Topics in the Theory of Solid Materials

J.M. Vail

Preface.

1 Strain and stress in continuous media
1.1 Introduction
1.2 Deformation: strain and rotation
1.2.1 The strain tensor
1.2.2 The rotation tensor
1.3 Forces and stress
1.4 Linear elasticity
1.4.1 Hooke’s law
1.4.2 Isotropic media
1.4.3 Elastic moduli
1.4.4 Stability conditions
1.5 Equilibrium

2
Wave propagation in continuous media
2.1 Introduction
2.2 Vector ?elds
2.3 Equation of motion
2.4 Wave propagation
2.4.1 Shear and rotational waves
2.4.2 Dilatational or irrotational waves
2.4.3 General discussion
Appendix to Chapter 2
3 Thermal properties of continuous media
3.1 Introduction
3.2 Classical thermodynamics
3.2.1 The Maxwell relations
3.2.2 Elastic constants, bulk moduli and speci?c heats
3.3 Thermal conduction and wave motion
3.4 Wave attenuation by thermal conduction


4 Surface waves
4.1 Introduction
4.2 Rayleigh waves
4.3 Boundary conditions
4.4 Dispersion relation
4.5 Character of the wave motion
5 Dislocations
5.1 Introduction
5.2 Description of dislocations
5.3 Deformation ?elds of dislocations
5.3.1 Screw dislocation
5.3.2 Edge dislocation
5.4 Uniform dislocation motion
5.5 Further study of dislocations
6 Classical theory of the polaron
6.1 Introduction
6.2 Equations of motion
6.3 The constant-velocity polaron
6.4 Polaron in a magnetic ?eld: quantization
7 Atomistic quantum theory of solids
7.1 Introduction
7.2 The hamiltonian of a solid
7.3 Nuclear dynamics: the adiabatic approximation
7.4 The harmonic approximation
7.5 Phonons
7.5.1 Periodic boundary conditions for bulk properties
7.5.2 The dynamical matrix of the crystal
7.5.3 The normal modes of crystal vibration
7.5.4 Electrons and phonons: total energy
7.6 Statistical thermodynamics of a solid
7.6.1 Partition function of the crystal
7.6.2 Equation of state of the crystal
7.6.3 Thermodynamic internal energy of the crystal;
phonons as bosons
7.7 Summary
8 Phonons
8.1 Introduction
8.2 Monatomic linear chain
8.3 Diatomic linear chain
8.4 Localized mode of a point defect

9 Classical atomistic modelling of crystals
9.1 Introduction
9.2 The shell model for insulating crystals
9.3 Cohesive energy of a crystal
9.4 Elastic constants
9.5 Dielectric and piezoelectric constants
10 Classical atomic di?usion in solids
10.1 Introduction
10.2 The di?usion equation
10.2.1 Derivation
10.2.2 Planar source problem
10.3 Di?usion as a random walk
10.4 Equilibrium concentration of point defects
10.5 Temperature dependence of di?usion: the Vineyard relation
Appendix to Chapter 10: Stirling’s formula
11 Point defects in crystals
11.1 Introduction
11.1.1 Crystals and defects
11.1.2 Modelling of point defects in ionic crystals
11.2 Classical di?usion
11.2.1 Copper and silver di?usion in alkali halides
11.2.2 Dissociation of the oxygen-vacancy defect complex
in BaF2
11.3 Defect complex stability
11.4 Impurity charge-state stability
11.4.1 Nickel in MgO
11.4.2 Oxygen in BaF2
11.5 Optical excitation
11.5.1 Frenkel exciton and impurity absorption in MgO
11.5.2 Cuþ in NaF
11.5.3 O- in BaF2
11.6 Spin densities
11.6.1 F center in NaF
11.6.2 F2þ center in NaF
11.6.3 F2þ * center in NaF
11.7 Local band-edge modi?cation
11.7.1 Valence band edge in NiO : Li
11.7.2 Conduction band edge in BaF2 : O-
11.8 Electronic localization
11.9 Quantum di?usion
11.10 E?ective force constants for local modes

11.11 Summary
Appendix to Chapter 11: the ICECAP method
12 Theoretical foundations of molecular cluster computations
12.1 Introduction
12.2 Hartree–Fock approximation
12.2.1 The approximation
12.2.2 Normalization
12.2.3 Total energy
12.2.4 Charge density and exchange charge
12.2.5 The single-particle density functional
12.3 The Fock equation
12.3.1 The variational derivation
12.3.2 Total energy algorithm
12.3.3 Solution of the Fock equation
12.4 Localizing potentials
12.5 Embedding in a crystal
12.5.1 Introduction
12.5.2 Approximate partitioning with a localizing potential
12.5.3 Summary
12.6 Correlation
12.7 One-, two- and N-particle density functionals
12.7.1 Introduction
12.7.2 Density functional of Hohenberg and Kohn
12.7.3 Reduced density matrices
12.7.4 The many-fermion system
12.7.5 The density functional and the two-particle density operator
13 Paramagnetism and diamagnetism in the electron gas
13.1 Introduction
13.2 Paramagnetism of the electron gas
13.2.1 The total energy
13.2.2 The magnetic susceptibility
13.2.3 Solution at low temperature
13.2.4 Solution at high temperature
13.3 Diamagnetism of the electron gas
13.3.1 Introduction
13.3.2 The Landau levels
13.3.3 The Fermi distribution
13.3.4 Energy considerations
13.3.5 Magnetization: the de Haas–van Alphen e?ect
13.3.6 Diamagnetism at T 0
Appendix to Chapter 13

14 Charge density waves in solids
14.1 Introduction
14.2 E?ective electron–electron interaction
14.3 The Hartree equation: uniform and periodic cases
14.3.1 The Hartree approximation
14.3.2 The uniform solution
14.3.3 The periodic solution
14.4 Charge density waves: the Mathieu equation
14.4.1 The Mathieu equation
14.4.2 Solution away from the band gap
14.4.3 Solution near the band gap
14.4.4 The self-consistency condition
14.4.5 The total energy
14.5 Discussion

References
Exercises
Answers
Author index
Subject index