Patterson methods - Revision history

MassimoNespolo: structure, not lattice

2019-06-19T16:02:52Z

structure, not lattice

BrianMcMahon: Tidied translations.

2017-11-17T09:40:27Z

Tidied translations.

BrianMcMahon: Style edits to align with printed edition

2017-05-16T12:21:01Z

Style edits to align with printed edition

MassimoNespolo: languages

2016-04-12T11:06:41Z

languages

AndreAuthier at 08:33, 3 April 2008

2008-04-03T08:33:05Z

BrianMcMahon at 09:55, 2 April 2008

2008-04-02T09:55:57Z

BrianMcMahon at 09:24, 2 April 2008

2008-04-02T09:24:20Z

BrianMcMahon at 09:13, 2 April 2008

2008-04-02T09:13:40Z

New page

== Definition ==

The family of methods employed in [[structure determination]] to derive relationships between the scattering centres in a crystal lattice when the diffraction phases are unknown. They depend upon interpretation of the Patterson function

<math>P(uvw) = (1/V)\sum_{h} \sum_{k} \sum_{l} {\{|F(hkl)|^2 \cos[2\pi(hu + kv + lw)]\}}</math>

to reveal interatomic vectors within the unit cell.

== Discussion ==

An electron density map can be constructed from an inverse Fourier transform of the [[structure factor]]s of a wave diffracted from a crystal. Diffracted intensities can be measured directly, and are related to the square of the amplitudes of the structure factors; but the diffraction phases cannot be determined by direct observation. The Patterson function represents a convolution of electron density with itself. It loses all phase information, but reduces to a function of <math>|F(hkl)|^2</math> alone, and is thus derivable from the measured intensities.

Without phase information, the Patterson map (''i.e.'' the Patterson function evaluated at points <math>u, v, w</math> throughout the unit cell) may be interpreted as a map of vectors between the scattering atoms. Vectors in a Patterson correspond to vectors in the real crystal cell, translated to the Patterson origin. Their weights are proportional to the product of electron densities at the tips of the vectors in the real cell.

The Patterson unit cell has the same size as the real crystal cell. The symmetry of the Patterson comprises the Laue point group of the crystal cell plus any additional lattice symmetry due to Bravais centring. The reduction of the real space group to the Laue symmetry is produced by the translation of all vectors to the Patterson origin and the introduction of a centre of symmetry.

For small structures, it may be possible to work out the original positions of the atoms that would give rise to the observed Patterson maxima (called ''deconvoluting'' the Patterson), but the procedure does not scale well for larger molecules. For <math>n</math> atoms in a unit cell, there will be <math>n^2</math> vectors (of which <math>n</math> self-vectors from an atom to itself accumulate to form a large origin peak).

Nevertheless, if other techniques are used to establish the position of one atom, the Patterson function becomes useful in determining the locations of other atoms.

== History ==

Patterson introduced the method of determining interatomic distances from a Fourier transform of intensities in 1934 [Patterson, A. L. (1934). [http://dx.doi.org/10.1103/PhysRev.46.372 ''Phys. Rev.'' '''46''', 372-376.] ''A Fourier series method for the determination of the components of interatomic distances in crystals''].

== See also ==

Patterson and molecular-replacement techniques.
M. G. Rossmann and E. Arnold. ''International Tables for Crystallography'' (2006). Vol. B, ch. 2.3, pp. 235-263 [http://dx.doi.org/10.1107/97809553602060000556 doi:10.1107/97809553602060000556]

← Older revision		Revision as of 09:40, 17 November 2017
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−	<~~Font~~ color="blue">Méthodes de Patterson</~~Font~~> (''Fr''). <~~Font~~ color="red">Patterson-Methoden</~~Font~~> (''Ge''). <~~Font~~ color="~~green~~">~~Métodos de~~ Patterson</~~Font~~> (''Sp''). <~~Font~~ color="~~black~~">~~Metodi di Patterson~~</~~Font~~> (''It''). <~~Font~~ color="~~purple~~">~~パターソン法~~</~~Font~~> (''Ja'').	+	<font color="blue">Méthodes de Patterson</font> (''Fr''). <font color="red">Patterson-Methoden</font> (''Ge''). <font color="black">Metodi di Patterson</font> (''It''). <font color="purple">パターソン法</font> (''Ja''). <font color="green">Métodos de Patterson</font> (''Sp'').

