The Crystal Lattice: Phonons, Solitons, Dislocations, Superlattices

Arnold M. Kosevich
The Crystal Lattice
Phonons, Solitons, Dislocations, Superlattices
Second, Revised and Updated Edition
WILEY-VCH Verlag GmbH & Co. KGaA
Arnold M. Kosevich
The Crystal Lattice
Arnold M. Kosevich
The Crystal Lattice
Phonons, Solitons, Dislocations, Superlattices
Second, Revised and Updated Edition
WILEY-VCH Verlag GmbH & Co. KGaA
Author
Arnold M. Kosevich
B. Verkin Institute for Low Temperature Physics and
Engineering
National Academy of Sciences of Ukraine
310164 Kharkov, Ukraine
e-mail: [email protected]
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Contents
Prefaces
IX
1
Part 1
Introduction
0
Geometry of Crystal Lattice
Translational Symmetry 3
Bravais Lattice 5
The Reciprocal Lattice 7
0.1
0.2
0.3
0.4
0.4.1
Part 2
3
Use of Penetrating Radiation to Determine Crystal Structure 10
Problems 12
Classical Dynamics of a Crystal Lattice
15
17
1
Mechanics of a One-Dimensional Crystal
1.1
1.1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
Equations of Motion and Dispersion Law 17
Problems 23
Motion of a Localized Excitation in a Monatomic Chain 24
Transverse Vibrations of a Linear Chain 29
Solitons of Bending Vibrations of a Linear Chain 33
Dynamics of Biatomic 1D Crystals 36
Frenkel–Kontorova Model and sine-Gordon Equation 39
Soliton as a Particle in 1D Crystals 43
Harmonic Vibrations in a 1D Crystal Containing a Crowdion (Kink) 46
Motion of the Crowdion in a Discrete Chain 49
Point Defect in the 1D Crystal 51
Heavy Defects and 1D Superlattice 54
2
General Analysis of Vibrations of Monatomic Lattices
Equation of Small Vibrations of 3D Lattice 59
2.1
59
VI
Contents
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.11.1
The Dispersion Law of Stationary Vibrations 63
Normal Modes of Vibrations 66
Analysis of the Dispersion Law 67
Spectrum of Quasi-Wave Vector Values 70
Normal Coordinates of Crystal Vibrations 72
The Crystal as a Violation of Space Symmetry 74
Long-Wave Approximation and Macroscopic Equations for the
Displacements Field 75
The Theory of Elasticity 77
Vibrations of a Strongly Anisotropic Crystal (Scalar Model) 80
“Bending” Waves in a Strongly Anisotropic Crystal 83
Problem 88
89
3
Vibrations of Polyatomic Lattices
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.8.1
Optical Vibrations 89
General Analysis of Vibrations of Polyatomic Lattice 94
Molecular Crystals 98
Two-Dimensional Dipole Lattice 101
Optical Vibrations of a 2D Lattice of Bubbles 105
Long-Wave Librational Vibrations of a 2D Dipole Lattice 109
Longitudinal Vibrations of 2D Electron Crystal 112
Long-Wave Vibrations of an Ion Crystal 117
Problems 123
125
4
Frequency Spectrum and Its Connection with the Green Function
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.8.1
Constant-Frequency Surface 125
Frequency Spectrum of Vibrations 129
Analysis of Vibrational Frequency Distribution 132
Dependence of Frequency Distribution on Crystal Dimensionality 136
Green Function for the Vibration Equation 141
Retarding and Advancing Green Functions 145
Relation Between Density of States and Green Function 147
The Spectrum of Eigenfrequencies and the Green Function of a Deformed
Crystal 149
Problems 151
5
Acoustics of Elastic Superlattices: Phonon Crystals
5.1
Forbidden Areas of Frequencies and Specific Dynamic States in such
Areas 153
Acoustics of Elastic Superlattices 155
