The theory of Barlow packings: Basic properties and cohesive energies from exact lattice summations within the sticky hard-sphere model

Abstract

The theory of periodic Barlow multi-lattices (X1X2…XN)∞ with Xi ∈ {A, B, C} and Xi ≠ Xi+1 of stacked two-dimensional hexagonal close-packed layers is presented and used to derive exact lattice sum expressions in terms of fast converging Bessel function expansions for inverse power potentials. We describe in detail the mathematical properties of Barlow sphere packings and demonstrate that only two basic lattice sums are required to describe all periodic packings. For the sticky hard-sphere model with an attractive inverse power law potential, we find a linear correlation between the cohesive energies of different Barlow packings and the face-centered cubic packing fraction. We introduce an efficient algorithm for enumerating the unique periodic Barlow sequences for any given period N. The theory and lattice sums introduced here pave the way for the future treatment of Barlow multi-lattices.