Misplaced Pages

The Patterson function is used to solve the phase problem in X-ray crystallography. 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 Patterson function is defined as

$${\displaystyle P(u,v,w)=\sum _{h,k,\ell \in \mathbb {Z} }\left|F_{h,k,\ell }\right|^{2}\;e^{-2\pi i(hu+kv+\ell w)}.}$$

It is essentially the Fourier transform of the intensities rather than the structure factors. The Patterson function is also equivalent to the electron density convolved with its inverse:

${\displaystyle P\left({\vec {u}}\right)=\rho \left({\vec {r}}\right)*\rho \left(-{\vec {r}}\right).}$

Furthermore, a Patterson map of N points will have N(N − 1) 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.

One-dimensional example

Consider the series of delta functions given by

${\displaystyle f(x)=\delta (x)+3\delta (x-2)+\delta (x-5)+3\delta (x-8)+5\delta (x-10).\,}$

The Patterson function is given by the following series of delta functions and unit step functions

${\displaystyle {\begin{aligned}P(u)={}&5\delta (u+10)+18\delta (u+8)+9\delta (u+6)+6\delta (u+5)+6\delta (u+3)\\&+18\delta (u+2)+45\delta (u)+18\delta (u-2)+6\delta (u-3)+6\delta (u-5)\\&+9\delta (u-6)+18\delta (u-8)+5\delta (u-10).\end{aligned}}}$

References

1. Patterson, A. L. (1934-09-01). "A Fourier Series Method for the Determination of the Components of Interatomic Distances in Crystals". Physical Review. 46 (5): 372–376. Bibcode:1934PhRv...46..372P. doi:10.1103/PhysRev.46.372. ISSN 0031-899X.
2. Patterson, A. L. (1935). "A direct method for the determination of the components of interatomic distances in crystals". Zeitschrift für Kristallographie. 90 (1–6): 517. doi:10.1524/zkri.1935.90.1.517. S2CID 102041995.

External links

• "Structural resolution. The Patterson function and the Patterson method".

v t e Crystallography
Key concepts	Timeline of crystallography Crystallographers Metallurgy Biocrystallography Structure Unit cell Bravais lattice Miller index Point group Reciprocal lattice Restriction theorem Periodic table Structure prediction Systems Cubic Hexagonal Monoclinic Orthorhombic Tetragonal Triclinic Growth Crystallite Equiaxed Twinning Fiveling Aperiodic crystal Quasicrystal Phase transition Phase diagram Eutectic Miscibility gap Polymorphism Liquid crystal Phase transformation crystallography Precipitation Segregation Spinodal decomposition Supersaturation GP-zone Ostwald ripening Defects Grain boundary Disclination CSL Growth Abnormal growth Perfect crystal Stacking fault Dislocation Burgers vector Partial dislocation Kink Cross slip Frank–Read source Cottrell atmosphere Peierls stress GND Lomer–Cottrell junction Slip Slip bands Interstitials Bjerrum defect Frenkel defect Wigner effect Vacancy Schottky defect F-center Stone–Wales defect Defects in diamond Bragg's law Bragg plane Ewald's sphere Friedel's law Hermann–Mauguin notation Structure factor Thermal ellipsoid
Characterisation	Electron Diffraction Scattering Neutron Diffraction Scattering Nuclear magnetic resonance X-ray Diffraction Scattering
Algorithms	Direct methods Isomorphous replacement Molecular replacement Molecular dynamics Patterson map Phase retrieval Gerchberg–Saxton Single particle analysis
Software	CCP4 Coot CrystalExplorer DSR JANA2020 MTEX OctaDist Olex2 SHELX
Databases	Bilbao Crystallographic Server CCDC CIF COD ICSD ICDD PDB
Journals	Crystal Growth & Design Crystallography Reviews Journal of Chemical Crystallography Journal of Crystal Growth Kristallografija Zeitschrift für Kristallographie – Crystalline Materials Zeitschrift für Kristallographie – New Crystal Structures
Awards	Carl Hermann Medal Ewald Prize Gregori Aminoff Prize
Organisation	IUCr IOBCr RAS DMG Associations Europe France Germany UK US Japan
Category Commons

• Crystallography