Phase Velocity of High-Frequency Phonons in Glass below 1 K

Precision measurements of the phase velocity of sound waves propagating in fused silica at temperatures between 0.03 and 1.0 K and at frequencies between 0.3 and 2.0 GHz reveal deviations from a $\ln T$ temperature dependence when $\frac{\hslash \omega }{\mathrm{kT}}\ll 1$ and the existence of a velocity minimum when $\frac{\hslash \omega }{\mathrm{kT}}\cong 2.2$. These results are analyzed in terms of the two-level tunneling model and yield quantitative information on the deviation from energy uniformity of the density of scattering states.