Phonons: Theory and Experiments I: Lattice Dynamics and Models of Interatomic Forces

ISBN-13: 9783642817830 / Angielski / Miękka / 2012 / 261 str.

Peter Br Esch

Opis

This two-volume treatment grew out of lectures the author gave at the "Ecole Poly technique Federale de Lausanne" during the years 1975-1980 for graduate students in experimental physics in their last year of study. It is written by an experimentalist with some interest in theory and is ad dressed mainly to experimentalists, but also to theoreticians interested in experiments. This treatment tries to bridge the gap between theory and experiments; it should assist experimentalists in the interpretation of their data in the vast field of lattice dynamics. An attempt has been made to provide not only the basic concepts but also a working knowledge in this field of solid-state physics. In this first volume, the basic concepts of the physics of phonons are developed and illustrated by many examples; it provides the background necessary for the interpretation of most experimental results. The second volume, which is in preparation, is devoted to experimental techniques, the interpretation of experiments, and discussion of phenomena which are directly related with phonons. The book is designed for introductory courses at the graduate level. It is believed that the book will also prove useful to those graduate students starting research in this or related fields, as well as to many workers already active in this branch of solid-state physics."

Spis treści

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.