From bcc to fcc: interplay between oscillating long-range and repulsive short-range forces

TL;DR

This paper investigates how oscillating long-range and repulsive short-range forces influence the ground state configurations of particles in continuous space, revealing transitions between bcc and fcc lattices with changing density.

Contribution

It extends previous work by characterizing ground states of combined pair interactions with specific Fourier and finite-range properties in arbitrary dimensions.

Findings

Ground states are unique Bravais lattices at certain densities.

Adding short-range interactions reduces degeneracy of ground states.

Transitions between bcc and fcc lattices occur at specific densities.

Abstract

This paper supplements and partly extends an earlier publication, Phys. Rev. Lett. 95, 265501 (2005). In $d$-dimensional continuous space we describe the infinite volume ground state configurations (GSCs) of pair interactions $\vfi$ and $\vfi+\psi$, where $\vfi$ is the inverse Fourier transform of a nonnegative function vanishing outside the sphere of radius $K_0$, and $\psi$ is any nonnegative finite-range interaction of range $r_0\leq\gamma_d/K_0$, where $\gamma_3=\sqrt{6}\pi$. In three dimensions the decay of $\vfi$ can be as slow as $\sim r^{-2}$, and an interaction of asymptotic form $\sim\cos(K_0r+\pi/2)/r^3$ is among the examples. At a dimension-dependent density $\rho_d$ the ground state of $\vfi$ is a unique Bravais lattice, and for higher densities it is continuously degenerate: any union of Bravais lattices whose reciprocal lattice vectors are not shorter than $K_0$ is a GSC. Adding $\psi$ decreases the ground state degeneracy which, nonetheless, remains continuous in the open interval $(\rho_d,\rho_d')$, where $\rho_d'$ is the close-packing density of hard balls of diameter $r_0$. The ground state is unique at both ends of the interval. In three dimensions this unique GSC is the bcc lattice at $\rho_3$ and the fcc lattice at $\rho_3'=\sqrt{2}/r_0^3$.