Symmetry does not rescue a Patterson

When the atoms are not all the same closes on the one simplification this argument has never removed. Homometry — two different structures with identical interatomic vectors, and therefore identical diffracted intensities at every reflection for ever — has been found here on a ring, on a torus, with one kind of atom and with two. Every one of those structures sits in no group at all, and a real crystal sits in a space group.

The essay also says why the answer is not obvious, and it is worth repeating because both halves are right. A group forces atoms into orbits, which cuts the number of configurations enormously and might leave no room for a coincidence. And a group makes the Patterson more symmetric than the structure, which is a loss of information and might manufacture one. Which effect wins is a measurement.

It is the second. Imposing a symmetry does not remove homometry, and it makes the ambiguity proportionally worse — at six atoms the general position loses 9.6 per cent of its structures to a twin it cannot be told from, and the three-fold groups lose between 21 and 30 per cent.

The search, and why the torus has six sites a side #

The setting is a 6 × 6 torus — thirty-six positions in a two-dimensional cell — with each of the seventeen plane groups acting on it. Six is not chosen for convenience: it is the smallest side on which all seventeen act. The translation parts of the seventeen are halves and thirds, so the side has to be divisible by six, and the three-fold and four-fold matrices are integral on any side at all.

A structure invariant under a group is a union of that group’s orbits, which is what makes the enumeration finite and complete: for each number of atoms, every union of orbits reaching that total is built. The orbit counts run from thirty-six for the group with no operations — every site its own orbit, so every subset is a structure — down to seven for the full hexagonal group.

Two structures count as the same if a translation or an inversion carries one onto the other, which are exactly the motions a Patterson cannot see. That quotient is the one the unconstrained search on the torus already uses, and sharing it is what makes the two numbers comparable: the constrained and unconstrained cases are one routine run with different inputs.

One consequence of the quotient is worth naming, because it is a place the constrained search could have gone wrong and did not. A translate of a p3-invariant structure is still p3-invariant, but it is a different union of orbits — the group’s orbits are not carried to themselves by a general translation of the torus. So the same structure arrives from several different unions, and counting unions rather than structures would inflate every count by a different factor for every group, making the rates incomparable. Reducing to the canonical form first is what removes it, and the check that it did is that the number of canonical forms equals the number of structures found.

The number of atoms is the other thing held fixed. Comparing a group at six atoms with the general position at six atoms is the only fair comparison available, because the rate moves with the size and the groups admit wildly different numbers of structures at each size. Every figure below that compares groups does so at one atom count.

Two hexagonal structures a diffraction pattern cannot separate #

The most symmetric of the seventeen plane groups is p6m, and on this torus it admits exactly four structures of six atoms. They have three Pattersons between them. Two of the four are genuinely different arrangements — no translation and no inversion carries either onto the other — and every one of the thirty-six interatomic vector counts is identical, so every diffracted intensity is identical and no measurement at any resolution separates them.

That is the answer to the question in one picture. Four structures is as constrained as this setting gets, and it is not constrained enough.

The pair is worth looking at for a second reason. Both structures put their six atoms on orbits of p6m, which on this torus means the special positions — the sites the hexagonal group fixes or nearly fixes — so the two are not merely symmetric, they are as constrained as a crystallographer’s idealised structure. This is not a coincidence hiding in the general position and dressed up; it is two entries of the sort a structure report would contain.

And a third. The ambiguity here is not the one a Patterson’s extra symmetry accounts for, where several plane groups share a vector set because the map is more symmetric than the structure. These two structures have the same group. A symmetry argument cannot tell them apart because there is no symmetry difference to find.

What an orbit costs, and what it buys #

Before the numbers, it is worth being precise about the two effects, because “symmetry reduces the configurations” is true in a way that sounds stronger than it is.

