Temperature Dependence of Phonons in Ele

Temperature Dependence of Phonons in Elemental Cubic Metals Studied by Inelastic Scattering of Neutrons and X-Rays - CaltechTHESIS
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Temperature Dependence of Phonons in Elemental Cubic Metals Studied by Inelastic Scattering of Neutrons and X-Rays
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Kresch, Max G.
(2009)
Temperature Dependence of Phonons in Elemental Cubic Metals Studied by Inelastic Scattering of Neutrons and X-Rays.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/5FYM-3M24.
Abstract
The vibrations of atoms in a crystal (phonons) make up the majority of its entropy (or heat capacity), and these vibrations are typically modeled by simple harmonic oscillators. Deviations from this harmonic oscillator model are responsible for such well known effects as thermal expansion and temperature dependent elastic constants. Other anharmonic effects, such as the temperature dependence of phonon energies and their linewidths, may be less well known, but also significant. Changes in phonon energies can impact the phonon entropy.
Measurements of the phonon spectra of aluminum, lead, nickel, and iron as a function of temperature are presented, and the anharmonic contributions to the entropies of these cubic metals are considered. These contributions are found to be of the same order of magnitude as those from independent electrons (discounting magnetic contributions). Trends in phonons and phonon-related properties of a wider array of face-centered- and body-centered-cubic (FCC & BCC) metals are also considered. The near-neighbor forces, spectral shapes, and anharmonic entropies of the BCC metals are shown to be far more varied than those of the FCC metals, and this is explained in terms of the crystal structures themselves. Finally, given the similarities in the FCC metals, experimental data and molecular dynamics simulations are used to investigate their phonon linewidths. Trends exist, and they imply similarities in the relative strengths of the harmonic and anharmonic forces in the FCC metals.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
anharmonicity; BCC; FCC; inelastic scattering; lattice dynamics; metals; molecular dynamics; neutron scattering; phonon; phonon lifetimes; phonon linewidths; temperature dependence
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Materials Science
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Fultz, Brent T.
Thesis Committee:
Fultz, Brent T. (chair)
Van de Walle, Axel
Jackson, Jennifer M.
Johnson, William Lewis
Sturhahn, Wolfgang
Defense Date:
5 December 2008
Record Number:
CaltechETD:etd-12082008-130722
Persistent URL:
DOI:
10.7907/5FYM-3M24
ORCID:
Author
ORCID
Kresch, Max G.
0000-0002-6990-8979
Default Usage Policy:
No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:
4884
Collection:
CaltechTHESIS
Deposited By:
Imported from ETD-db
Deposited On:
19 Dec 2008
Last Modified:
26 Nov 2019 19:14
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Temperature Dependence of Phonons in Elemental Cubic
Metals Studied by Inelastic Scattering of Neutrons and
X-Rays

Thesis by

Max G. Kresch

In Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy

California Institute of Technology
Pasadena, California

2009
(Defended 5 December 2008)

2009

Max G. Kresch

iii

To my wife, Agnes

Acknowledgments
Firstly, I thank my advisor, Brent Fultz. His advice was always insightful, and without his support
this thesis would not exist.
Secondly, I thank my research group. They have been a constant source of answers and support.
Former group members Olivier Delaire, Matt Lucas, Tabitha Swan-Wood, Alex Papandrew, Tim
Kelley, and Jason Graetz have provided friendship and guidance — many of them both during and
after their time at Caltech. Current group members Jiao Lin, Rebecca Stevens, Mike Winterrose,
Chen Li, Hongjin Tan, Mike McKerns, Justin Purewal, Jorge Munoz, and Nick Stadie continue to
make the Fultz lab a fun and fruitful place to work. Channing Ahn has always found time to lend
an ear, give a recommendation, or help me find some device that would otherwise be hopelessly lost
in the lab. To new and future members of the Fultz research group I wish the best of luck.
Thirdly, I thank my friends — both at Caltech and elsewhere. They have always been around
to keep me sane while my work was trying to do otherwise. Near-weekly dinners with Tom and
Susan Johnson, Mary Laura Lind, John McCorquodale, and Demetri Spanos have always been
events to look forward to. Moreover, I thank John and Demetri for having spent many hours
keeping me company, and for having made non-trivial contributions to my work. I also thank Jason
Schaefer, who has always had time to listen to my complaints and then remind me of their relative
unimportance. My friends have made my time at Caltech worthwhile.
Fourthly, I thank my family. Their love and support predates this work and will (I hope) persist
long after its completion. My mother, my father, Klara, Mike, Anyu, Apu, and Sabba always had
faith in me, and I could not have succeeded without it. In particular, my father always made time
to talk to me about my work, and his questions and comments were extremely valuable.
Fifthly, I thank my wife, Agnes. For more than four years she has patiently provided support
from halfway across the country. Without her love I would have never had the strength to complete
this work.
Finally, I thank Spork. His affectionate moments have far exceeded his bitey ones.

Financial support for this thesis was provided by the United States Department of Energy and
by the National Science Foundation.

Abstract
This thesis explores the temperature dependence of phonons in the cubic transition and nearly free
electron metals through both experiment and simulation. Particular attention is paid to the entropic
contributions of the phonons, and to their increased linewidths at high temperatures.
Measurements of the inelastic scattering of neutrons from face-centered-cubic (FCC) aluminum,
lead, and nickel were performed at temperatures ranging from near absolute zero to roughly 80%
of the melting temperature. Similarly, measurements of the nuclear resonant inelastic scattering of
x-rays from body-centered-cubic (BCC) iron were made at temperatures from near zero to roughly
50% of the melting temperature. These experimental techniques allowed access to the entire phonon
spectrum; and the experimental data were thus used to find the phonon spectra of these metals as a
function of temperature. Given the phonon spectra and previous measurements of the temperature
dependent thermal expansion and bulk modulus, the harmonic, quasiharmonic, and anharmonic
phonon contributions to entropy were evaluated. Further, detailed consideration was made of the
contributions of electrons, magnons, electron-phonon interactions, and vacancies to the entropy.
For the FCC metals aluminum and nickel, the anharmonic contributions of the phonons were
small and negative; that is, the phonons did not shift as much as expected given the thermal
expansion and bulk moduli. This was also the case for FCC lead; here, however, the anharmonic
phonon contributions were larger. For BCC iron, even though the temperatures measured were far
below the melting point, the anharmonic phonon contributions were found to be both large and
positive. The agreement of the sums of the contributions with measured values of the total entropy
was excellent for the nearly free metals aluminum and lead. For FCC nickel and BCC iron, the
differences between the total and the sums of electronic and vibrational contributions were used to
learn about the magnetic contributions to the entropy; and these contributions appear to be larger
than those of the anharmonic phonons. In all of these metals, the contributions of anharmonicity
were significantly larger than those of electron-phonon interactions and vacancies, and they were
not negligible on the scale of the electronic entropy. We found that any serious consideration of the
contributions to the total entropy of crystalline solids would be incomplete without consideration of
the anharmonicity.
In aluminum, the best agreement between the sum of the components and the total entropy was

vi
obtained using the damped phonon spectra (once corrected for instrument resolution). Tentatively,
it seems that models which account for phonon broadening may lead to more accurate estimations
of the phonon entropy than the widely accepted quasiharmonic model.
We have also considered some trends in FCC and BCC transition metals, as well as in FCC
nearly free electron metals; to do so, we have supplemented our experimental data with some from
other sources. Overall, we found relatively large variations in the anharmonic entropy of the BCC
metals; whereas for the FCC metals the anharmonic entropy tended to be small and negative. The
first nearest neighbors dominated the forces in the FCC metals, but for BCC metals the second
nearest neighbors made significant contributions. Further, there were instabilities in the transverse
forces in the first nearest neighbor shell for BCC metals, but not for FCC. Finally, we found that
the shapes of the phonon spectra of the FCC metals were more consistent than those of their BCC
counterparts — both across metals and as a function of temperature.
Given the similarities in the forces and anharmonic entropies in the FCC metals, we have performed a detailed study of their phonon linewidths using both experimental data and molecular
dynamics simulations. The linewidths were quantified using quality factors, and these were scaled
by the square root of the nuclear mass. For both the experimental and simulated data, we scaled
the temperature at which the phonon data were collected by the melting temperature, and plotted
the scaled qualities against this quantity. The curves so generated were all superlinear, and all fell
within roughly the same region on the plot. This indicated that the ratio of anharmonic to harmonic
forces was similar across the FCC metals. This was tested more directly using a ratio of anharmonic
to harmonic forces taken from the potentials used for the molecular dynamics simulations. The
simulations also indicated that the phonon linewidths were correlated with the thermal expansion
of the materials. Since the forces in the FCC metals — as measured by the linewidths — are similar despite differing electronic structures, and since they are tied to the nuclear potential through
their correlation with thermal expansion, we concluded that the phonon linewidths arise from the
nuclear potential. That is, phonon linewidths in the FCC metals were found to be determined by
phonon-phonon interactions, rather than by interactions of phonons with electrons or magnons.

vii

Contents
Acknowledgments

Abstract

1 Introduction

1.1

General background and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2

Present work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Background

2 Notation

2.1

Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Crystals and Phonons

3.1

Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.1

Periodic boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.2

Adiabatic approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.3

Expansion of the potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3

The quantum harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4

Harmonic lattice dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4.1

Decoupled modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4.2

Normal modes of a crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5

The quantum harmonic oscillator revisited: phonons . . . . . . . . . . . . . . . . . .

3.6

Anharmonic lattice dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Components of the Entropy of a Solid

viii
5 Neutron Scattering

5.1

General theory of neutron scattering . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2

Neutron scattering from crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2.1

The Debye-Waller factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2.2

Elastic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2.3

1-phonon scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2.4

Multiphonon scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2.5

Scattering from a damped harmonic oscillator . . . . . . . . . . . . . . . . . .

6 Time-of-flight Neutron Chopper Spectrometers

6.1

Spallation neutron sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2

Direct geometry, time-of-flight, chopper spectrometers . . . . . . . . . . . . . . . . .

6.3

Time-of-flight vs. triple-axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.4

Instruments used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Data Analysis and Computation

7 From Raw Data to S(Q,E)
7.1

52
53

Detector masking and efficiency corrections . . . . . . . . . . . . . . . . . . . . . . .

7.1.1

Masking bad detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.1.2

Energy-dependent detector efficiency . . . . . . . . . . . . . . . . . . . . . . .

7.1.3

Pressure-dependent detector efficiency . . . . . . . . . . . . . . . . . . . . . .

7.1.4

Solid-angle of the pixels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Determination of the incident energy . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.2.1

Using monitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.2.2

Using scattered data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.3

Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.4

Transformation to physical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . .

7.4.1

Rebinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.4.2

Analytical coordinate transformation . . . . . . . . . . . . . . . . . . . . . . .

7.2

7.5

Removal of background scattering

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

8 Processing S(Q,E)
8.1

8.2

71
73

Finding the phonon DOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.1.1

Fourier-log method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.1.2

Iterative method, with correction for multiple scattering . . . . . . . . . . . .

Shift and linewidth analysis of the DOS . . . . . . . . . . . . . . . . . . . . . . . . .

ix
8.3

8.4

III

Born–von Kármán fits to the DOS . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.3.1

Fitting real data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.3.2

Longitudinal and transverse force constants . . . . . . . . . . . . . . . . . . .

Finding the lattice parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Phonons in FCC Metals at Elevated Temperatures

9 Aluminum

92
93

9.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.2

Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.2.1

Sample preparation

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

9.2.2

Neutron scattering measurements . . . . . . . . . . . . . . . . . . . . . . . . .

Analysis and computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.3.1

General data reduction

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

9.3.2

Elastic scattering: in-situ neutron diffraction . . . . . . . . . . . . . . . . . .

9.3.3

Inelastic scattering: S(Q,E) and the phonon DOS . . . . . . . . . . . . . . . .

9.3.4

Phonon shifts and broadening . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.3.5

Born–von Kármán models of the lattice dynamics . . . . . . . . . . . . . . . 100

9.3.6

Ab-initio phonon calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

9.3

9.4

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

9.5

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

9.6

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

10 Lead

111

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
10.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
10.2.1 Sample preparation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

10.2.2 Neutron scattering measurements . . . . . . . . . . . . . . . . . . . . . . . . . 114
10.3 Analysis and computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
10.3.1 General data reduction

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

10.3.2 Elastic scattering: in-situ neutron diffraction . . . . . . . . . . . . . . . . . . 115
10.3.3 Background determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
10.3.4 Inelastic scattering: S(Q,E) and the phonon DOS . . . . . . . . . . . . . . . . 118
10.3.5 Phonon shifts and broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
10.3.6 Ab-initio electronic structure calculations . . . . . . . . . . . . . . . . . . . . 120
10.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
10.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

10.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
11 Nickel

129

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
11.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
11.2.1 Sample preparation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

11.2.2 Neutron scattering measurements . . . . . . . . . . . . . . . . . . . . . . . . . 131
11.3 Data analysis and computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
11.3.1 General data reduction

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

11.3.2 Elastic scattering: in-situ neutron diffraction . . . . . . . . . . . . . . . . . . 132
11.3.3 Inelastic scattering: S(Q,E) and the density of states . . . . . . . . . . . . . . 133
11.3.4 Phonon shifts and broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
11.3.5 Born–von Kármán models of lattice dynamics . . . . . . . . . . . . . . . . . . 135
11.3.6 Ab-initio calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
11.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
11.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

Phonons in BCC Metals at Elevated Temperatures

12 Iron

145
146

12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
12.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
12.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
12.3.1 Sample preparation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

12.3.2 Nuclear resonant inelastic x-ray scattering at HP-CAT . . . . . . . . . . . . . 150
12.4 Analysis and computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
12.4.1 S(E) and the phonon density of states . . . . . . . . . . . . . . . . . . . . . . 152
12.4.2 Phonon shifts and broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
12.4.3 Born–von Kármán models of lattice dynamics . . . . . . . . . . . . . . . . . . 154
12.4.4 Ab-initio calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
12.5 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
12.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
13 Chromium and Vanadium

165

Phonon Trends in Cubic Metals

169

14 Anharmonicity and the Shape of the Phonon DOS

170

15 Mean Phonon Linewidths in FCC metals

178

15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
15.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
15.2.1 Quality factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
15.2.2 Molecular dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
15.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
15.3.1 Qualities in conservative systems: The Fermi-Pasta-Ulam problem . . . . . . 186
15.3.2 Molecular dynamics with GULP . . . . . . . . . . . . . . . . . . . . . . . . . 189
15.4 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
15.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
15.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

Summary and Future Work

205

16 Summary

206

17 Future Work

209

17.1 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
17.2 Vibrations, magnetism, and superconductivity . . . . . . . . . . . . . . . . . . . . . . 211
17.3 Mean phonon lifetimes in FCC Metals . . . . . . . . . . . . . . . . . . . . . . . . . . 211
17.4 Vibrational entropy in the presence of damping . . . . . . . . . . . . . . . . . . . . . 212
Appendices

213

A Table of Symbols

214

B Entropy of Non-interacting Fermions and Bosons

219

C Analytical Reweighting — Algebra

222

D Summation Over Q

226

E Constraints on Force Constants in BvK Models

228

F Supplementary Material for Chapter 15

232

G Code

241

xii
Bibliography

243

xiii

List of Figures
1.1

Generic pressure-temperature phase diagram . . . . . . . . . . . . . . . . . . . . . . .

1.2

P-T phase diagram for Fe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3

Al-Ni Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.4

anharmonic entropy of K, Na, and Rb

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

1.5

Anharmonic entropy of a few transition metals . . . . . . . . . . . . . . . . . . . . . .

3.1

2D lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

3D lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3

Approximate fraction of atoms in the bulk . . . . . . . . . . . . . . . . . . . . . . . .

3.4

Approximate number of atoms in the bulk . . . . . . . . . . . . . . . . . . . . . . . . .

3.5

A torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.6

Coupled oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1

Scattering experiments

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

5.2

Scattering triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.1

Schematic of a spallation source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2

Direct-geometry, time-of-flight Chopper spectrometer . . . . . . . . . . . . . . . . . .

6.3

Neutron scattered into a detector module . . . . . . . . . . . . . . . . . . . . . . . . .

6.4

Lengths and times for a time-of-flight chopper spectrometer . . . . . . . . . . . . . . .

6.5

Region sampled in Q, E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.6

3-axis spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.7

Schematic of LRMECS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.8

Schematic of Pharos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.9

Schematic of ARCS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.1

Uncalibrated scattering from vanadium . . . . . . . . . . . . . . . . . . . . . . . . . .

7.2

Finding bad detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.3

Calibrated scattering from vanadium . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.4

Solid angle corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiv
7.5

Counts in the beam monitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.6

Incident energy from It . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.7

Comparison of incident energy determinations . . . . . . . . . . . . . . . . . . . . . .

7.8

Proton current versus monitor counts for Pharos . . . . . . . . . . . . . . . . . . . . .

7.9

Rebinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.10

Interpolation of LRMECS data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.11

Comparison of different I(Q,E) determinations . . . . . . . . . . . . . . . . . . . . . .

8.1

Previous multiphonon corrections

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

8.2

S inc (Q, E) for nickel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.3

Penalty functions for multiphonon/multiple scattering correction . . . . . . . . . . . .

8.4

Fit to scattering for nickel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.5

Non-commutation of broadening effects . . . . . . . . . . . . . . . . . . . . . . . . . .

8.6

Least squares error for fits of shift and quality factor . . . . . . . . . . . . . . . . . . .

8.7

BvK with anharmonic broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.8

Diffraction from nickel at varied EI

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

8.9

Nelson Riley analysis for nickel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.1

In-situ neutron diffraction from aluminum . . . . . . . . . . . . . . . . . . . . . . . . .

9.2

Phonon DOS of aluminum, comparison of LRMECS and Pharos . . . . . . . . . . . .

9.3

Phonon DOS of aluminum and fits from the low temperature DOS . . . . . . . . . . .

9.4

1/Q for aluminum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

9.5

Phonon DOS of aluminum and BvK fits . . . . . . . . . . . . . . . . . . . . . . . . . . 101

9.6

Longitudinal force constants of aluminum . . . . . . . . . . . . . . . . . . . . . . . . . 103

9.7

Averaged transverse force constants of aluminum . . . . . . . . . . . . . . . . . . . . . 104

9.8

Electronic DOS of aluminum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

9.9

Phonon DOS for aluminum with and without broadening . . . . . . . . . . . . . . . . 107

9.10

Contributions to the entropy of aluminum . . . . . . . . . . . . . . . . . . . . . . . . . 109

10.1

Phonons in lead from neutron scattering and electron tunneling . . . . . . . . . . . . . 113

10.2

Phonons in lead from electron tunneling and far-infrared reflectance . . . . . . . . . . 113

10.3

In-situ neutron diffraction from lead . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

10.4

Determination of the background for lead measured at ARCS . . . . . . . . . . . . . . 117

10.5

Phonon DOS of lead and fits from the low temperature DOS . . . . . . . . . . . . . . 119

10.6

1/Q for aluminum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

10.7

Electronic DOS of lead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

10.8

Phonon linewidths in lead from triple axis measurements . . . . . . . . . . . . . . . . 123

xv
10.9

Contributions to the entropy of lead . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

11.1

In-situ neutron diffraction from nickel . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

11.2

Phonon DOS of nickel and BvK fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

11.3

1/Q for nickel

11.4

Phonon DOS of nickel and fits from the low temperature DOS . . . . . . . . . . . . . 135

11.5

Phonon dispersions for nickel from BvK models . . . . . . . . . . . . . . . . . . . . . . 137

11.6

Non-magnetic electronic DOS of nickel . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

11.7

Spin-polarized and non-magnetic electronic DOS of nickel . . . . . . . . . . . . . . . . 139

11.8

Longitudinal force constants of nickel . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

11.9

Averaged transverse force constants of nickel . . . . . . . . . . . . . . . . . . . . . . . 141

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

11.10 Contributions to the entropy of nickel . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
12.1

Timing of NRIXS experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

12.2

S(E) for iron from NRIXS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

12.3

Phonon DOS of iron and fits from the low temperature DOS . . . . . . . . . . . . . . 152

12.4

Phonon DOS of iron and BvK fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

12.5

Longitudinal force constants of iron . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

12.6

Averaged transverse force constants of iron . . . . . . . . . . . . . . . . . . . . . . . . 156

12.7

Non-magnetic electronic DOS of iron . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

12.8

Spin-polarized and non-magnetic electronic DOS of iron . . . . . . . . . . . . . . . . . 158

12.9

Contributions to the entropy of iron . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

13.1

Phonon DOS of chromium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

13.2

Phonon DOS of vanadium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

14.1

Shapes of phonon spectra from BvK models for FCC and BCC metals . . . . . . . . . 174

14.2

Temperature dependence of phonon DOS for BCC and FCC metals from BvK models 176

14.3

First nearest neighbors for cubic lattices . . . . . . . . . . . . . . . . . . . . . . . . . . 177

15.1

S(Q, E) for aluminum, lead, and nickel . . . . . . . . . . . . . . . . . . . . . . . . . . 179

15.2

Potential energy of a damped harmonic oscillator with Q = 2 . . . . . . . . . . . . . . 181

15.3

Three atom chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

15.4

Displacement amplitude in the three atom chain . . . . . . . . . . . . . . . . . . . . . 187

15.5

Displacement amplitude in normal modes of three atom chain . . . . . . . . . . . . . . 187

15.6

Kinetic energy in normal modes of three atom chain . . . . . . . . . . . . . . . . . . . 188

15.7

Kinetic energy in the vibrational modes of the three atom chain . . . . . . . . . . . . 189

15.8

Lattice parameters and qualities for aluminum and lead from molecular dynamics . . 193

xvi
15.9

Cross-sections of super-cell for molecular dynamics . . . . . . . . . . . . . . . . . . . . 194

15.10 Amplitude in low energy mode for Cu and high energy mode for Ag . . . . . . . . . . 194
15.11 Energy response for modes in nickel and platinum . . . . . . . . . . . . . . . . . . . . 195
15.12 Phonon DOS of Al and Ir from molecular dynamics . . . . . . . . . . . . . . . . . . . 196
15.13 Phonon linewidths in copper from triple axis measurements . . . . . . . . . . . . . . . 197
15.14 Shifts vs. qualities for FCC metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
15.15 Universality of scaled quality factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
15.16 Ratios of anharmonic to harmonic forces in FCC metals from molecular dynamics . . 202
D.1

Integration region for determinig the phonon DOS . . . . . . . . . . . . . . . . . . . . 226

F.1

Lattice parameter and Q of Al from MD

. . . . . . . . . . . . . . . . . . . . . . . . . 232

F.2

Lattice parameter and Q of Ni from MD

. . . . . . . . . . . . . . . . . . . . . . . . . 232

F.3

Lattice parameter and Q of Cu from MD . . . . . . . . . . . . . . . . . . . . . . . . . 233

F.4

Lattice parameter and Q of Rh from MD . . . . . . . . . . . . . . . . . . . . . . . . . 233

F.5

Lattice parameter and Q of Pd from MD . . . . . . . . . . . . . . . . . . . . . . . . . 233

F.6

Lattice parameter and Q of Ag from MD . . . . . . . . . . . . . . . . . . . . . . . . . 233

F.7

Lattice parameter and Q of Ir from MD . . . . . . . . . . . . . . . . . . . . . . . . . . 234

F.8

Lattice parameter and Q of Pt from MD . . . . . . . . . . . . . . . . . . . . . . . . . 234

F.9

Lattice parameter and Q of Au from MD . . . . . . . . . . . . . . . . . . . . . . . . . 234

F.10

Lattice parameter and Q of Pb from MD . . . . . . . . . . . . . . . . . . . . . . . . . 234

F.11

Phonon DOS of Al from MD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

F.12

Phonon DOS of Ni from MD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

F.13

Phonon DOS of Cu from MD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

F.14

Phonon DOS of Rh from MD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

F.15

Phonon DOS of Pd from MD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

F.16

Phonon DOS of Ag from MD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

F.17

Phonon DOS of Ir from MD

F.18

Phonon DOS of Pt from MD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

F.19

Phonon DOS of Au from MD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

F.20

Phonon DOS of Pb from MD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

F.21

Projections onto normal modes for Al from MD

. . . . . . . . . . . . . . . . . . . . . 238

F.22

Projections onto normal modes for Ni from MD

. . . . . . . . . . . . . . . . . . . . . 238

F.23

Projections onto normal modes for Cu from MD . . . . . . . . . . . . . . . . . . . . . 239

F.24

Projections onto normal modes for Rh from MD . . . . . . . . . . . . . . . . . . . . . 239

F.25

Projections onto normal modes for Pd from MD . . . . . . . . . . . . . . . . . . . . . 239

F.26

Projections onto normal modes for Ag from MD . . . . . . . . . . . . . . . . . . . . . 239

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

xvii
F.27

Projections onto normal modes for Ir from MD . . . . . . . . . . . . . . . . . . . . . . 240

F.28

Projections onto normal modes for Pt from MD . . . . . . . . . . . . . . . . . . . . . 240

F.29

Projections onto normal modes for Au from MD . . . . . . . . . . . . . . . . . . . . . 240

F.30

Projections onto normal modes for Pb from MD . . . . . . . . . . . . . . . . . . . . . 240

xviii

List of Tables
6.1

Details about LRMECS, Pharos, and ARCS . . . . . . . . . . . . . . . . . . . . . . .

9.1

Lattice parameters and phonon energy shifts for aluminum . . . . . . . . . . . . . . .

9.2

Optimized tensorial force constants for aluminum

9.3

Comparison of longitudinal force constants for aluminum . . . . . . . . . . . . . . . . 105

10.1

Lattice parameters and phonon energy shifts for lead . . . . . . . . . . . . . . . . . . . 116

10.2

Characteristic phonon temperatures for lead . . . . . . . . . . . . . . . . . . . . . . . 122

10.3

Quality factors for lead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

10.4

Vacancy formation in lead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

11.1

Lattice parameters and phonon energy shifts for nickel . . . . . . . . . . . . . . . . . . 132

11.2

Optimized tensorial force constants for nickel . . . . . . . . . . . . . . . . . . . . . . . 136

12.1

Optimized tensorial force constants for iron . . . . . . . . . . . . . . . . . . . . . . . . 154

14.1

Anharmonic entropies at melting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

14.2

Longitudinal and transverse force constants for FCC metals . . . . . . . . . . . . . . . 172

14.3

Longitudinal and transverse force constants for BCC metals . . . . . . . . . . . . . . . 173

14.4

Shapes of phonon spectra from BvK models for FCC and BCC metals . . . . . . . . . 175

15.1

Optimized parameters of embedded atom potentials for FCC metals . . . . . . . . . . 191

15.2

Elastic constants of FCC metals from molecular dynamics . . . . . . . . . . . . . . . . 192

. . . . . . . . . . . . . . . . . . . . 102

Chapter 1

Introduction
1.1

General background and motivation

Despite great advances in both experimental and theoretical techniques, predicting phase diagrams
of even elemental materials remains an active area of research. [1–10] Even at zero temperature
and zero pressure, crystal structures are typically an input to first-principles calculations. The
most obvious difficulty is the roughly unlimited number of available spatial configurations.a This is
compounded as the number of constituents increases, and as more complicated phenomena, such as
magnetism or superconductivity, become involved.
At non-zero temperatures and pressures, the energy alone is insufficient to determine phase
stability. Thermal energy causes excitations from equilibrium (and the creation of quasi-particles),
and there are myriad possible distributions of this energy over the (possibly myriad) degrees of
freedom in the system. From basic thermodynamics, we know that the stability of a phase is
determined by its Gibbs or Helmholtz free energy, G or F respectively:

U + PV − TS ,

(1.1)

U − TS .

(1.2)

For elemental crystals, like those studied here, these energies determine temperatures and pressures
at which melting, boiling, and sublimation occur. Additionally, solid-solid phase transitions are
possible. In particular, it is fairly common for elemental metals to undergo phase changes under
high pressures or at elevated temperatures. [11–20] Fig. 1.1 shows a generic pressure-temperature
phase diagram for an element, with solid, liquid and gas phases. Fig. 1.2 shows the phase diagram
for iron, where there exist multiple solid-solid phase transitions.
In multi-component alloys, the variety of phases available is combinatorially greater. Again, the
a There are only 14 Bravais lattice in three dimension; but there are 230 space groups and non-periodic structures
as well.

perconductivity and magnetism are
ea of iron as a superconductor is ruled

e.
al

Temperature (K)

non-magnetic metals
when they areLiquid
cooled
2,000
Solid
to low enough temperatures for their confcc
duction electrons to pair up and flow withhcp
out resistance. However, tinyCritical
magnetic
Point
1,000
impurities can destroy superconductivity in
Incipient
conventional superconductors by breaking
antiferromagnetism?
bcc
up the electron pairs. And even when true
FerroSuperconductivity
Triple Point
100 Tc
magnetism
magnetic order does not occur, the dynamic
magnetic fluctuations that occur when a
20
60
40
Gas
material is on the verge of ferromagnetism
Pressure (gigapascals)
can be enough to suppress Temperature
superconductivi4
ty, as in palladium
. In addition to having Figure 1 The temperature–pressure phase
Figure 1.1: Generic phase diagram Figure 1.2: Phase diagram of iron at relatively low temzero electrical for
resistance,
superconductors
diagram showing
of iron. multiple
The low-pressure
a single component
system (like peratures
solid phases.and
[11] lowThe phase
an
element).
As
the
temperature
inboundaries
are
determined
not
only
by
energetics,
but also
expel magnetic fields from their interior — temperature phase of iron has a body-centred
creases the higher entropy gas and liq- by entropic contributions of the various degrees of freedom
the so-called Meissner
effect. (This dramatic cubic (bcc) structure and is strongly
uid phases are stabilized. Increas- of the system (electronic, nuclear, and magnetic). Intereffect allows a permanent
magnet
be leviferromagnetic.
Thedegrees
higher-pressure
phase
has a
ing pressure stabilizes
theto
lower
energy actions
between these
of freedom also
contribute,
solid
phase.
particularly
around
the
superconducting
region.
tated above a superconductor.) So it has long hexagonal close-packed (hcp) structure and
been thought that superconductivity and might be weakly or nearly antiferromagnetic at
ferromagnetismequilibrium
are incompatible.
But low temperatures. Shimizu et al.3 show that
phase is determined by minimization of the free energy. Phenomena such as martensitic
despite these widespread beliefs, supercon- below 2 K and at pressures above 10 GPa the hcp
phase transitions and spinodal decomposition also depend on the details of the energies and entropies
ductivity and ferromagnetism can mix when phase of iron becomes superconducting. (Here
of the phases. 5,6
phase diagram for aluminum and nickel, with a wide
the magnetization
is small [22–24]
. Fig. 1.3 shows a binary
the superconducting transition temperature, Tc,
variety of phases
including
FCC- andhas
BCC-type
structures. by a factor of about 100.)
In its ferromagnetic
bcc tate,
evenboth
at very
been magnified
It issuperconductivity
critical to understand thehas
contributions
of theofvarious
degrees
of freedom
(quasi-particles
low temperatures,
The shape
the phase
diagram
at low
and collective
excitations)
to the
free energy
(or entropy S,inorthe
heat
capacity
C...) of
materials,
never been observed
in iron.
This is
where
temperatures
border
region
between
theand
the high pressures
come
Applying
presbcc of
and
the superconducting
hcp
this too
is an in.
active
field of inquiry.
[13, 14,ferromagnetic
25–42] Calculations
electronic
structure using density
sure to iron changes
structure
hcp,
phases
has still
notinternal
been established
functional its
theory
frequentlyto
give
an accurate
account
of the
energy of a solid; however,
thereby destroying
its ferromagnetic
thirdmetal
structural
of
accounting
for the entropy isorder.
much moreexperimentally.
complicated. In The
a simple
at low form
temperatures,
In this situationthesuperconductivity
becomes iron shown here is fcc (face-centred cubic).
electrons dominate the entropy of the crystal. Defects and magnetism may also make sizable
Pressure

’s
os

is easily suppressed by strong magnetic fluctuations. So if this is the correct model, the
magnetic fluctuations must be quite weak, a
result that has implications for the Earth’s

contributions to the entropy, and in polyatomic NA
crystals,
is configurational
entropy as well.
TURE |there
VOL 412
| 19 JULY2001 | www.nature.com

At higher temperatures, the largest source of entropy is very commonly the vibrations of the nuclei
about their equilibrium positions.
Most of solid state theory relies on the assumption that the oscillations of the nuclei about their
equilibria are sufficiently small that the potential experienced by the nuclei is harmonic. Under
this assumption, the phonon spectrum may be determined from first principles calculations. [43–49]
Further, there exist closed form expressions for the free energy, entropy, and heat capacity of a
harmonic solid, given the phonon spectrum. We know, however, that this assumption is untrue.
Perhaps the most obvious contradiction is the existence of thermal expansion, which is precluded

Figure 1.3: Binary phase diagram for aluminum and nickel. [21] On either end, there are disordered
face-centered-cubic (FCC) solid solutions of the two constituents. Towards the center there are
ordered compounds at and around 3/4, 3/5, 1/2, 3/8, and 1/4 aluminum atoms. Individually, both
elements form are FCC; however, the equilibrium structures of their binary compounds are not
always FCC-like.
under the harmonic approximation. [50–52] Additionally, the frequency of vibration is independent
of the amplitude in a harmonic potential. [53] Therefore, if the primary effect of temperature on the
oscillations of the nuclei is to increase their amplitudes, the vibrational spectrum in a harmonic solid
would remain constant despite changes in temperature. Contrary to this prediction, experiments
have shown that the temperature can have a significant impact on the phonon spectrum. [35–42, 54–
60]
A relatively simple attempt to deal with this failure is called the quasiharmonic approximation. [36, 37] The idea, here, is that at fixed volume, the oscillations of the nuclei are indeed harmonic. It then remains only to find the appropriate harmonic potential at each volume (or at the
corresponding temperature). In this model, there is a closed form expression for the phonon entropy,
and it is simply the harmonic expression with the caveat that the phonon spectrum is now a function

Figure 1.4: Anharmonic phonon entropy of the alkali metals
taken from Wallace. [61] The total phonon entropy for these
metals is around 10kB /atom, so the anharmonic entropy is
less than 2% of the total phonon entropy. [61]

Figure 1.5: Anharmonic phonon
entropy for a few transition metals taken from Eriksson et al. [36]
In general, it is small relative to
the total phonon entropy; however,
this is not the case for Cr, Mo, and
W.

of volume (or temperature). [36, 37] Alternatively, the entropy of dilation, Sph,D , may be found by
taking the difference between the heat capacities at constant pressure and constant volume, CP and
CV :
Z T
Sph,D = Sph,D (T ) =

CP − CV
dT 0 =
T0

Z T

9KT α2 0
dT ,
ρN

(1.3)

where T is the temperature, KT is the isothermal bulk modulus, α is the linear coefficient, and ρN
is the number density. In the quasiharmonic model, these two methods of determining the entropy
should yield the same result; however, this is not always the case. Any entropy not accounted for
by the quasiharmonic model is called anharmonic phonon entropy. For example, Fig. 1.4 shows that
the quasiharmonic model is only a slight underestimate of the phonon entropy in the alkali metals
potassium, sodium and rubidium; and Fig. 1.5 shows that the model gives only a slight overestimate
of the entropy for vanadium, tantalum, palladium, and platinum. On the other hand, Fig. 1.5 also
shows that the model grossly underestimates the phonon entropy for chromium, molybdenum and
tungsten.
In addition to not accounting for all of the entropy, there are further failures of the quasiharmonic model. In spirit, the model is simple; however, accounting for the changes in energy of the
phonons can become quite involved. It is not always the case that all of the phonon energies shift

proportionally, or even in the same direction, as the volume of the crystal changes. [39, 55] Further, the quasiharmonic model does not account for the shortened lifetimes of the phonons through
electron-phonon and phonon-phonon interactions. Experimentally, these shortened lifetimes express
themselves as non-zero spreads in the measured phonon energies. There exists quantum field theoretical techniques which allow calculation of phonon shifts and linewidths, and there has been a
resurgence in their use in the last decade. [62–67] However, this method is very computationally
intensive, and its accuracy is still unclear. In particular, it is based on perturbation theory and assumes nearly harmonic behavior. As such, the temperature dependencies it predicts for the phonons
do not always match up with those seen in experiments.

1.2

Present work

The present work considers changes in the phonon spectrum of cubic metals as the temperature
increases from absolute zero to near melting. We consider both the shifts in the fundamental
energies, and the finite lifetimes of the phonons, as well as their effects on the thermodynamics.
In Part I we present the scientific background required to understand the rest of the text. Specifically, we start with a description of crystals and phonons, then move on to the various contributions
to the entropy of a solid. After that, we consider neutron scattering in general, and time-of-flight
chopper spectrometers in particular.
Part II describes the procedures used for analysis of the experimental data collected herein. This
includes new and/or modified methods for determining the incident energy (without use of beam
monitors), finding bad detectors, reducing experimental data to the the scattering function S(Q, E),
correcting for multiphonon- and multiple-scattering, analysis of phonon shifts and damping, and
determination of Born–von Kármán (BvK) force constants from a measured phonon spectrum.
In Part III, we present neutron scattering experiments to determine phonon spectra in facecentered-cubic (FCC) aluminum, lead, and nickel as a function of temperature. Lattice parameters,
interatomic force constants, phonon spectrum and entropies, mean energies, and mean quality factors
are all extracted from the data. For aluminum and nickel, the softening of the phonon spectrum
with increasing temperature is tied to force constants BvK models of the lattice dynamics. Thermal
expansion and bulk modulus data from the literature in conjunction with the phonon spectrum are
used to determine the harmonic, quasiharmonic, and anharmonic phonon entropies for these metals.
All other contributions to the entropy are also assessed, and comparisons are made to experimental
measurements of the total entropy. The effects of anharmonicity on the phonon spectrum and
the entropy are discussed, with particular attention to the phonon line broadening seen at higher
temperatures. Part IV presents the same sort of information for body-centered-cubic (BCC) iron,
in addition to less detailed presentation of experimentally determined phonon spectra for the BCC

metals chromium and vanadium.
Trends across FCC and BCC metals are considered in Part V. In particular, commonality in
the phonon linewidths of the FCC metals is explored. We find and present a simple empirical
model based on the damped harmonic oscillator that gives a reasonable estimate of the phonon
linewidths in the FCC metals studied. The experimental data is supplemented with data from
molecular dynamics simulations, and these provide some access to the interatomic forces involved
in the anharmonic phonon linewidths.
Finally, in Part VI, we summarize our results and offer some ideas for future avenues of research.

Part I

Background
In this part of the thesis, we examine some of the definitions and basic physics relevant
to the rest of the work. We discuss crystals and their vibrational excitations, the division
of the entropy into components, the theory of neutron scattering, and neutron scattering
instruments.

Chapter 2

Notation
2.1

Indices

We will often use a bold letter for some non-scalar quantity, and the same letter, not bold, as an
index. For example, we might have the component vv . We do this so that it will be easier to
remember to what the index refers, and because we believe that the reader will be able to identify
to which we are referring by context. Please note that any of these indices might be used for other
purposes in other parts of the text, though we have made some effort to avoid this.
In an (albeit weak) attempt to appease serious mathematicians, we also point out the following:
We will frequently use a dummy index i and a particular value of that dummy index interchangeably.
For example, we might say “Atoms a are indexed by i, as in ai .” This might be followed with
something like “The ith atom is at equilibrium,” or “Atom ai is at its equilibrium position.” These
statements should be taken to mean that for any particular value of i, the corresponding atom is at
its equilibrium position. If a specific value of i is intended, it will be made clear. E.g., “Atom a0 is
removed from the lattice.”

2.2

Matrices

In order to distinguish between indices that yield elements of matrices,a and those that yield submatrices or elements of sets of matrices, we have adopted the following convention: Square brackets, as
in [Mij ], indicate that Mij is not a scalar. Indices on the outside of square brackets, as in [Mij ]kl ,
do yield a scalar. The argument of a function of a continuous variable is given in parentheses, thus
[Mij (v)]kl is the klth element of the (non-scalar) ij th M which is a continuous function of v.
For those familiar with previous expositions of lattice dynamics, please compare: Using our
a Here, the term matrices also includes vectors and tensors.

variable names, Venkataraman et al. [68] have this definition of the dynamical matrix:
 
1 `
` 
 exp iq · x  ,
Dcc1   = p
Kcc1 
Ms Ms1 `
ss1
s s1
s 

and we have:
[Dss1 (q)]cc1 = p

X

K(1s)(`s1 ) cc exp iq · x(`s) ,
Ms Ms 1 `

where the index (ls) is a tuple, (l, s), and we have dropped the comma for brevity.

(2.1)

Chapter 3

Crystals and Phonons
This work is focused on the dynamics of nuclei in crystals, and we will have need of the following:
• A brief understanding of what constitutes a crystal.
• Some simplifications: periodic boundary conditions, the adiabatic approximation, and Taylor
expansion of the nuclear potential.
• An understanding of the 1D quantum harmonic oscillator, and of how to decouple systems
of classical, coupled harmonic oscillator. These concepts allow us to define phonons and the
phonon density of states (DOS).
• The effects of anharmonicity on the phonon spectrum.

3.1

Crystals

Crystalline materials have a regular, repeating structure, and physicists tend to describe this with
two sets of vectors. The elements of the first set are called lattice vectors, and in D dimensions,
there will be D of them. The lattice vectors describe the repeat unit of the crystal. The lattice is
the set of all integer linear combinations of the lattice vectors, and we denote elements of the lattice
[li ]. We also call elements of the lattice cell vectors, as the cell vector of an atom points to the unit
cell in which that atom is located. The elements of the second set are called site vectors or sites,
and they give the positions of the atoms relative to the lattice. We denote the sites [si ].a A crystal,
then, consists of a lattice and sites:
crystal = lattice + sites .

(3.1)

Examples of simple crystals in 2 and 3 dimensions are shown in Figs. 3.1 and 3.2 respectively.
a We avoid using the terms ‘basis’ or ‘basis-vectors’ commonly used by crystallographers and physicists, as these

terms already have clear meaning in linear algebra. In point of fact, the lattice vectors are a linear algebraic basis of
the lattice.

Figure 3.1: A 2D lattice, with lattice vectors
shown in dotted red, and a cell vector shown
in dash-dotted purple. The dashed green arrow shows the displacement of a particle from
its equilibrium position. The orange circle is a
site vector with zero length.

Figure 3.2: A 3D body centered cubic lattice,
with lattice vectors shown in dotted red and site
vectors shown in dashed green. (Two site vectors emanate from the same point on the lattice;
but, one of them has zero length.)

The equilibrium position, [xi ] of any atom indexed i is:
[xi ] = [li ] + [si ] ,

(3.2)

and its instantaneous position [ri ] is given by:
[ri ] = [xi ] + [ui ] = [li ] + [si ] + [ui ] ,

(3.3)

where [ui ] gives the displacement of the ith atom from its equilibrium.
More details about crystal lattices (and reciprocal lattices) are available elsewhere.[50, 68, 69]

3.2

Approximations

As much as we would like the theory of crystalline solids to be exact, it appears to be far too
complicated. The following approximations are instrumental to our understanding of the dynamics
in these materials.

3.2.1

Periodic boundary conditions

In a typical empirical model of a metal, the forces extend out to on the order of ten nearest neighbors
(10NN). Let us assume that we have a large cube with side length R filled with atoms, and that
the atoms are arranged in a simple cubic lattice with lattice parameter a. (This is the structure as
in Fig. 3.2 without the triangular markers.) Further, assume that any atom within 2a of the sides

1.0

1e+09

0.9

1e+08

0.8

1e+07

0.7

1e+06

AB (atoms)

AB/A (unitless)

0.6
0.5
0.4

1e+05
1e+04
1e+03

0.3
0.2

1e+02

0.1

1e+01

0.0

1e+00

100

1000

R/a (unitless)

100

1000

R/a (unitless)

Figure 3.3: Approximate fraction of atoms that Figure 3.4: Approximate number of atoms in the
are in the bulk as described in § 3.2.1.
bulk as described in § 3.2.1.

of the cube is part of the surface of the crystal, as opposed to being part of the bulk. The number
density of the atoms in the crystal is approximately 1/a3 , so the approximate number of atoms, A,
in the cube is given by:
A=

(3.4)

The number of atoms in the bulk AB is given by:
AB =

R − 2a

(3.5)

Note that these are exact whenever R is an integer multiple of a.
Figs. 3.3 and 3.4 show respectively the fraction of atoms that are in the bulk portion of the
crystal and the number of such atoms as a function of the length of the side of the crystal divided
by the lattice parameter. From the figure, we see that in order to have less than 1% of the crystal
be surface, we need around a billion atoms.
The point of all this is that when we need boundary conditions, it would be best if for any atom
in our crystal, they approximated the existence of many other atoms in any direction. It would also
be nice if we did not need to simulate billions of atoms. Periodic boundary conditions accomplish
both of these tasks, and are represented as follows:

r([l+Li ] s) = r(ls) ,

(3.6)

where Li is the number of unit cells in the direction of the ith lattice vector. The one and two
dimensional analogues are easy enough to visualize, where very long lines or planes of atoms become
rings or tori. The same principle applies to three dimensions, but visualization is more difficult. In

Figure 3.5: Imagining atoms at the intersections of the lines, the torus pictured above [70] is a
geometric representation of periodic boundary conditions in two dimensions.

any case, the periodic boundary conditions allow representation of an infinite crystal with a small
number of atoms.
More details about this approximation and its validity can be found in many standard texts on
solid state physics or lattice dynamics. [50, 51, 68, 71, 72]

3.2.2

Adiabatic approximation

A typical crystal has on the order of 1023 particles, and the problem of understanding the detailed
motions and interactions of all of them is hopelessly intractable. The greatest simplification possible would be a clean separation of the electronic and nuclear systems, with any (hopefully small)
interactions accounted for in perturbation. This separation is accomplished with the adiabatic or
Born-Oppenheimer approximation, which has been described in detail elsewhere. [50, 68, 73]
Briefly, the Hamiltonian, H, for the system may be written as follows:
H = Tel + Tph + V = Tel + Tph + Vel + Vph + Vel−ph ,

(3.7)

where T denotes kinetic energy, V denotes the potential energy, and el denotes electronic. The
subscript ph stands for ‘phonon’ (explained later) and it represents the contributions from the nuclei.

14
Finally, the subscript el−ph denotes interactions between the electrons and nuclei.
We now exploit the fact that the mass of an electron, me , is much less than the mass of one
−4
of the nuclei, M . For a crystal of hydrogen, we have m
, and for a more typical
M = 5.446 × 10
−6
. The Hamiltonian may
metallic crystal, say nickel, we have something more like m
M = 9.279 × 10
be expanded in powers of m
M , and it is then evident that the responses of the electrons occur on

a much shorter timescale than those of the nuclei. As a result, the electrons equilibrate quickly,
and the nuclei experience a roughly constant cloud of electrons about themselves. Therefore, when
considering the motion of the nuclei, we ignore the interactions between the electrons and other
particles (including themselves). This gives the following approximate equation for the nuclear
motion:
Hph

Tph + Vph .

(3.8)

The interactions Vel−ph may be treated in perturbation.

3.2.3

Expansion of the potential

In addition to simply not knowing the nuclear potential, we frequently can only successfully manipulate some portion of it. To this end, we employ a Taylor expansion:

Vph

= Vph
r=x

X

K(ls) c u(ls) c
(lsc)

X X

K(ls)(l1 s1 ) cc u(ls) c u(l1 s1 ) c

(lsc) (l1 s1 c1 )

X X

X

K(ls)(l1 s1 )(l2 s2 ) cc c u(ls) c u(l1 s1 ) c u(l2 s2 ) c
1 2

(lsc) (l1 s1 c1 ) (l2 s2 c2 )

+ ··· ,

(3.9)

where we have defined:
K(ls) c
K(ls)(l1 s1 ) cc

∂V
ph
∂ u(ls) c

r=x

∂2V
ph
∂ u(ls) c ∂ u(l1 s1 ) c

K(ls)(l1 s1 )(l2 s2 ) cc c

1 2

r=x

∂ 3 Vph

∂ u(ls) c ∂ u(l1 s1 ) c ∂ u(l2 s2 ) c

r=x

(3.10)

15
and so on. The derivatives of the potential may be rewritten as matrices (or tensors), and we will
use them in this form. As we will (presumably) not be dealing with strained samples, we assume
that the coefficients of the first order term are zero. Most of our work is done in the harmonic or
quasiharmonic approximations, dropping the third and higher order terms.

3.3

The quantum harmonic oscillator

The nuclear system has now been separated from the electronic one, and the next step is to analyze
the motion of the nuclei. To this end, we first consider the motion of a single nucleus in a harmonic
potential. This section borrows heavily from Griffiths. [74, pp. 31–37] Cohen-Tanoudji et al. [75]
also have a clear presentation of this material.
In 1D, we may write the Hamiltonian for a single harmonic oscillator:
p2
H=
+ Ku2 =
2M
2M
where ω ≡

~ ∂
i ∂u

+ M ω 2 u2 ,

(3.11)

M is the classical frequency of the oscillator. We may then write the time-independent

Schrödinger equation as follows:
"
2
~ ∂
Hψ =
+ (M ωu) ψ =
2M
i ∂u
~ ∂
~ ∂
− iM ωu √
+ iM ωu +
2M i ∂u
2M i ∂u

~ ∂
~ ∂
+ iM ωu √
− iM ωu ψ = Eψ .
2M i ∂u
2M i ∂u

(3.12)

We now define creation and annihilation operators:
â± ≡ √
2M

~ ∂
± iM ωu ,
i ∂u

(3.13)

which allow us to write Schrödinger’s equation for the oscillator in either this form: b

â± â∓ ± ~ω ψ = Eψ .

(3.14)

Assume that the λth state of the oscillator, ψλ , is known. When we apply the creation or annihilation
b Another, possibly prettier option is 1 (â â + â â ) ψ = Eψ .
+ −
− +

16
operators to our state, we get a new solution to the Schrödinger equation:
Hâ± ψλ

â± â∓ ± ~ω â± ψλ = â± â∓ â± ± ~ω ψλ
â± â∓ â± ∓ ~ω ± ~ω = â± (E ± ~ω) ψλ
(E ± ~ω) â± ψλ .

(3.15)

Applications of the creation and annihilation operators, then, create new states with energy shifted
by ~ω.
The overall energy must be positive, so we may apply the annihilation operator until we reach
the ground state, ψ0 . After that, we should get â− ψ0 = 0. Substituting ψ0 into the Schrödinger
equation, then, we get:

â+ â− + ~ω ψ0 = ~ωψ0 = E0 ψ0 ,

(3.16)

â+ â− ψ0 = â+ 0 = 0 .

(3.17)

where we have used:

Thus, the entire spectrum of energies is given by:
Eλ = λ +
~ω

3.4

λ∈N.

(3.18)

Harmonic lattice dynamics

We have now briefly studied the quantum mechanical oscillations of a particle (nucleus) in a harmonic
potential, and we wish to take our knowledge and apply it to a multi-dimensional, multi-particle
system. To do so, we first review the mathematics of decoupled and normal modes, which allow us
to take a single problem in many coupled variables and transform it into many problems, each in a
single, uncoupled variable.

3.4.1

Decoupled modes

Occasionally, a difficult integral may sometimes be greatly simplified by a change of variables. Similarly, the problem of a system of coupled harmonic oscillators can be made simple by an appropriate
coordinate transformation. First, we consider a system of two point particles of mass M connected
to each other and to nearby walls by Hookean springs of stiffness K.
If we say that the displacement of the first particle from its equilibrium position is given by u1 ,

Figure 3.6: Coupled point particles with mass M , connected by springs of stiffness K. The displacements from equilibrium are given by ui .
and that of the second particle is given by u2 , we may write the equations of motion:
M ü1 + Ku1 + K(u1 − u2 ) = 0 ,

(3.19)

M ü2 + Ku2 + K(u2 − u1 ) = 0 .

(3.20)

As with most differential equations, we may solve these by simply knowing the answer ahead of
time. If we take:
uj (t) = Aj exp (iωb t + φb ) ,

(3.21)

where the Aj are amplitudes, the ωb are frequencies, and the φb are phases,c and substitute it into
the equations of motion, we get:
−ωb2 M A1 exp (iωb t + φb ) + 2KA1 exp (iωb t + φb ) − KA2 exp (iωb t + φb ) = 0 ,

(3.22)

−ωb2 M A2 exp (iωb t + φb ) + 2KA2 exp (iωb t + φb ) − KA1 exp (iωb t + φb ) = 0 .

(3.23)

We may divide through by M times the exponential, and rewrite this as a matrix equation:



−ωb2 + 2K

−M

−ωb2 + 2K



A1
A2

=0.

(3.24)

This is equivalent to an eigenvalue problem with eigenvalues ωb2 :

−M




A1
A2

 = ωb2 

A1
A2

 .

(3.25)

At this point, it is clear that there are two ωb , which comes from DA = 1 · 2 = 2 (Again, D is the
dimension and A is the number of atoms). We may find both eigenvalues by setting the determinant
c Note that the index on the frequencies and phases is different than the index on the displacement or the amplitude.
This is because combined motion of the masses will give rise to the frequencies and phases.

18
of the matrix in Eq. 3.24 equal to zero:

2K
ωb2 −

2
=0.

(3.26)

Taking the positive values for ωb , this gives:

2K
3K
ω1 =
2K
ω2 = −

(3.27)
(3.28)

Substituting these eigenvalues back into the Eq. 3.25, we may find the eigenvectors:
ω1 → 

−1

ω2 → 

 .

