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# Patterson function

The Patterson function is used to solve the phase problem in X-ray crystallography. It was introduced by Arthur Lindo Patterson in 1934.

The Patterson function is defined as

$P(u,v,w) = \sum\limits_{h k l} \left|F_{h k l}\right|^2 \;e^{-2\pi i(hu + kv + lw)}.$

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

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

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

The peaks in the Patterson function are the interatomic distances weighted by the product of the number of electrons in the atoms concerned.

The Patterson always has central symmetry.

## One-dimensional example

Consider the series of delta functions given by

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

Then the Patterson function is

$P(u) = 5 \delta(u+10) + 18 \delta(u+8) + 9 \delta(u+6) + 6 \delta(u+5) + 6 \delta(u+3) + 18 \delta(u+2) + 45 \delta(u) \,$
${} + 18 \delta(u-2) + 6 \delta(u-3) + 6 \delta(u-5) + 9 \delta(u-6) + 18 \delta(u-8) + 5 \delta(u-10). \,$