Crystallography & Symmetry
patterns derived from their groups
Concept

Autocorrelation — where it appears

A function recording how much a structure overlaps a translated copy of itself, which for a crystal is the Patterson. It is what a diffraction experiment measures the transform of, which is why intensities give vectors and not atoms.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

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 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 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

Named alongside it

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

Interatomic vectorThe Patterson functionPhase problemCentrosymmetryHomometryStructure factorConvolutionEnumerationHeavy atom methodMeasurementDetectionExhaustive search

All concepts