Section 3.3.10.2.4. Permissible twin boundaries in twins with lattice index

In contrast to the mechanical compatibility of any composition plane in merohedral twins (lattice index ), twins of higher lattice index are more restricted with respect to the orientation of permissible twin boundaries. In fact, these special twins can be treated in the same way as the general non-merohedral twins described in Section 3.3.10.2.2 above. Again, we attribute equal but opposite spontaneous shear strain to the two twin domains 1 and 2. This `spontaneous' shear strain (referred to an intermediate state of zero strain) is half the shear deformation necessary to transform the orientation of domain 1 into that of domain 2. This also means that the lattice of domain 1 is transformed into the lattice of domain 2. The essential difference to the case in Section 3.3.10.2.2 is the fact that by this deformation only a subset of lattice points is `restored'. This subset forms the sublattice of index common to both domains (coincidence-site sublattice, twin lattice). With this analogy, the Sapriel formalism can be applied to the derivation of the mechanically compatible (permissible) twin boundaries. Again, the easiest way to find the permissible planes is the construction of the black–white symmetry symbol of the twin law, in which planes parallel to primed mirror planes or normal to primed twofold axes constitute the permissible W interfaces.

It is emphasized that the concept of a deformation from domain state 1 to domain state 2 is not always a mere mental construction, as it is for growth twins. It is physical reality in some deformation twins, for example in the famous deformation twins (spinel law) of cubic metals which are essential elements of the plasticity of these metals. During the deformation, the {100} cube (a rhombohedron) is switched from its `reverse' into its `obverse' orientation and vice versa, whereby the hexagonal P sublattice of index is restored and, thus, is common to both twin domains.

Exact lattice coincidences of twin domains result from special symmetry relations of the lattice. Such relations are systematically provided by n-fold symmetry axes of order , i.e. by three-, four- and sixfold axes. In other words: twins of lattice index occur systematically in trigonal, hexagonal, tetragonal and cubic crystals. This may lead to trigonal, tetragonal and hexagonal intersection symmetries (reduced eigensym­metries) of domain pairs. Consequently, if there exists one pair of permissible composition planes, all pairs of planes equivalent to the first one with respect to the intersection symmetry are permissible twin boundaries as well. This is illustrated by three examples in Table 3.3.10.1.

Table 3.3.10.1 | top | pdf | Examples of permissible twin boundaries for higher-order merohedral twins ([j]> 1)   growth and deformation twins of cubic crystals, twin mirror plane (111) (spinel law) growth twins of tetragonal rare-earth sulfides (SmS1.9), twin mirror plane (120) deformation twins of cubic galena (PbS), twin mirror plane (441)† Eigensymmetry Intersection symmetry ‡ parallel to [111] parallel to [001] parallel to Reduced composite symmetry ‡ Permissible twin boundaries Three pairs of perpendicular planes Two pairs of perpendicular planes One pair of permissible planes & & & & &   &     Reference system Cubic axes Tetragonal axes Cubic axes † The existence of this deformation twin is still in doubt (cf. Seifert, 1928). ‡The intersection symmetry and the permissible boundaries are referred to the coordinate system of the eigensymmetry; the reduced composite symmetries are based on their own conventional coordinate system derived from the intersection symmetry plus the twin law (cf. Section 3.3.4).

For the cubic and rhombohedral twins (spinel law), due to the threefold axis of the intersection symmetry, three pairs of permissible planes occur. The plane (111), normal to this threefold axis, is common to the three pairs of boundaries (threefold degeneracy), i.e. in total four different permissible W twin boundaries occur. These composition planes (111), , , are indeed observed in the spinel-type penetration twins, recognizable by their re-entrant edges (Fig. 3.3.6.6). They also occur as twin glide planes of cubic metals. For the tetragonal twin, two pairs of perpendicular permissible W composition planes result, (120) & ( and (310) & (), one pair bisecting the other pair under 45°. For the cubic twin [galena PbS, cf. Section 3.3.8.3, example (4)], due to the low intersection symmetry, only one pair of permissible W boundaries results.