to 1 PA) (G. Breuze, private communication, 1976). In Figs. 25 and 26, it can be seen thus the gain does not return exactly to its original value when the flux does. This hysteresis reflects an interaction between long- and short-term stability parameters. During prolonged operation, the higher the current the quicker the hysteresis tends to be taken up. Here, again, accelerated aging at high current has a favorable effect. Many authors, such as van Duyl and de Kruijk (1978), Yamashita (1980), or Coates, and Andrews (1981), have described test methods and given results of stability measurements on various types of PMTs.

D. Reliability The determination of a level of reliability for photomultipliers is useful where the number of tubes in one experiment or used for a series of equipments is high (for example, more than 1000 pieces). As usual, two types of failure can be determined:

(1) Catastrophic failure such as cracks in the glass envelope, electrodes no longer connected or short-circuited, resulting in useless or nonexistent signals at the output (2) Partial failures, such as high dark current (noisy tube), gain drifts, decrease of cathode sensitivity, resulting in incorrect or degraded signals at the output. Due to the long-term stability problems described in Section III,C, 1, an aging period has to be applied whenever tubes have not been operated for a long time. Photomultipliers which do not reach a fairly constant gain after that period can be regarded as failing. This clearly indicates that the reliability of photomultipliers is closely connected to the mode of operation (voltage, mean anode current, etc.) and to the specifications which are expected from these tubes. It is also clear that photomultipliers are more like systems than components as far as reliability is concerned due to the numerous characteristics affecting their operation (cathode sensitivity, noise, gain, etc.).

294

J . P. BOUTOT, J. NUSSLI, A N D D . VALLAT

Information about the reliability of photomultipliers has not been widely published. In recent years, high-energy physics has called for higher and higher quantities of photomultipliers per experiment, linked to the rapid increase of the size of calorimeters. One of the earliest large experiments seems to be WA 1 which came into operation at CERN in 1977 and which used 2450 tubes with the S11 cathode. As indicated by Holder ef al. (1978), the main failures met during the burn-in period were due to noise increases. After several years of operation, F. Dydak (private communication, 1981) reported that the failure rate, for the same reason, was about five tubes per year, when the replacement rate for catastrophic failure was less than one per year. It seems that the cause of that failure was identified as due to the coupling of the front window to an insulating plastic light guide with the photocathode at negative high voltage. Recently, 2500 bialkali tubes were added to the same experiment with the light-guide surface kept at cathode potential. First results show an even lower reject rate. It is interesting to translate the above figures into reliability figures, and especially to look for the mean time to failure (MTTF). Calculated over a period of time T , the MTTF is given by the formula MTTF

=

N(O)T/r

(31)

with N(0) is the number of tubes at the time t = 0 and r is the number of failures. Using the above failure rates leads to MTTF values of at least 4 x lo6 hr for partial failures and more than 2 x lo7 hr for catastrophic failures. These figures are rather high, but it should be remarked that in that experiment, all PMTs operate at a very low mean anode current which is, of course, favorable. Information related to the gamma-camera market suggests that the annual replacement rate for that more stringent application would be between a few tenths of apercent and one percent, depending on the type of tube. These give reliability MTTF figures between lo6 and 3 x lo6 hr. E. Operating Range of Photomultipliers

Determination of the operating range of a photomultiplier consists in looking for the minimum and maximum values of gain and anode current of the tube to be applied so that the expected performance (linearity, stability, detectivity) is met.

I . Continuous Operation If the illumination level can be set at choice, there is a temptation to set it high to obtain a good signal-to-noise ratio. How high it actually can

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be set depends on the cathode-current level at which the effects of cathode resistivity become significant; for SbKCs cathodes, that level is about 1 nA and for SbRbCs cathodes about 10 nA. With the cathode current determined by the working conditions, the required gain and anode current depend on the characteristics of the measuring or signal-processing circuits used. The minimum practical gain is that which corresponds to the minimum electrode voltages specified in the data sheets of the tube; these are based on considerations of linearity and minimum gain fluctuation. The maximum practical gain is usually determined by dark current and, consequently, signal-to-noise ratio considerations, as indicated by Jonas and Alon (1971). For good gain stability with time, the mean anode current must not exceed the limit specified in the data sheets. 2. Pulse Operation

In pulse operation, the following factors affect the choice of operating range: detection efficiency, energy or time resolution, linearity at pulse peaks, instrument triggering threshold, maximum count rate. Their significance varies according to whether the mean illumination is constant (small anode-current range) or widely varying (large anodecurrent range). a . Small Anode-Current Range. This is often encountered in spectrometry applications where optimum time or energy resolution is required. The important thing is to minimize response fluctuations by:

(1) Optimizing the voltages at the electron-optical input system and the first two multiplier stages to minimize transit-time and gain fluctuations (2) Optimizing the collection of light at the photocathode. (cathode-resistivity effects seldom figure in pulse operation) Beyond the threshold set by the first of these recommendations, the practical minimum for the gain depends on the sensitivity or detection threshold of the signal-processing circuits. The practical maximum is set either by dark-current considerations or by linearity limits that come into play at pulse peaks. Beyond a certain value of applied voltage, the dark pulse rate increases faster than the gain. The linearity limit may be due to either the photomultiplier or the circuits.

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In pulse-counting applications, the upper and lower gain limits can be found experimentally by plotting the variation of count rate as a function of applied voltage. This reveals a distinct counting plateau within which to set the operating point (Meade, 1981). In other small anode-current-range applications, the choice of operating point may depend on other criteria, such as the bandwidth and gain of the signal-processing circuits. b. Large Anode-Current Range. This commonly applies in high-energy physics where scintillation pulses often vary over a wide range. The same fundamental rules apply as for small anode-current range, but four additional criteria are now decisive: (1) The minimum number of photoelectrons per pulse that have to be detected (2) The sensitivity threshold of the electronics, which determines the amplitude or charge of the minimum detectable anode pulse (3) The maximum allowable anode pulse charge, which depends on the linearity limit of the tube at the chosen operating voltage (4) The ratio of maximum to minimum pulse amplitude, or number of photoelectrons per pulse. If this is referred to the anode, and the minimum number of photoelectrons is very small, it should be expressed in terms of pulse charge rather than amplitude. So long as the interval between successive photoelectrons is shorter than the response pulse width of the tube, the anode pulse resembles a single multielectron pulse. But when the interval is longer than the response pulse width, the tube resolves the individual electrons into discrete pulses. Under these conditions, it makes no sense to speak of amplitude.

Figure 27 illustrates how these criteria determine the practical boundaries of operation. The figure shows the relation between the mean number of photoelectrons per pulse, A K , the mean anode charge per pulse qa, and the gain G:

-

nK

=

Ga/Ge

(32)

with qa as parameter. The lines corresponding to each value of ia are called isocharge lines. The practical boundaries of operation are as follows : (1) A horizontal line corresponding to the minimum number of photoelectrons per pulse, nK(min) ( 2 ) An isocharge line corresponding to the charge-sensitivity threshold of the electronics, qacmin)

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Gain G

FIG.27. Diagram for determination of the operating range of a PMT (figures are given as an example).

If the threshold is given in terms of anode pulse voltage, Vacmin), rather than charge, the conversion is qacmin)

L-

Vatmin)twlR

(33)

where tw is the FWHM of the anode-current pulse and R is the anode load resistor.

(3) An isocharge line corresponding to the charge-linearity limit of This is usually given on the PMT data sheet as a pulse the tube, qacmax). current, at a specific operating voltage; the conversion is qacrnax)

Za(max)fw

(34)

This boundary cannot be found until the operating voltage is known or at least estimated. Its estimation depends on the fourth of the listed criteria: the pulse-charge ratio. How to arrive at a realistic estimate can best be explained by example. Consider a photomultiplier in pulse operation

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J . P. BOUTOT, J . NUSSLI, A N D D. VALLAT

under the following conditions: minimum number of photoelectrons per pulse, sensitivity threshold of the electronics, required max/min pulse-charge ratio, anode pulse FWHM, pulse-peak linearity limit at V,, = 2500 V,

10 4 PC 100 5 nsec 250 mA

In Fig. 27, the first of these conditions corresponds to the bold horizontal = 10; the second corresponds to the bold isocharge line line at EKcmin, qacmin) = 4 pC. Their intersection defines the minimum acceptable gain, about 2.5 x lo6. Assume that with the type of tube and voltage divider used, this is obtained at Vht = 2200 V. At that voltage, the pulse-peak linearity limit la(,,,)given above is reduced by the ratio /250

mA = [2200/2500]"

(35)

where the exponent n is between 2 and 3, depending on the type of tube (see Section III,B,2). For the present case, assume it is 2; then la(,,,) = 200 mA and from Eq. (34) and t, = 5 nsec, %(ma,)

=

1000 PC

which is indicated by the second bold isocharge line. This is the third boundary of the practical operating region. The fourth boundary is set by the required max/min pulse-charge ratio, which corresponds to 1000 photoelectrons/pulse. The qacmax) = 1000 pC isocharge line crosses the EK(max) = 1000 ordinate at a point corresponding to G = 6.25 x lo6. Thus, the practical boundaries of operation are defined by the two isocharge lines (%(,in) = 4 pC and qa(max) = 1000 pC) and by the minimum and maximum gain values, Gmin= 2.5 x lo6 and G,,, = 6.25 x lo6, respectively. If G < 2.5 x lo6, pulses containing less than 10 photoelectrons do not exceed the sensitivity threshold of the electronics. And if G > 6.25 x lo6, the required max/min ratio of 100 cannot be accomodated. It is then necessary to check that a gain in the indicated range can be obtained at the assumed V,, = 2200 V . If not, a new value of V,, will have to be assumed and the boundaries redetermined. Uncertainty about the value of the exponent n in Eq. (35) reflects on the accuracy with which the upper gain limit can be determined; however, this is seldom significant except when the operating range is narrow. In addition, the current-linearity limit given in the data sheets is a statistical value from which individual tubes may deviate; allowance should be made for this in the calculations. In some cases of large anode-current-range operation, other criteria

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may have to be taken into account. For instance, the high mean anode current due to a high count rate may affect stability, necessitating reconsideration of the initial parameters. IV. CONCLUSION Photomultipliers are meeting strong competition from many other techniques, but due to their specific characteristics plus continued improvements as described in this article, they are able to resist that competition and will retain a large place in nuclear physics. In the nuclear medicine market, considerable efforts are being made in the field of energy resolution and noticeable improvements have occurred during recent years. Numerous attempts have been made in the gamma-camera field to find an alternative to photomultipliers (Muehllehner, 1976), particularly by using solid-state detectors (Kaufman r t al., 1980), but because of the large interaction volume needed and other practical drawbacks, the price-performance ratio has remained unfavorable for its competitors. In the high-energy physics field, current demand is influenced by the new generation of collider rings at present being constructed o r projected. The detection of particles has to be made in 4.rr steradians, with a fine granularity which requires that the detection system used (calorimeter) should use larger and larger quantities of detectors (e.g., greater than lO,OOO), with resulting reductions in prices per channel. In addition, for electromagnetic calorimeters, there is a need for detectors able to withstand high magnetic fields (0.5- 1 T). Some compact tubes such as microchannel plate PMTs may to some extent meet that requirement, but their high price and limited lifetime make them unacceptable. A possible solution recently advanced is to use vacuum photodiodes if the number of photoelectrons per pulse is high enough combined with low-noise integrated amplifiers. It can be seen that the market needs are continually changing, and so the ideal phototube is a target which is also continually moving. This situation naturally makes manufacturing more complex, but at the same time, this stimulates innovation, resulting in the PMT being a continually improved device with openings for new applications.

ACKNOWLEDGMENTS The authors wish to thank the managements of their respective organizations: Laboratoires d’Electronique et de Physique Appliquee, RTC-La Radiotechnique Compelec, Hyperelec, who gave permission for this work to be undertaken.

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Oba, K., and Maeda, H. (1976). A d v . Electron. Electron Phys. MA, 123. Oba, K., and Rehak, P. (1981). IEEE Trans. Nucl. Sci. NS-28 (l), 683. Oba, K., Sugiyama, M., and Suzuki, Y. (1979). IEEE Trans. Nucl. Sci. NS-26 (I), 346. Olsen, G. H., Martinelli, R. U., and Ettenberg, M. (1977). U.S. Patent 4,019,082. Orphan, V., Polichar, R., and Ginaven, R. (1978). Trans. A m . Nucl. Soc. 28, 119. Ortec (1977). “Techniques for Improved Time Spectroscopy,” Appl. Note AN-41. Ortec Inc., Oak Ridge, Tennessee 37830. Panitz, J. A., and Foesch, J. A. (1976). R e v . Sci. Instrum. 47 (l), 44. Parkes, W., and Gott, R. (1971). Nucl. Instrum. Methods 95, 487. Patton, J . A., Rollo, F. D., and Brill, A. B. (1980). IEEE Trans. Nucl. Sci. NS-27 (3), 1066. Peifer, W. K. (1976). Proc. Tech. Program-Electro-Opt. Int. Laser Expo., 1976 p. 667. Persyk, D. E., and Moi, T. E . (1978). IEEE Trans. Nucl. Sci. NS-25 (l), 615. Persyk, D. E., Ibaugh, J. L., McDonie, A. F., and Faulker, R. D. (1976). IEEE Trans.Nuc1. Sci. NS-23 (I), 186. Persyk, D. E., Morales, J., McKeighen, R., and Muehllehner, G. (1979). IEEE Trans. Nucl. Sci. NS-26 (l), 364. Persyk, D. E., Schardt, M. A., Moi, T. E., Ritter, K. A., and Muehllehner, G. (1980). IEEE Trans. Nucl. Sci. NS-27 (l), 168. Petley, C. H., and Pook, R. (1971). Acta Electron. 14 (2), 151. Philips (1970). “Photomultipliers” (H. Kater and L. J. Thompson,eds.). Philips Electron. Components Mater. Div., Einhoven, The Netherlands. Philips (1971). “Fast Response Photomultipliers” (M. D. Hull and 0. Eng, eds.). Philips Electron. Components Mater. Div., Eindhoven, The Netherlands. Philips (1982). “Application Book on Photomultipliers.” Philips Electron. Components Mater. Div., Eindhoven, The Netherlands (to be published). Pietri, G. (1968). IEEE Trans. Nucl. Sci. NS-15 (3), 171. Pietri, G. (1973). Proc. Int. Conj: Instrum. High Energy Phys., 1973 p. 586. Pietri, G. (1975). Proc. ISPRA Nucl. Electron. S y m p . , 2nd, 1975 p. 397. Pietri, G. (1977). IEEE Trans. Nucl. Sci. NS-24 (I), 228. Pietri, G., and Nussli, J. (1968). Philips Tech. R e v . 29 (8/9), 267. Polaert, R., and Rodikre, J. (1974). Philips Tech. R e v . 34 (lo), 270. Pollehn, H., Bratton, J., and Feingold, R. (1976). A d v . Electron. Electron Phys. 40A, 21. Pook, R. (1971). Acta Electron. 14 (2), 135. Poultney, S. K. (1972). A d v . Electron. Electron Phys. 31, 39. Prydz, S. (1973). J . Phys. E 6, 189. RCA (1970). “Photomultiplier Manual,” Tech. Ser. m - 6 1 . Solid State Div., Electro-Optics and Devices, Lancaster, Pennsylvania 17604. RCA (1980). “Photomultiplier Handbook,” PMT-62. Lancaster, Pennsylvania 17604. Rees, D. M., Whirter, I., Rounce, P. A., and Barlow, F. E. (1981). J . Phys. E 14, 229. Roberts, A. (1980). Proc. 1980 D U M A N D Signal Processing Workshop, Hawaii, 79. DUMAND Center, Honolulu. Rome, M., Fleck, H. G., and Hines, D. C. (1964). Appl. O p t . 3 (6), 691. Roziere, G. (1977). French Patent 2,402,880. Rozikre, G., Verat, M., Rougeot, H., and Driard, B. (1981). IEEE Trans. Nucl. Sci. NS-28 (11, 60. RTC (1981). “Photomultiplicateurs,” Ref. 5482-07/1981. RTC La Radiotechnique Compelec, 130 avenue Ledru-Rollin, 75540 Paris Cedex 11, France. Ruggieri, D. J. (1972). IEEE Trans. Nucl. Sci. NS-19 (3), 74. Sandel, B. R., Broadfoot, A. L., and Shemansky, D. E. (1977). Appl. O p t . 16 (9,1435. Sandie, W. G., and Mende, S. B. (1982). IEEE Trans. Nucl. Sci. NS-29 (l), 212.

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Schagen, P. (1974). I n “Advances in Image Pickup and Display” (B. Kazan, ed.), Vol. 1, p. 1. Academic Press, New York. Schmidt, K. C., and Hendee, C. F. (1966). IEEE Trans. Nucl. Sci. NS-13 (3), 100. Science Applications Inc. (1980). Electro-Opt. Syst. Des. 12 (9), 29. Seib, D. H., and Aukerman, L. W. (1973). Adv. Electron. Electron Phys. 34, 95. Sharpe, J. (1975). G. B. Patent 15,126/75. Silzars, A,, Bates, D. J., and Ballonoff, A. (1974). Proc. IEEE 62 (8), 1119. Simon, R. E., Sommer, A. H . , Tietjen, J. J., and Williams, B. F. (1968). Appl. Phys. Lett. 13 (lo), 355. Sipp, B., and Miehe, J. A. (1974. Nircl. Instrum. Methods 114, 249. Sipp, B . , Miehe, J. A., and Clement G. (1975). J . Phys. E 8, 296. Sipp, B., Riehe, J. A., and Lopez-Delgado, R. (1976). Opt. Commun. 16 (l), 202. Smith, H. M . , Ruedy, J. E., and Morton, G. A. (1966). lEEE Trans. Nuel. Sci. NS-13(3), 77. Smith, L. G. (1951). Rev. Sci. Instrum. 22, 166. Sommer, A. H. (1973). Appl. Opt. 12 (I), 90. Stapleton, R. J., and Wright, A. G. (1979). Nucl. Instrum. Methods 167, 359. Stromswold, D. C. (1981). IEEE Trans. Nucl. Sci. NS-28 ( I ) , 290. Ter Pogossian, M., Raichle, M. E., and Sobel, B. E . (1980). Sci. A m . 243 (4), 141. Thomson-CSF, Departement Applications SpCciales de I’hstrumentation, 23-27 rue Pierre Valette, 92240 Malakoff, France. Timothy, A. F., and Timothy, J. G. (1971). Actu Electron. 14 (2), 159. Timothy, J. G., Mount, G. H., and Bybee, R. L . (1981). IEEE Trans. Nucl. Sc,i. NS-28 (l), 689. Uyttenhove, J., Demuynck, J . , and Deruytter, A. (1978). lEEE Trans. N M ’ ~Sci. . NS-25 (I), 566. Vallat, D. (1969). Onde Electr. 49 (7), 804. van Duyl, W. A,, and de Kruijk, A. (1978). Med. Prog. Techno/. 6, 29. Vasil’chenko, V. G., Lapshin, V . E., Melnikov, N. S . , Monich, E . A., Ronzhin, A. I., Rykalin, V. I., and Semonov, V. K. (1980). Nucl. Instrum. Methods 169, 389. Wahl, P., Auchet, J. C., and Donzel, B. (1974). Rev. Sci. Instrum. 45 ( I ) , 28. Washington, D., Duchenois, V., Polaert, R., and Beasley, R. M. (1971). Actu Electron. 14 (2), 201. Weinberg, S. (1981). Sci. A m . 244(6), 52. White, W. (1979). Radiology 132, 179. Wijnaendts van Resandt, R. W., den Harink, H . C., and Los, J. (1976). J . Phys. E 9, 503. Wilcox, D. A., Abraham, W. G., Bardas, D., Gwilliam, G. F., and Enck, R. S . (1979). Electro-Opt. Syst. Des. 11 (3), 41. Williams, B. F. (1972). IEEE Trans. Nucl. S c i . NS-19 (3), 39. Wiza, J. L. (1979). Nucl. Instrum. Methods 162, 587. Woodhead, A. W., and Ward, R. (1977). Radio Electron. Eng. 47 (12), 545. Wright, A. G. (1977). “Design of Photomultiplier Output Circuits for Optimum Amplitude or Time Response, EM1 Appl. Note R/P065. EM1 Industrial Electronics Ltd., Electron Tube Division, Middlesex, England. Yamashita, M. (1978). Re\,. Sci. Instrrrm. 49 (9), 1336. Yamashita, M. (1980). Rev. Sci. Instrum. 51, 768. Youngbluth, 0. (1970). Appl. Opt. 9, 321. Zonneveld, F. W. (1980). “Computed Tomography.” Philips Medical Systems, Eindhoven, The Netherlands. Zworykin, V. K., Morton, G. A,, and Malter, L. (1936). Proc. Inst. Radio Eng., Anst. 24 (3), 351.

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ADVANCES IN ELECTRONICS A N D ELECTRON PHYSICS, VOL. 60

Thermal and Electrothermal Instabilities in Semiconductors* M. P. SHAW

AND

N. YILDIRIM?

Drpurtment of Elec,tricml and Computer Engineering Wayne Stutc, Unitvr.\itv Detroit. Michigan

................................................... ................ A. Introduction ................................................. nductors . . . . . . . . . . .

I. Introduction

307 3 10 310 313 C. An RC Network Analog of the 3 16 Thermally Induced Negative Differential Conductance ...................... 321 A. Thermal Boundary Conditions. 32 1 B. The Effect of Inhomogeneities.. ....................................... 325 C. Critical Electric Field-Induced Switching Effects . . . . . . . . . . . . . . . . . 327 Thin Chalcogenide Films ........................................ 332 A. Introduction .......................................... 332 342 354 Vanadium Dioxide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 A. Introduction ..................................... 358 B. An Ideal Mo 361 Second Breakdown in Transistors 364 ...................................................... 369 Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 References . . . . . . . . . . . . . . . . . . . 382

11. The Thermistor..

111.

IV.

V.

VI.

I. INTRODUCTION Many important features of a wide variety of solid-state devices result from situations where a sufficiently high bias is applied so that the device either switches from one conductive state to another or oscillates between two different conductive states. Many of these phenomena are classified as electronic instabilities. Among devices that behave this way are the * Some parts of this critical review appear in print for the first time. t Present address: Middle East Technical University, Ankara, Turkey. 307 Copyright 0 1983 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-014660-6

308

M . P. S H A W A N D N . YILDIRIM

tunnel diode (Esaki, 1958); Gunn diode (Gunn, 1964); avalanche diode (Shockley, 1954); Josephson junction device (Josephson, 1962); thyristor (Fulop, 1963); and PIN diode (Prince, 1956). Others are classified as electrothermal instabilities, such as chalcogenide thin-film switching devices (DeWald et al., 1962; Ovshinsky, 1968), or thermal instabilities, such as vanadium dioxide, V O n(Anderson, 1954), and many cases of secondary breakdown in transistors (Thorton and Simmons, 1958). Although substantial evidence exists for the fundamentally electronic nature of threshold switching in chalcogenide films (see, e.g., Adler et al., 1978; Kotz and Shaw, 1982), the intrinsically thermal nature of memory and “forming” type processes, and the “delay time” for switching, make it important to emphasize both the thermal and electronic aspects of the problem. To this end, we will first discuss the important features of heat flow in solids that are pertinent to the development of thermal effects. Further, we will also review a reasonable cross section of models associated with both thermal and electrothermal breakdown (Klein, 1971; Berglund and Klein, 1971), switching, and memory effects. In order to analytically understand the electrical and thermal behavior of all the above structures, it is most often, but not always, convenient to characterize them in terms of an effective region of negative differential conductivity (NDC) in their current density ($electric field (%)characteristic. In this region d J / d 8 < 0 holds. Figure 1 shows the two major classes of NDC characteristics: SNDC and NNDC. As can readily be seen from the curves, the N and S stand for the shape of the J ( % ) [or % charac(.I)]

FIG. 1. Example SNDC (current controlled) and NNDC (voltage controlled) j ( $) curves.

THERMAL A N D ELECTROTHERMAL INSTABILITIES

.

309

I

$6 -

R t "\

I'

\

FIG.2 Load line (dcll) and current-voltage I(+) curve for a nonlinear element placed in the circuit shown in the inset.

teristic; in this article we concentrate on SNDC devices, with the realization that all NDC characteristics are functions of frequency, and this dependence must always be considered in determining the behavior of a particular device. The simplest circuit in which any of these devices operates is shown in the inset of Fig. 2. Here the NDC element is in series with a load resistor RLand bias battery that provides a voltage &. If I is the current in the circuit and 4 the voltage drop across the NDC element, then

4 B = IRL + 4

(1)

is the equation of the dc load line. This line is plotted in Fig. 2; its slope is - l/R,, and its intersection with the device characteristic I(+) defines the steady-state operating point. Intersections of the load line with the I(+) characteristic are stable as long as dZ/d4 > 0, which is the case depicted in Fig. 2. However, for situations where either dZ/d4 or d J / d 8 have regions of NDC, operating points at intersections in these regions are often unstable both against the formation of inhomogeneous field and/or current-density distributions (space-charge nonuniformities, see, e.g., Knight and Peterson, 1966, 1967; Kroemer, 1966, 1971; Shaw et al., 1979; Hilsum, 1962; Butcher, 1967; Conwell, 1970; Ridley, 1963) and/or circuitcontrolled oscillatory effects (see, e.g., Shaw el al., 1973a and the Appendix). In order to understand the detailed nature of these instabilities, we must ask two basic questions. First, what is the mechanism responsible for the NDC region? Second, how do we analyze the resultant phe-

3 10

M . P. SHAW A N D N . YILDIRIM

nomena that often occur? In this review we treat the case where the SNDC characteristic is produced by thermal or electrothermal means and limit ourselves to situations where the circuit plays no major role in controlling the behavior of the resultant instability. (For discussions of electronic SNDC elements see, e.g., Scholl, 1982; Barnett, 1969; Adler et al., 1980; Weber and Ford, 1970.) Circuit effects are discussed in the Appendix; it is primarily the mechanism and steady-state solutions we are going to concern ourselves with now. It is very tempting to begin the review by considering the VO, problem, since it is a reasonably clear-cut case. However, the myriad of other observed switching structures are not as well defined; alas, VO, is almost a special case. Indeed, for the reader who would like to begin with the simplest ideal model, wherein an analytical approach provides great insight into the problem iq general, we recommend reading Section V first. 11. THE THERMISTOR A . Introduction

Thermal effects in solids have been treated in great detail over the past 50 years (see e.g., Carslaw and Jaeger, 1959). Of particular interest has been the variety of phenomena associated with thermal runaway induced by Joule heating and the associated breakdown or switching processes often observed (see, e.g., Fock, 1927; Becker, 1936; Franz, 1956; Skanavi, 1958; Boer et af., 1961; Stocker et a/., 1970; Sousha, 1971; Altcheh et al., 1972; Thoma, 1976). These instabilities often result in regions of NDC appearing in the I(+) characteristics of a variety of materials. It is now well known that NDC can appear in the static and dynamic characteristics of common materials and devices in which the current level is determined not only by the applied voltage, but also by the temperature. One reason for this is that the Joule heating of the sample often causes the average temperature to rise above that of the ambient temperatures T , . Figure 3 shows how this might arise. Linear Z(4) characteristics are sketched for isothermal cases where the ambient temperature T,, > Ta3 > T,, > T,, . These are the characteristics that would result were the heat sinking sufficient to maintain the system at the ambient levels shown. However, when the heat sinking is insufficient to remove heat fast enough, then, e.g., if the ambient is Tal, it is possible that the steady-state average T can correspond to a point on the T,, line. The actual Z(4) characteristic might then appear as the thick solid line; a region of SNDC could occur. A device in which NDC is induced in this manner is called a thermistor. Note that every nonlinear point on the thermistor character-

THERMAL A N D ELECTROTHERMAL INSTABILITIES

311

,/

FIG.3. Nonlinear thermistor characteristics (heavy line) that might - arise in a material where linear I(4) curves at different ambient temperatures T, are shown at T4 > T3 > T2 > T I .

istic corresponds to a different average steady-state T distribution. The slope of the NDC characteristic will depend primarily upon the heat sinking, heating rate, pulse-repetition frequency, and pulse width. Hence, its detailed form is a variable that depends on the way in which the measurement is performed, and a major feature of its shape is the question of the existence and position of a “turnover” or threshold voltage &. Other types of thermal SNDC elements exhibit Z(4) characteristics that are determined by critical electric fields or critical temperatures at which appreciable changes in conductivity occur. We discuss them in later sections; first, we concentrate on the thermistor. Burgess (1955a,b,c, 1960) has made an extensive study of thermistor behavior in materials having a conductance of the form G = Go exp(- b / T ) , where Go and b are constants, which encompasses a large number of important semiconductor materials and devices. Independent of the form of the conductance, however, if Z = Z(+, T ) , then d+/dZ

=

- Y>/(Y +

“1

41

(2)

where R = +/Z, x = R(aZ/a+),, y = (+/B)(aZ/dT),, and B = a(Z+)/dT. To obtain Eq. (2) we write the total differential of Z = Z(+, T ) , divide by d+ and multiply both sides of the resulting equation by +/Z. We then use the linear relationship P

Z+

=

B(T - T,)

(3)

3 12

M . P. SHAW A N D N . YILDIRIM

where P is the power, B a constant, and T , the ambient temperature, to obtain an expression of d + / d T , which leads directly to Eq. ( 2 ) . It is clear from Eq. ( 2 ) that the turnover point occurs at y = 1, with NDC setting in for larger values of y . Thus, at turnover 1 = +T(dZ/dT),,T(dT/dP)

(4)

where TT is the temperature of the contact at turnover; +T for a given thermistor is a function only of T , . In order to achieve the condition y = 1 the conductivity of the semiconductor must have the proper temperature dependence. In particular, the common form

Z

A+ exp(- b / T )

=

(5)

where A and b are constants, is sufficient to achieve the turnover condition. In order to simultaneously solve both the heat-flow and current-flow equations, we require knowledge of the boundary conditions. In Shaw et al. (1979) we emphasized the electrical boundary conditions. We must now concentrate on the thermal boundary conditions. A common thermal boundary condition is given by Eq. (3), rewritten here as

T

-

T, = a P

(3)

where a is a constant relating the excess temperature to the power supplied to the material or device. Equations (3)-(5) yield for the temperature of the element at turnover

TT

=

+ b - [ib’ - bTa]”2

(6)

which shows that a requirement for turnover is b > 4T,. The power at turnover is

PT

=

a-’[jb

-

Ta - (fb’ - bT,)’”]

(7)

the power at turnover increases with increasing ambient temperature. This feature is characteristic of a semiconducting thermistor having the property shown in Eq. (4). For b >> T , , a good approximation of Eq. (7) is

PT

= Tz/ab

(8)

and the conductance at turnover is

ZT/VT

= GT

Go exp[l

+ (T,/b)]

(9)

i.e., the conductance at turnover is enhanced by about a factor of e from its isothermal value Go at the same T, (Burgess, 1955a). The potential of thermistor-type devices often hinges on their

THERMAL AND ELECTROTHERMAL INSTABILITIES

313

response times. Can the structure be heated and/or cooled fast enough for use as a high-speed switching device? The direction to take along these lines is to make the structure thin and small; thin-film technology then becomes crucial. It is important to examine materials having desirable thermistor properties in thin-film configurations (Hayes, 1974; Hayes and Thornburg, 1973). Hence, in what follows we emphasize that particular geometry.

