Temperature-dependent extended electron

Temperature-dependent extended electron energy loss fine structure measurements from K, L23, and M45 edges in metals, intermetallic alloys, and nanocrystalline materials - CaltechTHESIS
CaltechTHESIS
A Caltech Library Service
About
Browse
Deposit an Item
Instructions for Students
Temperature-dependent extended electron energy loss fine structure measurements from K, L23, and M45 edges in metals, intermetallic alloys, and nanocrystalline materials
Citation
Okamoto, James Kozo
(1993)
Temperature-dependent extended electron energy loss fine structure measurements from K, L23, and M45 edges in metals, intermetallic alloys, and nanocrystalline materials.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/kk1d-wm17.
Abstract
This dissertation developed the extended energy loss fine structure (EXELFS) technique. EXELFS experiments using the Al K, Fe L23 and Pd M45 edges in the elemental metals gave nearest-neighbor distances which were accurate to within ± 0.1 A. In addition, vibrational mean-square relative displacements (MSRD) derived from the temperature dependence of the EXELFS compared favorably with predictions from published force constant models derived from inelastic neutron scattering data. Thus, information about "local" atomic environments can be obtained not only from K edges, but from L23 and M45 edges as well. This opens up most of the periodic table to possible EXELFS experiments.

The EXELFS technique was used to study the local atomic structure and vibrations in intermetallic alloys and nanocrystalline materials. EXELFS measurements were performed on Fe3Al and Ni3Al alloys which were chemically disordered by piston-anvil quenching and high-vacuum evaporation, respectively. Chemical short-range order was observed to increase as the as-quenched Fe3Al and as-evaporated Ni3Al samples were annealed in-situ at 300 C and 150 C respectively. Temperature-dependent measurements indicated that local Einstein temperatures of ordered samples of Fe3Al and Ni3Al were higher than those of the corresponding disordered samples. Within a "pair" approximation, these increases in local Einstein temperatures for the ordered alloys corresponded to decreases in vibrational entropy per atom of 0.48 ± 0.25 kB for Fe3Al and 0.71 ± 0.38 kB for Ni3Al. In comparison, the decrease in configurational entropy per atom between perfectly disordered and ordered A3B alloys is 0.56 kB in the mean-field approximation. These results suggest that including vibrational entropy in theoretical treatments of phase transformations would lower significantly the critical temperature of ordering for these alloys.

EXELFS investigations were also performed on nanocrystalline Pd and TiO2. At 105 K, the MSRD in nanocrystalline Pd and TiO2 were found to be greater than that in the corresponding large-grained materials by 1.8 ± 0.3 x 10(-3) A2 and 1.8 ± 0.4 x 10(-3) A2, respectively. Temperature-dependent measurements were inconclusive in measuring differences in local atomic vibrations between the nanocrystalline and large-grained materials.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
Applied Physics
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Fultz, Brent T.
Thesis Committee:
Unknown, Unknown
Defense Date:
6 May 1993
Funders:
Funding Agency
Grant Number
U.S. Department of Energy
DE-FG03-86ER45270
Leila Clark Fellowship
UNSPECIFIED
NSF
DMR-8811795
Record Number:
CaltechETD:etd-12112006-073855
Persistent URL:
DOI:
10.7907/kk1d-wm17
Default Usage Policy:
No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:
4941
Collection:
CaltechTHESIS
Deposited By:
Imported from ETD-db
Deposited On:
22 Dec 2006
Last Modified:
16 Apr 2021 22:13
Thesis Files
Preview
PDF (Okamoto_jk_1993.pdf)
- Final Version
See Usage Policy.
9MB
Repository Staff Only:
item control page
CaltechTHESIS is powered by
EPrints 3.3
which is developed by the
School of Electronics and Computer Science
at the University of Southampton.
More information and software credits
Temperature-Dependent Extended Electron Energy Loss Fine
Structure Measurements from K, L23, and Mys Edges in Metals,

Intermetallic Alloys, and Nanocrystalline Materials

| Thesis by

James Kozo Okamoto

In Partial Fulfillment of the Requirements
for the Degree of

Doctor of Philosophy

California Institute of Technology

Pasadena, California

1993
(Defended May 6, 1993)

James Kozo Okamoto

To my parents

and my wife

Acknowledgements

First and foremost, | would like to express my gratitude to my advisor,
Professor Brent Fultz, for directing my research and providing me with constant
support and encouragement. My research would not have begun without Brent's
ideas, and it is a great pleasure to work in his group.

| would also like to acknowledge my deep indebtedness to Dr. Channing
Ahn for teaching me the art of electron microscopy and the science of electron
energy loss spectrometry. Channing gave this work a flying start by supervising
my initial experiments and generously sharing his expertise with me.

| am appreciative to Carol Garland for facilitating my work on the electron
microscope.

Finally, | would like to thank the various members of the Fultz group for
their help, especially Dr. Douglas Pearson, Zheng-Qiang Gao, and Lawrence
Anthony.

Financial support for this research was received from the United States
Department of Energy under grant DE-FG03-86ER45270 and from a Leila Clark
fellowship from Caltech. The Gatan model 666 parallel-detection EELS
spectrometer was acquired through a grant from Caltech's Program in Advanced
Technologies, supported by Aerojet, General Motors, and TRW. The transmission
electron microscopy facility was largely supported by the National Science

Foundation under grant DMR-8811795.

Abstract

This dissertation developed the extended energy loss fine structure
(EXELFS) technique. EXELFS experiments using the Al K, Fe Lo3, and Pd Mas
edges in the elemental metals gave nearest-neighbor distances which were
accurate to within + 0.1 A. In addition, vibrational mean-square relative
displacements (MSRD) derived from the temperature dependence of the EXELFS
compared favorably with predictions from published force constant models
derived from inelastic neutron scattering data. Thus, information about "local"
atomic environments can be obtained not only from K edges, but from L23 and
Mas edges as well. This opens up most of the periodic table to possible EXELFS
experiments.

The EXELFS technique was used to study the local atomic structure and
vibrations in intermetallic alloys and nanocrystalline materials. EXELFS
measurements were performed on FesAl and Nig3Al alloys which were chemically
disordered by piston-anvil quenching and high-vacuum evaporation,
respectively. Chemical short-range order was observed to increase as the as-
quenched FesAl and as-evaporated Ni3Al samples were annealed in-situ at 300
C and 150 C, respectively. Temperature-dependent measurements indicated
that local Einstein temperatures of ordered samples of Fe3Al and Ni3Al were
higher than those of the corresponding disordered samples. Within a “pair”
approximation, these increases in local Einstein temperatures for the ordered
alloys corresponded to decreases in vibrational entropy per atom of 0.48 + 0.25
kg for FegAl and 0.71 + 0.38 kg for NigAl. In comparison, the decrease in
configurational entropy per atom between perfectly disordered and ordered A3B

alloys is 0.56 kg in the mean-field approximation. These results suggest that

including vibrational entropy in theoretical treatments of phase transformations
would lower significantly the critical temperature of ordering for these alloys.
EXELFS investigations were also performed on nanocrystalline Pd and
TiOz. At 105 K, the MSRD in nanocrystalline Pd and TiO2 were found to be
greater than that in the corresponding large-grained materials by 1.8 + 0.3 x 10°3
A2 and 1.8 + 0.4 x 103 A2, respectively. Temperature-dependent measurements
were inconclusive in measuring differences in local atomic vibrations between

the nanocrystalline and large-grained materials.

vii

Table of Contents

Acknowledgements
Abstract
List of Figures

List of Tables

1 Historical Introduction
1.1. Electron Energy Loss Spectrometry (EELS)
1.2 Extended X-ray Absorption Fine Structure (EXAFS)
1.3 Extended Electron Energy Loss Fine Structure (EXELFS)
1.4 Physical Origin of Extended Fine Structure
1.5 Applications of EXELFS in Materials Science

2 Electron-Atom Scattering Theory
2.1 ‘Inelastic Scattering of Fast Electrons
2.1.1 Kinematics
2.1.2 lonization Cross Sections
2.1.3 Deconvolution of Multiple Inelastic Scattering
2.2 Elastic Scattering
2.2.1 Phase Shifts and Scattering Amplitudes
2.2.2 Theory of Extended Fine Structure

3 Instrumentation and Experimental Procedures

3.1. Specimen Preparation

oO Co fF W

12
12
14
16
18
21
21
26

42
42

viii

3.2 Characterization of Alloys and Nanocrystalline Materials
3.3 Control of Specimen Temperature

3.4 Parallel-Detection EELS (PEELS)

EXELFS Analysis of K, Lo3, and Mas Edges
4.1. Basic Analytical Procedures
4.2 Extension to Lo3 and Mas Edges
4.2.1 FeLog
4.2.2 Pd Mas
4.3. Effect of Multiple Inelastic Scattering on EXELFS

Temperature-Dependent EXELFS of Elemental Metals
5.1. Debye-Waller Type Factor for EXELFS
5.2 Vibrational Mean-Square Relative Displacement (MSRD)
5.3 Force Constant Model of Lattice Dynamics
5.4 Results from Al, Fe, and Pd

5.4.1 Einstein Analysis

5.4.2 Debye Analysis

5.4.3 Force Constant Analysis

Applications to Intermetallic Alloys and Nanocrystalline
Materials
6.1. Chemical SRO and Vibrational MSRD in FesAl and NisAl
6.1.1 FesAl
6.1.2 Ni3Al

Page
46
60
65

72
72
90
90
105
116

132
132
134
138
141
150
155
160

172

172
191

Page
6.2 Structural Disorder and Vibrational MSRD in Nanocrystalline 205

Pd and TiOe
6.2.1 Nanocrystalline Pd 206
6.2.2 Nanocrystalline TiO2 216
6.3. Conclusions and Perspective 222
Appendix A Electron-Atom Scattering Calculations 224
A.1—_ Energy-Differential Cross Sections for lonization 225

A.2. Central Atom Phase Shifts and Backscattering Amplitudes 239

Appendix B EXELFS Data Processing Software 264
B.1. Correction for Channel-to-Channel Gain Variations 265
B.2 Extraction and Normalization of EXELFS Oscillations 271
B.3 Fourier Band-Pass Filtering 279
B.4 _Least-Squares Fitting 287

Appendix C Software for Calculations of Vibrational MSRD 294

C.1 Correlated Einstein Model 295
C.2 Correlated Debye Model 299
C.3 Force Constant Model 304

References 319

List of Figures

Figure 1.1.

Figure 1.2.

Figure 2.1.

Figure 2.2.

Figure 2.3.

Figure 2.4.

Figure 2.5.

Figure 3.1.

Figure 3.2.

Figure 3.3.

Figure 3.4.

Figure 3.5.

Figure 3.6

Figure 3.7.

Page

EELS measurement of Al K edge from foil of pure aluminum. 6

Schematic illustration of (a) constructive and (b) destructive 9
interference at the central atom.

Classical picture of electron scattering by a single atom 13
(carbon) (After Egerton, 1986).

Vector relationship between q, Ko, and Ky due to conservation 14
of momentum (After Egerton, 1986).

Scattering of plane-wave packet from central potential (After 24
Cohen-Tannoudji et al., 1977).

Schematic illustration of the final state potential V(r) (After 31
Boland et al., 1982).

Diagrammatic representations of (a) zero-scattering, (b-c) 33
single-scattering, and (d-g) double-scattering processes
for three-atom system (After Boland et al., 1982).

Schematic illustration of piston-anvil quenching apparatus 43
(After Pearson, 1992).

Schematic illustration of high-vacuum evaporator. 45

Growth of superlattice diffracton peaks in initially piston-anvil 47
quenched FesAl annealed at 300 °C (Gao and Fultz, 1993).

Mossbauer spectra of Fe3Al as-quenched and after annealing 48
at 300 °C for 392 hours (Gao and Fultz, 1993).

Hyperfine magnetic field distributions for FesAl as piston-anvil 49
quenched and after annealing at 300 C for various times
(Gao and Fultz, 1993).

X-ray diffraction patterns from NigAl material as-evaporated 51
onto 84 K substrates, onto 300 K substrates, and from material
annealed in the DSC to 550 C (Harris et al., 1991).

DSC traces for NigAl material evaporated onto 300 K and 52
84 K substrates (Harris et al., 1991).

Figure 3.8.

Figure 3.9.

Figure 3.10.

Figure 3.11.

Figure 3.12.

Figure 3.13.

Figure 3.14.

Figure 3.15.

Figure 3.16.

Figure 3.17.

Figure 3.18.
Figure 3.19.
Figure 3.20.
Figure 3.21.

Figure 4.1.

Figure 4.2.

Figure 4.3.

xi
Bright field (BF) and dark field (DF) image pair and diffraction
pattern from as-evaporated thin-film of Pd.

X-ray diffraction measurement of (111) peak from as-
evaporated Pd.

Bright field (BF) and dark field (DF) image pair and diffraction
pattern from thin film of Pd after annealing at up to 550 C.

Bright field (BF) and dark field (DF) image pair from partially
compacted powder of Pd nanocrystals.

Bright field (BF) and dark field (DF) image pair and diffraction
pattern from as-prepared thin film of TiOo.

Bright field (BF) and dark field (DF) image pair and diffraction
pattern from thin film of TiO after annealing at 900 C for 11
hours.

Schematic diagram of liquid nitrogen cooled substrate holder
transmission electron microscopy.

Diagram of hypothetical situation used to estimate increases
in sample temperature due to heating from electron beam.

Change in temperature due to electron beam heating as
function of radial distance using Equation (3.6) for thin film
sample illustrated in Figure 3.15.

Schematic of electron energy loss spectrometer attached to
bottom of TEM.

Ray diagram of TEM operating in diffraction mode.
Schematic of PEELS spectrometer.

Typical gain calibration spectrum.

Illustration of gain averaging for Fe Log edge.

Power-law extrapolation (broken line) to remove pre-edge
background for Al K edge of Al metal.

Comparison between two periods of a sinusoid and a cubic
polynomial.

Cubic spline fit (broken line) for Al K edge of Al metal.

67
68
71
71
73

Figure 4.4.
Figure 4.5.
Figure 4.6.
Figure 4.7.

Figure 4.8.

Figure 4.9.

Figure 4.10.

Figure 4.11.
Figure 4.12.
Figure 4.13.

Figure 4.14.
Figure 4.15.

Figure 4.16.

Figure 4.17.
Figure 4.18.
Figure 4.19.

Figure 4.20.

xii

Al K-edge EXELFS from Al metal. 78
Partial energy-differential cross sections of Al K edge. 80
Al K-edge EXELFS from Al metal weighted by k? (solid line). 82

Magnitude of Fourier transform of Al K-edge EXELFS from Al 83
metal (solid line).

Al K-edge EXELFS from Al metal after Fourier filtering to 84
isolate 1nn shell data.

Theoretical (solid line) and experimental (dotted line) Al 86
K-edge EXELFS due to 1nn shell in Al metal.

Theoretical (solid line) and experimental (dotted line) 87
EXELFS on Al K-edge due to 1nn shell in Al metal after
weighting by k2.

Magnitude of FT of theoretical (solid line) and experimental 88
(dotted line) Al K-edge EXELFS due to 1nn shell in Al metal.

Theoretical (solid line) and experimental (dotted line) Al 89
K-edge EXELFS due to inn shell in Al metal after Fourier
filtering.

Background subtracted Fe L edge from foil of pure Fe metal. 91

Fe Lo3-edge EXELFS from Fe metal. 92
Fe La3-edge EXELFS from Fe metal weighted by k (solid 94
line).

Magnitude of Fourier transform of Fe Lo3-edge EXELFS from 95
Fe metal.

Partial energy-differential cross sections of Fe Log edge. 96
Energy-differential cross sections of Fe Lo3 and L; edges. 97

Theoretical Fe Lg (solid line), Lp (dashed line), and L; (dotted 99
line) EXELFS due to combined 1nn and 2nn shells in Fe
metal.

Sum of theoretical Fe Ls, Le, and L; EXELFS due to 100
combined 1nn and 2nn shells in Fe metal (solid line).

Figure 4.21.

Figure 4.22.
Figure 4.23.
Figure 4.24.

Figure 4.25.
Figure 4.26.
Figure 4.27.

Figure 4.28.
Figure 4.29.

Figure 4.30.

Figure 4.31.
Figure 4.32.
Figure 4.33.
Figure 4.34.
Figure 4.35.

Figure 4.36.

xiii

Theoretical (solid line) and experimental (dotted line) Fe
Lo3-edge EXELFS after Eo for experimental data shifted
by -15 eV.

Theoretical (solid line) and experimental (dotted line) Fe
Lo3-edge EXELFS weighted by k?.

Magnitude of FT of theoretical (solid line) and experimental
(dotted line) Fe Lo3-edge EXELFS.

Fourier filtered theoretical (solid line) and experimental
(dotted line) Fe Lo3-edge EXELFS.

EELS measurement of Pd M edge from foil of pure Pd metal.

Pd Mys-edge EXELFS from Pd metal (solid line).

Magnitude of Fourier transform of Pd Mgs-edge EXELFS
from Pd metal.

Partial energy-differential cross sections of Pd Mas edge.

Energy-differential cross sections of Pd Mas, M3, Ma, and My
edges.

Theoretical Pd Mas (thick line), Mg (thin line), Mz (dashed
line), and M, (dotted line) EXELFS due to inn shell in Pd
metal.

Theoretical (solid line) and experimental (dotted line) Pd
Mas-edge EXELFS.

Magnitude of FT of theoretical (solid line) and experimental
(dotted line) Pd Mys-edge EXELFS.

Fourier filtered theoretical (solid line) and experimental
(dotted line) Pd Mas-edge EXELFS.

Idealized low-loss spectra used to simulate the effect of
multiple inelastic scattering.

Simulated effect of multiple inelastic scattering on the
general shape of a hypothetical inner-shell edge.

Simulated EXELFS extracted from single-scattering (thin
solid) and multiple-scattering (thin dashed) spectra.

101

102

103

104

106
107
108

109
111

112

113

114

115

117

118

Figure 4.37.

Figure 4.38.

Figure 4.39.

Figure 4.40

Figure 4.41.

Figure 4.42.

Figure 4.43.

Figure 4.44.

Figure 4.45.

Figure 5.1.

Figure 5.2.

Figure 5.3.

Figure 5.4.

Figure 5.5.

Figure 5.6.

Xiv

Magnitude of FT of simulated EXELFS extracted from single-
scattering (thin solid) and multiple-scattering (thin dashed)
spectra.

Low loss region from multiple-scattering (solid line) and
single-scattering (dotted line) spectra of FesAl.

Fe Log edge from multiple-scattering (solid line) and single-
scattering (dotted line) spectra of FeAl.

Al K edge from multiple-scattering (solid line) and single-
scattering (dotted line) spectra of FesAl.

Background subtracted Al K edge from multiple-scattering
(solid line) and single-scattering (dotted line) spectra of
FesAl.

Fe Lo3-edge EXELFS from multiple-scattering (solid line)
and single-scattering (dotted line) spectra of FesAl.

Fourier transforms of Fe Lo3-edge EXELFS from multiple-
scattering (solid line) and single-scattering (dotted line)
spectra of FeAl.

Al K-edge EXELFS from multiple-scattering (solid line) and
single-scattering (dotted line) spectra of Fe3Al.

Magnitude of FT of Al K-edge EXELFS from multiple-
scattering (solid line) and single-scattering (dotted line)
spectra of FesAl.

Temperature-dependence of magnitude of FT of Al K-edge
EXELFS (3

Temperature-dependence of magnitude of FT of Fe Lo3
-edge EXELFS (7 < k < 13 A"1) from Fe metal.

Temperature-dependence of magnitude of FT of Pd Mas
-edge EXELFS (10.25

Fourier filtered 1nn shell EXELFS from Al metal at 97 K
(solid line) and 296 K (dashed line).

Change in 1nn MSRD for EXELFS from A! metal relative to
EXELFS at 97 K.

Change in 1nn MSRD for EXELFS from Fe metal relative to
EXELFS at 97 K.

119

123

124

125

126

127

128

130

131

142

143

144

146

147

148

Figure 5.7. Change in 1nn MSRD for EXELFS from Pd metal relative to 149
EXELFS at 98 K.

Figure 5.8. Einstein model fit to 1nn MSRD data from Al metal. 152
Figure 5.9. Einstein model fit to 1nn MSRD data from Fe metal. 153
Figure 5.10. Einstein model fit to 1nn MSRD data from Pd metal. 154
Figure 5.11. Debye model fit to 1nn MSRD data from Al metal. 157
Figure 5.12. Debye model fit to inn MSRD data from Fe metal. 158
Figure 5.13. Debye model fit to 1nn MSRD data from Pd metal. 159
Figure 5.14. Density of vibrational modes for Al metal determined from 162

interatomic force constants.

Figure 5.15. Density of vibrational modes for Fe metal determined from 163
interatomic force constants.

Figure 5.16. Density of vibrational modes for Pd metal determined from 164
interatomic force constants.

Figure 5.17. Projected density of vibrational modes for inn shell (dashed 166
line) compared with density of vibrational modes (solid line)
for Al metal.
Figure 5.18. Projected density of vibrational modes for inn shell (dashed 167
line) compared with density of vibrational modes (solid line)
for Fe metal.
Figure 5.19. Projected density of vibrational modes for inn shell (dashed 168
line) compared with density of vibrational modes (solid line)
for Pd metal.
Figure 5.20. Force constant model prediction of 1nn MSRD in Al metal. 169
Figure 5.21. Force constant model prediction of 1nn MSRD in Fe metal. 170
Figure 5.22. Force constant model prediction of inn MSRD in Pd metal. 171
Figure 6.1. Phase diagram for Fe-Al (Massalski, 1986). 173
Figure 6.2. DOs ordered structure of Fe3Al. 174

Figure 6.3. Theoretical Al K EXELFS signal from disordered FegAl. 177

Figure 6.4.

Figure 6.5.
Figure 6.6.

Figure 6.7.

Figure 6.8.

Figure 6.9.

Figure 6.10.

Figure 6.11.

Figure 6.12.

Figure 6.13.

Figure 6.14.

Figure 6.15.

Figure 6.16.

Figure 6.17.

Xvi

Magnitude of theoretical (a) Al K and (b) Fe Log EXELFS
from inn shell of completely disordered and perfectly
ordered FesAl.

EELS measurements of (a) Al K and (b) Fe L edges from
FesAl.

(a) Al K and (b) Fe Log EXELFS from as-quenched FesAl
at 296 K.

Magnitude of FT of experimental (a) Al K (5 and (b) Fe Lo3 (6.5 < k < 12 A!) EXELFS from as-quenched
FegAl and after annealing in-situ at 300 C for 10 minutes
and 30 minutes.

Change in 1nn EXELFS amplitudes as function of annealing
time at 300 C for piston-anvil quenched Fe3Al sample.

Temperature dependence of magnitude of FT of Al K
EXELFS (5 (b) after annealing at 300 C for 30 minutes.

Temperature dependence of magnitude of FT of Fe Log
EXELFS (6.5 (b) after annealing at 300 C for 30 minutes.

Einstein model fits to Al K EXELFS 1nn MSRD data from as-

quenched FesAl and after annealing at 300 C for 30 minutes.

Einstein model fits to Fe Lo3 EXELFS 1nn MSRD data from
as-quenched FesAl and after annealing at 300 C for 30
minutes.

Phase diagram for Ni-Al (Massalski, 1986).
L12 ordered structure of NisAl.

Magnitude of theoretical (a) Al K and (b) Ni Lag EXELFS
from 1nn shell of completely disordered and perfectly
ordered NisAl.

EELS measurements of (a) Al K and (b) Ni L edges from
NigAl.

(a) Al K and (b) Ni Log EXELFS from as-evaporated Ni3Al
at 105 K.

178

180

181

182

183

187

188

189

190

192

193
195

196

197

Figure 6.18.

Figure 6.19.

Figure 6.20.

Figure 6.21.

Figure 6.22.

Figure 6.23.

Figure 6.24.

Figure 6.25.

Figure 6.26.

Figure 6.27.

Figure 6.28.

Figure 6.29.

XVii

Magnitude of FT of experimental (a) Al K (4and (b) Ni Log (8.5 < k < 12.5 A!) EXELFS from as-
evaporated NisAl and after annealing in-situ at 150 C for

70 minutes.

Change in inn EXELFS amplitudes as function of annealing 199
time at 150 C for as-evaporated Ni3AIl sample.
Temperature dependence of magnitude of FT of Al K 200
EXELFS (4 (b) after annealing at 300 C for 60 minutes.

Temperature dependence of magnitude of FT of Ni La3 201
EXELFS (8.5 < k < 12.5 A"') from (a) as-evaporated NisAl
and (b) after annealing at 300 C for 60 minutes.

Einstein model fits to AlK EXELFS 1nn MSRD data from as- 202

evaporated NisAl and after annealing at 300 C for 60

minutes.

Einstein model fits to Ni Log EXELFS 1nn MSRD data from 203
as-evaporated NisAl and after annealing at 300 C for 60
minutes.

EELS measurements from (a) evaporated nanocrystalline 207

Pd at 105 K and (b) electropolished bulk Pd at 98 K.

Phase diagram for Pd-C (Massalski, 1986). 208

Temperature-dependence of magnitude of FT of Pd Mas 210
EXELFS (10.25 < k < 14.5 A“') from (a) as-evaporated
nanocrystalline Pd and after annealing in-situ at 550 C to grow
grains.

Change in 1nn MSRD for EXELFS relative to EXELFS at 211
105 K from as-evaporated nanocrystalline Pd and after
annealing in-situ at 550 C to grow grains.

Magnitude of FT of Pd Mas EXELFS (10.25 < k < 14.5 At) 212
from as-evaporated nanocrystalline Pd and after annealing

in-situ at 550 C to grow grains.

Magnitude of FT of Pd Mas EXELFS from partially 213

compacted powder of Pd nanocrystals and bulk Pd foil.

Figure 6.30.

Figure 6.31.

Figure 6.32.
Figure 6.33.

Figure 6.34.

Figure 6.35.

Figure 6.36.

Figure A.1.

Figure A.2.

Figure A.3.

Figure A.4.

Figure A.5.

Figure A.6.

Figure A.7.
Figure A.8.
Figure A.9.

xviii

Magnitude of FT of k? weighted EXAFS above Pd edge for
coarse-grained Pd foil, compacted nanocrystalline Pd, and
powder of uncompacted Pd nanocrystals (Eastman et al.,
1992).

EELS measurements of Ti L, O K, and Ti K edges from as-
prepared nanocrystalline TiOo.

Theoretical Ti K EXELFS from inn shell of TiOo.

Magnitude of FT of theoretical Ti K EXELFS from inn shell
of TiOo.

Ti K EXELFS from as-prepared nanocrystalline TiOo at
105 K.

Magnitude of FT of experimental Ti K EXELFS (7 A-1) from as-prepared nanocrystalline TiO» and after
annealing at 900 C for 11 hours to grow grains.

Change in 1nn MSRD for Ti K EXELFS relative to EXELFS
at 105 K from as-prepared nanocrystalline TiO2 and after
annealing at 900 C for 11 hours to grow grains.

Hartree-Slater atomic potential and 1s wavefunction for O
atom in its ground state.

Hartree-Slater atomic potential and 1s wavefunction for Al
atom in its ground state.

Hartree-Slater atomic potential and 1s wavefunction for Ti
atom in its ground state.

Hartree-Slater atomic potential along with 2s and 2p
wavefunctions for Fe atom in its ground state.

Hartree-Slater atomic potential along with 2s and 2p
wavefunctions for Ni atom in its ground state.

Hartree-Slater atomic potential along with 3s, 3p, and 3d
wavefunctions for Pd atom in its ground state.

Energy-differential cross section of O K edge.
Energy-differential cross section of Al K edge.

Energy-differential cross section of Ti K edge.

215

217

218
218

220

221

226

227

228

229

230

231

233
234
235

Figure A.10.
Figure A.11.
Figure A.12.

Figure A.13.

Figure A.14.
Figure A.15.
Figure A.16.
Figure A.17.
Figure A.18.
Figure A.19.
Figure A.20.
Figure A.21.
Figure A.22.

Figure A.23.

Figure A.24.

Figure A.25.

Figure A.26.

Figure A.27.

Figure A.28.

xix

Energy-differential cross section of Fe L edge.
Energy-differential cross section of Ni L edge.
Energy-differential cross section of Pd M edge.

Partial wave and corresponding free wave for relaxed C
atom with 1s core hole.

Central atom phase shift for C K edge.

Magnitude of backscattering amplitude for C neighbors.
Phase of backscattering amplitude for C neighbors.
Central atom phase shift for Al K edge.

Central atom phase shift for Ti K edge.

Central atom phase shifts for Fe L edge.

Central atom phase shifts for Ni L edge.

Central atom phase shifts for Pd M edge.

Magnitude of backscattering amplitude for O and Fe
neighbors.

Magnitude of backscattering amplitude for Al and Ni
neighbors.

Magnitude of backscattering amplitude for Ti and Pd
neighbors.

Phase of backscattering amplitude for O, Al, Ti, Fe, Ni, and
Pd neighbors.

Hartree-Slater calculations of central atom phase shifts for
K edges of very light elements not listed in Teo and Lee
(1979).

Hartree-Slater calculations of central atom phase shifts for
Mas edges of elements with 32 < Z < 38.

Hartree-Slater calculations of central atom phase shifts for
Mas edges of elements with 39 < Z < 48.

236
237
238
242

243
244
245
246
247
248
249
250
251

252

253

254

255

256

257

List of Tables

Table 1.1.

Table 3.1.

Table 5.1.

Table 6.1.

Table 6.2.

Table 6.3.

Page

Important advantages and disadvantages of EXELFS vs. 7
EXAFS.

Electrolytic solutions and approximate polishing temperatures 44
used to prepare thin foils of Al, Fe, Pd, and Feg3Al.

Interatomic (Born-von Karman) force constants (in N/m) for 161
the first several near-neighbor shells in Al (Cowley, 1974),

Fe (Minkiewicz et al., 1967), and Pd (Miiler and Brockhouse,
1971).

Average number of 1nn and 2nn Fe atoms surrounding Al 176
and Fe atoms in completely disordered and perfectly ordered
FesAl.

Fraction of each type of 1nn bond in completely disordered 185
and perfectly ordered FesAl (or NigAl).

Average number of 1nn Ni atoms surrounding Al and Ni 193
atoms in completely disordered and perfectly ordered NigAl.

1 Historical Introduction

§1.1 and §1.2 review the history of electron energy loss spectrometry
(EELS) and the history of extended x-ray absorption fine structure (EXAFS),
respectively. §1.3 introduces extended electron energy loss fine structure
(EXELFS) and discusses some practical differences between EXELFS and
EXAFS. §1.4 schematically explains the physical origin of extended fine
structure. §1.5 discusses applications of extended fine structure in materials

science.

1.1 Electron Energy Loss Spectrometry (EELS)

The history of electron energy loss spectrometry dates back to the work of
Franck and Hertz during the years 1914 to 1920. They showed that when a
fast-moving electron collides with an atom or molecule in a gas, it bounces off
with only a very small loss of kinetic energy, unless it has enough energy to
raise the atom or molecule to an excited electronic state, or to ionize the atom or
molecule.

The first report on the characteristic energy losses of electrons in solids
was made by Becker in an abstract printed in 1924. In his abstract, Becker
briefly described the energy distribution of electrons which were dispersed by a
magnetic field onto photographic film after being reflected from solid targets. A
more quantitative study on the energy losses of electrons reflected from the
surface of a solid was published by Rudberg in 1930. Rudberg prepared
samples of various metals and oxides in-situ by vacuum evaporation
immediately before his measurements.

In 1941, Ruthemann was the first to publish the energy spectrum of

electrons transmitted through thin solid specimens. In order to achieve

transmission, Ruthemann used incident electrons with energies of several keV.
His energy-loss spectrum from a thin film of aluminum revealed a series of
peaks which were later attributed by Bohm and Pines (1951) to multiple
plasmon excitations. Ruthemann (1942) also recorded an energy loss
spectrum from a thin film of collodion which showed the K shell ionization edge
from carbon.

Since these early measurements of electron energy losses, electron
energy loss spectrometry (EELS) has developed into an important technique for
materials characterization. Electron energy loss spectrometers are now
common analytical attachments to transmission electron microscopes. A
comprehensive text on the subject of EELS in the electron microscope was
published by Egerton in 1986. An up-to-date review of the applications of EELS
in materials science was given in the book by Disko et al. (1992).

EELS is well-known as a highly sensitive tool for elemental
microanalysis. There is no theoretical lower limit for the mass fraction one can
detect with EELS (Kruit, 1986). Recently, Atwater and Ahn (1991) used EELS in
the reflection geometry for the in-situ elemental analysis of semiconductor
surfaces during molecular-beam epitaxy. Alternatively, instead of using the
spectrometer to display a spectrum, the energy-selecting capabilities of the
spectrometer can be combined with the imaging capabilities of the microscope
to obtain energy-filtered images (Shuman and Somlyo, 1981).

Today, in addition to the capability of EELS for elemental microanalysis,
there is an increasing awareness that EELS can provide information about the
electronic and atomic structure of materials. Recently, Pearson et al. (1989)
used EELS measurements of near-edge fine structure to determine the

electronic occupancy of d states in transition metals. The present thesis uses

EELS measurements of extended fine structure to probe the local atomic

structure in metals and alloys.

1.2 Extended X-ray Absorption Fine Structure (EXAFS)

The first reports of fine structure on the high-energy side of ionization
edges were made by Fricke (1920) and Hertz (1920) using x-ray absorption
measurements. The structure that they observed was confined to strong
features within a few tens of electron volts (eV) of the edge onsets, in what today
is called the "near-edge" regime. These near-edge features were readily
attributed to bound excited electronic states using the theory of Kossel (1920).
Later, as experimental methods improved, the fine structure was observed to
extend up to several hundreds of eV past the edge. These "extended"
oscillations, now called EXAFS (extended x-ray absorption fine structure),
required a new physical explanation.

The first theory explaining the EXAFS was proposed by Kronig in 1931.
Kronig suggested that the structure could be attributed to variations in the
density of electronic states predicted by the zone theory of solids. This
description became known as a long-range order (LRO) theory of EXAFS
because it depended upon the periodicity of the solid. Kronig (1932) also
proposed a short-range order (SRO) theory to explain the observation of
EXAFS in molecules. SRO theories attributed EXAFS to variations in the final
state wavefunction caused by backscattering of the photoelectron from
neighboring atoms. Although LRO theories could not explain the EXAFS found
experimentally in molecules and amorphous solids, for many years confusion

existed as to which description, LRO or SRO, was appropriate (Azaroff, 1963).

The work of Sayers et al. (1971) elevated EXAFS from an obscure
phenomenon to a useful structural tool. Using single-scattering SRO theory,
they realized that a Fourier analysis of the EXAFS with respect to the
photoelectron wave number should peak at distances corresponding to
nearest-neighbor coordination shells of atoms. By separating the contributions
from the various atomic shells, the Fourier analysis technique made possible
the direct extraction of structural information. It suddenly became clear that
EXAFS could be used as a quantitative probe of SRO.

Following the work of Sayers et al., rapid advances were made in the
theory of EXAFS (Schiach, 1973; Stern, 1974; Ashley and Doniach, 1975; Lee
and Pendry, 1975). It quickly became well-established that single-scattering
SRO theory was an adequate description of EXAFS in most circumstances.

Meanwhile, the development of synchrotron radiation sources greatly
improved the statistical quality of experimental EXAFS data (Kincaid and
Eisenberger, 1975). Synchrotron sources became typically at least three orders
of magnitude more intense than standard x-ray tube sources, and now they are
even more intense.

These improvements in both theory and experiment made EXAFS a
practical tool for probing the atomic structure of materials. Since then, a large
number of EXAFS experiments have been performed. A recent review of
EXAFS and its applications was given in the book edited by Koningsberger and
Prins (1988).

1.3 Extended Electron Energy Loss Fine Structure (EXELFS)
Although the vast majority of extended fine structure measurements are

presently being made using x-ray absorption, it is also possible to measure

extended fine structure using EELS (Ritsko et al., 1974; Leapman and Cosslett,
1976; Colliex et al., 1976; Kincaid et al., 1978; Teo and Joy, 1981). When
extended fine structure is measured using EELS, the technique is called
EXELFS (extended electron energy loss fine structure). Figure 1.1 contains the
EELS spectrum from pure aluminum which clearly shows the EXELFS above
the ionization edge.

EXAFS and EXELFS originate from the same physical mechanism; they
are both caused by the backscattering of the excited electron from neighboring
atoms. The difference is that EXAFS uses a photoabsorption process which
completely transfers the x-ray photon energy to the excitation of the
photoelectron, while EXELFS involves partial energy transfers from the high-
energy incident electron beam. From this perspective, EXAFS is similar to an
infrared absorption experiment, while EXELFS is more analogous to a Raman
scattering experiment.

While EXAFS and EXELFS are basically the same physical
phenomenon, there are many significant differences between the experimental
techniques which are used to measure them. A list of important advantages
and disadvantages of EXELFS vs. EXAFS is given in Table 1.1. Unlike EXAFS
experiments which utilize x-rays from synchrotron or bremsstrahlung radiation
sources, EXELFS experiments are usually performed using the electron beam
in a transmission electron microscope (TEM). This makes it easy to combine
EXELFS experiments with the imaging, diffraction, and analytical capabilities of
the TEM. Other important advantages of the EXELFS technique are its
increased spatial resolution and its ability to measure extended fine structure in

elements with very low atomic number. Disadvantages of the EXELFS

5x10 TOT TTT Try ee rt

Intensity

o_o
Kee

roster tiny Litistepur teeta triritriri tire tia

TrTTpE rrr?

9) retest pti tits ti tt tt

1600 1700 1800 1900 2000
Energy Loss (eV)

Figure 1.1. EELS measurement of Al K edge from foil of pure aluminum.

technique include a greater likelihood of overlapping edges and the need for

very thin samples in order to avoid large multiple inelastic scattering effects.

van f EXELES v XAE

1. EXELFS can measure core edge fine structure in lower atomic
number elements (edges < 5 keV) than EXAFS (edges > 3 keV).

2. Very small electron probes can be used, allowing inhomogeneous
samples to be studied.

3. The instrumentation is more accessible and less expensive than
synchrotron sources.

4. EXELFS can be combined with electron diffraction and imaging in
the transmission electron microscope.

1. Overlapping edges are more likely to limit the data range or
complicate the analysis.

2. Samples must be very thin to limit multiple inelastic scattering
effects.
3. The electron beam may heat the samples, but this is shown not

to be a problem in §3.3.

Table 1.1. Important advantages and disadvantages of EXELFS vs. EXAFS.

Historically, EXELFS studies have been inhibited by the inherent
inefficiency of serial detection systems. With serial detectors, EXELFS data
have suffered from inadequate signal-to-noise ratios, resulting in very limited
data ranges in k-space (Csillag et al., 1981). The recent development of
parallel detectors has overcome this problem (Krivanek et al., 1987). Parallel

detection of a spectrum with 1000 discrete data channels is, in principle,

roughly 1000 times more efficient than serial detection of the same spectrum
would be.

Another factor inhibiting EXELFS studies has been their limitation mainly
to K edges. K-edge EXELFS is easy to interpret because of its simple structure.
L and M edges, on the other hand, are complicated by the variety of possible
transitions. The present thesis shows that useful EXELFS information can be
extracted from L23 (Leapman et al., 1982) and M4s edges, in spite of their more
complicated structure. The use of Lo3, M45 , and other similar edges opens up

most of the periodic table to EXELFS investigations.

1.4 Physical Origin of Extended Fine Structure

Using single-scattering SRO theory, the origin of extended fine structure
is illustrated schematically in Figure 1.2. The solid circles represent atomic
cores, and the rings represent electron-wave crests. An electron is excited from
the central atom core and can be thought of as an outgoing spherical wave
(solid rings). Note that the phase of the outgoing wave in Figure 1.2 is defined
so that there is a crest at the central atom core. The energy of the outgoing
wave is the energy loss in excess of the ionization energy. Some of the
outgoing wave is elastically scattered (dashed rings) from neighboring atoms.
From Fermi's Golden Rule, we know that it is only the interference in the region
of the initial state (i.e., at the central atom core) which changes the excitation
probability, and hence modifies the edge shape. One can visualize the
interference between outgoing and scattered waves at the central atom as
varying periodically with the wavelength of the excited electron (i.e., with the
distance between concentric rings in Figure 1.2). If constructive interference

occurs at the central atom, as in Figure 1.2a, then the excitation probability

increases, creating positive extended fine structure. For destructive
interference, shown in Figure 1.2b, the extended fine structure is negative.
Extended fine structure is thus a quantum interference phenomenon dependent
on the amplitude and phase of the backscattering from the local environment

surrounding the ionized atom.

(a) constructive (b) destructive

Figure 1.2. Schematic illustration of (a) constructive and (b) destructive
interference at the central atom.

1.5 Applications of EXELFS in Materials Science

Extended fine structure is useful because it can provide local information
which is difficult to obtain by diffraction techniques. Because of their sensitivity
to LRO, diffraction techniques are most powerful when applied to crystalline
materials. In contrast, extended fine structure is sensitive only to SRO.
Regardless of the amount of LRO in a material, extended fine structure can be
used to determine the identities and positions of nearest-neighbor atoms
surrounding the probe atom.

An important feature of extended fine structure is its ability to probe

independently the environments of different atomic species. This feature makes

EXELFS appropriate for studies of the atomic structure of alloys, especially
alloys with high concentrations of the probe species. To my knowledge, no
EXELFS studies of dilute alloys have been made; such experiments would
require extremely good signal-to-noise ratios, which are more easily achieved
with EXAFS using a synchrotron source than with EXELFS. This thesis work
shows, however, that EXELFS can be used to observe chemical short-range
order (CSRO) in non-dilute alloys. Measurements are presented in §6.1 which
show differences in CSRO between as-quenched and annealed alloys of Fe3Al
and Nig3Al.

Extended fine structure measurements are sensitive to disorder in the
local structure surrounding the probe atom. The disorder can be either
structural or vibrational in origin. Historically, extended fine structure has been
considered to be particularly suited to study the structural disorder in
amorphous materials (Sayers et al., 1971). The primary goal of such studies is
the determination of partial radial distribution functions (RDFs). The problem is
that it is difficult to differentiate between a reduction in coordination number and
an increase in disorder without assumptions about the partial RDFs in the first
place (Lee et al., 1981). Thus, in order to determine partial RDFs from
disordered systems, extended fine structure must be used in conjunction with
other techniques, such as x-ray and neutron RDF studies. There have been
many good reviews of the use of extended fine structure to study amorphous
materials (Lee et al., 1981; Gurman, 1982; Hayes and Boyce, 1982; Stearns
and Stearns, 1986; Crozier et al., 1988).

Recently the structural disorder in nanocrystalline materials has become
a topic of interest (Gleiter, 1989). EXAFS measurements have been used to

support the claim that the grain boundaries in some nanocrystalline materials

are highly disordered (Haubold et al., 1989). In this thesis work, EXELFS was
used to investigate the structural disorder in nanocrystalline Pd and TiOo.
Results presented in §6.2 indicate greater amounts of structural disorder are
present in the nanocrystalline Pd and TiOs than in large-grained materials.

Vibrational disorder results from the thermal vibrations of atoms in a
material. Extended fine structure measurements of vibrational disorder are
usually characterized with temperature-dependent mean-square relative
displacement (MSRD) data. Temperature-dependent MSRD data can be fit to
"local" Debye temperatures using the correlated Debye model (Beni and
Platzman, 1976). Local Debye temperatures indicate the stiffness of bonds
between the probe atom and its nearest-neighbor atoms. Data presented in
§5.3 give local Debye temperatures for the elemental metals Al, Fe, and Pd,
which correlate well with published force constant models derived from inelastic
neutron scattering data.

The vibrational entropy of a material can be estimated by a weighted
average of its local Debye temperatures. An important application of this thesis
was the measurement of the differences in vibrational entropy between
chemically disordered and ordered intermetallic alloys. EXELFS data
presented in §6.1 indicate that the differences in vibrational entropy between
chemically disordered and ordered alloys of Fe3Al and Ni3Al are almost as
large as the entropy of mixing.

In summary, previous applications of EXELFS to materials science have
been mostly meager and exploratory. This work is the first to apply the method

to contemporary problems.

2 Electron-Atom Scattering Theory

This chapter discusses the electron-atom scattering theory that underpins
the EXELFS technique. EXELFS utilizes both the inelastic and elastic scattering
of electrons by atoms.

The inelastic scattering of fast electrons by atoms is reviewed in §2.1.
§2.1.1 describes the kinematics of the problem. §2.1.2 outlines the calculation of
ionization cross sections in the Born approximation. Lastly, §2.1.3 discusses the
deconvolution of energy-loss spectra to remove multiple inelastic scattering.

The elastic scattering of electrons by atoms, and how it causes the
extended fine structure phenomenon, is reviewed in §2.2. First, §2.2.1
determines the phase shifts and scattering amplitudes associated with elastic
scattering. §2.2.2 then discusses the theory of extended fine structure. The
equation used to interpret extended fine structure is presented, and its derivation

is discussed in detail.

2.1 Inelastic Scattering of Fast Electrons

When electrons collide inelastically with atoms, the incident electrons may
be classified as either “fast or "slow" relative to the mean orbital velocity of the
atomic electrons involved in the interaction. For example, incident electrons with
1 keV of kinetic energy are fast with respect to any ionizations of He (22 eV), but
they are not fast with respect to the K-shell ionization of Al (1.56 keV). The
expression for the scattering cross section of fast collisions may be factored into
two distinct parts, one dealing with the incident electron only and the other
dealing with the target only. Because the characteristics of the incident electron

can be factored out, the study of fast collisions is effectively that of the scatterer

properties (Inokuti, 1971).

Inelastic scattering occurs when the incident electron interacts with either
outer-shell or inner-shell atomic electrons. Interaction with outer-shell electrons
can result in either a single-electron excitation or a collective excitation of
electrons in the specimen. In a single-electron excitation, a valence electron
makes a transition to a delocalized higher-energy state (Figure 2.1b). A
collective excitation can be described by the creation of a plasmon
pseudoparticle which represents an oscillation of the valence-electron density.
Interaction with an inner-shell electron results in the excitation of a core electron
to a delocalized higher-energy state (Figure 2.1c). These inner-shell interactions
cause the core edges observed in EELS, typically at energy losses of hundreds

or thousands of eV.

(a) (b) (c)

Figure 2.1. Classical picture of electron scattering by a single atom (carbon).
Gray dots represent atomic nuclei. Black dots represent electrons.
Rings represent classical electron orbitals. Lines represent
electron trajectories. (a) Elastic scattering caused by Coulomb
attraction of nucleus. Inelastic scattering from Coulomb repulsion
by (b) outer- and (c) inner-shell electrons (After Egerton, 1986).

2.1.1. Kinematics

Consider the scattering of a fast electron from an atom. The momentum
vectors of the electron before and after the collision are defined to be pj = Ak; and
pr = fikt, respectively. By conservation of momentum, the momentum supplied to
the atom is iq = hk; - Aky, where q is known as the scattering wavevector. The
vector relationship between q, kj, and kr is illustrated in Figure 2.2. For inelastic
scattering (kj ky), the magnitude, q, of the scattering wavevector depends on
both the scattering angle, 8, and the energy loss, E. The relationship between q,

8, and E is derived using conservation of both momentum and energy.

q min

Figure 2.2. Vector relationship of q, kj, and k¢ due to conservation of
. momentum (After Egerton, 1986).

