Large-wave-vector phonons in highly dispersive crystals: Phonon-focusing effects

The anisotropic propagation or the focusing of large-wave-vector acoustic phonons in highly dispersive crystalline Ge is studied in detail. Our analysis is entirely based on an elaborate Born---von K\'arm\'an model of the lattice dynamics which is constructed so as to reproduce the observed phonon dispersion curves very accurately. Effects of the lattice dispersion upon the phonon focusing are found to be discernible above 0.3 THz and become drastic at frequencies higher than 1 THz. These observations are made by studying complementarily the frequency dependences of the following objects: the shape of constant-frequency surfaces, the locations of phonon caustics, the distributions of phonons in real space, the angular dependences of phonon intensity, and the structure of group-velocity surfaces. A brief discussion is also given of the effects of isotopes which act to damp significantly the ballistic phonon intensity at frequencies in the 1-THz range.