Anharmonic Free Energy of Crystals at Low Temperatures and at Absolute Zero

The method of long waves is used to show that the anharmonic contribution to the Helmholtz free energy of crystals has a lowest order temperature dependence of $T^4$ at low temperatures. The anharmonic correction to the low-temperature Debye temperature is obtained in terms of the ratio of the anharmonic to harmonic free energies at low temperature. Approximate expressions are developed for the anharmonic zero-point energy for the case where all atoms in the crystal have the same mass. These expressions are shown to be simply related to the corresponding anharmonic contributions to the high-temperature free energy. Numerical evaluation of these approximations is carried out for fcc and hcp lattices for a model of two-body central forces represented by a Lennard-Jones form. The various contributions to the zero-point energy are compared for these two lattices, and an application to the inert gas crystals shows that the anharmonic contribution is sufficient to make the fcc structure stable at $T=0\mathrm{}$ for Ne and Ar, but not for Kr and Xe. An approximation is developed for the low-temperature anharmonic free energy for the case of one atom per unit cell, and this approximation is evaluated for the fcc lattice for the Lennard-Jones model. The results of the present paper are compared numerically with previously published calculations, and qualitative agreement is found in general.