Phonon Reflection from Crystal Interfaces and the Kapitza Problem

We have used the heat pulse technique to study phonon reflection from sapphire-vacuum and sapphire-liquid helium interfaces. The high resolution data presented here show more structure than has been observed in previous experiments of this type. In order to interpret the complex time-of-flight spectra, the problem of the reflection of elastic waves in an anisotropic medium is analyzed in detail. The analysis shows that there are, in general, nine phonon reflection processes, each with a different time of flight, which transfer energy from heater to detector via a single reflection. Iterative computer calculations are necessary to establish the trajectory of energy flow and the arrival time for each channel. The agreement between calculated and experimentally observed times of flight is very good. Although the sharp features in the reflection signal due to specular (k|| conserved) processes can be explained using anisotropic elastic theory, approximately half the energy which reaches the detector arrives via non-specular channels. The non-specular scattering, which may be due to surface roughness, gives rise to broad features in the signal. The main difference between crystal-vacuum and crystal-helium reflection signals is that the non-specular signal is much smaller for the helium covered surface. In contrast to previous works, we find that the specular signal is not affected by helium. Apparently, the non-specular processes are involved in the anomalous Kapitza conductance. In some crystallographic orientations of heater and bolometer, the non-specular signal is particularly large. The orientational dependence of the diffuse scattering is due to the extreme anisotropy of energy flow in crystals, an effect which is known as phonon focusing. We develop a new method of analyzing phonon focusing based on an asymptotic analysis of the phonon Green's function. Geometric arguments are used to show that certain singularities in the acoustic field called caustics can be expected in most crystals. The general features of caustics can be predicted using results from mathematical catastrophe theory. The caustics in sapphire were located by numerical calculation, and used to explain the results of several experiments.