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-<Font color="blue">Méthodes de Patterson</Font> (''Fr''); <Font color="red">Patterson-Methoden</Font> (''Ge''); <Font color="green">Métodos de Patterson</Font> (''Sp''); <Font color="black">Metodi di Patterson</Font> (''It''); <Font color="purple">パターソン法</Font> (''Ja'').
+<Font color="blue">Méthodes de Patterson</Font> (''Fr''). <Font color="red">Patterson-Methoden</Font> (''Ge''). <Font color="green">Métodos de Patterson</Font> (''Sp''). <Font color="black">Metodi di Patterson</Font> (''It''). <Font color="purple">パターソン法</Font> (''Ja'').
@@ Line 28: / Line 28: @@
 == See also ==
-Patterson and molecular-replacement techniques.
+* Patterson and molecular-replacement techniques. M. G. Rossmann and E. Arnold. ''International Tables for Crystallography'' (2006). ''Vol. B'',  [https://doi.org/10.1107/97809553602060000556 ch. 2.3, pp. 235-263]
 [[Category:Structure determination]]

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−	<Font color="blue"> Méthodes de Patterson </Font> (''Fr'').	+	<Font color="blue">Méthodes de Patterson</Font> (''Fr''); <Font color="red">Patterson-Methoden</Font> (''Ge''); <Font color="green">Métodos de Patterson</Font> (''Sp''); <Font color="black">Metodi di Patterson</Font> (''It''); <Font color="purple">パターソン法</Font> (''Ja'').
		+

	== Definition ==		== Definition ==

← Older revision		Revision as of 08:33, 3 April 2008
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		+	<Font color="blue"> Méthodes de Patterson </Font> (''Fr'').
		+
	== Definition ==		== Definition ==

← Older revision		Revision as of 09:24, 2 April 2008
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	Patterson and molecular-replacement techniques.		Patterson and molecular-replacement techniques.
	M. G. Rossmann and E. Arnold. ''International Tables for Crystallography'' (2006). Vol. B, ch. 2.3, pp. 235-263 [http://dx.doi.org/10.1107/97809553602060000556 doi:10.1107/97809553602060000556]		M. G. Rossmann and E. Arnold. ''International Tables for Crystallography'' (2006). Vol. B, ch. 2.3, pp. 235-263 [http://dx.doi.org/10.1107/97809553602060000556 doi:10.1107/97809553602060000556]
		+
		+	[[Category:Structure determination]]

@@ Line 4: / Line 4: @@
 == Definition ==
-The family of methods employed in [[structure determination]] to derive relationships between the scattering centres in a crystal lattice when the diffraction phases are unknown. They depend upon interpretation of the Patterson function
+The family of methods employed in [[structure determination]] to derive relationships between the scattering centres in a crystal structure when the diffraction phases are unknown. They depend upon interpretation of the Patterson function
 <math>P(uvw) = (1/V)\sum_{h} \sum_{k} \sum_{l} {\{|F(hkl)|^2 \cos[2\pi(hu + kv + lw)]\}}</math>

@@ Line 21: / Line 21: @@
 == History ==
-Patterson introduced the method of determining interatomic distances from a Fourier transform of intensities in 1934 [Patterson, A. L. (1934). [http://dx.doi.org/10.1103/PhysRev.46.372 ''Phys. Rev.'' '''46''', 372-376.] ''A Fourier series method for the determination of the components of interatomic distances in crystals''].
+Patterson introduced the method of determining interatomic distances from a Fourier transform of intensities in 1934 [Patterson, A. L. (1934). [http://dx.doi.org/10.1103/PhysRev.46.372 ''Phys. Rev.'' '''46''', 372-376.] ''A Fourier series method for the determination of the components of interatomic distances in crystals'']. The following year he introduced modifications to improve the practical application of the Patterson function: removal of the Patterson origin peak, [[sharpened Patterson function|sharpening]] of the overall function, and removing peaks due to atoms in special positions [Patterson, A. L. (1935). ''Z. Kristallogr.'' '''90''', 517–542. ''A direct method for the determination of the components of interatomic distances in crystals''].
 == See also ==