Table Of Content34 Springer Series in Solid-State Sciences
Edited by Peter Fulde
Springer Series in Solid-State Sciences
Editors: M. Cardona P. Fulde H.-J. Queisser
Volume 40 Semiconductor Physics - An Introduction By K Seeger
Volume 41 The LMTO Method By H.L. Skriver
Volume 42 Crystal Optics with Spatial Dispersion and the Theory of Excitations
By Y.M. Agranovich and V.L. Ginzburg
Volumes 1 - 39 are listed on the back inside cover
Peter Briiesch
Phonons: Theory
and Experiments I
Lattice Dynamics
and Models of Interatomic Forces
With 82 Figures
Springer-Verlag Berlin Heidelberg New York 1982
Dr. Peter Briiesch
Brown Boveri Research Center, CH-5405 Baden-Dattwil, and
Ecole Polytechnique Federal de Lausanne, CH-1015 Lausanne, Switzerland
Series Editors:
Professor Dr. Manuel Cardona
Professor Dr. Peter Fulde
Professor Dr. Hans-Joachim Queisser
Max-Planck-Institut fUr Festkorperforschung, HeisenbergstraBe 1
D-7000 Stuttgart 80, Fed. Rep. of Germany
ISBN-13:978-3-642-81783-0 e-ISBN-13:978-3-642-81781-6
001: 10.1007/978-3-642-81781-6
Library of Congress Cataloging in Publication Data. Briiesch, P. (Peter), 1934-Phonons :
theory and experiments. (Springer series in solid-state sciences; 34) Bibliography: p.
Includes index. 1. Phonons. I. Title. II. Series. QCI76.8.P5B78 539.7'217 81-21424 AACR2
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is
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© by Springer-Verlag Berlin Heidelberg 1982
Softcover reprint of the hardcover 1st edition 1982
The use of registered names, trademarks, etc. in this publication does not imply, even in the absence
of a specific statement, that such names are exempt from the relevant protective laws and regulations
and therefore free for general use.
2153/3130-543210
Dedicated to my wife
Preface
This two-volume treatment grew out of lectures the author gave at the
"Ecole Poly technique Federale de Lausanne" during the years 1975-1980 for
graduate students in experimental physics in their last year of study. It
is written by an experimentalist with some interest in theory and is ad
dressed mainly to experimentalists, but also to theoreticians interested
in experiments. This treatment tries to bridge the gap between theory and
experiments; it should assist experimentalists in the interpretation of
their data in the vast field of lattice dynamics. An attempt has been made
to provide not only the basic concepts but also a working knowledge in
this field of solid-state physics.
In this first volume, the basic concepts of the physics of phonons are
developed and illustrated by many examples; it provides the background
necessary for the interpretation of most experimental results. The second
volume, which is in preparation, is devoted to experimental techniques,
the interpretation of experiments, and discussion of phenomena which are
directly related with phonons.
The book is designed for introductory courses at the graduate level. It
is believed that the book will also prove useful to those graduate students
starting research in this or related fields, as well as to many workers
already active in this branch of solid-state physics.
The author is indebted to BBC Brown, Boveri & Company Limited, Baden,
Switzerland, for giving him the possibility to carry out this work. In par
ticular, its Research Center provided the necessary scientific atmosphere
to accomplish this goal. I am also grateful to the Ecole Poly technique
Federal de Lausanne for excusing me from my lecture duties during this period.
The entire manuscript was strongly influenced by the detailed criticism
and valuable suggestions of my colleagues Drs. J. Bernasconi, H.U. Beyeler,
T. Hibma, L. Pietronero, W.R. Schneider, S. $tr~ssler, H.J. Wiesmann and
H.R. Zeller from the Brown Boveri Research Center, and by Dr. W. BUhrer
VIII
from the Institut fUr Reaktortechnik ETHZ in WUrenlingen, Switzerland; my
debt to them is indeed great. I am also most grateful to Professor Ph.
Choquard for many interesting discussions during our common teaching in
Lausanne in 1979 and 1980. In addition, I must express very grateful thanks to
Mrs. E. Knotz for her never-ending patience in accurately typing the manu
script, to Mr. W. Foditsch for photographing the figures, and to Mrs. M.
Zamfirescu for the very skilful drawing of all the figures. Finally, I am
indebted to Professor P. Fulde for valuable suggestions and to Dr. H. Lotsch,
Springer-Verlag, for his patient and efficient cooperation.
