An analytic solution for three-dimensional axisymmetric equilibrium crystal shapes

A surface free energy polar plot contains two possible sources of singularity: the cusps that give facets on an equilibrium crystal, and the circular arcs connecting the cusps that can lead to missing orientations. Instead of specifying surface free energy, we model the surface stiffness so that both singularities can be treated precisely. Here, we apply our model to three-dimensional axisymmetric crystals and obtain for the first time an analytic solution to the nonlinear differential equation governing the crystal shape.