Space group symmetry symbols

The second part of a space group symbol consists of one, two or three entries after the initial letter symbol described above. At each position, the entry consists of one or two characters, describing a symmetry element, either an axis or a plane. 

Many space groups contain more symmetry elements than that given in the space group full symbol. The standard notation gives only the essential symmetry operators, which, when applied, allow all of the other symmetry operations to be recovered. The symbols used to represent the symmetry operations are set out in Table 5.3. 

As described previously, the position of the symmetry element in the overall symbol has a structural significance that varies from one crystal system to another as set out in Table 5.4. 

6 ra-fold rotation axis Rotation counter clockwise of 360°/n 

2(= m), 3, 4, 6 ra-fold inversion (rotoinversion) axis 360° jn rotation counter clockwise followed by inversion 

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Headline. Provides the short international (Hermann-Mauguin) space group symmetry symbol (Cmm2) followed by the SchOnflies symbol of... 

In referring to any particular space-group, the symbols for the symmetry elements are put together in a way similar to that used for the point-groups. First comes a capital letter indicating whether the lattice is simple (P for primitive), body-centred (I for inner), side-centred (A, B, or C), or centred on all faces (F). The rhombohedral lattice is also described by a special letter R. Following the capital letter for the lattice type comes the symbol for the principal axis, and if there is a plane of symmetry or a glide plane perpendicular to it, the two symbols... 

TABLE 11.6 Standard Symbols for Space Group Symmetry Elements... 

These rules not only reflect how the symbol of the space group symmetry... 

Fi re 1.45. The distribution of the inversion centers in the triclinic space group symmetry P1. Eight independent centers are labeled from a through h . Inversion centers that are equivalent to one another are marked using symbols of the same size and shading. The invisible centers are drawn using dotted lines. 

A description of these symmetry elements and their symbols may be found in Figs. 2.6-2.S and in Tables 2.4 and 2.5. This correspondence allows us to characterize space groups with symbols analogous to those used for the plane groups (Section 2.7.1). The international symbolfor a space group is composed of ... 

Table 2.6. Space Group Symmetry Element Symbolism...

FIGURE 9 Tetragonal glitter network of carbon, with Wells point symbol given by (6 8)2, and of space group symmetry (P42/mmc). 

Find out which symmetry elements are present in the structures of the following compounds. Derive the Hermann-Mauguin symbol of the corresponding space group (it may be helpful to consult International Tables for Crystallography, Vol. A). 

Every space group listed in the family tree corresponds to a structure. Since the space group symbol itself states only symmetry, and gives no information about the atomic positions, additional information concerning these is necessary for every member of the family tree (Wyckoff symbol, site symmetry, atomic coordinates). The value of information of a tree is rather restricted without these data. In simple cases the data can be included in the family tree in more complicated cases an additional table is convenient. The following examples show how specifications can be made for the site occupations. Because they are more informative, it is advisable to label the space groups with their full Hermann-Mauguin symbols. 

Note 3 The smectic C structure corresponds to monoclinic symmetry characterised by the symbol C2h, in the Schoenflies notation and the space group t Hm in the International System. 

The elementary unit cell can be quite easily described starting from the four mineral ion sites of the crystal F, Ca f+, Ca(ll)2+ and P04 , where the symbols I and II represent the two different crystallographic sites of the cations, with the application of all the symmetry operations relevant to the space group P63/m. Among the principal symmetry elements, one can cite mirror planes perpendicular to the c-axis (at z = 1/4 and 3/4), which contain most of the ions of the structure (F , Ca +(ll), P04 ), three-fold axes parallel to the c-axis (at x = 1/3, y = 2/3 and x = 2/3, y = 1/3) along which are located the Ca + (I) ions, screw axes 63 at the corners of the unit cell and parallel to the c-axis and screw axes 2i parallel to the c-axis and located at the midpoints of the cell edges and at the centre of the unit cell itself [3]. 

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