Kosevich A.M. The crystal lattice (2ed., Wiley, 2005)(ISBN 3527405089)(342s)_PSa_

Arnold M. Kosevich

The Crystal Lattice

Phonons, Solitons, Dislocations, Superlattices

Second, Revised and Updated Edition

WILEY-VCH Verlag GmbH & Co. KGaA

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	ISBN-13: 978-3-527-40508-4
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Contents

	Prefaces IX
Part 1	Introduction 1
0	Geometry of Crystal Lattice 3

0.1Translational Symmetry 3

0.2Bravais Lattice 5

0.3	The Reciprocal Lattice 7
0.4	Use of Penetrating Radiation to Determine Crystal Structure 10

0.4.1Problems 12

Part 2	Classical Dynamics of a Crystal Lattice	15
1	Mechanics of a One-Dimensional Crystal	17
1.1	Equations of Motion and Dispersion Law	17

1.1.1Problems 23

1.2	Motion of a Localized Excitation in a Monatomic Chain		24
1.3	Transverse Vibrations of a Linear Chain 29
1.4	Solitons of Bending Vibrations of a Linear Chain 33
1.5	Dynamics of Biatomic 1D Crystals	36
1.6	Frenkel–Kontorova Model and sine-Gordon Equation 39
1.7	Soliton as a Particle in 1D Crystals	43
1.8	Harmonic Vibrations in a 1D Crystal Containing a Crowdion (Kink) 46
1.9	Motion of the Crowdion in a Discrete Chain 49
1.10	Point Defect in the 1D Crystal 51
1.11	Heavy Defects and 1D Superlattice	54
2	General Analysis of Vibrations of Monatomic Lattices		59

2.1	Equation of Small Vibrations of 3D Lattice 59

VI	Contents
2.2		The Dispersion Law of Stationary Vibrations		63
2.3		Normal Modes of Vibrations 66
2.4		Analysis of the Dispersion Law 67
2.5		Spectrum of Quasi-Wave Vector Values	70
2.6		Normal Coordinates of Crystal Vibrations	72
2.7		The Crystal as a Violation of Space Symmetry		74

2.8Long-Wave Approximation and Macroscopic Equations for the

	Displacements Field 75
2.9	The Theory of Elasticity 77
2.10	Vibrations of a Strongly Anisotropic Crystal (Scalar Model) 80
2.11	“Bending” Waves in a Strongly Anisotropic Crystal 83
2.11.1	Problem 88
3	Vibrations of Polyatomic Lattices 89

3.1Optical Vibrations 89

3.2	General Analysis of Vibrations of Polyatomic Lattice 94
3.3	Molecular Crystals 98
3.4	Two-Dimensional Dipole Lattice	101
3.5	Optical Vibrations of a 2D Lattice of Bubbles			105
3.6	Long-Wave Librational Vibrations of a 2D Dipole Lattice				109
3.7	Longitudinal Vibrations of 2D Electron Crystal			112
3.8	Long-Wave Vibrations of an Ion Crystal 117
3.8.1	Problems 123
4	Frequency Spectrum and Its Connection with the Green Function					125
4.1	Constant-Frequency Surface 125
4.2	Frequency Spectrum of Vibrations	129
4.3	Analysis of Vibrational Frequency Distribution			132
4.4	Dependence of Frequency Distribution on Crystal Dimensionality					136
4.5	Green Function for the Vibration Equation		141
4.6	Retarding and Advancing Green Functions		145
4.7	Relation Between Density of States and Green Function				147

4.8The Spectrum of Eigenfrequencies and the Green Function of a Deformed

Crystal 149

4.8.1 Problems 151

5	Acoustics of Elastic Superlattices: Phonon Crystals 153

5.1Forbidden Areas of Frequencies and Specific Dynamic States in such

	Areas 153
5.2	Acoustics of Elastic Superlattices 155
5.3	Dispersion Relation for a Simple Superlattice Model 159
5.3.1	Problem 162

