Phonon conduction in elastically anisotropic cubic crystals

Striking differences in the boundary-scattered phonon conductivity and effective phonon mean free path are predicted along the principal axes of cubic crystals. The results are shown to be the result of phonon focusing arising from elastic anisotropy. Normalized curves of boundary-scattered effective phonon mean free path and phonon conductivity have been calculated for samples of square cross section as a function of the elastic anisotropy, $A=\frac{2C_{44}}{(C_{11}-C_{12})}$, and the elastic ratio, $\frac{C_{12}}{C_{11}}$. Normalized curves of phonon specific heat and effective velocity have been calculated as a function of the same variables. The effective phonon mean free path has also been calculated as a function of the sample side-face—thermal-length ratio, $\frac{D}{L}$. Anisotropies of more than 50% are possible for different rod axes. Silicon and calcium fluoride, materials in which this anisotropy was first reported, are shown to be very favorable materials to demonstrate this anisotropy. For silicon and calcium fluoride samples of rectangular cross section the thermal conduction is shown to depend upon the crystallogrphic orientation and width ratio of the side faces for samples with the same $〈110〉$ rod axis. Results are expressed in a convenient form for predicting the phonon conductivity of elastically anisotropic crystals, given the linear dimensions, the density, and the elastic constants.