Calculations of anharmonic and non-harmonic vibrational properties of a classical solid using a cluster-Bethe model

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Abstract
The model may be viewed both as giving the exact solution to the vibrational properties of a Bethe lattice of coordination number q, and also as giving a variational upper bound on the free energy of a Bravais lattice with coordination q. It does not rely on a perturbation treatment involving a Taylor expansion of the interaction potential. The method is applied to the two-dimensional square net (q=4) with Lennard-Jones 12-6 interactions between nearest-neighbour atoms, and the anharmonic free energy has been calculated by several different numerical methods for a series of values of temperature, T, and lattice parameter, a.