The Patterson Function
Transforming the structure factors needs their phases. Transform the intensities instead and you get a map of every interatomic vector in the cell — for nothing, from what a diffractometer actually records. A heavy atom stands out of it and can be read straight off.
- You supply
- One of the named structures, and how far the series should run. Everything else — the intensities, the vectors, the electron counts — is computed from the atoms.
- Reading it
- Summing over h alone gives the projected Patterson, so vectors sharing a u pile up — including on the origin, whose weight is then more than ΣZ2. And not every vector is its own maximum: a truncated series merges what it cannot resolve.
Worked examples: zirconia — the zirconium read straight off · aragonite, calcium at a quarter · quartz at 6 terms — the vectors merge · quartz at 30 terms — and separate · rock salt — where there is nothing to read
Input
Everything summed here is an intensity. No phase appears anywhere on this page, which is the whole reason the function exists.
What you can compute without the phases
A diffractometer measures |F|2. Transforming the structure factors gives the electron density, but that needs their phases, which are not measured. Transform the intensities instead and you get something you can have for nothing:
P(u) = Σh |F(h00)|2 cos(2πhu)
Patterson’s 1934 answer. It is the electron density convolved with itself, so it has a maximum at every interatomic vector — and the weight of each one is the product of the two atoms’ electron counts, which is why a single heavy atom stands out of the map and can be read straight off it.
Choose a structure. The page transforms its intensities, lists every interatomic vector derived from the atom positions, and shows where the two meet.
Try it
zirconia — the zirconium read straight off · aragonite, calcium at a quarter · quartz at 6 terms — the vectors merge · quartz at 30 terms — and separate · rock salt — where there is nothing to read