Phonons: Theory and Experiments I

1 Introduction.- 1.1 The Static Lattice Model and Its Limitations.- 1.2 Early History of Lattice Dynamics.- 1.3 The Adiabatic and Harmonic Approximations.- 1.4 Organization of the Book.- 1.5 Problems.- 1.5.1 Adiabatic Approximation.- 1.5.2 Harmonic Vibration of a Diatomic Molecule.- 2. Dynamics of the Linear Diatomic Chain.- 2.1 Classical Mechanics.- 2.1.1 Periodic Boundary Conditions and Dispersion Relations.- 2.1.2 Dynamical Matrix and Eigenvectors.- 2.1.3 An Illustration: The Linear NaCl-Chain; Transition to the Monoatomic Lattice.- 2.1.4 Normal Coordinates.- 2.2 Quantum Mechanics.- 2.2.1 The Schrödinger Equation of the Simple Harmonic Oscillator.- 2.2.2 The Schrödinger Equation of the Vibrating Chain.- 2.2.3 Creation and Annihilation Operators.- 2.2.4 Phonons.- 2.2.5 Specific Heat and Density of States.- 2.3 Problems.- 2.3.1 Monoatomic Chain.- 2.3.2 Chain with a Basis of Two Identical Atoms.- 2.3.3 Probability Densities of a Classical and Quantum Mechanical Oscillator.- 2.3.4 Density of States of the Monoatomic Chain with Nearest and Second-Nearest-Neighbour Interactions.- 3. Dynamics of Three-Dimensional Crystals.- 3.1 Equations of Motion and Atomic Force Constants.- 3.2 Dynamical Matrix and Eigenvectors.- 3.3 Periodic Boundary Conditions, Reciprocal Lattices and Brillouin Zones.- 3.4 Normal Coordinates, Phonons.- 3.5 Density of States and Specific Heat.- 3.5.1 Density of States.- 3.5.2 Specific Heat.- 3.6 Connection of Lattice Dynamics with the Theory of Elasticity.- 3.7 An Illustration: Phonon Dispersion of Monoatomic Crystals with fcc Structure.- 3.8 Problems.- 3.8.1 Brillouin Zone in Two Dimensions.- 3.8.2 Critical Points (c.p.) in the Density of States..- 3.8.3 Density of States in Two Dimensions.- 3.8.4 Debye Specific Heat in Two Dimensions.- 3.8.5 Elastic Waves in Continuous Media.- 3.8.6 Vibrations in Crystals with CsCl Structure.- 4. Interatomic Forces and Phonon Dispersion Curves.- 4.1 Lattice Dynamics of the Solid Inert Gases.- 4.2 The Rigid-Ion Model for Ionic Crystals.- 4.2.1 Definition of the Model and Dynamical Matrix.- 4.2.2 Coulomb Matrix and Electric Fields.- 4.2.3 Application to Crystals with NaCl Structure.- 4.2.4 Deficiencies of the Rigid-Ion Model.- 4.3 The Shell Model.- 4.3.1 The Essential Features of the Model.- 4.3.2 The Dielectric Constant and the Lyddane-Sachs-Teller Relation.- 4.3.3 Generalized Shell Model and Phonon Dispersions.- 4.4 The Adiabatic Bond Charge Model.- 4.5 The Valence Force Model.- 4.6 Internal and External Vibrations in Molecular Crystals.- 4.7 Phonons in Metals.- 4.7.1 Force Constant Models.- 4.7.2 Coulomb Interaction in the Uniform-Background Lattice Model.- 4.7.3 Bardeen’s Treatment of Screening.- 4.8 Problems.- 4.8.1 Lennard-Jones Parameters of the Linear Chain with Zero-Point Energy.- 4.8.2 Shell Model of the Linear Monoatomic Chain.- 4.8.3 Generalized Lyddane-Sachs-Teller Relation.- 4.8.4 Bending Coordinates: Application to the Linear Chain.- 4.8.5 Thomas-Fermi Screening.- 5. Anharmonicity.- 5.1 The Anharmonic Diatomic Molecule.- 5.2 The Anharmonic Linear Chain.- 5.2.1 Dynamical Aspects.- 5.2.2 The Free Energy of the Classical Anharmonic Chain.- 5.2.3 The Equation of State and Thermal Expansion in the Quasiharmonic Approximation.- 5.2.4 The Specific Heat.- 5.3 The Anharmonic Three-Dimensional Crystal.- 5.3.1 The Equation of State.- 5.3.2 Thermal Expansion.- 5.3.3 Anharmonic Effects on the Specific Heat and Elastic Constants.- 5.4 The Self-Consistent Harmonic Approximation (SCHA).- 5.4.1 General Remarks.- 5.4.2 The Diatomic Molecule.- 5.4.3 The SCHA for a Bravais Crystal.- 5.4.4 The Self-Consistent Isotropic Einstein Model.- 5.5 Response Function and Perturbation Theory of Phonon-Phonon Interactions.- 5.5.1 Response Function of Harmonic and Damped Harmonic Oscillators.- 5.5.2 Response Function for the Anharmonic Crystal.- 5.5.3 Frequency Widths and Shifts from Perturbation Theory.- 5.6 Problems.- 5.6.1 Thermal Expansion and Force Constant of Diatomic Molecules.- 5.6.2 Quantum Anharmonic Oscillator.- 5.6.3 Grüneisen Parameter, Thermal Expansion and Frequency Shift of a Monoatomic fcc Crystal.- 5.6.4 Grüneisen Parameter of TO and LO-Modes of Simple Diatomic Crystals.- 5.6.5 Displacement-Displacement Correlation Function ???(??’).- 5.6.6 Equations of Motion Including Damping and External Driving Forces.- A Periodicity of Eigenfrequencies and Atomic Displacements in Reciprocal Space.- B An Important Lattice Sum.- C Hamiltonian for the Diatomic Chain in Terms of Normal Coordinates.- D Commutator Relations for Normal Coordinates.- E The Occupation Number Representation.- F Restriction on Atomic Force Constants Which Follow from the Space Group Symmetry of the Crystal.- G Dynamical Matrix.- J Force Constants for Central Forces.- K Evaluation of the Coulomb Matrix Using Ewald’s Method.- L The Valence Force Model.- M The Saddle-Point Method.- N The Free Energy in the Harmonic or Quasiharmonic Approximation.- O The Self-Consistent Harmonic Approximation (SCHA).- P Expansion Coefficients of Anharmonic Terms.- Q Constants and Units.- General References.- References.