Determination of all misorientations of tetragonal lattices with low multiplicity; connection with Mallard's rule of twinning

Two congruent lattices are considered, which are misoriented in such a way that they have a fraction 1Σ of symmetry translations in common. Whereas for cubic lattices body or face centring does not affect the `multiplicity' or `twin index' Σ, this is not generally true for tetragonal lattices. Consider a fixed misorientation and let Σ_P and Σ_I be the multiplicities for tP and tI lattices with the same axial ratio ca. Grimmer [Mater. Sci. Forum (1993), 126–128, 269–272] has given an explicit formula for Σ_P (depending on the misorientation and the axial ratio) and showed that Σ_I = Σ_P2, Σ_P or 2Σ_P. Here stronger results on the occurrence of the three possibilities are presented. Lists of all axial ratios ca of tP and tI lattices admitting misorientations with Σ ≤ 5 are given. For each of these misorientations, the twin mirror planes and their normals are listed, so that a synopsis of all possible twin laws of tetragonal crystals by reticular merohedry with Σ ≤ 5 is obtained. It is shown that the two twin laws observed in β-Sn can be described by reticular pseudomerohedry with Σ_I = 2 and obliquity δ = 2.6134°.