International
Tables for Crystallography Volume F Crystallography of biological macromolecules Edited by M. G. Rossmann and E. Arnold © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. F. ch. 9.1, pp. 187-188
Section 9.1.7.4. Alternative indexing
a
National Cancer Institute, Brookhaven National Laboratory, NSLS, Building 725A-X9, Upton, NY 11973, USA, and bStructural Biology Laboratory, Department of Chemistry, University of York, York YO10 5DD, England |
If the crystal point-group symmetry is lower than the symmetry of its Bravais lattice, then the reflections can be indexed in more than one way. In other words, the symmetry of the reflection positions is higher than the symmetry of the distribution of their intensities. This situation typically arises for point groups with polar axes, such as groups 3, 4 or 6, which can be indexed with the c axis pointing in either one of two directions. The lattice does not define the directionality of such axes if its two remaining cell dimensions are equivalent. This problem does not occur in the monoclinic system, despite the polar twofold axis, as the two other axes are not equivalent. The most complex case is point group 3, which can be indexed in the 622 lattice in four non-equivalent ways. The other such groups have only two alternatives.
There is an analogous problem for cubic space groups within point group 23. Here the lattice possesses fourfold symmetry, but the intensity distribution has only twofold symmetry. Rotation by 90° leads to alternative, although perfectly permitted, indexing of reflections.
Each allowed scheme is permitted and self-consistent for a single crystal, since all possibilities will perfectly match the crystal lattice. However, under alternative indexing schemes, the same reflection will be given different indices, which can pose problems when data from more than one crystal are to be merged or compared. Merging is needed when more than one sample is required to record a complete data set. Comparison is needed when looking for heavy-atom derivatives or for ligand complexes with isomorphous crystals. For these, the reflections of one crystal must be selected as a standard, and it is easy to make other crystals consistent with this standard either by changing the orientation matrix at the time of intensity integration or by applying re-indexing to the integrated intensity set. The alternative indexing schemes are related by those symmetry operations present within the higher symmetry of the Bravais lattice but absent from the point-group symmetry. The point groups with alternative indexing systems are shown in Table 9.1.7.3, together with the necessary symmetry operations for re-indexing.
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Several experiments require the recording of multiple data sets from the same crystal. One example is the collection of more than one pass with different exposure times (see below), and a second is in multiwavelength anomalous dispersion (MAD) experiments. In these experiments, the software systems may independently choose any of the alternative systems for different sets, which may then be incompatible and need re-indexing. It is much simpler to ensure a common orientation matrix modified as appropriate for all sets at the time of intensity integration.