The Helmholtz free energy F of the hard-sphere bcc solid is calculated with use of two techniques: the weighted-density-functional approximation (WDA) and Monte Carlo simulations, using the coupling-constant technique of Frenkel and Ladd [J. Chem. Phys. 81, 3188 (1984)] to obtain F. Since the hard-sphere bcc structure is unstable to shear, the WDA calculations are carried out within a class of density profiles which exclude the shear mode and single-occupancy cell constraints are imposed in the simulation studies. The two results for F are in substantial agreement with each other and show that at sufficiently high densities the (shear-stabilized) bcc phase has a free energy lower than that of the liquid but higher than that of the fcc solid, while at lower densities the liquid is lowest in free energy followed by the bcc and then the fcc solid. Also, at fixed density the mean-square displacements of the bcc atoms are larger than those of the fcc atoms as a result of the relative openness of the bcc structure. The WDA results compare very favorably to other density-functional calculations of the bcc phase carried out within the same constrained class of density profiles which either fail to predict the existence of a bcc phase or predict a stable phase only at very high densities and having a free energy larger than even the liquid. In an appendix, we (i) present fcc and bcc free energies using improved liquid-state input data and (ii) allow for non-Gaussian density profiles in the solid. Both alterations yield only slightly lower (<2%) free energies than determined using Percus-Yevick liquid data and Gaussian profiles.

• Received 27 February 1987

DOI:https://doi.org/10.1103/PhysRevA.35.4755