(3.29)

The decoupled modes of the system, then, are given by these eigenvalues and eigenvectors. In this
case, the eigenvectors are orthogonal, so the decoupled modes are also normal modes. Let:
U 1 = u1 − u2

, Ü1 = ü1 − ü2 ,

(3.30)

U2 = u1 + u2

, Ü1 = ü1 + ü2 .

(3.31)

Then the sum and the difference of Eqs. 3.19 and 3.20 give respectively:
M (ü1 + ü2 ) + K(u1 + u2 ) = M U2 + KU2 = 0 ,

(3.32)

M (ü1 − ü2 ) + 3K(u1 − u2 ) = M U1 + 3KU1 = 0 .

(3.33)

Thus, the eigenvectors provide us with a coordinate transformation that allows us to rewrite our
original equations of motion in terms of two decoupled oscillators.
We now redo this for a more general system of oscillators. Let i, j label particles, and c, c1 label
Cartesian directions, then the displacement of the particle labeled i in the c direction is given by
[ui ]c . We now construct the mass matrix , M, such that:
[Mij ]cc1 = δij δcc1 Mi .

(3.34)

For the example given above, we have Mi = M , so:

= 

 .

(3.35)

19
Then it is clear that the force, F is given by Mü ,or in our example:

= 



ü1



ü2

M ü1

=

M ü2

 .

(3.36)

Next, we construct the stiffness matrix K. Each element [Kij ]cc1 gives the force constant linking
the motion of the ith particle in the c direction to a response of the j th particle in the c1 direction.
Going back to our two particle example, we have:
F = −Ku

−

−K




u1
u2

 = −

2Ku1 − Ku2
−Ku1 + 2Ku2

 .

(3.37)

The elements along the diagonal are frequently called self force constants, as they describe the forces
felt by a particle when it is the only one displaced from equilibrium. In our example, displacing only
the first mass leads to a restoring force −Ku1 from the compression of the spring connecting the
first mass to the second mass, and another restoring force −Ku1 from the extension of the spring
connecting the first mass to the wall.
Putting equations Eqs. 3.36 and 3.37 together, we recover Eqs. 3.19 and 3.20:

= −Ku ,
2Ku1 − Ku2
 .
= −
−Ku1 + 2Ku2

Mü

M ü1
M ü2

(3.38)
(3.39)

If we multiply the left and right of Eq. 3.38 by M-1 , we get:
ü = −M-1 Ku .

(3.40)

The decoupled modes of the system, then, are found by diagonalization of the matrix M-1 K:
M-1 K

= 

−K

 ,

(3.41)

as in Eq. 3.25.
So long as all of the forces are subject to Newton’s third law and are linear in the displacements,
a decomposition into decoupled modes is possible;d however, if we wish the modes to be normal in
the linear algebraic sense (i.e., for M-1 K to have orthogonal eigenvectors) we need to impose further
constraints. Fortunately, the symmetries of the crystal do this for us.
d Technically, other systems might also be decomposed into decoupled modes; however, this definition will cover all
cases of interest to us.

3.4.2

Normal modes of a crystal

We now take what we have learned of decoupled modes in the previous section, and apply it to
an arbitrary crystal structure, in arbitrary dimension. The mathematics is heavier, but the goal
remains the same. We start out with the equations for a large system of coupled oscillators, and
wish to end up with many equations, each for a single oscillator.
It is tempting to say that the problem is solved — why can’t we simply diagonalize M-1 K? This
approach poses a few problems. In a real crystal, we have on the order of 1023 atoms. Practically
speaking, the diagonalization of M-1 K is simply not possible. We will therefore exploit the fact that
we have a crystal — not just an arbitrarily distributed group of atoms — in order to reduce the
problem to manageable proportions. The periodic boundary conditions are critical in this regard.
Our D-dimensional crystal has A atoms, and they are indexed by i (or j).e x is the (AD)-vectorf
giving the equilibrium positions of the all the atoms in the crystal, and [xi ] is the equilibrium position
vector for the ith atom, a length D subvector of x. If the cell vectors for all the atoms in the crystal
are given by the (AD)-vector l, and similarly the site vectors by s, we have:
x=l+s,

(3.42)

[xi ] = [li ] + [si ] .

(3.43)

We will want to exploit the translational symmetry of the lattice; therefore, we will also describe
the ith atom by indices to its cell and site vectors (ls). There are L cell vectors and S site vectors,
thus we have A = LS. Using the indices for the cell and site vectors, we may write:
x(ls) = [ll ] + [ss ] ,

(3.44)

such that x(ls) is the position vector for the (ls)th atom, [ll ] its cell vector, and [ss ] its site vector.
Finally, u gives the displacement of all atoms from their equilibria, such that the instantaneous
positions of all the atoms are given by r, with:
r=x+u,

(3.45)

[ri ] = [xi ] + [ui ] ,
r(ls) = [ll ] + [ss ] + u(ls) .

(3.46)
(3.47)

Note that the instantaneous position and the displacement must be indexed by the tuple (ls), whereas
the cell vector and the site vector require only their respective indices, l and s, separately.
Let [Kij ] represent the D × D force constant matrix connecting the ith and the j th atoms, then
e We apologize that i is doing double duty as

−1 and as an index in this section.

f This is meant to indicate a vector of length AD.

21
we may write out K explicitly as follows:

[K11 ]

[K12 ]

···

[K1A ]

 [K21 ] [K22 ] · · · [K2A ] 
=
K=
..
..
..
..
[KA1 ] [KA2 ] · · · [KAA ]


K(11)(11)
K(11)(12)
···
K(11)(1S)


K
K(12)(12)
···
K(12)(1S)
 (12)(11)
..
..
..
..


 K(1S)(11)
K(1S)(12)
· · · K(1S)(1S)


 K(21)(11)
K(21)(12)
···
K(21)(1S)
..
..
..
..

K(LS)(12) · · · K(LS)(1S)
K(LS)(11)

(3.48)

K(11)(21)
K(12)(21)
..
K(1S)(21)
K(21)(21)
..
K(LS)(21)


K(11)(LS)

K(12)(LS) 
..

K(1S)(LS)  .

K(21)(LS) 
..
K(LS)(LS)

(3.49)

K(ls)(l1 s1 ) 1D
K(ls)(l1 s1 ) 2D 
 .
..
K(ls)(l1 s1 ) DD

(3.50)

···
···
..
···
···
..
···

Let us also make the submatrices explicit:

K(ls)(l1 s1 ) 11

 K(ls)(l1 s1 ) 21
[Kij ] = K(ls)(l1 s1 ) = 
..
K(ls)(l1 s1 ) D1

K(ls)(l1 s1 ) 12
K(ls)(l1 s1 ) 22
..
K(ls)(l1 s1 ) D2

···
···
..
···

We now wish to look at the equation of motion of the ith or (ls)th atom. This involves a
summation over a row of M-1 K and a column of u. In particular, we move in steps that are D
elements wide, performing matrix-vector multiplication of the D × D matrix M-1 K(ls)(l1 s1 ) onto
the D-vector u(l1 s1 ) and summing up the resulting vectors:
X

ü(ls) = −
M-1 K(ls)(l1 s1 ) u(l1 s1 ) .

(3.51)

(l1 s1 )

The mass depends only on the site vector and we may explicitly write into our equation the appropriate matrix element from M-1 :
ü(ls) = −
(l1 s1 )

1
K(ls)(l1 s1 ) u(l1 s1 ) .
Ms 1

(3.52)

We believe that the solutions will take the form of plane waves. From the symmetries of the
crystal, we know that for two atoms with the same site vector, only the phase of their oscillations
may differ. This means that the polarization vectors, [bs (q)] will not be indexed by l. Here, the
b is the branch index . As for b in §3.4.1, we see that we have b ∈ {1, 2, · · · , DS}. Similarly, the

22
amplitudes of the motions cannot depend on the cell vector. Thus, we make the following guess for
the solutions to the equation of motion:

u(ls)

Ms exp iq · x(ls) − ωb (q)t [bs (q)]
Ms exp (iq · ([ll ] + [ss ]) − ωb (q)t) [bs (q)] ,

(3.53)

where q is the wavevector for the normal mode and the amplitude of the mode is folded into the
polarization vector.
We may break up the exponential into a product:
p
u(ls) = Ms exp (iq · [ss ]) exp (iq · [ll ]) exp (−iωb (q)t) [bs (q)] .

(3.54)

In this form it is easy to see that the first and second time derivatives of the displacements are as
follows:
u̇(ls) = −iωb (q) u(ls) ,
ü(ls) = −ωb2 (q) u(ls) .

(3.55)
(3.56)

We now substitute these back into the equation of motion:
−ωb2 (q) Ms exp (iq · [ss ]) exp (iq · [ll ]) exp (−iωb (q)t) [bs (q)] =
X 1 X
p
Ms1 exp (iq · [ss1 ]) exp (iq · [ll1 ]) exp (−iωb (q)t) [bs1 (q)] .
K(ls)(l1 s1 )
Ms1

(3.57)

Dividing both sides of the equation by

Ms and the exponentials, and collecting terms, we get:

ωb2 (q) [bs (q)] =
X exp {iq · ([ss ] − [ss ])} X
K(ls)(l1 s1 ) exp {iq · ([ll1 ] − [ll ])} [bs1 (q)] .
Ms Ms1
s1

(3.58)

We note that any D × D submatrix of the force constant matrix [Kij ] connecting the atoms i
and j depends only on the distance between two atoms, not on their absolute positions. Therefore,
we fix one of the atoms into the first unit cell. g
` → 1,
K(ls)(l1 s1 ) → K(1s)(l1 s1 ) .

(3.59)
(3.60)

g In practice, there is sometimes reason to use a calculational cell that contains U repeat units of the crystal where
U > 1 (i.e., more than one primitive unit cell). In this case, we are adding up the same force constants U times, once
for each copy of the repeat unit that appears in the calculational cell. Thus, we must divide our result by U in order
to get the correct squared frequencies.

23
With periodic boundary conditions, or an infinite lattice, it is clear that the difference of any two
cite vectors is itself a site vector. Therefore, we have:

q · ([ll1 ] − [ll ]) →

q · [l` ] ,

(3.61)

and we may thus replace the cell vectors in the exponential with a new cell vector, [l` ]:
ωb2 (q) [bs (q)] =

X exp {iq · ([ss ] − [ss ])} X
K(1s)(`s1 ) exp (iq · [l` ]) [bs1 (q)] .
Ms Ms1
s1

(3.62)

At this point, it is traditional to define the dynamical matrix , [Dss1 (q)] as follows:
[Dss1 (q)] =

exp {iq · ([ss1 ] − [ss ])} X
K(1s)(`s1 ) exp (iq · [l` ]) .
Ms Ms1

(3.63)

Thus, the we have reduced our problem to one of solving the following equation:
ωb2 (q) [bs (q)] =

[Dss1 (q)] [bs1 (q)] .

(3.64)

For clarity, we explicitly write D in matrix form:

[D11 (q)]

[D12 (q)]

···

[D1S (q)]

 [D21 (q)]
D(q) = 
..
[DS1 (q)]

[D22 (q)]
..

···
..

[DS2 (q)]

···

[D2S (q)] 
 ,
..
[DSS (q)]

(3.65)

where:

[Dss1 (q)]11

[Dss1 (q)]12

···

[Dss1 (q)]1D

 [Dss1 (q)]21
[Dss1 (q)] = 
..
[Dss1 (q)]D1

[Dss1 (q)]22
..

···
..

[Dss1 (q)]D2

···

[Dss1 (q)]2D 
 ,
..
[Dss1 (q)]DD

(3.66)

and has eigenvectors:

[b1 (q)]

 [b2 (q)] 
 .
[b (q)] = 
..
[bS (q)]

(3.67)

24
Thus we may write:
D(q) [b (q)] = ωb2 (q) [b (q)] ,

(3.68)

and see that the problem of finding the decoupled modes is now a problem of diagonalizing D(q) at
a large number of q, where ~ωb (q) is the energy of the state associated with b (q).
For every q, we get DS squared frequencies; thus, in order to have the same number of eigenvalues
as M-1 K we must have DA
DS = S = L q-points. The exact constraint on the values of q comes from

the periodic boundary conditions described briefly in § 3.2.1. It may be shown that D is Hermitian,
and as a result its eigenvectors are orthogonal. [51]
For the purpose of practical calculations, we frequently have:
K=

[Kk ] ,

(3.69)

where the k indexes shells of neighbors. As all of the relevant operations performed in this section
have been linear, we may simply substitute the sum in wherever there is a K. This property can be
exploited to find the gradient of the dynamical matrix with respect to changes in a coefficient of K
as is described in § 8.3.
Practically speaking, we have converted the problem of diagonalizing a matrix with on the order
of 1023 × 1023 elements to one of diagonalizing a matrix with on the order of 10 × 10 or, for a
structure with more than 20 atoms per unit cell, 100 × 100 elements. For a crystal with only one
site, a very high quality calculation of the frequency spectrum might require diagonalization at
1,000,000 q-points.
For the interested reader, myriad texts about the theory of lattice dynamics, including derivations
of the dynamical matrix and its properties, are available. [51, 68, 71, 72]

3.5

The quantum harmonic oscillator revisited: phonons

We have shown that for an arbitrary crystal structure, we may decouple the oscillations into normal
modes. In D dimensions, with A atoms, we find the normal coordinates Uj , with j ∈ {1, 2, · · · , DA},
and then solve the 1D quantum harmonic oscillator for each of the Uj . If the classical frequencies
are ωj , the quantized spectrum is then given by:
Ejλ =

λ+

~ωj

λ∈N.

(3.70)

One way to look at this, is to say that there are DA modes, and at any time, each mode j may
be in any state λ, with energy given by Eq. 3.70. Another way follows. We note that for the j th

25
mode, the difference between successive energy levels is always ~ωj . We posit the existence of a
quantized excitation called a phonon (of type j), which has precisely this energy, and we say that
the λth excited state actually corresponds to the existence of λ phonons (of type j). The meanings
of the names ‘creation’ and ‘annihilation’ operator are now clear — respectively, these operators
create or annihilate a phonon.
At least in the case of a harmonic crystal, phonon thermodynamics is specified completely by
the distribution of phonon frequencies or the phonon density of states, g(E):
DA

g(E)

1 X
δ [E − ~ωj ] .
DA j

(3.71)

Or, using the q and the branch index b, we have:h

g(E)

1 XX
δ [E − Eb (q)] ,
DSL q

(3.72)

where Eb (q) ≡ ~ωb (q).
As we may have an arbitrary number of phonons in any state λ we see that phonons are bosons.i
In a crystal, at a fixed temperature and volume, phonons are created and destroyed so as to minimize
the Helmholtz free energy F. This may be expressed as follows:

∂F
∂PE

=0,

(3.73)

T,V

where PE is the number of phonons at energy E. We may identify the quantity on the left of Eq. 3.73
as the chemical potential, µ. Since this equation is true for phonons of all energies, we see that for
phonons, µ = 0. [76]
Finally, the mean thermal occupancy of any phonon state, nE (T ), is governed by Bose-Einstein
statistics, and is given as follows:
nE = nE (T ) =

eβE − 1

(3.74)

where β = kB1T , T is the temperature, and kB is Boltzmann’s constant.
h A reminder, the number of q-points is L.
i More technically, they are composite bosons; that is, they are made up of other elementary particles and the
agglomeration displays the statistics characteristic of bosons.

3.6

Anharmonic lattice dynamics

Thus far, we have only considered harmonic potentials; however, the failures of a harmonic model of
a crystal are both considerable and well known. [50, 52] Briefly, we paraphrase the summary given
by Brüesch. [51, pp. 162–163] Without anharmonicity:
• There is no thermal expansion.
• Force constants and elastic constants are independent of temperature.
• The specific heats at constant pressure or volume are equal.
• There is an infinite thermal conductivity, and measured phonon peaks would have no
linewidths, because the phonons do not interact.
The effects of anharmonicity can be treated, through perturbation theory, though the accuracy
of these techniques requires further testing. For the perturbed wavefunctions and energy levels
of a single anharmonic quantum oscillator, Cohen-Tanoudji et al. have an excellent account. [75]
Perturbing the potential of a 3D crystal is significantly more complicated, and we will only summarize
some of the results and point to some literature.
The result of an anharmonic perturbation to the potential is that the measured energies of the
phonons are shifted and broadened. The measured energy is given by:
Eb (q) = ~ [ωb (q) + ∆ωb (q)] ,

(3.75)

and it has a peak width of 2Γb (q).
We now summarize the various contributions to the shifts and linewidths; not because we will
use this level of math directly, but because we will discuss calculations of this type made by others.
A disinterested reader may skip to the next section with little or no cost to the readability of
the remainder of the manuscript, and the interested reader may look elsewhere for details.[51, 62–
64, 77, 78]
The shift may be broken into quasiharmonic, third, and fourth order contributions:
∆ωb (q) = ∆Q ωb (q) + ∆3 ωb (q) + ∆4 ωb (q) .

(3.76)

The quasiharmonic contribution to the shift is given as follows:
∆Q ω = −γω

∆V

(3.77)

27
where V is the volume and γ = γb (q) is the Grüneisen parameter:
X
1 X X
K(1s)(l1 s1 )(l2 s2 ) cc c [bs ]c [bs1 ]c1
1 2
2ω
(sc) (l1 s1 c1 ) (l2 s2 c2 )

× x(1s) c − x(l2 s2 ) c exp iq · x(1s) − x(l1 s1 )

γ=−

(3.78)

For the sake of readability, we have subsumed the qi and bi dependence of ω and γ into a single
number i. (E.g. γ = γb (q) and ω1 = ωb1 (q1 ).) We do this for the remainder of this section.
Likewise for the mean occupation factor for bosons, n (see Eq. 3.74). Further, we allow the qi
dependence of the polarization vectors to be indicated by the associated bi . (E.g. [b ] = [b (q)] and
[b1 ] = [b1 (q1 )].) The third order contribution to the shift, then, is given by:
2
18 X X
n1 + n2 + 1
∆3 ω = − 2
[Dbb1 b2 (q, q1 , q2 )]
ω1 + ω2 + ω
q1 b1 q2 b2

n1 + n2 + 1
n2 − n1
n2 − n1
+P
+P
+P
ω1 + ω2 − ω
ω1 − ω2 + ω
ω1 − ω2 − ω

(3.79)

where the P denotes the principal value,[79, p. 12] and is frequently approximated as follows:

= lim 2
→0 v + 2

(3.80)

The fourth order term is considerably simpler:
∆4 ω =

12 X
[Dbbb1 b1 (q, −q, q1 , −q1 )] {2n1 + 1} .

(3.81)

q1 b1

There is also a third order contribution to the linewidth:
Γ=−

18π X X
[Dbb1 b2 (q, q1 , q2 )]
~2

q1 b1 q2 b2

× {(n1 + n2 + 1) [δ(ω1 + ω2 − ω) − δ(ω1 + ω2 + ω)]
+ (n2 − n1 ) [δ(ω1 − ω2 − ω) − δ(ω1 − ω2 + ω)]} .

(3.82)

In the above equations, the n express the thermal occupations of the modes, and the delta
functions express conservation of energy. Most of the physics is hidden in the D, which, very
roughly, correspond to Fourier transforms of the terms in the Taylor expansion of the potential. For

28
the third order term, we have:
3
~ 2
3! 2N
X exp {i (q · [ss ] + q1 · [ss ] + q2 · [ss ])}
[Dbb1 b2 (q, q1 , q2 )] =

X X

(lsc) (l1 s1 c1 ) (l2 s2 c2 )

[bs ]c [b1 s1 ]c1 [b2 s2 ]c2
[ωω1 ω2 ]

(Ms Ms1 Ms2 ) 2

K(1s)(l1 s1 )(l2 s2 ) cc c exp {i (q · [ll ] + q1 · [ll1 ] + q2 · [ll2 ])} .
1 2

(3.83)

In addition to the Fourier transform, we see that we also have inner products of all the polarization
vectors of the modes, as well as weights relating to the masses of the nuclei and their frequencies of
oscillation. According to Brüesch, “an analogous expression holds for the coefficients of [the fourth
order term].” [51, p. 231]
The real challenge is to come up with the coefficients for the third and fourth order force constant
matrices. Information about them comes from a variety of sources, including thermal expansion;
temperature, pressure, and strain dependencies of elastic constants and moduli (higher order elastic
constants); experimentally determined Grüneisen parameters and Debye-Waller factors; etc. Even
with all of the above information their evaluation remains terribly complicated. See, for example,
Leibfried and Ludwig. [80]

Chapter 4

Components of the Entropy of a
Solid
To make the problem of finding the entropy of the solid tractable, we assume that the states of the
system may be split into combinations of independent subsystems of states such that:
S = kB ln Ω ≈ kB ln

Ωi

= kB

ln (Ωi ) =

Si ,

(4.1)

where Ω is the number of states of the system, Ωi is the number of states of the subsystem i, and Si
is the entropy of subsystem i. Especially at high temperatures, it may be necessary to correct for
the interactions between the subsystems. These corrections represent differences between simplified
models and reality, and may thus be positive or negative. They are called ‘entropy’ nevertheless.
For a non-magnetic, crystalline solid, the total entropy, S, may be broken up into contributions
from electrons, Sel , phonons, Sph , and their interactions, Sel−ph , with the phonon entropy usually
dominant at high temperatures. For a polyatomic solid the configurational entropy, Scf , is also
significant, but for crystals of a pure element the configurational contribution arises from defects,
and is usually quite small. In a magnetic solid, the spins contribute to the entropy through their
configurations as well as their dynamics; however we will group all of this under the electronic
entropy. Thus, we have:
S = Sph + Sel + Sel−ph + Scf .

(4.2)

First, we investigate the contributions of the phonons. The entropy for a group of non-interacting
bosons is derived in Appendix B. As the chemical potential for phonons is zero, their entropy is
given by:
Sph = Sph (T, T 0 ) = 3kB

dEgT 0 [(nT + 1) ln (nT + 1) − nT ln (nT )] ,

(4.3)

30
where both the phonon DOS at temperature T 0 , gT 0 , and the mean occupancy for bosons at temperature T , nT , are also functions of energy E. In a truly harmonic crystal, the phonon states would
remain unchanged with temperature; [50] therefore, the only temperature dependence in Eq. 4.3
would come from nT . Thus, we calculate the harmonic phonon entropy, Sph,H , at a temperature T
by using the T 0 = 0 K phonon spectrum and the occupancy at T into Eq. 4.3. In an experiment, we
may determine directly the temperature dependence of the phonon DOS. It can be shown that to
leading order, using Eq. 4.3 with the phonon spectrum from temperature T and the occupancy at
temperature T yields the total phonon entropy. [36, 81]
We know, that real solids are not perfectly harmonic (see § 3.6). For one, in a real solid, the
crystal volume changes with temperature, Classically, we may find the entropy associated with
dilation of the lattice, Sph,D by looking at the difference between the heat capacities at constant
pressure and at constant volume, CP and CV respectively: [82]
Z T
Sph,D = Sph,D (T ) =

CP − CV
dT 0 =
T0

Z T

9KT α2 0
dT ,
ρN

(4.4)

where KT is the isothermal bulk modulus, α is the linear coefficient of thermal expansion, and
ρN is the number density, all of which are temperature dependent. From Eq. 4.4, we see that the
increased entropy allows expansion despite the associated energy cost exacted by the elastic forces
(as represented by the bulk modulus) of the crystal.
The quasiharmonic entropy, Sph,Q is given by the sum of the harmonic and dilational contributions to the phonon entropy:
Sph,Q = Sph,H + Sph,D .

(4.5)

Any additional changes of the phonon entropy with increasing temperature are termed the anharmonic entropy, Sph,A , so
Sph = Sph,H + Sph,D + Sph,A = Sph,Q + Sph,A .

(4.6)

It is sometimes also useful to consider the nonharmonic phonon entropy, Sph,NH , which is the sum
of the dilation and anharmonic contributions
Sph,NH = Sph,D + Sph,A = Sph − Sph,H .

(4.7)

The entropy for non-interacting fermions is derived in Appendix B, and the electronic entropy is

31
given as follows:
Sel = Sel (T, T 0 ) = −kB

dEGT 0 [(1 − fT ) ln (1 − fT ) + fT ln (fT )] ,

(4.8)

where GT 0 = GT 0 (E) is the electron DOS at temperature T 0 and the mean occupation for Fermions
at temperature T is given by:
fT = fT (E) =

eβ(E−µ) + 1

(4.9)

The Fermi energy Ef is often a good approximation of the chemical potential, µ, of the electrons;
however, we will use:
Nel =

GT (E)fT (E)dE ,

(4.10)

where Nel is the number of electrons,a to find the chemical potential more precisely. This can make
a difference in metals like iron and nickel which have their Fermi level at the top of the d-band, for
example.
For a non-magnetic material, divisions similar to those made for the phonon entropy may also
be made for the electronic entropy:
Sel = Sel,G + Sel,D ,

(4.11)

where Sel,G is the ground state electron entropy and Sel,D is the electronic entropy associated with
changes in the electron DOS from dilation of the lattice. Both of these contributions to the electronic
entropy originate with non-interacting electrons. Sel,G is found using Eq. 4.8 with the T 0 =0 K
electron DOS and the temperature dependence coming only from fT . To find Sel,D , we use Eq. 4.8
with the electron DOS for the volume corresponding to T , and subtract the ground state term.
For a ferromagnetic material,b GT (E) is ferromagnetic at low temperatures, and paramagnetic at
high temperatures. Regardless, we will sometimes use a non-magnetic DOS at all temperatures, and
we denote this, GNM
T (E). The electronic entropy must also include a term for the entropy associated
with electron spins, Sel,M :
Sel = Sel,G + Sel,M + Sel,D .

(4.12)

Just as in the non-magnetic case, Sel,G is determined using G0 (E) in Eq. 4.8. The sum Sel,M + Sel,D
includes the effects of lattice dilation, spin dynamics, and the precipitous rise in the Fermi level at
a In practical calculations, this is frequently the number of valence electrons per unit cell, and is dependent on the
particular potential used.
b We will not be considering in detail antiferromagnets or more complicated magnetic materials in this thesis.

32
the Curie temperature. To within the errors of our calculations,c Sel,G gives a lower bound on the
electronic entropy. Spin dynamics and short range magnetic order make getting an upper bound
more difficult. The former would cause us to underestimate the electronic entropy, and the latter
to overestimate. In so much as these two errors cancel one another, the non-magnetic electronic
entropy, Sel,NM , calculated using Eq.4.8 with the non-magnetic electronic DOS, GNM
T (E), gives a
reasonable upper bound on the electronic entropy.d Thus, we have:
Sel,G . Sel . Sel,NM .

(4.13)

Here, Sel,NM includes the entropy due to dilation of the lattice, and we suggest the following approximation for the dilational contribution to the electronic entropy of a magnetic material:
Sel,D ≈ Sel,NM − Sel,G,NM ,

(4.14)

where Sel,G,NM is the entropy found by using GNM
0 (E) in Eq. 4.8. Roughly, this should give some
account of the effects of the expansion of the lattice as separated from spin fluctuations and the rise
in the Fermi level at the Curie temperature.
We now consider interactions between electrons and phonons. The electron-phonon entropy is
separated into two parts:
Sel−ph = Sel−ph,na + Sel−ph,ad .

(4.15)

The non-adiabatic electron-phonon entropy, Sel−ph,na , which dominates at low temperatures, is associated with the velocities of the nuclei and gives the mixing of the electron ground states from
the nuclear motion. At higher temperatures, the adiabatic electron-phonon entropy, Sel−ph,ad , dominates. It arises from the displacements of the nuclei, and accounts for the thermal shifts of electron
states caused by averaged nuclear motions. These contributions to the entropy have been discussed
in greater detail.[25, 26, 28–30, 37, 38] There should also be a contribution to the electron-phonon
entropy from the dilation of the lattice; however, we expect this to be negligible.
Putting this all together, we have:
S ≈ Sph,H + Sph,D + Sph,A + Sel,G + Sel,D + Sel,M + Sel−ph,ad + Sel−ph,na + Scf .

c According to Wallace, [37] these may be as large as 10%.
d We will examine this in greater detail in our discussions of iron and nickel.

(4.16)

Chapter 5

Neutron Scattering
The primary means of investigating the structure and dynamics of crystals is through their scattering
of incident radiation. A very basic picture of what goes on in a scattering experiment is shown in
Fig. 5.1. There is a source of radiation, a sample, and a detector. Depending on what we know
about the source, and on what we measure in the detector, we may garner information about the
sample by analyzing the differences between the incident and detected radiation.
Our study of iron was performed using x-rays from a synchotron source, and this type of experiment is discussed in Chapter 12. We treat neutron scattering separately and up front because it
applies to our studies of both aluminum and nickel, for which thermal neutrons from a spallation
source were used, and because it is a prerequisite for parts of the chapters on data analysis.

Detector

EF, QF

E, Q

Sample

Source

Figure 5.1: A very basic depiction of a scattering experiment. Radiation leaves the source,
and interacts with the sample. Radiation is
captured at the detector. The scattering angle Θ and the energy acquired by the sample
E = EI − EF are two things that might be
measured in such an experiment.

EI, QI

Figure 5.2: Depiction of the energies and wavevectors involved in a scattering experiment. The incident radiation has energy and wavevector EI and
QI , the outgoing EF and QF , and the energy and
wavevector transferred to the sample are given by
E = EI − EF and Q = QI − QF .

5.1

General theory of neutron scattering

The theory of neutron scattering has been discussed in great detail elsewhere; [83–85] and here we do
little more than briefly summarize some of the major results. Where applicable, equation numbers
for Squire’s book [83] are provided.
Modulo some details about particular instruments, the quantity measured in a neutron scattering
experiment is the number of neutrons that are scattered into some spread of solid angle dΩ centered
around Ω, with some spread of final energies dEF centered around EF . We expect the number of
neutrons arriving per unit time at any given solid angle Ω and energy EF to be proportional to the
flux of the incident neutrons, ΦI , as well as to the spreads in energy and solid angle:
number of neutrons scattered towards Ω , EF
unit time

d2 σ
dΩdEF

ΦI dΩdEF ,

(5.1)

number of neutrons scattered towards Ω , EF
unit time
ΦI dΩdEF

(5.2)

or:

d2 σ
dΩdEF

(Squires, Eq. 1.11), where

d2 σ
dΩdEF

, which has units of area per energy, is known as the double-

differential cross-section. Using Fermi’s golden rule [74, 75, 83] to describe transition probabilities
between states of the scattering system, and the Fermi pseudopotential [83, 84] to describe the
interaction of the neutron and the nucleus, we arrive at a very general expression for the double
differential scattering cross-section:

d2 σ
dΩdEF

QF 1 X
ai aj hexp {−iQ · [rj (0)]} exp {iQ · [ri (t)]}i e−iωt dt
QI 2π~ ij

(5.3)

(Squires, Eq. 2.59), where QI , QF , Q, ri , ai , ω, and t are the initial and final neutron wavevectors,
the wavevector transfer, the instantaneous position and scattering length of the atom indexed i,
angular frequency, and time, respectively. The angle brackets indicate a thermal average.
It is traditional to separate the scattering into coherent and incoherent contributions. Generally,
we do not deal with isotopically pure samples; therefore, we need to average over the scattering
lengths for different isotopes weighted by their natural frequencies. To do this, we:
• Break the sum into one over i = j and another over i 6= j.
• Use hai aj i = hai ihaj i for i 6= j.
• Add and subtract an i = j term to get rid of the i 6= j condition on the sum.

35
This yields:

d2 σ inc
dΩdEF

d2 σ coh
dΩdEF

QF 1 X σiinc
QI 2π~ i 4π

QF 1 X
hai ihaj i hexp {−iQ · [rj (0)]} exp {iQ · [ri (t)]}i e−iωt dt
QI 2π~ ij

hexp {−iQ · [ri (0)]} exp {iQ · [ri (t)]}i e−iωt dt ,

(5.4)
(5.5)

(Compare to Squires, Eqs. 2.68, 2.69), where we have used σiinc = 4π ha2i i − hai i2 . The first
equation depends only on the motions of a single atom i at a time. It thus gives the incoherent
scattering. The second equation, then, is the coherent scattering, and it depends on the motions of
both atoms i and j. We also have σ coh = 4πhai i2 ; however, we may not insert this into the above
equation for the coherent scattering unless we have a monatomic system.

5.2

Neutron scattering from crystals

Assuming a crystalline structure allows us to make further progress in developing theoretical expressions for the neutron scattering cross section. The exponentials from Eqs. 5.4 and 5.5 that involve the
instantaneous positions of the atoms may be broken up into time-dependent and time-independent
parts, using:
[ri (t)]

[xi ] + [ui (t)] = x(ls) + u(ls) (t) ,

(5.6)

where ri , xi , and ui are respectively the instantaneous position, equilibrium position, and displacement of atom i. Therefore, we may write:

d2 σ inc
dΩdEF

d2 σ coh
dΩdEF

QF 1 X σsinc
QI 2π~
4π

exp −iQ · u(ls) (0) exp iQ · u(ls) (t)

e−iωt dt , (5.7)

QF 1 X X
has ihas1 i exp −iQ · x(ls) − x(l1 s1 )
QI 2π~
(ls) (l1 s1 )
exp −iQ · u(l1 s1 ) (0) exp iQ · u(ls) (t)
e−iωt dt ,

(5.8)

where we have used the fact that the (isotopically averaged) scattering lengths must be a function
of the site alone. We may further simplify the equations by fixing one of the atoms to be in the first

36
cell:

d2 σ inc
dΩdEF

d2 σ coh
dΩdEF

QF L X σsinc
QI 2π~ s 4π

hexp {−iQ · [us (0)]} exp {iQ · [us (t)]}i e−iωt dt ,

QF L X X
has ihas1 i exp {−iQ · [l` ]} exp {−iQ · ([ss ] − [ss1 ])}
QI 2π~
` ss1
exp {−iQ · [us1 (0)]} exp iQ · u(`s) (t)
e−iωt dt ,

(5.9)

(5.10)

where we have dropped the cell index whenever it was fixed to be 1.
It is traditional, here, to define two operators, Û and V̂:
Ûs

V̂(`s)

−iQ · [us (0)] ,
iQ · u(`s) (t) .

(5.11)
(5.12)

Using the following results:

oE
exp Û(l1 s1 ) exp V̂(ls)

Û(l1 s1 ) + V̂(ls)
exp Û(l1 s1 ) V̂(ls) ,
exp

hÛ2s i = hV̂(`s)

(5.13)
(5.14)

(Compare to Squires, Eqs. 3.23–3.35), we may write the two cross-sections as follows:

d2 σ inc
dΩdEF

d2 σ coh
dΩdEF

D E
QF L X σsinc
exp Û2s
QI 2π~ s 4π

exp Ûs V̂s e−iωt dt ,

QF L X X
has ihas1 i exp {−iQ · [l` ]} exp {−iQ · ([ss ] − [ss1 ])}
QI 2π~
` ss1

× exp
Ûs1 + Ûs
exp Ûs1 V̂(`s) e−iωt dt

(5.15)

(5.16)

(Compare to Squires, Eqs. 3.127, 3.36).
We now expand exp Ûs1 V̂(`s) :

d2 σ inc
dΩdEF

d2 σ coh
dΩdEF

D E
QF L X σsinc
exp Û2s
QI 2π~ s 4π

Z X

EP
1 D
Ûs V̂s
e−iωt dt ,
P!

QF L X
exp {−iQ · [l` ]}
has ihas1 i exp {−iQ · ([ss ] − [ss1 ])}
QI 2π~
ss1

Z X 1 D
EP
× exp
Ûs1 + Ûs
Ûs1 V̂(`s)
e−iωt dt .
P!

(5.17)

(5.18)

Each term in the series represents scattering involving P phonons.
In the rest of this section, we give results for the Debye-Waller factor, Bragg and 1-phonon

37
scattering, and then develop the expression for multiphonon scattering, all for an arbitrary crystal
structure.

5.2.1

The Debye-Waller factor

The Debye-Waller factor is related to the mean-squared displacement, determines the ratio of elastic
to inelastic scattering intensity, and is given by exp (2Ws ), with:

= −

~ X |Q · [bs ]|
1 D 2E
Ûs =
h2nb + 1i
4Ms L
ωb

(5.19)

(Squires, Eq. 3.74). For a large enough crystal the spectrum of frequencies is continuous, and we
may convert the sum over branches to an integral:
Ws

DL
4Ms L

Z D

gs (ω)
h2nω + 1idω .

(5.20)

1 2
Q .

(5.21)

g(ω)
h2n + 1idω

(5.22)

|Q · [bs ]|

ω=ωb

In 3D, for a monatomic cubic crystal, we have:
|Q · [bs ]|

ω=ωb

Thus, Eq. 5.20 can be rewritten as follows:

~Q2
4M

(Compare to Squires, Eq. 3.66). This is frequently used as an approximation for other crystals as
well, including polyatomic ones:
Ws

5.2.2

~Q2
4Ms

gs (ω)
h2n + 1idω .

(5.23)

Elastic scattering

To find the elastic scattering, we set P = 0 in Eqs. 5.17 and 5.18:

d2 σ inc
dΩdEF

d2 σ coh
dΩdEF

QF L X σsinc −2Ws
QI 2π~ s 4π

e−iωt dt ,

QF L X
exp {−iQ · [l` ]}
has ihas1 i exp {−iQ · ([ss ] − [ss1 ])}
QI 2π~
ss1
×e−Ws1 −Ws e−iωt dt .

(5.24)

(5.25)

38
The 2π~
time the integrals yield delta functions in E, and the sum over ` also gives a delta function:

d2 σ inc
dΩdEF

d2 σ coh
dΩdEF

QF X σsinc −2Ws
δ(E) ,
QI
4π

QF (2π)D X
has ie−iQ·[ss ] e−Ws
δ(Q − q)
QI
Vq

(5.26)

δ(E) .

(5.27)

Finally, we integrate over EF , to get:
dσ inc
dΩ

dσ coh
dΩ

L X inc −2Ws
σ e
4π s s

(5.28)

(2π)D X
has ie−iQ·[ss ] e−Ws
= L
δ(Q − q)
Vq

(5.29)

(Compare to Squires, Eqs. 3.137, 3.75 & 3.76), where the delta-function forced QI = QF . The
first equation gives the incoherent elastic scattering, and it is closely related to the Debye-Waller
factor and the mean-squared displacement. The second is the coherent elastic scattering, or Bragg
scattering, and it carries information about the crystal structure.

5.2.3

1-phonon scattering

To find the 1-phonon scattering, we set P = 1 in Eqs. 5.17 and 5.18.

d2 σ inc
dΩdEF

d2 σ coh
dΩdEF

QF L X σsinc −2Ws
QI 2π~ s 4π

hÛs V̂s ie−iωt dt ,

(5.30)

QF L X
exp {−iQ · [l` ]}
has ihas1 i exp {−iQ · ([ss ] − [ss1 ])}
QI 2π~
ss1
−Ws1 −Ws
×e
hÛs1 V̂`s ie−iωt dt .

(5.31)

Using:
~ X
2L

hÛs1 V̂`s i =

Ms1 Ms

Q · [bs ] Q · [bs1 ]
ωb

× {exp (−iq · [l` ] + iωb t) hnb + 1i + exp (iq · [l` ] − iωb t) hnb i}

(5.32)

(Compare to Squires, Eqs. 3.105–3.108), we have:

d2 σ inc
dΩdEF

QF 1 X σsinc −2Ws
QI 2π s 8π

Z X

1 |Q · [bs ]|
Ms
ωb

× {exp (iωb t) hnb + 1i + exp (−iωb t) hnb i} e−iωt dt ,

(5.33)

d2 σ coh
dΩdEF 1

QF 1 X
exp {−iQ · [l` ]}
has ihas1 i exp {−iQ · ([ss ] − [ss1 ])}
QI 4π
ss1
21
Z X
Q · [bs ] Q · [bs1 ]
×e−Ws1 −Ws
Ms1 Ms
ωb

× {exp (−iq · [l` ] + iωb t) hnb + 1i + exp (iq · [l` ] − iωb t) hnb i} e−iωt dt . (5.34)
Here, 2π
times the integrals yield delta functions in ω, and the sum over ` also gives delta functions

in q and q1 :

d2 σ inc
dΩdEF

QF 1 X σsinc −2Ws
QI 8π s Ms

X |Q · [bs ]|2

d2 σ coh
dΩdEF

{δ(ω − ωb ) hnb + 1i + δ(ω + ωb ) hnb i} ,

ωb

QF (2π)D X X X has i
(Q · [bs ]) e−iQ·[ss ] e−Ws
QI 2Vq q

(5.35)

× {δ(Q − q − q1 ) hnb + 1i δ(ω − ωb ) + δ(Q + q − q1 ) hnb i δ(ω + ωb )} (5.36)
ωb
(Squires, Eqs. 3.138, 3.120). In both expressions, the first term corresponds to the creation of, and
the second term to annihilation of one phonon.
We may further simplify the expression for the incoherent cross-section. With x = ~ωβ, we use:
ex

hn(ω) + 1i =

ex − 1

−1
= −x
= −hn(−ω)i ,
−x
1−e
e −1

(5.37)

to write:

d2 σ inc
dΩdEF

QF DL X σsinc −2Ws gs (ω)
|Q
[
]|
bs
QI 8π s Ms
ω 1 − e−~ωβ
ω=ωb

(5.38)

where gs (ω) is the partial phonon DOS for vibrations of atoms at site s. We have also taken
gs (−ω) = gs (ω), and used the fact that for some function fs :

fs (ωb ) = DL

gs (ω)fs (ω)dω .

(5.39)

In 3D, for a monatomic cubic crystal, we have D = 3 and |Q · [bs ]|

d2 σ inc
dΩdEF

Again, we point out that |Q · [bs ]|

ω=ωb

QF Lσ inc g(ω)
Q2 e−2W .
QI 8πM ω 1 − e−~ωβ

ω=ωb

= 13 Q2 . Therefore:

(5.40)

= 13 Q2 is frequently not a bad approximation for other

40
crystal structures; therefore:

d2 σ inc
dΩdEF

X σ inc
QF L
Q2
e−2Ws gs (ω) .
−~ωβ
QI 8πω 1 − e

(5.41)

The factor modifying multiplying the DOS is a simplified version of the neutron weight factor , which
modulates the scattering from polyatomic crystals.

5.2.4

Multiphonon scattering

A neutron interacting with a sample may actually produce or annihilate (or some combination
thereof) more than one phonon at a time. Like the 0- and 1-phonon scattering, these processes can
be either coherent or incoherent. The incoherent approximation is the assumption that for numbers of
phonons P > 1, the coherent scattering is well approximated by the incoherent scattering multiplied
coh

by the ratio of the coherent and incoherent cross-sections, σσinc .[83, 84, 86]
From Eq. 5.17, we see that the incoherent scattering cross-section for site s and P phonons is
EP
proportional to the Fourier transform Ûs V̂s :
2π

D
EP
FT Ûs V̂s

(5.42)

D
EP−1 D
E
Ûs V̂s
Ûs V̂s

(5.43)

Z D
EP
Ûs V̂s
e−iωt dt =

For P ≥ 1, we may rewrite this in the following form:
2π

Z D

Ûs V̂s

e−iωt dt

= FT

The Fourier transform of the product may be rewritten as a convolution (denoted with ∗) of the
Fourier transforms of the multiplicands, yielding a recursion relation for the multiphonon scattering:
2π

D
EP−1
nD
Eo
Ûs V̂s
∗ FT Ûs V̂s
D
EP−1 1 Z D
Ûs V̂s e−iωt dt .
= FT Ûs V̂s
2π

Z D
EP
Ûs V̂s
e−iωt dt

= FT

(5.44)

We have already determined the integral on the far right, thus the problem of determining the multiphonon scattering is solved, given that we know the 1-phonon scattering. We denote P convolutions
with ∗P , so that:
2π

Z D
EP
Ûs V̂s
e−iωt dt =

2π

Z D
Z D
Ûs V̂s e−iωt dt ∗P
Ûs V̂s e−iωt dt . (5.45)
2π

41
Comparing Eqs. 5.30 and 5.40, we have for the case of a monatomic cubic lattice in 3D:
2π

Z D
ÛV̂ e−iωt dt

1 π~Q2 g(ω)
2π M
ω 1 − e−~ωβ

(5.46)

If we multiply the top and bottom of this equation by γ0 :
γ0 =

g(ω)
h2n + 1idω ,

(5.47)

~Q2
2M

(5.48)

then we have:
2π

Z D
ÛV̂ e−iωt dt =

g(ω)
g(ω)
h2n + 1idω
γ0 ω 1 − e−~ωβ

The factor in brackets is precisely the Debye Waller factor, 2W , as given in Eq. 5.22, and we define
the remaining factor to be S1 (ω). We thus rewrite this equation as:
2π

Z D
ÛV̂ e−iωt dt =

2W S1 (ω) .

(5.49)

If we make the following definition:
SP (ω) ≡

SP−1 (ω) ∗ S1 (ω) = S1 (ω) ∗P S1 (ω) ,

(5.50)

then, from Eq. 5.45, we have:
2π

Z D
EP
Ûs V̂s
e−iωt dt =

[2W S1 (ω)] ∗P [2W S1 (ω)]
= (2W )P S1 (ω) ∗P S1 (ω) = (2W )P SP (ω) .

(5.51)

Substituting this into the P th order term from Eq. 5.17 and rearranging, we have:
1 QI 4π
L QF σ inc

d2 σ inc
dΩdEF

1 (2W )P −2W
SP (ω)
~ P!
SP (Q)SP (ω)
SP (Q, ω) ≡ SP (Q, E) .

(5.52)

If we define S0 (E) = δ(E), then we may write:
1 QI 4π
L QF σ inc

d2 σ inc
dΩdEF

= e

−2W

(2W )P
P=0

SP (E) ≡ S(Q, E) .

(5.53)

S(Q, E) is called the scattering function, response function, scattering law, or the dynamic structure

42
factor, and is generally the quantity that we wish to extract from a given scattering experiment.
Finally, we note that the ratio of the intensity of the elastic to the inelastic scattering is given by
exp(2W )−1 .

5.2.5

Scattering from a damped harmonic oscillator

In the same way that we expressed changes in the phonon frequencies due to perturbations to the
harmonic potential in § 3.6, it is possible to find expressions for the double-differential scattering
cross-section in the presence of anharmonicity. The perturbed frequency response turns out to be
that of a damped harmonic oscillator, with the measured frequency given by ωb (q) + ∆ωb (q) and
with half width Γb (q).
For a single damped harmonic oscillator (DHO), we have the following expression for the response
function: [84]
= Q2 e−2W

S1DHO (Q, E)

M πQω 0 ω

ω0

ω 2

− ω0

1 2

1 − e−~βω

(5.54)

u̇ + ü = 0. Here, C is the
where the equation of motion for the oscillator was given by ω 02 u + M
damping coefficient, K is a harmonic force constant, ω ≡
M is the natural frequency of the

= 2Γ
is the quality factor.a
oscillator, and Q ≡ C/M

If we have L independent oscillators, each with the same quality factor, the response function is
given by:
S1DHO (Q, E)

Q2 e−2W

1X 1
−~βω
Mω 1 − e
L i πQωi ωi

Q2 e−2W

M ω 1 − e−~βω

g(ω 0 )
πQω 0

ω0

− ωωi
dω 0
ω 2

− ω0

1 2

(5.55)

That is, we measure an integral transform of the harmonic phonon spectrum with the damped
harmonic oscillator function, B:
B(Q, ω 0 , ω) =

πQω 0

ω0
ω 2
+ Q12
ω − ω0

(5.56)

The dependence on wavevector transfer of the response for an independent oscillator is identical
to that for a cubic crystal. Thus, provided that the quality factor is actually the same for all of
the modes, this expression should hold for a monatomic cubic crystal as well (modulo a constant of
proportionality). In general, this provision is not true; however, it does appear to be true on average
in some crystals.
a We apologize for the similarity between Q for wavevector transfer and Q for quality.

Chapter 6

Time-of-flight Neutron Chopper
Spectrometers
In this chapter, we briefly explain the concepts of direct-geometry, time-of-flight chopper spectrometry, make comparisons to the more widely use triple axis spectrometer, and describe in some detail
the two instruments used for the neutron scattering studies of aluminum and nickel presented in this
work.

6.1

Spallation neutron sources

Although chopper spectrometers may be built on reactor sources, they are more commonly found
at spallation sources. As all of the studies presented here were performed at spallation sources, we
focus our discussion on these types of instruments.
A schematic of a spallation source is given in Fig. 6.1. Generally, the process starts with the
acceleration of reasonably massive charge particles (usually H − ions) to energies on the order of 100
MeV or more in a linear accelerator . The ions are then stripped of electrons and bunched together

Moderator
Accumulator
Target
Instruments
Accelerator

Figure 6.1: Schematic of a spallation source. Charged particles are accelerated, accumulated, and
then directed into a heavy metal target. The particles cause spallation neutrons to come out in all
directions, passing through the moderator and into the instruments.

44
into pulses in an accumulator ring, and directed at a heavy metal target. As the protons hit nuclei
in the target, neutrons are spalled in all directions. In a circle about the target are moderators,
in which the fast neutrons interact inelastically with the moderator material and are thermalized .
Typically, moderators use materials rich in hydrogen, such as water or solid methane, because it is a
strong scatterer of neutrons and has a high average energy transfer per collision. On the other side
of the moderators are suites of instruments, optimized for different types of scientific experiments.
For all instruments, the neutrons exit the moderator in small pulses at known times, and this
timing is fundamental to the operation of a chopper spectrometer.

6.2

Direct geometry, time-of-flight, chopper spectrometers

Fig. 6.2 shows a schematic of a direct-geometry, time-of-flight chopper spectrometer. Each pulse of
neutrons leaves the moderator at a known time, τ0 . They then pass through the T0 and E0 (or
Fermi ) choppers. These are rotating cylinders with slits that are phased to allow only neutrons of
a certain velocity, vI , pass through. The T0 chopper stops fast neutrons (with MeV energies) and
γ-rays The Fermi chopper provides the monochromatization, occasionally with a contribution from
the T0 chopper. Having two choppers can stop some fast and slow neutrons that would otherwise
pass through chopper openings intended respectively for previous or subsequent pulses.

to
r2

ho
pp
Fe
er
rm
iC
ho
pp
er
on
ito
Sa 1
pl

od
er

The neutrons that pass through the choppers impinge upon the sample, where they might be

Figure 6.2: Schematic of a direct-geometry, time-of-flight chopper spectrometer. Neutrons of a wide
range of energies leave the moderator. The T0 and E0 (or Fermi ) choppers monochromatize the
beam. The neutrons then scatter off of the sample and travel into the detectors. The monitors may
be used to determine accurately the incident energy (spectrum) of the neutrons.

Detector

Row of pixels

Scattered neutron

Time channel

Figure 6.3: Schematic of a scattered neutron entering a typical detector module. The module
consists of a batch of detectors. Each detector, then, has a number of pixels. Finally, each pixel has
time channels. These are indexed by d, p, and t, respectively. Frequently, the detector modules are
positioned radially around the sample, forming a sort of cylinder. In this case, all the pixels in a
row in a given detector module are approximately equidistant from the sample. Pixels in different
rows, however, are at different distances from the sample. On older instruments the detectors are
not pixelated.
elastically or inelastically scattered — possibly more than once. Since the time the neutrons arrived
at the moderator and the distance from the moderator to the sample are known, and the velocities
of the neutrons are determined by the choppers, we may determine the time at which the neutrons
arrive at the sample, τI .
Some fraction of the neutrons that have made it to the sample will be scattered towards the
detectors. The arrival of a neutron into a time-channel in a pixel in a detector in a detector module
is depicted in Fig. 6.3 The modules are frequently arranged cylindrically about the sample, like the
detectors in Fig. 6.2.
We now know the time and velocity of the neutrons when they arrived at the sample, and the
time and location of their arrival in the banks of detectors, and we may use all of this information
to determine the initial and final energies and wavevectors of the neutrons. The initial values are

46
straightforward, we have:
EI

mn 2
v ,
2 I
mn
vI ,

(6.1)
(6.2)

where, |vI | is determined by the chopper frequencies and its direction is determined by the location
of the moderator.a For the final values, we consider an instrument that lacks pixelated detectors.
If we assume that a neutron arrives at a detector, d, in a time τd , and that the distance from the
sample to the center of the detector is Ld , we have:
EF

2
mn Ld
τd
mn Ld
~ τd

(6.3)
(6.4)

The angle, Θd at which the center of a particular detector is located is frequently tabulated along
with Ld , allowing us to find the direction of QF . Frequently, on an instrument that has pixelated
detectors, only the distance from the center of the detector to the pixel, Lp , is tabulated in addition
to Ld and Θd . In this case, we have:

where Ldp =

2
mn Ldp
τdp
mn Ldp
~ τdp

(6.5)
(6.6)

L2d + L2p is the distance and τdp is the time for the neutron to travel from the

sample to the pixel. The direction of QF is determined through the scattering angle, Θdp =
cos(Θd ) .b Fig. 6.4 is a schematic of some of the relevant angles, times, and distances
arccos LLdp
for a direct geometry, time-of-flight, chopper spectrometer.
The energy, E, and wavevector Q, transferred to the sample, then, are given by:

EI − EF ,

(6.7)

Q =

QI − QF .

