International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

International Tables for Crystallography (2006). Vol. A, ch. 3.1, pp. 46-51

Section 3.1.5. Diffraction symbols and possible space groups

A. Looijenga-Vosa and M. J. Buergerb§

aLaboratorium voor Chemische Fysica, Rijksuniversiteit Groningen, The Netherlands, and bDepartment of Earth and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA, USA

3.1.5. Diffraction symbols and possible space groups

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Table 3.1.4.1[link] contains 219 extinction symbols which, when combined with the Laue classes, lead to 242 different diffraction symbols. If, however, for the monoclinic and orthorhombic systems (as well as for the R space groups of the trigonal system), the different cell choices and settings of one space group are disregarded, 101 extinction symbols1 and 122 diffraction symbols for the 230 space-group types result.

Only in 50 cases does a diffraction symbol uniquely identify just one space group, thus leaving 72 diffraction symbols that correspond to more than one space group. The 50 unique cases can be easily recognized in Table 3.1.4.1[link] because the line for the possible space groups in the particular Laue class contains just one entry.

The non-uniqueness of the space-group determination has two reasons:

  • (i) Friedel's rule, i.e. the effect that, with neglect of anomalous dispersion, the diffraction pattern contains an inversion centre, even if such a centre is not present in the crystal.

    Example

    A monoclinic crystal (with unique axis b) has the diffraction symbol [1{\hbox to 2pt{}}2/m{\hbox to 2pt{}}1P1c1]. Possible space groups are P1c1 (7) without an inversion centre, and [P12/c1\;(13)] with an inversion centre. In both cases, the diffraction pattern has the Laue symmetry [1{\hbox to 2pt{}}2/m{\hbox to 2pt{}}1].

    One aspect of Friedel's rule is that the diffraction patterns are the same for two enantiomorphic space groups. Eleven diffraction symbols each correspond to a pair of enantiomorphic space groups. In Table 3.1.4.1[link], such pairs are grouped between braces. Either of the two space groups may be chosen for structure solution. If due to anomalous scattering Friedel's rule does not hold, at the refinement stage of structure determination it may be possible to determine the absolute structure and consequently the correct space group from the enantiomorphic pair.

  • (ii) The occurrence of four space groups in two `special' pairs, each pair belonging to the same point group: I222 (23) & [I2_{1}2_{1}2_{1}] (24) and I23 (197) & [I2_{1}3] (199). The two space groups of each pair differ in the location of the symmetry elements with respect to each other. In Table 3.1.4.1[link], these two special pairs are given in square brackets.








































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