The descent of symmetry is a lattice, not a tree

A crystal that loses symmetry ends up in a subgroup of the class it started in. Which subgroups are available is a finite question, and drawing the answer produces a picture that is worth having in front of anybody thinking about a phase transition.

For the cubic holohedry m3̅m the answer is twenty-five of the thirty-two classes. The seven that are missing are exactly the seven hexagonal classes — 6, 6̅, 6/m, 622, 6mm, 6̅2m and 6/mmm — because no group containing a six-fold rotation fits inside a cubic group, a cubic lattice having no six-fold axis for it to sit on. The five trigonal classes are all present, sitting on the body diagonals, which is the containment that makes a rhombohedral distortion of a cubic structure possible at all. Arranged by order, with an edge wherever one class is a maximal subgroup of another, the result is the diagram this essay is about.

What “maximal” buys #

A subgroup H of G is maximal when there is no group strictly between them — the same containment relation the fundamental domain of a subgroup is built on, tightened to admit nothing in the middle. That is a strong condition and it is what makes the diagram finite and legible: without it, m3̅m would be joined directly to all thirty-two of its subgroups and the picture would be a hedgehog.

With it, every descent is a path. A drop from m3̅m to mmm of index six is not one edge; it is a chain, and the chain has intermediate classes on it. Reading those intermediates is the practical value of the diagram, because each one is a phase that could exist between the two — and in many materials does.

The perovskites are the standing example. Their high-temperature form is cubic; on cooling, different members drop to tetragonal, to orthorhombic, to rhombohedral, and several go through two or three of those in sequence with a separate transition temperature for each. Every one of those sequences is a path down this diagram, and the sequence a given composition takes is a fact about the composition and not about the group.

Why it is not a tree #

The name in the literature is Bärnighausen tree, after Hartmut Bärnighausen, whose 1980 account made it the standard way to present a family of related structures.

It is not a tree in the graph-theorist’s sense, and the difference is not pedantry. A tree has one path from the root to each node. This diagram routinely has several: mmm can be reached from m3̅m by way of 4/mmm and by way of m3̅, by different chains of the same total index, and neither route is the correct one.

What it is, properly, is a lattice in the order-theoretic sense — a partially ordered set in which any two elements have a greatest lower bound and a least upper bound. The word collides unhappily with the crystallographic sense of lattice, which is why nobody in the field uses it, and the collision is worth noting rather than resolving.

The consequence is one to keep hold of: an index-six descent that happens in one step is not the same thing as two transitions of index two and three, and the diagram does not distinguish them. A single transition to a subgroup that is not maximal is perfectly possible. What it means is that the intermediate phases exist as groups without existing as phases of that material.

How far from a tree it is can be counted rather than described, and the count has to be made over the subgroups and not over their names. That distinction is the one thing here that is easy to get wrong. Class 2 sits inside m3̅m along a cube axis and along a face diagonal, and no operation of the group carries the first onto the second; drawing both as one node labelled 2 is what makes the diagram legible, and it also invents paths, because a step down to one of them followed by a step down from the other is a route through the picture that no chain of actual groups realises. Counted properly — ninety-eight subgroups and two hundred and ninety-nine maximal steps between them — the number of chains reaching a given class is a genuine quantity, and for most classes it is not one.

The tetragonal holohedry is smaller and shows the same feature more legibly, which is the reason to draw it beside the cubic one rather than instead of it. Fifteen classes is few enough to follow every edge by eye, and the multiple routes stop being a number in a column and become something visible in the picture.

Where the point-group picture is incomplete #

Everything above is about classes — point groups — and a real phase transition happens between space groups. The difference matters and it is the subject of the next rung, but the shape of it belongs here.

A space group has translations, so its subgroups come in kinds that a point group’s cannot:

• Translationengleiche, “with the same translations”: the lattice is unchanged and the point group shrinks. This is what a space group being an extension makes available at the point-group end. These are the descents this essay’s diagram draws, because the point-group picture sees exactly this kind.
• Klassengleiche, “with the same class”: the point group is unchanged and the lattice loses translations, so the cell grows — the reverse of what centring does, and the same sublattice arithmetic. Invisible in a point-group diagram — the class does not change, so both ends are the same node.
• Isomorphic: a klassengleiche subgroup of the same space-group type, with a cell some multiple of the original. Also invisible here, and there are infinitely many of them for any group.

So this diagram is a projection of the real thing, showing one of three kinds of step and collapsing the other two to nothing. A transition can be entirely real, produce domains, and appear nowhere on it — which is exactly what happens for an ordering transition that doubles a cell without changing the class, and is what the next rung is about.

Saying that plainly matters because the diagram is otherwise so persuasive. A picture that shows everything of one kind and nothing of another looks complete, and this site has met that failure before: a bounded search returning an answer correct as far as it went.

Landau’s condition, and what it does not say #

The rule quoted for continuous transitions is that the low-symmetry group must be a subgroup of the high-symmetry one. That is necessary and it is worth being clear that it is not sufficient.

