Solid-fluid phase transition of quantum hard spheres at finite temperatures.

Abstract

Path-integral Monte Carlo simulation is used to study a system of distinguishable quantum hard spheres at finite temperatures. Extensive tables of the energy and pressure in the fluid and fcc solid phases are presented along with a careful study of the convergence of these properties with ``time-slice discretization.'' The Helmholtz free energy in the solid phase is computed via a thermodynamic integration that continuously transforms the system into a harmonic Einstein crystal. The free energy in the solid is compared to that computed by the Debye model. Input data for the Debye model is provided by performing Monte Carlo computations of the elastic moduli, ${C}_{11}$, ${C}_{12}$, and ${C}_{44}$ at T=0. The two methods of computing the free energy are in qualitative agreement, with some uncertainty induced by the lack of dispersion in the phonon spectrum, \ensuremath{\omega}(k), in the Debye model. The solid-fluid phase transition is located along three isotherms, that correspond to 4, 10, and 20 K in a simple mapping of the hard-sphere system onto $^{4}\mathrm{He}$. From these data we postulate a generalization of Lindemann's melting criterion for quantum systems at finite temperatures. The hard-sphere free energy and pair distribution functions are used to predict the equations of state and freezing transition in $^{3}\mathrm{He}$ and $^{4}\mathrm{He}$ via a first-order perturbation theory. The liquid-vapor phase transition of $^{4}\mathrm{He}$ at 4 K is located as well. The agreement between these predictions and experiment is very good, except at very high density where a more sophisticated choice of hard-sphere reference system is required.