Patterson function

The Patterson function is used to solve the phase problem in X-ray crystallography. It was introduced in 1935 by Arthur Lindo Patterson while he was a visiting researcher in the laboratory of Bertram Eugene Warren at MIT.[1][2]

The Patterson function is defined as

P(u,v,w)=\sum _{h,k,\ell \in \mathbb {Z} }\left|F_{h,k,\ell }\right|^{2}\;e^{-2\pi i(hu+kv+\ell w)}.

It is essentially the Fourier transform of the intensities rather than the structure factors. The Patterson function is also equivalent to the electron density convolved with its inverse:

P\left({\vec {u}}\right)=\rho \left({\vec {r}}\right)*\rho \left(-{\vec {r}}\right).

Furthermore, a Patterson map of N points will have N(N − 1) peaks, excluding the central (origin) peak and any overlap.

The peaks' positions in the Patterson function are the interatomic distance vectors and the peak heights are proportional to the product of the number of electrons in the atoms concerned.

Because for each vector between atoms i and j there is an oppositely oriented vector of the same length (between atoms j and i), the Patterson function always has centrosymmetry.

One-dimensional example

Consider the series of delta functions given by

f(x)=\delta (x)+3\delta (x-2)+\delta (x-5)+3\delta (x-8)+5\delta (x-10).\,

The Patterson function is given by the following series of delta functions and unit step functions

{\begin{aligned}P(u)={}&5\delta (u+10)+18\delta (u+8)+9\delta (u+6)+6\delta (u+5)+6\delta (u+3)\\&+18\delta (u+2)+45\delta (u)+18\delta (u-2)+6\delta (u-3)+6\delta (u-5)\\&+9\delta (u-6)+18\delta (u-8)+5\delta (u-10).\end{aligned}}

References

Patterson, A. L. (1934-09-01). "A Fourier Series Method for the Determination of the Components of Interatomic Distances in Crystals". Physical Review. 46 (5): 372–376. doi:10.1103/PhysRev.46.372. ISSN 0031-899X.
Patterson, A. L. (1935). "A direct method for the determination of the components of interatomic distances in crystals". Zeitschrift für Kristallographie. 90 (1–6): 517. doi:10.1524/zkri.1935.90.1.517. S2CID 102041995.

External links

"Structural resolution. The Patterson function and the Patterson method".

Documentation

This template does not display in the mobile view of Wikipedia; it is desktop only. Read the documentation for an explanation.

This is a navigational template created using {{navbox}}. It can be transcluded on pages by placing {{Patterson function}} below the standard article appendices.

Initial visibility

This template's initial visibility currently defaults to autocollapse, meaning that if there is another collapsible item on the page (a navbox, sidebar, or table with the collapsible attribute), it is hidden apart from its title bar; if not, it is fully visible.

To change this template's initial visibility, the |state= parameter may be used:

{{Patterson function|state=collapsed}} will show the template collapsed, i.e. hidden apart from its title bar.
{{Patterson function|state=expanded}} will show the template expanded, i.e. fully visible.

TemplateData

A navigational box that can be placed at the bottom of articles.

Template parameters
Parameter		Description	Type	Status
State	`state`	The initial visibility of the navbox Suggested values `collapsed` `expanded` `autocollapse`	String	suggested

Template transclusions

Transclusion maintenance
Check completeness of transclusions

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] Patterson, A. L. (1934-09-01). "A Fourier Series Method for the Determination of the Components of Interatomic Distances in Crystals". Physical Review. 46 (5): 372–376. doi:10.1103/PhysRev.46.372. ISSN 0031-899X.

[2] Patterson, A. L. (1935). "A direct method for the determination of the components of interatomic distances in crystals". Zeitschrift für Kristallographie. 90 (1–6): 517. doi:10.1524/zkri.1935.90.1.517. S2CID 102041995.