Necessary. In a continuous transition the order parameter grows from zero, so arbitrarily close to the transition the structure is arbitrarily close to the parent. Any symmetry the child has must be a symmetry the parent had — which is what makes the index alone predict the domain count, with nothing else about the material needing to be known.

Not sufficient. Landau’s full analysis asks more of the pair: the order parameter must transform as a single irreducible representation of the parent group, there must be no third-order invariant in the free energy, and a further condition on the fourth-order terms decides stability. Pairs satisfying the subgroup relation and failing those tests exist, and their transitions are first-order — they jump, with latent heat.

So the diagram supplies the candidates and the representation theory prunes them. This site does not do the second half: an irreducible representation of a point group is a different computation from a subgroup enumeration, and claiming the diagram predicts which transitions are continuous would be exactly the over-claim its whole field is written against. The same care the character sums were stated with applies here, for the same reason.

One chain is worth following by hand, because it makes the arithmetic concrete. Take m3̅m of order forty-eight, down to 4/mmm of order sixteen, down to mmm of order eight, down to 2/m of order four, down to 1̅ of order two. The indices along the way are three, two, two and two, and every step loses operations and none gains any — which is what a subgroup relation is. A material passing through the whole chain would show four successive transitions with domain counts three, two, two and two, and the number of distinct orientation states at the bottom would be their product, twenty-four. That is the index of 1̅ in m3̅m, and it comes out the same whether it is computed step by step down this route or in one division at the ends, because the index is multiplicative. It is the same fact the last section of this essay puts to work in reverse.

How many edges there are, and why the number is not small #

The three diagrams in this essay have twenty-five nodes and fifty-six edges, fifteen nodes and twenty-nine edges, and twenty nodes and forty-seven edges. Those edge counts are larger than a reader expecting a family tree would guess, and the excess is exactly the multiple-route phenomenon.

A tree on n nodes has n − 1 edges. The cubic diagram has fifty-six on twenty-five nodes, which is more than twice as many. So on average each class in it has several distinct maximal supergroups, and a material sitting in that class could have arrived from any of them.

That is a fact about the classification and not about any material, and it has a practical use: given a low-symmetry structure and a suspicion that it is a distorted version of something, the number of candidate parents is small enough to test them all. Twenty-five nodes and fifty-six edges is a search space a computer exhausts instantly, which is why the pseudo-symmetry searches described below are routine rather than heroic.

Domains multiply along a path #

That last observation deserves its own statement because it is the practically important consequence of the diagram.

If a crystal descends from G to K through an intermediate H, the number of domain states at the end is the index of K in G — and the index is multiplicative, so it is the product of the two step indices. A crystal that goes cubic to tetragonal to orthorhombic in two transitions ends with the product of the two domain counts, and the domain structure it ends with has a hierarchy: the domains from the first transition are subdivided by the domains from the second.

That hierarchy is visible. In perovskite ceramics the coarse texture is set by the first transition and the fine lamellae within each coarse region by the second, and the microstructure is read as a record of the cooling path. The diagram is therefore a prediction about what a micrograph should look like, and it is one that a picture can falsify.

Reading it upwards #

The diagram is drawn downwards and it is at least as useful read the other way.

Given a structure, the question “what is its aristotype?” is a search for a group above it in the diagram, together with a small distortion that would carry the structure onto a higher-symmetry arrangement. That is a real procedure and it is now automated: the structure is tested against each candidate parent by asking whether the atoms would sit at special positions of that group after a small shift, and the ones that nearly do are reported with the size of the shift.

The output is the list of pseudo-symmetries a structure has, and it is worth having for two very different reasons.

One is that a pseudo-symmetry is a candidate high-temperature phase. A structure that is nearly cubic is a structure whose cubic parent may well be stable a few hundred degrees higher, and looking for it is a sensible experiment.

The other is that a pseudo-symmetry is a candidate mistake. A structure that is nearly centrosymmetric may in fact be centrosymmetric, solved in too low a group because a twin or a disorder was misread. Running the search upwards on a published structure is now a routine validation step, and it finds errors.

Both readings depend on the same arithmetic and on a threshold that is not in it. How near is nearly? The answer is a tolerance, chosen, and the whole of near-symmetry is about what choosing it costs.

The vocabulary, which is Megaw’s and Bärnighausen’s #

Helen Megaw introduced aristotype and hettotype for the two ends of one of these relations: the aristotype is the high-symmetry parent structure, the hettotypes are the derived lower-symmetry ones. Perovskite is the aristotype of a family with hundreds of members.

The vocabulary is doing real work rather than labelling. Calling two structures an aristotype and its hettotype is a claim that they are related by a group-subgroup relation and by a small distortion, and that claim can be wrong — two structures can look similar and have no such relation, in which case a continuous transition between them is impossible and the resemblance is a coincidence of packing.

Bärnighausen’s contribution was to insist that the relation be written down completely: the subgroup type, the index, the change of basis, the origin shift, and the correspondence between the atomic positions. A family tree in that format is checkable, and a great many published structural relationships turned out not to survive the check.

