New Approach to Lattice Dynamics Applied to SolidHe3at 0°K

Abstract

The expansion of a crystal potential in terms of a set of three-dimensional polynomials orthogonalized with respect to the weight function $f^2({\overrightarrow{\mathrm{r}}}_1, {\overrightarrow{\mathrm{r}}}_2, \cdots ,{\overrightarrow{\mathrm{r}}}_N)\exp \left[\right({\overrightarrow{\mathrm{r}}}_i-{\overrightarrow{\mathrm{R}}}_i\left)·{\mathrm{G}}_{\mathrm{ij}}·\right({\overrightarrow{\mathrm{r}}}_j-{\overrightarrow{\mathrm{R}}}_j\left)\right]$ is shown to be a logical generalization of the previously introduced self-consistent harmonic approximation which is particularly appropriate for highly anharmonic systems. Explicit expressions for the calculation of the ground-state energy and phonon spectrum of a crystal at 0°K are given and certain numerical results for solid ${\mathrm{He}}^3$ at 0°K are presented.