Crystallography & Symmetry
patterns derived from their groups
Concept

Homometry — where it appears

Two different arrangements of atoms with identical sets of interatomic vectors, and therefore identical diffraction intensities. It is not rare and not pathological: it appears at the smallest size that has room for it and gets commoner as structures grow.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

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 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 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 How many superstructures a diffraction pattern tells apart. For two-species superstructures of the square lattice with one to twelve sites in the primitive cell: the number of structures, and the number of classes that the positions of their superlattice spots separate them into, the positions together with the composition, and every intensity. At twelve sites 2,012 structures give 167 sets of spot positions, 504 with the composition and 1,815 sets of intensities. At five, seven and eleven sites the positions distinguish only the cells; the intensities distinguish everything. The bar shows the share resolved by positions, dark, and by intensities, light.

The spots say the cell and little else

Of the 2,012 two-species superstructures of the square lattice with twelve sites in the cell, more than half light every superlattice spot their cell allows, so the positions of the spots say which cell and nothing more. Positions separate the 2,012 into 167 classes, the composition takes that to 504, and every intensity to 1,815. At a prime number of sites no spot is ever missing, because a sum of distinct p-th roots of unity cannot vanish.

operations · Counting Almost no homometric pair on the torus is a reversed factor. For ten plane groups at several numbers of atoms on a 6 × 6 torus, the structures, the pairs of structures sharing one Patterson, how many of those pairs reversing one factor of a sumset explains, and how many are a patchwork of Galois classes. Factor reversal explains 2 of 3522; the patchwork explains all 3522.

A homometric twin is the structure moved class by class

On a ring, every homometric pair a small search found came from one construction: factor the structure as a sumset and reverse one factor. Imposing a plane group on a torus made homometry more common, and the obvious guess was that symmetry makes structures easier to factor. Tested pair by pair, factor reversal explains two of 3,522 homometric pairs on a 6 × 6 torus, and none at all at six atoms. What explains all 3,522 is a different mechanism: the partner is the structure's own Fourier transform moved independently on each Galois class of characters, by a shift, a sign or an inversion.

diffraction · Homometry Three-species arrangements whose sums are parallel at every spot, with and without a centre. Every arrangement of three species, all present, on a cyclic cell of four to nineteen sites. The third column counts those whose two phase sums are parallel at every superlattice spot, the fourth those among them with a centre of inversion, the fifth those without one, and the last how many kinds the exceptions make up to translation, inversion, relabelling the spots and renaming the species. Primes are shaded. At 5, 7 and 11 sites every parallel arrangement has a centre; at 13 sites 936 do not, and they are all one arrangement; at 17 there are 17 kinds and at 19 there are 48.

Thirteen sites obey Friedel's law without a centre

At five and seven sites the earlier census found that a three-species structure has a contrast that darkens a spot exactly when it has a centre, and asked whether that is a theorem. It is not. The two phase sums are parallel exactly when the A–B pairs are as common at every separation forwards as backwards, and exactly when no anomalous scatterer can produce a Bijvoet difference anywhere in the pattern. At 5, 7 and 11 sites only a centre achieves that. At 13 sites one arrangement does it without one.

operations · Counting

Named alongside it

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

Structure factorThe Patterson functionEnumerationInteratomic vectorPhase problemAutocorrelationExhaustive searchFriedel lawCentrosymmetryConvolutionCyclotomic polynomialsMeasurement

All concepts