Helmholtz free energy of an anharmonic crystal to O(λ4). III. Equation of state for the Lennard-Jones solid

Abstract

A complete calculation of the thermodynamic properties of a model of a rare-gas solid with a nearest-neighbor Lennard-Jones interaction has been carried out with use of the (${\lambda }^4$) anharmonic perturbation theory proposed by Shukla and Cowley carried to fourth order in the Van Hove ordering parameter (λ). The results are compared with the Monte Carlo method for the same model potential with three objectives in mind: first, to identify more precisely the breakdown of the lowest order (${\lambda }^2$) theory by showing where the ${\lambda }^4$ terms have an appreciable effect; and, second, to extend the valid temperature range for perturbation theory by including ${\lambda }^4$ terms; and, finally, to indicate the relative importance of the various terms of O(${\lambda }^4$). In addition, finite-temperature calculations were made for argon, krypton, and xenon for comparison with experiment. It is concluded that at high temperatures and at all volumes the Van Hove ordering scheme is accurate and none of the eight contributions of O(${\lambda }^4$) can be omitted on the grounds that it is much smaller than the rest. Among the other schemes [R. C. Shukla and E. R. Cowley, Phys. Rev. B 3, 4055 (1971)] of classifying the terms, such as the first-order self-consistent phonon theory and improved self-consistent phonon theory (ISC), only the ISC scheme produces results which are in better agreement with the Monte Carlo method and with experiment but similar agreement can also be achieved by omitting all but one of the terms of O(${\lambda }^4$), viz., 2h, a four-vertex ladder-type term or by including all the ${\lambda }^4$ terms but excluding the ladder term. As compared with the lowest-order theory (${\lambda }^2$), which is seriously in error above one-quarter of the melting temperature ($T_m$), the theory to O(${\lambda }^4$) converges well up to 40% of $T_m$.