InternationalMathematical, physical and chemical tablesTables for Crystallography Volume C Edited by E. Prince © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. C, ch. 9.2, pp. 762-763
## Section 9.2.2.2.4. Some geometrical properties of OD structures S. Ďurovič
^{a} |

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)*m*1. 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.

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** = **a**_{0}+ **b**/4 is the total operation (|**a**_{0}| is the distance between adjacent layers). It has basis vectors **a**_{1} = **a**_{0} + **b**/4, **b**_{1}= **b**, **c**_{1}= **c**, space group *P*111, Ramsdell symbol 1*A*,^{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 **a**_{2} = 2**a**_{0}, **b**_{2} = **b**, **c**_{2}= **c**, space group *P*1*a*1, Ramsdell symbol 2*M*, 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 **a**_{3} = 4**a**_{0}, **b**_{3} = **b**, **c**_{3}= **c**, space group *P*1*a*1, Ramsdell symbol 4*M*, 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, 1979*a*, *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)*m*1|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).

### References

Dornberger-Schiff, K. (1964).*Grundzüge einer Theorie von OD-Strukturen aus Schichten. Abh. Dtsch. Akad. Wiss. Berlin. Kl. Chem.*

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Dornberger-Schiff, K., Ďurovič, S. & Zvyagin, B. B. (1982).

*Proposal for general principles for the construction of fully descriptive polytype symbols. Cryst. Res. Technol.*

**17**, 1449–1457.

Fichtner, K. (1979

*a*).

*On the description of symmetry of OD structures (I). OD groupoid family, parameters, stacking. Krist. Tech.*

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Fichtner, K. (1979

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*On the description of symmetry of OD structures (II). The parameters. Krist. Tech.*

**14**, 1453–1461.

Fichtner, K. (1980).

*On the description of symmetry of OD structures (III). Short symbols for OD groupoid families. Krist. Tech.*

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Guinier, A., Bokij, G. B., Boll-Dornberger, K., Cowley, J. M., Ďurovič, S., Jagodzinski, H., Krishna, P., de Wolff, P. M., Zvyagin, B. B., Cox, D. E., Goodman, P., Hahn, Th., Kuchitsu, K. & Abrahams, S. C. (1984).

*Nomenclature of polytype structures. Report of the International Union of Crystallography Ad-Hoc Committee on the Nomenclature of Disordered, Modulated and Polytype Structures. Acta Cryst.*A

**40**, 399–404.