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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 [[Massachusetts Institute of Technology|MIT]].<ref>{{Cite journal |last=Patterson |first=A. L. |date=1934-09-01 |title=A Fourier Series Method for the Determination of the Components of Interatomic Distances in Crystals |url=https://link.aps.org/doi/10.1103/PhysRev.46.372 |journal=Physical Review |language=en |volume=46 |issue=5 |pages=372–376 |doi=10.1103/PhysRev.46.372 |bibcode=1934PhRv...46..372P |issn=0031-899X|url-access=subscription }}</ref><ref>{{cite journal|first=A. L. |last=Patterson |title=A direct method for the determination of the components of interatomic distances in crystals |journal= Zeitschrift für Kristallographie |volume=90 |page=517 |date=1935 |issue=1–6 |doi= 10.1524/zkri.1935.90.1.517|s2cid=102041995 }}</ref>

The Patterson function is defined as

<math display="block">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)}.</math>

It is essentially the [[Fourier transform]] of the intensities rather than the [[structure factor]]s. The Patterson function is also equivalent to the [[electron density]] [[convolution|convolved]] with its inverse:

:<math>P\left(\vec{u}\right) = \rho\left(\vec{r}\right) * \rho\left(-\vec{r}\right).</math>

Furthermore, a Patterson map of ''N'' points will have {{nowrap|''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 [[Dirac delta function|delta function]]s given by

:<math>f(x) = \delta(x) + 3 \delta(x-2) + \delta(x-5) + 3 \delta(x-8) + 5 \delta(x-10). \,</math>

The Patterson function is given by the following series of delta functions and [[Heaviside step function|unit step functions]]

:<math>\begin{align} 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{align}</math>

==References==
{{reflist}}

==External links==
* {{cite web|url=https://www.xtal.iqf.csic.es/Cristalografia/parte_07_3-en.html|title=Structural resolution. The Patterson function and the Patterson method}}

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Patterson function - Revision history

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