Surface contribution to the low-temperature specific heat of a cubic crystal

We have applied elasticity theory to determine the surface contribution to the low-temperature specific heat of a cubic crystal bounded by a (001) stress-free surface. The result has the form $\Delta C_{\nu }\mathrm{}\left(T\right)={\mathrm{BST}}^2$, where $S$ is the area of the crystal surface and $T$ is the absolute temperature. The coefficient $B$ is expressed in terms of an integral over the trace of a reduced, dynamical Green's tensor for a semi-infinite cubic elastic medium bounded by a (001) stress-free surface. The reduced Green's tensor is effectively the leading term in the large $k_{\parallel }$ expansion of the Fourier transform, in the coordinates parallel to the surface, of the full Green's tensor for this system, and its elements satisfy a set of nine coupled second-order ordinary differential equations in the coordinate normal to the surface, together with appropriate boundary conditions. These equations have been solved analytically, and the necessary integrals evaluated numerically for a range of values of the elastic moduli $c_{11}$, $c_{12}$, $c_{44}$ of the medium.