B . Heat Flow in Semiconductors The well-studied problem of heat conduction in solids (Carslaw and Jaeger, 1959; Kittel, 1976) is touched upon shortly; rather than be very detailed, we emphasize the specific problem at hand with reference to Fig. 4. The semiconductor has an electrical conductivity u , ( T ) = uOe-AE’kT, where AE is the thermal activation energy and k the Boltzmann constant; uois the conductivity in the limit T + CQ. For later use we also include the presence of inhomogeneities. For metallic inhomogeneities we take u2>> u1and assume for simplicity that u2is either constant or a slowly decreasing function of T. We also take the thermal conductivities as K , ( T ) = culT + pl, with a1 = dK,/dT > 0, and K2(T) = q T + p2, with a2 = dK2/dT < 0. The object of the exercise is to determine the Z(4) characteristics of the device and study the affect of the following: (1) temperature-dependent electrical conductivity; (2) temperaturedependent thermal conductivity; (3) thermal boundary conditions; (4) presence and morphology of inhomogeneities; ( 5 ) critical electric fields for the onset of impact ionization and carrier multiplication; (6) thermally induced phase changes and latent heats. To do the general analysis we must solve equations for the flow of both electric and thermal currents.

emiconducting Film: Ul(Tl; K1(Tl

Inhomogeneities: 0.C T I : K2( T I

FIG. 4. A thin-disk sample of semiconductor containing inhomogeneities placed in a resistive current. The symbols are defined in the text.

3 14

M. P. SHAW A N D N . YILDIRIM

Since we are emphasizing thermal effects in this review, we first concentrate primarily on the heat-flow equation. Later in the article we discuss the coupled thermal and electronic equations for a general semiconductor device containing both holes and electrons. We first consider the simplest system and search for general criteria for (1) thermistor behavior and (2) switching effects. To do this we remove the inhomogeneities, provide a constant current source by setting R L = a,and treat the general case where d u / d T > 0 . For the total electronic current J, with no sources or sinks we have div Je where

+ ata

7

niqi= 0

xiniqiis the total mobile charge density. In the steady-state div J,

0

( 1Ob)

grad C#I

(11)

=

For a uniform homogeneous medium, -u

J, =

we have for the stable steady state,

-

div(u grad 4 ) = 0

(12)

or

V2+

+ (grad u

grad 4)/c = 0

(13)

If Jh is the heat-current density, then the conversion of electrical energy into heat is given by -div[4Je

+ Jh]

=

aw/at

(14)

where w is the energy density and awlat the power density. Here we have assumed that the product of the electronic current and enthalpy per unit charge carrier per unit length is independent of position and that thermoelectric power and Peltier heating effects are negligibly small (Carslaw and Jaeger, 1959):

su

where W = w dv (joules) is the energy stored in the volume v enclosed by the surface s. The first term on the left-hand side (LHS) of Eq. (15) describes the electrical power flowing into the closed surface s: (16)

The second term on the LHS (without the minus sign) is the heat power

315

THERMAL A N D ELECTROTHERMAL INSTABILITIES

flowing out of s:

The right-hand side (RHS) is equal to the difference P , - Ph; i.e., dW/dt denotes the rate of increase of the energy stored in v ;

P, - P h

=

dW/dt

(18)

The energy stored in v is related to the heat capacity of v ,

C = dW/dT Thus, Eq. (18) becomes

dW= CdT

or

P, - P h

=

(19)

C(dT/dt)

(20)

Equation (20) states that the electrical input power P, is used in two ways: Part of the input power flows out of s in the form of heat current with density Jh ; the rest is used to increase the temperature of the system. The terms in Eq. (14) can also be interpreted in a similar way. Defining

P,

=

div[+J,]

=

(grad +) J,

+ +(div J,)

=

J,

*

grad

+

I

= - CTIV+~~ = - Je12/cr

(21)

which is the electrical input power density at a point, and P h

=

div

Jh

= - div(K

grad T ) = - K V2T - grad K

grad T

(22)

which is the heat power density flowing out of that point, Eq. (14) becomes

P, -

=

dw/dt

(23) The RHS of Eq. (23) is the power density required to alter the local temperature of a point. It can be rewritten as P h

dW w=-= - - - - C dT - C dT mg dv dv dv mg where mg is the mass in the volume v . Using the following definitions: p m = mg/dv = mass density = specific mass c = C/mg = heat capacity per unit mass (specific heat)

(25) (26)

Eq. (23) becomes Pe - Ph = prnc(dT/dt)

(27)

Thus, we have two power-continuity equations. One is in integral form, which can be used to relate the total input power, total heat power

316

M . P. S H A W A N D N . YILDIRIM

(efflux), and the rate of change of T:

C(dT/dt) + Ph

=

Pe

This equation can be used to study the system as a whole. The other equation is a point relation which can be used to study local regions of the material:

pmc(dT/dt) + P h = P e

(29) Both of these equations reemphasize the simple fact that the difference in input power and heat efflux goes into increasing the temperature of the system. When the external source is switched on, dT/dt will initially be greater than zero. That is, the temperature of the material will start to rise. The rise will continue until a sufficient temperature gradient is reached whereby, neglecting radiative losses, all the incoming electrical power will flow out as heat. When steady state is reached, dT/dt = 0, P , = Ph, and pe = p h . The differential equation (29) is a forced diffusion equation, an inhomogeneous, parabolic partial differential equation which can be rewritten in terms of 4, T , K , (+,and J e as p,c(dT/dt)

+ div(K(T) grad T ) = - IJe12/u

(30)

For a temperature-independent K it becomes

which has a diffusion constant Dh = K/p,c. Since the heat-flow process is diffusive, let us first discuss the transient state (where dT/dt # 0) in a qualitative manner. The most important feature is that the RHS of Eq. (31) is the heat power generated at a point. Considering that point alone (we use the superposition principle to study the other points in a similar way), it is seen that if & is large (good thermal conductivity, low pm and c ) , the temperature wave will move rapidly (higher diffusion velocity) and a local thermal pulse o r disturbance will propagate over a relatively long distance before being attenuated. A simple way of visualizing this process is offered next via the behavior of an analogous distributed RC network (Sousha, 1971). C . An RC Network Analog of the Heating Process

Heat flow in a solid is a diffusive phenomenon. The differential equation and solutions for T(t, r) are very similar to the solution describing current and voltage waves in a distributed RC network. Figure 5 shows a simple one-dimensional analog distributed RC circuit in which the voltage

317

THERMAL A N D ELECTROTHERMAL INSTABILITIES

2

2’0

I

-

d12

I I

I

FIG. 5 . (a) Thin-disk sample of thickness d ; (b) distributed RC analog network. The symbols are defined in the text.

is analogous to temperature and current is analogous to heat current. For simplicity and compactness we have considered heat flow only in one dimension (z)..A similar model readily accounts for heat flow in the radial direction. The R,’s are thermal resistances per unit length, which are inversly proportional to K ( T ) . The Ct’s are thermal capacitances per unit length, which are proportional to p m , c , and m g. The P ~ , ~are ’ s the “heat current sources,” the Joule heating at each point. The Tn’s are the node temperatures (voltages developed across the capacitors). The energy stored in a capacitor is analogous to the thermal energy stored in the thermal capacity of the system. During the transient state part of the current in R, is used to charge Ct and the rest flows to the load network, which simulates the thermal boundary conditions. At steady state all the capacitors are charged to their final values (determined by Rt and p e ) , and the input power flows to the load network. The temperature at the zeroth node represents the surface temperature T, . A constant temperature boundary condition (infinite heat sink) can be simulated by connecting an ideal voltage (temperature) source with a voltage (temperature) equal to the ambient temperature T,. Figure 6a shows the boundary condition. A boundary condition of the type Ih * Alat G(TS - T a ) (32) interface

Newton’s law of cooling, can be represented by an ideal currentdependent voltage source, Ts - T;, =

as shown in Fig. 6b.

(YJhl

(33)

M . P. SHAW AND N . YILDIRIM

318

Ts I

TS

I

2

-

z = dl)

dl*

(b)

(a)

FIG.6 . (a) RC network analog of a constant-temperature boundary condition; (b) RC network analog of Newton’s law of cooling boundary condition.

Let us first consider an infinite heat sink and suddenly impose an electrical power source on the system. Heat will be generated everywhere in accordance with

For a temperature-independent a,p e will be highest at points where the electric current density is highest. However, since u ( T ) also increases with temperature, the variation of p e with T must be inspected further, and we shall do this shortly. If we treat one current source at a time, we see that the heat power will diffuse toward the short-circuit load (electrodes) with a diffusion constant Dh

= Wpmc

(35)

and with a diffusion velocity proportional to K . Considering just the source pe,nfor the moment, at t = 0, C,,, acts as a short circuit and all the heat current flows through it. As C , , is charged, the temperature T J t ) will start to increase, which will in turn cause part of the heat current to flow to the neighboring circuit (Rt,n-lCt,,-l). This transient process will stop when sufficient temperature gradient ( T , , Tn-l, Tn-2, . . . , Ts) is developed such that all the heat power will flow through the Rt’s toward the boundary. That is, the Ct’s are charged to their limits and draw no more current. Thus, the T,’s will not increase further. The limiting temperature (voltage) for each C, is determined by the source strength be), Rt , and the boundary conditions. The heating process, which we just discussed for a single heat source, occurs for all sources simultaneously. Since there are sources at each point in the material, the steady state may be established in a time shorter than for the single-source case. Because of the symmetry, at steady state

THERMAL A N D ELECTROTHERMAL INSTABILITIES

319

the heat (current) flows only to the right toward the load. Therefore, > T, at steady state. T,,, > T, > Tnpl > Tnp2> It is also interesting to note that during the transient states that may arise due to changes in some heat sources, the direction of heat flow may reverse. For example, let us assume that all the sources except one are dead at an instant t = 0. Also assume that the capacitors have initial temperatures T,(O) > T,-,(O) > . . > T o . Let the strength of P , , ~suddenly increase, and let us kill all the other sources at t = 0. The source Pe,h will start to charge the nearest capacitors toward the new steady-state value, and all the other capacitors start to discharge (or charge) toward their new steady-state values as determined by the new steady-state network. In the above discussion we treated the case of an infinite-heat-sink boundary condition. Similar behavior will occur for a Newton-type boundary condition, which we discuss further in the next section. First, however, we must examine the role of a temperature-dependent electrical conductivity. Since CT increases with T , then the inner region of the example, which has the highest T , will have the highest CT. Hence, this region will draw the most current; a schematic is sketched in Fig. 7. The moderate “current-crowding’’ process shown in the figure is not self-acceleratingor divergent. That is, a larger J, does not necessarily mean more heat generation in this region. In fact, inspection shows that less heat is generated in

-

2

I

I I

I I Je(z=O,r) I

I I I

FIG. 7. Lines of electric current (top) and current profile (bottom) sketched for a sample with a “hot spot” near its geometrical center.

3 20

M . P. S H A W A N D N . YILDIRIM

the crowded region because p e = IJ,12/(+(T).Here an increase in J, at a point is the result of an increase in cr(T). Currents prefer to flow through the high-conductivity region to dissipate less power; the crowding process is a self-stabilizing one. The underlying process can be explained by considering two conducting plates connected to each other by two parallel resistors R and r ( T ) where r ( T ) is a decreasing function of T. Initially, we let R = r(To);equal currents will pass through R and r(To).The total power drawn will be

Po = Z2(rllR)= Z2R/2

(36)

As T increases r will decrease and draw more of the current. Let r(TJ << R ; almost all of Z will pass through r(Tl), and the power drawn from a constant current source will decrease to

P, = Z2r(T,) << Po

(37)

Therefore, it is not correct to assume that an increase in local temperature, which increases the local conductivity, will increase the local current density and lead to a further increase in temperature. That is, thermal runaway will not occur because of this process alone. It is important to note one further aspect of the effect of current crowding induced by the locally heated region. The channeled current will flow through the region in the z direction producing higher pe’s there. Hence, there will be a tendency for the hot region to expand in the direction of the electrodes. This expansion will stop when a sufficient temperature gradient is established that will allow all the heat power to flow out of the film, and this brings us back to the questions of “turnover,” the possible presence of NDC, and thermal runaway. All of these effects are related to the ability of the electrical resistance of the system to be reduced to a sufficiently low value by Joule heating. Ultimately, we must therefore find a mechanism by which the layers of the material adjacent to the electrodes can have their resistance lowered to sufficiently small values. The infinite-heat-sink boundary condition used above is unable to account for such effects since the material adjacent to the electrodes will remain cool and at a relatively high value of resistance. If turnover and NDC are to occur, we must have access to a mechanism that, e.g., will account for the fact that in the NDC region an increase in the current could lead to a decrease in the input power density, even though the temperature increases. To achieve this we can let the electrodes be heated (Newton’s law of cooling) or realize that the electric fields adjacent to the electrodes might be raised to sufficiently large values to induce the field stripping of carriers, carrier multiplication, or avalanche-breakdown effects (Shaw et al., 1973b).

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111. THERMALLY INDUCEDNEGATIVE DIFFERENTIAL CONDUCTANCE A . Thermal Boundary Conditions

The development of an NDC region will lead to switching and oscillatory effects for a sufficiently lightly loaded system. It could also result in thermally induced current filamentation (Stocker et al., 1970; Altcheh et al., 1972), thermal runaway, and permanent or alterable breakdown-type phenomena (memory, Ovshinsky, 1968). In order to understand how the above possibilities might occur, we return to the basic diffusion-free reiation J, = - c ( T ) grad 4. We see that J, can grow when either u,grad 4, or both increase. Let us imagine that we have established conditions for an infinite-heat-sink boundary condition, where substantial heating of the bulk material has occurred. The resulting current (a), temperature (b), potential (c), and electrical conductivity (d) profiles are shown in Fig. 8.

om

-+

''

FIG. 8. (a) Electric current lines; (b) isothermal lines (left), temperature profile (center), temperature gradient profile (right); ( c ) equipotential lines (left), potential distribution (center), electric field distribution (right); and (d) electrical conductivity profile.

322

M . P. SHAW AND N. YILDIRIM

With regard to Fig. 8, the following important points should be noted: (1) There are large temperature gradients near the electrodes (2) There are large electric fields near the electrodes caused by the low temperature of the electrodes (ambient) and nearby regions, and these produce “low conductivity” layers (3) The major voltage drop across the film is caused by the narrow low-temperature, low-conductivity layers (4) If no other factor is included, as the current increases further these narrow regions near the contacts continue to shrink, but they never vanish. Therefore, although the resistance of the structure continues to decrease, the voltage across the device continues to increase (PDC).

The situation outlined above cannot continue indefinitely. Eventually, at sufficiently high bias, the fields near the electrodes will cause a change in the transport properties of the semiconductor. Indeed, these important field effects shall be treated shortly. Now, however, we consider the effect that a more realistic boundary condition has on the profiles sketched in Fig. 8. Instead of keeping the electrodes at a constant ambient temperature, we employ a more realistic Newton-type thermal boundary condition, which allows the electrodes to heat up and thereby decreases the resistance of the thin layer near the electrodes. The appearance of the isothermal lines will now have the form shown in Fig. 9. We see that the electrode surface near the z axis will be heated first. This process can lead to current filamentation because the resistive barrier will be lowered in this region. The effect of this boundary condition is similar to the effect of a temperature-dependent electrical conductivity. That is, the electronic and

FIG.9. Schematic drawing of isothermal lines near the electrodes using Newton’s law of cooling boundary condition.

THERMAL A N D ELECTROTHERMAL INSTABILITIES

4 lines

323

,

Je lines

Jh lines

A z- Td(electrode heated)

le Ii

ines

Filamentation starts

FIG. 10. (a) Temperature (left), conductivity (center), and electric current and potential lines (right) for a constant-temperature boundary condition. (b) Heat current lines for a constant temperature boundary condition. (c) The same as (a) for a Newton’s law of cooling boundary condition. (d) The same as (b) for a Newton’s law of cooling boundary condition.

heat current densities are redistributed in accordance with the new boundary condition. The development of current filamentation will be a smooth, continuous process, with filamentation growing as the current increases. (This is to be contrasted with the critical field case, where the transition might be sudden.) Figure 10 shows how the filamentation develops. As discussed in Section II,A, the presence of a conductivity of the form u = moe-b’Tplus the realistic Newton boundary condition leads to turnover and NDC. However, for a sufficiently large load and proper circuit conditions (see the appendix and Shaw er al., 1973a), these thermally induced NDC points are stable, and the entire I(+) curve can in principle be mapped out (Jackson and Shaw, 1974). Of course, if the load is made sufficiently light, switching will occur from the point IT,+T (see Fig. 3) to another point at higher current and lower voltage, as determined by the load line. For very lightly loaded systems the final state may not be stable because of thermally induced phase changes in the material. Here melting

3 24

M . P . SHAW A N D N . Y I L D I R I M

and shorting or opens may occur, and we describe this event as a thermally induced breakdown or memory phenomenon (Kotz and Shaw, 1982). A word about stability is in order here. It is readily shown via Maxwell's equations (Shaw et al., 1973a, 1979) that bulk NDC points produced electronically are intrinsically unstable against the formation of both inhomogeneous field and current-density (Adler et al., 1980) distributions. When the system evolves into these inhomogeneous states, the domain or filamentary characteristics so produced often exhibit regions of NDC (conductance) themselves. These NDC points can he stahilized. The same situation holds for thermally induced NDC points: uniform, homogeneous NDC regions are unstable; filamentary NDC regions can be stabilized. The latter comprise essentially all the known thermally controlled conditionally unstable systems. Let us now review some major features of the development so far. The main principle that governs the behavior of electric current lines is that these lines prefer to traverse the easiest path from source to sink. An increase in T is the cause of current concentration; not the result. Therefore, this process is self-stabilizing, not self-accelerating. The major role of the Newton boundary condition is basically that, since the electrode is heated at its center (Y = 0, z = d ) , the electrical conductivity here will be increased and lead to current concentration. One effect of the increased conductivity is that the field here will not be as high as it was when the electrode was kept at ambient temperature. Therefore, if we combine a critical electric field concept (in the next section) with a Newton-type boundary condition, the boundary condition will inhibit the field at the electrodes from reaching its critical value. Of further consequence is that if the thermal conductivity is temperature-dependent and if d K / d T > 0, then the thermal resistance to the flow of heat (at the electrodes) will be reduced as the electrode is heated. This will cause a reduction in the overall temperature level of the system, making local heating more difficult to achieve. The thermal conductivity of the inner regions will increase and lead to a thermally short-circuited region. Thus, the heat generated in the inner regions will flow to the boundaries quite readily, and the thermal resistances at the boundaries will be more effective in shaping the thermal behavior of the system. In the equivalent RC network model the situation can be described by reducing the R,'s in the middle of the system compared to the R,'s near the electrodes (see Fig. 5 ) . At steady state, R,,, < Rt,n-l < Rt,n-2will be very small compared to Rt,l > Rt,2> Rt,3. We can exaggerate the situation by making Rt,n= Rt,n-l = Rt,n-2= 0. However, the Rt,,'s cannot decrease too much, otherwise the temperatures T,, Tnp1,TnP2will decrease because of the loss of heat. This in turn decreases the thermal conductivity, thus increasing Rt,,, Rt,,-l.

THERMAL AND ELECTROTHERMAL INSTABILITIES

325

That is, there is a lower limit for the Rt,,’s which provides a sufficient T , level to keep K(T,) high. The property d K / d T > 0 has a stabilizing effect on T. It prevents an excessive increase of T by allowing easier heat flow; when T tends to decrease, K ( T ) will decrease and prevent excessive heat loss. Since we have been emphasizing thin-film configurations, the most important boundary conditions are those at the metal electrodes; the axial boundaries have been fixed at the ambient temperature. It is important to realize, however, that when the geometry becomes that of, e.g., a cylinder, the axial boundary conditions also becomes important. An excellent example of its importance is the problem of the ballast resistor (Bedeaux et al., 1977a,b; Landauer, 1978). This device is simply a long metallic wire immersed in a gas kept at an externally controlled temperature TG, and is a useful example of a quasi-one-dimensional system exhibiting a variety of spatial and temporal thermal instabilities. The state of the wire at time t is described by only one variable, T(x, r), the temperature field along the length x. Under the proper set of conditions the system exhibits NNDC-type instabilities. The reader is referred to the articles by Bedeaux et al. (1977a,b) for further details.

B. The Effect of Inhomogeneities

The presence of a metallic-like inhomogeneity inside the semiconducting film (See Fig. 4) will influence the thermal and electrical transport processes and alter the conditions required for either turnover or switching. First, the inhomogeneity will cause current crowding. The highly conductive region draws the most current, therefore heat will be generated around the periphery of the inhomogeneity. Further, any sharp corners on the inhomogeneity will create large 8 fields leading to higher current densities and enhanced heating at these points. Although the heating process and concentration of current in the middle part of the film will also continue in the manner discussed above, the existence of a metallic inhomogeneity can affect the course of events in a number of ways; for example: (1) Local melting may occur at points surrounding the inhomogeneities, especially at sharp corners where the current density is high. (2) The points where the critical field, 8,, is exceeded may differ from the normal configuration at the electrodes. That is, if there were no inhomogeneity, would be exceeded at r = 0, z = 0, d ; on the electrodefilm boundaries. The existence of an inhomogeneity will cause distortion of both the shape of the equipotential lines and their densities.

326

M . P. SHAW A N D N . YILDIRIM

(3) The thermal conductivity of the inhomogeneity may be important. If we denote the thermal conductivity of the metallic inhomogeneity K,(T), then a wide range of temperatures will exist for which dK,(T)/dT < 0, whereas for the semiconductor dK,(T)/dT > 0 is typical. Since K , increases with T , the generated heat will flow out of the homogeneous material readily and inhibit excessive increases in temperature. With an inhomogeneity present, however, one must consider the system in somewhat more detail. Let us consider the region in the vicinity of the inhomogeneity. As the input power increases, the local T will increase for two reasons: (1) heat will be produced inside the inhomogeneity and (2) heat will be produced in the region surrounding the inhomogeneity. This will act to raise the T of the inhomogeneity via the thermal boundary conditions. Consider the situation sketched in Fig. 11. As the temperature inside the inhomogeneity increases K,(T) will decrease. The heat produced inside the inhomogeneity will see an increased thermal resistance to its flow out of this region. That is, a larger portion of the heat produced here will stay inside the region, which increases T further. This in turn leads to a further growth in T , which will reach a steady state at a higher level and in a longer time than if the thermal conductivity were independent of T . Further discussion of the role of inhomogeneities, including the case where several are

FIG. 11. Thin-disk sample (top) containing an inhomogeneity and the associated temperature distribution (bottom) induced in the steady state by Joule heating.

THERMAL A N D ELECTROTHERMAL INSTABILITIES

327

present, is deferred to the next section, after we discuss the concept of a critical electric field.

C . Critical Electric Field-Induced Switching Effects

The presence of a critical electric field 8c at which a precipitous increase in conductivity occurs will have a profound influence on the ability of a specific system to undergo a switching transition (Shaw et al., 1973b); 8,will play a role for either one or both of the following reasons: (1) it can act to short out the coolest region near the electrodes; (2) it can cause large current densities to flow which produce local heating and melting. As we have discussed, and will reemphasize analytically in Section V where we discuss VOz , a sudden change in u due either to 8,or a critical temperature T, will produce filamentary NDC points that can be stabilized. Systems that exhibit NDC due solely to thermistor-type effects can also be stabilized in their NDC regions by a proper choice of intrinsic inductance L , load resistance R , , and package capacitance C (Shaw et a l . , 1973a). In the Appendix we discuss these important circuit parameters in detail. When a critical electric field is present we may write for the conductivity aoe-AE/kT

u(T)=

7

8 < 8,

ah(>>aoe-AE’kT), 8 2

8,

In any thermistor-type system, as the turnover condition is neared the field near the electrodes becomes large, with the largest field occurring just outside the metallic electrodes near the center of the sample. If 8 is made to exceed 8,at that point, then the region adjacent the electrodes will become highly conducting. In essence this is similar to the sudden motion of a virtual electrode into the material over the warmest region, a shorting out of the highly resistive contact region (Kaplan and Adler, 1972) in this vicinity. This is a self-accelerating process. The expansion of the virtual electrode into the film increases the field just in front of it, which causes further penetration of the virtual electrode (under infinite heat-sink boundary conditions, for example). Thus the device resistance will decrease and the current will crowd into this region and nucleate a filament. When filamentation starts, less power will be drawn from the source. But since this power is dissipated in a narrower region, higher local temperatures will readily be produced. (Effects due to the variation of the thermal conductivity in this region are discussed shortly). We see

328

M . P. SHAW A N D N . YILDIRIM

that %, can lead to the occurrence of a switching event prior to the turnover condition being achieved. The above critical field effect is electronic in nature and is invoked as a mechanism by which a switching event can be electronically sustained. When metallic inhomogeneities are present, however, fields can be produced near sharp points that can lead to high local current densities and initiate a local melting event. Indeed, the two types of effects might work together to produce a switching event. A substantial increase in temperature (or melting) might occur near an inhomogeneity, which would lead to a field redistribution and the subsequent attainment of %, near an electrode; switching or breakdown would then occur. Other possible sequences and/or simultanieties are easy to visualize and categorize. The order of occurrence is not critical; a self-accelerating switching event will be induced. The above considerations will be modified somewhat when the variation of the local power density p e ( T )with temperature is considered. As T increases, v ( T ) will increase and P,(T) could decrease. If p , ( T ) decreases at a sufficiently rapid rate as T increases, the steady-state condition may be reached in a shorter time period and be at a lower T level. However, we have seen previously that P,(T) can be affected by things other than the presence of an inhomegeneity ( gC,T, , d K 2 / d T < 0, etc.). These features can make p e ( T ) increase with T . Because of this the projected steady-state T distribution may not be reached; a switching event will occur first. For example, a switching event can be lauched prior to turnover in a system where d K 2 / d T < 0 and an gCis present. Many common systems will have not only an 8,, but will also contain many inhomogeneities. It is important, therefore, to consider the effect that multiple metallic inhomogeneities might have on the above conclusions. First, it is clear that current crowding effects will be more pronounced in regions that contain more inhomogeneities per unit volume. Therefore, local heating effects will be prevalent here. Further, at the sharpest corners of the inhomogeneities the fields and current densities will be largest. If critical values are exceeded and melting occurs, the presence of a nearby inhomogeneity will aid in the development of the potential instability. This case can be seen simply by realizing that the presence of nearby inhomogeneities will lead to more current crowding about the specific inhomogeneity we are heating. Once the major hot spot is nucleated, the high conductance region will expand toward neighboring inhomogeneities and raise the local fields, for example, above & , thus reducing the time it takes to switch the sample, along with the voltage at switching.

