Crystallography & Symmetry
patterns derived from their groups
Series

The Patterson function — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A structure, and the vectors between its atoms. On the left, 4 atoms in a cell. On the right, every one of the 16 vectors between them, each drawn from a common origin: 13 distinct positions, with the 4-fold peak at the origin being each atom paired with itself. That right-hand picture is what a Patterson map shows, and it is the thing a diffraction experiment gives without phases. It has more peaks than the structure has atoms — n² against n — which is why interpreting one is hard, and why it is always symmetric about its centre.

    The map that needs no phases

    A diffraction experiment measures intensities and loses phases, so the electron density cannot be computed from it. One map can be: the transform of the intensities, whose peaks are not atoms but the vectors between them — every ordered pair, brought to a common origin.

    part 1 · diffraction
  2. P2₁2₁2₁: the sections its symmetry forces. The Patterson cell of P2₁2₁2₁ with the sections marked. Each operation (M, t) sends an atom at x to Mx + t, so the vector between them is (I − M)x − t; where I − M is singular that vector cannot leave a plane, and the plane's equation comes from the left null space in integers. This group has 3 such operations, giving the sections w = 0.5, u = 0.5, v = 0.5. A heavy atom's vector to its own image is somewhere on one of them, which is what made structure solution possible before computers: a plane can be searched by eye and a volume cannot.

    Where symmetry stacks the vectors

    A Patterson map of a real structure is a blur with thousands of overlapping peaks. A screw axis rescues it: the vectors between symmetry-related atoms cannot leave a plane, so the search for a heavy atom is a search of a section rather than of a volume — and which plane it is falls out of the operation's matrix in integers.

    part 2 · diffraction
  3. Every vector between every pair of atoms. The Patterson map of a four-atom structure: the transform of the intensities with every phase set to zero, so it is computable from a measurement and nothing else. Its peaks are not atoms but the vectors between them, and the ringed one is the strongest that is not the origin — taken here as an interatomic vector exactly as a crystallographer takes the vector between two heavy atoms. That it really is one of the structure's own vectors is checked rather than assumed: it lands within 0.0031 of a cell of a difference of two positions, and it sits 0.67 of a cell from the origin, well outside the skirt of the tall peak there. A candidate taken too close to the origin is the same peak seen again and the whole method fails quietly.

    Solving from the vector set

    A Patterson map contains a copy of the structure laid over every atom in turn. Shift it by one interatomic vector, take the pointwise minimum with itself, and the copies that fail to coincide are cut away — leaving the structure, together with its inverse, from a measurement that carries no phases at all.

    part 3 · diffraction
  4. pg scatters as pmm. The orbit of a motif under pg, and the set of all ordered differences between its points brought to a common origin. The vector set was handed to the detector with no indication of where it came from, and came back as pmm: 31 peaks from 6 atoms, on the same lattice.

    Seventeen groups, seven vector sets

    A map of interatomic vectors is more symmetric than the structure it came from, twice over: it always acquires a centre, and it loses every translation part. So a glide becomes a mirror, seventeen plane groups collapse onto seven — and the collapse is verified by handing the vectors to a detector that has never heard of Patterson.

    part 4 · diffraction
  5. The same structure, mapped from intensities and from differences. Left, the ordinary Patterson map of the structure: 14762 interatomic vectors, a continuous field of overlapping peaks, and the two vectors between the anomalous scatterers — circled — nowhere among its strongest. Right, the map from squared Bijvoet differences over the same reflections: two peaks after the origin, and they are those two vectors. The difference map is 7381 times smaller a problem to read.

    A map of the atoms that break the law

    Feed a Patterson synthesis the differences between the two halves of each Friedel pair instead of the intensities, and the map that comes back holds the vectors between the anomalous scatterers and nothing else. Two atoms among a hundred and twenty-two: 14,762 vectors become two.

    part 5 · diffraction

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