Re-examination of the Lattice Dynamics of White Tin Using a Modified Axially Symmetric Model

Experimental measurements of both the specific heat and the Debye-Waller factor for (white) Sn suggest the presence of low-lying phonon branches. In addition, recent experiments by Rowell et al. displayed structure in the Sn-Sn superconductors tunneling characteristic curve associated with very low-energy Van Hove singularities in the Sn vibrational spectrum. A previous theoretical calculation of the Sn phonon spectrum using the elastic constants of Mason and Bömmel does not predict the existence of such low-lying phonon branches. A subsequent calculation based on the elastic data of Rayne and Chandrasekhar, on the other hand, clearly indicates the presence of a very low-energy acoustic branch along the [110] direction. However, in this calculation complete elastic consistency was not possible because of the constraint imposed by the $A-S$ (axially symmetric) lattice dynamics model. Consequently, we have re-examined the lattice vibrational spectrum of Sn on the basis of a generalized $A-S$ lattice dynamics model which allows complete elastic consistency to be obtained. The specific heat, the magnitude of the Debye-Waller factor and the low-energy structure in the tunneling experiments calculated from the dispersion curves obtained from our modified A-S model are in quantitative agreement with experimental observations. The anisotropy factor $\epsilon$ of the Debye-Waller factor is increased from 1.2 to 1.56 and remains in disagreement with experiment. It appears to us that with the present elastic data it is not possible to obtain a mean-square displacement larger in the $z$ direction than that in the $x$ direction. Furthermore, we conclude that the Mason and Bömmel elastic data are incapable of explaining the present experimental data on white tin.