| Acknowledgement |
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v | |
| Foreword |
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vii | |
| Introduction |
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ix | |
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1 | (44) |
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Crystals: periodic arrays of atoms |
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1 | (3) |
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Mathematical description of crystal structures |
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4 | (22) |
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4 | (1) |
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5 | (1) |
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5 | (1) |
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Construction of subgroups |
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6 | (1) |
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7 | (1) |
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8 | (2) |
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Symmetry groups of a tetrahedron and a cube |
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10 | (2) |
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12 | (1) |
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12 | (2) |
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Types of Bravais lattices |
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14 | (5) |
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Crystallographic (conventional) unit cell |
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19 | (1) |
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20 | (2) |
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Symmorphic and nonsymmorphic crystal lattices |
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22 | (1) |
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22 | (1) |
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23 | (1) |
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*Matrix and operator representations of a group |
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23 | (1) |
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Operator representation of a group |
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23 | (1) |
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Matrix representation of a group |
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24 | (1) |
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Indexing of planes and directions: definitions |
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25 | (1) |
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*International Tables for X-Ray Crystallography |
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26 | (4) |
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Examples of crystal structures |
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30 | (7) |
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Cubic face-centred structures |
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31 | (1) |
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Space group O5h (Fm3m, No. 225) |
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31 | (1) |
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Space group O7h (Fd3m, No. 227) |
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32 | (1) |
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Space group T2d (F43m, No. 216) |
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32 | (1) |
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Cubic body-centred structures |
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32 | (1) |
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Space group O9h (Im3m, No. 229) |
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32 | (1) |
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Space group T7h (Ia3, No. 206) |
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33 | (1) |
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Structures with simple cubic lattice |
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33 | (1) |
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Space group O1h (Pm3m, No. 221) |
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33 | (1) |
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Space group T6h (Pa3, No. 205) |
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34 | (1) |
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34 | (1) |
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Space group D144h (P42/mnm, No. 136) |
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34 | (1) |
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Structures with trigonal lattice |
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35 | (1) |
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Space group D63d (R3c, No. 167) |
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35 | (1) |
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Structures with hexagonal lattice |
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36 | (1) |
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Space group D46h (P63/mmc, No. 194) |
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36 | (1) |
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Space group C46v (P63mc, No. 186) |
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36 | (1) |
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Space group D43 (P3121, No. 152) |
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37 | (1) |
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37 | (8) |
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Definition of order and quasicrystals |
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38 | (2) |
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40 | (1) |
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40 | (1) |
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40 | (3) |
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43 | (2) |
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The reciprocal lattice and X-ray diffraction |
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45 | (24) |
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45 | (2) |
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*Once again about crystal planes and Miller indices |
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47 | (3) |
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50 | (1) |
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Periodic functions: Fourier analysis |
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51 | (4) |
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Introduction to X-ray diffraction |
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55 | (14) |
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55 | (4) |
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59 | (1) |
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60 | (3) |
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Interpretation of diffraction experiments |
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63 | (1) |
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*X-ray diffraction of nonperiodic solids |
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63 | (1) |
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Density-density correlation function |
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64 | (1) |
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Periodic systems revisited |
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65 | (1) |
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Application to a binary alloy |
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66 | (2) |
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68 | (1) |
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69 | (32) |
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69 | (6) |
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Molecules: types of chemical bonding |
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75 | (13) |
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Simple example: a molecule with two atoms |
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75 | (2) |
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77 | (1) |
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77 | (1) |
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Hydrogen molecular ion H+2 |
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77 | (1) |
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Hydrogen molecule: MO method |
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78 | (4) |
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Hydrogen molecule: VB method |
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82 | (1) |
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Covalent bonds for elements having the (np)3 shells |
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83 | (1) |
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Covalent bonds for elements having the (ns)2(np)2 shells |
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83 | (1) |
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Some other examples of hybrid orbitals |
