
The Physics of Phonons
A.A. Maksimov
Annotation
In the first part of the course “The Physics of Phonons” the dynamics of atoms in one and three dimensional crystalline lattices is considered in detail.
The quasiparticle conception in solids, taking phonons as an example, is discussed; creation and annihilation operators of phonons, main properties of acoustic and optical phonons dispersion laws, density of phonon states.
In the second part the phonon contribution into the thermodynamics of solids is discussed, heat capacity of dielectrics at different temperatures, thermal expansion of solids, Gruneisen relation, instability of onedimensional and twodimensional crystals.
The main third part of the course is devoted to the study of different kinetic processes in phonon system of solids. Different phonon propagation modes are considered: ballistic regime, heat conductivity of dielectrics at low and high temperatures, independent diffusion mode, second sound.
Relaxation of highfrequency nonequilibrium acoustic phonons, the model of generations, quasiballistic and quasidiffusion phonon propagation regimes. Some problems of electronphonon interaction and phonon interactions with other excitations in solids are discussed as well.

Crystalline state.
Translation symmetry.
Bravais lattice, WignerSeitz cell.
Periodic functions in three dimensions.
Main properties of a reciprocal lattice, Brillouin zone.
Adiabatic approximation.

Dynamics of the linear chain.
Harmonic approximation.
Periodic boundary conditions. Geometrical interpretation in onedimensional case.
Normal coordinates.
Independent harmonic oscillators, phonons.
Phase and group velocity of phonons.
Creation and annihilation operators of phonons.
Complex onedimensional lattice.
Acoustic and optical phonons.
Movement of atoms in a longwave and shortwave limit.
Dynamics of a threedimensional crystalline lattice.

The general case of the threedimensional lattice with the basis.
Tensor of the force constants, its main properties.
BornKarman boundary conditions.
Bloch theorem.
Dynamic matrix, general properties of normal oscillations of a threedimensional crystalline lattice.
Polarization vectors.
Property of longwave acoustic and optical phonons in a threedimensional case.
Relationship between displacement operator and creation and annihilation operators of phonons.
Density of phonon states.
Van Hove singularities in the crystals of different dimensionalities.

Statistics of phonons.
Planck distribution.
Amplitude of the thermal oscillations of the atoms, applicability of the phonon description.
Instability of onedimensional and twodimensional crystals.
Contribution of phonons to thermodynamics.
Internal energy and a heat capacity of dielectrics at high and low temperatures.
Debye and Einstein models.

Anharmonicity.
Interaction forces of atoms in a crystal.
Gruneisen parameter.
Equation of state of a solid.
Gruneisen relation.
Thermal expansion of solids.
Phononphonon interaction and the lattice anharmonicity.
Conservation law of quasimomentum.
Probabilities of threephonon processes.
Selection rules for scattering of acoustic phonons.

Thermal conduction in solids.
Phonon kinetic equation.
Normal processes and umklapp processes, their temperature dependencies.
Planck distribution in moving coordinate system.
Relaxation method.
Cowley and GuyerKrumhansl models.
Heat conductivity of dielectrics at low and high temperatures.
Contribution of scattering on sample boundaries and defects.
Knudsen and Poiseuille modes of phonon propagation.

Nonequilibrium phonons.
Establishing of equilibrium at different initial phonon distributions.
Model of generations for the relaxation of nonequilibrium acoustic phonons. Estimation of equilibration times.
Spatially nonuniform distribution of nonequilibrium phonons.
Ballistic regime of phonon propagation, phonon focusing.
Diffusion modes: heat conductivity and independent diffusion mode.
Hydrodynamic regime, second sound.
Quasiballistic and quasidiffusion phonon propagation regimes in the case of strongly nonequilibrium distribution.

Electronphonon interaction.
Matrix element of onephonon process.
Macrofield and microfield.
Electron scattering kinematics by acoustic phonons.
Electronhole droplets and phonon wind.
Polarons.

Interaction of phonons with other excitations.
Infrared photon absorption.
Polaritons.
MandelstamBrillouin and Raman scattering of light.
"Folded" acoustic phonons in semiconductor superlattices.
Scattering of Xrays and gammaquanta.
Scattering of thermal neutrons on phonons
Literature

J.M. Ziman Electrons and Phonons, Cambridge, 1960

J.A. Reissland The physics of phonons, John Wiley and Sons ltd, 1973

A.I. Anselm Introduction to the semiconductor theory, in Rus., M.: Nauka, 1978

J.M. Ziman Principles of the theory of solids, Second Ed., Cambridge, 1972

C. Kittel Introduction to Solid State Physics, Fourth Ed., John Wiley and Sons, Inc.

V.L. Gurevich Kinetics of phonon systems, in Rus., M.: Nauka, 1980

R. Berman Thermal conduction in solids, Oxford, 1976

A.M. Kosevich Basis of crystalline lattice mechanics, in Rus., M.: Nauka, 1972

C. Kittel Quantum Theory of Solids, John Wiley and Sons, Inc. 1963

L.D. Landau, E.M. Lifshitz Statistical Physics, Third Ed., Part 1, (Course of Theoretical Physics, V.5), ButterworthHeinemann, 1980

V.F.Gantmakher, Y.B.Levinson Carrier scattering in metals and semiconductors, NorthHolland, Amsterdam, 1987

M. Born, Kun Huang Dynamical Theory of Crystal Lattices, Oxford, 1954
