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.
The Patterson function
halite (rock salt), Fm3m — 6 intensities out of 12, carrying detail to 0.47 Å. 8 atoms in the cell give 64 interatomic vectors, falling on 2 distinct values of u.
Every atom has a vector of length zero to itself, so the origin always carries the largest peak — and it says nothing about the structure. In three dimensions its weight would be ΣZ2 = 1,639, one term per atom. Here it is 6,269, because a projection also piles up every pair of different atoms that happens to share an x. The difference is exactly the information the projection threw away.
What went in
Every bar is above the line, because an intensity is a squared amplitude. The same reflections on the Fourier synthesis page are drawn with their signs, and the difference between the two pictures is exactly what a measurement loses.
The direct beam is not a reflection, so I(000) is never measured — but F(000) is the electron count of the cell, 112 here, which anybody who knows the formula can write down without measuring anything. It is the one coefficient that is free rather than missing, and it is why the mean of the curve comes out at F(000)2.
Gaps in the row of bars are systematic absences: a centred lattice or a glide plane makes whole classes of h00 vanish, so the series has fewer terms than its length suggests and the projection repeats more often than the cell does.
Nothing to read out of this one
There is nothing to read out of this map. The heaviest atom here is Cl, and it sits at x = 0 or ½ — a special position. Its Harker vector 2x therefore falls on the origin, buried under every self-vector in the cell, so the map confirms the atom is on a special position and tells you nothing further. Try zirconia, whose zirconium is at a general x.
The vectors themselves
| u | Å | Pairs | How many | Weight |
|---|---|---|---|---|
| 0.0000 | 0.000 | Na–Na, Na–Cl, Cl–Na, Cl–Cl | 32 | 6,269 |
| 0.5000 | 2.820 | Na–Na, Na–Cl, Cl–Na, Cl–Cl | 32 | 6,269 |
Derived from the atom coordinates, not from the curve above — this is the answer the map is trying to give you. Every vector appears in both directions, which is why the function is centrosymmetric however the crystal is built.
What was summed
| Cell edge a | 5.6400 Å |
|---|---|
| F(000) | 111.97 electrons |
| Σ|F|2 over the series | 15,107 |
| P(0) | 42,752 — F(000)2 + 2Σ|F|2, and the largest value anywhere |
| Mean of the curve | 12,538 — which is F(000)2 = 12,538, and also the total weight of every interatomic vector |
| Origin peak weight | 6,269 |
| ΣZ2, one term per atom | 1,639 — smaller, because this projection puts more than one atom on the same x |
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