Three-dimensional lattice ground states for Riesz and Lennard-Jones type energies

Abstract

The Riesz potential fs(r) = r −s is known to be an important building block of many interactions, including Lennard-Jones type potentials f LJ n,m (r) := ar −n − br −m , n > m that are widely used in Molecular Simulations. In this paper, we investigate analytically and numerically the minimizers among three-dimensional lattices of Riesz and Lennard-Jones energies. We discuss the minimality of the Body-Centred-Cubic lattice (BCC), Face-Centred-Cubic lattice (FCC), Simple Hexagonal lattices (SH) and Hexagonal Close-Packing structure (HCP), globally and at fixed density. In the Riesz case, new evidence of the global minimality at fixed density of the BCC lattice is shown for s < 0 and the HCP lattice is computed to have higher energy than the FCC (for s > 3/2) and BCC (for s < 3/2) lattices. In the Lennard-Jones case, the ground state among lattices is confirmed to be a FCC lattice whereas a HCP phase occurs once added to the investigated structures. Furthermore, phase transitions of type "FCC-SH" and "FCC-HCP-SH" (when the HCP lattice is added) as the inverse density V increases are observed for a large spectrum of exponents (n, m). In the SH phase, the variation of the ratio ∆ between the inter-layer distance d and the lattice parameter a is studied as V increases.

On minima of difference of theta functions and application to hexagonal crystallization

“…Many physical, chemical and number theoritic problems can be formulated to the following minimization problem on lattices: (1.1) See e.g. [2,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,28,29,38,39,36,34,40,42,43,44,45,46,47,4]. The summation ranges over all the lattice points except for the origin 0 and the function f denotes the potential of the system.…”