How many orientations a disorder needs

What a molecule gives up to sit in a crystal settles which way the compatibility runs — the site’s symmetry must be a subgroup of the molecule’s — and then names what a crystal does when the packing wants a symmetric site and the molecule is not symmetric enough. It disorders: the molecule occupies several orientations, each at fractional occupancy, and the average has the site’s symmetry although no individual molecule does.

That is where the previous rung stops, and it stops one question short. How many orientations?

The answer is not a quantity anybody measures. It is a group index, it is decided before any data is collected, and the occupancy a structure report prints for such a molecule is the reciprocal of a whole number for the same reason a coset count is.

The count is a coset count #

Write S for the group of operations that fix the site, and H for the part of it the molecule actually has. H is a subgroup of S, necessarily: the operations of S that leave the molecule alone are closed under composition and inverse, because being a symmetry of something is.

Now apply every operation of S to the molecule. Two operations produce the same orientation exactly when they differ by an element of H — that is what having H as its symmetry means — so the orientations are in one-to-one correspondence with the cosets of H in S, and there are

$$n = |S : H| = \frac{|S|}{|H|}$$

of them. The crystal contains all n, in equal proportion, because every one of them is carried to every other by an operation the crystal has; so each carries occupancy 1/n.

Both quantities in that fraction are orders of finite groups, so n is a whole number and 1/n is the reciprocal of one. That is the whole of the arithmetic, and its consequence is a constraint on what a refinement may legitimately report.

The occupancies that exist, and the ones that cannot #

Collecting every index available anywhere in the census gives the complete list of occupancies a site symmetry is able to require: one, a half, a third, a quarter, a sixth, an eighth, and so on downwards.

Nothing lies strictly between a half and one. A symmetry-imposed disorder is over two orientations or more, so its occupancy is at most a half; and there is no arrangement of a molecule over cosets that gives sixty per cent of one thing and forty of another, because the cosets of a subgroup all have the same size.

That is a diagnostic and it is worth stating as one. A structure report showing a molecule split 0.63 to 0.37 is describing something real, and whatever it is, the site’s symmetry did not impose it. The candidates are a genuine two-state disorder with an energy difference between the states — perfectly common, and it has a temperature dependence — or a substitutional mixture of two species, or a modelling artefact. What the arithmetic rules out is the reading that the site required it, and that reading is the default one.

The converse is weaker and worth being honest about. An occupancy that is a half is not evidence that symmetry imposed it, because two states of equal energy give a half as well. The arithmetic refuses; it does not confirm.

Lagrange permits and the group refuses #

The indices available at a given site are the indices of its subgroups, and that is a shorter list than the divisors of its order.

Lagrange’s theorem says a subgroup’s order divides the group’s. It does not say the converse, and the converse is false — a group need not have a subgroup of every order dividing its own. So a site whose group has order twelve does not automatically offer an orientation count of two, because that would require a subgroup of order six and there may not be one.

Exactly one row in nineteen behaves this way, and it is the cubic site of symmetry 23. Its group is the tetrahedral rotation group — the alternating group on four letters — and that group’s lack of a subgroup of index two is the classical fact that A₄ has no subgroup of order six, which is the standard first counterexample to the converse of Lagrange.

So a group-theoretic curiosity has a crystallographic consequence. A molecule sitting on a 23 site can be ordered, or disordered over three, four, six or twelve orientations — and cannot be disordered over two. There is no model of a half-and-half orientational disorder at that site that is consistent with the site symmetry, and a refinement that produces one has either moved the molecule off the special position or has the site symmetry wrong.

That the census contains exactly one such row is itself worth noting rather than passing over. Every other site symmetry this site builds — the orders forty-eight, twenty-four, sixteen, twelve, eight, six, four, three and two — offers every divisor of its order, which is why the constraint is invisible in ordinary practice and why it is worth knowing about when it bites.

What the average is, and what it is not #

The reason the disorder is tolerable at all is that the thing the experiment measures is not the molecule.

The superposition of the n orientations is invariant under all of S, because S permutes the cosets among themselves. So the object with the site’s symmetry exists — it is the average — and it is exactly what a diffraction experiment integrates over some 10²⁰ cells to obtain. The symmetry of an average is the general form of that statement and this is the case where the averaging is over a group rather than over a distribution.