Applying the "law of cosines" to the vector triangle in Figure 2.2 gives

q2 = k? +k? - 2kikcos0 (2.1a)

or
g2 = (ki- ky)? + 4kikisin® (9%) (2.1b)

Conservation of energy gives
W;- E = Ws (2.2)

where Wo and W, are the total energies of the high-energy electron before and

after the collision, respectively. From relativistic kinematics we know that the

total energy of an electron is given by W = [mec# + (Ak)@02]"/2, where Me is the

rest mass of the electron, k is its wavevector, and c is the speed of light. Using

this expression to substitute for Wo and W, in Equation (2.2) and solving for k?

gives

KPa k?{1- 22 4 =, (2.3)
P¥i (pic)

where vj is the speed of the incident electron. Using Equation (2.3) to substitute

for ky in Equation (2.1b) gives

V2
. 2E E?
+ sh (94 “oy * Es] (2.4)

Equation (2.4) gives q as a function of qand E. Since typically beam energies
are in the hundreds of keV while energy losses, E, are at most a few keV, we can

assume that E << piv; < pic. Therefore, we can make the approximation that

q? =k? (s) + esto) (2.5)

Furthermore, if 8 << 1, then

a? = ke [@e + 6°] (2.6)

where 6¢ = - It is shown geometrically in Figure 2.2 that for a given energy
Vi

loss, the minimum length, Gmin, for the scattering wavevector is at @ = 0. From

Equation (2.6) we see that min = Ki9e.

2.1.2 lonization Cross Sections

Energy loss experiments, in effect, measure the energy-differential cross
section, do/dE. In this section, the theoretical calculation of do/dE for the
ionization of an atom is reviewed.

In experiments, the scattered electrons are generally collected over a
range of 6. For our calculations, however, it is more convenient to use q, rather
than 6, as an independent variable. Therefore, we use Equation (2.6) to convert

from @ to q. The energy-differential cross section, do/dE, is then obtained by

dqdE

integrating the double-differential cross section over the appropriate range

in q.

Within the framework of nonrelativistic one-electron wavefunctions, we
assume that the collision affects only the wavefunction of the atomic electron

directly involved in the transition. The Hamiltonian for the system is then

2 2 2
H= Sy Pa iy wrg) + —o— (2.7)

where p and pa are the momenta for the incident and atomic electrons, r and ra
are their position vectors, and V is the atomic potential. The last term,

e*/ |r—r,|, is the interaction potential which perturbs the system during the
collision.

Before and after the collision, the system is assumed to be in energy
eigenstates of the unperturbed Hamiltonian, |k;)|Nolo) and |k;)|nl), respectively.

|k;) and |k;) are the initial and final planewave states of the incident electron.

|nglo) is the ground state of the atomic electron, and |nl) are excited states of

the atomic electron.
In the first Born approximation, in which the influence of the incident
particle upon the atom is regarded as a sudden and weak perturbation, the

differential cross section for the inelastic collision is

do 4e4m2k;
o ea > (nl|exp(iq er, Nolo) (2.8)
nl

dQ f4q4k

where dQ is the element of solid angle for the scattered electrons (Bethe, 1930;

Inokuti, 1971). When the final states of the atomic electron are unbound

continuum states, |el), rather than bound discrete states, |nl), the sum over the

final states is replaced with a density of final states, p(e), and we obtain a result

which is differential with respect to energy:

2 4e4m2k
aE n4q4k; i, Pe) | ellexp(iqera \Incto)/ (2.9)

2nqdq
Ke

Rewriting the element of solid angle with dQ = 2xsinéd6 = gives the

following expression for the cross section which is differential with respect to q

and E:

d°o 8xe4 me

dadE a4q3k?

P(e) [ellexp(iq ers }[Molo \f (2.10)

The energy-differential cross section is obtained by integrating Equation (2.10)

over q:

d 8xe4 més
cE nak? ple D> T sk ellexp(iqera)|Nolo \F dq (2.11)
Qmin

The matrix elements in Equation (2.11) are evaluated by expanding the operator
exp(iq*ra) in terms of spherical Bessel functions. Integrals over angular
coordinates are expressed as Wigner 3-j coefficients leaving a radial integral to

be evaluated numerically (Manson, 1972; Leapman, et al., 1980).

2.1.3. Deconvolution of Multiple Inelastic Scattering
This section reviews the deconvolution of energy-loss spectra to remove

multiple inelastic scattering. Multiple inelastic scattering can drastically affect the

overall shape of an ionization edge. Although deconvolution is reviewed here,
deconvolution is not necessarily required prior to EXELFS analysis. As shown in
§4.3, if the sample is reasonably thin, EXELFS oscillations are not radically
altered by multiple inelastic scattering.

While the probability of a transmitted electron causing more than one
inner-shell ionization is generally negligible, there is a significant chance that one
or more outer-shell excitations will occur in addition to the inner-shell ionization.
These additional outer-shell excitations change the observed shape of the inner-
shell edge. The edge that we measure is basically a convolution of the low-loss
distribution with the single inelastic scattering profile of the edge.

The effects of multiple inelastic scattering are commonly removed by
Fourier-transform methods of deconvolution. There are two schemes for Fourier
deconvolution: the Fourier-log method and the Fourier-ratio method. First, let us
discuss the Fourier-log method (Johnson and Spence, 1974). Assuming
independent scattering events that follow Poisson statistics, the measured

intensity in an energy loss spectrum, I{E), can be expressed as

I(E) = Z(E) ce) aa + Dik

S(E)* S(E) + - (2.12)
where Z(E) is the zero-loss peak, Ip is the area under the zero-loss peak, 5(E) is
a unit area delta function, S(E) is the single scattering distribution, and * denotes
convolution. Taking the Fourier transform of Equation (2.12) and solving for the

Fourier transform of the single scattering distribution gives

ay I'(v)
S'(v) = lo i 5 =| (2.13)

where primes denote the Fourier transforms.

Taking the inverse Fourier transform of Equation (2.13), in principle, gives
the single scattering intensity which is unbroadened by the instrumental
resolution. In practice, however, I'(v) contains noise, and the noise begins to
dominate the signal at high frequencies. Dividing I'(u) by Z'(u) preferentially
amplifies the high-frequency noise because Z'(v) generally falls with increasing v.
Thus, the direct use of Equation (2.13) results in the extreme amplification of
high-frequency noise in a spectrum. This noise amplification can be reduced by
multiplying S'(v) by a "reconvolution" function g(v) which has unit area and falls
rapidly with increasing v (Egerton, 1986). g(v) is basically a v-space filter.

Note that some ambiguity exists in Equation (2.13) because the logarithm
of a complex number is a multivalued function. In particular, In(z) = In(r) + i6+
i2mn, where z = r exp(i0) and n may be any integer. In practice, this ambiguity
becomes a problem only when the sample thickness, t, is about x times greater
than the mean free path for inelastic scattering, A (Spence, 1979).

The second method for spectrum deconvolution, Fourier-ratio method,
divides the energy-loss spectrum into the low-loss and core-loss regions. First,
the pre-edge background is subtracted to isolate the core edge. Deconvolution is
then accomplished by dividing the Fourier transform of the core edge by that of
the low-loss region. Unlike the Fourier-log method, which deconvolutes an entire
EELS spectrum, the Fourier-ratio method can remove multiple inelastic scattering
only from core edges.

in theory, deconvolution with respect to energy alone assumes that all of
the scattered electrons have been collected. in practice, only those electron
scattered within the spectrometer entrance aperture are collected. However,

recently Egerton and Wang (1989) have shown that the effect of the collection

aperture on deconvolution is relatively limited. Another assumption of the
previous deconvolution procedures is that the thickness of the sample is
constant. Johnson and Spence (1974) calculated that very little unwanted
multiple inelastic scattering would remain after the deconvolution of a slightly

wedge-shaped sample.

2.2 Elastic Scattering

An important difference between inelastic and elastic scattering is that
inelastic scattering is incoherent while elastic scattering is coherent. Because of
its coherency, elastic scattering results in interference effects. Diffraction is one
example of an interference effect caused by elastic scattering. Extended fine
structure is another.

Extended fine structure is an interference phenomenon caused by the
elastic scattering of an ionized electron by neighboring atomic cores. When an
electron is ionized from an isolated atom, the final state can be represented by an
outgoing electron-wave with spherical symmetry. In condensed matter, however,
the final state is perturbed by backscattering from the surrounding environment.
Elastically backscattered electron-waves coherently modify the amplitude of the
outgoing wave in the region of the initial atomic state, thus changing the

probability of excitation in the first place.

2.2.1. Phase Shifts and Scattering Amplitudes

In order to study quantitatively the extended fine structure phenomenon,
we must first understand phase shifts and scattering amplitudes. The following
discussion of phase shifts largely follows the one in Cohen-Tannoudji et al.,

(1977). In acentral (i.e., spherically symmetric) potential V(r), there exists

stationary states with well-defined angular momentum, i.e., eigenstates common
to the H, L?, and L,, where H is the Hamiltonian, and L is the orbital angular
momentum of the particle. The wave functions associated with these states are
called partial waves. Partial waves can be written as 4,,,,(r,0,0), where
h2k2/2me, | (1+ 1)#2, and mh are respectively the eigenvalues of H, L2, and L,.
The angular dependence of the partial wave 9,,,,(r,8,) is always given by the
spherical harmonic Yjm(8,6). However, the radial dependence of the partial wave
is influenced by the central potential V(r).

Consider the case where we are dealing with a free particle, i.e., V(r) = 0.

The stationary states with well-defined angular momentum are then called free

spherical waves of®) (1,0, 4). Free spherical waves are given by

ofG) (7.8.0) = jk) Yin(0.6) (2.14)

where ji(kr) is a spherical Bessel function. The asymptotic behavior of jy(kr) is

given by

_ exp(—ikr) exp(il 2/2) — exp(ikr)exp(—il 2/2)

2.15
kr—yoo 2ikr (

Therefore, the free spherical wave o{9) (r,8, 4) behaves asymptotically as the

superposition of an incoming wave exp(-ikr)/r and an outgoing wave exp(ikr)/r,
where the phase difference between the two waves is equal to In.

Assuming that V(r) = 0 for r > ro, the partial wave 9,,,,(r,8,0) also behaves
asymptotically as the superposition of an incoming wave exp(-ikr)/r and an

outgoing wave exp(ikr)/r, with a phase difference between the two waves.

However, the phase difference of the partial wave is not the same as that of the
free spherical wave. The potential V(r) introduces an additional phase shift 26)(k)
which is the only difference between the asymptotic behavior of 9,,,,(r,8,0) and
that of o{°) (r,8,o). The phase shift 28,(k) depends on both the orbital angular
momentum of the wave, through 1, and the energy of the wave, through k.

The phase shift can be interpreted in the following way. Suppose we have
an incoming spherical wave Y},(6,) exp(-ikr)/r. This incoming wave is perturbed
when it enters the zone of influence of the potential V(r). After turning back and
leaving the zone of influence, it is transformed into an outgoing wave which has
accumulated a phase shift of 25)(k) relative to the free outgoing wave that would
have resulted if V(r) had been zero. The additional phase factor exp[i28,(k)]
summarizes the total effect of the potential on the particle.

Next, we show how these phase shifts can be used to calculate the
scattering amplitude of a beam of particles with energy #°k*/2me from the central
potential V(r). The problem is illustrated in Figure 2.3. Initially, before a particle
in the beam reaches the influence of V(r), it is represented by the plane wave
state exp(ikz). When the plane wave collides with V(r), the structure and
evolution of the wave are modified in a complicated way. Nevertheless, when the
wave leaves the influence of V(r) it once again takes on a simple form. It
becomes split into a transmitted plane wave exp(ikz) which continues to
propagate along the z-direction and a scattered wave represented by f(0,k)
exp(ikr)/r. f(6,k) is called the scattering amplitude. Thus, for the steady-state
configuration described above, the stationary scattering state w(r,6) will have
asymptotic behavior of the form:

w(r,6) ~ exp(ikz) + f(6,k) exp(ikr)/r (2.16)

exp(ikz)

(a) before collision

f(0,k) exp(ikr)/r

exp(ikz)

(b) after collision

Figure 2.3. Scattering of plane-wave packet from central potential (After
Cohen-Tannoudji et al., 1977).

When V(r) is identically zero, w(r,8@) reduces to the plane wave exp(ikz).

The plane wave exp(ikz) can be expanded in terms of free spherical waves:

exp(ikz) = be Si ane 0°) (r,8) (2.17)

Note that because the plane wave is symmetric with respect to rotations around
the z-axis, its expansion includes only those free spherical waves with m = 0. If

we slowly turn on the potential V(r), then we intuitively expect that the free
spherical waves (9) (r,8) will slowly turn into the corresponding partial waves

Pigo(t,8). Therefore, in general, the stationary scattering state y(r,8) can be

expanded in terms of partial waves:
T yO.
w(r,8) = Ise Yi Y42(21+1) O49(r.9) (2.18)
1=0

Using the fact that, except for the additional phase shifts 26,(k), the asymptotic

behavior of partial waves is identical to that of free spherical waves, we find:

(2.19)

.[—oo

~ikr giln/2 ikr ~iln/2 12, (k) |

wit) ~ - Si ae(2i+4) ¥i9(0) 2
1=0

Rewriting the phase factor e2i51(k) = 1 + 2) ei8(K) sin ,(k), rearranging the terms,

and recognizing the asymptotic expansion of the plane wave exp(ikz):

rr r

w(t.8) ~ explikz) + se La) ei8i(k) sin &(k) Yio(®) (2.20)
“ 1=0

By comparing Equation (2.20) with Equation (2.16), we arrive at an expression

for the scattering amplitude f(@,k) in terms of the phase shifts 6(k):

4n(21+1) el51(k) sin &(k) Yio(@)

~~
eel
ll
Ms

(2141) ei5i(k) sin 8,(k) P\(cosé) (2.21)

iH
Me:

where P)(cos6) are the Legendre polynomials. Finally, note that because f(0,k) is

a complex function, it is often expressed in polar notation:
f(0,k) = |f(8,k)| exp[in(0,k)] (2.22)

2.2.2. Theory of Extended Fine Structure

The physical origin of extended fine structure was briefly explained in §1.4.
This section contains a more quantitative discussion of the theory.

Because the dipole rule does not strictly hold for EXELFS, transitions to
any angular-momentum channel are possible. However, for small scattering
angles the dipole approximation is generally valid. The dipole rule states that if Ip
is the angular-momentum quantum number of the initial state, then only
transitions to final states with angular-momentum quantum numbers lo + 1 are
allowed. Furthermore, as shown by calculations of partial energy-differential
cross sections presented in chapter 4, the transition to 19 + 1 dominates over that

to lg - 1. The dominance of the lo to 19 + 1 channel allows us to interpret EXELFS

in polycrystalline or amorphous materials with the following equation:

fi(x,k)| S(k) _or. ~267k?
x~(k)=(-1)0*" 5 Hee S) mo ( Me ARY/MK) Q-20)K" sinf2kRj + nj(m,k) + 28,,41(k)] (2.23)
J ]

x(k) represents the extended fine structure oscillations normalized to the non-
oscillatory intensity of the ionization edge. lo is the angular-momentum quantum
number of the initial state. The summation over j is over all atoms neighboring
the central (ionized) atom. The distance from the central atom to neighboring
atom j is denoted by Rj. |f(z,k)| and nj(x,k) are respectively the amplitude and
phase shift of the backscattered wave. 20)(k) is the central atom phase shift.
The factor S(k) approximately takes into account many-body effects such as
"shake up/off" processes at the central atom. The factor exp[-2R;/A(k)] is a
phenomenological term which accounts for the finite lifetime of the excited state,
where A(k) is the inelastic mean free path of the ionized electron. Finally, the
term exp(-20)2k2) is a Debye-Waller type factor due to vibrations between atoms,
where er is the mean-square relative displacement (MSRD) between the central
atom and neighboring atom j.

Equation (2.23) is basically the same as "the EXAFS equation” which is
commonly used to interpret EXAFS oscillations. The most serious approximation
made in Equation (2.23) is the plane-wave approximation. The plane-wave or
small-atom approximation assumes that the outgoing spherical wave can be
approximated by a plane wave in the vicinity of the scattering atom. This
approximation is valid if the effective size of the scattering atom is much smaller
than its distance from the center atom. At high k, say k > 4 A-1, this is generally
true because the electron penetrates deeply into the atom before scattering.
However, at low k the effective size of the atom can be about the same as the
interatomic distance. Therefore, in the low k region the curvature of the outgoing
wave and the finite size of the scattering atom must be taken into account.
Theories of extended fine structure that do not use the plane-wave approximation

are called curved-wave theories.

Another approximation made in Equation (2.23) is the single-scattering
approximation. The single-scattering approximation assumes that the outgoing
wave scatters only once from neighboring atoms before being combined with the
unscattered wave. Multiple-scattering processes are neglected. Like the plane-
wave approximation, the single-scattering approximation is valid in the high-k
region, again say k > 4 A-1. This is because scattering amplitudes generally
decrease with increasing k. In principle, multiple scattering should not have any
effect on extended fine structure oscillations from first nearest-neighbor (1nn)
atoms surrounding the center atom. This is because multiple-scattering path
lengths are always longer than the single-scattering path lengths to and from 1nn
atoms.

Since both the plane-wave and single-scattering approximations are valid
at high k, the use of Equation (2.23) is restricted to the high-k region. It is this
restriction that is responsible for the "extended" in the phrase "extended fine
structure.”

We now present a derivation of Equation (2.23). This derivation closely
follows that of Boland et al. (1982), except that we focus on EXELFS rather than
EXAFS. Although many other derivations of Equation (2.23) are published
(Stern, 1974; Ashley and Doniach, 1975; Lee and Pendry, 1975; Lee, 1976), the
derivation by Boland et al. is especially clear. For an efficient curved-wave
theory of extended fine structure, see either Gurman et al. (1984) or Schaich
(1984).

Since this thesis is concerned with EXELFS, we start with the energy-
differential cross section which is given by Equation (2.11). In the dipole

approximation, the matrix element in Equation (2.11) reduces to
(e(1o +1)iq er,|Nolo), where q is the scattering wavevector, ra is the position

vector of the atomic electron which undergoes the transition, Inglo) is the initial

state, and |e(lo +1)) are final states of energy «. The matrix element is then of

the same form as that for x-ray absorption. Fortunately, the dipole
approximation is generally valid for the experiments performed in this thesis. A
simple condition for the validity of the dipole approximation is that qrmax << 1,
where q is the magnitude of the scattering wavevector and Imax is the radial
extent of the initial core wavefunction. Consider our experiments on the Al K
edge which are performed using 200 keV incident electrons and collection angles
of roughly 5 mrad. For these experiments, q = 1 A-1 and tmax = 0.1 A, $0 Ofmax
= 0.1 << 1. The validity of the dipole approximation for our experiments is shown
more precisely in §A.1. §A.1 calculates the cross sections for excitation into the
various angular momentum channels and shows that dipole transitions to lo+ 1
dominate over non-dipole transitions.

Moreover, as mentioned previously, calculations of partial energy-
differential cross sections presented in chapter 4 show that the lp + 1 transition

dominates over all others. Therefore, the matrix element further reduces to
(e(1o +1)|\q*ra|Nolo). To simplify our notation, we now substitute r for ra, |i) for

|Mlo), and |f) for |e(Io +1)). In this notation, the energy-differential cross section

can be written as

min

do 8re4 m2 a “i2
e = — Pe) S22" Jar (2.24)

where Gmax is the maximum scattering wavelength experimentally collected.
The initial and final states of the system are both eigenfunctions of the

Hamiltonian H:

H =- i ye + U(r) + V(r) (2.25)

where U(r) is the attractive atomic potential primarily felt by the electron in its
initial state, and V(r) is the total scattering potential seen by the final-state
electron. We represent V(r) with a muffin-tin potential, i.e., a sum of
nonoverlapping, spherically symmetric potentials centered around each atomic
site of the alloy. The potential of the center atom seen by the final-state electron
is that of a "relaxed" ion with a core hole. This is because the transit time for the
ejected electron to travel to a neighboring atom and back is much shorter than
the lifetime of the core hole, but it is generally much longer than the relaxation
time for the remaining core electrons (Teo, 1986).

In particular, we consider the scattering from two neighbors about an
atom. This problem is illustrated in Figure 2.4. The atom undergoing ionization
is at the center of the coordinate system and is labeled "c." The two neighbors
labeled "a" and "b" are located respectively at Ra and Rp.

To calculate the matrix element in Equation (2.24), it is necessary to find
the initial and final states. These states must be eigenfunctions of H. At the
large negative energy corresponding to the initial state, the scattering potential
V(r) may be ignored. The resulting Hamiltonian has eigenfunctions which are the
usual core wavefunctions obtained from atomic structure calculations.

For final-state electrons of sufficiently high energy, the attractive atomic
potential U(r), which determined the initial state, becomes negligible. The

resulting Schrédinger equation for the final state is as follows:

(e - H®) |f+) = V |f+) (2.26)

Figure 2.4. Schematic illustration of the final state potential Vir) (After Boland
et al., 1982).

where H? is the free-particle Hamiltonian. This equation is inverted to give the

Lippman-Schwinger equation:

|f+) = |k) + G5 V |f+)
= |k) + G5 T* |k) (2.27)

where (r|k) are the normalized eigenfunctions of H°. Because we want (r|k) to

correspond to the outgoing asymptote of the scattering process, we use the

minus form of the free-particle Green and T operators.

The full T operator can be expanded in terms of operators tj associated
with the individual scattering centers located at r = Rj.

jze j~m jem,mezn

Note that successive scattering by the same potential is not permitted. The first-
order terms in the expansion correspond to single-scattering processes, second-
order terms to double-scattering processes, and so on. Figure 2.5 diagrams the
zero-scattering, single-scattering , and double-scattering processes for our three
atom system. Note that the free-particle Green functions represent free
propagation between two neighboring atom potentials.

One might assume that only processes (a) through (c) would be used in
the single-scattering approximation of extended fine structure. That assumption,
however, would be incorrect. In addition to zero-scattering and single-scattering
processes, the correct single-scattering approximation also includes some select
double-scattering processes as well. In particular, (d) and (e) in Figure 2.5
represent double-scattering processes for which the second scattering center is
the center atom potential. In such processes, the scattering path lengths to the
center atom are identical with those of the single-scattering processes (b) and
(c), respectively. Since it is the path length back to the center atom which is
important for extended fine structure, (d) and (e) must also be included in the
single-scattering approximation.

Thus, in the single-scattering formalism, the terms corresponding to
diagrams (a) through (e) in Figure 2.5 are substituted into the matrix element of

Equation (2.24):

b b b b
a a & a a &
© a Qu
Cc Cc Cc Cc
Gp tpG5 t¢GotiG5
(a) (b) ‘(c) (d)
b b b
ey D 2
a } a @ ag
© © ©
Cc Cc Cc
teGotgGo thG5tzG5 tzG5tpG5

(e) (f) (g)

Figure 2.5. Diagrammatic representations of (a) zero-scattering, (b-c) single-
scattering, and (d-g) double-scattering processes for three-atom
system (After Boland et al., 1982).

(f=laerli) = (klaerli) + D(kltGsaerli) + Dik |eagrrasgerli) (2.29)
J J

where, of course, we have taken the complex conjugate of Equation (2.27). To
proceed further, we must determine the effective values at the origin of all the
matrix elements in the right-hand side of Equation (2.29).

The first matrix element on the right-hand side of Equation (2.29) is
responsible for the unscattered outgoing electron and can be evaluated using the

addition theorem for spherical harmonics to become of the form:

(k|qerli) = M(k) keg (2.30)

where M(k) = (22) 9? 4n(—i) f ig(kr)(r[i)r2dr (2.31)

and k is the direction of propagation of the electron as it originally leaves the

center atom. Intuitively, the term keq makes sense because it means that the

electron is most likely to be ejected in the direction of the scattering vector.

Boland et al. (1982) determines M(k) explicitly for the case where (r|i) is a

hydrogenic wave function.

The single-scattering and double-scattering terms can be expanded:

dX (klt7God eri) = DH Pik ltels)rslGslryaer(rlijdrdr, — (2.32)
J J

¥ (kteGstP Osa eri) =

Assuming that the scattering potentials are due mostly to the core electrons
which are very close to the center of the atoms, the free-particle Green function

may be approximated using:

m_exp(ik|r,—r|)

(1G5 |r) =-5743 mn olkie(s-¥)

\r;-r| anh R;

(2.34)

where kj = kR; is the wavevector of the outgoing electron as it heads towards

atom j. Equation (2.34) is equivalent to the plane-wave or small-atom
approximation. Boland et al. (1982) shows that substituting Equation (2.34) into
Equations (2.32) and (2.33) allows one to perform the space integrals, with the

results:

¥ (klt?G3q ori) = - ae a M(t aeF) (KI) (2.35)

&(klteGst} Gadel) = ee MUN) aeA)(aslK,)( I) (2.36)

where k; = -k R; is the wavevector of the backscattered electron.

The matrix element (k/t; (Rj )/k; ) represents the scattering of the electron
i he Bae

by an atom at Rj. We can relate (k|tf (Ry)|k;) to (k|tf (0)|k;), which represents

the same scattering problem but is centered at the origin:

(k|t? (Fy)|kj) = exp[i(k;-k) °F] (k{t} (0)|k;) (2.37)

Because we are dealing only with elastic scattering, the matrix elements
(kt? (0)|k;) may be expressed in terms of the scattering amplitude j(6),k):

(k|t (0)|k;) = moe (0k) | (2.38)

where §j is the angle between kj and k.

Equations (2.35) and (2.36) may now be rewritten:

> (klt#OG4 eri) = YSMik) )(4 °F, }f(@, k)exp| ikR; (1- cos ;)| (2.39)
J J

SAKIteGSHGS orl) = Daz 2 MIKI(a* Ry )fo(n-6,k)f(n,.k)exp(2ikR) (2.40)

The complete matrix element of Equation (2.24) is the sum of the three
terms Equations (2.30), (2.39), and (2.40) corresponding respectively to the
unscattered outgoing electron, single scattering by neighboring atoms, and

secondary scattering by the center atom:

(t-faerlof = Meakoa+S (ey esaeri) + Zkieeseyesaeriy (24

Equation (2.41) emphasizes the interference nature of the extended fine structure
phenomenon in which the probability amplitude for ionization is given by the sum
of the amplitudes of three independent scattering processes. Such a sum is
required due to the indistinguishability of the individual events; the ejection of the

electron in some direction k upon ionization is indistinguishable from a process

whereby the ejected electron scatters off of an adjacent atom into the same
direction k.

The development above treats a single ionization event. Experimentally,
however, a large number of such ionizations will occur, and the ejected electrons
will be scattered into many directions k. In order to compute the average
energy-differential cross section of such a macroscopic system, it is necessary to

average over all such directions k in Equation (2.41):

SS. l(t -Jger))f’ 2
aa

= f MUk) (Ke) + 3 (kit G54 onli) +O (kigGst Sa onli} S Ok (2.42)
] j

The four lowest order terms in Rj in this spherical average are evaluated in

Boland et al. (1982) with the results:

JIMik)P (Keay Sa = SIM) (2.43)

- \2
(°F) nfoxp(2ikr )t)(m,k) + fj (0,k)} (2.44)

wey! =

ie iresaen| —k = [MP pes a J F;(8; a (2.45)

J on 4) X(klteGot; Go Geri fee =

- \2
aye Me sy (ei A) Im [exp(2i8, a)- 1}f;(7,k) exp(2ikR;)| (2.46)
Rj

In summing the above expressions, the forward-scattering term f\(0,k) in Equation
(2.44) cancels with Equation (2.45) because of the optical theorem. The rest of
Equation (2.44) cancels with part of Equation (2.46). Thus, the macroscopic

energy-differential cross section is of the form:

2 cs =IMP + (-1)°*'IMP So (Gey) im ti(x,k)exp(2iKR, +216, 1) (2.47)

Equation (2.47) gives the energy-differential cross section for ionization from the
initial state |noly) into the final states le(Ip +1)). By convention, extended fine

structure oscillations are normalized to gM:

= (ety 3 (aeR,) fi(a.k)|sinf 2kR; +nj(1.k) +25, 44(k)] (2.48)

kR?

where the scattering amplitude fj(z,k) has been decomposed according to

Equation (2.22).

If the sample has no angular dependence, such as a polycrystalline

material with no preferred orientation or an amorphous solid, then Equation
(2.48) should be averaged over all directions R. Averaging in three dimensions:

J(aeF,) dp, _1
An 3

(2.49)

Therefore, for samples with no angular dependence, Equation (2.48) becomes

x(k) = (1) ffi (wk)]sin[ 24; +nj(mk)+23,,44(k)] (2.50)
J J

With Equation (2.50) we have derived the basic form of Equation (2.23).
However, Equation (2.23) also contains three additional factors.

The first factor S(k) is an amplitude reduction factor due to many-body
effects during the excitation of the central atom. Equation (2.50) simply assumes
that a single electron is excited from a core state to the continuum. In reality, the
(Z — 1) "bystander" electrons may also be excited in so-called shake-up
(excitation to a bound state) and shake-off (excitation to continuum) processes.
When these additional excitations occur, the final state consists of the ionized
electron and a partially relaxed ion with (Z — 1) electrons. In these cases, the
ionized electron ends up a kinetic energy less than (E — Eo). This tends to "wash
out" the extended fine structure signal since shake-up and shake-off processes
generally have broad energy spectra (Teo, 1986). Shake-up and shake-off
processes do not turn on until the excess energy is several times the binding

energies of the outer electrons. Thus, S(k) = 1 for low k values and S(k) < 1 fork

greater than about 5 A-1. It has been shown that generally 0.6 < S(k) < 0.8 fork
greater than about 7 A-1 (Martin and Davidson, 1977; Stern et al., 1980).

The second factor exp[-2Rj/A(k)], where 2(k) is the inelastic mean free
path of the ejected electron, is an exponential damping term which approximates
the inelastic losses due to the excitation of other electrons or plasmons in the
neighboring environment. Actually, this exponential term can only roughly
approximate these inelastic losses. A more general expression would be
Lj(k)Lm(k,Rj)L¢(k), where Lj(k) represents inelastic losses due to electrons on the
neighboring atom, L¢(k) losses due to electrons on the central atom, and Lm(k,R))
losses due to the electronic medium in between the two (Eisenberger and
Lengeler, 1980). Within the exponential damping approximation, A(k) can be

roughly approximated by

A(k) = C (2) + | | (2.51)

where C and D are constants, 1

materials, C = 1 and D = 3 (Powell, 1974; Penn, 1976; Seah and Dench, 1979;

Teo, 1986).
The third factor exp(-207k*), where oF is the mean-square relative

displacement (MSRD) between the central atom and neighboring atom j, is a

Debye-Waller type factor which is used to account for disorder in the interatomic
distance between the two atoms. Generally speaking, oF has two components:

2 2 2
Oj = Oj vib + Oj struct (2.52)

where OF vip is due to vibrational disorder and SF struct is due to structural or static

disorder. Changes in the vibrational MSRD SF vib can be measured by varying

the temperature of the sample. Theoretical calculations of vibrational MSRD

using various models are discussed in Chapter 5.

Chapter 3 Instrumentation and Experimental Procedures

This chapter discusses the instrumentation and experimental procedures
used for the measurements in this dissertation. §3.1 reviews specimen
preparation techniques. §3.2 discusses the characterization of alloys and
nanocrystalline materials. §3.3 describes the equipment used to control the
temperature of the specimens and calculates the amount of electron beam
heating during a typical experiment. Finally, §3.4 describes the parallel-
detection electron energy loss spectrometer and outlines the procedure which

was used to mitigate its channel-to-channel gain variations.

3.1 Specimen Preparation
Specimen preparation began with elemental metals of at least 99.99%
purity. Alloys of FesAl and NisAl were synthesized from the elemental metals
using an Edmund-Buehler arc-melting apparatus. The apparatus melts metals
on a water-cooled copper hearth in an argon atmosphere. Since the mass
losses after melting were negligible, the stoichiometry of the alloys was
assumed to be that of the initial mixture of elements. The stoichiometry was
also checked by energy-dispersive x-ray (EDX) analysis and EELS.
Foils of the elemental metals Al, Fe, and Pd were prepared by cold
rolling. Foils of chemically disordered Fe3Al were prepared using an Edmund-
Buehler piston-anvil quenching (splat cooling) apparatus. The apparatus
levitates and melts a small piece of metal in an argon atmosphere using a radio
frequency power supply connected to a conical copper coil. When the radio
frequency current to the coil is stopped, the molten droplet falls, and two copper
disks are triggered to rapidly quench the droplet into a foil. Figure 3.1 depicts

schematically the piston-anvil quenching apparatus.

RF
power

supply

detector | ~* light source

copper
disks

Figure 3.1. Schematic illustration of piston-anvil quenching apparatus
(After Pearson, 1992).

For my energy loss experiments, it was necessary to have specimens
which were thin enough to be transparent to the 200 keV electrons in the
transmission electron microscope. Specimens approximately 1000 A or less in
thickness were required. The Al, Fe, Pd, and Fe3Al foils were thinned using a
Fishione twin-jet electropolisher. The specimen is mounted between two jets of
electrolyte, and a voltage is applied across the electropolishing cell. When it is
necessary to cool the electropolishing solution below room temperature, the
apparatus is immersed in a bath of methanol cooled by liquid nitrogen. A light
source and a photo-detector are used to stop polishing at the moment of
perforation (Schoone and Fischione, 1966). Table 3.1 lists the conditions at

which the specimens were successfully electropolished.

Specimen | [Electrolytic Solution Temperature
Al 30% nitric acid, 70% methanol -30 C
Fe 20% perchloric acid, 80% methanol -30 C
Pd 20% perchloric acid, 80% acetic acid +20 C
Fe3Al 20% perchloric acid, 80% methanol -30 C

Table 3.1. Electrolytic solutions and approximate polishing temperatures
used to prepare thin foils of Al, Fe, Pd, and FegAl.
Chemically disordered Niz3Al was prepared in a Denton Vacuum model
502 high vacuum evaporator. A piece of the arc-melted NisAl ingot was placed
in a tungsten wire basket. In high vacuum, current was run through the tungsten
wire until the NizAl was evaporated onto substrates of either rock salt or copper.
The substrates were at room temperature. Figure 3.2 schematically depicts the

high vacuum evaporator. Thin films approximately 1000 A thick were floated in

water off the rock salt substrates onto copper TEM grids. Larger quantities of
Ni3Al were scraped off the copper substrates. Some of the thin films of NigAl

were annealed at 300 C in a heating holder of the TEM to develop L1o order.

RNS] Substrate

Tungsten
Coil

a Ingot

Vacuum
Chamber

Figure 3.2 Schematic illustration of high-vacuum evaporator.

Nanocrystalline Pd was also prepared using the high vacuum
evaporator. Thin films of Pd were evaporated onto substrates of rock salt at
room temperature and subsequently floated in water onto copper TEM grids.
Some of the thin films of Pd were annealed at 600 C to develop larger grains.

| also used a partially compacted powder of Pd nanocrystals synthesized
by inert gas condensation. Pd was evaporated into He gas. A holey carbon
TEM grid was held at the temperature of liquid nitrogen to collect some of the
particles. The powder was partially compacted in atmosphere at room

temperature using a hand-powered compaction device (courtesy Z.Q. Gao).

The preparation of TiO2 samples started with the evaporation of thin films
of Ti metal onto rock salt substrates. The Ti metal was then oxidized by heating
the substrates in air in a furnace at 500 C. At 500 C, the shiny film of Ti metal
transformed into a transparent film of TiO2. Some of the films of TiO2 were
subsequently sealed in an evacuated quartz tube and annealed for 30 minutes
at 850 C to develop larger grain sizes. To make TEM samples, the rock salt
substrates were placed in water and the thin films of TiO2 were floated onto

copper TEM grids.

3.2 Characterization of Alloys and Nanocrystalline Materials

Gao and Fultz (1993) performed x-ray diffractometry measurements on
the Fe3Al foils prepared by piston-anvil quenching. They found an absence of
superlattice peaks in the as-quenched foils. This indicated that the FegAl did
not have significant amounts of B2 or DO3 long-range order. Figure 3.3
presents the growth of the (222) and (100) superlattice peaks as the FeAl
samples were annealed for increasing times at 300 C.

Gao and Fultz also performed Méssbauer spectrometry measurements
on the FeaAl foils. Two of their Méssbauer spectra are presented in Figure 3.4.
These spectra are basically composed of overlapping sextets of peaks which
are caused by the nuclear Zeeman effect. The distribution of 5’Fe hyperfine
magnetic fields (HMF) are obtained from these spectra using the method of Le
Caér and Dubois (Le Caér and Dubois, 1979). The HMF distributions from an
Fe3Al sample as it was annealed for increasing times at 300 C are shown in
Figure 3.5. The numbers of the peaks in Figure 3.5 correspond to the number of
Al atoms in the inn shell of an 5’Fe atom. The intensities of these peaks

correspond approximately to the probability of each 1nn environment. As the

6x10
5 =
AL

a }

i | { I i

a .
A+ 2800 min

? des 120 min

’ ris a
epoF by
t H

“Ty

Figure 3.3.

Two-Theta Angle

Growth of superlattice diffraction peaks in initially piston-anvil
quenched Fe.Al annealed at 300 C (Gao and Fultz, 1993).

1.62
1.60

- 1.58

(o)

r= 1.56

(ep)

fab)

2 1,52

fa)

c 1.50
1.48
1.46

Figure 3.4.

T l | I
annealed at 300°C for 392 hours

as piston-anvil quenched

| I | J | | | | |

-8 -6 -4 -2 0 2 4 6 8
Velocity (mm/s)

Mossbauer spectra of Fe,Al as-quenched and after annealing
at 300 C for 392 hours (Gao and Fultz, 1993).

-2
1.8x10 : | | |

Probability

Figure 3.5.

1.4}

1.2 O.5h

1.0-

0.8 -

Pay
ot
ny
aan
no
Yd
hd
Nn
as
Wik
oO,
ak
if % “4
0 6 1 it ‘ A —
. — li iM “ a4
fe rae
it Veen
fy Viet
fa ‘ f
port Lee,
Gay “
— 4‘ ommend
. -
O.2- [nr
iS id
4 °
i % ¢
ie ny

0.0% | at kd |

0 50 100 150 200 250 300 350
HME (kG)

Hyperfine magnetic field distributions for Fe,Al as piston-anvil
quenched and after annealing at 300 C for various times.

Numbers at top of figure identify resonances from "Fe atoms
with different numbers of Al neighbors (Gao and Fultz, 1993).

sample is annealed, there is significant growth in the peaks corresponding to
four and zero inn Al atoms. These changes in the local environment of Fe
atoms are consistent with the DO3 ordered structure.

The evaporated Ni3Al was shown to be both stoichiometric and
chemically disordered by the x-ray diffractometry, calorimetry, and energy-
dispersive x-ray analysis performed by Harris et al. (1991). Figure 3.6
compares the x-ray diffraction patterns of the as-evaporated and annealed
material. Average grain sizes of approximately 5 nm were determined using x-
ray diffractometry data and transmission electron microscopy dark field images.
The differential scanning calorimetry (DSC) traces of the as-evaporated
material are displayed in Figure 3.7. The DSC traces show an initial exothermic
relaxation beginning near 100 C and a larger exothermic relaxation starting
near 300 C. Harris et al. found that the large relaxation near 300 C is due to
both long-range ordering and grain growth. They speculated that the
relaxation near 100 C might be due to chemical short-range ordering.

Transmission electron microscopy was performed on the thin films of Pd.
Figure 3.8 presents a typical bright and dark field image pair and a diffraction
pattern from the as-evaporated material. The images indicate an average grain
size of roughly 5 nm. Figure 3.9 displays the x-ray diffraction measurement of
the (111) peak from the as-evaporated Pd. A simple Scherrer analysis of the
line broadening gives a grain size of 6.5 nm.

Some of these thin films of Pd were annealed in situ in the heating holder
of the electron microscope. Rapid growth of the grains was observed when the
annealing temperature reached approximately 550 C. Figures 3.10 displays a

bright and dark field image pair and a diffraction pattern from an annealed film

PEVEPUVETPUCeTparrup rere per erpreereprerepreerypryrrry et

Spe Reda Wh ae re

Strat

84K

apoose

Intensity

; i Annealed i

~" f .4
soerirrretbercretecrebrrritersrlsssrtssyy 'WEYETSEREREES

20 40 60 80 100 120
Two-Theta Angle

omnen a ee

Figure 3.6. X-ray diffraction patterns from NigAl material as-evaporated onto
84 K substrates, onto 300 K substrates, and from material
annealed in the DSC to 550 C (Harris et al., 1991).

1.0 T T T T T T T q T
oO
LZ 0.0
4)
oO
Cj
om
Ao)
-1.0
@ :
-2.0 | { t { i { { |

100 200 300 400 500
Temperature (°C)

Figure 3.7. DSC traces for NigAl material evaporated onto 300 K and 84 K
substrates. 5.7 mg of material was used for the upper trace, 8.7
mg of material for the lower trace (Harris et al., 1991).

Bright field (BF) and dark field (DF) image pair and diffraction

Figure 3.8.

pattern from as-evaporated thin film of Pd. DF image taken using

portion of (111) diffraction ring.

1200 Gre

1000

800

Counts

600

POP rrr rye rare rrrryprrry perry yprrerr

Bilitistlrrisptiisptirrptirrslisgri tis

400

PePPpPrerrypee
perl yi

rrrpttrirrptuyrpzlrrpitirr pry trriritrtritiiryg

44 46 48 50
Two- Theta Angle

Figure 3.9. X-ray diffraction measurement of (111) peak from as-evaporated
Pd. Smooth line is Lorentzian fit to lineshape.

eke)

Figure 3.10. Bright field (BF) and dark field (DF) image pair and diffraction
pattern from thin film of Pd after annealing at up to 550 C. DF
image taken using portion of (111) diffraction ring. Streaking in DF
image due to sample drift in microscope.

of Pd. The images show that the average grain size in the annealed films is
approximately 30 nm.

Transmission electron microscopy of the partially compacted powder of
Pd nanocrystals, prepared by inert gas condensation, showed that the material
had an average grain size of roughly 6 nm. Figure 3.11 gives the bright and
dark field image pair.

Transmission electron microscopy was also performed on the thin films of
TiOz. Figure 3.12 presents a typical bright and dark field image pair and a
diffraction pattern from the as-prepared material. Analysis of the diffraction
pattern indicates that the as-prepared thin films of TiO2 are dominated by the
rutile phase but also contain some of the anatase phase. The images show that
the as-prepared film has an average grain size of roughly 7 nm.

After some of the thin films of TiOz2 were annealed at 900 C for 11 hours,
transmission electron microscopy was again performed. A bright and dark field
image pair and a diffraction pattern from the annealed material are presented in
Figure 3.13. Analysis of the diffraction pattern indicates the presence only of the
rutile phase. The images show that the grains have grown to an average size of
approximately 20 nm. This grain growth is consistent with that seen in TiO2
prepared by the gas-condensation method after annealing at temperatures

above 800 C (Siegel et al., 1988).

Figure 3.11. Bright field (BF) and dark field (DF) image pair from partially
compacted powder of Pd nanocrystals.

Figure 3.12. Bright field (BF) and dark field (DF) image pair and diffraction
pattern from as-prepared thin film of TiOo.

ahh!

Figure 3.13. Bright field (BF) and dark field (DF) image pair and diffraction
pattern from thin film of TiOzs after annealing at 900 C for 11
hours. Streaking in DF image due to sample drift in microscope.

3.3 Control of Specimen Temperature

During the EXELFS measurements a liquid nitrogen (LN2) cooled
substrate holder with a heating element was used to control the temperature of
the specimens. The holder, Gatan model 636, is depicted in Figure 3.14. It has
a temperature range of approximately -175 C to +150 C. Intermediate
temperatures are maintained by the feedback-controlled heating of a copper
transfer rod between the specimen and the LN» reservoir. A silicon diode is

used to sense the temperature at the specimen cradle.

specimen
cradle

\ LN 5» reservoir

copper transfer rod

Figure 3.14. Schematic diagram of liquid nitrogen cooled substrate holder for
transmission electron microscopy.

The substrate holder measures the temperature at the edge of the
specimen. The temperature of the material being sampled may be higher due
to heating from the electron beam. The amount of beam heating may be
estimated as follows: Consider the specimen to be a self-supporting film of

uniform thickness t and thermal conductivity « that, for simplicity, lies over a

copper support grid with a circular hole of radius Igrig. Assume that the electron
beam has a uniform current density of Jo and falls on a circular area of radius
lbeam which is centered over the hole in the grid. Furthermore, assume that the
copper grid is held at temperature Tgrig by the temperature control unit of the

substrate holder. The situation is illustrated in Figure 3.15.

thin film
uk

iluminated by
electron beam

Figure 3.15. Diagram of the hypothetical situation used to estimate increases in
sample temperature due to heating from the electron beam.

Assume that the film is thin enough so that the problem becomes two-
dimensional. in other words, temperature varies in the plane of the film, but it is
constant within the thickness of the film. Furthermore, the problem actually
becomes one-dimensional because of its circular symmetry. Our goal is to

determine the radial distribution of the temperature T(r).

Through inelastic collisions, the electron beam acts as a heat source.
Assuming that all energy lost by the beam is eventually converted to heat within

the thin film, the amount of heating per unit area is given by

s(t) =So=do fP(E)GE —, if’ S Theam

=0 _ if foeam

where P(E) is the energy-loss probability distribution, and {P(E)dE gives the

average energy loss for an electron transmitted through the sample. Of course,
P(E) is determined from EELS measurements.

Assuming thermal equilibrium, the three-dimensional heat diffusion
equation becomes Poisson's equation, which has solutions that are well-known

from electrostatics. The appropriate analog of Gauss's Law for heat diffusion is
(heat flux out of surface S) = (heat generated within surface S) (3.3)

where S is any closed surface. Applying Equation (3.3) to cylindrical surfaces

appropriate to the problem gives

Jens = So(m teoam) If Tbeam <0 S Sorid (3.4)

and

[-x ae \eenre = So(mr) lf < tbheam (3.5)

Equations (3.4) and (3.5) depend on the assumption that no heat transfer
occurs due to convection or radiation from the top and bottom surfaces of the
thin film. Applying the boundary condition T(grig) = Tgrid allows us to solve for

T(r):

So beam "grid j
T(t) = Tgrig + Det In . If fbeam Sofbeam grid So_/,2 2 :
= T grid + In + (reeam Tf » lf < theam (3.6)
2KT lheam AKT

Reasonable values for the EXELFS experiments in this thesis are grid =

20 UM, beam = 10 uM, So = 0.16 SS K = 1 We (metal) or 0.05 WW
cm cmkK cmK

(ceramic), and t = 0.1 um. To obtain the value for s,, Equation (3.1) was applied

using dg = 2 x 1016 a and [P(E)dE = 50 eV = 8x 10°18 J, as determined
seccm 0

from a typical low-loss spectrum.

Figure 3.16 shows the result of substituting these values into Equation
(3.6). The increase in temperature is seen to be negligible, even for the ceramic
thin film. The calculated effect of beam heating is so small because the electron
beam is rather spread-out during the EXELFS measurements. If, for instance,
"beam were ten times smaller, then the increase in temperature due to beam
heating would be about 100 times larger. In contrast, the increase in
temperature is less sensitive to changes in the current density, Jo, the thermal
conductivity of the sample, x, or the distance from the grid, rgrig. Changes in

sample thickness, t, should have almost no effect on the temperature because

the average energy loss, JP(E)dE, iS proportional to t.

