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

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 $\vfi+ψ$, 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 $r_0\le {\gamma }_d/K_0$, 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 $\sim \cos (K_{0}r+\pi /2)/r^3$ 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 $({\rho }_d,{\rho }_d^{\prime })$, where ${\rho }_d^{\prime }$ 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 ${\rho }_3^{\prime }=\sqrt{2}/r_0^3$.

Published version

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