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84 | (1) |
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85 | (1) |
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Van der Waals interaction |
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85 | (3) |
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88 | (1) |
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88 | (13) |
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Cohesive and lattice energies |
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88 | (1) |
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89 | (1) |
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89 | (1) |
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*Ewald method: electrostatic potential |
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90 | (2) |
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92 | (1) |
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*Ewald method: electrostatic energy |
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93 | (1) |
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The Madelung energy of a finite large crystal sample |
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94 | (1) |
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94 | (1) |
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95 | (2) |
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97 | (1) |
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98 | (1) |
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98 | (1) |
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99 | (2) |
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101 | (102) |
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Lagrangian and Hamiltonian method |
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101 | (2) |
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103 | (19) |
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103 | (1) |
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Lagrangian and equation of motion |
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104 | (1) |
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105 | (1) |
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106 | (1) |
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106 | (1) |
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107 | (1) |
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107 | (1) |
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Lagrangian and equations of motion |
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108 | (1) |
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109 | (1) |
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Limiting case of identical atoms |
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110 | (2) |
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Why optical and acoustic? |
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112 | (1) |
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112 | (2) |
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114 | (1) |
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Discrete Fourier transform |
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115 | (1) |
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Matrix form for the eigenvalue problem and the dynamical matrix of the chain |
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116 | (2) |
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Diagonal representation for the kinetic and potential energies of the chain |
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118 | (2) |
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Normal coordinates of the chain |
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120 | (1) |
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Density of states for the ID chain |
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121 | (1) |
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Three dimensional lattice: classical |
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122 | (23) |
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123 | (2) |
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125 | (1) |
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Hamiltonian and equations of motion |
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125 | (1) |
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125 | (1) |
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126 | (1) |
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Eigenvalue and eigenvector problem for lattice vibrations |
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126 | (1) |
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127 | (1) |
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128 | (1) |
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129 | (1) |
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Limiting case of long waves |
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129 | (1) |
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130 | (1) |
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130 | (1) |
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Example: a crystal with central forces |
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131 | (1) |
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Crystal with central forces |
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131 | (2) |
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Oscillations of a binary fcc crystal |
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133 | (2) |
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Phonon density of states (DOS) |
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135 | (1) |
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Contribution from long acoustic waves: Debye model |
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136 | (1) |
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137 | (2) |
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139 | (1) |
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3D discrete Fourier transform |
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139 | (2) |
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Formal introduction of normal coordinates |
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141 | (1) |
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Diagonalisation of the kinetic and potential energies using complex coordinates |
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142 | (2) |
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Introduction of real coordinates |
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144 | (1) |
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Lattice stability at zero temperature |
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145 | (1) |
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Three dimensional lattice dynamics: quantum |
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145 | (10) |
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A single harmonic oscillator |
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145 | (1) |
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Introduction of creation and destruction (annihilation) operators |
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146 | (1) |
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Introduction to algebra of operators a and a† |
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147 | (2) |
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*Some useful operator identities |
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149 | (3) |
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Crystal vibrations in the harmonic approximation |
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152 | (1) |
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153 | (2) |
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Thermal properties of crystals |
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155 | (24) |
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Equilibrium statistical mechanics |
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155 | (1) |
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Classical statistical mechanics |
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155 | (1) |
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Quantum statistical mechanics |
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156 | (1) |
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156 | (2) |
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Phonons as quasiparticles |
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158 | (1) |
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*Some useful statistical averages |
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158 | (3) |
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*Displacement-displacement correlation function |
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161 | (1) |
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Internal energy and specific heat |
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162 | (1) |
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162 | (1) |
