Sound propagation and relaxation times for a fermi liquid

Abstract

The techniques for solving exactly the linearized Boltzmann transport equation of a Fermi liquid are applied to sound propagation. By separating out the collision integral into spherical harmonics, we show that different circular polarizations propagate independently; the present derivation is believed to be more general than previous ones. As an aid to calculation, a set of relaxation times is defined, there being one time for each spherical harmonic of the disturbance. The relation between these relaxation times and the single relaxation time introduced by Khalatnikov and Abrikosov is discussed. Using the relaxation times, we set up equations by which the dispersion of sound (longitudinal waves) can be calculated without approximating the collision integral. In the hydrodynamic limit, these equations have a simple solution, and we show that the use of a single relaxation time is rigorously valid in this limit. A relation is derived between this relaxation time and the viscosity, and the attenuation of sound is calculated without appeal to hydrodynamics. A similar analysis for transverse waves shows that in the hydrodynamic limit a single relaxation time is again valid, and this time is shown to be equal to the one that describes sound.