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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]] (1902–1966) while he was a visiting researcher in the laboratory of [[Bertram Eugene Warren]] (1902–1991) at MIT.<ref>A. L. Patterson ''Z. Krist''. '''A90''', 517 (1935).</ref>
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| The Patterson function is defined as
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| :<math>P(u,v,w) = \sum\limits_{h k l} \left|F_{h k l}\right|^2 \;e^{-2\pi i(hu + kv + lw)}.</math>
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| 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:
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| :<math>P(\vec{u}) = \rho(\vec{r}) * \rho(-\vec{r}).</math>
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| Furthermore, a Patterson map of ''N'' points will have ''N(N−1)'' peaks, excluding the central peak and any overlap.
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| The peaks in the Patterson function are the interatomic distance vectors weighted by the product of the number of electrons in the atoms concerned.
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| 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]].
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| ==One-dimensional example==
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| Consider the series of [[Dirac delta function|delta function]]s given by
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| :<math>f(x) = \delta(x) + 3 \delta(x-2) + \delta(x-5) + 3 \delta(x-8) + 5 \delta(x-10). \,</math>
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| Then the Patterson function is
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| :<math>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) \, </math> | |
| ::::<math> {} + 18 \delta(u-2) + 6 \delta(u-3) + 6 \delta(u-5) + 9 \delta(u-6) + 18 \delta(u-8) + 5 \delta(u-10). \, </math> | |
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| ==References==
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| {{reflist}}
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| {{DEFAULTSORT:Patterson Function}}
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| [[Category:Crystallography]]
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