A geodesic octonion metric for grain boundaries

Abstract We propose a representation of grain boundaries on the unit octonion 7-sphere. This representation comes with algebraic and computational tools for comparing and interpolating between grain boundaries. We derive a geodesic metric based on the inner product between unit octonions, which themselves consist of pairs of quaternions describing the grain orientations in the reference frame of the grain boundary plane. We show how the metric behaves under the application of a number of symmetries: grain exchange symmetry; U ( 1 ) symmetry around the grain boundary normal; the no-boundary condition; and crystal symmetry. We show that this metric reproduces the Mackenzie distribution in the no-boundary case, and correctly determines the angular distances between grain boundaries with a common normal or misorientation. We provide a graphical representation of the mapping of quaternion pairs onto the octonion unit sphere, and describe a number of practical computations in the Supplementary Material accompanying this paper.