Patterson method

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

The Patterson method is defined as the Fourier transform of the squares of the structure factor amounts. Lindo Patterson himself therefore called his process the series. It does not directly provide the positions of the atoms in the unit cell , but rather interatomic vectors are the result of the Patterson method. The length of the vector is the interatomic distance, the direction the interatomic direction. The height of the diffraction reflex depends on the number of electrons in the two atoms involved. The greater the number of electrons, the higher the reflex. In the crystal structure analysis of the Patterson method is therefore used in preference when the crystal structure of a few heavy atoms and many light atoms. The highest reflections of the diffractogram then indicate the interatomic vectors between the heavy atoms. Once the position of the heavy atoms has been determined, their partial structure factor can be determined and subtracted from the calculated structure factor . In this way the position of the other atoms can be determined. ${\ displaystyle | F | ^ {2}}$

Definition of the Patterson function

${\ displaystyle P (U, V, W) = {\ frac {1} {V_ {EZ}}} \ sum _ {h = - \ infty} ^ {\ infty} \ sum _ {k = - \ infty} ^ {\ infty} \ sum _ {l = - \ infty} ^ {\ infty} | F (hkl) | ^ {2} \ cdot {\ text {exp}} [- 2 \ pi {\ text {i} } (hU + kV + lW)]}$

with = volume of unit cell = indexed structure factor = position vector within the unit cell

${\ displaystyle V_ {EZ}}$
${\ displaystyle F (hkl)}$
${\ displaystyle U, V, W}$

According to the convolution theorem of the Fourier transformation, the Patterson function can also be written as a pair correlation function

${\ displaystyle P (U, V, W) = P (\ mathbf {U}) = \ int \ limits _ {0} ^ {a} \ int \ limits _ {0} ^ {b} \ int \ limits _ {0} ^ {c} \ rho (\ mathbf {R}) \ rho (\ mathbf {R + U}) {\ text {d}} \ mathbf {R}}$

with = electron density at location x

${\ displaystyle \ rho (x)}$

Properties of the Patterson function

If the structure is made up of atoms, then the Patterson function predicts diffraction reflections. ${\ displaystyle N}$ ${\ displaystyle N ^ {2}}$
The translational symmetry of the electron density and the Patterson function are the same. In other words: both unit cells are the same size. However, the unit cell of the electron density has peaks, the unit cell of the Patterson function has peaks. ${\ displaystyle N}$ ${\ displaystyle N ^ {2}}$
The maximum of the Patterson function is always at the origin (0,0,0) and represents the interatomic vector of an atom with itself.
The Patterson function is always centrosymmetric, even if the crystal symmetry and thus the electron density is not centrosymmetric. If there is a vector between atoms A and B, then there is also the reverse vector between B and A.
The reflexes from the Fourier transformation of the structure factors are much sharper than the Patterson reflexes (from the amounts of the structure factors).

Harker cuts and lines

The original Patterson publication in 1934 referred to the triclinic crystal system , which is the lowest symmetry. David Harker expanded the concept of the Patterson method by introducing the symmetry operations of higher space groups . He found out that one often only has to carry out one or two-dimensional Fourier transformations in order to obtain the relevant structural information. In times without electronic computers, this was very advantageous because the three-dimensional Fourier transform is very computationally intensive. Even today, the one and two-dimensional Harker lines and Harker cuts are still used for large crystal structures ( protein crystals ).

Sharpened Patterson function

Because the normal Patterson function provides many blurry reflections, often sharpened Patterson functions (English: sharpened Patterson functions ) are used, resulting in sharper reflexes. Mostly these procedures are based on normalized structural factors . These values are derived from the structure factors so that they correspond to point atoms or atoms in the resting state. So they contain a correction for thermal movement. The sharpened Patterson function is then calculated as a Fourier transform of or better . ${\ displaystyle E}$ ${\ displaystyle E}$ ${\ displaystyle F}$ ${\ displaystyle \ left | E \ right | ^ {2}}$ ${\ displaystyle \ left | EF \ right |}$

Other methods of generating sharp Patterson reflexes appear regularly in the literature.

Fragment search

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

When searching for fragments, the Patterson function of the X-ray intensities is calculated first. Then the molecule fragment (or the intramolecular distance vectors of the fragment) is rotated and shifted until it fits optimally into the Patterson map. Various computer algorithms have been developed for this process.

literature

AL Patterson: A Fourier Series Method for the Determination of the Components of Interatomic Distances in Crystals . In: Phys. Rev. 46. 1934, 372-376
AL Patterson: A direct method for the determination of the components of interatomic distances in crystals . In: Z. Krist. (A) 90. 1935, 517-542