International
Tables for Crystallography Volume A Space-group symmetry Edited by Th. Hahn © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. A, ch. 2.2, pp. 26-27
Section 2.2.9. Symmetry operations^{a}Institut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany, and ^{b}Laboratorium voor Chemische Fysica, Rijksuniversiteit Groningen, The Netherlands |
As explained in Sections 8.1.6 and 11.1.1 , the coordinate triplets of the General position of a space group may be interpreted as a shorthand description of the symmetry operations in matrix notation. The geometric description of the symmetry operations is found in the space-group tables under the heading Symmetry operations.
The numbering of the entries in the blocks Symmetry operations and General position (first block below Positions) is the same. Each listed coordinate triplet of the general position is preceded by a number between parentheses (p). The same number (p) precedes the corresponding symmetry operation. For space groups with primitive cells, both lists contain the same number of entries.
For space groups with centred cells, to the one block General position several (2, 3 or 4) blocks Symmetry operations correspond. The numbering scheme of the general position is applied to each one of these blocks. The number of blocks equals the multiplicity of the centred cell, i.e. the number of centring translations below the subheading Coordinates, such as .
Whereas for the Positions the reader is expected to add these centring translations to each printed coordinate triplet himself (in order to obtain the complete general position), for the Symmetry operations the corresponding data are listed explicitly. The different blocks have the subheadings `For (0,0,0) set', `For set', etc. Thus, an obvious one-to-one correspondence exists between the analytical description of a symmetry operation in the form of its general-position coordinate triplet and the geometrical description under Symmetry operations. Note that the coordinates are reduced modulo 1, where applicable, as shown in the example below.
Example: Ibca (73)
The centring translation is . Accordingly, above the general position one finds and . In the block Symmetry operations, under the subheading `For set', entry (2) refers to the coordinate triplet . Under the subheading `For set', however, entry (2) refers to . The triplet is selected rather than , because the coordinates are reduced modulo 1.
In space groups with two origins where a `symmetry element' and an `additional symmetry element' are of different type (e.g. mirror versus glide plane, rotation versus screw axis, Tables 4.1.2.2 and 4.1.2.3 ), the origin shift may interchange the two different types in the same location (referred to the appropriate origin) under the same number (p). Thus, in (129), (p) = (7) represents a and a 2_{1} axis, both in , whereas (p) = (16) represents a g and an m plane, both in .
An entry in the block Symmetry operations is characterized as follows.
Details of this symbolism are presented in Section 11.1.2 .
Examples