On energy ground states among crystal lattice structures with prescribed bonds

Abstract

We consider pairwise interaction energies and we investigate their minimizers among lattices with prescribed minimal vectors (length and coordination number), i.e. the one corresponding to the crystal's bonds. In particular, we show the universal minimality -i.e. the optimality for all completely monotone interaction potentials -of strongly eutactic lattices among these structures. This gives new optimality results for the square, triangular, simple cubic (SC), face-centred-cubic (FCC) and body-centred-cubic (BCC) lattices in dimensions 2 and 3 when points are interacting through completely monotone potentials. We also show the universal maximality of the triangular and FCC lattices among all lattices with prescribed bonds. Furthermore, we apply our results to Lennard-Jones type potentials, showing the minimality of any universal minimizer (resp. maximizer) for small (resp. large) bond lengths, where the ranges of optimality are easily computable. Finally, a numerical investigation is presented where a phase transition of type "square-rhombic-triangular" (resp. "SC-rhombic-BCC-rhombic-FCC") in dimension d = 2 (resp. d = 3) among lattices with more than 4 (resp. 6) bonds is observed.

Optimality of the triangular lattice for Lennard-Jones type lattice energies: a computer-assisted method

“…the Laplace transform of a nonnegative Radon measure), by Montgomery in [32] where he showed the optimality of the triangular lattice for any Gaussian function f (r) = e −αr 2 , α > 0. Concerning the Lennard-Jones type potentials f (r) = ar −α −br −β , several results have been derived in [2,3,5,9,10], especially the optimality of the triangular lattice among lattices with fixed high density as well as the minimality of a triangular lattice for E f , with small exponents, among all possible two-dimensional lattices. We have also shown in [9] that the type (or "shape") of the global minimizer of E f is independant of (a, b) ∈ (0, ∞) (see also Theorem 2.6).…”