Section 9.2.2.2.6. The vicinity condition

A polytype family contains periodic as well as non-periodic members. The latter are as important as the former, since the very fact that they can be non-periodic carries important crystallochemical information. Non-periodic polytypes do not comply with the classical definition of crystals, but we believe that this definition should be generalized to include rather than exclude non-periodic polytypes from the world of crystals (Dornberger-Schiff & Grell, 1982b). The OD theory places them, together with the periodic ones, in the hierarchy of the so-called VC structures. The reason for this is that all periodic structures, even the non-polytypic ones, can be thought of as consisting of disjunct, two-dimensionally periodic slabs, the VC layers, which are stacked together according to three rules called the vicinity condition (VC) (Dornberger-Schiff, 1964, pp. 29 ff., Dornberger-Schiff, 1979; Dornberger-Schiff & Fichtner, 1972):

If the stacking of VC layers is unambiguous, traditional three-dimensionally periodic structures result (fully ordered structures). OD structures are VC structures in which the stacking of VC layers is ambiguous at every layer boundary (Z > 1). The corresponding VC layers then become OD layers. OD layers are, in general, not identical with crystallochemical layers; they may contain half-atoms at their boundaries. In this context, they are analogous with unit cells in traditional crystallography, which may also contain parts of atoms at their boundaries. However, the choice of OD layers is not absolute: it depends on the polytypism, either actually observed or reasonably anticipated, on the degree of symmetry idealization, and other circumstances (Grell, 1984).