Crystallographic groups, symmetry notations

Table 3-1. Symmetry Notations of the Crystallographic and a Few Limiting Groups...

Obviously, much of the development of crystallography predates the discovery of diffraction of X-rays by crystals. Early studies of crystal structures were concerned with external features of crystals and the angles between faces. Descriptions and notations used were based on these external features of crystals. Crystallographers using X-ray diffraction are concerned with the unit cells and use the notation based on the symmetry of the 230 space groups established earlier. 

Table 1-4 lists the point symmetry elements and the corresponding symmetry operations. The notation used by spectroscopists and chemists, and used here, is the so-called Schoenflies system, which deals only with point groups. Crystallographers generally use the Hermann-Mauguin system, which applies to both point and space groups. 

From Appendix 4, we can observe that for (space group 63) the following site symmetries are possible 2C2h(2) C2V(2) C,-(4) C2(4) 2Q(4) Ci(8). With the number of molecules in the unit cell equal to two, we must place two Pu3+ ions on a set of particular sites and six Br on other sets of sites. We observe that two site symmetries are available for the two Pu3+ ions—either C 2h or C2v, each having two equivalent sites per set to place the metal ions. An unambiguous choice cannot be made with the data available. For the six Br ions, no site symmetry has six equivalent sites available. Thus, we must conclude that the six Br- ions must be nonequivalent, and some are on one site and others on another site. At this point one must consult the Wyckoff tables (see Appendix 5) on published crystallographic data, and when this is done, we find the notation tabulated here. 

Table 7.8 The 32 Crystallographic Point Groups, Listed by Main Symmetry Axes or Plane, Using Both the Schoenflies Notation (S, e.g., C2v) and the Hermann-Mauguin or International Notation (HM, e.g., mm2)3...

Most theoretical studies of the electronic properties of haems and haemoproteins have assumed that the chromophore has >4 symmetry. The consequences of the crystallographic results on earlier theoretical treatments have been discussed (9, 10) and the modifications which have to be made are not too serious. We shall frequently use notation, such as symmetry labels, which are appropriate to D 4 symmetry, although sometimes we shall have to consider the effects of a lower symmetry, such as Ci or C%v. It is important to appreciate that the extent of the displacement of the iron atom out of the haem plane is likely to determine the electronic properties of the group to some extent. Moreover, the exact position of the iron atom is likely to be dependent on the axial ligands. 

Thus, if the rotation, inversions and planes of reflections are combined, keeping as sub- indiees the indicative of the rotation axes symmetry order in the group, there is naturally dedueed the complete list of crystallographic class-groups relations in Schoenflies notation. 

It is often said that group 432 is too symmetric to allow piezoelectricity, in spite of the fact that it lacks a center of inversion. It is instructive to see how this comes about. In 1934 Neumann s principle was complemented by a powerful theorem proven by Hermann (1898-1961), an outstanding theoretical physicist with a passionate interest for symmetry, whose name is today mostly connected with the Hermann-Mau-guin crystallographic notation, internationally adopted since 1930. In the special issue on liquid crystals by ZeitschriftfUr Kristal-lographie in 1931 he also derived the 18 symmetrically different possible states for liquid crystals, which could exist between three-dimensional crystals and isotropic liquids [100]. His theorem from 1934 states [101] that if there is a rotation axis C (of order n), then every tensor of rank rcubic crystals, this means that second rank tensors like the thermal expansion coefficient a, the electrical conductivity Gjj, or the dielectric constant e,y, will be isotropic perpendicular to all four space diagonals that have threefold symme-... 

Network morphologies in linear ABC are summarized in detail in the review paper by Meuler et al. (2009). Three types of network morphologies have been known for the equilibrium microdomain structures of linear ABC. Here, we follow their notation and call them and Q °. This naming is based on the crystallographic symmetry of the repeat units. The capital letter O stands for orthorhombic unit cell and Q for cubic unit cell. The numbers in the superscripts indicate the number of the space group and and have Fddd,... 

The comparison of the Hermann-Mauguin notation for crystallographic point groups with the Schoenflies notation for symmetry point groups (Section 3) is apparent fi-om Table 5. 

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