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

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. F. ch. 15.2, pp. 326-327   | 1 | 2 |

Section 15.2.3.3. General treatment of the structure-factor distribution

R. J. Reada*

a Department of Haematology, University of Cambridge, Wellcome Trust Centre for Molecular Mechanisms in Disease, CIMR, Wellcome Trust/MRC Building, Hills Road, Cambridge CB2 2XY, England
Correspondence e-mail: rjr27@cam.ac.uk

15.2.3.3. General treatment of the structure-factor distribution

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The Wilson, Sim, Luzzati and variable-error distributions have very similar forms, because they are all Gaussians arising from the application of the central limit theorem. The central limit theorem is valid under many circumstances; even when there are errors in position, scattering factor and B factor, as well as missing atoms, a similar distribution still applies. As long as these sources of error are independent, the true structure factor will have a Gaussian distribution centred on [D{\bf F}_{C}] (Fig. 15.2.3.2),[link] where D now includes effects of all sources of error, as well as compensating for errors in the overall scale and B factor (Read, 1990[link]). [p({\bf F}\hbox {;}\ {\bf F}_{C}) = (1/\pi \varepsilon \sigma_{\Delta}^{2}) \exp \left(-|{\bf F} - D{\bf F}_{C}|^{2}/\varepsilon \sigma_{\Delta}^{2}\right)] in the acentric case, where [\sigma_{\Delta}^{2} = \Sigma_{N} - D^{2}\Sigma_{P}], ɛ is the expected intensity factor and [\Sigma_{P}] is the Wilson distribution parameter for the model.

[Figure 15.2.3.2]

Figure 15.2.3.2| top | pdf |

Schematic illustration of the general structure-factor distribution, relevant in the case of any set of independent random errors in the atomic model.

For centric reflections, the scattering differences are distributed along a line, so the probability distribution is a one-dimensional Gaussian. [p({\bf F}\hbox{;}\ {\bf F}_{C}) = [1/(2 \pi \varepsilon \sigma_{\Delta}^{2})^{1/2} ]\exp \left(-|{\bf F} - D{\bf F}_{C}|^{2}/2 \varepsilon \sigma_{\Delta}^{2}\right).]

References

First citation Read, R. J. (1990). Structure-factor probabilities for related structures. Acta Cryst. A46, 900–912.Google Scholar








































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