Dispersion Relation for a Simple Superlattice Model 159
Problem 162
5.2
5.3
5.3.1
153
Contents
Part 3
Quantum Mechanics of Crystals
163
165
6
Quantization of Crystal Vibrations
6.1
6.2
6.3
6.4
6.5
6.6
Occupation-Number Representation 165
Phonons 170
Quantum-Mechanical Definition of the Green Function 172
Displacement Correlator and the Mean Square of Atomic
Displacement 174
Atomic Localization near the Crystal Lattice Site 176
Quantization of Elastic Deformation Field 178
7
Interaction of Excitations in a Crystal
7.1
7.2
Anharmonicity of Crystal Vibrations and Phonon Interaction 183
The Effective Hamiltonian for Phonon Interaction and Decay
Processes 186
Inelastic Diffraction on a Crystal and Reproduction of the Vibration
Dispersion Law 191
Effect of Thermal Atomic Motion on Elastic γ-Quantum-Scattering 196
Equation of Phonon Motion in a Deformed Crystal 198
7.3
7.4
7.5
183
203
8
Quantum Crystals
8.1
8.2
8.3
8.4
Stability Condition of a Crystal State 203
The Ground State of Quantum Crystal 206
Equations for Small Vibrations of a Quantum Crystal 207
The Long-Wave Vibration Spectrum 211
Part 4
Crystal Lattice Defects
9
Point Defects
9.1
9.2
9.3
Point-Defect Models in the Crystal Lattice 215
Defects in Quantum Crystals 218
Mechanisms of Classical Diffusion and Quantum Diffusion of
Defectons 222
Quantum Crowdion Motion 225
Point Defect in Elasticity Theory 227
Problem 232
9.4
9.5
9.5.1
10
10.1
10.2
10.3
10.4
10.5
10.5.1
213
215
Linear Crystal Defects
Dislocations 233
233
Dislocations in Elasticity Theory 235
Glide and Climb of a Dislocation 238
Disclinations 241
Disclinations and Dislocations 244
Problems 246
VII
VIII
Contents
247
11
Localization of Vibrations
11.1
11.2
11.3
11.4
11.5
11.6
11.7
Localization of Vibrations near an Isolated Isotope Defect 247
Elastic Wave Scattering by Point Defects 253
Green Function for a Crystal with Point Defects 259
Influence of Defects on the Density of Vibrational States in a Crystal 264
Quasi-Local Vibrations 267
Collective Excitations in a Crystal with Heavy Impurities 271
Possible Rearrangement of the Spectrum of Long-Wave Crystal
Vibrations 274
Problems 277
11.7.1
279
12
Localization of Vibrations Near Extended Defects
12.1
12.2
12.3
Crystal Vibrations with 1D Local Inhomogeneity 279
Quasi-Local Vibrations Near a Dislocation 283
Localization of Small Vibrations in the Elastic Field of a Screw
Dislocation 285
Frequency of Local Vibrations in the Presence of a Two-Dimensional
(Planar) Defect 288
12.4
297
13
Elastic Field of Dislocations in a Crystal
13.1
13.2
13.3
13.4
13.5
13.6
13.6.1
Equilibrium Equation for an Elastic Medium Containing Dislocations 297
Stress Field Action on Dislocation 299
Fields and the Interaction of Straight Dislocations 303
The Peierls Model 309
Dislocation Field in a Sample of Finite Dimensions 312
Long-Range Order in a Dislocated Crystal 314
Problems 319
321
14
Dislocation Dynamics
14.1
14.2
14.3
14.4
14.5
Elastic Field of Moving Dislocations 321
Dislocations as Plasticity Carriers 325
Energy and Effective Mass of a Moving Dislocation 327
Equation for Dislocation Motion 331
Vibrations of a Lattice of Screw Dislocations 336
Bibliography
Index
343
341
Prefaces
Preface to the First Edition
The design of new materials is one of the most important tasks in promoting progress.
To do this efficiently, the fundamental properties of the simplest forms of solids, i. e.,
single crystals must be understood.
