Minimizing lattice structures for Morse potential energy in two and three dimensions

Abstract

We investigate the local and global optimality of the triangular, square, simple cubic, facecentred-cubic (FCC), body-centred-cubic (BCC) lattices and the hexagonal-close-packing (HCP) structure for a potential energy per point generated by a Morse potential with parameters (α, r 0 ). In dimension 2 and for α large enough, the optimality of the triangular lattice is shown at fixed densities belonging to an explicit interval, using a method based on lattice theta function properties. Furthermore, this energy per point is numerically studied among all twodimensional Bravais lattices with respect to their density. The behaviour of the minimizer, when the density varies, matches with the one that has been already observed for the Lennard-Jones potential, confirming a conjecture we have previously stated for differences of completely monotone functions. Furthermore, in dimension 3, the local minimality of the cubic, FCC and BCC lattices are checked, showing several interesting similarities with the Lennard-Jones potential case. We also show that the square, triangular, cubic, FCC and BCC lattices are the only Bravais lattices in dimensions 2 and 3 being critical points of a large class of lattice energies (including the one studied in this paper) in some open intervals of densities, as we observe for the Lennard-Jones and the Morse potential lattice energies. More surprisingly, in the Morse potential case, we numerically found a transition of the global minimizer from BCC, FCC to HCP, as α increases, that we partially and heuristically explain from the lattice theta functions properties. Thus, it allows us to state a conjecture about the global minimizer of the Morse lattice energy with respect to the value of α. Finally, we compare the values of α found experimentally for metals and rare-gas crystals with the expected lattice ground-state structure given by our numerical investigation/conjecture. Only in a few cases does the known ground-state crystal structure match the minimizer we find for the expected value of α. Our conclusion is that the pairwise interaction model with Morse potential and fixed α is not adapted to describe metals and rare-gas crystals if we want to take into consideration that the lattice structure we find in nature is the ground-state of the associated potential energy.

Optimal and non-optimal lattices for non-completely monotone interaction potentials

“…We have already studied the above kind of questions for some more specific choices of f and in some related problems, in [12,4,6,11,5,8], again with special emphasis on "physical" dimensions d ∈ {2, 3}. The first author and Knüpfer have treated the case of two-dimensional interaction of masses located on lattice sites in [9,7], and the second author and Serfaty have treated the case of Jellium-type energies in 2 -dimensions for power-law interactions in [54].…”

An Extremal Property of the Hexagonal Lattice

“…Problems in this direction have been considered e.g. in [1,2,4,5,6,7,13,14,17,19,23]. These problems tend to be fairly difficult: even the special case of f being a Gaussian is not yet fully understood (see [3,12]).…”

Minimal Soft Lattice Theta Functions

“…α ≤ 1). Corollary 5.7 and the fact that the only three-dimensional "volume-stationary" lattices are Z 3 , D 3 and D 3 * (see [10,Section 3]) supports this conjecture for the soft lattice theta function. and D * 3 (resp.…”