A stack with no space group

Every structure in this collection so far has had a space group. That is close to a definition of what the collection is about: a crystal is a pattern, a pattern has a symmetry group, and the group is what the whole apparatus computes, checks and draws.

Close packing is where that stops working, and it stops working in a way worth understanding rather than avoiding. A layer of spheres has two sets of hollows; the next layer sits in one of them; and either choice is equally well packed. So a stacking is a word in three letters with no letter immediately repeated, every such word is a legal structure, and the number of them doubles with every layer.

Most of those structures have no period at all. A structure with no period in the stacking direction has no lattice in that direction, and a structure with no lattice has no space group — not an unusual one, not a large one, none. And yet nothing is wrong with it: it is close-packed everywhere, its density is the same as the cubic and hexagonal stackings’, and no local measurement can tell it from either.

One kind of contact #

The first column of that table is the fact everything else rests on, and it is worth proving rather than asserting.

There are six ordered pairs of distinct registrations — AB, AC, BA, BC, CA, CB — and they fall into two families by the shift between the layers: three step forward round the cycle and three step back. The layer’s own symmetry closes the gap. Its three-fold rotation cycles the registrations A → B → C, which carries the three forward steps onto one another; a mirror of the layer reverses the sense, which carries the forward family onto the backward one.

So every adjacent pair of layers in every close-packed structure is congruent to every other, in every one of the structures, whatever the word. There is one local configuration and there are 2ⁿ structures.

That is the defining condition of an order–disorder family in Dornberger-Schiff’s sense: a set of structures built from one layer, all of whose local configurations are equivalent, which are nonetheless different structures. It is not a statement about close packing in particular — micas, silicon carbide, the clays and a long list of organic crystals are OD families for the same reason — but close packing is the case where every step of the argument is short enough to compute.

Operations that are defined somewhere #

If a family cannot have a single symmetry group, the question is what it has instead, and the answer is a familiar object used in a way this collection has not needed before.

A symmetry operation of a crystal maps the whole structure onto itself. A partial operation maps one layer onto another and is defined nowhere else. The operation taking layer A to layer B is a real isometry — a translation by a third of the cell diagonal — and it is not a symmetry of any structure, because applying it to the whole stack moves every layer and lands most of them nowhere.

Partial operations compose, but only sometimes. The step ending on B composes with a step starting from B and with nothing else.

Twelve of the thirty-six composites exist. A set of operations closed under composition wherever composition is defined is a groupoid rather than a group, and the empty cells are the entire difference between the two.

Why that is the right object here, and not a piece of vocabulary. The groupoid is the same for every member of the family. The cubic stacking, the hexagonal stacking, a polytype of period nine and a stack of two hundred layers faulted at random all have exactly these six partial operations composing in exactly this pattern — because they are all built from the same layer with the same contacts. The space group, where there is one, differs from member to member and is not a property of the family at all.

So the groupoid answers the question a reader of a disordered diffraction pattern actually has, which is not “what is the symmetry of this specimen” but “what family is this a member of”.

Where the ordering lives #

A family whose pairs are all equivalent is not a family with no structure in it. Go up one level, to triples.

A layer’s two neighbours either agree or differ. ABA has them agreeing and ABC has them differing, and no operation of the layer carries one to the other — so a triple has two kinds where a pair had one, and the ordering of an OD family lives there. Jagodzinski’s notation writes h for the first and c for the second, and a stacking becomes a word in those two letters.

A structure in which every triple is alike is at maximum degree of order, and there are exactly two of them: all h, which is ABAB — the hexagonal stacking — and all c, which is ABC — the cubic one. Nothing else, at any period.

That is a real explanation of a fact every mineralogist knows. Silicon carbide has hundreds of polytypes and two of them are common; cobalt is hexagonal or cubic and rarely anything else. The reason is not that other stackings are impossible — they are all equally well packed — but that only two of them are locally uniform one level up, and a growth process that makes the same local decision at every layer produces one of those two.