The molecule keeps H and the average keeps S, and the difference between them is the fractional atoms. A report that lists positions at occupancy 1/3 is listing one orbit of the average and calling it a molecule; the molecule is one third of what is written down, and which third is not determined by anything in the file.

What the average cannot show is whether the disorder is static or dynamic. A molecule hopping between the three orientations and a crystal containing three fixed populations give the same average, hence the same intensities, hence the same refinement. Distinguishing them takes a measurement that is not diffraction — a relaxation time, a heat capacity anomaly, a linewidth — and the site symmetry has nothing to say about it. The fullerene case the previous rung describes is exactly this ambiguity resolved by cooling until the answer changes.

When the intersection is trivial #

The index is largest when the molecule and the site share nothing, and that case is the one the crystallographic restriction creates.

A molecule whose only symmetry is a five-fold axis has a group of order five. No site in any space group contains a five-fold operation, so the intersection of the two is the identity alone — |H| = 1 inside the site group — and the orientation count is the full order of the site. A ferrocene ring at a site with a three-fold axis is disordered over three orientations at a third each, and it is not disordered because the ring is awkward: it is disordered because five and three are coprime and there is no shared subgroup to sit in.

That is a fivefold axis in an ordinary crystal arriving as an occupancy. The molecule’s five-fold symmetry is exact and the crystal has no use for it, and the arithmetic here says precisely how much use: none, and the cost is an orientation count equal to whatever the site offers.

The general statement is worth having because it inverts the usual reading. A molecule is not disordered because it is too symmetric for its site or too asymmetric for it. It is disordered by the amount the two groups fail to share — the index of the intersection — and a molecule with an enormous symmetry group can sit perfectly ordered at a site whose symmetry it happens to contain, while one with a modest group disorders badly at a site whose operations it happens to miss. The most of an icosahedron a crystal can keep is the extreme case of the first half, and this is the second.

The same integer, measured with a calorimeter #

There is an independent route to n that involves no diffraction at all, and it is the reason this arithmetic was first taken seriously.

A crystal in which every molecule independently takes one of n orientations has n choices per molecule, so a mole of them has n^{N} arrangements of equal energy and a residual entropy of R ln n — an entropy that survives to absolute zero because the crystal cannot decide. That is measurable: integrate the heat capacity from near zero upwards, compare with the entropy the gas-phase statistics predict, and the shortfall is the residual.

So the occupancy a refinement reports and the entropy a calorimeter measures are two readings of the same integer. An occupancy of a third and a residual entropy of R ln 3 are the same statement, and a disagreement between them is informative — it usually means the orientations are not independent, because a molecule’s choice constrains its neighbours.

Pauling’s account of ice is the case that established the method, and it is a case where the constraint matters: the residual entropy of ice is not R ln n for any small n but R ln(3/2), because the hydrogen positions are correlated by the requirement that each oxygen keeps two close protons. The correlation reduces the count below the free one, and the gap between the naive number and the measured one is exactly the amount of local ordering present. That is configurational entropy doing the same work a diffuse pattern does, from a thermodynamic direction.

For the symmetry-imposed case above there is no such correlation to worry about only when the orientations are genuinely independent from site to site, which is an assumption and not a theorem. Where they are not, the crystal usually orders on cooling — and the ordering shows up as a superstructure, with new reflections at positions the disordered cell had nothing at.

Where the numbers come from #

Computed here: every site symmetry of each of the forty-five space groups this site builds, by asking each operation whether it moves a point of a grid of twelfths; the linear parts of each, deduplicated; every subgroup of each, by taking each subgroup found so far together with each element and closing under multiplication until nothing new appears; the indices; the divisors; and the difference between the two lists. Then, for one site, the averaged position set and the operations that carry it onto itself.

The subgroup enumeration is exhaustive because every subgroup is generated by its own elements, so adding one element at a time to a subgroup already found reaches every subgroup eventually. That is the argument for completeness; the alternative — enumerating subsets — is not available at order forty-eight and would be the wrong shape anyway.

