Ultrasonic Attenuation in Dirty Dielectric Crystals

The attenuation of a sound wave by anharmonic interaction with the thermal phonons in a dielectric crystal has been calculated using the Boltzmann-equation approach of Woodruff and Ehrenreich. Contributions to the thermal-phonon collision rate due to elastic scattering of phonons by crystal defects (relaxation time ${\tau }_e$) and due to intrinsic inelastic anharmonic processes (relaxation time ${\tau }_i$) are considered separately. In the case of longitudinal waves the attenuation remains finite as ${\tau }_e\to 0$, whereas for transverse waves propagating in high-symmetry directions the attenuation tends to zero in this limit. For $\Omega {\tau }_i\ll 1$, an estimate is made of the ratio of the longitudinal-wave attenuation in a dirty crystal ($\Omega {\tau }_e\to \infty$) to that in a clean crystal ($\Omega {\tau }_e\to 0$). This ratio is found to be a sensitive function of the Grüneisen constants of the crystal and of the temperature.