On the cubic and hexagonal close-packed lattices

Abstract

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.

Cyril Domb: A Personal View and Appreciation

“…With his Oxford student Hugh Barron, inspired by a polymorphic fcc-to-hcp lattice transition then recently discovered in solid helium by Dugdale and Simon, he discussed the corresponding zero-point quantum-mechanical dynamics [18]. With Dugdale he later wrote an instructive review of solid helium and its phases [19] and, more generally, of the theory of melting [20].…”

“…First, in terms of the variable z = exp(−2J/k B T ), which vanishes when T → 0, he undertook the calculation of the exact expansion coefficients of M 0 (T ) to order z 18 . Then in 1949 [7], from an insightful analysis of the many exact coefficients and an intelligent asymptotic extrapolation, he concluded that β was close to or somewhat less than 0.16.…”

100 Years of the Lennard-Jones Potential

“…If we sum over all the interactions between the atoms in a (Bravais) lattice using an ( n , m )-LJ potential, we obtain the following expression for the cohesive energy for a monatomic lattice: ,, E coh = n m italicϵ 2 ( n − m ) [ L n n true( r normale R true) n − L m m true( r normale R true) m ] where the so-called lattice sums L n can be defined such that the distance R reflects the nearest-neighbor distance in the crystal. The lattice sums L n contain expressions for quadratic forms or functions and depend on various cell parameters, i.e., L n ( x i ) where x i are the lattice constants (and Wyckoff positions in the case of multilattices). , Lattice sums for inverse-power potentials are related to the more general Epstein zeta function: ,− scriptZ ( c ; B ; u⃗ , v⃗ ) = ∑ ′ z ⃗ ∈ Z N normale 2 π normali u⃗ · B z⃗ | B...…”

“…Lattice Sums. If we sum over all the interactions between the atoms in a (Bravais) lattice using an (n, m)-LJ potential, we obtain the following expression for the cohesive energy for a monatomic lattice: 3,77,78…”

Influence of an Anharmonicity and Electron–Phonon Interaction on a Frequency Spectrum of Crystals with the Hexagonal Close-Packed Lattice

“…В роботах [102][103][104] досліджувалися деякі властивості спектру частот в зв'язку з задачами тепломісткости, стійкости ґратниці та нульової енергії. Проте, простота структури «компенсується» труднощами одержання хороших монокристалів, які б можна було досліджувати експериментально.…”