Patterson function

The Patterson method is a method for solving the phase problem of X-ray diffraction. It goes back to Lindo Patterson (1902-1966), who introduced the method in 1934.

Description

Patterson, the method is defined as the Fourier transform of the squares of the structure factor amounts. Lindo Patterson himself called his method therefore the series. It supplies not directly the positions of the atoms in the unit cell, but the result of the Patterson method are interatomic vectors. The length of the vector is the interatomic distance, the direction of the interatomic direction. The height of the peak is dependent on the number of electrons of the atoms involved. The greater the number of electrons, the higher is the peak. In the crystal structure analysis of the Patterson method is therefore used in preference when the crystal structure consists of a few heavy atoms, and many light atoms. The highest peaks then specify the interatomic vectors between the heavy atoms. If the position of the heavy atoms determines their partial structure factor can be determined and subtracted from the calculated structure factor. In this way, the location of other atoms can be determined.

Definition of the Patterson function

With   = Volume of the unit cell   = Indexed structure factor   = Radius vector in the unit cell

According to the convolution theorem of the Fourier transform we can also write the Patterson function as a pair correlation function

With   = Electron density at x

Properties of the Patterson function

• If the structure is composed of atoms, then has the Patterson function peaks.
• The translational symmetry of the electron density and the Patterson function are the same. In other words, both the unit cells are equal. However, the unit cell of the electron density peaks, the unit cell of the Patterson function peaks.
• The maximum of the Patterson function is always at the origin ( 0,0,0) and represents the interatomic vector of an atom dar. with itself
• The Patterson function is always centrosymmetric, even if the crystal symmetry, and hence the electron density is not centrosymmetric. If there is a vector between the atoms A and B, then there is also the reverse vector between B and A.
• The peaks of the Fourier transform of the structure factors are much sharper than the Patterson - peaks ( in the amounts of the structure factors).

Harkerschnitte and lines

The original publication of Patterson from 1934 referred to the triclinic crystal system, ie the lowest symmetry. David Harker extended the concept of the Patterson method by the symmetry operations earned higher space groups. He noted that one often has to perform only one or two dimensional Fourier transforms to obtain the relevant structural information. This was at a time without electronic computer very advantageous because the three-dimensional Fourier transformation is very computationally intensive. Even today (protein crystals ), the one - and two-dimensional Harkerlinien and Harkerschnitte used in large crystal structures.

Sharpened Patterson function

Because the normal Patterson function provides many blurry peaks sharpened Patterson functions are often (English: sharpened Patterson functions ) are used, leading to sharper peaks. Most of these methods are based on normalized structure factors. These values ​​are derived from the structure factors so that they correspond to point atoms or atoms at rest. So you include a correction of the thermal motion. The sharpened Patterson function is calculated as the Fourier transform or more.

In the literature, appear regularly also other methods to produce sharp Patterson peaks.

Fragment search

As explained above, the Patterson method is only bad if the crystal structure consists exclusively of light atoms, such as in organic molecules. However, if it is known, the molecular structure, the fragment search can be used. It does not have the complete molecule to be known, a large molecular fragment is sufficient. This molecular structure can be obtained by quantum chemical calculations or derived from known molecular fragments from databases.

In the fragment Search the Patterson function of the X-ray intensities is calculated first. Then the molecular fragment (or the intramolecular distance vectors of the fragment ) is kept rotated and translated until it fits perfectly in the Patterson map. Various computer algorithms have been developed for this procedure.