A function, denoted P(x,y,z), used in the analysis of the results of X-ray crystallography. P(x,y,z) is defined by

where h, k, and l are the Miller indices of the crystal. This function is plotted as a contour map with maxima of the contours at vector distances from the origin, i.e. vectors from the origin to the point (x,y,z), where the maximum occurs, corresponding to vector distances between pairs of maxima in electron density. Thus, the Patterson function enables interatomic vectors, which give the lengths and directions between atomic centres, to be determined. This technique, introduced by Arthur Lindo Patterson (1902–66) in 1934–35, is called Patterson synthesis and is most useful if the unit cell contains heavy atoms.