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The Patterson Function

https://xraytools.com/patterson

Single-crystal data

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 corrected, scaled measurement gives. A heavy atom stands out of it and can be read straight off.

Before this
This map is built from intensities, not from structure factors, which is exactly why it needs no phases. The transform that does need them is next door.
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 · albite — a Patterson with no line to read

Earlier on the path: Fourier Synthesis and the Phase Problem Next on the path: Direct Methods and the Sign Relation On From intensities to a structure, step 3 of 6

Notation here: I — what each one means here

Terms here: zone

See also: Fourier Synthesis and the Phase Problem · Direct Methods and the Sign Relation

What each input changes
Terms
How many intensities the sum runs over. The map is of vectors, so adding terms sharpens the vector peaks, not the atoms — there are no atoms in it.
Teaching with this page
Objective
After this page a learner can say what a Patterson map contains, and why it needs no phases.
Start from
this worked example
Ask first
A Patterson map has a strong maximum away from the origin. Is there an atom at those coordinates?
Watch for
“Yes — a maximum in a map is where the density is”
Then
Direct Methods and the Sign Relation
Check yourself: A Patterson map has a strong maximum away from the origin. Is there an atom at those coordinates?

No — a Patterson maximum is a vector between two atoms Yes — a maximum in a map is where the density is

The Patterson function is the autocorrelation of the electron density, so what it holds is interatomic vectors rather than positions. Its largest maximum is always at the origin, which is every atom paired with itself. That is why it needs no phases and why reading it as a structure is the first mistake everybody makes with it.

Input

up to h =

How far the series runs. Blank means 12. Fewer terms is lower resolution — the vectors broaden and neighbouring ones merge.

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 records intensities, and correcting and scaling them gives |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 · albite — a Patterson with no line to read

Where this comes from