Fundamentals for the description of hexagonal lattices in general and in coincidence orientation

The connection between the rotation matrix in hexagonal lattice coordinates and an angle-axis quadruple is given. The multiplication law of quadruples is derived. It corresponds to multiplying two matrices and gives the effect of two successive rotations. The relation is given between two quadruples that describe the same relative orientation of two lattices owing to their hexagonal symmetry; a unique standard description of the relative orientation is proposed. The restrictions satisfied by rotations generating coincidence site lattices (CSL's) are derived for any value of the axial ratio ρ = c/a. It is shown that the law for cubic lattices, where the multiplicity Σ of the CSL is equal to the lowest common denominator of the elements of the rotation matrix, does not always hold for hexagonal lattices. A generalization of this law to lattices of arbitrary symmetry is given and another, quicker, method of determining Σ for hexagonal lattices is derived. Finally, convenient algorithms are described for determining bases of the CSL and the DSC lattice.