The Patterson function is used to solve the
phase problem
In physics, the phase problem is the problem of loss of information concerning the phase that can occur when making a physical measurement. The name comes from the field of X-ray crystallography, where the phase problem has to be solved for the de ...
in
X-ray crystallography
X-ray crystallography is the experimental science of determining the atomic and molecular structure of a crystal, in which the crystalline structure causes a beam of incident X-rays to Diffraction, diffract in specific directions. By measuring th ...
. It was introduced in 1935 by
Arthur Lindo Patterson while he was a visiting researcher in the laboratory of
Bertram Eugene Warren at
MIT
The Massachusetts Institute of Technology (MIT) is a private research university in Cambridge, Massachusetts, United States. Established in 1861, MIT has played a significant role in the development of many areas of modern technology and sc ...
.
The Patterson function is defined as
It is essentially the
Fourier transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
of the intensities rather than the
structure factor
In condensed matter physics and crystallography, the static structure factor (or structure factor for short) is a mathematical description of how a material scatters incident radiation. The structure factor is a critical tool in the interpretation ...
s. The Patterson function is also equivalent to the
electron density
Electron density or electronic density is the measure of the probability of an electron being present at an infinitesimal element of space surrounding any given point. It is a scalar quantity depending upon three spatial variables and is typical ...
convolved with its inverse:
:
Furthermore, a Patterson map of ''N'' points will have 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
In crystallography, a centrosymmetric point group contains an inversion center as one of its symmetry elements. In such a point group, for every point (x, y, z) in the unit cell there is an indistinguishable point (-x, -y, -z). Such point group ...
.
One-dimensional example
Consider the series of
delta function
In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real lin ...
s given by
:
The Patterson function is given by the following series of delta functions and
unit step functions
:
References
External links
*
{{DEFAULTSORT:Patterson Function
Crystallography