Thermal Conductivity, Second Sound, and Phonon Hydrodynamic Phenomena in Nonmetallic Crystals

Abstract

A variety of phonon-gas phenomena in nonmetals are discussed in a unified manner using a set of macroscopic equations developed from the solution of the linearized phonon Boltzmann equation. This set of macroscopic equations, appropriate for the description of a low-temperature phonon gas, is solved for a cylindrical sample in the limit ${\ensuremath{\lambda}}_{N}\ensuremath{\ll}R$; ${\ensuremath{\lambda}}_{N}\ensuremath{\lambda}_{R}^{}{}_{}{}^{z}\ensuremath{\gg}{R}^{2}$. Here ${\ensuremath{\lambda}}_{N}$ is the normal-process mean free path, $\ensuremath{\lambda}_{R}^{}{}_{}{}^{z}$ is the mean free path for momentum-loss scattering calculated in the Ziman limit, and $R$ is the radius of the sample. The solution in this limit exhibits Poiseuille flow of the phonon gas as first discussed by Sussmann and Thellung. An equation for the thermal conductivity which correctly includes this phenomenon is found. Using this equation, the possible outcomes of steady-state thermal-conductivity measurements are discussed in terms of the microscopic scattering rates. Heat-pulse propagation is discussed from a similar point of view. The existence of Poiseuille flow in steady-state thermal-conductivity measurements bears directly on the possibility of observing second sound in solids. A quantitative analysis of available data on LiF suggests that the chemical purity of these samples sets very stringent limits on the observation of either of these effects. The observation of Poiseuille flow in solid ${\mathrm{He}}^{4}$ samples by Mezov-Deglin strongly suggests that this material is a prime subject for investigations of second-sound propagation.