What is computed here, and what is checked #

The nodes and edges come from the subgroup lattice rather than from a table.

Every subgroup of the parent is enumerated by closure, each is named by its element signature — how many operations of each of the ten types it has — and the naming rests on a result rather than on a hope: the enumeration of the thirty-two classes proves that the signatures of the thirty-two are distinct, so a signature identifies a class outright.

Maximality is then decided by search rather than assumed: for each pair with one contained in the other, everything else in the list is checked for lying strictly between them. That is quadratic in the number of subgroups and the number is under a hundred, so it costs nothing and it is a decision rather than a lookup.

Two things are asserted. Every subgroup of a crystal class is itself a crystal class — which is not obvious and is what makes the diagram’s nodes a subset of the thirty-two. And the diagram runs from the parent down to class 1, so nothing has been lost at either end.

Why a maximal step is never mixed #

The essay observes that a space group’s subgroups come in two kinds and that the point-group diagram shows only one of them. There is a theorem making that division sharp, and it is what allows a Bärnighausen tree to label every edge with one letter rather than two.

Hermann’s theorem says that a maximal subgroup of a space group is either translationengleiche or klassengleiche, never a mixture. A subgroup that loses some operations and some translations is never maximal: there is always a group properly between it and its parent, obtained by making one of the two losses and not the other.

So every maximal step is pure, and a chain of maximal steps is a sequence of pure steps in some order. A descent that loses operations and translations together is a path of length at least two, and the diagram’s job is to say which intermediate groups the path can pass through.

That has a practical consequence for reading a structural family. Two structures related by a loss of both kinds are not related by a single step, so there is always a third structure implied between them — a group that may or may not be realised by any material, and which is worth looking for. Several structural families have been completed that way: the intermediate was predicted from the diagram and then found.

How long a chain can be, and why the cubic case is different #

There is a bound on the depth of these diagrams, it is exact, and it comes from group theory rather than from crystallography. Every crystallographic point group is solvable, which is a consequence of the orders available: the largest is forty-eight, and no group of order under sixty is simple except the cyclic ones of prime order. For a solvable group, the longest maximal chain from the group down to the identity has exactly as many steps as the order has prime factors counted with multiplicity. So the cubic holohedry, of order 48 = 2⁴ × 3, has a longest chain of five steps; the tetragonal, of order 16 = 2⁴, four; the hexagonal, of order 24 = 2³ × 3, four.

It is very natural to go one word further and say that every maximal chain has that length, which would make the depth of a class an invariant and the rows of the diagram exact. That is a different theorem and it is about a different class of group. Chains of equal length characterise the supersolvable groups, not the solvable ones, and the difference is whether a maximal subgroup may have composite index. In a supersolvable group every maximal subgroup has prime index, so every step drops one prime factor and every chain is the same length. A solvable group is allowed a maximal subgroup of index four or six, and a chain through one of those spends a step where another chain spends two.

The cubic holohedry is exactly that case, and it is not an obscure corner of it. 3̅m is a maximal subgroup of m3̅m at index four — order twelve inside order forty-eight, with no group of order twenty-four containing it. That is the rhombohedral distortion of a cubic structure, which the perovskites do routinely, and it is a single step rather than a pair of them. Counting over all ninety-eight subgroups, twenty of the two hundred and ninety-nine maximal steps below m3̅m have composite index, and the consequence is that five of the twenty-four classes sit at two different depths depending on the route taken.

The other holohedries do not do this. 4/mmm, 6/mmm and mmm have no maximal subgroup of composite index at all, so within each of them a class does sit at one definite depth, reached in the same number of steps by every route. The check the figures carry ties the two facts together rather than testing them separately: chains of unequal length exist precisely when a maximal step of composite index exists, and the assertion fails if either half is found without the other.

So the rows of the descent diagram are a drawing convention, and on the cubic diagram they are only that. They are ordered by group order, which is a real quantity, and reading them as a count of transitions is safe on the tetragonal and hexagonal diagrams and wrong on the cubic one. That distinction was not obvious and was not visible in any drawing of the tree; it came out of counting the chains.

Where the ladder goes next #

The gap this essay named is the one to close: a transition can change the lattice rather than the class, and the point-group diagram cannot see it at all.

Those transitions produce domains too — as many as the index of the new translation lattice in the old one — and the domains they produce differ from each other by a translation rather than by an orientation. Two such regions have the same shape, the same optical properties, the same everything a polarising microscope can measure, and where they meet the ordering is out of step. They are the next rung.

The two facts fit together, and they fit together more carefully than they first appear to. Hermann’s theorem says every edge is of one pure kind, so the edges are labelled. The solvability bound says how long the longest chain is, so the diagram has a definite depth. What it does not say is that every route to a class takes that many steps — that would need supersolvability, which the cubic holohedry does not have — so the rows are ordered by group order and by nothing stronger. Within those limits the edges are labelled, the depth is bounded, and the only freedom left is which route a material takes. That freedom is the whole physical content, and it is what the diagram is built to display rather than to resolve.