Volume dependence of self consistent phonon energies

The integral equation for the Gruneisen parameter that governs the volume dependence of self consistent phonon energies is solved numerically for a model approximating to solid Ar at 0 and 80K and Ne at 0 and 23K. The equation is solved for several wave vectors that lie in the irreducible portion of the face centred cubic lattice's Brillouin zone. Exact solutions are compared with an approximation suggested by Horner (1967) and by Gotze and Michel (1968). For Ar at 0 and 80K respectively the true solution is found to be about 91 and 64% of the approximate one. For Ne at 0 and 23K respectively the corresponding numbers are 76 and 63%. The implication of this result for the calculation of self consistent elastic constants is discussed briefly.