On the Cubic and Hexagonal Close-Packed Lattices

• Published in: Proceedings of the Royal Society of London. Series A, Mathematical and physical sciences Mathematical and physical sciences, 1955-02, Vol.227 (1171), p.447-465
• Main Authors: Barron, T. H. K., Domb, Cyril
• Format: Article
• Language: English
• Subjects: Approximation; Close packed lattices; Crystal lattices; Crystals; Cubic crystals; Cubic lattices; Energy; Helium; Mathematical lattices; Molecules
• ISSN: 1364-5021; 0080-4630; 1471-2946; 2053-9169
• DOI: 10.1098/rspa.1955.0023

The present paper was stimulated by the discovery by Dugdale & Simon (1953) of a polymorphic transition in solid helium. A discussion is given of the relative stability of the cubic and hexagonal close-packed lattices assuming central forces of the Mie—Lennard-Jones type. Taking static lattice energy alone into account the usual laws of force favour the hexagonal close-packed lattice, the difference in energy being about 0·01%. However, lattice dynamics indicates that the equivalent Debye Θ at the absolute zero is smaller for the cubic lattice, the difference being about 1%. Hence, ignoring zero-point energy, we should expect a transition to occur from hexagonal to cubic at an elevated temperature. The estimated temperature and energy of the transition are of the same order of magnitude as those observed experimentally in solid helium. An estimate is made of the effect of zero-point energy; the results can be applied with confidence to the heavier inert gases, but can only be considered as giving a qualitative indication for helium, since anharmonic effects are of great importance in this case. For the other inert gas solids it is concluded that the experimentally observed cubic close-packing at all temperatures must be due to non-central forces.