First and Second Sound in Crystals

A semiphenomenological theory for pure, insulating Bravais lattices is developed to clarify in a unified fashion the problems of sound propagation, sound damping, second sound, and heat conduction. The crystal is described as a gas of interacting quasiparticles coupled to an anisotropic elastic medium. The thermodynamic functions are expressed by functional derivatives of the total energy with respect to the quasiparticle distribution and the lattice deformations. The nonequilibrium behavior is described by a Boltzmann equation for the quasiparticles and an equation for the elastic waves. For low temperatures, the collision operator is written as a direct sum of one part due to normal scattering and a second part due to the relevant umklapp processes. In the hydrodynamic regime, the general equations are reduced to two coupled equations for the lattice deformation and for the local temperature. The solutions of these equations are studied exhaustively. Correlation functions are calculated to discuss the intermediate regime between second sound and heat conduction, and are applied to a brief discussion of sum rules.