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

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) tan(delta) is a product of two factors, one of which is sin(phi), the other depends only on the relative deviation of r from s; (b) tan(delta) =~(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. Mallardʹs criterion requires that twin laws by (reticular) pseudo-merohedry have (sigma)<=5 and (delta)<=6°. 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 =~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.