Minimizing atomic configurations of short range pair potentials in two dimensions: crystallization in the Wulff shape

Abstract

We investigate ground state configurations of atomic systems in two dimensions interacting via short range pair potentials. As the number of particles tends to infinity, we show that low-energy configurations converge to a macroscopic cluster of finite surface area and constant density, the latter being given by the density of atoms per unit volume in the triangular lattice. In the special case of the Heitmann–Radin sticky disc potential and exact ground states, we show that the macroscopic cluster has a (unique) Wulff shape. This is done by showing that the atomistic energy of crystalline configurations, after subtracting off a bulk part and re-scaling, Gamma-converges to a macroscopic anisotropic surface energy.