64

0.20 LJ LJ J t | t T q T | i q t Li ] t t UJ q | ' LU t q

0.15- beam film grid 4

ZV 010+ _
0.05- _
a metal a
0.00 L oo
i j 1 1 1 | i 1 j l | { I Ll 1 | 1 l i 1 | | | | 1 |
0) 5 10 15 20 25

Figure 3.16. Change in temperature due to electron beam heating as function
of radial distance using Equation (3.6) for thin film sample
illustrated in Figure 3.15. Values used for parameters in Equation

(3.6) are given on p. 63.

65

3.4 Parallel-Detection EELS (PEELS)
The experiments in this thesis were performed using a Gatan PEELS
model 666 mounted beneath a Philips TEM model EM430. The experimental

configuration is schematically illustrated in Figure 3.17.

_ - TEM column

—_—
Lene

— - Incident electron beam

~ Specimen

_ Transmitted electrons

— Spectrometer
/“ entrance aperture

c Magnetic prism
Y y spectrometer

Detector

AE >0O

AE =0

Figure 3.17. Schematic of electron energy loss spectrometer attached to
bottom of TEM.

66

EELS measurements can be made with the TEM in either its imaging or
diffraction mode. Measurements in this thesis were made using the TEM
diffraction mode, and this mode is diagrammed in Figure 3.18. In this mode, a
diffraction pattern is visible on the microscope viewing screen, and the
spectrometer entrance aperture collects the electrons which are scattered within
the collection semi-angle B. B is determined by the diameter of the
spectrometer entrance aperture d, and the camera length of the microscope L.
d/2

tan(p) = (3.7)

The Gatan PEELS is a magnetic-prism spectrometer which utilizes a
one-dimensional array of photodiodes to record the electron energy-loss
spectrum in parallel. Figure 3.19 schematically illustrates the spectrometer.

The transmitted electron beam enters the magnetic prism through the
spectrometer entrance aperture. The shape of the magnetic sector allows for
bending of about 90 °. Bending of the electron beam occurs because electrons
travel in circular orbits within a perpendicular magnetic field. The radius of

curvature of the circular orbits is given by

Ra Mey (3.8)
eB
where y= 1-~—+> is a relativistic factor, me is the rest mass of an electron, e

is the electronic charge, B is the strength of the magnetic field, and v is the

velocity of the electron.

67

electron source

condenser lens

object

objective lens

back focal plane

front focal plane

intermediate lens

viewing screen

“*+ spectrometer entrance
aperture

Figure 3.18. Ray diagram of TEM operating in diffraction mode.

68

ee mum ENtrance aperture

Pre-sector lenses

Magnetic sector

Scintillator Fiber-optic
\ . )
\ /

~~ AHR Se888888808

Post-sector
lenses /

N“

Photodiodes

Thermoelectric
Peltier-effect
cooler

Figure 3.19. Schematic of PEELS spectrometer.

69

The strength of the perpendicular magnetic field, B, can be set so that the
electrons with the zero-loss velocity vo are bent onto the detector. Electrons
with velocities lower than vo will have smaller R and so leave the magnetic
prism with a slightly larger deflection angle. This is the source of the dispersion
in the spectrometer. The dispersion is magnified with the use of post-sector
quadrupole lenses.

The magnified spectrum of electrons falls on the detector and is
converted to light by a scintillator disk. A fiber-optic plate channels the light onto
the active area of a linear photodiode array. The linear photodiode array
consists of 1024 independent channels, each one with an active area 25 um
high and 2.5 mm wide. The total active area is 25 mm high and 2.5 mm wide.
The back of the array is cooled by a thermoelectric cooler.

Although the collection of EELS core loss data is relatively efficient when
compared to the collection of energy-dispersive x-ray (EDX) emission data, the
EXELFS oscillations superposed on these core losses are comparatively weak,
only a few percent of the signal amplitude. Consequently, data of high
Statistical quality are required.

EXELFS data from serial detectors generally suffers from inadequate
signal-to-noise ratios, resulting in very limited data ranges in k-space (Csillag et
al., 1981). The advent of parallel detectors has greatly improved the statistical
quality of EELS data (Krivanek et al., 1987). Unfortunately, the EXELFS signal
is often overwhelmed by the gain variations of the linear photo-diode arrays
used in parallel detectors.

Fortunately, there are ways to mitigate the effects of these variations in
gain. A particularly effective method involves dividing by a gain calibration

spectrum followed by gain averaging over many data channels (Shuman and

70

Kruit, 1985). Figure 3.20 shows a gain calibration spectrum collected in the so-
called “uniform illumination mode" of a Gatan model! 666 parallel EELS
detector. Gain averaging involves collecting several spectra, each shifted by a
few data channels, as illustrated in Figure 3.21. These spectra are then aligned
using a feature in the data as a marker, and subsequently added together. Gain
averaging over a large energy range is particularly important in obtaining
reliable EXELFS data to large momentum transfers, because detector uniformity

over a larger energy range is required.

71

1.15 Gee
T T T l T T T | T Ta

1.10
1.05
1.00
0.95
0.90

thos i treet tipi tira lig

Normalized Gain

0.85

0.80 peoitrrip tir r tipi t iii tirtrtiti stipe t typi
0 200 400 600 800 1000

Photodiode Channel

PETEDPUTTTpUrreryp rr yr yp rere yet rt ryt

Figure 3.20. Typical gain calibration spectrum. Note offset of vertical axis.

2.0x10° T T T T T T T T Tt T T T T T T T T T TT
1.56 -

2 r 1

Cc Ss

3S 1.0

O E 4

O a J
0.5L +
0.0 a L ! 1 L L ! 1 1 r 1 ! n l 1 1 | l |

50 “400 150 200
Photodiode Channel

Figure 3.21. Illustration of gain averaging for Fe Lo3 edge. Although only
4 spectra, each shifted by about 20 channels, are shown,
gain averaging in this thesis is actually performed using about
20 spectra, each shifted by about 3 channels.

72

Chapter 4 EXELFS Analysis of K, Lo3, and Mas Edges

In this chapter the analysis and interpretation of the EXELFS data are
discussed. §4.1 describes the procedures used to isolate, normalize, and Fourier filter
the EXELFS oscillations for K edges. §4.2 describes how these procedures can be
extended to Log and Mgs edges. §4.3 discusses the effect of multiple inelastic

scattering on EXELFS.

4.1 Basic Analytical Procedures

The EXELFS signal, x, is the oscillatory part of the edge intensity, AJ(E),

normalized to the non-oscillatory part, J9(E):

J(E) = Jo(E) _ Ad(E)
X(E) = SE) = Jo(E)

(4.1)

where J(E) is the experimental edge intensity, and J,(E) is the smooth edge
intensity which would be observed in the absence of backscattering. In
principle, J(E) and J,(E) should both be single inelastic scattering intensities
(Egerton, 1986). However, as discussed in §4.3, if the sample is sufficiently
thin, EXELFS analysis can be performed without prior deconvolution of the
EELS data.

Subtraction of the pre-edge background removes counts that are not due
to the particular atomic edge of interest. One method commonly used in EELS
for performing such background subtraction involves fitting an energy range
preceding the edge to a power-law energy dependence, AE-8, where A and B
are the parameters (Egerton, 1986). The power law is then extrapolated
through the edge, as shown in Figure 4.1 for the Al K edge of Al metal. Ideally,

this determines the general shape of the normalizing intensity Jo(E), which is

73

5x10 ORR00S OR ee
4p 4
> 3 q
= LF 4
“” r a"
Cc = _|
rad) F |
Pa) L 4
Cc Dr a
2 I
TR 4
ke a

1500 1600 1700 1800 1900 2000
Energy Loss (eV)

Figure 4.1. Power-law extrapolation (broken line) to remove pre-edge
background for Al K edge of Al metal. Spectrum was not
deconvoluted. Sample thickness about 0.4 times mean
free path for inelastic scattering.

74

the denominator in Equation (4.1). Unfortunately, the power-law extrapolation
into the extended region is not always accurate. It is especially inaccurate
when the ratio of the edge jump to the background is much less than one.

Instead of determining the general shape of J, (E) by subtracting the pre-
edge background, a more robust alternative is to use background subtraction
simply to define the height of the edge jump, while assuming a theoretical form
for the energy dependence of J,(E) (Sayers and Bunker, 1988). Such an
approach was used in this thesis to determine the normalizing intensity J,(E).
Theoretical ionization cross sections, as calculated in §A.1, were used for the
energy dependence of J,(E).

In principle, the edge onset energy Ep is the minimum energy needed to
free the core electron. Eo is known to be affected by the chemistry of a material.
Unfortunately, there is no unique way to determine Eo from the experimental
spectrum. Fortunately, in the analysis of extended fine structure, it is not usually
necessary to know the exact value of E>. Any reasonable choice for Eo is
usually sufficient. It is important, however, to be consistent about the choice of
E. when comparing the fine structure between chemically similar compounds.

Once Ep is determined, transformation from energy loss, E, to the
wavevector of the outgoing electron, k, is accomplished by the equation

h2ke :
E - Eo = 3m, = (3.81 A? eV) k? (4.2)

From Equation (4.2), it is apparent that the choice of E, affects the positions of
the extended fine structure oscillations in k-space. This is especially true in the

low-k regime but is less important in the high-k regime.

75

The most popular method for isolating the oscillating intensity AJ(E) from
the rest of the core edge is a polynomial spline fit. A polynomial spline function
is composed of a series of consecutive intervals, each containing a polynomial
of some order. The intervals are "tied together" by making the function and its
first derivative continuous across the boundaries or "knots" (Sayers and Bunker,
1988). If the spline intervals and the orders of their polynomials are chosen
well, a spline fit can remove the low frequency components due to the smooth
atomic edge shape, without affecting the higher frequency EXELFS signal. Too
many intervals or a polynomial of too high an order will result in the removal of
part of the EXELFS oscillations. Not enough intervals or too low an order
results in a large peak in the low-r region (r below about 1.0 A) in the Fourier
transform (Teo, 1986).

In k-space, the EXELFS oscillations have periods which are
approximately 7/Rinn or less, where Ry_”n is the inn interatomic spacing in the
material. Figure 4.2 compares two periods of a sinusoidal oscillation with a
cubic polynomial. Clearly, the cubic polynomial does not have enough degrees
of freedom to simulate well the behavior of the sinusoid over two periods.
Therefore, a reasonable first attempt may be to use a cubic spline fit with knots
spread about 2n/Rynn apart in k-space. Figure 4.3 presents a spline fit to the Al
K edge of Al metal. Figure 4.4 displays the resultant EXELFS interference
function x(k), which was normalized by using the method described previously
to determine J,(E).

Recall from §2.2.2 that x(k) is interpreted in the plane-wave

approximation using Equation (2.23):

k)| Sj(kK) — 22
7(k)=(- eee BRIM) 20K" sinf2kRj + ni(7k) + 28, ,,(k)] (2.23)

76

two periods of a sinusoid

cubic polynomial

Figure 4.2. Comparison between two periods of a sinusoid and a
cubic polynomial.

77

TT TT Tt Tt TY TT y rl TT ToT Try Fr TT ToT TT y rT tt
4.0x10° + 4
3.54 J
3.04 J
2 [ :
7 i 4
Cc L 4
AY 2.51 _
20L _
1.5/4 =
1.0 L rut corte retirrrr torr iriterrira bree te J

1600 1700 1800 1900
Energy Loss (eV)

Figure 4.3. Cubic spline fit (broken line) for Al K edge of Al metal.

78

PUeTPpPPrrrryprerrypeerrperreryprrrrprrreprerryprereyl

0.05

0.00

-0.05

-0.10

Litetirisrtipirrtrirttriprtirritrrp st tiprr tri py

2 4 6 8 10
k (A")

Figure 4.4. Al K-edge EXELFS from Al metal. Data taken at 97 K.

79

The symbols in Equation (2.23) have already been defined in §2.2.2.

The use of Equation (2.23) is justified only if the lp to (lp + 1) transition
dominates over ail others. Figure 4.5, as calculated by the method in §A.1,
shows this holds true for the Al K edge. Given realistic experimental
parameters, the partial energy-differential cross section for transitions to final
states with p symmetry is seen to be at least 100 times larger than that for all
other transitions combined. This result is interesting because it is well-known
that non-dipole transitions are not strictly forbidden in EELS.

To compensate for its attenuation at high-k values, x(k) is usually
multiplied by kK", where n = 1, 2, or 3. This prevents the low-k data from
dominating the high-k data in the determination of interatomic distances which
depend only on the frequency, and not the amplitude, of the oscillations (Teo,
1986). In general, when the neighboring atoms are light elements, n=3 should
work well. Heavier neighbors require smaller n values (Teo and Lee, 1979). In
practice, the value of n which best compensates for the attenuation is chosen.

Fourier band-pass filtering of the EXELFS data is the most common
method used to isolate the structural information from individual atomic shells.

Fourier transformation (FT) is performed on k"y(k) using Equation (4.3).

Kmax Kmax
FT (ky) = — | W(k)kMx(k)cos(2kr)dk +i f W(k)kMy(k)sin(2kr)dk
Kmin Kmin
= Re[FT(khy)] +i Im[FT(k"y)] (4.3)

where W(k) is a window function whose edges are smoothed by Gaussian
lineshapes to reduce ringing effects. The magnitude of the FT is given by

Equation (4.4):

10°
10"
7p)
BS 19?
LU
oS
\o)
=e)
10°
107
Figure 4.5.

80

TOU RUT

T | Porrriyey

t PUrrr'y

PRPrprrrrprrrrprerrprrrry;rrerrprrrrperrryprrery

LioELiiti

| ! reritl l toupiull

rith tri tirre tipper tir tti tt t t ly tt

1600 1800 2000 2200 2400
Energy Loss (eV)

Partial energy-differential cross sections of Al K edge. Letters
indicate angular momentum of final state. Energy of incident
beam = 200 keV. Collection semiangle = 10 mrad.

81

FT(K"x) = {Re2[FT(k"y)] + Im2[FT(k"y)]} 1/2 (4.4)

Peaks in FT(K"x) correspond to shells of nearest-neighbor atoms, although

their positions are shifted slightly from the actual radial distances because of the
k-dependence of the scattering phase shifts nj(z,k) and 26),,1(k) in Equation
(2.23). The reverse transform of the data within a selected window in r-space
isolates the EXELFS oscillation due to a particular atomic shell. The reverse

transform of the data is given by Equation (4.5):

1 max

FT [FTI )I=— J w(r){Re[FT(K"x)]}cos(2kr) —Im[FT(k"x) ]sin(2kr) lr (4.5)
Fin

where w(r) is a window function whose edges are smoothed by Gaussian
lineshapes, like the window for the forward transform.

Figures 4.6 through 4.8 present the Fourier filtering for the Al K edge of Al
metal. Figure 4.6 displays the EXELFS data weighted by k* and the window in
k-space for the forward transform. Figure 4.7 shows the magnitude of the FT
and the window in r-space for the reverse transform. Finally, Figure 4.8
presents the oscillation, due to the 1nn shell, which was isolated with this
Fourier filtering process.

In Figure 4.7, the 1nn peak is located at r = 2.34 A. The actual distance to
the 1nn shell in Al metal is 2.86 A. As mentioned previously, this shift in the
peak position is due to the dependence of the scattering phase shifts on k. The
peak near r= 1.6 Acan be identified with Al-O bonds in surface oxide. Note that
some of the 1nn peak was removed by the window in r-space. Therefore,

theoretical calculations of the 1nn shell oscillation must also be put through the

82

PPEPPeyPrrrrprrrrprrrryprrrrprrrrprrereypey

1.55 ee | ee . 41.0
1.0F | 1 |
E A \ ~ 0.5

0.5 ‘7

>= «0.0

fa]

45 -1.0

-1.0F

rhrritterertisirsrtirrirrtrrirtrriprtipris ly

4 6 8 10
k (A)

Figure 4.6. Al K-edge EXELFS from Al metal weighted by ke (solid line).
Also shown is window in k-space for Fourier transform
(dashed line). Data taken at 97 K.

83

2.5

2.0

1.5

|FT(k*x)|

1.0

0.5

0.0

Figure 4.7.

a ee ee

Pr pr rer prt

1nn 7

ary, + 1.0

rt oda + 0.5

ae 1 Lone eee eee eee ee------] 0.0
oxide 1

4 -0.5

-1.0
fr)

2 4 6 8
k (A")

Magnitude of Fourier transform of Al K-edge EXELFS from Al
metal (solid line). Also shown is window in r-space to select
1nn shell data for inverse Fourier transform (dashed line).
Data taken at 97 K.

84

0.4

0.2

PEETPUPPrPprerrryprrerryprrrryprerri

rhirrrtprrrtitipztlitrirptiuprrtlrrirr tripple

Pissitrtridipp rt lata tiy chociitrpri tars t tipi lige

4 6 8 10

k (A")

Figure 4.8. Al K-edge EXELFS from Al metal after Fourier filtering to
isolate 1nn shell data. Note filtered oscillation is still

weighted by k*. Data taken at 97 K.

85

same Fourier filtering process before they can be compared to the experimental
oscillation in Figure 4.8.

The first principles calculation of phase shifts and scattering amplitudes
in the plane-wave approximation is discussed and presented in §A.2. These
calculated phase shifts and scattering amplitudes can be used to determine
theoretical EXELFS oscillations using Equation (2.23). Figure 4.9 displays the
theoretical oscillation on the Al K edge due to the 1nn shell in Al metal and
compares it with the measured EXELFS from Figure 4.4. The calculation of the
theoretical oscillation used phase and amplitude functions from Teo and Lee
(1979) and furthermore assumed: S(k) = 0.7; A(k) followed Equation (2.51) with
C =1,D=3, andn = 1.2; and o2 = 0.006 A2. Given the experimental
temperature of 97 K, the results of §5.3 were used to choose the value for o2.

Figures 4.10 through 4.12 display the result of applying the Fourier
filtering process on the theoretical 1nn oscillation. Figure 4.10 shows the
theoretical oscillation weighted by k?, along with the measured EXELFS
weighted by k2 from Figure 4.6, and the window for the forward transform.
Figure 4.11 presents the magnitude of the FT of the theoretical oscillation, along
with the measured data from Figure 4.7, and the window for the reverse .
transform. The theoretical data shows a shift in the 1nn peak position of -0.43 A
which compares reasonably well with the measured shift of -0.52 A. The
discrepancy between the two values is less than 0.1 A and is probably due to
my arbitrary choice of edge onset energy, Eo. Lee and Beni (1977) suggest
choosing Eo using the requirement that the imaginary part and the absolute
value of the Fourier transform should peak at the same distance. | simply chose
Eo to be at the location of the maximum edge height. Lastly, Figure 4.12 gives

the theoretical oscillation after Fourier filtering, along with the measured

86

O10

0.05

33

-0.05

0.10 eters bisistessi list i titre tirei tists

4 6 8 10
k (A)

Figure 4.9. Theoretical (solid line) and experimental (dotted line) Al K-edge
EXELFS due to 1nn shell in Al metal.

87

PEePPrerrprrrrperrreprrrryprrrryprerroprrreges

1.5 10

i yl i ‘
1.0 —— i Hi ‘\ “4

: } | . 405

> 0.0K fF \ 4 0.0
NI al fy
Ve 5 /

-1.0

-1.0

~The.
1 | Lt i L | ; a |

-1.5

petinrrtiurirtirrirrtisittrisrrttippitiristirtis

4 6 8 10
k (A)

Figure 4.10. Theoretical (solid line) and experimental (dotted line) EXELFS

on Al K-edge due to inn shell in Al metal after weighting by kK?
Also shown is window for FT (dashed line).

88

PUerryprerrrvyprrrrprrvryprrr ry errr pr rrt yr rr

2.5

2.0L
1.5b

|FT(K°x)|

1.0

0.5

aad ET | t oT fT fT | re ee

0.0

oO

% | ;
0 2 4 6

Figure 4.11. Magnitude of FT of theoretical (solid line) and experimental
(dotted line) Al K-edge EXELFS due to 1nn shell in Al metal.
Also shown is window for inverse FT (dashed line).

89

PP PTET pereryperry pr errr ype rrr y rrr ry rrr ryt

0.4

0.2

rhitsttipritipiep titi lai

EET EVURE TOT TTT rrp trt

Fiseilisitttuit tiga

ao
“e

phitirrtirrriiripgtippr yr tippy lip pip tipi ily

4 6 8 10

bw
bee
be
oad
be
anal
bo
ba

Figure 4.12. Theoretical (solid line) and experimental (dotted line) Al K-edge
EXELFS due to inn shell in Al metal after Fourier filtering.

90

oscillation after Fourier filtering from Figure 4.8. Comparing the amplitudes of
the two oscillations and neglecting the presence of Al oxide gives an
experimental coordination number of 11.9 + 1.4 for the inn shell in Al metal.
Given the relatively crude and somewhat arbitrary nature of the approximations
made, this is in coincidentally good agreement with the known value of 12 for

the fcc structure.

4.2 Extension to Lo3 and Mas Edges

EXELFS occurs above all the ionization edges in a condensed matter
sample, but the analysis of EXELFS is usually performed only for K-edge data.
K edges are simple to analyze because they correspond to transitions from 1s
core states to only those unbound final states with p symmetry (assuming the
dipole selection rule holds, as is typical in these cases). L and M edges, on the
other hand, are complicated by the variety of possible initial and final angular
momentum states. L edges, for instance, have both 2s and 2p initial states, and
transitions from the 2p initial states can result in final states with either s ord
symmetry. L edges have been previously used for EXELFS by Leapman et al.
(1982), but | believe the present work is the first time that M edges have been
used. In particular, | show that nearest-neighbor distances can be obtained by
comparing first principles calculations with the experimental Fe Log and Pd Mas

EXELFS data.

4.2.1 Fe Lo3

Figure 4.13 displays the experimentally measured EELS spectrum of the
Fe La3 edge from Fe metal. Figure 4.14 presents the extracted Fe Log EXELFS
signal. As with my Al K-edge EXELFS data, Eo was chosen to be at the location

91

3.5x10° PEPPER perrey reer prreryprerryerrey errr rerry errr yer

3.0

2.5

2.0

Intensity

1.5

1.0

Peprrtirrir tire er trr rp pla p pp diay

0.5

PEP prererprrrrprrrryprrrryprrrry rr

0.0 SP STITT TTT
700 800 900 1000 1100 1200

Energy Loss (eV)

Figure 4.13. Background subtracted Fe L edge from foil of pure Fe metal.
Spectrum was not deconvoluted, so it contains multiple inelastic
scattering. Sample thickness roughly 1/2 times mean free path
for inelastic scattering.

92

0.06 E t | TF TC Uf | TOT F fF | yO OF F OF | TY ff | t Fg | TT 7 fF | T 4
0.046 =
0.02F 4
s 0.00F
0.02 | 4
-0.04E J
a L I | os | | | os co | i | is on Gone ] a oe co | | Louee ¢ | Lisft | 1 4
7 8 9 10. 11 12. +13
k (A")

Figure 4.14. Fe L,,-edge EXELFS from Fe metal. Data taken at 97 K.

93

of the maximum edge height. Fourier transformation of the EXELFS signal is
illustrated in Figures 4.15 and 4.16. Figure 4.15 displays the signal weighted by
k and the window in k-space. Figure 4.16 shows the magnitude of the FT. Note
that the main peak contains data from both 1nn and 2nn shells because the
distance to the 2nn shell (2.86 A) in bcc Fe is relatively close to that of the 1nn
shell (2.48 A).

Consider the analysis of the Fe Lo3-edge EXELFS. First, | show that the
Fe Log edge is dominated by the 2p to d channel. Figure 4.17 contains
calculated partial energy-differential cross sections for the excitation of 2p
electrons in Fe into final states of s, p, d, or f character. It is seen that, in the
EXELFS region, the 2p to d channel dominates over the sum of all others by a
factor of about 25. This domination by the 2p to d channel makes possible the
interpretation of the Fe Lag EXELFS using Equation (2.23) with lg = 1.

Having shown that transitions to final states of d character dominate over
all others for the Fe Lo3 edge, now consider the complications arising from the
presence of different initial states in the Fe L edge. The spin-orbit splitting
between the L3 and Lo edges of about 13 eV is not a major problem because it
is small compared with the spacing between EXELFS maxima far above the
ionization threshold energy, so EXELFS oscillations from Lg and Lo edges will
be nearly in phase in this energy range. The presence of the Fe L; edge can
complicate the analysis of the Fe Log EXELFS for two reasons. First, the Ly
edge, which occurs as a relatively sharp jump near E = 846 eV (corresponding
to k = 6.0 A” for the Lo3 edge), interrupts the Lo3 EXELFS signal. This problem
can be eliminated by using only L23 EXELFS data sufficiently past the Ly edge
jump. Second, the L; EXELFS signal overlaps with the Lo3 EXELFS. However,

as shown in Figure 4.18, the differential cross section of the Fe L; edge in the

94

F T | tort ff | vor YT Ff | roirTg | ee a | | TT fT } tT gt | T a
0.4 _ Cn , ae V — 1.0
0.2F 4 | 0.5
~ oof 0.0
-0.2E 0.5
0.46 }-4.0

E, Let | Leet I ee | | | oe oe | | Ltd i} | ie oe | | l ql

7 8 9 10 11 12 13

Figure 4.15. Fe L,,-edge EXELFS from Fe metal weighted by k (solid line).
Also shown is window for FT (dashed line). Data taken at
97 K.

0.7

0.6

0.5

0.4

|FT(kx)|

0.3

0.2

0.1

0.0

95

PETUePrreryerrerprerveyrrrryprreryrerevyprerr pr rrr yp errr yr erry re

inn + 2nn

rotor Vir er tise tie tists t ili tay Lipititey

ims

rortrrirrtirrrtirirr tis sp lpr gp tags

1 2 3
r (A)

Figure 4.16. Magnitude of Fourier transform of Fe L,,-edge EXELFS from

Fe metal. Data taken at 97 K.

96

Lm
10 - 4
—~ TE =
> F
& 5 4
D l
S 5 4
~" O.1k& =
Lu E 3
5S C
2) . 4
xe) —_ -_
0.01 b =
0.001 I po te tt th ritirirt

800 1000 1200 1400
Energy Loss (eV)

Figure 4.17. Partial energy-differential cross sections of Fe L,, edge.

Letters indicate angular momentum of final state. Energy of
incident beam = 200 keV. Collection semiangle = 5 mrad.

97

10-
7L
6L
5.
— 4
@ 3r
OD
Cc 2F
Pant L
rao) be
wm te
O .
— Th
\o) 6L
<@) 5L
4b
3e
2r
0.1 Fre ee | ee

600

Figure 4.18. Energy-differential cross sections of Fe L,, and L, edges. Energy of

800

1000 1200 1400
Energy Loss (eV)

incident beam = 200 keV. Collection semiangle = 5 mrad.

98

region of interest is about four times smaller than that of the Fe Lo3 edge.
Moreover, transforming the data from the Ly edge to the k-space corresponding
to the L3 edge effectively raises the frequencies of the L; EXELFS oscillations
and makes them somewhat incoherent.

Figure 4.19 displays, in energy-loss space, the theoretical Ls, Lo, and Ly
EXELFS signals from the combined 1nn and 2nn shells in Fe metal. The
theoretical EXELFS were generated using phase and amplitude functions from
Teo and Lee (1979) and additionally assumed S(k) = 0.7; A(k) followed
Equation (2.51) with C=1, D=3, and n=1.2; and o%,, = o$,,= 0.003 A2. The
results of §5.3 were used to choose the value for o2.

Figure 4.20 presents the sum of the three theoretical EXELFS signals
and superimposes the measured EXELFS for comparison. The general shape
of the theoretical and experimental oscillations compare well, especially in the
range 8 A1main discrepancy seems to be a phase shift between the two. This is probably
due to my arbitrary choice of Eo for the experimental data. Figure 4.21
compares the two oscillations after Eo for the experimental data was shifted by
-15 eV. After this adjustment of Eo, the two oscillations match very well.

Fourier filtering of the EXELFS is displayed in Figures 4.22 through 4.24.
Figure 4.22 displays the EXELFS weighted by k and the window for the Fourier
transform. Figure 4.23 shows the magnitude of the FT and the window for the
inverse FT. The main peak in the theoretical spectrum is at r = 2.29 A which
compares well with the experimental peak position of r = 2.23 A. The secondary
peak near r = 3 A can be attributed to the higher-frequency L; EXELFS signal.
This peak does not greatly affect the combined 1nn and 2nn peak from the Lo3

EXELFS, but it does interfere with the weaker 3nn peak which, after accounting

99

PECPUTrrPEPreyrrrrprecrperrrypeeeryrrurprreyyureerprrery cerry ere

0.04

0.02

TPIT TTT TTI Trip ttt yr rity ry

-0.02

-0.04

534 tame JET,
phitsiteriitipyi tial CLACHEEREREEEOEEEEREREEEREOE

900 1000 1100 1200 1300 1400 1500
Energy Loss (eV)

Figure 4.19. Theoretical Fe L, (solid line), L, (dashed line), and L, (dotted
line) EXELFS due to combined 1nn and 2nn shells in Fe metal.

100

| a | toast J |e ee ee | Tt T fF | it TT | TE T fT | rT fT | ul i

0.06 _
0.04 ft _
-0.02/; ney fa
-0.04b ¥ 4
-0.06 L a
F L | | ie 2 | Lt ji | Let | a a | | Ltt | | a a a | | i ul

7 8 9 10 14 12 13

°°

k (A)

Figure 4.20. Sum of theoretical Fe L,, L,, and L, EXELFS due to combined

1Tnn and 2nn shells in Fe metal (solid line). Also shown is
experimental EXELFS (dotted line).

101

frvt PPPPePrerepeereperernryurvuryprrery errr per er yrereprrrry rr rey rey ]

-0.02

-0.04

-0.06

pocberr Wiper t eet ttt it
7 8 9 10 11 12 13

k (A")

Figure 4.21. Theoretical (solid line) and experimental (dotted line) Fe L,,-
edge EXELFS after E, for experimental data shifted by -15 eV.

102

PeUPUPUEeeperreprereyp errr pr eeeprererererperery errr errr erry erry

F ut :
F a i rE E i
OeF j { vt 4-0.5
r i :
-0.4F i
E +-1.0
Pebisiteebiseitirselistibe te lititee tial
7 8 9 10 11 12 13

9°

k (A)

Figure 4.22. Theoretical (solid line) and experimental (dotted line) Fe L,,-

edge EXELFS weighted by k. Also shown is window for FT
(dashed line).

103

prevrrrrrryrrrrrrrer yp rrr rye perry rrr
: 4 1.0
0.8F Ts J
r iy Jos
0.66 ' \ 1
ee E : ‘ 1
= F------ bonne enn - eee -- +--+ 4 0.0 3
204k .
E -0.5
0.0 oa ; hee LI i Pee ee area Foo ha ud

0 2 4 6

ee)

Figure 4.23. Magnitude of FT of theoretical (solid line) and experimental
(dotted line) Fe L,,-edge EXELFS. Also shown is window

to select 1nn and 2nn data for inverse FT (dashed line).

104

0.2 PEEYPEVUTETUCEPerre peer ry rer rp rere ye rer perry r erry rrr errr yet

0.1

Lit rprrrryprry?

ibe tt td ttt

tititirryr, [ry iy dy

foo)
—_
PErTUPETrTTyTTTrryttrrt

-0.2 citttiriitrrritiriitisribe rt tis tips tit litt tility
7 8 9 10 11 12 13

k (A')

Figure 4.24. Fourier filtered theoretical (solid line) and experimental (dotted
line) Fe L,,-edge EXELFS.

105

for phase shifts, should be located near r = 3.2 A. Figure 4.24 displays the
EXELFS oscillations after the inverse FT. Comparing the amplitudes of the two
oscillations and neglecting the presence of Fe oxide gives an experimental
coordination number of about 8.5 + 0.8 for the combined 1nn and 2nn shells in
bec Fe metal. This is 40% less than the known value of 14, but this level of
accuracy is reasonable considering the somewhat arbitrary normalizations of
both the theoretical and experimental EXELFS signals. Also, accounting for the
presence of Fe oxide would raise the experimentally determined coordination
number. This is because Fe atoms in the oxide contribute significantly to the
edge height, but only slightly to the peak corresponding to the 1nn and 2nn

shells in Fe metal.

4.2.2 Pd Mas

Figure 4.25 displays the EELS measurement of the Pd Mas edge from Pd
metal. Note the large number of counts in the spectrum. Figure 4.26 presents
the extracted Pd Mas EXELFS signal and the window for the Fourier
transformation. Eo was chosen to be near the bottom at the very beginning of
the Pd Mgs edge. Figure 4.27 shows the magnitude of the FT.

The EXELFS analysis of the Pd Mas edge parallels that of the Fe Log
edge. First, | show that the Pd Mgs edge is dominated by the 3d to f channel.
Figure 4.28 contains the calculated partial energy-differential cross sections for
the excitation of 3d electrons in Pd into final states of various angular
momentum. The 3d to f channel is seen to dominate over the sum of all others
by a factor of about 20. This domination by the 3d to f channel makes possible

the interpretation of the Pd Mags EXELFS using Equation (2.23) with lp = 2.

106

2.5x10 a OO Oe De

Mas

2.0

1.5

Intensity

1.0

0.5

ae | ee es | l | as On ee | l Lt | je ee ee | | ae os ee |

O.Q bore tiri sr tirii tis er tepid Pa
300 400 500 600 700 800 900 1000

Energy Loss (eV)

Figure 4.25. EELS measurement of Pd M edge from foil of pure Pd metal.
Spectrum was not deconvoluted. Sample thickness roughly
0.6 times mean free path for inelastic scattering.

0.03

0.02
0.01
< 0.00
-0.01
-0.02
-0.03

107

Te

PErTrperveypereryrrrrypyt

PERPVOrrypereryrerrprerry ire iy

1.0

| l I i l | L

(SE REROE SE SEE CER REE Ce eee Pe Re

-1.0

risissi tipi yy la
nn

10

11 12 13 14 15
k (A")

Figure 4.26. Pd M,-edge EXELFS from Pd metal (solid line). Also shown
is window for FT (dashed line). Data taken at 98 K.

108

PPrrryprrrryprrrryprrrryprrrrprrrryprerrrpreree

. inn a
0.020

0.015
--
i 0.010
0.005
0.000 Lepitripzrtirpirtirirctizrirrtirirrtirep tis it
2 4 6 8
r (A)

Figure 4.27. Magnitude of Fourier transform of Pd M,.-edge EXELFS from
Pd metal. Data taken at 98 K.

109

CS
100 4
— 10 3
> F
& 5 4
Py) 5
= r 7
Ss Te 3
Lu E 7
Ko) . i
& - 4
0.1 &
0.01
Fil Alli sii tay Peper ta tre teri titi prp rtp rp ites yy
400 600 800 1000 1200

Energy Loss (eV)

Figure 4.28. Partial energy-differential cross sections of Pd My, edge.

Letters indicate angular momentum of final state. Energy of
incident beam = 200 keV. Collection semiangle = 5 mrad.

110

Having shown that transitions to final states of f character dominate over
all others for the Pd Mgs edge, now consider the complications arising from the
presence of different initial states in the Pd M edge. The spin-orbit splitting
between the Ms and Mg edges of about 5 eV has very little effect because it is
much smaller than the spacing between the EXELFS maxima far above the
ionization threshold energy. The Mg3 and M, edge jumps are removed from the
Mas EXELFS signal by transforming only data sufficiently past the M; edge
jump. Figure 4.29 compares the energy-differential cross sections of the Pd
Mas, Mo3, and M; edges. In the experimental EXELFS region, the Pd Mas edge
is about three times larger than the Mz edge, six times larger than the Mz edge,
and eight times larger than the M; edge.

Figure 4.30 displays the theoretical Mas, M3, Mz, and M; EXELFS signals
from the inn shell in Pd metal. The theoretical EXELFS were generated using
f(x,k) from Teo and Lee (1979), and 84(k), 52(k), and 53(k) from my Hartree-
Slater calculations presented in §A.2. The following were also assumed: S(k)
= 0.7; A(k) followed Equation (2.51) with C=1, D=3, and n=1.2; and o? = 0.002
A2. The results of §5.3 were used to choose the value for 02.

Figure 4.31 presents the sum of the four theoretical EXELFS signals and
superimposes the experimental EXELFS for comparison. The periodicity and
phase of the theoretical and experimental oscillations compare reasonably well.
Figure 4.32 displays the magnitude of the FT. The main peak in the theoretical
spectrum is at r = 2.76 A which is close to the experimental peak position of r=
2.68 A. The theoretical spectrum also has a smaller peak near 3.9 A which
overlaps with the expected position of the 2nn peak. Figure 4.33 displays the
EXELFS oscillations after the inverse FT. The amplitude of the theoretical

oscillation is seen to be considerably greater than that of the experimental

114

a 4

100 E -

6 a

St -

4b 4
3h.
— OL
> '

cad)
P75)

wo C
6e
2 5L
Lu 4r
Ss J
oO 2b
Te
6h
5k
4e

Sheets tht tr htt dip tial 7

400 600 800 1000 1200
Energy Loss (eV)

Figure 4.29. Energy-differential cross sections of Pd My, M3, M,, and M, edges.
Energy of incident beam = 200 keV. Collection semiangle = 5
mrad.

112

PEPeprrrryrerrryprrrryrrryyprrrryprrrryprrerryprrrry rrr

0.04

0.02

Hep Ane!

bores tipi Th

XN

-0.02

-0.04

PerprereuyrrrryrvriTry ry

Poppe trite dirs ittirii ter rp tari dip tt tepid

700 800 900 1000 1100 1200

Energy Loss (eV)

Figure 4.30. Theoretical Pd Mg, (thick line), Mg (thin line), M, (dashed line),
and M, (dotted line) EXELFS due to 1nn shell in Pd metal.

113

0.06 PEETERTUReeyrrerpr rere rrr yer ery rrr ry rr ery rr rr erry

ul

1.0

0.04

0.5

wre
P00
‘Oe, -
atte
ite,

rlats 1p) ]lip pp il,

-0.02

-0.04
-1.0

Oo
Nh
USE TVETYDTTERTPUTUTPT UTI ITT TST TTT TT rp try rt

-0.06 Piper terri tipi tips tetrad lips tie

10 11 12 13 14 15

Figure 4.31. Theoretical (solid line) and experimental (dotted line) Pd Mas-
edge EXELFS. Also shown is window for Fourier transform
(dashed line).

114

0.10 PEPEPEPerprrrrprrrryprrrryprrrryprrrryerry

0.08
0.5

0.06

PETEPrerrVyprervyrrrry ire
| L i i I l l

0.0

IFT (x)|

0.04

0.02

l l 1 1 it | i l l L

aaron,
COL Pere Lenae ry
tay,

0 2 4 6 8
r (A)

0.00

Figure 4.32. Magnitude of FT of theoretical (solid line) and experimental
(dotted line) Pd M,.-edge EXELFS. Also shown is window

for inverse FT (dashed line).

1.0

115

0.03

PPEUrreprrerrvyrrrryprerrperrryererrypr rr ry rrrryprerry rrr

0.02

0.01

PEVPPPrepererypertryprrres

titertispebarsstrrtilipitiie

0.00

FT '[FT(x)]

-0.01

-0.02

TET TTIT TTT ITT rrr parti
Deserticertarealipai te

-0.03 oe Fee eee ee eee Pe eee eee Fee

10 11 12 13 14
k (A")

o1

Figure 4.33. Fourier filtered theoretical (solid line) and experimental (dotted
line) Pd M,,-edge EXELFS.

116

oscillation. This disparity is not surprising because of the somewhat arbitrary
normalizations of both the theoretical and experimental EXELFS signals.

In conclusion, the cross-section for high energy electron scattering
makes EELS possible only for core edges at energy losses below about 5 keV
(Ahn and Krivanek, 1983). Only elements lighter than vanadium (Z = 23) have
K edges below 5 keV. This does not, however, limit EXELFS experiments to
only elements of low atomic number. As this section has shown, useful
EXELFS information can be extracted from Los and Mas edges as well. The use
of Lag and Mas edges opens up most of the periodic table to possible EXELFS
experiments. |

Nearest-neighbor distances in Al, Fe, and Pd have been determined
using EXELFS which agree with x-ray diffraction results (Ashcroft and Mermin,
1976) to within + 0.1 A. Distances to more distant neighbor shells, however, are
probably not reliable. It should be pointed out that diffraction is, of course, far
superior than EXELFS for determining distances in crystalline solids, which
have long-range order. EXELFS is useful because it has the ability to measure

short-range order.

4.3 Effect of Multiple Inelastic Scattering on EXELFS

§2.1.3 described the use of Fourier transform deconvolution methods to
remove multiple inelastic scattering from energy-loss spectra. However, this
section shows that unless the TEM sample is exceedingly thick, useful EXELFS
information can be obtained without prior deconvolution of the energy-loss
spectrum.

The simulation presented in Figures 4.34 through 4.37 demonstrates the
effect of multiple inelastic scattering on a hypothetical inner-shell edge and its

extended fine structure. To simplify the simulation, perfect instrumental!

Intensity (units of I,)

1.0
0.8
0.6
0.4
0.2
0.0

PERL ETVperryprrryrrry

ie eee
t/A=05 J
| 2 ec

0 40 80

Energy Loss (eV)

117

Intensity (units of |.)

1.0
0.8
0.6
0.4
0.2

t/A=1

srlisrtirrtitslisa ls

0.0

Teryprrrperryprrryprrryt

Le

0 4

0 80
Energy Loss (eV)

Figure 4.34. Idealized low-loss spectra used to simulate the effect of multiple
inelastic scattering. Low-loss spectra for two different sample

thicknesses are shown.

6x1 0 F Tt if TY ut | TT tT | Le ee ee | 5
~ OF t/A=05 J
= 4 FE multiple 4
ra ms q
S 3 4
> F&F :
C of 4
= . :
= E :
TE sing! . :
E Cc J
@) : es Ge ae | ys Ee ae l | a oon See l je oes ae | :
400 500 600 700 800

Energy Loss (eV)

2.0x1 04 TTT TTT Ty
3 15h 4
‘O " A
2 + 4
=> 1.0F a
> 5 4
Cc rs 7
E 05- “a——single 7
0.0 r porrtiritrtipi re
400 500 600 700 800

Energy Loss (eV)

Figure 4.35. Simulated effect of multiple inelastic scattering on the general
shape of a hypothetical inner-shell edge. Multiple-scattering
(solid line) and single-scattering (dotted line) spectra are
shown for two different sample thicknesses.

118

0.02

multiple

original convoluted

0.01

Liferitlerritirpgy

_~

-0.01

Peppa ts wt

PTTT TTT rr ryt

One

titstrisrtirsrsrtisrit itis tirpp trip tur rp aly

6 7 8 9
k (A"')

-0.02

Figure 4.36. Simulated EXELFS extracted from single-scattering (thin
solid) and multiple-scattering (thin dashed) spectra. Also
shown is EXELFS originally superimposed on edge (thick
solid) and original EXELFS convoluted with low-loss (thick

dashed). t/A = 0.5 assumed.

PEE TTEE SS TEEPE TEREST EPre pene) PEGS SPE rr hs Pebrrpr rsh T Rehr yrnn'|
. rial 4
0.025. original J
r convoluted 7
0.020 F 4
L ,
SB 0.015- 4
-- i a
ah L J
0.010F 4
0.005 + J
0.000
0 1 2 3 4 5 6
r (A)
Figure 4.37. Magnitude of FT of simulated EXELFS extracted from single-

119

scattering (thin solid) and multiple-scattering (thin dashed)
spectra. Also shown is magnitude of FT of EXELFS originally
superimposed on edge (thick solid) and original EXELFS
convoluted with low-loss (thick dashed). Data in range from

5.25
120

resolution was assumed, i.e., Z(E) = lp 5(E). A hypothetical single-scattering
distribution, S(E), was constructed. The low-loss region was assumed to
contain a single outer-shell scattering process with an energy of exactly 20 eV.
The inner-shell edge and the background were calculated from the power law
AE-4, and the edge-to-background ratio in the single-scattering distribution was
assumed to be unity. Extended fine structure from a single nearest-neighbor
shell of atoms was superimposed upon the edge in the single-scattering
distribution. The extended fine structure was calculated using the following

simple equation:

x(k) = <4 sin(2kFinn) (4.6)

where x(k) is the EXELFS oscillation normalized to the non-oscillatory part of
the edge intensity, Rinn = 2 Ais a hypothetical 1nn peak position, and 0.1 is an
arbitrarily chosen factor.

Figure 4.34 displays the idealized low-loss spectrum using two different
sample thicknesses. The hypothetical inner-shell edge with and without
multiple inelastic scattering is displayed in Figure 4.35. Figure 4.35 shows that
the primary effect of multiple inelastic scattering on inner-shell edges is the
presence of successively smaller "steps" in intensity above the edge. The first
step above the edge (at 20 eV past the edge onset) is due to double inelastic
scattering processes, the second step above the edge is due to triple inelastic
scattering processes, and so on. Of course, for actual spectra these steps are
rounded because the low-loss peaks are considerably broadened.
Nevertheless, the simulation shows that while the multiple inelastic scattering

steps affect strongly the near-edge structure, the steps are negligible in the

121

region more than 100 eV beyond the edge onset (corresponding to k > 5 A-1). It
is possible, however, that the extended fine structure superimposed upon the
steps may contribute incoherent higher-frequency oscillations to the EXELFS
signal.

Using the procedure detailed in §4.1, EXELFS signals were extracted
from the multiple-scattering spectrum with t / = 0.5 and from the single-
scattering distribution. Figure 4.36 displays the two extracted EXELFS signals,
along with the EXELFS that was originally superimposed upon the edge and
the original EXELFS convoluted with the low-loss. The periodicity and phase of
the four signals are seen to be very similar. From the figure, it can be deduced
that the multiple-scattering signal is, in effect, the single-scattering signal
convoluted with the low-loss. This results in a reduced signal at low k, where
the multiple-scattering and single-scattering signals are out of phase, and an
enhanced signal at higher k, as the signals become more in phase. Figure 4.37
shows the Fourier transforms of the four EXELFS signals. Each transform has a
peak centered near 2 A. The peak extracted from the single-scattering
spectrum is shifted only by about +0.1 A from the peak corresponding to the
original oscillation. Since the EXELFS technique can generally determine
radial distances to only within approximately +0.1 A, this small shift in peak
position is within the expected error. The peak extracted from the multiple-
scattering spectrum is shifted by about +0.1 A from the peak extracted from the
single-scattering spectrum.

The preceding simulation demonstrated that useful EXELFS data can, in
principle, be extracted without prior deconvolution of the EELS spectrum. The
following analysis of actual experimental data shows that this is true in practice

as well. Experimentally, EELS spectra covering the range below about 2 keV in

122

energy loss were collected from a relatively thick sample of FesAl. The sample
thickness was approximately 1.1 times the mean free path for inelastic
scattering, i.e., A = 1.1. Channel-to-channel gain variations in the parallel
detector were compensated using the procedure given in §3.4. Multiple
inelastic scattering was removed using a Fourier-log deconvolution procedure.
In the deconvolution procedure, high-frequency noise amplification was
reduced by reconvolving the single-scattering distribution with a unit area
Gaussian function whose full width at half maximum (FWHM) was 4 eV. The
FWHM was chosen to be approximately equal to the instrumental resolution.