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329

Next, it is very possible that the switching event may also start at regions near the electrode interface where the inhomogeneities are often densest. Further, since dKJdT < 0, the power generated inside the inhomogeneities will tend to remain there. Although negligible power should develop there because of the high values of u ( T ) ,p , ( T ) [= IJe12/(r(T)]may become large due to substantial increases in current density. The existence of regions with high thermal resistances (metallic inhomogeneities at elevated temperatures) will also narrow the path of the flow of heat current, which will then act to raise the average temperature of the inhomogeneous region. Finally, the heating of the electrodes (Newton’s law of cooling) may be enhanced in regions where & is reached, and this will also tend to accelerate the switching process. It is useful to conclude this section with a general description of the possible types of thermal behavior that might occur in a system where the electrical conductivity is a function of both temperature and electric field. A qualitative analytical approach, with support from numerical calculations, leads to the following observations and conclusions. Consider the nonlinear eigenvalue problem AU

=

-Af(x, u ) ,

au

x ED

+ p(au/an) = 0

(384 (38b)

+

where D S is the active region, S the boundary surface, and Eq. (38b) a general boundary condition; Afis the source of u. Some well-known results on the existence, uniqueness, and stability of the solutions of Eqs. (38a,b) are the following (Joseph, 1965; Joseph and Sparrow, 1970; Keller and Cohen, 1967; Simpson and Cohen, 1970): (A) If af/au = f , 2 0 and f ( u ) < F (identical to a function independent of u ) for 0 5 u s 03, then a finite unique and stable solution u > 0 exists for all A in the interval 0 d A d w. (B) Iff, > 0 and unbounded as u += m, then a solution u > 0 exists only in a range 0 < A < A*, where A* is a limit determined byf(u), the boundary conditions and geometry of the system. This solution is unique if f ( u ) is concave ( f u ( u z )< fu(ul) for u2 > u l ) and nonunique if f(u) is convex V,(uz>> fU(Ul>). Equations (38a,b) are in a form similar to (but more general than) Eq. (30) under steady-state conditions div[K(T) grad TI = - g 2 u ( T , 8 )

(39)

330

M. P. S H A W A N D N . YILDIRIM

where T is analogous to u , u ( T , @ to f(u) and %? to A. Therefore, the well-known features of Eqs. (38a,b) may be used to study the solutions of Eq. (39). Examples are the following: (1) If the conductivity u has no 8 dependence, then Eq. (39) has the same form as Eqs. (38a,b); for a conductivity of the form u(T) = u0e?lT (or any other form which remains finite for all T ) , Eq. (39) has a unique solution because u ( T ) has the property A stated above. Under these conditions a SNDC region can be generated by using Newton's law of cooling as the boundary condition for T . The filamentary solution can be stabilized in the NDC region using a source with a sufficiently high resistance and the proper local circuit environment (Shaw et al., 1973a). (2) For a conductivity of the form limT,, u(T)cc T" ( n > 0), which goes to infinity as T + 00, property B above holds, and solutions do not exist beyond a certain local electric field gC(analogous to Ac). Also, for 0 < n < 1 , the existing T solutions are unique and stable; for n > 1, the solutions are nonunique and only one (the lowest) is stable. (3) For an electric field-dependent conductivity u(T, 8), the problem is slightly more complicated because the two coupled equations must be considered together in considering the existence, uniqueness, and stability of the solutions div[K(T) grad TI = - u ( T , 8)g2 div[u(T, 8)grad

41 = 0

(39) (12)

Experimentally, it is known that in materials such as the chalcogenides the electrical conductivity increases with electric field, d u / d 8 > 0. This property has a limiting effect on the maximum field value 8,,, which occurs at the electrode-film interface. As the material is heated, steadily increases. However, this increase is decelerated by the increase in conductivity, which tends not to support high 8 fields. Thus, the RHS of Eq. (39) remains finite, and property A will hold again, yielding a unique T solution for all possible values of field. The best fits between numerical and experimental results are obtained with such conductivities. For example, Kaplan and Adler (1972) have obtained a switching phenomenon with a conductivity of the form u ( T , 8) = uoe-alTewlwo. Shaw and Subhani (1981) have obtained a better fit by assuming a discontinuous u(8) variation given by

Both conductivity functions satisfy the condition du/de > 0, lead to cur-

THERMAL A N D ELECTROTHERMAL INSTABILITIES

33 1

rent filamentation, and cause a discontinuous jump from a low- to a high-conductance state under sufficiently high bias, independent of the external circuit. With such conductivity functions no stabilizable uniform NDC region apparently exists. We have an inherently bistable system independent of the external circuit. Considering all of these features and the numerical results, electrothermal switching phenomena in semiconductors can be described, in general, as follows. As the applied voltage is increased, the temperature of the central region grows faster than the regions close to the boundaries. At low heating levels the electric field is not very effective in controlling the conductivity, hence the conductivity can be approximated by c o e d T . Therefore, the conductivity of the central region will be larger than the regions close to the electrodes. This produces a low-8 field in the central region and a high-8 field near the electrodes. As the voltage is increased further, if c were not a function of 8, very high %fieldswould be produced near the electrodes. Since the heat generation rate is (rg2,increasingly large power will be generated near the electrodes. To keep the power density at the electrode-material interface finite, the conductivity here will increase by some mechanism, which in turn will decrease the local field. This can be accomplished in a number of ways. For example, Newton’s law of cooling is a realistic thermal boundary condition. It allows a certain amount of heating at the interface, thus increasing the conductivity and lowering the field. Newton’s law of cooling introduces a higher thermal resistance at the electrodes, which elevates the temperature level of the material, thus increasing the conductivity everywhere. However, the electrodes are still cooler than the central region, resulting in higher 8 fields at the electrodes. Therefore, most of the heat generation again occurs near the electrodes. The temperature distribution over the electrode surface will be such that it will be maximum at the center of the electrode if there is no inhomogeneity. However, if a metallic inhomogeneity exists, Twill be maximum at the point nearest to the inhomogeneity. At that point, where the low-conductivity barrier is reduced, current filamentation starts. This confines the dominant heat generation near the electrodes to these weak points, which reduces the electrical resistance further. The dependence of (T on 8 with do-/d8 > 0 can also lead to a limitation on the power generation near the electrodes. For example, we have pointed out that as the Joule heating increases, the e-=lTdependence produces a higher conductivity in the central regions compared to the regions near the electrodes, leading to low 8’s in the central region and high 8’s near the electrodes. The e(p’xO or gC-typedependence starts to be effective after a certain field level, thus increasing the conductivity more at a point on the electrode where 8 is maximum. In the presence of a metallic inho-

332

M. P. SHAW A N D N . YILDIRIM

mogeneity, this point will be nearest the inhomogeneity. Here is where the filamentation begins, leading to higher 8 fields, which in turn produces a higher local conductance. This process is equivalent to the expansion of the electrode into the material at the point of interest. This is a selfaccelerating event, because as the virtual electrode pushes into the material (Kaplan and Adler, 1972), the entire voltage drops across a shorter distance, which means a higher 8 field and further expansion of the virtual electrode into the material. It is clear from what we have discussed so far that the switching process in thin films is a very rich and complex subject, and the presence of inhomogeneities can play a very important role. Indeed, in an experimental study, Thoma (1976) provided evidence that bias-induced reversible-switching transitions in a wide variety of thin insulating and semiconducting films between 2 and 100 p m thick occurred when a critical amount of power per unit volume was dissipated in the samples. The materials investigated-crystalline and polymeric, as well as amorphous-included ZnS, mica, A1,0,, anthracene crystals, Mylar, polystyrole foils, crystalline LiF, ZnO, CdS, Si, and GeAsTl glasses. He concluded that in order to explain this ubiquitous phenomenon, one must assume that many real insulating materials contain defects or inhomogeneities arranged in chainlike patterns which give rise to higher mobility and/or higher carrier concentration paths through the films, a view taken and exploited analytically for inhomogeneous multicomponent chalcogenide films by Popescu (1975). Under bias, these inhomogeneities lead to very narrow current filaments that extend throughout the thickness of the films. Since the switching effect in thin amorphous chalcogenide films has been explored in great detail, we treat it first. We shall see that here a critical electric field is fundamental to the switching process. Further, we shall find that the switching process is electronically initiated and sustained, and that thermal effects are important with regard to: (1) the “delay” time for switching; (2) the capacitive effects that occurs upon switching; (3) the “forming” and memory process.

FILMS IV. T H I NCHALCOGENIDE A . Introduction

I . Scope ctfthe Problem As we have already pointed out, the application of sufficiently high electric fields to any material sandwiched between metal contacts almost always results in departures from linearity in the observed current-voltage

THERMAL A N D ELECTROTHERMAL INSTABILITIES

333

characteristics (Shaw, 1981). With further increases in bias in the nonlinear regime, either a breakdown, switching, or oscillatory event eventually occurs. Breakdown usually results in local “opens” or “shorts,” whereas switching is often involved with local changes in morphology that are not as catastrophic as those that result from a breakdown event; here reversible changes in conductance are induced (Kotz and Shaw, 1982, 1983). In many thin films, after a switching event from an “OFF” to an “ON” state occurs, when the ON state is maintained for a sufficiently long time, a “setting” or memory can occur such that when the bias is reduced the sample will not switch back to the OFF state until it is subjected to further treatment such as the application of high-current pulses. There are two classes of explanations for the above array of complex phenomena: thermal and electronic. In general, we have stressed that both effects must be considered in any quantitative analysis, and the two can produce a coupled response called “electrothermal.” In a discussion of the physical mechanisms involved with a particular specimen, the major parameters controlling its operation must be identified and separated out from the less significant features. In this section we do this for bias-induced switching effects in amorphous chalcogenide films (DeWald et al., 1962; Ovshinsky, 1968; Pearson and Miller, 1969) typically 0.5-10.0-pm thick. It is the purpose of this section to review the major experimental features of these phenomena; present the results of numerical calculations that model the first-fire event in homogeneous films and compare favorably with experiment; and discuss switching in inhomogeneous films that have become so because of the morphological changes induced by prior switching events-a process known as “forming.” Here, we suggest that specific inhomogeneous films can show a switching transition initiated by an instability that nucleates at a critical local-power density (Thoma, 1976; Shaw, 1979). On the other hand, specific homogeneous films can be induced into a switching event at sufficiently high electric fields (Shaw el al., 1973b), but rather than resulting in an open or short, intermediate (inhomogeneous) states are formed which serve as basis states for subsequent switching events (Kotz and Shaw, 1982, 1983). The differences and similarities between virgin and formed films and their electronic behavior are emphasized throughout the section. It has been common practice among some investigators to separate switching in thin amorphous chalcogenide films into two classes, threshold and memory (Adler et ul., 1978, 1980), according to whether the OFF state can be resuscitated after the ON state has been maintained for a given length of time. In fact, in most of these materials a memory effect will occur when the ON state is held by direct current for sufficiently long times. Specimens called memory switches usually are of relatively high

334

M . P. S H A W A N D N . YILDIRIM

conductivity [e.g., Ge,,Te,,Sb,S, (Buckley and Holmberg, 1975; Kotz and Shaw, 1982)], where the room-temperature conductivity is about lop5 R-’ cm-l) and “set” in a matter of milliseconds. Specimens called threshold switches usually are of relatively low conductivity [e.g., Te,,As,,Si,,Ge,P, (Petersen and Adler, 1976)], where the roomtemperature conductivity is about 1 O-, R-’ cm-’ ) and require times much longer than a millisecond to set. On some occasions relatively lowconductivity films can still be returned to the OFF state (without additional treatment) by reducing the applied dc bias after being kept ON for times on the order of 10 hr. Under ac excitation the transition can be made reversible for an indefinitely large number of cycles. It is probably the case that the major difference between memory and threshold switches is the time required to set when excited by pulsed or continuous dc. However, in what follows we delineate between the two in deference to common practice. Since our emphasis here is on threshold switching, the memory effect is only discussed when its understanding helps elucidate the threshold effect. The mechanism for the initiation of the switching event in both cases is the same (Kotz and Shaw, 1982).

2 . Switching Parameters The Z(4) characteristics of a typical threshold switch are shown in Fig. 12. At low currents a high resistance is observed (- lo’ R); this is called the OFF-state regime. When a threshold voltage +T, typically 10- 100 V, is exceeded, the sample switches to a low-resistance operating point on the load line, with the dynamic resistance falling to about 1-100 R; this regime is known as the ON state. As long as a minimum current I h called the holding current is maintained, the sample remains in the ON state. However, if the current falls below I h , the sample either switches back to an OFF-state operating point on the load line or, as shown in the Appendix, undergoes relaxation oscillations between the ON and OFF states, depending on the value of the load resistance (Shaw et a l . , 1973a). Since Ih depends on the circuit conditions, it is better to treat an essentially circuit-independent parameter, the holding voltage C#Jh as more fundamental; f#Jh is usually about 1-2 V. Voltage-pulse experiments (Buckley and Holmberg, 1975; Petersen and Adler, 1976; Pryor and Henisch, 1972; Shaw ef a l . , 1973b; Kotz and Shaw, 1982) provide a major source of information about threshold switching and lead to the introduction of other parameters of interest. After a voltage pulse is applied, a delay time t d , typically less than 10 p e c , elapses before the onset of switching. The switching time to has proven to be faster than any means found of measuring it, but is known to be less than 1.5 x sec. It has also been convenient to define a

THERMAL A N D ELECTROTHERMAL INSTABILITIES

335

I(mA)

3 --

2 -I

I

1

I I

I I

--

I

I

I

I

I

I

I

I

2

4

6

-

I

I

1

)

+

(V)

1-

FIG. 12. Current as a function of voltage for a I-pm-thick film of amorphous Te,,As,,Si,,Ge,P sandwiched between Mo electrodes. This is a trace from a Tektronix curve-tracer oscilloscope, which implies a 60-Hz ac signal. (After Petersen and Adler, 1976.)

pulse-interruption time t, (Pryor and Henisch, 1972) as the time between removal of 4 h and application of an ensuing pulse with 4 2 4,, . After (Ph is removed for a time tsm, the maximum benign interruption time, only $h is required to restore the ON state (tsmis typically about 250 nsec, but varies with the original ON-state operating point). For longer values of t , , the voltage required to reswitch the sample approaches the original threshold, 4T; the latter being completely restored in a recovery time t,. Figure 13 shows some of these parameters for two pulses, each of width tP.

3. OFF-Stutr Chuructrristics of u Homogeneous Film For conciseness, we emphasize a typical switching material, Te39A~36Si17Ge7P1, which has been perhaps the most thoroughly studied sample (Petersen and Adler, 1976). At low fields, less than about lo3 V/cm, the Z(4) characteristics are linear, and the resistivity varies with temperature as p ( T ) = 5 x lo3 exp(0.5 [eV]/kT) R-cm. This yields a room-temperature resistivity of the order of lo7 R-cm. The optical energy gap is approx. 1 . 1 eV, or about twice the thermal activation energy, a result typical of amorphous chalcogenide semiconductors, which are usually p type in nature (see, e.g., Tauc, 1974). When, e.g., Mo electrodes are put in contact with the chalcogenide,

336

M . P. S H A W A N D N . YILDIRIM

0

td td

t

to

tp tp

f

tsm

2tp

+

tsrn

Time

FIG.13. Voltage across a sample as a function of time for two pulses of width t, separated in time by t,, . All parameters are defined in the text.

the bands in the latter bend upward by approximately 0.15 eV. Under extremely low applied bias the characteristics are linear, but as the applied bias is increased into the field range 103-105 V/cm, the Schottky barrier manifests itself and the current becomes contact limited; it is controlled by various tunneling contributions from field and thermionic field emission (Shaw, 1981). However, in the 105-V/cm range a “high-field” characteristic appears in which the conduction is bulk limited and of the form u = A exp(8/ga), where 8 is the electric field and A and 8, are constants. In fact, in the field region above about lo4 V/cm, we shall show that the OFF-state I($) characteristics of a virgin, homogeneous film can be fit rather well by using an expression for the conductivity given by q = uoexp[

-

AEk;”

-

3 1 8

where p is a constant representing a field-dependent decrease in activation energy, and gois a constant associated with carrier multiplication; uo is the conductivity as T + 00 in the absence of a field effect (Reinhard e t ul., 1973). 4 . The Switching Transition

When thin amorphous chalcogenide films containing tellurium are homogeneous, uniform, and virgin, under sufficiently short pulse conditions, the initial (first-fire) switching event is a completely electronic event (Shaw e t al., 1973b; Buckley and Holmberg, 1975). Although the firstfire event is classified as a switching phenomenon, it is sometimes also

THERMAL A N D ELECTROTHERMAL INSTABILITIES

337

useful to treat it as a breakdown-type process, since it is often the case that the voltage at threshold is substantially lower after the first few fire events. Furthermore, after further firings the voltage at threshold often continues to drop slowly until, after a sufficient number of firings, in many cases it stabilizes at a “running” value. The drop in threshold voltage upon firing is associated in most cases with a forming process (Shaw et al., 1973c; Kotz and Shaw, 1982, 1983) wherein either crystalline (Bosnell and Thomas, 1972), morphological (Allinson et al., 1979), or amorphous imperfections are produced locally. In a formed or inhomogeneous sample the instability that develops at a critical value of local power density has some features that are somewhat different from the first-fire event in homogeneous films. In the latter case, for sufficiently short pulses a critical electric field is reached isothermally over the entire sample, independent of the thickness (Buckley and Holmberg, 1975). As we shall discuss, we expect that this field strips trapped carriers off local defects, and then the significantly increased Joule heating often dominated by the subsequent capacitive discharge (Kotz and Shaw, 1982) can cause morphological changes at the weakest point in the film. Further, in order to obtain a switchback effect (voltage drop), we must do more than just create excess carriers. We must also have their presence alter some transport property of the film, such as the mobility (see, e.g., Adler er al., 1978, 1980). sec the common delay-time For pulse widths greater than about mode is observed in formed samples of all thicknesses. (To our knowledge, single-shot data showing the existence of a delay-time event occurring during first-fire in a thin homogeneous sample is not available.) There is abundant experimental evidence (Thoma, 1976; Balberg, 1970; Reinhard, 1977) that the delay-time mode produces a switching event at a critical local-power input. It was first shown (Balberg, 1970) that intimate double-inverted pulses produced identical delay times and later (Reinhard, 1977) that a critical rms voltage switched samples after identical times in a study of their response to pulse-burst waveforms. These results are evidence that t d is associated with the time it takes for a local hot spot to grow and rearrange the field in the sample (Shaw et al., 1973c; Homma, 1971). As we have emphasized above, voltage-pulse measurements have been very useful in elucidating several important aspects of t d . In fact, for sufficiently short pulses (Shaw et al., 1973b), t d can be made comparable to the time it takes for the voltage across the sample to collapse from & to &,, the switching time t o . Whereas fd is thermal in origin, to is due to an electronic process in both virgin and formed films. Models for both are discussed shortly.

338

M . P. SHAW A N D N . YILDIRIM

in 25-pm- A diameter pore A~

+

/o

/ A/

01

6/'

Pore saturation

. PI

k

PI o

kkrpore p).

/

J

11

2

5

10

20

M

loo

rf ( p m I Filament Radius

FIG. 14. Filament radius as a function of steady-state current determined by four methods. The solid line represents the results of the velocity-saturation analysis, the data points are the TONC results, and the dashed line is calculated from Shanks' (1970) carbon/chalcogenide/carbon results. Two pore-saturation points are also indicated. TONC d = 2.0 p m (0, 0 , A). data points calculated with pon= 0.1 Cl cm; d = 1.5 p m (V, A, 0); (After Petersen and Adler, 1976.)

5 . ON-State Characteristics

The forming process produces a local inhomogeneous region typically 1-5 pm in diameter. It is through this relatively high-conductance region

that the major portion of the current flows when the sample is in the ON state. This filamentary current-carrying path has a radius rf and its major features have been described for samples that did not exhibit forming (Petersen and Adler, 1976). In the studies of Petersen and Adler (1976) the current dependence of rf was experimentally exposed by several independent methods. The results are shown in Fig. 14. First, a study of velocity saturation in crystalline Si/amorphous chalcogenide heterojunctions provided a means of determining the current density in the chalcogenide in the 2-9-mA range (prior to avalanche breakdown in the Si depletion region). It was found

339

THERMAL A N D ELECTROTHERMAL INSTABILITIES

that the area of the current filament Af increased more or less proportionally to the increase in current, indicating that the current density remains constant in the filament over a wide range of current. Second, the transient-ON-Z(+) characteristic (TONC) technique was used to analyze the ON-state behavior (see Fig. 15). It was found that the TONC was stable for only about 50 nsec, after which the response gradually relaxed to the steady ON-state I(+) characteristic. Therefore, for TONC pulses less than 50 nsec (Kotz and Shaw, 1982), we expect that the area of the current filament remains the same as in the steady state; the shape of the TONC should then depend upon the value of the operating steady ONstate current for which a particular TONC is taken. This is in fact the case. In general, we expect three contributions to the voltage drop across the sample in the ON state: the resistance of the ON-state material; the contact resistance R , ; and the interfacial barrier +B. The TONC curves should obey ~ T O N C= +B

-k

I[&

-k

pod/Adzdc)]

(41)

where ponis the ON-state resistivity and Adzdc)the area of the current filament at the steady-state operating point. Extrapolation of the sub-50-nsec TONC curves should yield the same value for + B , and this value should be the same as the metal/amorphous chalcogenide barrier measured by other means. The agreement is good. The TONC slopes then determine the variation of Af with steady-state current.

m

'TONC

"'in

200 180 160

- 140 4

-.

Time

-E

E 120 "TONC

100

0' 80 60 Time

40

-.

20 Ib )

0

0.4

0.8

1.2

$J~

Ic)

Voltage

1.6

FIG. 15. (a) Transient ON-state characteristic (TONC) measurement. (b) A typical result. (c) Different Mo/amorphous/Mo sample TONC curves taken from different I,, , v,, points. The sample is 50 p m in diameter. (After Petersen and Adler, 1976.)

3 40

M . P. SHAW A N D N . YILDIRIM

Further, the steady ON-state voltage is +dc

=

+B

+ IRc + Pon dJ(Z)

(42)

If, as is expected, J is independent of I , extrapolation of the steadyON-state characteristics to Z = 0 will yield an effective barrier voltage +Bteff)

= +B

+

Pon

dJ

(43)

and this should vary linearly with the thickness of the chalcogenide film. Experiments show this correlation rather well. Furthermore, extrapolation of +B(eff) to Z = 0 should yield +B. Again, this is the experimentally observed situation. These results yield pon = 0.07 R cm. Shanks’ results (1970) for the ON-state I(+) characteristics of a formed chalcogenide film having pyrolytic graphite electrodes were also used to determine the area of the current filament as a function of current. The results are shown as the dotted line in Fig. 14. Analysis of the gain observed in an N (ON state) PN amorphous crystalline heterojunction transistor (Petersen et al., 1976) as a function of the cystalline-Si-base doping concentration showed that the free-carrier concentration in the ON state is of the order of 1019 CII-~.This implies that the ON-state carrier mobility is about 10 cm2/V-sec. The total of the cathode- and anode-interface barriers for MO/amorphous MO samples is 0.4 eV. If this is distributed evenly between cathode and anode, and if in the ON state fields above lo5 V/cm must be maintained near both electrodes, the band bending will then extend about 30-70 A into the amorphous material. This is sufficiently narrow such that it is possible that the ON state can be maintained by either strongfield-emission or thermionic-field-emission tunneling through the electrode barriers. However, if the barriers are asymmetric the depletion regions can be larger in extent, and tunneling processes become less likely. Alternatively, the ON state can be maintained from carrier generation in the high-field regions themselves. Since the potential drop in these regions is less than Eg , such generation would have to be from localized states rather than from across-the-gap excitation. In either event, it is likely that both electrons and holes contribute to the ON-state current, just as they do in the OFF state. However, just as holes predominate in the O F F state, there is evidence that electrons predominate in the ON state (Petersen et al., 1976).

6. Recovery Properties When the current is reduced below Zh,the sample switches back to the O F F state. One possible mechanism for the initiation of this transition suggests that there might exist a minimum rf for which radial diffusion

THERMAL A N D ELECTROTHERMAL INSTABILITIES

I

0.0 0.0

I

I

I

I

I

I

I

1.0

2.0

3.0

4.0

5.0

6.0

7.0

34 1

1, (jlsecl

FIG.16. Recovery of threshold voltage as a function of interruption time t , for several values of ON-state current. The double pulse sketched on top is as shown in Fig. 2, but with t , > t,, so that V , > V,. Curves: A, 5 mA; B, 15 mA; C, 60 mA; D, 100 mA. (After Petersen and Adler, 1976.)

would break the filament. This would set an absolute minimum value for the current that can be maintained in the ON state, z h m . However, observation of z h m is normally difficult to achieve because of the reactive components in the circuit. If we define I h as that current below which circuit-controlled relaxation oscillations occur (Shaw et ul., 1973a), then for most sample configurations there will always be a range of currents between z h and z h m that are unstable against relaxation oscillations. The package capacitance C and intrinsic plus package inductance L will always produce > z h m . On the other hand, if an intimate double-pulse technique is employed (Hughes et a / . , 1975), where the sample is forced to remain in the ON state after switching by first rapidly reducing the applied bias, then by minimizing C and maximizing the load resistor R , , values of z h as low as 10 pA can be observed. (These are currents that would produce relaxation oscillation were the ON state not “held” by the second pulse.) For current densities in the filament in the range of lo3 A/cm2, such low values of current imply that filament radii in the 0.5-pm regime can be stabilized. In this case, & is rather high since the ON-state (filamentary) characteristic itself exhibits a long, stabilizable NDC region for these currents. The fields are therefore sufficiently high so that the ON state is maintained in an extremely narrow filament; as + c $ ~rf, ap-

342

M . P. SHAW A N D N . YILDIRIM

proaches its minimum value. There is evidence that the minimum radius of the current filament may, in fact, be in the fractional micrometer regime; thus there is a possibility that Zhm exists. However, for essentially all circumstances where the battery voltage is kept constant after switching occurs, Zh should be treated as a completely circuit-controlled parameter. Once the voltage across the sample is removed, the recovery curve can be studied. As shown in Fig. 16, the recovery process depends upon the steady-state operating point. One explanation of the data is that after the voltage is removed, the field at the anode adjusts almost instantaneously but the cathode field decays slowly (Frye et al., 1980), maintaining carrier generation or tunneling near that contact. Since the applied voltage is now zero, a counterfield will be built up near the anode within a dielectric relaxation time: this explains the symmetry of the TONC results shown in Fig. 15b. The limiting feature of the recovery process is then the ambipolar diffusion of carriers radially out of the conducting filament. As the diffusion proceeds, the radius of the filament decreases. As long as any filament remains, only &, is required to resuscitate the ON state. However, after a time which depends on the original rf (and thus I,,), the filament shrinks to zero radius and the contact barriers begin to decay (this is the origin of the parameter t,, discussed in Section IV,A,2). Once the equilibrium contact barriers are reestablished, the original is completely restored. An alternative explanation of the recovery data is that t,, is the time it takes for the contact barriers to widen to lengths insufficient to sustain the motion of large numbers of carriers through them. At this point a sharp increase in resistance occurs as the contact-to-contact path is broken; the remainder of the recovery process involves diffusion of heat out of the filament.

