Twinning by reticular pseudo-merohedry in trigonal, tetragonal and hexagonal crystals.

Abstract

Twin laws for trigonal, tetragonal and hexagonal crystals describing twins with principal axes inclined by an angle Phi > 0 are analysed. Twins by reticular merohedry (i.e. obliquity delta = 0) are possible only for certain values s of the axial ratio c/a. For any other axial ratio r, the laws describe twinning by reticular pseudo-merohedry, i.e. with obliquity delta > 0. It is shown that (a) tandelta is a product of two factors, one of which is sinPhi, the other depends only on the relative deviation of r from s; (b) tandelta approximately epsilon, where epsilon denotes the deformation parameter introduced by Bonnet & Durand [Philos. Mag. (1975), 32, 997-1006]. The angle Phi is listed for all cases of reticular merohedry of trigonal, tetragonal and hexagonal (i.e. optically uniaxial) crystals with twin index Sigma </= 5. Mallard's criterion requires that twin laws by (reticular) pseudo-merohedry have Sigma </= 5 and delta </= 6 degrees. Le Page [J. Appl. Cryst. (2002), 35, 175-181] has written a program determining laws with twin index Sigma </= Sigma(max) and obliquity delta </= delta(max) for any given lattice geometry. Here those solutions are analysed and completed for optically uniaxial crystals. Their lattices are characterized by the Bravais class (tP, tI, hP or hR) and the axial ratio c/a = r. For small delta(max), most solutions are related to (reticular) merohedry for an appropriate value s approximately r of the axial ratio. It is argued that other solutions, which are not related to (reticular) merohedry, are not needed to explain observed laws of growth twinning but may be important to interpret observed laws of deformation twinning.