Kirkwood-Monroe approximation for quantum solids

The energy of a Boltzmann quantum solid is calculated using the Kirkwood-Monroe liquid ansatz for the solid pair correlation function. Approximate liquid pair correlation functions generated from the Born-Bogoliubov-Green-Kirkwood-Yvon, Kirkwood-superposition-approximation (BBGKY-KSA) equation and from the hypernetted-chain (HNC) equation are used to calculate the energy of solids with $\frac{{\hslash }^2}{m\epsilon {\sigma }^2}=\eta$ of 0.1815 (${}_{}{}^4{}_{}{}^{}\mathrm{He}$) and 0.200. The calculations are done over the physical density range ($0.024<~\rho <~0.035$ part. /${\mathrm{\AA }}^3$). The energy of the liquid is also calculated. Results are obtained for solid and liquid energies which depend critically on the choice of liquid correlation function and on $\eta$. For $\eta =0.1815\mathrm{}$ and the BBGKY-KSA pair correlation function the system has a liquid → solid → liquid phase transition, whereas for $\eta =0.1815\mathrm{}$ and the HNC pair correlation function the system can exist as a zero-pressure solid.