B . Numerical Culculations of the First-Fire Event in Homogeneous Films 1 . Introduction

Shaw et al. (1973b) and Buckley and Holmberg (1975) have presented experimental data on the first-fire event in both virgin threshold and memory material, with the latter work being more extensive in that, among other things, a range of samples thicknesses were explored. Both sets of experiments showed that for sufficiently short pulses, a critical electric field exists that initiates a breakdown-like switching process which, in these experiments, leads to forming and a substantial drop in threshold voltage upon subsequent firings. Although the workers cited above did not explore the formed filamentary region via scanning electron micros-

THERMAL A N D ELECTROTHERMAL INSTABILITIES

343

copy, others (see, e.g., Bosnell and Thomas, 1972) have done so in great detail, and different types of inhomogeneities have been shown to be present. We discuss one type shortly. In this section we present the results of extensive numerical electrothermal calculations for both threshold- and memory-type material, and compare our results with the experiments discussed above. The details of the calculations can be found in Subhani (1977). We study both timeindependent and time-dependent processes, incorporating a critical electric field into the model in order to obtain agreement with experiment (Shaw and Subhani, 1981). It is important to note in what follows that the only difference between the two types of samples involves the setting or lock-on in the memory ON state. The mechanism for the initiation of the ON state (prior to memory lock-on, if it occurs) is the same for both. 2 . Memory-Type Samples: Calculations of the Steady State We first solve an electrothermal model for the steady-state Z(+) and T ( r ) characteristics of a virgin memory sample (Fritzschi and Ovshinsky , 1970), and then compare our calculated I(+) characteristics with the experimental results of Buckley and Holmberg (1975). By electrothermal we mean solutions of the heat-balance equation explicitly including nonohmic contributions such as a field-dependent conductivity and /or a critical electric field. The geometry of the sample under analysis is that of a homogeneous thin circular disk of radius R and thickness d sandwiched between metallic electrodes (see inset, Fig. 17). Because switching occurs primarily along a central axial path, the temperature far from the center of the sample remains at ambient. Hence, rather than apply the boundary condition (aT/ar)\,, = 0, the radial surface of the sample is kept at a temperature T(r = R ) = T a , where T, is the ambient temperature. However, the axial surfaces of the sample (the amorphous/electrode interfaces) have finite heat losses and are modeled using the electrode boundary condition previously described,

K , grad T * A

= - G,(T -

Ta)

(32) Newton’s law of cooling, where a = e for the electrodes and a = a for the chalcogenide material; A is a unit vector normal to the boundary. Here both G, and K , are taken as finite and independent of T (G is the Newton coefficient). For simplicity, K, is also taken as a constant, although it is a slowly increasing function of T . The electrical conductivity of the amorphous material is taken as thermally activated and of the form CT = CT,,

exp

I-

AEkT -

-“I

k?

344

M . P. SHAW AND N . YILDIRIM

+Radius

I 0

I

I

1

I

105 2 x lo5 Average Electric Field across Sample ( V Icm I

I

I

3 x 10'

FIG. 17. Comparison of the experimental and calculated dc OFF-state characteristics for a memory-type sample whose parameters are given in Table I. Four different sizes are modeled. The insert shows the geometry of the sample; experiment (-), numerical calculations (---). (After Shaw and Subhani, 1981.)

where the AE - pa term represents a field-induced decrease of thermal activation energy and the last term in the bracket represents carrier multiplication effects [in the actual numerical calculation, this term is written as a,,/($ + a') in order to yield a finite conductivity at zero field]. For the steady-state calculations the inhomogeneous elliptic partial differential equation of heat conduction [Eq. (31) with aT/at = 01,

K V 2 T + sL,/u = 0

(39)

together with the coupled electrical and thermal processes in the sample are solved on a grid in the finite-difference approximation. It is important to appreciate that for well heat-sunk electrodes, the solution of this equation for a temperature-independent K cannot successfully account for the observed virgin Z(4) characteristics of either memory or threshold-type material unless a critical electric field ZYC at which a precipitous increase in conductivity occurs, is included in the calculations. The OFF-state characteristics used for memory material are given in Table I. To compare with experiment, calculations are performed for four cases: 20- and 40pm-pore diam.; 1.40- and 2.55-pm thickness. Figure 17 shows a comparison of the numerical results with the experimental values reported by Buckley and Holmberg (1975). The agreement

THERMAL AND ELECTROTHERMAL INSTABILITIES

345

TABLE I

PARAMETERS USED FOR MEMORY-TYPE SAMPLES go= 7 x lo3 V/cm gC= 3.1 X lo5 V/cm

AE

p

=

0) T, K,,,, G,/K,

u ( ~ a re =

0.43 eV

e cm

= 1.5 X

1 . 1 x 10-5 (n crn-1 297 K = 3.0 mW/”C cm = 3.5 x lo4 cm-l = =

is quite good. We determined steady-state values using a constantcurrent source. Starting at low-current density in the OFF state, the current was incremented slowly until the sample underwent a large change in conductance. The steady-state voltage was recorded for each current density. Figure 18 shows the temperature as a function of radial position on a plane through the center of the memory sample just after breakdown, 140,

h

mL Glass Transition Temp. ( l 3 5 O C 1

-

r = A r ( i l), A r - l#m

FIG.18. Temperature above ambient versus radial position for the ON state of a memory switch. Using the experimental fact that the microcrystalline ON-state filament is about 2-4 pm in diameter, and the glass transition temperature in the material is about 135”C,we inserted the filament diameter and found that the best fit to the I(+) characteristic was ob= 10 n-l cm-’. The sample is 2.55 p m thick and tained for Kon/Koff= loz. We also used uoan 40 p m in diameter; R L = 10 kn. (After Shaw and Subhani, 1981.)

346

M . P. S H A W A N D N . YILDIRIM

where the steady ON-state current is equivalent to that produced with a load resistor RL = 10 k a . Note that (1) in Fig. 17, the J(@ characteristics exhibit a slight region of thermistor-type SNDC before the onset of switching, and (2) in Fig. 18 the temperature distribution and current density define a sharp filamentary conducting path after switching. Prior to switching, the maximum temperature in the O F F state is calculated to be about 20°C above ambient. Just after switching, the maximum temperature at the center of the filament is 126°C above ambient, which is above the glass transition temperature for this material. However, to develop a crystalline filament, substantially higher temperatures must be developed. It appears that the energy required to do this comes from the capacitive discharge induced during the switching transition (Shaw et al., 1973a; Kotz and Shaw, 1982). The microcrystalline filament eventually formed is approximately 2-4-pm diam., in agreement with experimental observations. This demonstrates clearly that electrothermal numerical calculations, modified with a critical electric field, can quantitatively mimic the breakdown-type switching characteristics observed in memot-y-type chalcogenide films. In general, our calculations yield for virgin memory-type samples: (1) The calculated and experimental I($) values are coincident at low fields for the range of film thicknesses and radii investigated. This indicates that a uniform field and current-density distribution is present at low fields and thermal effects are unimportant here. (2) The I($) characteristics diverge from a common line near the threshold voltage, indicating that the electric field or the current density, or both, become nonuniform within the sample because of local heating. (3) As the film thickness increases, the Characteristics diverge from the common curve at lower fields and current densities, again because of local heating. (4) As the sample diameter increases, the breakdown voltage decreases. ( 5 ) The current density at breakdown is unaffected by variations in diameter for a given sample thickness (in the range investigated). (6) As the film thickness decreases, the effects of the diameter variations are sharply diminished, thus making the I($) characteristics less sensitive to diameter for very thin samples. (7) For all the memory samples under investigation there is a tendency for the OFF-state characteristic to bend back upon itself near the onset of switching (thermistor-type behavior). The resulting effect is to decrease the average applied field because of the NDC region.

THERMAL AND ELECTROTHERMAL INSTABILITIES

347

(8) Complete thermistor-type behavior (see Fig. 19) can be observed by decreasing the thermal conductivity of the electrodes o r eliminating gC. When G J K , is increased, thermistor behavior with a turnover voltage either above o r below & can be induced. Here, in many cases gCis never reached in the sample. For the case where gCis removed from the calculations, a turnover voltage without breakdown occurs; the turnover voltage is much higher than &. (It is important to reemphasize here that for a sufficiently light load thermistor characteristics produce a thermal runaway event.) (9) The current-density and temperature distributions define sharp filamentary conducting paths connecting the electrodes for currents above threshold. Our calculations reveal the fact that for any one sample thickness, the switching current density is independent of radius. This is also evident from a current-density-versus-radius plot in the immediate preswitching region, where prethreshold heating is observed. In other words, any divergence from the common curve at high fields (near switching) is due to a nonuniform field distribution along the axis of the sample.

-

1. 6

r

1.2

-

0.8

-

0.4

-

II

'5

4

-0'4r

9 0.0-

I

-0.8

I

I

I

106

I

I

2 x 106

1

I

I

3 x 10'

Average Electric Field across Sample ( V I cm)

FIG. 19. Calculated prethreshold I ( 4 ) characteristics of a typical memory sample having a 20-pm radius and 1.40-prnthickness as a function of G , / K , , the heat-conducting properties of the electrodes. (After Shaw and Subhani, 1981.)

348

M . P. SHAW A N D N. YILDIRIM

go= 7 x 103 V/cm gC= 7.0 x lo5 V/cm

AE = 0.55 eV B = 1.5 x

e cm

u(T,,

0) T, K,,,,, G,/K, E =

(acm)-' 2 x 297 K = 3.0 mW/"C cm = 3.5 x lo4 cm-'

= =

We can also expect a somewhat radially nonuniform current distribution as a result of nonmetallic inhomogeneities (film imperfections). We have verified this expectation by simulating an inhomogeneous conductivity model wherein we allow for small conductivity perturbations that model imperfections having the same value of thermal conductivity as the amorphous material. Relatively small changes in the virgin I(+) characteristics are observed for conductivity variations across the sample of up to 30%. The general shape of the I(+) curve shown in Fig. 17 is maintained. 3 . Threshold-Type Samples: Calculations of the Steady State

Now that is clear that the OFF-state conditions of a memory sample can be modeled, we turn to the threshold case (Shaw et al., 1973b), making use of those ON-state parameters which have emerged from the best fit for the memory behavior. We assume that the thermal properties of the memory and the threshold material are the same, and in our calculations use the same ratio of ON-to-OFF-state thermal conductivity for the threshold sample as for the memory sample. All other parameters are obtained directly from observations on the threshold material. The parameters are given in Table 11. The major features of the I(+) and T(r) calculations shown in Figs. 20 and 21 are similar to that of memory-type samples. However, note that in Fig. 20 the departure from the common curve at high fields is very small compared to that of the memory samples. Thus, according to the calculation, there should be no phase change induced in threshold-type samples. Forming should be absent and the switching initiation and maintenance should be electronic processes with only minor thermal overtones. Although this is sometimes the case, most experiments, however, produce different results. First, scanning electron microscopy reveals the presence of both crystalline and morphological imperfections (Bosnell and Thomas, 1977; Allinson et al., 1979) in formed films, and these are the most common films encountered in practice. Hence, we suggest that '& causes a switching event that produces high temperatures often because of the capacitive

THERMAL AND ELECTROTHERMAL INSTABILITIES

349

Thickness

0.0

-

-

-0.5

-

-1.0

-

-1.5

-

n

'5

u

:-

9

-2.0

-2.5

-

-3.0 0

7 x 10'

3 x 10' 5 x 10' Averwe Electric Field across Sample V I cm)

10'

FIG.20. OFF-state J ( 4 ) characteristics for a threshold-type sample whose parameters are given in Table 11. Four different sizes are modeled. (After Shaw and Subhani, 1981.)

I

I

I

3

5

7

I

9 r*Arli

I

-

I

I

I

11 13 I5 1): A r - Ipm

I

17

19 -i

21

FIG.21. Temperature above ambient versus radial position for an electronic model of the ON state of a threshold switch. The sample is 2.55 p m thick and 40 p m in diameter (for a 1.40-pm-thick sample the maximum temperature is 18°C above ambient); R , = 10 k n . No overvoltage is applied. These results assume that the carriers induced by '& produce no additional Joule heating. (After Shaw and Subhani, 1981.)

350

M . P. SHAW A N D N . YILDIRIM

discharge in both memory and threshold samples, and also because of operation at high ON-state currents (Kotz and Shaw, 1982). In order to explain the divergence from the common curve at high fields for a given (memory or threshold) film thickness with different diameters, we study the temperature profile of the sample in the OFF state. Investigation reveals that for low fields the power input is small, causing negligible heating effects, and the curves are coincident for all geometries. As the current density increases, heating effects are observed if the power input approaches the power-dissipation capacity of the sample. Generally, the diameter of the sample is much larger than its thickness and heat is dissipated primarily along the axial direction. Thicker films will develop higher internal temperatures than thinner films under the same conditions because the heat transfer is limited primarily by the low thermal conductivity of the material. Furthermore, the conductivity expression descriptive of the material is a temperature-activated type; a small change in temperature will result in a comparatively large change in conductivity. Hence, the temperature gradient will redistribute the applied voltage across the colder regions of the film, causing an axially nonuniform field distribution. Therefore, it is reemphasized that aside from electronic contact effects, the highest fields will occur next to the electrodes, where the film is coolest. When the local field exceeds ‘i&, the sample will switch. Our calculations indicate that for R >> d, radial heat transfer is small. However, it is not neglible for R = d (thick films or small diameters) because the ratio of diameter to thickness is reduced and the relative contribution of the radial heat transfer is increased. This can result in a lower internal temperature rise and higher calculated average breakdown field. The experimental results as well as the calculations provide good evidence of internal heating in the immediate preswitching region for memory material. A sample will usually undergo breakdown at higher average fields and higher average current densities if heating effects are reduced by changing the sample geometry or material composition (conductivity) for a given set of thermal boundary conditions. In comparing the memory-type virgin I(+) characteristics with the threshold-type characteristics, we see that the threshold sample has (1) a lower OFF-state conductance; (2) a higher breakdown voltage; (3) almost no departure form the “common” curve; (4) essentially the same breakdown voltage for different diameters. We therefore reemphasize that prethreshold heating in virgin threshold-type samples is not important in producing the breakdown-type switching event. Rather, it is the critical field that initiates the switching event and the ensuing capacitive discharge pro-

THERMAL A N D ELECTROTHERMAL INSTABILITIES

35 1

duces changes in the nature of the material and a formed filamentary region suitable for reversible switching events upon subsequent firings. We suggest how this might happen in Section IV,C. 4 . Culculutions of Time-Dependent Processes

Although to our knowledge no direct data exists on delay-time effects involved in switching events in virgin samples, we calculate the average field at breakdown as a function of pulse width t,, which can be compared with experiments of this type (Shaw et ul., 1973b; Buckley and Holmberg, 1975). To do this, we solve the time-dependent heat equation [Eq. (31) with J, = (~81, cpa(dT/dt)= K V 2 T + (+g2

(44)

subject to the boundary conditions previously described. The results of the calculations for memory-type samples are shown in Fig. 22. Pulse widths in the range 2 x lop9 5 t, 5 sec were investigated. Comparison of Fig. 22 with Fig. 23 shows that good qualitative agreement exists between experiment and the numerical calculations. We see that the sec, in approxisec and t 2 threshold voltage saturates for t, 5 mate agreement with the data. However, the difference in the average fields at which the long- and short-pulse results saturate, which we call A g T , is generally not as great in the numerical calculations as it is in the experimental data. Furthermore, the experimental value of gT is about

-8

1 0 ~ mradius 20 rm 10 rrn 20 I r m

-7

-6 Log Pulse Width (sec)

-5

FIG.22. Calculated variation of the average switching field versus log pulse width (seconds) of a memory-type sample modeled after that producing the data shown in Fig. 17. The arrowheads denote that 4t was independent oft, in both directions for all larger and smaller values of t,. (After Shaa and Subhani, 1981 .)

352

M . P. SHAW A N D N . YILDIRIM

1.0 Irm

- - - - - t 0I- 9

-8

-7

-6

-5

-4

-3

Log Pulse Width (sec)

FIG. 23. Variation of the average switching field with voltage pulse width for virgin samples of GE,,Te,,Sb,S, having a 20-pm pore diameter and three different thicknesses. Note that for the shortest pulse the switching field asymptotically approaches the same value independent of sample thickness. (After Buckley and Holmberg, 1975.)

20% higher than the value required to obtain the precise I(+) fit in the memory material shown in Fig. 17. The closeness of 2Yc in these two cases is, in fact, evidence that our model applies rather well to this memorytype virgin material. Finally, the typical t d ’ s of about 5 psec predicted from the calculations for virgin samples are sufficiently close to those observed experimentally in inhomogeneous “running” samples to support a model where t d is thermally induced in formed memory-type samples for sufficiently long pulses. In this standard model, a “hot spot” nucleates in the center or high-conductance region of the sample and the conductivity there increases, thereby reducing the voltage in the central region and increasing it near the electrodes where 2Yc is eventually reached and switching occurs. This model is basically the one we outlined in Section II1,C. The agreement between experiment and numerical calculations is not nearly as good for the threshold-type virgin material (Shaw and Subhani, 1981) [here the data are sparse since only a single thickness was studied (Shaw er ul., 1973b)l. The experiments revealed no clear cut saturation for tp 5 lo-’ sec. Furthermore, the experimental value of A2YT is substantially larger than the predicted value. Finally, the predicted t d is less than

THERMAL A N D ELECTROTHERMAL INSTABILITIES

353

lop6 sec, a value much below those experimentally observed in inhomogeneous samples, which is of the same order of magnitude as for formed memory-type samples, less than about lov5sec. This is an important point. Experimental values of t,, are typically the same for both formed (inhomogeneous) threshold and memory-type material. The numerical calculations for virgin samples, however, show that td should be about an order of magnitude longer in memory-type material. This result is in harmony with the switching model that we shall discuss in the next section and is based on the precept that gCinitiates a switching event in both types of materials. This event causes forming in both, and the mechanism for the switching effect observed in subsequent firings is the same in both-an electronic instability that is thermally modified and electronically sustained. One other important point, most clearly seen with reference to Fig. 23, is that as t, decreases below about lop7sec, the rate of rise of %‘T first tends to diminish and then eventually saturates. These data are contrary to the behavior expected from a model of the first-fire event based solely on heating with weakly heat-sunk electrodes. For long values of t, (Z sec), gT decreases with increasing film thickness and is almost independent of t, for any one thickness. Furthermore, as previously discussed, the axial nonunformity that causes gTto diverge with thickness is consistent with our model over the range of studied f,’s. However, gT is independent of film thickness for short t,’s. Calculated values deviate slightly in the sense that the increase of gTwith thickness disappears as t, decreases, whereas a small slope is seen in the experimental data. This may be due to the fact that in our model the voltage drop near the electrodes is symmetric, which is probably not the case for actual samples. However, the fact that gTis independent of film thickness in this time regime suggests that the field is only slightly nonuniform along the axial direction and the breakdown event is a bulk effect; i.e., the threshold voltage equals %& for sufficiently short pulses. We have seen that the first-fire event in homogenous films can be interpreted as a switching event induced by a critical electric field. In the next section we discuss how this event might lead to forming and how formed samples might act as reversible switches. In what follows we make use of the several experimental observations showing that differences exist between virgin and formed films; e.g., virgin films show a short-pulse critical field and a dc critical voltage-formed films seem to show a critical local power density; virgin films show large prethreshold currents for short pulses-formed films only do so when the temperature is lowered substantially.

354

M . P . SHAW A N D N . YILDIRIM

C. Electrothermal Switching Mechanisms in Formed- Chalcogenide Films

I . Introduction In the next section we discuss switching effects in vanadium dioxide (VO,). In this material a structural phase change occurs at a critical temperature T, ; at T, the conductivity rises precipitously by several orders of magnitude. By modeling this phenomenon with a step-function change in conductivity at T, , we can treat the problem analytically and predict the observed Z(4) curves successfully. A similar phenomenon occurs for inhomogeneous chalcogenide films. Here, however, it is again 8, which causes u to rise precipitously, and it is reached locally in many cases because of thermal effects. In the last few sections we have discussed switching effects in uniform homogeneous chalcogenide films. Thermal theories attempting to explain this phenomenon have been presented by many authors (see, e.g., Stocker rt al., 1970; Croitoru and Popescu, 1970; Sousha, 1971; Duchene et al., 1971; Altcheh et ul., 1972; Kaplan and Adler, 1972; Warren, 1973; Kroll, 1974; Owen et a f . , 1979; Shaw and Subhani, 1981). From these studies, it has been made clear that for nonthermistor-type switching to occur, an electronic mechanism must also be operative in order to short out the low-conductance regions adjacent to the cool electrodes. A critical electric field will certainly suffice, and it is this assumption that we have used to provide good agreement with the first-fire-event data for memory samples discussed in the last section. Experiments show that during the first few firing events a breakdown-type process occurs that is driven by the capacitive discharge and/or high operating currents (Kotz and Shaw, 1982). An open, and most often, forming or a short (memory) occurs if the sample is kept ON for a sufficiently long time. In general, the first-fire event produces an intermediate state that is a narrow (< 5-pm diam.) filamentary region containing crystalline o r morphological imperfections. [Sometimes several firings are required to develop a formed state that is amenable to easy observation by scanning microscopy, but it is the first-fire event that often results in the largest change in threshold voltage (Allinson et al., 1979).] For a threshold switch the formed region is of higher conductance than the surrounding homogeneous film (Coward, 1971), but still of substantially lower conductance than the ON state. We can imagine the intermediate state as being formed in the following manner. Consider, for example, the case where sufficiently short pulses are applied such that 8,is reached isothermally over the entire sample, independent of its thickness (Buckley and Holmberg, 1975). In this region of pulse width ( 5lo-* sec) the power P dissipated in the sample because of Joule heating is given by P = u R T g 2 , where uRT

THERMAL A N D ELECTROTHERMAL INSTABILITIES

355

denotes the the room-temperature (ambient) conductivity. When gCis reached, the current increases by orders of magnitude (Buckley and Holmberg, 1975) at constant voltage. (This large increase in current at constant voltage prior to breakdown or switching has only been observed at room temperature in virgin samples for sufficiently short pulses. Formed samples show this effect at low temperatures.) In this region, P = ai, 8:where crh >> uRT. The large increase in conductivity induced by gc can be due to either the field stripping of trapped carriers and/or avalanching. Once these carriers are generated, because of the ensuing capacitive discharge, the significantly increased Joule heating causes a breakdown at the weakest point in the film. The sequence leading to forming in virgin samples subjected to short pulses is first electronic, then thermal. As previously stated, the outcome of the switching event can be (1) an open; (2) a short (e.g., the memory state); and (3) an inhomogeneous formed region (threshold switch). In what follows we support the view that a formed or intrinsically inhomogeneous region is common in conventional threshold switches (Popescu, 1975) made from thin amorphous chalcogenide films. We also suggest that the mechanism for the switching event has features that are somewhat different from that of the first-fire switching event in a homogeneous film. In the latter case gc is either reached isothermally over the entire sample for short pulses or, for longer pulses, some thermal modification allows for switching to occur when '& is reached only over part of the sample. In either case the switch occurs very rapidly when a critical field or voltage is reached. Formed samples, however, show a switching transition, after a delay time t d , when a critical local power density is reached (Balberg, 1970; Thoma, 1976; Reinhard, 1977; Shaw and Subhani, 1981). In the following section a model for these effects is presented.

2. An Electrothermul Model f o r Threshold Su3itching in Inhomogeneous Films Popescu (1975) has provided a detailed analytical model of how switching can occur in inhomogeneous chalcogenide films. It is our view that his arguments center correctly on the properties of the formed region and the nature of the current instabilities possible in the vicinity of such paths. In what follows we offer a simple supplement to Popeseu's work by suggesting possible means by which an electronic instability can be encouraged in such systems. Figure 24a shows the geometry under analysis. As in Fig. 17, the sample is a thin cylindrical disk of radius R composed of material having a thermal conductivity K,(T) and electrical conductivity cr,(T) that increase with increasing temperature. These are the conductivities associated with

356

M . P. SHAW AND N . YILDIRIM

a +B

r

(a) b

FIG. 24. (a) Geometry under analysis; all parameters are defined in the text. (b) Heating and cooling curves as a function of temperature. To is the ambient temperature.

the homogeneous parts of the film. Now, however, we have imbedded in the material an array of inhomogeneities (shaded) confined to the region r < RI. (In general, we expect these inhomogeneities to be near or attached to one of the electrodes.) For the specific but common case where the inhomogeneities are Te-rich crystallites, they have a thermal conductivity Ki ( T ) that decreases with increasing temperature (over the temperature range of interest) and an electrical conductivity gi( T ) substantially higher than ua( T ) and weakly dependent on temperature. A bias voltage 4 is applied across its thickness d and current Z flows in the external circuit. Because of the properties of the system outlined above, the current density in the region r < RI is greater than in the surrounding homogeneous medium. The conventional thermal instability (Landauer, 1978)that can occur at a critical value of local power density in such systems has been outlined by Laundauer and Woo (1972)and treated in detail by Popescu (1975).It can be understood most simply by considering Fig. 24b. Here the cooling curve represents the rate at which heat can be taken away from the region r < R , , - ( d / 2 ) < z < (d/2),when it is excited by Joule heating. The heating curve is sketched for the case u ( T ) = goexp(- A E / k T ) . A stable solution exists at the lower intersection of the heating and cooling curves. As the input power is increased the heating curve shifts to the right, and an instability results when no lower intersection point between the two curves is possible. The upper intersection point represents another stable state of the system, and switching occurs between these two stable states; a sudden increase in local temperature can occur.

THERMAL A N D ELECTROTHERMAL INSTABILITIES

357

This type of thermal instability does not seem to be operative in amorphous chalcogenide films. Rather, the dominant thermal aspect of the event initiation is simply the delay time, t d ,which is the time it takes a hot spot to spread through the formed region, approach both electrodes (Shaw et a[., 1973c; Homma, 1971; Newland, 1975), and cause ‘& to be reached near an electrode. (The delay time is a consequence of a thermal process, in good agreement with experiment; see, e.g., Balberg, 1970; Reinhard, 1977.) When the field near an electrode reaches a magnitude sufficient to sustain field stripping within, or tunneling through, the amorphous regions adjacent the contacts, switching occurs along with the concommittant capacitive discharge. If the entire formed region melts, this picture would be in harmony with the observations of Pearson and Miller (1969). Here, upon turning the switch off, the molten region could revitrify. The subsequent switching event could then initiate at a different spot; the conducting path could “jump around” from cycle to cycle. However, if partial crystallization occurred, then the same spot could initiate the switch upon consecutive firings. These two modes of switching are discussed in detail by Kotz and Shaw (1983). The switching transition electronically stabilizes the filamentary region, which can sustain relatively high temperatures in its center. The electrode temperature, however, is cooler. The regions near the electrodes are amorphous, maintain a large temperature gradient, and have an average energy band gap that depends upon the temperature gradient. In the narrow amorphous regons carriers are being supplied by fields on the order of lo5 V/cm. Recombination radiation is being emitted near 0.5 eV (Walsh er al., 1978, 1979); it could be originating from either (1) defect transitions in the amorphous layers or (2) band-to-band transitions in the core of the filament. A blackbody spectrum has not been observed in these recent studies, although it has been in others. In the above switching model the width of the current filament in the ON state is largely constrained to the width of the formed region. Typical formed regions in threshold- and memory-type samples have been measured by scanning microscopy and found to be 1-5 pm in diameter. We expect that formed threshold-type samples will generally have highly conducting ON-state paths of this size. Thus, for a given load line that produces an ON state below current saturation of the formed region, the current filament will be smaller than the formed region. As the load is lightened or the current increased at fixed load, the current filament will widen until it fills the formed region. Further increases in current will result primarily in heating of the current-carrying path rather than its continued spatial expansion (Kotz and Shaw, 1982).

