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International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. C. ch. 1.4, p. 21
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In the absence of dispersion (`anomalous scattering'), the intensities of the reflections hkl and ![[\bar {h}\bar {k}{\bar l}]](/teximages/cbch1o4/cbch1o4fi252.svg)
are equal (Friedel's law), and statements about the symmetry of the weighted reciprocal lattice and quantities derived from it often rest on the tacit or explicit assumption of this law – the condition underlying it being forgotten. In particular, if dispersion is appreciable, the symmetry of the Patterson synthesis and the `Laue' symmetry are altered.
In Volume A of International Tables, the symmetry of the Patterson synthesis is derived in two stages. First, any glide planes and screw axes are replaced by mirror planes and the corresponding rotation axes, giving a symmorphic space group (Subsection 1.4.2.1![[link]](/graphics/greenarr.gif)
). Second, a centre of symmetry is added. This second step involves the tacit assumption of Friedel's law, and should not be taken if any atomic scattering factors have appreciable imaginary components. In such cases, the symmetry of the Patterson synthesis will not be that of one of the 24 centrosymmetric symmorphic space groups, as given in Volume A, but will be that of the symmorphic space group belonging to the arithmetic crystal class to which the space group of the structure belongs. There are thus 73 possible Patterson symmetries.
An equivalent description of such symmetries, in terms of 73 of the 1651 dichromatic colour groups, has been given by Fischer & Knop (1987); see also Wilson (1993).
Similarly, the eleven conventional `Laue' symmetries [International Tables for Crystallography (2005), Volume A, Section 3.1.2
and elsewhere] involve the explicit assumption of Friedel's law. If dispersion is appreciable, the `Laue' symmetry may be that of any of the 32 point groups. The point group, in correct orientation, is obtained by dropping the Bravais-lattice symbol from the symbol of the arithmetic crystal class or of the Patterson symmetry.
References
Fischer, K. F. & Knop, W. E. (1987). Space groups for imaginary Patterson and for difference Patterson functions in the lambda technique. Z. Kristallogr. 180, 237–242.Google Scholar
International Tables for Crystallography
(2005). Vol. A. Space-group symmetry, fifth ed., edited by Th. Hahn. Heidelberg: Springer.Google Scholar
Wilson, A. J. C. (1993). Laue and Patterson symmetry in the complex case. Z. Kristallogr. 208, 199–206.Google Scholar
