Boundary-limited thermal conduction of crystalline rods oriented near phonon-focusing caustics

•Casimir thermal conduction in anisotropic rods oriented near phonon-focusing caustics is studied theoretically.•In the direction of an external conical refraction caustic the Casimir limit does not exist.•For fold and cusp caustics the Casimir conductivity remains finite.•Analytical expressions describing the Casimir conductivity in the vicinity of caustics are obtained.

We apply the Casimir model for boundary-limited heat conduction to single-crystal rods oriented near phonon-focusing caustics. We show that rods with axes close to the direction of an external conical refraction caustic, a highly degenerate caustic that exists for certain hexagonal crystals, exhibit a thermal conductivity that diverges logarithmically on approaching the caustic. For rods with axes close to the directions of the more generic fold and cusp caustics, the conductivity remains finite, but displays singular behavior with a 1/2- or 1/3-power law falloff with angular deviation from the caustic. Moreover, in the direction of a fold caustic, the Casimir conduction is not necessarily a maximum. Numerical results are presented for zinc, with the quasi-transverse branch providing examples of the external conical refraction and fold caustics, and in a certain sense, also the cusp caustic.