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Centric and acentric reflections: Difference between revisions
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Centric reflections in space group P2 and P2<sub>1</sub> resulting from operator R_g = (-h, k, -l) are thus those with h,0,l. There exist space groups without centric reflections, like R3. | Centric reflections in space group P2 and P2<sub>1</sub> resulting from operator R_g = (-h, k, -l) are thus those with h,0,l. There exist space groups without centric reflections, like R3. | ||
Other definitions of centric reflections:<br> | Other (equivalent) definitions of centric reflections:<br> | ||
Rupp: Centric structure factors are centrosymmetrically related reflections that are additionally related by the point group symmetry of the crystal.< | Rupp: Centric structure factors are centrosymmetrically related reflections that are additionally related by the point group symmetry of the crystal.<br><br> | ||
Properties: centric reflections have only two phase possibilities, e.g. 0° and 180° (but in any case 180° apart), and centric reflections do not have an anomalous signal (can these properties be easily derived here?). | Properties: centric reflections have only two phase possibilities, e.g. 0° and 180° (but in any case 180° apart), and centric reflections do not have an anomalous signal (can these properties be easily derived here?). | ||
Furthermore, the | Furthermore, with E being the structure factor, the statistical distribution [2] of acentric reflections is | ||
<math> P(|E|) = 2 |E| e^{-|E|^2} </math> | |||
[[file:I_acentrics.png]] | [[file:I_acentrics.png]] | ||
which is different from those of centric reflections; these follow <math> P(|E|) = \sqrt{\frac{2}{\pi}} e^{-|E|^2/2} </math> | |||
[[file:I_centrics.png]] | [[file:I_centrics.png]] | ||
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Centric reflections have a special role in experimental [[phasing]]. | Centric reflections have a special role in experimental [[phasing]]. | ||
The moments of intensities (centric <math><I^2>/<I>^2=3</math> ; acentric <math><I^2>/<I>^2=2</math>) can be calculated from the above formulas [3]; the first result can e.g. be obtained with [https://wolframalpha.com Wolframalpha] using <code>(sqrt(2/Pi) * integral ( x^4 exp(-0.5x^2) from 0 to inf )) / (sqrt(2/Pi) * integral ( x^2 exp(-0.5x^2) from 0 to inf ))^2</code>, where the x stands for the E in the formula above. | |||
== References == | == References == | ||
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[2] U. Shmueli and A. J. C. Wilson, "Statistical properties of the weighted reciprocal lattice", page 190-209, Chapter 2.1, International Tables for Crystallography Volume B, Kluwer Publishers (2006) | [2] U. Shmueli and A. J. C. Wilson, "Statistical properties of the weighted reciprocal lattice", page 190-209, Chapter 2.1, International Tables for Crystallography Volume B, Kluwer Publishers (2006) | ||
[3] [https://www.ccp4.ac.uk/html/pxmaths/bmg10.html Basic maths for crystallographers] | |||