The occupancy does not name the disorder

How many orientations a disorder needs answers a question with a number. A molecule sitting on a site whose symmetry S is more than it can honour keeps a subgroup H of S, takes one orientation for every coset of H, and so occupies each at 1/n, where n is the index of H in S. The count is decided before any data are collected, and an occupancy that is not the reciprocal of a whole number is not a symmetry-imposed disorder.

It ends by naming what the number leaves out. “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.”

That is correct, and it understates the problem in one direction and overstates it in another. There are more than three such disorders at an mmm site, and at some sites many more share one occupancy. But the averaged structure a diffraction experiment refines is not blind to which subgroup was kept, and there is a precise sense in which it always has enough information to say.

Two subgroups can be one disorder #

The first step is to count the disorders correctly, and counting subgroups does not do it.

Suppose a molecule keeps a subgroup H of its site’s symmetry S, and a second molecule keeps H′ = sHs⁻¹ for some operation s of the site. Turn the first molecule by s. The turned molecule is carried onto itself by exactly the operations sHs⁻¹, so it is a molecule keeping H′. And turning by s changes nothing about the disorder, because s belongs to the site: it carries the set of orientations of the first molecule onto itself, permuting them. The two molecules produce the same set of orientations, in the same proportions, around the same site.

So two subgroups that are conjugate in S describe one disorder, and the disorders at a site are the conjugacy classes of its subgroups. That is conjugation as the test of sameness, applied to a molecule rather than to an operation, and it is the same quotient that takes the ninety-eight subgroups of the cubic holohedry to thirty-three classes.

The four-fold site shows the difference most plainly. A molecule keeping the half-turn about a, turned by a quarter, is a molecule keeping the half-turn about b. The two are not two disorders that happen to agree; they are the same disorder described from two starting orientations, and the averaged structure puts atoms from that model on both axes whichever description is used.

Counting subgroups instead of classes overcounts at every site where some subgroup is not normal, and the error is not small. A 4/mmm site has thirty-five subgroups and twenty-seven disorder models; a site of symmetry m3̅m has ninety-eight subgroups and thirty-three models. Only where every subgroup is normal — the abelian sites, mmm among them — do the two counts agree.

Seven disorders at a quarter #

An mmm site is the example the question was asked about, and it is abelian, so every subgroup is its own model. It has sixteen of them.

One is the whole group, at full occupancy, and one is the identity alone, a molecule with no symmetry disordered over all eight orientations. The other fourteen split evenly: seven models keep a subgroup of order four and occupy a half, and seven keep a subgroup of order two and occupy a quarter. The three the question named — a two-fold along a, along b or along c — are three of the second seven.

The other four are a molecule keeping a mirror perpendicular to a, to b or to c, and a molecule keeping only a centre of inversion. The three mirrors and the centre have nothing obvious in common with the two-folds except the order of their group, which is two. That is all the index sees, and it is why seven structures a chemist would never confuse arrive at a refinement with one occupancy between them.

The drawings record what does separate them. A molecule with a mirror has its symmetry fixed along a whole plane, and an atom of the molecule lying in that plane stays in it; a molecule with a two-fold has an axis; a molecule with only a centre has a single point, the site itself. Those loci are where a disordered model’s atoms can sit on its own symmetry, and they are different for each of the seven.

The orientations themselves are arranged differently too, and it is worth seeing how, because it is the structure a refinement is actually fitting. The four orientations of a molecule are the four cosets of the subgroup it keeps, and each coset is a pair of site operations that carry the molecule to the same orientation. A molecule keeping the two-fold about a has its four orientations reached by the pairs {1, 2 about a}, {2 about b, 2 about c}, {mirror across a, centre} and {mirror across b, mirror across c}. A molecule keeping the mirror across a has them reached by {1, mirror across a}, {2 about a, centre}, {2 about b, mirror across c} and {2 about c, mirror across b}. The same eight operations, grouped into pairs in two different ways.

Read as structure, the grouping says which orientations are mirror images of which. In the first model, two of the four orientations are reached from the first by operations that reverse handedness, so a chiral molecule keeping a two-fold is disordered over two right-handed and two left-handed copies. In the second, every coset contains one operation of each kind — the mirror across a is in the subgroup itself — so every orientation is reached by a rotation, and all four copies have the same handedness as the molecule, which is achiral anyway because it has a mirror. A molecule keeping only the centre is achiral for the same reason. Seven models at a quarter differ in whether the disorder mixes the hands, and that is a difference a chemist would care about more than any other, carried by no number in the report.