Figures 4.38 through 4.40 display the three relevant regions in the EELS
data both before and after Fourier-log deconvolution. The low loss region is
displayed in Figure 4.38, the Fe Lo3 edge in Figure 4.39, and the Al K edge in
Figure 4.40. After deconvolution, one can more clearly see the Fe Moa3 edge at
54 eV in Figure 4.38 and the "white lines" on the Fe Lo3 edge in Figure 4.39.
The small edge-to-background ratio makes it difficult to see any details on top of
the Al K edges in Figure 4.40. To better resolve the structure of the Al K edges,
Figure 4.41 displays the Al K edge data atter background subtraction. Without
the background intensity, the Al K edges are effectively magnified. As Figure
4.41 shows, although the overall shapes of the Al K edges before and after
deconvolution are very different, the EXELFS oscillations superimposed on the
edges are remarkably similar.

For a more quantitative analysis, EXELFS data were extracted from the
multiple-scattering and single-scattering spectra using the procedure explained
in §4.1. Figure 4.42 presents the Fe Lo3-edge EXELFS data. Notice that the
two sets of data follow the same general pattern, regardless of whether they

were extracted from multiple-scattering or single-scattering spectra. Apparently,

123

t v | Ly q i q | q t i q | i i LI i

single

6x10°

«————-multiple

Intensity
wo

TUTTPTTVTPTVITETITTP TUT epee perry per erp eer ye erry rey rr try
pisterestiristirsi list testes titi liiitiplirtel

Energy Loss (eV)

Figure 4.38. Low loss region from multiple-scattering (solid line) and single-
scattering (dotted line) spectra of FeAl.

124

1.0x10° PEETPETUUPTUUTeperrrypr erry errr per ery erry yp rrr rp erry rrr

0.8

multiple

{UPreperrvryprrrryrrr

0.6

sac as eeN EPCS

poset tA tissr tipi tepp tarp tii lipep tines

£7) C
Cc =
Re) C
= C
= 0.46
0.2F
C Eveeebesetes ests pe toes des ett
700 800 900 1000 1100 1200

Energy Loss (eV)

Figure 4.39. Fe Lp3 edge from multiple-scattering (solid line) and single-
scattering (dotted line) spectra of Fe3Al.

125

1.0x10 ror ft | ror tT Tt | es oe ee | | es bee ee | | Le ee ee | | Le ee ee |

0.8 multiple

op)

PESTTETYYPTCTTTTT TTT TTT per er yp rrr rp rrr ty tity yy qT

Intensity

0.4

borer tiry ely ritrrtrtisritiitrlipietiat

0.2

Peta or eeetnnes,
TO eheeeeunsasenge,
tere nenteetaceens,

| es eae eee | | | a ee eee | | | es en ee | | | Ca Tees Goa | ! |e Se | i se a ee |

Energy Loss (eV)

Figure 4.40. Al K edge from multiple-scattering (solid line) and single-
scattering (dotted line) spectra of FesAl.

126

E T T qT T | qT T Ul qT { T qT UJ qT i qT t LI Lf | Lj J qT qT I q T t i 3
6x10° E 4
5E =
E multiple 7
4-- J
a _ x
97) E =
c a ite q
2 3 oa ; Nereepeteedtt Mertns, J
= F ‘ Na te 4
2E =
- single =
1E =
@) Fat pe ie | | | H l | 4d _f_t | 1 Lt | lL l | Cn 4 | l 1 l 1

1500 1550 1600 1650 1700 1750 1800

Energy Loss (eV)

Figure 4.41. Background subtracted Al K edge from multiple-scattering (solid
line) and single-scattering (dotted line) spectra of Fe3Al.

127

pT ET ETUNPTUTTperrepereryp rr reper rey trer yp errry errr y rrr y try
a .

0.4

#.——single

0.2

EPrurrtisittri ri ty

multiple

pititrrrs lip ite

TrPeprrrryrrrryrTt

-0.4 rence tepir tipi terri dastatitii tits lips tipi lis titsiliias

7 8 9g 10 11 12
k (A"')

Figure 4.42. Fe Lo3-edge EXELFS from multiple-scattering (solid line)
and single-scattering (dotted line) spectra of Fe3Al. Both
signals have been "smoothed" to somewhat reduce noise.

128

0.35 PYrryprrrryprrrrprrrryprrrryprrrryprrrryprrry

ft

0.30 single

0.25

—_ ’ multiple :
xe C :
}— L. 4
Le O.45P a
0.10F 4
0.05 H a Vn ee Y n\n -|
0.00 Cirestirrri tir itiiirtirirtiit 7

0 2 4 6 8

r (A)

Figure 4.43. Fourier transforms of Fe Lo3-edge EXELFS from multiple-
scattering (solid line) and single-scattering (dotted line)

spectra of Fe3Al. Data in the range 7transformed.

129

the main difference between them is that the single-scattering data has a
greater amount of high-frequency noise. As discussed in §2.1.3, a side-effect of
the Fourier-log deconvolution procedure is the amplification of high frequency
noise in the single-scattering spectrum.

Figure 4.43 displays the magnitude of the FT of the Fe L23-edge EXELFS
data. Notice the similarities between the transforms of both the multiple-
scattering and single-scattering data. Each transform has a inn peak near 2.1
A.

Figure 4.44 presents the Al K-edge EXELFS data. Both signals follow
the same general pattern, although a greater amount of high-frequency noise is
present in the single-scattering signal.

Figure 4.45 displays the magnitude of the FT of the Al K-edge EXELFS
data. Both transforms contain a 1nn peak near 2.2 A. In addition, the smaller
peaks in the two transforms match well.

In conclusion, the analysis of both simulated and experimental data has
shown that, unless the sample is exceedingly thick, useful EXELFS information
can be obtained without first deconvolving the EELS spectrum. This is
especially true for the experiments in this thesis which aim to measure only
relative changes in the amplitude of the EXELFS oscillations as either the
temperature or the state of SRO of the sample is varied, but the thickness is held
constant.

On the other hand, deconvolution is important when comparing data from
samples of different thicknesses. Deconvolution is also important when

comparing EXELFS data with EXAFS data.

130

0.6 per eet

0.4E single E
0.2E 4
< 0.08
0.2F 4
ed Se ees PTE PEST TEET FPSET TTT ET:
5 6 7 8 9 10
k (A’’)

Figure 4.44. Al K-edge EXELFS from multiple-scattering (solid line) and
single-scattering (dotted line) spectra of Fe3Al. Both signals

have been "smoothed" to somewhat reduce noise.

131

ee ee TI
0.6 7 it ——single F
055 multiple E
O46 4
x F F
a OS E
0.2E =
0.16 “4
0.0 Py pheririrtirirrtirrir ty py 11 MT Lu bis i Ltt \A
0 2 4 6 8

r (A)

Figure 4.45. Magnitude of FT of Al K-edge EXELFS from multiple-
scattering (solid line) and single-scattering (dotted line)

spectra of FegAl. Data in range 5.5 transformed.

132

Chapter 5 Temperature-Dependent EXELFS of Elemental Metals

This chapter discusses the interpretation of my temperature-dependent
EXELFS data from Al, Fe, and Pd metals. As temperature increases, vibrations
between atoms in the sample increase. This causes a decrease in amplitude of
the EXELFS oscillations which is accounted for in Equation (2.23) by the
Debye-Waller type factor exp(-2.07k2), where of is the vibrational mean-square
relative displacement (MSRD).

§5.1 contains a brief derivation of the Debye-Waller type factor. §5.2
then derives an expression for the vibrational MSRD as a function of the
“projected” density of vibrational modes and contrasts the vibrational MSRD
with the vibrational mean-square displacement (MSD). §5.3 discusses the
force constant model of lattice dynamics. Finally, §5.4 presents my
experimental data on elemental metals and analyzes them within the Einstein,
Debye, and force constant models. Debye temperatures from my MSRD
measurements are compared with published Debye temperatures from heat

capacity measurements.

5.1 Debye-Waller Type Factor for EXELFS
This section briefly derives the Debye-Waller type factor for EXELFS. For

simplicity consider "half" of the sine term in Equation (2.23):

(exp(i2k|r, —to )) (5.1)

where ro and rj are the instantaneous position vectors of the central and

neighboring atoms, respectively. The brackets ( ) represent averaging over an

ensemble of systems. The amplitude-reducing terms of Equation (2.23) which

133

depend on the bond length |r; — r9| can be neglected because they are less
sensitive to small changes in the bond length. The sine term, on the other hand,
is very sensitive to changes in the bond length because such changes affect the
phase of the sinusoidal oscillation.

|tj — fo] can be approximated to first-order by Ryo(tj — fo), where Rj is the
equilibrium direction between the central and neighboring atoms. Substituting

this approximation into Equation (5.1) gives

(exp(i2k[r -to\)) = (exp[i2kR, (tj -t)|) (5.2a)
= (expfi2kR; «(Rj +uj - 0-up)]) (5.2b)
= exp(i2kR)) (exp[i2kR; e(uj— Up ))) (5.2c)

where Up and uj; are the instantaneous displacements of the central and
neighboring atoms from their equilibrium positions at 0 and Rj, respectively.
The second factor on the right-hand side of Equation (5.2c) can be

expanded into a series:
(expli2kt, «(uj —uo)}) = 1 + i2k (A, «(uj—uo)) - F(A +(uj-uo})° +. 6.3)

The first-order term on the right-hand side of Equation (5.3) vanishes because
the ensemble averages of the displacements are zero. Thus, the second-order

term is the lowest-order correction:

(expfiakA, «(u;—up)]) = 1 ~ 242(Rj +(uj up) (5.4)

134

The right-hand side of Equation (5.4) is approximately equal to

exp(-2k? 07) (5.5)
where of = (Ry e(u, —ug))”.

5.2 Vibrational Mean-Square Relative Displacement (MSRD)

This section derives an expression for the vibrational MSRD, o8, asa
function of the "projected" density of vibrational modes, gr(w). This is done
using the quantum theory of lattice dynamics. For contrast, the vibrational MSD,
02, is also discussed.

Consider a monatomic Bravais lattice. Let up denote the displacement of
the atom whose lattice site is associated with the Bravais lattice vector R. From
the quantum theory of lattice dynamics, it is well known that ug can be

expressed as a function of annihilation Ags and creation alas operators:

una Te y Vata gs (a+ at.) 6gs exp(iq-R) (5.6)
qs s

where @gs is the frequency and 6g; is the polarization vector of the phonon with
wavevector q and polarization s, N is the number of atoms in the crystal, and M
is the atomic mass. The summation is over all allowed wavevectors q in the first
Brillouin zone and over the three independent polarizations s (Ashcroft and
Mermin, 1976).

As shown in §5.1, the vibrational MSRD between atoms at 0 and Ris

given in a first-order approximation by

‘) (5.7)

where the brackets ( ) indicate time (or thermal) averaging. From Equation

(5.6), we find:

ae Oa ~ (age + at.) qs *R [expliqR)-1] (5.8)
qs s

Squaring the magnitude of Equation (5.8):

= aN 1 (&qs*F)2|[exp(iaqeR)—1](aq. +alys)
®gs

h 1-cosqe R
= NM py On (Eqs°R R)? (aqs4-as + al al. + agsal, + at .2-gs} (5.9)
qs Ss

(ur - u)*Al

where we have used the fact that operators on different modes (q,s) commute.
In the Heisenberg picture, the time dependences of the annihilation and

creation operator are determined to be

Ags(t) = Ags exp(-iat) (5.10a)
al.(t)= al, exp(iat) (5.10b)

Therefore, the ags@-gs and al “gs al. terms vanish when time averaged, leaving

of = arn oa (6qsF)2(2(ngs) +1) (5.11)

136

where (ng, ) is the time-averaged phonon occupancy of the vibrational mode
qs

with wavevector q and polarization s. For phonons, the distribution function
(ngs) is well known to be

= 5.12
(Ngs) EXP(AWgs/kgT) — 1 (5.12)
In this way, phonons are like bosons whose chemical potential is h@g;/2.
Substituting Equation (5.12) into Equation (5.11) gives
qs = gs
or equivalently
th(iw/2kgT
of = Fe oe ga(o) — ( BT) (5.14a)
where 9r(o) = = ¥,(1-cosqeR)(Eqs Rh)? 5(@—Wgs) (5.14b)

qs

Yr(o) is called the "projected" density of vibrational modes. gp(w) weights the
contribution of each mode to the mean-square compression of the bond
distance between the atoms at 0 and R.

For contrast, consider the vibrational mean-square displacement (MSD),
represented by o?, which is used in the Debye-Waller factor for Bragg peaks in
x-ray diffraction. In x-ray diffraction, the intensities of Bragg peaks are reduced

by the factor exp(-o7k2), where k is the magnitude of the scattering vector.

137

Similarly, in Méssbauer spectrometry, the recoil-free fraction is also given by
exp(-o7k2) (Gonser, 1975).
The vibrational MSD of an atom from its equilibrium lattice position is

defined as

uck

where u is the instantaneous displacement of the atom, k is the direction of the

*) (5.15)

scattering vector, and the brackets indicate time (or thermal) averaging. It turns

out that
h coth(Aw/2kgT
02 = OM Ido g(a) o (5.16a)
where g(a) = A) (Bqs* k)2 8(co-t0qs) (5.16b)
qs

g(@) is equivalent to the normalized density of vibrational modes since (8gs*k)2
averages to 1/3 and there are a total of 3N vibrational modes. By normalized, |

mean that

[do g(o) =1 (5.17)

Equation (5.16) should be contrasted with Equation (5.14). The most important
difference is that the additional term cosqeR in Equation (5.14b) insures that
only the out-of-phase motion of the atoms in the direction of R contributes to the

MSRD (Beni and Platzman, 1976).

138

5.3 Force Constant Model of Lattice Dynamics

This section determines the vibrational modes of a lattice within the force
constant model of lattice dynamics. This is done for a monatomic crystal using a
classical description of the atomic vibrations which largely follows the one given
in Ashcroft and Mermin (1976).

Assume that the equilibrium position of each atom in our monatomic
crystal is a Bravais lattice vector R. Define r(R) to be the instantaneous position
of the atom whose equilibrium position is R. The total potential energy or

cohesive energy of the crystal can then be written as

U= o{r(R) — r(R")] =2r E SY) o[R + u(R) — R' — u(R’)] (5.18)

rol

R R'zR
where ¢(r) is the interaction energy between atoms separated by r, and u(R) is
the deviation of the atom from its equilibrium position R, i.e., r(R) = u(R) + R.
From now on it will be implicitly assumed that the summations over R' exclude
R.

Assume that the deviations u(R) are small compared with the interatomic
spacing. The potential energy U can then be expanded about its equilibrium

value, using Taylors theorem in three dimensions:

U=3DD o(R — R’) 2d DIM R)—u(R')]}« Vo(R—R’)
R R'
+E {[uR) wer Je v}* o(R — R')+ ... (5.19)

The zero-order term is simply the equilibrium potential energy. The coefficient

of u(R) in the linear term is >) Vo(R—R’), but this is simply the force exerted on
R’

139

the atom at R by all the other atoms, when each is placed at its equilibrium
position. Clearly, by definition of the equilibrium position, this coefficient and
therefore the linear term must vanish.

Since the linear term vanishes, the quadratic term is the lowest order
correction to the equilibrium potential energy. In the harmonic approximation

only this term is retained:

U = Ueq + Yharm (5.20)

where U®9 is the equilibrium potential energy, and Uharm is the harmonic
approximation to the extra potential energy due to the deviations of the atoms
from their equilibrium positions.

Changing notation and rearranging the expression for Uharm:

Uram = “yy {(u(R)—u(R’)]°V}°o(R-R)
RR
= EDD E[H AG po(R-F WU (R)—Uy(R)oo(R—-F')Uy(R)] (6.21)
aren
_ 9° (r) |
where 9, (Fr) = ror. and the summations over p and v are over x,y,z. The
BY’ vo

summations over the first term in Equation (5.21) can be manipulated to give

2X Xd Duy (R) op» (R—-R' uy (R) = YY Yu, (R)O,»(R—R")u, (R)

RRA v RR" yp v

= 2 » py » ORR Un (R)o.»(R ~ R")u, (R')

R'R" p v

140

Therefore, Equation (5.21) can be written simply as

Uharm —

No| —

LED Ly (R)Cyy(R—R')uy(R') (5.23)
ines

where the force constant matrix Cy»(R-R') = oar), Oy» (R-R")— yy (R-R’).
Equation (5.23) is analogous to the familiar U = shoe for a single spring.

Now consider the 3N equations of motion for the system. In analogy with

the familiar F = ma = -< = —kx, the force on the atom at site R in the p-

direction is

Ma,(R) = -> °C, (R-R')u, (R) (5.24)
R' v

where M is the atomic mass, and double counting cancels the factor of 1/2 in
Equation (5.23). Thus, —C,..(R-R')u,(R') is the force on the atom at site R in the
u-direction when the atom at site R' is displaced by Uy(R’) in the v-direction.

Equation (5.24) can be rewritten in matrix notation as

Ma(R) = —55C(R—R')u(R') (5.25)

Consider solutions to Equation (5.25) of the form u(R,t) = Aéexp[i(qeR — ot)]:

—Ma?A é exp(iqeR) = -Aé b C(R-R')exp(iqe P|

-Aé| > C(R")exp[iqe (a-r")| (5.26)
R"=R-R’

—Mw2é = D(q) é (5.27)

141

where the dynamical matrix D(q) = } C(R)exp(—iq*R). The dynamical matrix

can be thought of as a Fourier transform of the force constant matrix. Using

inherent symmetries of the force constant matrix C(R) (Ashcroft and Mermin,

1976), the dynamical matrix can be rewritten as

-_ in2( 428
D(q) = 25, C(R) sin? ; (5.28)

Equation (5.28) shows that the dynamical matrix must be real and an even
function of q.

For each of the N allowed wavevectors q in the first Brillouin zone,
solving Equation (5.27) gives three orthonormal eigenvectors gs and three
corresponding eigenvalues wgs. Of course, the eigenvectors and eigenvalues
correspond respectively to the polarization vectors and the frequencies of the
normal modes of vibration. The normalized density of vibrational modes, g(a),
is simply the probability distribution of frequencies Wgs- The projected density of

vibrational modes, gr(w), can be determined by applying Equation (5.1 4b).

5.4 Results from Al, Fe, and Pd

This section presents temperature-dependent EXELFS measurements
from Al, Fe, and Pd foils and analyzes the results within the Einstein, Debye,
and force constant models. Debye temperatures obtained from MSRD and heat
Capacity measurements are compared.

Figures 5.1 through 5.3 present the temperature dependence of the
magnitude of the Fourier transform of the Al K-edge, Fe L23-edge, and Pd Mgs-

edge EXELFS. In this section, for simplicity, it is reasonable to consider the

142

2.5

2.0

1.5

|FT(k*x)|

1.0

0.5

I a ee

TUTTI TTT Try rrr rrr ry ry erry errr errr yr rrr

riba rit Liens Ly pp yp ly yyy ly

» *
oy

SAN! /

Pamkied |

coe

A ey >» ark
on ofhing 5 . WF -.
nat oe coe ee i ee

i iY
ay
roritfirrtisii ds

Figure 5.1.

1 2 3 4 5

°o

r (A)

Temperature dependence of magnitude of FT of Al K-edge
EXELFS (3
143

PPE EP OUUPEUPTPRPeep Pree prreryp rer ypeeerypererprerepeerryper ets

0.6 97 K 4

0.56 =

0.4E =

0.2E =

O.1£ 4
0.0§

0 1 2 3 4 5 6

r (A)
Figure 5.2. Temperature dependence of magnitude of FT of Fe L,,-edge

EXELFS (7
144

PTUTT TVET TY TIT Tye erry rer peer rp er rrp rere preety rrr ry erry rire

0.020 J

0.015

IFT (x)

0.010

0.005

0.000 portipistrere teri tip rtorertis tere bette te
1 2 3 4 5 6
r (A)

Figure 5.3. Temperature dependence of magnitude of FT of Pd Mg-edge
EXELFS (10.25 < k < 14.5 A’) from Pd metal.

145

mayor peak in the FT of the Fe Lo3-edge EXELFS to be due solely to the 1nn
shell although the 2nn shell also contributes to it.

In general, the 1nn peaks are seen to decrease in size with increasing
temperature. This effect is usually represented by the Debye-Waller type factor
exp(-2 o%,,k2), where of, is the mean-square relative displacement (MSRD)
between the central atom and the 1nn shell.

Figure 5.4 compares the Fourier filtered 1nn shell EXELFS from Al at 97
K and 296 K. Using a simple least-squares routine given in §B.4, the difference
in MSRD between the two oscillations is determined. This difference in MSRD
is denoted by Ao%,,. To show the quality of the fit, Figure 5.2 also displays the
97 K data multiplied by exp(-2A.0%,,k2), where Aod,, = 5.3 x 10°3 A2.

Figure 5.5 through 5.7 display Act, for the EXELFS from Al, Fe, and Pd
metals relative to the EXELFS at the lowest temperature. The error bars were
obtained from values of Aotn at which the variance of the least-squares fit
increased by 20%. As expected, Act, is seen to increase with increasing

temperature. The temperature dependence of Act, can be interpreted within

the Einstein, Debye, and force constant models.

146

PYPPrrrprrrryperyr {rRrrrprrrrprorrryprrreryt

Porsitirestisiiteri iter

0.4E
7 i
: j
C H
be {
0.2- i
r H
— C i
~ -
af .
i 0.0F =
L :
u p :
-O0.2F ‘ as
C \ 7
' \
0.46 4
a a
4 6 8 10
k (A)

Figure 5.4. Fourier filtered 1nn shell EXELFS from Al metal at 97 K (solid
line) and 296 K (dashed line). Also shown is 97 K data

multiplied by exp[-2(5.3x10~A’) k7] (dotted line).

147

10 3
an 8E LJ
a po ot:
oo " a
(on) - 7
~~ 6F q
NI 7 qa
E E 5
©) - q
Pf tt tt i
100 150 200 250 300 350 400 450

Temperature (kK)

Figure 5.5. Change in 1nn MSRD for EXELFS from Al metal relative to

EXELFS at 97 K. Error bars obtained from values of AG tn’ at
which variance of least-squares fit increased by 20%.

148

a ee ee ee 5

4 7

3f -
tO :
© E 7
(/) r 4
a 2k =
Ef :
©) - 4
a F- t .
iE i 3
Ofte ttt tis tt tipi sti yyii,,,,4,4

100 150 200 250 300 350

Temperature (K)

Figure 5.6. Change in 1nn MSRD for EXELFS from Fe metal relative to
EXELFS at 97 K. Error bars obtained from values of AG," at
which variance of least-squares fit increased by 20%.

149

4 _ | qT T qT Li t t q LU t | T LU U U ij U qT LI Li } Lm
E 7
3h 3
oct a 7
ro) : 7
2 ' 7
— 2b 4
Oo L 4
= : 7
oO = 4
1f ;
r J
6) - __t.. I 1 |. I l l I I | | iT l | | l i i ] ] 7

100 150 200 250 300

Temperature (K)

Figure 5.7. Change in 1nn MSRD for EXELFS from Pd metal relative to
EXELFS at 98 K. Error bars obtained from values of AG inn at
which variance of least-squares fit increased by 20%.

150

5.4.1 Einstein Analysis
The Einstein model is the simplest. In solid-state theory, a solid of N
atoms is considered to have 3N vibrational modes. The Einstein model

assumes that all 3N modes have the same characteristic frequency wg. In other
words, the Einstein model assumes that the density of vibrational modes is

simply
g(@) = 5(@ — We) (5.29)

The projected density of vibrational modes in the correlated Einstein model is

also a delta function:
QA(@) = 8(@ — w_) (5.30)

Substituting Equation (5.30) into Equation (5.14a) gives the following

expression for the MSRD within the Einstein model:

of = —— coth(hae/2kgT) (5.31)

Mae

The Einstein frequency w_ and Einstein temperature 0¢ are related by the

simple equation

h@_e = kpOe (5.32)

151

Using the computer program listed in §C.1, Einstein temperatures can be

determined from AoZ,, vs temperature data. Allowing the value of 07, at the

lowest temperature to float, the program fits the temperature-dependent data to
Equation (5.31). Figures 5.8 through 5.10 display the Einstein model fits to the
A OF, data from Al, Fe, and Pd metals. The fits gave 6¢ = 318 + 10 K for Al, 306

+ 16 K for Fe, and 223 + 30 K for Pd.

152

PRETPUTTrpPrrrryprrrrypurrryprrrryprrrryprrrryprrrryrt

18

16

14

12

Sinn” (1 0° A?)

10

A De ee

riritirsrrtitrrtiyrtrltisittisrip tipi p tippy tei i dy

0 100 200 300 400

Temperature (K)

Figure 5.8. Einstein model fit to 1nn MSRD data from Al metal. Absolute
offset of data was allowed to float. Fit gave 8. = 318+ 10 K.

153

- TT € | | a or oe | tT FT fF | TT Ff f | tT TT CT fT | | om | Le | Tf T FT { qT i]
7E J
6E 4
am O& E
oe : :
‘S OF 4
6 4 = 4
3E 3
2 - I L | | oe oe | ji f Ff | ; ee os ee | 77 ¢ Ff | 7s] |; 7 [ Lijit | I F

0 50 100 150 200 250 300 350

Temperature (K)

Figure 5.9. Einstein model fit to 1nn MSRD data from Fe metal. Absolute
offset of data was allowed to float. Fit gave 8, = 306 + 16 K.

154

- LJ qT T T | T t qT t | tT U UJ T | qT qT | ee | qT t ul T | Lj T qT 1 { is
6E :
BF 4
< Ff
of 7 7
oe r q
= 4E 3
uv. o£ :
6 F£ :
3E a

oF =

E l 1 l l | ] l 1 l | i | | i I 1 L ] ] I 1 1 1 1 [ ] L 1 1 | 4

0 50 100 150 200 £250 300

Temperature (K)

Figure 5.10. Einstein model fit to 1nn MSRD data from Pd metal. Absolute
offset of data was allowed to float. Fit gave 9, = 223 + 30 K.

155

5.4.2. Debye Analysis
The Debye model is slightly more sophisticated than the Einstein model.

The Debye model assumes a linear dispersion relation w = cq and that the

density of vibrational modes is given by

Va? . kp@p
g(a) = Onecs ' if @<@p= h
=0 , ifo>a@ (5.33)
, kp@p
where V = atomic volume, c = Figp 7 = (622/V)'3, 8p = Debye temperature.

In the correlated Debye model, the summation in Equation (5.14b) over
all allowed q in the first Brillouin zone is replaced by an integral in q-space over

a sphere of radius qp. Furthermore, since the polarization directions are
orthonormal, ¥(6q,s°f)? = 1. The projected density of vibrational modes
Ss

becomes

Qp at =
faaznq2®8-2 [aot1-cos(arcosesing

9R(@) (5.34)
An 3
3 4D
The integral over @ works out to
T ina'R
Jastt-cos(qrcose)]sing = a(t - “an | (5.35)

Therefore, Equation (5.34) becomes

156

Qp
3 ' ot sing'R
onto) = 598 Joa’ 8(a-a Ja?(1 - oR
_ 3¢? 1- singR
eal qR
sine
307 Cc.
=—~|1- — (5.36)
Cp oR
Cc
Substituting Equation (5.36) into Equation (5.14a) gives
OD
. oR
3h sino
2=—— h(h T 1- .37
Of Mag dw coth(im/2kgT) w oR (5.37)
Cc

(Sevillano et al., 1979).

The Debye frequency mp and the Debye temperature @p are related by

hop = Kg@p (5.38)

Using the computer program listed in §C.2, Debye temperatures can be
determined from Actin vs temperature data. Figures 5.11 through 5.13 display

the Debye model fits to the Ao,2,, data from Al, Fe, and Pd metals. The fits gave
Op = 438 + 13 K for Al, 417 + 22 K for Fe, and 306 + 40 K for Pd. These values

for 8p are approximately 0.73 times the corresponding values for @¢. Disko et

157

TeT TTT ey reer perryrt rer perry rrr ry rrr yr
ig a

: T
i6L _
an 14 a
ot a -_
° L
= 12b 4
NO i 4
= + -
oO 10- =
gL _
5 4
gL _
Pratt dep eplipipti partir ert iti ttt rit litis tad

0 100 200 300 400

Temperature (kK)

Figure 5.11. Debye model fit to inn MSRD data from Al metal. Absolute offset
of data was allowed to float. Fit gave 9, = 438+ 13 K.

158

Peryprrrryprrrr per rrperrrprrrryprerrerpey

. =

7E 2

— & :
a a 3
o(op) m -
A 7 7
NO a 3
Fo r
b&b 4eE -
E ;

; :

3E 4

r 4

Ss -_

r i"
OErrritiiirit irri trp er te PP

50 100 150 200 250 300 8 350

oO

Temperature (K)

Figure 5.12. Debye model fit to 1nn MSRD data from Fe metal. Absolute offset
of data was allowed to float. Fit gave 9, = 417 + 22 K.

159

qT J tT [ T Tf fF t | t T TT | | oe a | T qT t 1 | qT T ui t | is
6b 4
5E 4
—< UE =
9 . :
= 4f 3
an:
o£ E
SE 4
: l i 1 iT | I Lse4Le | l i Lut | L I 1 L Ls i L 1 | L 1 I ] | oq

0 50 100 150 200 250 300

Temperature (K)

Figure 5.13. Debye model fit to 1nn MSRD data from Pd metal. Absolute offset
of data was allowed to float. Fit gave 9, = 306 + 40 K.

160

al. (1989) obtained a value of 6p = 415 + 30 K from the temperature-dependent
EXELFS of Al.

@p derived from MSRD measurements are expected to be different from
@p derived from MSD or heat capacity measurements. Each measurement can
be thought of as placing emphasis on different regions of the frequency
distribution of vibrational modes. MSD data emphasizes the lower-frequency
modes more than MSRD or heat capacity data. Moreover, 6p are usually
derived from heat capacity data by matching the data near the point where the
heat capacity is about half the Dulong and Petit value. Obviously, 8p must be
determined from MSRD and MSD data using a completely different
methodology. Despite these differences, the values for 6p determined from
these measurements should be roughly comparable. From heat capacity
measurements, @p = 394 K for Al, 420 K for Fe, and 275 K for Pd (Seitz and
Turnbull, 1956); these values are roughly comparable to those from my MSRD

measurements, which were 438, 417, and 306 K, respectively.

5.4.3. Force Constant Analysis

The force constant model discussed in §5.3 uses interatomic force
constants from inelastic neutron scattering experiments to determine the
frequencies and polarizations of the 3N vibrational modes in a crystal. Unlike
the Einstein and Debye models, the force constant model does not have any
"free" parameters because all the necessary parameters are determined from
the neutron scattering data.

Table 5.1 contains interatomic force constants for Al, Fe, and Pd metals
which were derived from neutron scattering data. The density of vibrational

modes, g(w), can be determined from these force constants using my program

161

fcc bec
force const __Al Pd force const Fe
110 XX 10.46 19.76 111 XX 16.88
ZZ -2.63 ~-2.51 XY 15.01
XY 10.37 23.19 200 XX 14.63
200 XX 2.43 0.92 YY 0.55
YY -0.14 0.42 220 XX 0.92
211 XX 0.099 0.91 ZZ -0.57
YY -0.24 0.13 XY 0.69
YZ -0.29 0.61 311 XX -0.12
XZ -0.18 0.91 YY 0.03
220 XX 0.14 -1.04 YZ 0.52
ZZ 0.19 -0.13 XZ 0.007
XY 0.38 -1.86 222 XX -0.29
310 XX -0.30 0.09 XY 0.32
YY 0.18 -0.23
ZZ 0.26 -0.27
XY -0.32 0.12
222 XX -0.14 0.22
XY 0.20 0.15
Table 5.1. interatomic (Born-von Karman) force constants (in N/m) for the first

several near-neighbor shells in Al (Cowley, 1974), Fe (Minkiewicz
et al., 1967), and Pd (Miiller and Brockhouse, 1971).

162

0.6 TIUPTUUTepereryurerypereryp errr prre rye rere ypererpereryrrreyrrrryy

- 4

= ( a

F J

0.56 4

F 7

ro} 0.45 4

fa) “+ T- a

— * =~

— be ~

oO = 4

@ C J

Yn C 4

2 O3E 4

o) c 4

boa 3 a

~—— base ~~

— EC 7

f 0.2E- 4

oO 7 2

0O.1E- +
0.0 Hemet Ta ela et cori

« (10'° rad/sec)

Figure 5.14. Density of vibrational modes for Al metal determined from
interatomic force constants. Breakdown into longitudinal
and two transverse branches is indicated.

163

0.74

0.6

0.5

0.4

0.3

g(@) (10°'? sec/rad)

0.2

0.4

0.0!

pi PPrprrrrp ert

PRreryprrrryprrrryprrrst

PEVPSVUr PU ere prr ery err rp er eeprre rp rr erp errr pr rrr perry yp rr rry rt

piretiririrtirrsritiprrtirir tp pps tia,

c (10'° rad/sec)

Figure 5.15. Density of vibrational modes for Fe metal determined from

interatomic force constants. Breakdown into longitudinal
and two transverse branches is indicated.

164

PPP rp rer ryprrrryperrryprrrrprrrryprreryrrrvyprrrrprryr ro
- a
0.6£ E
0.5E- J
- q
So FE ;
S 046 =
Oo O3E J
- : 7
= F E
E E
O.1F£ 4
E J
a z
0.0¢ J

w (10'° rad/sec)

Figure 5.16. Density of vibrational modes for Pd metal determined from
interatomic force constants. Breakdown into longitudinal
and two transverse branches is indicated.

165

listed in §C.3. Figures 5.14 through 5.16 display g(@) for Al, Fe, and Pd metals.
in each case, the breakdown into the longitudinal and two transverse branches
is indicated.

Figures 5.17 through 5.19 show the projected density of vibrational
modes, 91nn(@), for the 1nn shell in Al, Fe, and Pd metals. In comparison to the
density of modes, the projected density of modes weights more heavily the
higher frequency modes. In particular, the high frequency (or equivalentiy the
short wavelength) longitudinal modes are most heavily weighted. This is as
expected because the short wavelength longitudinal modes contribute most
heavily to the MSRD between inn atoms.

Applying Equation (5.14a), these gjnn(@) can be used to determine the
vibrational MSRD 6,2, as a function of temperature. Figures 5.20 through 5.22
shows 6,2, calculated from the force constant models for Al, Fe, and Pd. My
experimental data are superimposed for comparison, and they match well with

the predictions of the force constant models.

0.7

0.3

0.2

9(), G4nn(@) (10°'? sec/rad)

0.1

0.0

166

bee
bee

POTPUPerpereryperreyrerepererperrryrrreyp rere perv rrperergpreregag

Erstr terri sr tirirrtrrrir tip pp typ pp ity ty

ow
1 Lisette Cis tittle

1 2 3 4 5 6
w (10°° rad/sec)

Figure 5.17. Projected density of vibrational modes for inn shell (dashed

line) compared with density of vibrational modes (solid line)
for Al metal.

167

pT ES SSTTETTTTTETTTUCTTTETUT TTT eee rrr errr ry errr prt rr yt

lid

1.2

1.0

0.8

0.6

0.4

G(®), Ginn() (10°'? sec/rad)

0.2

etrsitirritiiritiiretissitissibossstereebesstiptiliseetiris

SRERERSEa

bo} +

0.0

1 2 3 4 5 6
wo (1 0° rad/sec)

Figure 5.18. Projected density of vibrational modes for 1nn shell (dashed
line) compared with density of vibrational modes (solid line)
for Fe metal.

168

PRVrVyprreryperrrperrryurrrprrrrypererperrvyprrereprery

1.0

0.8

0.6

0.4

9(®), J4an() (10°'? sec/rad)

ho

TETUTETUPTP UTP TTT APT e Try errr perry errr yr rrr irre

veritrsritisritieer distr tare bts ttespttasettees dias

Lindt et rirtrisr tres tir tr trp trtisrtr tsa Litt

1 2 3 4
@ (10'° rad/sec)

ba

ad
ro)

© mrt
oT

Figure 5.19. Projected density of vibrational modes for 1nn shell (dashed
line) compared with density of vibrational modes (solid line)

for Pd metal.

169

rerrprrrryprrrrprrrryprrrrprerryprrrryp rrr yprrrry ry
18- 4
: 1
16b »
arn 14h ~ _
ot ‘on 4
oO ~ 4
= 12E =
A L a
= - 4
& 10F +
BE J
[ =
6- _
Peseta dtd tt lipid ilar ty

0 100 200 300 400

Temperature (K)

Figure 5.20. Force constant model prediction of 1nn MSRD in Al metal.
Experimental data superimposed for comparison, after adjusting
absolute offset.

170

SE ee ee
= “
7E a
of F :
©? 5k 4
ae = :
a -£
E C 7
o 46 4
3E 4
OErsri teres tipi te de te td
0 50 100 150 200 250 300 350

Temperature (K)

Figure 5.21. Force constant model prediction of 1nn MSRD in Fe metal.
Experimental data superimposed for comparison, after adjusting

absolute offset.

171

ne ee Le
6F =
F :
- a
5E rm
~ &£ :
< :
<< OE :
° ' q
= 4E :
& -
Ne 5 0
Cc = |
bob &F :
3F =
a J
Y q
2eb a
ee ee Dc
0 50 100 150 200 250 300

Temperature (K)

Figure 5.22. Force constant model prediction of 1nn MSRD in Pd metal.
Experimental data superimposed for comparison, after adjusting
absolute offset.

172

Chapter 6 Applications to Intermetallic Alloys and Nanocrystalline
Materials
EXELFS can be applied to problems in materials science which utilize its
sensitivity to local atomic environments. §6.1 presents measurements of
chemical short-range order (SRO) and vibrational MSRD in FesAl and NisAl
using EXELFS. Differences in vibrational entropy between the ordered and
disordered alloys are discussed. §6.2 presents measurements of structural

disorder and vibrational MSRD in samples of nanocrystalline Pd and TiOo.

6.1 Chemical Short-Range Order (SRO) and Vibrational MSRD in
Fes3Al and NisAl
This section presents EXELFS measurements from FesAl and NisAl
alloys which were chemically disordered by piston-anvil quenching and high-
vacuum evaporation, respectively. Chemical SRO was observed to increase as
the as-quenched samples were annealed. Temperature-dependent
measurements indicated that the local environments of the annealed samples

were "stiffer" than those of the as-quenched samples.

6.1.1 Feg3Al

Figures 6.1 displays the phase diagram for Fe-Al. The phase diagram
shows that near the Fe-25at%Al composition, the equilibrium phase for the alloy
below about 500 C is the intermetallic compound FesAl. Intermetallic Fe3Al has
the DOs ordered structure which is displayed in Figure 6.2. The DOs structure
can be thought of as consisting of four interpenetrating fcc sublattices, one of
which is occupied by Al atoms. Basically, the Al atoms tend to repel each other

as either first or second nearest neighbors.

Temperature °C

173

Weight Percent Aluminum

1600- 10 20 30 40 S060 70-80-90 100.
* 31598°c T T T T T rT T T —t
; 9,
1400-41394°C L a
1200 ° ;
{p95 E1157°c 7
; (7Fe) ‘
1000-1 otis E
dy. t & é
Hott
je12°c Ho etn 4
q t t i
HoTE
800-4770°C uote :
; u tte
] Magnetic ~~ ftw
| Transformation “S. oats 652°C 660.452°C
600- ‘¥ Coherent rH ite 00.1F
“Equilibria uote F
t 1
} £ P| t {
4004 OUST ES (al) E
; . cae hoot he ;
7 é t * 4 t et «# r
7 tot Ly ‘ ' fr oat 8 r
3 oot i 4 ‘ en q
1 tot 4 { H Notte ’
200 peeeeerre preceded — aaa ~ T Ay wer He pert verTY T
0 10 20 30 40 50 60 70 80 90 100
Fe Atomic Percent Aluminum Al

Figure 6.1. Phase diagram for Fe-Al (Massalski, 1986).

174

Figure 6.2. DOs ordered structure of Fe3Al.

175

Piston-anvil quenching, described in §3.1, cools metals at rates on the
order of 106 K per second. This cooling rate is rapid enough to preserve a
significant amount of chemical disorder in the as-quenched bcc Fe3Al samples.
As shown in §3.2, the lack of superlattice peaks in x-ray diffraction spectra
indicate a lack of long-range order in the as-quenched samples, while
Méssbauer spectrometry shows a lack of short-range order. Both short-range
and long-range order evolved when the as-quenched samples were annealed
at 300 C.

Since EXELFS is sensitive to the chemical composition of the near-
neighbor environment surrounding the central atom, Table 6.1 lists the average
number of 1nn and 2nn Fe atoms surrounding either Al or Fe central atoms in
disordered and ordered FesAl. The number of neighboring Fe atoms is
important because the backscattering in FesAl is dominated by the heavier Fe
atoms. It is interesting to note that for both 1nn and 2nn shells, when going from
disordered to ordered Fe3Al, the average number of Fe neighbors surrounding
Al central atoms increases by one third, and the average number surrounding
Fe central atoms decreases by one ninth.

To show the sensitivity of EXELFS to chemical SRO, the theoretical
contribution to the EXELFS from the 1nn shell was calculated for completely
disordered and perfectly ordered Fe3Al. The calculations were made simply by
substituting phase shifts and scattering amplitudes from Teo and Lee (1979)
into Equation (2.50). For purposes of illustration, Figure 6.3 displays the
theoretical Al K EXELFS signal from completely disordered Fe3Al which has an
average of 6 Fe and 2 Al inn atoms. It is seen that the 2 Al component of the
signal destructively interferes with the dominant 6 Fe component. Fourier

transforms of the theoretical EXELFS were taken over ranges in k-space which

176

Average number of 1nn Fe atoms

Al central atom Fe central atom
disordered FesAl 6 6
ordered FesAl 8 5.333

Average number of 2nn Fe atoms

Alcentralatom § Fe central atom
disordered Fes3Al 4.5 4.5
ordered FesAl 6 4

Table 6.1. | Average number of inn and 2nn Fe atoms surrounding Al and Fe
atoms in completely disordered and perfectly ordered FesAl.

Figure 6.3.

0.6

0.4

177

5 6 Fe—*"* 4
1 é

P - ras 6 Fe + 2Al 4
bg 4

- ' 1 r ~

S t t A 1 ay ~q

5 we\ ' Va a

‘ ' \ ‘ \ «

m7 t ‘ 4 \ oun

ma U 1 1 ‘ 14

LA | 1 1 1 4

\\ ! 4 a

Ls 8 om, 4

fr 7

5 ry t f f -

\ t 4 t f

rT 1 ’ j t f 7

- 1 ! fi p 4

= 1 ' 2 Al ! f Jt —

a ‘oa 1 ' t v/é -

toe \ }! \ ft -
5 va \ /! ~ 7
~ Na iW! Vi oe |
qt ~
jp 1 —
. i
= _
v/
L. ~
sotrlirertirrr ter erterer teri titte tit tt

5 6 7 8 9 10
k (A)

Theoretical Al K EXELFS signal from disordered Fe,Al.

The 1nn shell consists of an average of 6 Fe and 2 Al
atoms. Signal is broken down into its two components.

178

are similar to the corresponding ranges in the experimental data. Figure 6.4
presents the magnitude of the FT of the theoretical Al K and Fe Lo3 EXELFS
signals. In going from disorder to order, the height of the 1nn peak increases for
the Al K EXELFS (which correspond to Al central atoms) and decreases by a
smalier amount for the Fe Lo3 EXELFS (which correspond to Fe central atoms).
This result makes sense intuitively when one considers the 1nn shell
occupancies given in Table 6.1.

Figure 6.5 displays EELS measurements of the Al K and Fe L edges from
an electropolished sample of piston-anvil quenched FesAl. Figures 6.6 shows
the Al K and Fe L23 EXELFS from a sample of the as-quenched FesAl at 296 K.
Figure 6.7 compares the magnitude of the FT of the EXELFS from the sample
as-quenched and after it was annealed in situ at 300 C for 10 and 30 minutes.
The positions of the experimental nearest-neighbor peaks are in good
agreement with the theoretically calculated positions for the 1nn peaks shown
in Figure 6.4. Moreover, after annealing, the increases in the height of the inn
peak of the Al K EXELFS is accompanied by smaller decreases in the height of
the 1nn peak of the Fe Lo3 EXELFS. Figure 6.8 displays the change in the
EXELFS amplitudes as a function of annealing time. The quantitative
determination of order parameters from these results is complicated by the
changing vibrational characteristics of the local environment as the alloy orders.
If the local environment stiffens as the alloy orders, then the size of the nearest-
neighbor peaks for the annealed samples would increase. Taking this effect
into account, my results are consistent with the results from the Méssbauer
spectrometry experiments discussed in §3.2. My EXELFS results indicate that
the piston-anvil quenched FegAl develops partial short-range order after

annealing at 300 C for 10 and 30 minutes.

|FT(kx)|

|FT(kx)|

Figure 6.4.

179

2.0 Li T 7 qT J T 7 T Lj | , T Li T | i LJ T qT I t t T T | qT T T T
' \—~ordered

1.5
1.0

0.5

pirtirisrtiri ir ti iy, |

0.0

oO

ee

1.2
1.0
0.8
0.6
0.4
0.2

0.0 =
0 2 4 6

A)

rohoritrsitirr tis tiii ty

STEP Prep rrryp rr ry rrr ytrty

cot

~~

Magnitude of FT of theoretical (a) Al K and (b) Fe L,, EXELFS
from inn shell of completely disordered and perfectly ordered
FeAl. Transformation range 5
6.5
180

1.4x10" rrp rrr pre ry es TT perre | ae ry rq Le
1.2- 4
1.0F 4
= . 7
o c
E O8F F
0.2F
Fela ian reortbrprertirrarrtu rity ft bed

1500 1600 1700 1800 1900 2000 2100

Energy Loss (eV)

MAS BR Eee eee eee
8x10° 4
> bf (b) Fe L 7
oO at 1
= ' 7
[ A
al a “
eta tp let pt tit lisp tte plete ta

800 1000 1200 1400
Energy Loss (eV)

Figure 6.5. EELS measurements of (a) Al K and (b) Fe L edges from Fe.Al.
Spectra were not deconvoluted.

181

Peprrerprrrrperrreprerrr perryrt

0.4 (a) AIK

0.2

0.0

kx(k)

pose tir y, |

ee ee

poeraetirrisrtisrrsrtrrirr fyi, fy

5 6 7 8 9 10
k (A")

PEEP Peer prrrrprurrryrr rr yr ryt

(b) Fe Log

IN)

hk

Trprerryprrrryprrty

kx(k)

Oo 5
iD
Tt rrt

Petia tay

pore tiritrtirirrirtirre tp pip tp fp yp py ly

7 8 9 10 11 12
k (A!)

Oo) TTP

Figure 6.6. (a) Al K and (b) Fe L,, EXELFS from as-quenched Fe,Al
at 296 K.

|FT(kx)|

|FT(kx)|

Figure 6.7.

