Evaluation of the Free Energy of an Anharmonic Crystal

Abstract

The anharmonic contributions of order ${\eta }^4$ to the free energy of a crystal have been evaluated for a central-force nearest-neighbor model in the Ludwig approximation. The accuracy of the Ludwig approximation in evaluating the Brillouin-zone sums occurring in the expression for the anharmonic contributions to the free energy has been investigated in some detail. It is found that the Ludwig approximation gives exact results for some simple sums. While most of the remaining sums are underestimated by about 18%, the sums occurring in the expression for the second-order quartic anharmonic contribution to the free energy are overestimated by about 20%. The coefficient of the $T^2$ term in the specific heat of lead at high temperatures is also estimated for a Morse potential and is found to be of the same order of magnitude as the experimental value obtained by Leadbetter.