Spot prediction

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, pp. 218-219 | 1 | 2 |

Section 11.3.2.2. Spot prediction

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

11.3.2.2. Spot prediction

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It is assumed here that accurate values of all parameters describing the diffraction experiment are available, permitting prediction of the positions of all diffraction peaks recorded in the data images. Let $[{\bf p}^{*}_{0}]$ denote any arbitrary reciprocal-lattice vector if the crystal has not been rotated, i.e., at rotation angle $[\varphi = 0^{\circ}]$ . $[{\bf p}^{*}_{0}]$ can be expressed by its components with respect to the orthonormal goniostat system as $[{\bf p}^{*}_{0} = {\bf m}_{1} ({\bf m}_{1} \cdot {\bf p}^{*}_{0}) + {\bf m}_{2}({\bf m}_{2} \cdot {\bf p}^{*}_{0}) + {\bf m}_{3}({\bf m}_{3} \cdot {\bf p}^{*}_{0}).]$ Depending on the diffraction geometry, $[{\bf p}^{*}_{0}]$ may be rotated into a position fulfilling the reflecting condition. The required rotation angle φ and the coordinates X, Y of the diffracted beam at its intersection with the detector plane can be found from $[{\bf p}^{*}_{0}]$ as follows.

Rotation by φ around axis $[{\bf m}_{2}]$ changes $[{\bf p}^{*}_{0}]$ into $[{\bf p}^{*}]$ . $[\eqalign{{\bf p}^{*} &=D({\bf m}_{2}, \varphi){\bf p}^{*}_{0} = {\bf m}_{2}({\bf m}_{2} \cdot {\bf p}^{*}_{0}) + [{\bf p}^{*}_{0} - {\bf m}_{2}({\bf m}_{2} \cdot {\bf p}^{*}_{0})]\cos \varphi\cr&\quad+ \;{\bf m}_{2} \times {\bf p}^{*}_{0} \sin \varphi\cr &= {\bf m}_{1}({\bf m}_{1} \cdot {\bf p}^{*}_{0} \cos \varphi + {\bf m}_{3} \cdot {\bf p}^{*}_{0} \sin \varphi) + {\bf m}_{2}{\bf m}_{2} \cdot {\bf p}^{*}_{0}\cr &\quad+\; {\bf m}_{3}({\bf m}_{3} \cdot {\bf p}^{*}_{0}\cos \varphi - {\bf m}_{1} \cdot {\bf p}^{*}_{0}\sin \varphi)\cr &= {\bf m}_{1}({\bf m}_{1} \cdot {\bf p}^{*}) + {\bf m}_{2}({\bf m}_{2} \cdot {\bf p}^{*}) + {\bf m}_{3}({\bf m}_{3} \cdot {\bf p}^{*}).}]$ The incident and diffracted beam wave vectors, $[{\bf S}_{0}]$ and S, have their termini on the Ewald sphere and satisfy the Laue equations $[{\bf S} = {\bf S}_{0} + {\bf p}^{*}, \quad {\bf S}^{2} = {\bf S}_{0}^{2}\;\Longrightarrow\; {\bf p}^{*2} = -2{\bf S}_{0} \cdot {\bf p}^{*} = {\bf p}^{*2}_{0}.]$ If $[\rho = [{\bf p}^{*2}_{0} - ({\bf p}^{*}_{0} \cdot {\bf m}_{2})^{2}]^{1/2}]$ denotes the distance of $[{\bf p}^{*}_{0}]$ from the rotation axis, solutions for $[{\bf p}^{*}]$ and φ can be obtained in terms of $[{\bf p}^{*}_{0}]$ as $[\eqalign{{\bf p}^{*} \cdot {\bf m}_{3} &= [-{\bf p}^{*2}_{0}/2 - ({\bf p}^{*}_{0} \cdot {\bf m}_{2})({\bf S}_{0} \cdot {\bf m}_{2})]/{\bf S}_{0}\cdot {\bf m}_{3}\cr {\bf p}^{*} \cdot {\bf m}_{2} &= {\bf p}^{*}_{0} \cdot {\bf m}_{2}\cr {\bf p}^{*} \cdot {\bf m}_{1} &= \pm [\rho^{2} - ({\bf p}^{*} \cdot {\bf m}_{3})^{2}]^{1/2}\cr \cos \varphi &= [({\bf p}^{*} \cdot {\bf m}_{1})({\bf p}^{*}_{0} \cdot {\bf m}_{1}) + ({\bf p}^{*} \cdot {\bf m}_{3})({\bf p}^{*}_{0} \cdot {\bf m}_{3})]/\rho^{2}\cr \sin \varphi &= [({\bf p}^{*} \cdot {\bf m}_{1})({\bf p}^{*}_{0} \cdot {\bf m}_{3}) - ({\bf p}^{*} \cdot {\bf m}_{3})({\bf p}^{*}_{0} \cdot {\bf m}_{1})]/\rho^{2}.}]$ In general, there are two solutions according to the sign of $[{\bf p}^{*} \cdot {\bf m}_{1}]$ . If $[\rho^{2} \lt ({\bf p}^{*} \cdot {\bf m}_{3})^{2}]$ or $[{\bf p}^{*2}_{0}\gt 4{\bf S}_{0}^{2}]$ , the Laue equations have no solution and the reciprocal-lattice point $[{\bf p}^{*}_{0}]$ is in the `blind' region.

If $[F{\bf S} \cdot {\bf d}_{3} \gt 0]$ , the diffracted beam intersects the detector plane at the point $[\eqalign{F{\bf S}/{\bf S} \cdot {\bf d}_{3} &= (F{\bf S} \cdot {\bf d}_{1}/{\bf S} \cdot {\bf d}_{3}){\bf d}_{1} + (F{\bf S} \cdot {\bf d}_{2}/{\bf S} \cdot {\bf d}_{3}){\bf d}_{2} + F{\bf d}_{3}\cr &= (X - X_{0}){\bf d}_{1} + (Y - Y_{0}){\bf d}_{2} + F{\bf d}_{3},}]$ which leads to a diffraction spot recorded at detector coordinates $[\eqalign{X &= X_{0} + F{\bf S} \cdot {\bf d}_{1}/{\bf S} \cdot {\bf d}_{3},\cr Y &= Y_{0} + F{\bf S} \cdot {\bf d}_{2}/{\bf S} \cdot {\bf d}_{3}.}]$

References

International Tables for Crystallography (2006). Vol. F. ch. 11.3, pp. 218-219