(6.8)

The relationship of the initial, final, and transferred wavevectors was depicted in Fig. 5.2. If the
sample is a single crystal, the direction of Q is related to the actual wavevector transferred to the
sample through the orientation of the crystal. For a polycrystalline sample, like those used for the
a More accurate methods of determining the initial energy of the neutrons are discussed in Chapter 7.
b We have assumed that the detectors are oriented perpendicular to the initial velocities of the neutrons. If this is

not the case, more involved geometry is required, as well as additional information about the experimental setup.

pi
xe

, τ dp
L dp

τ0
Θ = Θd p

Θd

pl
sa

at
or

LΙ, τΙ

Figure 6.4: Another schematic of a direct geometry, time-of-flight, chopper spectrometer; this one
showing some of the times, distances and, angles relevant to the analysis of a measurement. The
neutron leaves the moderator at τ0 = 0, and travels a distance LI in time τI , in order to arrive at
the sample. Here, it is scattered an angle Θ = Θdp in the direction of a pixel in a detector that
is located at an angle Θd to the incident neutron beam in the plane containing the centers of the
detectors. The distance from the sample to theqcenter of the detector is Ld , and from there to the
pixel Lp , so that the total distance is Ldp = L2d + L2p ; and the neutron travels this distance in
time τdp . From this information and the initial neutron energies, we may determine energies and
wavevectors of the scattered neutrons and, in turn, of the excitations in the sample.
studies presented here, the direction of Q is not measurable. Rather, we measure an average over
neutrons that have the same Q = |Q|. Fig. 6.5 shows an example of the region in Q−E space sampled
in a time-of-flight neutron chopper experiment. The large portion of reciprocal space sampled in a
time-of-flight experiment makes it a superior technique for measuring the phonon DOS.

6.3

Time-of-flight vs. triple-axis

Pioneered by Bertram Brockhouse,c the triple-axis spectrometer has been the workhorse of inelastic
neutron scattering since its advent in the mid 1950s. If the purpose of an experiment is to measure
the dispersion relations for phonons or magnons, there is no better technique. Similarly if the
c He won the 1994 Nobel Prize for his work.

Figure 6.5: Region sampled in Q and E on LRMECS (described in § 6.4) for an incident energy of
60.0 meV. Each light blue point in the plot corresponds to one time-of-flight in one detector. The
spacing of the points in time-of-flight was constant; however, in energy the spacing gets larger as we
move from positive to negative energy transfers, i.e., there is better resolution for phonon creation
than for phonon annihilation. The darker black points are contours of constant angle. Specifically,
they are at 2.4◦ , 21.0◦ , 37.8◦ , 64.2◦ , 92.4◦ , and 117.6◦ from left to right. The gaps are either spaces
between detector modules or missing/broken detectors.
purpose of the experiment is to investigate the properties of some particular phonon, magnon, or
excitation.
A schematic of a triple-axis spectrometer is shown in Fig. 6.6. Typically, neutrons from a nuclear
reactor reach the moderator where they are thermalized. They then travel to a crystal monochromator, which selects neutrons with a particular energy that continue along to the sample. The neutrons
are scattered in the sample, and some fraction of those scattered proceed to the analyzer. At the
analyzer, neutrons with a particular final energy are scattered toward the detector, where they are,
one hopes, detected. The sample may be rotated about the monochromator, the analyzer about
the sample, the detector about the analyzer, and the sample about its center, with those angles
(respectively Θ0 , Θ1 , Θ2 , and Θs as shown in Fig. 6.6) defining the incident and final energies and
wavevectors of the neutrons.
For a sample which is a single crystal, the technique thus provides a direct measurement of the
scattering intensity for excitations with some fixed energy E = EI − EF and wavevector transfer
Q = QI − QF . Typically, scans are performed in either constant energy or constant wavevector, the

Θ0

er
An
al
yz

pl
Sa

od
er

on
oc
hr
om

at
or

De
te
ct
or

Θs
Θ1

Θ2

Figure 6.6: Schematic of a 3-axis spectrometer. Neutrons exit the moderator, and are monochromatized by a single crystal. They then enter the sample, where they are scattered. The analyzer
crystal then monochromatizes the outgoing neutrons before they head to the detector. Typically, the
sample may be rotated about the monochromator, the analyzer about the sample, and the detector
about the analyzer. Additionally, the sample might be rotated in place.
latter providing the dispersion in some particular direction in the reciprocal space of the sample.
Measurements of the phonon dispersions in some small number of directions are thus feasible, and
may be used to optimize some model of the lattice dynamics in the sample — usually to allow a
calculation of the phonon DOS. Since the phonon DOS is an average over the entire reciprocal space,
a time-of-flight chopper spectrometer provides a much more direct measurement.

6.4

Instruments used

The three instruments used in this work were the Low Resolution Medium Energy Chopper Spectrometer (LRMECS) at the Intense Pulsed Neutron Source (IPNS) at Argnonne National Lab (ANL),
Pharos (not an acronym, but rather a reference to the lighthouse of Alexandria) at the Los Alamos
Neutron Science Center (LANSCE) at Los Alamos National Laboratory (LANL), and the wide-Angle
ChoppeR Spectrometer (ARCS) at the Spallation Neutron Source (SNS) at Oak Ridge National Laboratory (ORNL). LRMECS and Pharos were used for both the aluminum and nickel experiments,
and ARCS for the lead experiment. Schematics are given in Figs. 6.7 and 6.8, and some of the more
general details are given in Table 6.1.

Figure 6.7: Schematic of LRMECS.

Figure 6.8: Schematic of Pharos.

Figure 6.9: Schematic of ARCS.

Quantity
moderator type
detector type
number of detectors
detector heights
pixels per detector
pixel size
LI
hLd i
T0 chopper position
E0 chopper position
1st monitor position
2nd monitor position
beam size
h∆Θd i
angular range

LRMECS
Liquid CH4 at 100 K
He
148
Θd < 25◦ → 0.229 m
Θd ≥ 25◦ → 0.457 m
NA
NA
8.1 m
2.5 m
∼ 6.5 m
7.6 m
7.65 m
11.34 m
0.05 m × 0.10 m
0.6◦
−7.2 < Θd < −2.4◦ ,
2.4◦ < Θd < 117.6◦

Pharos
Liquid H2 O at 283 K
He
376
1.0 m
0.25 m
40
0.025 m
20.00 m
4.13 m
14.0 m
18.0 m
NA
NA
0.05 m × 0.075 m
0.4◦
−10.9 < Θd < −1.7◦
1.77◦ < Θd < 147.07◦

ARCS
Liquid H2 O, ambient
He
920
high angles → 1.0 m
low angles → 0.25 m
128 or 256
or / 256
128
13.6 m
3m
9m
11.6 m
11.825 m
18.5 m
0.05 m × 0.05 m
0.48◦
−28.1 < Θd < −3◦
3◦ < Θd < 135◦

Table 6.1: Some details about LRMECS, Pharos, and ARCS. hLd i is the modal distance from the
sample to the detectors. All other positions are given relative to the moderator. hΘd i is the modal
angular step between detectors. The ARCS water moderator is decoupled, and the number of pixels
in its detectors may be varied.

Part II

Data Analysis and Computation
Analytical and computational techniques for simplifying and interpreting data from timeof-flight chopper spectrometers are presented here. New or modified procedures are
presented in the following sections: §7.1.1, §7.2.2, §7.4.2, §8.1.2, §8.2, and §8.3.1.2.

Chapter 7

From Raw Data to S(Q,E)
Up to this point, we have considered some general formulas relating the double differential crosssection or the dynamic structure factor to the elementary excitations in a solid. In this chapter, we
present methods by which the actual measured quantities are related to the theoretical quantities
presented in Chapter 5.
As discussed in § 6.2, the information available to us in a direct geometry, neutron time-of-flight
chopper spectrometer experiment are the time-of-flight τ of the neutron, the distance from the
moderator to the sample, LI , and the distance, Ldp , from sample to a pixel indexed p (in a detector
indexed d). Using these, we must determine the initial and final energy of the neutrons, EI and
EF , and the amount of energy absorbed by the sample E = EI − EF . We know approximately the
direction of the incident neutron beam, so given its energy we may also determine its wavevector,
QI . If we also consider the scattering angle, Θ, which is determined by the locations of the pixel
and sample, we may determine the final wavevector of the neutron QF , and the wavevector transfer ,
Q = QI − QF .
The goal of this chapter is to describe in detail the process by which the raw data acquired
on a direct geometry, time-of-flight, chopper spectrometer is converted to a physically meaningful
quantity, such as the dynamic structure factor S(Q, E). We present both commonly used and new
techniques. At a minimum, the following operations (or analogous ones) must be performed:
• Detector masking and efficiency corrections
• Determination of the incident energy
• Normalization of the data
• Transformation to physical coordinates
• Removal of background scattering
In practice, these operations are usually performed once for each data set. In reality, the results from

54
some of these operations affect the others, and iterating over the operations to find a self consistent
solution might be more accurate.
A final note before we begin: Technically, the removal of multiply scattered neutrons belongs to
this chapter, as this type of scattering involves the geometry of the instrument, and is not part of
the dynamic structure factor. Corrections for multiple scattering are frequently skipped all together,
and the multiple scattering correction presented in this text is coupled to the determination of the
phonon DOS. As such, we delay discussion of this correction until § 8.1.

7.1

Detector masking and efficiency corrections

On LRMECS and Pharos, the neutron detectors are based on the helium conversion reaction:
n + 3 He → 3 H + 1 H + 0.764 MeV .

(7.1)

The detectors are hollow tubes with a voltage across them, filled with 3 He. When a neutron hits
the detector, it undergoes the above reaction, which in turn causes a cascade of charged particles.
These charged particles are then detected as a current flow at the terminals on the detector.
In order to calibrate the detectors, it is very helpful to perform an experiment where the same
number of neutrons will reach each detector. Fig. 7.1 shows a measurement of the scattering from
vanadium made on LRMECS. Pure vanadium is a highly incoherent scatterer of thermal neutrons,
with coherent and incoherent neutron scattering cross-sections of σ coh = 0.0184 barns and σ inc =
5.08 barns, respectively. As such, measurements of the scattering from vanadium are well suited to
the purpose of neutron instrument calibration. Although it is possible to measure the vanadium
with a fixed incident neutron energy, all of the detector calibrations in this text were performed
using a white beam measurement of vanadium.
The efficiency of the detector depends on its electronics, at least in so much as faulty electronics
may cause it to cease working altogether. We assume that this is completely accounted for by the
elimination of bad detectors.a The efficiency of the detector also depends on the pressure of the 3 He
gas that it contains, as well as on the energy of the particle it is detecting. We assume that these
two contributions are independent. Thus, the probabilities, Υdt , of detecting neutrons in detectors
indexed by d and time channels indexed by t are given by:b
Υdt = Υt Υd ,

(7.2)

a It is likely that other effects from the electronics get taken care of fortuitously when we correct for the pressuredependence of the detector efficiency.
b Given that neutrons have arrived at the detector at some times and that they should be detected. We will not
be considering probabilities that neutrons get registered in the wrong detectors or time channels.

Figure 7.1: Vanadium data from LRMECS. Because vanadium is a purely incoherent scatterer, the
angular dependencies of the data must come from somewhere other than the sample. (Technically,
the inelastic scattering increase with increasing Q; however, the elastic scattering completely dominates this contribution.) The black bar around Θ = 0 is because there are no detectors there. The
near horizontal streaks at the higher angles are probably Bragg scattering from somewhere in the
sample well. The vertical streaks are the variations in detector efficiency for which we must correct.
where Υt gives the energy-dependent probabilities and Υd gives the pressure-dependent ones.
Here, we consider a scheme for automatically determining which detectors are unusable, as well
as methods for correcting for the energy- and pressure-dependencies of the detectors.

7.1.1

Masking bad detectors

Instrument scientists sometimes provide a list (called a mask ) of the detectors that are not working
properly, but these lists can get lost, or be out of date. Further, the bad detectors are usually found
by hand, which is only feasible so long as the numbers of detectors on an instrument are sufficiently
small. Thus, automating the task of finding the bad detectors is desirable. The problem here is
that it is not at all obvious a-priori which detectors are good and which are bad. Operating under
the assumption that the data should vary relatively smoothly from detector to detector, we may
automate the removal of some bad detectors. Thus, the measurement of vanadium is used for finding
the detector mask.
Before doing anything more involved, we make note of any detectors with exactly zero counts.
This is a highly unlikely measurement in a facility geared towards the production of neutrons. This

Mask
Vanadium Data

4.0e+06

Counts

3.0e+06
2.0e+06
1.0e+06
0.0e+00

100

120

Θ (degrees)
Figure 7.2: Vanadium data from LRMECS, summed over time-of-flight, is shown in black. Two hot
detectors are clearly visible at around 12◦ , and absent or broken detectors are visible throughout.
The lighter, purple bars show masked detectors. (Note that the the vanadium plate is oriented at
145◦ to the beam, and that this is why the detectors at high angles have decreasing intensity. Dark
angle corrections based on simple models of neutron absorption are commonly used, but the details
are not presented here.)
part of the correction should be performed directly on both the scattering from vanadium, and from
the sample of interest; in case a detector has failed suddenly in between the measurements.
We now attempt to iteratively find detectors with unreasonable numbers of counts. We choose
a final threshold for keeping a detector — for example, we might require that no detector has more
than 10% more counts than the mean number of counts in a detector.c We should not immediately
find the mean number of counts in a detector and throw out all detectors which deviate from that
value by more than 10%. For an extreme example, imagine that there are 10 detectors and that a
perfect experiment would yield 10 counts per detector. Further, imagine that the measured number
of counts in each detector is 10, with the exception of one detector where we found 1 count, and
another where we found 1,000 counts.d The mean number of counts per detector is 108, and the
c Unlike finding detectors with zero counts, it is not possible to perform this part of the corrections with data from
the sample of interest unless it is a totally incoherent scatterer. If it is coherent, the Bragg scattering will not vary
smoothly with angle.
d This example is not absurd. The collected counts are stored digitally, and an error in a single bit may change the
number of counts by orders of magnitude.

57
percent errors between the 1 count, the 1,000 count, and the 10 count detectors and the mean are
-100%, 825%, and -91%, respectively. If we blindly apply our 10% criterion, we have no detectors
left.
Instead, we iterate, each time finding the detector with the maximum and the minimum counts.
If these deviate from the mean by more than our threshold, they are masked, if neither of them
deviate by this amount, our mask is complete. In our example, the 1 and 1,000 count detectors are
eliminated in the first round. Once that is done, the mean number of counts goes to 10, and we
mask no other detectors.
In practice, a threshold of about 20% deviation from the mean number of counts seems to work
quite well for LRMECS data, as seen in Fig. 7.2. The technique has yet to be applied to Pharos
data, where the detectors are pixelated; however, it seems that it should work either by summing
the counts over the pixels, or by treating each individual pixel as a detector was treated here.e This
scheme is by no means perfect, but it is often a good starting point for the removal of bad detectors.

7.1.2

Energy-dependent detector efficiency

As a rule of thumb, the probabilities of a neutron being detected, Υt , are proportional to the
amount of time the neutron spends in the detector. This, in turn, is inversely proportional to the
final velocity of the neutron, vF . If the mean distance through the detector is given by hLdet i, then
we have approximately:
Υt ∝

hLdet i
hLdet i
∝ √
vF
EF

(7.3)

dp
where vF is the final neutron velocity. Noting that vF = τdp
, where τdp is the time-of-flight for the

neutron to travel from the sample to a pixel (in a detector), we have:
Υt ∝ τdp .

(7.4)

This may also be seen in the following way. The probabilities of detection are the same as the
probabilities of absorption, which are approximately given as follows:
Υt

1 − exp −ρHe σ abs hLdet i
1
2
1 − 1 − ρHe σ abs hLdet i +
ρHe σ abs hLdet i + · · ·
ρHe σ abs hLdet i ,

(7.5)

e The former is the proper choice if the failure of a pixel indicates that the detector in which it resides has also
failed.

58
where ρHe is the number density of the 3 He atoms in the detector, and σ abs is their absorption
cross-section. Since σ abs ∝ v1F ∝ √E
, [83, 84] we find again that the absorption of neutrons in the

detector is proportional to τdp or inversely proportional to the square root of the energy.
In principle, the efficiency of the detectors as a function of energy could be measured;f however,
it is often provided by the detector manufacturer. In any case, given Υt , the measured intensity as
00
a function of time-of-flight, detector ID, and pixel ID, Idpt
, is scaled accordingly:

Idpt

00
Idpt
Υt

(7.6)

where Idpt
, then, is the intensity corrected for the energy-efficiency of the detectors.

7.1.3

Pressure-dependent detector efficiency

As seen in Eq. 7.5, the efficiency of the detectors depends upon the density, and thus, the pressure of
the 3 He in the tubes. Here, we correct the data for variations of the pressure in the detectors. The
measurement of a vanadium sample gives intensity as a function of detector, pixel, and time-channel;
but we are not interested in the time-of-flight as we have already corrected for the energy-efficiency
of the detectors. Two simple possibilities for reducing the data to a function of detector only are to
(1) Sum over the time-of-flight coordinate, or (2) Select the time-of-flight, τe , that corresponds to
the neutrons traveling from the moderator to the pixels with energy EI :
τe ≡

LI + Ldp
τI ,
LI

(7.7)

where τI = τ − τdp is the time-of-flight for the neutron to travel from the moderator to the sample.
The supposed advantage of the latter method is that EI is most relevant to the experiment. In
reality, we will be measuring neutrons with a variety of energies, and, if we are doing an inelastic
experiment, we will care in particular about those with energies not equal to the incident energy.
Additionally, the sum over time-of-flight means that we have a much larger set of data to work with.
For these two reasons, the sum over all time channels is likely a better choice.
LRMECS has un-pixelated detectors, and on Pharos there may not be sufficient intensity in a
pixel for its efficiency to be corrected individually. Further, the pixels in a detector all share the
same 3 He gas. Because of this, we assume that the efficiency depends only on the detector, and not
f One way to do this would be to make many measurements of vanadium at different incident energies; however
this is generally considered too costly. As a result, the efficiencies provided by the manufacturer are commonly
used. An alternative is to make a single measurement of the vanadium spectrum with either a monochromatic or
white-beam incident neutron spectrum. (Presumably the monochromatic beam would have the same energy as for
subsequent measurement of the sample.) In so much as the inelastic scattering spectrum of vanadium approximates
the inelastic spectrum of the sample to be measured, the monochromatic run will weight the counts at different energies
appropriately. In practice, it seems more likely that the benefits of the greatly increased intensity of the white beam
far outweigh the benefits derived from similarities between the spectrum of the sample and that of vanadium.

Figure 7.3: Vanadium data from LRMECS, as seen in Fig. 7.1, but corrected for detector efficiencies.
The black bar around Θ = 0 is because there are no detectors there. The Bragg scattering at the
higher angles is still visible, but the other variations have been accounted for. The correction derived
from and applied to the vanadium data will be applied to our experimental data as well.
on the pixel. The probabilities, then, for the pressure-efficiency of the detectors are given as follows:
XX

00van
Idpt

Υd = X X X

00van
Idpt

(7.8)

and the intensity, corrected for both energy- and pressure-dependencies, Idpt , is given by:
Idpt =

00
00
Idpt
Idpt
Idpt
Υd
Υd Υt
Υdt

(7.9)

The vanadium data from LRMECS, shown uncorrected in Fig. 7.1, are shown in Fig. 7.3, once
the detector efficiency corrections have been applied.

7.1.4

Solid-angle of the pixels

Since the detectors are straight tubes, and the pixels have equal length along those tubes, pixels at
different heights cover different amounts of solid-angle. The solid angles covered may be calculated
and corrected for analytically; or they may be corrected for by performing the procedure for the

1.1

Normalized Counts

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
-0.5 -0.4 -0.3 -0.2 -0.1

0.1 0.2 0.3 0.4 0.5

Offset from center of the detector (m)
Figure 7.4: Counts from vanadium as a function of the offset of a pixel from the center of the
detector, normalized such that the mean number of counts per pixel is 1. Presumably there are
differences because of the different solid angle subtended by each pixel; however, in reality, this
effect is quite small. The drop in intensity at the ends of the detector is probably due to shielding
by the detector mounts, rather than differences in solid-angle coverage.

pressure-efficiency of the detectors, but with sums over detectors at equal distances to the sample,
rather than over pixels. In either case, this is typically a very small correction on LRMECS and
Pharos, as seen in Fig. 7.4, which shows normalized intensity as a function of the offset of the pixel
from the center of the detector. The correction can be more significant on ARCS, where the detectors
are stacked three rows tall.

7.2

Determination of the incident energy

As seen in § 6.2, the incident energy is nominally determined by the rotation frequencies of the
T0 and Fermi choppers; however, in practice it is more accurate to determine the incident neutron
energy from the data itself. Here, we present two methods for doing this.

7.2.1

Using monitors

The traditional way to find the incident energy is very simple. Two special detectors, called monitors,
are used to measure the incident neutron spectrum. They are located at different points along the
path the incident neutrons would take were they not scattered, and we assume that the neutrons
counted in the monitors have traveled there from the moderator without interacting with anything
along the way. Presumably, if the choppers are working properly, the spectrum in either monitor
consists of a large peak, as is seen in Fig. 7.5
Some sort of fit to the peaks is performed, and the time required for the neutrons to reach
the monitor is extracted from the fit. In Fig. 7.5, the fit to the peaks was parabolic, but fits with
Gaussian or other lineshapes are also possible. Given the fits, we may calculate a single value that
is our estimate of EI , as follows:
EI =

Lmon
τmon

(7.10)

where mn is the neutron mass, Lmon is the distance between the two monitors, and τmon is the
difference between the two times-of-flight determined by the fits. For the LRMECS data shown
in Fig. 7.5, τmon = 1107.98 µs, and Lmon = 3.732 m; thus the experimentally determined incident
energy is given by:
1.67 × 10−27 kg
EI =

3.73 m
1107.98 × 10−6 s

0.1

= 9.50 × 10−21 J = 59.30 meV .

(7.11)

0.1
Vertex: 2268.38 µs
0.08
Normalized Counts

Normalized Counts

0.08

0.06

0.04

0.02

2200

0.06
Vertex: 3376.36 µs
0.04

0.02

2220

2240

2260

2280

Time-of-flight (µs)

2300

2320

2340

3250

3300

3350

3400

3450

3500

Time-of-flight (µs)

Figure 7.5: Counts in the first and second beam monitors on LRMECS. The data, shown with
markers, were fit to parabolas, shown by the dashed line. The vertices of the parabolas allow us to
make an estimate of the incident energy.

62
The nominal incident energy (from the chopper settings) was 60.00 meV, which is in error by a little
over 1%.

7.2.2

Using scattered data

Some instruments have no monitors, or have broken monitors, and the following procedure allows
determination of the incident energy using the scattered neutron counts. More generally, we should
be able to use the scattered neutrons to help with determination of the incident energy even if there
are monitors, or to help with refinement of the instrument and sample geometry.
We wish to determine the incident energy of the neutrons from the corrected intensity, Idpt . Our
procedure rests primarily on the assumptions that an elastic peak exists at all angles and that the
scattering will be the strongest in the elastic peak — both of which are usually satisfied. On most
instruments, the detectors are grouped together in modules, like the one seen in Fig. 6.3. Here, we
will assume that the centers of all the detectors are equidistant from the sample, and that all the
pixels in a given row are also equidistant from the sample. This is generally not the case; however,
it is easy to adjust this method to work detector bank by detector bank, where these positional
constraints are usually met.
In the past, where monitors were wanting, the incident energy was determined by finding the
intensity as a function of time-of-flight only, without regard to the varying sample to pixel distances:
It =

Idpt .

(7.12)

This spectrum was then fit to a Gaussian, or some other function, and the incident energy was taken
to be the maximum (or the mean, etc.). This is shown in Fig. 7.6. The vertex is at 6587.260µs, and
given the distance from the moderator to the detectors is LI + Ldp ∼ 24.013m, we get an incident
energy:

EI =

1.675 × 10−27 kg

24.013 m
6587.260 × 10−6 s

= 1.112 × 10−20 J = 69.461 meV .

(7.13)

To make a more accurate estimate, we will consider sums over the intensity at constant distance
from the sample. Given our assumption about detector banks, this means we may sum over detectors
to get intensity as a function of time-of-flight and pixel:
Ipt ≡

Idpt .

(7.14)

In a fixed pixel, p = p∗ , we find the time-of-flight channel that corresponds to the maximum counts
and then fit a parabola (or some other function) to a few of the closely surrounding points. The

80000
vertex at 6587.26 µs

70000

Counts

60000
50000
40000
30000
20000
10000
6400

6600

6800

7000

7200

Time-of-flight, τ (µs)
Figure 7.6: Intensity summed over detectors and pixels, It , from nickel at room temperature as
measured on Pharos. The red triangles are the data, and the black, dashed line shows a fit of a
Gaussian to the peak. The inset shows a closeup of the fit, and the time-of-flight at the vertex is
taken to correspond to the incident energy.

time-of-flight at the vertex of the parabola, [τe ]p∗ , is taken to correspond to the incident energy,
[Ee ]p∗ , as seen in the row of pixels:

[Ee ]p∗ =

Lp∗
[τe ]p∗

(7.15)

where Lp∗ is the distance from the moderator to the row of pixels. Finally, we take the incident
energy, EI to be the average of these over all rows of pixels:
EI = h[Ee ]p i =

p [Ee ]p

mn
p 2

Lp
[τe ]p

(7.16)

1000

60,000
50,000

800

40,000

Counts

30,000

600

20,000
10,000

400

-4

-2

E (meV)

200
10

E (meV)
Figure 7.7: Scattering as a function of the energy transferred to the sample. The dashed black curve
was found with the incident energy determined by sums over both detectors and pixels, the solid red
curve by sums over detectors only. The differences in the inelastic scattering (5+ meV) seem small;
however, they are systematic, and will have an effect on the phonon DOS. In particular, the errors
in the estimation of the incident energy are likely to cause an underestimate of the high energy
cutoff of the phonon spectrum, which will impact our calculations of thermodynamic quantities like
the phonon entropy. At elastic energies (inset) the differences are more apparent; with the solid red
curve clearly centered around 0 meV, and the dashed black curve incorrectly shifted.

where the denominator gives the number of rows of pixels. In this fashion, we get:
EI = 69.517 meV .

(7.17)

In this case, the change in the incident energy is relatively small; however, the effects on the calculated
spectrum are larger. The data here are quite high quality, and larger discrepancies can be expected
for lower quality data sets. Fig. 7.7 shows the scattering as a function of the energy transferred to
the sample, E = EF − EI , with EI determined by sums over detectors and pixels, or by sums over
detectors only.
Since most newer instruments (like ARCS) have monitors, which are probably better suited for
the determination of the incident spectrum, this sort of procedure could be adopted for fine-tuning
our understanding of the instrument geometry. Specifically, it could be used to find the distances

65
from the sample to the pixels, given an incident energy determined via the scattering in the monitors.
To do so, we invert Eq. 7.15, and use the calculated incident energy in order to determine the sample
to pixel distance:
Lp∗ =

2EI
mn

12
[τe ]p∗ .

(7.18)

Although there is probably not enough intensity on Pharos to actually perform this procedure pixel
by pixel (e.g., not summed over detectors), there likely will be on newer instruments (again, like
ARCS).

7.3

Normalization

Frequently, we will collect data on more than one sample, and we will wish to compare these
measurements. In order to do so, we need a way to estimate the number of neutrons that have hit
the sample, whether they were scattered or not.
Normalization of the data is generally quite straight forward. On an instrument with a monitor,
we simply divide the data through by the summed intensity in said monitor. On an instrument
without a monitor, the integrated proton current is often used as a proxy. Fig. 7.8 indicates that
this works reasonably well.

7.4

Transformation to physical coordinates

The data measured measured in a typical neutron scattering experiment conflates information about
the instrument with information about the sample. We wish now to try and separate these two pieces
of information. In this section, we will discuss the conversion of the data to physical coordinates such
as the scattering angle Θ, the energy transfer E, or the momentum transfer Q, that are independent
of the geometry of the instrument.

7.4.1

Rebinning

The most commonly used method of transforming the data from instrument to physical coordinates
is through a technique called rebinning. In particular, the raw data is rebinned in multiple steps;
first into counts as a function of detector, pixel, and energy, IdpE ; then of scattering angle and energy
transfer, IΘE ; and finally of momentum and energy transfer, IQE .g The first two of these steps are
roughly orthogonal, so we might group them and say that the data is rebinned twice.h We will give
g With a sample that is a single crystal, we get I

QE , where the measured wavevector transfer is a vector quantity.
This does not have a large effect on the procedures outlined here, other than to make their explanation more tedious.
h Remember, the data is already binned in hardware.

7e+06

Monitor 1
Monitor 2

6e+06

Counts

5e+06
4e+06
3e+06
2e+06
1e+06

Integrated proton current (A h)
Figure 7.8: Integrated proton current versus integrated counts in the monitors for Pharos. The red
diamonds are counts from monitor 1, which is between the T0 and E0 choppers, and the blue triangles
are from monitor 2, which is between the E0 chopper and the sample. The dashed black lines are
linear fits to the data, constrained to go through 0. The dependence is quite linear, indicating that
the integrated proton current is a good proxy for use in normalization of data.

a 2D example of rebinning, but this technique is also used for higher dimensional data sets.
The data is considered to be a histogram on a regular rectangular grid in the current coordinate
system, with axes x and y. We measure the counts within some rectangle (bin) in the x-y system,
and wish to know how many counts that corresponds to for a bin in the target u-v coordinate
system. To estimate this, we transform each of the 4 corners of our bin in x and y into the u-v
coordinate system, and then connect the transformed corners with straight lines, forming a 4-sided
polygon. This is shown in Fig. 7.9. Each bin in the u-v system contains some fraction of the area
of the polygon, and those bins receive counts in proportion to that fraction. If the transformation
is highly structured — which is usually the case in a physical experiment — this procedure leads to
systematic errors in the placement of counts in the new coordinate system. This is also shown in
Fig. 7.9.

7.4.2

Analytical coordinate transformation

In this section, we propose an alternative method of treating neutron scattering data from a powder
sample as measured on a direct-geometry, time-of-flight, chopper spectrometer. Proof of principle
has been performed using LRMECS data; however, the technique should be generalizable to other
instruments — there is nothing in the method that precludes generalization to single crystals and/or
inverse geometry instruments. The approximations involved are superior to those made in rebinning.
Throughout the section, data from LRMECS are used because the instrument is relatively simple
for a time-of-flight chopper spectrometer. In particular, its detectors are not pixelated, and are all
approximately the same distance from the sample.
The method involves two steps. We first account for the differences between a count in Θ-τ space
and one in Q-E-space. Once this reweighting is complete, we take as input pairs of Q and E and

Figure 7.9: Rebinning from x and y to u and v. The short-dashed black rectangles depict bins in
the x-y coordinate system, and the dotted blue in the u-v system. The solid red arrows show the
transformation of the corners of a bin in the x-y coordinate system to points in the u-v system.
Connecting those transformed corners with straight lines forms the long-dashed, shaded, gray polygon in the u-v system; whereas an exact transformation would yield the black dashed curve, which
shares the same corner points. Each bin in the u-v system contains some fraction of the area of the
gray polygon, and those bins receive counts in proportion to that fraction. Here, then, the rebinned
intensity is being systematically shifted to lower v and to higher u.

68
linearly interpolate to give I(Q, E).i
7.4.2.1

Reweighting

Here, we will consider data only as a function of detector and time-of-flight, because the detectors on
LRMECS are not pixelated. Further, for LRMECS, there is a one-to-one correspondence between a
detector, and a scattering angle so we may write:
Idt = Iθt ,

(7.19)

where θ indexes scattering angle. On LRMECS, some of the detectors are located at negative
scattering angles. For a powder sample, the sign of the scattering angle is irrelevant.j Further, on
LRMECS, it so happens that for each detector at negative scattering angle Θ− there is an identical
detector at Θ+ = |Θ− |. We assume these to be independent measurements of the same quantity,
and thus take:
Iθt =

Iθ − t + Iθ + t

(7.20)

for all detectors that have matching positive- and negative-angle instances.k
The intensity Iθt , when properly normalized, gives the probability of a neutron arriving in the
area of Θ-τ space covered by the detector(s) indexed by θ and t. We wish to know the probability
of finding a neutron arriving in the associated region of Q-E space. For LRMECS, it is (at least
approximately) true that all of the scattering angle and time-of-flight bins cover the same area in
Θ-τ space; however, it is by no means true that they all cover the same area in Q-E space. The first
thing to do, then, is to reweight the probabilities by the areas in the two coordinate systems:
IQE =

|Rθt |
Iθt ,
|RQE |

(7.21)

where |Rθt | gives the area of a bin in Θ-τ space, and |RQE | its area in Q-E space.l We will continue
to use abbreviated notation in the subscripts: Q = Qθt and E = Et .
The area in Θ-τ space is a known quantity. If ∆τ is the width of a time channel, and ∆Θ is the
width of a detector (in scattering angle), then the area in Θ-τ space is given by their product:
|Rθt | = ∆Θ ∆τ ,

(7.22)

i Here, we use I(Q, E) as opposed to I
QE because the the linear interpolation provides a continuous function of
inputs Q and E.
j For a single crystal, the sign matters, and we simply skip these first steps.
k In a sense, the repeated detector angles are a complication rather than a simplification. Without them, there is
no need to combine counts from different detectors.
l Care must be taken here that none of the area in Θ-τ space, |R |, has been accounted for in the detector efficiency
θt
corrections — in particular, in a correction for the solid angle subtended by the pixels.

69
In theory, the area in Q-E space is also easy to find, it is given by:
ZZ
|RQE | =

dQdE .

(7.23)

RQE

The problem is that we don’t know how to write the region of integration, RQE , explicitly. Therefore,
we use:
ZZ

|RQE | =

|J|dΘdτ =
RΘτ

RΘτ

dQ
dΘ

dQ
dτ

dE
dΘ

dE
dτ

dΘdτ ,

(7.24)

where |J| is the determinant of the Jacobian matrix , [87] J, and RΘτ is the region of integration
in φ-T space, which is the rectangle formed by the points (Θ, τ ), (Θ, τ + ∆τ ), (Θ + ∆Θ, τ ), and
(Θ + ∆Θ, τ + ∆τ ). Integrating Eq. 7.24 is feasible, but messy and uninstructive. Thus, the details
are given in Appendix C. Here we present the result of the indefinite integral:
QF
cos(Θ)
−~2 Q3I
|RQE | =
3 cos(Θ) sin2 (Θ)arcsinh
6mn
sin(Θ)
"
# 12 "
#
2
2
QF
Q2F
QF
QF
cos(Θ) + 1
cos(Θ)
cos
(Θ)
−2
QI
QI
Q2I
QI

(7.25)

RΘτ

where in terms of τ and τI , we have
QF
Ld τI
QI
LI (τ − τI )

(7.26)

where Ld is the distance from the sample to the detector.m The vertical bar at the end of Eq. 7.25
indicates that the function must be evaluated over the appropriate region in Θ-τ .
7.4.2.2

Coordinate mapping

Now that we have the area |RQE |, we may reweight our data using Eq. 7.21. What then remains is
to generate a map that connects the axes of the old coordinate system, Θ and τ , to the axes in the
new coordinate system, Q and E. Considering that many analysis codes require data in a certain
format, and that we believe that our data informs us about the space in between the hardware bins
as well as the bins, we would like a general way to get I(Q, E) for any pair of Q and E within the
overall region sampled. Moreover, we would like to do so without any loss of resolution. (Nor with
any gain.) A relatively simple way to do this is to take a pair of Q and E as user input, map them
back to the instrument coordinates, find the smallest simplex which encloses the point of interest,
and linearly interpolate between those points.
m For pixelated detectors, we would use L
dp

70
This method assumes that the data has been preprocessed, so that for any given τ and Θ pair,
there exists only one value of Iτ Θ . (On LRMECS, for example, we took care of this when we combined
counts in detectors at equivalent positive and negative angles.) Given that this preprocessing has
been done, we use:
Θ = arccos

Q2I + Q2F − Q2
2QI QF

(7.27)

to map Q back to Θ and its index θ∗ . Then, we use:
τ=

√I +√d
EI
EF

(7.28)

to map E back to τ , and its index t∗ . Determining t∗ and Θ∗ and finding the enclosing simplex
efficiently is a matter of having the original data in an appropriate data structure, such as a kD-tree
or an R-tree. [88–90]
If the data is highly structured in τ and Θ, care needs to be taken to avoid cases where the
smallest enclosing simplex is not well defined. For example, if the data points form a rectangle
centered about τ and Θ, which three points should be used to form an enclosing simplex? Here,
careless choices might lead to a discontinuous representation of the data.
In point of fact, the geometry and electronics on LRMECS are such that τ and Θ have a regular,
gridded structure.n In this case, the native data structure proves to be the most efficient for mapping
from the user’s Q and E to the instrument coordinates. As such, we implemented a minor variation
on the scheme outlined above. Since τ and Θ form a regular grid, we may transform the Q-E pair
into τ and Θ, and use simple arithmetic to determine the nearest neighboring points:
t∗
θ∗

∗
τ − τ1
floor
∆τ
∗
Θ − Θ1
= floor
∆Θ

(7.29)
(7.30)

where τ1 and Θ1 are the first elements arrays containing time-of-flight and scattering angle, respectively. The four corners of a rectangle that encloses the point of interest, then, are given by (t∗ , θ∗ ),
(t∗ , θ∗ + 1), (t∗ + 1, θ∗ ), and (t∗ + 1, θ∗ + 1).
The astute reader will have noticed that linearly interpolating over a rectangle is not strictly
possible.o One solution would be to quadratically interpolate. Instead, we calculate the mean value
of the counts at the four corners of the rectangle, and take that to be the number of counts at the
center of the rectangle. We then interpolate over the triangles, as is shown in Fig.7.10.

Fig. 7.11

n Actually, the data are gridded slightly irregularly, but the required modifications of the algorithms are not very
instructive.
o Simply because three points define a plane.

(t*, θ∗+1)

(t*+1,θ∗+1)

(Q*,E*)

(τ∗,Θ∗)
(t*,θ∗)

(t*+1,θ∗)

Figure 7.10: Interpolation for LRMECS data. The point of interest is given in red, and it is shown
in the Θ-τ and the E-Q coordinate systems on the left and the right respectively, with the longdashed, red arrow representing the transformation. The bins from the hardware are shown by the
short-dashed, black rectangle in Θ-τ , and the short-dashed, black curvy figure in Q-E. The two
solid gray lines cross at the center of the rectangle, and we assign the mean of the counts at the
corners of the rectangle to this point. The yellow, shaded region represents the linear interpolation
surface which gives the value of the intensity for the point of interest.
shows nickel data, measured on LRMECS, that has been analyzed with both the rebinning and the
analytical coordinate transformation methods described above.

7.5

Removal of background scattering

Occasionally, it is thought that a measurement of the background is unnecessary because the sample
container and mount will not scatter enough neutrons to merit it. On the contrary, it is not only the
sample container and mount that contribute to the background scattering, but also the instrument
itself.
For this reason, it is always critical to make a measurement of the background. Assuming such
a measurement, I b has been made, we simply subtract it from the measurement of the sample, I s :
I(Q, E) = I s (Q, E) − f I b (Q, E) .

(7.31)

As some fraction of the neutrons which were scattered by the instrument and sample environment are
now scattered by the sample it is common to subtract only a fraction, f of the measured background.
Usually, samples are designed to scatter between 10% and 20% of neutrons; thus, we normally have

Figure 7.11: Comparison of I(Q, E) for nickel as measured on LRMECS and as calculated with with
the analytical coordinate transformation on the left and with rebinning on the right. In so much as it
was possible, the two analyses were performed so as to be comparable. For example, the plotted data
have the same numbers of points in Q and E. The plot on the left should be considered no more than
proof-of-principle, and there are noticeable defects in the analysis. Namely, the calculation of the
incident energy was not quite correct, and the resulting spectrum is skewed off center. Regardless,
the plot on the right shows a great deal more streaking along lines of constant angle (See Fig. 6.5.
For LRMECS, this corresponds to counts in single detector).
0.8 < f < 1.0.
Although the removal of background scattering could be performed with the raw data; it is
frequently only done at this stage in the analysis process. A disadvantage of this strategy is that
it might compound certain types of systematic errors in the data analysis. An advantage is that
the intensity, up to now, has always remained positive, which can aid in the speed and ease of the
analysis process.
Another variation is to remove separately the measured background I b and the time-of-flightindependent background.p Changes in the background may occur on time scales similar to a single
measurement. For example, the neighboring instrument may be running while I s is being measured,
but not during measurement of I b . The correction for this is often performed even before the
corrections for detector efficiencies, and it entails selecting a region in time of flight where the counts
are only ambient noise, finding the average number of counts in this region, and subtracting that
from the data. The rest of the removal of background scattering remains the same.
p This is frequently called the time-independent background; however, that is a gross misnomer. It is precisely the

time dependence of the background for which it attempts to correct; albeit for times that are on a longer scale than
the time-of-flight.

Chapter 8

Processing S(Q,E)
Once S(Q, E) is determined to the best of our abilities,a we frequently wish to extract further information about the structure or dynamics of our sample. In this chapter, we present methods for
determining the phonon DOS and the lattice parameter from the scattering. We also consider methods for characterizing phonon linewidths and the interatomic potential, given a phonon spectrum.

8.1

Finding the phonon DOS

There are two commonly used methods for extracting the phonon spectrum from inelastic neutron
scattering data. Both methods are designed to separate 1-phonon from multiphonon scattering,
which is the simultaneous creation or annihilation of many phonons by 1 neutron. This is to be
distinguished from multiple scattering, in which a neutron is scattered by a phonon (or phonons) and
then at some later time is scattered again. The two are qualitatively different, as multiphonon scattering is a property of the sample alone, but multiple scattering may involve the sample environment
and the instrument.
The first method presented is called the Fourier-log method and corrects only for the multiphonon scattering. The second is an iterative procedure, and it has been modified to also correct
approximately for multiply scattered neutrons.

8.1.1

Fourier-log method

Here, we assume that the measured scattering I(Q, E), is given by the dynamic structure factor,
S(Q, E), with the energy dependent portion convolvedb with some resolution function Z(Q, E). We
a Since we have not yet corrected for multiple scattering, we are technically still dealing with I(Q, E). [91–93] We
will only concern ourselves with this distinction in § 8.1.2, where we will actually attempt to make such a correction.
b This must really be a convolution — as opposed to some other integral transform — in order for this method
to work. In particular, this means that the resolution function must be independent of the energy transferred to the
sample, E 0 ; e.g., a Gaussian with σ = 2 meV, as opposed to a Gaussian with σ = √1 0 .

74
use Eq. 5.15 to express this as follows:
I(Q, E) ∝

oi
Z(Q, E) ∗ e−2W FT exp hÛV̂i ,

(8.1)

where we have taken advantage of the fact that S(Q, E) is proportional to FT hÛV̂i . We may
pull the Debye-Waller factor through, and then take the inverse Fourier transform of both sides,
converting the convolution into a product:
FT -1{I(Q, E)}

oo
∝ e−2W FT -1 Z(Q, E) ∗ FT exp hÛV̂i
n n
oo
∝ e−2W FT -1{Z(Q, E)}FT -1 FT exp hÛV̂i
∝ e−2W FT -1{Z(Q, E)} exp hÛV̂i .

(8.2)

We now take the logarithm of both sides:
log FT -1{I(Q, E)} ∼

−2W + log FT -1{Z(Q, E)} + hÛV̂i .

(8.3)

We subtract log FT -1{Z(Q, E)} − 2W from both sides, and take the Fourier transform:
FT hÛV̂i

FT -1{I(Q, E)}
FT log
2W
FT -1{Z(Q, E)}

FT -1{I(Q, E)}
FT log
+ 2W δ(E) .
FT -1{Z(Q, E)}

(8.4)

In the case that Z(Q, E) = Z(Q)Z(E) = e2W Z(E) — that is, the Q dependent part of the resolution
function is determined only by the Debye-Waller factor — we have:
FT hÛV̂i

-1

FT {I(Q, E)}
FT log
+ 2W δ(E) .
2W
FT -1{Z(E)}

(8.5)

The advantage of this method is that it is direct, and that for sufficiently simplec resolution
functions, you may correct for Z(Q, E) as you find the DOS. This is not the case for time-of-flight
chopper spectrometers; however, this is the type of procedure used for the nuclear resonant inelastic
x-ray scattering data presented in Chapter 12.

8.1.2

Iterative method, with correction for multiple scattering

Here, we wish to deal with only inelastic scattering, so we must first remove any elastic scattering
from the data. This is done by assuming an E 2 energy dependence of the phonon DOS, which leads
c Read, energy transfer independent.

Figure 8.1: A typical determination of the multiphonon-scattering using data that has been binned
into angle banks. The green dotted line labeled “Raw-Data” shows I(E) as determined by experiment. The solid black line labeled “Fitted 1-5 Phonon Scattering” is the best fit to the data. (This
should be compared to the fit in Fig. 8.4.) Figure taken from Swan-Wood. [94]

to inelastic scattering at low energies of the following form:
I(E) =

ζ0 E
1 − exp (−βE)

(8.6)

where the constant ζ0 is determined by matching the function to the experimentally measured
scattering, just past the edges of the elastic peak. The peak, then, is replaced with a scattering in
the form of Eq. 8.6.
Once the elastic scattering has been removed we analyze the remaining, inelastic scattering to
determine the phonon DOS. This method of finding the DOS is iterative, and it is the approach we
have used for the neutron measurements of aluminum and nickel presented herein. The process has
been described previously,[92, 95] but we present some modifications here. Namely, in the past, the
correction has either been used for incoherent scattering measured on a triple-axis spectrometer at
a single value of Q,[92] or for time-of-flight neutron scattering data, where the data has been binned
into angle-banks. [13, 14, 22–24, 54, 95–99] Fig. 8.1 shows a typical best fit to data obtained in this
fashion. Here, the correction is performed with data that is a function of Q, not angle bank; and an
approximate correction for multiple scattering has also been added.

76
Corrections for multiple scattering have been performed in many ways, from subtracting a constant from the data, [92] to full Monte-Carlo simulations. [100] At high temperatures, the former
does not account for the slope of the scattering past the high energy cutoff of the phonon DOS.
The latter can be computationally intensive, and requires details of the shape of the sample and the
instrument. Here we take an approach of intermediate complexity.
In § 5.2.4, we developed an expression for the multiphonon scattering, which showed that the
P-phonon scattering can be written as a convolution of the 1- and the (P − 1)-phonon scattering.
Briefly, we consider the physical meaning of this. Let ΥP (E) be the probability of an event where
the neutron loses energy E to P phonons. If all of the 1-phonon scattering processes are statistically
independent, and if a 2-phonon process is made up of two 1-phonon processes, then we have the
probability of a 2-phonon process where a neutron loses E 0 to one phonon and E 00 to another is
given by:
Υ2 (E 0 , E 00 ) = Υ1 (E 0 )Υ1 (E 00 ) .

(8.7)

In an experiment, however, we only see the resulting neutron energy loss, E = E 0 + E 00 , which gives
E 00 = E − E 0 . There may be many combinations of two 1-phonon processes that produce this energy
loss, and the intensity that we see will correspond to a sum over all of these possibilities:
Υ2 (E) =

Υ1 (E 0 )Υ1 (E − E 0 ) .

(8.8)

In the limit of a continuous energy expression, we have a convolution. For P-phonons, this is
expressed by the recursion relation:
ΥP (E) =

Υ1 (E 0 )ΥP−1 (E − E 0 )dE 0 .

(8.9)

The point of this discussion is that we have actually said very little about the underlying scattering
processes — simply that we believe them to be statistically independent. As such, we expect the
same reasoning to apply to multiply scattered neutrons. That is, for both multiple scattering and
multiphonon scattering, a P-phonon scattering profile involves a convolution of the single-phonon
scattering profiles with the (P − 1)-phonon profile.
Additionally, the P-phonon probability function for multiple scattering will have position and
momentum dependencies. (These do not appear for multiphonon scattering processes.) Sears,
et al., [92] argue that the integrals for multiple scattering are related to those for the multiphonon
scattering through slowly varying functions of Q and E. Here we take these functions to be constants,
ζP . In essence, we make the approximation that the position and momentum dependencies can be

77
factored out. Thus:
I(Q, E) = N

" ∞

(1 + ζP )SP (Q, E)

(8.10)

P=1

where I(Q, E) is the experimentally-determined total scattering (including multiple scattering),
SP (Q, E) is the P-phonon scattering (both creation and annihilation), and N 0 is a normalization
constant.d (When we stripped the elastic peak from the data, the dominant multiple elastic scattering is removed, so the index P in Eq. 8.10 starts at 1 rather than 0.)
We now make the incoherent approximation: [83]
coh
SP
(Q, E) =

σ coh inc
S (Q, E) .
σ inc P

(8.11)

Note that this does not really apply to the 1-phonon scattering; nevertheless, it is common to apply
this equation to the 1-phonon terms as well as all higher orders. The last step in our procedure will
be to assess any error this has introduced into our analysis. This allows us to write:
"X
coh
inc
I(Q, E) = N 0 1 + inc
(1 + ζP )SP
(Q, E) .

(8.12)

P=1

Our next assumption is that ζP = ζms
for all P ≥ 2, where ζms
is a single constant that relates

the multiple scattering to the multiphonon scattering. Thus:

I(Q, E) = N

σ coh
1 + inc

(1 + ζ1 )S1inc (Q, E) +

inc
(1 + ζms
)SP
(Q, E)

(8.13)

P=2

Since the multiphonon scattering drops off rapidly with increasing P, this approximation will only
have a small effect on our results. Collecting some terms into the normalization constant, we have:
inc
(Q, E) ,
I(Q, E) = N S1inc (Q, E) + (1 + ζms )S2+

(8.14)

)/(1+
where N = N 0 1 + σ coh /σ inc (1+ζ1 ) is the new normalization constant, and 1+ζms ≡ (1+ζms
ζ1 ). Also, for notational convenience, we have introduced:
Sj+ (Q, E) ≡

SP (Q, E) .

(8.15)

P=j

The reason that the 1-phonon scattering has a different constant than the higher order scattering
is that there are much stronger kinematic restrictions on the 1-phonon scattering than on the higher
d Again, we note that I(Q, E) is distinct from the scattering function, S(Q, E), which does not include multiple
scattering.

Figure 8.2: S inc (Q, E) for nickel at 300 K calculated from the phonon DOS. This includes multiphonon processes of all orders, but not multiple scattering. The white polygon borders the region
sampled if the incident energy is roughly 70 meV, and if there is no multiple scattering. The 1phonon scattering — including a transverse and a longitudinal peak — can be seen on top of the
multiphonon background at −40 < E < 40. It is quite dim for Q < 10, becomes bright red and
white for 10 < Q < 30, then fades out again at Q > 30. Clearly the ratio of 1-phonon to multiphonon scattering decreases dramatically with increasing Q. If we allow for multiple scattering, a
neutron might be involved in processes outside the white polygon such that sums over the energy
and momentum transferred would allow us to detect it. This could change the ratio of 1-phonon
to multiphonon scattering as seen in an experiment from that expected when considering only the
region in the white polygon.

orders. That is, a neutron may not lose more energy than it has in a single scattering event;
however, a neutron may gain energy in a first event, and then lose this energy in addition to its
initial energy in a subsequent event. This is shown in Fig. 8.2, where the region outlined in white
are the total energy and momentum transfers that we are capable of measuring. Processes from
within the outlined region may scatter a neutron out of our detection range; but, another process
may return that neutron to the outlined region, where we detect it.
For a cubic crystal, and a fixed value of ζms , we can now find the DOS by solving Eq. 8.14 in the
manner described by Sears, et al. [92] Since we do not know the value of ζms a-priori, we generate a
list of possible values, and solve for the DOS at each one. For nickel, as seen in Chapter 11, values
of ζms between 0.0 and 2.0 were tested. It then remains to select the best DOS from those generated
with the different ζms . This is done by minimizing a penalty function constructed to find the DOS

Figure 8.3: Penalty functions for nickel at 300 K, as defined in the text. The dash-dotted line (1)
relates to the overall fit, the dotted line (2) relates to the noise near the incident energy, and the
dashed line (3) relates to the slope near the incident energy. The solid line is the sum of these three
contributions (offset).

that produces S(E) that best satisfies the following conditions:
(1)
X
1 X
inc
I(Q, E) =
S1inc (Q, E) + (1 + ζms )S2+
(Q, E) ,

(8.16)

where the sum over Q is basically the application of the incoherent approximation to the data.
(Details of the summation over Q are given in Appendix D.) With simplified notation:
I(E)

inc
S1inc (E) + (1 + ζms )S2+
(E) .

(8.17)

(2) The experimental noise at energy transfers near the incident energy oscillates about the value
inc
(1 + ζms )S2+
(E).

(3) At energy transfers near the incident energy, the slope of a linear fit to the experimental noise

I(E)/N

I(E)

Sinc(E)

(1+ζms)Sinc2+(E)
Sinc2+(E)
ζmsSinc2+(E)
-60

-40

-20

E (meV)

Figure 8.4: Best fit to scattering for nickel at 300 K. The triangles are the normalized experimental
inc
scattering, I(E)/N . The solid line shows the fit, S1inc (E) + (1 + ζms )S2+
(E). The dashed line is the
inc
inc
multiple scattering, ζms S2+ (E). The dash-dotted line is the multiphonon scattering, S2+
(E). The
inc
dotted line is the sum, (1 + ζms )S2+ (E). The point at E = 0 is not used in the fitting procedure.

inc
matches the slope of a linear fit to (1 + ζms )S2+
(E).

