the Patterson function
/ PAT-er-son /
The Patterson function is a trick map you can compute even though the phases are missing, because it deliberately uses only the intensities — the one thing you actually measured. Instead of showing you where the atoms are, it shows you all the vectors between pairs of atoms. Every peak in a Patterson map says: there exists a pair of atoms this far apart, in this direction. It is a map of separations rather than positions.
The mechanism is a Fourier sum with a clever choice of ingredients: P(u) = (1/V) times the sum over hkl of |F(hkl)|^2 cos(2 pi (h u + k v + l w)). Notice it uses |F|^2, which the experiment gives directly, and it needs no phases at all (they are effectively set to zero). Each peak appears at an interatomic vector r_i minus r_j, and its weight is proportional to the product of the two atoms' scattering powers, roughly Z_i times Z_j. So vectors between heavy atoms tower over the rest, which lets you find the heavy atom's position first and then bootstrap the lighter atoms and their phases from there.
This was the classic route around the phase problem for structures containing a few heavy atoms (the heavy-atom method), and it remains the foundation of isomorphous replacement used to phase proteins. Be honest about the crowding: for N atoms there are about N squared interatomic vectors, so the Patterson map is far busier than a real structure and its peaks overlap heavily. Untangling it is straightforward when a few heavy atoms dominate, but genuinely hard when many atoms of similar weight all contribute comparable vectors.
In a molecule with one heavy platinum atom among light carbons, the Patterson map is dominated by the Pt-to-Pt vector, whose position immediately reveals where the platinum sits — the anchor from which the rest of the structure is phased.
It maps vectors between atoms, not atoms; heavy-atom vectors dominate and give the starting anchor.
A Patterson peak is a distance-and-direction between two atoms, not an atom position. With about N squared vectors, the map gets hopelessly crowded when many similar atoms contribute — it shines only when a few heavy atoms stand out.