Crystalline ground states for classical particles -: art. no. 265501

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 cosK(0)r/r(4). The result is obtained for densities p >= p(d), where p(1) = K-0/2 pi, p(2) = (root 3/8)(K-0/pi)(2), and p(3) = (1/8 root 2) x bcc lattice for d = 1,2,3, respectively. For p > p(d), the GSC is nonunique and the degeneracy is continuous: Any periodic configuration of density p 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 p >= (1/6 root 3) x (K-0/root pi)(3).