The zero-point vibrational energy of cubic crystals

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Abstract
The zero-point vibrational energies of the three cubic lattices are calculated using the Bethe approximation. The harmonic and the quartic anharmonic energies are in good agreement with the results of standard lattice dynamical methods. However, the cubic and some of the non-leading anharmonic terms appear to be considerably overestimated. The results are applied to the inert gas solids to investigate the following models: (i) central nearest-neighbour interactions only; (ii) central all-neighbour interactions using a Lennard-Jones m-6 potential function; (iii) central all-neighbour and non-central interactions (the latter using the potential function given by Axilrod and Teller). Non-nearest-neighbour and non-central interactions were taken into account by using the Einstein approximation while central nearest-neighbour interactions were calculated using the Bethe approximation. It is found that the approximations (ii) and (iii) decrease the harmonic energy and increase the anharmonic energy with respect to approximation (i). The results indicate that the approximation made in (i) is accurate to approximately 5% for calculating the harmonic and 4% for calculating the anharmonic energies of neon. For xenon the accuracy is approximately 11% and 15% respectively. It is therefore concluded that calculations of the properties of the inert gas solids, especially the heavier gases, should include non-central forces and all-neighbour interactions.

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