Standard spot shape

Kabsch, W.

doi:10.1107/97809553602060000676

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

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

Section 11.3.2.3. Standard spot shape

W. Kabsch^a ^*

^a Max-Planck-Institut für medizinische Forschung, Abteilung Biophysik, Jahnstrasse 29, 69120 Heidelberg, Germany
Correspondence e-mail: kabsch@mpimf-heidelberg.mpg.de

11.3.2.3. Standard spot shape

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A reciprocal-lattice point crosses the Ewald sphere by the shortest route only if the crystal happens to be rotated about an axis perpendicular to both the diffracted and incident beam wave vectors, the `β-axis' $[{\bf e}_{1} = {\bf S} \times {\bf S}_{0}/|{\bf S} \times {\bf S}_{0}|]$ , as introduced by Schutt & Winkler (1977). Rotation around the fixed axis $[{\bf m}_{2}]$ , as enforced by the rotation camera, thus leads to an increase in the length of the shortest path by the factor $[1/|{\bf e}_{1} \cdot {\bf m}_{2}|]$ . This has motivated the introduction of a coordinate system $[\{{\bf e}_{1}, {\bf e}_{2}, {\bf e}_{3}\}]$ , specific for each reflection, which has its origin on the surface of the Ewald sphere at the terminus of the diffracted beam wave vector S, $[\displaylines{{\bf e}_{1} = {\bf S} \times {\bf S}_{0}/|{\bf S} \times {\bf S}_{0}|, \quad {\bf e}_{2} = {\bf S} \times {\bf e}_{1}/|{\bf S} \times {\bf e}_{1}|,\cr \quad {\bf e}_{3} = ({\bf S} + {\bf S}_{0})/|{\bf S} + {\bf S}_{0}|.}]$ The unit vectors $[{\bf e}_{1}]$ and $[{\bf e}_{2}]$ are tangential to the Ewald sphere, while $[{\bf e}_{3}]$ is perpendicular to $[{\bf e}_{1}]$ and $[{\bf p}^{*} = {\bf S} - {\bf S}_{0}]$ . The shape of a reflection, as represented with respect to $[\{{\bf e}_{1}, {\bf e}_{2}, {\bf e}_{3}\}]$ , then no longer contains geometrical distortions resulting from the fixed rotation axis of the camera and the oblique incidence of the diffracted beam on a flat detector. Instead, all reflections appear as if they had followed the shortest path through the Ewald sphere and had been recorded on the surface of the sphere.

A detector pixel at X′, Y′ in the neighbourhood of the reflection centre X, Y, when the crystal is rotated by φ′ instead of φ, is mapped to the profile coordinates $[\varepsilon_{1}, \varepsilon_{2}, \varepsilon_{3}]$ by the following procedure: $[\eqalign{{\bf S}' &= [(X' - X_{0}){\bf d}_{1} + (Y' - Y_{0}){\bf d}_{2} + F{\bf d}_{3}]\cr &\quad\times \{\lambda \cdot [(X' - X_{0})^{2} + (Y' - Y_{0})^{2} + F^{2}]^{1/2}\}^{-1}\cr \varepsilon_{1} &= {\bf e}_{1} \cdot ({\bf S}' - {\bf S})180/(|{\bf S}|\pi ),\cr \varepsilon_{2} &= {\bf e}_{2} \cdot ({\bf S}' - {\bf S})180/(|{\bf S}|\pi )\cr \varepsilon_{3} &= {\bf e}_{3} \cdot [D({\bf m}_{2}, \varphi' - \varphi){\bf p}^{*} - {\bf p}^{*}]180/(|{\bf p}^{*}|\pi ) \simeq \zeta \cdot (\varphi' - \varphi)\cr \zeta &= {\bf m}_{2} \cdot {\bf e}_{1}.}]$ ζ corrects for the increased path length of the reflection through the Ewald sphere and is closely related to the reciprocal Lorentz correction factor $[L^{-1} = |{\bf m}_{2} \cdot ({\bf S} \times {\bf S}_{0})|/(|{\bf S}| \cdot |{\bf S}_{0}|) = |\zeta \cdot \sin \angle ({\bf S},{\bf S}_{0})|.]$

Because of crystal mosaicity and beam divergence, the intensity of a reflection is smeared around the diffraction maximum. The fraction of total reflection intensity found in the volume element $[\hbox{d}\varepsilon_{1} \hbox{d}\varepsilon_{2} \hbox{d}\varepsilon_{3}]$ at $[\varepsilon_{1}, \varepsilon_{2}, \varepsilon_{3}]$ can be approximated by Gaussian functions: $[\displaylines{\omega(\varepsilon_{1}, \varepsilon_{2}, \varepsilon_{3})\hbox{d}\varepsilon_{1} \hbox{d}\varepsilon_{2} \hbox{d}\varepsilon_{3}\hfill\cr \quad= {\exp(- \varepsilon_{1}^{2}/2\sigma_{D}^{2}) \over (2\pi )^{1/2} \sigma_{D}} \hbox{d}\varepsilon_{1} \cdot {\exp(- \varepsilon_{2}^{2}/2\sigma_{D}^{2}) \over (2\pi )^{1/2} \sigma_{D}} \hbox{d}\varepsilon_{2} \cdot {\exp(- \varepsilon_{3}^{2}/2\sigma_{M}^{2}) \over (2\pi )^{1/2} \sigma_{M}} \hbox{d}\varepsilon_{3}.}]$

References

Schutt, C. & Winkler, F. K. (1977). The oscillation method for very large unit cells. In The rotation method in crystallography, edited by U. W. Arndt & A. J. Wonacott, pp. 173–186. Amsterdam, New York, Oxford: North-Holland.Google Scholar

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