Overlooked problems in manifold twins: twin misfit in zero-obliquity TLQS twinning and twin index calculation

It is shown that the twin index n calculated, according to Friedel, as a function of the indices (hkl) and [uvw] of the lattice plane and lattice direction defining the cell of the twin lattice applies only to twofold twins, i.e. twins where the twin element is of order 2. For manifold twins, where the twin operation is a three-, four- or sixfold (direct or inverse) rotation, it is shown that the generalized formula becomes n = NΞ/ξ, where N is the number of lattice planes of the (hkl) family passing within the cell of the twin lattice, Ξ the two-dimensional coincidence index for a plane of the (hkl) family and ξ the number of planes out of N of that family that are partially restored by the twin operation. The existence of twin lattice quasi-symmetry (TLQS) twins with zero-obliquity in manifold twins leads to the introduction of a new parameter as a general measure of the pseudo-symmetry of TLQS rotation twins: the twin misfit δ, defined as the distance between the first nodes along the two shortest directions in the plane of L_T (quasi-)perpendicular to the twin axes that are quasi-restored by the twin operation. Taking the example of staurolite twins, several inconsistencies in the treatment of manifold twins are pointed out.