# Patterson methods

The family of methods employed in structure determination to derive relationships between the scattering centers in a crystal lattice when the diffraction phases are unknown. They depend upon interpretation of the Patterson function

$P(uvw) = \dfrac{1}{V} \sum_h\sum_k\sum_l { | F(hkl) | ^2\cos[2\pi(hu + kv + lw)]}$

to reveal interatomic vectors within the unit cell.

### Discussion

An electron density map can be constructed from an inverse Fourier transform of the structure factors of a wave diffracted from a crystal. Diffracted intensities can be measured directly, and are related to the square of the amplitudes of the structure factors; but the diffraction phases cannot be determined by direct observation. The Patterson function represents a convolution of electron density with itself. It loses all phase information, but reduces to a function of $$| F(hkl) | ^2$$ alone, and is thus derivable from the measured intensities.

Without phase information, the Patterson map (i.e. the Patterson function evaluated at points u,v,w throughout the unit cell) may be interpreted as a map of vectors between the scattering atoms. Vectors in a Patterson correspond to vectors in the real crystal cell, translated to the Patterson origin. Their weights are proportional to the product of electron densities at the tips of the vectors in the real cell.

The Patterson unit cell has the same size as the real crystal cell. The symmetry of the Patterson comprises the Laue point group of the crystal cell plus any additional lattice symmetry due to Bravais centering. The reduction of the real space group to the Laue symmetry is produced by the translation of all vectors to the Patterson origin and the introduction of a center of symmetry.

For small structures, it may be possible to work out the original positions of the atoms that would give rise to the observed Patterson maxima (called deconvoluting the Patterson), but the procedure does not scale well for larger molecules. For $$n$$ atoms in a unit cell, there will be $$n^2$$ vectors (of which n self-vectors from an atom to itself accumulate to form a large origin peak).

Nevertheless, if other techniques are used to establish the position of one atom, the Patterson function becomes useful in determining the locations of other atoms.

### History

Patterson introduced the method of determining interatomic distances from a Fourier transform of intensities in 1934 [Patterson, A. L. (1934). Phys. Rev. 46, 372-376. A Fourier series method for the determination of the components of interatomic distances in crystals]. The following year he introduced modifications to improve the practical application of the Patterson function: removal of the Patterson origin peak, sharpening of the overall function, and removing peaks due to atoms in special positions [Patterson, A. L. (1935). Z. Kristallogr. 90, 517–542. A direct method for the determination of the components of interatomic distances in crystals].