Contents VII

Part 3	Quantum Mechanics of Crystals 163
6	Quantization of Crystal Vibrations	165
6.1	Occupation-Number Representation	165

6.2Phonons 170

6.3	Quantum-Mechanical Definition of the Green Function 172

6.4Displacement Correlator and the Mean Square of Atomic

	Displacement 174
6.5	Atomic Localization near the Crystal Lattice Site 176
6.6	Quantization of Elastic Deformation Field 178
7	Interaction of Excitations in a Crystal 183
7.1	Anharmonicity of Crystal Vibrations and Phonon Interaction 183

7.2The Effective Hamiltonian for Phonon Interaction and Decay Processes 186

7.3Inelastic Diffraction on a Crystal and Reproduction of the Vibration

	Dispersion Law 191
7.4	Effect of Thermal Atomic Motion on Elastic γ-Quantum-Scattering 196
7.5	Equation of Phonon Motion in a Deformed Crystal 198

8Quantum Crystals 203

8.1	Stability Condition of a Crystal State	203
8.2	The Ground State of Quantum Crystal	206
8.3	Equations for Small Vibrations of a Quantum Crystal 207
8.4	The Long-Wave Vibration Spectrum	211
Part 4	Crystal Lattice Defects 213

9Point Defects 215

9.1	Point-Defect Models in the Crystal Lattice 215
9.2	Defects in Quantum Crystals 218

9.3Mechanisms of Classical Diffusion and Quantum Diffusion of

	Defectons 222
9.4	Quantum Crowdion Motion 225
9.5	Point Defect in Elasticity Theory 227

9.5.1Problem 232

10	Linear Crystal Defects 233

10.1Dislocations 233

10.2	Dislocations in Elasticity Theory	235
10.3	Glide and Climb of a Dislocation	238

10.4Disclinations 241

10.5	Disclinations and Dislocations 244
10.5.1	Problems 246

VIII	Contents
11		Localization of Vibrations 247
11.1		Localization of Vibrations near an Isolated Isotope Defect	247
11.2		Elastic Wave Scattering by Point Defects 253
11.3		Green Function for a Crystal with Point Defects 259
11.4		Influence of Defects on the Density of Vibrational States in a Crystal 264
11.5		Quasi-Local Vibrations 267
11.6		Collective Excitations in a Crystal with Heavy Impurities	271

11.7Possible Rearrangement of the Spectrum of Long-Wave Crystal

	Vibrations	274
11.7.1	Problems	277
12	Localization of Vibrations Near Extended Defects		279
12.1	Crystal Vibrations with 1D Local Inhomogeneity		279
12.2	Quasi-Local Vibrations Near a Dislocation 283

12.3Localization of Small Vibrations in the Elastic Field of a Screw Dislocation 285

12.4Frequency of Local Vibrations in the Presence of a Two-Dimensional

(Planar) Defect 288

13	Elastic Field of Dislocations in a Crystal	297
13.1	Equilibrium Equation for an Elastic Medium Containing Dislocations 297
13.2	Stress Field Action on Dislocation 299
13.3	Fields and the Interaction of Straight Dislocations		303
13.4	The Peierls Model 309
13.5	Dislocation Field in a Sample of Finite Dimensions		312
13.6	Long-Range Order in a Dislocated Crystal	314

13.6.1Problems 319

14	Dislocation Dynamics 321
14.1	Elastic Field of Moving Dislocations 321
14.2	Dislocations as Plasticity Carriers	325
14.3	Energy and Effective Mass of a Moving Dislocation 327
14.4	Equation for Dislocation Motion	331
14.5	Vibrations of a Lattice of Screw Dislocations 336

Bibliography 341

Index 343

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-

XPrefaces

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 XI

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

Part 1 Introduction

The Crystal Lattice: Phonons, Solitons, Dislocations, Superlattices, Second Edition. Arnold M. Kosevich Copyright c 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ISBN: 3-527-40508-9