Geminography: the crystallography of twins

The geometric theory of twinning was developed almost a century ago. Despite its age, it still represents the fundamental approach to the analysis and interpretation of twinned crystals, in both the direct and the reciprocal space. In recent years, this theory has been extended not only in its formalism (group-subgroup analysis, chromatic symmetry) but also in its classification of special cases that were not recognized before. The geometrical theory of twinning is thus reviewed here with emphasis on lattice aspects and recent developments. The classification of various types of twins starts with Friedel's well-known scheme, which distinguishes four cases according to whether the twin index n is equal to or larger than 1 and whether the obliquity omega is equal to or larger than 0: the computation of these quantities is discussed in detail. It is shown that the concept of obliquity is not sufficient to characterize the pseudo-symmetry of a lattice, and the consequent twinning, in the case of manifold twins. The application of the theory of coincidence-site lattices to twinning is presented. Finally, the effect of twinning on the diffraction pattern is illustrated with a number of examples.