Entropy difference between the face-centred cubic and hexagonal close-packed crystal structures

SPHERES can be stacked into two close-packed crystalline arrangements, face-centred cubic (f.c.c.) and hexagonal closed-packed (h.c.p.), which have identical close-packed volumes1 and very similar equations of state2. But they are structurally distinct, implying that they might have different thermodynamic properties and stabilities. Finding a difference in free energy between the two structures has been the objective of much theoretical3,4 and computational5–7 effort, but without a conclusive resolution. Here I report that a significant difference in the pressure–volume (P–V) behaviour can be detected in the vicinity of a mechanical instability point within a single-occupancy cell model8 of these packings. This model provides an exact thermodynamically reversible path between the two structures, and so the P–V isotherms can be integrated to obtain the Gibbs free-energy difference. I find that the f.c.c. phase is more stable by around 0.005RT per mol (where R is the universal gas constant); as the enthalpy difference is negligible, this implies that the entropy difference is of the order of 0.005R for all temperatures up to the melting point.