A group of order $|G|$ acting freely would cut the number of structures of $k$ atoms by roughly a factor of $\binom{36}{k}$ down to $\binom{36/|G|}{k/|G|}$ — an enormous reduction, and that is the effect the first half of the argument has in mind. But the reduction is in the number of structures, and what a diffraction experiment has to separate is not structures but Pattersons. The question is which of the two sets shrinks faster, and there is no reason in the reduction itself for the answer to favour the experiment.

Against that, the constraint on the Pattersons is at least as strong. The Patterson of a $G$-invariant structure is itself $G$-invariant — every operation of $G$ permutes the atoms, hence permutes the pairs, hence permutes the vectors — and it is centrosymmetric as well. So the target set shrinks by the same order and by one more factor besides. Written that way it stops being surprising that the ambiguity rises; what is surprising is only that anyone expects otherwise, and the expectation comes from thinking about structures rather than about what distinguishes them.

The rate, group by group #

The general position sits at 9.6 per cent — of the 27,564 distinct six-atom structures on this torus, the ones sharing a Patterson with somebody account for that share. Six groups are worse: cm at 12 per cent, p6 at 17, p3 at 21, p3m1 and p6m at 25, and p31m at 30.

Twelve of the thirteen groups searched at larger sizes have a pair somewhere. The one that does not is p4g, and the picture says why rather than hiding it: at these sizes p4g admits between two and five structures in total, so “no pairs” is a statement about a very small sample and not about the group.

The ordering has a pattern worth reading and one exception worth not explaining away. The three-fold and six-fold groups occupy five of the six places above the general position, which is the strongest signal in the table; cm is the sixth and is a rectangular group with a single mirror. Below the line sit the groups built on a half-turn, and the lowest of all are pmm at 2.6 per cent and p2 at 8.3. The exception is cmm, which contains the half-turn and still runs at 7 per cent at six atoms and 10 at twelve — higher than several groups without one. Whatever is happening there is not the centrosymmetry argument, and this search does not say what it is.

There is one more reading available and it cuts against the intuition that drives the whole question. The groups where a crystallographer would feel safest — the high-symmetry hexagonal ones, where a structure is nearly determined by a handful of coordinates — are precisely the worst. p6m on this torus admits four six-atom structures and cannot tell two of them apart. Having almost no freedom left is not the same as having no ambiguity left.

Plotting the two counts against each other is the clearest way to see what symmetry does and does not do. Four orders of magnitude separate the general position from the most symmetric groups, so the reduction in configurations is exactly as large as the argument for it suggested. What does not happen is the points climbing onto the diagonal as they fall. The gap below it is the ambiguity, and it widens.

Why the count falls faster than the ambiguity #

The mechanism has two parts and only one of them is a fact about groups.

The part that is about groups is that a Patterson is centrosymmetric whatever the structure is — Friedel’s law in its simplest form, since reversing every interatomic vector leaves the multiset alone. So a structure that already has a centre loses nothing to that, and a structure without one loses a factor.

Split that way the means come out at 6 per cent against 8, in the direction the argument predicts. It is a tendency over seven groups rather than a law and the picture says so: the two columns hold three groups and four, and cmm sits on the wrong side of it.

There is a cleaner way to say what the factor is. A structure with no centre and its own inversion are two different structures with the same Patterson, automatically and for free — the trivial homometric pair every non-centrosymmetric arrangement has. The quotient used here divides that out, so it is not counted; what the centrosymmetric groups avoid is not that pair but the other coincidences that the loss of handedness makes available, where two structures differ by more than an inversion and still agree. Being already centrosymmetric closes a door that the quotient only papers over.

The part that is not about groups at all is a pigeonhole. The Pattersons of structures on a 6 × 6 torus are multisets of thirty-six counts summing to the square of the atom count, and imposing a symmetry constrains them twice over — the Patterson of a p3-invariant structure is itself p3-invariant, and centrosymmetric besides. So the number of Pattersons available shrinks along with the number of structures, and there is no reason for it to shrink more slowly.

