Diffuse reflection of phonons and the anomalous Kapitza resistance

High-resolution time-of-flight phonon-reflection spectra in sapphire crystals are analyzed with the aid of detailed numerical calculations. It is argued that spectra from vacuum interfaces may be understood if about half the incident phonons are assumed to be reflected specularly, the remainder diffusely. A simple model of the effect on spectra of diffuse scattering is presented and compared to experimental data. We show that the specularly reflected signal is unaffected by placing helium at the interface, but that the diffuse part is transmitted almost entirely, accounting for the anomalous Kapitza resistance. It is shown that conventional calculations of phonon focusing are not suitable for interpreting our data, and a new, more correct theory of phonon focusing, aided by formal catastrophe theory, is introduced. We argue that the experimental resolution and theoretical analysis we have presented are essential for understanding phonon-reflection experiments.