Dispersion Relation for Second Sound in Solids

Abstract

In this paper the dispersion relation for second sound in solids is derived. The starting point of the analysis is a Boltzmann equation for a phonon gas undergoing a temperature perturbation $\delta T_0{e}^{i(\mathbf{k}·\mathbf{x}-\omega t)}$; the Callaway approximation to the collision term is employed. We obtain a dispersion relation which explicitly exhibits the need for a "window" in the relaxation time spectrum. Further, the dispersion relation shows that measurement of the attenuation of second sound as a function of frequency is a direct measurement of the normal process and umklapp process relaxation times. We derive macroscopic equations for energy density and energy flux and show their relation to the macroscopic equations with which Chester has treated second sound.