Filling the cell makes it worse, which is the same pigeonhole seen from the other side. The lines are jagged because at each size the number of structures with a given symmetry is small enough that one extra coincidence moves the share by several points; what the picture is for is the direction.

pg is the clearest run: 6.5 per cent at six atoms, 2.4 at eight, 11.8 at ten and 25.3 at twelve. Twelve atoms on thirty-six sites is a third of the cell occupied, which for a two-dimensional crystal is not a strange density, and at that point a quarter of all the structures with that symmetry have a twin no intensity measurement separates. p3 runs 21, 20, 27 and 37 per cent over the same sizes.

What this settles and what it does not #

It settles the question that was asked, which was whether homometry is a hazard of the general position or of crystals as such. It is of crystals as such: the coincidence is not an artefact of allowing atoms anywhere, and constraining them to a group’s orbits makes a measurement less able to separate structures rather than more.

That is worth stating in the form a crystallographer would use. Knowing a structure’s space group is knowing a great deal, and it is the first thing a determination establishes; the result here is that it does not help with this particular failure at all, and makes it commoner. What breaks a homometric ambiguity is not more symmetry and not more reflections — both structures produce the same reflections — but information from outside the intensities: chemistry, a bond length that one arrangement makes absurd, or an anomalous signal that measures something the vector set does not carry.

A comparison with the searches that came before puts the size of the effect in place. The first search of all found the smallest cyclic pair at four atoms on eight sites, and moving to a torus established that dimension is not what decides whether pairs exist. Allowing two kinds of atom made them commoner — six pairs where there had been none. Symmetry is the fourth thing varied and the third to make it worse; the only variation that has ever reduced the ambiguity is reducing the number of atoms, which is not a variation a crystal offers.

It does not settle the rate for real crystals. A thirty-six-site cell with six atoms is a dense structure by crystallographic standards, and the rates here are for a torus of that size. Whether the share falls towards zero as the cell grows at fixed density, or settles, is a question this search cannot reach: the enumeration is exhaustive and exhaustive searches stop.

It says nothing about three dimensions. A space group’s orbits in a cell of the same relative size are larger and fewer, the point groups are bigger, and the Patterson’s extra symmetry is the Laue group rather than a single inversion — all of which push in the same direction as the plane result but none of which is measured here. The plane is where an exhaustive search of this kind is still finite, and the honest reading is that the plane result is a reason to expect the same in space rather than evidence for it.

And the small-sample warning applies to most of the table. Six of the seventeen groups admit fewer than twenty structures at six atoms, so their percentages are one or two coincidences expressed as a fraction. The comparison that carries weight is between the general position, with tens of thousands of structures, and the groups with dozens — pg, cm, p3 and p31m — and those are the ones the argument rests on.

And the result is about counting, not about any particular pair being important. A rate of 25 per cent does not mean a quarter of real structures are wrong; it means a quarter of the arrangements in this exhaustively enumerated family have a partner, and real structures are not drawn uniformly from that family. What the rate is good for is the comparison with the general position, where the same sampling objection applies equally and therefore cancels.

The Patterson here is the exact vector multiset, not a measured map. Two structures with identical multisets are indistinguishable by intensities however good the data, which is the strongest form of the statement; two with nearly identical ones are indistinguishable in practice, which is a weaker and much commoner condition that this search does not count at all. The real rate at which a real experiment cannot decide between two structures is therefore higher than anything here, not lower.

Still open: whether the pairs come from the same place #

Where the pairs come from explains most of the pairs on a ring by a construction: write the structure as a polynomial with a term per atom, factor it, reverse one factor, and the interatomic vectors do not notice. Every cyclic pair the search finds is accounted for by it, and the construction survives the move to a torus unchanged.

What nothing here asks is whether the constrained pairs come from that construction too. A structure invariant under p3 that factors would have to factor into pieces the three-fold respects, which is a real condition on the factors and might be the reason the three-fold groups are the worst of the seventeen — or might have nothing to do with it, and the coincidences might be of a kind the factorisation never produces. Checking it is the same exhaustive search with one more test inside the loop, and it would turn the measurement above into an explanation.