Bragg's law for non-ideal conditions: mosaicity

Otwinowski, Z.; Minor, W.

doi:10.1107/97809553602060000677

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. 11.4, p. 232 | 1 | 2 |

Section 11.4.6.2. Bragg's law for non-ideal conditions: mosaicity

Z. Otwinowski^a ^* and W. Minor^b

^a UT Southwestern Medical Center at Dallas, 5323 Harry Hines Boulevard, Dallas, TX 75390-9038, USA, and ^bDepartment of Molecular Physiology and Biological Physics, University of Virginia, 1300 Jefferson Park Avenue, Charlottesville, VA 22908, USA
Correspondence e-mail: zbyszek@mix.swmed.edu

11.4.6.2. Bragg's law for non-ideal conditions: mosaicity

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The Bragg condition [equation (11.4.2.1)] assumes ideal crystals and a parallel X-ray beam. In reality, crystals are mosaic and the beam has some angular spread. The value of mosaicity describes the range of orientations of the crystal lattice within a sample. As the impacts of mosaicity and the beam's angular spread on the angular width of reflections are equivalent, the keyword mosaicity describes the sum of both effects.

DENZO assumes the following model of angular shape of diffraction peaks: $[M = {\hbox{mos} \over \pi} \left[1 + \cos {\pi (\varphi - \varphi_{c}) \over \hbox{mos}}\right] \eqno(11.4.6.8)]$ for $[\varphi_{c}]$ in the range ( $[\varphi_{c} - \hbox{mos}/2; \varphi_{c} + \hbox{mos}/2), \hbox{ otherwise } M = 0]$ , $[P_{\varphi_{1} \varphi_{2}} = {\textstyle\int\limits_{\varphi_{1}}^{\varphi_{2}}} M (\varphi)\ \hbox{d} \varphi, \eqno(11.4.6.9)]$ where mos is mosaicity, $[\varphi_{c}]$ is the predicted angle and P is the predicted partiality of data collected by oscillating from $[\varphi_{1}]$ to $[\varphi_{2}]$ .

Partiality is a number that represents what fraction of the reflection intensity is present in one image. If partiality is 1, such reflections are called fully recorded; otherwise they are called partials. For partials, predictions of partiality can be compared with the observed fraction $[P_{0}]$ of the reflection intensity present in one image. The partiality model contributes the following term to the refinement: $[\chi_{P}^{2} = (P_{\varphi_1 \varphi_{2}} - P_{0})^{2}/{\sigma{_{P_{0}}^2}}. \eqno(11.4.6.10)]$ The combined positional and partiality refinement used in DENZO is both stable and very accurate. The power of this method is in proper weighting (by estimated error) of two very different terms – one describing positional differences and the other describing intensity differences. Both detector and crystal variables are uniformly treated in the refinement process.

References

International Tables for Crystallography (2006). Vol. F. ch. 11.4, p. 232