182

re a oe | | Tre | TTT T q Tryst | Ln et ee t TT iPgei | orgs J Lae i | "7
0.5F {300 C for 30 min 7
a ro 3
0.4f 1 OT 300 C for 10 min 4
E i j as-quenched :
0.3F ! ae
r (a) Al K 3
0.2F i 4
" us! Ot en J

1 7N l \ \ ? Vv \ (ke Lt a

S v WA A vA “4
0.0 & aes were a ee i re 2 Se
0 2 4 6 8

r (A)
0.35 pepe epee
y - hed
0.30L as-quenc |
3 300 C for 10 min

0.25 - 300 C for 30 min -
0.20} =
0.15 (b) Fe Ly, -
0.10 _
0.05 _
0.00 eee Trey
0 2 4 6 8

r (A)

Magnitude of FT of experimental (a) Al K (5(b) Fe Lz (6.5
and after annealing jn situ at 300 C for 10 minutes and 30
minutes. Data taken at 296 K.

183

40 F Por oF | Tt fT | TT tT fT if Tort ] TT TF | oF FT T | or oe | gy

So 30F =
= : Al K——__—_> :
< F :
“) 20- 4
LL o£ i
| _ =
we 10 :
cC _ =
= : ' :
& 7
Fe L —_—-f :

-10 Lt | | i oe ee | ‘a a | Ll! | oon ee | | Lot tt tL | on oe ce Lo it :

0 5 10 15 20 25 30 35

Annealing Time at 300 C (minutes)

Figure 6.8. Change in inn EXELFS amplitudes as function of annealing
time at 300 C for piston-anvil quenched Fe,Al sample. Error

bars obtained from values at which variance of least-squares
fit increased by 20%.

184

Temperature-dependent EXELFS measurements can be used to probe
vibrational characteristics of local atomic environments. Measurements of local
vibrational characteristics can be used to estimate the vibrational entropy of a
material.

Consider comparing a state, a, of a material having 3N vibrational modes

(of, @3, ..., @S) to another state, B. In the classical (high temperature) limit, the

difference in vibrational entropy between the two states is:

SN 3N
ASvibr = SB. -—S,% = kp In nN = kp In oN (6.1)

vibr
To} (oP

7a

where the correspondence between characteristic frequencies w and
characteristic temperatures 0 is made using Aw = kp@.

From my temperature-dependent EXELFS experiments on FesAl and
NisAl, | obtain local Einstein temperatures of the each atomic species in the two
states (disordered and ordered) of the material. In the Einstein model of a solid,
each atom behaves like three independent harmonic oscillators and so
contributes three of the total 3N vibrational modes of a solid. Therefore, within

the Einstein model, Equation (6.1) applied to Fe3Al or NisAl becomes

dis _ cord 3, (Oren) 1, (ORF
ASvibr = Syibr — Svibr = SNkg | Zin ais |t 4!" ais (6.2)
Fe/Ni Al

185

Since the correspondence between local Einstein frequencies and the
frequencies of the normal modes is very rough at best, Equation (6.2) is
expected to be only qualitatively useful.

The above approach is a mean-field approximation. Another approach
would be to interpret my EXELFS results within a pair approximation. Instead of
considering individual atoms, a pair approximation considers the interatomic
bonds between each pair of 1nn atoms.

For a binary A-B alloy, there are three different types of bonds: A-A, B-B,
and A-B bonds. For an alloy that develops chemical order, we expect the A-B
bonds to be stiffer than the A-A and B-B bonds (i.e. @,4, > @aa@pp).

Table 6.2 gives the fraction of each type of 1nn bond in completely

disordered and perfectly ordered FesAI (or Ni3Al).

Fraction of 1nn bond type
disordered FesAl (or Ni3Al) 3/8 9/16 1/16
ordered FesAl (or Ni3Al) 1/2 1/2 0

Table 6.2. Fraction of each type of 1nn bond in completely disordered and
perfectly disordered FesAl (or NisAl).

Allowing the frequencies @aiFe, OFeFe, 2Nd Maia; to be dependent on the state of
order in the alloy, then the change in vibrational entropy between perfectly

disordered and ordered FesAl becomes in the classical limit

186

(0%) (08tte)

, , | (6.3)
(On. ye (0%$-. ye" ° (Off yn °

EXELFS is more sensitive to the heavier Fe neighbors than lighter Al
neighbors. In fact, a first-order approximation would be to ignore the
backscattering from the Al neighbors. In that case, the temperature-dependent
Al K EXELFS measures values for 8,,-,, and the temperature-dependent Fe Lo3
EXELFS measures values for @Fere. Only values for Oaia) are not measured by
EXELFS and must therefore be estimated.

Figures 6.9 and 6.10 presents the magnitude of the FT of the Al K and Fe
Log EXELFS from as-quenched and annealed samples of FesAl at temperatures
from 97 K to 348 K. Figures 6.11 and 6.12 display Einstein temperature fits to
the Al K and Fe L23 1nn MSRD data from as-quenched and annealed samples.
The MSRD data indicate that the local environments of both Al and Fe atoms in

Fe3Al become "stiffer" as the alloy orders.
Using the mean-field approach, | substitute 6%¢ = 460 + 50 K, es = 391 +

45 K, 6? = 430 + 30 K, and eds = 369 + 20 K into Equation (6.2) to obtain

ASvipr = [0.12 + 0.12 (Al)] + [0.34 + 0.20 (Fe)] kp/atom
= +0.46 + 0.23 kp/atom (6.4)

Using the pair approach, | substitute 007% = 460 + 50 K, ogis = ods, = 391 + 45

K, O28, = 430 + 30 K, and 6,45, = 369 + 20 K into Equation (6.3) to obtain

ASvibr = +0.48 + 0.25 kp/atom (6.5)

187

0.6 Ln ae ee a | tTrrtT purr i] Pe Try pure s | prYe pry

0.5 4

_ 04 4

=< 0.3 fi 4

Tae i (a) aS-quenched 4

— 0.2 : =

O1F fe 4 NN J

0.0 a o poiuriztias j aA “A poirt a bent eit ees

0 2 4 6 8

r (A)
0 7 i rary oe ee Tt prriyproreri 1! TRrporeig | ure fl md
. 97 K

x<
on
Land

Figure 6.9. Temperature dependence of magnitude of FT of Al K
EXELFS (5(b) after annealing at 300 C for 30 minutes.

188

0.30 = rerypuedre | ie J TTrgTe | teTig¢ t Tern | Ln ae cyt tq "y
0.256 4
_ 0.20E 4
coy " 7
<= 0.15E 4
Ld 0.10E (a) as-quenched 2
0.05E 4 2
0.00 Ex wea
0 2 4 6 8

r (A)
0.46 193 K 4
: 296 K :
Be O8p 348 K 7
SS c 1
t 7 4
ue 0.25 (b) 300C for 30min 7
ot i
0.0 S ne TS emits itn :
0 2 4 6 8

r (A)

Figure 6.10. Temperature dependence of magnitude of FT of Fe L,,

EXELFS (6.5 < k < 12 A’) from (a) as-quenched Fe,Al and
(b) after annealing at 300 C for 30 minutes.

189

_' rrryprrry if Prrrgeodgred | PrergryT € tT eT | Trrtyg gigi "44
cet F @ as-quenched a Z
© a 4
=) c :
~ SE =
N c : 2
= 7 4
6 : :
poe ° = 300 C for 30min]
3E q
7 perterr rp torr ir trprr tip ypr tipi rt tise ly Lt

0 100 200 300 400

Temperature (K)

Figure 6.11. Einstein model fits to Al K EXELFS 1nn MSRD data from as-
quenched Fe,Al and after annealing at 300 C for 30 minutes.

Absolute offsets of data were allowed to float. Fits gave 0, =

391 + 45 K for as-quenched Fe,Al and 9, = 460 + 50 K after
annealing.

190

. T. as
SE eo” 7
J “ 7
. ¢
9 4r | @ as-quenched|] “
mT J =
“ s _
= 5 J
6 ' 4
3h of 7
Fee m 300 C for 30 min] 7
a 7
eet er tt tp tt tt tt hl ll
0 100 200 300 400

Temperature (K)

Figure 6.12. Einstein model fits to Fe L,, EXELFS inn MSRD data from as-
quenched Fe,Al and after annealing at 300 C for 30 minutes.

Absolute offsets of data were allowed to float. Fits gave 0. =

369 + 20 K for as-quenched Fe,Al and 0, = 430 + 30 K after
annealing.

191

Note that in the above calculation, 6,/§, was estimated to be the same as 04s.

Both the mean-field and pair approaches give approximately the same
value for the difference in vibrational entropy between disordered and ordered
Feg3Al. While these values ASvyipr are slightly less than the configurational
entropy of mixing for the A3B alloys (+0.56 kp/atom), they are large enough to
affect the relative thermodynamic stabilities of the disordered and ordered
states of FegAl.

The sign of AS vip; indicates that the vibrational entropy of the disordered
state is greater than the ordered state. This would suppress the critical
temperature for ordering in theoretical calculations of phase diagrams because
the reduced entropy of the ordered phase would make it less stable at higher

temperatures.

6.1.2 NisAl

Now consider my EXELFS measurements from Nig3Al. Figure 6.13
displays the phase diagram for Ni-Al. The phase diagram shows the
intermetallic compound NisAl near the Ni-25at%Al composition. Intermetallic
NizAI has the L12 ordered structure which is displayed in Figure 6.14. The L12
structure is an fcc lattice where the Ni atoms occupy the face sites and the Al
atoms occupy the corner sites. This maximizes the number of unlike 1nn atoms.
Table 6.3 lists the average number of 1nn Ni atoms surrounding either Al or Ni
central atoms in chemically disordered and ordered Ni3Al.

The critical temperature for the L12 ordering of NizAl has been estimated
to be near its melting temperature of 1385 C (Corey and Lisowsky, 1967; Cahn

et al., 1987a; Bremer et al., 1988). This high critical temperature prevents the

192

Weight Percent Nickel

Figure 6.13. Phase diagram for Ni-Al (Massalski, 1986).

0 20 30 40 50 60 70 80 90 100
1800 ; 1 t 1 7 I 1 H 7 4 1 iT . 7
1600 L E
1455°C
14004 .
j AINi 1995
is)
fo}
- 2 12004
3 0,
—_
oO
fu
a.
10004
Pa
854°C
] 660.452
800-4
; ~700°C
; 639,9°C
6004 _ ~
400 t T T T T T rT
0 10 20 30 40 50 60 70
Al Atomic Percent Nickel

193

O Ni
@ A

Figure 6.14. L1». ordered structure of Ni3Al.

Average number of 1nn Ni atoms

Alcentralatom § Nicentral atom
disordered NisAl 9 9
ordered NigAl 12 8

Table 6.3. | Average number of 1nn Ni atoms surrounding Al and Ni atoms in
completely disordered and perfectly ordered NigAl.

194

preparation of disordered fcc Ni3AI by piston-anvil quenching from the melt
(Inoue et al., 1983; Horton and Liu, 1985; Cahn et al., 1987b). High-vacuum
evaporation, described in §3.1, cools metals at extremely high effective
quenching rates. As discussed in §3.2, high-vacuum evaporation can
successfully prepare disordered samples of fcc NisAl.

Figure 6.15 presents the magnitude of the FT of theoretical Al K and Ni
Log EXELFS from completely disordered and perfectly ordered NisAl. In going
from disorder to order, the height of the 1nn peak increases for the Al K EXELFS
and decreases by a smaller amount for the Ni Lo3 EXELFS. This result makes
sense intuitively given Table 6.3 and is similar to the calculation for Fe3Al
shown in Figure 6.4.

Figure 6.16 displays EELS measurements of the Al K and Ni L edges
from a sample of evaporated Ni3Al. Figure 6.17 shows the Al K and Ni Los
EXELFS from the sample of as-evaporated NisAl at 296 K. Figure 6.18
compares the magnitude of the FT of the EXELFS from the sample as-
evaporated and after it was annealed in-situ at 150 C for 70 minutes. The
increase in the height of the 1nn peak of the Al K EXELFS is accompanied by a
smaller decrease in the height of the 1nn peak of the Ni Loz; EXELFS. Figure
6.19 displays the change in EXELFS amplitudes as a function of annealing
time. My EXELFS results indicate that the evaporated Ni3Al undergoes short-
range ordering at the relatively low temperature of 150 C. This evidence
supports the hypothesis that the relaxation observed near 150 C in Figure 3.7 is
associated with the onset of short-range ordering.

Figures 6.20 and 6.21 present the magnitude of the FT of the Al K and Ni
Lo3 EXELFS from as-evaporated and annealed samples of NisAl at

temperatures from 105 K to 295 K. Figures 6.22 and 6.23 display Einstein

195

T T | { t i oo | Lf | rot i om if qT t i | if qT T T J | Lm i a

@ 20 E 1 ordered E

= : i ' :

| i ! 1 4

Te) - H isordered 7

@& r ! J

= 10 = ' 4

a C ; (a) Al K J

- r ! q

— C y 7

oF :

0 1 2 3 4 5 6

r (A)

i 1 LJ tT | | oo | LI T ] , 7 | roe if Ll t tot ] Tf tT Li if TF TT T J

@ job a

Cc _ 4

aad wal -_

2 L 1

& L 4

ROT 7

wf ]

0 L 4

0 1 2 3 4 5 6

Figure 6.15. Magnitude of FT of theoretical (a) Al K and (b) NiL,, EXELFS
from inn shell of completely disordered and perfectly ordered
Ni,Al. Transformation range 4
8.5
196

3.0x10° FPPUPUCreprrenprervepurerprreegerecprereyrrevyrernyygy a
25b 4
: q
- a) AIK J
20k (a)
2 155 :
g Ee F
£ C J
1.06 aa
0.5 =
0.0 Feeestebertrtet tt
1400 1600 1800 2000 2200 2400
Energy Loss (eV)
3.5x10 TOT Ty
ary
‘DO
O.Q teed te te

800 1000 1200 1400
Energy Loss (eV)

Figure 6.16. EELS measurements of (a) Al K and (b) Ni L edges from
Ni,Al. Spectra were not deconvoluted.

197

F Lj { ite vy tc TT 7 if TT £ i tt rt 1 TU t t Fegre ] TT

0.046 (a) AIK
0.02 j
<= 0.00F J
-0.02F J
-0.04 is! [no oe | I £op op | Liss | | on on | | Lieu ft I [on on ra | LL

4 5 6 7 8 9 10
k (A")

phirri tise tir di sired lt

9 10 11 12 13
k (A)

Figure 6.17. (a) Al K and (b) Ni L,, EXELFS from as-evaporated Ni.Al
at 105 K.

|FT(x)| (arb. units)

FT (x) (arb. units)

Figure 6.18. Magnitude of FT of experimental (a) Al K (4
id
oO

Trrprirye | org ty t on oe | TrPrgprrere if Cree qT cre a

c i 150 C for 70 min :
1.5 : -
r as evaporated :
1.0 =
0.5 a
* 4

a ea ee 7

0.0 Marte MY =
0 2 4 6 8

r (A)

TrTrry rg a | PRrrigprerere if PRrrryprrede if PRrrrperyrid.
0.4- -
r 1

he +

. -
0.3- —
r |
0.2F 4
0.1F 4
r =
0.0 re Fee de ee eee toe: 4
2 4 6 8

r (A)

198

(b) Ni Lo; (8.5Ni,Al and after annealing in situ at 150 C for 70 minutes.
Data taken at 97 K.

199

S& 30E 4
C 4

s F :
= C Al K > 7
20 =

< r
op) C q
Le — 4
—! r 4
Gj TOF 1
- c 7
c - 4
< F a :
) Om- .
> Ff
£ P 4 4 Ni Log - 3
) : 1
-1Q Cosestrrirtirirtiriitipietisir tipi rtiiny

0 20 40 60 80

Annealing Time at 150 C (minutes)

Figure 6.19. Change in inn EXELFS amplitudes as function of annealing
time at 150 C for as-evaporated Ni,Al sample. Error bars

obtained from values at which variance of least-squares fit
increased by 20%.

200

I a ee

0.08 - 105 K
0.06 F
S r
Fe 0.04
ad F (a) as-evaporated
0.02 F
r Ni
.e Yor
0.00 put, MM, iy there S
0 1 2 3 4 5
r (A)
0.08 L 105 K
0.06
R fT
tr 0.04 (b) 300 C for 60 min
0.02
- YY“ Sw
0.00 Lee TM tt tes
0 1 2 3 4 5
r (A)

Figure 6.20. Temperature dependence of magnitude of FT of Al K
EXELFS (4and (b) after annealing at 300 C for 60 minutes.

201

a , FF | tT Tt Ff | 1 TTF i T T TT | TUR Tg TT TOT fo OF 7

0.06 ; 105K q
0.05 F 198 K 7

= 0.04F 4
cos 5 95 K :
im 0.03F (a) as-evaporated4
0.02 F- 4
O01E ’ \ f\ ~~

a e * ‘ XQ farm

0.00 rap pt tt tl ty th a LY
0 1 2 3 4 5 6

r (A)

ei rerprnrye ry rrrry rer vy rrTrry ree | TTrrTpery®e M™

0.06—E 105 K 4

C ? 3

0.05— i 148K 2

' 233 K 4

= 0.046 MI 4
RE i cit —296 k :
— E / FA (0) 300 C for6O min 43
0.02 = ‘ 3 ‘ 7

C ,o"* 4 ‘y a

ria { Soh a prryin ahs Woes. & _

0.00 Exsoetoris boii bur Lope pdeies “Le —
0 2 4 6 8

r (A)

Figure 6.21. Temperature dependence of magnitude of FT of Ni Lz,

EXELFS (8.5 and (b) after annealing at 300 C for 60 minutes.

Ginn (10° A’)

202

PEEP Eee rprreryprrrvryr rr rp rrr ry rrr ryt

“4
“4
~q
~q
«4

@ as-evaporated aa

pherirtirrr tip ri tipped,

m 300 C for 60 min

proiterirtiy

l I i l ] 1 L | i | j

—_
Oo
oO

200 250 300

Temperature (K)

Figure 6.22. Einstein model fits to Al K EXELFS 1nn MSRD data from as-

evaporated Ni,Al and after annealing at 300 C for 60 minutes.
Absolute offsets of data were allowed to float. Fits gave 6, =

312 + 35 K for as-evaporated Ni,Al and 9, = 453 + 30 K after
annealing.

203

a q qT i lj | t J li t | mo tT i t | t T q qT 7
7h :
of. wo te

on o£ ® as-evaporated ae Z
—t OF :
°2 c 7
= 5b 3
rat] C J
Ee Ff J
bo OC :
ab j

5 m 300 C for 60 min E
3 -

100: 150 200 250 300

Temperature (K)

Figure 6.23. Einstein model fits to Ni L,, EXELFS inn MSRD data from as-
evaporated Ni.Al and after annealing at 300 C for 60 minutes.

Absolute offsets of data were allowed to float. Fits gave 0. =

279 + 45 K for as-evaporated Ni,Al and 9, = 304 + 20 K after
annealing.

204

temperature fits to the Al K and Ni Lo3 1nn MSRD data from as-evaporated and
annealed samples. The MSRD data indicate that the local environments of both
Al and Ni atoms in Ni3Al become "stiffer" as the alloy orders. The "stiffening" of
local environments is similar to that which | observed in FesAl.

First, using the mean-field approach, | substitute 6% = 453 + 30 K, ods =

312+ 35K, @9(¢ = 304 + 20 K, and efi$ = 279 + 45 K into Equation (6.2) to

obtain

ASvibr = [0.28 + 0.10 (Al)] + [0.19 + 0.39 (Ni)] kp/atom
= +0.47 + 0.40 kp/atom (6.6)

Second, using the pair approach, | substitute 09"5. = 453 + 30 K, Ogis, = esis. =
312+35K, O04 = 304 + 20 K, and 6,7S. = 279 + 45 K into the appropriate form

of Equation (6.3) to obtain

AS vipr = +0.71 + 0.38 kp/atom (6.7)

Like the analogous calculation for FesAl, the above calculation for Ni3Al
assumes that 64/5, can be estimated by 64/5.

The pair approach gives a larger value for the difference in vibrational
entropy between disordered and ordered NisAl than the mean-field approach.
This is because the pair approach puts greater emphasis on the contribution
from the Al K EXELFS data. While the mean-field approach puts three times
more weight on the Ni Leg EXELFS than on the Al K EXELFS, the pair approach
weights the Al K and Ni L23 EXELFS almost equally. | believe that the pair

approach is more correct.

205

Anthony et al. (1993) measured ASvjip, for disordered and ordered NisAl
using low temperature calorimetry and temperature-dependent x-ray
diffractometry. The calorimetry measurement gave ASvip; = 0.3 kp/atom and the
x-ray measurement gave ASyip; = 0.7 Kp/atom. The accuracy of the x-ray
measurement was questionable, however.

It may be expected that ASyip, would increase with the enthalpy of
ordering for an alloy. Using this criterion, one would expect ASvip; to be larger
for NigAl than for Fe3Al. While this was the observed trend, the uncertainties of

my measurements are too large to discern definitely such a correlation.

6.2 Structural Disorder and Vibrational MSRD in Nanocrystalline

Pd and TiO.

Nanocrystalline materials, both metals and ceramics, have recently
become a topic of great interest (Gleiter, 1989). Nanocrystalline materials are
defined as materials whose grains are on the order of several (typically 5 - 15)
nanometers in length. EXAFS measurements have been used to support the
claim that grain boundaries in some nanocrystalline materials are highly
disordered (Haubold et al., 1989). This section compares EXELFS
measurements from nanocrystalline Pd and TiO2 with those from the larger
grained materials. Low temperature measurements indicated that the
nanocrystalline materials contained a significant amount of structural disorder.
Temperature-dependent measurements did not show any strong differences

between bulk and nanophase materials.

206

6.2.1 Nanocrystalline Pd

Currently, inert gas condensation is the most popular technique used to
synthesize nanocrystals. Such nanocrystals, however, must be consolidated in
order to form a bulk nanocrystalline material.

Inert gas condensation and compaction have been used previously to
synthesize nanocrystalline Pd (Birringer et al., 1984). As discussed in Chapter
3, | found that dense nanocrystalline Pd can also be synthesized by high-
vacuum evaporation. The evaporated Pd formed a thin film whose average
grain size was approximately 6 nm, as determined by x-ray diffraction and TEM.
To grow the grains, some of the evaporated Pd samples were annealed in-situ
in the heating holder of the TEM at approximately 550 C. The annealed Pd had
an average grain size of approximately 30 nm.

Figure 6.24 compares EELS measurements from the nanocrystalline Pd
and from electropolished bulk Pd. The spectra were deconvolved to remove the
effects of sample thickness. In comparison with the spectrum from the bulk Pd,
the spectrum from the nanocrystalline Pd has a significant edge near an energy
loss of 284 eV which is the location of the C K edge. Although elemental
analysis with EELS is only approximate, the relative size of the edge
nevertheless indicates that the nanocrystalline Pd contains on the order of 10%
C atoms. Furthermore, notice the edge near 532 eV which is a combination of
Pd M3 and O K edges. This edge is slightly larger for the evaporated Pd than for
the bulk Pd. Analyzing this change in edge size indicates that the nanocrystal-
line Pd may also contain on the order of 10% O atoms.

Figure 6.25 displays the Pd-C phase diagram. The phase diagram

shows that the atomic solubility limit of C in Pd at room temperature is

207

2.0x10°

Intensity

0.5

5b

rrp meet

PePprrrrprrrrprrrrprerrprrrryprrrryprrrr

Pd Mg and O K

| ee

(a) evaporated Pd

pote rir tipi rtrrrrtrrrr tire tip yp tapi

0.0

4x10°

Intensity

300 400 500 600 700 800 900 1000
Energy Loss (eV)

ja ee 1] rari? 1 TT tt I 1rgry4 | a tcirgT { TT Ff Py tiirggT | t U
— —
r _ Ms :
. 1
r 5
r (b) bulk Pd 7
r 1
ra tet ran i oe Oe | | [i oo oo | l [a i Ll |e oe me | | Lt i ; a ae ii Lt

300 400 500 600 700 800 900 1000

Energy Loss (eV)

Figure 6.24. EELS measurements from (a) evaporated nanocrystalline
Pd at 105 K and (b) electropolished bulk Pd at 98 K.
Spectra were deconvolved to remove thickness effects.

208

Weight Percent Carbon

0 1 2 3 4 5 6 7 9
1800 + rerperertnerpre attr epee perder pty rt !
P| Pd
q a
3 ?
; L / L + Graphite
1600 4 ‘ L
555% ; / ; '
Wee ee cli Tori tttte-2-- ee, if 1504416°C
q re r
; F t
1 ; r
O 1400 (Pd) Pf 7
[e) 4 4 S
4 O
4 é
2 4 i
be | 7 ( t
ee) i
§ 1200 5 t
i ]
a, ]
F |
; ' [
jo! H
soo, ; 7
7! '
} ! t
4 f b
{! }
600 4 a —— —— — —— + [
0 5 10 15 20 25 30 35 40 45 50
Pd Atomic Percent Carbon

Figure 6.25. Phase diagram for Pd-C (Massalski, 1986).

209

approximately 1%. This means that only one atomic percent of C is soluble in
the fcc matrix of bulk Pd. Nevertheless, more than one atomic percent of C may
be soluble in my thin film of nanocrystalline Pd, especially in grain boundaries.

Figure 6.26 presents the magnitude of the FT of the Pd M4s EXELFS from
nanocrystalline and annealed samples of Pd at temperatures from 105 K to 295
K. Note that because C and O are much lighter than Pd, these EXELFS signals
must be dominated by Pd neighbors. Figure 6.27 displays the 1nn MSRD data
from nanocrystalline and annealed samples. Debye model fits yielded @p = 357
+ 60 K for the nanocrystalline Pd and 0p = 273 + 35 K for the annealed Pd. The
behavior of the annealed Pd matches well that of the bulk Pd given in Chapter
5. Surprisingly, the nanocrystalline material seems slightly "stiffer" in
comparison. The scatter in the MSRD data, however, makes this conclusion
unreliable.

Figure 6.28 compares the magnitude of the FT of the EXELFS from the
nanocrystalline and annealed Pd at 105 K. The amplitude of the 1nn peak from
the nanocrystalline material is significantly suppressed. Interpreting my
measurements in terms of a Debye-Waller type factor indicate that the MSRD
between Pd atoms in the nanocrystalline Pd is greater than that in the annealed
Pd by 1.8+0.3 A2. This increase in MSRD would give the structural MSRD of
the nanocrystalline Pd if the vibrational MSRD of the two materials were equal
at 105 K. However, my temperature-dependent measurements indicate that
the structural MSRD of the nanocrystalline Pd may be even greater because
some of it is masked by the greater vibrational MSRD in the annealed Pd.

| also used a partially compacted powder of Pd nanocrystals, synthesized

by inert gas condensation, to make EXELFS measurements. Figure 6.29

210

reer prcrrprrr rp} err pre rrp ert

0.035 4
(a) as-evaporated

0.030
0.025
0.020
0.015
0.010
0.005
0.000

|FT(x)|

{ 2 3 #4 #5 6
r (A)

rrr por er pr rep rrp erry ree

0.06
0.05
0.04
0.03
0.02
0.01
0.00

(b) annealed

PRETTPRrery tr

[FT (x)|

at ee Sen ate
aie t ee
: es ae ee | | ps ees es | pet [an onan 8 it ; an ee 6 r! | Ge Se | t js Ce et

1 2 3 4 5 6
r (A)

© PUETT TERT TUTE rrr

Figure 6.26. Temperature dependence of magnitude of FT of Pd M,, EXELFS

(10.25 and after annealing in-situ at 550 C to grow grains.

211

a [ oe | q UJ | if t UJ qT I qT T
“EF
: -m - annealed
an 3E
oct a
oO -
-) a
Ne 2b —
£ FE :
© a 4
r ;
+f ~
. —® as-evaporated q
6) F a | I l 1 1 rl | l l 1 l | Lt L jl :
100 150 200 250 300

Temperature (K)

Figure 6.27. Change in 1nn MSRD for EXELFS relative to EXELFS at
105 K from as-evaporated nanocrystalline Pd and after
annealing in-situ at 550 C to grow grains.

212

a rryperrryprreryprrerprrrrprrrryprerrprrrrprrrtper spe

0.06 E we 7

E fy annealed :

E § \ =

- ‘ 4

0.05— no4 3

7 ! \ 7

E ' ‘ q

a I 1 _

= F yoy ‘ :
= t ' ua

= : 1 as-evaporated :
ui. O.0O3E ' -
— F 7
- i x

~ i 4

0.02F ; 4

: 3

. U \ ~

0.01 a:

C a

E 4

0.00 & roprtrprttirrtlicsttopirtirir tira te te
0 1 2 3 4 5 6

r (A)

Figure 6.28. Magnitude of FT of Pd M,, EXELFS (10.25
from as-evaporated nanocrystalline Pd and after annealing
in-situ at 550 C to grow grains. Data taken at 105 K.

213

6 Cry TErryrenry4e rye PUTT oe mi TTT CETTUPECTULT ict Titi Ure
T | T | TITTY] T T TTITT] T

bulk

|FT(kx)| (arbitrary units)

Lossstrrrrbrrrstrrrebiristapttletistiteebepettics

oo
PEESTPPerepererpererepererperery recep errep rere yp rr rrp rire

ft
ya Crp

2 3 4 5 6
r (A)

Onmm
—k

Figure 6.29. Magnitude of FT of Pd M,, EXELFS from partially compacted
, powder of Pd nanocrystals and bulk Pd foil. Data taken at 96 K.

214

compares the magnitude of the FT of the EXELFS from the powder with that of
bulk Pd. The EXELFS amplitude for the powder is greatly suppressed, even
more so than for the nanocrystalline thin film. This result, however, may be
complicated by problems due to the low thermal conductivity of the powder.
The measurements shown in Figures 6.28 and 6.29 are consistent with
previous EXAFS measurements of Haubold et al. (1989) and Eastman et al.
(1992). Both Haubold et al. and Eastman et al. observed large reductions in
EXAFS amplitudes from compacted Pd nanocrystals in comparison to bulk Pd.
Moreover, as shown in Figure 6.30, Eastman and co-workers also observed a
slightly larger reduction in EXAFS amplitude from the uncompacted powder of

Pd nanocrystals.

215

~—~ Foil

Compact
i — Powder :

Fourier Transform Mag.
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

l 2 38 4°5 #6 7 8
Radial Coordinate

Figure 6.30. Magnitude of FT of k? weighted EXAFS above Pd edge for coarse-
grained Pd foil, compacted nanocrystalline Pd, and powder of
uncompacted Pd nanocrystals (Eastman et al., 1992).

216

6.2.2 Nanocrystalline TiO2

Nanocrystalline TiOz is one of the most studied nanocrystalline materials.
Previous studies have synthesized nanocrystalline TiOz via various methods
(Gleiter, 1989). One popular method involved inert gas condensation of Ti
powder, followed by oxidation, then compaction (Siegel et al., 1988).
Nanocrystalline TiOz has shown interesting mechanical properties, including a
potential for significant ductility for a ceramic material (Mayo et al., 1990).

As described in Chapter 3, | synthesized nanocrystalline TiO2 by
evaporating a thin film of Ti metal on a substrate of rock salt, then oxidizing the
Ti metal by heating the substrates in air in a furnace at 500 C. The as-prepared
TiOz2 had an average grain size of roughly 7 nm and was dominated by the rutile
phase, but also contained some of the anatase phase. After annealing in
vacuum at 900 C for 11 hours, the grains grew to approximately 20 nm and
consisted of only the rutile phase.

Both rutile and anatase have tetragonal symmetry. The unit cell of rutile
contains two Ti atoms and four O atoms. The larger unit cell of anatase has four
Ti atoms and eight O atoms. In both structures, the 1nn shell surrounding Ti
atoms consists of six O atoms, two of them being slightly closer (1.95 + 0.01 A)
than the other four (1.97 + 0.01 A). Rutile is known to be harder than anatase
(Kepert, 1972).

Figure 6.31 displays the EELS measurements of the Ti L, O K and Ti K
edges from the as-prepared TiO2. The overlap between Ti L and O K edges
precludes the straightforward EXELFS analysis of these edges. Therefore, |
analyzed only the EXELFS on the Ti K edge.

Figure 6.32 displays the theoretical Ti K EXELFS signal from the inn

shellin TiO2. This calculation was made by substituting the relevant phase and

- 217

1.2x10 v ] t t tT ui | UJ T t ij | qt t t q | C i T
1.0
0.8

0.6

Intensity

0.4

PerPyPrryperryprurryprerr

0.2 i Los

0.0 Berta a Pa

beri terriflisr pr tp pp tir ity. 44

450 500 550 600
Energy Loss (eV)

ol
Oo
ao

PPPrrreprrperrrprerrrrrrprrprr ry

Intensity

spittihbrrtrrirtdirirttiges

PRPUprrrrprryryrrrryrt

0) poder rs ta te ta ttt ti tt tt LY

4800 5200 5600 6000 6400
Energy Loss (eV)

Figure 6.31. EELS measurements of Ti L, O K, and Ti K edges from as-prepared
nanocrystalline TiO,. Spectra were not deconvoluted.

218

20

PPreerprrerprerryrernrprrreryy
od

ee ee

porirtrrirsrtirrrrtipyirtiri ily

it
7 8 9 10 11 12

k (A)

Figure 6.32. Theoretical Ti K EXELFS from inn shell of TiO,.

Figure 6.33.

PUETUpPPrreprrrryprercryprrerrypreer

Lt

TT

|FT(k*x)|
mM
TUtt PEEL EEOTPereeprerryprrerrreryy

AWARSUERORCUECSUERCERTERERROLEEES

Magnitude of FT of theoretical Ti K EXELFS from 1nn shell of
TiO,. Transformation range 7
219

amplitude functions from Teo and Lee (1979) into Equation (2.50). The
magnitude of the FT of the theoretical signal is shown in Figure 6.33. Theory
predicts the 1nn peak to be nearr=1.6A.

Figure 6.34 compares the EXELFS from as-prepared and annealed TiO>.
A relatively high range in k-space was chosen to avoid any distortions due to
multiple-inelastic scattering in the low-k region. The period and phase of my
measured EXELFS match well with those of the theoretical EXELFS given in
Figure 6.32. Figure 6.35 presents the magnitude of the FT of the EXELFS from
as-prepared and annealed TiO2. The 1nn peaks are near r = 1.45 A, which is
within 0.02 A from the theoretical calculation. The difference may be due to the
choice of edge onset energy, Eo. The signal of the as-prepared sample is seen
to be smaller than that of the annealed sample. Interpreting the damping in
terms of MSRD, the 1nn MSRD is 1.8 + 0.4 x 10-3 A2 greater in the as-prepared
sample. |

Although single-crystal rutile has a Young's modulus of about 490 GPa,
Mayo et al. (1990) found moduli as low as 50 GPa for nanocrystalline TiO2
using nanoindenter (Doerner and Nix, 1986) measurements. This suggests that
the Debye temperature of their nanocrystalline TiO2 is much lower than that of
their large-grained TiOo.

My temperature-dependent Ti K EXELFS measurements, however, were
not able to determine local Debye temperatures because the amplitude of the
EXELFS did not damp appreciably for either sample. Figure 6.36 displays the
temperature-dependent 1nn MSRD data. Relative to the uncertainty in my
measurements, the changes in MSRD between 105 K and 295 K are near zero.
This result is not particularly surprising because the ionic Ti-O bonds are

expected to be very strong.

220

ry rreg Pree TRE? cure erry ur
i l i i ! Lj

4b

oe |

rh
PE PPrPrPeprrr

por rer tien tir pr tip ptr

7 8 9 10 11 #=12
k (A)

Figure 6.34. Ti K EXELFS from as-prepared nanocrystalline TiO, at 105 K.

5 _ Loe | | U UJ if vTrie | Ln oe eS if coreg { cit hs
C ; .
c i
4b hn annealed J
= Ff WAY :
ba i 4 -
a SF tf \ as-prepared +
— 5 i q
if 2F } z
— r Od +
rf 7
1 ay - a
LY iv NS wy :

0 a Ltd | Lilt | l Amun I 4 Pan L ara 1 Lai fe
0 1 2 3 4 5 6

r (A)

Figure 6.35. Magnitude of FT of experimental Ti K EXELFS (7 from as-prepared nanocrystalline TiO, and after annealing at

900 C for 11 hours to grow grains. Data taken at 105 K.

221

1.0

0.5

-—e® annealed 7

a,
<< ~J
oO “4
= 0.0
N c a
6 =
-0.5- _ 4
—® aS-prepared 4
-1 me) |e a Ce OS A OS SD | |
100 150 200 250 300
Temperature (K)
Figure 6.36. Change in 1nn MSRD for Ti K EXELFS relative to EXELFS

at 105 K from as-prepared nanocrystalline TiO, and after

annealing at 900 C for 11 hours to grow grains. Error bars
obtained from values at which variance of least-squares fit

increased by 20%.

222

6.3. Conclusions and Perspective

With a parallel detector, EXELFS is a viable structural probe with certain
advantages over EXAFS. EXELFS can generally measure core edge fine
structure in lower atomic number elements than EXAFS. Very small electron
probes can be used, allowing inhomogeneous samples to be studied. The
instrumentation is more accessible and less expensive than synchrotron sources.
Finally, EXELFS can be combined with electron diffraction and imaging in the
TEM.

| have shown that EXELFS can measure characteristics of local atomic
“environments using not only K edges, but Lo3 and Mys edges as well. Central
atom phase shifts for outgoing f-waves were calculated which are needed to
analyze Mas-edge EXELFS. This opens up most of the periodic table to EXELFS
experiments.

An important feature of the technique is its ability to probe independently
the environments of different atomic species. | have presented EXELFS
measurements of chemical short-range order and local atomic vibrations in
intermetallic alloys. Chemical short-range order was observed to evolve as
samples of chemically disordered Fes3Al and NisAl were annealed in-situ in the
electron microscope. Temperature-dependent measurements indicated that the
local atomic vibrations in the disordered alloys were significantly greater than
those in the ordered alloys. These results suggested that including vibrational
entropy in theoretical treatments of phase transformations would lower
significantly the critical temperature of ordering for these alloys.

| have also presented EXELFS measurements of local structural disorder
and atomic vibrations in nanocrystalline Pd and TiOz. The nanocrystalline

materials were observed to have significantly greater amounts of local structural

223

disorder than the large-grained materials. Temperature-dependent
measurements were inconclusive in measuring differences in local atomic
vibrations between the nanocrystalline and large-grained materials.

While EXELFS can give results which are comparable to those of EXAFS,
previous investigations of EXELFS have been mostly meager and exploratory.
This dissertation was an in-depth discussion of EXELFS and was the first to
apply the technique to contemporary problems in materials science. There are
many other important problems in materials science to which EXELFS can be

applied.

224

Appendix A_ Electron-Atom Scattering Caiculations

The electron-atom scattering theory discussed in Chapter 2 can be used
to make first principles calculations of important quantities in the EXELFS
technique. This appendix discusses the computer software which implements
these calculations and presents the results.

Energy-differential cross sections for ionization were discussed
theoretically in §2.1.2 within the framework of the Born approximation. §A.1
briefly discusses the computer software used to implement this theory. Energy-
differential cross sections are displayed for the elements relevant to the
experiments in this thesis.

Phase shifts and scattering amplitudes were discussed in §2.2.1. §A.2
presents the calculation of central atom phase shifts and backscattering
amplitudes for EXELFS. Phase shifts and scattering amplitudes relevant to the
experiments in this thesis are presented, along with many additional phase
shifts which should be useful for future EXELFS experiments. The computer

program which | wrote to make the calculations is listed.

225

A.1___Energy-Differential Cross Sections for lonization

This section briefly outlines the computer programs which | used to
calculate energy-differential cross sections for ionization. Calculations relevant to
the experiments in this thesis are presented.

First, Hartree-Slater atomic potentials and wavefunctions are calculated
using a program that was originally written by Herman and Skillman (1963).
These calculations assume a spherically symmetric atomic potential V(r) which is
the sum of the nuclear Coulomb potential, the total electronic Coulomb potential,
and the exchange potential. The exchange potential Voxcn(r) is approximated by
0.7 times the free-electron exchange potential, which is proportional to the cube

root of the total charge density of the atomic electrons:

if (A.1)

Voxch(t) = -(0.7)6| © lpr

where energies are expressed in Rydberg units and distances in Bohr units. The
factor of 0.7 is attributed to Kohn and Sham (1965).

The electronic wavefunctions are expressed as products of radial
wavefunctions Rp\(r) and spherical harmonics Y}m(8,o). Applying the method of
separation of variables, the three-dimensional Schrédinger equation is reduced to

the one-dimensional radial wave equation:

|-3 + uo + ver) Ren = ER(r) (A.1)

Figures A.1 through A.6 display the calculated atomic potentials and radial

wavefunctions relevant to O K, Al K, Ti K, Fe L, Ni L, and Pd M edges.

226

100 TTT TTT TTT TTT TT 1 TTT TTT T
[ oo i i | | i | E 9
5 ‘ ‘. Rye :
SOT ‘ O 1372s°2p* 41
— r 4y Dp
Ss 13
ir 0 ~ 10 Y
— | a >
> 7 ™ |
50L qo
L + -2
-100 pods Pay | ta tl tl
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

r (Bohr)

Figure A.1. Hartree-Slater atomic potential and 1s wavefunction for O atom
in its ground state.

227

100 rt OO J
Pe ‘ rRy, F
50 |e ‘ q
. Al [Ne]3s73p' |
~ } 4 oo
Co - 7 —
ae 0 40 9
> i 2 ~ |
4-1
50K 3
-100 i poutl irs tert tae pt ta tt ty a
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

r (Bohr)

Figure A.2._ Hartree-Slater atomic potential and 1s wavefunction for Al atom
in its ground state.

228

100 x OD v7
i 4 3
Lu \ rRy. 4
sok. | 42
rN Ti [Ar]3d*45? /
Tr 1 +
a Ji.
—~ y ‘ J op
Sf 1) B
rc 0 = 0 w
— 5 7 >
> i 7 5,
: 414"
“50 - 4.2
5 |-3
-100 rooutihor sis Lyi st Li iy Ly ritirartyr., | i
0.0 0.2 04 O06 0.8 1.0 1.2 1.4

r (Bohr)

Figure A.3. Hartree-Slater atomic potential and 1s wavefunction for Ti atom
in its ground state.

229

100 TTT TTT TTT rr TTT TTT TTT T
pT TTT | l l EP
sol / \. Fe [Ar]3d74s' 1,
j Dg
10 8
4 a
4-1
4.2
-10Q beets ti Pa
0.0 02 04 O06 08 1.0 1.2 1.4
r (Bohr)

Figure A.4. Hartree-Slater atomic potential along with 2s and 2p wavefunctions
for Fe atom in its ground state.

230

100 T Lf U | TT T | UJ UJ q | Lj t lj | q tT | lj q Ul | LJ T T | roo
50) is Ni [Arjsd°4s' :
Lf ‘ :
= pi * { D
oR 408
> 7
Fy PRs NN 7
vA V 14
“50-1 , cn
-100 perl rap Lr Lat tp ba Lt tl
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

r (Bohr)

Figure A.5. Hartree-Slater atomic potential along with 2s and 2p wavefunctions
for Ni atom in its ground state.

V (Ryd)

231

NO

100 rer yp rrp rrr pr rrp rrr perp ere yt

La “~~

mim,

50 — ; ' ‘, ‘

Q.
ss
aay
Q.
oO
oO
wn
possetrrir terri turret

tee

oO
—_
BLY

oO

rrtrtiprr tippy tri y
—_

Jyog) YJ

0b

-{OQOL.e 1th ba te
0.0 0.2 04 06 08 1.0 1.2 1.4

r (Bohr)

Figure A.6. Hartree-Slater atomic potential along with 3s, 3p, and 3d
wavefunctions for Pd atom in its ground state.

232

Using the Hartree-Slater atomic wavefunctions, energy-differential cross
sections for ionization are calcuated using two programs provided by Professor
Peter Rez of Arizona State University (Leapman et al., 1980). The first program
calculates continuum wavefunctions or partial waves by solving Equation (A.2)
with E > 0 and uses them, along with the initial atomic wavefunction, to calculate
the double-differential cross section via Equation (2.10). The double-differential
cross section is a function of both energy loss and momentum transfer. The
second program implements Equation (2.11) which integrates the double-
differential cross section over the appropriate range of momentum transfer to
determine the energy-differential cross section for ionization. Figures A.7
through A.12 display the calculated energy-differential cross sections of the O K,
AIK, Ti K, Fe L, Ni L, and Pd M edges. Note that spin-orbit splitting is included in
the figures. On the basis of the (2j + 1) degeneracy of the initial core states, the
ratio of L3-to-L2 or Mg-to-Me intensities is assumed to be 2:1, and the ratio of Ms-

to-M, intensities is assumed to be 3:2.

233

PUPRUErPrrnrryperrryprerrrvyprrrryprrrryprrrrprrrryrr4ry

shitirritossetipitiseiterrsleii tip lipitiiiliiiys

phirrr terre terir tre rt titi tte ret tity lip tay

10E

BF
Ss E
@ 7
7) C
£ 6E
© 7
Q C
WE
6 46
O 7
2E

@) F l

500
Figure A.7.

600 700 800 900 1000
Energy Loss (eV)

Energy-differential cross section of O K edge. Energy of incident
beam = 200 keV. Collection semiangle = 5 mrad.

234

EO
0.8F 4
_& Al K edge 1
S 06F 3
® . a
Oo c 7
= - a
o C q
2 c 1
| O4E 4
AS . :
© - a
1e) a “4
0.26 4
0.0 Ex cetbepeetopetlipeetr ri hepp deeded

1500 1600 1700 1800 1900 2000

Energy Loss (eV)

Figure A.8. Energy-differential cross section of Al K edge. Energy of incident
beam = 200 keV. Collection semiangle = 10 mrad.

235

Perryprrrryprrrrprererryprrrrprerrprorrryprerey

rertirprr trp rrp tbe pap yr trp,

0.035
0.030
0.025
s 0.020
ig)
tu =. 0.015
AS
is)
0.010
0.005
0.000
Figure A.9.

PEPrperrryprrrryprrrryprrrryprrryryrry

poherprrtrtrrrtipzprartizrirtitizttirrr tipi y

5000 5200 5400 5600

Energy Loss (eV)

Energy-differential cross section of Ti K edge. Energy of incident
beam = 200 keV. Collection semiangle = 35 mrad.

236

V6
= L _
14 oo
r Fe Ledge :
2E L, J
> L \ :
@ 10F -
ep) - 4
= r i
& 8 7
Ww f
6 Fr 7
=e) 5 4
4 _
2b J
eb te tt tt ti li
600 800 1000 1200 1400

Energy Loss (eV)

Figure A.10. Energy-differential cross section of Fe L edge. Energy of incident
beam = 200 keV. Collection semiangle = 5 mrad.

237

8 LOPES ELUEERUPUrreyerenpererperery rr er prev ry Terry rrr rp rriry errr yer ir yr

Lo

do/dE (barns/eV)
ff

phorir tires tiuriitipip tsp rp pip pi pty

ba
ben
be
be
baw

0) WiiWeee Gees POSER ERROR SERRE REOEE CERER ERED CEERERREES REEROREREO REECE EROEE

800 900 1000 1100 1200 1300 1400 1500

Energy Loss (eV)

Figure A.11. Energy-differential cross section of Ni L edge. Energy of incident
beam = 200 keV. Collection semiangle = 5 mrad.

