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Centric and acentric reflections: Difference between revisions
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== Reciprocal space == | == Reciprocal space == | ||
As mentioned above, a reflection is centric if there is a reciprocal space symmetry operator which maps it onto its Friedel mate (-h,-k,-l). Reciprocal space symmetry operators can be obtained from the real space symmetry operators by following two rules: | |||
a) take the rotation matrix and transpose it | a) take the rotation matrix and transpose it | ||
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b) omit the translation vector | b) omit the translation vector | ||
Whereas rule b | Whereas rule b makes things slightly easier in reciprocal space, we must be aware that in reciprocal space we have additional symmetry, namely Friedel symmetry. This means for each reciprocal symmetry operator we also have to consider the Friedel-related operator (all elements of the matrix multiplied by -1). | ||
To find centric reflections, we just solve the Eigenvalue problem A v = -v, now considering each reciprocal space symmetry operator in turn. | To find centric reflections, we just solve the Eigenvalue problem A v = -v, now considering each reciprocal space symmetry operator in turn.<br> | ||
Centric reflections in space group P2 and P2<sub>1</sub> are thus those with 0, | 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 (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.<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]] | |||
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]] | |||
. | . | ||
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] | |||