Acoustic Attenuation in Dielectric Solids

The solution of the Boltzmann equation for a phonon gas in terms of the eigenfunction spectrum of the linearized normal-process collision operator has been investigated by Guyer and Krumhansl. This treatment is extended to the case when the phonon gas is driven by an acoustic wave. An expression for the acoustic attenuation $\Gamma$ is obtained in the form $\Gamma \propto \chi \left(\mathbf{k}\Omega \right)$, where $\chi \left(\mathbf{k}\Omega \right)$ is the dynamic thermal response coefficient of Griffin. $\chi \left(\mathbf{k}\Omega \right)$ depends upon the thermal conductivity $\kappa \left(\mathbf{k}\Omega \right)$, which in turn depends upon the wave vector k and frequency $\Omega$. An approximation is made for $\kappa \left(\mathbf{k}\Omega \right)$ which leads to expressions for $\Gamma$ in the limit $\Omega {\tau }_R<1\mathrm{}$ and $\Omega {\tau }_R>1\mathrm{}$ in agreement with previous investigations. In addition, in the temperature range where second sound can exist we find three frequency ranges separated by the conditions $\Omega {\tau }_N=1\mathrm{}$ and $\Omega {\tau }_R=1\mathrm{}$ instead of the usual two separated by $\Omega {\tau }_R=1\mathrm{}$. (${\tau }_R$ and ${\tau }_N$ are the relaxation times for non-momentum-conserving phonon scattering and normal-process scattering, respectively). The acoustic attenuation in the intermediate range, $\tau _R^{}{}_{}{}^{-1\mathrm{}}<~\Omega <~\tau _N^{}{}_{}{}^{-1\mathrm{}}$, is qualitatively different from that in the ranges $\Omega {\tau }_N>~1$ and $\Omega {\tau }_R<~1$. Further, in this range the possibility of a resonance between first and second sound occurs. The implications of these results for acoustic attenuation experiments and the light-scattering experiment of Griffin are discussed.