Baden. December 1981 Peter Bruesah
Contents
1. Introduction
1.1 The Static Lattice Model and Its Limitations
1.2 Early History of Lattice Dynamics 2
1.3 The Adiabatic and Harmonic Approximations 6
1.4 Organization of the Book 11
1.5 Problems 13
1.5.1 Adiabatic Approximation .....•..•..•..•.•.......•.. 13
1.5.2 Harmonic Vibration of a Diatomic Molecule ••••.••.. 13
2. Dynamics of the Linear Diatomic Chain ......................... 14
2.1 Classical Mechanics ...................................... 15
2.1.1 Periodic Boundary Conditions and Dispersion
Relations ..••...•..•.••.....•.•...•.••..•••.•.•••. 15
2.1.2 Dynamical Matrix and Eigenvectors ••.•...•••••.•.•• 19
2.1.3 An Illustration: The Linear NaCl-Chain; Transition
to the Monoatomic Lattice ••••..•.••.••••.•••••.••• 22
2.1.4 Normal Coordinates ................................ 27
2.2 Quantum Mechanics 33
2.2.1 The Schrodinger Equation of the Simple Harmonic
Osci1lator ...••..•••.••••.•.•••••••.•••..••••.•.•• 33
2.2.2 The Schrodinger Equation of the Vibrating Chain 36
2.2.3 Creation and Annihilation Operators •.•••.••.•.•••• 39
2.2.4 Phonons ....•........•........••••.•.••••.••.....•• 42
2.2.5 Specific Heat and Density of States •••••..•••••••. 45
2.3 Problems 51
2.3.1 Monoatomic Chain ................................. . 51
2.3.2 Chain with a Basis of Two Identical Atoms ••.•.•••• 52
2.3.3 Probability Densities of a Classical and Quantum
Mechanical Oscillator •.•.......•.••..•.••••.••••.. 54
2.3.4 Density of States of the Monoatomic Chain with
Nearest and Second-Nearest-Neighbour Interactions 54
x
3. Dynamics of Three-Dimensional Crystals .•................•..... 55
3.1 Equations of Motion and Atomic Force Constants ........... 56
3.2 Dynamical Matrix and Eigenvectors 60
3.3 Periodic Boundary Conditions, Reciprocal Lattices and
Brillouin Zones 64
3.4 Normal Coordinates, Phonons ...•...•.....•...•.••......... 69
3.5 Density of States and Specific Heat ............•..•...... 72
3.5.1 Density of States ...•.....•...•...•..•.•.......•.. 72
3.5.2 Specific Heat ..................................... 78
3.6 Connection of Lattice Dynamics with the Theory of
Elasticity ••..•.......•.•.........•..•....•....•.....•... 84
3.7 An Illustration: Phonon Dispersion of Monoatomic Crystals
with fcc Structure .•.........•................•.......... 93
3.8 Problems 97
3.8.1 Brillouin Zone in Two Dimensions .•..............•. 97
3.8.2 Critical Points (c.p.) in the Density of States 97
3.8.3 Density of States in Two Dimensions 98
3.8.4 Debye Specific Heat in Two Dimensions 98
3.8.5 Elastic Waves in Continuous Media ................. 98
3.8.6 Vibrations in Crystals with CsCl Structure ........ 99
4. Interatomic Forces and Phonon Dispersion Curves 100
4.1 Lattice Dynamics of the Solid Inert Gases 102
4.2 The Rigid-Ion Model for Ionic Crystals ......••........... 106
4.2.1 Definition of the Model and Dynamical Matrix 106
4.2.2 Coulomb Matrix and Electric Fields 109
4.2.3 Application to Crystals with NaCl Structure •...... 113
4.2.4 Deficiencies of the Rigid-Ion Model •.............. 119
4.3 The Shell Model • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.3.1 The Essential Features of the Model............... 119
4.3.2 The Dielectric Constant and the Lyddane-Sachs-Teller
Relation ••...................•.................... 123
4.3.3 Generalized Shell Model and Phonon Dispersions 127
4.4 The Adiabatic Bond Charge Model •................•.....•.. 129
4.5 The Valence Force Model.................................. 133
4.6 Internal and External Vibrations in Molecular Crystals 138
4.7 Phonons in Metals ...........•............................ 142
4.7.1 Force Constant Models ................•..••........ 143
4.7.2 Coulomb Interaction in the Uniform-Background
Lattice Model..................................... 144
XI
4.7.3 Bardeen's Treatment of Screening 145
4.8 Problems 149
4.8.1 Lennard-Jones Parameters of the Linear Chain with
Zero-Poi nt Energy ....•....•..•••...••..••..•...••• 149
4.8.2 Shell Model of the Linear Monoatomic Chain •.•...•• 149
4.8.3 Generalized Lyddane-Sachs-Teller Relation 150
4.8.4 Bending Coordinates: Application to the Linear Chain 150
4.8.5 Thomas-Fermi Screening 151
5. Anharmonicity ..•..••.•..••....•.•..•..............•.•.•••..... 152
5.1 The Anharmonic Diatomic Molecule ....•.................••• 153
5.2 The Anharmonic Linear Chain ........•..•......•........... 156
5.2.1 Dynamical Aspects ...............•.•..•.•.......... 156
5.2.2 The Free Energy of the Classical Anharmonic Chain 157
5.2.3 The Equation of State and Thermal Expansion in the
Quasiharmonic Approximation ..•.•...•.•.••........• 163
5.2.4 The Specific Heat ............•.....•.••........... 166
5.3 The Anharmonic Three-Dimensional Crystal .......•......... 166
5.3.1 The Equation of State •.........•..•.•...•••.•..... 166
5.3.2 Thermal Expansion .•...•.....•...•......•....•..... 170
5.3.3 Anharmonic Effects on the Specific Heat and
Elastic Constants ..............•......•.....•...•. 174
5.4 The Self-Consistent Harmonic Approximation (SCHA) •....... 175
5.4.1 General Remarks ..•.............•...•....••...•.•.. 175
5.4.2 The Diatomic Molecule 176
5.4.3 The SCHA for a Bravais Crystal 178
5.4.4 The Self-Consistent Isotropic Einstein Model 182
5.5 Response Function and Perturbation Theory of Phonon-
Phonon Interactions ..••.................•..........•..•.. 184
5.5.1 Response Function of Harmonic and Damped Harmonic
Oscillators ..•.......•......•.......•.•...•....... 185
5.5.2 Response Function for the Anharmonic Crystal 187
5.5.3 Frequency Widths and Shifts from Perturbation Theory 188
5.6 Problems 194
5.6.1 Thermal Expansion and Force Constant of Diatomic
Molecules .•.••..........•.........•...•..•......•. 194
5.6.2 Quantum Anharmonic Oscillator .......•.••..••...•.. 195
5.6.3 GrUneisen Parameter, Thermal Expansion and Frequency
Shift of a Monoatomic fcc Crystal ......•...•.•..•• 196
5.6.4 GrUneisen Parameter of TO and LO-Modes of Simple
Diatomic Crystals ••.....•...•..•..•.......••.....• 197
