Theory of Acoustic Attenuation in Insulators

A Green's-function theory of the attenuation of an acoustic phonon by its interaction through the anharmonic lattice potential with thermal phonons is derived. A general mechanism scattering the thermal phonons is included by the use of the full Green's function for the thermal phonons. The theory thus provides a unified treatment for the entire range of values of the thermal-phonon lifetime. When the acoustic period is long compared to the thermal-phonon lifetime, our result reduces to the formula obtained by Akheiser only for a certain class of mechanisms scattering the thermal phonons (e.g., normal three-phonon processes). For other scattering mechanisms (e.g., random point defects), we obtain a qualitatively different result which reflects the fact that the acoustic phonon now interacts most strongly with thermal phonons of energy considerably below $\mathrm{kT}$.