  In the crater function approach to the erosion of a solid surface by a broad
ion beam, the average crater produced by the impact of an ion is used to
compute the constant coefficients in the continuum equation of motion for the
surface. We extend the crater function formalism so that it includes the
dependence of the crater on the curvature of the surface at the point of
impact. We then demonstrate that our formalism yields the correct coefficients
for the Sigmund model of ion sputtering if terms up to second order in the
spatial derivatives are retained. In contrast, if the curvature dependence of
the crater is neglected, the coefficients can deviate substantially from their
exact values. Our results show that accurately estimating the coefficients
using craters obtained from molecular dynamics simulations will require
significantly more computational power than was previously thought.
