The derivation of twin laws in non-merohedric twins. Application to the analysis of hybrid twins.

An algorithm is presented to derive the twin laws in non-merohedric twins through the systematic search for quasi-perpendicular lattice planes/directions. The twin lattice, i.e. the sublattice common to the whole crystalline edifice built by the twinned individuals, is based on a supercell of the individual defined by a pair of quasi-perpendicular lattice elements (hkl)/[uvw]. Starting from a (real or supposed) twin element, (hkl) or [uvw], the set of quasi-perpendicular lattice elements with user-defined limits on the twin index and obliquity is explored. The degree of lattice quasi-restoration is commonly measured by the classical twin index but in some cases, especially for large supercells, this index represents only a first approximation of the degree of lattice quasi-restoration, because more than one pair (hkl)/[uvw] may exist, and more than one concurrent sublattices of nodes, based on the same twin element, may be quasi-restored, although within different obliquities. These twins, whose existence has been recently recognized, are termed hybrid twins. In hybrid twins, the degree of lattice quasi-restoration is measured by the effective twin index n(E), a generalization of the classical twin index: in the limiting case of only one (quasi-)restored sublattice, the effective twin index and the classical twin index coincide. A number of examples previously reported simply as 'non-Friedelian' twins (with a twin index higher than the empirical limit of 6 established by Friedel) are analysed and reinterpreted as hybrid twins. A Fortran program is made available, which derives the possible twin laws according to this algorithm and analyses the pseudosymmetry of the concurrent sublattices defined by each pair (hkl)/[uvw]. The occurrence of hybrid concurrent components in twinning does not affect the normal procedures of dealing with diffraction patterns of twinned crystals.