Three-dimensional space groups

In Section 11.3 it was shown how the 17 2D space groups could be built up by combining, in a nonredundant fashion, all the possible symmetry opera- 

We cannot present here the complete derivation, as it is very lengthy. However, we shall discuss some representative cases in detail and these should serve to convey the essential ideas. For each illustrative case, we shall give the space group symbol and the conventional diagrams and tables used by X-ray crystallographers. On the basis of these specific examples the general rules for notation and diagrams will be relatively easy to appreciate. 

Triclinic Space Groups. The triclinic crystal system allows no axis of rotation of order higher than one, namely, 1 or 1. Since neither of these can give rise to any additional symmetry, there are just two triclinic space groups. Both are primitive and are designated FI and FI. 

This group is so trivial, having no symmetry apart from the three independent lattice translations, that we shall not present the diagrams or coordinate tables. 

The action of the lattice translations (i.e., the symmetry of the lattice itself) upon any one inversion center (1) that we introduce is to generate others (cf. the 2D group / 2). It is conventional to place one inversion center at the origin of the unit cell. The translational symmetry of the lattice then generates another one at the center of the cell (i,U), three more at face centers (e.g., 0,, i), and three at the midpoints of the edges (e.g., 2,0,0), for a total of eight inversion centers, none of which are equivalent. 

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Then he came to extending the division of continuous two-dimensional space into the third dimension. He restricted his examinations to polyhedra and found one of the five space-filling parallelohedra, which were discovered by E. S. Fedorov as capable of filling the space in parallel orientation without gaps or overlaps. Fedorov was one of the three scientists who determined the number (230) of three-dimensional space groups. The other two were Arthur Schoenflies and the amateur William Barlow. 

Considerations of complementarity in molecular packing culminated in the works of Kitaigorodskii. His most important contribution was the prediction that three-dimensional space groups of lower symmetry should be much more frequent than those of higher symmetry among crystal structures. This was a prediction at a time when few crystal structures had been determined experimentally. 

The 230 three-dimensional space groups are combinations of rotational and translational symmetry elements. A symmetry operation S transforms a vector r into r ... 

If we combine the 32 crystal point groups with the 14 Bravais lattices we find 230 three-dimensional space groups that crystal structures can adopt (i.e., 230... 

The site symmetry of each atom must be one of the 32 crystallographic point groups shown in Fig. 10.7, since these are the only point groups compatible with three-dimensional space groups. 

Objects or patterns which are periodic in one, two, and three directions will have one-, two-, and three-dimensional space groups, respectively. The dimensionality of the object/pattem is merely a necessary but not a satisfactory condition for the dimensionality of their space groups. We shall first describe a planar pattern after Budden [3] in order to get the flavor of space-group symmetry. Also, some new symmetry elements will be introduced. Later in this chapter, the simplest one-dimensional and two-dimensional space groups will be presented. The next Chapter will be devoted to the three-dimensional space groups which characterize crystal structures. 

The statistical analysis has also been applied separately to the data on inorganic and organic crystals. In both cases the extrapolated estimate for the total number of three-dimensional space groups was smaller than when all data had been considered together. The total... 

The criteria for the suitability as well as incompatibility of plane groups for achieving molecular six-coordination have been considered. The next step is to apply the geometrical model to the examination of the suitability of three-dimensional space groups for densest... 

Kitaigorodskii [94] analyzed all 230 three-dimensional space groups from the point of view of densest packing. Only the following space groups were found to be available for the densest packing of molecules of arbitrary form ... 

For molecules with symmetry centers, there are even fewer suitable three-dimensional space groups, namely ... 

There are some crystal structures in which further symmetries are present in addition to those prescribed by their three-dimensional space groups. The phenomenon is called hypersymmetry [102], Thus, it refers to symmetry features not included in the system of the 230 three-dimensional space groups. For example, phenol molecules, connected by hydrogen bonds, form spirals with threefold screw axes as indicated in Figure 9-55. This screw axis does not extend, however, to the whole crystal, and it does not occur in the three-dimensional space group characterizing the phenol crystal. 

Figure 9-55. The molecules in the phenol crystal are connected by hydrogen bonds and are forming spirals with a threefold screw axis. This symmetry element is not part of the three-dimensional space group of the phenol crystal. After Zorky and Koptsik [103],...

A prerequisite for hypersymmetry is that there should be chemically identical (having the same structural formula), but symmetrically independent, molecules in the crystal structure—symmetrically independent, that is, in the sense of the three-dimensional space group to which the crystal belongs. The question then arises as to whether these symmetrically independent but chemically identical molecules will have the same structure or not. Only if they do have the same structure, conformation as well as bond configuration, can we talk about the validity of the hypersymmetry operations. Here, preferably, quantitative criteria should be introduced, which is the more difficult since, for example, with increasing accuracy, structures that could be considered identical before, may no longer be considered so later when more accurate data become available. 

Figure 4 Relative positions along c axis of the alternating double and single bonds of the two chains in the cell, for the two possible three-dimensional space groups.

This illustrates a type of symmetry only seen in crystals and other extended arrays. That is, the symmetry operation combines both elements of point symmetry (as seen in molecules) and translation (which generates arrays). Here you can see that the repeat of this operation yields a vertical translation of one unit. The two-and three-dimensional space groups are realizations of the more general topic of group theory, which has been one of the tremendous scientific achievements in the last two centuries in the field of pure mathematics. 

The problem of combining the point groups with Bravais lattices to provide a finite number of three-dimensional space groups was worked out independently by Federov and by Schoenflies in 1890. Since the centred cells contain elements of translational symmetry new symmetry elements, not of the point-group type are generated in the process. 

The three-dimensional space groups are characterized by numbers from 1 to 230 and symbols, standardized by the International Union of Crystallography. The Union also publishes International Tables that summarize in detail all of these groups and their interrelationships. 

Harker, D. The three-colored three-dimensional space groups. Acta Cryst. A37, 286-292 (1981). 

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