Patterson maps give structural information in X-ray crystallography, even if the phases of the reflections are not known (see Phase Problem). With the Patterson function P(uvw), a vector map is calculated. Vectors between atoms in the real structure show up as vectors from the origin to maxima in the Patterson map (Fig. 1). The size and shape of the unit cell in this map is identical to the crystal unit cell.
where u, v, and w are relative coordinates in the Patterson cell; h, k, and ‘ are the indices of the reflections; Fis the volume of the unit cell; and5 the amplitude of the structure factor of reflection h k i. Calculation of the Patterson function from Eq. 1 does not require knowledge of the phase angles of the reflections and can always be calculated from the experimental diffraction data because, apart from correction factors,, where I(hk £) is the intensity of reflection hkt.
Figure 1. Example of a simple Patterson map. (a) A two-dimensional unit cell with three atoms. (b) The corresponding 1
For very simple chemical compounds, it is possible to derive the real structure from the Patterson map. A protein Patterson map, however, has too many overlapping vectors to extract any structural information. However, protein Patterson maps are extremely useful for understanding the molecular replacement technique. They are also employed in isomorphous replacement to locate the attached heavy atoms. The difference between the amplitudes of the reflections for the native and heavy atom crystals is used in the calculation, to produce a difference Patterson map. Such a map should result only from the heavy atoms, and it should be interpretable because the number of heavy atoms attached is usually small and therefore they form a simple structure.