Section 3.3.8.7. Conclusions

(i) The lattice theory represents one of the first systematic theories of twinning; it is based on a clear and well defined concept and thus has found widespread acceptance, especially for the description, characterization and classification of `triperiodic' (merohedral and pseudo-merohedral) twins.

(ii) The concept, however, is purely geometrical and has as its object a mathematical, not a physical, item, the lattice. It takes into account neither the crystal structure nor the orientation and energy of the twin interface. This deficit has been pointed out and critically discussed by Buerger (1945), Cahn (1954, Section 1.3), Hartman (1956) and Holser (1958, 1960); it is the major reason for the limitations of the theory and its low power of prediction for actual cases of twinning.

(iii) The relations between twinning and lattice (pseudo-)symmetries, however, become immediately obvious and are proven by many observations as soon as structural pseudosymmetries exist. Twinning is always facilitated if a real or hypothetical `parent structure' exists from which the twin law and the interface can be derived. Here, the lattice pseudosymmetry appears as a necessary consequence of the structural pseudosymmetry, which usually involves only small deformations of the parent structure, resulting in small obliquities of twin planes and twin axes (which are symmetry elements of the parent structure) and, hence, in twin interfaces of low energy. These structural pseudosymmetries are the result either of actual or hypothetical phase transitions (domain structures, cf. Chapter 3.4 ) or of structural relationships to a high-symmetry `prototype' structure, as explained in Section 3.3.9.2 below.

(iv) On the other hand, twinning quite often occurs without recognizable structural pseudosymmetry, e.g. the (100) dovetail twins and the (001) Montmartre twins of gypsum, as well as the (101) and (301) reflection twins of rutile and some further examples listed in Table 3.3.8.2. In all these cases, it can be concluded that the lattice theory of twinning is not the suitable tool for the characterization and prediction of the twins; in the terminology of Friedel: the twins are not `triperiodic' but only `diperiodic' or `monoperiodic'.