Theory of the Compressibility of SolidHe4andHe3at 0°K

Abstract

A quantum mechanical variational method is used to calculate several properties of a close-packed lattice phase of solid ${\mathrm{He}}^{4}$ and ${\mathrm{He}}^{3}$ at 0\ifmmode^\circ\else\textdegree\fi{}K. The method consists in expressing the expectation value of the interatomic potential energy, with respect to a Heitler-London trial wave function, as a series of powers of the mean square deviation of an atom from its lattice site. Due to the small mass of a helium atom this mean square deviation is relatively large and the series converges slowly. Three sets of numerical results are obtained by truncating the series after the first, second, and third term, respectively. A comparison of these results with the experimental data shows that the final results, i.e., after minimizing $〈H〉$ with respect to the variational parameter, converge much faster than the expectation value series itself. The results include values for: cohesive energy, sound velocity, compressibility, Debye temperature, and Gr\"uneisen constant. The calculations are repeated for a body-centered cubic lattice, and no indication of a crystallographic phase transition is found.