Section 9.2.2.2.4. Some geometrical properties of OD structures

As already pointed out, all relevant geometrical properties of a polytype family can be deduced from its symmetry principle. Let us thus consider a hypothetical simple family in which we shall disregard any concrete atomic arrangements and use geometrical figures with the appropriate symmetry instead.

Three periodic polytypes are shown in Fig. 9.2.2.2 (left-hand side). Any member of this family consists of equivalent layers perpendicular to the plane of the drawing, with symmetry P(1)m1. The symmetry of layers is indicated by isosceles triangles with a mirror plane [.m.]. All pairs of adjacent layers are also equivalent, no matter whether a layer is shifted by +b/4 or −b/4 relative to its predecessor, since the reflection across [.m.] transforms any given layer into itself and the adjacent layer from one possible position into the other. These two positions follow also from the NFZ relation: N = 2, F = 1 [the layer group of the pair of adjacent layers is P(1)11] and thus Z = 2.

Figure 9.2.2.2| top | pdf | Schematic representation of three structures belonging to the OD groupoid family P(1)m1|1, y = 0.25 (left). The layers are perpendicular to the plane of the drawing and their constituent atomic configurations are represented by isosceles triangles with symmetry [.m.]. All structures are related to a common orthogonal four-layer cell with a = 4a0. The hk0 nets in reciprocal space corresponding to these structures are shown on the right and the diffraction indices refer also to the common cell. Family diffractions common to all members of this family (k = 2) and the characteristic diffractions for individual polytypes (k = 2k + 1) are indicated by open and solid circles, respectively.

The layers are all equivalent and accordingly there must also be two coincidence operations transforming any layer into the adjacent one. The first operation is evidently the translation, the second is the glide reflection. If any of these becomes total for the remaining part of the structure, we obtain a polytype with all layer triples equivalent, i.e. a MDO polytype. The polytype (a) (Fig. 9.2.2.2) is one of them: the translation t = a₀+ b/4 is the total operation (|a₀| is the distance between adjacent layers). It has basis vectors a₁ = a₀ + b/4, b₁= b, c₁= c, space group P111, Ramsdell symbol 1A,⁴ Hägg symbol |+|. This polytype also has its enantiomorphous counterpart with Hägg symbol |−|. In the other polytype (b) (Fig. 9.2.2.2), the glide reflection is the total operation. The basis vectors of the polytype are a₂ = 2a₀, b₂ = b, c₂= c, space group P1a1, Ramsdell symbol 2M, Hägg symbol |+ −|. The equivalence of all layer triples in either of these polytypes is evident. The third polytype (c) (Fig. 9.2.2.2) is not a MDO polytype because it contains two kinds of layer triples, whereas it is possible to construct a polytype of this family containing only a selection of these. The polytype is again monoclinic with basis vectors a₃ = 4a₀, b₃ = b, c₃= c, space group P1a1, Ramsdell symbol 4M, and Hägg symbol |+−−+|.

Evidently, the partial mirror plane is crucial for the polytypism of this family. And yet the space group of none of its periodic members can contain it – simply because it can never become total. The space-group symbols thus leave some of the most important properties of periodic polytypes unnoticed. Moreover, the atomic coordinates of different polytypes expressed in terms of the respective lattice geometries cannot be immediately compared. And, finally, for non-periodic members of a family, a space-group symbol cannot be written at all. This is why the OD theory gives a special symbol indicating the symmetry proper of individual layers (λ symmetry) as well as the coincidence operations transforming a layer into the adjacent one (σ symmetry). The symbol of the OD groupoid family of our hypothetical example thus consists of two lines (Dornberger-Schiff, 1964, pp. 41 ff.; Fichtner, 1979a, b): where the unusual subscript 2 indicates that the glide reflection transforms the given layer into the subsequent one.

It is possible to write such a symbol for any OD groupoid family for equivalent layers, and thus also for the close packing of spheres. However, keeping in mind that the number of asymmetric units here is 24 (λ symmetry), one has to indicate also 24 σ operations, which is instructive but unwieldy. This is why Fichtner (1980) proposed simplified one-line symbols, containing full λ symmetry and only the rotational part of any one of the σ operations plus its translational components. Accordingly, the symbol of our hypothetical family reads: P(1)m1|1, y = 0.25; for the family of close packings of equal spheres: P(6/m)mm|1, x = 2/3, y = 1/3 (the layers are in both cases translationally equivalent and the rotational part of a translation is the identity).

An OD groupoid family symbol should not be confused with a polytype symbol, which gives information about the structure of an individual polytype (Dornberger-Schiff, Ďurovič & Zvyagin, 1982; Guinier et al., 1984).