Anisotropic heat conduction in cubic crystals in the boundary scattering regime

The thermal conductivities of single crystals of silicon and of calcium fluoride have been measured in the temperature range 3-40 degrees K and 2.5-30 degrees K, respectively. For samples in the form of square cross-section rods, the conductivity in the boundary scattering regime was found to depend on the orientation of the rod axis, the variation being as much as 50% for silicon. This anisotropy is accounted for in terms of phonon focusing due to the fact that in elastically anisotropic crystals the phonon phase and group velocities are, in general, not collinear. Casimir's theory of the thermal conduction in the boundary scattering regime has been generalized to include the effects of focusing; the predictions of this generalized theory are in quantitative agreement with the experimental results. It is predicted that anisotropic thermal conductivity in the boundary scattering regime is a general property of elastically anisotropic cubic crystals.