Crystallography & Symmetry
patterns derived from their groups
Concept

The Patterson function — where it appears

The transform of the measured intensities with every phase set to zero, whose peaks sit at the vectors between pairs of atoms. It is the one map a diffraction experiment gives away for free, and its symmetry is higher than the crystal's in two separate ways.

Named by 12 essays across 2 fields — each of them below, with the objects they name alongside it.

The phase problem. The same structure rebuilt from its diffraction three ways: with the true amplitudes and phases, with the phases scrambled, and with the amplitudes discarded but the phases kept. The atoms survive the loss of the amplitudes and do not survive the loss of the phases, which is what an experiment throws away.

The phase problem

A detector records how much light arrives and not when it arrives, so half of every diffraction measurement is thrown away before it is written down. The half that is lost turns out to be the half that carries the structure.

diffraction · Diffraction A 5-fold cluster in a crystal that has no 5-fold axis. A cluster of 10 points with an exact 5-fold axis at the centre of each cell, repeated by the lattice. Two measurements, on the same points. The cluster is carried onto itself by a turn of 72° to within 2e-16 of a cell — exact, as far as the arithmetic goes. The pattern is not: applying the same turn about a lattice point sends some atoms 1.19 of a cell from the nearest atom, which is most of the way across it. Both are true at once. The axis is a symmetry of the contents of one cell and not of the crystal, which is what non-crystallographic symmetry means and why a virus with a sixty-fold capsid can crystallise in an ordinary space group.

A fivefold axis in an ordinary crystal

A virus with sixty-fold symmetry crystallises in a space group that has none of it. The restriction forbids a fivefold axis to the lattice and says nothing about what sits inside one cell — so the axis is exact, the crystal genuinely lacks it, and both statements are measurable on the same set of atoms.

restriction · Local symmetry 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.

diffraction · The Patterson function 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.

diffraction · The Patterson function 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.

diffraction · The Patterson function Two structures on 8 sites with the same vectors. Two arrangements of 4 atoms on a ring of 8 positions. They are not the same arrangement — no rotation of the ring and no reflection carries one onto the other — and every interatomic vector occurs the same number of times in both. The bars below are the shared vector counts, which is the Patterson function of each: the tall one at the origin is the atom count and carries no information, and everything else is what a diffraction experiment measures. Their diffraction patterns are identical in every intensity, so no measurement of intensities, at any resolution, distinguishes them.

Two structures, one Patterson

Eight arrangements of four atoms on a ring of eight sites, and only seven distinct sets of interatomic vectors between them. Two of the arrangements are genuinely different and no measurement of intensities can tell them apart — at any resolution, for ever.

diffraction · Homometry Where a homometric pair comes from. A set that factors as a sumset gives its own partner. If every point of A is a sum b + c with b in B and c in C, and every sum arises once, then reversing C produces a different set with the same vectors — because reversing a factor and reversing its conjugate cancel in the product that the vector set is. Both factors must be asymmetric, which is the constraint that decides where the construction can be used: a two-point set is its own reflection up to a translation, so the smallest useful factorisation is three points by three points, and the smallest structure it builds has nine atoms.

Where the pairs come from

A structure whose atoms are the sums of two smaller sets has a partner: reverse one factor and the interatomic vectors do not notice. The construction is Patterson's own, it explains why homometry exists, and the smallest structure it can build has nine atoms for a reason worth following.

diffraction · Homometry 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.

diffraction · The Patterson function Two plane structures with one Patterson. Two arrangements of 4 atoms on a 4 by 4 torus. No translation and no half-turn carries one onto the other, so they are different structures; every interatomic vector occurs the same number of times in both, so no measurement of intensities distinguishes them. The plane case is not the chain case with an extra index: a mirror in the plane sends a vector set to its mirror image rather than to itself, which is the one place the analogy with a cycle breaks.

Two structures on a torus, and one Patterson

Homometry was settled here on a ring of positions, which is a crystal in one dimension. Moving the same exhaustive search to a torus asks whether the coincidence is commoner or rarer when the vectors have a plane to land in — and the honest answer is that dimension is not what decides it.

diffraction · Homometry Two kinds of atom, and more ambiguity rather than less. Exhaustive searches on rings of four sizes. The third column counts homometric groups when every atom is identical; the fourth counts them when each atom may be one of two kinds. The fourth is larger at every size, and at nine and ten sites the third is nothing at all — there is no pair of arrangements of four identical atoms that a diffraction experiment cannot separate, and there are six and four once the atoms may differ. Distinguishing the atoms adds information to the structure and adds ambiguity to the measurement.

When the atoms are not all the same

Every homometric pair found so far is a pair of point sets, where an atom is a point and counts once. Give the atoms different scattering powers and the ambiguity does not go away — it grows. On a ring of nine there is no pair of four identical atoms that diffraction cannot separate, and there are six once two kinds of atom are allowed.

diffraction · Homometry 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.

diffraction · The Patterson function Two symmetric structures a diffraction pattern cannot separate. Two arrangements of 6 atoms on a 6 × 6 torus, each invariant under the plane group p6m, drawn beside the Patterson they share. No translation and no inversion carries one onto the other, so they are different structures; every one of the thirty-six interatomic vector counts is the same, so every diffracted intensity is the same and no measurement at any resolution separates them. Of the 4 structures with this symmetry and this many atoms, there are only 3 Pattersons — so imposing the most symmetric of the seventeen plane groups has not removed the ambiguity.

Symmetry does not rescue a Patterson

Every homometric pair found so far sits on a bare ring with no operations imposed, and a real crystal sits in a space group. Impose one and the ambiguity does not go away: 12 of the 13 groups searched still have pairs, and at six atoms the hexagonal groups are indistinguishable two to three times as often as the general position.

diffraction · Homometry

Named alongside it

The objects these essays reach for when they reach for this one.

Interatomic vectorPhase problemStructure factorAutocorrelationHeavy atom methodHomometryCentrosymmetryConvolutionEnumerationFriedel lawCentrosymmetricExhaustive search

All concepts