358

M. P. SHAW A N D N . YILDIRIM

The model presented here satisfactorily explains the phenomenology of threshold switching. It is consistent with the experimental observation that the instability initiates at approximately zero time for any overvoltage (Shaw et a / . , 1973c) and the inference, taken from the data, that is like a convective instability. Furthermore, it explains the behavior of f d in the “statistical” regime just at threshold. Here, very long t d ’ S can be observed, where the current is not observed to rise until within a microsecond or two prior to the switching event. We suggest that the instability is triggered by a thermal fluctuation. Slightly past threshold, t d is usually 1-2 psec in a 1-pm-thick film. This is the time it takes for the hot spot to grow. As this occurs, the current increases with time. In the statistical regime we must wait for the thermal fluctuation that will trigger an event in a material that will be slightly different each cycle. There will be no rise in current while we wait. The model also explains the results of Henisch et af. (1974) and Rodgers et a / . (1976). The former group found that the voltage at threshold was insentitive to light intensity at low excitation levels, even though the current at threshold increased due to the enhanced conductivity of the material. The latter group showed that the voltage at threshold decreased with intensity at high levels where the material is heated by the optical pulse. A straightforward explanation can now be given for these effects. The Te-rich crystallites are essentially unaffected by the light. At low intensities the conductivity of the region surrounding the crystallites is increased, but the local field is thereby decreased, and the local power density remains essentially unaltered. The critical condition is local, and if the temperature of the surrounding medium is unchanged, the instability will occur at the same value of local power density. Once the temperature increases locally, however, the threshold power density will drop. The excess currents observed in these experiments with increasing light intensity are due primarily to the enhanced conductance in those (major) parts of the films that remain homogeneous. One final point should be made. Many electron models for threshold switching have been put forth after Ovshinsky’s paper appeared in 1968 (see, e.g., Walsh and Vizzoli, 1974; Fritzsche, 1974).

V. VANADIUM DIOXIDE A . Introduction

Vanadium dioxide (VO,) exhibits a first-order phase transition at 68°C from a high-temperature tetragonal structure to a low-temperature mono-

THERMAL A N D ELECTROTHERMAL INSTABILITIES

359

clinic structure (Anderson, 1954; Berglund, 1969; Duchene et d., 1972a,b,c, 1972; Fisher, 1975; Jelks rt al., 1975). Accompanying this phase change is a change in conductivity by a factor near lo4. The high-temperature phase is metallic; the low-temperature phase is akin to an intrinsic semiconductor. When the high-temperature phase is induced locally by Joule heating, switching is observed in the I(+) characteristics, as shown in Fig. 25. Further, narrow high-temperature filaments are easily produced and studied (Berglund, 1969; Duchene et ul., 1971b). A substantial measure of the understanding of the high-current-density filament has come from a symbiosis between approximate calculation, numerical analysis, and experiment. In this section we outline the results of exact calculations of the current-voltage characteristics and stability for ideal one-dimensional models (Jackson and Shaw, 1974). We solve, analytically, systems having (1) parallelopiped and cylindrical geometries, (2) heat flow J h parallel and perpendicular to current flow I , and (3) abrupt conductivity increases and decreases at a critical temperature T, . We find that for a given direction of Jh, the steady-state I(+) characteristics are completely determined by a single parameter, the conductivity ratio E = u o / u swhere , r0is the value of u below T, and v Sthe value of CT above T, . The derived I(+) characteristics for the parallelopiped slab geometry of Fig. 26, where J h is restricted to the plane of the slab, are shown in Fig. 27. The various characteristics for the parameters chosen are symmetric about the prethreshold (+ < + T , Z < IT)characteristic and NDR only results for E < 1 with Jh I I and for E > 1 with J h 11 I . We

c

B

LL

3 LL

" 5

0 VOLTAGE

( #)

FIG.25. Static Z(4) characteristic for two values of load resistor RLfor a coplanar sample. (After Duchene rt al., 1971b.)

3 60

M . P. SHAW AND N . YILDIRIM

prove that a filamentary (high electric field domain) SNDR (NNDR) characteristic (44) is stable only if RL + d4/dZ > 0 (<0), in agreement with experiment. An important consequence of these criteria is the following: if there is only one possible intersection between the load line and Z(4) curve, it must be stable if we neglect the effect of the circuit.

FIG.26. Sample geometry under analysis; 4

=

4B- ZR,.

FIG. 27. I(+) characteristic for the case w = 1. The threshold point [&Zc](= 1, 1) is common to all curves. Solid lines, stable; dashed lines, conditionally stable. (a) E < I , J h l z . A high-current-density filament and SNDC results; NDC points stable only if RL + d4/dZ < 0. (b) E < 1, Jh I(I . A low-electric-field domain and no NDC results; always stable. (c) E > 1, JhlZ.A low-current-density filament and no NDC results; always stable. (d) E > 1, J h 11 I . A high-electric-field domain and NNDC results. NDC points stable only if RL + d4/dZ < 0. E = & for curves a and b and E = 10 for curves c and d .

THERMAL A N D ELECTROTHERMAL INSTABILITIES

361

Although the thermally induced NNDR case ( d ) has applications in the area of superconductivity (flux flow in the intermediate state, microbridges, etc.) we do not emphasize it in this section. By way of example, we choose the case where E < 1. For the geometry of Fig. 26 with Jh I I , the results are in harmony with the major experimental features of switching in V O z films (Duchene et al., 1971b).

B . An Ideal Model f o r Thermal Switching in Thin VOnFilms We first derive a closed-form expression for the steady-state I(+) characteristics and then outline the details of a full analysis of the stability of the sample when it is in series with RL(Jackson and Shaw, 1974). Assuming that the thermal conductivity K is a constant and the electric field is uniform in the sample, the heat equation now reads pC-

aT at

=

Rs K -a2T + u ( T ) -4; 8x2 l2 (RL + RS)’

(45)

where R s , the nonlinear sample resistance, is given by l[h J!!gs2 u ( T ( x ) )dx1-l. We do not include a latent heat of transformation at T, ; its inclusion will not alter the results. The boundary condition is that T ( + i w ) = Ta (the ambient temperature). The electrodes, top, and bottom of the sample are perfectly insulated. We first seek steady-state solutions of Eq. (45). As long as the maximum temperature is less than T, , there is a solution

When C#I is increased to the point where Eq. (46) would yield a value greater than T, at x = 0 , the equation becomes invalid and we look for a “two-phase solution” with an internal hot filamentary region for which T > T, and u = us. The critical voltage & at which Eq. (46) becomes invalid is

A two-phase solution satisfies

with T = T, at x = +$f.If these equations are integrated with the additional conditions that the temperature and heat current are continuous at

M. P. SHAW AND N . YILDIRIM

362

+f,we obtain the width of the hot filamentary region as a function of +:

If1w

- - 1 - E k-

2--E

+

1

2--E

[ l -@(26

4Jz

-

€2)

1

1’2

(49)

For > only the plus sign in Eq. (49) applies and T > Tc in all but a small strip near the surface; as -+ m, If1 + w . We see from Eq. (49) that there are also two-phase solutions for < &. Defining +h = ( 2 -~ E ~ ) ~ / ~there + ~ , are two values off for voltages in the region f$h < < +T. The current in the two-phase state is

+

+

+

where R, = l/a,wh. A typical Z(+) curve is shown in Fig. 27, curve a; voltage reached in the filamentary state. At +h , lfl/w

= (1 -

4/(2

-

4

+h

is the minimum (51)

Furthermore, independent of E and the relative dimensions of the paral= Z(&)&; the powers dissipated at &, and the lowlelopiped, Z(&)& current +T are the same. To investigate stability we consider a small perturbation about the time-independent solutions which we have found and see whether Eq. (45) causes the perturbation to grow or decay. If T,(x) is a timeindependent solution and q(x, t ) the perturbation, we write T(x, t ) = To(x) ~ ( xt,) , insert it into Eq. ( 4 9 , and linearize to terms of first order in ~ ( xt,) . The equation for r)(x, t ) is then

+

We seek a solution for the form q(x, t> = exp(at)X(x)

(53)

If there is a solution of this form with a positive a , the perturbation grows and the time-independent To(x) is unstable; if not, it is stable. Inserting Eq. (53) into Eq. (52), we find that there is no positive a (and hence stability) when

THERMAL A N D ELECTROTHERMAL INSTABILITIES

363

For RL + 0 , Eq. (54) becomes V ( / w 2 (1 - 4/(2 - E ) . Comparing this result with Eq. (51), we see that NDR points are unstable for the unloaded case; PDR points are always stable. As RL increases from zero, additional NDR states will stabilize and finally as R, -+m, Eq. (54) shows us that If[-w 2 0. All NDR states can be stabilized with an infinite load. The condition for stability in the NDR region can be shown to be RL + d+/dZ > 0 . Identical results are obtained for a right circular cylinder of radius y o , which attests to the general validity of the stability criterion. For the cylinder we find that in two-phase region

and

where

Although all NDR states can in principle be stabilized, characteristic (a) in Fig. 27 approaches the point (&, IT) from the NDR region with zero slope [Eqs. (49) and (SO)]. Prohibitively large resistive loads are therefore required to stabilize all the NDR points and an “open” region should always be present in the experimental characteristics. This behavior is in fact what is observed. The model also predicts the major features of the observed I(+) characteristics including the narrow filaments that are observed even for relatively low (<<+T) voltages. The calculated filamentary I(+) characteristics produced by a thermally induced switch when a critical temperature is reached somewhere in the sample can also be compared with I(+) characteristics produced by a thermally induced switch when a critical electric field is reach somewhere in the sample, as for the amorphous chalcogenide films discussed in the last section (Hughes rt al., 1975). The similarities are striking. It is important to appreciate how these results fit into the general framework on the NDC stability problem. Two aspects are important: the role of reactive components and the stability of uniform isothermal bulk NDC points. It is well known that uniform isothermal bulk NDC points are unstable against both (1) current filamentation (Shaw et al., 1973a)and electric field domain formation (Shaw er al., 1979); and (2) circuit-

3 64

M . P . S H A W A N D N . YILDIRIM

controlled relaxation oscillations (Solomon et a / ., 1972) (if particular conditions are satisfied). Our solutions are the nonuniform filamentary and domain configurations; we need only enquire about their stability in a circuit containing reactive elements. which we do in the Appendix. It is easily shown that if the above stability criteria are met (RL+ dS/dZ 2 0 ) , then the nonuniform NDR points will be circuit stable if the circuit response allows those points to be reached (Shaw et a/., 1979). The above analysis is a simple model for an ideal case. In most real systems, the change in (T is never as abrupt as we have made it (in most semiconductors it is thermally activated, a situation we have discussed in prior sections). Furthermore, the heat flow will be in all directions. For the case < 1 , the filamentary Z(+) characteristic will be an admixture of curves a and b in Fig. 27. Nevertheless, the basic physics involved in a wide variety of switching phenomena is contained in this model. For example, if a local hot spot occurs due to Joule heating, and the hot region has a higher conductivity than the cooler material, then a switching event will occur for a thin film even with Jh 11 Z if the cool regions near the electrodes are shorted by the effect of a critical electric field. Section IV,C outlines phenomena of this type.

VI. SECONDBREAKDOWN I N TRANSISTORS When a sufficiently large reverse bias +RB is applied to a diode or either one of the two or more junctions of a transistor-type, structure, avalanche or Zener breakdown is induced (Sze, 1969). In this mode the device acts as a voltage limiter with little or no NDC observed. Once avalanche or Zener breakdown is achieved, the application of a sufficiently large current often causes a second breakdown (SB) to occur (Thornton and Simmons, 1958). In this mode there is a large “switchback” effect to a holding voltage +h << + R B . Typical transistor characteristics are shown in Fig. 28 (Schafft and French, 1966a,b). In some cases SB is due to purely electronic effects, such as avalanche injunction at the collector n-n+ junction (Hower and Reddi, 1970). In this section we concern ourselves with SB effects in transistors due solely to thermal instabilities [SB in diodes is discussed by Tauc and Abraham (1957) and Oha and Oshima (1962)l. As is the case for the devices discussed previously, once a sufficiently large input power is provided, a switching effect (SB) will occur after a time td (Schafft and French, 1962); a triggering energy, or local power density of sufficient magnitude must be present in the transitor before SB will occur. Some portion of the structure must therefore reach a

THERMAL A N D ELECTROTHERMAL INSTABILITIES

365

+ce

FIG.28. Swept c $ ~ ~Zc characteristics of a transistor with forward (F), zero (O), and reverse (R) constant-base current drive. The characteristics are drawn for only the first half of the sweep cycle. The initiation of second breakdown is indicated by the abrupt drop in ~ C E (After . Alwin et al., 1977; copyright @ 1977 IEEE.)

critical temperature T, before SB will occur. The switching phenomenon here is similar to those previously discussed. Nucleation of the hot spot can be caused by (1) material nonuniformities; (2) base-width nonuniformities; (3) heat-sinking irregularities; (4) localized regions of breakdown (microplasmas); and ( 5 ) application of reverse-base currents. Indeed, the current constriction that results from any of the above mechanisms (Bergmann and Gerstner, 1966) can cause local melting to occur and extend through the base to create a collector-to-emitter short circuit, the characteristic failure mechanism of SB. Scarlett et al. (1963) and Bergmann and Gerstner (1963) were the first to study the thermal stability of a uniform current distribution in a transistor. The main feature of these linear models is that any transistor structure will develop a lateral current instability when its temperature rises above that of the heat sink by a sufficiently large amount. Later work indicated that a very important factor in initiating and sustaining SB is the distribution of minority carrier current injected into the base (Schafft and French, 1966b).

3 66

M. P. S H A W A N D N . YILDIRIM

0 COLLECTOR -EMITTER VOLTAGE (V)

FIG.29. The transistor steady-state collector current versus collector-emitter voltage characteristic at 4BE = 0.9 V. Also indicated are load lines corresponding to RL = 0 and RL = 10 n/cm, with a circuit battery voltage hcc = 20.9 V. (After Alwin et a / . , 1977; copyright @ 1977 IEEE.)

A symbiosis between experiment, simple analysis, and numerical modeling is an optimum way of understanding the behavior of semiconductor devices. Since the complexities involved with thermal switching (SB) effects often make it extremely difficult to obtain quantitative fits between experiment and the prediction of simple models, numerical analysis is often called for (see, e.g., Gaur and Navon, 1976; Alwin et af., 1977; Shaw and Subhani, 1981). Typical techniques involve the use of the finite-difference method to solve the nonlinear partial differential equations of both heat [Eq. (30)] and current flow. The latter are

V24= - q ( p

-

n

div J p

=

-q[R - G

div J ,

=

q[R - G

+ N& - N x )

(58)

+ (dp/dt)]

(59)

+ (an/dt)]

(60)

Here p and n are the mobile hole and electron densities, N & and N X are the number of ionized donors and acceptors, R is the net carrier recombination rate, and G is the carrier generation rate. The auxiliary equations are Jp

= - pp4p

grad 4 - qDp grad p - qpDg grad T

J,

= - p,qn

grad 4

J

=

Jp +

Jn

+ qDn grad n

-

qnDi grad T

(61)

(62) (63)

367

THERMAL A N D ELECTROTHERMAL INSTABILITIES

3.0

-

--

2.5

-

Q

2.0-

= O' p: B

1.5

g

1.0-

k

c z

Y

CL

8 2

0.5

-

-

FIG.30. Curves showing the turn on of the collector current along a load line: RL = 0 and RL = 10 Q/cm. In each case the voltage applied to the base contact is 0.9 V and the battery voltage is 20.9 V. The base currents are also shown. The rise in collector current beginning at about 0.1 psec is the electrical rise time, whereas the rise beginning at about 0.02 msec corresponds to the current increase caused by thermal effects; collector current (4 base ,current (---). (After Alwin et a / . , 1977; copyright @ 1977 IEEE.)

I DISTANCE ALONG COLLECTOR - BASE JUNCTION (fltml

FIG.3 1. Temperature distribution along the collector-base junction at various points in time indicating the heat effect due to the flow of emitter current with a load resistance RL = 0. Curves: a, 16.5 psec; b, 94.4 psec; c, 0.456 msec; d, 1.56 msec. (After Alwin efa/.. 1977; copyright @ 1977 IEEE.)

368

M . P. S H A W A N D N . YILDIRIM 500

4w

-

I

1W

0

2W

DISTANCE ALONG

300

EMITTER ( ~ m l

FIG.32. Longitudinal electron current density along the emitter-base junction showing the current buildup in time. Curves a (0.075 psec) and b (1.07 psec) reflect primarily the electrical buildup, whereas curves c (16.5 psec), d (94.4 psec), and e (1.56 msec) reflect the effect of heating. All curves show current crowding to the emitter edge. (After Alwin ef al., 1977; copyright @ 1977 IEEE.)

- --_

-a-

-

----

-- - --

N

- ----

- -

--

- - - ---N+

COLLECTOR

+

FIG.33. Plot of the electrostatic potential = 20.5 V for the transistor at +BE = 0.9 V and +CE = 20.9 V at various points in time, indicating the current buildup in the collector region for the transistor operating in the circuit with R , = 0. Curves: a, 0.0 sec; b, 0.12 psec; c , 0.26 psec; d, 0.47 psec; e , 1.07 psec; f, 4.36 psec. (After Alwin et a/., 1977; copyright @ 1977 IEEE.)

THERMAL A N D ELECTROTHERMAL INSTABILITIES

369

Here p is the mobility and D is the diffusion coefficient. The boundary conditions and the circuit interactions are defined next and the problem then reduces to solving the equations for 4, T , and Z as functions of time and position. [The use of quasi-Fermi potentials is often useful; see, e.g., Alwin et al., 1977).] Typical results are shown in Figs. 29-33. These figures do not demonstrate the postswitching effects. Rather, they display the role of heating in the pre-SB regime and characterize the current, voltage, and temperature profiles in the stable steady state.

VII. SUMMARY We have reviewed the basic thermal and electrothermal instabilities that lead to breakdown, switching, or memory effects in semiconductor materials and devices. By way of introduction we first discussed the thermistor, a device that relies on the development of a temperature gradient induced by Joule heating in a material having the proper variation of electrical conductivity with temperature. The thermistor can be stabilized at all of its I(+) points for the proper set of circuit parameters. We then discussed the general problem of heat flow in semiconductors, with emphasis on thermally induced NDC, critical fields and temperatures, boundary conditions, and inhomogeneities. Our understanding was then applied to three specific configurations: thin chalcogenide films, where critical electric field effects are important; VO, , where a critical temperature dominates; and second breakdown in transistors. We compared theory with experiment as much as possible, relying primarily on numerical calculations and simple analytical models for special situations such as VOz . APPENDIX

Our task is to determine the conditions necessary for either a highcurrent-density filament (switching to the ON state) or circuit-controlled oscillations to occur. We examine the situation where oscillations occur in the absence of filament formation and then establish the additional limitation imposed by filamentation, which is demonstrated for an idealized situation. The techniques used are applicable to any SNDC element. We find that either switching or relaxation oscillations can occur for the same SNDC element in the same circuit with appropriate circuit parameters. A major aspect of the investigation is the fact that the circuit theory transforms directly between the S and N case (Shaw et al., 1973a; Schuller and Gartner, 1961). The important circuit parameters in each case produce a

370

M . P. SHAW A N D N . YILDIRIM

‘‘dual’’-circuit system where the voltage across the NNDC element in its “primary” circuit behaves exactly as the current through the SNDC element in its primary circuit. A crucial aspect of the present study is the identification of the primary circuit for an SNDC element. We regard this as having been established by Shaw and Gastman (1971). The primary circuit is shown in Fig. A l . Here C (= C,,) is the package capacitance and L (Li+ &) is the sum of the intrinsic and package inductance. For convenience, we study the cylindrical SNDC element (of length 1, radius a , and cross-sectional area S) shown in Fig. A2 in the circuit shown in Fig. A3, where the SNDC element is modeled by the series combination of an appropriate Li with an appropriate S-shaped conductioncurrent curve,of low current resistance R , . The electrical behavior of an SNDC element is obtained from a simultaneous solution of the circuit equations and Maxwell’s equations with the appropriate boundary conditions. For the cylindrical symmetry shown in Fig. A2, where all quantities are independent of z and 8, Poisson’s equation and Maxwell’s “curl B” equation yield

r-’[d(rEr)/dr] = E-’(p(r, t ) j,

+ (d?&/dt>

=

- po(r, t ) )

0

(All (A21

where 2?r is the radial electric field, E the permittivity,jr the radial current density, and p(v, t ) and Po(‘, t ) the mobile and background charge densities, respectively. Equation (A2) is an expression of zero total radial C

FIG.A l . Circuit under analysis; the “primary” circuit for an SNDC element; i is the conduction current and iDis the displacement current; R , is the lead plus load resistance; and q5B is the battery voltage. The inductance L is the sum of the package inductance L, and intrinsic inductance L i . The contact resistance R , , which is generally nonlinear, is included in the nonlinear conduction current characteristic +,(i). The intrinsic capacitance is ignored; q5 is the voltage drop across C , which is the package capacitance C,; I = i + iD. (After Shaw et al., 1973a; copyright @ 1973 IEEE.)

THERMAL A N D ELECTROTHERMAL INSTABILITIES

371

current. For longitudinal fields ‘8‘z and current densities j,, we eliminate the magnetic field from the curl B equation and Faraday’s equation, yielding

where po is permeability. The constitutive current equations are

.A

= p(r, t)p& -

D[ap(r, t)ldrI

j , = p(r, t ) p $ ,

(A41

(A51

where p is the mobility and D the diffusion coefficient, both of which are chosen to be constant. Equation (AS) is the SNDC curve. Equations (Al)-(A5) are the governing equations of the system. They are coupled to the external circuit through the equations for total current

FIG.A2. A cylindrical shaped inhomogeneous SNDC element. A difference of potential &,(t) is applied across the endplates. In general, for radial inhomogeneities only, the electric field 8 = ‘&Jr, t ) and the transport current densityj = j&, t ) :Bn(r, t ) is the magnetic field produced by j,. (After Shaw et ul., 1973a; copyright @ 1973 IEEE.)

FIG.A3. Circuit containing the cylinder of Fig. A2.

372

M . P. S H A W A N D N . YILDIRIM

through the SNDC element

iW

=

I

M-,t ) + [a %AT,t)/atl>dS,

(-46)

and total voltage across the SNDC element +,(t)

(A71

gZ(r= a, t)l

=

where 8,(r = a , t ) is the value of the longitudinal electric field on the cylindrical surface of the SNDC element and d S , is an element of surface area perpendicular to z. Equations (Al)-(A7), when solved simultaneously with the external circuit equations, completely specify the problem. When current flows along a cylindrical wire, energy flows into the wire radially through the surface (Feynman et al., 1964). Conservation of energy with displacement effects neglected yields -

I

( Z ? x H ) . dS,

=

j

-

8 d7

+2 dt

& B . B d7

(A8a)

where d S , is a surface (sheath) element and d7 a volune element B = p,,H. The left-hand side represents the total power driving the electron stream. For the symmetry of the problem -

I

(8x H) - dS,

=

gZ(r= a)l

I

H , ds

=

8,(r = a)li

(A8b)

where i is the total current flowing in the SNDC element and ds is an element of circumferential length. The right-hand side of Eq. (A8a) represents the Joule heating loss and the time rate of change of magnetic energy. Defining resistance and inductance as

Eqs. (A8a) and (8b) become [%(r = a) = g Z ( r= a)] 8(r

= a)/ =

iR

I d + 217Lii2 dt

and %(r = a)l is identified as the total voltage drop +,,(t)across the SNDC element and Lidepends on the distribution of current. Rather than deal with resistances, it is more convenient and instructive to define an effective conductive voltage +,(t) as

THERMAL A N D ELECTROTHERMAL INSTABILITIES

3 73

Equation (A10) then becomes

In the absence of current-density filamentation, +Jt) is a single-valued function of i and can be written as +&), and Li is a constant. In this case we can apply standard techniques to solve for the circuit response (Shaw and Gastman, 1971). However, when filamentation is present (Rockstad and Shaw, 1973), & ( i ) is multivalued and Li varies over an oscillatory cycle. We shall shortly show how & ( i ) is obtained for this case. We plan to represent the inhomogeneous NDC element by an appropriate SNDC conductive voltage curve &(i) (nonlinear resistor) in series with an appropriate inductor Li . We must therefore investigate the circuit response under these conditions. The total voltage drop across C in Fig. A3 is +(t) = c#~~(t) + L,(di/dt)

=

&(t)

+ L(di/dt)

(A131

where L = L, + L i ; L, is the package inductance, which includes the geometrical inductance of the SNDC element associated with magnetic fields for Y > a. With reference to Fig. A3, we seek solutions of the following coupled equations:

L(di/dt) = +(t) - + c ( t > , C[d+(t)/dtl= [+B - +(t>- ~ R L I / R L Solutions to the above equations come in many forms. We may obtain the current-time (or voltage-time) profiles and the current-voltage Lissajous figures. We require both in the following arguments. We therefore transform the content of the above equations into the two differential equations:

Equation (A14) is the dual of the circuit equation for NNDC elements used by Shaw et al. (1979). Thus, the arguments of this section are in one-to-one correspondence with those. In Eqs. (A14) and (A15), A = -/ROC, t' = t / a , and we have written &(t) as I#J~(~). The significance of the transformation on is that in Eq. (A14) the bracketed non-linear damping term is usually of the order of unity or less. Thus, the strength of the damping term is determined primarily by the value of A . Equation (A14) is a generalization of Van der Pol's free running oscillator. For every small A the nonlinear damping term is strong and the solutions

314

M . P. S H A W A N D N . YILDIRIM

are well-defined relaxation oscillations. For large A the damping term is small and the i ( t ) solutions are nearly sinusoidal. Using the three-piece linear approximation for +,(i) shown in Fig. A4, i(t) solutions can be obtained in the three regions i < i,, i, < i < i,, and i, < i, which correspond to d + / d i = R , , -R, , and 0 , respectively, where R , is the magnitude of the negative differential resistance. The individual solutions are joined smoothly from one region to the next. The current waveform begins with a slow exponential rise with time constant - ( R , / L + I/RLC)-', followed by a sharp spike in current when i = i, . The spike is composed of a fast exponential transit through the region of negative slope (i, < i < i,) with time constant -(I/RLC - R,/L)-l, where IR,/LI > 1/RLC, followed by a damped sine wave ror i > i, , and another exponential transit for i, < i < i,. An exponential decay for i < ip completes the cycle. The time required to reach i = i, during the initial slow exponential rise depends on the applied bias b . Thus, the frequency of the relaxation oscillations is voltage tunable. Returning to Eq. (A15), we integrate it over that portion of the cycle where &(i) = &, assuming +,JRL >> +/RL:

which is the equation of an ellipse. For a particular SNDC element the

FIG.A4. Three-piece linear approximation for &.(i).

375

THERMAL A N D ELECTROTHERMAL INSTABILITIES

shape of the ellipse is determined by A . Plotting +(i) versus R,i, the trajectory is a circle when A = 1, an ellipse with the major axis along R,i for A < 1, and an ellipse with the major axis along 4 for A > 1. For small A the current amplitude is large and for large A the voltage amplitude is large. If we assume the validity of extending the ellipse below i, until it intersects the positive resistance part of $&), then the +(i)curve is deterThe mined back to this point. At threshold (i = ip), $(ip) = +c(ip) = constant in Eq. (A16) may then be evaluated to give

+,.

