Absorption of Sound in Insulators

The theory of sound attenuation in structurally perfect dielectric crystals is extended and applied to recent experiments on the absorption of acoustic waves in crystalline quartz at frequencies from ${10}^9$ cps to 2.4×${10}^{10}$ cps. The sound wave is assumed to vary the frequencies of the thermal phonons adiabatically, and the complete Boltzmann equation is used to determine the response of the thermal phonon distribution to this disturbance. The rate of energy transfer from the thermal phonons to the temperature bath is computed. In the steady state, energy is supplied by the driving sound wave to the thermal phonons at the same rate, which gives the attenuation. Relaxation times are assumed for $N$ and $U$ processes. Since the effect of the sound wave on a thermal phonon depends on the relative polarization and wave-number vectors of both, the phonon distribution in a small spatial region tends to relax to a new temperature $T^{\prime }$ which is determined by requiring local conservation of the total energy to first order. The present treatment leads to better understanding of the rapid decrease in attenuation with decreasing temperature in the range in which the sound-wave period becomes comparable to the average relaxation time of the thermal phonons.