We study an atomistic pair potential-energy E(n)(y) that describes
the elastic behavior of two-dimensional crystals with n 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 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:
${\rm min}_y E^{(n)}(y) = n \, E_{\mathrm{bulk}}+ \sqrt{n} \, E_\mathrm{surface} +o(\sqrt{n}), \qquad n \to \infty.$
The bulk energy density Ebulk is given by an explicit
expression involving the interaction potentials. The surface energy
Esurface 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.