238

SERRDE DERE S RUBE SREB E EERE

120 | | | i I Z

E Mas F

100F ioe

_ E Pd M edge q
@ 80F =
~~, Ss -
ep) - a
c 3 q
o . 3
2 60F
LU F J
12) yy a
ry . =
Oo 40E- =
20 E -

Fira tares taped tists lett litt tate li tia

300 400 500 600 700 800
Energy Loss (eV)

Figure A.12. Energy-differential cross section of Pd M edge. Energy of incident
beam = 200 keV. Collection semiangle = 5 mrad.

239

A.2._ Central Atom Phase Shifts and Backscattering Amplitudes

This section contains calculations of central atom phase shifts and
backscattering amplitudes for EXELFS. Results relevant to the experiments in
this thesis are presented, along with some results which should be useful for
future EXELFS experiments. The computer program "PHASE," which | wrote to
make the calculations, is given.

My calculations start with Hartree-Slater atomic wavefunctions from a
computer program originally written by Herman and Skillman (1963). Hartree-
Slater atomic wavefunctions are self-consistent wavefunctions which assume that
exchange between atomic electrons can be accounted for by the exchange
potential given in Equation (A.1). Here, | assume that exchange between the
external electron and the electrons in the scattering atom can also be described
by the exchange potential given in Equation (A.1), where p(r) is the total charge
density of the atomic electrons.

To calculate the central atom phase shift, a completely relaxed central
atom with a core-hole was used. The central atom is assumed to be relaxed
because the relaxation time (~10-18 s) is much shorter than the transit time of the
ionized electron as it travels to a neighboring atom and back (~10°17 s). The
lifetime of the core hole (~10-15 s), on the other hand, is much longer than the
transit time (Teo, 1986). To calculate the backscattering amplitude, a neutral
neighboring atom was assumed. In either case, beyond two times the covalent
radius of the atom, the potential was gradually reduced to zero over the distance
of 1 Bohr.

Using the Hartree-Slater potential V(r), partial waves rRy,j(r) are calculated
by solving Equation (A.2) with E = k2. These partial waves are compared with

the corresponding free waves rj)(kr) to determine the phase shifts 8)(k), where

240

ji(kr) is the spherical Bessel function of the first kind of order 1. For the central
atom phase shift, only 3),,1(k) needs to be calculated, where lo is the orbital
angular momentum quantum number of the initial core state. For the
backscattering amplitude, all 5;(k) which have 1 < kftunc need to be calculated,
where [max is the maximum radius of the atomic potential. The backscattering
amplitude, f(x,k) = |f(x,k)|exp[in(z,k)], is then determined using Equation (2.21).
Note that multiples of 2x can be added or subtracted from (x,k) without
changing the physical consequences.

For the purpose of illustration, Figure A.13 presents an example of a
partial wave for a relaxed C atom with a 1s core hole, along with the
corresponding free wave. The figure shows that the partial wave experiences a
phase shift 5,.;(k=10 A-1) of a bit more than +n/2 with respect to the free wave.
In this way, by varying k, the k-dependence of the phase shifts, 8(k), can be
determined. Figure A.14 displays the central atom phase shift for the C K edge
as calculated both by myself using the Hartree-Slater potential and by Teo and
Lee (1979). Teo and Lee's calculation used a simple energy-dependent
approximation for the exchange and correlation potentials between the external
electron and electrons in the scattering atom (Lee and Beni, 1977). Figures A.15
and A.16 display calculations of the magnitude and phase of the backscattering
amplitude from a neutral C atom.

Figures A.14 and A.16 show that my calculations predict slightly larger
phase shifts than those calculated by Teo and Lee. The k-dependence of the
phase shifts, however, are remarkably similar, and it is this k-dependence which
determines the change in peak positions in the Fourier transform of the EXELFS.

This thesis measures and analyzes the EXELFS on the AI K, Ti K, Fe L, Ni
L, and Pd M edges. Figures A.17 through A.21 present the central atom phase

241

shifts for these edges. Figures A.22 through A.25 display the magnitude and
phase of backscattering amplitudes from neighboring O, Al, Ti, Fe, Ni, and Pd
atoms.

In particular, note the central atom phase shift 53(k) for the Pd M edge in
Figure A.21. 6g(k) is the phase shift for a partial wave with f-symmetry. Although
knowledge of 53(k) is necessary for the EXELFS analysis of Mas or Nas edges,
values for 63(k) are not published in Teo and Lee (1979) because x-ray
absorption cannot easily measure those edges.

In principle, EXELFS experiments may be performed using any atomic
edge which can be measured by EELS. However, most published calculations of
central atom phase shifts are for those atomic edges which can be measured by
x-ray absorption spectrometry. To help remedy this situation, Figures A.26
through A.28 present central atom phase shifts for some edges that, while not
easily measurable by EXAFS, are possible candidates for EXELFS experiments.
Specifically Figure A.26 gives 54(k) for K edges of very light elements. Figures
A.27 and A.28 give 53(k) for Mas edges of elements with 32 calculations, the valence configuration for the 4d transition metal series is
assumed to be 4d2:375s1.

Figures A.27 and A.28 show that, in the region of interest, the central atom
phase shifts d3(k) have relatively small k-dependences. This implies that the
nearest-neighbor peak positions in the Fourier transform of the EXELFS from Mas
edges are only slightly changed by the central atom phase shifts. In comparison,
the central atom phase shifts 5;(k) for K edges have relatively large k-

dependences, and thus have greater effect on the peak positions.

242

~ ‘ \ s
= 7 \ 7 =
_ \ 4
_ ‘i ry —
ba é \ +
5 4 ‘ =
_ U \ 4
U ‘
= ! 1 _
. ' \ _
rc ! \ -
- t \ -
rep) Le f 7 \ ~~
Cc rf! “
[o) wh \ 5
5 Fi ‘ 4
< Of 7
=) 5 4
——
® r ~
> =
S bee
pot i trp i yt Lp tp Ly yp yy ft tp fp
0.0 0.5 1.0 1.5 2.0 2.5 3.0

Figure A.13. Partial wave and corresponding free wave for relaxed C atom with
1s core hole.

243

3.0
tO
A 25
Y)
ig)
oO
S 20
1.5

TUT TTT TTY TTY yer ryt yr
ros C central atom ,
| oo. |

= — Hartree-Slater ~
L a m Teo & Lee (1979) 4
Posto trie terrible tt tt
4 6 8 10 12 14 16

k (A)

Figure A.14. Central atom phase shift for C K edge.

244

—— Hartree-Slater
m Teo & Lee (1979)

0.6 Rn T! PEPePprereyererypreereyerrerpervryperrrprrrry rrr ey rere errr yt rr
O.5— m

E C neighbors
0.4E

Fld

Backscattering Magnitude |f(z,k)|

pitrrerterrstirirlesirtiritesittipitlsri telat

0.3
0.2E
0.1
0.0 Feerebrretereibiseete petits iitibti tiie
4 6) 8 10 12 14 16
k (A)

Figure A.15. Magnitude of backscattering amplitude for C neighbors.

245

EPPEPePerreprrerprerreprereyp reer Pecrep rere cere perrerr erry

C neighbors

— Hartree-Slater
m Teo & Lee (1979)

Backscattering Phase 1(z,k)
1O7]
PEETPEPrreprrrryprerrvryprrrryprreryprrrryprrery

3 retitirir tier tipi tittr tite tt

4 6 8 10 12 14
k (A")

Figure A.16. Phase of backscattering amplitude for C neighbors.

16

246

6 eT Te ey

S r Al central atom ;
so °F 3
& E = — Hartree-Slater 4
o i m Teo & Lee (1979) :
” i a 7
& C 4
a 46 7
£ C J
2 } q
© C :
o $F _
O P 4
BA eWeeT OWONTNTTS POTTY TESTS HOTTT EVENS FOTET EVENT FOTTTSTTTS TTT TTT

4 6 8 10 12 14 16

k (A")

Figure A.17. Central atom phase shift for Al K edge.

247

rt

Liirltus phisertirertdrrie terri tipi tpt dias

4 6 8 10 12 14

k (A"')

Figure A.18. Central atom phase shift for Ti K edge.

8 pe TT
se 7 Ti central atom
wo 7
= |
ow . — Hartree-Slater
@ 6E mw Teo & Lee (1979)
% C
= -
0. r
= r
£ 5E
< Ff
© E
ro C
(ab) .
O 4b
: "
Cu SeeWE FUQOTEWETS FOUCUETOTE FUTOEOTERERUESTEETE OTSTSTOS CVOTE TTT

lop)

248

—— Hartree-Slater

Central Atom Phase Shifts

on
PUPCPUPrrprreryprrryy PETPPPPrPrerrPrrrryprrrryprrey

TUTPUTTTPRerT per eee ere rp rere yp rere pre eer perry rere ypereryeerry re

Fe central atom

m Teo & Lee (1979)

corrtrrritrrirtererteres lists testi tip ttt te

beneptirig | reitirriatererdartidiprp tits tiga

Litt

4 6 8 10 12 14

k (A")

Figure A.19. Central atom phase shifts for Fe L edge.

—_

249

8 PP TTT

oF Nicentralatom 4

“ E .
= 7 a
Zo = -
op) c — Hartree-Slater :
® 6F m Teo & Lee (1979) =
£ 7 3
Oo. C 7
§ 56 4
< S 7
© E J
& 46 q
3E zs

aWOae EUEWESENTE CUUUE ESET CUOTEROOTE FOOTE OETT PORTE RTTOE ROTTS ETE
4 6 8 10 12 14 16

k (A)

Figure A.20. Central atom phase shifts for Ni L edge.

250

EPR EESPECUEPTUUTypUrrryp eee eyprrrryprver errr e yer ery rer ry Terry rr

LI

10 Pd central atom

—— Hartree-Slater
m Teo & Lee (1979)

tislitiptyyratgy ritiiy

Central Atom Phase Shifts
PUPrprrrryprureryprrrryt

upp tire startet dst lll it title

4 6 8 10 12 14 16

k (A’)

Figure A.21. Central atom phase shifts for Pd M edge.

251

1.2 EVTUU PUTT PUT erp reer yprrer pert rper repre rperir per eryp errr piri :
S 10E% 4
R _—_ 7
= E Neighbors 4
® E | :
S O8F H-S T&l
e a O nesses A J
=> O6F =
D F :
i 7 =
= F 7
= 04E E
fa) . q
Oo - =
ie) — —
x< a a
® O02 =
Oo CE :
FE ay See eg. F
0.0 F lestepbste lteter le A A
4 6 8 10 12 14 16

k (A")

Figure A.22. Magnitude of backscattering amplitude for O and Fe neighbors.

252

1.2 SUPT PEUPSPEPeryprrreypeeery errr yprereyrreeyperre cv eeerprreryereryrre

Le
fe

1.0 Neighbors

0.8

PEPryprrenvpereverrerry ty

0.6

fesortosrilitietrrsphrritrsiilipsites

0.4

Backscattering Magnitude |f(z,k)|

TUT PTET PTT TTT trey rrr rrr

_ A cS
A ‘A Sates gene, weed
0.0 Prosi detirtirriterietiristisiistiter titi tits te tit
4 6 8 10 12 14 16
k (A)

Figure A.23. Magnitude of backscattering amplitude for Al and Ni neighbors.

253

ETT T TTT RBER ERB ERURSESS RBBRURER ES ERREUBREAE BRED BLEED REAR EELS

_ ny: Neighbors :
= 10E 4
) P- 3
3 : :
e O8F 7
o - —
g E E
om 0.6E 4
= - 3
ay 0.4E
© ™ on q
E Aes, :

aa) O26 A Meee, wees ce
A‘

0.0 PNT a POSE E TOTS PU ETOETETS FOTETETETS FOTTT EET FOTET TOTES POTEET TTT

4 6 8 10 12 14 16

k (A"')

Figure A.24. Magnitude of backscattering amplitude for Ti and Pd neighbors.

Backscattering Phase n(z,k)

25

20

15

10

254

T I | 1 Le i | Li Ly Li

PEPUPEETEPPUPePer erp errr pereryp ever ypreer perry erry ere rye rrey errr year yee

eh fof ow ew a Pa |

Neighbors

Hartree-Slater

» Teo & Lee (1979)

s a Ni

1 T t ] '

so. Fe

Ti

qT t T | i! t q T |

“4
~4

Al

~4

PETE EEUAEORESRUREERNOOE CECE REREERGERER ORES CEEROREEROERERERROOERREEEE

4 6 8 10 12 14 16
k (A)

Figure A.25. Phase of backscattering amplitude for O, Al, Ti, Fe, Ni, and Pd

neighbors. Note phases plotted as increasing function of Z
merely to separate them.

255

LEPURPUUEEPEerryerreyparreypeerrgpreereerreperee errr ypreeryererypereryrrrey ey

3.0

2.5
Central atoms

Central Atom Phase Shifts 6, (k)

0.0 EUGG GERER EERE CEREEREREE FEUER RRUCROCORESREERECREOOECRRRERRRERREREGREER EE

4 6 8 10 12 14 16

k (A")

Figure A.26. Hartree-Slater calculations of central atom phase shifts for
K edges of very light elements not listed in Teo and Lee
(1979).

256

ORBADRRBEBEREASRASORESOAERESEU LEEDS LEEEELESOSESLSRSSES LOLS BELO REESE EERE S

3.4f 4

: Central atoms J

3.2F .

x< F 7
“Ss 5 =
“« a 4
on C 7
5 3.0F J
rab) a a
WY P 4
w g a
— 5 -
Oo C 7
= 286 4
Oo C 4
< 7 J
© c
ce 266 4
® 5 4
O r .
- Kr 7

' Br 3

2.4- +

r Se 3

Z As 3

E Ge 4

2.2 CS TAEUENE RSERROUESE SURED PEROU REESE CORRE RROROEUOURGEEROEERECERED GER TSET EOS
4 6 8 10 12 14 16 18

k (A)

Figure A.27. Hartree-Slater calculations of central atom phase shifts for
Ms edges of elements with 32 < Z < 38.

257

3.8 TRUTPUTePPeP rep eee reer reer Perec recy rere yeereypeereprrreypeere yp eraryperel

Central atoms

3.6

3.4

3.0 |

ox

Zr

SPETA CYSTS PTETS FOTET EET FTUTTETTST TET ETSTOOSTOTETOTACTOTETST SS FET STEREOS
4 6 8 10 12 14 16 18

k (A")

Central Atom Phase Shift 5,(k)
'?)
Q.
rhirittispirtdiaip iti ly rertrrtrtiritrtirperp tat yr pty Ld

se

2.8

Figure A.28. Hartree-Slater calculations of central atom phase shifts for
M,; edges of elements with 39 < Z < 48.

258

PROGRAM PHASE

CG CALCULATION OF PHASE SHIFTS WHEN PARTIAL WAVES ARE SCATTERED
C FROMA CENTRAL POTENTIAL.

C READS POTENTIAL FROM HERMAN-SKILLMAN OUTPUT FILE CONVERTED TO
C "IGOR" FORMAT.

C JAMES K OKAMOTO 02FEB93

C CONSTANTS
REAL P|
DATA PI/3.1415927/

C VARIABLES
CHARACTER*40 TEXTO,TEXT
REAL Z,R1,H
INTEGER NDATA,NWAVS
INTEGER IM
REAL A,B,C
REAL UD(0:512),OCC(20), EORB(20), WAV(20,0:512)
CHARACTER*3 ORB(20)
INTEGER ANGMOM(20)

REAL RHOD(0:512)

REAL RAD,CUT,RCUT,RAMP,RMAX,DR
INTEGER 10,11 ,l2,NEXTRA,NR

INTEGER LL

REAL RD(0:512),RB(5000)
REAL DK,KA(100),KB(100)

REAL U(5000),V(5000),RHO(5000)

INTEGER LK,NK,IK,KK
REAL X

INTEGER L,MAXL(100),LHIGH

REAL N(0:199,2),J(0:399,2),J1J2

INTEGER M1,M2,M3,RSIGN, REX,JSIGN,JEX
REAL F(5000),FJ(5000), RWAV(5000),JWAV(5000)
REAL RRWAV(5000),JJWAV(5000)

REAL G,T,D(0:199,100)

REAL RESUM,IMSUM,REAMP(100),IMAMP(100)
REAL ABSAMP(100), PHAMP(100),FIX(100)

DATA DR,CUT,RAMP,NEXTRA/0.002,2.0, 1.0, 100/
DATA DK,LK,NK,KK/0.2, 15,80,50/

C READ PARAMETERS, U=-RV/2 (BOHR*RYD), AND ORBITALS
OPEN(UNIT=3,NAME='phase.in', TYPE='OLD’)
READ(3,5) TEXTO
WRITE(6,5) TEXTO
READ(3,5) TEXT
WRITE(6,5) TEXT

5 FORMAT(A20)

READ(3,10) Z,R1,H,NDATA,NWAVS
WRITE(6, 10) Z,R1,H,NDATA,NWAVS
IF (NDATA.EQ.0) STOP

10 FORMAT(F3.0," '.F6.4," ',F6.4,"',13," 12)

DO I=0,NDATA-1
READ(3,50) UD(I)
50 FORMAT(E12.5)

259

ENDDO
DO M=1,NWAVS
READ(3,55) TEXT
IF (M.EQ.1) WRITE(6,55) TEXT
55 FORMAT(A20)
READ(3,60) ORB(M), ANGMOM(M),OCC(M),EORB(M)
WRITE(6,60) ORB(M), ANGMOM(M),OCC(M),EORB(M)

60 FORMAT(' ,A2,' 12," ‘,F5.2,"',F9.2)
DO I=0,NDATA-1
READ(3,70) WAV(M,!)
70 FORMAT (E12.5)
ENDDO
ENDDO
CLOSE(3)

C INPUT COVALENT RADIUS, DETERMINE CUTOFF, MAX, AND INTEGRATION RADII
WRITE(6,*) ‘COVALENT RADIUS OF ELEMENT IN ANG (F5.3):
READ(5,100) RAD

100 FORMAT(FS.3)

WRITE(6, 105) CUT*RAD

105 FORMAT(‘CUTOFF RADIUS(ANG) =',F5.3)
RCUT=CUT*RAD ‘1.89
RMAX=(CUT)*RAD*1.89 + RAMP
lo=INT(RCUT/DR)+1
I1=INT(RMAX/DR)+1
I2=114+NEXTRA
NR=l2

C CREATE RDATA(BOHR), R(BOHR AND ANG), AND K(INV ANG AND INV BOHR)
DO I=0,NDATA-1
RD(I)=R1*EXP(H*REAL(I))
ENDDO
DO I=1,NR
RB(I)=REAL(I)*DR
RA(I)=RB(I)*0.5292
ENDDO
DO I=LK,NK
KA(I}=REAL(I)*DK
KB(I}=KA(1)*0.5292
ENDDO
WRITE(6,200) RA(I1),RA(I2)
200 FORMAT(‘RADIUS1(ANG)=",F5.3,' RADIUS2(ANG)=",F5.3)

C INPUT PARTICULAR ANG MOM LL
WRITE(6,*) ‘PARTICULAR ANG MOM L FOR WAVE(I2):"
READ(5,210) LL

210 FORMAT(I2)

C TOTAL CHARGE DENSITY OF ATOM FROM DATA
DO I=0,NDATA-1
RHOD(i)=0.0
DO M=1,NWAVS
RHOD(I)=RHOD(1)+OCC(M)*WAV(M,|)*WAV(M,|)
ENDDO
RHOD(1)=RHOD(1)/(4.0°P*RD(I)*RD(I)
ENDDO

C INTERPOLATE TO DETERMINE ATOMIC POTENTIAL V(RYD)
C AND TOTAL CHARGE DENSITY OF ATOM RHO(ELECTRONS/BOHR"*3)
WRITE(6,*) ‘STARTING INTERPOLATION’
DO l=1,NR
U(I)=0.0
DO M=0,NDATA-1

260

A=UD(M+1)-UD(M)
B=RB(I)-RD(M)
C=RD(M+1)-RD(M)
IF ((RB(!).GE.RD(M)).AND.(RB(1).LT.RD(M+1))) U(l)=UD(M)+A*B/C
ENDDO
V(I)=-2.0*U(I)/RB(I)
ENDDO
DO I=1,NR
RHO(1)=0.0
DO M=0,NDATA-1
A=RHOD(M+1)-RHOD(M)
B=RB(I)-RD(M)
C=RD(M+1)-RD(M)
IF ((RB(I).GE.RD(M)).AND.(RB(1).LT.RD(M+1))) RHO(I)=RHOD(M)+A*B/C
ENDDO
ENDDO

C CUTOFF POTENTIAL
990 DOl=2,NR
A=RB(l) - RB(I0)
B=RB(I1) - RB(I0)
IF (ILGT.10) V(}=V(I)"(1.0-A/B)
IF (.GE.11) V(I)=0.0
ENDDO

CC MAIN LOOP OVER ENERGIES
WRITE(6,*) ‘STARTING MAIN LOOP"
WRITE(6, 1000)

1000 FORMAT(‘K(INV ANG) MAXL’)

DO IK=LK,NK
C DETERMINE MAXIMUM L NEEDED TO CALCULATE
A=KB(IK)*RMAX
MAXL(IK)=INT(A) + 2
IF (MAXL(IK).GT.199) MAXL(IK)=199
C WRITE SOME INFO TO SCREEN
A=REAL((IK-1)/10)
B=REAL(IK/10)
IF (A.NE.B) WRITE(6, 1010) KA(IK), MAXL(IK)
1010 | FORMAT(F4.1," 12)
C CALCULATE SPHERICAL BESSELS AT 2 RADII
DO |=1,2
X=KB(IK)*RB(I1)
IF (I.EQ.2) X=KB(IK)*RB(I2)
N(O,l)= -COS(X)/X
N(1,I)= -COS(X)/(X*X) - SIN(X)/X
DO L=2,MAXL(IK)
N(L,)=REAL(2*(L-1)41)*N(L-1,1)/X = N(L-2,1)
IF (ABS(N(L,I)).GT.(1.0E5)) THEN
DO M=0,L
N(M,I)=N(M,|)*1.0E-5
ENDDO
ENDIF
ENDDO
LHIGH=INT(X+100.0)
J2=0.0
J1=1.0E-25
DO L=LHIGH,O,-1
A=REAL(2*(L+1)41)*J1/X - J2
IF (L.LE.MAXL(IK)) J(L,)=A
J2=J1
JisA
IF (ABS(A).GT.(1.0E5)) THEN
J2=J2"1.0E-5

261

Jt=J1*1.0E-5
DO M=L,MAXL(IK)
J(M,1=J(M, 1)"1 .0E-5
ENDDO
ENDIF
ENDDO
A=J(0,1)/(SIN(X)/X)
DO L=0,MAXL(Ik)
J(LN=u(L,IVA
ENDDO
ENDDO
C ZERO SUMS
RESUM-=0.0
IMSUM=0.0
C LOOP OVER ANG MOM
DO L=0,MAXL(iK)
C REAL FUNCTION FROM SCHROD EQN
DO I=1,NR
F(I)=-V(I)-REAL(L*(L+1))(RB(I)*RB(1))+KB(IK)*KB(IK)
Fu(I)= -REAL(L*(L+1))/(RB(I)*RB(I)) + KB(IK)*KB(IK)
ENDDO
C INITIAL VALUES FOR NUMEROV ALGORITHM
RWAV(1)=1.0E-25
RWAV(2)=(2.0 - DR*DR*F(1))*RWAV(1)
JWAV(1)=1.0E-25
JWAV(2)=(2.0 - DR*DR*Fu(1))*UWAV(1)
C INTEGRATE USING NUMEROV ALGORITHM
RSIGN=0
REX=0
JSIGN=0
JEX=0
DO 1=3,NR
A=1.0 + DR*DR*F(1)/12.0
B=2.0°(1.0 - 5.0‘DR*DR*F(I-1)/12.0)
C=1.0 + DR*DR*F(I-2)/12.0
RWAV(1)=(B*RWAV(I-1)-C*RWAV(I-2))/A
A=0.0
Mi=1
M2=1
IF (RWAV(I).LT.A) M1 =-1
IF (RWAV(I-1).LT.A) M2=-1
M3=M1*M2
IF (M3.LT.0) RSIGN=RSIGN + 1
Mi=1
M2=1
IF ((RWAV(I)-RWAV(I-1)).LT.A) M1=-1
IF ((RWAV(I-1)-RWAV(I-2)).LT.A) M2=-1
M3=M1*M2
IF (M3.LT.0) REX=REX + 1
IF (ABS(RWAV(1)).GT.(1.0E5)) THEN
DO M=1,!
RWAV(M)=RWAV(M)*1.0E-5
ENDDO
ENDIF
A=1.0 + DR*DR*Fu(1)/12.0
B=2.0*(1.0 - 5.0°DR*DR*Fu(I-1)/12.0)
C=1.0 + DR*DR*FJ(I-2)/12.0
JWAV(1)=(B*JWAV(I-1)-C*JWAV(I-2))/A
A=0.0
M1=1
M2=1
IF (JWAV().LT.A) M1 =-1
IF (JWAV(I-1).LT.A) M2=-1

262

M3=M1*M2
IF (M3.LT.0) JSIGN=JSIGN + 1
M1=1
M2=1
IF ((SWAV(1)-SWAV(I-1)).LT.A) Mt=-1
IF ((JWAV(I-1)-JWAV(I-2)).LT.A) M2=-1
M3=M1*M2
IF (M3.LT.O) JEX=JEX + 1
IF (ABS(JWAV(I)).GT.(1.0E5)) THEN
DO M=1,I
JWAV(M)=JWAV(M)*1.0E-5
ENDDO
ENDIF
ENDDO

C SAVE RWAV AND JWAV FOR IK=KK AND L.LE.LL

IF (IK.NE.KK) GOTO 1200
IF (L.NE.LL) GOTO 1200
DO I=1,NR
RRWAV(!)=RWAV(1)
JJWAV(I)=JWAV(1)
ENDDO

C USING TWO RADII FOR MATCHING WITH FREE WAVES

1200

1300

G=RB(I1)*RWAV(I2)/(RB(I2)*RWAV(I1))
T=(G*J(L,1) - J(L,2))/(G*N(L,1) - N(L,2))
A=ATAN(T)
B=0.0
M1=RSIGN+REX-JSIGN-JEX
M2=M1/2
IF (A.LE.B) M2=(M1+1)/2
D(L,IK)=A + REAL(M2)*PI
A=REAL(2*L+1)
B=SIN(D(L,IK))
C=COS(D(L,IK))
RESUM=RESUM + A*B*C*REAL((-1)**L)
IMSUM=IMSUM + A‘B*B*REAL((-1)**L)
ENDDO
REAMP(IK)=RESUM/KA(IK)
IMAMP(IK)=IMSUM/KA(IK)
ABSAMP(IK)=(REAMP(IK)**2 + IMAMP(IK)**2)**0.5
PHAMP(IK)=ATAN(IMAMP(IK)/REAMP(IK))
IF (PHAMP(IK).GT.(P1/2.0)) WRITE(6,*) ‘WARNING: ATAN'
IF (PHAMP(IK).LT.(-P1/2.0)) WRITE(6,*) ‘WARNING: ATAN'
IF (REAMP(IK).LT.(0.0)) PHAMP(IK)=PHAMP(IK)-PI
ENDDO
FIX(NK)=0
A=5.0
DO IK=NK-1,LK,-1
FIX(IK)=FIX(IK+1)
B=PHAMP(IK)+FIX(IK)-PHAMP(IK+1)-FIX(IK+1)
IF (B.LT.(-A)) FIX(IK)=FIX(IK)+2.0°P I
IF (B.GT.A) FIX(IK)=FIX(IK)-2.0*PI
IF ((B.LT.(-A)).OR.(B.GT.A)) GOTO 1300
ENDDO

C WRITE THE DATA IN IGOR ASCII FORMAT

1950

OPEN(UNIT=4,NAME='phase.out',STATUS='NEW’)
WRITE(4,*) TEXTO
WRITE(4, 1950) CUT*RAD
FORMAT(‘CUTOFF RADIUS(ANG) = ‘,F4.2)
WRITE(4,*) ‘K(INV ANG) AMP(ANG) PHA(RAD)'
DO IK=LK,NK
WRITE(4,2000) KA(IK),ABSAMP(IK), PHAMP (1K) +FIX(IK)

2000

2005

2007

2010

263

FORMAT(F6.3,' 8.4," ',F8.4)
ENDDO
WRITE(4,*) ‘DL(RAD): L = 012345"
DO IK=LK,NK
WRITE(4,2005) D(0,IK),D(1,IK),D(2,1K),D(3,IK),D(4,1K),D(5, 1K)
FORMAT(F8.4,' \F8.4,' ',F8.4," ',F8.4," ,F8.4,"',F8.4)
ENDDO
WRITE(4,2007) LL,KA(KK)
FORMAT(‘LL=',I2,"_ KK(INV ANG)=",F6.3)
WRITE(4,*) 'R(BOHR) V(RYD) RWAVLL(BOHR’*-1/2) JWAVLL'
DO |=1,NR
M1=(1-1)/5
M2=1/5
IF (M1.NE.M2) WRITE(4,2010) RB(|),V(1), RRWAV(I),JJWAV(1)
FORMAT(F5.3," ,E10.3,' ',E10.3,",E10.3)
ENDDO
CLOSE(4)
END

264

Appendix B EXELFS Data Processing Software

The procedures used to extract EXELFS information from the
unprocessed PEELS spectra were outlined in §3.4 and §4.1. These
procedures are implemented in the computer programs documented in this
appendix.

First of all, the PEELS spectra are corrected for gain variations in the
parallel-detection system. §B.1 documents my program for direct normalization
and gain averaging of PEELS spectra. §B.1 also contains my program which
changes the format of the data.

After correcting for gain variations in the parallel detection system, the
oscillatory EXELFS data are extracted from the EELS spectrum. §B.2
documents my program which uses polynomial spline fits to extract and
normalize the EXELFS oscillations.

After the EXELFS oscillations are isolated, Fourier band-pass filtering is
used to select information from a particular nearest-neighbor shell. §B.3
documents my program for Fourier band-pass filtering.

Finally, after data from a particular nearest-neighbor shell has been
selected, least-squares fitting is used to determine the difference in
backscattering amplitude or mean-square relative displacement between two
experimental measurements. §B.4 documents my simple program for least-

squares fitting of EXELFS data.

265

B.1 Correction for Channel-to-Channel Gain Variations

This section has a listing of my program "CORR" for the direct
normalization and gain averaging of PEELS spectra. Direct normalization is
performed by dividing by a “uniform illumination” spectrum, which is obtained by
illuminating the linear diode array with a nearly uniform electron beam. Gain
averaging is accomplished by using a common feature to align the spectra, then
adding them together.

This section also lists my program, "EIC" (which stands for EL/P to Igor
conversion), which converts the data files from "EL/P" format to "Igor" format.
EL/P is the data collection program supplied by Gatan with the spectrometer, and

Igor is a data analysis program.

PROGRAM CORR

C DIRECT NORMALIZATION AND GAIN AVERAGING OF ENERGY LOSS SPECTRA
C FOR EXELFS ANALYSIS.

C OPTIONAL TO FIRST SU8TRACT A REPRESENTATIVE NOISE SPECTRUM FROM
C THE EXPERIMENTALLY OBTAINED SPECTRA. THE NOISE SPECTRUM MUST

C BE NAMED "noise."

C DIRECT NORMALIZATION IS PERFORMED BY DIVIDING SPECTRA BY A UNIFORM
C ILLUMINATION SPECTRUM, COLLECTED USING A SPECIAL MODE ON THE

C GATAN MODEL 666 PEELS. THE UNIFORM ILLUMINATION SPECTRUM

C MUST BE NAMED “uni.”

C GAIN AVERAGING IS PERFORMED BY FIRST CHOOSING A REGION IN THE FIRST
SPECTRUM WHICH IS TO BE MATCHED. EACH SUBSEQUENT SPECTRUM IS
REALIGNED TO THE FIRST SPECTRUM BY MINIMIZING THE ABSOLUTE VALUE
OF THE DIFFERENCE BETWEEN THE TWO SPECTRA. BEFORE THE COMPARISON
BETWEEN THE SPECTRA IS MADE, EACH SPECTRUM IS NORMALIZED BY THE
TOTAL NUMBER OF COUNTS IN THE SELECTED REGION.

NAAANAANO

C JAMES K OKAMOTO 120CT92

REAL*4 TEMP(1024),PEELS(22,1024),MAX(22),NORM(1024),ZERO
REAL*4 NORMLIM,VAR(22),AREA1 ,AREA

REAL*8 INTEG,SUMDATA(1024)

INTEGER NSPECTRA,UNIFORM,NOISE,NFILES, VRSION

INTEGER UPPER,LOWER,MINSH, MAXSH,SHIFT(22)

INTEGER I,J,K,N

CHARACTER*80 CHRBUF

CHARACTER*14 ELFILE(22),SUMFILE

DATA MINSH,MAXSH /-99,99/

266

DATA NORMLIM /0.1/

DATA ZERO /0.0/

DATA ELFILE / 'noise’,‘uni’,

1 'a.01','a.02','a.03','a.04','a.05',
2 'a.06','a.07','a.08','a.09','a.10',
1 'a.11','a.12','a.13','a.14'5'a.15',
2 'a.16','a.17','a.18','a.19','a.20'/

C ASK HOW MANY DATA FILES AND IF NOISE SPECTRUM EXISTS

10
15

WRITE(6,10) 'HOW MANY SPECTRA TO ADD (MAX 20): "
READ(5,15) NSPECTRA

WRITE(6,10) 'SUBTRACT NOISE SPECTRUM (O=NO, 1=YES) ?'
READ(5,15) NOISE

WRITE(6,10) ‘DIVIDE BY UNI ILLUM SPECTRUM (O=NO, 1=YES) ?'
READ(5,15) UNIFORM

WRITE(6,10) 'LOWEST CHANNEL OF REGION TO MATCH (MIN 100)'
READ(5,15) LOWER

IF (LOWER.LE.99) LOWER=100

WRITE(6,10) "HIGHEST CHANNEL OF REGION TO MATCH (MAX 900)'
READ(5,15) UPPER

FORMAT(A45)

FORMAT(I3)

IF (UPPER.GE.901) UPPER=900

IF (UPPER.LE.LOWER) UPPER=900

NFILES=NSPECTRA+2

C OPEN TAGGED ASCII DATA FILES FROM EL/P

DO N = 2-NOISE,NFILES
OPEN(UNIT=7,FILE=ELFILE(N),STATUS="OLD')

C READ PRECEDING TEXT

20

30
35

40

READ(7,20) CHRBUF
FORMAT(A18)
IF (CHRBUF.NE. Tagged ASCII Data’) GOTO 120
READ (7,30) VRSION
FORMAT(I2)
CONTINUE
READ(7,40) CHRBUF
FORMAT(A4)
IF (CHRBUF.NE.'EELS') GOTO 35

C READ EELS DATA

110

120

READ(7,*,END=110) (TEMP(J),J=1,1024)
DO J=1,1024
IF (TEMP(J).EQ.0) TEMP(J)=1.
PEELS(N,J)=TEMP(J)
END DO
CLOSE(7)
DO J = 1,1024
IF (PEELS(N,J).GT.MAX(N)) MAX(N)=PEELS(N, J)
END DO
WRITE(6,50) ‘FILE = ',ELFILE(N),"MAX DATA = ',MAX(N)

267
50 FORMAT(A8,A15,A12,F10.0)
END DO

C IF NEEDED THEN SUBTRACT NOISE SPECTRUM
IF (NOISE.EQ.0) GOTO 130

DON = 3,NFILES
WRITE(6,53) 'SUBRACTING NOISE SPECTRA FROM '",ELFILE(N)
53 FORMAT(A31,A15)
DO J = 1,1024
PEELS(N,J) = PEELS(N,J) - PEELS(1,J)
END DO
END DO

C IF REQUESTED THEN NORMALIZE UNIFORM ILLUMINATION SPECTRUM
130 — IF (UNIFORM.EQ.0) GOTO 140
INTEG = 0.0
DO J = 1,1024
INTEG = INTEG + PEELS(2,J)
END DO
WRITE(6,55) ‘AVERAGE COUNTS IN UNIFORM SPECTRUM = ',INTEG/1024.0
55 FORMAT(A38,F16.0)
DO J = 1,1024
NORM(J) = PEELS(2,J)*(1024.0/INTEG)
IF (NORM(J).LT.NORMLIM) NORM(J)=1.0
END DO

C NORMALIZE ENERGY-LOSS SPECTRA
DO N = 3,NFILES
WRITE(6,56) ELFILE(N)
56 FORMAT('DIRECT NORMALIZATION OF ',A15)
DO J = 1,1024
PEELS(N,J) = PEELS(N,J)/NORM(J)
END DO
END DO

C FIND SHIFTS BY MINIMIZING ABS DIFF IN REGION BETWEEN LOWER AND UPPER CHANNELS
140 AREA1=0.0
DO J=LOWER,UPPER
AREA1=AREA1+PEELS(3,J)

ENDDO

DO N = 4,NFILES
VAR(N)=0.0
DO I=MINSH,MAXSH

AREA=0.0
DO J=LOWER,UPPER
AREA=AREA+PEELS(N, J+)
ENDDO
SUM=0.0
DO J=LOWER,UPPER
SUM=SUM+ABS(PEELS(3,J)/AREA1 -PEELS(N, J+1)/AREA)

60

65

268

ENDDO
IF ((SUM.LT.VAR(N)).OR.(VAR(N).EQ.ZERO)) SHIFT(N)=I
IF ((SUM.LT.VAR(N)).OR.(VAR(N).EQ.ZERO)) VAR(N)=SUM
ENDDO
WRITE(6,65) ELFILE(N),SHIFT(N), VAR(N)
FORMAT('FILE:',A10,’ SHIFT=",13,' ABS DIFF=',F8.6)
ENDDO

C SET ANY SHIFTS?

400
80

85

88

89

WRITE(6,80)
FORMAT(‘SET SHIFT OF WHICH FILE (none=0,a.01=1,...,a.20=20):")
READ(5,85) J

FORMAT(I2)

IF (J.LE.1) GOTO 500

IF (J.GT.NSPECTRA) GOTO 500
WRITE(6,88)'SHIFT:'

FORMAT(A7)

READ(S5,89) |

FORMAT(I4)

SHIFT (J+2)=I

GOTO 400

C SHIFT AND ADD SPECTRA

500

67

DO J = 1,1024
SUMDATA(J) = PEELS(3,J)
END DO
DO N = 4,NFILES
WRITE(6,67) ‘SHIFTING AND ADDING ‘,ELFILE(N)
FORMAT(A21,A15)
DO J = 1,1024
IF ((J+SHIFT(N).GE.1).AND.(J+SHIFT(N).LE.1024))
1 SUMDATA(J) = SUMDATA(J) + PEELS(N,J+SHIFT(N))
END DO
END DO

C DIVIDE SUMMED SPECTRUM BY NUMBER OF SPECTRA

70

DO J = 1,1024
SUMDATA(J)=SUMDATA(J)/NSPECTRA

END DO

WRITE(6,70) 'SUMMED SPECTRA DIVIDED BY ',NSPECTRA

FORMAT(A27,I3)

C SAVE SUMMED SPECTRUM

75

1010

10712

1014

WRITE(6,75) ‘INPUT FILE NAME FOR SUMMED SPECTRUM: '
FORMAT(A38)

READ(5,1010) SUMFILE

FORMAT(A14)

OPEN(UNIT=9, FILE=SUMFILE,STATUS='NEW')
WRITE(9,1012)

FORMAT(‘Tagged ASCil Data’)

WRITE(9,1014)

FORMAT('2')

269

WRITE(9,1015)

1015 FORMAT()
WRITE(9,1016)

1016 FORMATC('EELS f 1024')

DO!= 1,8
DO J = 1,16
WRITE(9,1050) (SUMDATA(128*(I-1)+8*(J-1)+K),K=1,8)

1050 FORMAT(8F8.0)

END DO

WRITE(9,*)

END DO

CLOSE (9)

C END
9999 END

KEKKKKKEEKREEEKRERERKEKEKEKKEKRKEKEKKEREKEKEKKEKERERKERERKEEKREEEKKK KKK KKK

PROGRAM EIC

C ASCil TO ASCIl CONVERSION OF EL/P TEXT DATA TO FORMAT

C FOR IGOR INPUT.

C THE INPUT FILE BASICALLY HAS DATA IN ROWS OF 8 COLUMNS, AS

THE EL/P PROGRAM OUTPUTS, BUT THE HEADER MUST BE EDITED.

THE 1ST LINE IN THE HEADER SHOULD CONTAIN INFORMATION

ABOUT THE EXPERIMENT. TO CALIBRATE THE ENERGY SCALE, THREE
ADDITIONAL VALUES ARE NEEDED. THE 2ND AND 3RD LINES MUST HAVE AN
ENERGY LOSS VALUE (REAL, EV) AND ITS CORRESPONDING DATA
CHANNEL (INTEGER). THE 4TH LINE MUST HAVE THE DISPERSION (REAL,
EV/CH) OF THE SPECTRUM.

C THIS PROGRAM USES "PGPLOT" GRAPHICS WHICH WAS WRITTEN

C BY THE CALTECH ASTRONOMY DEPARTMENT. GRAPHICS IMPLEMEN-

C TATION RESIDES ON A SYSTEM FILE.

QAANAAAAD

C JAMES K OKAMOTO 170CT92

C VARIABLES
REAL*4 EL(4096),MAX,B
REAL*4 C(8),CTS(4096)
INTEGER 1,J,N
INTEGER CALCH
REAL*4 EVCH,CALEV
REAL*4 EOFF
CHARACTER*40 TEXT1

C READ ENERGY-SCALE AND DATA
OPEN(UNIT=13,NAME='eic.in', TYPE='OLD')
READ(13,100) TEXT1

100 FORMAT(A40)
READ(13,110) CALEV
110 FORMAT(F8.3)

270

READ(13,120) CALCH

120 FORMAT(I4)
READ(13,130) EVCH
130 FORMAT(F5.3)
EOFF=CALEV-(CALCH-1)*EVCH
N=0
DO 500 J=1,512
READ(13,FMT=*,END=530) (C(1),l=1,8)
DO 470 |=1,8
N=N+1
CTS(N)=C(1)
470 CONTINUE
500 CONTINUE
530 CLOSE(13)
NCH=N
WRITE(*,510) NCH
510 FORMAT('READ ',14," DATA POINTS.')
C DETERMINE ENERGY SCALE AND MAX NUMBER OF COUNTS
EL(1)=EOFF
MAX=CTS(1)
1400 DO 1465 Il=2,NCH
EL(D=EL(I-1 )+EVCH
IF(CTS(I).LT.(1.0)) CTS()=1.0
IF(MAX.LT.CTS(1)) MAX=CTS(1)
1465 CONTINUE
C PLOT DATA
B=1.2*MAX

CALL PGBEGIN(O,'/tek’,1,1)

CALL PGENV(EL(1),EL(NCH),0.,B,0,0)

CALL PGLABEL(’CHANNELS','COUNTS',TEXT1)
CALL PGLINE(NCH,EL,CTS)

CALL PGPOINT(1,CALEV,CTS(CALCH),4)
CALL PGEND

C WRITE THE DATA IN IGOR ASCII FORMAT

3005

3007

3010

3020

OPEN(UNIT=14,NAME='eic.out', STATUS='NEW')
WRITE(14,3005) TEXT1
FORMAT(A40)
WRITE(14,3007)
FORMAT('ELOSS COUNTS’)
DO 1=1,NCH
WRITE(14,3010) EL(1),CTS(I)
FORMAT(F8.3,F10.0)
ENDDO
WRITE(14,3020) 0.0,0.0
FORMAT(F8.3,F10.0)
CLOSE(14)

STOP
END

271

B.2 Extraction and Normalization of EXELFS Oscillations

After correcting for gain variations in the parallel detection system, the
oscillatory EXELFS data is extracted from the EELS spectrum. This section
contains a listing of my program "EXT" for the extraction and normalization of
the EXELFS oscillations. Utilized in "EXT" is a subroutine called "PSPLIN" from
an EXAFS software package developed at the University of Illinois (Scott,
1983). "PSPLIN" fits a polynomial spline function to data.

First, the pre-edge background is subtracted by fitting a relatively stiff
polynomial spline to the post-edge region, then subtracting a constant offset
from the spline so that it matches the data in the pre-edge region. In practice,
this determines an experimental edge jump and removes most of the non-
oscillatory curvature in the post-edge data. A polynomial spline fit is then used
to extract the EXELFS oscillations from the post-edge data. The experimental

edge jump and theoretical edge shapes are used to normalize the oscillations.

PROGRAM EXT

C EXTRACTION AND NORMALIZATION PROGRAM.

C POLYNOMIAL SPLINE FITS TO EXTRACT EXELFS OSCILLATIONS AND
C DETERMINE EDGE JUMP HEIGHT. USES EXPERIMENTAL EDGE

C JUMP HEIGHT AND THEORETICAL EDGE SHAPES TO NORMALIZE

C EXELFS.

C READS FILE FORMATTED AS AN OUTPUT FILE FROM THE PROGRAM
C CALLED GIC.F WHICH CONVERTS AND ENERGY-LOSS SPECTRUM

C FROM "GATAN" FORMAT TO "IGOR" FORMAT. THE "IGOR"

C FORMAT IS BASICALLY AN ASCII FILE WITH ENERGY LOSS IN THE

C 1ST COLUMN AND COUNTS IN THE 2ND COLUMN.

C PROGRAM USES SUBROUTINE CALLED PSPLIN.

C PROGRAM ALSO USES PGPLOT GRAPHICS WHICH WAS WRITTEN

C BY THE CALTECH ASTRONOMY DEPARTMENT. GRAPHICS

C IMPLEMENTATION RESIDES ON A SYSTEM FILE.

