Phonon focusing in cubic crystals

Abstract

The nature of the phonon intensity pattern associated with low-temperature heat-pulse propagation is examined theoretically for a wide range of cubic crystals. In general, huge anisotropies in the thermal flux are predicted due to phonon focusing. The phonon-focusing patterns of cubic crystals divide naturally into two major groups, as distinguished by the sign of their anisotropy factor Δ=$C_{11}$/$C_{44}$-$C_{12}$/$C_{44}$-2. Experimental and theoretical intensity patterns of ${\mathrm{CaF}}_2$ and Si in the long-wavelength, dispersionless limit are examined in detail to illustrate the typical focusing-pattern features characteristic of positive and negative Δ. For both regimes, the evolution of particular pattern structures is studied systematically as the elastic constants are varied. Investigation of the flux-intensity pattern and slowness-surface topology for various elastic constants reveals distinct trends in the phonon-focusing structures. Based on this study, we have defined a small number of angular dimensions (one or two for each phonon mode and anisotropy regime) which quantitatively determine the principal directions of singular flux. The variations of these angles are mapped onto the general elastic-parameter space. We thus present a means for quickly determining the angular positions of the principle phonon-focusing caustics for an arbitrary set of elastic constants.