Contribution of anharmonic free energy F 4 to the specific heat of a simple lattice at arbitrary temperature

Abstract

It has been shown recently by Maradudin that the quartic anharmonic contribution F 4 to the free energy of a face-centered cubic lattice can be expressed as the square of a single summation instead of the usual complex double summation over the Brillouin zone provided only nearest neighbor interactions are considered. Using this result we express the specific heat contribution C 4 corresponding to this F 4 in exact form in terms of the reduced vibration spectrum. We then calculate C 4 versus T for arbitrary T between 0°K and 404°K for a nearest-neighbor-only lattice model which roughly represents copper. The results displayed in the form C 4/C v (harmonic) versus T disclose new structural features of the quartic anharmonic specific heat contribution. This curve, which has a sharp minimum at about 0.067 Debye θ ∞ and a broad maximum at about θ ∞/3, is shown together with previous results obtained at 0°K and in the high-T limit by Maradudin and co-workers.