Coordinate systems and parameters

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. 218 | 1 | 2 |

Section 11.3.2.1. Coordinate systems and parameters

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

11.3.2.1. Coordinate systems and parameters

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In the rotation method, the incident beam wave vector $[{\bf S}_{0}]$ of length $[1/\lambda]$ (λ is the wavelength) is fixed while the crystal is rotated around a fixed axis described by a unit vector $[{\bf m}_{2}]$ . $[{\bf S}_{0}]$ points from the X-ray source towards the crystal. It is assumed that the incident beam and the rotation axis intersect at one point at which the crystal must be located. This point is defined as the origin of a right-handed orthonormal laboratory coordinate system $[\{{\bf l}_{1},{\bf l}_{2},{\bf l}_{3}\}]$ . This fixed but otherwise arbitrary system is used as a reference frame to specify the setup of the diffraction experiment.

Diffraction data are assumed to be recorded on a fixed planar detector. A right-handed orthonormal detector coordinate system $[\{{\bf d}_{1},{\bf d}_{2},{\bf d}_{3}\}]$ is defined such that a point with coordinates X, Y in the detector plane is represented by the vector $[(X - X_{0}){\bf d}_{1} + {(Y - Y_{0}){\bf d}_{2} + F{\bf d}_{3}}]$ with respect to the laboratory coordinate system. The origin $[X_{0}, Y_{0}]$ of the detector plane is found at a distance [|F|] from the crystal position. It is assumed that the diffraction data are recorded on adjacent non-overlapping rotation images, each covering a constant oscillation range $[\Delta_{\varphi}]$ with image No. 1 starting at spindle angle $[\varphi_{0}]$ .

Diffraction geometry is conveniently expressed with respect to a right-handed orthonormal goniostat system $[\{{\bf m}_{1}, {\bf m}_{2}, {\bf m}_{3}\}]$ . It is constructed from the rotation axis and the incident beam direction such that $[{\bf m}_{1} = ({\bf m}_{2}\times {\bf S}_{0})/|{\bf m}_{2} \times {\bf S}_{0}|]$ and $[{\bf m}_{3} = {\bf m}_{1} \times {\bf m}_{2}]$ . The origin of the goniostat system is defined to coincide with the origin of the laboratory system.

Finally, a right-handed crystal coordinate system $[\{{\bf b}_{1}, {\bf b}_{2}, {\bf b}_{3}\}]$ and its reciprocal basis $[\{{\bf b}_{1}^{*}, {\bf b}_{2}^{*}, {\bf b}_{3}^{*}\}]$ are defined to represent the unrotated crystal, i.e., at rotation angle $[\varphi = 0^{\circ}]$ , such that any reciprocal-lattice vector can be expressed as $[{\bf p}_{0}^{*} = h{\bf b}_{1}^{*} + k{\bf b}_{2}^{*} + l{\bf b}_{3}^{*}]$ where [h, k, l] are integers.

Using a Gaussian model, the shape of the diffraction spots is specified by two parameters: the standard deviations of the reflecting range $[\sigma_{M}]$ and the beam divergence $[\sigma_{D}]$ (see Section 11.3.2.3). This leads to an integration region around the spot defined by the parameters $[\delta_{M}]$ and $[\delta_{D}]$ , which are typically chosen to be 6–10 times larger than $[\sigma_{M}]$ and $[\sigma_{D}]$ , respectively.

Knowledge of the parameters $[{\bf S}_{0}]$ , $[{\bf m}_{2}]$ , $[{\bf b}_{1}]$ , $[{\bf b}_{2}]$ , $[{\bf b}_{3}]$ , $[X_{0}]$ , $[Y_{0}, F]$ , $[{\bf d}_{1}]$ , $[{\bf d}_{2}]$ , $[{\bf d}_{3}]$ , $[\varphi_{0}]$ and $[\Delta_{\varphi}]$ is sufficient to compute the location of all diffraction peaks recorded in the data images. Determination and refinement of these parameters are described in the following sections.

References

International Tables for Crystallography (2006). Vol. F. ch. 11.3, p. 218