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163 | (1) |
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Debye model for acoustic branches |
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163 | (3) |
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Einstein model for optical branches |
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166 | (1) |
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166 | (1) |
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*Quasiharmonic approximation |
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166 | (2) |
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168 | (1) |
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169 | (1) |
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170 | (2) |
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Thermal conductivity and anharmonicity |
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172 | (1) |
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Elementary kinetic theory of thermal conductivity |
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172 | (1) |
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173 | (3) |
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176 | (2) |
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Elastic and inelastic phonon processes |
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178 | (1) |
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*Elementary theory of elasticity and stability |
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179 | (24) |
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Main ideas of the classical theory of elasticity |
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179 | (1) |
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External and Lagrangian strain |
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180 | (2) |
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182 | (2) |
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184 | (1) |
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Shear and normal strain and stress |
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184 | (1) |
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185 | (1) |
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186 | (1) |
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Elastic properties of crystals |
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186 | (1) |
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187 | (1) |
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187 | (2) |
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189 | (2) |
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191 | (2) |
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193 | (2) |
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195 | (1) |
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Method of homogeneous deformation |
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196 | (1) |
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General description of a homogeneous deformation in a crystal |
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196 | (1) |
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General expressions for the isothermal elastic constants |
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197 | (3) |
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Example: a crystal with pairwise central interactions |
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200 | (3) |
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Electrons in a periodic potential |
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203 | (96) |
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Model of a free electron gas |
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203 | (15) |
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203 | (1) |
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Energies and wavefunctions |
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204 | (1) |
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Periodic boundary conditions |
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204 | (1) |
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Orthogonality and completeness of plane waves |
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205 | (2) |
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Distribution of electrons on energy levels. Fermi sphere |
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207 | (1) |
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208 | (1) |
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Quantum statistics: Fermi-Dirac distribution |
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208 | (3) |
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*Heat capacity and chemical potential of the electron gas |
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211 | (1) |
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211 | (1) |
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212 | (1) |
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213 | (1) |
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Comparison with experiment |
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214 | (1) |
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214 | (1) |
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215 | (1) |
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215 | (1) |
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Motion in magnetic field. Hall effect |
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216 | (2) |
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218 | (1) |
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218 | (1) |
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218 | (24) |
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219 | (1) |
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The meaning of vector q and periodic boundary conditions |
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220 | (1) |
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221 | (1) |
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Electronic band structure via plane waves |
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222 | (1) |
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*Density-functional theory and Kohn-Sham potential |
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223 | (2) |
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*Calculation of the Hartree potential |
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225 | (2) |
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Approximation of a nearly free electron gas |
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227 | (1) |
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Empty lattice approximation: reduced zone scheme |
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227 | (2) |
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Model of a nearly free electron gas |
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229 | (2) |
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231 | (2) |
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233 | (1) |
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Assembling a crystal from atoms |
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234 | (1) |
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234 | (2) |
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236 | (3) |
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239 | (3) |
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Transport properties: electrical and thermal conductivity revisited |
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242 | (24) |
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242 | (1) |
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243 | (1) |
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244 | (1) |
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244 | (1) |
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245 | (3) |
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248 | (2) |
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250 | (2) |
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252 | (2) |
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Collision term and the detailed balance |
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254 | (1) |
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Relaxation time approximation |
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255 | (1) |
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256 | (2) |
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258 | (1) |
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*Quantum description of transport processes |
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259 | (1) |
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Nonequilibrium quantum statistical mechanics |
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259 | (2) |
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Kubo's linear response theory |
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261 | (2) |
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Generalised susceptibilities |
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263 | (1) |
