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. 11.5, pp. 236-237
Section 11.5.2. Generalization of the Hamilton, Rollett and Sparks equations to take into account partial reflections^{a}Department of Biological Sciences, Purdue University, West Lafayette, IN 47907-1392, USA |
11.5.2. Generalization of the Hamilton, Rollett and Sparks equations to take into account partial reflections
When a Bragg reflection is completely exposed within the oscillation range of one frame, a so-called `full reflection', it gives rise to the `full intensity'. In general, a Bragg reflection will occur on a number of consecutive frames as a series of partial reflections, and the full intensity can only be estimated from the measured intensities of the partial reflections. Let represent the intensity contribution of reflection recorded on frame m; if all the parts of are available in the data set, then In practice, there will always be reflections that do not have all their parts available. In such cases, the only way to estimate the full intensity of a reflection is to apply an estimated value of the partiality to the measured reflection intensities.
Various models have been proposed to calculate the reflection partiality. Here we use Rossmann's model (Rossmann, 1979; Rossmann et al., 1979) with Greenhough & Helliwell's (1982) correction. This model treats partiality as a fraction of a spherical volume swept through the Ewald sphere. The coordinates of the reciprocal-lattice point are defined by the Miller indices of the reflection, the crystal orientation matrix and the rotation angle. The volume of the sphere around the reciprocal-lattice point accounts for crystal mosaicity and beam divergence. Alternative geometrical descriptions of a reciprocal-lattice point passing through the Ewald sphere have been given by Winkler et al. (1979) and Bolotovsky & Coppens (1997).
Provided the reflection partiality, , is known, the full intensity is estimated by This expression can produce as many estimates of as there are parts of reflection , while expression (11.5.2.1) produces only one estimate of when all parts of reflection are recorded. Having defined the relationships between measured intensities of partial reflections and estimated full reflections by expressions (11.5.2.1) and (11.5.2.2), two methods of generalizing the HRS equations can be considered.
The scale factor can be generalized to incorporate crystal decay (Gewirth, 1996; Otwinowski & Minor, 1997): where is a parameter describing the crystal disorder while frame m was recorded, is the Bragg angle of reflection and λ is the X-ray wavelength.
Method 1 only allows the refinement of the scale factors while method 2 allows refinement of the scale factors, crystal mosaicity and orientation matrix, as the latter two factors contribute to the calculated partiality. Furthermore, method 2 is essential for scaling of data sets with low redundancy (e.g. data collected from low-symmetry crystals or data collected over small rotation ranges). When a reflection spans more than one frame, but there are no other reflections with the same reduced Miller indices h in the data set, the contribution of any partial reflection to expression (11.5.2.3) will be zero, as in this case will be the same as . In contrast, in method 2 the reflection can be used for scaling because the estimates of the full intensity are calculated independently from every frame spanned by reflection .
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