The symmetry of an average

A fivefold axis in an ordinary crystal makes the case that a molecule may carry symmetry the crystal has not: the restriction forbids a fivefold axis to the lattice and says nothing about what sits inside one cell. This is the other half of the same subject, and it is the half that causes trouble, because the error runs the other way.

A molecule may appear to carry symmetry it has not, and the reason is that nobody measures a molecule. A diffraction experiment measures a crystal, a crystal is some 10²⁰ unit cells, and what comes out is the average of all of them. If different cells make different choices, the average is the superposition of the choices — and a superposition is an orbit, so it has the symmetry of the group that generated it whether or not anything in the crystal does.

The arithmetic, which is orbits run backwards #

Put a molecule whose own symmetry is H at a site whose site symmetry is S, with H a subgroup of S of index k. The molecule can be oriented in k ways, one for each coset of H in S, and each is as good as the others — nothing in the structure prefers one, or the site symmetry would not be what it is.

So each cell chooses, the choices are made independently, and in a large crystal all k occur in equal proportion. The structure a measurement reports is the superposition of the k orientations, with every atom present in a fraction 1/k of the cells. That fraction is the occupancy, and it is the number a refinement reports when it is describing disorder honestly.

The superposition is the orbit of the molecule under S, and its symmetry is S — the full site symmetry, which the individual molecule does not have. So the average has more symmetry than the thing being averaged, exactly, and by a factor which is the index k.

This is Wyckoff positions with the implication reversed. That essay observes that an atom at a special position has the symmetry of the position forced on it, and notes that a molecule with no threefold symmetry apparently sitting on a threefold centre is a sign of disorder. Here that sign is the whole subject: the structure is an average over orientations, and finding a molecule at a site whose symmetry it cannot have is a statement about the measurement, not about the molecule.

Detected rather than argued #

Both groups in the figure above are found by the detector that has never heard of either. A point set is generated by applying the smaller group to a motif, handed to a routine that enumerates every operation the lattice permits and keeps those that map the set to itself, and the answer is required to be the smaller group exactly. Then the same motif is orbited under the larger group and the same routine returns the larger one.

The assertion in the middle is the one that makes the demonstration honest: the ordered arrangement must have exactly the smaller group and not more. A motif that happened to be invariant under something extra would make both answers agree and the demonstration vacuous, and this site has a standing rule about that — the motif must be a comma, three asymmetric points, so that nothing is accidentally invariant.

Which sites can do this, and how often #

The condition for the trouble is that a site’s symmetry exceeds the molecule’s, and Wyckoff positions says how many such sites a group has. Two of the seventeen plane groups — p1 and pg — have none at all: their operations are translations and glides, neither of which fixes a point, so every position is general, every molecule sits where nothing constrains it, and orientational disorder of this particular kind cannot arise.

Every other group has special positions, and the higher the group’s order the more of them and the larger their stabilisers. p6m has sites of order twelve. A molecule with twofold symmetry placed at one of them has six orientations, an occupancy of one sixth, and an average with twelve operations where the molecule has two.

There is a converse worth stating, because it is the reason crystallographers place molecules where they do. A molecule with the site’s symmetry sits there in one orientation, ordered, with full occupancy — so a structure in which every molecule’s own symmetry matches its site’s is a structure with no disorder of this kind anywhere. Symmetry-matching is what makes a structure tidy, and mismatching is what makes it interesting.

The worst case: a centre from nowhere #

The general statement becomes a specific and famous danger when the index is two and the extra operation is an inversion.

A chiral molecule has no improper operation. Put one at a site, let the two orientations related by inversion both occur, and the average is centrosymmetric — while every molecule in the crystal is of one hand and the bulk material may be optically active. A structure refined against that average is refined in a centrosymmetric space group, which is the wrong group for what is actually there.

This is the commonest route to a structure published in the wrong space group, and it is the accidental symmetry this site’s round trip was built to catch — arriving from physics rather than from a careless drawing. The pattern figures on this site refuse to be drawn when the point set has more symmetry than the caption claims. A crystal has no such refusal built in, and the extra symmetry arrives in the data.

Two things make it hard to catch. The averaged structure fits the data well, because it is what the data measure. And the individual molecules are entirely ordinary — there is no strained geometry, no impossible bond, nothing an internal check would flag. The evidence is elsewhere: half-occupied sites, anisotropic displacement parameters elongated along the direction the two orientations differ in, and a physical property such as optical rotation that a centrosymmetric structure cannot have.