C JAMES K. OKAMOTO 3 170CT92
C VARIABLES:

C GENERIC
CHARACTER*40 TEXT

272

INTEGER I,J,FINI
REAL RZERO,A,B,C,D,XX(2), YY (2)
DATA RZERO /0.0/
C FOR READ DATA
CHARACTER*40 TEXT1,TEXT2
INTEGER NPTS
REAL EL(1025),RCTS(1025), MAXRCTS
C FOR READ SPLINE PARAMETERS
INTEGER PREREG,PREORD(9)
INTEGER POSTREG,POSTORD(9)
REAL PREXL(9),PREXH(9),EPRE
REAL POSTKL(9),POSTKH(9), ONSET
REAL POSTXL(9), POSTXH(9)
C FOR READ THEORETICAL EDGE SHAPE
CHARACTER*40 TEXT3
INTEGER NTH,LASTTH
REAL ETH(1025),TH(1025), MAXTH
C FOR DETERMINE CHANNELS
INTEGER ONSETCH,KNOT(10)
C FOR PRE-EDGE BACKGROUND SUBTRACTION
REAL BKFIT(1025)
C FOR SUBROUTINE PSPLIN
INTEGER NREG,NORD(9)
REAL XL(9),XH(9), WGHT(1025)
REAL XSPL(1025),YDAT(1025),YSPL(1025)
C COMMONS FOR SUBROUTINE PSPLIN
COMMON/XY/EL,XSPL,YDAT,YSPL
COMMON/SPLINE/NREG, XL,XH,NORD,WGHT
C FOR DETERMINE EDGE SHAPE
REAL J0(1025)
C FOR EDGE JUMP
REAL JUMP
C FOR (K, UNNORMALIZED FINE STRUCTURE)
INTEGER NOUT
REAL K(1025),FS(1025)
C FOR NORMALIZED OF FINE STRUCTURE
REAL CHI(1025)
C FOR TOTAL SPLINE FIT TO EELS DATA
REAL FIT(1025)

C READ EELS DATA

OPEN(UNIT=2,FILE="ext.in' STATUS="OLD’)
READ(2,1005) TEXT1
1005 FORMAT(A40)
READ(2,1007) TEXT2
1007 FORMAT(A40)
FINI=0
I=0
DO WHILE (FINI.EQ.0)
l=1+1

273

READ(2,*) EL(1), RCTS(I)
IF (MAXRCTS.LT.RCTS(I)) MAXRCTS=RCTS‘(1)
IF (EL(l).EQ.RZERO) FINI=1

ENDDO

NPTS=I-1

CLOSE (2)

C READ POLYNOMIAL SPLINE PARAMETERS

OPEN(UNIT=2,FILE="ext.poly', STATUS='OLD’)
READ(2,1060) TEXT2
1060 FORMAT(A40)
READ(2,1070) PREREG
1070 FORMAT(I1)
IF (PREREG.EQ.0) GOTO 1100
NREG=PREREG
DO I=1,PREREG
READ(2,1075) PREXL(I)
1075 FORMAT(F8.3)
ENDDO
READ(2,1080) PREXH(PREREG)
1080 FORMAT(F8.3)
DO I=1,PREREG-1
PREXH(I)=PREXL(I+1)
ENDDO
DO I=1,PREREG
READ(2,1083) PREORD(I)
1083 FORMAT(I1)
ENDDO
1100 READ(2,1105) EPRE
1105 FORMAT(F8.3)

READ(2,1110) POSTREG
1110 FORMAT(I1)
DO I=1,POSTREG
READ(2,1120) POSTKL(1)
1120 FORMAT(F8.3)
ENDDO
READ(2,1130) POSTKH(POSTREG)
1130 FORMAT(F8.3)
DOl=1,POSTREG
READ(2,1140) POSTORD(|)
1140 FORMAT(I1)
ENDDO
READ(2,1150) ONSET
1150 FORMAT(F8.3)
DO I=1,POSTREG
POSTXL(I)=ONSET+3.81 *POSTKL(I)*POSTKL(I)
ENDDO
POSTXH(POSTREG)=ONSET+3.81*POSTKH(POSTREG)*POSTKH(POSTREG)
CLOSE(2)

274

C READ THEORETICAL EDGE SHAPE

OPEN (UNIT=2,FILE='ext.shape’,STATUS='OLD’')
READ(2,1155) TEXTS
1155 FORMAT(A40)

FINi=0
l=0
DO WHILE (FINI.EQ.0)

l=1+1

READ(2,*) ETH(l), TH(l)

IF (ETH(I).EQ.RZERO) FINI=1
ENDDO
CLOSE(2)
NTH=I-1
MAXTH=0.0
DO [=1,NTH

IF (TH(I).GT.MAXTH) MAXTH=TH(l)
ENDDO

C DETERMINE CHANNEL OF ONSET ENERGY AND CHANNELS OF
C POST-EDGE KNOTS

ONSETCH=0
DO J=1,NPTS

IF (ONSET.GE.EL(J)).AND.(ONSET.LE.EL(J+1))) ONSETCH=J+1
ENDDO

DO I=1,POSTREG
KNOT(I)=0
DO J=1,NPTS
IF ( (POSTXL(I).GE.EL(J)).AND.(POSTXL(I).LE.EL(J+1)) )
1 KNOT (I)=J+1
ENDDO
ENDDO
KNOT(POSTREG+1)=0
DO J=1,NPTS
IF ((POSTXH(POSTREG).GE.EL(J)). AND.(POSTXH(POSTREG).LE.EL(J+1)))
1 KNOT(POSTREG#1)=J+1
ENDDO
KNOT(POSTREG+1)=KNOT(POSTREG#1)-1

C PRE-EDGE BACKGROUND SUBTRACTION
C NOTE THAT LAST STEP REPLACES "YDAT"
C WITH PRE-EDGE SUBTRACTED DATA.

IF (PREREG.EQ.0) GOTO 1190

DO I=1,NPTS
XSPL(I)=EL(I)
WGHT(I)=1.0

ENDDO

DO I=1,NPTS

275

YDAT(I)=RCTS(1)
ENDDO
NREG-PREREG
DO I=1,PREREG
XL(I)=PREXL(I)
ENDDO
XH(PREREG)=PREXH(PREREG)
DO |l=1,PREREG-1
XH(I)=PREXL(I+1)
ENDDO
DO I=1,PREREG
NORD(I) = PREORD(|)
ENDDO

CALL PSPLIN(NPTS)

PRECH=0
DO I=1,NPTS

IF ((EL(I).GT.EPRE).AND.(PRECH.EQ.0)) PRECH=!
ENDDO
OFFSET = YSPL(PRECH) - RCTS(PRECH)
DO i=1,NPTS

BKFIT(I) = YSPL(1)-OFFSET

YDAT(I) = RCTS(I) - BKFIT(1)

IF (YDAT(I).LT.RZERO) YDAT(I)=1.0
ENDDO

C DISPLAY EELS DATA AND PRE-EDGE BACKGROUND SUBTRACTION

1190

1192

1193
1195

A=1.1*MAXRCTS

CALL PGBEGIN(O,'/tek', 1,1)

CALL PGENV(XSPL(1),XSPL(NPTS),0.,A,0,0)

CALL PGLABEL('ENERGY-LOSS (eV),,'EELS',TEXT1)
CALL PGLINE(NPTS,XSPL,RCTS)

CALL PGPOINT(1,XSPL(ONSETCH),RCTS(ONSETCH),4)

IF (PREREG.EQ.0) GOTO 1193
CALL PGLINE(NPTS,XSPL,YSPL)
CALL PGLINE(NPTS,XSPL,BKFIT)
DO 1=1,14
WRITE(*,1192)

FORMAT()
ENDDO
READ(*,1195) TEXT
FORMAT(A1)
CALL PGEND

C POST-EDGE BACKGROUND SUBTRACTION

NREG-POSTREG
DO |=1,POSTREG
XL(Il)=POSTXL(I)

276

ENDDO
XH(POSTREG)=POSTXH(POSTREG)
DO |=1,POSTREG-1
XH(I)=POSTXL(I+1)
ENDDO
DO I=1,POSTREG
NORD(I) = POSTORD(!)
ENDDO

CALL PSPLIN(NPTS)
C DETERMINE EDGE SHAPE

FINI=0
DO I=ONSETCH,NPTS
IF (EL(1).GT.ETH(NTH)) FINI=1
IF (FINI.EQ.0) LASTTH=I
ENDDO

J=1
DO I-ONSETCH,LASTTH
JeJ-1
1430 J=J+1
IF ((EL(1).LE.ETH(J)).OR.(EL(I).GT.ETH(J+1))) GOTO 1430
IF ((EL(1).GT.ETH(J)). AND.(EL(1).LE.ETH(J+1)))
1 Jo(I)=TH(J)+(EL(I)-ETH(J))*
2 (TH(J+1)-TH(J))/(ETH(J+1)-ETH(J))
1440 CONTINUE
ENDDO

C DETERMINE EDGE JUMP

WRITE(6,1450) YSPL(ONSETCH)
1450 FORMAT('DEFAULT EDGE JUMP HEIGHT = ',E10.4)
WRITE(6, 1460)
1460 FORMAT(INPUT EDGE JUMP HEIGHT, IF DIFFERENT FROM ABOVE:’)
READ(5,1470) JUMP
1470 FORMAT(E10.4)
IF (JUMP.LE.(0.0)) JUMP=YSPL(ONSETCH)

C NORMALIZE EDGE SHAPE TO EDGE JUMP TO GET Jo

DO I=ONSETCH,LASTTH
JO(1)=JO(1)*(JUMP/MAXTH)
ENDDO

C DISPLAY POST-EDGE SPLINE AND JO

IF (PREREG.NE.0) A=1.2*OF FSET
IF (PREREG.EQ.0) A=1.2*MAXRCTS

CALL PGBEGIN(O,'/tek’,1,1)

CALL PGENV(XSPL(1),XSPL(NPTS),0.,A,0,0)

277

CALL PGLABEL('E-LOSS (eV),
1 ‘POST-EDGE DATA, SPLINE, THEORETICAL EDGE SHAPE’,
2 TEXT1)
CALL PGLINE(NPTS, XSPL,YDAT)
CALL PGLINE(NPTS,XSPL,YSPL)
CALL PGLINE(NPTS,XSPL,JO)
YY(1)=0.0
DO I=1,NREG+1
XX(1)=XSPL(KNOT(I))
XX(2)=XSPL(KNOT(I))
YY(2)=YDAT(KNOT(I))
CALL PGLINE(2,XX,YY)
ENDDO
READ(*,1400) TEXT
1400 FORMAT(A1)
CALL PGEND

C DETERMINE kK, UN-NORMALIZED FINE STRUCTURE, FS,
C AND NORMALIZED FINE STRUCTURE, CHI.
C ALSO DETERMINE NUMBER OF POINTS OF FINE STRUCTURE
C_ DATA, NOUT, BETWEEN FIRST AND LAST KNOTS.
NOUT=KNOT(POSTREG+1)-KNOT(1)+1
J=0
DO I=KNOT(1),KNOT(POSTREG+1)
J=J+1
K(J)=0.512*((EL(I)-ONSET)**0.5)
FS(J)=YDAT(I)-YSPL()
IF (JO(I).LE.RZERO) GOTO 1670
CHI(J)=FS(J)/JO(1)
1670 CONTINUE
ENDDO

C DISPLAY CHI VS K
A=0.0
B=0.0
DO l=1,NOUT
IF (CHI(I).LT.A) A=CHI(1)
IF (CHI(I).GT.B) B=CHI(I)

CALL PGBEGIN(O,‘/tek’,1,1)
CALL PGENV(C,D,A,B,0,0)
CALL PGLABEL('‘K','CHI', TEXT1)
CALL PGLINE(NOUT,K,CHI)
YY(1)=A
DO I=1,POSTREG+1
J=KNOT(N)
XX(1)=0.512*((EL(J)-ONSET)**0.5)

278

XX(2)=XX(1)
YY(2)=(YDAT(J)-YSPL(J))/JO(J)
CALL PGLINE(2,XX,YY)
ENDDO
CALL PGEND

C WRITE TOTAL SPLINE FIT AND THEN (K,CHI) TO OUTPUT FILE

OPEN(UNIT=3, FILE='ext.out’, STATUS='NEW’)
WRITE(3,1900) TEXT1

1900 FORMAT(A40)
WRITE(3, 1901)
1901 FORMAT(KNOT ORDER’
IF (PREREG.EQ.0) GOTO 1906
DO |=1,PREREG
WRITE(3, 1903) PREXL(I), PREORD(I)
1903 FORMAT(F8.3,13)
ENDDO
WRITE(3,1905) PREXH(PREREG)
1905 FORMAT(F8.3)
1906 DOl=1,POSTREG
WRITE(3,1907) POSTKL(I), POSTORD(1)
1907 FORMAT(F8.3, 13)
ENDDO
WRITE(3, 1908) POSTKH(POSTREG)
1908 FORMAT(F8.3)
WRITE(3,1909) ONSET
1909 FORMAT(F8.3)
WRITE(3,1910) JUMP
1910 FORMAT('EDGE JUMP=',E10.4)
WRITE(3,1912) TEXT2
1912 FORMAT(A40)
WRITE(3, 1925)
1925 FORMAT('ELOSS COUNTS TOTALSPLINE’)
DO |=1,NPTS
IF (XSPL(I).LT.EPRE) GOTO 1995
IF (XSPL(I).GT.XH(NREG)) GOTO 1995
FIT(I)=BKFIT(1)+YSPL(1)
WRITE(3,1990) XSPL(I), RCTS(1), FIT(I)
1990 FORMAT(F8.3,2F 10.0)
1995 CONTINUE
ENDDO
WRITE(3,2000)
2000 FORMAT(‘K CHI’)
DO l=1,NOUT
WRITE(3,2010) K(I),CHI(I)
2010 FORMAT(F8.5,F10.5)
ENDDO
CLOSE(3)

END

279

B.3 Fourier Band-Pass Filtering

After the EXELFS oscillations are isolated and normalized, Fourier band-
pass filtering can be used to select information from a particular nearest-
neighbor shell. This section contains a listing of my program "FOUR" for Fourier
band-pass filtering.

First, an optional polynomial spline fit can be applied to reduce any low
frequency curvature remaining after the previous polynomial spline fits. The
data is then multiplied by a window whose ends are smoothed to reduce the
possibility of false peaks due to "ringing" in the transform. After Fourier
transformation, a band-pass window is applied to isolate the EXELFS
oscillations from a particular nearest-neighbor shell. These particular nearest-
neighbor shell EXELFS oscillations are then contained in the real part of the

inverse Fourier transformation.

PROGRAM FOUR

C FOURIER FILTERING.

C SLOW FOURIER TRANSFORM (FT) AND INVERSE FT OF

C EXELFS DATA.

C READS FILE FORMATTED AS THE OUTPUT FILE FROM THE

C THE BACKGROUND SUBTRACTION PROGRAM CALLED EXT.F.

C MAIN PROGRAM
C FORTRAN 77
C JAMES K. OKAMOTO 170CT92

C VARIABLES:

C GENERIC
CHARACTER*40 TEXT
INTEGER I,J
REAL A,B
C FOR READ DATA
CHARACTER*40 TEXT1
INTEGER NDATA
REAL TEMP(2050),KDATA(1025), CHIDATA(1025)
C FOR K-LIMITS AND WEIGHTING
REAL KLO,KHI,KHW,KMIN,KMAX,KLT,KRT
REAL N

280

REAL WCHIDATA(1025)
C FOR (SECONDARY) SPLINE FIT

INTEGER NREG,NORD(9)

REAL XL(9),XH(9)

REAL XSPL(1025),WGHT(1025), YSPL(1025)
C COMMON WITH SPLINE SUBROUTINE

COMMON/XY/KDATA,XSPL,WCHIDATA, YSPL

COMMON/SPLINE/NREG, XL,XH,NORD,WGHT
C FOR FOURIER TRANSFORM

INTEGER NR

REAL WCHI(1025)

REAL DK(1025),K(1025),WK(1025)

REAL WWCHI(1025)

REAL MAXMAG, MINIMFT

REAL R(0:160),DR

REAL REFT(0:160),IMFT(0:160), MAGFT(0:160)

DATA NR,DR /160,0.05/
C FOR R-LIMITS

REAL RLO,RHI,RHW,RMIN, RMAX,RLT,RRT
C FOR INVERSE FOURIER TRANSFORM

INTEGER NKK

REAL WR(0:160),PLOTWR(0:160)

REAL DKK,KK(301)

REAL REIFT(301),IMIFT(301)

REAL MAXREIFT,MINREIFT

REAL PI

DATA NKK,PI /300,3.1415927/

C READ DATA, FIGURE OUT NDATA
OPEN(UNIT=2,FILE='four.in', STATUS='OLD’)

READ(2,105) TEXT1
105 FORMAT(A40)
108 READ(2,109) TEXT
109 FORMAT(A4)
IF (TEXT.NE.'K ') GOTO 108
READ(2,*,END=115) (TEMP(I),l=1,2050)
115 DO I=1,2050
A=REAL(I)/2.0
B=1/2
IF (A.NE.B) KDATA(I/2+1)=TEMP (|)
IF (A.EQ.B) CHIDATA(I/2)=TEMP(I)
END DO
CLOSE (2)

NDATA=0
DO 1=1,1025
IF (KDATA(I).NE.0.0) NDATA=I
ENDDO
WRITE(*,117) TEXT1
117 FORMAT(A40)

120

281

WRITE(*,120) NDATA
FORMAT('NUMBER OF DATA POINTS READ:'I4)

C INPUT (SECONDARY) SPLINE PARAMETERS

150

155

157

160

162

165

168

170

WRITE(*,150)
FORMAT(‘# OF REGIONS FOR SECONDARY SPLINE FIT (11) 2’)
READ(*,155) NREG
FORMAT(I1)
DO I=1,NREG
WRITE(*,157) |
FORMAT(‘KNOT #',I1,' (REAL) 2‘)
READ(*,160) XL(I)
FORMAT(F8.3)
ENDDO
IF (NREG.NE.O) WRITE(*,162) NREG+1
FORMAT(‘KNOT #11," (REAL) ?')
IF (NREG.NE.0) READ(*,165) XH(NREG)
FORMAT(F8.3)
DO I=1,NREG-1
XH(I)=XL(1+1)
ENDDO
DO I=1,NREG
WRITE(*,168) |
FORMAT(‘ORDER OF REGION #',I1," (11) 2)
READ(*,170) NORD(I)
FORMAT(I1)
ENDDO

DO I=1,NDATA
WGHT(I)=1.0
XSPL(I)=KDATA(I)

ENDDO

C INPUT K-LIMITS AND WEIGHTING, WEIGHT CHIDATA

180

185

190

WRITE(*,180) ' KLO (REAL) ?"
FORMAT(A15)

READ(*,185) KLO

FORMAT(F8.5)

WRITE(*,180) ' KHI (REAL) ?'
READ(*,185) KHI

WRITE(*,180) ‘ KHW (REAL) ?'
READ(*,185) KHW

WRITE(*,190) ' WEIGHTING N (REAL) ?'
FORMAT(A22)

READ(*,185) N

KMIN=KLO-KHW
KMAX=KHI+KHW
KLT=KLO+KHW
KRT=KHI-KHW

282

DO l=1,NDATA
WCHIDATA(1)=(KDATA(1)**N)*CHIDATA(I)
ENDDO
C SECONDARY SPLINE FIT:
C CALL (SECONDARY) SPLINE FIT

CALL PSPLIN(NDATA)

DO l=1,NDATA
WCHI(1)=WCHIDATA(I)-YSPL(1)
ENDDO

C DISPLAY SECONDARY SPLINE FIT

A=0.0
B=0.0
DO I=1,NDATA
IF (KDATA(I).LT.KMIN) GOTO 205
IF (KDATA(I).GT.KMAX) GOTO 205
IF (WCHIDATA(|).LT.A) A=WCHIDATA(|)
IF (WCHIDATA()).GT.B) B=WCHIDATA(!)
205 CONTINUE
ENDDO
A=1.1°A
B=1.1°B
CALL PGBEGIN(0,'/tek’,1,1)
CALL PGENV(KMIN,KMAX,A,B, 0,0)
CALL PGLABEL(‘K',"SECONDARY SPLINE FIT',TEXT1)
CALL PGLINE(NDATA,KDATA,WCHIDATA)
CALL PGLINE(NDATA,KDATA,YSPL)
CALL PGLINE(NDATA,KDATA,WCHI)
CALL PGEND

READ(*,220) TEXT
220 FORMAT(A1)

C FOURIER TRANSFORM:

C DETERMINE K, DK, WINDOW-K FOR EACH DATA INTERVAL
A=KHW**2/LOG(2.0)
DO J=1,NDATA-1

DK(J)=KDATA(J+1)-KDATA(J)
K(J)=(KDATA(J)+KDATA(J+1))/2.0

WK(J)=1.0
IF ((K(J).LT.KLT).AND.(K(J).GT.KMIN))
1 WK(J)=EXP(-(K(J)-KLT)**2/A)

IF ((K(J).GT.KRT).AND.(K(J).LT.KMAX))
1 WK (J) =EXP(-(K(J)-KRT)**2/A)

283

IF ((K(J).LT.KMIN).OR.(K(J).GT.KMAX)) WK(J)=0.0
ENDDO

C MULTIPLY DATA BY WINDOW

DO I=1,NDATA
WWCHI(1)=WK(I)*WCHI(1)
ENDDO

C PERFORM FT

MAXMAG=0.
MINIMET=0.
DO I=0,NR
Ril) = REAL(I)*DR
DO J=1,NDATA-1
B=2*K(J)*R(I)
REFT(I)=REFT(1)+WWCHI(J)*COS(B)*DK(J)
IMFT(1)=IMFT(1)+WWCHI(J)*SIN(B)*DK(d)
ENDDO
MAGFT(1)=(REFT(1)**2.0+IMFT(I)**2.0)**0.5
IF (MAGFT(I).GT.MAXMAG) MAXMAG=MAGFT(1)
IF (IMFT(I).LT.MINIMFT) MINIMET=IMFT(I)
ENDDO

C DISPLAY FT MAGNITUDE AND IMAGINARY PART

A=1.1*MINIMET
B=1.1*MAXMAG
CALL PGBEGIN(O,'/tek',1,1)
CALL PGENV (0.,8.,A,B,0,0)
CALL PGLABEL('R’,'FT MAG AND IMAG‘,TEXT1)
CALL PGLINE(NR,R,MAGFT)
CALL PGLINE(NR,R, MFT)
WRITE(*,250) KLO,KHI,KHW

250 FORMAT(‘KLO='F5.2,' KHI=',F5.2," KHW=',F4.2)
WRITE(*,260) N

260 FORMAT('WEIGHTED BY K**,F3.1)

C INVERSE FOURIER TRANSFORM:
C ASK FOR R-LIMITS

WRITE(*,410) ' RLO (REAL) ?'
READ(*,415) RLO
WRITE(*,410) ' RHI (REAL) ?"
READ(*,415) RHI
WRITE(*,410) ‘ RHW (REAL) ?'
READ(*,415) RHW

410 FORMAT(A15)

415 FORMAT(F8.5)

420 FORMAT(A22)

284

RMIN=RLO-RHW
RMAX=RHI+-RHW
RLT=RLO+RHW
RRT=RHI-RHW

C DETERMINE DKK AND WINDOW-R, APPEND WINDOW-R
C TO DISPLAY
DKK=(KMAX-KMIN)/REAL(NKK)

A=RHW**2/LOG(2.0)

DO J=0,NR
WR(J)=1.0
IF ((R(J).LE.RLT).AND.(R(J).GE.RMIN))
1 WR(J)=EXP(-(R(J)-RLT)**2.0/A)
IF ((R(J).GE.RRT).AND.(R(J).LE.RMAX))
1 WR(J)=EXP(-(R(J)-RRT)**2.0/A)
IF ((R(J).LT.RMIN).OR.(R(J).GT.RMAX)) WR(J)=0.0
ENDDO
DO J=0,NR
PLOTWR(J)=WR(J)*B
ENDDO

CALL PGLINE(NR,R,PLOTWR)

CALL PGEND
READ(*,430)
430 FORMAT()

C PERFORM INVERSE FT

MAXREIFT=0.
MINREIFT=0.
DO I=0,NKK
A=l
KK(1) = KMIN+A*DKK
DO J=0,NR
B=-2.0*°KK(1)*R(J)
REIFT(|)=REIFT(I)+(REFT(J)*COS(B)-IMFT(J)*SIN(B))
1 *2.0°DR*WR(J)
IMIFT(I)=IMIFT(I) + (REFT(J)*SIN(B) + IMET(J)*COS(B))
1 *2.0°DR*WR(J)
ENDDO
REIFT(1)=REIFT(1)/(2.0*P1)
IMIFT(I)=!MIFT(1)/(2.0*P1)
IF (REIFT(I).GT.MAXREIFT) MAXREIFT=REIFT(|)
IF (REIFT(I).LT.MINREIFT) MINREIFT=REIFT(1)
ENDDO

285

C DISPLAY REAL PART OF INVERSE FT

A=1.1*MINREIFT
B=1.1*MAXREIFT
CALL PGBEGIN(0,"/tek',1,1)
CALL PGENV (KMIN,KMAX,A,B,0,0)
CALL PGLABEL(K', INVERSE FT',TEXT1)
CALL PGLINE(NKK,KK,REIFT)
CALL PGEND
WRITE(*,550) RLO,RHI,RHW
550 FORMAT('RLO=',F5.2,' RHI=',F5.2,' RHW=',F4.2)

C SAVE RESULTS:
OPEN(UNIT=3,FILE='four.out'", STATUS='NEW’)
C SAVE INVERSE FT REAL PART

WRITE(3,600) TEXT1
600 FORMAT(A40)
WRITE(3,602) RLO,RHI,RHW
602. FORMAT('RLO=',F5.2,’ RHI=',F5.2,' RHW=',F4.2)
WRITE(3,603)
603 FORMAT('K_FILT CHI_FILT’)
DO | = 0,NKK
WRITE(3,605) KK(!), REIFT(I)
605 FORMAT(F8.5,F10.5)
END DO

C SAVE FT MAGNITUDE, REAL, IMAGINARY, WINDOW-R

WRITE(3,640) KLO,KHI,KHW

640 FORMAT('KLO=',F5.2," KHI=',F5.2," KHW=',F4.2)
WRITE(3,650)

650 FORMAT('R MAGFT REALFT IMAGFT WINDOW-R’)
DO1=0,NR
WRITE(3,660) R(I), MAGFT(I), REFT(I),IMFT(I), WR(1)

660 FORMAT(F8.5,3F 10.5,F8.5)
END DO

C SAVE WEIGHTED CHI (AFTER SECONDARY SPLINE,
C BUT BEFORE WINDOW) AND WINDOW-K

WRITE(3,672)
672 FORMAT(KNOT ORDER’)

DO I=1,NREG

WRITE(3,674) XL(I), NORD(I)
674. FORMAT(2F8.3)

ENDDO

IF (NREG.NE.O) WRITE(3,676) XH(NREG)
676 FORMAT(F8.3)

WRITE(3,678) N

286

678 FORMAT(‘K CHI*K**,F3.1,', WINDOW-K’)
DO | = 1,NDATA-1
IF (WK(I).LT.(0.01)) GOTO 690
WRITE(3,680) K(I), WCHI(), WK(1)
680 FORMAT(F8.5,F10.5,F8.5)
690 CONTINUE
END DO

CLOSE (3)
9999 END

287

B.4 Least-Squares Fitting

After Fourier band-pass filtering, least-squares fitting can be used to
quantify the differences between two experimental measurements. This section
lists my program "LEAST" for such least-squares fitting.

The edge onset energy, Eo, is variable in the fits. Either the change in
MSRD or the change in backscattering amplitude between two sets of data can

be determined.

PROGRAM LEAST

C LEAST-SQUARES FIT TO DETERMINE CHANGE IN MSRD

C OR CHANGE IN AMPLITUDE BETWEEN TWO EXELFS SPECTRA.
C READS FILE IN FORMAT OF OUTPUT FILE FROM THE

C FOURIER FILTERING PROGRAM CALLED FOUR.

C THE DATA FROM INPUT FILE #1 1S ADJUSTED UNTIL IT

C BEST FITS THE DATA FROM INPUT FILE #2.

C JAMES K. OKAMOTO = 190CT92
C UNITS: ANGSTROMS, EV.
C VARIABLES:

C GENERIC
INTEGER L
INTEGER I,J
REAL A,B,C,D
C FOR READ DATA
CHARACTER*40 TEXT(4)
INTEGER*4 NDATA(2),NDAT
REAL KDATA(301,2),XDATA(301,2)
REAL K(301),X(301)
REAL TEMP(604)
C FOR LEAST-SQUARES FIT
INTEGER KIND
INTEGER LL
REAL DEL
REAL E(301),DELTAE(40)
REAL KNEW(301),ENEW(301),X1E(301)
REAL VARE(301)
INTEGER LOC(301)
INTEGER N,NE,MINE(10)
INTEGER MINS(10)
REAL X1(301)
REAL SIGMA2(301),VARS(301)

288

REAL LIMIT,SIGN,UPPER,LOWER
REAL TOTDELTAE, TOTSIGMA2, TOTUPPER, TOTLOWER
DATA LL /3/
C FOR ERROR BARS
REAL PERCEN
DATA PERCEN /20.0/
C CONSTANTS
REAL ZERO,HBAR2ME
DATA ZERO,HBAR2ME /0.0,14.409/

C READ DATA

DO I=1,2
IF (1.EQ.1) OPEN(UNIT=2,FILE="least.in1',STATUS='OLD')
IF (1.EQ.2) OPEN(UNIT=2,FILE="'least.in2', STATUS='OLD')
READ(2,105) TEXT(1)
105 FORMAT(A40)
108 READ(2,109) TEXT(3)
109 FORMAT(A6)
IF (TEXT(3).NE."K_FILT') GOTO 108
READ(2,*,END=115) (TEMP(J),J=1,602)
115 DO J=1,602
A=J
A=A/2.0
B=J/2
IF (A.NE.B) KDATA(J/2+1,1)=TEMP(J)
IF (A.EQ.B) XDATA(J/2,1)=TEMP(J)
ENDDO
CLOSE(2)
ENDDO

C DETERMINE NDAT

DO |=1,2
NDATA(I)=0

DO J=1,301

K(J)=KDATA(J,I)

IF (KDATA(J,I).NE.ZERO) NDATA(I)=J
ENDDO
ENDDO
NDAT=NDATA(1)

C DISPLAY ORIGINAL DATA

A=0.

B=0.

DO J=1,NDAT
IF (XDATA(J,1).LT.A) A=XDATA(J,1)
IF (XDATA(J,1).GT.B) B=XDATA(J,1)

ENDDO

A=1.1*A

B=1.1*B

289

C=K(1)
D=K(NDAT)
CALL PGBEGIN(0,'/tek',1,1)
CALL PGENV(C,D,A,B,0,0)
CALL PGLABEL('K','ORIGINAL DATA (#1 PLUSES, #2 DOTS)',")
DO J=1,NDAT
X(J)=XDATA(J,1)
ENDDO
CALL PGPOINT(NDAT,K,X,2)
DO J=1,NDAT
X(J)=XDATA(J,2)
ENDDO
CALL PGPOINT(NDAT,K,X,1)
CALL PGEND

C ASK WHETHER TO VARY DELTA MSRD OR AMPLITUDE

200 WRITE(*,205)

205 FORMAT('VARY 1) DELTA MSRD OR 2) AMPLITUDE?')
READ(*,210) KIND

210 FORMAT(I1)
IF ((KIND.NE.1).AND.(KIND.NE.2)) GOTO 200

C *** LEAST-SQUARES FIT ***
C INITIALIZE X1 VECTOR

DO J=1,NDAT
X1(J)=XDATA(J,1)
ENDDO

C FOR LL ITERATIONS
DO L=1,LL

C * LEAST-SQUARES FIT FOR DELTA Eo PARAMETER *
C FOR A RANGE OF DELTA Eo VALUES
DO l=1,40
DEL=REAL(I)
DELTAE(1)=DEL*1.0 - 20.0
C CALCULATE NEW K VECTOR PER Eo
DO J=1,NDAT
E(J)=HBAR2ME*K(J)*K(J)/2.0
ENEW(J)=E(J)+DELTAE()
KNEW(J)=SQRT(2.0*ENEW(J)/HBAR2ME)
ENDDO
C INTERPOLATE TO FIND NEW EXELFS VECTOR (WHICH
C CORRESPONDS TO NEW K VECTOR) PER Eo
NE=NDAT
DO J=1,NDAT
X1E(J)=0.0
LOC(J)=0

290

DO N=1,NDAT
IF ((K(J).GE.KNEW(N)).AND.(K(J).LT.KNEW(N+1))) LOC(J)=N
ENDDO
N=LOC(J)
IF (N.EQ.0) GOTO 700
B=(X1(N+1)-X1(N))/(KNEW(N+1)-KNEW(N))
X1E(J)=X1(N)+(K(J)-KNEW(N))*B
700 CONTINUE
ENDDO
C CALCULATE SQUARE VARIATION OF FIT PER Eo
VARE(I)=0.0
N=NDAT
DO J=1,NDAT
IF (LOC(J).EQ.0) N=N-1
IF (LOC(J).EQ.0) GOTO 750
VARE(1)=VARE()+(X1E(J)-XDATA(J,2))**2.0
750 CONTINUE
ENDDO
VARE(1)=VARE(1)/REAL(N)
ENDDO
C DETERMINE DELTA Eo THAT GAVE BEST FIT
MINE(L)=1
A=VARE(1)
DO I=1,40
IF (VARE(I).LT.A) MINE(L)=1
IF (VARE(I).LT.A) A=VARE(I)
ENDDO
C RECALCULATE KNEW VECTOR FOR BEST FIT
DO J=1,NDAT
KNEW(J)=0.0
E(J)=HBAR2ME*K(J)*K(J)/2.0
ENEW(J)=E(J)+DELTAE(MINE(L))
IF (ENEW(J).LT.ZERO) GOTO 770
KNEW(J)=SQRT(2.0*ENEW(J)/HBAR2ME)
770 CONTINUE
ENDDO
C REINTERPOLATE TO FIND X1E VECTOR FOR BEST FIT
DO J=1,NDAT
X1E(J)=0.0
LOC(J)=0
DO N=1,NDAT
IF ((K(J).GE.KNEW(N)).AND.(K(J).LT.KNEW(N+1))) LOC(J)=N
ENDDO
N=LOC(J)
IF (N.EQ.0) GOTO 800
B=(X1(N+1)-X1(N))/(KNEW(N+1 )-KNEW(N))
X1E(J)=X1(N)+(K(J)-KNEW(N))*B
800 CONTINUE
ENDDO

C *** LEAST-SQUARES FIT DELTA MSRD OR AMPLITUDE PARAMETER

291

DO l=1,300
IF (KIND.EQ.1) SIGMA2(1)=REAL(1)*0.0001 - 0.015
IF (KIND.EQ.2) AMP(I)=REAL(I)*0.005

ENDDO

DO I=1,300
VARS(I)=0.0
N=NDAT
DO J=1,NDAT

IF (LOC(J).EQ.0) N=N-1
IF (LOC(J).EQ.0) GOTO 900
IF (KIND.EQ.1) X1(J)=X1E(J)*EXP(-2.0*K(J)*K(J)*SIGMA2(I))
IF (KIND.EQ.2) X1(J)=X1E(J)*AMP(I)
VARS(1)=VARS(I)+(X1(J)-XDATA(J,2))**2.0
900 CONTINUE
ENDDO
VARS(I)=VARS(I)/REAL(N)
ENDDO
C DETERMINE BEST FIT PER ITERATION

MINS(L)=0

A=VARS(1)

DO I=1,300
IF (VARS(I).LT.A) MINS(L)=I
IF (VARS(1).LT.A) A=VARS(I)

ENDDO

DO J=1,NDAT
IF (KIND.EQ.1) X1(J)=X1E(J)*EXP(-2.0*K(J)*K(J)*SIGMA2(MINS(L)))
IF (KIND.EQ.2) X1(J)=X1E(J)*AMP(MINS(L))

ENDDO

C DISPLAY BEST FIT PER ITERATION

A=0.

B=0.

DO J=1,NDAT
IF (XDATA(J,1).LT.A) A=XDATA(J,1)

IF (XDATA(J,1).GT.B) B=XDATA(J,1)

ENDDO

A=1.1*A

B=1.1*B

C=K(1)

D=K(NDAT)

CALL PGBEGIN(0,'/tek’,1,1)

CALL PGENV(C,D,A,B,0,0)

CALL PGLABEL('K','FITTED DATA #1 (PLUSES), ORIGINAL DATA #2 (DOTS)',")

CALL PGPOINT(NDAT,K,X1,2)

DO J=1,NDAT
X(J)=XDATA(J,2)

ENDDO

CALL PGPOINT(NDAT,K,X,1)

CALL PGEND

WRITE(6,910) L,DELTAE(MINE(L))

910 FORMAT(‘ITERATION #',I1,' DEL Eo=',F3.0)
IF (KIND.EQ.1) WRITE(6,920) L,SIGMA2(MINS(L))
IF (KIND.EQ.2) WRITE(6,930) L,AMP(MINS(L))

292

920 FORMAT(‘ITERATION #',I1,' DEL SIG=",F7.4)
930 FORMATC‘ITERATION #',11,' FACTOR AMP="/F6.3)

C END ITERATION LOOP
ENDDO

c¢ DETERMINE TOTAL DEL Eo,
C AND TOTAL DEL MSRD OR TOTAL AMP CHANGE
TOTDELTAE=0.0
TOTSIGMA2=0.0
TOTAMP=1.0
DO L=1,LL .
TOTDELTAE=TOTDELTAE+DELTAE(MINE(L))
IF (KIND.EQ.1) TOTSIGMA2=TOTSIGMA2+SIGMA2(MINS(L))
IF (KIND.EQ.2) TOTAMP=TOTAMP*AMP(MINS(L))
ENDDO

c DETERMINE ERROR BAR
LIMIT=VARS(MINS(LL))*(1.0+PERCEN/100.0)
DO J=2,300
SIGN=(VARS(J-1)-LIMIT)*(VARS(J)-LIMIT)
IF (KIND.EQ.1) GOTO 950
IF (KIND.EQ.2) GOTO 960
950 IF ((SIGN.LT.ZERO).AND.(J.LT.MINS(LL))) LOWER=SIGMA2(J)
IF ((SIGN.LT.ZERO).AND.(J.GT.MINS(LL))) UPPER=SIGMA2(J-1)
GOTO 970
960 IF ((SIGN.LT.ZERO).AND.(J.LT.MINS(LL))) LOWER=AMP(J)
IF ((SIGN.LT.ZERO).AND.(J.GT.MINS(LL))) UPPER=AMP(J-1)
970 ENDDO
IF (KIND.EQ.1) TOTLOWER=TOTSIGMA2-SIGMA2(MINS(LL))+LOWER
IF (KIND.EQ.1) TOTUPPER=TOTSIGMA2-SIGMA2(MINS(LL))+UPPER
IF (KIND.EQ.2) TOTLOWER=(TOTAMP/AMP(MINS(LL))) * LOWER
IF (KIND.EQ.2) TOTUPPER=(TOTAMP/AMP(MINS(LL))) * UPPER

C WRITE TO OUTPUT FILE:

C BEST DELTA MSRD (OR AMPLITUDE), TOTAL MSRD (OR TOTAL AMPLITUDE) VS VARIANCE
C VECTORS, TOTAL DELTA EO VS VARIANCE VECTORS, AND FINAL X1 VS K

OPEN(UNIT=3,FILE='least.out’, STATUS='NEW')
DO l=1,2
WRITE(3,1000) I, TEXT(I)
1000 — FORMAT(‘FILE #°,11,": ',A40)
ENDDO
WRITE(3, 1002) PERCEN
1002 FORMAT('LIMITS AT ',F4.1,'% LARGER AVE FUNCTIONAL.’)
IF (KIND.EQ.1) WRITE(3,1003) TOTSIGMA2, TOTLOWER, TOTUPPER
IF (KIND.EQ.2) WRITE(3,1004) TOTAMP, TOTLOWER, TOTUPPER
1003. FORMAT('SIGMA‘2:',F7.5," LOWER:',F7.4,' UPPER:',F7.4)
1004 FORMAT(‘AMP:',F6.3," LOWER:',£6.3,' UPPER:',F6.3)
IF (KIND.EQ.1) WRITE(3,1006)

293

1006 FORMAT('TOT SIGMAA2 VARIANCE’)
IF (KIND.EQ.2) WRITE(3,1007)
1007 FORMAT('TOT AMP VARIANCE’)
DO | = 1,300
IF (KIND.EQ.1) WRITE(3,1008) TOTSIGMA2-SIGMA2(MINS(LL))+SIGMA2(I), VARS(I)
IF (KIND.EQ.2) WRITE(3,1009) (TOTAMP/AMP(MINS(LL))) * AMP(1), VARS(I)
1008 —FORMAT(F8.5," ',E10.4)
1009 FORMAT(F6.3,' ',£10.4)
END DO
WRITE(3,1010) TOTDELTAE
1010 FORMAT('DEL Eo:',F4.0)
WRITE(3,1015)
1015 FORMAT('TOT DEL Eo VARIANCE’)
DO I=1,40
WRITE(3,1020) TOTDELTAE-DELTAE(MINE(LL))+DELTAE(!), VARE(!)
1020 FORMAT(F4.0,' ',F8.5)
ENDDO
WRITE(3,1100)
1100 FORMAT('BEST FIT: K X1(K)')
DO J = 1,NDAT
WRITE(3,1105) K(J),X1(J)
1105 FORMAT(F8.5,' F8.5)
END DO
CLOSE (3)

CEND |.
9999 END

294

Appendix C Software for Calculations of Vibrational MSRD

Various models used to calculate vibrational MSRD were discussed in
§5.1. This appendix documents the computer software which implements these
calculations.

The Einstein model is a very simple model, but it adequately
parameterizes all of the MSRD data presented in this thesis. The software used
to fit AMSRD vs temperature data to Einstein temperatures is documented in
§C.1.

The correlated Debye model is slightly more sophisticated than the
Einstein model. The software used to fit AMSRD vs temperature data to Debye
temperatures is documented in §C.2.

The force constant model uses interatomic force constants to calculate
the "projected" density of vibrational modes, which determines the vibrational

MSRD. §C.3 contains computer software for such calculations.

295

C.1 Correlated Einstein Model
This section contains a listing of my program "EIN" which fits AMSRD vs

temperature data to an Einstein temperature. The reduced mass of the two
atoms of interest, i.e., the central and neighboring atoms, must be input. The

AMSRD data are then fit to predictions of the correlated Einstein model, allowing

the value of the lowest-temperature experimental MSRD to float.

PROGRAM EIN

C INPUT DELMSRD (SQ ANGSTROMS) VS TEMPERATURE (KELVIN) DATA.
C OUTPUT EINSTEIN TEMPS WITH VARIANCE AND OFFSET VECTORS
C ALSO OUTPUT MSRD VS TEMP FOR BEST FIT.

C JAMES K OKAMOTO 200CT92

C VARIABLES:
CHARACTER*40 TEXT
INTEGER NSETS,NPTS(6),I,J,K,N
REAL T(6,20),DELMSRD(6,20)
REAL MRED
REAL LOWEST,HIGHEST
REAL TEIN(101),OFF(300),TOUT(100)
REAL SUM, DIFF,CONST,ARG,FUNCT,MSRD
REAL VAR(6), TOTVAR(101 ), OFFSET(101,6)
REAL MINTOTVAR,BESTTEIN,BESTOFFSET(6),MSRDOUT(100)
REAL PERCEN,LIMIT,SIGN, LOWER,UPPER
DATA PERCEN /100.0/

C CONSTANTS:
REAL H2MKA2
DATA H2MKA2 /48.46/

C READ DATA
OPEN(UNIT=2,FILE="'ein.in', STATUS='OLD')
READ(2,100) TEXT

100 FORMAT(A40)

WRITE(*,105) TEXT
105 FORMAT(A40)
READ(2,107) NSETS
107 — FORMAT(i2)
DO N=1,NSETS
READ(2,110) NPTS(N)
110 FORMAT(I2)
DO l=1,NPTS(N)
READ(2,120) T(N,I), DELMSRD(N, 1)

296

120 FORMAT(F8. 1,F8.4)
ENDDO
ENDDO
CLOSE(2)

C INPUT REDUCED MASS, RANGE OF TEIN
WRITE(*,150)

150 FORMATC'INPUT AVE REDUCED MASS OF BOND (AMU):')
READ(5,160) MRED

160 FORMAT(F8.4)
WRITE(*,170)

170 = FORMAT(‘INPUT LOWEST TEIN TO TRY (K):')
READ(5,180) LOWEST

180 FORMAT(F6.2)
WRITE(*,190)

190 FORMAT(C'INPUT HIGHEST TEIN TO TRY (K):')
READ(5,200) HIGHEST

200 FORMAT(F6.2)

C SET UP TEIN, OFF, AND TOUT VECTORS
DO |=1,101
TEIN(1I)=LOWEST+(I-1 )*(HIGHEST-LOWEST)/100.0
ENDDO
DO l=1,300
OFF(1)=REAL(1)*0.0001 - 0.0150
ENODDO
DO I=1,100
TOUT(I)=REAL(I)*10.0
ENDDO

C MAIN LOOPS

C FOR EACH EINSTEIN TEMP
MINTOTVAR=1.0
DO Il=1,101
TOTVAR(I)=0.0
CONST=H2MKA2/(2.0*MRED*TEIN(I))
C FOR EACH SET OF TEMPERATURE-DEPENDENT DATA
DO N=1,NSETS
VAR(N)=1.0
OFFSET(I,N)=0.0
C FOR EACH OFFSET
DO J=1,300
SUM=0.0
C FOR EACH TEMPERATURE DATA POINT
DO K=1,NPTS(N)

Cc CALCULATE MSRD, DETERMINE SUM OF SQ DIFF
ARG=TEIN(I)/T(N,K)
FUNCT=(2.0/(EXP(ARG)-1.0)) + 1.0
MSRD=CONST*FUNCT
DIFF=(OFF(J)+DELMSRD(N,K)) - MSRD

297

SUM=SUM+DIFF*DIFF
ENDDO
IF (SUM.LT.VAR(N)) OFFSET(I,N)=OFF(J)
IF (SUM.LT.VAR(N)) VAR(N)=SUM
ENDDO
TOTVAR(I)=TOTVAR(I)+VAR(N)
ENDDO
IF (TOTVAR(I).LT.MINTOTVAR) BESTTEIN=TEIN(1)
DO N=1,NSETS
IF (TOTVAR(I).LT.MINTOTVAR) BESTOFFSET(N)=OFFSET(I,N)
ENDDO
IF (TOTVAR(I).LT.MINTOTVAR) MINTOTVAR=TOTVAR(I)
ENDDO

C CALCULATE MSRD VS TEMP FOR BEST TEIN
DO l=1,100
CONST=H2MKA2/(2.0*MRED*BESTTEIN)
ARG=BESTTEIN/TOUT(I)
FUNCT=(2.0/(EXP(ARG)-1.0)) +1.0
MSRDOUT(I)=CONST*FUNCT
ENDDO

C DETERMINE LOWER AND UPPER LIMITS OF ERROR BAR FOR TEIN
LIMIT=MINTOTVAR*(1.0+PERCEN/100.0)
DO l=2,101
SIGN=(TOTVAR(I-1 )-LIMIT)*(TOTVAR(1)-LIMIT)
IF ((SIGN.LT.ZERO).AND.(TEIN(I).LT.BESTTEIN)) LOWER=TEIN(I-1)
IF ((SIGN.LT.ZERO).AND.(TEIN(I).GT.BESTTEIN)) UPPER=TEIN(I)
ENDDO

C WRITE OUTPUT
OPEN(UNIT=3,FILE="ein.out', STATUS='NEW’')
WRITE(3,1000) TEXT
1000 FORMAT(A40)
DO N=1,NSETS
DO l=1,NPTS(N)
WRITE(3,1002) T(N,I), DELMSRD(N,I)
1002 FORMAT(F6.1," ',F8.4)
ENDDO
ENDDO
WRITE(3,1004) MRED
1004 FORMAT('REDUCED MASS OF BOND = ',F8.4)
WRITE(3,1005) PERCEN
1005 FORMATC('LIMITS AT ',F5.1,'% GREATER VARIANCE’)
WRITE(3,1010) BESTTEIN,LOWER,UPPER
1010 FORMAT('TEIN: BEST=',F6.1,' LOWER=",F6.1,' UPPER=",F6.1)
WRITE(3,1015) (BESTOFFSET(N),N=1,NSETS)
1015 FORMAT('BEST OFFSETS:',6F8.5)
WRITE(3,1017) MINTOTVAR
1017 FORMAT('MIN TOTAL VARIANCE=',£10.4)
WRITE(3, 1020)

1020

1100

1110

1200

298

FORMAT(‘TEIN VARIANCE OFFSETS')
DO l=1,101
WRITE(3,1100) TEIN(I), TOTVAR(I),(OFFSET(I,N),N=1,NSETS)
FORMAT(F6.1,X,E10.4,X,6(F8.5,X))
ENDDO
WRITE(3,1110)
FORMAT('TEMP MSRDOUT')
DO l=1,100
WRITE(3,1200) TOUT(I),MSRDOUT(I)
FORMAT(F6.1,' ',F8.6)
ENDDO
CLOSE(3)

END

299

C.2 Correlated Debye Model
This section contains a listing of my program "DEB" which fits AMSRD vs

temperature data to a Debye temperature. The required inputs are as follows:

1) reduced mass of the central atom and neighbor of interest
2) neighbor distance

3) atomic density of solid

The AMSRD data are then fit to predictions of the correlated Debye model,

allowing the value of the lowest-temperature experimental MSRD to float.

