Orientation relationships, orientational variants and the embedding approach

For phase transformations within polycrystalline materials, the connection between the crystal orientations of parent grains and those of child grains is usually expressed in terms of (theoretical or measured) orientation relationships. This paper introduces a new approach to various problems associated with orientation relationships: (i) estimation, (ii) whether or not a single orientation relationship fits the data adequately, (iii) whether or not a set of children comes from a common parent, and (iv) reconstruction of a parent or of grain boundaries. The approach is an extension to the crystallographic context of the well established embedding approach to directional statistics. It is inherently statistical, producing precise probabilistic statements. Explicit coordinate systems are not used and arbitrary thresholds are avoided.