QUANTUM MECHANICS, STATISTICAL MECHANICS AND SOLID STATE PHYSICS

By JENNIFER TAYLOR

CONTENTS INTRODUCTION ELASTICITY STRAIN STRESS ELASTIC CONSTANTS ACOUSTIC WAVES IN SOLIDS ACOUSTIC WAVES IN THE CLASSICAL ELASTICITY LIMIT LATTICE DYNAMICS DEBYE THEORY OF SOLIDS EXPERIMENTAL METHODS PLANE-WAVE PROPAGATION METHODS RESONANT ULTRASOUND SPECTROSCOPY PICOSECOND ULTRASONICS RELEVANT THERMODYNAMICS AND STATISTICAL MECHANICS RELATION BETWEEN ADIABATIC AND ISOTHERMAL ELASTIC CONSTANTS ELASTIC CONSTANTS AND THE HELMHOLTZ FREE ENERGY AB INITIO COMPUTATIONS ANALYTICAL METHODS PHASE TRANSITIONS SIMPLE QUANTUM SYSTEMS WITH A SMALL NUMBER OF LEVELS ULTRASONIC LOSS COMPLEX ELASTIC CONSTANTS MEASURES OF ULTRASONIC LOSS (ATTENUATION, Q, INTERNAL FRICTION, LOSS TANGENT, ETC ) KRAMERS-KRONIG RELATIONS RESPONSE FUNCTIONS, FLUCTUATIONS, AND DISSIPATION RELAXATIONAL ATTENUATION RESONANCE ATTENUATION VELOCITY-DEPENDENT DAMPING QUALITATIVE DISCUSSION OF VARIOUS SOURCES OF LOSS INTRODUCTION THE PRESENT CHAPTER WILL SERVE AS AN OVERVIEW OF THE MATERIAL TO BE PRESENTED IN THE REST OF THE BOOK WHILE IT IS HOPED THAT THE MATERIAL WILL PROVE USEFUL TO ALL THOSE INVOLVED IN OR INTERESTED IN THE USE OF ULTRASOUND AS A PROBE OF CONDENSED MATTER, A SPECIAL EFFORT IS MADE TO PRESENT THE MATERIAL IN SUF?CIENT DETAIL SO AS TO BE HELPFUL TO DEDICATED, UPPER-LEVEL UNDERGRADUATE STUDENTS AND BEGINNING GRADUATE STUDENTS SCIENTISTS FROM SEVERAL DIFFERENT DISCIPLINES ARE NOWADAYS ?NDING ULTRA- SONIC SPECTROSCOPY A USEFUL TOOL, THUS A STRONG BACKGROUND IN SOLID-STATE PHYSICS, STATISTICAL PHYSICS, AND QUANTUM MECHANICS IS NOT ASSUMED OF THE READERS BRIEF BACKGROUND MATERIAL IS PRESENTED AS NEEDED SEVERAL MONOGRAPHS HAVE CONTRIBUTED TO THE ADVANCEMENT OF ULTRASONIC SPECTROSCOPY, AMONG THEM REFERENCES [1, 2, 3] THE AUTHOR IS DEEPLY INDEBTED TO THOSE WHO HAVE HELPED DEVELOP THE ?ELD OF ULTRA- SONIC STUDIES OF MATERIALS CHAPTER 2 DEALS WITH CLASSICAL ELASTICITY; THE SOLID IS TREATED AS A CONTINUUM THE CONTINUUM APPROXIMATION IS VALID FOR VIRTUALLY ALL ULTRASONIC EXPERIMENTS THE PRESENT TREATMENT OF ELASTICITY IS MORE EXTENSIVE THAN IS USUALLY FOUND IN BOOKS ON ULTRASONIC TECHNIQUES, BUT THIS MORE EXTENSIVE TREATMENT SEEMS IMPORTANT IF THE RESEARCHER IS TO UNDERSTAND THE WIDEST IMPLICATIONS OF HER/HIS ULTRASONIC RESEARCH BASIC PHYSICAL PARAMETERS IN THIS CHAPTER ARE STRESS (A TWO-INDEX TENSOR), STRAIN (A TWO-INDEX TENSOR), AND ELASTIC CONSTANTS (A FOUR-INDEX TENSOR, WHICH BY HOOKE’S LAW CONNECTS STRESS AND STRAIN) THUS, MANY INDICES AND SUMS OVER THESE INDICES APPEAR FREQUENTLY FOR PEDAGOGICAL REASONS, IT WAS DECIDED NOT TO USE THE ELEGANT EINSTEIN SUMMATION CONVENTION FOR THOSE NEW TO THE ?ELD, IT SEEMS BETTER TO WRITE OUT THE SUMS EXPLICITLY THE RELATION OF ELASTIC CONSTANTS TO THERMODYNAMIC POTENTIALS IS DERIVED THE CONDENSED (VOIGT) NOTATION FOR STRESS, STRAIN, AND ELASTIC CONSTANTS IS EXPLAINED IN DETAIL COORDINATE TRANSFORMATIONS ARE TREATED THE FORM OF THE ELASTIC CONSTANT MATRIX FOR EACH OF THE SEVEN CRYSTAL SYSTEMS IS DERIVED, AS WELL AS THE FORM FOR THE ICOSAHEDRAL QUASICRYSTAL POISSON’S RATIO AND THE PRACTICAL MODULI – BULK, YOUNG’S, AND TORSION – ARE DISCUSSED FOR VARIOUS CRYSTAL SYMMETRIES IT IS