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

TL;DR

This paper analytically and numerically investigates the minimality of various three-dimensional lattice structures for Riesz and Lennard-Jones energies, revealing phase transitions and the conditions under which specific lattices are optimal.

Contribution

It provides new evidence for the global minimality of BCC lattices for certain Riesz potentials and confirms FCC as the ground state for Lennard-Jones energies, also exploring phase transitions among lattices.

Findings

BCC lattice is globally minimal for s<0 in Riesz energies.

FCC lattice is the ground state for Lennard-Jones energies with exponents 3<m<n.

Phase transitions between FCC, SH, and HCP structures are observed as density varies.

Abstract

The Riesz potential $f_s(r)=r^{-s}$ is known to be an important building block of many interactions, including Lennard-Jones type potentials $f_{n,m}^{\rm{LJ}}(r):=a r^{-n}-b r^{-m}$, $n>m$ 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 with exponents $3<m<n$, 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 $\Delta$ between the inter-layer distance $d$ and the lattice parameter $a$ is studied as $V$ increases. In the critical region of exponents $0<m<n<3$, the SH phase with an extreme value of the anisotropy parameter $\Delta$ dominates. If one limits oneself to rigid lattices, the BCC-FCC-HCP phase diagram is found. For $-2<m<n<0$, the BCC lattice is the only energy minimizer. Choosing $-4<m<n<-2$, the FCC and SH latices become minimizers.