Centric and acentric reflections - Revision history

Kay at 18:45, 19 May 2022

2022-05-19T18:45:28Z

← Older revision		Revision as of 20:45, 19 May 2022
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	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 ~~"intensity statistics"~~ [2] of acentric reflections		Furthermore, with E being the structure factor, the statistical distribution [2] of acentric reflections is
	(<math> P(\|E\|) = 2 \|E\| e^{-\|E\|^2} </math> )		<math> P(\|E\|) = 2 \|E\| e^{-\|E\|^2} </math>

	[[file:I_acentrics.png]]		[[file:I_acentrics.png]]

	~~are~~ different from those of centric reflections		which is different from those of centric reflections; these follow <math> P(\|E\|) = \sqrt{\frac{2}{\pi}} e^{-\|E\|^2/2} </math>
	(<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 ==

Kay: /* References */

2022-05-19T18:30:02Z

References

← Older revision		Revision as of 20:30, 19 May 2022
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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]

Eaberry: /* Reciprocal space */

2019-03-14T17:10:39Z

Reciprocal space

← Older revision		Revision as of 19:10, 14 March 2019
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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.<br><br>		Rupp: Centric structure factors are centrosymmetrically related reflections that are additionally related by the point group symmetry of the crystal.<br><br>

Eaberry: /* Reciprocal space */ get rid of
which disrupted formatting

2019-03-06T18:05:21Z

Reciprocal space: get rid of <p> which disrupted formatting

← Older revision		Revision as of 20:05, 6 March 2019
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	Other definitions of centric reflections:<br>		Other 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.<p><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?).

Eaberry: /* Reciprocal space */ explain centrics in monoclinic space group

2019-03-05T16:15:24Z

Reciprocal space: explain centrics in monoclinic space group

← Older revision		Revision as of 18:15, 5 March 2019
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	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).		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 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 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>		Rupp: Centric structure factors are centrosymmetrically related reflections that are additionally related by the point group symmetry of the crystal.<p><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?).

Eaberry: /* Reciprocal space */ Correct centric zone for B-setting monoclinic crystals.

2019-03-04T17:58:41Z

Reciprocal space: Correct centric zone for B-setting monoclinic crystals.

← Older revision		Revision as of 19:58, 4 March 2019
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	== Reciprocal space ==		== Reciprocal space ==

	~~In reciprocal space the situation is quite similar. A~~ reflection is centric if there is a reciprocal space symmetry operator which maps it onto ~~itself (or rather~~ its Friedel mate). Reciprocal space symmetry operators can be obtained from the real space symmetry operators by following two rules:		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) 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).		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.
	Centric reflections in space group P2 and P2<sub>1</sub> are thus those with 0,~~k,0~~. There exist space groups without centric reflections, like R3.		Centric reflections in space group P2 and P2<sub>1</sub> are thus those with h,0,l. There exist space groups without centric reflections, like R3.

			Other 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>

	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?).

Karsten at 15:34, 8 March 2016

2016-03-08T15:34:40Z

← Older revision		Revision as of 17:34, 8 March 2016
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	Furthermore, the "intensity statistics" [2] of acentric reflections		Furthermore, the "intensity statistics" [2] of acentric reflections
	(<math> P(\|E\|) = 2 \|E\| e^{-\|E\|^2} </math> )		(<math> P(\|E\|) = 2 \|E\| e^{-\|E\|^2} </math> )
	~~<gnuplot>~~
	~~set xrange~~ [0:5]		[[file:I_acentrics.png]]
	~~plot 2xexp(-x*x)~~
	~~</gnuplot>~~
	are different from those of centric reflections		are different from those of centric reflections
	(<math> P(\|E\|) = \sqrt{\frac{2}{\pi}} e^{-\|E\|^2/2} </math> )		(<math> P(\|E\|) = \sqrt{\frac{2}{\pi}} e^{-\|E\|^2/2} </math> )
	~~<gnuplot>~~
	~~set xrange~~ [0:5]		[[file:I_centrics.png]]
	~~plot sqrt(2/3.14)exp(-xx/2)~~
	~~</gnuplot>~~
	.		.