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General expression for electrical conductivity |
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264 | (1) |
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Relaxation time approximation |
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265 | (1) |
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Electron-electron interaction |
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266 | (33) |
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Qualitative consideration |
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266 | (2) |
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*Elementary theory of ``plasma'' oscillations |
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268 | (2) |
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Excitations of plasmons by fast electrons |
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270 | (1) |
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Interaction with electromagnetic waves |
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271 | (2) |
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Interaction with longitudinal electrostatic field |
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273 | (1) |
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*Theory of plasma oscillations based on density fluctuations |
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274 | (1) |
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Electron Hamiltonian in the jellium model |
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274 | (2) |
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Classical treatment of plasma oscillations |
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276 | (2) |
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Quantum treatment of plasma oscillations |
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278 | (1) |
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*Screening in the electron gas |
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279 | (1) |
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Screening Coulomb potential of a point charge |
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280 | (3) |
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*Dielectric function of the electron gas |
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283 | (1) |
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Maxwell equations for zero magnetic field |
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284 | (1) |
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Tensor of the microscopic dielectric function |
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285 | (3) |
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General expression for electronic susceptibility |
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288 | (2) |
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Self-consistent consideration of the electronic response |
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290 | (1) |
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Susceptibility in the independent particles approximation |
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291 | (3) |
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Application to a free electron gas |
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294 | (5) |
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299 | (68) |
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Magnetic moment in classical electrodynamics |
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299 | (8) |
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Magnetic field of a system of moving charges far away from them |
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299 | (2) |
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Relation between the magnetic moment and angular momentum |
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301 | (1) |
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Movement of a charged particle in a magnetic field |
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301 | (2) |
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Magnetic field in matter and magnetic permeability |
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303 | (4) |
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Magnetic moment in quantum mechanics |
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307 | (14) |
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*Relativistic description of an electron |
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307 | (1) |
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307 | (2) |
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Solution of the Dirac equation for a free relativistic electron |
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309 | (1) |
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310 | (2) |
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*An electron in electro-magnetic field |
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312 | (2) |
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Magnetic moment of an electron |
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314 | (1) |
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Quasi-relativistic approach |
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314 | (1) |
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An electron in a magnetic field |
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315 | (1) |
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*One electron atom in a homogeneous magnetic field |
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316 | (1) |
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Magnetic moment of an atom |
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317 | (1) |
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317 | (2) |
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319 | (1) |
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Hund rules and physical reasons for permanent localised magnetic moments |
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320 | (1) |
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Thermodynamics of magnetic materials |
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321 | (1) |
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Para- and diamagnetism of localised electrons |
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322 | (7) |
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322 | (1) |
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Almost classical theory of diamagnetism |
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323 | (1) |
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Quantum theory of diamagnetism |
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324 | (1) |
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Almost classical theory of paramagnetism (Langevin) |
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325 | (2) |
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Quantum theory of paramagnetism |
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327 | (2) |
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Para- and diamagnetism of the electron gas |
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329 | (7) |
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329 | (2) |
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*Magnetism of electrons in metals: Landau diamagnetism and the de Haas-van Alphen effect |
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331 | (1) |
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General expression for the grand potential |
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331 | (4) |
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Pauli paramagnetism versus Landau diamagnetism |
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335 | (1) |
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The de Haas-van Alphen effect |
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336 | (1) |
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336 | (21) |
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Interaction between localised magnetic moments |
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336 | (1) |
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336 | (1) |
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337 | (1) |
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338 | (1) |
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339 | (3) |
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342 | (1) |
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Hysteresis and domain structure |
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343 | (1) |
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343 | (1) |
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344 | (1) |
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344 | (2) |
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Wall motion and rotation versus reversibility and irreversibility |
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346 | (1) |
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346 | (1) |
