2D Theta Functions and Crystallization among Bravais Lattices

TL;DR

This paper investigates energy minimization among Bravais lattices, proving triangular lattice optimality for certain potentials and densities, and identifying cases where it is not, with implications for crystallization patterns.

Contribution

It establishes conditions under which the triangular lattice minimizes energy among Bravais lattices, including for Lennard-Jones and Yukawa potentials, and introduces new analytical methods.

Findings

Triangular lattice minimizes energy for completely monotonic functions at fixed density.

Counterexamples where triangular lattice is not a minimizer for some convex decreasing potentials.

Global minimality of triangular lattice for specific Lennard-Jones and Yukawa potential parameters.

Abstract

In this paper, we study minimization problems among Bravais lattices for finite energy per point. We prove that if a function is completely monotonic, then the triangular lattice minimizes energy per particle among Bravais lattices with density fixed for any density. Furthermore we give an example of convex decreasing positive potential for which triangular lattice is not a minimizer for some densities. We use Montgomery method presented in our previous work to prove minimality of triangular lattice among Bravais lattices at high density for some general potentials. Finally we deduce global minimality among all Bravais lattices, i.e. without density constraint, of a triangular lattice for some parameters of Lennard-Jones type potentials and attractive-repulsive Yukawa potentials.