Crystalline Ground States for Classical Particles

Pair interactions whose Fourier transform is non-negative and vanishes above a wave number $K_0$ are shown to give rise to periodic and aperiodic infinite volume ground state configurations (GSCs) in any dimension $d$. A typical three-dimensional example is an interaction of asymptotic form $\cos K_{0}r/r^4$. The result is obtained for densities $\rho \ge {\rho }_d$, where ${\rho }_1=K_0/2\pi$, ${\rho }_2=(\sqrt{3}/8)(K_0/\pi {)}^2$, and ${\rho }_3=(1/8\sqrt{2})(K_0/\pi {)}^3$. At ${\rho }_d$ there is a unique periodic GSC which is the uniform chain, the triangular lattice, and the bcc lattice for $d=1,2,3$, respectively. For $\rho >{\rho }_d$, the GSC is nonunique and the degeneracy is continuous: Any periodic configuration of density $\rho$ with all reciprocal lattice vectors not smaller than $K_0$, and any union of such configurations, is a GSC. The fcc lattice is a GSC only for $\rho \ge (1/6\sqrt{3})(K_0/\pi {)}^3$.