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Exchange interaction and the phenomenological theory of ferromagnetism |
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347 | (1) |
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*Hydrogen molecule revisited |
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348 | (1) |
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349 | (1) |
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350 | (1) |
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351 | (1) |
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*Band theory of ferromagnetism |
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352 | (1) |
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Exchange interaction in metals: exchange hole |
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352 | (1) |
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Stoner model: general equations |
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353 | (2) |
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Stoner model: paramagnetism |
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355 | (1) |
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Stoner model: ferromagnetism |
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356 | (1) |
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Stoner model: specific heat |
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357 | (1) |
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Symmetry breaking and order parameters |
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357 | (10) |
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358 | (2) |
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The Landau theory of second order phase transition |
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360 | (2) |
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362 | (5) |
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367 | (54) |
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367 | (4) |
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Critical magnetic field and critical current |
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368 | (1) |
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Meissner-Ochsenfeld effect |
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368 | (2) |
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Superconducting phase transition |
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370 | (1) |
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370 | (1) |
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371 | (1) |
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Phenomenological theory of superconductivity |
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371 | (6) |
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Thermodynamics of superconductors |
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371 | (2) |
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373 | (3) |
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376 | (1) |
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Main ideas of the microscopic theory of superconductivity |
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377 | (32) |
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Attraction between electrons |
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377 | (1) |
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377 | (5) |
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*Ground state of the metal in the superconducting state |
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382 | (1) |
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Creation and annihilation operators for electrons in a normal metal |
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382 | (2) |
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Variational wavefunction for a superconductor |
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384 | (2) |
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Calculation of the ground state using a variational method |
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386 | (5) |
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391 | (1) |
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Correlation and coherence lengths |
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391 | (2) |
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*Excitation energies in the superconducting state |
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393 | (4) |
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397 | (2) |
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399 | (2) |
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401 | (2) |
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Existence of the critical magnetic field |
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403 | (1) |
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*The Meissner-Ochsenfeld effect |
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404 | (1) |
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404 | (1) |
|
Coordinate representation of the wave function |
|
|
405 | (1) |
|
Derivation of the second London equation |
|
|
406 | (1) |
|
Quantisation of magnetic flux |
|
|
407 | (2) |
|
*Ginzburg-Landau theory of superconductivity |
|
|
409 | (6) |
|
Order parameter and the free energy |
|
|
410 | (2) |
|
Ginzburg-Landau equations |
|
|
412 | (2) |
|
|
|
414 | (1) |
|
|
|
415 | (1) |
|
|
|
416 | (5) |
|
|
|
416 | (3) |
|
|
|
419 | (2) |
|
|
|
421 | (50) |
|
|
|
422 | (4) |
|
Phonon contribution to the dielectric function |
|
|
426 | (21) |
|
|
|
426 | (3) |
|
Optical vibrations of a binary ionic crystal |
|
|
429 | (1) |
|
|
|
430 | (4) |
|
Dispersion formula for the dielectric function |
|
|
434 | (2) |
|
|
|
436 | (2) |
|
*General consideration: classical |
|
|
438 | (5) |
|
*General consideration: quantum |
|
|
443 | (4) |
|
Thermodynamics of dielectrics |
|
|
447 | (8) |
|
Contribution of the field to thermodynamic potentials |
|
|
447 | (1) |
|
|
|
448 | (1) |
|
|
|
448 | (1) |
|
*Pyroelectrics and crystal symmetry |
|
|
449 | (1) |
|
Dielectric tensor and crystal symmetry |
|
|
449 | (1) |
|
*Effect of the elastic deformation |
|
|
450 | (1) |
|
|
|
451 | (1) |
|
Crystal symmetry allowing piezoelectricity |
|
|
452 | (1) |
|
Statics and dynamics of a piezoelectric crystal |
|
|
453 | (1) |
|
Elastic waves in piezoelectrics |
|
|
454 | (1) |
|
|
|
455 | (16) |
|
General description of ferroelectrics |
|
|
455 | (5) |
|
Landau theory of the ferroelectric phase transition |
|
|
460 | (1) |
|
|
|
461 | (1) |
|
|
|
462 | (3) |
|
*Microscopic consideration: Effective field model of Lines |
|
|
465 | (6) |
|
*Modern methods of electronic structure calculations |
|
|
471 | (124) |
|
Many-electron wavefunction |
|
|
472 | (21) |
|
|
|
472 | (2) |
|
|
|
474 | (3) |
|
|
|
477 | (1) |
|
Creation and annihilation operators |
|
|
478 | (2) |
|
Slater rules: matrix elements between determinants |
|
|
480 | (4) |
|
Non-orthogonal spin-orbitals |
|
|
484 | (1) |
|
Operators in second quantisation |
|
|
484 | (1) |
|
|
|
485 | (2) |
|
|
|
487 | (1) |
|
|
|
488 | (1) |
|
|
|
488 | (1) |
|
|
|
488 | (2) |
|
|
|
490 | (2) |
|
Natural orbitals and occupation numbers |
|
|
492 | (1) |
|
Quantum chemistry methods |
|
|
493 | (25) |
|
Configuration Interaction (CI) method |
|
|
494 | (2) |
|
|
|
496 | (3) |
|
|
|
499 | (2) |
|
Hartree-Fock energy and electronic density |
|
|
501 | (1) |
|
Variational method: Hartree-Fock equations |
|
|
502 | (4) |
|
Koopman's theorem and physical significance of εi |
|
|
506 | (1) |
|
Closed shell electron system |
|
|
507 | (1) |
|
Hartree-Fock-Roothaan method |
|
|
508 | (2) |
|
HF theory of the homogeneous electron gas |
|
|
510 | (4) |
|
|
|
514 | (4) |
|
Density Functional Theory |
|
|
518 | (21) |
|
|
|
518 | (3) |
|
The Levy constrained search method |
|
|
521 | (2) |
|
|
|
523 | (1) |
|
Relation to a fictitious noninteracting electron gas |
|
|
523 | (1) |
|
Derivation of the KS equations from the variational principle |
|
|
524 | (2) |
|
Matrix form of the KS equations |
|
|
526 | (1) |
|
Local density approximation (LDA) and beyond |
|
|
527 | (3) |
|
More about the KS equations |
|
|
530 | (1) |
|
Meaning of the KS eigenvalues |
|
|
531 | (2) |
|
|
|
533 | (3) |
|
Other extensions of the DFT |
|
|
536 | (1) |
|
|
|
536 | (1) |
|
|
|
537 | (1) |
|
Time dependent DFT (TDDFT) |
|
|
538 | (1) |
|
|
|
539 | (14) |
|
|
|
539 | (3) |
|
Periodic boundary conditions and k-point sampling |
|
|
542 | (4) |
|
Pseudopotentials: ``hard'' and ``soft'' |
|
|
546 | (5) |
|
``Exact'' pseudopotentials: PAW method |
|
|
551 | (1) |
|
|
|
552 | (1) |
|
|
|
553 | (42) |
|
Static properties: energies and forces |
|
|
554 | (1) |
|
|
|
555 | (2) |
|
|
|
557 | (2) |
|
|
|
559 | (1) |
|
|
|
560 | (1) |
|
Adsorption of atomic and molecular oxygen on the MgO (001) surface |
|
|
561 | (7) |
|
Ab initio molecular dynamics simulations |
|
|
568 | (3) |
|
Hydrolysis at stepped MgO surfaces |
|
|
571 | (2) |
|
Calculation of free energies |
|
|
573 | (3) |
|
Ab initio lattice dynamics: direct method |
|
|
576 | (4) |
|
Density functional perturbation theory |
|
|
580 | (5) |
|
|
|
585 | (10) |
| Bibliography |
|
595 | (14) |
| Index |
|
609 | |
