Anharmonic Free Energy of Crystals at High Temperatures

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Abstract
The general expressions for the high-temperature contributions to the anharmonic free energy at constant volume are simplified for the case of two-body forces for lattices with one or two atoms per unit cell. The quadratic, cubic, and quartic potential-energy coefficients are derived for any lattice for the case of two-body central forces. Accurate calculations of the free-energy contributions, for face-centered cubic and hexagonal close-packed lattices, are described. The calculations are based on two-body central-force interactions, with various ranges of the forces, represented by a Lennard-Jones form. The nearest-neighbor distance was fixed at the value which minimizes the static lattice potential energy. The results for the two lattice types are quite similar, and show a negative anharmonic contribution to the high-temperature specific heat at constant volume. A very simple method of approximating the complicated free-energy expressions is formulated in general and evaluated for the same cases for which accurate calculations are presented. This approximation is based on the replacement of each dynamical matrix of the harmonic lattice dynamics problem by a constant multiple of the unit matrix. Previously published approximations are also compared to the present results, and it is concluded that the approximation developed here is the best one currently available in terms of accuracy and simplicity.

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