Formation of phonon-focusing caustics in crystals.

Abstract

A great variety of caustics are observed in the phonon images of crystals. These structures are the result of phonon focusing, i.e., the crowding together in direction of phonon ray vectors that is brought about by elastic anisotropy. Many of the elementary catastrophes can be identified in these caustics. It is argued in this paper that a more penetrating understanding of phonon caustics is arrived at if the ratios of the elastic constants are regarded as control parameters supplementing the two that arise in the specification of the direction of the phonon flux. In this way one can explain the presence in phonon images of slightly unfolded sections of higher-order catastrophes such as the butterfly and hyperbolic umbilic. The main thrust of this paper is to draw attention to a variety of extraordinary caustics occasioned by the particular form of the elastic wave equation for crystals, and by the constraints of symmetry and degeneracy between the different phonon branches. Examples are given of the global development of caustics, and the phenomenon of external conical refraction and its unfolding under a symmetry-breaking perturbation is treated. The close link between degeneracy and the existence of caustics is explored. Conical, tangential, line, surface, and triple degeneracies are considered, and the transformations brought about in the caustics by perturbations that change the form of the degeneracy, are described. These findings suggest a route by which phonon imaging could be used in the study of phase transitions and for measuring certain hard-to-obtain elastic constants. It is anticipated that these results should also assist in the classification of the phonon-focusing patterns of lower-symmetry crystals.