Surface energies in a two-dimensional mass-spring model for crystals

We study an atomistic pair potential-energy E((n))(y) that describes the elastic behavior of two-dimensional crystals with n atoms where y is an element of R(2xn) characterizes the particle positions. The main focus is the asymptotic analysis of the ground state energy as n tends to infinity. We show in a suitable scaling regime where the energy is essentially quadratic that the energy minimum of E((n)) admits an asymptotic expansion involving fractional powers of n: min(y) E((n)) (y) = n E(bulk) + root n E(surface) + o(root n), n -> infinity. The bulk energy density E(bulk) is given by an explicit expression involving the interaction potentials. The surface energy E(surface) can be expressed as a surface integral where the integrand depends only on the surface normal and the interaction potentials. The evaluation of the integrand involves solving a discrete algebraic Riccati equation. Numerical simulations suggest that the integrand is a continuous, but nowhere differentiable function of the surface normal