Space groups, symmetry diagrams

Fig 2. Schematic diagram indicating the structural relationships between two structures. The structure with the highest space group symmetry is placed on the uf er part of the diagram whereas the parent structure with a lower symmetry is indicated in the lower part of the diagram. 

This structure can be directly derived from the rutile aristotype with space group symmetry P42/mnm. From the data given above and in the appendix, sketch briefly the structure of CoRe04 and establish the group-subgroup diagram relating the two structures. 

The extraordinary bonding properties in 16 can also be seen in a diagram of 16 (Fig. 34). In 16, an almost perfect five-numbered axis is attained, distorted only by the central Ga2 unit. This is the first time that a symmetry close to the five-numbered symmetry is observed for molecular metalloid clusters. A few solid-state modifications with five-numbered symmetry have, however, been found for compounds involving Group 13 elements. Because there is no crystallographic space group with a five-numbered axis, these compounds are summarized under the collective name quasi-crystals .83 For a better understanding of quasi-crystals,... 

We have now introduced all of the symmetry elements required to build up the 3D space groups, and in the next section, we shall introduce these space groups. Before doing so, it is convenient to define the complete set of symbols that are required to represent all of the necessary symmetry elements, as seen from all possible directions, when representing them in a diagram of a unit cell. Table 11.6 displays these symbols. 

Figure 11.19. Diagrams showing symmetry elements and general point positions for space groups PI, P2, /42, and /42, (which is not different from A2 except for placement of the origin).

Shown below are the symmetry diagrams for C222 and (2222, the only two C-centered orthorhombic space groups. C2,2 2 does not exist because C222 already contains the sets of 2, axes parallel to the a and b axis. (22,2,2, is, in fact, (2222, possibly with a shift of origin. Finally, C2,22 and C22,2 are simply C222, with relabeled axes, while (222,2, and (22,22, are relabeled versions of (22,2,2 (which, as already noted, is (2222). 

The 2D space group p6 arises by explicitly introducing one set of sixfold axes. Show with a diagram the other symmetry elements that arise automatically. 

There is no space group that could be called Pna2. Show why. (Hint draw the symmetry diagram implied and examine its effect on a general point.)... 

The unit cell parameters a and were 5.76 X and 13.20 respectively which agrees well with published data (14,15) based on fiber diagrams. Clearly the diffractogram is an hkO reciprocal lattice net as expected if the symmetry axis of the molecular helix is perpendicular to the crystal face. Since only even reflections are observed along the hOO and OkO directions the pgg base plane symmetry is confirmed in keeping with the proposed P2j2i2i space group for the orthorhombic unit cell (14,15). 

The symbols are given at the upper comer of the space group diagrams. A fraction h attached to a symbol indicates two symmetry planes with height h and h + 1/2 above the plane of projection e.g. 1/8 stands for h — 1/8 and 5/8. No fraction means h — 0 and 1/2. 

In the following discussion, for simplicity we make use of the pair of symmetry diagrams that constitute a representation of space group P2 jc (No. 14) in International Tables for X-Ray Crystallography Volume /, as displayed in Fig. 9.3.4. 

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