Not so long ago, materials science implied the development, experimental investigation, and theoretical description, of primarily construction materials with given elastic, plastic and resistive properties. In the last few decades, however, new materials,
primarily crystalline, have begun to be viewed differently: as finished, self-contained
devices. This is particularly true in electronics and optics.
To understand the properties of a crystal device it is not only necessary to know its
structure but also the dynamics of physical processes occurring within it. For example,
to describe the simplest displacement of the crystal atoms already requires a knowledge of the interatomic forces, which of course, entails a knowledge of the atomic
positions.
The dynamics of a crystal lattice is a part of the solid-state mechanics that studies
intrinsic crystal motions taking into account structure. It involves classical and quantum mechanics of collective atomic motions in an ideal crystal, the dynamics of crystal
lattice defects, a theory of the interaction of a real crystal with penetrating radiation,
the description of physical mechanisms of elasticity and strength of crystal bodies.
In this book new trends in dislocation theory and an introduction to the nonlinear
dynamics of 1D systems, that is, soliton theory, are presented. In particular, the dislocation theory of melting of 2D crystals is briefly discussed. We also provide a new
treatment of the application of crystal lattice theory to physical objects and phenomena
whose investigation began only recently, that is, quantum crystals, electron crystals on
a liquid-helium surface, lattices of cylindrical magnetic bubbles in thin-film ferromagnetics, and second sound in crystals.
In this book we treat in a simple way, not going into details of specific cases, the
fundamentals of the physics of a crystalline lattice. To simplify a quantitative descrip-
X
Prefaces
tion of physical phenomena, a simple (scalar) model is often used. This model does
not reduce the generality of qualitative calculations and allows us to perform almost
all quantitative calculations.
The book is written on the basis of lectures delivered by the author at the Kharkov
University (Ukraine). The prerequisites for understanding this material are a general
undergraduate-level knowledge of theoretical physics.
Finally, as author, I would like to thank the many people who helped me during the
work on the manuscript.
I am pleased to express gratitude to Professor Paul Ziesche for his idea to submit
the manuscript to WILEY-VCH for publication, and for his aid in the realization of
this project.
I am deeply indebted to Dr. Sergey Feodosiev for his invaluable help in preparing a
camera-ready manuscript and improving the presentation of some parts of the book. I
am grateful to Maria Mamalui and Maria Gvozdikova for their assistance in preparing
the computer version of the manuscript. I would like to thank my wife Dina for her
encouragement.
I thank Dr. Anthony Owen for his careful reading of the manuscript and useful
remarks.
Kharkov July 1999
Arnold M. Kosevich
Preface to the Second Edition
Many parts of this book are not very different from what was in the first edition (1999).
This is a result of the fact that the basic equations and conclusions of the theory of the
crystal lattice have long since been established. The main changes (“reconstruction”)
of the book are the following
1. All the questions concerning one-dimensional (1D) crystals are combined in
one chapter (Chapter 1). I consider the theory of a 1D crystal lattice as a training
and proving ground for studying dynamics of three-dimensional structures. The 1D
models allow us to formulate and solve simply many complicated problems of crystal
mechanics and obtain exact solutions to equations not only of the linear dynamics but
also for dynamics of anharmonic (nonlinear) crystals.
2. The second edition includes a new chapter devoted to the theory of elastic superlattices (Chapter 5). A new class of materials, namely, phonon and photon crystals has
recently been of the great interest, and I would like to propose a simple explanation of
many properties of superlattices that were studied before and known in the theory of
normal crystal lattices.
3. New sections are added to the new edition concerning defects in the crystal
lattice.
Prefaces
Finally, I would like to thank the people who helped me in the preparation of the
manuscript.
I am indebted to Dr. Michail Ivanov and Dr. Sergey Feodosiev for their advise
in improving the presentation of some parts of the book. I express many thanks to
Alexander Kotlyar for his invaluable help in preparing the figures and electronic version of the manuscript. The author is grateful to Oksana Charkina for assistance in
preparing the manuscript. I would like to thank my wife Dina for her encouragement.
Kharkov March 2005
Arnold M. Kosevich
XI
Part 1 Introduction