PROGRAM DEB

C INPUT DMSRDDAT (SQ ANGSTROMS) VS TEMPERATURE (KELVIN) DATA.
C IN GENERAL, CALCULATIONS USE UNITS OF ANG,AMU,PICOSEC,KELVIN.

C OUTPUT CORRELATED DEBYE TEMPS WITH MSRD OFFSET AND LOWEST
C VARIANCE VECTORS

C ALSO OUTPUT MSRD VS TEMP FOR BEST FIT.

C JAMES K OKAMOTO OSMAR93

C LOOPS:
INTEGER 1,J,K,L,N

C CONSTANTS:
REAL Pi, HBAR,KB
DATA PI,HBAR,KB /3.14159,6.35,0.831/

C VARIABLES:
CHARACTER*40 TEXT
INTEGER NPTS(6)
REAL TDAT(6,20),DMSRDDAT(6,20)

REAL MRED,RNN,DENS,KD, LOWEST,HIGHEST

REAL TDEB(51),WDEB(51),OFF(150), TOUT(100), WOUT(100),DTOUT
DATA DTOUT/10.0/

REAL BESTTOTVAR,BESTTDEB, BESTOFFSET(6)
REAL VAR(6), TOTVAR(51),OFFSET(51,6),C,DW
INTEGER LMAX

300

DATA DW /0.5/

REAL SUM

REAL INTEG

REAL ARG,W(2000),COTH,WRC,PROJDOS
REAL MSRD(20),DIFF

REAL BESTWDEB,MSRDOUT(100)

REAL PERCEN,LIMIT,SIGN,LOWER,UPPER
DATA PERCEN /100.0/

C READ DATA
OPEN(UNIT=2,FILE="deb.in', STATUS="OLD')
READ(2,100) TEXT

100 FORMAT(A40)

WRITE(*,105) TEXT
105 FORMAT(A40)
READ(2,107) NSETS
107. FORMAT(I1)
DO N=1,NSETS
READ(2,110) NPTS(N)
110 FORMAT(I2)
DO l=1,NPTS(N)
READ(2,120) TDAT(N,I), DMSRDDAT(N,!)

120 FORMAT(F8.1,F8.4)

ENDDO
ENDDO
CLOSE(2)

C INPUT REDUCED MASS, ATOMIC DENS, RANGE OF TDEB
WRITE(*,140)

140 FORMAT(‘ALL INPUTS MUST BE FLOATS (NOT INTEGER)')
WRITE(*,150)

150 FORMAT(INPUT AVE REDUCED MASS OF BOND (AMU):')
READ(5,160) MRED

160 FORMAT(F8.4)
WRITE(*,161)

161 FORMAT(‘INPUT NEIGHBOR DISTANCE (ANG)')
READ(5,162) RNN

162 FORMAT(F8.4)
WRITE(*,163)

163. FORMAT(‘INPUT ATOMIC DENSITY (ANGA-3):")
READ(5,165) DENS

165 FORMAT(F8.4)
KD=(6.0*PI*PI*DENS)**0.333
WRITE(*,170)

170 FORMAT(‘INPUT LOWEST TDEB TO TRY (K):’)
READ(5,180) LOWEST

180 FORMAT(F6.2)
WRITE(*,190)

190 FORMAT(‘INPUT HIGHEST TDEB TO TRY (K):')
READ(5,200) HIGHEST

301

200 FORMAT(F6.2)

C SET UP TDEB,WDEB, OFF,W, TOUT, WOUT VECTORS
WRITE(*,250)
250 ~—FORMAT('SETTING UP VECTORS’)
DO |=1,51
TDEB(I)=LOWEST+REAL(I-1 )*(HIGHEST-LOWEST)/50.0
WDEB(!)=TDEB(I)*KB/HBAR
ENDDO
DO l=1,150
OFF(I)=REAL(I)*0.0001
ENDDO
DO I=1,2000
W(1)=(REAL(1)+0.5)*DW
ENDDO
DO l=1,100
TOUT(I)=REAL(I)*DTOUT
WOUT(I)=TOUT(I)*KB/HBAR
ENDDO

C MAIN LOOPS
WRITE(*,300)
300 — FORMAT(‘IN MAIN LOOP’)
WRITE(*,400)
400 FORMAT('TDEB TOTVAR OFFSETS’)
BESTTOTVAR=1.0E20
C FOR EACH DEBYE TEMP
DO I=1,51
C=KB*TDEB(1)/(HBAR*KD)
LMAX=INT(WDEB(I)/DW+0.5)
TOTVAR(I)=0.0
C FOR EACH SET OF TEMP-DEP DATA
DO N=1,NSETS
OFFSET(I,N)=0.0
C FOR EACH OFFSET
DO J=1,150
SUM=0.0
C FOR EACH TEMPERATURE DATA POINT
C CALCULATE MSRD, DETERMINE SUM OF SQ DIFF
DO K=1,NPTS(N)
INTEG=0.0
C INTEGRATION LOOP OVER PHONON FREQUENCIES
DO L=1,LMAX
ARG=HBAR*W(L)/(KB*TDAT(N,K))
COTH=(2.0/(EXP(ARG)-1.0)) + 1.0
WRC=W(L)*RNN/C
PROJDOS=3.0*W(L)*W(L)*(1.0-SIN(WRC)/WRC)/(WDEB(1)**3.0)
INTEG=INTEG + DW*PROJDOS*COTH/W(L)
ENDDO
MSRD(K)=INTEG*HBAR/(2.0*MRED)
DIFF=(OFF(J)+DMSRDDAT(N,K)) - MSRD(K)

302

SUM=SUM+DIFF*DIFF
ENDDO
IF (J.EQ.1) VAR(N)=SUM
IF (SUM.LE.VAR(N)) OFFSET(I,N)=OFF(J)
IF (SUM.LE.VAR(N)) VAR(N)=SUM
ENDDO
TOTVAR(D=TOTVAR(I)+VAR(N)
ENDDO
WRITE(*,490) TDEB(I), TOTVAR(I), (OFFSET(I,N),N=1 sNSETS)
490 FORMAT(F6.1,3X,E10.4,6(X,F8.4))
IF (TOTVAR(I).LE.BESTTOTVAR) BESTTDEB=TDEB(I)
DO N=1,NSETS
IF (TOTVAR(I).LE.BESTTOTVAR) BESTOFFSET(N)=OFFSET(I,N)
ENDDO
IF (TOTVAR(1).LE.BESTTOTVAR) BESTTOTVAR=TOTVAR(I)
ENDDO
WRITE(*,500) BESTTDEB
500 ~=FORMAT('best fit Debye temp: ',F6.1)

C CALCULATE MSRD VS TEMP FOR BEST TDEB
BESTWDEB=BESTTDEB*KB/HBAR
LMAX=INT(BESTWDEB/DW+0.5)
C=KB*BESTTDEB/(HBAR*KD)

DO I=1,100
INTEG=0.0
DO L=1,LMAX
ARG=W(L)/WOUT(I)
COTH=(2.0/(EXP(ARG)-1.0)) +1.0
WRC=W(L)*RNN/C
PROJDOS=3*W(L)*W(L)*(1.0-SIN(WRC)/WRC)/(BESTWDEB**3)
INTEG=INTEG + DW*PROJDOS*COTH/W(L)
ENDDO
MSRDOUT(I)=INTEG*HBAR/(2.0*MRED)
ENDDO

C DETERMINE LOWER AND UPPER LIMITS OF ERROR BAR FOR TDEB
LIMIT=BESTTOTVAR*(1.0+PERCEN/100.0)
DO 1l=2,51
SIGN=(TOTVAR(I-1 )-LIMIT)*(TOTVAR(I)-LIMIT)
IF ((SIGN.LT.ZERO).AND.(TDEB(I).LT.BESTTDEB)) LOWER=TDEB(I-1)
IF ((SIGN.LT.ZERO).AND.(TDEB(I).GT.BESTTDEB)) UPPER=TDEB(I)
ENDDO

C WRITE OUTPUT
OPEN(UNIT=3,FILE="'deb.out', STATUS='NEW’')
WRITE(3,1000) TEXT

1000 FORMAT(A40)

DO N=1,NSETS
DO l=1,NPTS(N)
WRITE(3,1002) TDAT(N,!), DMSRDDAT(N,!)

1002 FORMAT(F6.1," ‘,F8.4)

1005
1010
1015
1017

1020

1100

1110

1200

303

ENDDO
ENDDO
WRITE(3,1005) PERCEN
FORMAT(‘LIMITS AT ',F5.1,'% GREATER VARIANCE’)
WRITE(3,1010) BESTTDEB,LOWER,UPPER
FORMAT(‘TDEB: BEST=',F6.1," LOWER="',F6.1,' UPPER=',F6.1)
WRITE(3,1015) (BESTOFFSET(N),N=1,NSETS)
FORMAT('BEST OFFSETS='/,6F8.5)
WRITE(3,1017) BESTTOTVAR
FORMAT('MIN TOTAL VARIANCE="',E10.4)
WRITE(3,1020)
FORMAT(‘TDEB VARIANCE OFFSET’)
DO Il=1,51
WRITE(3,1100) TDEB(I), TOTVAR(I),(OFFSET(I,N),N=1 iNSETS)
FORMAT(F6.1,X,E10.4,X,6(F8.5,X))
ENDDO
WRITE(3,1110)
FORMAT('TEMP MSRDOUT")
DO l=1,100
WRITE(3,1200) TOUT(I), MSRDOUT(!)
FORMAT(F6.1,' ',F8.6)
ENDDO
CLOSE(3)

END

304

C.3. Force Constant Model

This section documents my computer software to calculate the "projected"
density of vibrational modes from the interatomic force constants of a monatomic
Bravais lattice. The program outputs the density of vibrational modes and the
projected density of vibrational modes for the three phonon branches: one
longitudinal and two transverse.

The particular program presented is tailored for fcc lattices. The program
was modified for bcc lattices, but that version of the program is not shown. The
program can also be easily modified for other crystal lattice structures and other
neighboring shells.

Also included is the program "FCMSRD" which uses the projected density

of modes to calculate the vibrational MSRD.

/* projected phonon DOS calculation
for L,T2,T1 branches
version fcc, Inn shell
James K. Okamoto 1/15/91 */

#include
#include
#include “alforce.h"

#define NQ 50
#define DX 0.02
#define PI 3.14159

main()
int i,j,k;
int n,x2,nn[25];

int Lm;
float s,ctemp;

int r[730},x[730][4];

float g[4],q0,q1;

305

float qr,dyn[4][4],d[4][4];
int thesign;

int nrot;

float e[4],v[4][4],test[4][4];

int iq, ip;

float tresh,theta,tau,t,sm,h,g,c,b[4],z[4];
float under;

int first,longitudinal;

int origbranch[4],oldbranch[4],koldbranch[4], joldbranch[4],branch[4];
float olde[4], kolde[4] jolde[4],olddel[4],kolddel[4],jolddel[4];

float del[4][4],long1 ,long2,long3,same,reverse;

float origv[4][4],cross[4],orighand,hand;

int nfreq{4};

float polardot[4],freq[4],dot,unnormg[4][100],unnormproj[4][100];
int nn1[13],i1,i2;

float areag,normg[4][100],normproj[4][100];
FILE *fp,*fopen();
/* determines allowed r for fcc lattice up to 8nn shell */

printf("determining allowed r in fcc lattice\n");
for (i=1;i<=8;i++)
nn[i]=0;
for (i = 39; i <= 689; i++)
for G = 1; j <= 3; j++)
switch (j)
case 1:
n=i/81;
break;
case 2:
n=i/9;
break;
case 3:
n=i;
break;
default:
break;
x[i]Uj] = (n%9) - 4; /* x{i]{f] is 2 times the jth coordinate of atom i
e.g. if atomi is at 0.5(110) then x{ij[1 J=1,
x{ij[2]=1,xf[i][3]=0. */

x2 = xfiJ(1]*xfi)(1] + xfif2)*xfi)[2] + xfi][3)*xfi[3);
if ((x2%2)==0)

306

r[i] = x2/2; /*atom i is in the r[i]th nearest neigbor shell */
nn{[r[i]]++;
if (r[fiJ==1)

nn1 [nn[r[i]]J=i;

_ else
r[i] = 0;
for (i=1;i<=8;i++)
printf("nn[%d]=%d\n" i,nn[i]);

/* main loop over all q allowed in 1st BZ needed by symmetry */

first=1;
printf("in main loop\n");
for (i= 1; i <= NQ; i++)
for Gj = 1; j <= NQ; j++)
for (Kk = 1; k <= i; k++)
q[1 ]=((float)i - 0.5)*DX; /* q=(qx,qy,qz) is a wavevector in reciprocal space */
q[2]=((float)j - 0.5)*Dx;
q[3]=((float)k - 0.5)*Dx;
qO0=0.0;
q1=0.0;
for (l= 1; 1 <= 3; l+4)
q0+=q[I!]*q[I]; /* distance from origin of recip space */
q1+=(q[I]-1.0)*(q[l]-1.0); /* distance from recip latt pt at (111) */

if (qO < ql)
if (G==1) && (k==1))
printf("i = %d\n",i);
/* printf("qx=%f qy=%f qz=%f\n",q[1],q([2],q[3]); */

/* calculate dynamical matrix dyn given q */

/* printf("calculating dynamical matrix\n"); */
for (m= 1; m<=3; m++)
for (n = 1; n<=3; n++)
dyn[m][n] = 0.0;

for (Il = 0; | <= 728; l++)

if (r[l] '= 0)

qr=0.0;
for (m = 1; m<=3; m++)
gr += q[m]*(float)x{[l][m];
s = 2.0*sin(0.5*Pl*qr)*sin(0.5*Pl*qr);
for (m= 1; m <= 3; m++)

307

for (n = 1;1n <= m;n++)
thesign = 1;
if (x[}][m]<0)
thesign = -thesign;

if (x[f][n]<0)
thesign = -thesign;
ctemp = 0.0;
switch (r[I])
case 1:
if (m ==n)
{ .
if (x{1][m]==0)
ctemp = C12Z;
else
ctemp = C1Xx;
else
if ((x[][m]!=0) && (x[I][n]!=0))
ctemp = C1XY;
break;
case 2:
if (m==n)
if (x[I][m]==0)
ctemp = C2YY;
else
ctemp = C2Xx;
break;
case 3:
if (m==n)
if (abs(x[I][m])==2)
ctemp = C3Xx;
if (abs(x[I][m])==1)
ctemp = C3YY;
break;
case 4:
if (m==n)
if (x[I][mJ!=0)
ctemp = C4Xx;
if (x[][m]==0)
ctemp = C42ZZ;
else

if ((x(1][m]!=0) && (x[!][n]!=0))
ctemp = C4XY;

308

break;

case 5:
if (m==n)
if (abs(x[I][m])==3)
ctemp = C5Xx;
if (abs(x[I][m])==1)
ctemp = CSYY;
if (x[1][m]==0)
ctemp = C522;
else
if ((x[][m]!=0) && (x[I][n]}!=0))
ctemp = C5XY;
break;
case 6:
if (m==n)
ctemp = C6Xx;
else
ctemp = C6XY;
break;
case 7:
if (m==n)
if (abs(x[I][m])==3)
ctemp = C7Xx;
if (abs(x[I][m])==2)
ctemp = C7YY;
if (abs(x[1][m])==1)
ctemp = C7ZZ;
else
if (Cabs(x[I][m])!=3) && (abs(x{[I][n])!=3))
ctemp = C7YZ;
if (Cabs(x[I][m])!=2) && (abs(x[I[n])!=2))
ctemp = C7XZ;
if ((abs(x[I][m])!=1) && (abs(x[I][n])!=1))
ctemp = C7XY;
break;
case 8:
if (m==n)
if (x[!][m]!=0)
ctemp = C8XxX;
if (x{i][m]==0)
ctemp = C8YY;

309

break;
default:
ctemp = 0.0;
break;
dyn[{m][n] += (float)thesign*ctemp*s;

for (m= 1; m <= 2; m++)
for (n = m+1; n<=3; n++)
dyn[m][n] = dyn[n][m];
for (m= 1; m <= 3; m++)
for (n = 1; n<=3; n++)
d[m][n] = dyn[m][n];
/* printf("dyn[%d][%d] = %e ",m,n,dyn[m][n]); */
/* printf('\n"); */

/* find eigenvalues e and eigenvectors v of dynamical matrix d
section adapted from subroutine jacobi in numerical recipes in c */

/* printf(“in eigen-routine jacobi\n");fflush(stdout); */
n= 3;

/* Computes all eigenvalues and eigenvectors of a real symmetric matrix
d[1...n][1...n]. On output, elements of a above the diagonal are destroyed.
e[1...n] returns eigenvalues of a. v[1...n(direction)][1 ..-N(eigenvector)]}

is a matrix whose columns contain, on output, the normalized eigenvectors of
d. nrot returns the number of Jacobi rotations that were required, */

for (ip=1;ip<=njip++) — /* Initialize identity matrix. */
for (iq=1;iq<=n;iq++)
vip] [iq]=0.0;
Vlip] [ip]=1.0;

for (ip=1;ip<=n;ip++)
blip]=efip]=d[ip][ip]; /* init b and e to the diagonal of d. */
z[ip]=0.0; /* This vector accumulates terms of the form
t*d[ip]{iq] as in equation (11.1.14). */

nrot=0;

1=0;

under=999.0;

while (under != 0.0) /* Exit of loop relies on quadratic

310

convergence to machine underflow. */

I++;
sm=0.0;
for (ip=1; ip<=n-1; ip++) /* Sum off-diagonal elements. */
for (iq=ip+1 ;iq<=n;iq++)
sm += fabs(d[ip][iq]);
/* printf("sum off-diagonal elements = %e\n",sm);fflush(stdout); */
under=sm;
if (sm!=0.0)
if (<4) /* on the first 3 sweeps ... */
tresh=0.2*sm/((float)n*(float)n);
else /* thereafter ... */
tresh=0.0;
for (ip=1;ip<=n-1;ip++)
for (iq=ip+1 ;iq<=n;iq++)
g=100.0*fabs(d[ip][iq]);
if
((l>4)&& ((fabs(e[ip])}+g)==fabs(e[ip]))&& ((fabs(e[iq])+g)==fabs(e[iq])))
d[ip][iq]=0.0;
else if (fabs(d[ip][iq]) > tresh)

h=e[iq]-e[ip];

if ((fabs(h)+g) == fabs(h))
t=(d[ip] [iq])/h;

else

theta=0.5*h/(dfip] [iq]);
t=1.0/(fabs(theta)+sqrt((1.0+theta*theta)));
if (theta < 0.0)

t=-t;

c=1.0/sqrt(1+t*t);

s=t*c;

tau=s/(1.0+c);

h=t*d[ip] [iq];

z[ip] -= h;

z[iq] += h;

e[ip] -= h;

e[iq] += h;
d[ip][iq]=0.0;

for (m=1;m<=ip-1; m++)

g=d[m][ip]; h=d[m][iq]; d[m] [ip]=g-s*(h+g*tau);
d[m]{iq]=h+s*(g-h*tau);

for (m=ip+1;m<=ig-1;m++)

311

g=d[ip][m]; h=d[m] [iq]; d{ip][m]=g-s*(h+g*tau);
d[m]fiq]=h+s*(g-h*tau);

for (m=iq+1;m<=njm++)
g=d[ip][m]; h=d[iq][m]; d[ip][m]=g-s*(h+g*tau);
d[iq][m]=h+s*(g-h*tau);
for (m=1;m<=n;m++)
g=vim] [ip]; h=v[m] [iq]; vim} [ip]=g-s*(h+g*tau);
vim] [iq]=h+s*(g-h*tau);
++nrot;
for (ip=1;ip<=n;ip++)
blip] += zip];
e[ip]=b[ip];
z[ip]=0.0;

/*
for (m=1;m<=3;m++)
for (n=1;n<=3;n++)
test[n][m]=0.0;
for(l=1;l<=3;l++)
test[n][m] += dyn[n][!]*v[I][m];

for (m=1;m<=3;m++)
printf("e[%d] = %e ",m,e[m]);
printf("\n");

for (n=1;n<=3;n++)
for (m=1;m<=3;m++)
printf("v[%d][%d] = %f ",n,m,v[n][m]);
printf("\n");

for (n=1;n<=33n++)

for (m=1;m<=3;m++)

312

printf("test[%d][%d] = %f ",n,m,test[n][m]/e[m]);

printf("\n");
*/

/* determining to which branch each eigenvector belongs */
/* calculate how parallel to the q vector */
n=0;
for (l=1;l<=3;l++) /* for each eigenvector */

polardot[!]=0.0;
for (m=1;m<=3;m++) /* for each direction */
polardot[I]+=v[m][!]*q[m]/sqrt(q0);
/* assign original branches 1=L,2=T1,3=T2 */
if (first==1)
first=0;
n=0;
for (l=1;l<=3;l++)
if ((polardot[I]>0.95)&&(polardot[I]<1.01))
origbranch[1 J=1;
else
origbranch[2+n]=1;
n++;
for(m=1;m<=3;m++)
origv[I][mJ=v[I][m];
/* set olde,kolde,jolde,oldbranch,...,olddel,...*/
for (l=1;l<=3;1++)
olde [!]=kolde[I]=jolde[!]=e[I];
oldbranch[!]=koldbranch[!]=joldbranch[I]=origbranch[I];
olddel[I]=kolddel{i]=jolddei{l]=0.0;
/* determine original handedness */
/* for (l= 1; | <= 3; 144)
cross[I]=0.0;

for (m= 1; m <= 3; m++)
for (n = 1; n <= 3; n++)
if (l=m)&&(ml=n)&&(n!=1))
cross[I]+=pow(-1 -0,(m-1+2)%3)*origv[m][2]}*origv[n] [3];
orighand=0.0;
for (l=1;l<=351++)

313

orighand+=cross[!]*origv({l}[1];
*/

/* assign new branches */
if (k==1)
for (l=1;l<=3;1++)
if G==1)
olde[l]=joldef[I];
oldbranch[I]=joldbranch[I];
olddel[!]=jolddel[I];
else
olde[!]=kolde[I];
oldbranch[l]=koldbranch[}];
olddel[!]=kolddel[!];
n=0;
longitudinal=0;
/*
for (l=1;l<=3;14++)
printf("polardot[%d] = %f ",l,polardot[I]);
printf("\n");
*/
for (1 = 1; 1 <= 3;1++) /* for each eigenvector */
if ((polardot[I]>0.95)&&(polardot[I]<1.01))
branch[1 J=1;
longitudinal++;
else
branch[2+n]=I;
n++;
if (longitudinal!=1 )

printf(“warning: %d longitudinal modes\n", longitudinal);

/* check for min change in derivs of long branch */

for (l= 1; 1 <= 3;1++) /* for each possible long branch */
delf[1][1]=e[branch[1 ]]-olde[oldbranch{1 ]];

tong1=fabs(del[1}[1 ]-olddel[1 ]);

long2=fabs(del[1 ][2]-olddel[1]);

long3=fabs(delf 1 ][3 }-olddel[1 ]);

if (longi < long2) && (long1 < long3))

314

branch[1]=1;
branch[2]=2;
branch[3]=3;
if ((long2 < long1) && (long2 < long3))
branch[1]=2;
branch[2]=1;
branch[3]=3;
if ((long3 < long1) && (long3 < long2))
branch[1]=3;
branch[2]=1;
branch[3]=2;

/* check for minimum change in derivatives of trans branches */
for (Il = 2; 1 <= 3; 1++) /* for each transverse branch */

for (m=2;m<=3;m++) /*for each new transverse branch */

del{I][m]=e[branch{m]]-olde[oldbranch[I]];

same=fabs(del[2][2]-olddel[2])+fabs(del[3][3]-olddel[3]);
reverse=fabs(del[2][3]-olddel[2])+fabs(del[3][2]-olddel[3]);
if (reverse < same)

l=branch[2];

branch[2 ]=branch[3];

branch[3]=!;

/* check handedness */
/* for (l= 1; 1 <= 3; 144+)
cross[l]=0.0;
for (m= 1; m <= 3; m++)
for (n = 1; n <= 3; n++)
if ((l!=m)&& (ml=n)&&(n!=1))
cross[!]+=pow(-1.0,(m-l+2)%3)*v[m][2]*v[n][3];
hand=0.0;
for (l=1;l<=3;l++)
hand+=cross[I]*v[1][1];
if ((hand*orighand) < 0.0)
l=branch[2];
branch[2]=branch[3];
branch[3]=I;

/* assign olde,kolde,jolde,oldbranch,...,olddel,... */

*/

315

for (l=1jl<=3;l4++)
olde[I]=e[I];
oldbranch[l]=branch[I];
olddel[!]=del[!] [branch[I]];
if (k==1)
kolde[l]=e[I];
koldbranch[l]=branch[I];
kolddel(!]=del [I] [branch[I]];
if G==1)
jolde[!]=e[!];
jJoldbranch[I]=branch[I];
jolddel[!]=del[!][branch[1]];

/* from each dynamical matrix, find 3(eigenvectors)*1 2(1nns)
= 36 projections and put them into histogram, unnormg(nfreq) */

for (l= 1; 1 <= 3314+) /* for each branch */
freq[!] = 1.0e-12 * sqrt(e[branch[I]]/MASS) / (2.0*PI);
/* eigenfrequency in THz */
/* printf("freq(in THz) = %f ",freq[l]); */
nfreq[I] = (int)(freq[I]*10.0);
if ((nfreq[I]> 0) && (nfreq[l]<100))
unnormg[I][nfreq{!]]+=1.0;
for (m= 1; m <= 12; m++) /* for each of 12 Inn */
qr=0.0;
for (n = 1; n<= 3; n++) /* for each direction */
qr += Pi*q[n]*x[nn1 [m]][n];
dot=0.0;
for (n = 1; n <= 3;n++) /* for each direction */
dot += v[n][branch{I]}*(float)x[nn1 [m]][n]/sqrt(2.0);
if (dot>1.01)
printf("error: dot>1 in code.\n");

unnormproj[!] [nfreq{!]]+=dot*dot*(1.0-cos(qr))/12.0;

/* printfC'\n"); */

316

/* write output to file "proj.out" */
/* contains 7 columns: freq in 104-13 rad/sec, 3 norm phonon dos,
3 proj norm phonon dos */

areag=0.0;
for (l=1; l<=3; 1++)
for (m=0;m<=99;m++)
areag += (float)unnormg[I][m]*0.01*2.0*PI;
printf("areag = %f\n",areag);
for (l=1; <=3; l++)
for (m=0; m<=99; m++)
normg[!][m] = (float)unnormg[!][m]/areag;
normproj[l][m] = 3*unnormproj[!] [m]/areag;

printf("writing output to proj.out\n");fflush(stdout);
fp = fopen("proj.out", "w");
for (m=0; m<=99; m++)
fprintf(fp, "Yf\t%F\t%F\LHFA\LHA\ EMAL",
((float)m+0.5)*0.01 *2.0*Pl,normg[1][m],normg[2][m],
normg[3][m],normproj[1 ][m],normproj[2][m],normproj[3][m]);

BERNER REE EERE KRERREREKREKEREREKEKEREKREKEKKKEKKKKKKEK KEK KKK

The following is a listing of the file "alforce.h" which is "included" in the C
program above. This file contains the atomic mass in kg and the force constants
in dyn/cm for the first 8 nn shells. These particular force constants were obtained
from a fit by Cowley (1974) to inelastic neutron scattering data taken by Stedman

et al. (1967):

#define MASS 4.48e-26

#define C1XX 10.4578
#define C1ZZ -2.6322
#define C1XY 10.3657

#define C2XX 2.4314
#define C2YY -0.1351

#define C3XX 0.0986
#define C3YY -0.2366

317

#define C3YZ -0.2862
#define C3XZ -0.1819

#define C4XX 0.1363
#define C4ZZ 0.1854
#define C4XY 0.3753

#define CSXX -0.3003
#define CSYY 0.1842
#define C5ZZ 0.2603
#define C5XY -0.3239

#define C6XX -0.1412
#define C6XY 0.1990

#define C7XX 0.1828
#define C7YY -0.2207
#define C7ZZ -0.0173
#define C7YZ -0.0214
#define C7XZ -0.0747
#define C7XY 0.0397

#define C8XX -0.0681
#define C8YY -0.0202

KKKKKEKERERERK EERE ER KEKEIKERERKEKEKRERKKEKEKRREIEKREKKEKREREKRAEKKAKK RE

PROGRAM FCMSRD
C PROGRAM TO DETERMINE MSRD FROM PROJECTED DOS.
C JAMES K. OKAMOTO + 24MAR93

C VARIABLES:
C GENERIC
INTEGER 1,J
C CONSTANTS (AMU, ANG, 104-13 SEC,KELVIN)
REAL HBAR,KB
DATA HBAR,KB /0.635,0.00831/
C FOR READ DATA .
INTEGER NPTS
DATA NPTS /100/
REAL W(200),G1(200),G2(200),G3(200),P1(200),P2(200),P3(200)
C FOR INPUT
REAL M
C FOR SETUP
REAL T(100),DW
REAL G(200),P(200),GNORM,PNORM
C FOR DETERMINE MSRD
REAL SUM,ARG,COTH,MSRD(100)

C READ DATA

318

OPEN(UNIT=2,FILE='fcmsrd.in’",STATUS='OLD')
DO l=1,NPTS
READ(2,1010) W(1),G1(1),G2(1),G3(1),P1(1),P2(I),P3(I)
1010 — FORMAT(7(F8.6,X))
ENDDO
CLOSE(2)

C SETUP TEMPERATURE VECTOR, DW,PROJDOS VECTOR
DO l=1,100
T(I)=REAL(1)*10.0
ENDDO
DW=W(2)-W(1)
GNORM=0.0
PNORM=0.0
DO I=1,NPTS
G(I)=G1 (1)+G2(1)+G3(1)
P(I)=P1(1)+P2(I)+P3(1)
GNORM=GNORM+G(1)*DW
PNORM=PNORM+P(1)*DW
ENDDO
WRITE(6,1200) GNORM,PNORM
1200 FORMAT('GNORM=",F8.6,' PNORM=',F8.6)

C INPUT ATOMIC MASS
WRITE(6,1300)

1300 FORMAT(‘INPUT ATOMIC MASS (AMU):')
READ(5,1310) M

1310 FORMAT(F8.4)

C DETERMINE MSRD FROM PROJECTED DOS, DOUBLE LOOP
DO l=1,100
SUM=0.0
DO J=1,NPTS
ARG=HBAR*W(J)/(2.0*KB*T(1))
COTH=(EXP(ARG) + EXP(-ARG))/(EXP(ARG) - EXP(-ARG))
IF (ARG.GT.(10.0)) COTH=1.0
SUM=SUM + DW*P(J)*COTH/W(J)
ENDDO
MSRD(1)=HBAR*SUM/M
ENDDO

C WRITE TO OUTPUT FILE
OPEN(UNIT=3,FILE="fcmsrd.out', STATUS='NEW')
WRITE(3,2000)

2000 FORMAT(‘Temp(K) — Vibr MSRD(ANGA2)')

DO I=1,100
WRITE(3,2010) T(I),MSRD(I)

2010 FORMAT(F6.1,X,E10.4)

ENDDO

CLOSE(3)
END

319

References

Ahn, C.C. and Krivanek O.L. (1983) EELS Atlas, Center for Solid State Science,
Arizona State University, Tempe, Arizona 85287.

Anthony, L., Okamoto, J.K., and Fultz B. (1993) "Vibrational Entropy of Ordered
and Disordered NisAl," Phys. Rev. Lett. 70, 1128-1130.

Ashcroft, N.W. and Mermin, N.D. (1976) Solid State Physics, Holt, Rhinehart &
Winston, New York.

Ashley, C.A. and Doniach, S. (1975) "Theory of extended x-ray absorption edge
fine structure (EXAFS) in crystalline solids," Phys. Rev. B 11, 1279-1288.

Atwater, H.A. and Ahn, C.C. (1991) "Reflection electron energy loss
spectroscopy during initial stages of Ge growth on Si by molecular beam
epitaxy,” Appl. Phys. Lett. 58, 269-271.

Azaroff, L.V. (1963) "Theory of extended fine structure of x-ray absorption
edges,” Rev. Mod. Phys. 35, 1012-1022.

Becker, J.A. (1924) "Velocity distribution of secondary electrons," Phys. Rev. 23,
664.

Beni, G. and Platzman, P.M. (1976) "Temperature and polarization dependence
of extended x-ray absorption fine-structure spectra," Phys. Rev. B 14,
1514-1518.

Bethe, H. (1930) "Zur Theorie des Durchgangs schneller Korpuskularstrahlen
durch Materie," Ann. Phys. 5, 325-400.

Birringer, R., Gleiter, H., Klein, H-P., and Marquardt, P. (1984) "Nanocrystalline
materials: An approach to a novel solid structure with gas-like
disorder," Phys. Lett. 102A, 365-369.

Bohm, D. and Pines, D. (1951) "A collective description of electron interactions:
Il. Collective vs. individual particle aspects of the interactions,”
Phys. Rev. 85, 338-353.

Boland, J.J., Crane, S.E., and Baldeschwieler, J.D. (1982) "Theory of extended x-
ray absorption fine structure: Single and multiple scattering formalisms,"
J. Chem. Phys. 77, 142-153.

Cohen-Tannoudji, C., Diu, B., and Laloé, F. (1977) Quantum Mechanics, Wiley,
New York and Hermann, Paris.

320

Colliex, C. (1984) "Electron energy-loss spectroscopy in the electron microscope"
in Advances in Optical and Electron Microscopy, ed. by V.E. Cosslett and
R. Barer, Academic Press, London, vol. 9, pp. 65-177.

Colliex, C., Cosslett, V.E., Leapman, R.D., and Trebbia, P. (1976) "Contribution
of electron energy-loss spectroscopy to the development of analytical
electron microscopy," Ultramicroscopy 1, 301-315.

Cowley, E.R. (1974) "A Born-von Karman Force Constant Model for
Aluminum," Can. J. Phys. 52, 1714-1715.

Crozier, E.D., Rehr, J.J., and Ingalls, R. (1988) "Amorphous and liquid systems,"
chap. 9 in X-ray Absorption, ed. by D.C. Koningsberger and R. Prins,
Wiley, New York, pp. 373-442.

Csillag, S., Johnson, D.E., and Stern, E.A. (1981) "EXELFS analysis--the useful
data range,” Analytical Electron Microscopy--1981, ed. by R.H. Geiss, San
Francisco Press, San Francisco, pp. 221-224.

Disko, M.M., Meitzner, G., Ahn, C.C., and Krivanek, O.L. (1989) “Temperature-
dependent transmission extended electron energy-loss fine-structure of
aluminum," J. Appl. Phys. 65, 3295-3297.

Disko, M.M., Ahn C.C., and Fultz B., eds. (1992) Applications of Transmission
EELS in Materials Science, The Minerals, Metals & Materials Society,
Warrendale, Pennsylvania 15086.

Egerton, R.F. (1986) Electron Energy-Loss Spectroscopy in the Electron
Microscope, Plenum Press, New York and London.

Egerton, R.F. and Wang, Z.L. (1990) "Plural-scattering deconvolution of electron
energy-loss spectra recorded with an angle-limiting aperture,”
Ultramicroscopy 32, 137-148.

Eisenberger and Lengeler (1980) "Extended x-ray absorption fine-structure
determinations of coordination numbers: Limitations,"Phys. Rev. B 22,
3551-3562.

Franck, J. and Hertz G. (1914) Verhandl. Phys. Ges. 16, 457.

Fricke, H. (1920) "The K-characteristic absorption frequencies for the chemical
elements magnesium to chromium,” Phys. Rev. 16, 202-215.

Gao, Z.Q. and Fultz, B. (1993) "Transient B32-like order during the early stages
of ordering in undercooled FesAl," Phil. Mag. (in press).

Gleiter, H. (1989) "Nanocrystalline materials," Prog. Mat. Sci. 33, 223-315.

321

Gottfried, K. (1966) Quantum Mechanics, Benjamin, New York.

Gurman, S.J. (1982) "Review, EXAFS studies in materials science," J. Mat. Sci.
17, 1541-1570.

Gurman, S.J., Binsted, N., and Ross, I. (1984) "A rapid, exact curved-wave
theory for EXAFS calculations," J. Phys. C17, 143-151.

Harris, S.R., Pearson, D.H., and Fultz, B. (1991) "Chemically disordered Ni3Al
synthesized by high vacuum evaporation," J. Mater. Res. 6, 2019-2021.

Haubold, T., Birringer, R., Lengeler, B., and Gleiter, H. (1989) "Extended X-ray
absorption fine structure studies of nanocrystalline materials exhibiting a
new solid state structure with randomly arranged atoms,” Phys. Lett. 135,
461-466.

Hayes, T.M. and Boyce, J.B. (1982) "Extended x-ray absorption fine structure
spectroscopy,” Solid State Phys. 37, 173-351.

Herman, F. and Skillman, S. (1963) Atomic Structure Calculations, Prentice Hall,
Englewood Cliffs, New Jersey.

Hertz, G. (1920) "Uber die Absorptionsgrenzen in der L-Serie," Z. Phys. 3, 19-25.

Inokuti, M. (1971) "Inelastic collisions of fast charged particles with atoms and
molecules--The Bethe theory revisited," Rev. Mod. Phys. 43, 297-347.

Johnson, D.W. and Spence, J.C.H. (1974) "Determination of the single-scattering
probability distribution from plural-scattering data,” J. Phys. D 7, 771-780.

Kepert, D.L. (1972) The Early Transition Metals, Academic Press, London and
New York.

Kincaid, B.M. and Eisenberger, P. (1975) "Synchrotron radiation studies of the K-
edge photoabsorption spectra of Kr, Bra, and GeCl4: A comparison of
theory and experiment," Phys. Rev. Lett. 34, 1361-1364.

Kincaid, B.M., Meixner, A.E., and Platzman, P.M. (1978) "Carbon K-edge in
graphite measured using electron energy-loss spectroscopy," Phys. Rev.
Lett. 40, 1296-1299.

Kohn, W. and Sham, L.J. (1965) "Self-consistent equations including exchange
and correlation effects,"Phys. Rev. 140, A1133-A1138.

Koningsberger, D.C. and Prins, R., eds. (1988) X-ray Absorption, Wiley, New
York.

322

Kossel, W. (1920) "Zum Bau der Réntgenspektren," Z. Phys. 1, 119-134.

Krivanek, O.L., Ahn, C.C., and Keeney R.B. (1987) "Parallel detection electron
spectrometer using quadrupole lenses," Ultramicroscopy 22, 103-115.

Kronig, R. de L. (1931) "Zur Theorie der Feinstruktur in den
Réntgenabsorptionsspektren," Z. Phys. 70, 317-323.

Kronig, R. de L. (1932) "Zur Theorie der Feinstruktur in den
Rontgenabsorptionsspektren. Ill," Z. Phys. 75, 468-475.

Kruit, P. (1986) "Pushing toward the limits of detectability in electron energy loss
spectrometry," in Microbeam Analysis--1986, pp. 411-416.

Landau, L.D. and Lifshitz, E.M. (1965) Quantum Mechanics; Non-Relativistic
Theory, vol. 3 of Course of Theoretical Physics, second edition,
Pergamon, Oxford and New York.

Leapman, R.D. and Cosslett, V.E. (1976) "Extended fine structure above the x-
ray edge in electron energy-loss spectra," J. Phys. D9, L29-L32.

Leapman, R.D., Rez, P., and Mayers, D.F. (1980) "K, L, and M shell generalized
oscillator strengths and ionization cross sections for fast electron
collisions," J. Chem. Phys. 72, 1232-1243,

Leapman, R.D., Grunes, L.A., and Fejes, P.L. (1982) "Study of the Los edges in
3d transition metals and their oxides by electron-energy-loss spectroscopy.
with comparisons to theory,” Phys. Rev. B 26, 614-635.

Le Caér, G. and Dubois, J.M. (1979) "Evaluation of hyperfine parameter
distributions from overlapped Méssbauer spectra of amorphous alloys,"
J. Phys. E 12, 1083-1090.

Lee, P.A. (1976) "Possibility of absorbate position determination using final-state
interference effects," Phys. Rev. B 13, 5261-5270.

Lee, P.A. and Beni, G. (1977) "New method for the calculation of atomic phase
shifts: Application to extended x-ray absorption fine structure (EXAFS) in
molecules and crystals," Phys. Rev. B 15, 2862-2883.

Lee, P.A. and Pendry, J.B. (1975) "Theory of extended x-ray absorption fine
structure," Phys. Rev. B 11, 2795-2811.

Lee, P.A., Citrin, P.H., Eisenberger, P., and Kincaid, B.M. (1981) “Extended x-ray
absorption fine structure--its strengths and limitations as a structural tool,”
Rev. Mod. Phys. 53, 769-806.

323

Manson, S.T. (1972) "Inelastic collision of fast charged particles with atoms:
lonization of the aluminum L shell," Phys. Rev. A 6, 1013-1024.

Martin, R.L. and Davidson, E.R. (1977) "Halogen atomic and diatomic 1s hole
states," Phys. Rev. A 16, 1341-1346.

Marton, L., Leder, L.B., and Mendlowitz, H. (1955) "Characteristic energy losses

of electrons in solids" in Advances in Electronics and Electron Physics VII,
Academic Press, New York, pp. 183-238.

Massalski, T.B., ed. (1986) Binary Alloy Phase Diagrams, American Society for
Metals, Metals Park, Ohio 44073.

Mayo, M.J., Siegel, R.W., Narayanasamy, A., and Nix, W.D. (1990) "Mechanical
properties of nanophase TiO2 as determined by nanoindentation,” J.
Mater. Res. 5, 1073-1082.

Miiler, A.P. and Brockhouse, B.N. (1971) "Crystal dynamics and electronic
specific
heats of palladium and copper," Can. J. Phys. 49, 704-723.

Minkiewicz, V.J., Shirane, G., and Nathans R. (1967) "Phonon dispersion relation
for iron," Phys. Rev. 162, 528-531.

Pearson, D.H. (1992) "Measurements of White Lines in Transition Metals and
Alloys using Electron Energy Loss Spectrometry,” doctoral dissertation,
California Institute of Technology, Pasadena, California 91125.

Pearson, D.H., Ahn, C.C., and Fultz B. (1988) "Measurements of 3d state
occupancy in transition metals using electron energy loss spectrometry,"
Appl. Phys. Lett. 53, 1405-1407.

Pearson, D.H. (1992) "Measurements of White Lines in Transition Metals and
Alloys using Electron Energy Loss Spectrometry,” doctoral dissertation,
California Institute of Technology, Pasadena, California.

Penn, D.R. (1976) "Electron mean free paths for free-electron-like materials,"
Phys. Rev. B 13, 5248-5254.

Powell, C.J. (1974) "Attenuation lengths of low-energy electrons in solids,"
Surface Sci. 44, 29-46.

Ritsko, J.J., Schnatterly, S.E., and Gibbons, P.C. (1974) "Simple calculation of
Lj, absorption spectra of Na, Al, and Si," Phys. Rev. Lett. 32, 671-674.

Rudberg, E. (1930) "Characteristic energy losses of electrons scattered from
incandescent solids," Proc. R. Soc. London A127, 111-140.

324

Ruthemann, G. (1941) "Diskrete Energieverluste schneller Elektronen in
Festkérpern," Naturwissenschaften 29, 648.

Ruthemann, G. (1942) "Elektronenbremsung an Réntgenniveaus,"
Naturwissenschaften 30, 145.

Sayers, D.E., Stern, E.A., and Lytle, F.W. (1971) "New technique for investigating
noncrystalline structures: Fourier analysis of the extended X-ray
absorption fine structure," Phys. Rev. Lett. 27, 1204-1207.

Sayers and Bunker (1988) "Data Analysis" in X-ray Absorption, ed. by
Koningsberger, D.C. and Prins, R., Wiley, New York.

Schiach, W. (1973) "Comment on the Theory of Extended X-Ray-Absorption Fine
Structure," Phys. Rev. B 8, 4028-4032.

Schiach, W.L. (1984) "Derivation of single-scattering formulas for X-ray-
absorption and high-energy electron-loss spectroscopies," Phys. Rev. B
29, 6513-6519.

Schoone, R.D. and Fischione, E.A. (1966) "Automatic unit for thinning
transmission electron micrograph specimens of metals," Rev. Sci. Inst.,
37, 1351.

Scott, R.A. (1983) “EXAFS Software Documentation,” School! of Chemical
Sciences, University of Illinois.

Seah, M.P. and Dench, W.A. (1 979) “Quantitative electron spectroscopy of
surfaces: A standard data base for electron inelastic mean free paths in
solids,” Surf. Inter. Anal. 1, 2-11.

Seitz, F. and Turnbull, D., eds. (1956) Solid State Physics, vol. 2, Academic
Press, New York.

Sevillano, E., Meuth, H., and Rehr J.J. (1979) "Extended x-ray absorption fine
structure Debye-Waller factors. |. Monatomic Crystals," Phys. Rev. B 20,
4908-4911.

Shuman, H. and Somlyo, A.P. (1981) "Energy filtered ‘conventional' transmission
imaging with a magnetic sector spectrometer," in Analytical Electron
Microscopy--1981, ed. R.H. Geiss, San Francisco Press, San Francisco,
California, pp. 202-204.

Siegel, R.W., Ramasamy, S., Hahn, H., Li, Z., Lu, T., and Gronsky, R. (1988)
"Synthesis, characterization, and properties of nanophase TiOo," J. Mater.
Res. 3, 1367-1372.

325

Spence, J.C.H. (1979) "Uniqueness and the inversion problem of incoherent
multiple scattering," Ultramicroscopy 4, 9-12.

Stearns, D.G. and Stearns, M.B. (1986) "Extended x-ray absorption fine
Structure,” chap. 6 in Microscopic Methods in Metals, ed. by U. Gonser,
vol. 40 of Topics in Curent Physics, Springer, Berlin, pp. 153-192.

Stedman, R., Almqvist, L., and Nilsson, G. (1967) “Phonon-frequency
distributions and heat capacities for aluminum and lead," Phys. Rev. 162,
549-557.

Stern, E.A. (1974) "Theory of extended x-ray-absorption fine structure," Phys.
Rev. B 10, 3027-3037.

Stern, E.A., Bunker, B.A., and Heald, S.M. (1 980) “Many-body effects on
extended x-ray absorption fine structure amplitudes," Phys. Rev. B 21,
5521-5539.

Teo, B.K. and Lee, P.A. (1979) "Ab initio calculations of amplitude and phase
functions for extended x-ray absorption fine structure spectroscopy,"
J. Am. Chem. Soc. 101, 2815-2832.

Teo, B.K. and Joy, D.C., eds. (1981) EXAFS spectroscopy, techniques and
applications, Plenum Press, New York and London.

Teo, B.K. (1986) EXAFS: Basic Principles and Data Analysis, Springer-Verlag,
Berlin and New York.