A-2(4(i> -

$SI2

+ Rg[(+B/Rd - iI2 = A-2(& - +A2 + R ~ [ ( ~ B / R &I2 L)

(A171

Under the assumptions leading to Eq. (A17), the complete +(i) trajectory is obtained by joining the ellipse equation (A17) to the positive-resistance segment of &(i). This approximation is best for small A . An elliptical +(i) trajectory is plotted in Fig. A5. In the $(I) plane [replace the abscissa in Fig. A5 by Z.= i + C ( d + / d t ) ] ,the trajectory collapses to the load line. Equation (A17) shows that besides the parameters A and &/RL,the circuit response is determined by the SNDC element parameters R, , i, , 4* and 4s. As we shall show, the form and nature of the circuit response is a dominant factor in the formation and quenching of current-density filaments. In particular, the maximum voltage +M and minimum current im(Fig. A5) reached during the first cycle are of major importance. From the ellipse equation (A17) it follows that for large A (sinusoidal oscillations), &I and im are high, whereas for small A (relaxation oscillations), 4~ and im are low. We have now analyzed the response of a circuit containing an SNDC element with a uniform distribution of current. We next ask the following: (1) How does filamentation affect the circuit response?; and (2) How does the circuit response affect filamentation? We consider the second question first. The results of the uniform current case indicated that the +(i) trajectory was determined primarily by the parameters R, , i, , + p , and and was relatively insensitive to the slope of the NDC region. Therefore, if we assume that the major effect of filamentation when sustained circuit-controlled oscillations are present is to change the slope of the NDC region, then the circuit will simply control the extent of filamentation. As we shall see, this is in fact the case. To answer the first question we note that when uniform currents flow and circuit-controlled oscillations occur, a specific current minimum is reached once each cycle. When filamentation occurs during sustained circuit-controlled oscillations, we 9

M . P. SHAW A N D N. YILDIRIM

376

FIG.A5. The +(i) trajectory for a relaxation oscillation. The load line in this plane, 4B- iRLis the line d+/dr = 0. The &(i) curve is the curve di/dr = 0. The voltage and conduction current extrema on the +(i) trajectory are therefore the points of intersection of +(i) with d4/dt = 0 and di/dt = 0, respectively. The extrema in voltage are denoted (pM and I#+,, . The current extrema are i M and im. (After Shaw er a / . , 1973a; copyright @ 1973 IEEE.)

4

=

require filament quenching once each cycle. If we assume that there is a minimum sustaining current for filamentation, then the filament-quenching criterion will impose an additional limitation on the range of circuit parameters €or which circuit-controlled oscillations will occur. We illustrate both conclusions below where we draw upon an approximate scheme for computing &(i) and obtaining a quenching criterion. The model neglects the skin effect, filament formation times, and spatial derivatives. We divide the cylindrical SNDC element into two subelements: (1) a core cylinder of radius ai ; and (2) a surrounding cylindrical shell of inner radius aiand outer radius a, as shown in Fig. A6. The subelements have different SNDC curves and within each subelement j and % are uniform. For the configuration shown in Fig. A6, Eqs. (A9) and ( A l l ) yield +c(i) =

where i

=

i,

+ i2, i,

i-l(j1

= jlSl, and

%,S, 1

+ j 2 g2S21)

i2 = j2S2.The computation of &(i)

(A 18)

for a

THERMAL A N D ELECTROTHERMAL INSTABILITIES

377

FIG.A6. Two subelement model of an inhomogeneous cylinder.

relaxation oscillation with L, = 0 [here the $(i) = 90(i)] is illustrated in Fig. A7 where, for ease in plotting, we compute (bc(T) rather than (bc(i). Here 7 = i / S , where S = S, + S, . We consider the case S, = S,, thus a; = 2a:, j = Hjl

+ j,)

(A20)

+ E,/(jl + j,)I+c(j2) (A21) where &(jJ = 8,l and 4c(j2)= 8,l. To determine & ( j ) we first obtain the trajectory from A, , i, , and +s using the results of the circuit analysis for the uniform cylinder. Next, we consider the response of each subelement and determine &(j)by making use of Eqs. (A20) and A21). From Eq. (A12) [c#47)= & ( j ) - (S/27)/(d/dt)Li7], we see that the induced voltage drop in the cylinder is just the difference 4x7)- &G). In Fig. A7a we show +&) for a circuit-controlled relaxation oscillation I#&), & ( j 2 ) ,and c $ ~ = !i(&(j,) +- &(j,)). In Fig. A7b we show computed values of &(j) (crosses) at four instants of time: a , b, c , and d. The conductive voltages in subelement 1 (solid circles) and subelement 2 (open circles) are also shown. At each instant of time an assumption is made about the time rate of change of current or voltage. At time a , 4,o) has reached its maximum. If at this point we make the reasonable assumption_ of neglecting the inductive voltage drop in subelement 2, then 40(j)M = q d j , is determined. From Eq. (A20) we obtainj, , which in turn yields & ( j ) from Eq. (A21); $A,,(?)at time a is indicated in Fig. A7b This as the cross a. At time b , j is a maximum, hence here 6x7)= +,(j).

4c(J?

=

[jl/(jl

+,,

&(7,)

+j,)I+C(~l)

378

M . P. SHAW AND N . YILDIRIM

i

&o)

FIG.A7. Illustration of the way for a relaxation o_scillation is obtained using the two subelement model. At time a , and @.j, =)@,jn), etc. For clarity we have dropped the subscript 0 on 4.In (b) the position of subelement 1 is designated by t_he closed circles, the position of subelement &by the open circles, and the values of I#&) by the cross_es.In (c) we plot a complete 4J.j) curve estimated point by point throughout one cycle of I#J(~).(After Shaw et al., 1973a; copyright @ 1973 IEEE.)

7 =in

point is the cross b . Similar determinations are made at c and d . A sketch of &(J) throughout the cycle is shown in Fig. A7c. It is of interest to examine the system in the two limiting cases: (1) a, >> ai and (2) a, ai . If, for a, >> a i , we also consider the situation where the narrow central region has a much larger conductivity than the rest of the SNDC element, then the situation presumably corresponds quite closely to an electrothermal threshold switch. Here we see from Eq. (A19) that the ln(a,/ai) term of Li becomes quite important. The term has

-

THERMAL A N D ELECTROTHERMAL INSTABILITIES

379

no upper limit and is produced by the flux in subelement 2 produced by current in subelement 1. For the case a , a i , as i, vanishes Eqs. (A18) and (A19) produce the results required of a uniform cylinder where Li = p01/8r. Performing computations on these systems similar to those discussed with regard to Fig. A7 reveals several general features of the problem. We find that (1) the curves can be regarded as members of a family of curves characterized by similar values of i, ,(bp, and 4s; (2) the &(j> trajectory reflects the current-density evolution; and (3) when inhomogeneous current-density distributions occur and sustained circuit oscillations are maintained, the major effect of filament formation will be to alter the shape of the region of negative slope. Since the circuit response is relatively insensitive to this parameter, the waveforms will be essentially indistinguishable from the uniform current-density case. We have been describing the oscillation in terms of the parameter A = m / R , C . When filamentation is present A may be regarded as circuit dependent insofar as Li depends on the current distribution. As shown by Eq. (A19), filamentation yields values of Li greater than its value for uniform current densities. Therefore, A is larger above than below threshold. From the arguments for the uniform current density case, a larger A would yield higher values of ,i (minimum current). The implications of filamentation are therefore clear; if the SNDC element is part of a circuit that for the uniform-field case yields a value of ,i substantially below a “filament-sustaining current” i f , the effect of filamentation on the circuit oscillations will be negligible. On the other hand, if ,i Iif, then small increases in Lidue to filamentation may well increase ,i so that it exceeds i f . In this case the circuit oscillations will damp and switching will occur. The above calculations illustrated filamentation when at threshold only one subelement entered its region of negative slope. The presence of relatively uniform current densities implies that at some time during the cycle both subelements are in their regions of negative slope. For the example considered above, when subelement 1 enters the NDC region its total voltage drop ceases to rise significantly, and the voltage drop across subelement 2 soon begins to decrease. If we imagine a situation where +,(J) rises significantly after threshold, then subelement 2 can also be pushed into its NDC region. The current distribution will initially be relatively uniform. Therefore, an important parameter in the growth of current filaments is the maximum voltage &, reached during the cycle. From Eq. (A17) we see that $M is a strong function of A . For relaxation oscillations (small A ) , 4Mbarely exceeds the threshold voltage for the subelement with the highest carrier concentration. This leads to the highly non-

-

&c)

3 80

M . P. SHAW A N D N . YILDIRIM

+L i

FIG. A8. Estimated 4(i),&(i), and i ( t ) curves [(a), (b), and (c), respectively] for the case where ,i does not fall low enough to quench the inhomogeneities. The circuit oscillations damp, a filament forms, and switching occurs to the ON state. The ON state is the heavy dot. As + B is varied the dots fall on a “filament characteristic,” which is sketched (dark) in (d). When the load line intersects the SNDC element at points i > i, (see Fig. A4), a switch will occur to that point without relaxation oscillations (unless for some reason the stable point at i > i, is never reached). The switch will also take place, however, in a damped oscillatory fashion. A “spiraling in” to the ON state is fundamental to the switching process in general. In the current-time profile this appears as a damped “ringing” oscillation at a frequency determined in part by

a

uniform situation where a filament appears immediately upon reaching threshold. For near sinusoidal oscillations (large A ) , +M may be much higher than the threshold voltage, producing a more uniform initial current-density distribution. In order for circuit-controlled oscillations to be sustained it is necessary that all current-density nonuniformities be quenched at the end of each cycle. When this occurs, the 4,,G) trajectory is _closedand completed on the low-current line of positive slope of +,G). If, however, the current-density minimum Jm is not sufficiently low (im>?,), the trajectory is open and the nonuniformities are enhanced. The circuit oscillation damps and a filament remains as a steady-state solution. This behavior is shown schematically in Fig. A8.

THERMAL A N D ELECTROTHERMAL INSTABILITIES

38 1

It is possible to make some qualitative predictions about how small ,i must be to ensure filament quenching. Certainly, +,(i) will not form a closed trajectory unless ,i is sufficiently below i, . In general, we expect that i, must typically be below is (see Fig. A4) to ensure circuit-controlled oscillations. According to Eq. (A17), the elliptical part of the +,(i) trajectory will pass through the line of positive slope at i, when A = [

1,

- 1s

1

1'2 ( i+~ For A less than this critical value filaments will quench. Consider the case where i, = 24. Here, since +B/RL> i,, a necessary condition for quenching is A S 1. For a realistic bias C#JB/RL- 2i, and A = t is the critical value. Note that for a given A , as +B increases, a transition from a relaxation oscillation to the ON state can occur as the critical value of A is exceeded. Furthermore, if we design the SNDC element such that A is significantly greater than unity, filament quenching can be avoided. It is, therefore, possible to construct a filament-forming SNDC element that will only turn ON and not exhibit relaxation oscillations for any +B and > ROC. RL. To achieve this we require Throughout this discussion we have assumed that the package inductance L, was zero. For a finite L, , L = Li + L, and some inductive volttrajectory, age will be drained off the SNDC element. For a given will be below its L, = 0 value. therefore, the maximum value of Since a large +M is necessary for relatively uniform current densities, increasing the package inductance enhances the possibility of filamentation. The two circuit factors that control filamentation are thus L, and A , with filamentation optimized for large L, and small A . We conclude by considering the presence of a primary filament nucleation site. In modeling a thermal filament, the primary site is often at the center of the cylinder. However, wherever the site is, its role may be likened to that of a subelement with a high carrier concentration compared to the other subelements. Its presence will thus act to decrease C#JD - &, which from Eq. (A17) can be seen to reduce the amplitude of C#J(i). This will increase ,i and lead to the damping of the relaxation oscillations and the domination of a filament. Thus, if a primary site exists, a filament will not quench unless A >> 1 . Switching to the ON state will dominate. Amorphous chalcogenide threshold switches almost always exhibit relaxation oscillations for sufficiently large R and also have a highconductivity central region (Coward, 1971; Bosnell and Thomas, 1972; Allinson et al., 1979). At first glance these two features seem incompati-

(2+B/Rd

-

+,c)

$6)

3 82

M . P. S H A W A N D N . YILDIRIM

-

ble. However, the high resistances of these devices ( R , lo6- lo7 a)and values of C , F produce values of A much less than unity, even for reasonably large estimates of the inductance. Hence, filament quenching and relaxation oscillations are still favored.

-

ACKNOWLEDGMENT The research component of this review was supported in part by the National Science Foundation grant number ENG 7817666.

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Author Index Numbers in italics indicate the pages on which the complete references are given. A

Armstrong, B., 59, 60, 93 Amtz, F. O., 336,384 Abdullakhatov, M. K., 59,87 Arsenin, V. Ya., 53, 93 Abdurazakov, A. A., 23, 63,84 Artamonova, K. P., 51,84 Abiko, K., 57,89 Arvay, Z., 24, 26,84 Abraham, A., 365,384 Asam, A. R., 255,300 Abraham, W. G., 258, 261,305 Auchet, J. C., 272,305 Adam, G., 359, 360, 361, 383 Audier, M., 251, 255, 256,300 Adam, J., 27, 32, 44,84, 93, 249,300 Augier, D., 383 Adams, J. E., 111, 124, 156 Adler, D., 327, 330, 332,334,335, 336, 338, Aukerman, L. W., 224, 263,305 339, 340, 341, 342, 354, 358, 383, 384, Aurouet, C., 227,300 Avila, R., 148, 149, 157 385 Adler, D. A., 310, 324, 333, 337,382 Adriaens, M. A., 158 Afanas’ev, V. P., 23,84 B Ahmed, N., 200, 219 Akiba, M., 63, 66, 93 Backe, H., 24, 25, 63,84 Akopov, G. A., 59,87 Baev, A. S., 74, 93 Akkerman, A. F., 46,84 Baverstam, U., 46, 90 Albert, R. D., 37,84 Bainbridge, K. T., 2, 71,84 Alder, K., 2 , 6 , 9 , 10, 11, 12, 13,67, 70,84, Balberg, I., 337, 355,382 91 Ballard, R. E., 48,84 Alexeev, V. L., 63,84 Ballonoff, A., 259,305 Alldredge, G. P., 125, 153, 158 Ballu, Y., 29, 52,84, 87 Allemand, R., 275, 276,300 Bambynek, W., 19, 59,84, 88 Allen, R. E., 147, 156 Band, I. M.,2,7,9, 10, 11, 12, 13,14,15,16, Allinson, D. L., 337, 348, 354, 381,382 18, 19,20,60,67,68,69,70,81,84,85, Allison, J., 358,385 90,92 Ahnodovar, I., 293,301 Barash-Schmidt, N., 59,60,93 Alon, Y., 295,302 Barbaro-Galtieri, A., 59, 60, 93 Altcheh, L., 310, 321, 354,382 Barcellona, A., 98, 99, 157 Alwh, V. C., 365, 366, 367, 368, 369,382 Bardas, D., 255, 258, 261,302, 305 Amus’ya, M. Ya., 18,84 Barish, B., 227,300 Anderson, E. M., 9, 19, 21,84, 85 Barlow, C. A., Jr., 310, 321, 354,384 Anderson, F., 359,382 Barlow, F. E., 255,304 Andrews, H., 164, 198,219 Barnett, A. M., 310,382 Andrews, H. C., 192,220 Barrett, H., 219 Andrews, 3. W., 293,301 Barrow, H., 215,219 Anitin, I. V., 69, 89 Bany, T. I., 337, 348, 354, 381,382 Apsimon, R. J., 252, 253,300 Bateman, J. E., 252, 253,300 Arakawa, R., 71,91 Bates, D. J., 259,300, 305 Armand,G.,97,116,117,122,124,125, 148, Baumgartner, W., 255,300 157, 159 Baverstam, U., 29,85 387

388

AUTHOR INDEX

Beasley, R. M., 249,305 Beaver, E. A., 260,300 Beck, G., 268,300 Becker, G. E., 133, 134, 157 Becker, L., 310,382 BeCvBf, F., 15,60, 61,87 Bedeaux, D., 325,382 Beeby, J. L., 122, 123, 124,157 Behrens, H., 19,84 Bell, R. L., 262,300 Belyaev, L. D., 59,87 Belykh, V. Ts., 59,87 BenC, R. W., 359,383 Benedek, G., 123, 124, 125, 149, 153,157 Bengston, B., 263, 274, 275, 276,300, 303 Benot, M.,253,300 Berknyi, D., 39,85 Bergkvist, K.-E., 28, 29, 30, 39,48, 53,M Berglund, C. N., 308, 359,382 Bergman, 0..54,88 Bergmann, F., 366,382 Bergmark, R., 35,89 Bergstriim, I., 40,85 Berlman, I. B., 227,300 Berry, M. V., 114, 157 Bertand, F. E., 63,85 Bertrand, J. C., 253,300 Beyer, G., 40,85 Beyer, G.-J., 23, 93 Bhalla, C. P., 10,85 Bieber, E., 54, 93 Birkhoff, R. D., 47, 52,85 Birks, J. B., 225,300 Birstein, L., 124, 159 Bishara, M. N., 97, 158 Blackstock, A. W., 47, 52,85 Blakeway, S. J., 41, 45, 56, 60,61,87 Bledsoe, J. R., 152, 158 Blumenfeld, H., 227,300 Boato, G., 109, 110, 115, 126, 130, 135, 139, 141, 142, 143, 157 Bocquet, J.-P., 2, 76, 81,85 Bodek, K., 227,300 Bodlund-Ringstrom, B., 29,85 Boer, K. W., 310,382 Bomer, H. G., 63, 65,84, 86 Boersch, H., 52,85 BogdanoviC, M., 65,86 Bohg, A., 52,85 Bohm, C., 29,85

Bohr, A., 2,84 Bonchev, Zw., 28,85 Borisoglibskii, L. A., 9, 17,85 Bosc, G., 227,300 Bosnell, J. R., 337, 343, 381,382 Botvin, V. A., 46,84 Bourdinaud, M., 227,300 Boutot, J. P., 247, 251, 252, 253, 254, 255, 256, 257, 268, 269, 276,300, 301, 302 Boyle, J., 261,300 Brabec, V., 9, 15,27, 32,49, 50, 60,61, 63, 67, 68,86, 87, 91, 93 Bracewell, R., 212,219 Bratton, J., 255,304 Braumadl, F., 3, 24, 54, 58, 90 Breedlove, J., 164, 165,220 Brenner, R., 240,300 Brethon, J. P., 59, 88 Breuze, G., 287,300 Brianson, C., 24, 32, 51,85, 93 Bricman, C., 59, 60, 93 Brill, A. B., 239,304 Broadfoot, A. L., 257,304 Brooks, F. D., 227, 274,300 Brown, B. C., 234,300 Brown, D. B., 46,M Brown, F., 40,85 Brown, H. T., 261,300 Browne, E., 69,85 Bruch, L., 143,157 Brudanin, V. B., 27,85 Brunner, G., 10, 16, 18, 21,88, 89, 91 Brusdeylins, G., 152, 153, 154, 157 Brust, R., 45, 60,92 Buckely, W. D., 334, 336, 337, 342, 343, 344, 351, 352, 354, 355,382 Budinger, T. F., 221, 248,301 Bulgakov, V. V., 60, 61,85 Bunaciu, D., 9 , 8 5 Burgess, R. E., 311, 312,382 Burin, K., 29, 32, YO, 91 Bums, J. E., 59, 94 Butcher, R. N., 309,382 Bybee, R. L., 254, 255, 257,305

C

Cabrera, N., 97, 114, 116, 117, 157, 158 Cahoon, J. L., 248,301

389

AUTHOR INDEX

Caldwell, L., 261,300 Caldwell, S. E., 269,303 Calligaris, F., 273,301 Campbell, J. L., 15, 85 Campion, P. J., 59, 94 Camplan, J., 40, 92 Cannon, T. M., 183, 190, 191,220, 221 Cantal, R.,234,300 Cantarell, I., 293,301 Cantini, P., 106, 109, 110, 115,126, 130, 135, 138, 139, 141, 142, 143, 150, 151, 157

Cardillo, M. J., 133, 134, 135,157 Carley, A. F., 53,M Carlos, W. E., 112, 121, 139, 140, 143, 144, 145,157, 158

Carlson, C. W., 36,89 Carlson, P. J., 227,301 Carlson, T. A., 9, 52, 76,86, 90 Carnall, W. T., 79, 93 Carney, E., 252, 254, 255,301 Carroll, C. O., 7, 91 Carslaw, H. S., 310, 313, 314,382 Casten, R. F., 41, 45, 56, 60, 61, 65,86, 87 Castleman, K., 221 Catchpole, C. E., 253, 254, 276,301 Cekowski, D. H., 251, 252, 254, 255, 261, 301

Celen, E., 59,88 Celli, V., 97, 114, 116, 118, 119, 120, 121, 124, 126, 132, 142, 143, 157, 158, 159, 160 Chaban, E. E., 36, 93 Chabrier, G., 241,301 Chaloupka, V., 59, 60,93 Charpak, G., 247,301 Charvet, A., 62,90 Chen, M. H., 19,84 Chen, T. S., 125, 153,158 Chen, W., 203,219 Cheng, J., 269,303 Cheng, Y., 189,219 Cheshkov, A. A., 70,87 Chevalier, P., 259, 260, 276,301 Chistyakov, L. V., 34,45,52,55,56,57,65, 76,87 C h o k e r , J. P., 260,301, 302 Chou, H. P., 240,300 Chow, H., 114, 116, 117, 121,158 Christman, S. B., 36, 93 Chu, Y. K., 227,300

Chu, Y. Y., 2, 61, 76, 81,85, 86 Chumin, V. G., 23, 27,85, 93 Chumin, V. M., 23, 32, 65, 93 Church, E. L., 2, 7, 12,86 Ciuti, P., 273,301 Clement, G., 267,301, 305 Clinton, D. J., 337, 348, 354, 381,382 Coates, P. B., 234, 288, 293,301 Cadona, J., 145,160 Cohen, D. S., 329,383, 384 Cole, M. W., 97,98, 112, 121, 143, 144, 145, 158, 160

Colella, R., 135, 157 Colson, W. B., 251,301 Conwell, E. M., 309,382 Cory, C., 245,301 Cothern, C. R.,54,88 Coulthard, M. A., 15,86 Cowan, R. D., 19,86 Coward, L. A., 354, 381,382 Cowin, J. P., 139,158 Crandall, D. G., 269,303 Crasemann, B., 71,86 Crasemann, R. L., 19,84 Craven, P. G., 260,301 Cretu, Tr., 29,86 Croitoru, N., 354,300 Csorba, I. P., 257,301 Czerny, I., 27, 32, 93

D Daniel, H., 29, 30, 51, 72,86, 91, 92 Dautov, I. M., 82,86 Dautov, L. M., 82,86 Davidonis, R. I., 30, 55,86, 89 Davidson, W. F., 65,86 Davidson, W. F., 63,84 Davies, J. A., 40,85, 86 Davis, B., 252, 266, 269,302 Dedieu, M., 24, 32,85 Deeney, F. A,, 28,86 de Kruijk, A., 293,305 de La Barre, F., 270,271,272,273,274,275, 301

Del Mannol, P., 74, 75, 80,91 Delmotte, J. C., 251,253,254,256,269,300, 302

Demuynck, J., 277,305

3 90

AUTHOR INDEX

Dench, W. A., 48,92 den Harink, H. C., 255,305 Dex, R., 2, 20, 21,88 Derenzo, S. E., 248,301 De Rost, E., 59,88 Derry, G., 106,93 Deny, G. N., 138, 139, 140, 141, 143, 147, 148,158, 159 Deruytter, A., 277,305 Devonshire, A. F., 97, 104, 121,159 de Waard, H., 19, 71, 72, 82,91, 92 De Wald, J. F., 382 de Wette, F. W., 125, 153, 158 Dhawan, S., 250,301 Dingus, R. S., 10,86 Dmitriev, V. D., 34,86 Doak,R. B., 152, 153, 154, 157 DobriloviC, L. J., 39,86 Dolizy, P., 241,301 Doll, J., 114,158 Dondi, M. G., 98, 99,157 Donzel, B., 272,305 Dostal, K.-P., 20, 21, 71, 76,86, 89, 90 Douglas, D. G., 38,86 Dragoun, O., 2, 7, 9, 10, 13, 14, 15, 20, 27, 32, 39,45,49, 50,54, 55,57,60,61,63, 67, 68, 77, 78,86, 87, 91, 92, 93 DragounovB, N., 7, 10,87 Draper, J. E., 29, 41, 53,87 Driard, B., 261,304 Dubovik, V. M., 70,87 Duchene, J., 359, 360, 361,383 Duchenois, V., 249, 251,300, 305 Duckett, S. W., 249,301 Duda, F., 49, 50,86 Duda, R., 221 DupCik, J., 66,88 Dupe, D., 29,87 Dupuy, J., 259,302 Dwyer, S., 216,220 Dzhelepov, B. S., 65, 67,87

E Ebel, H., 39,87 Ebel, M.F., 39,87 Eberhardt, E. H., 251, 252, 254, 255, 256, 301 Eckardt, V., 227,301 Eglais, M.D., 9, 15, 21,84, 94

Eglais, M.O., 9, 15, 21,84, 94 Egorov, A. I., 63,84 Egorov, Yu. S., 59,87 Ehrlich, G., 109, 110, 159 Eichler, D., 245,303 Eifrig, Ch., 81,88 Ejiri, H., 63, 91 Ekdahl, T., 29, 46,85, 90 Ekilebei, S., 24, 32,85 Ekstrom, M.,173,219 Elgin, R. L., 145, 158 Ellis, T. H., 136, 158 Eltze, T. W., 54,93 Emberson, D. L., 249,301 Emelianov, B. A., 63,84 Emery, G. T., 71,87 Enck, R. S., 258, 261,305 Engel, R., 136, 137,160 Engel, T., 98, 131, 136, 137, 158, 160 Engstrom, R. W., 241,301 Esaki, L., 308,383 Esbjerg, N., 129, 158 Eschard, G., 241, 249, 251,300, 301 Estermann, J., %, 97, 158 Ettenberg, M.,262,304 Evrard, P., 227,300 Ewan, G. T., 51, 53,87 Ewbank, W. B., 10, 11, 12,87 Ewins, J., 299,302 F Famworth, P. T., 246,301 Farukhi, M. R., 228,301 Faulker, R. D., 230, 241,304 Fegan, D. J., 260,301 Feifrlik, V., 49, 50,86 Feingold, R., 255,304 Felcher, G. P., 110, 126, 130, 138, 139, 141, 142, 143, 150, 157 Femenie, F. R., 69,85 Fenster, A., 286,302 Fenyes, T., 24,26,84 Feoktistov, A. I., 60,61, 66,85, 87 Feresin, A. P., 9, 12, 18, 19, 68, 69, 70,84, 90,93 Fertin, J., 259,302 Feuerbacher, B., 158 Feynman, R. P., 372,383 Ficke, D. C., 228, 275, 276,303

AUTHOR INDEX

Fillot, J. M., 247,301 Finger, M., 66,88 Finzel, H. U., 99, 138, 147, 158, I60 FiSer, M., 20, 92 Fisher, B., 359,383 Fisher, D. G., 262, 302 Fisher, S. S., 52, 97, 158 Fitzpatrick, M. L., 19,84 Flasck, R., 357,383 Fleck, H. G., 246,304 Florescu, V., 9,85 Fluerasu, D., 9,85 Foesch, J. A., 255,304 Folkes, J. R., 262,302 Fomenko, V. N., 17,89 Fomichev, V. I., 81,84 Fominykh, M. I., 66,88 Fominykh, V. I., 66,88 Ford, G. W., 310,385 Fock, W. A., 310,383 Frank, H., 99, 126, 138, 147, 158, 160 Frankl, D. R., 97, 98, 106, 107, 138, 139, 140, 141, 143, 145, 147, 148, 158, 159, 160

Franks, L. A., 227,302 Frantsev, Yu. E., 60, 61, 66,85, 87 Franz, W., 310,383 Freedman, M. S., 10, 41, 42, 51, 71, 76,85, 87, 91 French, J. C., 365, 366,384 Frieden, B. R., 188,219 Fries, H. M., 6, 10, 11, 12, 13, 67,91 Friesen, J., 40,86 Frisch, R., 96,97, 104,158 Fritzschi, H., 343,383 Frye, R., 342,383 Fujioka, M., 28, 31, 32, 45, 54, 55, 63, 66, 77, 78, 81,82, 92, 93 Fujita, Y.,34, 87 Fu, K. S., 217,219 Fiile, K., 24, 26,84 Fukunaga, K., 206,220 Fulop, W., 308,383 G

Gabrielli, I., 273,301 Calanti, M., 35, 87 Garcia, N., 116,118, 119, 120, 121, 124, 125, 126, 130, 132, 142, 143, 157, 158, 159

