Surface Contribution to the Specific Heat of Crystals at Low Temperatures

Abstract

A lattice dynamical calculation of the low-temperature specific heat of a crystal possessing free surfaces is presented. A pair of adjacent [100] free surfaces is created in a nearest and next-nearest neighbor central force model of a simple cubic crystal by setting all atomic interactions which link the two planes equal to zero. The negatives of these interactions are treated as a perturbation on the Hamiltonian of the uncut crystal. The change in the specific heat of the crystal resulting from the introduction of the pair of free surfaces is expressed as a contour integral over the Green’s function of the perturbed crystal. In the limit of low temperatures, only the long wavelength components of the Fourier transform of the Green’s function are required, and these can be evaluated analytically if the relation on the atomic force constants which corresponds to elastic isotropy is imposed. The result for the surface contribution to the specific heat C s has the form C s = bST 2 , where S is the surface area, T is the absolute temperature, and b is a numerical constant. The value for b obtained in the present calculation differs somewhat from the value obtained earlier by Dupuis, Mazo, and Onsager, who studied a semi-infinite isotropic elastic continuum.