These three criteria are based on our experiences trying to fit data generated from known phonon
spectra with known values of ζms . These three criteria are correlated, but are not identical. In
particular, (1) will be well satisfied whenever the DOS has no cutoff, even though this is undesirable.
On the other hand, if there is no cutoff, the noise at energy transfers near the incident energy will
not oscillate about the sum of the multiphonon and multiple scattering contributions — e.g. (2)
will not be satisfied. This, in turn, does not imply that (3) is not satisfied. If the two linear fits are
parallel, (3) is satisfied, regardless of (2). For nickel at 300 K, these three contributions and their
sum are shown in Fig. 8.3.
Fig. 8.4 shows the best fit to the normalized scattering, I(E)/N for nickel at 300 K, which had
ζms = 0.6. Finding a DOS from experimental scattering always involves some art; this procedure
attempts to limit that art to the construction of a suitable penalty function. The phonon DOS

81
obtained this way were fit with a Born–von Kármán model, from which all phonon contributions
to the scattering, both coherent and incoherent, were calculated. With these results, and with the
final value for ζms , the calculation was checked against the measured scattering, which showed that
the effects of assuming all the scattering to be incoherent were negligible. The procedure has been
applied successfully to cubic metals, including nickel and aluminum as studied in this work.

8.2

Shift and linewidth analysis of the DOS

As mentioned in § 3.6, the phonons in a metallic solid are expected undergo a shift to lower energies,
and to broaden, with increasing temperature. The shift might be approximated as a constant
multiplier, ∆s , applied to all phonon energies E:
E → ∆s E .

(8.18)

Broadening of the phonons is expected to take the form of a damped harmonic oscillator function,
B = B(Q, E 0 , E) centered about energy E 0 :
B(Q, E 0 , E) =

πQE 0

E0
E 2
+ Q12
E − E0

(8.19)

Note that the quality factor, Q, is related to the full width at half maximum, 2Γ of the phonon
peaks as follows:
2Γ

(8.20)

Using Eqs. 8.18 and 8.19, we see that the high temperature phonon DOS may be approximated as
a function of the low temperature DOS, with only two free parameters, ∆s and Q:
gT (E) = B

g0 (∆ E ) =

B(Q, E 0 , E)g0 (∆s E 0 )dE 0 ,

(8.21)

where gT is the phonon DOS at temperature T and g0 is the zero temperature DOS. The integral
transform of the DOS is very similar to a convolution and we denote it with
in an experiment is Z

. What we measure

g0 , but assuming that there is no broadening at 0 K, Eq. 8.21 gives

g0 ). Thus, the equation is valid for use with the experimentally determined phonon DOS

in so far as we believe the following:

B≈B

(8.22)

82
0.016

Z B δ(10)
Z B δ(50)
B Z δ(10)
B Z δ(50)

0.012
0.01

Q = 4.0
Z B δ(10)
Z B δ(50)
B Z δ(10)
B Z δ(50)

0.01
P(E) (unitless)

0.014

P(E) (unitless)

0.012

Q = 40.0

0.008
0.006

0.008
0.006
0.004

0.004
0.002

0.002

60
E (meV)

100

120

100

120

E (meV)

Figure 8.5: Solid gray lines show Z B for g(E) = δ(E = 10 meV) = δ(10) and g(E) = δ(E =
50 meV) = δ(50). B Z for g(E) = δ(10) and g(E) = δ(50) are shown by the dashed red and
the dotted blue curves, respectively. The left-hand plot was produced with a relatively high quality
factor, Q = 40.0, such that most of the broadening is given by Z. In this case B Z and Z B
are almost identical. In the right-hand plot, Q = 4.0, which is representative of some of the lower
quality phonon spectra presented here. Here, the differences between B Z and Z B are just
beginning to be noticeable.

This is approximately true when Q is large. This can be seen in Fig. 8.5, which shows B

Z and

B for Q ∈ {40, 4}.
At any temperature, the best Q and ∆s for the experimental DOS may be determined through a

least squares algorithm. At least within reasonable ranges for the parameters — 0.7 ≤ ∆s ≤ 1.1 and
0.1 ≤ Q ≤ 1000 — the problem has only one minimum. A representative plot of the least squares
error is given in Fig. 8.6. As a result, we may thus easily find the optimal shift and the quality
factor.
The shifts ∆s may be compared to the ratios of the mean phonon energies, hEiT /hEi0 as determined from the DOS:
EgT dE
hEiT
= R
hEi0
Eg0 dE

(8.23)

Experimentally, we do not have access to 0 K, so we use in its place the lowest temperatures we can
measure. For neutron measurements, this frequently works out to be about 10 K.

8.3

Born–von Kármán fits to the DOS

Once the phonon DOS has been determined, we may try and learn about the interatomic forces
which give rise to it. A relatively simple way to do this is with a Born–von Kármán (BvK) model of

Figure 8.6: Close-up of least squares error as a function of Q and ∆s for nickel using Eq. 8.21 with
300 K the lower and 1275 K the higher temperature. The lighter red corresponds to larger errors
and the darker blue to smaller, the optimum values being roughly Q = 5.1 and ∆s = 0.92. The
same trends continue for (at least) 0.1 ≤ Q ≤ 1000.0 and 0.7 ≤ ∆s ≤ 1.1, thus, we may easily take
the path of steepest decent to the optima.

the lattice dynamics.e The theory underlying these models was described in § 3.4.2. Here we discuss
methods for fitting these models to real data and simple ways of interpreting the results.

8.3.1

Fitting real data

Although the statistical noise of a neutron measurement of the DOS can make fitting difficult,
neither this, nor any of the other problems quite generally associated with modeling of data, will
be the subject here. Rather, we focus on the fact that on its own, a BvK model cannot reproduce
instrument or anharmonic broadening of the phonon spectrum, and on means for getting around
this shortcoming. Here, we present both derivative-free and gradient methods for fitting the DOS.
8.3.1.1

Derivative-free methods

If we are willing to forgo the use of derivatives, adjusting a BvK model to account for instrument and
anharmonic broadening of the phonons is quite simple. The penalty function, Y , for optimization
is constructed by direct comparison of the experimentally determined phonon DOS, g exp and the
computed DOS, which is found through an integral transformation of the DOS from BvK, g bvk .
e These models are also useful for the so-called neutron-weight correction[23, 98, 101] that is required when interpreting the scattering from a polyatomic sample.

84
Specifically:

exp
g (E) − Z

Y =

g bvk (E)

dE .

(8.24)

In practice, the experimental and computed DOS are frequently manipulated as histograms, in which
case the integral is replaced with a sum.
8.3.1.2

Gradient methods

If we wish to cut down on function evaluations and use a gradient in our optimization, we write the
dynamical matrix as a sum of matrices Dk multiplied by independent force constants Kk , as was
briefly discussed in the text around Eq. 3.69:
D(q) =

Kk [Dk (q)] .

(8.25)

The force constants, Kk , are actually the parameters to be varied, whereas the matrices Dk will
remain unchanged.f
We may exploit this structure with a result from linear algebra. If the dynamical matrix has
eigenvectors [b (q)] and eigenvalues Λb (q) = ωb2 (q), then the derivative of the eigenvalue with respect
to a small change in the force constants is given by[102]:
∂Λb
∂Kk

= T
b Dk b ,

(8.26)

where we have dropped the q dependencies for clarity.
In theory, we could now take the derivative of the DOS with respect to a change in an eigenvalue,
and the penalty with respect to a change in the DOS, apply the chain rule, and have the derivative
of the penalty function with respect to a change in a force constant:
∂Y
∂Kk

∂Y
∂g bvk (E)

∂g bvk (E)
∂Λbvk

∂Λbvk
∂Kk

(8.27)

where the first partial derivative on the right can be found by inspecting Eq. 8.24. The second partial
derivative on the right can be found by differentiating Eq. 3.71, which gives the density of states as
a sum over delta functions. In practice, however, the derivative of a delta-function is poorly defined;
thus, using Eq. 8.27 successfully can be quite difficult.
An alternative is to think of the experimentally determined phonon DOS as a probability distribution, and to draw a uniform sample of squared frequencies that follow it. This still leaves us with
f The number of independent constants K and the shapes of the matrices D are largely determined by the
point-group symmetries of the lattice. A method for numerically determining these constraints on the force constants
is outlined in Appendix E.

85
the problem of modeling instrument and anharmonic broadening of the spectrum, and two methods
of handling this follow.
The first method is to undo the broadening of the experimentally determined phonon DOS. That
is, we assume knowledge of Z and B (or, at least, of Z
g 0exp (E) = (Z

B)-1 , so that:

B)-1 g exp (E) .

The problem with this sort of technique is that B
fixed Z

B), and we then find (Z

(8.28)

Z is usually very poorly conditioned, i.e., for

B, there exist many choices of g 0exp (E) that satisfy Eq. 8.28 equally well. In so much

as the integral transformation represented by

is like a convolution, this is no surprise. One way

around this difficulty is with regularization techniques such as Tikhonov regularization.[103]
Assuming that our deconvolution is successful,g we may then draw a uniform sample of experimentally determined eigenvalues, Λexp from the distribution (Z

B)-1 g exp (E), and construct the

following penalty function:
Y =

X exp
2
Λb (q) − Λbvk
b (q)

(8.29)

q,b

Then the change in the penalty with a change in a force constant is given by:
∂Y
∂Kk

bvk
∂Λb
∂Y
bvk
∂Kk
∂Λ
Xb
T
bvk
Λb (q) − Λexp
b (q) b Dk b .

(8.30)

q,b

Note that in an experiment we may not actually know to which branch b and momentum transfer q
a particular eigenvalue Λexp belongs. We circumvent this difficulty by sorting both the experimental
and simulated eigenvalues before comparison. Let there be an index i such that Λi < Λi+1 , and such
that each i corresponds to one and only one pair (q , b), then we may write:
∂Y
∂Kk

X
T
Λbvk
− Λexp
i D k i .

(8.31)

In the second method, we apply the broadening Z

B to the eigenvalues determined with the

BvK model. The advantage of this method is that it allows use of a gradient, but requires no matrix
inversion. Because the method is somewhat involved, we give the steps in bullet form and motivate
them below. We:
• find the eigenvalue Λbvk
b (q) from simulation
• assume that the square root of the eigenvalue ωbbvk (q) =
g Actually, an inverse integral transform.

Λbvk
b (q) represents the argument

86
to a delta-function, δ ωbbvk (q)
• apply the instrument and anharmonic broadening to the delta function to generate a distribu
tion Z B δ ωbbvk (q)
bvk
• draw from the distribution a sample with elements ωbj
(q) (Note the new index, j)
i2
bvk
• square the elements of the sample to get Λbvk
(q)
(q)
bj
bj

The complexity of the method stems almost entirely from the need to apply Z
simulated results. Firstly, we must apply the broadening Z

B to the

B to a specific value rather than to a

distribution. As mentioned before, this is necessary because we taking a derivative of the DOS with
respect to a change in an eigenvalue is non-trivial. Secondly, the broadening Z

B is known for a

frequency ω, not for an eigenvalue ω 2 . As a result, we have to take a square root before applying
the broadening. Squaring things afterward simplifies finding the change in the penalty with respect
to a change in a force constant.
We may now construct a penalty function by comparing Λbvk
bj (q) with a sample drawn from the
experimentally determined phonon DOS, Λexp
bj (q):
Y =

i2
X h exp
Λbj (q) − Λbvk
bj (q)

(8.32)

q,b,j

The change in the penalty with respect to a change in a force constant is given by:
∂Y
∂Kk

∂Y
∂Λbvk
bj

∂Λbvk
bj
∂Λbvk

∂Λbvk
∂Kk

(8.33)

As in the previous method, we assume an index i such that Λi < Λi+1 , and such that each i
corresponds to one and only one pair (q , b). We also assume an index l such that Λl < Λl+1 , and
such that each i corresponds to one and only one triplet (q , b , j). We may then write:
∂Y
∂Kk

bvk bvk
∂Y
∂Λl
∂Λi
bvk
bvk
∂Kk
∂Λl
∂Λi
bvk
X
∂Λl
= 2
Λbvk
− Λexp
T
i Dk i .
bvk
∂Λ
l,i

(8.34)

It remains to determine the derivative of one of the resample eigenvalues Λl with respect to a
change in one of the Λi . One way to do this is to evaluate it numerically. In order to be able to
compare the simulation to the experiment, the Λi must be quite densely packed; therefore:
∂Λbvk
∂Λbvk

bvk
Λbvk
l+1 − Λl
bvk
Λbvk
i+1 − Λi

(8.35)

87
8.3.1.3

Comparison of methods

The fitting methods presented fall into two major classes, those that used analytical derivatives and
those that do not. The advantages of the former are directness and the fact that only one function
evaluation is required per optimization step. The disadvantage is that analytical derivatives can be
complicated, costly or even impossible determine. In the past, only the derivative-free methods have
been used for optimizing BvK models to a phonon DOS.
More specifically, for a BvK model, a gradient based technique requires determination of the
eigenvectors as well as the eigenvalues. In practice, the extra time spent determining the eigenvectors
is made up by the decreased number of function evaluations per optimization step. Additionally, if
we are trying to do an optimization of S(Q, E) rather than the DOS, or if we have a polyatomic
system, we have to calculate the eigenvectors anyways.
Regardless of the choice of methods, the problem of fitting force constants to a DOS is not
convex.[65] Any procedure, then, only guarantees a local optimum. It is currently too computationally intensive to search the entire force constant parameter space, which may include well over
ten parameters. That said, simulated annealing, genetic algorithms, and other global optimization
chicanery will probably yield better results than simply running any of these local optimizations on
its own.
The last point we make is about the broadening of the spectrum, which we have done by transforming the DOS, and also by transforming each phonon frequency separately. Fig. 8.7 shows that
the two methods yield similar, but by no means identical results. At least part of the difference is
related to the operation of binning the eigenvalues into a density of states. Although transforming the DOS is somewhat more intuitive than broadening each phonon frequency individually, the
latter is probably more reflective of the underlying physics. For example, in the case of a neutron
experiment, a single neutron will be scattered by a particular phonon with some linewidth.

8.3.2

Longitudinal and transverse force constants

Each force constant tensor connects two atoms with a bond , and we will frequently want to look
at the force projected onto the bond direction. Unlike the generalized tensorial force constants,
the longitudinal force has a clear physical interpretation — it tells us about how strongly the two
bonded atoms attract each other. Moreover, from model to model there is often great variation in
the generalized constants; however, if the models are at all realistic, the longitudinal force constants
are very likely to be similar. (See Table 9.3, in § 9.5, for example.) The transverse force constants
are less robust to changes in model, but are also sometimes of interest.
Given an interatomic force constant tensor, K, and the bond with equilibrium separation, x, to

0.08
0.07

g(E) (1/meV)

0.06
0.05
0.04
0.03
0.02
0.01

20
30
E (meV)

Figure 8.7: The solid, light gray line in the background shows a phonon DOS, g, from a BvK model
of aluminum with no anharmonic broadening. The blue dashed line is Z B g. The solid red line
is also from the same BvK model but with the eigenvalues have been broadened as in the second
method presented in § 8.3.1.2. In order to focus on the anharmonic broadening, the calculations
were performed assuming an instrument with perfect resolution, i.e., Z = 1. The differences in the
2 methods of broadening the DOS are clearly visible in the taller peaks; but will have only a small
impact on thermodynamic calculations that depend on the phonon spectrum. Also, at least some
of the disparity between the spectra has to do with the binning of the phonon frequencies that was
done to determine their density.

which it belongs, we may find the longitudinal force constant as follows:
K=

xT Kx
xT x

(8.36)

Since there is an entire 2D subspace that is orthogonal to the bond direction, finding a transverse
force constant is more complicated. Often, we may skirt this difficulty by comparing K to the
eigenvalues of K. If the forces are axial, then one of the eigenvectors will be in the longitudinal
direction and its eigenvalue will match K. We then take the remaining D −1 (two, in 3D) eigenvalues
to be transverse, and average them to obtain a single transverse force constant.

Figure 8.8: Indexed diffraction patterns from polycrystalline nickel at room temperature with incident neutron energies of 30, 50, and 70 meV. The corrected intensity, I(Θ, E), was summed over
energies from −2.5 to 2.5 meV. Each peak can be fit and, the vertex of the fit used in Bragg’s Law
to determine a lattice parameter. The resulting parameters for the 70 meV pattern were used in
a Nelson-Riley plot shown in Fig. 8.9. As an aside, the fact that the ratios of the peak heights
are similar for the different incident energies indicates that the nickel sample was not textured
crystallographically.

8.4

Finding the lattice parameter

Along with the inelastic scattering which yields the phonon spectrum, a measurement of a polycrystalline sample on an time-of-flight chopper spectrometer also yields Bragg scattering.h This can be
seen in Fig. 7.11, where the bumps in the red line at E = 0 are actually Bragg peaks. The quality
of the diffraction patterns is much less than those from dedicated neutron or x-ray diffractometers.
Nevertheless, it is still possible to obtain reasonable estimates of quantities such as lattice parameters, given a known structure. A relatively simple method for finding the lattice parameter follows.

First, we wish to separate the elastic and the inelastic scattering. To do so, we simply sum over
a range of energies that we consider to be in the elastic region, for example, from −2.5 to 2.5 meV:

I (Θ) =

+2.5

I(Θ, E) .

E=−2.5
h This requires that the sample is a coherent scatterer of neutrons.

(8.37)

90
I el (Θ), then, is our diffraction pattern, like the ones shown in Fig.8.8.
Once we have the diffraction pattern, we may fit the peaks with parabolae, or some other function,
and use the Θ-coordinate at the vertex in Bragg’s law:
(h2 + k 2 + l2 )
a(Θ) =
sin (Θ/2)

(8.38)

where λI is the wavelength of the incident (and outgoing, since it is elastic scattering) neutrons, and
h, k, l are the Miller indices [50, 52, 104] for the peak.
Unfortunately, the different peaks are very likely to yield different lattice parameters. Thusfar,
the largest errors in the estimation of the lattice parameter have come from displacements of the
sample from its assumed position.i To some extent, we may correct for this by noting that for
scattering of Θ = 180◦ , the displacements of the sample have no effect on the peak position. In fact,
the relative error in the lattice parameter should go linearly to zero as follows:[104]
cos2 (Θ/2)
∆a
sin (Θ/2)

(8.39)

Nelson and Riley,[104, 105] suggested a slightly modified relationship:
cos2 (Θ/2)
cos2 (Θ/2)
∆a
+2
sin (Θ/2)

(8.40)

which is the form used in Fig. 8.9 in order to find lattice parameters for nickel at 10, 300, 575, 875,
and 1275 K.
A more robust method of diffraction analysis would be to create a parametrized model of the sample and its environment that includes sample positioning, lattice parameters, crystallite orientation
and strain distributions, etc. and to then perform an optimization of the model to the data. This
sort of procedure, when applied to diffraction patterns, is commonly called Rietveld refinement.[106]
It is the technique of choice for diffraction data taken on dedicated neutron or x-ray diffractometers;
however, it is rarely (if ever) applied to diffraction data from chopper spectrometers.

i The displacements may also be due simply to the non-zero thickness of the sample.

1275 K, a = 3.585 Å
875 K, a = 3.559 Å
575 K, a = 3.540 Å
300 K, a = 3.521 Å
10 K, a = 3.513 Å

3.64
3.62
a(Θ) (Å)

3.6
3.58
3.56
3.54
3.52
3.5

0.5

1.5

2.5

3.5

cos (Θ/2)/sin(Θ/2)+2cos (Θ/2)/Θ
Figure 8.9: Nelson Riley plot for nickel from data measured at 10, 300, 575, 875 and 1275 K on
Pharos. The corrected data, I(Θ, E), were summed over energies from −2.5 to 2.5 meV to get the
diffraction patterns. (Shown in Fig. 11.1) The peaks were fit to parabolas, and the Θ coordinates
of the vertices were used as Θ for the markers. The nominal lattice parameter for each vertex was
determined using Bragg’s law. The lines are linear fits to the markers at a given temperature. Error
due to displacement of the sample goes to zero at Θ = 180◦ , thus, the a(Θ)-intercept represents the
best estimate of the lattice parameter. The 10 K was taken in a displex refrigerator, at 300 K on a
sample stick, and for the remaining temperatures in a furnace. For the three furnace measurements,
the sample was not moved, and the similar slopes of the lines reflect this fact.

Part III

Phonons in FCC Metals at
Elevated Temperatures
Inelastic neutron scattering measurements of polycrystalline aluminum, lead, and nickel
are presented here, with specific attention paid to high temperature effects on phonons.
The three sections on analysis and computation are quite similar, with the following
exceptions: § 10.3.3 describes problems only relevant to the measurement of lead and
§,11.3.6 describes spin polarized electronic sctructure calculations relevant only to nickel.

Chapter 9

Aluminum
9.1

Introduction

On account of its abundance and the favorable thermal and mechanical properties of its alloys,
aluminum is one of the most widely used metals for industrial and engineering applications. Its
melting temperature of 933 K is relatively low, and the thermodynamic stability of aluminum at
elevated temperatures is of technological and scientific importance. Because of its simple electronic
structure, aluminum metal is frequently used as a test case for theoretical models of crystals and
their thermodynamics. [15, 43, 44, 66, 107–112]
Aluminum is non-magnetic, so the majority of its entropy comes from phonons. In turn, the
majority of its phonon entropy, Sph , can be attributed to harmonic oscillations of the nuclei about
their equilibrium positions. The ‘quasiharmonic’ phonon entropy Sph,Q includes both the harmonic
phonon entropy, and the entropy due to a decrease in phonon frequencies (softening) as the crystal
expands. Measurements of phonon dispersions in body centered cubic metals [16, 39, 54, 55, 113, 114]
have shown that the quasiharmonic model is often insufficient to explain the temperature dependence
of the phonon entropy. For example, Sph,Q is an overestimate of the phonon entropy in both niobium
[36] and vanadium [54], but a severe underestimate for chromium. [36] There is less experimental
data on the high temperature trends in phonons and phonon entropy in face-centered cubic metals,
although recent work has shown that the phonons in nickel are slightly stiffer than predicted by the
expansion of the lattice against the bulk modulus. [35]
Aluminum has one natural isotope, 27 Al, and scatters thermal neutrons coherently. There have
been a number of measurements of its phonon dispersions using neutron triple-axis spectrometers
[56, 115–118], and other work using x-ray diffuse scattering. [119, 120] The phonon dispersions at
80 and 300 K measured with inelastic neutron scattering by Stedman et al. [118] have been used
frequently, sometimes to generate phonon DOS. [43, 44, 65, 121, 122]
Previous measurements of phonons in aluminum at temperatures above 300 K were limited to
small numbers of momentum transfers. Because aluminum is a coherent scatterer, such measure-

94
ments sampled only small numbers of phonon states, and are not optimal for determining the phonon
density-of-states that is so important for thermodynamics. Nevertheless, Larsson et al. [56] found
that some phonon frequencies shifted by approximately 15% between temperatures of 298 K and
932 K. They found phonon linewidths to increase at temperatures above 600 K. Peterson et al. [120]
also suggested that the longitudinal modes are anharmonic.
Energy shifts and lifetime broadening of phonons in aluminum and other FCC metals have been
studied theoretically. [62–64, 123–125] Björkman et al., using a pseudopotential model, [124] show
how electron-phonon interactions shorten phonon lifetimes in aluminum. Using Born–von Kármán
fits to neutron data in conjunction with measurements of second and third order elastic constants,
Zoli et al. calculated the shifts [62] and lifetime broadening [64] of the phonons in aluminum at
temperatures below room temperature. More generally, understanding the contributions of quasiparticles and collective excitations to the free energy, entropy, and heat capacity of crystalline solids
is an active area of research. [25, 26, 31–33, 36–38, 126–128]
Here we present results from inelastic neutron scattering measurements of the phonon DOS of
aluminum at temperatures of 10, 150, 300, 525, and 775 K. We use these results to determine the
phonon contributions to the entropy of aluminum, and we assess the other entropic contributions,
finally obtaining excellent agreement with the total thermodynamic entropy. The overall softening
of the phonons is found to be caused by a monotonic temperature dependence of the 1NN, 2NN, and
3NN force constants, with the 1NN force constants decreasing approximately 10% over the temperature range of measurement. The purely anharmonic part of the phonon entropy, not accounted for
by the expansion of the lattice, is approximately −0.07 kB /atom at 775 K. Additionally, we quantify
the temperature dependence of the energy widths and shifts of the phonons, and we attribute the
anharmonic effects to phonon-phonon interactions.

9.2

Experiment

9.2.1

Sample preparation

Clean aluminum shot of 99.99% purity was arranged to cover maximally the interior of a thin walled,
rectangular, aluminum pan, whose height, width and depth were approximately 10.0, 7.0 and 0.5 cm.
The ratio of singly- to multiply-scattered neutrons was designed to be approximately 10:1.

9.2.2

Neutron scattering measurements

The first set of inelastic neutron scattering measurements was performed on LRMECS. The aluminum pan was mounted at 45◦ to the incident neutron beam, and measurements were made at
10, 150, 300, 525, and 775 K. Measurements were also made on the empty aluminum pan at all

95
temperatures, to allow for some removal of background scattering. For the lower temperatures, 10,
150, and 300 K, the sample was mounted in a displex refrigerator. For the higher temperatures,
the sample was mounted in a low background, electrical resistance furnace designed for vacuum
applications. In both cases, temperature was monitored with several thermocouples, and is believed
accurate to within 5 K over the bulk of the sample.
In order to achieve a nominal energy of 60 meV for the incident neutrons, the T0 chopper was
set to 90 Hz, and the E0 chopper to 150 Hz. The counts from neutrons taking between 2400 and
6000 µs to reach the sample from the moderator were stored in bins of width 4 µs. Incident neutron
energies for the different temperatures determined from scattering in the monitors ranged from
59.0 to 59.2 meV (details in § 7.2.1). The experimentally determined full width at half maximum
resolution in angle was approximately 1◦ . All measurements were roughly 8 hours in duration, and
included approximately 750,000 counts.
A second set of measurements was made on Pharos. Another displex refrigerator was used for
a measurement at 10 K, and the same furnace was used for a measurement at 775 K. The same
aluminum sample was used, again mounted at an angle of 45◦ to the incident neutron beam. The
data acquisition system was set to fill time bins spaced every 2.5µs from 5500 to 9000 µs. Data
were collected for a minimum of four hours at each temperature, giving on the order of one million
counts. The T0 and E0 choppers were set to 40 Hz and 300 Hz respectively, for a nominal incident
energy of 70 meV. There are no monitors on Pharos, but the incident energies calculated from the
data ranged from 69.3 to 69.4 meV (details in § 7.2.2).
For both instruments, the energy resolution function, Z(E, E 0 ), was assumed to be Gaussian in
energy:
(
2)
1 E − E0
√ exp −
Z(E, E ) =
2 σi (E 0 )
σi (E 0 ) 2π

(9.1)

The standard deviations, σi , were assumed to decrease linearly from the elastic line to the incident
energy, and our expressions for two instruments are given here:
σLRMECS (E 0 ) ≈

−0.0071E 0 + 1.2744, EI = 60.0 meV ,

σPharos (E 0 ) ≈

−0.0091E 0 + 1.0620, EI = 70.0 meV ,

where E 0 and the σi (E 0 ) are in meV.

(9.2)

96
Table 9.1: Experimentally determined lattice parameter, a, of aluminum and shifts of the aluminum
phonon energies as a function of temperature. Fits of the 10 K DOS to the high temperature
DOS by scaling of the energy and integral transformation with the damped oscillator function yield
the relative frequency shifts ∆s (Eq. 9.6). hEiT /hEi10 are ratios of the first moments of the DOS
(Eq. 9.8). Values in square brackets are from Pharos data.
T (K)
10
150
300
525
775

9.3

a ± 0.01 (Å)
4.041
4.045
4.056
4.079
4.111

∆s
1.000
0.995
0.977
0.954
0.952

hEiT /hEi10
1.000
0.990
0.964
0.941
0.941

Sph ± 0.03
0.001
1.628
3.462
5.146
6.332

(kB /atom)
[0.001]
[6.306]

Analysis and computation

Details of the data reduction procedures used here are given in Chapter 7, with rebinning described
in § 7.4.1. The correction for multiple- and multiphonon scattering was described in § 8.1.2. For
diffraction, determination of the quality factors, and fitting of BvK models, see § 8.4, § 8.2, and
§ 8.3.1.2, respectively.

9.3.1

General data reduction

The raw data from both instruments, in time-of-flight and scattering angle Θ were first normalized
using the counts in the beam monitor for LRMECS, and the integrated proton current for Pharos.
Bad detectors were identified and masked, and the data were corrected for detector efficiency using
a measurement of vanadium, an incoherent scatterer. At each temperature, the measured scattering
from the empty aluminum pan was subtracted from the data, reduced by 10% to account for the
self-shielding of the sample. The data were then binned to get intensity, I(Θ, E), as a function of
scattering angle, Θ, and energy, E, transferred to the sample. Approximately, Θ ranged from 10◦ to
120◦ with a bin width of 0.75◦ , and E ranged from −60.0 to 60.0 meV with a bin width of 0.5 meV
for both instruments.

9.3.2

Elastic scattering: in-situ neutron diffraction

By summing the LRMECS data from −5.0 to 5.0 meV, in-situ neutron diffraction patterns were
obtained. Lattice parameters were determined from these data using Nelson–Riley [104, 105] plots,
and are listed in Table 9.1. Their thermal trends are consistent with thermal expansion data.

200
111

311

400 420
333+511
222 331 422
440

220

I(Θ)

775 K
525 K
300 K
150 K
10 K
10

90 100 110

Θ (degrees)
Figure 9.1: Diffraction patterns from aluminum at temperatures as indicated. Despite their appearance, these were used to determine the lattice parameters shown in Table 9.1.

9.3.3

Inelastic scattering: S(Q,E) and the phonon DOS

Data reduced to I(Θ, E) were then rebinned into intensity, I(Q, E), where Q is the momentum
transferred to the sample. For both instruments, Q ranged from 0.0 to 12 Å−1 , with a binwidth of
about 0.075 Å−1 . The elastic peak was removed below 8 meV and replaced by a function of the
following form:
I(E) ≈

ζ0 E
1 − exp (−βE)

(9.3)

where the constant ζ0 was determined from the inelastic scattering just past the elastic peak. Here,
we have assumed that the phonon DOS is proportional to E 2 in the low energy regime, as in a Debye
model. The phonon DOS were then extracted from the scattering, making the thermal corrections
and corrections for multiphonon and multiple scattering, as described in § 8.1.2. These phonon DOS
are shown by markers and lines in Fig. 9.2, and by the markers in Figs. 9.3.

0.1

g(E) (1/meV)

0.08
LRMECS

775 K

Pharos

0.06
0.04
LRMECS
Pharos

0.02

10 K

E (meV)
Figure 9.2: Phonon DOS of aluminum at temperatures as indicated. The markers show the experimentally determined DOS from LRMECS, and the lines show the DOS from Pharos.

9.3.4

Phonon shifts and broadening

With increasing temperature, the phonon peaks in a metallic solid typically broaden and undergo a
shift to lower energies. These shifts were approximated as a constant multiplier, ∆s , applied to all
phonon energies E:
E → ∆s E .

(9.4)

The broadening of the phonons was assumed to take the form of a damped harmonic oscillator
function, B(Q, E 0 , E) centered about energy E 0 : [84]
B(Q, E 0 , E) =

πQE 0

E0
E 2
+ Q12
E − E0

(9.5)

0.18
0.15
775 K
g(E) (1/meV)

0.12
525 K
0.09
300 K
0.06
150 K
0.03
10 K

E (meV)
Figure 9.3: Phonon DOS of aluminum at temperatures as indicated. The markers show the experimentally determined DOS, and the lines the best fits of the 10 K DOS to the high temperature
DOS by scaling of the energy and intrgral transform with the damped oscillator function as a kernel
(Eq. 9.6).
Using Eqs. 9.4 and 9.5, the high temperature phonon DOS was approximated as a function of the
low temperature DOS, with only two free parameters, ∆s and Q:
gT (E) = B(Q, E 0 , E)
where gT is the phonon DOS at temperature T , and

g10 (∆s E 0 ) ,

(9.6)

denotes an integral transform that is similar

to a convolution. (The subscript 10, as in 10 K, refers to the lowest temperature data from this set
of experiments.)
At each temperature, the best Q and ∆s for the experimental DOS were determined through a
least squares algorithm. The Q so determined are shown in Fig. 9.4, and the fits to the phonon DOS
are shown in Fig. 9.3. The inverse of the quality factor was well described by a quadratic function
of T :
≈ 3.523 × 10−7 T 2 ,

(9.7)

100

0.25
1/Q (unitless)

0.2
0.15
0.1
0.05

100

200

300

400

500

600

700

800

T (K)
Figure 9.4: Markers show the inverse of the quality factor, 1/Q, as a function of temperature for
aluminum phonons, circles are LRMECS data, and the square is Pharos data. The line is a parabolic
fit (Eq. 9.7).
where T is in degrees Kelvin. The shifts ∆s are given in Table 9.1, along with the ratios of the mean
phonon energies, hEiT /hEi10 , as determined from the DOS:
E gT (E) dE
hEiT
=R
hEi10
E g10 (E) dE

9.3.5

(9.8)

Born–von Kármán models of the lattice dynamics

In a second analysis, the phonon DOS were fit with Born–von Kármán models of the lattice dynamics.
[68, 71, 72, 129] Tensorial force constants to the 3NN shell were determined with a gradient search
method. For the higher temperatures, where there is significant anharmonic broadening, these
models were sufficiently accurate. At lower temperatures, however, they were unable to reproduce
the shape of the DOS. For the DOS at 10, 150, and 300 K, axially-symmetric force constants from
4NN through 8NN shells were also optimized. These showed little change with temperature, so they
were averaged and kept constant for a final round of optimization.
To account for the thermal and instrument broadening, the frequencies, ω, calculated as the
square root of each eigenvalue of the dynamical matrix, were taken to be the arguments of delta
functions in energy, δ (E − ~ω). Each delta function was transformed using the damped harmonic
oscillator function of Eq. 8.19, and the Gaussian instrument resolution function given by Eqs. 9.1

101

0.18
0.15
g(E) (1/meV)

775 K
0.12
525 K
0.09
300 K
0.06
150 K
0.03
10 K

E (meV)
Figure 9.5: Phonon DOS of aluminum at temperatures as indicated. The markers show the experimentally determined DOS, and the lines the best BvK models found by fitting the data.
and 9.2. The force constants so determined are listed in Table 9.2, and the best fits to the DOS at
all temperatures are shown in Fig. 9.5.
The longitudinal force constants were found by projecting the tensor onto the bond vectors
hxyzi. The 3 × 3 force constant tensors were then diagonalized. The longitudinal force constant was
matched to one of the eigenvalues and the average transverse constant was taken to be the mean
of the remaining eigenvalues. Longitudinal force constants to 3NN are shown in Fig. 9.6, and the
averaged transverse force constants out to 3NN are shown in Fig. 9.7. The XNN longitudinal force
constants, KX (T ), decrease with increasing temperature approximately as:
K1 (T ) = 21.022 − 2.559 × 10−3 T ,

(9.9)

K2 (T ) = 2.463 − 7.384 × 10−4 T ,

(9.10)

K3 (T ) =−0.862 − 2.066 × 10−4 T ,

(9.11)

where T is in degrees Kelvin and KX (T ) is in N/m. The fits are also shown in Fig. 9.6.

102
Table 9.2: Optimized tensor force constants in N/m as a function of temperature for FCC aluminum. They are given in a Cartesian basis, where < xyz > is the bond vector for the given tensor
components.
[K1 ]xx
[K1 ]xy
[K1 ]zz
[K2 ]xx
[K2 ]yy
[K3 ]xx
[K3 ]xy
[K3 ]yy
[K3 ]yz
[K4 ]xx
[K4 ]xy
[K4 ]zz
[K5 ]xx
[K5 ]xy
[K5 ]yy
[K5 ]zz
[K6 ]xx
[K6 ]xy
[K7 ]xx
[K7 ]xy
[K7 ]xz
[K7 ]yy
[K7 ]yz
[K7 ]zz
[K8 ]xx
[K8 ]yy

9.3.6

< xyz >
< 110 >

< 200 >
< 211 >

< 220 >

< 310 >

< 222 >
< 321 >

< 400 >

10 K
10.112
11.148
−1.356
2.454
−0.532
−0.634
−0.185
−0.298
−0.149
0.273
−0.051
0.324
0.469
0.090
0.229
0.199
0.144
−0.110
−0.061
0.032
0.016
−0.088
0.011
−0.105
−0.536
−0.117

150 K
9.708
10.697
−1.201
2.408
−0.508
−0.636
−0.301
−0.183
−0.147
0.273
−0.051
0.324
0.469
0.090
0.229
0.199
0.144
−0.110
−0.061
0.032
0.016
−0.088
0.011
−0.105
−0.536
−0.117

300 K
9.708
10.378
−2.059
2.224
−0.367
−0.635
−0.294
−0.181
−0.148
0.273
−0.051
0.324
0.469
0.090
0.229
0.199
0.144
−0.110
−0.061
0.032
0.016
−0.088
0.011
−0.105
−0.536
−0.117

525 K
10.542
9.232
−3.370
1.972
−0.148
−0.707
−0.301
−0.225
−0.151
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

775 K
10.112
8.970
−3.463
1.956
−0.144
−0.706
−0.299
−0.222
−0.151
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

Ab-initio phonon calculations

In a third computational effort, we used the plane-wave code VASP [130, 131] to calculate the
electronic DOS of aluminum as a function of unit cell volume. The calculations used projector
augmented plane waves and the Perdew-Burke-Ernzerhof generalized gradient approximation. [133]
A conventional FCC cell was used, and it was relaxed using the ‘accurate’ setting for the kinetic
energy cutoff, with a 20 × 20 × 20 Monckhorst-Pack q-point grid. [134] The relaxed volume (which
matched the experimentally determined volume to better than 0.2%) was taken to be the 0 K volume
of the unit cell, and the values of the linear coefficient of thermal expansion from Wang et al. [107]
were used to determine the volumes at temperatures corresponding to our experiments. At each
of these volumes, the electronic DOS was determined on a larger, 70 × 70 × 70 q-point grid. The
electronic DOS at 10 and 775 K are shown in Fig. 9.8.
The interatomic forces and the phonons were calculated from first principles in the direct method

103

21.00

1NN

Longitudinal Force Constant (N/m)

20.00
19.00
18.00
17.00
2.40

2NN

2.10
1.80
-0.75
-0.85

3NN

-0.95
-1.05

100 200 300 400 500 600 700 800
T (K)

Figure 9.6: Longitudinal force constants for FCC aluminum as a function of temperature. Filled
markers and solid lines are from BvK fits, unfilled markers and dashed lines are from calculations
using VASP [130, 131] and PHON [132]. The fits to the BvK force force constants are given
by Eqs. 9.9, 9.10, and 9.11. For the BvK models, the higher order force constants were fixed as
described in the text.
using VASP in conjunction with the program PHON [132]. At volumes consistent with the temperatures 0, 300, and 775 K, the total energy was minimized for a 4 × 4 × 4 (64 atom) super-cell with
a 4 × 4 × 4 electronic q-point grid. A single displacement whose length was 1% of the interatomic
separation was used, and testing showed that the effects of the size of the displacement on the calculated force constants was negligible. The longitudinal and average transverse force constants were

Average Transverse
Force Constant (N/m)

104

3NN

0.00

2NN

-0.50

1NN

-1.00
-1.50

100

200

300

400 500
T (K)

600

700

800

Figure 9.7: Averaged transverse force constants for FCC aluminum as a function of temperature.
Filled markers and solid lines are from BvK fits, unfilled markers and dashed lines are from calculations using VASP [130, 131] and PHON [132].

G(E) (states/atom/eV)

0.5
0.4
0.3
10 K
0.2
0.1
-12

775 K
-10

-8

-6

-4
-2
E - Ef (eV)

Figure 9.8: Electronic DOS of FCC aluminum at 10 K and 775 K, showing the effects of the expansion
of the lattice with increasing temperature. Ef is the Fermi energy.

105
determined as for the BvK models, and are shown in Figs. 9.6 and 9.7 respectively. The values of
the longitudinal force constants to 3NN at 0 K are given in Table 9.3.

9.4

Results

The force constants obtained from BvK models need not be unique, especially for more distant
nearest-neighbor shells and for off-bond directions. [65] Nevertheless, Wallis et al. found that
their BvK models for aluminum showed reasonable agreement with pseudopotential calculations for
longitudinal force constants of the first three nearest-neighbor shells. [65] Table 9.3 presents the
longitudinal force constants from the pseudopotential and empirical models (at 0 and 80 K, respectively) of Wallis et al., from the empirical model (at 80 K) of Gilat et al., from our empirical model
(at 10 K), and from our plane-wave calculations (at 0 K). The table shows remarkable agreement of
the major longitudinal force constants, giving confidence in our values of these force constants as a
function of temperature.
As seen in Fig. 9.6, thermal changes in the 1NN force constants are dominant, and are expected
to have the largest effect in shifting the DOS to lower energies with increasing temperature. The
transverse force constants shown in Fig. 9.7 have small changes with temperature. They have negative signs, indicating some instability in the off-bond directions. At all temperatures, the magnitudes
of both the longitudinal and transverse force constants decrease rapidly with increasing distance in
the first two or three shells. The behavior of the fit force constants for longer bonds is less structured,
possibly due to noise in the data and the difficulty in fitting parameters that have smaller impacts
on the DOS. Nevertheless, fluctuations of sign could be consistent with Friedel oscillations in Al.
[65]
The frequencies of the transverse modes and longitudinal modes have slightly different temperature dependencies, leading to modest differences in the second and third columns of Table 9.1.
Stedman et al. [121] reported hEi300 /hEi80 = 0.98, which seems in reasonable agreement with
the value of 0.969, obtained as the average of our values at 10 and 150 K. A previous study by

Table 9.3: Longitudinal force constants, KX for the XNN shell in units of N/m as determined by
BvK models and ab-initio calculations. Data from Wallis et al. [65] and Gilat et al. [122] are also
tabulated.
Force
Constant

K1
K2
K3

Wallis
Pseudopotential
(0K)

Empirical
( 80 K )

Gilat

21.70
2.60
-0.86

24.60
2.68
-0.68

( 80 K )

Present Study
BvK
( 10 K )

Plane wave
(0K)

21.55
2.45
-0.92

21.26
2.45
-0.82

20.69
2.07
-0.75

106
Larsson et al. [56] found significantly larger shifts in the mean frequency at higher temperatures:
hEi775 /hEi300 = 0.925a where we find hEi775 /hEi300 = 0.976, perhaps because these early results
were based on the central energies of broadened phonon peaks, and also because the much greater
region of Q-space sampled in the present measurements gives a better average of the phonon softening.
From the values of 1/Q shown in Fig. 9.4, at the highest energy of the phonons, 38.0 meV, we
find maximum values of the full width at half maximum 2Γ = E/Q to be approximately 0, 0.8, 0.9,
2.6, and 7.5 meV at the temperatures 10, 150, 300, 525, and 775 K respectively. This broadening
seems consistent with the experimental values reported by Larsson et al. [56] Linewidths due to
phonon-phonon interactions for aluminum at 300 K were calculated by Zoli et al. using an empirical
force constant model. [64] They find a maximum 2Γ of about 1.5 meV for the longitudinal modes in
the [111] direction, which is also in reasonable agreement with our data. In all cases, the linewidths
appear to increase with increasing phonon energy.
Examples of phonon DOS from BvK models, damped BvK models, and reduction of experimental
data are shown in Fig. 9.9. Especially at high temperatures, these DOS yield slightly different phonon
entropies. To leading order in anharmonic perturbation theory, the phonon entropy is given by the
quasiharmonic formula (Eq. 4.3) with the shifted energies. [37, 38] This would correspond to using
our undamped BvK models. Calculating the entropy using the undamped BvK models gives a total
entropy that is larger than that obtained from reduced experimental data. As the phonon linewidths
increase, a particular phonon can be created or annihilated over a wider spectrum of energies. The
damping function of Eq. 8.19, causes an increase in the mean phonon energy, and, thus, a decrease in
phonon entropy. To minimize data manipulations, we report the entropy from reduced experimental
data. We did not correct for the effects of instrument resolution broadening, which causes us to
overestimate the phonon entropy by as much as 0.03 kB /atom. This is not included in our estimates
of the error.

9.5

Discussion

The harmonic and quasiharmonic contributions to the phonon entropy, Sph,H and Sph,Q were determined using Eq. 4.3. For the former, the spectrum measured at 10 K was assumed a good approximation of the 0 K phonon DOS. To find the contribution of lattice dilation to the phonon entropy,
Sph,D , we used Eq. 4.4 with the temperature dependent isothermal bulk modulus found by He et al.,
[135] and the temperature dependent lattice parameter and linear coefficient of thermal expansion
a Here, we have used the formula from Ref. [38] with hEi
300 from our experiment.

107

0.14

Resolution broadened, damped BvK

0.12
g775(E) (1/meV)

Exp.
0.1
0.08
Damped BvK
0.06
0.04
BvK
0.02

E (meV)
Figure 9.9: Phonon DOS for aluminum at 775 K. Markers show the experimental data. Lines show
BvK models without damping, with damping, and with both damping and instrument resolution
broadening, in the bottom, middle and top curves respectively.
for a ‘real crystal’ found by Wang et al.. [107] Specifically:
Sph

Sph (T )

= Sph (T, T ) ,

Sph,H

Sph,H (T )

= Sph (T, T0 ) ,

Sph,D

Sph,D (T )

= Sph − Sph,H ,

(9.12)

with T0 = 10 K.
The total and ground state electronic entropy, Sel,G were found using Eq. 4.8 and the T0 =0 K
electronic DOS calculated from first principles. The electronic entropy of dilation, Sel,D , was taken
to be the difference:
Sel

Sel (T )

Sel (T, T ) ,

Sel,G

Sel,G (T )

= Sel (T, T0 ) ,

Sel,D

Sel,D (T )

(9.13)

Sel − Sel,G ,

The electron-phonon entropy, described briefly in Chapter 4, was determined from the adia-

108
batic and non-adiabatic electron-phonon free energies calculated by Bock et al. [26, 33], using the
thermodynamic relationship:
S=−

∂F
∂T

(9.14)

We expect the only configurational entropy to be from vacancies, thus we set Scf = Svac . Forsblom
et al. [128] have found the entropy and enthalpy of formation of a vacancy in aluminum to be
∆Svac,f = 2.35 kB and ∆Hvac,f = 0.75 eV, respectively. The configurational entropy Svac,c and the
formation entropy Svac,f of sets of vacancies are given by:
c =

c(T )

exp [−β(∆Hvac,f − T ∆Svac,f )] ,

Svac,f

Svac,f (T )

= c∆Svac,f ,

Svac,c

Svac,c (T )

= −kB [c ln (c) + (1 − c) ln (1 − c)] ,

(9.15)

where c is the concentration of vacancies.
All contributions to the entropy are shown in Fig. 9.10. The total entropy of aluminum was obtained with Eq. 4.16. Values for the total entropy were taken from the NIST-JANAF thermochemical tables, [136] and agreement is excellent, with a root-mean-square deviation of 0.046 kB /atom.
The root-mean-square deviations for the entropy as determined from the BvK models and from
the damped BvK models were 0.066 and 0.039 kB /atom respectively. The differences between these
three models are small; nevertheless, it may be that using a damped DOS (without resolution broadening) more accurately represents the phase space covered by the anharmonic oscillators. We also
see excellent agreement between our (time-of-flight based) values for the phonon entropy and those
derived using triple axis data at 80 and 300 K. (The data were taken from Stedman et al. [118] and
used by Gilat et al. [122] to generate phonon DOS which we then used to find the phonon entropy.)
The harmonic phonon entropy accounts for most of the entropy of aluminum. The next largest
contribution is from the phonon entropy of dilation, but this is already an order of magnitude smaller.
The quasiharmonic model is a useful one for aluminum; nevertheless, the anharmonic phonon entropy
is non-negligible, and comparable in magnitude to the the electronic entropy. The adiabatic electronphonon interaction is another order of magnitude smaller, followed by the dilation correction to the
electronic entropy. The vacancy contribution to the entropy is primarily configurational, and is very
small.
Over the temperatures measured, the anharmonic phonon entropy, Sph,A , ranges from −0.10
to +0.08 kB /atom (where we have incorporated errors of approximately ±0.03 kB /atom). At our
highest temperature of 775 K, we have −0.10 < Sph,A < −0.04 kB /atom. The anharmonic entropy
is positive up to around 625 K, after which it becomes negative and decreasing.
The anharmonic contribution to the entropy of aluminum is dominated by phonon-phonon inter-

109

7.000
S = Sph+Sel+Sel-ph+Svac

6.000
5.000
4.000

S (kB/atom)

Sph

SJanaf
Sph,H

3.000

0.0021

2.000

0.0014

1.000

Sph,3-axis
0.0007

Svac
Svac,c
Svac,f

0.000
Sph,D
0.250

Sph,NH = Sph - Sph,H

0.150

Sel

0.050
-0.050

Sph,A = Sph - Sph,H - Sph,D

0.016

Sel-ph,ad
Sel,D

Sel-ph

0.008
0.000

Sel-ph,na

-0.008

100 200 300 400 500 600 700 800
T (K)

Figure 9.10: Contributions to the entropy of aluminum as a function of temperature. The temperature labels at the bottom of the plot apply to all panels and the inset. Open triangles are total
entropy data taken from the NIST-JANAF thermochemical tables and open squares are phonon
entropy derived from triple-axis data. Closed markers are data from the current experiment; lines
are either calculations or interpolations.
actions over the much smaller contribution from electron-phonon interactions. [124] It is interesting
that the shape of the anharmonic entropy curve is similar to that reported recently for the FCC nickel

110
[35] and the superlinear temperature dependence of 1/Q is also similar. Aluminum is a simple metal,
and nickel is a magnetic d-band metal with a complex electronic structure, so it would be surprising
if their similar anharmonic behaviors originated from electron-phonon interactions. Phonon-phonon
interactions are the probable source of the large increase in phonon linewidth with temperature
in both metals. It is interesting that this increase in linewidth is approximately quadratic with
temperature, as opposed to the linear effect predicted by the quasiharmonic approximation and
perturbation theory. [38, 137, 138] The quasiharmonic approximation is expected to be most appropriate when the phonon frequency shifts are small and the phonon lifetimes are long. The latter
condition may not apply well to aluminum or nickel.

9.6

Summary

Measurements of the inelastic scattering of neutrons by phonons in aluminum were made at temperatures of 10, 150, 300, 525, and 775 K. Phonon DOS were obtained from the reduced experimental
data, and were used to determine the harmonic, nonharmonic and total phonon entropy of aluminum.
The sum of the phonon, electronic and vacancy contributions to the entropy agree exceptionally well
with accepted values for the total entropy over the entire range of temperatures studied. The anharmonic entropy obtained from the shifts of phonon frequencies was small, but the broadening of the
phonon DOS was significant, and scaled superlinearly with temperature. The anharmonic behavior
was attributed to phonon-phonon interactions. The experimental phonon DOS were fit to BvK
models of the forces in the solid. A linear decrease with temperature was found for the the 1NN,
2NN and 3NN force constants, with the 1NN force constants decreasing by approximately 10% over
the range of temperatures measured.

111

Chapter 10

Lead
10.1

Introduction

Lead has a melting temperature of TM = 600 K, and a Debye temperature of TD = 100 K. Temperatures from near absolute zero to the melting point are easily accessible experimentally, and we
expect to see signs of anharmonicity for all temperatures greater than TD . As such, lead is nearly
ideal for studies of phonon anharmonicity in metals, and its properties have been a common subject
of interest for both experimentalists [57, 139–143] and theorists. [144–147]
Practically speaking, mining and processing of lead is inexpensive, and lead is highly malleable
and resistant to corrosion. Consequently, the metal has myriad industrial uses, most notably in
bullets, solder, radiation shielding, and lead-acid batteries. Perhaps less practically, the prospect of
converting lead to gold has garnered the attention of some great scientific minds.a More recently,
much of the scientific interest in lead has focused on the interactions between electrons and phonons.
This over the entire temperature range of the solid phase.
At moderate and high temperatures, where lead is a nearly free electron metal, a great deal of
attention has been paid to the presence of Kohn anomalies in the phonon dispersions. [148] Briefly,
let q1 and q2 be electron wavevectors lying on the Fermi surface such that the planes tangent to
the surface at these points are parallel. There may be many such points on the Fermi surface —
for example, this would be the case for any pair of points directly opposite one another on the
spherical Fermi surface of a free-electron metal. All such pairs contribute a logarithmic singularity
to the electron screening at wavevector Q = q1 − q2 , and this may be seen as a kink in the
phonon dispersions at wavevector Q. As lead is a strong coherent scatterer of neutrons, and has
a comparatively high superconducting transition temperature, it is a good candidate for a neutron
scattering measurement of Kohn anomalies. Indeed, both Brockhouse et al. [149] and Stedman et
al. [150] have made such measurements.
a For example, Roger Bacon, Tycho Brahe, and Isaac Newton, all expended significant effort on the study of
alchemy.