391

Gardier, S., 32, 34, 90 Garibaldi, U., 111, 115, 116, 130, 157, 158 Gartner, W. W., 370,384 Gastman, I. J., 309, 323, 324, 327, 330, 334, 341, 346, 364, 370, 371, 373, 376, 378, 384 Gatti, E., 263,302 Gaur, S. P., 366,383 Gaveilyuk, V. I., 60, 61, 66,85, 87 Gedcke, D. A., 274,302 Geidel’man, A. M., 59,87 Geiger, J., 52,85 Geiger, J. S., 38, 40, 49, 52,85, 87, 88 Gelletly, W., 41,45,49,56,60,61,65,86,87 Genevey, J., 62, 90 Gentillon, C. D., 256,303 Genz, H., 19,84 Gerasimov, V. N., 34,45,52,55,56,57,65, 76,87 Gerber, R. B., 111, 158 Gerstner, D., 366,382 Gex, J. P., 269,302 Ghio, E., 138,159 Giacomich, R., 273,301 Ginaven, R., 260,302, 304 Gilliard , B ., 255,300 Gobleau, O., 24, 32,85 Goertzel, G. H., 2, 7, 91 Gol’danskii, V. I., 70, 71,87, 88 Goldhaber, M., 2, 71,84, 88 Gono, Y., 63,88 Gonzalez, R., 221 Goodman, F. O., 97, 111, 114, 116, 121, 124, 132, 142,157, 158, 159 Goodman, J. W., 163, 191,220 Goodstein, D. L., 145, 158, 160 Gorozhankin, V. M., 23,93 Gott, R., 35,87, 251,304 Goudonnet, J. P., 241,301 Graf, J., 249, 250,302 Graham, R. L., 38,40,49,52,53,67,85,87, 88

Grant, I. P., 15,88 Grant Rowe, R., 109, 110, 159 Gray, P. A., 261,300 Grechukhin, D. P., 2, 34,73,76, 77,78,80, 88

Greif, J. M., 145, 158 Greiner, L., 121,159 Gresset, C., 228, 276, 289,220, 300

392

AUTHOR INDEX

Grigor’ev, E. P., 51, 84 Grigor’ev, V. N., 70, 90 Grinberg, B., 59,88 Gromova, E. A., 59,87 Gromova, I. I., 51, 66, 84, 88 Gromov, K. Ya., 23, 27, 29, 63,84, 85, 86 Groshev, L. V., 63,88 Grosse, G., 59,88 Grubin, H. L., 309, 312, 323, 324, 327, 330, 334, 341, 346, 364, 370, 371, 374, 376, 378,384 Guest, A. J., 250, 252,302 Guidi, C., 130, 139, 141, 142, 143, 157 Gunn, J. B., 308,383 Gwilliam, G. F., 258, 261,305

H Habibi, A., 204,220 Hagberg, E., 40,92 Hager, R. S.,2,11, 12, 13,17,60,66,69,70, 88

Hagino, M., 262,302 Hahn, O., 2,88, 93 Haitz, R. B., 366,384 Hall, C., 220 Hall, D., 216,220 Hall, E., 221 Hamann, D. R., 133, 159 Hamilton, J. H., 3, 23, 32, 43, 67,88 Hanna, B., 234,300 Hansen, H. H., 32, 43, 59, 65, 67,88, 91 Hardy, J. C., 40, 92 Harr, J., 2, 7,91 Hart, P., 221 Hartmann, E., 2, 20, 21, 81,88, 90 Hartz, R., 275,303 Harvie, C. E., 117, 121, 159, 160 Haskell, B. G., 220 Haskey, J., 245,301 Hayakawa, T., 234,302 Hayashi, T., 234, 245,302 Hayes, T. M., 313,383, Heanni, D. R., 63,88 Hearing, R., 245,302 Heath, R. L., 226, 238, 246, 275,302 Helvy, F. A., 259,302 Hemingway, R. J., 59, 60,93

Hendee, C. F., 249,305 Henisch, H. K., 333,334,337,358,382,383, 384 Henkel, P., 251,302 Hennecke, H. J., 54,88 Herman, F., 9 , 8 8 Herman, E., 40,85 Heuser, G., 7, 10,86, 87 Hietanen, M., 59, 91 Hill, N. R., 116, 132, 158, 160 Hilsum, C., 309,383 Hines, D. C., 246,304 Hinman, G. W., 9, 90 Hinneburg, D., 10, 16, 18, 21,88, 89, 90 Hirasawa, M., 31,87 Hirokawa, K., 57,89 Hisatake, K., 28, 63, 64, 66, 81,87, 89, 92, 93 Hocker, L. P., 252, 266, 269,302, 303 Hoekstra, H. R., 79, 93 Hofstadter, R., 226, 238, 246, 275,302 Hoinkes, H., 97, 98, 99, 102, 112, 121, 122, 124, 126, 138, 139, 147, 158, 159, 160 Holder, M., 294,302 Holland, P. A., 341, 364,383 Holloway, P. H., 39,89 Holmberg, S. H., 312, 320, 324, 327, 333, 334, 336, 337, 342, 343, 344, 348, 351, 352, 354, 355, 364, 374,382, 384 Homma, K., 337, 357,383 Hons, Z., 44,84 Honusek, M., 44,84 Horl, E. M., 155, 160 Home, J. M., 131, 152, 159 Hosier, K., 299,302 Hosoda, M., 34,87 Hower, P. L., 365,383 Howie, J. M., 253,300 Howorth, J. R., 262,302 Huber, O., 68,86 Hughes, A. J., 337, 341, 348, 354, 364, 381, 382, 383 Hughes, E. B., 226, 238, 246, 275,302 Hulme, H. R., 2,89 Hunt, B. R., 164, 165, 170, 177, 181, 183, 184,219, 220, 221 Hurlbut, C. R., 269,303 Hurst, J. E., 139, 158 Hutchinson, J. S., 118, 119, 120, 121, 126, 143,158, 159 Huus, T., 2,84

AUTHOR INDEX

I Iannotta, S., 136, 158 Ibaugh, J. L., 230,304 Ignat’ev, S. V., 27,85 Ingebretsen, R. B., 190, 191,221 Inteman, R. L., 19, 84 Ionov, S. P., 82,86 Ishioka, S., 358,385 Ishizuka, T., 39, 93 Islamov, T. A., 63,84 Isozumi, Y., 32,89 Ito, S., 32,89 Ivanov, A. I., 23, 93 Ivanov, R. B., 23, 93 Izawa, Ya., 70,89

J Jackson, J. L., 323, 357, 359, 361,383 Jadmy, R., 35,89 Jaeger, J. C., 310, 313, 314,382 Jahn, P., 30, 51, 54,815 Jahne, E., 310,382 Jamar, J., 32, 34, 90 Jatteau, M., 239, 240, 243, 244,302 Jeanney, C., 227,300 Jech, C., 89 Jelks, E. C., 359,383 Jelley, J. V., 260,302 Jenkin, J. G., 48,93 Jenkens, G., 196, 201,220 Jeuch, P., 3, 24, 54, 58,89, 90 Johannsen, B., 71,89 Johansson, A., 52,89 Johns, H. E., 286,302 Johnson, C. B., 261,302 Johnson, K. H., 21,89 Jonas, M., 295,302 Jordanov, A., 28,85 Joseph, D. D., 329,383 Josephson, B. D., 308,383 Joyner, R. W., 53,85

K Kabina, L. P., 63,84 Kaarman, H., 121,159 Kaczmarczyk, J., 29,89 Khdhr, I., 27, 32, 93 Kadykenov, M. M., 82,86 Kaipov, D. K., 82,86 Kak, A., 214,220, 221

Kalibjian, R., 259,302 Kakiuchi, S., 21, 90 Kalbitzer, S., 40, 41,89 Kalinauskas, R. A., 23, 30, 55,86, 89 Kalinnikov, V. G., 23, 44,84, 93 Kamada, H., 35,89 Kaminker, D. M., 63,84 Kanbe, M., 32, 45, 55, 77, 78,87 Kankeleit, E., 24, 25, 63,84 Kaplan, T., 327, 330, 332, 354,383 Karakhodzhaev, A,, 63,84 Karlsson, L., 35,89 Katano, R., 32,89 Katz, I. N., 310, 321, 354,382 Kaufman, L., 299,302 Kaul, R., 384 Kawakami, H., 31, 34, 64,87, 89 Keller, H. B., 329,383 Kellog, E. M., 255,302 Kelly, M. A., 34, 77,89 Kelly, R., 40,85 Kelly, R. L., 59, 60, 93 Kerns, C. R., 234, 292,300, 302 Kershulene, M. Yu., 30,86 Keski-Rahkonen, O., 23, 51,89 Khan Khen Mo., 66,88 Khanonkind, M. A,, 19,84 Khazov, Yu. L., 63,84 Khol’nov, Yu. V., 59,87 Kholmov, Yu. V., 59,87 KiMdi, T., 24, 26,84 Kienlen, M., 267,302 Kigawa, M., 28, 93 King, F. T., 251,301 Kimura, H., 57,89 Kinoshita, M., 262,302 Kistner, 0. C., 2, 76, 81,85 Kitahara, T., 32,89 Kittel, C., 313,383 Klein, N., 308, 310, 321, 354,382, 383 Kleinknecht, K., 227,300 Klyuchnikov, A. A., 60, 66,85, 8 7 Knapp, G., 255,302 Knight, B. W., 309,383 Knight, R. I., 259,300 Knipper, A,, 62,90 Knispel, G., 267,302 Knyazkov, 0. M., 17,89 Koitki, S., 65,86 Kondurov, I. A., 63,84 Konicek, J., 66,88

393

3 94

AUTHOR INDEX

Kostylev, S. A,, 320, 327, 333, 334, 336, 337, 342, 348, 351, 352, 357, 358,384 Kotz, J., 324, 333, 334, 337, 339, 346, 350, 354, 358,383 Kouri, D. J., 111, 158 Kozik, V. S., 24, 48,90 Kovkr, A., 27, 32, 93 Kovalik, A., 27, 32, 93 Kowalski, G., 239,302 Krall, H. R., 259, 263, 266, 266,302, 303 Krane, K. S., 66,89

Kratsikova, T. I., 66,88 Krause, M. O., 23, 51,8Y Krishnasevamy, S. V., 106, 138, 147, 148, 158, 159

Kroemer, H., 309,383 Kroll, D. M., 354,383 KrpiC, D. K., 69,89 Kruger, R. P., 216,220 Krutov, V. A., 17,89 Kudo, M., 35,89 Kugler, E., 40, 91 Kuhlthau, A. R., 97, 158 Kuklik, A., 49, 50,86 Kulakov, V. M., 2,34,45,52,55,56,57,65, 74, 76, 77, 78, 79, 80,87, 88, 93, 94

Kuntze, M.,30, 51,86 Kuphal, E., 24, 25, 63,84 Kupryashkin, V. T., 60,66,85, 87 Kurakado, M., 32,89 Kuznetsova, M. Ya., 23, 93 Kuznetsov, V. V., 29, 44, 63,84, 86 Kupsch, H., 21,91 Kuroda, K., 248,303

Larson, P. E., 56, 91 Larysz, J., 3, 24, 54, 58, 90 Lashko, A. P., 61,85 Uszl6, S., 24, 26,84 Latuszyfiski, A., 40,89 Laughlin, R. B., 133, 159 Lawrence, R. S., 256,303 Leamer, R. D., 11, 43, 90 Learned, J. G., 245,301, 303 Lebeder, N. A., 23, 63, 66,84, 88, 93 Leblanc, J. C., 286,302 Lecante, J., 52,84 Leckey, R. C. G., 47, 48,93 Lecomte, P., 250, 253, 276,303 Le Cruer, Y., 110, 159 Lederer, C. M., 11, 43, YO Ledingham, K. W. D., 19,84 Lee, C., 29, 53,87 Lee, J., 227,300 Lee-Whiting, G. E., 31, 90 Lefort, M., 110, 159 Le Gellic, Y., 59,88 Legrand, B., 24, 32,85 Legrand, J., 59,88 Lei, T., 206,220 Leighton, R. B., 372,383 Lejay, Y.,131, 132, 159 Lelong, P., 239, 243, 244,302 Lennard-Jones, J. E., 97, 104, 121,159 Leroi, J. K., 24, 32,85 Leskovar, B., 250, 253, 257, 263, 268, 274, 275, 276, 277,303

Lettington, A. H., 341, 364,383 Leushkin, E. K., 63,84 Levi, A. C., 111, 115, 116, 122, 123, 124, 125, 130, 148, 157, 158, 159

L

Lach, B., 259,302 Lafond, C., 227,300 Lagomarsino, V., 98,. 99, 157 Lagos, M., 124, 148, 149,157, 159 Lagoutine, F., 59,88 Lam, D. J., 79,Y3 Lampert, M. A., 383 Lampton, M., 36,89, 255,303 Landauer, R., 325, 356,383 Lapshin, V. E., 248,305 Lapujoulade, J., 97, 110, 117, 122, 124, 131, 132, 148,157, 159

Lewitt, R., 214,220 L’Hermite, P., 259,302 Lifshitz, E. M., 102, 159 Liljequist, D., 29, 46,85, 90 Lin, Y. W., 124,159 Lindgren, I., 14,90 Lindhard, J., 40, 90 Linsay, P. S.,227,300 Lipsovskii, A. A., 59,87 Lisegang, J., 48,93 Listengarten, M. A., 6, 7,9, 11, 12, 13, 14, 15, 16, 18, 19,67,68,69,70,84,85, 93 Litt, J., 253,300

90,

AUTHOR INDEX

395

Litthgton, A. H., 337, 348, 354, 381,382 Makovetskii, Yu. V., 60,66,85, 87 Liu, W.S., 111,159 Malik, F. B., 9 , 9 0 Lizurej, H. I., 29,86 Malmsten, G., 52,89 Lo, C. C., 253,257,263,268,274,275, 276, Maker, L., 258,305 277,303 Malv, L., 67,91 Lodge, J. A., 258,303 Mampe, W., 3, 24, 54, 58,89, 90 Lodwick, G., 216,220 Manalio, A. A., 32,90 Loeweneck, P., 68,90 Manley , B. W., 249,300 Logan, R. M., 109,159 Manson, J. R., 97, 114, 116, 117, 121, 124, Loginov, Yu. E., 63,84 125,157, 159 Lombard, F. J., 255,303 Manthuruthil, J. C., 54,88 Long, D. C., 253,303 Marelius, A., 52,89 Looma, J., 59, 91 Marguier, G., 62, 90 Lopez-Delgado, R., 272, 275,305 Mark, P., 383 Los, J., 255,305 Markham, J., 275,303 Losty, M. J., 59, 60,93 Martin, B., 2, 13, 14, 15, 24, 25, 30, 39, 51, Loty, C., 255, 267,301, 303 55, 57, 63, 67, 68,84, 85, 86, 90 Lovtsyus, A. V., 59,87 Martin, F., 255,303 Lu, C. C., 9 , 9 0 Martinelli, R. U., 262,302, 304 Ludwig, J., 227,300 Martynov, V. V., 63,84 Lukyanov, C. M., 34,86 Marubayashi, K., 33,94 Luschka, M., 99, 138,158 Marvin, A., 116, 160 Lutz, S., 227,302 Marvin, A. M., 124, 126,159 Lyons, P. B., 227, 252, 266, 268, 269, 276, Masel, R. I., 116,159 302, 303 Mashirov, L. G., 74, 93 Lynch, F. J., 274,303 Mason, B. F., 148, 149, 155, I59 Lyubimov, V. A., 24,48,53,90 Massenet, O., 33, 90 M W , Z., 24, 26,84 Matsui, M., 21, 90 Matsumoto, Y., 33, 94 M Mattera, L., 109, 110, 115, 126, 143, 155, McCann, K. J., 114, 159 157, 159, 160 McCarthy, P. J., 28,86 Mattsson, L., 35,89 McDonald, R. J., 41,87 Maurel, E., 110,159 McDonie, A. F., 230,304 Maurer, A., 253, 254, 257,300, 303 McIntyre, J. D., 40,86 Mazaki, H., 21,90 McKeighen, R., 247,304 Mazur, P., 325,382 McClure, J. D., 103, I59 Meade, M. L., 296,303 McPherson, 251,301 Medved’, S. V., 27,85 Macarie, G., 29,86 Medvedev, A. I., 64,90 Macau, J. P., 32, 34,90 Mees, Q . , 164,220 Machovh, A., 66,88 Meitner, L., 2,88, 90 Madrid, J., 252, 266, 269,302 Melnikov, I. V., 74, 93 Maeda, H., 250, 252,303, 304 Melnikov, N. S., 248,305 Maier, B. P., 3, 24, 54, 58,90 Mende, S. B., 254,304 Main, C., 354,383 Merkert, D., 13, 14, 57,86 Majka, R., 253,303 Merrill, R. P., 116, 159 Makariunas, K. B., 30, 55,86, 89 Memtt, J. S., 52,87 Makanonas, K. V., 18,90 Mersereau, R., 209, 213,220 Makarov, E. F., 82,86 Meuleman, J., 259,302

3 96

AUTHOR INDEX

Meunier, P., 254, 257,303 Meunier, R., 253,300 Meyer, H. D., 122, 148, 160 Meyers, J. A., 106, 107,160 Miehe, J. A., 267, 269, 270, 272, 300, 302, 305

Mihelich, J. W., 2, 90 Mikhailova, M. A., 23, 93 Mikhailov, V. M., 69, YO Millar, I. C. P., 249,303 Miller, C. F., 333, 357,383 Miller, D. R., 97, 131, 133, 134, 152, 157, 159, 160

Miller, R. C., 242, 247,303 Miller, W. H., 114, 116, 159, 160 Miminov, A. I., 23, 93 Minkova, A., 28,85 Mladjenovic, M., 3, 16, 23, 43, 60, 90, 91 Moi, T. E., 228, 242, 243,304 Monich, E. A., 248,305 Morales, J., 247,304 Morales, J. J., 242, 247,303 Morii, T., 59, 91 M6rik, G., 27, 32, 93 Mbrik, Gy., 24, 26,84 Morita, M., 70, 71, 91 Morton, G. A., 258, 262,303, 305 Morton, J., 192,220 Mosetti, R., 273,301 Moss, S. C., 337, 357, 358,384 Mostovoi, V. I., 34, 65, 76, 77, 79,Y4 Moszynski, M., 228,263,272,273,274,275, 276,300, 303

Mott, N., 2, 93, 333, 337,382 Mottelson, B., 2, 84 Mouchel, D., 32, 59,88, 91 Mount, G. H., 254, 255, 257,305 Muehllehner, G., 228, 247, 299,220, 304 Miiller, J. W., 59, 91 Muff, P., 258,303 Mukoyama, T., 21, 90 Mullani, N. A., 228, 275, 276,220 Miinze, R., 71,8Y Muminov, T. M., 44,84 Murray, G., 38,88 Murray, S. S., 255,302 Mutterer, M., 19, 32,84, 88 Muzalev, P. N., 60, 61, 66,85, 87 Myasoedov, N. F., 24, 48,90

N Nagai, Y., 63,91 Nagel, M., 2, 10, 16, 18, 20, 21, 71, 76,86, 88, 89, 91 Nahr, H., 99, 138, 147, 158, 160 Nakayama, S., 63,91 Nakayama, Y.,24, 25, 63,84 Namiot, V. A., 70, 71,87, 88 Natarazan, T., 200,219 Navon, D. H., 365, 366, 367,368, 369,382, 383

Neal, W. H., 359,383 Nefedov, V. I., 39,91 Neganov, B. S., 66,88 Nestor, C. W.,Jr., 9, 90 Neve de Mevergnies, M., 73, 74, 75, 80,91 Nevzorov, N. B., 74, 93 Newbauer, E., 310,382 Newland, F. J., 357,383 Newman, R., 234,300 Nguen Kong Chang, 63,84 Nieschmidt, E. B., 256, 303 Nikii, Y.,35,89 Nikitushev, Yu. M., 46,84 Nishida, R., 264,302 Noak, M., 27,85 Nohr, H., 98, 122, 124, 147, 159 Normand, G., 239,302 Norseev, Yu. V., 23,93 Neirskov, J. K., 129, 158 Northover, W. R., 308, 333,382 NovAk, D., 24, 26,84 Novgorodov, A. F., 23, 93 Novikov, A. I., 53,92 Novikov, E. G., 24,48, 53, YO Nozik, V. Z., 24, 48, 53, 90 Nussli, J., 263, 265,304 Nylandsted Larsen, A., 59,88, 91 Nyman, B., 52,89

0

Oba, K., 250, 252, 253, 254, 257, 277, 303, 304

O’Connell, R. F., 7 , 91 Odinov, B. V., 2, 34, 65, 76, 77, 78, 79, 80, 88, 94

AUTHOR INDEX

Odru, R.,228,276,220 Oetzmann, H., 40, 41,89 O’Gorman, T. J., 106, 147, 148, 158, 159 Oha, H., 365,383 Okamoto, K., 70, 71,91 O’Keefe, D. R., 97, 160 Okerlund, M., 299,302 Oku, M., 57,89 Olsen, G . H., 262,304 Omanov, Sh., 23, 63,84, 93 Ohvier, S., 24, 32,85 Oppenheim, A., 164, 165, 166,194,209,213, 220 Orlov, V. I., 59,87 Orphan, V. J., 260,302, 304 Orr, G. B., 65,86 Ortec, 274,304 Ortendahl, D., 299,302 Oshima, S., 365,383 Osipenko, B. P., 23, 32, 65,93 Ota, J., 66,88 Otozai, K., 71, 91 Ott, J., 239,302 Ovshinsky, S. R., 308, 310, 321, 324, 333, 343,382, 383 Owen, A. E., 354,383 Owen, R. B., 258,303

P Pabst, D., 20, 21, 71, 76,86, 91 Pailly, M., 359, 360, 361,383 Palmer, I. C., 262,302 Palmer, R. L., 97,160 Panitz, J. A., 255,304 Papoulis, A., 175, 211,220 Parellada, J., 20, 29, 91, 92 Parker, W. C., 39, 91 Parkes, W., 251,304 Pasmanter, R. A., 325,382 Patton, J. A., 239,304 Pauli, H. C . , 2 , 6 , 8 , 9 , 10, 11, 12, 13, 17, 19, 67, 69, 70,86, 91 Pauvert, J., 239,302 Pavlov, V. N., 66,88 Pchelin, V. A., 34,45,52,55,56,57,65,76, 87

397

Pearson, A. D., 308, 333, 357,382, 383 Peck, W. F., 308, 333,382 Peifer, W. K., 257,304 Penionzhkevich, Yu. E., 34,86 Penn, D. R.,47, 57, 91 Perez-Mendez, V., 250,303 Pergrale, J., 239,302 Perlman, M. L., 2, 61, 76, 81,85, 86 Persyk, D. E., 228, 230, 242, 243, 247, 259, 263, 266, 267,302, 304 Peters, T., 214,220 Petersen, K. E.,334,335,338,339,340,341, 384 Peterson, G. A., 309,383 Petev, P., 27,85 Petley, C. H., 249,304 Petit, R.,259,302 Pettersson, H., 52, 89 Philips, 235, 259,304 Pierson, W. R., 64,91 Piktri, G., 247, 250, 252, 254, 259, 263, 265, 268, 276,300, 301, 304 Pinston, J. A., 63,84 Piotrowski, A., 40,85 Plajner, Z., 2, 9, 39, 55, 67,86, 91 Plch, J., 63, 68,86 Pleiter, F., 19, 71, 72, 81, 82,91, 92 Polaert, R.,249,250,251,255,300,302,304, 305 Polcan, M. R., 29, 91 Polichar, R.,260,304 Pollehn, H., 255,304 Pook, R.,249, 257,304 Pool, P. J., 261,300 Pooladdej, D., 358,385 Popescu, C., 332, 354, 355, 356, 357, 382, 384 Potempa, A., 40,89 Porter, F. T., 10, 41, 42, 51, 76,85, 91 Poultney, S. K., 224, 263,274,304 Powell, C. J., 47, 56, 91, 92 Pratt, R. H., 9,85 Pratt, W., 163, 167, 174, 196, 197, 199, 200, 203, 205, 206,220 PraZkk, F., 44,84 Preobrazhenskaya, L. D., 59,87 Prince, M. B., 308,384 Prochkzka, I., 66,88 Proykova, A., 46, 91

398

AUTHOR INDEX

Prydz, S., 248,304 Pryor, R. W., 334,384 Pytkowski, S., 29,89

R Raff, U., 17, 69,91 Raichle, M. E., 275,305 Raiko, V. I., 40,85 Ralston, A., 165,220 Rao, K. R., 200,219 Rathburn, L., 109, 110,159 Ravn, H. L., 40,91 Rayleigh, Lord J. W. Strutt, 115, 160 Reddi, V. G. K., 365,383 Reehal, H. S., 358,384 Rees, D. M., 255,304 Rehak, P., 252,253, 257, 277,304 Reinhard, D. K., 336, 337, 355,384 Rengan, K., 64,91 Renaud, J. F., 35,87 Ribordy, C., 68,86 Richard-Serre, C., 62, 90 Richter, L., 24, 25, 63,84 Ridley, B. K., 309,384 Rieckeheer, R., 239,302 Rieder, K. H., 98, 131, 136, 137, 158, 160 Rig6, A., 21, 91 kfkovskB, J., 66,88 Ritchie, R. H., 47, 52,85 Rittenberg, A., 39, 60, 93 Ritter, K. A., 228,304 Robertson, J. M., 354,383 Roberts, A., 234,304 Robinson, G., 206,220 Robinson, R., 2,91 Rockstad, H. K., 357,383, 384 Rodgers, D., 358,384 Rodiere, J., 255,304 Rode, B., 30,91 Roese, J., 206,220 Rollo, F. D., 239,304 Rome, M., 246,304 Ronzhin, A. I., 248,305 Roos, M., 39, 59, 60, 91, 93 Rosel, F., 6, 10, 11, 12, 13, 67, 91 Rose, M. E., 2, 7, 9, 16, 59,91 R o d n, A., 14,90 Rosenfeld , A . , 221 Rosenfeld, A. H., 39, 60,93

Rothberg, G. M., 20, 29, 32,90, 91, 92 Rougeot, H., 261,304 Rounce, P. A., 255,304 Rousseau, H., 52,84 Rowan, W., 299,302 Rowe, J. E., 36,93 Roy, D., 29,87 Roy, R., 251,302 Rozi&re,G., 261,304 Rud, N., 10,86 Ruedy, J. E., 262,305 Ruggieri, D. J., 249, 257,304 Rumiantsev, V. L., 63,84 Rutherford, E., 2,91 Rykalin, V. I., 248,305 RySavq, M., 9, 10, 15,20,32,45,55,60,61, 63, 67, 68, 77, 78,86, 87, 91, 92 Ryzhinskii, M. V., 59,87 S

Saito, T., 71,91 Sakharov, S. L., 63,84 Saltsburg, H., 97, 160 Salvat, F., 20, 92 Salvo, C., 106, 138, 143, 150, 155,157, 159, 160 Sandel, B. R., 257,304 Sandie, W. G., 254,304 Sands, M., 372,383 Sar-El, H. Z., 23, 92 Sattarov, D. K., 34,86 Sauneuf, R., 269,302 Savage, J. A., 337, 348, 354, 381,382 Sawicka, B. D., 28, 92 Sawicky, J. A., 28,92 Schafer, R., 164, 165, 166, 194,220 Schafft, H. A., 365, 366,384 Schagen, P., 249,305 Schardt, M. A., 228,304 Scharff, M., 40,90 Scheinberg, N., 206, 220 Schilling, D., 206,220 Schiott, H. E., 40, 90 Schliiter, P., 4, 92 Schmeing, H., 40, 92 Schmidt, K. C., 249,305 Schmutzler, F., 2, 9, 29,86, 92 Scholl, E., 310,382 Schonfeld, E., 45, 60, 92

