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 determining the atomic and molecular structure of a crystal, in which the crystalline structure causes a beam of incident X-rays to diffract into many specific directions. By measuring the angles ...
. It was introduced in 1935 by
Arthur Lindo Patterson
Arthur Lindo Patterson (23 July 1902, Nelson, New Zealand - 6 November 1966, Philadelphia, Pennsylvania) was a pioneering British X-ray crystallographer. Patterson was born to British parents in New Zealand in 1902. Shortly afterwards the famil ...
while he was a visiting researcher in the laboratory of
Bertram Eugene Warren
Bertram Eugene Warren (June 28, 1902 – June 27, 1991) was an American crystallographer. His studies of X-rays provided much knowledge and understanding of both crystalline and non-crystalline materials. He also worked on changing amorphous soli ...
at
MIT
The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the m ...
.
The Patterson function is defined as
:
It is essentially the
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
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
In quantum chemistry, 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 va ...
convolved
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
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 mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
s given by
:
then the Patterson function is
:
References
External Links
*
{{DEFAULTSORT:Patterson Function
Crystallography