International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 9.7, p. 897

## Section 9.7.1.1. Kitajgorodskij's categories

A. J. C. Wilson,a V. L. Karenb and A. Mighellb

aSt John's College, Cambridge CB2 1TP, England, and bNIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA

#### 9.7.1.1. Kitajgorodskij's categories

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In his book2 Organicheskaya Kristallokhimiya, Kitajgorodskij (1955) treated the triclinic, monoclinic and orthorhombic space groups in considerable detail, analysing the possibility of (a) forming close-packed layers (six-point contact), and (b) close stacking of the layers. On this basis, he divided the layers and the space groups into four categories each. For the layers they are:

 (1) Coordination close-packed layers. A coordination close-packed layer is one in which molecules of arbitrary shape and symmetry can be packed with six-point coordination. (2) Closest-packed layers. A closest-packed layer is one in which one can select the orientation of molecules of given shape and symmetry so as to produce a cell of minimal dimensions. (3) Limitingly close-packed layers. A limitingly close-packed layer for a given symmetry is a closest-packed layer in which a molecule retains inherent symmetry; in other words, in which it occupies a special position. (4) Permissible layers. A permissible layer is coordination close packed but neither closest packed nor limitingly close packed.

The categories of space groups are:

 (1) Closest-packed space groups are those that permit the closest stacking of closest-packed layers – the packing can be made no denser by varying the cell parameters and the orientation of the molecules. Closest stackings can be made by a monoclinic displacement (a translation making an arbitrary angle with the layer plane), a centre of symmetry, a glide plane, or a screw axis. (2) Limitingly close-packed space groups are those that contain limitingly close-packed layers stacked as closely as possible. (3) Permissible space groups fall into three subcategories: (a) Those containing closest-packed layers that can be closely stacked if the layer relief is suitable; this group contains layers stacked by centring (C, I, F) or by diad axes. (b) Those containing limitingly close-packed layers that can be most closely stacked if the layer relief is suitable. (c) Those containing permissible layers stacked in the densest fashion. (4) Impossible space groups fall into two subcategories: (a) Those containing any layers (even closest-packed layers) that are related by mirror planes and translations normal to the layer plane. (b) Those containing permissible coordination close-packed layers not stacked in the densest possible way.

Kitajgorodskij expected the frequency of space groups to decrease in the order (1) > (2) > (3 > (4). In particular, permissible space groups should be found but rarely, as exceptions'. The categorization is summarized in Table 9.7.1.1, based on Table 8 of Kitajgorodskij (1955).

 Table 9.7.1.1| top | pdf | Kitajgorodskij's categorization of the triclinic, monoclinic and orthorhombic space groups, as modified by Wilson (1993a)
 Wilson's additions are enclosed in square brackets [] and the original positions of space groups transferred by him by round brackets (). Space groups not listed belong to the impossible' category.
Molecular symmetry12m2/m222mmmmm
Closest packed P P
P21
P21/c P21/c
[C2/c] C2/c [C2/c]
P212121
Pca21
Pna21
[Pbca] Pbca
Limitingly close packed     (C2/c)   C2/m
[P21212]   P21212     C222
F222
I222
Pmc21     Fmm2
Cmc21
[Pbcn]   Pbcn Pnma     Pmma, Pmmn
Cmca Ccca   Cmmm
Fmmm
Immm
Permissible P1
C2   C2
[Pc]
Cc     Cm Pbam
[P2/c]   [P2/c] P21/m
(P21212)
[C2221]   [C2221]
Pmn21
Aba2 Ama2
[Fdd2] Ima2
(Pbca)     Pbcm
[Pccn] Pccn
Kitajgorodskij (1961) includes Pnnm at this position, but this is inconsistent with the text of either the Russian or the English version.
Kitajgorodskij (1961) correctly includes Pbcm at this position.

Kitajgorodskij's categorization proved very successful in broad outline, but Wilson's (1993b,c) detailed statistics revealed about a dozen anomalous space-group types. The anomalies were of two kinds. The first was the frequent occurrence of molecules in general positions in space groups in which Kitajgorodskij expected molecules to use inherent symmetry in special positions. Wilson (1993a) pointed out that in such cases structural dimers3 can be formed, with two molecules in general positions related by the required symmetry elements – both enantiomers would be required if the element were or m. Such space groups could therefore be added to Kitajgorodskij's table, in the column for molecular symmetry 1'. The second kind of anomaly was the fairly frequent occurrence of structures with the impossible' space groups Pc and P2/c. These could be transferred from impossible' to permissible', subgroup (a), by the same packing argument that Kitajgorodskij had used for P1. These and a few other reclassifications are indicated in Table 9.7.1.1, the new entries being enclosed in square brackets for distinction. Where the change is a transfer to a higher category, the original position of the space group is indicated in round brackets.

### References

Kitaigorodskii, A. I. (1961). Organic chemical crystallography. New York: Consultants Bureau.Google Scholar