On the Crystallization of 2D Hexagonal Lattices

It is a fundamental problem to understand why solids form crystals at zero temperature and how atomic interaction determines the particular crystal structure that a material selects. In this paper we focus on the zero temperature case and consider a class of atomic potentials V = V ₂ + V ₃, where V ₂ is a pair potential of Lennard-Jones type and V ₃ is a three-body potential of Stillinger-Weber type. For this class of potentials we prove that the ground state energy per particle converges to a finite value as the number of particles tends to infinity. This value is given by the corresponding value for a optimal hexagonal lattice, optimized with respect to the lattice spacing. Furthermore, under suitable periodic or Dirichlet boundary condition, we show that the minimizers do form a hexagonal lattice. Dedicated with admiration to Professor Tom Spencer on occasion of his 60th birthday