Why the average is not a structure #

There is a temptation to say the crystal “is” centrosymmetric because that is what the measurement reports, and it is worth resisting precisely.

The average structure is a real object in the sense that it is what the Bragg reflections measure — that much is not in doubt. What it is not is a possible arrangement of atoms: it has two half-atoms at positions a bond length apart, which no cell contains. Every cell contains one or the other, and the average contains both at half weight.

So the average is a description of an ensemble rather than a description of a structure, and asking for its space group is asking a question about the ensemble. The ensemble genuinely has the larger symmetry — it is invariant under the operation that exchanges the orientations, because the two occur equally. The individual cells do not, and no experiment measuring only the Bragg intensities can see them.

What can see them is the scattering the Bragg peaks leave out. The disorder is not random in each cell independently: a molecule’s neighbours generally prefer a particular relative orientation, so the choices are correlated over a few cells, and correlated disorder produces diffuse scattering between the Bragg peaks. That signal carries the local arrangement, it is weak and broad, and it is thrown away by an experiment set up to integrate peaks. Everything about the individual cells lives there.

The near-symmetric case, and where the tolerance comes back #

Everything above assumes the k orientations are exact copies of one another, related by the site’s own operations. Real disorder is untidier: the molecule distorts a little differently in each orientation, so the superposition has nearly the site symmetry and not exactly it.

That puts the question straight back into the territory of near-symmetry and the tolerance, and with it the problem that essay is about. On this site a symmetry is decided by integer arithmetic and there is no threshold to choose. A measured, disordered structure has no such luxury: whether the average is centrosymmetric is decided by whether a residual is small enough, the residual depends on the data quality, and the answer stops being a fact about the structure and becomes a fact about the experiment.

Two consequences follow. The published space group of a disordered structure is a judgement, and different investigators with different data have reached different judgements about the same material. And the standard tests — refining in both groups and comparing R factors, or examining the distribution of intensities — are statistical rather than decidable, which is why they are argued about.

The same trap in three other places #

The pattern — an average acquiring symmetry no individual has — turns up across this site under four names, and they are worth putting together because the arithmetic is identical each time.

Merohedral twinning. Two orientations of one structure, related by an operation of the lattice that the crystal lacks, superposed in reciprocal space rather than in real space. What a twin does to diffraction is exactly this computation performed on intensities: nothing splits, nothing moves, and the measured pattern has the symmetry of the twin law whether or not the crystal does.

The symmetry diffraction adds. Friedel’s law gives every measured pattern a centre of inversion whatever the structure is — an average over ±h rather than over orientations, and the same conclusion: the measurement is more symmetric than the object.

A dot as a motif. The motif must be a comma: a single point at a general position acquires an inversion centre, because the midpoint between it and its own lattice translate is one. That is an average over the lattice rather than over orientations, and the extra symmetry is just as real and just as misleading.

A powder pattern. Grinding a crystal averages over every orientation at once, and what a powder pattern loses is precisely the information the averaging destroys.

Four averages, four apparent symmetries, one piece of arithmetic. The general statement is that averaging over a group’s orbit produces something invariant under the group, which is a triviality, and the difficulty is entirely in noticing which average an instrument is taking.

What is being counted, exactly #

Three quantities in this essay are easy to run together and are different.

The index k is the number of orientations, which is the site symmetry divided by the molecular symmetry. It is a group-theoretic integer and it decides the occupancy.

The occupancy is 1/k, and it is what a refinement reports per site. It is the fraction of cells in which a given atom of a given orientation is present, and it is exactly 1/k only if the orientations are equally likely — which is what the site symmetry guarantees, and which fails the moment an ordering interaction or an applied field prefers one.

The apparent atom count is k times the true one. A disordered structure looks as though it has more atoms than it does, each with a fraction of an electron’s worth of scattering, and a formula computed from the apparent atoms without the occupancies is wrong by the factor k. That is the same arithmetic the asymmetric unit in space uses for multiplicities, doing damage rather than bookkeeping.

What this does not decide #

Three refusals, all of them about the same line.

Nothing here says a crystal is disordered. The computation says that if a molecule of symmetry H sits at a site of symmetry S, the average has symmetry S and the occupancy is 1/k. Whether any real crystal does that is a question about a material, and this site does not answer questions about materials.