Hypsu: /* Real space */

2014-03-13T13:30:17Z

Real space

← Older revision		Revision as of 15:30, 13 March 2014
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	== Real space ==		== Real space ==

	Let's first look at real space. A special position results if there exists one or more symmetry operators, other than the trivial operator {x,y,z}, which map this position upon itself. As a example: take spacegroup P2 with its symmetry operators {x,y,z} and {-x,y,-z}. Now consider any point with x=0 and z=0, and some value of y. Obviously this point, when transformed with -x,y,-z , yields 0,y,0 - just the same point! Thus this is a special position. Generally, positions on n-fold symmetry axes are special.		Let's first look at real space. A special position results if there exists one or more symmetry operators, other than the trivial operator {x,y,z}, which map this position upon itself. As an example: take spacegroup P2 with its symmetry operators {x,y,z} and {-x,y,-z}. Now consider any point with x=0 and z=0, and some value of y. Obviously this point, when transformed with -x,y,-z , yields 0,y,0 - just the same point! Thus this is a special position. Generally, positions on n-fold symmetry axes are special.

	Mathematically, to find special positions we have to solve the Eigenproblem A v = v where A is the symmetry operator (expressed as rotation matrix and translation vector), and v, the Eigenvector, represents the special position(s). For a given space group, we need to check all symmetry operators.		Mathematically, to find special positions we have to solve the Eigenproblem A v = v where A is the symmetry operator (expressed as rotation matrix and translation vector), and v, the Eigenvector, represents the special position(s). For a given space group, we need to check all symmetry operators.

	An atom at a special position usually has (at most) an occupancy of 0.5. However, it may happen that more than one symmetry operator maps the special position upon itself; in that case the occupancy is 1/(number of symmetry operators ~~mapping~~ point onto itself). ~~Ths~~, a point on a n-fold rotation axis has (maximum) occupancy of 1/n.		An atom at a special position usually has (at most) an occupancy of 0.5. However, it may happen that more than one symmetry operator maps the special position upon itself; in that case the occupancy is 1/(number of positions generated by all symmetry operators that map the point onto itself). Thus, a point on a n-fold rotation axis has (maximum) occupancy of 1/n. Disorder or partial occupation will result in lower occupancy.

	In space group P2<sub>1</sub>, there are no special positions - the Eigenproblem has no solution.		In space group P2<sub>1</sub>, there are no special positions - the Eigenproblem has no solution.

Kay: /* Reciprocal space */

2013-01-20T18:20:05Z

Reciprocal space

← Older revision		Revision as of 20:20, 20 January 2013
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	== Reciprocal space ==		== Reciprocal space ==

	In reciprocal space the situation is quite similar. A reflection is centric if there is a reciprocal space symmetry operator which maps it onto itself. Reciprocal space symmetry operators can be obtained from the real space symmetry operators by following two rules:		In reciprocal space the situation is quite similar. A reflection is centric if there is a reciprocal space symmetry operator which maps it onto itself (or rather its Friedel mate). 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
Line 33:		Line 33:
	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).		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.
	Centric reflections in space group P2 and P2<sub>1</sub> are thus those with 0,k,0. There exist space groups without centric reflections, like R3.		Centric reflections in space group P2 and P2<sub>1</sub> are thus those with 0,k,0. There exist space groups without centric reflections, like R3.

Kay: /* Reciprocal space */

2011-10-01T19:06:28Z

Reciprocal space

← Older revision		Revision as of 21:06, 1 October 2011
Line 38:		Line 38:
	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 "intensity statistics" of acentric reflections		Furthermore, the "intensity statistics" [2] of acentric reflections
	(<math> P(\|E\|) = 2 \|E\| e^{-\|E\|^2} </math> )		(<math> P(\|E\|) = 2 \|E\| e^{-\|E\|^2} </math> )
	<gnuplot>		<gnuplot>

Centric and acentric reflections - Revision history

Kay at 18:45, 19 May 2022

Kay: /* References */

Eaberry: /* Reciprocal space */

Eaberry: /* Reciprocal space */ get rid of which disrupted formatting

Eaberry: /* Reciprocal space */ explain centrics in monoclinic space group

Eaberry: /* Reciprocal space */ Correct centric zone for B-setting monoclinic crystals.

Karsten at 15:34, 8 March 2016

Hypsu: /* Real space */

Kay: /* Reciprocal space */

Kay: /* Reciprocal space */

Eaberry: /* Reciprocal space */ get rid of
which disrupted formatting