Topics in the Theory of Solid Materials

Topics in the Theory of Solid Materials......Page 1
Contents......Page 4
Preface......Page 9
References......Page 13
Exercises......Page 18
Answers......Page 25
1.1 Introduction......Page 32
1.2 Deformation: strain and rotation......Page 33
1.2.1 The strain tensor......Page 35
1.2.2 The rotation tensor......Page 37
1.3 Forces and stress......Page 38
1.4 Linear elasticity......Page 41
1.4.1 Hooke’s law......Page 43
1.4.2 Isotropic media......Page 44
1.4.3 Elastic moduli......Page 47
1.4.4 Stability conditions......Page 49
1.5 Equilibrium......Page 51
2.2 Vector fields......Page 53
2.3 Equation of motion......Page 55
2.4.1 Shear and rotational waves......Page 58
2.4.2 Dilatational or irrotational waves......Page 59
2.4.3 General discussion......Page 61
Appendix to Chapter 2......Page 62
3.2.1 The Maxwell relations......Page 65
3.2.2 Elastic constants, bulk moduli and specific heats......Page 66
3.3 Thermal conduction and wave motion......Page 69
3.4 Wave attenuation by thermal conduction......Page 72
4.1 Introduction......Page 79
4.2 Rayleigh waves......Page 80
4.3 Boundary conditions......Page 81
4.4 Dispersion relation......Page 82
4.5 Character of the wave motion......Page 85
5.2 Description of dislocations......Page 88
5.3 Deformation fields of dislocations......Page 93
5.3.1 Screw dislocation......Page 94
5.3.2 Edge dislocation......Page 96
5.4 Uniform dislocation motion......Page 100
5.5 Further study of dislocations......Page 102
6.1 Introduction......Page 104
6.2 Equations of motion......Page 105
6.3 The constant-velocity polaron......Page 109
6.4 Polaron in a magnetic field: quantization......Page 117
7.2 The hamiltonian of a solid......Page 121
7.3 Nuclear dynamics: the adiabatic approximation......Page 122
7.4 The harmonic approximation......Page 124
7.5.1 Periodic boundary conditions for bulk properties......Page 125
7.5.2 The dynamical matrix of the crystal......Page 128
7.5.3 The normal modes of crystal vibration......Page 131
7.5.4 Electrons and phonons: total energy......Page 133
7.6 Statistical thermodynamics of a solid......Page 134
7.6.1 Partition function of the crystal......Page 135
7.6.2 Equation of state of the crystal......Page 136
7.6.3 Thermodynamic internal energy of the crystal; phonons as bosons......Page 138
7.7 Summary......Page 139
8.1 Introduction......Page 140
8.2 Monatomic linear chain......Page 141
8.3 Diatomic linear chain......Page 147
8.4 Localized mode of a point defect......Page 152
9.2 The shell model for insulating crystals......Page 157
9.3 Cohesive energy of a crystal......Page 160
9.4 Elastic constants......Page 162
9.5 Dielectric and piezoelectric constants......Page 166
10.1 Introduction......Page 171
10.2.1 Derivation......Page 172
10.2.2 Planar source problem......Page 175
10.3 Diffusion as a random walk......Page 178
10.4 Equilibrium concentration of point defects......Page 182
10.5 Temperature dependence of di.usion: the Vineyard relation......Page 185
Appendix to Chapter 10: Stirling’s formula......Page 192
11.1.1 Crystals and defects......Page 194
11.1.2 Modelling of point defects in ionic crystals......Page 196
11.2.1 Copper and silver diffusion in alkali halides......Page 199
11.2.2 Dissociation of the oxygen-vacancy defect complex in BaF2......Page 202
11.3 Defect complex stability......Page 204
11.4.1 Nickel in MgO......Page 207
11.5 Optical excitation......Page 208
11.5.1 Frenkel exciton and impurity absorption in MgO......Page 209
11.5.3 O  in BaF2......Page 210
11.6.1 F center in NaF......Page 212
11.6.3 ðFþ2 Þ  center in NaF......Page 213
11.7.1 Valence band edge in NiO:Li......Page 214
11.7.2 Conduction band edge in BaF2 :O......Page 215
11.8 Electronic localization......Page 216
11.9 Quantum diffusion......Page 218
11.10 Effective force constants for local modes......Page 220
11.11 Summary......Page 221
Appendix to Chapter 11: the ICECAP method......Page 224
12.1 Introduction......Page 227
12.2.1 The approximation......Page 228
12.2.2 Normalization......Page 232
12.2.3 Total energy......Page 233
12.2.4 Charge density and exchange charge......Page 235
12.2.5 The single-particle density functional......Page 238
12.3.1 The variational derivation......Page 239
12.3.3 Solution of the Fock equation......Page 245
12.4 Localizing potentials......Page 249
12.5.1 Introduction......Page 252
12.5.2 Approximate partitioning with a localizing potential......Page 254
12.5.3 Summary......Page 258
12.6 Correlation......Page 260
12.7.1 Introduction......Page 269
12.7.2 Density functional of Hohenberg and Kohn......Page 270
12.7.3 Reduced density matrices......Page 272
12.7.4 The many-fermion system......Page 276
12.7.5 The density functional and the two-particle density operator......Page 279
13.1 Introduction......Page 281
13.2.1 The total energy......Page 282
13.2.2 The magnetic susceptibility......Page 285
13.2.3 Solution at low temperature......Page 288
13.2.4 Solution at high temperature......Page 297
13.3.1 Introduction......Page 300
13.3.2 The Landau levels......Page 301
13.3.3 The Fermi distribution......Page 304
13.3.4 Energy considerations......Page 312
13.3.5 Magnetization: the de Haas?van Alphen effect......Page 314
13.3.6 Diamagnetism at T = 0......Page 322
Appendix to Chapter 13......Page 325
14.1 Introduction......Page 329
14.2 Effective electron?electron interaction......Page 330
14.3.1 The Hartree approximation......Page 334
14.3.2 The uniform solution......Page 337
14.3.3 The periodic solution......Page 340
14.4.1 The Mathieu equation......Page 344
14.4.2 Solution away from the band gap......Page 347
14.4.3 Solution near the band gap......Page 349
14.4.4 The self-consistency condition......Page 353
14.4.5 The total energy......Page 359
14.5 Discussion......Page 362