Helmholtz Free Energy of an Anharmonic Crystal toO(λ4)

All of the diagrams contributing to the Helmholtz free energy to $O\left({\lambda }^4\right)$, for a crystal in which every atom is on a site of inversion symmetry, have been evaluated. The expressions given are valid for finite temperatures, and separate expressions are given for all cases where the double occurrence of phonon lines is important. Numerical calculations have been carried out for all the diagrams for a nearest-neighbor central-force model of a face-centered-cubic lattice in the high-temperature limit and in the leading-term approximation. Two separate techniques, a scanning method and the plane-wave expansion method, were used for the evaluation of each diagram. When applied to a Lennard-Jones potential it is found that none of the diagrams makes a negligibly small contribution and that the convergence of the perturbation expansion appears poor for $T>\mathrm{}\frac{1}{3}{T}_m$.