Nothing here computes an energy. Which orientation a cell prefers, how strongly, and whether the disorder freezes out at low temperature into an ordered superstructure, are thermodynamic questions. The symmetry argument gives the list of possibilities and says nothing about which is taken — a rule the applied field of this site keeps and this essay keeps too.

Nothing here is a statistical test. The claim is exact for exact superpositions of exact copies. The real diagnostic problem — deciding from noisy data whether an apparent centre is genuine or an artefact of averaging — needs statistics that are outside this site’s arithmetic entirely, and the essays that touch it say so.

The index, seen in the point count #

The cleanest signature of the whole business is a count, and it can be read straight off the figures.

An ordered arrangement at a site of index two in p4 has six atoms in the drawn cell; the average has twelve, each at half occupancy, so the total scattering power is the same. That equality is not a nicety — it is what makes the average fit the data. A refinement that placed twelve atoms at full occupancy would predict twice the scattering and fail immediately; one that places twelve at half occupancy predicts exactly what six full atoms predict, and is indistinguishable from the ordered model on the Bragg intensities alone.

So the ordered and disordered models are not competing explanations of the same data — they are the same explanation, written twice. What separates them is not the fit but the interpretation: whether the crystal contains six atoms in one arrangement or twelve half-atoms in two. Deciding that needs something outside the Bragg data, and the something is usually a physical property, a chemical impossibility, or the diffuse scattering.

Where the ladder goes next #

The local-symmetry anchor now has two rungs: symmetry a crystal has locally and not globally, and symmetry an average has that no cell does. The obvious third is correlated disorder and its diffuse scattering — the signal that carries the local arrangement, which is where the information the average throws away actually is, and which would need a different computation from anything on this site: a pair distribution function rather than a group.

The nearer rung is partial ordering: what happens when the k orientations are not equally likely, so the average is neither the ordered structure nor the fully disordered one. The symmetry then depends on the occupancies, which is a genuinely uncomfortable statement — a group deciding on a continuous parameter — and pinning down exactly what is meant by it would be worth a rung on its own.

Where the discarded information went #

An average throws information away, and in this case the discarded part is not gone. It is somewhere else in the same experiment, and knowing where turns the whole difficulty into a measurement.

Bragg peaks are the average and nothing else. A reflection is sharp because every cell contributes in phase, and cells contribute in phase only in as much as they are identical. So the sharp part of the pattern is built from the mean contents of a cell — precisely the superposition this essay is about — and no refinement against Bragg intensities alone can recover more.

The deviations scatter too, and not into the peaks. Whatever each cell does differently from the mean scatters with a phase that varies from cell to cell, so it contributes nowhere in particular: it spreads as diffuse intensity between the reflections.

And its shape carries the correlations. If the orientations really are chosen independently in each cell, the diffuse intensity is smooth and structureless. If a molecule’s orientation is correlated with its neighbours’ — the usual case, since neighbours touch — the diffuse intensity has structure: streaks along a direction in which the correlation runs, broad maxima where the ordering is nearly periodic. What a diffuse pattern measures is that reading.

So the two halves of the pattern answer two different questions. The peaks say what the average cell contains; the diffuse says how the cells differ and how those differences are arranged. A structure determination that records only the first has not measured the crystal wrongly — it has measured half of it and reported that half.

Static, or merely fast #

A superposition of orientations has two quite different causes, and they are distinguished by one experiment.

The molecules may each be fixed and differently oriented, with the choice frozen in as the crystal grew. That is static disorder, and the average is over positions in the crystal.

Or each molecule may be turning, visiting every orientation faster than the measurement, so that the average is over time as well as over cells. That is dynamic disorder, and no individual site has a fixed orientation to speak of.

Diffraction cannot tell them apart at one temperature. The measurement is an average either way, and the averaged density is the same.

Cooling separates them. A dynamically disordered crystal slows as it cools, and at some temperature the motion stops — usually by the molecules ordering into a definite arrangement, which multiplies the cell, adds superlattice reflections, and lowers the symmetry to a subgroup. A statically disordered crystal does none of that: it is already frozen, and cooling only sharpens the peaks it already has.

So the temperature dependence is the diagnostic. A structure that looks too symmetric at room temperature and resolves into an ordered subgroup on cooling was dynamically disordered; one that keeps the same average down to the lowest temperature reachable was disordered when it grew.