The ultrasonic attenuation of A1,0, along the a axis at low temperatures is explained by taking into account the finite lifetimes of the thermal phonons. The longitudinal attenuation proceeds via unallowed transitions. The slow transverse is normal attenuation. The fast transverse is a mixed case.
IntroductionThe basic processes of low temperature ultrasonic attenuation have been well understood since the early paper by Landau and Rumer [l]. They calculated the loss of energy from the U.S. wave by collisions with thermal phonons. Taking into account all collisions which could conserve energy and momentum they arrived at a T4 temperature dependance for the attenuation.Many anomalous temperature dependances have been observed however where the dependance of the attenuation differs markedly from T4. In addition it was soon realized that attenuation of longitudinal waves could not be explained by the allowed scattering of thermal phonons. By allowed scattering we mean three phonon scattering in which energy and momentum are conserved.Gradually, an understanding of the role of unallowed three phonon scattering developed [2]. In these processes, energy is not strictly conserved but the scattering may take place with some probability due to the uncertainty in the energy of the thermal phonons. This uncertainty is caused by the short lifetime of the thermal phonons. A consideration of the way the thermal phonon lifetime effects the ultrasonic attenuation can explain most of the ultrasonic attenuation anomalies observed. This paper will describe the application of this method to a previously anomalous case of attenuation, that of A1,0, along the a axis at low temperature.
Three-Phonon Scattering in the Presence of Energy UncertaintyBy using simple time dependant perturbation theory, the probability for making a transition from initial state i to final statefper unit time is