112
Below the critical temperature, TS ≈ 7.2 K, the coupling of electrons and phonons in lead results
in superconductivity. A number of measurements have been made investigating the changes in
the energies and linewidths of the phonons around TS . Mikhaĭlov et al. found that transverse
phonons in the (111) direction in lead had energies that increased and linewidths that decreased as
expected upon lowering the temperature from 300 to 20.4 K; however, they found these trends to be
reversed upon lowering the temperature further, to 4.2 K.[151] Youngblood et al., on the other hand,
measured the same (111) transverse phonons and found no effect of the superconducting transition
on their energies and lifetimes. [152] More recently, Habicht et al. found no anomalous changes
in the phonon linewidths above and below TS as measured with high resolution neutron-resonance
spin-echo spectroscopy. [58] In a later measurement, however, theyb did find changes in the phonon
linewidths at TS . [153] The later measurement had increased intensity, and this is the ostensible
cause of the discrepancy. None of these measurements, however, cover enough of reciprocal space to
give a phonon DOS.
Brockhouse et al. made triple-axis measurements of the phonon dispersions in lead in high
symmetry directions at 100 K. [154, 155] They found that the forces in lead were exceptionally long
range; nevertheless, Gilat used this data to find a Born–von Kármán model of the lattice dynamics
in lead. [156] Subsequent triple-axis work by Stedman et al. [121, 157] covered a significantly larger
fraction of reciprocal space and did not match the model very well, particularly in off-symmetry
directions.c There is strong evidence that a measurement that samples a larger volume of reciprocal
space is preferable for determination of the phonon spectrum of lead. [121, 158–162]
The superconducting state allows for determination of the phonon spectrum via measurements
of electron tunneling [162–164] or of far-infrared reflectance. [165, 166] These measurements yield
η 2 (E)g(E), where η 2 (E) is an effective electron-phonon coupling function for phonons with energy
E. The factor η 2 (E) is fairly smooth as a function of energy [167] with the possible exception of a
feature at roughly 1.6 meV. [161] This variation is insufficient to explain the differences between the
phonon spectra derived from tunneling and those derived from neutron scattering seen in Fig. 10.1
The far-infrared reflectance measurements seen in Fig. 10.2 show more of the features seen in the
neutron scattering results; however, there remain issues of the overall energy scale and of the precise
energy dependence of η(E). Moreover, these measurements may only be made at temperatures
below the superconducting transition.
Here, we present measurements of the inelastic scattering of neutrons by phonons in lead at
temperatures of 18, 38, 63, 88, 113, 137, 163, 188, 300, 390, and 500 K. The measurements were
performed on a time-of-flight chopper spectrometer at a spallation neutron source so they sample
all of reciprocal space. They represent a fairly direct determination of the phonon spectrum of lead,
b Only B. Keimer, T. Keller, and K. Habicht are authors on both Refs. [58] and [153]. T.Keller is actually the first
author of the second reference.
c A more detailed comparison of our results with those of Stedman et al. is presented in § 10.4.

113

Figure 10.1: Phonon spectrum of lead
derived from neutron scattering (solid
line) and electron tunneling (dashed
line) experiments. The spectrum from
tunneling lacks all but the major features visible in the spectrum from neutron scattering. Figure taken from
Ref. [165].

Figure 10.2: Electron-phonon coupling function weighted
phonon spectrum of superconducting lead derived from
far-infrared reflectance (solid line) and from electron tunneling (dashed line) experiments. The letters are meant to
indicate common features in the reflectance and neutron
derived spectra. The energy for the tunneling measurement been scaled so the spectra match at “C”. Figure
taken from Ref. [166].

which we use to determine the phonon contributions to the entropy of lead. We also assess the
other entropic contributions, and compare to the total thermodynamic entropy. We find a purely
anharmonic contribution to the phonon entropy, unaccounted for by the expansion of the lattice,
of approximately −0.23 kB /atom at 500 K. Additionally, we find a superlinear dependence of the
phonon linewidths on temperature and discuss the measured shifts in phonon energy.

10.2

Experiment

10.2.1

Sample preparation

Clean sheets of lead foil of 99.998% purity were arranged in a thin-walled aluminum pan, whose
height, width and depth were approximately 7.5, 5.0 and 0.1 cm, covering maximally the incident
neutron beam The ratio of singly- to multiply-scattered neutrons was designed to be approximately
24:1.

114

10.2.2

Neutron scattering measurements

Inelastic neutron scattering measurements were performed on the ARCS spectrometer at the Spallation Neutron Source at Oak Ridge National Laboratory. The sample was mounted at 45◦ to the
incident neutron beam, and measurements were made at 18, 300, 390, and 500 K. For the higher
temperatures, 300, 390, and 500 K, the sample was mounted in a low background, electrical resistance furnace designed for vacuum applications. For the lower temperatures (including a second
measurement at 300 K) the sample was mounted in a closed-cycle helium refrigerator (CCR). In
both cases, temperature was monitored with several thermocouples, and is believed accurate to
within 5 K over the bulk of the sample. As a measurement on ARCS takes roughly 20 minutes, and
the CCR requires 4 hours to cool from room temperature to 21 K, measurements were also taken
during the cooling process at median temperatures of 38, 63, 88, 113, 137, 163, and 188 K. All of
these were used to determine lattice parameters, and the measurements at 88 and 163 K were given
a complete analysis.d Due to a shortage of time, the only background measurement taken was of
the empty CCR at room temperature (300 K).
The E0 chopper was set at 480 Hz giving a nominal energy of 30 meV for the incident neutrons.
The counts from neutrons taking between 5500 and 8500 µs to reach the sample from the moderator were stored in bins of width 3 µs. Incident neutron energies for the different temperatures
determined from the elastic peaks ranged from 29.63 to 29.70 meV (details in § 7.2.2). The experimentally determined full width at half maximum resolution in angle was approximately 0.1◦ . The
measurements at 21, 300, 390, and 500 K were roughly 30 minutes in duration, and included approximately 20,000,000 counts. The 88 and 163 K measurements were shorter, with roughly 9,000,000
and 16,000,000 counts respectively.

10.3

Analysis and computation

Details of the data reduction procedures used here are given in Chapter 7, with rebinning described
in § 7.4.1. The correction for multiple- and multiphonon scattering was described in § 8.1.2. For
diffraction and determination of the quality factors see § 8.4 and § 8.2 respectively.

10.3.1

General data reduction

The raw data in time-of-flight and scattering angle Θ were first normalized using the integrated
proton current. Bad detectors were identified and masked, and the data were corrected for detector
efficiency using a measurement of vanadium, an incoherent scatterer. The data were then binned to
get intensity, I(Θ, E), as a function of scattering angle, Θ, and energy, E, transferred to the sample.
d The measurements at 38 and 63 K were deemed to have insufficient counts for analysis. The 113, 137, and 188 K
measurements will be analyzed in the future.

115

200
111

311
220

400
420
222 331

422

500 K

I(Θ)

390 K
300 K
163 K
88 K
18 K
10

90 100 110 120 130

Θ (degrees)
Figure 10.3: Diffraction patterns from lead at temperatures as indicated. These were used to
determine the lattice parameters shown in Table 10.1.
Approximately, Θ ranged from 0◦ to 145◦ with a bin width of 0.1◦ , and E ranged from −30.0 to
30.0 meV with a bin width of 0.1 meV. The data, now reduced to, I(Θ, E) were then rebinned again
into intensity, I(Q, E), where Q is the momentum transferred to the sample. Values of Q ranged
from 0.0 to 8 Å−1 , with a binwidth of about 0.025 Å−1 .

10.3.2

Elastic scattering: in-situ neutron diffraction

By summing I(Θ, E) from −2.1 to 2.1 meV, in-situ neutron diffraction patterns were obtained, and
are shown in Fig. 10.3. Lattice parameters were determined using Nelson–Riley [104, 105] plots,
and these are listed in Table 10.1. These are systematically larger than accepted values of the
lattice parameter by roughly 0.02 Å; but the trends are consistent with thermal expansion data
from Touloukian et al. [168]

116
Table 10.1: Experimentally determined lattice parameter, a, of lead and shifts of the lead phonon
energies as a function of temperature. Fits of the 18 K DOS to the high temperature DOS by
scaling of the energy and integral transformation with the damped oscillator function yield the
relative frequency shifts ∆s (Eq. 10.5). hEiT /hEi18 are ratios of the first moments of the DOS
(Eq. 10.6).
T (K)
18
38
63
88
113
137
163
188
300
390
500

10.3.3

a ± 0.03 (Å)
4.934
4.934
4.939
4.944
4.948
4.953
4.953
4.957
4.971
4.986
5.004

∆s
1.000
1.006
1.003
1.005
1.007
1.021

hEiT /hEi18
1.000
1.017
1.019
1.023
1.039
1.061

Sph ± 0.2 (kB /atom)
0.558
4.040
5.872
7.731
8.542
9.319

Background determination

As mentioned in the previous section, the only background measurement taken was of the empty
CCR at room temperature (300 K). On LRMECS or Pharos, this might have been sufficient for
all temperatures; however, ARCS is still in the process of acquiring shielding, a T0 chopper, and
various other components. As such, the single background measurement was not even acceptable for
the measurement of lead at room temperature in the CCR. Fig. 10.4 shows the measured scattering,
CCR
(E).e
ITPb (E), from lead at T = 18, 88, 163, and 300 K, as well as the measured background, I300

The high energy cutoff of the phonon spectrum of lead is known from previous studies to be roughly
10 meV. [154–166] In order for us to have measured a similar cutoff, we would have needed to measure
background scattering at 18 K with intensity similar to the curve marked I b in Fig. 10.4; however,
CCR
our measurement, I300
, has nowhere near this intensity.
mph
be the multiphonon scattering from lead at 18 K. Then, in order to overcome the
Let I18

difficulties with the background, we have done the following (all energies are given in meV):
• Assume that the cutoff of the phonon spectrum is 10 meV at 18 K.
• Take:
Pb
I18
mph
Pb
I b + I18
≡ I18
(Q, 10)
I CCR
300

E > 10
E < E < 10 ,

(10.1)

e The intensities have been summed over Q. The functional dependence of the intensity on E and Q will be dropped
whenever this can be done without causing confusion.

117

I(E)

IPb
300
IPb
163
IIPb
88
IIPb
18

mph

II + II18

II
-5

II18

mph

ICCR
300

E (meV)
Figure 10.4: Determination of the background for lead measured at ARCS. ITPb (E) is the scattering
CCR
from lead at temperature T . I300
is the measured scattering from the empty CCR at 300 K.
mph
I +I18 is the assumed combination of background and multiphonon scattering at 18 K as described
mph
in § 10.3.3. I18
is the multiphonon scattering determined using the phonon DOS found by taking
mph
I + I18 to be the background, and I b is the background finally used in our analysis.
Pb
where E ∼ 2.1 is the point where a line of constant intensity emanating from I18
(Q, 10) and
CCR
heading towards E = 0 crosses I300
. (The crossing is on the elastic peak.)
mph
Pb
• Find a phonon DOS for lead at 18 K, g18
, by taking I18
as the scattering and I b + I18
as

the background and using the iterative technique from § 8.1.2.
mph
, and separate it from background
• Use g18
to determine the multiphonon scattering, I18

scattering I b .
The apparent deviations from linearity in the 2.1 . E < 10 region are caused by the kinematical
restrictions on Q and E. For measurements at or below 300 K, I b was used as the background. For
temperatures above 300 K, we took the background to be the sum of I b and the difference of the

118
scattering from lead as measured in the furnace at 300 K and in the CCR at 300 K, I b + δI. These
were then subtracted from ITPb .
There is some justification for this procedure in that the intensity of inelastic scattering is temperature dependent. That the features at ∼ 14 meV in the alleged background scattering appear to
be constant as a function of temperature, and that they are completely absent from the measurement
of the empty CCR, suggests that some neutrons are scattering elastically in the lead sample, hitting
another object in the instrument, and then and then rescattering elastically into the detectors.

10.3.4

Inelastic scattering: S(Q,E) and the phonon DOS

To determine the inelastic scattering, the elastic peak was removed below 2.1 meV and replaced by
a function of the form:
I(E) =

ζ0 E
1 − exp (−βE)

(10.2)

where the constant ζ0 was determined from the inelastic scattering just past the elastic peak. Here,
we have assumed that the phonon DOS is proportional to E 2 in the low energy regime, as in a Debye
model. The phonon DOS were then extracted from the scattering, making the thermal corrections
and corrections for multiphonon and multiple scattering, as described previously. [35] These phonon
DOS are shown by markers in Fig. 10.5.

10.3.5

Phonon shifts and broadening

(10.3)

The broadening of the phonons was assumed to take the form of a damped harmonic oscillator
function, B(Q, E 0 , E) centered about energy E 0 : [84]
B(Q, E 0 , E) =

πQE 0

E0
E 2
+ Q12
E − E0

(10.4)

Using Eqs. 10.3 and 10.4, the high temperature phonon DOS was approximated as a function of the
low temperature DOS, with only two free parameters, ∆s and Q:
gT (E) = B(Q, E 0 , E)

g18 (∆s E 0 ) ,

(10.5)

119

0.70
0.65
0.60
500 K

0.55
0.50

390 K

g(E) (1/meV)

0.45
0.40

300 K

0.35
0.30

163 K

0.25
0.20

88 K

0.15
0.10

18 K

0.05
0.00

E (meV)

Figure 10.5: Phonon DOS of lead at temperatures as indicated, offset by integer multiples of
0.1 meV−1 . The markers show the experimentally determined DOS, and the lines the best fits
of the 18 K DOS to the high temperature DOS by scaling of the energy and intrgral transform
with the damped oscillator function as a kernel (Eq. 10.5). The two dotted-vertical lines serve to
illustrate the fact that the positions of the longitudinal and transverse peaks are relatively constant
as a function of temperature.
where gT is the phonon DOS at temperature T , and

denotes an integral transform that is similar

to a convolution. (The subscript 18, as in 18 K, refers to the lowest temperature data from this
experiment.)
At each temperature, the best Q and ∆s for the experimental DOS were determined through a
least squares algorithm. The Q so determined are shown in Fig. 10.6, and the fits to the phonon
DOS are shown in Fig. 10.5. The shifts ∆s are given in Table 10.1, along with the ratios of the mean

120

1/Q (unitless)

0.25
0.2
0.15
0.1
0.05

100

200

300

400

500

600

T (K)
Figure 10.6: Markers show the inverse of the quality factor, 1/Q, as a function of temperature for
lead phonons The dashed and solid lines are linear and parabolic fits, respectively. Neither of these
seems to capture the temperature dependence of 1/Q.
phonon energies, hEiT /hEi18 , as determined from the DOS:
E gT (E) dE
hEiT
hEi18
E g18 (E) dE

10.3.6

(10.6)

Ab-initio electronic structure calculations

We used the plane-wave code VASP [130, 131] to calculate the electronic DOS of lead as a function
of unit cell volume. The calculations used projector augmented plane waves and the Perdew-BurkeErnzerhof generalized gradient approximation. [133] A conventional FCC cell was used, and it was
relaxed using the ‘accurate’ setting for the kinetic energy cutoff, with a 41 × 41 × 41 MonckhorstPack q-point grid. [134] The relaxed volume (which matched the experimentally determined volume
within 2.3%) was taken to be the 0 K volume of the unit cell, and the values of the linear coefficient
of thermal expansion from Touloukian et al. [168] were used to determine the volumes at 0, 100,
200, 300, 400, 500, and 600 K. At each of these volumes, the electronic DOS was determined. The
electronic DOS at 0 and 600 K are shown in Fig. 10.7.

121

G(E) (states/atom/eV)

1.0
600 K
0.8
0.6
0.4
0K
0.2
0.0
-12

-10

-8

-6

-4
-2
E - Ef (eV)

Figure 10.7: Electronic DOS of FCC lead at 0 K and 600 K, showing the effects of the expansion of
the lattice with increasing temperature. Ef is the Fermi energy.

10.4

Results

Given the difficulties with the measurement of the background outlined in § 10.3.3, we will begin by
comparing information culled from the current experiment with other experimental data.
Brockhouse et al. made extensive measurements of the phonons in lead at 100 K [154], and less
extensive measurements at 213, 296, 425, and 570 K. [155] They found that the phonon energies in
lead decreased with increasing temperature. This is at least somewhat at odds with the values of ∆s
or hEiT /hEi18 presented in Table 10.1, which seem to increase slowly, but steadily, with increasing
temperatures. On the other hand, Brockhouse et al. find interplanar force constants for the 1NN
planes that only decrease slightly as a function of temperature. These force constants are frequently
most responsible for phonon softening in metals, as was the case in aluminum, and as we will see
in nickel, iron, and many other metals. (See Chaps. 9, 11, 12 and 14.) More generally, they find
that with increased temperature, the longitudinal modes decrease in frequency more slowly than
the transverse modes. Looking at the fits in Fig. 10.5 we can see a similar effect at the highest
temperatures.
Stedman et al. have measured phonons in lead at 80 and 300 K, [157], although the lower
temperature measurement was much more extensive. To compare our measurements with theirs, we

122
Table 10.2: Characteristic temperatures from the phonon moments of lead, calculated using Eq. 10.7,
Eq. 10.8, and the experimentally determined phonon spectra. Values are all given in degrees Kelvin.
Wallace’s values [37] are taken from the experimental data of Stedman et al. [121].
Characteristic
temperature
T0
T1
T2

Current
work
63.7
91.3
93.5

Wallace &
Stedman
64.1
91.3
93.4

introduce moments and characteristic temperatures of the phonon spectrum:
ωj

ω0

1
ω j g(ω)dω j ,
R∞
exp 13 0 ln(ω)g(ω)dω ,
j+3 R ∞

~ω j
kB ,
~ω 0 − 13
kB e

j > −3, j 6= 0 ,

(10.7)

(10.8)

Values of T 0 , T 1 , and T 2 from the 88 K DOS of the current experiment and from the DOS determined
by Stedman et al. at 80 K are shown in Table 10.2.

The agreement is excellent. The difference

in T 0 is explained by the fact that both measurements are most error prone at low energies. In
particular, the current measurements require removal of a large elastic peak to find the inelastic
scattering at low energies. In a few specific directions, Stedman et al. found that the frequencies
of the high Q (usually higher energy) modes tended to increase with temperature, whereas those at
lower Q (usually lower energy) tended to decrease. This corresponds with what we see in Fig. 10.5,
where the modes to the left of the first dashed vertical line appear to be moving to lower energies,
and those to the right of the second dashed line are moving to higher energies.
On the other hand, agreement with the measurements of Furrer and Hälg [158] is not as good
— at least with respect to the phonon shifts. They have measured phonons in lead over the entire
(110) plane at 5, 80, and 290 K. We find ratios hEi88 /hEi18 = 1.017 and hEi300 /hEi18 = 1.019 as
compared to hEi80 /hEi5 = 0.982 and hEi290 /hEi5 = 0.940 found by found by Furrer and Hälg. It
should be noted that the measurements of Furrer and Hälg covered only a single plane in reciprocal
space, whereas ours cover its entirety. Further, since our 88 K DOS is in excellent agreement with
that of Stedman et al., and our estimate of the background is probably best at 18 K, it seems that
the ratio of the mean phonon energies for these temperatures is probably not off by the 3% required
for agreement with Furrer and Hälg.
In the same experiment, Furrer and Hälg also measured phonon linewidths. Additionally, Habicht
et al. made neutron spin-echo measurements of linewidths of a few low energy modes at 5, 10, 15,

123

290 K
200 K
80 K

1.2

2Γ (meV)

0.8
0.6
0.4
0.2

E (meV)

Figure 10.8: Markers show phonon full width at half maximum, 2Γ as a function of phonon energy,
at temperatures as given in the key, taken from the triple axis and neutron spin echo measurements
of Furrer and Hälg and of Habicht et al. [58, 158] The solid lines are linear fits to the (combined)
data, the slope of which give 1/Q. The dotted lines have slopes 1/Q taken from fits of the data from
the current experiment at 88, 163, and 300 K using Eqs. 10.3, 10.4 and 10.5.
Table 10.3: Quality factors for lead from the linear fits in Fig. 10.8
Current work
T (K)
88
163
300

64.9
29.6
13.6

Furrer & Hälg,
Habicht et al.
T (K)
80
200
290

45.8
28.4
13.6

50, 100, 150, 200, and 300 K. [58] We have combined the measurements from the two sources at 80
and 100 K, and at 290 and 300 K. These and the linewidths at 200 K are plotted as a function of
phonon energy in Fig. 10.8, as are linear fits to the data. These fits give the quality factors, Q, via
the relationship:
2Γ

(10.9)

The values of Q for 88, 163, and 300 K are shown in Table 10.3 At the lowest temperature, the

124
agreement is fair. This is mainly because the triple axis and neutron spin echo measurements are
sensitive to the intrinsic phonon linewidths, whereas ours are not. At higher temperatures, the
intrinsic contributions to the phonon linewidths become negligible, and the agreement is excellent.
On the other hand, at temperatures above 300 K we have no data with which to compare ours.
Further, our high temperature measurements were made in the furnace rather than the CCR, and
the assumed background is even more problematic.
Zoli has performed a theoretical investigation of the shifts and widths of phonons in lead at a
few temperatures. [63] He finds shifts between 5 and 80 K that are even larger than those found by
Furrer and Hälg; i.e., in even greater disagreement with our measurements. At room temperature,
he finds 2Γ/E ≈ 0.2, which is almost 3 times the value we find, 0.074. Additionally, he finds a
linear temperature dependence of the phonon linewidths between 80 and 300 K; whereas we find a
temperature dependence of 1/Q that is somewhere between linear and quadratic. This can be seen
in Fig. 10.6. The temperature dependence here is not nearly as strong as that seen in aluminum in
Chapter 9.

10.5

Discussion

Fig. 10.9 shows the various contributions to the entropy of lead. In increasing order of magnitude, we
have contributions of monovacancies and divacancies, negative contributions from electron-phonon
interactions, contributions from noninteracting electrons and anharmonic phonons, the contribution
of harmonic phonons, and the sum total of all these.
As expected, the largest contribution to the total comes from the phonons, and this is shown in
the top panel of Fig. 10.9. This contribution was calculated by interpolating between the spectra
measured at 18, 88, 163, 300, 390, and 500 K and then using the interpolated gT in the quasiharmonic
formula for the phonon entropy:
Sph = 3kB

gT [(nT + 1) ln(nT + 1) − nT ln(nT )] dE ,

(10.10)

where nT = eβE1−1 is the mean occupation number for bosons. The vast majority of this entropy
comes from the harmonic oscillations of the lead atoms about their equilibria, and this contribution
can be calculated by using the DOS from 18 K, g18 , instead of gT in Eq. 10.10 This is also shown in
Fig. 10.9, and it makes up more than 98% of the total phonon entropy at melting.
The next largest contributions to the entropy are the phonon entropy of dilation and the nonharmonic phonon entropy, Sph,D and Sph,NH , respectively. Both are shown in the center panel of

125

10.000
9.000

S = Sph+Sel+Sel-ph+Svac

8.000

SJanaf

7.000

Sph
Sph,H

S (kB/atom)

6.000
5.000

1.2e-3

4.000

9.6e-4

3.000

7.2e-4

2.000

4.8e-4

1.000

2.4e-4

Svac

0.000
0.400
Sph,D

0.300
0.200

Sph,NH

0.100
0.000

Sel

-0.100
-0.200

Sph,A

-0.300
Sel,D

0.004
0.002

Sel-ph,na

0.000

Sel-ph

Sel-ph,ad

-0.002
-0.004

100

200

300

400

500

600

T (K)
Figure 10.9: Contributions to the entropy of lead as a function of temperature. The temperature
labels at the bottom of the plot apply to all panels and the inset. Open triangles are total entropy
data taken from the NIST-JANAF thermochemical tables. Closed markers are data from the current
experiment; lines are either calculations or interpolations.
Fig. 10.9. We have determined the former using the thermodynamic relationship:
Z T
Sph,D =

9KT α2 0
dT ,
ρN

(10.11)

126
with values for the thermal expansion, α, of lead from Touloukian et al. [168] and the isothermal
bulk modulus, KT , from Cordoba and Brooks. [169] The latter is determined by taking the difference
of the total phonon and harmonic phonon entropies, Sph,NH = Sph − Sph,H . If the only effects of
anharmonicity are the expansion of the lattice against the bulk modulus, these two contributions
should be equal. This is clearly not the case in lead, as can be seen by looking at the difference
between these two, the anharmonic phonon entropy Sph,A = Sph,NH −Sph,D . We find the anharmonic
entropy to be negative at all temperatures. At higher temperatures, it is of moderate magnitude,
reaching −0.23kB /atom at 500 K. Through an analysis of heat capacity measurements and other
bulk thermodynamic properties, Leadbetter finds values of the anharmonic entropy of roughly −0.06
or −0.15 kB /atom depending on whether he uses the 0- or ∞-temperature values for the electronphonon enhancement factor. [34] Either of these is smaller than our value; however, the latter is
of the same order of magnitude, and both have the same sign. Cordoba and Brooks get Sph,A ≈
−0.2 kB /atom at 500 K, which is in closer agreement with our measurement. [170] Both of these
papers find a decrease in the magnitude of the anharmonic entropy between 500 K and melting;
therefore, the extrapolations between 500 and 600 K in Fig. 10.9 should be regarded with some
suspicion. That said, it seems likely that the anharmonic entropy at melting found by Wallace,
Sph,A = −0.04 kB /atom, is too small in magnitude. [37]
At roughly the same order of magnitude, we have the entropy for non-interacting electrons, which
can be determined using the electron DOS shown in Fig. 10.7 and the following formula:
Sel = −kB

GT [fT ln(fT ) + (1 − fT ) ln(1 − fT )] dE ,

(10.12)

where fT = eβ(E−µ)
is the mean occupation number for bosons, and the chemical potential µ
+1
was determined using Nel = GT (E)fT (E)dE. This contribution is also shown in the central

panel of Fig. 10.9. As expected in a nearly free electron metal, this contribution to the entropy is
approximately a linear function of temperature.
Nearly an order of magnitude smaller are the contributions to the entropy from the interactions of
the electrons and phonons, and these are shown in the bottom panel of Fig. 10.9. These were found
using the free energies for non-adiabatic and adiabatic electron-phonon interactions calculated by
Bock et al. and the identity S = − ∂F
∂T . [26, 28] The non-adiabatic contribution, Sel−ph,na , accounts
for the effects of the finite velocities of the nuclei, and is dominant at the lowest temperatures. This
is a positive contribution to the entropy, and is frequently quantified as a multiplier on the electronic
heat capacity at low temperatures. The adiabatic contribution, Sel−ph,ad , accounts for the displacements of the nuclei from their equilibria, and dominates at higher temperatures. This provides a
negative contribution to the entropy. The total contribution of electron-phonon interactions is their
sum, Sel−ph = Sel−ph,na + Sel−ph,ad , and this is also shown in the bottom panel Fig. 10.9, along with

127
Table 10.4: Formation energies and entropies for monovacancies and divacancies in lead after Cordoba and Brooks. [170]
Defect
monovacancy
divacancy

∆Hvac,f (eV)
0.52
0.96

∆Svac,f ( kB )
1.0
1.5

the contributions of the expansion of the lattice to the electronic entropy, Sel,D . Given that lead
is a superconductor up to 7.2 K, it is perhaps surprising that these contributions are so small, and
further work verifying the size of this contribution is desirable.
The smallest contribution to the entropy comes from vacancies, and this is shown in the inset of
Fig. 10.9. This contribution was determined using the values of the formation energies and entropies
for monovacancies and divacancies from Cordoba and Brooks which are shown in Table 10.4. [170]
Equations for the heat capacities of these defects are also given by Cordoba and Brooks, [171] and
these were integrated to get the following expressions for the entropy:
∆Svac,f,1
exp (−β∆Hvac,f,1 ) ,
= kB (β∆Hvac,f,1 + 1) exp
kB
∆Svac,f,2
exp (−β∆Hvac,f,2 ) ,
= 6kB (β∆Hvac,f,2 + 1) exp
kB

Svac,1
Svac,2

(10.13)
(10.14)

where the value 6 in the second equation is half the coordination number for FCC lead. The value
shown in the figure, Svac , is the sum of these two contributions. We note that the contribution from
divacancies is negligible at all temperatures.
Finally, we discuss the total entropy of lead. The values of the total entropy shown as triangles in
the top panel of Fig. 10.9 are taken from the NIST-JANAF thermochemical tables. [136] Also plotted
is the sum of the phonon, electron, electron-phonon, and vacancy contributions to the entropy as
determined in the current work, S = Sph + Sel + Sel−ph + Svac . The agreement is excellent up to
300 K, and is quite good all the way to melting, with a maximum deviation of -1.6% at melting.
Here we will consider currently the discrepancy at 500 K to avoid extrapolation. The anharmonic
entropy at 500 K is Sph,A = −0.23 kB /atom, and the difference between our total entropy and that
given by NIST-JANAF is ∆S = −0.12kB /atom. This suggests that if our determination of the
anharmonic entropy is off by a roughly factor of two, the discrepancy in the total entropy vanishes.
It is difficult to rule out this possibility, given the difficulties with the background outlined in § 10.3.3.
Another possible source of error is the electron-phonon interaction at high temperatures. This is
plausible, as lead has a fairly high critical temperature for an elemental superconductor, and is thus
known to have reasonably strong electron-phonon interactions at low temperatures. In particular,
it is only recently that large, adiabatic electron-phonon effects at high temperatures have been seen
experimentally in vanadium and its alloys. [40, 41] In these studies, the electron-phonon interactions

128
were expressed as a smearing of the electron DOS at the Fermi level; however, looking at Fig. 10.7
its clear that the smearing would have to be very large in order to have a significant impact.f

10.6

Summary

Measurements of the inelastic scattering of neutrons by phonons in lead were made at temperatures
of 18, 38, 63, 88, 113, 137, 163, 188, 300, 390, and 500 K, and detailed comparisons to previous
measurements were made. Phonon DOS were obtained from the reduced experimental data, and
were used to determine the harmonic, nonharmonic and total phonon entropy of lead. The sum
of the phonon, electronic and vacancy contributions to the entropy agree well with accepted values
for the total entropy up to half the melting temperature, and there is fair agreement for the rest
of the range of temperatures studied. The anharmonic entropy obtained from the shifts of phonon
frequencies, Sph,A = −0.23kB /atom, was negative and larger than expected. Finally, the broadening
of the phonon DOS was significant, scaling superlinearly with temperature.

f The formalism used in the cited papers applies a Lorentzian broadening to the electron DOS. In order for this to
have a significant effect, the full width at half max for the Lorentzian would have to be larger than an electron-volt.

129

Chapter 11

Nickel
11.1

Introduction

Nickel is a ferromagnetic 3d transition metal commonly used in industrial and consumer products
including stainless steels, corrosion resistant alloys, magnets, coins, and batteries. In addition to general scientific interest in nickel, [67, 172–179], its magnetic properties have been the subject of a large
number of experimental [27, 180–187] and theoretical [45, 188–192] studies; particularly because the
nature of magnetism in nickel is only partially explained by either localized or itinerant electrons
models; [193–200] At low temperatures, observations of the d-electron Fermi surfaces in nickel show
that the magnetic electrons are itinerant in nature. [193] The behavior at high temperatures, however, is not easily understood in an itinerant electron model. Specifically, it is well known that spin
excitations persist well above the Curie temperature of nickel, TC = 631 K, [201–205] and that they
make significant contributions to the entropy and heat capacity at high temperatures. [31, 36, 37]
In their studies of the entropy of nickel at high temperatures, Wallace [37] and Eriksson et al. [36]
divided the electronic entropy into a contribution from the non-interacting, non-magnetic electrons,
Sel,NM , and a contribution from magnetism, Sel,M ,a such that:
Sel = Sel,NM + Sel,M .

(11.1)

They determined Sel,G through an ab-initio electronic structure calculation, and used it to make an
estimate of the sum of the anharmonic phonon and the magnetic entropies:
Sph,A + Sel,M = S − Sph,H − Sph,D − Sel,NM .

(11.2)

They are, however, unable to separate the two terms on the left hand side of this equation. Meschter,
et al., have performed a similar analysis of the heat capacity of nickel, also estimating the anharmonic
a They call these S

el or SEP

and δSM , respectively.

130
contributions to the entropy.[31]
By measuring the phonon DOS at high temperatures, we are able to determine precisely the
anharmonic contribution to the total entropy of nickel. Previous measurements of the phonon dispersions in nickel were reported by Birgeneau, Cordes, Dolling and Woods; Hautecler and Van
Dingenen; and de Wit and Brockhouse;[206–208] but all these studies were confined to temperatures between 296 K and 676 K. The purpose of de Wit and Brockhouse’s measurements was not
specifically to investigate phonon thermodynamics, but rather to look for changes in the phonon
modes as the metal went through the Curie transition at TC = 631 K. They reported little change
in the phonons through the magnetic transition, and shifts in the phonon energies that were largely
consistent with a quasiharmonic model. They also reported significant broadening of the phonon
peaks with increasing temperature, which they suggested may be due to interactions of magnetism
with the lattice.[208]
Zoli, et al., investigated the broadening of phonons in face-centered cubic (FCC) noble metals
and aluminum.[64] Using force constants from Born–von Kármán fits to neutron data and thirdorder elastic constants, they calculated the full width at half maximum, 2Γ, of the phonon peaks.
For aluminum and the noble metals, broadening of the phonon peaks is of course not caused by
magnetism, but by phonon-phonon interactions that shorten phonon lifetimes. The coupling of
phonon modes is also responsible for the expansion of the lattice; however, the quasiharmonic model
does not necessarily offer a method of rationalizing the observed decrease in phonon lifetimes.
In the present research we measured the inelastic scattering of neutrons from elemental nickel
from 10 K to 1275 K, which is about 75% of the melting temperature. We discuss the modest shifts
of the phonon energies as well the large broadening of phonon peaks at elevated temperatures. The
phonon DOS are used to calculate the anharmonic contribution to the entropy, and this is used to
bound the value of the magnetic entropy at high temperature. Finally, we evaluate other phonon
and electron contributions to the entropy.

11.2

Experiment

11.2.1

Sample preparation

Ingots of 99.98% pure nickel were cold-rolled to a thickness of 0.45 mm. At this thickness, 10%
of the incident neutrons are scattered by the sample. The cold-rolled pieces were then cut into
strips, and annealed at 1075 K in evacuated quartz tubes for 16 hours to relieve stress and induce
recrystallization. There were no signs of oxidation on the annealed strips.

131

11.2.2

Neutron scattering measurements

Inelastic neutron scattering measurements were performed with the Pharos time-of-flight directgeometry chopper spectrometer at the Los Alamos Neutron Science Center at temperatures of 10 K,
300 K, 575 K, 875 K, and 1275 K. For the 10 K and 300 K measurements, the strips of nickel were
laid flat in a thin-walled aluminum pan, which was then mounted on a closed-cycle refrigerator. For
higher temperatures a niobium pan was used, and the sample was mounted in a vacuum furnace built
by A.S. Scientific. Several thermocouples were used to monitor the temperature of the sample, and
it is estimated that the temperature deviations in the sample were no more than 5 K. Measurements
of the empty sample pans were also performed at all temperatures.
Details about the Pharos spectrometer at Los Alamos have been given in § 6.4. At the time
of the experiment, the 6 atm pressure detectors were located at the lower, and the 10 atm at the
higher scattering angles. The time bins were spaced every 2.5 µs from 5500 µs to 9000 µs. Data
were collected for a minimum of four hours at each temperature, giving on the order of one million
counts. The nominal incident energy was 70 meV, and the incident energies calculated from the data
ranged from 69.3 meV to 69.6 meV. The experimentally determined resolution of the instrument
(full width at half maximum) was approximately 2.5 meV at the elastic line, and 1.0 meV at the
high energy cutoff of the phonon DOS (∼40 meV).

11.3

Data analysis and computation

11.3.1

General data reduction

The measured spectra, in time of flight, detector number, and pixel were first corrected for the
efficiencies of the detectors. This was done using a room temperature measurement of pure vanadium,
a fully incoherent scatterer, for calibration. Next, the time of flight independent background was
estimated as an average over a region in time of flight having no appreciable scattering from the the
sample or the environment and subtracted. The corrected data were normalized by the integrated
proton current and converted to intensity, I(Θ, E), by rebinning into scattering angles, Θ, ranging
from 5◦ to 145◦ with a binwidth of 0.5◦ and energy transfers, E, from −65 meV to 65 meV with a
bin width of 0.5 meV. The scattering from the empty pans was subtracted from the data, scaled by
90% to account for the self-shielding of the sample.

132

200
111

331
400

311
220

222

333+511
422
420

440

I(Θ)

1275 K
875 K
575 K
300 K
10 K
10

90 100 110 120 130

Θ (degrees)
Figure 11.1: Diffraction patterns from nickel, taken in-situ at temperatures as labeled. The quality
of the patterns taken above room temperature is reduced due to the increased background of the
furnace.

11.3.2

Elastic scattering: in-situ neutron diffraction

The scattering with energy transfers between −2.5 meV and 2.5 meV was used to obtain diffraction
patterns from nickel, as shown in Fig. 11.1. Using the Nelson–Riley plots [104, 105] shown in Fig. 8.9,
the lattice parameter, a, was found at all measured temperatures. These are listed in Table 11.1,
and agree with values of the lattice parameter calculated using the accepted temperature dependent
linear coefficient of thermal expansion[168] and room temperature lattice parameter.[209]
In addition to those shown in Fig. 11.1, diffraction patterns were obtained without the furnace
at incident neutron energies of 30 meV, 50 meV, and 70 meV, with the 222 peak at 108◦ , 78◦ , and

Table 11.1: Experimentally determined lattice parameter, a, of nickel and shifts of the nickel phonon
energies as a function of temperature. Fits using a damped oscillator function and Eq. 11.4 yield
∆s , hET i/hE10 i was calculated with Eq. 11.6.
T (K) a ± 0.005 (Å)
∆s
hEiT /hEi10
10
3.513
1.000
1.000
300
3.521
0.988
0.985
575
3.540
0.970
0.963
875
3.559
0.947
0.945
1275
3.585
0.920
0.913

133
64◦ in 2θ respectively. These are shown in Fig. 8.8. The ratios of peak intensities remained largely
unchanged, showing that the sample did not have substantial crystallographic texture. Sample
texture should not affect lattice parameters as determined with Nelson-Riley plots. Regardless,
effects of the texture are addressed briefly in Sec. 11.3.5.

11.3.3

Inelastic scattering: S(Q,E) and the density of states

The data were rebinned again to obtain the intensity, I(Q, E), with momentum transfer Q ranging
from 0.0 Å−1 to 13.5 Å−1 with a binwidth of 0.0675 Å−1 . Since nickel is ferromagnetic up to the
Curie temperature, we excluded scattering at lower momentum transfers, where magnetic scattering
is present, to ensure that the scattering from the phonons was dominant. The elastic peak was
removed below ∼5 meV, and replaced with a straight line, corresponding to the continuum limit at
low energies. The data were then corrected for multiple and multiphonon scattering simultaneously
as described in § 8.1.2. The resulting DOS at all temperatures are shown in Fig. 11.2.

0.18
0.15
g(E) (1/meV)

1275 K
0.12
875 K
0.09
575 K
0.06
300 K
0.03
10 K

E (meV)
Figure 11.2: Phonon DOS for nickel at temperatures indicated. Markers are experimental data, lines
are Born–von Kármán fits. The increase in phonon lifetime broadening and the shifting of modes
to lower energies with increasing temperature is evident. The DOS are offset by integer multiples of
0.03 meV−1 .

134

11.3.4

Phonon shifts and broadening

We expect the broadening of the phonons to take the form of a damped harmonic oscillator function,[16,
113, 114] B(Q, E 0 , E), with central energy E 0 and width proportional to the phonon energy E:[54]
B(Q, E 0 , E) =

πQE 0

E0
E 2
+ Q12
E − E0

(11.3)

The quality factors, Q, were determined by a least-squares fit. First, the energies of the 10 K DOS
were scaled by a factor ∆s . Candidate fits of the hight temperature DOS were then found using:
gT (E) ≈ B(Q, E 0 , E)

g10 (∆s E 0 ) ,

(11.4)

where g10 is the phonon DOS at the temperature of 10 K, and

denotes an integral transform

similar to a convolution. The Q so determined at all temperatures are shown in Fig. 11.3, and the
fits to the DOS are shown in Fig. 11.4. Approximately, we find:
2Γ
≈ 6.816 × 10−8 T 2 ,

(11.5)

where T is in Kelvin. The shifts ∆s and the ratios of the mean phonon energies:
E gT (E) dE
hEiT
hEiT0
E gT0 (E) dE

(11.6)

0.14
1/Q (unitless)

0.12
0.1
0.08
0.06
0.04
0.02

200

400

600

800

1000

1200

1400

T (K)
Figure 11.3: Markers show the inverse of the quality factor, 1/Q, as a function of temperature for
nickel phonons. The line is a fit from Eq. 11.5.

135

0.18
0.15
g(E) (1/meV)

1275 K
0.12
875 K
0.09
575 K
0.06
300 K
0.03
10 K

E (meV)
Figure 11.4: Phonon DOS of nickel at temperatures as indicated. Markers are experimental data.
The lines are fits to the experimental data, acquired by shifting the 10 K DOS and performing an
integral transform the damped oscillator function as a kernel, as described by Eqs. 11.3 and 11.4.
The shifts are listed in Table 11.1. The DOS are offset by integer multiples of 0.02 meV−1 .
are presented in Table 11.1.

11.3.5

Born–von Kármán models of lattice dynamics

The DOS were fit with a Born–von Kármán model of the lattice dynamics.[71, 72] Force constants
out to fifth-nearest-neighbors were optimized using a gradient search method. The fits at all temperatures, after integral transforms with the damped oscillator and the Gaussian instrument resolution
functions as kernels, are shown in Fig. 11.2. The optimized force constants are listed in Table 11.2.
As a check on our calculated force constants, and on the effects of sample texture on our
determination of the DOS, comparisons were made to the dispersions measured by de Wit and
Brockhouse.[208] These are plotted in Fig. 11.5 Our 300 and 575 K models are in reasonable agreement with their 295 and 673 K models, respectively, as are trends in the dispersions with respect to
temperature.

136

11.3.6

Ab-initio calculations

In order to investigate electronic contributions to the entropy of FCC nickel, we used the planewave code VASP [130, 131] to calculate the electronic DOS as a function of unit cell volume in the
non-magnetic state, and as a function of unit cell volume and magnetization in the magnetic state.
The non-magnetic and spin-polarized (magnetic) calculations both used projector augmented plane
waves and the Perdew-Burke-Ernzerhof generalized gradient approximation. [133] A conventional
FCC cell was used, and it was relaxed using the ‘accurate’ setting for the kinetic energy cutoff, with
a 51 × 51 × 51 Monckhorst-Pack q-point grid. [134]
For the non-magnetic case, the relaxed lattice parameter was within 0.05% of the experimentally
determined lattice parameter, and was taken to be the 0 K lattice parameter of the unit cell. The
values of the linear coefficient of thermal expansion from Touloukian et al. [168] were used to
determine the corresponding volumes at 631 and 1500 K. The electronic DOS in the non-magnetic
state were determined at these two volumes, and are shown in Fig. 11.6.
For the spin polarized case, a ferromagnetic ground state was found. The relaxed lattice parameter was within 0.25% and the magnetic moment was within 1.3% of the experimentally determined
values. These were taken to be the 0 K lattice parameter and magnetic moment, and the values of
the linear coefficient of thermal expansion from Touloukian et al. [168] and of the relative magnetization from Crangle et al. [210] were used to fix the corresponding volumes and magnetizations at
0, 480, 556, 594, 618, 625, 628, 630, 631, 1200, and, 1500 K.b The electronic DOS of the majority
b For example, our values for the lattice parameter and magnetic moment of a conventional FCC Ni unit cell at
0 K were a0 = 3.52415 Å and µ0 = 2.4948 Bohr magnetons, respectively. At 480 K, we are at 76% of TC = 631 K.

Table 11.2: Optimized tensor force constants in N/m as a function of temperature for FCC nickel.
A Cartesian basis is used, where hxyzi is the bond vector for the given tensor components.
hxyzi
10 K
300 K
575 K
875 K 1275 K
[K1 ]xx
h110i
17.584 17.545 16.584 15.910 13.975
[K1 ]xy
18.976 18.253 18.822 17.670 16.915
[K1 ]zz
−0.391 −0.274 −0.384 −0.316 −0.345
[K2 ]xx
h200i
0.975
0.885
1.235
0.920
1.009
[K2 ]yy
−0.610 −0.993 −0.551 −0.559 −0.644
[K3 ]xx
h211i
0.593
0.442
0.518
0.440
0.850
[K3 ]xy
0.378
0.340
0.368
0.441
0.357
[K3 ]yy
0.302
0.133
0.220
0.157
0.325
[K3 ]yz
−0.120 −0.128 −0.105 −0.092 −0.106
[K4 ]xx
h220i
0.386
0.331
0.314
0.262
0.400
[K4 ]xy
0.517
0.412
0.502
0.444
0.466
[K4 ]zz
−0.218 −0.167 −0.127 −0.153 −0.217
[K5 ]xx
h310i −0.085 −0.065 −0.093 −0.078 −0.092
[K5 ]xy
−0.039 −0.047 −0.031 −0.035 −0.028
[K5 ]yy
0.006
0.003
0.007
0.004
0.006
[K5 ]zz
0.014
0.014
0.014
0.016
0.021

137
40
673 K
35
30
25
20

575 K

15
875 K

E (meV)

10
35

295 K

30
25
20
15
10

300 K

(000)

(001)

(011)

(000)

(111)

Figure 11.5: Phonon dispersions for nickel from BvK models. Along the bottom of the figure, points
(qx qy qz ) are marked such that zero corresponds to the zone center and 1 to a zone boundary. Paths
between these points are linear. Markers show the models of de Wit and Brockhouse[208] at 295 and
673 K, and lines show our models at 300, 575, and 875 K. At the lower temperature, the agreement
is quite good with the exception of the phonons near (111). This improves at higher temperatures;
however, the phonons between (001) and (011) develop some discrepancies.
and minority spin electronsc were determined at these volumes and magnetizations, and those at 0,
625, and 1500 K are shown in Fig. 11.7.
It should be noted that our assumption that the size of the magnetic moments are changing
with temperature is wrong. Actually, the size of the magnetic moments is largely temperatureindependent, and it is solely their orientations that change with temperature. Nevertheless, this
greatly simplified model may help guide our thoughts about the magnetic and non-magnetic contributions of the electrons to the entropy.
Crangle et al. report that the magnetization should be 0.7938 µ0 and Touloukian that the lattice parameter should be
1.005 a0 , so the 480 K simulation was run with the lattice parameter fixed to be 3.5418 Å and the magnetic moment
fixed to be 1.9804 Bohr magnetons.
c “Majority” or “minority” indicate the spin (up or down) that is carried by the majority or minority of the
electrons.

138

GNM(E) (states/atom/eV)

5.0
1500 K

4.0
3.0

631 K

2.0

1.0
0.0
-8

-7

-6

-5

-4
-3
-2
E - Ef (eV)

-1

Figure 11.6: Non-magnetic electronic DOS of nickel at temperatures as indicated, calculated as
described in the text.

11.4

Results and discussion

At all temperatures, both the longitudinal and the transverse force constants decrease rapidly with
nearest-neighbor distance. (These constants were found by diagonalizing the force constant tensors
[KX ], where X corresponds to the XN N shell. The longitudinal force constant was determined by
comparison to the projection of [KX ] onto the bond vector hxyzi, and the transverse modes were
taken to be the remaining two eigenvalues.) Only the first-nearest-neighbor (1NN) longitudinal force
constants show a monotonic decrease with temperature, and are almost solely responsible for the
shift of the DOS to lower energies with increasing temperature. All longitudinal force constants are
plotted in Fig. 11.8.
The softening of the measured phonon DOS is consistent with that found by de Wit and Brockhouse. They found hEi573 /hEi295 = 0.976,[208] and we find hEi575 /hEi300 = 0.978. The values
of Q shown in Fig. 11.3 are related to the full widths at half maximum, 2Γ, of the phonon peaks,
through the equation 1/Q ≈ 2Γ/E. Values of a similar magnitude were found experimentally for
BCC titanium, zirconium, and hafnium,[16, 113, 114] and our 2Γ values are also comparable in magnitude to the values found by Zoli, et al., for phonon-phonon interactions in aluminum and the noble
FCC metals.[64] The quadratic form of 1/Q seen in Fig. 11.3 is also consistent with phonon-phonon
interactions, if we assume that the damping is proportional to the number of phonons, and that the

139

5.0
4.5

1500 K

4.0
Total
(Minority + Majority)

3.5
3.0
2.5

G(E) (states/atom/eV)

2.0
1.5

625 K

1.0
0.5
0.0
2.5
2.0

Majority spin

1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5

(Minority spin)

-2.0
-2.5
-8

-7

-6

-5

-4 -3 -2
E - Ef (eV)

-1

Figure 11.7: Spin-polarized and non-magnetic electronic DOS of nickel calculated as described in the
text. In both the top and bottom panels, the red-dotted, yellow, and purple-dashed lines correspond
to 0, 625, and 1500 K respectively. The bottom panel shows the DOS for majority and minority
spin electrons. As the temperature and volume increase and the magnetization decreases, electronic
density accumulates at the Fermi energy. This can be seen in the top panel, where the total density
of states at the Fermi level rises by a factor of 2 going from 0 to 1500 K. (Actually, this occurs going
from 0 to 631 K, the Curie temperature, as can be seen by comparing the 1500 K DOS shown here
with the very similar 631 K DOS shown in Fig. 11.6).

number of phonons is linear in T.
To find the entropy from the softening of the DOS, we use the following expression:

Z ∞
gT 0 [(nT + 1) ln(nT + 1) − nT ln(nT )] dE ,

Sph (T, T ) = 3kB

(11.7)

140

36.000

Longitudinal Force Constant (N/m)

34.000
32.000

1NN

1.200
2NN
1.000

3NN
4NN

0.800
-0.090
-0.095

5NN

-0.100

200

400

600
T (K)

800

1000

1200

Figure 11.8: Longitudinal force constants for first through fifth-nearest neighbors as indicated.

where gT 0 is the phonon DOS at temperature T 0 and nT is the mean occupancy for bosons at
temperature T (both gT 0 and nT are functions of E). We seek the change in entropy due to changes
in the phonon states, not due to changes in phonon occupancy. We calculate the difference between
the total phonon entropy and the harmonic phonon entropy as:
Sph − SH = Sph (T, T ) − Sph (T, T0 ) .

(11.8)

We now compare the entropy of phonon softening to the entropy of dilation. With tabulated
data for the elastic constants of nickel,[211] we use:
Z T
Sph,D = Sph,D (T ) =

CP − CV
dT 0 =
T0

Z T

9KT α2 0
dT ,
ρN

(11.9)

with T0 =10 K, to calculate, Sph,D , the entropy of dilation. The center panel of Fig. 11.10 shows

141

Averaged Transverse Force Constant (N/m)

0.021
0.019
5NN

0.017
0.015

3NN

0.000
-0.300

4NN

-0.600
2NN

-0.900
-0.500

1NN
-1.000
-1.500

200

400

600
T (K)

800

1000

1200

Figure 11.9: Averaged transverse force constants for first through fifth-nearest neighbors as indicated.

the entropy of dilation and Sph − Sph,H as determined with Eqs. 11.7 and 11.8. The agreement
is generally good. The anharmonic entropy, shown as a solid line, is the difference between these
two contributions. Over the entire temperature range, the anharmonic entropy is bounded by
−0.08 < Sph,A < 0.05 k B /atom, where we have already incorporated our errors of ±0.02 k B /atom.
Negative values indicate phonons that are slightly stiffer than they would be if their energies were
determined by lattice expansion against the bulk modulus alone. This is nominally the case above
700 K where we have −0.08 < Sph,A < 0.03 k B /atom. The crossover from positive to negative
appears to occur somewhere in the vicinity of the Curie temperature, and the trends in the sign of
Sph − Sph,H appear to be consistent over either the ferromagnetic or the paramagnetic regions.
We will now consider the high temperature contributions of the magnetism to the entropy: First,
by using values for S −Sph and for Sel taken from Wallace [37] and second by using the NIST-JANAF
values for the total entropy, our measurements of the phonon entropy, and the electronic entropy as
determined from the electronic structure calculations described in § 11.3.6.

142

11.0
SJanaf

10.0
9.0
8.0
Sph+Sel,NM

7.0

Sph Sph,H

6.0

Sph+Sel,G
Sph+Sel

5.0
4.0
3.0
Sel,NM

2.0

Sel,G

1.0
S (kB/atom)

0.0

Sel

-1.0
0.6
0.5
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
0.6

Sph,D

Sph,NH
Sph,A
S-Sph-Sel,G

0.4

S-Sph-Sel

0.2
0.0

S-Sph-Sel,NM

-0.2

250

500

750 1000 1250 1500 1750
T (K)

Figure 11.10: Contributions to the entropy of nickel, as labeled in the plot. The vertical gray line
denotes the highest temperature at which the phonon spectrum was measured. Values for the total
entropy, SJanaf , were taken from the NIST-JANAF thermochemical tables[136] and are shown as
open triangles. Closed markers are data from the current experiment; lines are either calculations
or interpolations. Calculation of the various contributions is described in the text.
From Wallace’s plots of Sel and S − Sph ,[37] we find the sum of the magnetic and anharmonic contributions to the entropy to be approximately 0.23 k B /atom at 1275 K. Wallace found

143
that the sum of the magnetic and anharmonic entropies at the melting temperature, 1728 K, was
0.31 k B /atom. Wallace assumes that the anharmonic contribution is zero, and attributes the entire quantity to magnetic entropy. It appears that the magnetic entropy is slightly larger. At
1275 K, we find −0.08 < Sph,A < −0.04 k B /atom, subtracting these values from Sph,A + Sph,M ,
we obtain 0.27 < Sel,M < 0.31 k B /atom. If we take this to be the value at melting, we get
0.35 < Sel,M < 0.39 k B /atom. The center panel of Fig. 11.10 suggests that the anharmonic entropy
will decrease linearly with temperature above 1273 K, so the magnetic entropy at melting may be
larger than was previously suggested. If we assume that this trend continues out to 1728 K, we get
Sph,A (1728) ≈ −0.11 k B /atom, and Sel,M (1728) ≈ 0.42 k B /atom.
Using the electronic DOS shown in Figs. 11.6 and 11.7, we may determine various electronic
entropies as follows: First, with Nel the number of electrons and fT (E) the mean occupancy for
fermions at temperature T , we use:
Nel =

GT (E)fT (E)dE ,

(11.10)

to find the chemical potential, µ. Then we have:
Sel

= −kB

dEGT [(1 − fT ) ln (1 − fT ) + fT ln (fT )] ,

(11.11)

dEG0 [(1 − fT ) ln (1 − fT ) + fT ln (fT )] ,

(11.12)

dEGNM
[(1 − fT ) ln (1 − fT ) + fT ln (fT )] .

