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

International Tables for Crystallography (2006). Vol. A, ch. 3.1, pp. 51-53

Section 3.1.6. Space-group determination by additional methods

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

3.1.6. Space-group determination by additional methods

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3.1.6.1. Chemical information

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In some cases, chemical information determines whether or not the space group is centrosymmetric. For instance, all proteins crystallize in noncentrosymmetric space groups as they are constituted of L-amino acids only. Less certain indications may be obtained by considering the number of molecules per cell and the possible space-group symmetry. For instance, if experiment shows that there are two molecules of formula [A_{\alpha}B_{\beta}] per cell in either space group [P2_{1}] or [P2_{1}/m] and if the molecule [A_{\alpha}B_{\beta}] cannot possibly have either a mirror plane or an inversion centre, then there is a strong indication that the correct space group is [P2_{1}]. Crystallization of [A_{\alpha}B_{\beta}] in [P2_{1}/m] with random disorder of the molecules cannot be excluded, however. In a similar way, multiplicities of Wyckoff positions and the number of formula units per cell may be used to distinguish between space groups.

3.1.6.2. Point-group determination by methods other than the use of X-ray diffraction

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This is discussed in Chapter 10.2[link] . In favourable cases, suitably chosen methods can prove the absence of an inversion centre or a mirror plane.

3.1.6.3. Study of X-ray intensity distributions

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X-ray data can give a strong clue to the presence or absence of an inversion centre if not only the symmetry of the diffraction pattern but also the distribution of the intensities of the reflection spots is taken into account. Methods have been developed by Wilson and others that involve a statistical examination of certain groups of reflections. For a textbook description, see Lipson & Cochran (1966)[link] and Wilson (1970)[link]. In this way, the presence of an inversion centre in a three-dimensional structure or in certain projections can be tested. Usually it is difficult, however, to obtain reliable conclusions from projection data. The same applies to crystals possessing pseudo-symmetry, such as a centrosymmetric arrangement of heavy atoms in a noncentrosymmetric structure. Several computer programs performing the statistical analysis of the diffraction intensities are available.

3.1.6.4. Consideration of maxima in Patterson syntheses

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The application of Patterson syntheses for space-group determination is described by Buerger (1950[link], 1959[link]).

3.1.6.5. Anomalous dispersion

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Friedel's rule, [|F(hkl)|^{2} = |F(\bar{h}\bar{k}{\hbox to 1pt{}}\bar{l})|^{2}], does not hold for noncentrosymmetric crystals containing atoms showing anomalous dispersion. The difference between these intensities becomes particularly strong when use is made of a wavelength near the resonance level (absorption edge) of a particular atom in the crystal. Synchrotron radiation, from which a wide variety of wavelengths can be chosen, may be used for this purpose. In such cases, the diffraction pattern reveals the symmetry of the actual point group of the crystal (including the orientation of the point group with respect to the lattice).

3.1.6.6. Summary

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One or more of the methods discussed above may reveal whether or not the point group of the crystal has an inversion centre. With this information, in addition to the diffraction symbol, 192 space groups can be uniquely identified. The rest consist of the eleven pairs of enantiomorphic space groups, the two `special pairs' and six further ambiguities: 3 in the orthorhombic system (Nos. 26 & 28, 35 & 38, 36 & 40), 2 in the tetragonal system (Nos. 111 & 115, 119 & 121), and 1 in the hexagonal system (Nos. 187 & 189). If not only the point group but also its orientation with respect to the lattice can be determined, the six ambiguities can be resolved. This implies that 204 space groups can be uniquely identified, the only exceptions being the eleven pairs of enantiomorphic space groups and the two `special pairs'.

References

Buerger, M. J. (1950). The crystallographic symmetries determinable by X-ray diffraction. Proc. Natl Acad. Sci. USA, 36, 324–329.
Buerger, M. J. (1959). Vector space, pp. 167–168. New York: Wiley.
Lipson, H. & Cochran, W. (1966). The determination of crystal structures, Chaps. 3 and 4.4. London: Bell.
Wilson, A. J. C. (1970). Elements of X-ray crystallography, Chap. 8. Reading, MA: Addison Wesley.








































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