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Entropy of hard spheres in the close-packing limit

The Helmholtz free energies of the face-centred cubic (FCC) and hexagonal close packed (HCP) hard-sphere solids in the close-packing limit have been evaluated using two different approaches based on the Einstein crystal method. Different system sizes and orientations of the crystal with respect to the simulation box have been investigated, both methods giving free energies that are consistent within statistical uncertainty. Our results show that for a given orientation of the crystal and system size, the FCC crystal is always slightly more stable than the HCP, the free-energy difference remaining practically constant with the number of particles up to the thermodynamic limit. In agreement with previous calculations, it is found that the free-energy difference between the HCP and FCC crystals at close packing in the thermodynamic limit is 0.001 164(8) NkBT.