Temperature Dependence of Microwave Phonon Attenuation

Abstract

Measurements of the temperature dependence of the attenuation of 9-Gc/sec phonons are presented. They include measurements on crystals of quartz, CdS, GaAs, Ge, Si, Ca${\mathrm{F}}_2$, ${\mathrm{Al}}_2$ ${\mathrm{O}}_3$, and MgO in several directions and modes of polarization. The attenuation in most of the materials is proportional to $T^n$, where the average value of $n=4.1\mathrm{}\pm 1.2$ for the fast transverse waves, 4.0±1.7 for the slow transverse waves, 4.8±1.4 for the longitudinal waves. For MgO and ${\mathrm{Al}}_2$ ${\mathrm{O}}_3$ the attenuation is proportional to the frequency of the sound between 3 and 9 Gc/sec. An empirical correlation of the data is that the attenuation of 9-Gc/sec phonons is 3 dB/cm when $\left(\frac{T}{\Theta }\right)\approx 0.1$, where $\Theta =\mathrm{Debye}\mathrm{temperature}\mathrm{of}\mathrm{the}\mathrm{crystal}$. Values of the third-order elastic constants are used to make absolute comparisons of the theory of attenuation by three-phonon processes with the data. The agreement is good in some cases but not in others.