(11.13)

Sel,G

= −kB

Sel,NM

= −kB

The values of the electronic entropy used by Wallace [37] are derived from a non-magnetic determination of the electron DOS. This should be analogous to our Sel,NM , determined with the
electronic DOS in Fig. 11.6. Using this, the experimentally determined phonon entropy Sph , the
values of the total entropy from the NIST-JANAF, SJanaf — all of which are shown in the top panel
of Fig. 11.10 — we may determine the unaccounted for entropy. Assuming that all unaccounted for
entropy is magnetic in origin, we find Sel,M = 0.14 kB /atom at 1275 K. Extrapolating to the melting
temperature, TM = 1728 K, we get Sel,M = 0.23 kB /atom. This is considerably smaller than the
value found using plots from Wallace [37] and our value for the anharmonic phonon entropy.
Part of this discrepancy stems from our differing estimates of Sel,NM . Again, looking at plots from
Wallace [37], we see that at 1275 K, he has Sel,NM ≈ 0.88 kB /atom, whereas we get 0.95 kB /atom.
The difference, δSel,NM ≈ 0.07 kB /atom, is only about half of the difference in the estimates of the
magnetic entropy, δSel,M = 0.14kB /atom at 1275 K. Given the information presented by Wallace,
the rest of the difference is much more difficult to sort out; however, it seems likely that it stems
more from values of the total entropy and of the phonon entropy of dilation than from values of the
phonon entropy, Sph . Specifically, values of the total entropy at 1275 K from Meschter et al. [31] or

144
from NIST-JANAF [136] vary by roughly 0.3 kB /atom. Measurements of temperature dependence
of the thermal expansion coefficient and the bulk modulus are also subject to variation, and this will
impact the fraction of the nonharmonic entropy, Sph,NH = Sph,D + Sph,A that goes into the dilation
contribution as opposed to the anharmonic contribution to the phonon entropy.

11.5

Summary

Phonon spectra of FCC nickel were measured by time-of-flight neutron spectrometry over a wide
range of temperatures spanning from 10 K to 1275 K. The softening of the DOS was generally
consistent with the softening expected from expansion of the lattice against the bulk modulus, but the
softening is less than expected at high temperatures. We are able to bound the entropic contribution
from phonon anharmonicity to −0.08 < Sph,A < −0.04 kB /atom at 1275 K. Taking values of Sph,A +
Sel,M from Wallace, [37] this bounds the contribution of the magnetic entropy to 0.27 < Sel,M <
0.31 kB /atom at 1275 K. Additionally, we found there to be significant broadening of the phonons
with increased temperature, which we tentatively attribute to phonon-phonon interactions.

145

Part IV

Phonons in BCC Metals at
Elevated Temperatures
Nuclear resonant inelastic x-ray scattering (NRIXS) measurements of polycrystalline iron
are presented in detail here, and inelastic neutron scattering measurements of polycrystalline chromium and vanadium are briefly summarized. With the exception of § 12.4.1,
the analyses and computations for iron are quite similar to the corresponding ones for
nickel described in § 11.3.

146

Chapter 12

Iron
12.1

Introduction

Due to its abundance, technological value, importance to geology, myriad solid-solid phase transitions, and itinerant electron magnetism, the fundamental physics of iron and its phase diagram is
and has been the focus of a great deal of theoretical [17–20, 46, 212–229] and experimental [230–241]
research.
Iron comprises roughly 5% of the Earth’s crust, and its use in steel alone makes it one of the most
widely used metals in industry. Additionally, due to its low cost and relatively strong magnetism, it
is frequently used in magnetic components. [242] Much of our understanding of geodynamics comes
from studying seismic waves. Accurate values of the sound velocity of iron at high temperatures
and pressures are critical to refining our understanding of the composition and dynamics of Earth’s
iron core. At 0 K and ambient pressures, iron is a ferromagnet which takes the body-centeredcubic (BCC) structure. It undergoes a transition from ferromagnet to paramagnet at TC = 1043 K,
transforms to a face-centered-cubic (FCC) structure at 1185 K, and then transforms back to a
BCC structure at 1667 K. At room temperature, iron undergoes a phase transition to a hexagonalclose-packed (HCP) structure somewhere between 10 and 16 GPa.[243] Understanding these phase
transitions requires detailed knowledge of the electronic structure of iron — including its magnetism
— as well as its vibrational dynamics.
According to Moriya and Takahashi, [193] the debate over whether magnetism in iron is due to
localized or itinerant electrons has spanned nearly half a century. [194–200] Briefly, we paraphrase
their summary of the situation: A purely localized electron magnet has magnetic excitations that
are localized in real space; whereas those of a purely itinerant electron magnet are localized in reciprocal space. For magnetic 3d transition metals, the former fails to explain nonintegral values
(in Bohr magnetons) of the saturization magnetization and predicts values of the low temperature
cohesive energy and specific heat that are too small. The latter, on the other hand, predicts Curie
temperatures, TC , that are too high, and fails to predict the Curie-Weiss behavior of the magnetic

147
susceptibility at high temperatures. Thus, it is clear that neither of these extremes offers a satisfactory explanation of the magnetism in iron; rather, spin excitations in iron are somewhat localized
(and somewhat delocalized) in both real and reciprocal space. The summary is from 1984; however, despite great progress, our understanding of the electronic structure of magnetic transition
metals — particularly in their high temperature paramagnetic states — is still less than satisfactory. [32, 45, 47] For example, spin-polarized electronic structure calculations including adiabatic
spin-dynamics only successfully predict the magnetization of iron to roughly T2C . [191, 192]
Iron is a strong, coherent scatterer of neutrons; therefore, its phonon dispersions have been
measured using triple-axis neutron spectrometry, [244–247] including studies of the temperature
dependence [59, 248–250] and the pressure dependence. [251, 252] Many of these studies are targeted
at determining whether or not the phonons play a role in the various solid-solid phase transitions.
There have also been a number of neutron measurements of the critical scattering [253, 254] and of
the magnetic scattering including its temperature dependence. [255–258] Most of these experiments
explored the nature of the magnetic excitations in iron, as well as to determine their role in the
solid-solid phase transitions. Finally, there is a good deal of theoretical interest in the interactions
between the vibrational and magnetic excitations. [45, 259–261]
From measurements of magnetic scattering, we know that only the lowest energy magnetic excitations contribute to the thermodynamics of iron. [191, 192] At low energies, the spin dispersions
of iron are expected to be isotropic in q; [258] therefore, a measurement in a single direction in
q-space may yield a picture of the spin dynamics that is sufficient for thermodynamic use. Phonons,
on the other hand, may vary significantly with q. Born–von Kármán models provide a means of
interpolating data from a few high-symmetry directions over the whole Brillouin zone; however, the
phonon spectra from these models often differ slightly from the spectra found by experiments that
sample a larger portion of the Brillouin zone. As such, a measurement of the phonons over the entire
Brillouin zone is desirable, as it answers directly any questions about the phonon contributions to
the thermodynamics.a
Iron is a Mössbauer isotope, and with the advent of synchotron x-ray sources, it has become a
test material for nuclear forward and resonant inelastic x-ray scattering instruments. [262–275] As
the requirements on sample size are smaller i for x-ray scattering than for neutron scattering, practitioners of NRIXS have paid particular attention to iron under pressure. [274–282] NRIXS provides
a direct measurement of the phonon spectrum, and analysis of NRIXS data typically requires only
the assumption of harmonic phonons, and has no need of BvK models.b Additionally, NRIXS is an
a Most determinations of phonon spectra are made under the assumption of harmonicity. This might also lead to
inaccuracies in our interpretations of the vibrational thermodynamics; perhaps of the same order of magnitude as
using BvK models for the vibrational dynamics.
b Analysis of data from neutron time-of-flight chopper spectrometers is also frequently done without BvK models;
however, this involves the incoherent approximation. Quite generally, the analysis of data from neutron time-of-flight
chopper spectrometers is significantly more involved than for NRIXS, and a brief description of this is given towards
the end of § 12.2.

148
extremely low noise measurement; therefore, the phonon spectra determined by this technique are
likely of the highest quality currently available.
Here, we present NRIXS measurements of the phonon spectrum for the BCC phase from near
absolute zero to near the Curie point. We find that there is a large differential softening between the
first and second transverse modes. Specifically, the former soften dramatically upon approaching the
Curie temperature; whereas the latter seem to soften largely in proportion to the longitudinal modes.
We also consider the phonon contributions to the entropy of iron, and find that the phonons in iron
are highly anharmonic. Finally, we will comment on the electronic and magnetic contributions to
the entropy.

12.2

Background

Nuclear resonant inelastic x-ray scattering (NRIXS), also known as the phonon-assisted Mössbauer
effect, has previously been described in great detail. [270–275, 283–286] Here we present only a brief
summary. Additionally, many of the principles are similar to those for the scattering of neutrons by
phonons, which were presented in Chapter 5.
A free nucleus may enter an excited state when a photon of the right energy is incident upon
it. After some time, the nucleus relaxes back to the lower energy state and emits a γ-ray. In both
cases, conservation of momentum forces the nucleus to recoil, and the recoil energy ER is given by:
ER =

Et2
2M c2

(12.1)

where M is the mass of the nucleus, c is the speed of light, and Et is the energy of the nuclear
transition. For a free nucleus, the recoil energy can be a significant fraction of the linewidth of
the nuclear excited state; causing the emitted photon to lack the energy required to excite another
(similar) nucleus. If the nucleus is in a crystal, however, the recoil energy may be taken up by the
entire lattice. The mass in Eq. 12.1, then, becomes the mass of the crystal, and the recoil energy
becomes negligible relative to the linewidth. The emitted photon thus retains the energy required
to excite another nucleus, and a resonance occurs. Rudolph Mössbauer discovered this process, for
which he won the 1961 Nobel prize.
In a NRIXS experiment, the incident x-rays are detuned from resonance by energies that may be
made up for by the creation or annihilation of phonons (or other excitations coupled to the nuclei)
in the sample. The resulting counts are proportional to the excitation probability density for the
1X
resonant nuclei in the sample, Sr (E). This is precisely
SP (ω) as defined around Eqs. 5.48, and
5.50, and thus gives access to the partial phonon spectrum for the resonant nucleus.
With inelastic neutron scattering, a lack of incident neutron flux, insufficient energy resolution,

Nuclear resonant spectroscopy

S505
149

Figure 12.1: A schematic of the importance of timing in a nuclear resonant inelastic x-ray scattering
Figure 3. Scattered intensity versus time after excitation. At time zero, a SR pulse excites a
experiment. The x-ray pulse arrives at time t = 0, and is followed by a huge amount of electronic
a nuclear
from
is prompt, i.e.,
scattering.material
After a containing
few picoseconds
this hasresonant
subsided,isotope.
and afterThe
a fewscattering
nanoseconds
theelectrons
counts from
almost
immediately
after
the
pulse
arrived.
The
response
of
the
resonant
nuclei
is
delayed. Time
the nuclear resonant scattering may be collected without background. Figure taken from “Nuclear
resonant spectroscopy”
W. Sturhahn.
discriminationbypermits
one to [284]
distinguish nuclear and electronic scattering.

and a wide variety of unwanted scattering create challenges for the experimentalist. For NRIXS, none
of these problems exist provided that an appropriate resonant nucleus is used. Synchotron sources
generate huge numbers of x-rays, and the tuning of the incident x-ray beam may be accomplished
with modern high resolution monochromators. [287–291] The background may be suppressed entirely
using what is sometimes called the time discrimination trick . When a pulse of x-rays hits the sample,
a huge intensity of electronic scattering occurs on a picosecond timescale. If the linewidth of the
excited state of the resonant nucleus is sufficiently small, the γ-rays emitted by the nuclei upon
relaxation may be delayed significantly. This is shown schematically in Fig.12.1. The disadvantage
of NRIXS, then, is that it is only feasible for nuclei that have excited states at accessible energies
with appropriate lifetimes.c Further, the technique is only sensitive to the properties of the resonant
nuclei; therefore, it is only able to give a partial account of the vibrational dynamics for most
materials.d
Fortunately, the first excited state of 57 Fe is at 14.413 keV, and its linewidth is 4.66 neV, which
c Neutron scattering is also not feasible for all elements, as some of them absorb too many neutrons. That said, the
number of elements suitable for neutron scattering far exceeds the number of isotopes currently suitable for NRIXS.
d Conversely, neutrons generally give information about the vibrational dynamics of all the nuclei in the sample;
Figure
4. NRIXS
of an
iron
foil neutron
at ambient
conditions.
Thecan
graph
events
albeit weighted
by their
scatteringspectrum
cross-sections.
The
varying
scattering
cross-sections
makeshows
sortingmeasured
out
which vibrations
come from
whichphotons
nuclei extremely
for delayed
x-ray
versusdifficult.
the energy of the incident SR. Energy zero corresponds to the

nuclear transition energy of 14.4125 keV. Positive energies indicate the region of phonon creation;
i.e., the energy of the x-ray is too large to excite the nuclear resonance directly, and phonons must
be created simultaneously. In the region of negative energies, the x-ray energy is too small, and
phonons must be annihilated to produce resonant excitation.

150
corresponds to a lifetime of 141 ns. Of these energies, the former is easily achievable with modern
synchotrons and monochromators, and the latter is sufficiently small to allow for use of the time
discrimination trick. This makes iron an ideal nucleus for NRIXS experiments.

12.3

Experiment

12.3.1

Sample preparation

Samples were made of 96.06% 57 Fe enriched ingot of 99.99% chemical purity rolled to a foil of 27 µm
thickness. To remove strain and crystallographic texture, they were sealed in an evacuated quartz
tube, heat treated at 1173 K for 30 minutes, and quenched in iced brine. The resulting samples
showed no visible signs of texture, preferred orientation, or oxidation.

12.3.2

Nuclear resonant inelastic x-ray scattering at HP-CAT

The NRIXS experiment was performed at beam line 16-IDD of the High Pressure Collaborative
Access Team at the Advanced Photon Source at Argonne National Laboratory. The synchotron
ring was operated in top-up singlet mode with x-ray bunches separated by 153 ns. The crystal
monochromator at this beamline was diamond 111, and the focused beam size was 20 µm× 50µm.
The incident x-ray energy was tuned to 14.413 keV to match the 57 Fe resonance. Data were collected
in scans of the incident energy from -120 to 120 meV with a monochromator resolution of 2 meV.
A room temperature (295 K) measurement was made with the sample placed at a grazing angle
with respect to the incident photon beam, and the delayed signal was detected by two avalanche
photodiode (APD) detectors. These were positioned opposite each other, avoiding the forward beam,
with a separation of approximately 5 mm. For measurements at high temperatures, the sample was
mounted in a custom furnace as described below. The 57 Fe foil was cylindrically wrapped around a
single resistive heating element as the primary radiation shield. Several other niobium and copper
radiation shields were utilized, each with a small slit allowing passage of the incident and scattered
photons. The temperature was monitored with a single thermocouple attached to the 57 Fe foil, and
was maintained using a temperature controller and DC power supply. The entire assembly was held
under vacuum in an aluminum tube with a Kapton window giving access to the sample. Two APDs
were positioned approximately 2 cm from the sample. Measurements were taken at 523, 773, 923,
and 1023 K, after which the sample was allowed to cool. No visible oxidation was present after the
heating. Finally, the sample was remounted in a He-flow cryostat equipped with a Be dome, with
a single APD positioned several centimeters (∼ 4 cm) from the sample for a measurement at 21 K.
The normalized spectra are shown in Fig. 12.2.

151

12.4

Analysis and computation

Details of the determination of the quality factors and fitting of BvK models are given in § 8.4, § 8.2,
and § 8.3.1.2, respectively.

12.4.1

S(E) and the phonon density of states

The phonon DOS were extracted from the scattering using PHOENIX. [292] The program performs
all the necessary corrections including removal of the elastic peak and multiphonon scattering using

4000
3500

S(E) (Counts)

3000
1023 K
2500
923 K
2000
773 K
1500
523 K
1000
295 K
500
21 K
-100

-50

100

E (meV)
Figure 12.2: Phonon probability densities, S(E) of iron at temperatures as indicated. The elastic
peaks extend well above the figure. The absence of all background scattering is particularly evident
at 21 K. As the temperature increases, the phonon annihilation (left) side of the spectrum increases
relative to the phonon creation (right) side as dictated by detailed balance. The multiphonon
scattering — above ∼35 meV also becomes more significant with increasing temperature.

152

0.21
0.18
1023 K
0.15
g(E) (1/meV)

923 K
0.12
773 K
0.09
523 K
0.06
295 K
0.03
21 K

E (meV)
Figure 12.3: Phonon DOS of iron at temperatures as indicated. The markers show the experimentally
determined DOS, and the lines the best fits of the 21 K DOS to the high temperature DOS by scaling
of the energy and integral transform with the damped oscillator function as a kernel. (Eq. 12.3).

the Fourier-log technique described in § 8.1.1. The phonon spectra thus determined are shown by
markers in Figs. 12.3 and 12.4.

12.4.2

Phonon shifts and broadening

Fits of the quality factor Q were performed as described in § 8.2, and are shown in Fig. 12.3. Briefly,
the phonon broadening was assumed to take the form:
B(Q, E 0 , E) =

πQE 0

E0
E 2
+ Q12
E − E0

(12.2)

The high temperature phonon DOS was then approximated as a function of the low temperature
DOS with only two free parameters, ∆s and Q:
gT (E) = B(Q, E 0 , E)

g21 (∆s E 0 ) ,

(12.3)

153

0.21
0.18
1023 K
0.15
g(E) (1/meV)

923 K
0.12
773 K
0.09
523 K
0.06
295 K
0.03
21 K

E (meV)
Figure 12.4: Phonon DOS for iron at temperatures as indicated. Markers show experimental data,
lines show Born–von Kármán fits. There is a large overall shift of the modes to lower frequencies,
and the changes in the shapes of the first and second transverse peaks and the longitudinal peak are
strong indications of anharmonicity. The DOS are offset by integer multiples of 0.03 meV−1 .

where

denotes an integral transform over the variable E 0 , ∆s rescales the phonon energy, gT is

the phonon DOS at temperature T , and g21 is the phonon DOS at 21 K — the lowest temperature
for which we have a measurement. At 923 and 1023 K, the Q are of order 100. As such, the
transformation with the damped harmonic oscillator function had a negligible effect on the DOS,
and it is mainly the effects of the shift, ∆s , that is seen in the solid lines in Fig. 12.3. Because of
this, and because of the failure of this model at high T, we will subsequently take B = 1 — i.e., we
assume oscillators of infinite quality.

154
Table 12.1: Optimized tensor force constants in N/m as a function of temperature for BCC iron. A
Cartesian basis is used, where hxyzi is the bond vector for the given tensor components.
hxyzi
21 K
295 K
523 K
773 K
923 K 1023 K
[K1 ]xx
h111i
17.263 16.213 15.641 13.887 14.020 13.336
[K1 ]xy
14.910 14.941 14.489 14.354 13.729 13.652
[K2 ]xx
h200i
15.314 14.568 15.103 15.289 12.726 12.829
[K2 ]yy
0.115
0.311
0.652
0.943
1.128
0.943
[K3 ]xx
h220i
1.020
1.289
0.961
0.843
0.528
0.352
[K3 ]xy
0.273
0.329
0.054 −0.749 −0.812 −1.847
[K3 ]zz
−0.393
0.450 −0.385 −0.320 −0.256 −0.077
[K4 ]xx
h311i −0.286 −0.191 −0.340 −0.426 −0.600 −0.362
[K4 ]xy
−0.067
0.016 −0.101 −0.179 −0.331 −0.398
[K4 ]yy
0.048 −0.001
0.033
0.002
0.050 −0.167
[K4 ]yz
0.566
0.891
0.654
1.006
0.700
1.332
[K4 ]zz
0.048 −0.001
0.033
0.002
0.050 −0.167
[K5 ]xx
h222i −0.382 −0.411 −0.316 −0.201 −0.485 −0.419
[K5 ]xy
0.090
0.465
0.575
1.166
0.674
1.053

12.4.3

Born–von Kármán models of lattice dynamics

The phonon spectra were then fit with Born–von Kármán models of the lattice dynamics. Tensorial
force constants out to the 5NN shell were determined using a Nelder-Mead simplex algorithm as
implemented in SciPy [293] as per the technique discussed briefly in § 8.3.1.1. The experimentally
determined resolution function was used for Z, and we assumed B = 1 as explained above. The force
constants so determined are listed in Table 12.1, and the best fits to the DOS at all temperatures
are shown in Fig. 12.4.
The longitudinal and average transverse force constants were determined in the manner described
in § 8.3.2. Briefly, the longitudinal force constants were found by projecting the tensor onto the
bond vectors hxyzi. The 3 × 3 force constant tensors were then diagonalized. The longitudinal force
constant was matched to one of the eigenvalues and the average transverse constant was taken to
be the mean of the two remaining eigenvalues. Longitudinal force constants to 3NN are shown in
Fig. 12.5, and the averaged transverse force constants out to 3NN are shown in Fig. 12.6. The 1NN
longitudinal force constants, K1 (T ), decrease with increasing temperature approximately as:
K1 (T ) = 47.689 − 6.640 × 10−3 T ,

(12.4)

where T is in degrees Kelvin and K1 (T ) is in N/m. The fit is also shown in Fig. 12.5.

12.4.4

Ab-initio calculations

To investigate electronic contributions to the entropy of BCC iron, we used the plane-wave code
VASP [130, 131] to calculate the electronic DOS of iron as a function of unit cell volume in the

155

47.00
Longitudinal Force Constant (N/m)

45.00
43.00
1NN
41.00
16.00
15.00
14.00

2NN

13.00
1.50
0.50
-0.50

3NN

-1.50

200

400

600
T (K)

800

1000

Figure 12.5: Markers show longitudinal force constants of iron for first through third nearest neighbors as indicated. The lines show linear fits, which appear to match reasonably well only for the
1NN shell.

non-magnetic state, and as a function of unit cell volume and magnetization in the magnetic state.
The non-magnetic and spin-polarized (magnetic) calculations both used projector augmented plane
waves and the Perdew-Burke-Ernzerhof generalized gradient approximation. [133] A conventional
BCC cell was used, and it was relaxed using the ‘accurate’ setting for the kinetic energy cutoff, with
a 31 × 31 × 31 Monckhorst-Pack q-point grid. [134]
For the non-magnetic case, the relaxed lattice parameter was within 4% of the experimentally

Average Transverse Force Constant (N/m)

156

2.50
1NN
2.00
1.50
1.00
3NN
0.50
2NN
0.00
-0.50

200

400

600
T (K)

800

1000

Figure 12.6: The markers and lines show averaged transverse force constants of iron. In the 1NN
shell, the transverse forces appear to go rapidly to zero by roughly 750 K. The transverse forces
for the second and third neighbor shells, on the other hand, appear to increase with increasing
temperature.

determined lattice parameter, and was taken to be the 0 K lattice parameter of the unit cell. The
values of the linear coefficient of thermal expansion from Touloukian et al. [168] were used to
determine the corresponding volume at 1043 K. The electronic DOS in the non-magnetic state were
determined at these two volumes, and are shown in Fig. 12.7.
For the spin polarized case, a ferromagnetic ground state was found. The relaxed lattice parameter was within 1% and the magnetic moment was within 0.5% of the experimentally determined
values. These were taken to be the 0 K lattice parameter and magnetic moment, and the values of
the linear coefficient of thermal expansion from Touloukian et al. [168] and of the relative magnetization from Crangle et al. [210] were used to fix the corresponding volumes and magnetizations at

157

GNM(E) (states/atom/eV)

5.0
4.0

1043 K

3.0
2.0

1.0
0.0
-7

-6

-5

-4

-3
-2
-1
E - Ef (eV)

Figure 12.7: Non-magnetic electronic DOS of iron at temperatures as indicated, calculated as described in the text. The increased volume at 1043 K is accompanied by a modest increase of the
density of states at the Fermi level.

522, 793, 918, 981, 1022, 1033, 1038, 1041, 1043, and 1150 K. e The electronic DOS of the majority
and minority spin electronsf were determined at these volumes and magnetizations, and those at 0,
1038, and 1150 K are shown in Fig. 12.8.
As in Chapter 11, we point out that our tacit assumption that the size of the magnetic moments
are changing with temperature is wrong. Rather it is the orientations of the spins that change with
temperature. Regardless, this model can give us some guidance as we consider the magnetic and
non-magnetic contributions of the electrons to the entropy.

12.5

Results and discussion

The best fits of the quality factor and shift as a function of temperature shown in Fig. 12.3 fail
quite dramatically for temperatures 773 K and higher. Specifically, they are unable to capture the
differential shifting of the two transverse peaks. Even at 523 K the deviations in the two transverse
e For example, our values for the lattice parameter and magnetic moment of a conventional BCC Fe unit cell at
0 K were a0 = 2.8346 Å and µ0 = 4.4153 Bohr magnetons, respectively. At 522 K, we are at 50% of TC = 1043 K.
Crangle et al. report that the magnetization should be 0.9400 µ0 and Touloukian that the lattice parameter should be
1.005a0 , so the 522 K simulation was run with the lattice parameter fixed to be 2.8488 Å and the magnetic moment
fixed to be 4.1504 Bohr magnetons.
f “Majority” or “minority” indicate the spin (up or down) that is carried by the majority or minority of the
electrons.

158

Total
(Minority + Majority)
1150 K

5.0
4.0

1038 K

G(E) (states/atom/eV)

3.0
0K
2.0
1.0
0.0
3.0
Majority spin
2.0
1.0
0.0
-1.0
-2.0
(Minority spin)
-3.0
-6

-5

-4

-3

-2 -1
E - Ef (eV)

Figure 12.8: Spin-polarized and non-magnetic electronic DOS of iron calculated as described in the
text. In both the top and bottom panels, the red-dotted, yellow, and purple-dashed lines correspond
to 0, 1038, and 1150 K respectively. The bottom panel shows the DOS for majority and minority
spin electrons. As the temperature and volume increase and the magnetization decreases, electronic
density accumulates at the Fermi energy. This can be seen in the top panel, where the total density
of states at the Fermi level rises by a factor of 4 going from 0 to 1150 K. (Actually, this occurs going
from 0 to 1043 K, the Curie temperature, as can be seen by comparing the 1150 K DOS shown here
with the nearly identical 1043 K DOS shown in Fig. 12.7).

peaks are noticeable in the fit; nevertheless, at 295 and 523 K the fits are much better than at the
higher temperatures. The first transverse peak shifts to lower energies much more rapidly than
the second transverse or the longitudinal peaks. This trend has been seen in triple-axis neutron

159
scattering measurements by Neuhaus et al. [248] and Satija et al.,g [261] as well as in elastic constants
measured by Alves and Vâllera. [260] The effect can also been seen in theoretical calculations of the
phonon spectrum of iron by Hasegawa et al. [214] These temperature effects are somewhat in contrast
to measurements by Klotz and Braden [252] of the pressure dependence of the phonons, which
show no differential shifting of the first transverse modes. Similarly for calculations of the pressure
dependence by Sha and Cohen. [243] The longitudinal peak appears to broaden with temperature;
however, it lacks the damped harmonic oscillator-like tails seen in the spectra of the FCC metals, Al,
Pb, and Ni as seen in Chaps. 9, 10, and 11. [35, 42] At room temperature, Minkiewicz et al. found
that the measured phonon linewidths could be attributed to their instrument resolution. [247] For
the [110]T1 zone boundary phonon, Satija et al. report negligible linewidths at room temperature,
and a linewidth on the order of 1 meV at TC . [261] Our assumption for our BvK models that B = 1,
then, is reasonable but not perfect. We note that the anharmonicties in the FCC metals are different,
as the damping of the modes is far and away their most obvious manifestation. Here, the effects
of anharmonicity on the phonon spectra are much more noticeable in the energy shifts, which are
larger than expected.
The longitudinal constants for the first two shells of neighbors decrease with increasing temperature, and are thus largely responsible for the large softening of the phonon modes. The decrease
in the 1NN longitudinal force constant is quite linear from room temperature up, and likely makes
the largest contribution to the softening. The 3NN shell also contributes to the softening up to
around 750 K, after which it becomes negative and begins increasing in magnitude. Room temperature force constants found by Brockhouse et al., [245] Bergsma et al., [246], and Minkiewicz
et al. [247], as well as force constants found by Klotz and Braden [252] at room temperature and
both ambient pressure and 9.8 GPa all agree with our finding that [K1 ]xx > [K1 ]xy from 21-523 K,
implying that at these temperatures the 1NN transverse forces are bonding rather than repulsive.
Zaretsky and Stassis [250] find [K1 ]xx < [K1 ]xy for the FCC phase, at 1428 K, indicating repulsive
transverse forces. From 773 K up we find that [K1 ]xx − [K1 ]xy is scattered about zero, which is
at least consistent with the idea that the 1NN transverse forces go from bonding to repulsive with
increasing temperature. Also around 750 K, the averaged transverse constant in the 1NN shell stops
decreasing in magnitude and appears to be scattered about zero. The instability in the longitudinal
forces in the 3NN shell, the switch from bonding to repulsive transverse forces in the 1NN shell,
and the overall decrease in the 1NN transverse forces might be related to the strong anharmonicity
seen in the transverse peaks of the phonon DOS, and may contribute to the BCC to FCC phase
transition that occurs at 1185 K. Indeed, Neuhaus et al. cite the shifts in the transverse branches
as an indication of a low potential energy barrier for displacements towards an FCC structure, and
g These authors speak of softening in the T2 or second transverse modes; however it is clear that they are actually
referring to the same set of modes as we are.

160
thus can be thought of as a dynamical precursor to the phase transition. [248]
Looking at the spectra from 923 and 1023 K in Fig.12.4, the slow oscillation of the background
— with trough at ∼45 and crest at ∼65 meV — is another indication of anharmonicity. Specifically, it seems that the harmonic model fails to correctly predict the multiphonon scattering seen
experimentally. As the crest is at approximately twice the energy of the longitudinal peak, it seems
likely that the 2-phonon processes are sampling anharmonic parts of the potential. Interference
effects between 1-phonon and multiphonon scattering are known to exist for sufficiently anharmonic
solids, [294] and this seems a possibility here.
Perhaps the clearest indicator of anharmonicity in iron is the deviation of its phonon entropy from
that predicted by the quasiharmonic model of a solid. As discussed in Chapter 4, in a quasiharmonic
solid, the entropy of dilation, Sph,D , is determined by the expansion of the lattice against the bulk
modulus:
Z T
Sph,D =

9KT α2 0
dT ,
ρN

(12.5)

where we have used the coefficients of thermal expansion, α, from Touloukian et al., [168] and
the temperature dependent isothermal bulk modulus, KT from Rayne and Chandasekhar [295] and
Fukuhara and Sanpei. [296] We may compare this to the nonharmonic entropy, Sph,NH , which may
be determined directly from the phonon spectra:
Sph

3kB

dEgT [(nT + 1) ln (nT + 1) − nT ln (nT )] ,

(12.6)

dEgT0 [(nT + 1) ln (nT + 1) − nT ln (nT )] ,

(12.7)

Sph,H

3kB

Sph,NH

Sph − Sph,H .

(12.8)

The difference between these two is the anharmonic entropy, Sph,A :
Sph,A = Sph,NH − Sph,D .

(12.9)

All of these are shown in the center panel of Fig. 12.9. The anharmonic entropy is quite large,
comprising over 3.5% of the total entropy of iron at 1023 K. If the measured trends hold, it would
be close to 5% at the BCC to FCC phase transition — typical values for other solids at melting are
less than 1%. [37]
Given the electronic DOS as a function of temperature, we may determine the various electronic
entropies as follows: First, with Nel the number of electrons, we use:
Nel =

GT (E)fT (E)dE ,

(12.10)

161

Sph+Sel

9.0
8.0
7.0
6.0

SJanaf

Sph+Sel,G

Sph+Sel,NM

5.0

Sph

4.0

Sph,H

3.0
2.0

Sel S
el,G

Sel,NM

S (kB/atom)

1.0
0.0
0.7
0.6
0.5

Sph,NH

0.4
0.3

Sph,D

0.2
0.1

Sph,A

0.0
S-Sph-Sel,G
S-Sph-Sel

0.8
0.6
0.4
0.2
0.0
-0.2

S-Sph-Sel,NM

200

400

600

800

1000

1200

T (K)
Figure 12.9: Contributions to the entropy of iron. The vertical gray line denotes the highest temperature at which the phonon spectrum was measured. The open markers show accepted values
of the total entropy taken from the NIST-JANAF thermochemical tables, [136] filled markers show
experimental data points. The various contributions are discussed in the text.

to find the chemical potential, µ. Then:
Sel

= −kB

dEGT [(1 − fT ) ln (1 − fT ) + fT ln (fT )] ,

(12.11)

dEG0 [(1 − fT ) ln (1 − fT ) + fT ln (fT )] ,

(12.12)

dEGNM
[(1 − fT ) ln (1 − fT ) + fT ln (fT )] .

(12.13)

Sel,G

= −kB

Sel,NM

= −kB

162
Our best estimate of the electronic entropy, Sel , is found using the temperature dependent electronic
DOS including the scaled magnetization. We get the ground state electronic entropy, Sel,G , by using
the 0 K ferromagnetic electron DOS, G0 (E). This is in reasonable agreement with values found by
Weiss and Tauer [297] and Grimvall, [199]. Alternatively, we get Sel,NM by using the temperature
dependent non-magnetic electron DOS, GNM
T (E). The three corresponding curves are shown in the
top panel of Fig. 12.9. The phonon and harmonic phonon entropy, Sph and Sph,H determined with
the phonon DOS, gT (E) and g21 (E) from experiment are also shown, where the subscript 21 is for
21 K, our lowest temperature measurement.
If contributions from spin dynamics, electron-phonon interactions, and defects are negligible,
the sum of the electronic and phonon entropies should give the total entropy.h The top panel of
Fig. 12.9 shows values for the total entropy, SJanaf , taken from the NIST-JANAF thermochemical
tables [136] (as markers) as well as the sums of the phonon entropy and the ground state and nonmagnetic electronic entropies. For all temperatures the former underestimates the total entropy and
the latter overestimates:
Sph + Sel,G < S < Sph + Sel,NM .

(12.14)

This is particularly noticeable below the Curie point, where spin excitations are actually playing a
large role in the entropy.
The bottom panel of Fig. 12.9 shows the differences between SJanaf and the three sums Sph +
Sel,G , Sph + Sel , and Sph + Sel,NM . The solid-yellow curve in the bottom panel of Fig. 12.9 comes
from assuming a non-magnetic electronic DOS at all temperatures. The negative values at low
temperatures show that the electron DOS at the Fermi level for the non-magnetic state is significantly
higher than it is in the (actual) ferromagnetic state. Starting around T ≈ 800 K, the curve begins to
increase, finally leveling off at the Curie temperature. The Fermi level is almost certainly increasing
as iron passes through the Curie point; however, some of the increase seen here is due to short range
magnetic excitations.
At low temperatures, the black-dashed curve, S − (Sph + Sel,G ), is approximately zero. That
is, the contributions from the phonons and electrons are probably well understood here, and other
contributions to the entropy are negligible. As the temperature increases, a variety of effects begin
to contribute to the entropy. There are adiabatic electron-phonon and defect contributions to the
entropy; however, we will assume that these are negligible at these temperatures. What remains
are changes in the electronic DOS at the Fermi level, and increasing magnon entropy. Looking at
Fig. 12.8, it appears that the electronic DOS at the Fermi level is indeed increasing as we approach
the Curie point. The black-dashed curve, however, continues increasing past the Curie temperature.
h It is well established that spin dynamics do contribute to the entropy of iron, and we will discuss this further;
however, the other two contributions are likely to be quite small.

163
One interpretation would be that the ferromagnet to paramagnet transition comes about when spin
excitations have disrupted the long range order; however, some short range order persists above the
Curie temperature. The continued increase in the black-dashed curve above the Curie temperature,
then, corresponds to the increasing disorder at short ranges.
Using the electronic DOS calculated by fixing both the volume and magnetization, we get the
entropy shown by the red-dotted curve in Fig. 12.9. This curve is nearly identical to the blackdashed one up to about 750 K, after which it levels off, followed by a precipitous drop at the
Curie temperature. In this case, the changes in the electronic states at the Fermi level have been
incorporated into the electronic entropy, and the only contribution to the red-dotted curve is the
dynamics of the spins. As we expect the disorder in the magnetic degrees of freedom as measured
by the entropy to be a monotonically increasing function of the temperature, the shape of the reddotted curve cannot be correct. It is clear, then, that the changes in the electronic DOS at the Fermi
level have been overestimated by our simple model of the electronic structure.

12.6

Summary

Nuclear resonant inelastic x-ray scattering measurements of the phonon spectrum of BCC iron were
made at temperatures of 21, 295, 523, 773, 923, and 1023 K. We found little broadening of the
phonons at increased temperature but large shifts in phonon energies at increased temperatures.
Further, a large differential shifting of the transverse modes was apparent. BvK models of the lattice dynamics showed a fairly linear decrease in the 1NN longitudinal force constant as a function of
temperature which likely dominated the softening of the phonons. Additionally, we found a transition from bonding to repulsive transverse forces in the 1NN shell at roughly 750 K. Combining this
with an instability of the longitudinal forces in the 3NN shell and decreases in magnitude of the 1NN
transverse forces also occurring around 750 K, we have found strong evidence for dynamical precursors to the BCC-FCC phase transition at 1185 K. The phonon DOS were also used to determine
the various components of the phonon entropy of iron as a function of temperature up to TC . The
anharmonic entropy obtained from the shifts of phonon frequencies was quite large, approaching 4%
of the total entropy at the Curie point. Electronic contributions to the entropy were also calculated,
and by comparison to the total entropy we concur that the magnetic contributions to the entropy
must continue to increase above TC , indicating some persistent short range order.

164

Chapter 13

Chromium and Vanadium
Here we consider inelastic neutron scattering measurements of the phonons in BCC chromium and
vanadium. The treatment is significantly less detailed than those of Chapters 9–12. The information
presented here will be used in subsequent discussion of trends in the phonons of cubic metals.
Extensive high temperature measurements of the phonon spectra of chromium have been made
by Trampenau et al. [39] and of vanadium by Bogdanoff et al. [54] and Delaire et al. [41]; however, as
we have made our own measurements of the former and have direct access to the data for the latter,
we will very briefly present these data here. We will only consider the anharmonic phonon entropy,
and fits of the high temperature phonon spectra using the low temperature DOS. Briefly, these fits
consist of shifting the energy E → ∆s E of the low temperature spectrum and then performing an
integral transform using the damped harmonic oscillator function as a kernel:
B(Q, E 0 , E) =

πQE 0

E0
E 2
+ Q12
E − E0

(13.1)

where Q is a quality factor. That is:
gT (E) = B(Q, E 0 , E)

g10 (∆s E 0 ) ,

(13.2)

with ∆s and Q determined through least squares analysis. The subscript 10 refers to 10 K, the
lowest temperature measurement of the phonon spectra for vanadium and chromium. Unless noted
otherwise, all experimental and data analysis procedures are the same as those detailed in Chapter 9
and its references.
For both the chromium and the vanadium, a difference that should be pointed out between the
fits shown here and those in Chapters 9, 10, and 11 is that these fits rely on data from two different
instruments. That is, the 10 K measurements from LRMECS were used to fit high temperature
data from Pharos. Presumably, the latter instrument is higher resolution; and this may have an
impact on the optimized values of Q. At 300 K, we have measurements from both instruments, and

165

0.33
0.3
1275 K
0.27
875 K
0.24
775 K
0.21
575 K
0.18
525 K
0.15
320 K
0.12
300 K
0.09
240 K
0.06
140 K
0.03
10 K

Figure 13.1: Phonon DOS of chromium at temperatures as indicated. Unfilled markers are from
LRMECS measurements, solid markers from Pharos. The solid black line at 10 K simply connects the
experimental data points. Solid blue lines and dashed orange lines are respectively fits to LRMECS
or Pharos data using Eqs. 13.1 and 13.2.

these are shown in Figs. 13.1 and 13.2.

At least for chromium, the phonon DOS from LRMECS

appears to be a very slightly broadened version of the Pharos measurement. For vanadium, the
LRMECS spectrum actually appears to be sharper. Regardless, fits to both 300 K measurements
are also shown in Figs. 13.1 and 13.2, and the differences are negligible. We take this as an indication
that the fitting procedure gives reasonable results for Q and ∆s , despite the differences in the two
instruments.
Chromium samples were prepared by crushing 99.995% pure chromium ingot into a coarse powder
powder. The powder was then sealed in quartz tubes and annealed at 1200 K for 24 hours to remove

166

0.24

0.21
1275 K
0.18
875 K
0.15
775 K
0.12
525 K
0.09
300 K
0.06
150 K
0.03
10 K

Figure 13.2: Phonon DOS of vanadium at temperatures as indicated. Unfilled markers are from
LRMECS measurements, solid markers from Pharos. The solid black line at 10 K simply connects
the experimental data points. Solid blue lines and dashed orange lines are respectively fits to
LRMECS or Pharos data using Eqs. 13.1 and 13.2.

strain and induce recrystallization. There were minimal signs of oxidation on the sample surface;
however x-ray and in-situ neutron diffraction patterns indicated that the bulk samples were largely
free of oxidation. Vanadium samples were made by cutting and rolling a vanadium slab of 99.998%
purity.a The rolled vanadium strips were then sealed in quartz tubes and annealed at 1200 K for 24
hours to remove strain and induce recrystallization. There were no visible signs of oxidation.
The inelastic scattering of neutrons from chromium was measured in experiments at LRMECS at
10, 140, 240, 300, 320, 575, and 775 K, and at Pharos at 300, 575, 875, and 1275 K. For vanadium,
a Just for kicks, we point out that the slab was purchased on E-bay.

167
measurements were made at 10, 150, 300, 525, and 775 K on LRMECS and at 300, 875, and 1275 K
on Pharos. At all temperatures, the measured scattering was reduced to a phonon spectrum, and
these are shown as markers in Figs. 13.1 and 13.2, and the lines are fits using Eqs. 13.1 and 13.2.
For chromium, the fits have failed noticeably by the time the temperature reaches 1275 K, and
there is evidence of the failure already at 525 K. Specifically, the low transverse modes appear
to move to lower energies at a higher rate than the high transverse or longitudinal modes. More
generally, Trampenau et al. found chromium to be highly anharmonic, and our measurements agree.
Using the data from Trampenau et al., Eriksson et al. [36] find Sph,A ≈ 0.83 at melting. Finally, we
note that the melting temperature of chromium is 2180 K, thus the failure at our model is apparent
by T /TM ≈ 0.24
Similar to chromium, the low transverse modes in vanadium shift faster than the high transverse
or longitudinal modes. The first clear signs of our model failing appear at about 525 K, and it is
still worse by 1275 K. As the melting temperature of vanadium is 2183 K, we see the model failing
by T /TM ≈ 0.24. The anharmonic entropy of vanadium is bit odd, with a minimum of Sph,A ≈
−0.13kB /atom at roughly 1200 K, rising to a maximum of Sph,A ≈ 0.17kB /atom at melting. [36]
In summary, phonons in chromium and vanadium appear to broaden less than the FCC metals.
At sufficiently high temperatures — TTM & 0.3 — characterizing the phonon linewdiths becomes
somewhat problematic because of the differential motion of the longitudinal, low transverse, and
high transverse modes.

168

Part V

Phonon Trends in Cubic Metals

169

Chapter 14

Anharmonicity and the Shape of
the Phonon DOS
Here we will outline similarities and differences in the phonon spectra of BCC and FCC transition
metals, as well as of the nearly free electron (NFE) metals lead and aluminum. Specifically, we
consider the phonon entropy and the force constants from BvK models, and we will argue that there
are greater similarities amongst the FCC metals than amongst the BCC.
To get a quick sense of the temperature dependencies of the phonons in these materials, we first
consider anharmonic contributions to the phonon entropy at melting. Table 14.1 shows Sph,A at
melting for BCC and FCC transition metals and the NFE FCC metals aluminum and lead. The difference in the range of values is remarkable, with the BCC metals spanning −0.09 to 0.72 kB /atom,
whereas the FCC metals span only −0.15 to 0.00 kB /atom.a The FCC metals are largely quasiharmonic, with expansion against the bulk modulus accounting for nearly all of the nonharmonic
phonon entropy. This is also true for the BCC metals niobium and tantalum and to some extent
vanadium; however, for iron, chromium, tungsten, and molybdenum, the anharmonic entropy is
quite significant.
Looking now at the forces, for any BvK model, we may determine the longitudinal and transverse
force constants for a given shell of neighbors using the procedure described in § 8.3.2. The longitudinal force constants for first- and second-nearest neighbor (1NN and 2NN) and the transverse
force constants for 1NN so determined are shown for FCC metals in Table 14.2 and for BCC metals
in Table 14.3.

There are some rather striking trends in these data. For all of the FCC metals,

the 1NN longitudinal force constants are at least 3.8 times larger than the 2NN longitudinal force
constants. If we discard iridium — this is the metal with the least available experimental data —
then the 1NN longitudinal forces are at least 5.8 times as large as the 2NN ones. For the BCC
metals, on the other hand, only the 1NN forces in tantalum are this large relative to the 2NN ones.
In fact, the 2NN forces are larger than their 1NN counterparts in molybdenum and in chromium at
a We have taken our adjusted value for aluminum to be reliable, but not so our value for lead.

170
Table 14.1: Anharmonic entropies at melting from Wallace [37] and from the present work. For the
latter, we give the value at the highest measured temperature, and assume that the trend at that
temperature will continue to melting. This is probably quite reasonable for aluminum, lead, and
nickel. For iron, however, there remains one magnetic and two structural phase transitions before
melting, making an estimate of Sph,A more difficult.
Structure

Element

FCC

Ni
Al
Cu
Pb
Ag
Au
Pd
Pt
Rh
Ir
Nb
Mo
Ta
Cr
Fe

BCC

Sph,A at T = TM (kB /atom)
Wallace
Present Work
0.00
≤ -0.06
0.01
≤ -0.07
0.00
-0.04
≤ -0.20
-0.07
-0.09
-0.03
-0.15
0.59
-0.09
0.72
0.16
-0.07
0.58

0.28

all but the highest temperature.
Another trend can be seen in the signs of the 1NN transverse force constants. The average of
these constants is negative for all of the FCC metals. With the exceptions of aluminum at 525 and
775 K this is true also of the unaveraged transverse force constants. The BvK models for aluminum
at high temperatures were perhaps overly simplified, as they included forces only to the 3NN shell
despite the fact that the forces in NFE aluminum are known to be fairly long range. This could
very well explain why these two models are an exception to the rule. For the BCC metals, with the
exceptions of chromium at 1773 K, and of Fe at 773 and 1023 K, all of the transverse constants are
positive. For chromium, the negative values are likely not statistically different from zero, or from
small positive values, and we believe that the negative values in iron above 773 K are precursors to
the BCC to FCC transition at 1185 K. Physically, this means that whereas the FCC metals have
repulsive forces in the transverse directions, the BCC transition metals appear to have attractive
ones. Very closely related is the trend found by Brockhouse et al. [245] that [K1 ]xx < [K1 ]xy for
FCC transition metals and [K1 ]xx > [K1 ]xy for BCC transition metals. For non-transition BCC
metals, such as sodium and potassium, they find the opposite inequality.b They concluded that this
suggests that the d-electrons are involved in some sort of covalent bonding in the BCC transition
b Roughly, these trends are the same, and are related through the Gershgorin circle theorem. [298]

171
Table 14.2: 1NN and 2NN Longitudinal force constants, K1 and K2 , as well as 1NN transverse
force constants, K1T1 and K1T2 , from Born–von Kármán models for FCC metals. The ratio of the
longitudinal force constants in the 2NN and 1NN shell, K2 /K1 , and the mean value of the transverse
force constants in the 1NN shell, hK1Tj i, are also shown. All force constants given in N/m.
Symbol
Al

Rh
Pd

Ag
Ir
Pt
Au
Pb

T (K)
10
150
300
525
775
10
300
575
875
1275
49
295
296
298
673
973
1336
297
120
296
673
853
296
90
296
80

K1
21.26
20.41
20.09
19.77
19.08
36.56
35.80
35.41
33.58
30.89
27.91
27.92
28.37
27.75
26.34
26.07
25.37
40.89
42.95
41.76
38.95
38.15
23.03
46.89
55.62
36.36
9.00

K2
2.45
2.41
2.22
1.97
1.96
0.97
0.89
1.24
0.92
1.01
-0.04
0.36
0.29
0.53
0.70
1.55
0.24
6.96
0.92
1.42
1.39
2.00
0.06
12.15
3.94
4.04
1.41

K2 /K1
0.12
0.12
0.11
0.10
0.10
0.03
0.02
0.03
0.03
0.03
-0.00
0.01
0.01
0.02
0.03
0.06
0.01
0.17
0.02
0.03
0.04
0.05
0.00
0.26
0.07
0.11
0.16

K1T1
-1.36
-1.20
-2.06
-3.37
-3.46
-1.39
-0.71
-2.24
-1.76
-2.94
-1.35
-1.72
-1.81
-1.50
-1.79
-2.70
-1.94
-4.03
-3.43
-3.09
-3.75
-3.38
-1.75
-2.53
-6.92
-6.54
-2.49

K1T2
-1.04
-0.99
-0.67
1.31
1.14
-0.39
-0.27
-0.38
-0.32
-0.34
-1.35
-1.42
-1.25
-1.50
-1.32
-1.42
-1.79
-0.69
-2.51
-2.83
-2.41
-2.88
-1.61
-2.04
-3.95
-3.50
-0.35

hK1Tj i
-1.20
-1.09
-1.36
-1.03
-1.16
-0.89
-0.49
-1.31
-1.04
-1.64
-1.35
-1.57
-1.53
-1.50
-1.55
-2.06
-1.86
-2.36
-2.97
-2.96
-3.08
-3.13
-1.68
-2.29
-5.44
-5.02
-1.42

metals.
There has been some theoretical work on determining the shape of phonon spectra based on the
space group symmetries of a crystal. [72, 299–302] In particular, Rosenstock [300] determined the
locations of the critical points of the phonon spectra for simple-, body-centered-, and face-centeredcubic lattices, given forces that went out to only 2NN. This is an underestimate of the range of
the forces in typical metals; in particular, the forces in aluminum and lead are known to be very
long range. Regardless, in this model, the ratio of the 2NN to 1NN force constants determines the
shape of the phonon spectrum. For FCC metals, the phonon spectrum has different shapes for the

172
Table 14.3: 1NN and 2NN Longitudinal force constants, K1 and K2 , as well as 1NN transverse
force constants, K1T1 and K1T2 , from Born–von Kármán models for BCC metals. The ratio of the
longitudinal force constants in the 2NN and 1NN shell, K2 /K1 , and the mean value of the transverse
force constants in the 1NN shell, hK1Tj i, are also shown. All force constants given in N/m.
Symbol
Cr

Mo
Ta

T (K)
293
673
1073
1473
1773
21
295
523
773
923
1023
293
773
1773
2223
296
296
298

K1
28.21
32.38
30.20
30.22
29.41
47.08
46.09
44.62
42.59
41.48
40.64
33.08
32.70
32.82
31.84
40.07
39.38
59.90

K2
37.70
27.52
27.44
23.16
19.10
15.31
14.57
15.10
15.29
12.73
12.83
13.33
13.32
12.11
12.46
44.57
1.42
45.70

K2 /K1
1.34
0.85
0.91
0.77
0.65
0.33
0.32
0.34
0.36
0.31
0.32
0.40
0.41
0.37
0.39
1.11
0.04
0.76

K1T1
7.42
2.41
1.85
1.42
-0.05
2.35
1.27
1.15
-0.47
0.29
-0.32
3.23
3.62
2.74
1.63
4.73
5.78
3.20

K1T2
7.42
2.41
1.85
1.42
-0.05
2.35
1.27
1.15
-0.47
0.29
-0.32
3.23
3.62
2.74
1.63
4.73
5.78
3.20

hK1Tj i
7.42
2.41
1.85
1.42
-0.05
2.35
1.27
1.15
-0.47
0.29
-0.32
3.23
3.62
2.74
1.63
4.73
5.78
3.20

following cases:
K2 /K1 < −0.067
K2 /K1 = −0.067
−0.067 < K2 /K1 < 0.0

(14.1)

K2 /K1 = 0.0
0.0 < K2 /K1
For BCC metals, we have the following cases:
K2 /K1 < 0.0
K2 /K1 = 0.0

(14.2)

0.0 < K2 /K1
Taking the longitudinal force constants to be most representative, and ignoring forces past the 2NN
shell, we may calculate these ratios from the force constants for the BvK models shown in Fig. 14.1.
The ratios for the FCC metals are given in Table 14.2 and for BCC metals in Table 14.3. Despite
the variety of shapes available, particularly for FCC metals, the values all indicate the same shape

173
30.0

C D

EF GH

I J

27.5

17.5
15.0
12.5
10.0
7.5
5.0
2.5

E GH

18.0
Pb, 80 K

16.5

Al, 300 K

15.0

Au, 296 K
Ag, 296 K
Cu, 296 K

g(E/EC) (unitless)

20.0

D/F

19.5

25.0
22.5

21.0

13.5

Nb, 293 K

12.0
10.5

W, 298 K

9.0

Pt, 90 K

7.5

Pd, 296 K

6.0

Ni, 300 K

Ta, 296 K

4.5

Mo, 296 K

Cr, 293 K

3.0
Ir, 5 K
1.5
Rh, 297 K

0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
E/EC (unitless)

Fe, 295 K
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
E/EC (unitless)

Figure 14.1: Phonon DOS near room temperature from BvK models for FCC and BCC metals in the
left and right hand panels, respectively. Note that the cutoff energy, EC , is temperature dependent.
The spectra for the FCC metals appear to have much more in common than those for the BCC
metals. The meanings of the vertical lines (marked by letters) are described in Table 14.4 and in
the text.

for either FCC or BCC metals.c The values for the FCC metals are distributed over a small range,
from 0.003 to 0.259, whereas the BCC metals range from 0.030 up to 1.112.
Looking at Fig. 14.1 and the associated Table 14.4, we may identify some trends in the shapes
of the phonon spectra for the cubic metals. First, consider a general description of these phonon
spectra, where we have renormalized the energy using the cutoff energy, EC . Starting at E/EC = 0,
the spectra rise monotonically to a critical point (CP). For sufficiently small E/EC , the rise is
quadratic as in a Debye model of a crystal. For larger E/EC , higher order polynomial terms are
required. The first transverse modes dominate between this first CP and another CP, after which the
second transverse modes begin to dominate. The second transverse modes then decline in density
until a local minimum is reached. After the minimum, the longitudinal modes begin to dominate
until, finally, the density of longitudinal modes drops off rather sharply and the spectrum goes to
zero at E/EC = 1.
The biggest difference between the shapes of the spectra for the FCC and BCC metals concerns
c The model of the 49 K copper DOS actually gives K2 = −0.001, which would indicate a different shape of the
K1
DOS. This value, however, is very small, relative to the errors.

