The diffraction patterns from twins and allotwins of the 4 basic polytypes (1M, 2M1, 3T, 2M2) were analyzed in terms of the so-called minimal rhombus. This was a geometrically asymmetrical unit, in reciprocal space, which was defined by 9 translationally independent reciprocal-lattice rows. The minimal rhombus contained the necessary information to decompose the reciprocal lattice of twins or allotwins into the reciprocal lattices of the individuals. The 9 translationally independent reciprocal-lattice rows were divided into 3 types. A symbolic representation of the absolute orientation of the individuals, similar to that used for layers in polytypes, was introduced. The polytypes 1M and 2M1 underwent twinning via reticular pseudo-merohedry, with 5 pairs of twin laws. They produced 12 independent twins, of which 9 could be distinguished by using the minimal rhombus analysis. The 2M2 polytype had 2 pairs of twin laws, according to pseudo-merohedry. This gave a single diffraction pattern, which was geometrically indistinguishable from that of the single crystal, plus 3 pairs of twin laws by reticular pseudo-merohedry. This gave a single diffraction pattern which was different to that of the single crystal. The 3T polytype had 3 twin laws. One corresponded to complete merohedry, and the other 2 to selective merohedry. Selective merohedry produced only partial restoration of the weighted reciprocal lattice built on the family rows, and the presence of twinning could be seen from the geometry of the diffraction pattern.

Twins and Allotwins of Basic Mica Polytypes: Theoretical Derivation and Identification in Reciprocal Space. M.Nespolo, G.Ferraris, H.Takeda: Acta Crystallographica, 2000, A56[2], 132-41