Crystallography & Symmetry
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P2₁2₁2₁: the sections its symmetry forces

P2₁2₁2₁: the sections its symmetry forces
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.

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.

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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. How it is known

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.

 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. How it is known

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.

 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. How it is known

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.

 Two candidates, and the sign that chooses. The phase of reflection (2, 3), recovered from three measurements and no model. The circle is every complex number of the measured amplitude; the isomorphous difference fixes the cosine of the angle between the unknown phase and the heavy atom's, leaving the two candidates marked; the anomalous difference fixes the sine, which picks one. The recovered phase agrees with the true one to fifteen decimal places, and the true phase was never used in the calculation. How it is known

One experiment gives the cosine, the other gives the sine

Friedel's law holding exactly is what makes the phase unreachable. Its breaking is what hands it back: an isomorphous difference fixes the cosine of the phase and leaves two candidates, and the anomalous difference fixes the sine, which chooses.

 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. How it is known

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.

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