A grid of twelfths is what decides which sites are seen. A site whose coordinates are not twelfths is not on the grid and does not appear, so the census is a census of the site symmetries this grid finds rather than of all of them. Raising it to twenty-fourths adds no new signature to any group here, which is evidence and not proof; a site type invisible at both would be invisible at both.

And the orientation count is about the site, not the compound. Nothing above says a particular molecule will disorder rather than move to a general position, or that a crystal will form at all. What is settled is the shorter statement: given that a molecule of symmetry H sits at a site of symmetry S, the number of orientations is the index and nothing else is available.

When the occupancy comes back wrong #

A refinement that returns something other than 1/n for a molecule on a special position is saying one of three things, and telling them apart is ordinary practice rather than a puzzle.

The site is not the site. The commonest cause is that the molecule is not on the special position at all but a little off it, and the structure has been refined in a space group with more symmetry than the crystal has. The occupancies then drift towards whatever the true arrangement is, and the repair is to descend to a subgroup — which is near-symmetry with no tolerance in it read from the practical end: the higher symmetry was a hypothesis and the occupancies are the evidence against it.

The two states differ in energy. A genuine two-orientation disorder whose states are not related by any operation of the site has no reason to be equal, and its ratio is a Boltzmann factor rather than an index. Such a disorder is real, its occupancies move with temperature, and no symmetry argument constrains them at all.

Or the model has too many parameters. Occupancy and displacement parameter are strongly correlated in a least-squares refinement, so a fitted occupancy near a half is often a fitted occupancy near a half plus a compensating change somewhere else. Fixing it at the index the symmetry requires and letting everything else move is the standard response, and it is standard precisely because the index is not negotiable.

The reason all three are worth separating is that only the first is an error. The second is a physical fact the crystal is reporting and the third is an artefact of the fit; the first is a wrong space group, and a wrong space group makes every subsequent statement about the structure suspect. An occupancy off 1/n is therefore a cheap test of an assignment made much earlier, and it is available in every report that contains a disordered molecule on a special position.

The same index, in three other places #

The quantity |S : H| has appeared in this collection several times already under other descriptions, and the resemblance is not a coincidence — it is the same theorem each time.

The multiplicity of a Wyckoff position is |G : S|, the index of the site symmetry in the whole group, and it counts how many equivalent points the orbit has. That is the orbit-stabiliser theorem, and the count here is the same theorem applied one level down, with the molecule’s own group in place of the site and the site in place of the crystal.

The number of descriptions of a structure is |N : G|, the index of the group in its Euclidean normaliser, and it counts coordinate lists rather than orientations. Same arithmetic, different pair of groups, and the same consequence: the answer is a whole number and a non-integer is a proof that the computation was wrong.

And Z′ — the formula units in the asymmetric unit — is the reciprocal of |G : S| when a molecule sits on a special position, which is why it comes out as a half, a third or a sixth and never as 0.4.

Reading all four together gives the general shape. Whenever a smaller symmetry sits inside a larger one, the ratio of their orders counts something a crystallographer prints: equivalent positions, coordinate lists, formula units, orientations. The arithmetic is one line and the work is deciding which two groups the question is about — which, in the disorder case, means deciding what the molecule actually keeps rather than what it has in isolation.

What this ladder still owes: which subgroup, not how big #

The count above assumes the molecule’s retained symmetry is a subgroup of the site’s, which is the previous rung’s compatibility read at the site. It says nothing about which subgroup. A molecule with a two-fold axis at a site of symmetry mmm might keep the axis along a, along b or along c, and those are three different subgroups of the same order giving the same orientation count and completely different structures. Distinguishing them is a matter of what the molecule looks like, not of how big its group is, and no index will do it.

Nor does anything here compute which disorder a real crystal chooses. The available counts are a list of possibilities and the crystal takes one. What decides is packing energy, which this collection does not compute anywhere and which is the standing boundary between what symmetry permits and what matter does — the same boundary the polar classes run into, where a permission is not a prediction.

The direction this leaves open is the one the missing index points at. A site whose group refuses an index is a site where the ordinary reasoning about disorder fails, and the census found one such site among nineteen by enumerating subgroups. Whether the full two hundred and thirty groups contain more — sites of order twenty-four or forty-eight with subgroup lattices that skip an index — is a question this grid of forty-five groups cannot answer, and the answer would be a small table worth having.