Universal Jump of Gaussian Curvature at the Facet Edge of a Crystal

Novel universal behavior of the equilibrium crystal shape is reported: The Gaussian curvature, a product of two principal curvatures, assumes a universal jump across the facet contour at any temperature below the roughening temperature. This behavior is shown to be a consequence of a universal relation between the coefficients ${\gamma }_s$ and $B$ in the small-p expansion (p is the surface gradient) of the interface free energy, $f\left(\mathbf{p}\right)=f\left(0\right)+{\gamma }_s\left|\mathbf{p}\right|+B{\left|\mathbf{p}\right|}^3+O\left({\left|\mathbf{p}\right|}^4\right)$. Both exact results on a solvable model and Monte Carlo calculations support this behavior—universal Gaussian-curvature jump at the facet edge.