Parameterization of rotations in reference frames with redundant crystallographic axes

Abstract

Parameterizations of rotations in three-dimensional space are fundamental for description of crystallographic textures. Such parameterizations (e.g., Rodrigues parameters) are usually specified using orthonormal coordinate systems, whereas bases of crystal lattices are generally non-orthogonal. In the case of crystals with hexagonal or rhombohedral lattices, the reference frames involve redundant crystallographic axes, and hence a question arises about feasibility of the generalization of the rotation parameterizations to such frames. The definition of Rodrigues parameters can be extended so they are directly linked to non-Cartesian bases of crystal lattices. The new Rodrigues parameters are contra- or covariant components of vectors specified with respect to exactly the same lattice basis as atomic positions in a unit cell. The generalized formalism allows for using redundant crystallographic axes. Also the orientation matrices can be represented in such frames. The Rodrigues parameterization in non-Cartesian coordinate frames is convenient for crystallographic applications because the generalized parameters are directly related to indices of rotation-invariant lattice directions and to Miller indices of rotation-invariant lattice planes. In the case of the hexagonal and rhombohedral lattices, the frames with redundant axes are used to account for lattice symmetry, but one may apply such frames for other reasons. They can be convenient for handling arbitrary symmetries, in particular symmetries arising in description of some phenomena or symmetries of physical processes. The practicality of frames is illustrated by an alternative description of body-centered lattices, a formula for lattice rotation in deformation by slip, and a new interpretation of indexing of single crystal diffraction patterns.