Minimization of energy per particle among Bravais lattices in R^2 : Lennard-Jones and Thomas-Fermi cases

TL;DR

This paper investigates the minimal energy configurations of Bravais lattices in two dimensions for Lennard-Jones and Thomas-Fermi models, showing triangular lattices are optimal at high density for Lennard-Jones and always for Thomas-Fermi.

Contribution

It proves the optimality of triangular lattices for Lennard-Jones at high density and for Thomas-Fermi models, using number theory techniques.

Findings

Triangular lattice minimizes Lennard-Jones energy at high density.

Triangular lattice is always optimal for Thomas-Fermi energy.

Different lattice structures are optimal at different densities for Lennard-Jones.

Abstract

We study the two dimensional Lennard-Jones energy per particle of lattices and we prove that the minimizer among Bravais lattices with sufficiently large density is triangular and that is not the case for sufficiently small density. We give other results about the global minimizer of this energy. Moreover we study the energy per particle stemming from Thomas-Fermi model in two dimensions and we prove that the minimizer among Bravais lattices with fixed density is triangular. We use a result of Montgomery from Number Theory about the minimization of Theta functions in the plane.