Self-Consistent Theory of Second-Order Elastic Constants with an Application to Noble-Gas Crystals

Formal expressions for the first two strain derivatives of the first-order self-consistent free-energy density are rederived, and presented in a form suitable for numerical computation. The first strain derivative is the first-order self-consistent stress tensor and the second derivatives are the corresponding elastic constants. Because of the self-consistency condition, these elastic constants contain thermally averaged third-and fourth-order force constants. Special reference is made to an approximation first introduced by Horner in 1967. The expressions are applied to solid Ne, Ar, Kr, and Xe using a (12-6) Mie-Lennard-Jones potential. Calculations are carried out for the temperature range 0°K to their respective melting points at zero pressure. The calculations are presented for the 0°K volume, the experimental volume at zero pressure, and the volume produced by first-order self-consistent theory (SC). The volume effect is often large. However, at the same volume, the bulk moduli derived from $F_{\mathrm{ISC}}$ and $F_{\mathrm{SC}}$ differ by at most a few percent. This is taken to indicate the probable accuracy of our results.