Crystallography crystallographic point groups

Fig. 10.7. The crystallographic point groups arranged according to their order, ms, shown on the left, and linked to show sub- and supergroup relations (adapted from International Tables for Crystallography Vol. A, (1996) Table 10.3.2).

Table 1.17. The 230 crystallographic space groups arranged according to seven crystal systems and 32 crystallographic point groups as they are listed in the International Tables for Crystallography, vol. A. The centrosymmetric groups are in bold, while the non-centrosymmetric groups that do not invert an object are in italic. The remaining are non-...

For a given molecule the combination of symmetry elements is known as its point group. There are 32 crystallographic point groups that are possible. The asymmetric unit volume can contain an object which may or may not have symmetry. It is quite common in protein crystallography that this object has symmetry of its own, known as non-crystallo-graphic symmetry (examples include two-fold, three-fold, four-fold, five-fold etc, i.e. any point symmetry is possible in the asymmetric unit because it need not obey the translational symmetry of the crystal). 

When the symmetry elements are introduced into the possible lattices, the symmorphic elements combine to yield a total of 32 so-called point groups, while inclusion of symmorphic and nonsymmorphic elements leads to the 230 crystallographic space groups. Space groups, symmetry elements, and other aspects of crystal symmetry are tabulated in great detail in the International Tables for Crystallography. ... 

A serious problem in analyzing the structures of ordered mesoporous materials is, however, the fact that typically X-ray diffraction patterns are obtained which have only few reflections. Thus, often only space groups can be assigned, but no full structure solution is possible. So far, most structural models are just that, i.e. models, and they still have to be confirmed by either X-ray or electron crystallography. The 2-dimensionally ordered MCM-41 and the cubic MCM-48 have until recently probably been the two examples which were structurally well characterized by XRD and TEM [5,6]. One of the most notable developments, from the structure characterization point of view as well as with respect to the structure of the materials itself, was the structure solution of SBA-1, SBA-6, and SBA-16 with a novel electron crystallographic technique [7]. 

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