174
Table 14.4: Shapes of phonon spectra from BvK models for FCC and BCC metals. EC is the
cutoff energy, CP1 stands for first critical point, L for longitudinal, T for transverse, and TX for
Xth transverse, X∈ {1, 2}. “First” and “Last” refer respectively the lowest and highest energy
occurrences of some feature for the set of phonon DOS in question.
Label

Description
First CP1
First T1 peak to “end”
Last CP1
Last T1 peak to “end”
First minimum between T and L peaks
First CP starting L peak
Last minimum between T and L peaks
Last CP ending L peak

E/EC
FCC BCC
0.000 0.000
0.385 0.480
0.505 0.540
0.570 0.655
0.745 0.765
0.775 0.655
0.845 0.830
0.855 0.875
0.950 0.970
1.000 1.000

Comments

Unclear for FCC Pb

Unclear for BCC Ta

Unclear for FCC Ir

the existence of a single local minimum between the transverse and longitudinal peaks. For the FCC
metals, shown in the left hand panel of Fig. 14.1, this minimum is easy to find, is the only local
minimum after the first transverse peak has died out, and lies between 77 and 86% of the cutoff
energy, as marked by the letters F and H. Further, the longitudinal peaks for all of the FCC spectra
are concentrated between the lines marked G and I, at energies higher than all of these minima.d For
the BCC metal tantalum, shown in the right hand panel of Fig. 14.1 it is not even clear where this
minimum lies, or if it exists at all. The letter F marks the only obvious local minimum after the first
transverse peak in tantalum; however, it seem likely that this is before the second transverse peak,
rather than after it. The density in the longitudinal modes for tantalum is almost gone by the point
H at which the minimum appears in the chromium spectrum. There are no obvious minima between
the first and second transverse modes for the FCC metals; however, the BCC metals sometimes
show such minima and sometimes do not. At lower energies, the spread in the starting points for
the the first transverse modes in the FCC metals is larger than for the BCC metals, as marked by
the letters B and D in the left and right hand panels of Fig. 14.1. Similarly for the ends of the first
transverse modes, as marked by the letters C and E. Overall, the FCC metals appear to have more
similarities in the shape of their spectra than do the BCC metals — certainly at medium and high
energies.
A similar trend is visible as a function of temperature, as shown in Fig. 14.2. Once normalized
for the cutoff energy, EC , the changes in the phonon spectra with temperature for the FCC metals
aluminum, nickel, copper, and palladium is much less noticeable than the changes for the BCC
metals chromium iron and niobium.e In particular, the transverse modes in the BCC metals —
d There is a tiny overlap between the first CP of the longitudinal peak in palladium at G and the minimum between
the peaks in gold at H.
e There appear to be more changes in aluminum than in the other FCC metals; however, some of this may be do

175
15.0
10.0
13.5
12.0

10.5

Cu
5.0

g(E/EC) (unitless)

7.5

9.0
7.5
Nb
6.0
4.5
Fe

2.5
3.0

Al
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
E/EC (unitless)

1.5
Cr
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
E/EC (unitless)

Figure 14.2: Temperature dependence of phonon DOS for BCC and FCC metals from BvK models
are shown in the left and right hand panels, respectively. Note that the cutoff energy, EC , is a
function of temperature. The shape of the phonon spectra for the FCC metals are much more
consistent than those of the BCC metals. (From cold to hot, the curves are sparsely-dotted red,
dashed orange, solid yellow, dash-dotted green, densely-dotted blue, and dash-double-dotted purple;
and the relevant temperatures for the FCC and BCC metals are listed in Tables 14.2 and 14.3,
respectively.)

especially for chromium and niobium — appear to change much more with temperature than do
those in the FCC metals. For niobium, there also appear to be significant shifts in the longitudinal
modes.
In so much as differences between the phonon spectra of the FCC and BCC metals may be
attributed to differences in the transverse modes (as opposed to the longitudinal ones), we may
understand their differences in terms of the structural stability of the two crystal lattices. [303, 304]
Fig. 14.3 shows a central atom and its nearest neighbors in the simple cubic (SC), BCC, and FCC
lattices. Assuming that only there are only central forces between 1NN, for all three lattices, we
may not move an atom towards its 1NN without inducing a force on those atoms. For the case of
the SC lattice, consider the 1NN that are not on the pictured plane. For any small rotation of these
atoms about the central atom, there is no restoring force.f The SC lattice, then, has some structural
to shorter range of the forces in the models for the highest temperature. For example, this also seems to have had an
effect on the transverse force constants as seen in Table 14.2 and discussed in the text.
f Consider planes parallel to the one pictured, that pass through the out-of-plane atoms. If these planes pivot so
as to remain parallel while the out-of-plane atoms undergo their rotation, then there is no restoring force throughout
the entire lattice.

176

Figure 14.3: First nearest neighbors for SC, BCC, and FCC lattices are shown in the left, center,
and right panels, respectively. The 1NN bonds are given by thin cylinders connecting the atoms.
The planes pass through the central atom and as many 1NN atoms as possible. For the SC and
BCC lattices, the only 1NN bonds visible are those connecting to the central atom; however, for the
FCC lattice some of the 1NN are also 1NN of each other.

instability in transverse directions. There are fewer degrees of freedom for the BCC lattice; however,
we may still rotate one plane about the line where the planes intersect without inducing a restoring
force. In the FCC lattice, there are no such degrees of freedom. That is, structurally, the FCC
lattice is stable to all displacements, longitudinal or transverse. This is directly related to the trends
we see in the non-central, 1NN force constants for the FCC and BCC metals. Specifically, the
1NN transverse forces in the BCC metals are necessarily bonding in nature to compensate for the
structural instability, whereas those force in the FCC metals can be repulsive. Structural differences
are also responsible for the trend of 2NN force constants being more significant in BCC than in FCC
metals, simply because the ratio of 1NN to 2NN distances is 0.87 as opposed to 0.71. That is, the
2NN in a BCC lattice are not significantly farther away than the 1NN, which is not the case for
FCC lattices.
Finally, we consider the phonon linewidths as a function of temperature. Looking at Figs. 9.3,
10.5, and 11.4, we see that for the FCC metals aluminum, lead, and nickel, our simple model of
the phonon linewidths begins showing the first signs of failure at roughly T /TM = 525/933 = 0.56,
T /TM = 390/600 = 0.65, and T /TM > 1275/1728 = 0.74, respectively. From Figs. 12.3, 13.1, and
13.2, we see the first signs of failure at roughly T /TM = 775/1811 = 0.43, T /TM = 525/2180 = 0.24,
and T /TM = 525/2183 = 0.24 respectively for the BCC metals iron, chromium, and vanadium. In
short, this simplified model of the phonon broadening appears to be much more robust in the FCC
metals than in the BCC metals. As such, we consider phonon linewidths in the FCC metals in
greater detail.

177

Chapter 15

Mean Phonon Linewidths in FCC
metals
15.1

Introduction

Anharmonic effects in solids cause non-zero phonon linewidths in addition to more widely known
phenomena such as shifts in phonon energies, thermal expansion, and temperature-dependent elastic
constants. [50–52] Phonon broadening can be a prominent feature in inelastic scattering experiments,
as seen in the plots of S(Q, E) for aluminum, lead, and nickel in Fig. 15.1, and in the plots of the
phonon densities of states (DOS) in Figs. 9.3, 10.5, and 11.4. The shifts in phonon energy can
sometimes be predicted by a quasiharmonic model [36–38] in terms of the bulk modulus and the
temperature-dependent lattice parameter; however, to our knowledge, a simple model to predict
phonon linewidths doesn’t exist, despite a number of systems that appear to have simple trends
(see, for example, Chaps. 9, 10, and 11).
In Chapter 14, we saw that a wide variety of behaviors exists for phonons in BCC metals;
whereas there is greater uniformity in the phonons in the elemental FCC metals. Table 14.1, shows
similarities in their anharmonic entropies, which may be understood in terms of the shifts of phonon
energies. [36–38] There are further likenesses in the shape of their phonon spectra, and in their nearneighbor force constants. Because of these similarities, we have focused our study on FCC metals,
and we hypothesize and explore a simple model for predicting phonon linewidths in FCC metals
based on damped harmonic oscillators.
In this chapter, we give a brief review of damped, driven harmonic oscillators and the relationship
between quality factors, linewidths and lifetimes. Also a brief review of molecular dynamics is given.
A simple molecular dynamics simulation of a linear chain is used to illustrate the possible meaning
of quality factors in conservative systems. We then examine our simple model for predicting the
phonon linewidths in FCC metals in light of the results of a series of simulations and experiments.
Namely, we consider molecular dynamics simulations — including determinations of phonon spectra

178

Figure 15.1: S(Q, E) for aluminum, lead, and nickel at temperatures as marked. At low temperatures, dispersive information is visible on the phonon creation side of the spectrum. As the
temperature rises, in addition to increased scattering, the dispersive features broaden significantly.

and associated quality factors — for the FCC transition metals nickel, copper, rhodium, palladium,
silver, iridium, platinum, and gold, and the nearly free electron (NFE) metals aluminum and lead.

179
We also consider experimental data for FCC aluminum, lead, and nickel that was presented in
Part III, and experimental data for FCC copper taken from triple-axis measurements by Larose and
Brockhouse. [60, 305] We find that our simple model is remarkably successful in accounting for the
phonon qualities as a function of temperature, and it suggests a deep commonality in the interatomic
potentials of the FCC metals.

15.2

Background

15.2.1

Quality factors

We use the notion of a quality factor for oscillatory motion extensively in this chapter, so we review
it briefly with respect to some features of the damped harmonic oscillator. More detailed treatments
are widely available. [53, 306]
15.2.1.1

Damped driven harmonic oscillators

We have considered the harmonic oscillator and coupled harmonic oscillators in § 3.3 and 3.4. We
now add a dissipative term, C u̇, to the 1D harmonic oscillator, giving the following equation of
motion:

= M ü + C u̇ + Ku = ü +

ω0
u̇ + ω 02 u ,

(15.1)

where u is the displacement of the mass M , K is a spring constant, and C a damping constant. We
have also defined the undamped frequency ω 0 = M
and the quality factor Q = M
C . For Q > 2 ,
a solution to the equation is:
ω0
exp −
t cos (ωt) ,
2Q
ω0
ω0
ω0
u̇ = −
exp −
t cos (ωt) − ω exp −
t sin (ωt) ,
2Q
2Q
2Q
02
ωω
ω0
ü =
exp
cos
(ωt)
exp
sin (ωt) ,
4Q2
2Q
u =

(15.2)
(15.3)
(15.4)

where we have defined:
ω ≡ 1−
ω 02 .
4Q2

(15.5)

The oscillatory solutions to this equation are modulated by an exponential decay as is illustrated in
Fig. 15.2.

180

1 period

2 periods

Potential Energy

1.0e+00

cos2(ωt)

1.0e−01

e−ω0/Q t

1.0e−02
−2π

1.0e−03

e−ω0/Q tcos2(ωt)
1.0e−04

2π

3π

4π

5π

Time
Figure 15.2: The potential energy of a damped harmonic oscillator with ω = 1 and Q = 2. The
dense-dotted gray line shows the potential energy for the oscillations in the absence of damping
(Q = ∞) with ω 0 = ω. The dashed red line shows decay envelope, and the solid blue line shows
the potential energy for the damped oscillator. After approximately Q = 2 periods, the potential
energy in the oscillations has decreased by a factor of e2π .

15.2.1.2

Effects of damping: shifts, quality factors, linewidths, and lifetimes

To compare with anharmonic phonons, we consider the frequency shift, ∆ω ≡ ω − ω 0 , caused by the
damping term in the equation of motion:
∆ω
ω0

ω − ω0
ω0

1−

−1≈− 2 .
4Q
8Q

(15.6)

In this model, the frequency shift and the quality factor are inversely related.a
The quality factor is unitless, and approximately equals the number of oscillations before the
energy in the oscillations decays by a factor of e2π . Systems for which Q > 12 are called underdamped,
and this classification applies to all of the oscillations considered here. If Q < 12 the system does
not oscillate at all, and such systems are called overdamped. Finally, if Q = 21 the system is said to
be critically damped.
The quality factor is related to the linewidth of the damped harmonic oscillator as measured in
a ∆ω should not be confused with a linewidth, an equation for which we will derive shortly.

181
a scattering experiment. We demonstrate this by considering a driving force F = F (t):

ü +

ω0
u̇ + ω 02 u .

(15.7)

We now take the Fourier transform:
FT {F}

−ω 2 FT {u} + iω

ω0
FT {u} + ω 02 FT {u} ,

(15.8)

which yields the transfer function:
FT {u}
FT {F}

M −ω 2 + ω 02 + iωω

ω 02 − ω 2
+i
ω0
02
02 ω 2 ω 0
ω 2
ω 2
Mω ω
− 0 + 2
− 0 + 12

−ω + ω 02 + iωω

−ω 2 + ω 02 − iωω

ω0 ω

(15.9)

The fluctuation-dissipation theorem [307–312] relates the spectrum of fluctuations (related, in turn,
to the incoherent inelastic scattering) to the imaginary (or dissipative) part of the transfer function,
and we consider this here:

FT {u}
FT {F}

M Qωω 0 ω0 − ω0 2 + 12

(15.10)

For large Q, we have ω ≈ ω 0 , and we may write:b
ω0
− 0

2 (ω 0 − ω)
(ω 0 − ω) (ω 0 + ω) ≈
ωω
ω0

(15.11)

Substituting this into Eq. 15.10, we have:

FT {u}
FT {F}

h 0
i2
02
M Qω
2(ω −ω)
ω0

+ Q12

~2
4M Q (E 0 − E)2 +

.
E0 2
2Q

(15.12)

This is a Lorentzian with full width at half maximum given by 2Γ ≈ EQ . 2Γ, then, is the linewidth
measured in a measurement of the spectrum of the oscillator.
The Heisenberg uncertainty principle for time and energy is 2Γ∆t ≥ ~2 ; therefore, we expect the
lifetime of a quantum mechanical state associated with the driven damped harmonic oscillator to be
b Though somewhat disconcerting, this partial substitution of ω 0 for ω is commonplace, and leads to the expected
Lorentzian behavior for high quality resonances.

182
related to its linewidth and quality factor:
2Γ =

E0
2∆t

(15.13)

That is, a phonon with a large linewidth (relative to its energy) also has a short lifetime and a low
quality factor.

15.2.2

Molecular dynamics

Molecular dynamics (MD) is a computational technique that takes as input a set of particles and an
analytical description of their interactions and produces as output the particle trajectories through
configuration space. Generally, the interactions are given as potential energies or force-fields. The
work of the simulation consists of evaluating the forces on the particles and integrating the classical
equations of motion, and the interest to the scientist is in calculating various physical properties
from the trajectories. Here, we give a very brief description of some of the key components of an
MD engine. There are many texts that provide more detail. [313–315]
15.2.2.1

Equations of motion and numerical integration

We have, to some extent, already discussed equations of motions for atoms in a crystal in § 3.4.2;
however, we limited ourselves to a harmonic potential. One of the great advantages of molecular
dynamics, and of simulations in general, is that any potential is amenable.
The positions of all the atoms in the crystal are given by r = r(t), then the potential energy is
some function of the positions, V(r). The force on the j th atom in the cth direction, then, is given
by:
[Fj ]c = [Mjj ]cc [r̈j ]c = −

∂V(r)
∂ [rj ]c

(15.14)

We may rewrite the second order differential equations as pairs of coupled first order ones:
[ṙj ]c
[v̇j ]c

[vj ]c ,
∂V(r)
= − M-1 jj cc
∂ [rj ]c

(15.15)
(15.16)

Given the positions and velocities, r(t0 ) and v(t0 ) at some time t0 , we may integrate the equations of
motion numerically. The simplest technique for doing this is the forward Euler method. We choose
a discrete step in time, ∆t, and let tm = t0 + m∆t, which allows us to write:
r(tm )

= r(tm−1 ) + ∆t

∂r
= r(tm−1 ) + ∆t v(tm−1 ) .
∂t t=tm−1

(15.17)

183
This particular integration technique does not enforce conservation of energy. Here, as for most
algorithms, the rate of error accumulation is largely dependent on the size of the time step relative
to the magnitudes of the velocities of the particles being simulated. For the purpose of molecular
dynamics, the Verlet integrators [315–317] are enormously popular because they are time reversible
and conserve volume in phase space.c These properties, in turn, are related to conservation of energy.
15.2.2.2

Thermostats and barostats

At best, the integration procedures discussed in the previous section conserve energy and particle
number. Volume can be conserved through appropriate boundary conditions — for example, periodic
boundary conditions are typically used for studies of crystals. These conserved quantities correspond
to the microcanonical or NVE ensemble. Experiments, however, are typically conducted at constant
temperature and constant pressure, corresponding to the isothermal isobaric or NPT ensemble.
By adding fictitious particles to the equations of motion, we may dynamically reproduce trajectories consistent with an NPT ensemble. [318–322] One possible set of equations is given below:
[ṙj ]c
[v̇j ]c
v̇T

[vj ]c + vP [rj ]c ,

-1 ∂V(r)
= − M jj cc
1+
vP + vT [vj ]c ,
∂ [rj ]c
[Mjj ]cc [vj ]c + MP vP2 − (N + 1) kB T ,
MT 

V̇

v̇P

(15.18)
(15.19)

(15.20)

DV vP ,
DV
D XX
(PI − P ) +
[vj ]c − vP vT ,
MP
N MP j c
∂V(r)
∂V(r) 
[vj ]c −
[rj ]c ·
− DV
DV  j c
∂ [rj ]c
∂V 

(15.21)
(15.22)

(15.23)

The desired temperature and pressure are given by T and P , the dimension of the problem by D,
and the number of degrees of freedom by N = DA, where A is the number of atoms (particles) in
the simulation. The internal or instantaneous pressure is given by PI . The velocity and mass of
the fictitious particle responsible for pressure regulation are vP and MP , and vT and MT are those
for the particle involved in temperature regulation. These masses are constants that impact, for
example, the frequencies and amplitudes of the volume and temperature fluctuations experienced by
the particles in the simulation, and choosing optimal values for a particular simulation is somewhat
of a black art.
Simulation in the canonical or NVT ensemble is also possible, and the so-called Nosé–Hoover
c They are only time reversible in theory.

reverse trajectories.

In practice, numerical error will cause differences in the forward and

184
thermostat[320–325] represents the most common set of equations used to these ends:
[ṙj ]c
[v̇j ]c
v̇T

[vj ]c ,
∂V(r)
= − M-1 jj cc
− vT [vj ]c ,
∂ [rj ]c
[Mjj ]cc [vj ]c − N kB T .
MT 

(15.24)
(15.25)

(15.26)

These are the same as Eqs. 15.18–15.20 above, with vP ≡ 0 and (N + 1) → N .
15.2.2.3

Projections onto normal modes

At the end of a molecular dynamics simulation, we know the trajectory for each particle as a function
of time, r(t). In a crystal, we likely know the equilibrium positions of the atoms, x, and we may
therefore calculate the displacements:
u = u(t) = r(t) − x .

(15.27)

In general, finding the normal modes of a crystal is rather involved (as seen in § 3.4.2); however,
frequently some modes may be found by inspection. Given the displacement pattern for some normal
mode, we may project the particle displacements onto that pattern and look at the amplitude in
that mode as a function of time, A = A (t):
A

u(t) ·
|u(t)| |(t)|

(15.28)

By taking the Fourier transform of the amplitude with respect to time, we may see the frequency
response in the particular mode, .
15.2.2.4

Phonon DOS

The phonon density of states of a simulated material may be found by looking at the Fourier
transform of the velocity autocorrelation function: [83, 315, 326–330]

hv(0) · v(t)i

g(E)

1 XX
[vj (m∆t + t)] · [vj (m∆t)] ,
MDA m j
hv(0) · v(t)i
hv(0) · v(t)i
FT
dteiωt
hv(0) · v(0)i
2π~
hv(0) · v(0)i

where M gives the total number of time steps in the simulation.

(15.29)
(15.30)

185

15.3

Simulation

15.3.1

Qualities in conservative systems: The Fermi-Pasta-Ulam problem

In the 1950s, Fermi et al. performed a numerical study of a linear chain of oscillators subject to nonlinear forces. [331] Their goal was to witness the rate of approach to equipartition of energy amongst
the degrees of freedom in the chain; however, they found instead long-term oscillatory behavior.
Here, we have performed the same style of numerical study but with a much shorter chain, and with
the intent of looking at behavior that could be considered analogous to dissipation.
Clearly, an anharmonic crystal is a conservative system, and conservative systems do not have
dissipative terms. Here, we justify our use of a quality factor in conservative systems by investigating
a simple anharmonic system that displays behavior that looks like damping. Specifically, the system
we investigated is depicted in Fig. 15.3 and consisted of three atoms, all with mass M , connected
to one another by non-linear springs. This simple calculation shows basic effects of anharmonicity.
The initial positions and velocities of the atoms were selected at random, subject to the constraint
that the center of mass for the system be stationary. With the definition ujk ≡ uj − uk , we used the
following equations of motion:
M ü0

= −K(u01 + u02 ) − K2 (u201 + u202 ) − K3 (u301 + u302 ) ,

M ü1

= −K(u12 + u10 ) − K2 (u212 + u210 ) − K3 (u312 + u310 ) ,

M ü2

= −K(u20 + u21 ) − K2 (u220 + u221 ) − K3 (u320 + u321 ) .

(15.31)

Atom 0

F = -Ku - K2u - K3u
Atom 1

Atom 2

u1
Figure 15.3: Periodic linear chain of 3 atoms connected by non-linear springs used in the simple
simulations as described in the text.

186
2.5

Atom 2
2.5

Mode 1 (Vibration)
1.

-0.5

-0
.5

Mode 0 (Translation)

0.5

Atom 0

-0
.5

0.5

1.5

2.
1.
0.

Position

Atom 1

Mode Amplitude

Mode 2 (Vibration)

1.5

-0.5

500

1000

1500
Time

2000

2500

3000

500

1000

1500

2000

2500

3000

Time

Figure 15.4: Displacement amplitude for the
three atoms in the chain. The inset shows the
behavior on a much smaller time scale, where the
atoms appear to be undergoing a normal sort of
harmonic motion. On the larger time scale of the
figure, however, it is clear that energy is being
passed back and forth from atom to atom. The
curves for the three atoms are offset by 1.

Figure 15.5: Displacement amplitude projected
onto normal modes of the harmonic system. By
construction, there is no displacement in the
translational mode. As in Fig. 15.4, the inset
shows that the behavior on a smaller time scale
might be mistaken for regular harmonic motion.
Again, on the larger time scale of the figure it is
clear that energy is being passed back and forth
from mode to mode. They are thus, not normal
modes at all. The curves for the three modes are
offset by 1.

These were integrated numerically using the forward Euler method described in § 15.2.2.1, and the
displacements of the masses as a function of time are shown in Fig. 15.4.
In the harmonic case, the equations of motion reduce to:
M ü0

= −K(u01 ) − K(u02 ) = −K(2u0 − u1 − u2 ) ,

M ü1

−K(u12 ) − K(u10 ) = −K(2u1 − u2 − u0 ) ,

M ü2

−K(u20 ) − K(u21 ) = −K(2u2 − u0 − u1 ) .

(15.32)

Or:
Mü = −Ku ,

(15.33)

where we have:

M= 0

1 0 M ,
0 1

K =  −1
−1

−1
−1

−1

−1  K .

(15.34)

This system has three modes: the first is translational, with frequency 0, and the remaining two are
degenerate and vibrational, with frequency 3ω 0 . So long as we choose orthogonal eigenvectors that

187

Mode 2 (Vibration)
1.5

τm

Mode 1 (Vibration)

Mode Kinetic Energy

2.5

75
25

50
25

25
25

Mode 0 (Translation)

0.5

500

1000

1500
Time

2000

2500

3000

Figure 15.6: Kinetic energy projected onto the normal modes of the three atom chain. By construction and conservation laws, there is no energy in the translational mode. The inset shows behavior
on a smaller time scale, with a sort of periodicity, τm . Here, fluctuations in the local maxima and
minima of the kinetic energy are clearly visible, and the motion cannot be mistaken for regular
harmonic motion. This is even more clear on the larger time scale, where almost all of the energy is
quite visibly being passed back and forth from mode 1 to mode 2. The curves for the three modes
are offset by 1.

span the degenerate subspace, we find that the energy in a particular mode is constant.
In either the harmonic or anharmonic case, we may project the trajectories onto the eigenvectors
of the harmonic system, as described in § 15.2.2.3. These projections are shown in Fig. 15.5. With
the non-linear springs, it is clear that energy moves in and out of the eigenmodes of the harmonic
system. Figs. 15.6 and 15.7 show the kinetic energy for the two vibrational modes,d the former
showing the short time scale behavior and the latter showing the autocorrelation function over large
time scales. The energy starts out roughly split between the two degenerate modes. As time passes,
nearly all of the energy passes from one mode to the other, and the time scale upon which this
occurs is given clearly by the first maximum after the first minimum in the autocorrelation, marked
as τC in Fig. 15.7. Also marked are the minimum and maximum kinetic energy in each mode, Tmin
and Tmax . Finally, we note that the frequency, ω 0 , for displacement in the modes may be found by
looking at the distance between subsequent local minima (or maxima) in the kinetic energy. This is
d For an anharmonic oscillator, there is no way of breaking the potential energy up amongst the different modes as
can be done in the harmonic case.

188

τc
Autocorrelation &
Mode Kinetic Energy

1.50
Tmax,2

1.25
Mode 2 (Vibration)

1.00

Tmin,2

0.75

Tmax,1

0.50
Mode 1 (Vibration)

0.25
0.00

500

1000

1500
Time

2000

2500

Tmin,1
3000

Figure 15.7: Kinetic energy in the vibrational modes of a periodic linear chain of 3 atoms connected
by non-linear springs. The black curves are the autocorrelation functions for the two modes, and
they show that the overall periodicity in the energy transfer is roughly τC , which is marked with the
vertical, red dashed line. The maximum and minimum kinetic energy, Tmax and Tmin are shown by
the horizontal, dashed yellow lines.

shown as τm in the inset of Fig. 15.6.
Using these sorts of parameters, we may define a quality factor for the anharmonic linear chain.
For example, we might be motivated by the decay of the kinetic energy in a damped harmonic
oscillator to propose the following relationship:
ω 0 τc
Tmax exp −
Q 2

Tmin .

Tmax
Tmin

(15.35)

, this yields:
Taking ω 0 = τ2π

τc
= π
ln
τm

(15.36)

The point of this discussion is not whether or not Eq. 15.36 is in any sense right — in fact, Fig. 15.7
does not appear to have an exponential decay at all. Rather, the point is that there are any
number of ways to define an effective quality factor for the vibrations in a crystal — a system that
is conservative but not harmonic — and that the quality somehow quantifies the rate of energy
transfer in and out of the vibrational modes.

15.3.2

Molecular dynamics with GULP

Here we provide a description of the molecular dynamics simulations performed as a part of the
current study. Specifically, using the program GULP [332–335] as our molecular dynamics engine we

189
have simulated FCC aluminum, nickel, copper, rhodium, palladium, silver, iridium, platinum, gold,
and lead using published interatomic potentials as well as our own, modified ones. The trajectories
were analyzed with the program nMoldyn [314, 336, 337], allowing us to find simulated phonon
spectra for the metals as a function of temperature. Figures showing the results for the ten metals
are given in a more systematic fashion in Appendix F. Only a representative fraction of the results
are shown here.
15.3.2.1

Optimizing the potential

As a starting point for our simulations, we adopted embedded atom potentials for FCC metals from
Cleri and Rosato: [338]
) 21
jk
= −
K02 exp −2K1
−1
r0

rjk
= −
K2 exp −K3
−1
r0
VjB + VjR ,

VjB
VjR

(15.37)
(15.38)
(15.39)

where VjB gives the bonding and V R the repulsive contribution to the potential energy for the j th
atom. The equilibrium 1NN distance is r0 , and the distance between the j th and k th atoms is rjk :
rjk

|[rj ] − [rk ]| .

(15.40)

The parameters K0 , K1 , K2 , K3 , and r0 may be determined by fits to experimental data.
Cleri and Rosato optimized their potentials to reproduce experimental values for the enthalpy,
bulk modulus, elastic constants, and lattice parameters of the FCC metals. As such, the potentials
were reasonably well suited to calculations of low temperature properties of the pure metals, and
succeeded in reproducing some high temperature properties as well. Further, the fit to the enthalpy
allowed for use of the potentials in simulations of alloys. The authors indicate that the potentials
tend to overestimate both thermal expansion and the Grüeisen parameter; therefore, they tend to
overestimate the anharmonicity in these metals.
To obtain potentials better suited to our studies of anharmonicities, we optimized the parameters
K0 , K1 , K2 , K3 , and r0 to reproduce experimental values for the elastic constants, and for the
temperature dependent lattice parameter. We did not try to match experimental values of the
enthalpy as they are not relevant to studies of phonon dynamics in pure metals. As a result, our
potentials may reproduce anharmonic effects in the pure metals better than the originals; however,
they are not currently suitable for use in models of alloys. We describe our fitting procedure in
slightly greater detail here.

190
Table 15.1: Optimized parameters for embedded atom potentials for FCC metals. The parameters
for nickel, rhodium, and iridium are separated from the others due to the problems described in the
text in § 15.3.2.1.
Symbol
Al
Cu
Pd
Ag
Pt
Au
Pb
Ni
Rh
Ir

K0 (eV)
1.464
1.346
2.564
1.399
3.872
2.305
1.729
0.905
1.340
2.307

K1
2.378
1.983
2.894
2.732
3.223
3.282
2.273
0.326
0.370
0.259

K2 (eV)
0.142
0.085
0.192
0.098
0.341
0.180
0.159
0.074
0.109
0.058

K3
7.494
10.377
8.449
9.790
8.224
8.484
6.614
11.817
15.033
14.486

The potentials from Cleri and Rosato were used as initial guesses at the parameters of the
embedded atom potential, and the values for the elastic constants and isothermal bulk modulus,
C11 , C12 , C44 and KT , were taken from Simmons and Wang.[338, 339] Values for the temperature
dependent thermal expansion were taken from Touloukian et al. [168], except for aluminum, which
was taken from Wang and Reeber. [107] Values of the lattice parameter at room temperature were
taken from various sources [340–348] and used in conjunction with the thermal expansion to produce
experimental values for the lattice parameter as a function of temperature. In particular, lattice
parameters at 61 , 26 , 36 , 46 , and 65 of the highest temperature for which values of the thermal expansion
coefficient were available were used in the optimization. As the molecular dynamics simulations are
classical in nature, the lowest temperatures were not included in the optimization to avoid quantum
effects impacting the lattice expansion.
For each function evaluation in our optimization, the low temperature elastic constants and bulk
modulus were determined analytically by GULP. At each temperature, a 1 ps molecular dynamics
simulation was run in the NPT ensemble at a pressure of 1 atmosphere. The first 0.49 ps were
used for equilibration and the last 0.51 ps were used to determine the lattice parameter at pressure
and temperature. These values were then compared to the experimental ones, and a weighted least
squares penalty was minimized using the program “fmin” from the SciPy python package.[293] A
convergence study showed that a 3 × 3 × 3 conventional FCC unit cell gave a sufficiently accurate
lattice parameter, so this cell size was used for our potential optimizations. The potentials were cut
off after 12.0 Å, which is the default setting for the Cleri-Rosato potentials in GULP. Roughly, this
captures interactions out to 5NN in nickel and copper; 4NN in aluminum, rhodium, palladium, silver,
iridium, platinum, and gold; and 3NN in lead. The optimized parameters for the potentials are shown
in Table 15.1 and the values of the bulk modulus and elastic constants from experiment, the CleriRosato potential, and our optimized potentials are shown in Table 15.2. The top panels of Fig. 15.8

191
Table 15.2: Elastic constants of FCC metals from experiment (Exp.), [339] the potentials of Cleri
and Rosato (Clr.), [338] and from our optimized potentials (Opt.) — % errors relative to the
experimental values are given in parentheses. The parameters for nickel, rhodium, and iridium are
separated from the others due to the problems described in the text in § 15.3.2.1.
KT
Al

Exp.
Clr.
Opt.
Exp.
Clr.
Opt.
Exp.
Clr.
Opt.
Exp.
Clr.
Opt.
Exp.
Clr.
Opt.
Exp.
Clr.
Opt.
Exp.
Clr.
Opt.
Exp.
Clr.
Opt.
Exp.
Clr.
Opt.
Exp.
Clr.
Opt.

74.7
81.3
76.0
141.7
142.4
142.0
195.7
196.3
195.0
108.4
108.3
108.0
288.6
296.0
288.0
166.3
165.4
166.0
40.9
42.6
41.0
186.5
222.1
188.0
268.7
294.5
269.0
369.5
415.4
370.0

C11
(-8.9)
(-1.8)
(-0.5)
(-0.2)
(-0.3)
( 0.4)
( 0.0)
( 0.3)
(-2.6)
( 0.2)
( 0.6)
( 0.2)
(-4.4)
(-0.3)

(-19.1)
(-0.8)
(-9.6)
(-0.1)
(-12.4)
(-0.1)

84.4
95.0
107.0
174.2
176.7
176.0
221.6
231.9
234.0
129.1
131.7
131.0
317.9
341.1
358.0
184.0
187.4
187.0
44.8
48.4
46.0
256.4
298.2
261.0
374.2
399.0
422.0
524.4
554.5
599.0

C12

(-12.5)
(-26.7)
(-1.4)
(-1.0)
(-4.6)
(-5.6)
(-2.0)
(-1.5)
(-7.3)
(-12.6)
(-1.8)
(-1.6)
(-8.1)
(-2.7)
(-16.3)
(-1.8)
(-6.6)
(-12.8)
(-5.7)
(-14.2)

69.8
74.5
61.0
125.5
125.2
125.0
182.8
178.5
176.0
98.0
96.6
97.0
273.9
273.4
254.0
157.5
154.4
155.0
38.9
39.7
38.0
151.6
184.1
151.0
216.0
242.3
192.0
292.0
345.8
256.0

C44

(-6.8)
(12.6)
( 0.2)
( 0.4)
( 2.3)
( 3.7)
( 1.4)
( 1.0)
( 0.2)
( 7.3)
( 2.0)
( 1.6)
(-2.2)
( 2.3)
(-21.5)
( 0.4)
(-12.2)
(11.1)
(-18.4)
(12.3)

30.2
37.0
29.0
84.3
82.2
82.0
71.9
72.6
71.0
51.9
50.6
51.0
78.4
90.6
77.0
47.0
44.7
45.0
13.9
12.8
14.0
134.7
157.2
132.0
196.4
202.0
194.0
271.3
261.4
269.0

(-22.7)
( 3.8)
( 2.4)
( 2.7)
(-1.0)
( 1.2)
( 2.4)
( 1.7)
(-15.6)
( 1.7)
( 4.8)
( 4.2)
( 8.5)
(-0.4)
(-16.7)
( 2.0)
(-2.8)
( 1.2)
( 3.6)
( 0.8)

show the experimental, Cleri and Rosato, and our optimized lattice parameters for aluminum and
lead. In both cases, our optimized potential reproduces the experimental data significantly better
than does the original Cleri-Rosato potential. The discrepancy in the lattice parameter for aluminum
at the highest temperature is due to its having melted, and the optimized potential quite consistently
underestimates the thermal expansion for lead.
15.3.2.2

Generating trajectories

Using GULP with both the original potentials from Cleri and Rosato and the optimized potentials,
we ran 70 ps simulations of aluminum, nickel, copper, rhodium, palladium, silver, iridium, platinum,
gold, lead using 864-atom supercells (6 × 6 × 6 conventional FCC cells). For each combination of

192
4.16

5.08
Exp.
Clr.
Opt.

4.14

5.04
5.02

4.10

a (Å)

4.12

4.08

5.00
4.98
4.96

4.06

4.94

4.04

4.92

4.02
3.00
Exp.
Clr.
Opt.

2.76 (T/TM)4/3

1.62 (T/TM)2

1.50
1.00

2.00
1.50
1.00

0.50

0.00

0.00
0.2

0.4

0.6

0.8

T/TM (unitless)

2.76 (T/TM)4/3

Exp.
Clr.
Opt.

2.50

2.00

4.90
3.00

M1/2/Q (amu1/2)

2.50
M1/2/Q (amu1/2)

Exp.
Clr.
Opt.

5.06

1.62 (T/TM)2

0.2

0.4

0.6

0.8

T/TM (unitless)

Figure 15.8: Lattice parameters and inverse qualities for aluminum (left) and lead (right). In the top
panels, the thick gray line shows the experimental values for the lattice parameter, the red line with
circular points the values from the potential of Cleri and Rosato, and the blue line with triangular
points the values from the optimized potential. The bottom panels show the inverse quality scaled
by the square root of the mass. The gray region corresponds roughly to the region in which lie our
experimental data, the red lines with circular points are from the potential of Cleri and Rosato, and
the blue lines with triangular points are from the optimized potential. The qualities were found
from fits of the phonon spectra as described in § 15.3.2.4. In the bottom panel of the right hand
plot for lead, the black points are the experimental data from Chapter 10 — these points were used
to find the upper boundary of the gray region.

potential, metal, and temperature, we performed a 1 ps simulation at NPT at a pressure of one
atmosphere with the 864-atom cell to find the temperature dependent lattice parameter.e This
was, in turn, used to fix the size of the supercell, and the simulations were then run at constant
temperature and volume (NVT). For the longer simulations, the use of an NVT ensemble rather than
NPT was necessary because the most effective operating frequencies for the barostats significantly
overlapped with the frequencies of the phonon spectrum (perhaps this is not surprising).
15.3.2.3

Projections onto normal modes

The normal modes of an 864 atom crystal are fairly complicated; however, we may pick out some
simple ones and investigate some of their properties. Fig. 15.9 shows cross sections of the supercell
in equilibrium, displaced in a low energy mode, and displaced in a high energy mode. We may
project the trajectories of the atoms in our crystal onto these displacement patterns as described
e Cross checks with the temperature dependent lattice parameters from the optimizations showed solid agreement
for all the metals despite change in cell size, including nickel, rhodium and iridium with their problematic potentials.

193

Figure 15.9: Cross-sections of the super-cell used for the molecular dynamics simulations. The left
panel shows the cell in equilibrium, the center displaced in a low energy mode, and the right displaced
in a high energy mode. The low energy mode has a wavelength of one cell width (twelve planes with
different displacements), and the high energy mode a wavelength one sixth that (displacements in
pairs of planes).
Low energy mode in Cu

High energy mode in Ag
956 K

777 K

717 K

518 K

478 K
A(t)

A(t)

1036 K

259 K

239 K

10 K

30
40
t (ns)

Figure 15.10: Amplitude as a function of time and temperature of a low energy mode in copper
(left) and of a high energy mode in silver (right). The local minima and maxima are visible on the
right, but not on the left, consistent with their differing frequencies. In both panels, the reduction
in quality with increased temperature is visible in the progression from smoother to more jagged
curves.
in § 15.2.2.3, and examples of these projections for the low energy mode in Cu and the high energy
mode in Ag are shown in Fig. 15.10. The difference in frequency of the modes is visible in the
relative density (in time) of the amplitude curves for the two modes. In both cases, there appears

194
150

Q=13.2

1495 K

150

Q=4.1

Q=5.2

1895 K
Q=4.2

150

Q=14.9

1196 K

200

Q=7.6

Q=14.8

100

1516 K
Q=6.6

300

Q=38.3

897 K
Q=7.1

150
250

Q=28.2

598 K

125

Q=10.2

Normalized Amplitude

600

Q=36.0

1137 K
Q=6.5

300
600

Q=33.5

300

758 K
Q=7.1

400

Q=53.4

299 K

200

700

Q=51.4

Q=34.4

350

379 K
Q=14.8

1200

Q=113.2

600

10 K

1100

Q=282.0

Q=72.9

550

10 K
Q=216.1

20
30
E (meV)

10
15
E (meV)

Figure 15.11: Energy response for the lower and higher energy modes in nickel (left) and platinum
(right). The points in the plot are a downsampling of the Fourier transformed trajectories, and the
lines are fits to the data using the damped harmonic oscillator function of Eq. 15.10. The quality
factors, Q, are displayed in the plot, and they generally decrease with increasing temperature.
Linewidths for the high energy mode tend to be larger than for the low energy one, as would be the
case with constant Q. The plots look very similar, despite difficulties with the optimized potential
in nickel.
to be longer time-scale structure at lower temperatures that is largely broken up into random fits
and spurts at higher temperatures.
The energy response in the two modes may be assessed more directly by taking the Fourier
transform of the amplitude. Fig. 15.11 show the energy response for the lower and higher energy
modes in nickelf and platinum, respectively. The points in the plot are a downsampling of the
Fourier transformed trajectories, and the lines are fits to the data using the damped harmonic
oscillator function of Eq. 15.10 where the fit parameters were ω 0 and Q, the latter of which are
displayed in the plot. Generally, the quality factors decrease with increasing temperature. Also, the
linewidths tend to be larger for the high energy mode than for the low energy ones. This would also
be the case if Q was constant at a given temperature.
f Note that these look very similar, despite the aforementioned difficulties with the optimized potential in nickel.

195
Al

0.270

0.360
895 K

2495 K

0.225

0.300
1996 K
g(E) (1/meV)

g(E) (1/meV)

716 K
0.180
537 K
0.135

0.240
1497 K
0.180

358 K

998 K

0.090

0.120
179 K

499 K

0.045

0.060
0K

0.000

20 25
E (meV)

20
25
E (meV)

Figure 15.12: Phonon DOS of aluminum (left) and iridium (right) from molecular dynamics. The
black lines with points were found using nMoldyn as described in the text, the colored solid lines
are fits using Eqs. 15.41 and 15.42. In the simulation, aluminum has melted at 895 K, as is visible
by the non-zero value of the DOS at E = 0. (This is related to the diffusion constant.) The noise
in the spectra at low temperatures is largely due to the reduced thermal motion, which reduces the
sampling of vibrational states. At higher temperatures, the persistent small peaks are largely due
to finite-size effects. Note that the spectra for the two metals are similar, despite problems with the
optimized potential for iridium.

15.3.2.4

Phonon spectra and quality factors

Once the simulations were completed, the program nMoldyn was used to obtain the velocity autocorrelation function and the phonon DOS from the particle velocities as described in § 15.2.2.4.
Phonon DOS of aluminum and iridium from molecular dynamics are shown in Fig. 15.12. As seen
in Chapters 9–13 , we may describe the damping in terms of a damped harmonic oscillator response
function:
B(Q, E 0 , E) =

πQE 0

E0
E 2
+ Q12
E − E0

(15.41)

where Q is an average quality for the phonons. We then model the temperature dependent energy
shifts and broadening of the DOS with:
gT (E) = B(Q, E 0 , E)

gT0 (∆s E 0 ) ,

(15.42)

where Q and ∆s are parameters to be determined, and gT (E) and gT0 (E) are the phonon DOS
at high and low temperatures, respectively. Fits of this variety are shown as solid colored lines in

196

1336 K
973 K
673 K
293 K

2 Γ (meV)

E (meV)
Figure 15.13: Markers show phonon full width at half maximum, 2Γ as a function of phonon energy,
at temperatures as given in the key. Data are from the triple axis neutron scattering measurements
of copper taken by Larose and Brockhouse. [60, 305] The solid lines are linear fits to the data, the
slope of which give 1/Q.
Fig. 15.12, and work quite well for all of the FCC metals studied here.

15.4

Experiment

Measurements of the inelastic scattering of neutrons by aluminum, lead and nickel were made over
a wide range of temperatures, and these experiments are described in great detail in Chapters 9, 10,
and 11 respectively. At each temperature, the phonon DOS was determined, and these were used
to get quality factors as a function of temperature for each metal using Eqs. 15.41 and 15.42.
Using the same procedure as was outlined for the lead data on phonon linewidths taken from
triple-axis measurements (see § 10.4), the quality factors for copper at elevated temperatures were
also determined using triple-axis data from Larose and Brockhouse. [60, 305] The fits to the triple
axis measurements are shown in Fig. 15.13.

15.5

Discussion

In § 15.2.1.2, we found a relationship between the shifts and the qualities for a damped harmonic
oscillator, ∆ω
ω 0 = − 8Q2 (Eq. 15.6). This is shown for experimental data from aluminum, nickel,

copper, and lead in Fig. 15.14. For lead, the dependence appears to be superlinear; however, we
have discussed at length in Chapter 10 the fact that the values we have found for the shifts in lead do
not agree with previous results, and are suspicious because of problems with the background. With

( T - 0 ) / 0 (unitless)

197

Al
Ni
Cu
Pb

0.05

0.00

-0.05

-0.10
0.00

0.04

0.08

0.12

0.16

0.20

0.24

1/Q (unitless)
Figure 15.14: Experimental values for shifts plotted against quality factors for FCC metals aluminum, nickel, copper, and lead. For a given metal, points farther to the right are at higher
T −hEi0
on
temperatures. A damped harmonic oscillator should show a quadratic dependence of hEihEi
Q ; however, the dependence appears to be more like a square root.

the possible exception of lead, the dependence of the scaled shift on the quality is not quadratic
as suggested by Eq. 15.6. Rather, the dependence for aluminum, nickel, and copper appears to be
more like a square root. Nevertheless, the sign of the relationship appears to be correct for these
metals, and there does appear to be some structure in the relationship between the two anharmonic
quantities.
Shifts in phonon energy with change in temperature are sometimes quantified through a mode
Grüneisen parameter, γb (q), which gives the frequency shift for a particular mode given a change in
volume. (See Eq. 3.78) The thermodynamic Grüneisen parameter γ gives an average over all modes:
b γb (q) ∂T nωb (q) (T )
P P ∂
b ∂T nωb (q) (T )

P P

(15.43)

where nωb (q) is the mean occupancy for bosons with energy Eb (q) = ~ωb (q). Clearly, the collection
of all mode Grüneisen parameters provides more information than the thermodynamic one; however,
as the name implies, the latter provides valuable thermodynamic information. For example, in the
quasiharmonic model, the Grüneisen relation gives the thermal expansion in terms of the volume,
V , the heat capacity at constant volume, CV , and the isothermal bulk modulus, KT :
α(T ) =

γCV (T ) .
3KT V

(15.44)

198
Similarly, the quality factors for all modes are not the same; however, we may consider an
average over the phonon spectrum. The mode dependence of the linewidths can be seen in a variety
of experimental measurements. Figs. 15.13 and 10.8 show the varied values of the linewidths for
specific phonons in copper and lead as determined by triple-axis neutron spectrometry. [58, 60, 152,
158, 349, 350] The phenomenon is also clearly visible in Raman spectra. [351–354] The molecular
dynamics simulations presented here also display this property, with the qualities for the low and
high energy modes in Ni and Pt shown in Fig. 15.11 having different quality factors even at a fixed
temperature. That said, at fixed temperature the energy response appears to be resonance-like, and
the linewidths of the individual modes do tend to be larger at larger energies, as seen in Fig. 15.11,
and as would be the case if the quality factor were constant. Further, Figs. 15.13 and 10.8 show
the approximation that the individual phonon linewidths have a linear dependence on energy is
plausible. We suspect that average qualities may have thermodynamic importanceg and we hope
that the average quality may have a more structured temperature dependence than the qualities of
individual modes. Thus, average quality factors are shown in Fig. 15.15 and are considered here.

In § 15.2.1 we defined the quality factor, Q ≡

KM
C , and in § 15.3.1 we showed how it might

apply to a conservative system, such as phonons in a crystal. Here, we test far this analogy can take
us. From the definition of the quality factor, we have:

=√ .

(15.45)

The scaled quality factors are shown as a function of scaled temperature, TTM , in Fig. 15.15. The
experimental values of the melting temperatures, TM , were used. The left hand side of Eq. 15.45 is
known — the qualities from experiment or simulation, the mass from experiment. The right hand
side is not known, and is some ratio of the damping coefficient to the harmonic force constant. Thus,

in so much as the damping coefficient represents the anharmonic part of the potential,

gives

the relative strengths of the harmonic and anharmonic forces. This ratio is strikingly similar for
aluminum, nickel, copper, and lead as determined by experiment, and for all the FCC metals when
using the optimized interatomic potential.
To aid comparison with experiment, we have fit the scaled quality factors from aluminum, nickel,
2
43
and from lead to QM = c TTM . We find:
and copper to a function QM = c TTM

Pb :
Al, Ni, & Cu :

≈ 2.76

1.62

TM
TM

43
2

(15.46)

where the mass is given in AMU. These are shown as lines in the top panel of Fig. 15.15, and they
g See, for example, Chapter 9, where the entropy determined with phonon spectra that include anharmonic phonon
broadening best reproduce the total entropy of aluminum.

199

2.5
2.0
1.5

Exp.
Al
Ni
Cu
Pb

2.76 (T/TM)4/3

1.0
0.5

m1/2/Q ( AMU1/2 )

0.0
5.0
4.0
3.0
2.0

1.62 (T/TM)
Opt.
Al
Ni
Cu
Pb
Rh

Pd
Ag
Ir
Pt
Au

Clr.
Al
Ni
Cu
Pb
Rh

Pd
Ag
Ir
Pt
Au

1.0
0.0
7.5
6.0
4.5
3.0
1.5
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
T/TM (unitless)

Figure 15.15: Scaled qualities, QM as a function of scaled temperature TTM for FCC metals from
experiment (top) and molecular dynamics with potentials from Cleri and Rosato (bottom) and
optimized potentials (center) as described in the text. In all three panels, the top of the gray region
comes from a fit to the experimental lead data, and the bottom comes from a fit to the experimental
data for aluminum, nickel, and copper. For aluminum with the optimized potential, the simulated
crystal has actually melted at the highest temperature point, TTM ≈ 0.95.

form the border of the light gray region in all three panels of the figure.
For the molecular dynamics simulations, we may investigate a sort of ratio of anharmonic to
harmonic forces more directly. To do so, we have found the potential energy as a function of volume
(lattice parameter). We fit this to a high order polynomial over a wide range, and to a second order
polynomial close to the zero temperature lattice parameter, a0 . Taking the derivatives of these with
respect to the lattice parameter gave a force F (a) and a harmonic force FH (a). At each temperature,

200
the force was integrated over the region, R, for which the potential energy was less than the energy
at twice the root mean squared displacement:h

o
a|V(a) ≤ V a0 + 2 hu2 i
o
a|VH (a) ≤ VH a0 + 2 hu2 i

(15.47)
(15.48)

where the mean squared displacement was calculated from the phonon DOS in the harmonic approximation:
u2

3~
2M

g(ω)
h2n + 1idω .

(15.49)

This was then used to determine root mean squared forces:
sR
rms(F ) = hF 2 i =

rms(FH ) =

F 2 (a)da
da

sR
2i
hFH

2 (a)da
FH

(15.50)

(15.51)

2i
hF 2 i − hFH
2i
hFH

(15.52)

The ratio of the anharmonic to harmonic forces, then, is given by:
rms(F ) − rms(FH )
rms(FH )

For both the Cleri and Rosato and optimized potentials, this ratio is plotted as a function of the
scaled temperature, TTM in Fig. 15.16 In both cases, the structure is rather remarkable, but this is
particularly true for the optimized potentials where all but the lead data fall precisely on a line. A
linear fit to the data from the optimized potential, excluding lead, gives:
rms(F ) − rms(FH )
rms(FH )

= −0.046 − 0.225

(15.53)

The deviation for lead is at least partially explained by the inferior quality of the fit to its lattice
parameter, as seen in the right panel of Fig. 15.8. The sign of the errors in the lead data is consistent
with this explanation. Anecdotally, in optimizing the potential for lead, it seemed that improvements
in the lattice parameter tended to come at a large cost in accuracy of the elastic constants.
The idea of universality in the forces in metals is by no means new. Perhaps the most famous
work in this regard is that from Rose et al. [355–362] who propose a universal binding energy curve
for metals and alloys. Their formulation requires knowledge of the Wigner-Seitz radius, rWS ≡
1
3 3
, the equilibrium binding energy, and the bulk modulus. A correlation between the binding
16π a0
h For the harmonic case, this is equivalent to R =

a| − a0 − 2 hu2 i ≤ a ≤ a0 + 2 hu2 i .