The divisibility is a small pleasure of the same argument. All-h needs an even period because ABAB repeats in two, and all-c needs a multiple of three because ABC repeats in three. So a stacking of period five or seven cannot be at maximum degree of order at all, however it is arranged — a statement about the arithmetic of the period rather than about the packing.

Why the count is a colouring #

The last column of the first table is worth a paragraph, because it turns a question about stackings into one about graphs and the answer is exact.

A stacking of period n is a sequence of n registrations in which adjacent ones differ — and the last is adjacent to the first, since the sequence repeats. That is precisely a proper colouring of a ring of n vertices with three colours, and the number of those is the chromatic polynomial of a cycle evaluated at three: 2ⁿ + 2(−1)ⁿ.

The enumeration and the formula are computed separately here and agree at every period from two to eight. Nothing about that agreement is deep; what it buys is a check on the enumeration, which is the kind of check this collection prefers to a stated total — and it makes the doubling in the table a fact with a reason rather than an observation about a column.

The count is of words, not of structures. Two words describing the same stack seen from a different starting layer, or with the registrations relabelled, are one structure, and the polytype count divides those out. Both numbers are worth having and they answer different questions: the word count is how many ways the choices can be made, and the structure count is how many distinguishable crystals result.

A local rule cannot fix a global structure #

There is a general statement underneath all of this, and stating it is most of what an order–disorder family is for.

A rule about neighbours does not determine an arrangement. Close packing is a rule about neighbours — no two adjacent layers in the same registration — and it leaves an exponentially growing number of arrangements satisfying it. That is not a peculiarity of spheres. It is the same situation as the ice rule, where a purely local condition on each oxygen leaves a residual entropy a calorimeter can measure, and the same as the arrangements a local rule allows, where counting them is the whole difficulty.

What is unusual about close packing is how completely the local rule is satisfied: not merely legal at every contact but congruent at every contact, so that no local measurement whatever distinguishes two members of the family. In the ice rule the local configurations come in several kinds and counting them is the subject; here there is one kind, and the subject is that one kind is not enough.

That is why the groupoid is the right object and a group is not. A group is what describes an arrangement. A groupoid is what describes a rule about neighbours — arrows with ends, composing when the ends match — and a rule about neighbours is all an OD family has.

The stack that has nothing #

Generate two hundred layers with a twelve per cent chance of taking the wrong hollow, and the word has no period. Not a long one: none, checked over the whole word rather than over a window.

So the structure has no translation in the stacking direction, no lattice in three dimensions, and no space group. Its symmetry is the layer’s own group together with the in-plane translations, which is a layer group — one of the eighty this collection enumerates elsewhere — and a layer group is not a space group missing a few operations; it is a symmetry group of a different kind of object.

And nothing about it is defective. Every one of its hundred and ninety-nine contacts is one of the six, and the six are one kind. Its density is exactly that of the cubic and hexagonal stackings. A measurement over any two adjacent layers gives the same answer it would give on a perfect crystal, and a measurement over three gives a mixture of the two answers a perfect crystal gives.

That is what makes it interesting rather than merely broken. The disorder is not damage to a structure; it is a choice made independently at each layer, in a situation where the choices are equally good, and it is why a faulted stack diffracts as streaks rather than as spots: the reflections that can see the sequence have no lattice to sharpen them and the ones that cannot are as sharp as ever.

What the account refuses #

The one to keep is the fourth. A groupoid whose composition happened to be total would be a group, and the whole framing of this essay would be a longer way of saying something a group already says. Twelve composites of thirty-six is the measurement that makes the distinction real, and it is computed rather than described.

The last is the control that makes the sixth mean anything. A period test that returned “none” on every input would agree with the faulted stack for the wrong reason, so the same test is run on an unfaulted one and has to find a period of two.