What the average records, and what it cannot #

The quantity a refinement actually produces is the averaged structure: the symmetry of an average is the site’s, because the average over all the orientations is carried onto itself by every operation of S. The question is whether the average remembers which H produced it.

Most of it does not. An atom of the molecule at a general position is carried by S to |S| positions, and those positions are the same whatever subgroup the molecule kept — the orbit of a point under S does not depend on H. Everything that distinguishes one model from another is carried by atoms lying on the molecule’s own symmetry elements. Such an atom is fixed by some operation of H; in the average it appears on the locus of that operation and on the loci of every operation conjugate to it in S, and nowhere else special.

So what the average can report is a record of which classes of the site’s operations the molecule keeps — the half-turns about the four-fold axis, or the pair about a and b, or the mirrors across the diagonals — together with how many of each. Whether that record is enough to name the model is not obvious, and it is not true in general. There are finite groups with two subgroups that are not conjugate and yet contain exactly the same number of operations from every conjugacy class; the phenomenon is Gassmann’s, from 1926, and such pairs are indistinguishable by any measurement that sees only which kinds of operation are present.

No crystallographic site symmetry has such a pair. At each of the nineteen site symmetries in the census, every two models with the same occupancy keep different classes of the site’s operations, and the same holds for all thirty-two crystal classes taken as site symmetries. The plainer record — which classes appear at all, without how many of each — separates them too. So an average in which every symmetry element of the molecule carries an atom determines the model, and the occupancy never does.

That is a statement with a hypothesis, and the hypothesis is not idle. A molecule with no atom on any of its own symmetry elements leaves the same average under every model of its occupancy, because then every atom is at a general position of H and its orbit under S is blind to H. Such a molecule can be disordered in seven different ways at an mmm site and no refinement of the average will ever say which. What decides it then is not the average but the correlations between neighbouring molecules, which a diffuse pattern measures and a set of occupancies does not contain.

Eleven at one tetragonal site #

A four-fold site crowds the models harder than an orthorhombic one.

Eleven models share the occupancy of a quarter, seven share a half and seven share an eighth. The quarter is the worst, and the eleven are worth reading as structures. A molecule keeping the four-fold rotations and a molecule keeping the 4̅ operations both leave the four-fold axis, and the second leaves the centre as well. Two models keep a group of three half-turns, one with its two-folds along the cell edges and one along the diagonals. Four models keep a mirror group of order four, in four different arrangements against the site’s axes. Three keep a half-turn with the mirror perpendicular to it and therefore a centre.

None of those differences touches the occupancy, and every one of them changes where the averaged structure has atoms from the molecule’s own symmetry. The m3̅m site has the most models in the census, thirty-three over ten occupancies, with nine of them sharing an occupancy of a twelfth — fewer crowded onto one value than at the tetragonal site, because its models are spread over twice as many.

There is a connection here worth stating, because it runs in an unexpected direction. The set of orientations a disordered molecule takes, with the site’s operations permuting them, is a transitive action of the site group, and two models are the same exactly when those actions are the same. The table that decides every action is the complete invariant for that question — a table of fixed-coset counts, which in general is strictly finer than any record of which operations a subgroup contains. For the crystallographic groups the coarser record already suffices, and it is exactly the record an average of atoms on symmetry elements can carry.

Where the occupancy does name the disorder #

The crowding is not universal, and the sites where it is absent are worth knowing exactly, because at them the ordinary reading of an occupancy is complete.

Running the same count over all thirty-two crystal classes as site symmetries, thirteen of them have exactly one model at every occupancy: 1, 1̅, 2, m, 3, 3̅, 4, 4̅, 6, 6̅, 23, 32 and 3m. At those sites, knowing how many orientations a molecule takes is knowing which subgroup it keeps, and a refinement that gets the occupancy right has got the model right with it.

The reason is a property of the groups rather than of the sites. A cyclic group has exactly one subgroup of each order that divides its own, so eight of the thirteen — every class generated by a single operation — cannot have two models at one occupancy. The other five are not cyclic, and they escape for a different reason: their subgroups of each order are all conjugate. In 32 the three half-turns about the three horizontal axes are carried onto one another by the three-fold rotation, so a molecule keeping any one of them is the same model; in 23 the same happens to the three half-turns and to the four three-fold axes. What makes the occupancy sufficient is not that a site has few subgroups but that it has few classes of them.

