Analytical criteria for missing orientations on three-dimensional equilibrium shapes

The equilibrium shape of a crystal is the shape that minimizes its anisotropic interfacial free energy subject to the constraint of constant volume. This shape can be determined geometrically by using the Wulff construction; it can have missing orientations, sharp edges and corners, and its faces can be rounded or flat (facets). In two dimensions, when the surface free energy, γ, is a function of a single angle θ, the analytical criterion for the onset of missing orientations is that γ+γθθ changes sign from positive to negative. In three dimensions, for which γ depends on two angles, θ and ϕ, no such analytical criterion is known. A geometrical criterion for missing orientations can be based on the Herring sphere construction. By means of inversion through the origin, an equivalent criterion becomes whether any portion of a polar plot of the reciprocal of γ (1/γ-plot) lies outside a tangent plane. The onset of missing orientations occurs when the 1/γ-plot changes from convex to concave. An equivalent analytical criterion is obtained by showing that the normal to the 1/γ-plot is proportional to the ξ vector of Hoffman and Cahn and then using an invariant representation of the Gaussian curvature of the 1/γ-plot to test its convexity. Results are illustrated for cubic symmetry. As the magnitude of anisotropy is increased, the equilibrium shape loses orientations and tends to an octahedron (positive anisotropy) or a cube (negative anisotropy). A similar formalism can be used to find an analytical criterion for missing orientations on growth shapes of crystals that are growing under the control of anisotropic interface kinetics.