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

We study an atomistic pair potential-energy () that describes the elastic behavior of two-dimensional crystals with atoms where $y\in {\mathbb{R}}^{2\times n}$ characterizes the particle positions. The main focus is the asymptotic analysis of the ground state energy as tends to infinity. We show in a suitable scaling regime where the energy is essentially quadratic that the energy minimum of admits an asymptotic expansion involving fractional powers of : ${\mathrm{min}}_{y}{E}^{\left(n\right)}\left(y\right)=n\phantom{\rule{0.166667em}{0ex}}{E}_{\mathrm{bulk}}+\sqrt{n}\phantom{\rule{0.166667em}{0ex}}{E}_{\mathrm{surface}}+o\left(\sqrt{n}\right),\phantom{\rule{2.0em}{0ex}}n\to \infty .$ The bulk energy density ...