Tables for
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:

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(] 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(] This expression can produce as many estimates of [I_{hi}] as there are parts of reflection [h_{i}], while expression ([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 ([link]) and ([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(] Using expression ([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(]

  • 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(] and, using expression ([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(] When all reflections in the data set are fully recorded, expressions ([link]) and ([link]) reduce to the `classical' HRS expression ([link]), and expressions ([link]) and ([link]) reduce to expression ([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(] 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 ([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}].


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.
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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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