AUTHOR INDEX

Schrechkenbach, K., 3, 24, 41, 45, 54, 56, 58,60,61,63, 65,84, 86,87, 90, 92 Schubert, W. K., 52,92 SchuK, R., 15,90 Schuller, M., 370,384 Schulyakovskii, G. S., 9, 17,85, 92 Schumtzler, F., 2,86 Schvantsenberg, F., 27,85 Schwartz, C., 143, 160 Schwarz, K., 14, 21,92 Scarlett, R. M., 366,384 Sciulli, F., 227,300 Scoles, G., 136,158 Scott, J. E., Jr., 97, 158 Seah, M. P., 48,92 Seib, D. H., 224, 263,305 Seifert, G., 2,21,88 Semerad, E., 155, 160 Seltzer, E. C., 2, 11, 12, 13, 17, 60, 66, 69, 70,88 Semonov, V. K., 248,305 Sergeenkov, Yu. V., 68,92 Sergeev, V. O., 13, 15, 16, 51,84, 90 Seriani, G., 124, 157 Sessler, A. M.,29,85 Sevier, K. D., 5 , 38, 47, 51, 52,92 Shaevitz, M., 227,300 Shanks, R. L., 338, 340,384 Shapiro, S. G., 18, 63,84, 88 Sharpe, J., 249,305 Shaw, M. P., 309, 312, 320, 323, 324, 327, 330, 333, 334, 336, 337, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 354, 355, 357, 358, 364, 366, 370, 371, 373, 374, 376, 378, 383, 384 Shchus, A. F., 23,93 Shemansky, D. E., 257,304 Shenoy, G. K., 81,92 Shestopalova, S. A., 32,64, 65,87, 90, 92 Shibata, T., 63,91 Shimoni, Y., 111,158 Shinohara, T., 54, 81,87, 92 Shirley, D. A., 48, 92 Shirley, V. S., 11, 43,90 Shockley, W., 308, 366,384 Shur, M. S., 310, 324, 333,382 Sibener, S. J., 133, 134, 139,157, 158 Sidorov, V. T., 27,85 Siegbahn, H., 29, 35, 39,48, 53, 63,89, 92

399

Sillou, D., 248,303 Silva-Moreira, A. F., 145, 160 Silver, M., 310, 324, 333,382 Silzars, A., 259,305 SimiC, J., 65,86 Simmons, C. D., 308, 365,384 Simmons, D., 252, 266, 268, 269,302 Simon, R. E., 259,305 SimoviC, M., 39,86 Simpson, R. B., 329,384 Sinaev, A. N., Sipp, B., 267, 269, 270, 272, 275,300, 302, 305 Skanavi, G. I., 310,384 Skillman, S., 9,88 Slack, L. A., 337, 357, 358,384 Sliv, L. A., 2, 7, 9, 14,92 Smith, C., 203,219 Smith, D., 245,301 Smith, H. M., 262,303, 305 Smith, J. N., Jr., 97, 160 Smith, L. G., 258,305 Smith, W. R., 358,383 Smout, D., 258,303 Sobel, B. E., 275,305 Soff, G., 4,92 Soldatov, A. A., 2,34,45,52,55,56,57,65, 73, 76, 77, 78, 80,87, 88 Solomon, P. R., 309, 312, 324, 364,374,384 Soloschenko, B. A., 48,92 Sommer, A. H., 287,305 Sondhi, M., 164,221 Sousha, A. M., 310, 316, 354,384 Spadacini, R., 111, 115, 116, 130,157 Spalek, A., 9, 15, 46,86, 92 Sparrow, E. M., 329,383 Spemol, A., 59,88 Spinela, S., 259,300 Spijkervet, W. J. J., 81, 82,92 Spinrad, B. I., 2, 7,91 Srapenyants, R. A., 53,92 Stapleton, R. J., 230,305 Satis, H., 39, 91 Steele, W. A., 112, 144,160 Steffen, R. M., 6, 9, 12, 19, 70,91 Stepanov, E. K., 23,93 Stapanov, A. V., 59,87 Stem, O., 96, 97, 104,158 Stickney, R. E., 131,160 Stocker, H. J., 310, 321, 354,384

400

AUTHOR I N D E X

Stockham, T. G., 190, 191,221 Stolyarova, E. L., 53,92 Straws, M. G., 240,300 Streltsov, V. A., 74, 93 Stromswold, D. C., 226,305 Strong, P., 2, 7,91 Subba Rao, B. N., 43, 69, 92 Subbarao, R. B., 97, 160 Subhani, K. F., 330,333,343,344,345,347, 349, 351, 352, 354, 355, 366,384 Sugihara, T. T., 63,88 Sugiyama, M., 254,304 Suglobov, D. H., 74, 93 Suhl, H., 122, 123, 124, 148, 159 Sundell, S., 40, 91 Sunyar, A. W., 2,88 Sushkov, P. A., 63,84 Suzuki, S., 57,89 Suzuki, Y., 254,304 Stelts, M. L., 65,86 StojanoviC, M., 65,86 Svahn, B., 52,89 Svelto, V., 263,302 Swanson, N., 47, 92 Swindell, W., 219 Szab6, B., 18,93 Szajman, J., 47, 48,93 Szczepihki, L., 29,89 Sze, S. M., 365,384

T Takashima, M., 32, 45, 5 5 , 77, 78,87 Takeuchi, F., 248,303 Tan, T. H., 252, 268, 269,303 Tan, W. K., 124, I59 Tkrkknyi, F., 24, 26,84 Tatarek, R., 110, 126, 130, 138, 139, 141, 142, 143, 150, 151, 157 Tauc, J., 335, 365,384 Taylor, E. A., 31, 90 Taylor, H. M., 2, 93 Taylor, W. B., 286,302 Televinova, T. M., 27,85 Tendulkar, D. V., 131, 160 Tenebaum, M., 215,219 Ter-Nersesyants, V. E., 64,90 Ter Pogossian, M. M., 228, 275, 276, 303, 305 Terraillon, M., 359, 360, 361,383

Terreni, S., 106, 143, 150, 155,157, 159, 160 Tesevich, B. I., 17,85 Teterin, Yu. A., 2, 34,74,76,77,78,80,88, 93 Thevenin, J. C., 227,300 Thieme, K., 23,93 Thoma, P., 310, 332, 333, 337, 355,384 Thomas, C. B., 337, 343, 358, 381,382, 384 Thompson, E. D., 114, 116, 117, 121, 158 Thompson, M. S., 358,385 Thompson, W., 216,220 Thornburg, D. D., 313,383 Thornton, C. G., 308, 365,384 Thuis, H., 158 Tietjen, J. J., 259,305 Tikhonov, A. N., 53,93 Timothy, A. F., 249,305 Timothy, J. G., 249, 254, 255, 257,305 Tirsell, G., 269,303 Toennies, J. P., 152, 153, 154, 157 Toigo, F., 116, 124, 126, 159, 160 Tommasini, F., 98, 99, 138, 143, 155, 157, 159, 160 Tommei,G. E., 111, 115, 116, 130,157, 158 Toriyama, T., 28, 63, 66,93 Tothill, H. A. W., 245,301 Townes, J. R., 216,220 Tracy, J. C., 47,93 Tretyakov, E. F., 24, 48, 53,90 Tret’yakov, F. F., 23, 93 Trippe, T. G., 39, 60, 93 Trusov, V. F., 9, 11, 15, 21,84, 85, 93, 94 Trussell, H. J., 184,221 Trzhaskovskaya, M. B., 7, 9, 11, 12, 13, 14, 16, 20, 67,84, 85 Tsuchida, A., 114, 160 Tsui, E., 221 Tsuji, K., 33, 94 Tsupko-Sitnikov, V. M., 66,88 Tucker, T. C., 9,90 Turgeon, L. J., 365, 366, 367, 368, 369,382 Turner, A. F., 216,220 Tyler, C. E., 34, 77,89 Tyroff, A., 40,85

U Uchevatkin, I. E., 64,90 Umarov, G. Ya., 23,84 Umesaki, S., 33, 94 Uslamov, R. R., 44,84

AUTHOR INDEX

Uyttenhove, J., 277,305 Uwamino, Y., 39,93

40 1

Washington, D., 249,305 Watanabe, H., 143, 157 Watts, D., 196, 201,220 Weare, J. H., 117, 121, 124, 160 V Weber, W. H., 310,385 Vacher, J., 228,272, 273,275, 276,220,300 Weeks, S. P., 36, 93 Valadares, M., 51, 93 Weinberg, S., 245,305 Valbusa, U.,98,99, 136, 138, 155,157, 158, Weirauch, D. F., 310, 321, 354,384 159, 160 Weiss, H. M., 59,88 Vallet, D., 274,305 Weneser, J., 2, 7, 12,86 van der Eijh, W., 39, 93 Wensel, F., 385 van Duyl, W. A., 293,305 Wesner, D., 106, 138, 139, 140, 141, 143, Van Hove, L., 122, 160 147, 148, 158, 159 Vanenbroukx, R., 59,88 Whirter, I., 255,304 Van Trees, H. L., 182,221 White, W., 240,305 Varga, D., 27, 32, 93 Wierzbowski, P., 29,89 Vasil’chenko, V. G., 248,305 Wijnaendts van Resandt, R. W., 34,94, 255 Vatai, E., 18, 93 Wilcox, D. A., 258, 261,305 Veal, B. W., 79,93 Wild, R. K., 39,94 Vegors, S. H., 256,303 Williams, A., 59, 94 Verat, M., 261,304 Williams, A. H., 252, 268, 269,303 Verbinski, V. V., 260,302 Williams, B. R., 108, 126,148,149, 155,159, Vernier, P. J., 241,301 160, 259, 262,305 Vidali, G., 143, 160 Williams, C. W., 274,302 VinduSka, M., 10, 13, 14, 57,86, 92 Willwater, R., 24, 25, 63,84 Vizzoli, G., 385 Wilson, E., 2, 71,84 Vobecky, M., 32,93 Wilsch, H., 98, 121, 122, 124, 126, 138, 147, Voikhanskii, M. E., 70, 93 158, 159, 160 von Baeyer, O., 2, 93 Winiecki, A. L., 240,300 von Egidy, T., 3, 24, 54, 58, 90, 93 Winther, A., 2,84 Vukanovid, R., 16, 43, 60,69,89, 90, 91 Wintz, P., 198,221 Vylov, Ts., 23, 27, 32, 63, 65,84, 85, 93 Wittwer, N. C., 258,303 Wiza, J., 251,302 Wiza, J. L., 34, 35, 36, 94, 249,250,305 w Wohl, W., 358,383 Wagner, F., 10,85 Wolf, E. L., 52, 92 Wagner, F., Jr., 41, 51, 91 Wolfe, K. L., 117, 121, 160 Wagner, F. E., 81, 92 Wolken, G., Jr., 124, 159, 160 Wagner, W., 239,302 Wong, G., 275,303 Wahl, P., 272,305 Wonka, U., 99, 138,158 Walen, R. J., 24, 32, 51,85, 93 Woo, J. W., 356,383 Walser, R. M., 359,383 Woodhead, A. W., 257,305 Walsh, J., 385 Wright, A. G., 230, 245, 274,301,302, 305 Walsh, P. H., 358,385 Wu, C. S., 37,84 Walter, G., 62, 90 Wuest, C., 245,301 wang, L., 221 Wapstra, A. H., 59,88 Ward, R., 257,305 Y Wardley, J., 245,301 Warner, D. D., 65,86 Yaffe, L., 39,94 Warren, A. C., 354,385 Yamanaka, C., 70,89

402 Yamashita, M., 288, 293,305 Yamatera, H . , 39,93 Yavor, S . Ya., 23,84 Yinnon, A . T., 111,158 Yoshida, Y.,33,94 Yoshizaki, S . , 262,302 Yost, G . P., 39, 60, 93 Youngbluth, O . , 288,305 Yu, C . F., 139,158 Yushkevich, Yu. V . , 23,93

AUTHOR INDEX

Zelenkov, A. G . , 2,34,45,52,55,56,51,65, 1 4 , 1 6 , 1 1 , 18, 80,87, 88, 93, 94 Zhelev, Zh., 23,93 Zhirgulyavichyus, R. K., 30, 55,86, 94 Zhudov, V. I., 2, 34, 6 5 , 1 6 , 1 1 , 1 8 , 1 9 ,80, 88, 94 Zhuravlev, N. I., 21,M Zilitis, V. A., IS, 94 Zolnowski, D.R., 63,88 Zolotavin, A. V . , 51,84 Zonneveld, F. W.,239,305 Zuber, K., 40,89 Z Zuk, W . , 40,89 Zagarino, P. A . , 252,266,268,269,302,303 hpanciC, M., 16, 43, 60,90, 91 ZderadiCka, J . , 63, 68,86 Zworykin, V. K . , 258,305

Subject Index

A

C

A X , see Analog-to-digital converter Adsorbate-covered surfaces, elastic diffraction, 135-138 Alkali antimonide photocathode, 226 Aluminum foils, in conversion-electron spectroscopy, 40 Analog image processing, 162 Analog-to-digital converter (ADC), 286-287 Anode load, photomultiplier, 281 APD, see Avalanche photodiode Armand effect, in surface diffraction, 123 Atom diffraction, see Diffraction of atoms and molecules from crystalline surfaces Atomic screening, in internal conversion coefficient calculations, 8-9, 14- 16 Avalanche photodiode (APD), 260 Axial-mirror analyzers, in internal conversion studies, 29

Cathode resistivity, photomultiplier, 283284 CAT scanning, see Computer-aided tomography scanning Centrifuge method of target fabrication, for in-beam electron spectroscopy, 42 cerenkov radiator, 227, 245 Chalcogenides electrical conductivity, 330 thin films, 332-359 electrothermal switching, 354-359 first-fire event, 342-353 Channeltrons, see Electron-channel multipliers Charge linearity, photomultiplier, 277 Chemical effects, on decay-rate variations, 74 Circulant matrix, 170 Circular-cage structure, photomultiplier, 237-238, 267 B Color-image compression, in digital image BBQ, spectrum, scintillator, 227-228 processing, 204-207 Beeby effect, in surface diffraction, 123, 146 Computer-aided tomography (CAT) scanBismuth germanate (BGO) scintillator, 228 ning, 207-215 Bit allocation, in digital image processing, scintallators, 228, 238-239 200-202 Computers Blind deconvolution, in image restoration, conversion-electron spectra, analysis of, 189- 192 53-57 Block circulant matrix, 172 digital image processing, 161-221 Block Toeplitz matrix, 172 Computer vision, 215-219 Conversion-electron lines, in internal conBlue-sensitive photocathode, 229-23 1 Bound-state resonance (BSR), in surface version-electron spectroscopy, 51 59 diffraction, 105-107,117,120-121,126Conversion-electron Mossbauer spectros128, 138, 140-143, 150-152 COPY,28-29, 33 Box-and-grid structure, photomultiplier, Conversion electrons, energy of, 4-5 237, 242 Conversion-electron spectra, variations in, BSR, see Bound-state resonance 75-81 Bulls-eye EM1 tube, 245-246 403

404

SUBJECT INDEX

Coming blue sensitivity, photocathode, 228-229 Corrugated hard-wall potential, in elastic diffraction, 115- 117, 129- I30 Corrugated surface with a well, in elastic diffraction, 117- 121 Coulomb field, internal conversion coefficients, 8-9 Crossed-field photomultiplier, 258 Cryogenic vacuum, in surface diffraction studies, 99 Crystalline surfaces, diffraction from, see Diffraction of atoms and molecules from crystalline surfaces Current linearity, photomultiplier, 277, 285, 298 Cylindrical-mirror analyzers, in internal conversion studies, 29

D Damping resistor, photomultiplier, 281 Deblumng, see Image restoration Debye-Waller (D-W) factor, 108, 122, 146-148, 155 Decay-rate variations, 72-75 Deexcitation modes, in electromagnetic transitions in nuclei, 3-4 Deformed surfaces, elastic diffraction, 132135 Detectors in internal conversion studies, 31-37 in surface diffraction studies, 99 Diffraction of atoms and molecules from crystalline surfaces, 95- 160 conclusions, 155-156 elastic diffraction, 100- 106 experimental techniques, 98-99 inelastic diffraction, 106- 109 introduction, 95-98 quantum theory of atom-surface scattering, 111-128 rotational diffraction, 109- 111 structural information from elastic diffraction, 129-138 surface lattice dynamics, 146- 155 surface potential well, 138- 145 DIGICON (digital image tube), 260 Digital image processing, 161-221 image analysis (computer vision), 215219

image data compression, 192-207 image restoration, 163-192 introduction, 161- 163 reconstruction from projections, 207- 2 15 Divider current, photomultiplier, 278-280 D-W factor, see Debye-Waller factor

E EBS diode, see Electron-bombarded semiconductor diode Eikonal approximation, 116, 124- 126, 129130 Elastic diffraction, 100- 106 static potentials, 112-121 structural information from, 129- 138 surface potential well, 138- 145. Electric monopole transitions, 63 Electromagnetic mass separation, in internal conversion studies, 40 Electromagnetic transitions in nuclei, 3-5 Electron-bombarded semiconductor (EBS) diode, 258-261 Electron-channel multipliers (channeltrons), 32-33, 249-258 Electron detectors, in internal conversion studies, 31-37 microchannel-plate detectors, 34-37 position-sensitive proportional counters, 33-34 Electron energy losses, in conversion-electron spectroscopy, 44-49, 52 Electronic instabilities, solid-state devices, 307-308 Electron multiplier, of photomultiplier, 236238 Electron spectrometers, in internal conversion studies, 22-31, 45 data-collection efficiency, improvements of, 30-31 Electron spectroscopy for chemical analysis (ESCA), 35 Electroplating, in internal conversion studies, 39 Electrostatic spectrometer, in internal conversion studies, 22, 24, 26-27, 32 exposure times, 49 microchannel plates, 35 Electrothermal instabilities, semiconductor, see Semiconductors, thermal and electrothermal instabilities

SUBJECT INDEX

Energy loss, in surface diffraction, 152- 155 Environmental effects on internal conversion, 71-82 ESCA, see Electron spectroscopy for chemical analysis F

Focus blur, in image degradation, 191 G

Gain drift, photomultiplier, 284, 287-293 Gallium arsenide surface, reconstruction of, 133- 134 transmission-mode secondary emitter, 262 Gamma-ray scintillator camera, 238-244 Gamma-target structure, photomultiplier, 242 Gas-surface interaction, 97-98; see also Helium-graphite system; Helium-lithium fluoride system Ge(Li) detectors, 65 Geiger-Muller counters, 31-32 Graphite in internal-conversion studies, 41 surface diffraction, 101-102, 131, 139145, 151-152 GR method, to obtain scattering amplitudes, 116 H

Hard corrugated surface, see Corrugated hard-wall potential HCS (hard corrugated surface), see Corrugated hard-wall potential Helium, surface diffraction, 129-132, 136, 138-139, 146-148 Helium-graphite system, surface diffraction, 101-102, 131, 139-145, 151-152 Helium-lithium fluoride system, surface diffraction, 101, 103, 114, 152-154 Hydrogen, diffraction studies, 98, 109- 111, 138- 139, 146- 147

I ICC, see Internal conversion coefficient Image analysis (image understanding), see Computer vision

405

Image data compression, in digital image processing, 192-207 correlatioddecorrelation, 192- 193 DPCM compression, 193- 197 hybrid compression, 203-207 transform domain compression, 197- 203 Image processing, 162; see also Digital image processing Image restoration, in digital image processing, 163-192 blind deconvolution, 189- 192 matrix derivation of image-restoration algorithms, 174-189 matrix forms of convolutions, 164-174 Inelastic diffraction, 106-109 vibrating surfaces, 121-128 Inelastic scattering mean free path, in internal conversion studies, 47-49 Inorganic scintillator, 226 Internal conversion coefficient (ICC), 2-4 atomic screening, 14- 16 calculations, accuracy of, 11 - 14 comparisons of calculated and measured ICCS, 60-61 environmental effects, 72-73 measurements of, 42-43 nuclear size, effects of, 7-8 numerical calculations, 9- 14 perturbation theory, higher orders of, 16-19 point nucleus approximation, 5-7 tables, 10- 11 valence atomic shells, 19-22 Internal conversion-electron spectroscopy, 1-94 environmental effects on internal conversion, 71 -82 calibration of the Mossbauer isomer shift, 81-82 conversion-electron spectra, variations in, 75-81 decay-rate variations, 72-75 experimental data, treatment of, 43-59 measured spectra, analysis of, 49-59 spectrum quality, factors limiting, 4349 uncertainty of measured quantities, 59 experimental methods, 22-43 electron detectors, 31-37 electron spectrometers, 22-31 radioactive sources, 37-42

406

SUBJECT INDEX

introduction and historical remarks, 1-3 nuclear spectroscopy, role in, 61-71 summary and outlook, 82-83 theory, 3-22 theory and experiments, comparison of, 60-61 Inverse internal conversion, 70-71 Ion implantation, in internal conversion studies, 39-40 Iron-core magnetic spectrometer, 24 Iron-free spectrometer, 23-24, 28, 31

K Karhunen- Loeve transformation, in digital image processing, 198, 206

L Lattice, surface, 146- 155 LED, see Light-emitting diode LEED, see Low-energy electron diffraction Levi effect, in surface diffraction, 123 Light-emitting diode (LED), 273-274, 285, 290-292 Linear focusing dynode structure, photomultiplier, 237-238 Lithium fluoride, surface diffraction, 101, 1 0 9 - 1 1 1 , 114, 121, 130, 138-139, 147149, 152-155 Low-energy electron diffraction (LEED), 95

M Magnetic spectrometer, in internal conversion studies, 22-24, 28, 31-32 exposure times, 49 Matrix analysis, in digital image processing, 164- 189 Maximum-entropy method, in digital image processing, 185-189 MCA, see Multichannel analyzer MCP detectors, see Microchannel-plate detectors MCP photomultiplier, see Microchannelplate photomultiplier Mean gain deviation (MGD), photomultiplier, 289-291

Mean time to failure (MTTF), photomultiplier, 294 Memory effect, in thin films, 333 Memory switch, in thin chalcogenide films, 333-334, 342-348, 353 MGD, see Mean gain deviation Microchannel-plate (MCP) detectors, 34- 37 Microchannel-plate (MCP) photomultiplier, 249-258 pulse response, 268-269 timing characteristics, 276-277 transit time, 263 Molecule diffraction, see Diffraction of atoms and molecules from crystalline surfaces Mossbauer spectroscopy, 28-29, 33, 81-82 Motion blur, in image degradation, 191 M'ITF, see Mean time to failure Multichannel analyzer (MCA), photomultiplier, 285 Multichannel detector, 35

N Negative differential conductivity (NDC), semiconductors, 308-309, 320, 370382 thermally induced, 321-332 critical electric field, 327-332 effect of inhomogeneities, 325-327 thermal boundary conditions, 321-325 thermistor, 310-320 vanadium dioxide, 360-361, 363-364 Newton-Cotes quadrature formulas, 165 Noise, in digital image processing, 174- 178, 184-187 No-penetration model, of internal conversion coefficients, 7-8, 14 Nuclear size, effects of, in internal conversion coefficient calculations, 7-8 Nuclear spectroscopy, role of internal conversion in, 61-71 inverse internal conversion, 70-71 nuclear structure effects, 68-70 transition energies, 63-65 transition multipolarities, 65-68 Nuclear structure (penetration) effect, in internal conversion, 2, 7-8, 68-70 Nyquist sample distance, in image processing, 166

407

SUBJECT INDEX

0

Optical image processing, 162 Optics, input optics system of photomultiplier, 235-236, 240-241 Organic scintillator, 227 Orthogonal transforms, in digital image processing, 199-200 Oscilloscope, 266-267, 285

P Penetration effect, in internal conversion, see Nuclear structure effect Periodically deformed surfaces, elastic diffraction, 132- 135 Perturbation theory, higher orders of, in calculation of internal conversion coefficients, 16-19 PET, see Position-emission tomography Photocathode, 226-231 hemispherical, 245 Photomultiplier (PMT), 223-305 collection space, 238 conclusion, 299 electron multiplier, 236-238 input optics system, 235-236 introduction, 223-224 main characteristics, present situation on, 262-299 linearity, 277-287 operating range, 294-299 reliability, 293-294 stability, 287-293 timing performance, 262-277 state of the art, 224-262 conventional structures, 235-248 photocathodes, 226-23 1 secondary emission, 231-234 unconventional structures, 249-262 PMT, see Photomultiplier Point nucleus approximation, in calculation of internal conversion coefficient, 5-7 Position-emission tomography (PET), 275276 Position-sensitive detectors, 35-37 Position-sensitive photomultipliers, 245248, 253-255 Position-sensitive proportional counters, 33-34

Power supply, photomultiplier, 277-281 Proportional counters, 32-34 Pulse response, photomultiplier, 263-269

Q QMP, see Quadrant-multiplier phototube QPD, see Quadrant photosil detector Quadrant-multiplier phototube (QMP), 246247 Quadrant photosil detector (QPD), 260 Quantum mechanical phenomena, in conversion-electron spectroscopy, 44 Quantum scattering theory, 111- 128 static potentials and elastic diffraction theories, 112-121 vibrating surfaces and inelastic scattering, 121-128 Quantum surface rainbow, 103, 115-116

R Radioactive sources, for internal conversion studies, 37-42 Rapid-scan counting techniques, in conversion-electron spectroscopy, 49 Rayleigh-phonon dispersion relation, 148149 Recoil shadow method, in internal conversion studies, 24 Reconstructed surfaces, elastic diffraction, 132-135 Reservoir capacitor, photomultiplier, 280281 Rotational diffraction, 109- 111 S

SCEM photomultiplier, see Single-channel electron-multiplier photomultiplier Scintillator, 225-229 timing characteristics, 274-276 Secondary emission, photomultiplier, 231234 Selective absorption, 97, 104, 120, 150 Semiconductors detectors, in internal conversion studies, 22-24 electrical conductivity, 313 heat flow, 313-316

408

SUBJECT INDEX

inhomogeneities, effect of, 325-327 R C analog network of heating process, 316-320 surface reconstruction, 133 switching effects, critical electric field-induced thermally based, 327-332 thermal and electrothermal instabilities, 307-385 introduction, 307-310 negative differential conductivity, 321 332, 370-382 summary, 369-370 thermistor, 310-320 thin chalcogenide films, 332-359 transistors, second breakdown in, 364369 vanadium dioxide, 359-364 SEPHD, see Single-electron pulse-height distribution SER, see Single-electron response Si(Li) detectors, 32 Si(Li) electron spectrometer, 24, 26 Signal transit time, photomultiplier, 263 Simply corrugated surfaces, elastic diffraction, 130-132 Single-channel electron-multiplier (SCEM) photomultiplier, 249 Single-electron pulse-height distribution (SEPHD), photomultiplier, 233 - 234 Single-electron response (SER), photomultiplier, 266 Space-charge effects, photomultiplier, 281 283 Spectrometers, in internal conversion studies, 22-31, 45 data-collection efficiency, improvements Of, 30-31 Specular inelastic selective adsorption, 150 Star-tracking system, 246 Static crossed-field multiplier structure, 258 gain-bandwidth factor, 268 Static potentials, in elastic diffraction theories, 112-121 Surface-current model, of internal conversion coefficients, 7, 14 Surface Debye temperature, 107 Surface lattice dvnamics. 146- 155 Surface potential well, 138-145 Surface scattering, 95; see also Diffraction of atoms and molecules from crystalline surfaces

Switching thin chalcogenide films, 332-359 vanadium dioxide, 359-364

T Tea-cup structure, photomultiplier, 241 -242 Temporal compression, in digital image processing, 204, 206 Thermal instabilities, semiconductor, see Semiconductors, thermal and electrothermal instabilities Thermistor, 310-320 Thin films chalcogenides, 332 - 359 electrothermal switching, 354-359 first-fire event, 342-353 switching process, 332 thermistor properties, 313 vanadium dioxide, 361-364 Three-wire proportional counter, 33 Threshold switch, in thin chalcogenide films, 334, 342, 348-352, 355-359 Time-of-flight (TOF) analysis photomultiplier, 274-276 in surface diffraction studies, 152- 155 Toeplitz matrix, 169 TOF analysis, see Time-of-flight analysis Transistors, second breakdown in, 364-369 Transition energies, 63-65 Transition multipolarities, 65-68 Transit-time differences, photomultiplier, 269-277 Transmission secondary electron (TSE) image-intensifier tube, 248 Transmission secondary-emission (TSE) dynode structure, 261-262 TSE dynode structure, see Transmission secondary-emission dynode structure TSE image-intensifier tube, see Transmission secondary electron image-intensifier tube Two-dimensional position-sensitive detectors, 36-37 IJ -

235Uranium,decay-rate variations, 72-75

v Vacuum deposition, in internal conversion studies, 39

409

SUBJECT INDEX

Valence atomic shells, in internal conversion coefficient calculations, 19-22 Vanadium dioxide, 359-364 Venetian-blind structure, photomultiplier, 237, 241-242, 293 Vibrating surfaces, inelastic scattering, 121 128 W Weiner filter theory, 179 Whitening transformation, in digital image processing, 206

Windows gamma-ray scintillation camera, 240 photocathode, 230-23 1

X X-ray computerized tomography scanning, see Computer-aided tomography scanning X-ray image-analysis system, 216

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