Attenuation of Hypersound by Collinear Phonon Interactions in Quartz

Abstract

Low-temperature measurements of the attenuation of microwave phonons of longitudinal polarization along the $x$ axis and of transverse polarization along the $\mathrm{ac}$ axis of natural quartz at 1.25 and 2.1 GHz are compared with calculations for a dispersive anisotropic continuum. By separating the analysis of the data into two temperature zones whose extent is determined by the relative magnitudes of $\ensuremath{\Omega}\ensuremath{\tau}$ and of the dispersion factor $\ensuremath{\beta}=\frac{v}{c}$, it was possible to obtain, respectively, estimates for the generalized Gr\"uneisen parameters when ${(1+\ensuremath{\beta})}^{\ensuremath{-}1}<\ensuremath{\Omega}\ensuremath{\tau}<{(1\ensuremath{-}\ensuremath{\beta})}^{\ensuremath{-}1}$ and for the strength of the thermal-phonon interactions at low temperatures when $\ensuremath{\Omega}\ensuremath{\tau}>{(1\ensuremath{-}\ensuremath{\beta})}^{\ensuremath{-}1}$. Very good agreement is obtained for the temperature and frequency dependence of the absorption coefficient by assuming that the transverse-wave attenuation is due to collisions with collinear transverse phonons. For longitudinal waves, evidence is in accordance with the assumption that the main contribution to the attenuation is from collisions with longitudinal and transverse thermal phonons. We find however, that the former process becomes increasingly predominant as the temperature is reduced. The experimental and theoretical Gr\"uneisen parameters agree to better than a factor of 5. For large $\ensuremath{\Omega}\ensuremath{\tau}$ calculations predict a reduction in the temperature dependence of the absorption with moderate concentration of point impurities.