Metrics, measures, and parametrizations for grain boundaries: a dialog

Analysis of experimental data on grain boundaries (GBs) can involve putting data into angle “bins.” An example is Brandon’s classification: if rotation angle and axis are each within $$15^\circ/\sqrt{\Sigma}$$ of a perfect coincidence site lattice (CSL) with density 1/Σ, the GBs can also be considered to be in that CSL relationship, and not if otherwise. Other examples of binning are studies of GB distributions in the full 5D angle space. To determine the size of a bin (necessary for densities, gradients, etc.) one must find a useful way, respecting symmetry, of determining metrics and measures on the full 5-dimensional space of both misorientation and interface normal. For a pair of low-angle GBs, the issue of metric is complicated by the fact that both their rotation axes and their GB normals can stay far apart as their rotation angles approach zero. We address all these issues as a dialog, and provide a framework for choosing metrics.