Theory of Quantum Crystals

The ground-state properties of quantum crystals (crystals of the isotopes of He and ${\mathrm{H}}_2$) are studied by means of a variational calculation using cluster-expansion techniques. The cluster expansion for such a crystal is derived in a very general form and applied to a trial wave function that is assumed to be a product of one- and two-particle functions. The energy minimum is calculated by first truncating the expansion to retain only the one-particle and part of the two-particle term. In this form the calculation can be viewed as a Hartree calculation in which the true interaction is replaced by an appropriate effective interaction. The validity of the truncation is assessed by computing all of the leading correction terms. Calculations of the ground-state energy, pressure, and compressibility for the bcc and hcp structures of both ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ and ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ are presented as a function of molar volume. Although the calculated energies are too high, they represent a considerable improvement over previous work and the calculated pressures and compressibilities agree with experiment to about 10%. Tables of the variational parameters which give the minimum energy are presented. A discussion of the accuracy of the approximations used to treat the correlations in the system is given.