Disorientation between any two lattices

The concept of disorientation, previously used for studying the statistical distribution of the relative orientation of identical cubic crystals, is defined in this work for any two lattices. Using the proposed definition, an algorithm is presented, allowing all the known relative orientations between the two lattices to be conveniently classed. As an example, a unified classification of the numerous mutual orientations of the Al and CuAl₂ crystals is suggested. The unit quaternion method used by Grimmer [Acta Cryst. (1974), A30, 685-688] for identical cubic lattices is here proved efficient for discussing the pair axis/angle disorientations in more complicated cases: cubic 1/cubic 2; tetragonal 1/tetragonal 2; hexagonal 1/hexagonal 2; cubic/tetragonal; cubic/orthorhombic and cubic/hexagonal. The general expressions of equivalent quaternions are given for any point group of lattice 1 or lattice 2.