Topics in the Theory of Solid Materials

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