International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by M. G. Rossmann and E. Arnold

International Tables for Crystallography (2006). Vol. F, ch. 11.5, pp. 236-237   | 1 | 2 |

Section 11.5.2. Generalization of the Hamilton, Rollett and Sparks equations to take into account partial reflections

C. G. van Beek,a R. Bolotovskya§ and M. G. Rossmanna*

aDepartment of Biological Sciences, Purdue University, West Lafayette, IN 47907-1392, USA
Correspondence e-mail:  mgr@indiana.bio.purdue.edu

11.5.2. Generalization of the Hamilton, Rollett and Sparks equations to take into account partial reflections

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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 [I_{him}] represent the intensity contribution of reflection [h_{i}] recorded on frame m; if all the parts of [h_{i}] are available in the data set, then [I_{hi} = {\textstyle\sum\limits_{m}} (I_{him}/G_{m}). \eqno(11.5.2.1)] 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[link].

Various models have been proposed to calculate the reflection partiality. Here we use Rossmann's model (Rossmann, 1979[link]; Rossmann et al., 1979[link]) with Greenhough & Helliwell's (1982[link]) 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[link]) and Bolotovsky & Coppens (1997[link]).

Provided the reflection partiality, [p_{him}], is known, the full intensity is estimated by [I_{hi} = I_{him}\big/p_{him} G_{m}. \eqno(11.5.2.2)] This expression can produce as many estimates of [I_{hi}] as there are parts of reflection [h_{i}], while expression (11.5.2.1[link]) produces only one estimate of [I_{hi}] when all parts of reflection [h_{i}] are recorded. Having defined the relationships between measured intensities of partial reflections and estimated full reflections by expressions (11.5.2.1[link]) and (11.5.2.2[link]), two methods of generalizing the HRS equations can be considered.

  • Method 1. If a reflection [h_{i}] occurs on a number of consecutive frames and all parts of [I_{him}] are available in the data set, the generalized HRS target equation takes the form [\psi = {\textstyle\sum\limits_{h}} {\textstyle\sum\limits_{i}} {\textstyle\sum\limits_{m}} W_{him} \left\{I_{him} - G_{m} \left[I_{h} - {\textstyle\sum\limits_{m' \neq m}} (I_{him'}\big/G_{m'})\right]\right\}^{2}. \eqno(11.5.2.3)] Using expression (11.5.1.2)[link], the best least-squares estimate of [I_{h}] will be [{I_{h} = {{\textstyle\sum_{i}} \left[{\textstyle\sum_{m}} (I_{him}/G_{m})\right] \left({\textstyle\sum_{m}} W_{him} G_{m}^{2} \right) \over {\textstyle\sum_{i}} {\textstyle\sum_{m}} W_{him} G_{m}^{2}} = {{\textstyle\sum_{i}} I_{hi} {\textstyle\sum_{m}} W_{him} G_{m}^{2} \over {\textstyle\sum_{i}} {\textstyle\sum_{m}} W_{him} G_{m}^{2}}.} \eqno(11.5.2.4)]

  • Method 2. If the theoretical partiality, [p_{him}], of the partially recorded reflection [h_{im}] can be estimated, the generalized HRS target equation takes the form [\psi = {\textstyle\sum\limits_{h}} {\textstyle\sum\limits_{i}} {\textstyle\sum\limits_{m}} W_{him} (I_{him} - G_{m} p_{him} I_{h})^{2} \eqno(11.5.2.5)] and, using expression (11.5.1.2[link]), the best least-squares estimate of [I_{h}] will then be [I_{h} = {{\textstyle\sum_{i}} {\textstyle\sum_{m}} W_{him} G_{m} p_{him} I_{him} \over {\textstyle\sum_{i}} {\textstyle\sum_{m}} W_{him} G_{m}^{2} p_{him}^{2}}. \eqno(11.5.2.6)] When all reflections in the data set are fully recorded, expressions (11.5.2.3[link]) and (11.5.2.5[link]) reduce to the `classical' HRS expression (11.5.1.1[link]), and expressions (11.5.2.4[link]) and (11.5.2.6[link]) reduce to expression (11.5.1.3[link]).

The scale factor [G_{m}] can be generalized to incorporate crystal decay (Gewirth, 1996[link]; Otwinowski & Minor, 1997[link]): [G_{him} = G_{m} \exp \left\{ - 2B_{m} \left[\sin (\theta_{hi})/\lambda\right]^{2}\right\}, \eqno(11.5.2.7)] where [B_{m}] is a parameter describing the crystal disorder while frame m was recorded, [\theta_{hi}] is the Bragg angle of reflection [h_{i}] and λ is the X-ray wavelength.

Method 1[link] only allows the refinement of the scale factors while method 2[link] allows refinement of the scale factors, crystal mosaicity and orientation matrix, as the latter two factors contribute to the calculated partiality. Furthermore, method 2[link] 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 [h_{i}] 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 [h_{im}] to expression (11.5.2.3[link]) will be zero, as in this case [I_{h}] will be the same as [I_{hi}]. In contrast, in method 2[link] the reflection [h_{i}] can be used for scaling because the estimates of the full intensity [I_{hi}] are calculated independently from every frame spanned by reflection [h_{i}].

References

Bolotovsky, R. & Coppens, P. (1997). The ϕ extent of the reflection range in the oscillation method according to the mosaicity-cap model. J. Appl. Cryst. 30, 65–70.
Gewirth, D. (1996). The HKL manual. A description of the programs DENZO, XDSPLAYF and SCALEPACK, 5th ed., pp. 87–90. Yale University, New Haven, USA.
Greenhough, T. J. & Helliwell, J. R. (1982). Oscillation camera data processing: reflecting range and prediction of partiality. I. Conventional X-ray sources. J. Appl. Cryst. 15, 338–351.
Otwinowski, Z. & Minor, W. (1997). Processing of X-ray diffraction data collected in oscillation mode. Methods Enzymol. 276, 307–326.
Rossmann, M. G. (1979). Processing oscillation diffraction data for very large unit cells with an automatic convolution technique and profile fitting. J. Appl. Cryst. 12, 225–238.
Rossmann, M. G., Leslie, A. G. W., Abdel-Meguid, S. S. & Tsukihara, T. (1979). Processing and post-refinement of oscillation camera data. J. Appl. Cryst. 12, 570–581.
Winkler, F. K., Schutt, C. E. & Harrison, S. C. (1979). The oscillation method for crystals with very large unit cells. Acta Cryst. A35, 901–911.








































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