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

Abstract

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 theoretical^3,4 and computational^5–7 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.