Crystalline Ground States for Classical Particles

Pair interactions whose Fourier transform is non-negative and vanishes above a wave number K₀ 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⁡K₀r/r⁴. The result is obtained for densities ρ≥ρ_d, where ρ₁=K₀/2π, ρ₂=(√3/8)(K₀/π)², and ρ₃=(1/8√2)(K₀/π)³. At ρ_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 ρ>ρ_d, the GSC is nonunique and the degeneracy is continuous: Any periodic configuration of density ρ with all reciprocal lattice vectors not smaller than K₀, and any union of such configurations, is a GSC. The fcc lattice is a GSC only for ρ≥(1/6√3)(K₀/π)³.