The Patterson Function, P(u,v,w) is defined as

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

which is essentially the Fourier transform of the intensities rather than the structure factors. But using some quick relations you find that the Patterson function is also equivalent to the electron density convoluted with its inverse:

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

Further a Patterson map of N points will have N(N-1) peaks NOT including 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.

1D Example

Consider the series of delta functions

$f(x)\; =\; \backslash delta(x)\; +\; 3\; \backslash delta(x-2)\; +\; \backslash delta(x-5)\; +\; 3\; \backslash delta(x-8)\; +\; 5\; \backslash delta(x-10)$

then the Patterson is:

$P(u)\; =\; 5\; \backslash delta(u+10)\; +\; 18\; \backslash delta(u+8)\; +\; 9\; \backslash delta(u+6)\; +\; 6\; \backslash delta(u+5)\; +\; 6\; \backslash delta(u+3)\; +\; 18\; \backslash delta(u+2)\; +\; 45\; \backslash delta(u)\; +\; 18\; \backslash delta(u-2)\; +\; 6\; \backslash delta(u-3)\; +\; 6\; \backslash delta(u-5)\; +\; 9\; \backslash delta(u-6)\; +\; 18\; \backslash delta(u-8)\; +\; 5\; \backslash delta(u-10)$

Note: the Patterson always has central symmetry.

Fourier analysis