201

0.00

[ rms(F) − rms(FH) ] / rms(FH) (unitless)

−0.08
−0.16
−0.24
−0.32
−0.40

Clr.
Al
Cu
Pd
Ag
Pt
Au
Pb

−0.08
−0.16
−0.24
−0.32
−0.40

Opt.
Al
Cu
Pd
Ag
Pt
Au
Pb
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
T/TM (unitless)

Figure 15.16: Ratios of anharmonic to harmonic forces in FCC metals from molecular dynamics
simulations. For the potential from Cleri and Rosato (top), the simulated data show a good deal of
spread about the linear fit (dotted gray line). The solid gray line is a linear fit to the simulated data
from the optimized potential, and these data (bottom) are very near linear, with the exception of
lead. The deviation of this metal may be explained by a relatively large underestimate of its thermal
expansion, as seen in the top right hand panel of Fig. 15.8.

energy and the melting temperature is well known,[362] and somewhat intuitive, and this may
explain its presence in Figs. 15.15 and 15.16. Another possible interpretation is that the melting
temperature is related to the mean squared displacement, as suggested by Lindemann. [362, 363]
Rose’s universal equation of state is formulated in terms of a pressure-volume relationship, which
would have implications primarily for the longitudinal modes. Here, we see that the transverse
phonon modes are also showing universality.
Looking at Fig. 15.8 (and the similar figures in Appendix F), it is clear that the thermal expansion
has an effect on the quality factors in the molecular dynamics simulations. The lattice parameters
themselves are of little consequence, but their slope as a function of temperature (thermal expansion)
is important. For TTM < 0.8 in aluminum, the lattice parameter has basically been shifted by a
constant (∼ −0.02 Å); however, the slope remains unchanged going from the potential of Cleri and
Rosato to the optimized potential. The quality factors for aluminum are also unchanged. For lead,

202
the optimization reduced the slope significantly, and the quality factors are reduced as well.
It should be noted that the electronic structures of these FCC metals are quite varied: ranging
from the nearly free electron metal aluminum, to the magnetic d-band metal nickel, to the nonmagnetic d-band metal iridium. It would be surprising, then, if the similarities in the phonon
damping arose from electron-phonon or magnon-phonon interactions. Rather, it seems that the
similarities in the phonon linewidths arise from phonon-phonon interactions. The impact of the
thermal expansion on the qualities and the strong similarities in the ratio of anharmonic to harmonic
forces both lend further support to this thesis.

15.6

Summary

Inelastic neutron scattering measurements were made of the phonon spectra of FCC aluminum,
lead, and nickel, and were used in conjunction with previous neutron measurements of copper to
evaluated trends in phonon linewidths as a function of temperature. Additionally, we performed
molecular dynamics simulations of FCC aluminum, nickel, copper, rhodium, palladium, silver, iridium, platinum, gold, and lead, finding simulated phonon spectra and phonon linewidths. Overall,
we see strong trends in the linewidths when scaling the temperature by the melting temperature,
and the linewidths by the square root of the mass, the latter being suggested by the equations for
the damped harmonic oscillator. From experiment, we find:
1.62

. 2.76

(15.54)

where the mass is in AMU. We also find that this equation becomes increasingly accurate for
the simulations as improvements are made to the interatomic potential. The relationship implies
similarity in the ratio of anharmonic to harmonic forces of FCC metals, and we have found a strong
linear trend in this ratio for the simulated FCC metals:
rms(F ) − rms(FH )
rms(FH )

= −0.046 − 0.225

(15.55)

This may be thought of as an extension to the Grüneisen model of a solid. That is, we have
developed a scheme that, taking a low temperature measurement of a phonon spectrum as input,
allows a rough prediction of the phonon linewidths as a function of temperature. Combining this
with a Grüneisen parameter, the overall change in the phonon spectrum may be approximated.
Finally, we conclude that the phonon broadening is likely related to the interatomic potentials
because:
• The phonon linewidths are similar despite widely varying electronic structure in FCC metals

203
• Molecular dynamics simulations indicate that broadening is tied to thermal expansion, which
is known to be related to the interatomic potential.
• Molecular dynamics simulations indicate trends in the ratio of anharmonic to harmonic forces.
That is, the broadening of the phonon spectra with increased temperature is due to phonon-phonon,
not electron-phonon or magnon-phonon interactions.

204

Part VI

Summary and Future Work

205

Chapter 16

Summary
In Part II of this thesis, we considered the analysis of data from time-of-flight chopper spectrometers
in some detail. The outline of a new technique for converting the raw data to S(Q, E) was presented,
as well as improved methods for automatically generating a mask and for finding the incident neutron
energy for instruments lacking beam monitors. Given S(Q, E), a new procedure for estimating the
multiple scattering based on an analytical calculation of the multiphonon scattering was presented,
as were updates to methods for fitting BvK models to phonon spectra, and quantifying the overall
broadening of phonon spectra with temperature.
In Parts III and IV, we considered in great detail the various contributions to the total entropy
of the FCC nearly free electron metals aluminum and lead from near absolute zero to near melting,
and of the magnetic transition metals FCC nickel and BCC iron from near zero to temperatures
well above, and just below, their Curie transitions. Specifically, we used time-of-flight chopper
spectrometers to make measurements of the inelastic scattering of neutrons by phonons in aluminum,
lead, and nickel from less than 3% to 83%, 73%, and 83% of their respective melting temperatures.
For BCC iron, we used nuclear resonant inelastic x-ray scattering to measure the incoherent inelastic
scattering at temperatures from less than 1% to 57% of its melting temperature. The reduced
experimental data were used to obtain phonon spectra. Both neutron scattering with a time-of-flight
chopper spectrometer and nuclear resonant inelastic x-ray scattering have great potential as methods
for learning about anharmonic effects in solids in terms of the entire vibrational spectrum, rather than
some limited set of modes. The experimentally determined phonon spectra were used to determine
the harmonic, nonharmonic and total phonon entropy. Data for the temperature dependent thermal
expansion and bulk modulus were then used to evaluate the purely anharmonic contributions to
the phonon entropy. For the FCC metals aluminum and nickel, these contributions were small and
negative, indicating that the phonons did not shift as much as expected, given the thermal expansion
and bulk modulus. The anharmonic contributions for FCC lead were also negative, but slightly
larger. For BCC iron, however, the anharmonic entropy was large and positive, even relatively far
below the melting temperature. In all cases, anharmonicity made a larger contribution to the total

206
entropy than did vacancies or electron-phonon interactions, and the contributions were of the same
order of magnitude as the electronic entropy. In iron and nickel, the anharmonic entropy was likely
smaller than the magnetic contributions; however, they were by no means negligible. Overall, the
anharmonic phonon entropy simply cannot be ignored in a serious evaluation of the contributions
to the entropies of solids.
For aluminum in particular, we found that the sum of the various components of the entropy
was in excellent agreement with the accepted value of the total entropy — particularly if the spectra
broadened by phonon damping (but not by instrument resolution) were used. Very tentatively, we
suggest that the quasiharmonic approximation, which states that the central frequencies are all that
matters in determining the phonon entropy, may be less accurate than a model in which the entire,
broadened spectrum is used.
In Part V, we have considered some trends in the forces in FCC and BCC transition metals, and
the FCC nearly free electron metals aluminum and lead. (Henceforth referred to simply as “FCC
metals” or “BCC metals”.) The trends are:
• The 1NN longitudinal forces are dominant in FCC metals, but in BCC metals the 2NN longitudinal forces are also important.
• The transverse forces in the 1NN shell of the FCC metals are repulsive, whereas those in the
BCC metals are attractive.
• The shapes of the phonon spectra for the FCC metals show less variation than those of the
BCC metals, both across metals at fixed temperature, and for a particular metal as a function
of temperature.
• The anharmonic entropy of the FCC metals tends to be small and negative, whereas there is
a large variation for the BCC metals.
It seems that many of these differences can be attributed to the structures themselves. Specifically,
the relative strength of 2NN longitudinal forces in BCC metals is likely related to the fact that
the 2NN are not significantly further away than the 1NN, as they are in FCC metals. Also, the
more open BCC structure may give more room for the constituent atoms to move about, providing
increased possibilities for anharmonicity.
Given the similarities of the FCC metals, we were motivated to look for similarities in their
phonon linewidths. Using experimental data and molecular dynamics simulations, we saw strong
trends in the linewidths (when scaling the temperature by the melting temperature, and the linewidths
by the square root of the mass). If we model the atoms in our solid as damped harmonic oscillators,
this relationship implies similarity in the ratio of anharmonic to harmonic forces of FCC metals.
We found a linear trend in this ratio for the simulated FCC metals. The simulations also indicated

207
that the broadening is related to the thermal expansion. Because of all of these similarities and the
fact that the electronic structures of the FCC metals are quite varied, we have concluded that the
phonon broadening in the FCC metals is likely related to the interatomic potentials. That is, the
broadening of the phonon spectra with increased temperature is due to phonon-phonon interactions
as opposed to interactions of phonons with electrons or magnons.

208

Chapter 17

Future Work
Here we present work that could improve upon the techniques and results presented in this thesis.

17.1

Data analysis

Procedures for converting the raw data — counts as a function of time-of-flight, detector, pixel —
to the scattering function S(Q, E) were discussed in § 7.4. In particular, the analytical reweighting
and coordinate mapping technique has only been applied to LRMECS data because of the relative
simplicity of the instrument. This technique should be generalized for use with instruments that have
pixelated detectors and detectors at varying distances from the sample. Basically, the generalization
would involve a method for ordering the pixels and time-channels such that given a (Q, E) pair it is
not too difficult to find the neighboring points required for interpolation. A kD-tree might provide
such a structure, for example.
When we analyze our neutron data, we do not take into account all of the information we have
available. For example, we suspect that a neutron that appears at one time-of-flight cannot also
appear at another. We also expect our data to be continuous and — with the possible exceptions
of Bragg peaks and van Hove singularities — differentiable. A variety of methods exist for taking
this sort of information into account when analyzing an experiment, perhaps the most popular
being Bayesian techniques. [364–367] In fact, these methods have already been applied to neutron
data, [368, 369] and even to data from time-of-flight chopper spectrometers; [255] though their use
is not wide spread. In addition to offering solutions to the problems mentioned above, these sorts
of methods also provide clean formalisms for handling instrument resolution, which would allow
an experimentalist to make measurements using more than one incident neutron energy (thus with
different resolutions) and later combine his data.
The procedures for obtaining a phonon spectrum from a measurement of S(Q, E) are also in
need of further improvement, particularly for measurements at elevated temperatures. The technique presented in § 8.1.2 does not account at all for instrument resolution, nor does it account for

209
anharmonic phonon broadening in so much as this should produce a Lorentzian style tail for the
phonon DOS.
Recasting the problem as a constrained optimization may provide a solution to these difficulties.
First, a representation of the instrument resolution is required. Specifically, given a true value of
S(Q, E), we need to be able to calculate the S(Q, E) that we would measure with the instrument.
Second, a quality factor might be estimated by a procedure similar to the one presented in § 8.2,
but perhaps using the scattering S(Q, E) or S(E) instead of the phonon DOS. Third, the phonon
spectrum should be parametrized in some economical fashion. For example, a polynomial spline with
roughly 20 free parameters can frequently account for the detail in a phonon spectrum that requires
120 parameters when represented as a histogram.a A first guess at the DOS — and consequently,
the parameters of the spline — may be obtained by assuming that there are no contributions from
multiphonon or multiple scattering.
We may then perform an optimization in which the polynomial spline is constrained to be a
phonon DOS — that is, to be positive and to integrate to 1. Given a phonon spectrum, the
incoherent scattering S inc (Q, E) may be calculated, including multiphonon scattering and multiple
scattering (in the approximation of § 8.1.2). We then account for the anharmonic phonon broadening
by convolvingb S inc (Q, E) with the damped harmonic oscillator function, using the estimated quality
factor. We convolve this result with the instrument resolution, and compare to the experimental
data. The parameters may then be adjusted according to some optimization algorithm. As with
any optimization, the art and difficulty is in designing the appropriate penalty function. This sort of
procedure should account for instrument resolution in so far as it is known, and can at least provide
some guess at the anharmonic broadening.
More generally, undoing stuff is hard. Put more technically, matrix multiplication (or convolution) is more robust than matrix inversion (or deconvolution). If a model can be devised which —
through a convolution with a resolution function — allows the data to be analyzed without deconvolution, the analysis will be much more robust. The optimization technique for finding a phonon
spectrum given above is one example. Another might be to use Born–von-Kármán models to produce a phonon DOS or the scattering S(Q, E), again convolving the model data with anharmonic
and instrument broadening functions, rather than deconvolving the experimental data. Going even
further, it may be that there are advantages to converting the model data to time-of-flight, detector,
and pixel rather than performing the inverse operation on the experimental data. A Monte Carlo
simulation of the entire instrument [370] and sample assembly would be one method by which counts
a A more physical picture might arise if the knots in the spline were aligned with the van Hove singularities in the
phonon spectrum.
b For readability, we will use the terms convolution and deconvolution here, even though we are actually discussing
integral transformations that are merely similar to convolutions.

210
in time-of-flight, detector, and pixel could be generated for comparison with experimental data.c d

17.2

Vibrations, magnetism, and superconductivity

For aluminum, we found exceptional agreement between the experimentally determined total entropy
and the sum of many independent components. These sorts of analyses worked quite well for lead,
nickel, and iron as well; however, in these cases there exists clear room for improvement.
For lead, the first step is to remeasure the phonon spectra as a function of temperature with
accompanying measurements of the background. This experiment is already under way on the
Pharos spectrometer at the Lujan center of Los Alamos National Lab for temperatures between
8 K and 550 K. Likely, this will explain the vast majority of the discrepancy between the total and
componentized entropy for the metal. Further measurements above and below the superconducting
phase transition at 7.2 K are of interest as well, as these may shed more light on the interaction
of electrons and phonons in lead. If high temperature experiments do not yield a components of
the entropy that sum to the accepted value of the total entropy, the entropy from electron-phonon
coupling is the next most likely source of an explanation.
For nickel and iron, questions remain about the magnetic entropy, particularly above the Curie
temperatures. Measurements of the phonon spectra directly above and below the Curie temperature
would be of interest, even though previous such experiments on nickel concluded that there was not
a significant effect on phonons in certain high symmetry directions. [208] The largest impacts for
these metals, however, would be in a better assessment of their electronic structure, including their
magnetism. Fixing the magnetization in order to account for the effects of temperature, as we did
in Chapters 11 and 12 is not a good approximation. Modern first principles techniques allow for
accurate determination of the magnetization to at least T2C , [192] and calculations with larger cells
and more degrees of freedom may do even better. Further, modern first-principles techniques allow
determinations of the magnon spectra, which might be used to find the magnon contributions to the
entropy. Alternatively, a method might be devised for using the experimentally determined magnon
dispersions to estimate the magnon entropy.

17.3

Mean phonon lifetimes in FCC Metals

We have collected or found experimental data on phonon lifetimes in the FCC metals aluminum,
nickel, copper, and lead; however, there still remain rhodium, palladium, silver, iridium, platinum,
and gold to measure. In particular, phonon spectra for rhodium and iridium have only been measured
at room temperature. Time-of-flight chopper spectrometer measurements of these metals would be
c The buzzphrase for this general approach to data analysis is generative modeling.
d These last two procedures have the added advantage of using both coherent and incoherent scattering.

211
of the greatest use in refining and solidifying the model of the phonon linewidths presented in
Chapter 15.
Triple-axis measurements are also of great interest, as they allow consideration of specific phonons;
however, the amount of time required to get triple-axis measurements over a significant fraction of
the Brillouin zone is prohibitive. A good compromise may be to take measurements using single
crystals and time-of-flight chopper spectrometers. In this way, dispersions may be rapidly measured
over large swaths of reciprocal space without the loss of directional information that accompanies
measurements of polycrystals.
With respect to the molecular dynamics simulations, a great deal of work is left to be done.
Perhaps the most obvious step is to redo the optimizations of nickel, rhodium, and iridium allowing
for longer range interactions. Beyond this, the potentials developed in this thesis were the results of
a very few number of optimizations with the parameters from Cleri and Rosato always taken as a
starting point. A larger search with a greater number of randomized initial conditions might yield
significantly better results.
Here, we fit to elastic constants and thermal expansion; however, there are many other relevant
parameters to which the potentials might be optimized. In particular, fitting to experimental phonon
spectra would be of interest; however this is almost certainly too time consuming. On the other
hand, we might be able to use specific features of the spectra. For example, the high energy cutoff
of the phonon DOS might be found by considering only a handful of carefully selected Q-points in
a BvK style simulation, or possibly even by a very short molecular dynamics simulation. Similar
procedures may be possible for finding van Hove singularities. Another exciting possibility is to use
ab-initio molecular dynamics simulations. One advantage of this technique is that it should be able
to capture quantum effects at low temperatures.

17.4

Vibrational entropy in the presence of damping

For all of the metals studied, the question of how to determine the entropy given a damped phonon
spectrum is still open. Here, we have used the quasiharmonic approximation with the damped
phonon spectrum. For aluminum, the damped phonon spectrum, first corrected for effects of instrument resolution, then used in the quasiharmonic approximation was found to give the better
agreement with the total entropy than either the uncorrected or unbroadened spectra. Is this generally true? Are there still better techniques?
Molecular dynamics simulations may provide a way to get at an answer to these questions.
Specifically, for a sufficiently long simulation with a sufficiently large supercell, the phonon entropy
may be calculated directly from the trajectory, as well as from the phonon spectrum with and
without damping. That is, we may compare the entropy as determined through a probability density

212
in configuration space with that found through the velocity correlation function. We may then see
which phonon DOS — with or without anharmonic broadening — leads to the best agreement with
the direct determination, and whether or not other approximations yield still better agreement.

213

Appendices

214

Appendix A

Table of Symbols
Symbol
Operations, not quite variables:
————–
ẋ
ẍ
∆x
hxi
|x|
x∗
x-vector
FT {f (t)}

Meaning

Partial derivative
Time derivative of x
Second time derivative of x
Increment or small change in x
Mean value of x
Measure of x: magnitude, area, determinant, etc.
A particular value of x
Vector with x elements
Fourier transform of f (t)
Convolution
Integral transform — similar to convolution

215

Symbol
ai
â+
â−
B(Q, E 0 , E)
c, c1 , c2 , · · ·
CP
CV
EF
EI
[Ee ]p∗
E0
Eλ
hEiT
Ef
fT (E)
g(E) = gT (E)
G(E) = GT (E)
h, k, l
Hph
I 00
I0
I 00van
Ib
Is
I el

Meaning
Lattice parameter
Scattering length
Creation operator
Annihilation operator
Example atom vector
Amplitude of oscillation
Number of atoms
Branch index
Damped harmonic oscillator function
Speed of light (299,792,458 m/s)
Index Cartesian directions
Heat capacity at constant pressure
Heat capacity at constant volume
Index for detectors
Dynamical matrix (Various orders)
Dimension (Number of dimensions)
2.7182818284590451
Energy, measured phonon energy (Energy transfer)
Final neutron energy (From sample to pixel)
Initial neutron energy (From moderator to sample)
Energy for elastically scattered neutron as seen in a row of pixels
Ground State Energy
Energy of the λth state
Mean phonon energy at temperature T
Fermi energy
Fraction of background scattering to be removed
Mean occupation for Fermions at temperature T
Force
Helmholtz free energy
Phonon density of states at temperature T
Electron density of states at temperature T
Gibbs free energy
Planck’s constant (6.626068 × 10−34 J·s)
Miller indices (only used together)
Planck’s constant over 2π
Hamiltonian
Nuclear Hamiltonian
Index ; Imaginary number, −1
Intensity before pressure- and energy- efficiency corrections
Intensity before pressure- and after energy- efficiency correction
Intensity after pressure- and energy- efficiency corrections
Intensity from vanadium (uncorrected)
Intensity from measurement of background
Intensity from measurement of sample
Intensity from elastic scattering
Index
Jacobian matrix

216

Symbol
kB
KT
[li ]
LI
Ldp
Ld
Lp∗
Lmon
hLdet i
Li
me
mn
nT (E)
N, N 0
PE
QI
QF
[ri ]
|R|
[si ]

Meaning
General index, index for force constant matrices
Boltzmann’s constant (1.3806503 × 10−23 J/K)
Force constant or force constant matrix
Longitudinal force constant
Isothermal bulk modulus
Index
Stack of all cell vectors
Cell vector
Distance from moderator to the sample
Distance from sample to pixel
Distance to a detector (used for un-pixelated detectors)
Distance to a row of pixels
Distance between first and second monitors.
Mean distance traveled by a neutron while in a detector
Number of cell vectors (number of q-points too)
Number of cells along ith direction.
Electron mass
Neutron mass
Nuclear mass
Example matrix
Mean occupancy for bosons at temperature T
Normalization constants for multiphonon correction
Counting numbers
Indexes pixels ; Momentum
Number of phonons
Number of phonons at energy E
Principal value
Reciprocal lattice vector
Wavevector transfer or phonon wavevector
Initial neutron wavevector (from moderator to sample)
Final neutron wavevector (from sample to pixel)
Quality factor
Stack of all instantaneous position vectors
Instantaneous position vector
Cube edge
Region of integration (like a set)
Area of region R (like the measure of the set)
Stack of all site vectors
Site vector

217

Symbol
S(Q, E)
S P (Q, E)
S inc
S coh
Si
Scf
Sel
Sel,G
Sel,D
Sel,M
Sph
Sph,H
Sph,D
Sph,Q
Sph,A
Sph,NH
Sel−ph
Sel−ph,ad
Sel−ph,na
SJanaf
T̄j
Tph
Tel
[ui ]
Û
vF
vv , v
Vq
Vph
Vel
Vel−ph
V̂
[xi ]
Z(Q, E)

Meaning
Angle averaged response function
P-phonon angle averaged response function
Incoherent response function
Coherent response function
Entropy
Component of the entropy
Configurational entropy
Electronic entropy (non-magnetic)
Ground state electronic entropy
Dilation electronic entropy
Magnetic electronic entropy
Phonon entropy
Harmonic Phonon entropy
Dilation Phonon entropy
Quasiharmonic Phonon entropy
Anharmonic Phonon entropy
Nonharmonic Phonon entropy
Electron-phonon entropy
Adiabatic electron-phonon entropy
Non-adiabatic electron-phonon entropy
Total entropy taken from NIST-Janaf
Number of site vectors
Time ; Index for time-of-flight
Temperature
Characteristic temperature of phonon spectrum
Nuclear kinetic energy
Electronic kinetic energy
Stack of all displacement vectors
Displacement vector
Normal mode vector
Displacement operator for neutron scattering (Squires p. 29)
Number of primitive unit cells
Final neutron velocity
Example vector and variable
Volume
Reciprocal space volume
Potential energy
Nuclear potential energy
Electronic potential energy
Potential energy of electron-phonon interactions
Displacement operator for neutron scattering (Squires p. 29)
1/2 argument of Debye-Waller factor, exp (2W )
Stack of all equilibrium position vectors
Equilibrium position vector
Penalty function
Instrument resolution function

218

Symbol
δij
δ(v)
∆i
∆s
ζ0
ζP , ζms
, ζms
b (q)
Θ1
λI
Λb q
ξ =E−µ
ρHe
ρA
σ inc
σ coh
σ abs
τI
τe
τdp
τmon
[τe ]p∗
τ1
Υdt
Υt
Υd
ΥP (E)
ψ0
ψλ
ω̄j
Ωi

Meaning
Linear coefficient of thermal expansion
kB T

Grüneisen parameter
Halfwidth of a phonon peak
Kronecker delta function
Dirac delta function
Shift of type i ∈ {3, 4, Q}
Shift from fit for quality factor
Small number ...
Constant for removal of elastic peak
Multiple scattering constants
Effective electron phonon coupling function
Polarization vector
Index for scattering angles
Scattering angle
First scattering angle in array
Energy level index
Incident neutron wavelength
Eigenvalue of the dynamical matrix
Chemical potential
Substitution variable, energy less chemical potential
Grand canonical ensemble
3.1415926535897931
Number density of 3 He in a detector
Atomic number density
Incoherent neutron cross-section (for vanadium, in Chapter 7)
Coherent neutron cross-section (for vanadium, in Chapter 7)
Absorption cross-section (for helium, in Chapter 7)
Time-of-flight
Time-of-flight at which neutrons arrive at the sample
Time-of-flight at which elastically scattered neutrons arrive at a detector (or pixel)
Time-of-flight for neutron to travel from sample to pixel
Time-of-flight for neutron to go from first to second monitor.
Time-of-flight for elastically scattered neutron as seen in a row of pixels
First time-of-flight in array
Probabilities of detecting a neutron
Energy-dependent probabilities of detecting a neutron
Pressure-dependent probabilities of detecting a neutron
Probability of P-phonon scattering with energy E
Phase of oscillation
parameters for potentials
Wavefunction
Ground state wavefunction
Wavefunction for λth energy level
Angular frequency
Moment of phonon spectrum
Number of microstates corresponding to macrostate ; Solid angle
Number of microstates... for subsystem i

219

Appendix B

Entropy of Non-interacting
Fermions and Bosons
In order to evaluate the entropy of systems of bosons or fermions, we place ourselves in the grand
canonical ensemble. We have the following partition function, and expression for the entropy:

i±1
Yh
1 ± e−β(Eλ −µ)

(B.1)

= kB ln Ξ + kB T

∂ ln Ξ
∂T

(B.2)

where µ is the chemical potential, Eλ is an energy, β ≡ kB1T and kB is Boltzmann’s constant. The
plus signs correspond to fermions, and the minuses to bosons. The logarithm of a product may be
rewritten as a sum of logs, so:
ln Ξ

h
i±1
X h
1 ± e−β(Eλ −µ)
=±
ln 1 ± e−β(Eλ −µ) .

(B.3)

Applying the chain rule, we have:
∂ ln Ξ
∂T

βX
Eλ − µ
eβ(Eλ −µ) ± 1

(B.4)

220
To simplify the development, we define ξλ = Eλ − µ, so:
kB

X
ln 1 ± e−βξλ +

βξλ
eβξλ ± 1

X βξλ
−βξ
1±e λ
eβξλ ± 1
X eβξλ X βξλ
ln βξ
e λ ±1
eβξλ ± 1
X ∓1 + eβξλ ± 1 X βξλ
ln
eβξλ ± 1
eβξλ ± 1
X
X ∓1
βξλ
ln βξ
+1 +
e λ ±1
eβξλ ± 1
X ∓1
βξλ
ln βξ
+ 1 ∓ βξ
e λ ±1
e λ ±1
[ln (1 ∓ nFB ) ∓ nFB βξλ ] ,

(B.5)

where the nFB = eβξλ1 ±1 is either the mean fermion or boson occupation number, also denoted fT
and nT , respectively. Continuing, we have:
kB

= ∓

X

ln (1 ∓ nFB ) ∓ nFB ln eβξλ

X

= ∓
ln (1 ∓ nFB ) ∓ nFB ln (1 ∓ nFB ) ± nFB ln (1 ∓ nFB ) ∓ nFB ln eβξλ

X

= ∓
(1 ∓ nFB ) ln (1 ∓ nFB ) ± nFB ln (1 ∓ nFB ) − ln eβξλ

X
= ∓
(1 ∓ nFB ) ln (1 ∓ nFB ) ± nFB ln 1 ∓

eβξλ ± 1

− ln e

βξλ

βξλ
X
±1
βξλ
− ln e
(1 ∓ nFB ) ln (1 ∓ nFB ) ± nFB ln βξ
e λ ± 1 eβξλ ± 1

βξλ
X
βξλ
− ln e
(1 ∓ nFB ) ln (1 ∓ nFB ) ± nFB ln βξ
e λ ±1

X
(1 ∓ nFB ) ln (1 ∓ nFB ) ± nFB ln βξ
e λ ±1
[(1 ∓ nFB ) ln (1 ∓ nFB ) ± nFB ln (nFB )] .
(B.6)

For fermions, we take the upper sign, and we have:
kB

= −

[(1 − fT ) ln (1 − fT ) + fT ln (fT )] .

(B.7)

221
For bosons, we take the lower sign, and we have:
kB

[(1 + nT ) ln (1 + nT ) − nT ln (nT )] .

(B.8)

For phonons, we may use Eq.3.71, to convert the expression for the boson entropy into an integral
using the density of states:
Sph

kB D

g(E) [(1 + nT ) ln (1 + nT ) − nT ln (nT )] dE .

(B.9)

g(E) [(nT + 1) ln (nT + 1) − nT ln (nT )] dE .

(B.10)

Or, in 3D:
Sph

3kB

Likewise, for electrons we may write:
Sel

−kB

G(E) [(1 − fT ) ln (1 − fT ) + fT ln (fT )] dE .

(B.11)

222

Appendix C

Analytical Reweighting — Algebra
In § 7.4.2 we came across the problem of finding the solution to the following integral:
ZZ

ZZ
|J|dΘdτ =

|RQE | =

RΘτ

dQ
dΘ

dQ
dτ

dE
dΘ

dE
dτ

dΘdτ ,

(C.1)

where |J| is the determinant of the Jacobian matrix , J, and RΘτ is the region of integration in
φ-T space, which is the rectangle formed by the points (Θ, τ ), (Θ, τ + ∆τ ), (Θ + ∆Θ, τ ), and
(Θ + ∆Θ, τ + ∆τ ).
The determinant is given by |J| = J11 J22 − J12 J21 , and our first task is to find the components
of J. Since:
J21 =

dE
=0,
dΘ

(C.2)

we see that we don’t need to find J12 in order to evaluate the integral.a From Fig.5.2, we see that:
J11 =

dQ
dΘ

dΘ

Q2I + Q2F − 2QI QF cos(Θ)

−QI QF sin(Θ)
QI + Q2F − 2QI QF cos(Θ)

(C.3)

Finally, we have:
2 #
dE
mn
Ld
mn L2d
J22 =
(EI − EF ) =
EI −
3 ,
dτ
dτ
dτ
τ − τI
(τ − τI )

(C.4)

where Ld is the distance from the sample to the detector indexed by d.b
a Technically, the detectors at different angles might also be at different distances. As such J
21 is not identically
zero; however, our bins in Θ-τ space will never cross detector boundaries, and thus we will always have J21 = 0 while
we are evaluating Eq. C.1
b On a pixelated instrument, this distance should be L .
dp

223
We are thus trying to evaluate the following integral:
mn L2d

−QI QF sin(Θ)
dΘdτ
QI + Q2F − 2QI QF cos(Θ)
RΘτ (τ − τI )
"Z
mn L2d
−QI QF sin(Θ)
dΘ dτ .
Q2I + Q2F − 2QI QF cos(Θ)
Rτ (τ − τI )
RΘ
ZZ

|RQE | =

(C.5)

The integral over Θ may be evaluated immediately. In fact, it consists in simply undoing the
derivative taken on the previous page:
mn L2d

|RQE | =

Q2I + Q2F − 2QI QF cos(Θ) dτ .

(C.6)

QF =

mn Ld
~(τ − τI )

(C.7)

~3
m2n Ld

Rτ (τ − τI )

We note that:

and rewrite Eq. C.6 in this form:
|RQE | =

Q3F

Rτ

Q4F

Q2I + Q2F − 2QI QF cos(Θ) dτ

Rτ

QI
QF

2
+1−2

QI
cos(Θ) dτ .
QF

(C.8)

QI
and z = cos(Θ), then dx = m~Q
If we let x = Q
dτ , and we have:
n Ld

|RQE | =

~ Q3I
mn

x2 + 1 − 2xz
x4

Rτ

~ Q3I p
mn

1 − z2

Rτ

~ Q3I

1 + x12 − 2z

mn
2

dx =
1

−z
√x
1−z 2

Rτ

dx .

(C.9)

Let:
−z
y = √x
1 − z2

(C.10)

which also gives:
dy = √

−1 dx
1 − z 2 x2

(C.11)

224
Then:
|RQE | =

~2 Q3I p
1 − z2
mn
−~2 Q3I
mn

1 + y2
(−x2 1 − z 2 )dy
Rτ
Z p
1 + y2
1 − z2
dy .
Rτ

(C.12)

We may solve Eq. C.10 for x1 , and substitute this into our integral:
= y 1 − z2 + z ,

(C.13)

Z p
p
−~2 Q3I
1 + y 2 y 1 − z 2 + z dy
1 − z2
mn
Rτ
Z p
2 3
−~ QI
1 − z2 2
1 + y2 y + √
dy
mn
1 − z2
Rτ
Z
Z p
3
−~2 Q3I
2 2
1−z
y 1 + y dy +
1 + y dy .
mn
1 − z 2 Rτ
Rτ

|RQE | =

(C.14)

We may evaluate these two integrals with any standard software for symbolic manipulation (Maxima,
Maple, Mathematica...); but, since we have come this far, we might as well do it the old-fashioned
way.
For the first integral, we let u = 1 + y 2 , du = 2ydy, then:
Rτ

3
1 3
u 2 du = u 2 =
1 + y 2 dy =
1 + y2 2 .
2 Rτ

(C.15)

For the second integral, we let y = sinh(v), dy = cosh(v)dv, then:

1 + y 2 dy

Rτ

1 + sinh (v) cosh(v)dv =

cosh2 (v)dv

Rτ

[exp(2v) + exp(−2v) + 2] dv
Rτ
1 1
exp(2v) + exp(−2v) + 2v
4 2
1h p
[2 cosh(v) sinh(v) + 2v] =
y 1 + y 2 + arcsinh(y) .

(C.16)

Putting this all together, we have:
|RQE | =

i
3 1
3
−~2 Q3I
1h p
2 2
2 2
1−z
1+y
+√
y 1 + y + arcsinh(y)
mn
1 − z2 2
h p
i
3
3
−~2 Q3I
3z
1 − z2 2 2 1 + y2 2 + √
y 1 + y 2 + arcsinh(y)
. (C.17)
6mn
1 − z2

225
We may now simplify and back-substitute:

|RQE | =

1
3
3
−~2 Q3I
3z
1 + y2 2
2 2
2 2
1−z
+√
arcsinh(y)
2 1+y
+ 3zy
6mn
1 − z2
1 − z2
3
−~2 Q3I n
2 1 + y2 1 − z2 2
6mn
1
+3yz 1 − z 2 1 + y 2 1 − z 2 2 + 3z 1 − z 2 arcsinh(y)
ip
−~2 Q3I nh
(1 + y 2 ) (1 − z 2 )
2 1 + y 2 1 − z 2 + 3yz 1 − z 2
6mn
+3z 1 − z 2 arcsinh(y) .
(C.18)

Finally, we use the following relationships:
(1 − z 2 )(1 + y 2 ) =
yz 1 − z 2 =

2z
+1
x2
− z2 ,

(C.19)

which yield:

|RQE | =

−~2 Q3I
6mn

(

4z
3z
+2+
− 3z 2
x2

12
2z
+1
x2
+3z 1 − z

(

x −z

arcsinh

1 − z2

−~2 Q3I
6mn

−z
+3z 1 − z arcsinh √x
1 − z2
(
12
Q2
QF
QF
Q2F
−~2 Q3I
2 F2 −
cos(Θ) − 3 cos2 (Θ) + 2
cos(Θ)
6mn
QI
QI
Q2I
QI
!)
QF
QI − cos(Θ)
+3 cos(Θ) sin (Θ)arcsinh
sin(Θ)

− + 2 − 3z 2
x2

2z
+1
x2

(C.20)

The last expression may be rewritten in terms of τ and τI , rather than QI and QF ; however, that is
left as an exercise to the reader who is truly bored.

226

Appendix D

Summation Over Q
Inelastic neutron scattering data from a time-of-flight chopper spectrometer includes both coherent
and incoherent scattering. As the phonon DOS is closely related to the incoherent scattering, and
summing the coherent scattering over Q yields a good approximation of the incoherent scattering, we
sum over Q to determine the phonon DOS. Fig. D.1 shows the region of the data used for integration.

Figure D.1: Integration region for determination of the phonon DOS from inelastic neutron scattering
data. Explanation given in the surrounding text.

The top panel shows the scattering, converted from time-of-flight and detector (and pixel) to

227
the scattering function, S(Q, E). The energy, E, marked “longE” is an estimate of the highest
single-phonon energy, and that marked “eStop” is the highest energy considered. From longE, a
rectangular area of integration is found by looking along E =longE for the values of the momentum
transfer, Q, at which the data starts and stops. The data above E =eStop is thrown away, as is
the data below E = 0, the former because of issues with noise as the final velocity of the neutrons
approaches zero, the latter because the resolution gets significantly worse as the final velocity of
the neutrons increases. The bottom panel shows the phonon density of states as determined from
the data, and it is clear that the chosen value of longE indeed corresponds to the highest phonon
frequencies.

228

Appendix E

Constraints on Force Constants in
BvK Models
Consider a pair of atoms in a crystal whose separation is given by some vector. The forces one atom
experiences as the other is displaced may be represented with a force constant tensor. Assuming
our crystal resides in three dimensions, we write:

F11

F =  F12
F13

F12
F22
F32

F13

F23  .
F33

(E.1)

Here, we have 9 degrees of freedom (DOF).
The point group symmetries of the crystal may be used to reduce the DOF. We take one of the
3 × 3 representations of the point group symmetries, Ss , and apply it to F , requiring that F remain
unchanged:

S11

 s
Ss =  S21
S31

S12

S13

S22

S23

S32

S33

 .

(E.2)

Or:
SsT F Ss = F .

(E.3)

SsT F − F Ss−1 = 0 .

(E.4)

We may rewrite this equation:

229
This is a special case of the Lyapunov equation:
AX + XB = C ,

(E.5)

where A, B, C and X are all square matrices of dimension N . It is clear that all the terms on the
left-hand side of the equation are linear in the components of X, thus, their sum must be linear in
the components of X. This means that we may rewrite the equation as follows:
Mx + b = 0 ,

(E.6)

where M is a N 2 × N 2 matrix, and y and b are N 2 -vectors. The order in which we choose to map
the components of the matrix X into the vector x is arbitrary; however, once we choose an order,
we must be consistent. Let us work through an example. Let:

x11

 x12 
 x13 
 x21 
x =  x22  ,
 x23 
 x31 
 x32 
x33

(E.7)

which would imply the same ordering for b, where the components would be given by −Cij .
Let us construct the matrix Z ≡ AX + XB. For the 3x3 case, we have:
Z=
A X + A12 X21 + A13 X31
 11 11
 A21 X11 + A22 X21 + A23 X31
A31 X11 + A32 X21 + A33 X31

A11 X12 + A12 X22 + A13 X32
A21 X12 + A22 X22 + A23 X32
A31 X12 + A32 X22 + A33 X32

A11 X13 + A12 X23 + A13 X33

A21 X13 + A22 X23 + A23 X33 
A31 X13 + A32 X23 + A33 X33

X11 B11 + X12 B21 + X13 B31

 X21 B11 + X22 B21 + X23 B31
X31 B11 + X32 B21 + X33 B31

X11 B12 + X12 B22 + X13 B32
X21 B12 + X22 B22 + X23 B32
X31 B12 + X32 B22 + X33 B32

X11 B13 + X12 B23 + X13 B33

X21 B13 + X22 B23 + X23 B33  .
X31 B13 + X32 B23 + X33 B33

We see that Z12 depends on the 1st row of A, the 2nd column of X, the 1st row of X, and the 2nd

230
column of B. More generally:
Zij =

Aik Xkj + Bkj Xik .

(E.8)

Because of our choice of mapping from X to x, the Zij tell us what row in M we are dealing with.
The indices on the variables Xkj or Xik tell us which column in M . All that is left is to add the Aik
and Bkj into the slots in M , so given. This is difficult to show explicily, but we will try anyways:

{Z11 , X11 }

M11

M12

...

M19

{Z11 , X12 }

...

{Z11 , X33 }

 M21
M =
 ..
 .
M91

M22
..

...
..

M92

...

M29 
 {Z21 , X11 } {Z21 , X12 }
←
..
..
.. 
. 
{Z33 , X11 } {Z33 , X12 }
M99

{Z21 , X33 } 
 .
..
{Z33 , X33 }

...

(E.9)

The left and center of the equation are M and the components of M . The thing on the right is
supposed to indicate that any time we have Zij on the left in Eq. E.8, and an Xmn next to one of
the coefficients Apq or Brt on the right, we add that coefficient at the slot marked {Zij Xmn } in M .
The problem AX + XB = C has now been reduced to M x + b = 0, which linear algebra tells us
how to solve.
For our particular case, we wish to find the constraints on the components of F . Simply take
A → SsT , B → Ss−1 , X → F in Eq. E.5, and C = 0 and then put M into reduced row echelon form.
Reading off the rows gives us the constraints on the components of F . For example, we may end up
with something that looks like this:
xx

[ 0.

0.]

[ 0.

0.]

[ 0.

0.]

[ 0.

0. -1.]

[ 0.

0.]

[ 0.

0.]

[ 0.

0.] ,

231
which we can rewrite like so:
xy

= 0
xz

= 0
yx

= 0
yy

-zz = 0
yz

= 0
zx

= 0
zy

= 0 .

We see, then, that there are two DOF. Both yy = zz, and xx may be varied independently This
particular force constant tensor is axially symmetric.
If there are n symmetry elements, Ss , that can transform our bond vector back onto itself, we
simply stack up the n Ms :

 M2 
M = . 
 ,
 .. 
Mn

(E.10)

find the reduced row echelon form of M , and read off the constraints on the force constants as
above.

232

Appendix F

Supplementary Material for
Chapter 15
Figs. F.1–F.10 show lattice parameters and quality factors for the FCC metals as a function of
normalized temperature T /TM . In the top panels, the thick gray line is the experimental data
(sources are given in § 15.3.2.1). In all panels, the red circles and lines were calculated using the
potentials from Cleri and Rosato [338], and the blue triangles and lines were calculated using our
optimized potentials. The light gray region in the lower panels comes from fits to experimental data
as described in § 15.5. The black squares in the bottom panel show the experimental data, when
available.
4.16

3.60
Exp.
Clr.
Opt.

4.14

3.56

4.10

a (Å)

4.12

4.08

3.54
3.52

4.06

3.50

4.04
Al

4.02
3.00
Exp.
Clr.
Opt.

3.48
3.00

2.76 (T/TM)4/3

2.00

1.62 (T/TM)

1.50
1.00
0.50

2.76 (T/TM)

Exp.
Clr.
Opt.

2.50
M1/2/Q (amu1/2)

Exp.
Clr.
Opt.

3.58

2.00
1.50
1.00
0.50

0.00

4/3

1.62 (T/TM)2

0.00

0.2

0.4

0.6

0.8

0.2

0.4

0.6

T/TM (unitless)

Figure F.1: Aluminum

Figure F.2: Nickel

0.8

233

3.72

3.92
Exp.
Clr.
Opt.

3.70

3.88

3.66

a (Å)

3.68

3.64
3.62

M1/2/Q (amu1/2)

1.50
1.00

2.76 (T/TM)4/3

Clr.
Opt.

2.50

2.00

0.50

3.78
3.00

2.76 (T/TM)4/3

Exp.
Clr.
Opt.

2.50
M1/2/Q (amu1/2)

3.84

3.80
Cu

3.58
3.00

2.00
1.50
1.00
0.50

1.62 (T/TM)2

0.00

1.62 (T/TM)2

0.00

0.2

0.4

0.6

0.8

0.2

0.4

0.6

T/TM (unitless)

Figure F.3: Copper

Figure F.4: Rhodium

4.15

0.8

4.24
Exp.
Clr.
Opt.

4.10

Exp.
Clr.
Opt.

4.22
4.20
4.18
a (Å)

4.05
a (Å)

3.86

3.82

3.60

4.00
3.95

4.16
4.14
4.12
4.10

3.90

4.08

3.85
3.00

Clr.
Opt.

M1/2/Q (amu1/2)

2.50

2.00
1.50
1.00
0.50

4.06
3.00

2.76 (T/TM)4/3

Clr.
Opt.

2.50
M1/2/Q (amu1/2)

Exp.
Clr.
Opt.

3.90

2.76 (T/TM)4/3

2.00
1.50
1.00
0.50

1.62 (T/TM)2

0.00

1.62 (T/TM)2

0.00

0.2

0.4

0.6

T/TM (unitless)

Figure F.5: Palladium

0.8

0.2

0.4

0.6

T/TM (unitless)

Figure F.6: Silver

0.8

3.93
3.92
3.91
3.90
3.89
3.88
3.87
3.86
3.85
3.84
3.83
3.82
3.00

Exp.
Clr.
Opt.
a (Å)

a (Å)

234

Ir
Clr.
Opt.

2.00
1.50
1.00
0.50
0.00

0.2

2.76 (T/TM)4/3

1.62 (T/TM)2

0.4

0.8

0.6

Clr.
Opt.

2.00
1.50
1.00

0.00

0.2

2.76 (T/TM)4/3

1.62 (T/TM)2

0.4

0.8

0.6

T/TM (unitless)

Figure F.7: Iridium

Figure F.8: Platinum

5.08
Exp.
Clr.
Opt.

4.25

Exp.
Clr.
Opt.

5.06
5.04
5.02

4.20

a (Å)

0.50

4.30

4.15

5.00
4.98
4.96
4.94

4.10

4.92

4.05
3.00

M1/2/Q (amu1/2)

1.50
1.00
0.50
2.76 (T/TM)

0.2

0.4

4/3

0.6

T/TM (unitless)

Figure F.9: Gold

2.76 (T/TM)4/3

Exp.
Clr.
Opt.

2.50

2.00

0.00

4.90
3.00

Clr.
Opt.

2.50
M1/2/Q (amu1/2)

Exp.
Clr.
Opt.

2.50
M1/2/Q (amu1/2)

M1/2/Q (amu1/2)

2.50

4.10
4.08
4.06
4.04
4.02
4.00
3.98
3.96
3.94
3.92
3.90
3.00

2.00
1.50
1.00
0.50

1.62 (T/TM)2

0.00
0.8

0.2

0.4

0.6

T/TM (unitless)

Figure F.10: Lead

0.8

235
Figs. F.11–F.20 show phonon spectra as a function of temperature (as marked) for the FCC
metals as determined with molecular dynamics simulations using our optimized potentials in black
lines and points. The colored, solid lines are fits to the data using Eqs. 15.41 and 15.42.
Al

0.270

0.270
895 K

1495 K

0.225

537 K
0.135

1196 K
g(E) (1/meV)

g(E) (1/meV)

716 K
0.180

0.180
897 K
0.135

358 K

598 K

0.090

0.090
179 K

299 K

0.045

0.045
0K

0.000

20 25
E (meV)

Figure F.11: Aluminum

0.270
1295 K

1595 K

0.300

0.225

0.240
777 K
0.180

1276 K
g(E) (1/meV)

1036 K
g(E) (1/meV)

Figure F.12: Nickel

0.360

20
30
E (meV)

0.180
957 K
0.135

518 K
0.120

638 K
0.090

259 K
0.060

319 K
0.045

0K
0.000

20
25
E (meV)

Figure F.13: Copper

20
30
E (meV)

Figure F.14: Rhodium

236

0.450

0.540
1195 K

1195 K

0.375

0.450

717 K
0.225

956 K
g(E) (1/meV)

g(E) (1/meV)

956 K
0.300

0.360
717 K
0.270

478 K

0.150

0.180
239 K

239 K

0.075

0.090
0K

0.000

15
E (meV)

Figure F.15: Palladium

15
E (meV)

0.630
2495 K

1895 K

0.300

0.525

0.240
1497 K
0.180

1516 K
g(E) (1/meV)

1996 K
g(E) (1/meV)

Figure F.16: Silver

0.360

0.420
1137 K
0.315

998 K
0.120

758 K
0.210

499 K
0.060

379 K
0.105

0K
0.000

20
25
E (meV)

Figure F.17: Iridium

10
15
E (meV)

Figure F.18: Platinum

237

0.720

0.810
1295 K

595 K

0.600

0.675

777 K
0.360

476 K
g(E) (1/meV)

g(E) (1/meV)

1036 K
0.480

0.540
357 K
0.405

518 K
0.240

238 K
0.270

259 K
0.120

119 K
0.135

0K
0.000

10
E (meV)

Figure F.19: Gold

E (meV)

Figure F.20: Lead

238
Figs. F.21–F.30 show projections of molecular dynamics trajectories onto the high and low energy normal mode displacement patterns shown in Fig. 15.9. For each FCC metal, our optimized
potential was used for the simulation, and the projections were performed as described in § 15.3.2.3
at temperatures as marked.
600

Q=0.0

300

895 K

150

Q=2.9

Q=13.2

1495 K
Q=4.1

200

Q=15.7

100

716 K

150

Q=3.9

Q=14.9

1196 K

Q=7.6

100

Q=7.7

537 K
Q=4.0

50
500

Q=64.6

250

358 K
Q=6.3

Normalized Amplitude

300

Q=38.3

897 K
Q=7.1

150
250

Q=28.2

598 K

125

Q=10.2

450

Q=43.4

225

179 K
Q=13.0

400

Q=53.4

299 K

200

Q=51.4

800

Q=50.1

400

10 K
Q=120.3

1200

Q=113.2

600

10 K
Q=282.0

20
25
E (meV)

Figure F.21: Aluminum

20
30
E (meV)

Figure F.22: Nickel

239
120

Q=7.4

1295 K

150

Q=3.2

1595 K
Q=2.6

120

Q=7.0

1036 K

200

Q=3.5

Q=22.0

1276 K

100

Q=3.4

300

Q=26.1

777 K
Q=4.4

150
400

Q=48.1

518 K

200

Q=5.7

Normalized Amplitude

400

Q=43.0

957 K
Q=5.0

200
500

Q=80.2

638 K

250

Q=4.9

800

Q=72.3

259 K

400

900

Q=12.3

Q=74.0

319 K

450

Q=16.6

1200

Q=116.2

10 K

600

900

Q=57.9

Q=48.2

10 K

450

Q=188.0

20
25
E (meV)

Figure F.23: Copper

300

Q=30.3

20
30
E (meV)

Figure F.24: Rhodium

1195 K

150

120

Q=7.4

Q=3.5

1195 K
Q=4.0

500

Q=41.7

956 K

250

500

Q=11.4

Q=48.1

250

956 K
Q=5.4

250

Q=14.7

717 K
Q=18.9

125
600

Q=47.0

478 K

300

Q=17.4

Normalized Amplitude

Q=10.5

Normalized Amplitude

700

Q=51.7

717 K
Q=8.2

350
400

Q=27.6

200

478 K
Q=10.4

1000

Q=55.5

239 K

500

Q=10.1

600

Q=31.8

300

239 K
Q=13.7

1100

Q=78.0

10 K

550

Q=60.3

1400

Q=72.4

700

10 K
Q=341.4

15
E (meV)

Figure F.25: Palladium

15
E (meV)

Figure F.26: Silver

240
250

Q=20.7

2495 K

125

150

Q=2.9

1895 K
Q=4.2

250

Q=16.0

1996 K

125

200

Q=4.4

Q=14.8

1516 K

100

Q=6.6

300

Q=29.1

1497 K
Q=4.4

150
600

Q=48.3

998 K

300

Q=6.4

Normalized Amplitude

600

Q=36.0

1137 K
Q=6.5

300
600

Q=33.5

758 K

300

Q=7.1

350

Q=31.6

499 K

175

700

Q=13.8

Q=34.4

379 K

350

Q=14.8

1200

Q=89.6

10 K

600

1100

Q=136.6

Q=72.9

10 K

550

Q=216.1

20
25
E (meV)

Figure F.27: Iridium

300

10
15
E (meV)

Figure F.28: Platinum

Q=18.8

150

1295 K

500

Q=0.5

Q=13.8

595 K

250

Q=4.2

160

Q=7.2

1036 K

500

Q=4.5

Q=16.0

476 K

250

Q=4.0

450

Q=18.5

777 K
Q=5.1

225
300

Q=11.5

150

518 K
Q=8.7

Normalized Amplitude

Q=5.2

Normalized Amplitude

1000

Q=17.9

357 K
Q=5.1

500
1200

Q=27.6

238 K

600

Q=8.4

800

Q=50.7

400

259 K
Q=13.0

1300

Q=35.4

119 K

650

Q=15.8

1200

Q=54.2

600

10 K
Q=255.4

1300

Q=47439.6

650

10 K
Q=358.3

10
E (meV)

Figure F.29: Gold

E (meV)

Figure F.30: Lead

241

Appendix G

Code
Some of the code that was produced for and used in this thesis is available on the web. None of it is
complete, and in particular, much of it lacks decent documentation. It is here so that an extremely
interested scientist may be able to more accurately reproduce my work.
The files are named C.L.tbz2 where C ∈ {bvk,mph,qfit,gcon} is a program “title” and L ∈
{gpl,bsd } refers to the license. The programs are very briefly explained here:
• bvk — Refers to the BvK code (without the optimization components) that was used for fits
to aluminum, nickel, and iron spectra in Chapters 9, 11, and 12.
• mph — Refers to the multiphonon correction code described in § 8.1.2 and used for the analysis
of the aluminum, lead, nickel, chromium, and vanadium data in Chapters 9–11 and 13.
• qfit — Refers to the quality factor and shift finding code described in § 8.2 and used for the
analysis of the quality factors for aluminum, lead, nickel, iron, chromium, and vanadium in
Chapters 9–13; as well as for the quality factors for the spectra from molecular dynamics shown
in Chapter 15.
• gcon — Refers to the code for numerically determining the constraints on force constants given
point group symmetries.
The URLs for the BSD licensed programs are:

The URLs for the GPLv3 licensed programs are:

242

243

244

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