Group-theoretical consideration of the CSL symmetry

A theoretical analysis for the computation of the coincidence site lattice (CSL) symmetry is presented. It is shown that three types of symmetry elements can exist and each one can be found by properly using the CSL's rotation matrix of the smallest-angle description. Thus, from the existence of the subgroup H₁, the order of which is directly connected with the number of the different orientations that the sublattice Λ¹₁ can have, a low-symmetry H₁ group implies more possibilities for the formation of the corresponding CSL. From the existence of the symmetry elements of the second type, the smallest-angle rotation matrix can be a symmetry element but only of the fourth or sixth order. From the third type of elements a connection between CSLs of different Σvalues can exist. Since the analytical form of this smallest-angle rotation matrix can be deduced for every crystallographic system, the procedure described here is of general use. Thus a new classification of the different CSLs is possible according to their symmetry group. This allows the study of the CSL model from the symmetry point of view.