What a diffraction experiment sees of all this #

The account above is about structures, and the reason it matters is that the structures are told apart — and not told apart — by an experiment.

A reflection with h − k a multiple of three picks up a phase of exactly one per layer, whatever the registration, so it is blind to the sequence. Those reflections are as sharp on a disordered stack as on a perfect one, and their positions give the layer’s own two-dimensional cell exactly. A reflection with h − k not a multiple of three picks up a third of a turn per step, so it sees the sequence and has no lattice to sharpen it: it spreads into a rod.

So a diffraction pattern of an OD family reads as a set of sharp spots with streaks between them, and the two halves are answering different questions. The sharp part measures the layer, which every member of the family shares. The diffuse part measures the sequence, which is what distinguishes them. That division is exactly the division between the groupoid and the space group: the sharp reflections see the part of the structure the groupoid describes, and the streaks are where the missing group would have been.

It is also why a family like this is so often misindexed. A pattern with sharp spots on a hexagonal net looks like a hexagonal crystal, and a crystallographer who indexes the sharp part and ignores the streaks gets a perfectly consistent small cell for a structure that has no cell at all in the third direction.

Where the exactness stops #

Computed here: the six ordered pairs of registrations with their shifts, and the identification of all six under the layer’s own operations; the composition table of the six partial operations; every stacking of period up to eight, against the chromatic polynomial of a cycle; the Jagodzinski letter of every layer of every one of them; the structures with every triple alike, up to rotation of the stack and relabelling of the registrations; and a faulted sequence of two hundred layers with its period, its faults and its contacts.

A period is a period of the word as generated. An infinite aperiodic sequence and a very long periodic one are indistinguishable inside any finite window, so “no period” here means no period in two hundred letters and not a theorem about an infinite structure. That is the same caveat the faulted-stack essay carries and it applies unchanged.

The groupoid computed here is the one close packing has. An OD family in general has partial operations of several kinds — Dornberger-Schiff’s λ-operations within a layer and σ-operations between layers — and the classification of OD families by their groupoids is a subject this essay does not enter. What is computed is one family’s, in full.

Nothing here derives which structures actually grow. Two structures are at maximum degree of order and hundreds of others exist; which of them a given substance forms depends on the energy difference between an h triple and a c triple, which is chemistry and is not in this file. What the symmetry decides is that the two are the only candidates for a locally uniform structure, and that everything else is a mixture.

Who found it, and what they were looking at #

Käte Dornberger-Schiff set the theory out between 1956 and 1964, and the structures that forced it were not close-packed metals but layer silicates and a scatter of organic crystals whose diffraction patterns had streaks in them that nobody could index.

The situation she named is the one above: a family of structures built from one layer, in which every way of adding the next layer is equivalent to every other, so that the family has a description and its members individually have less of one than a crystallographer expects. Her contribution was to insist that the family is the object — that “this is an OD family with these partial operations” is a complete statement about a substance, and that asking which member a particular specimen is may have no answer, because a specimen with no period is not a member of a list.

That is a stronger claim than it sounds. Crystallography’s ordinary move on a disordered pattern is to model the disorder as a departure from an ideal structure, which presumes the ideal structure is what the substance is. An OD family says there may be no ideal structure to depart from: the two ordered polytypes are not the “real” arrangement with the rest as damage, they are the two members that happen to be locally uniform one level up, and the family is what the substance is.

Where the ladder goes next #

Back, to the family this one describes: how many polytypes, where the stackings of each period are counted up to equivalence, and two stackings, one density, where the two maximum-degree-of-order structures first appear.

Sideways, to what a disordered stack does to an experiment: the streaks a faulted stack makes, where the reflections that can see the sequence are exactly those whose indices differ by something other than a multiple of three, and the order a diffuse pattern measures.

Onward, to the symmetry a structure with no space group does have: a layer is not a wallpaper, where the layer groups are enumerated, and what a cleave leaves, which is the same object arrived at from a surface.