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

Abstract

This paper supplements and partly extends an earlier presentation [Phys. Rev. Lett. 95, 265501 (2005)]. In $d$-dimensional continuous space we describe the infinite volume ground state configurations (GSCs) of pair interactions $\ensuremath{\varphi}$ and $\ensuremath{\varphi}+\ensuremath{\psi}$, where $\ensuremath{\varphi}$ is the inverse Fourier transform of a non-negative function vanishing outside the sphere of radius ${K}_{0}$, and $\ensuremath{\psi}$ is any non-negative finite-range interaction of range ${r}_{0}\ensuremath{\le}{\ensuremath{\gamma}}_{d}∕{K}_{0}$, where ${\ensuremath{\gamma}}_{3}=\sqrt{6}\ensuremath{\pi}$. In three dimensions the decay of $\ensuremath{\varphi}$ can be as slow as $\ensuremath{\sim}{r}^{\ensuremath{-}2}$, and an interaction of asymptotic form $\ensuremath{\sim}\mathrm{cos}({K}_{0}r+\ensuremath{\pi}∕2)∕{r}^{3}$ is among the examples. At a dimension-dependent density ${\ensuremath{\rho}}_{d}$ the ground state of $\ensuremath{\varphi}$ 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 $\ensuremath{\psi}$ decreases the ground state degeneracy which, nonetheless, remains continuous in the open interval $({\ensuremath{\rho}}_{d},{\ensuremath{\rho}}_{d}^{\ensuremath{'}})$, where ${\ensuremath{\rho}}_{d}^{\ensuremath{'}}$ 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 ${\ensuremath{\rho}}_{3}$ and the fcc lattice at ${\ensuremath{\rho}}_{3}^{\ensuremath{'}}=\sqrt{2}∕{r}_{0}^{3}$.