Twins and allotwins of basic mica polytypes: theoretical derivation and identification in the reciprocal space

The geometry of the diffraction pattern from twins and allotwins of the four basic mica polytypes (1M, 2M1, 3T, 2M2) is analysed in terms of the 'minimal rhombus', a geometrical asymmetric unit in reciprocal space defined by nine translationally independent reciprocal-lattice rows. The minimal rhombus contains the necessary information to decompose the reciprocal lattice of twins or allotwins into the reciprocal lattices of the individuals. The nine translationally independent reciprocal-lattice rows are divided into three types (S, D and X): rows of different type are not overlapped by the n x 60 degrees rotations about c*, which correspond to the relative rotations between pairs of twinned or allotwinned individuals. A symbolic representation of the absolute orientation of the individuals, similar to that used for layers in polytypes, is introduced. The polytypes 1M and 2M1 undergo twinning by reticular pseudo-merohedry with five pairs of twin laws: they produce twelve independent twins, of which nine can be distinguished by the minimal rhombus analysis. The 2M2 polytype has two pairs of twin laws by pseudo-merohedry, which give a single diffraction pattern geometrically indistinguishable from that of the single crystal, and three pairs of twin laws by reticular pseudo-merohedry, which give a single diffraction pattern different from that of the single crystal. The 3T polytype has three twin laws: one corresponds to complete merohedry and the other two to selective merohedry. Selective merohedry produces only partial restoration of the weighted reciprocal lattice built on the family rows and the presence of twinning can be recognized from the geometry of the diffraction pattern.