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 | Kosevich A.M. The crystal lattice: phonons, solitons, dislocations, superlattices. - 2nd, rev. & updated ed. - Weinheim: Wiley-VCH, 2005. - 345 p. - ISBN 3-527-40508-9
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Prefaces ....................................................... IX
Part 1. Introduction ............................................ 1
0. Geometry of Crystal Lattice .................................. 3
0.1. Translational Symmetry .................................. 3
0.2. Bravais Lattice ......................................... 5
0.3. The Reciprocal Lattice .................................. 7
0.4. Use of Penetrating Radiation to Determine Crystal
Structure .............................................. 10
0.4.1. Problems ........................................ 12
Part 2. Classical Dynamics of a Crystal Lattice ................ 15
1. Mechanics of a One-Dimensional Crystal ...................... 17
l.1. Equations of Motion and Dispersion Law ................. 17
1.1.1. Problems ........................................ 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 ID Superlattice ..................... 54
2. General Analysis of Vibrations of Monatomic Lattices ........ 59
2.1. Equation of Small Vibrations of 3D Lattice ............. 59
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
2.8. Long-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
3.1. Optical 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
4.8. The 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
5.1. Forbidden 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
Part 3. Quantum Mechanics of Crystals ......................... 163
6. Quantization of Crystal Vibrations ......................... 165
6.1. Occupation-Number Representation ...................... 165
6.2. Phonons ............................................... 170
6.3. Quantum-Mechanical Definition of the Green Function ... 172
6.4. Displacement 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
7.2. The Effective Hamiltonian for Phonon Interaction and
Decay Processes ....................................... 186
7.3. Inelastic 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
8. Quantum 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
9. Point Defects .............................................. 215
9.1. Point-Defect Models in the Crystal Lattice ............ 215
9.2. Defects in Quantum Crystals ........................... 218
9.3. Mechanisms of Classical Diffusion and Quantum Diffusion
of Defectons .......................................... 222
9.4. Quantum Crowdion Motion ............................... 225
9.5. Point Defect in Elasticity Theory ..................... 227
9.5.1. Problem ........................................ 232
10. Linear Crystal Defects .................................... 233
10.1. Dislocations ........................................ 233
10.2. Dislocations in Elasticity Theory ................... 235
10.3. Glide and Climb of a Dislocation .................... 238
10.4. Disclinations ....................................... 241
10.5. Disclinations and Dislocations ...................... 244
10.5.1. Problems .................................... 246
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
11.7. Possible 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 ID Local Inhomogeneity ...... 279
12.2. Quasi-Local Vibrations Near a Dislocation ........... 283
12.3. Localization of Small Vibrations in the Elastic Field
of a Screw Dislocation .............................. 285
12.4. Frequency 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
13.6.1. Problems .................................... 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
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