The other nineteen classes are the ones where a site has two subgroups of the same order that no operation of the site relates — a mirror and a half-turn, or half-turns about axes that no operation of the site permutes — and those are where the number printed beside an occupancy stops being the whole story. Every class containing both a mirror and a two-fold axis is among them, which includes most of the classes molecular crystals actually adopt.

The two lists sort the census’s own sites the same way. Of the nineteen site symmetries in the space groups built here, eight — 23, 6, 3m, 4, 3, 1̅, 2 and m — have one model at every occupancy, and the other eleven crowd.

Why the index cannot be refined into an answer #

It would be natural to hope that something other than the occupancy but still a number — a displacement parameter, a refined multiplicity, a residual — would stand in for the subgroup. None can, for a reason that is about the arithmetic rather than about the experiment.

The orientation count depends on H only through its order. The residual entropy of a crystal in which each molecule chooses independently among its orientations is the logarithm of that count, and the same integer a calorimeter reads is therefore also blind to which subgroup. The multiplicity of the averaged atoms at general positions is |S|, again independent of H. Every scalar derived from the orientations as a set of equally likely choices inherits the index and nothing else.

What is not a scalar is the geometry of where the special atoms sit, and that geometry is a list of loci, one for each class of operation kept. The subgroup is recovered from positions, not from any count, which is the same shape of answer what diffraction cannot tell apart keeps arriving at: a measurement that averages can separate what differs in where things are, and cannot separate what differs only in how the average was assembled.

What the count does not settle #

Which model a crystal takes. The census lists the disorders a site permits and says nothing about which one occurs. That is decided by packing energy and by the correlations between neighbours, neither of which is computed here — the same boundary the question of what a molecule gives up meets at every turn, where symmetry supplies the list and matter makes the choice.

Sites the census did not visit. The nineteen site symmetries are the ones the space groups built here happen to contain. The separation test was also run on all thirty-two crystal classes as abstract groups, each of which is the site symmetry of the origin in its symmorphic space group, so the conclusion does not depend on which space groups were built.

Molecules with non-crystallographic symmetry. A molecule’s own group may be larger than anything a site offers, and only its intersection with the site matters for the disorder. The census counts subgroups of the site, so a five-fold molecule enters as whatever of it the site symmetry can hold, and the most of an icosahedron a crystal can keep is where the size of that intersection is worked out.

What was computed, and how. For each site symmetry, every subgroup by closing subsets under multiplication; their conjugacy classes, by conjugating each subgroup by every operation of the site; the conjugacy classes of the site’s operations; and, for every model, how many operations of each class it keeps. Then the test at every occupancy that no two models keep the same census, with multiplicities and without, at the nineteen sites and at all thirty-two classes. The drawings are the fixed loci of every operation in every kept class.

Where the two ideas came from #

Conjugacy classes of subgroups are as old as group theory, and the observation that conjugate subgroups give the same action on their cosets goes back to its beginnings. The warning that counting a subgroup’s contents can fail to identify it came from number theory: Fritz Gassmann constructed non-conjugate subgroups with identical counts in 1926, to show that two number fields could share every local invariant a prime could detect and still differ.

Orientational disorder on special positions entered crystallography through the plastic crystals, where molecules nearly spherical in shape rotate or hop on a lattice long before the solid melts, and through the routine discovery, in refinement after refinement, that a molecule had been placed on a site more symmetric than itself. The site-symmetry tables of the International Tables were the tool, and the question they answer — what the site offers — is the question above with the molecule left out.

Where this goes: whether the averaged structure is refined against the right model #

The census says that every disorder model is in principle legible from the averaged structure, provided the molecule’s symmetry elements carry atoms. It does not say that the routine practice of refinement tests the alternatives. A structure refined with one model keeps its occupancies at 1/n by construction, and a second model with the same occupancy and different special positions may fit nearly as well, differing only in a handful of partial atoms near the site. Whether the models the census distinguishes are distinguished in practice, at the precision a diffraction experiment reaches, is a question about data and not about groups, and nothing computed here can answer it.