The purpose of this lecture is to give the participants a solid introduction to the analysis of the point and reticular (psuedo) symmetry of twinned crystals.

Fundamental concepts (such as twin law, twin operation, twin element, twin lattice) and categories (TLS vs. TLQS and their subcategories) are introduced and the use of intersection groups of point groups and their chromatic supergroups is illustrated with some examples using mainly the stereographic projection. Participants will learn, through guided exercises, to calculate the obliquity and the twin index of a twin, the cell parameters and the pseudo-symmetry of the twin lattice. Finally, a brief introduction to the concept of hybrid twins will be presented.

The use of BCS is illustrated mainly through some space-group routines whose results are adapted to point-group problems in twinning: in particular, the coset decomposition is illustrated to find the twin laws and the twin operations.

Supplementary Material