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

International Tables for Crystallography (2006). Vol. F, ch. 11.3, p. 219   | 1 | 2 |

Section Spot centroids and partiality

W. Kabscha*

aMax-Planck-Institut für medizinische Forschung, Abteilung Biophysik, Jahnstrasse 29, 69120 Heidelberg, Germany
Correspondence e-mail: Spot centroids and partiality

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The intensity of a reflection can be completely recorded on one image, or distributed among several adjacent images. The fraction [R_{j}] of total intensity recorded on image j, the `partiality' of the reflection, can be derived from the distribution function [\omega (\varepsilon_{1}, \varepsilon_{2}, \varepsilon_{3})] as [\eqalign{R_{j} &= {\textstyle\int\limits_{- \infty}^{\infty}} \hbox{d}\varepsilon_{1} {\textstyle\int\limits_{- \infty}^{\infty}} \hbox{d}\varepsilon_{2} {\textstyle\int\limits_{\zeta [\varphi_{0} + (j - 1)\Delta_{\varphi} - \varphi]}^{\zeta (\varphi_{0} + j\Delta_{\varphi} - \varphi)}} \hbox{d}\varepsilon_{3} \;\omega(\varepsilon_{1}, \varepsilon_{2}, \varepsilon_{3})\cr &= \{[1/(2\pi )^{1/2} \sigma_{M}]/|\zeta|\} \cr&\quad\times{\textstyle\int\limits_{\varphi_{0} + (j - 1)\Delta_{\varphi}}^{\varphi_{0} + j\Delta_{\varphi}}} \exp[- (\varphi' - \varphi)^{2}/2(\sigma_{M}/|\zeta|)^{2}] \;\hbox{d}\varphi'\cr &= \left(\hbox{erf}[|\zeta|(\varphi_{0} + j\Delta_{\varphi} - \varphi)/(2)^{1/2} \sigma_{M}]\right. \cr&\left.\quad-\;\hbox{erf}\{|\zeta|[\varphi_{0} + (j - 1)\Delta_{\varphi} - \varphi]/(2)^{1/2} \sigma_{M}\}\right)\big/2.}] The integral is evaluated by using a numerical approximation of the error function, erf (Abramowitz & Stegun, 1972[link]).

While the spot centroids in the detector plane are usually good estimates for the detector position of the diffraction maximum, the angular centroid about the rotation axis, [Z = \varphi_{0} + \Delta_{\varphi} \cdot {\textstyle\sum\limits_{j = -\infty}^{\infty}} (j - 1/2) R_{j} \approx \varphi,] can be a rather poor guess for the true ϕ angle of the maximum. Its accuracy depends strongly on the value of ϕ and the size of the oscillation range [\Delta_{\varphi}] relative to the mosaicity [\sigma_{M}] of the crystal. For a reflection fully recorded on image j, the value [Z = \varphi_{0} + {(\;j - 1/2) \cdot \Delta_{\varphi}}] will always be obtained, which is correct only if ϕ accidentally happens to be close to the centre of the rotation range of the image. In contrast, the ϕ angle of a partial reflection recorded on images j and [j + 1] is closely approximated by [Z = \varphi_{0} + [\;j + (R_{j + 1} - R_{j})/2] \cdot \Delta_{\varphi}]. If many images contribute to the spot intensity, [Z(\varphi)] is always an excellent approximation to the ideal angular position ϕ when the Laue equations are satisfied; in fact, in the limiting case of infinitely fine-sliced data, it can be shown that [\lim_{\Delta_{\varphi} \rightarrow 0}Z(\varphi) = \varphi].

Most refinement routines minimize the discrepancies between the predicted ϕ angles and their approximations obtained from the observed Z centroids, and must therefore carefully distinguish between fully and partially recorded reflections. This distinction is unnecessary, however, if observed Z centroids are compared with their analytic forms instead, because the sensitivity of the centroid positions to the diffraction parameters is correctly weighted in either case (see Section[link]).


Abramowitz, M. & Stegun, I. A. (1972). Handbook of mathematical functions. New York: Dover Publications.Google Scholar

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