Two hundred and forty-seven descents, or two hundred and twelve

Every symmetry-lowering transition is a pair: a parent class and a child class inside it. Asking how many such transitions symmetry permits is asking how many such pairs there are, which is a finite question with a definite answer.

Enumerating the subgroups of each of the thirty-two classes, quotienting each list by conjugacy inside its own parent, and discarding the case where the child is the parent gives 247.

The number the ferroics literature records is 212. Both are right, and the gap between them is the most interesting thing in this essay.

What “the same transition” has to mean #

A count is only as meaningful as its equivalence, and this site has now had to say that about three separate numbers — the space groups, the two-colour groups and the crystal forms. Here the equivalence is the whole subject, so it is worth stating both candidates precisely.

Consider class 4/mmm as a parent, and a child of order two generated by a single two-fold rotation. There are two kinds available: a two-fold along ⟨100⟩, and a two-fold along ⟨110⟩. Both are operations of 4/mmm. Both generate a subgroup isomorphic to class 2.

Are they the same transition?

Under conjugacy in the parent: no. No operation of 4/mmm carries a ⟨100⟩ two-fold onto a ⟨110⟩ one — the four-fold rotation cycles ⟨100⟩ among itself and ⟨110⟩ among itself, and nothing crosses between them. They are two conjugacy classes, so they count twice.

Under the equivalence the ferroics literature uses: yes. A rotation through forty-five degrees about c carries the first onto the second, and it also carries 4/mmm onto itself — so it is an operation of the parent’s normaliser, and two subgroups related by a normalising operation describe the same physical situation seen in a rotated frame.

That single example is the whole of the difference between 247 and 212.

The example is not exceptional, and the way to see that is to ask the same question of every child class of 4/mmm at once. There are fourteen distinct classes below it, and seven of those fourteen names occur more than once: mm2 in four inequivalent placements, and 2/m, 2 and m in three each, and 4̅2m, mmm and 222 in two. Class 2 splits three ways rather than two because there are three families of two-folds in a tetragonal holohedry — along c, along ⟨100⟩ and along ⟨110⟩ — and no operation of the group crosses between them. Counting each placement separately is how fourteen names become twenty-six descents, so twelve of the tetragonal census’s twenty-six entries exist only because of that distinction, made once for every class that has a version of it.

The operation doing the merging is one a lattice forbids #

Here is the part that makes this more than bookkeeping.

A rotation through forty-five degrees is an eight-fold operation. No crystal lattice has one. The crystallographic restriction is exactly the statement that a lattice permits rotations of order 1, 2, 3, 4 and 6 and no others, and the proof takes one line: a rotation mapping a lattice to itself is an integer matrix in the lattice basis, so its trace is an integer, so 2cos(2π/n) is an integer, so n is one of five values.

Eight is not among them.

So the equivalence that merges the two subgroups of 4/mmm is witnessed by an operation that is not a symmetry of any lattice, cannot be written as an integer matrix in a lattice basis, and is invisible to every piece of machinery this collection has built on lattice arithmetic.

That is why this site computes 247 and not 212. It is not a shortcut and not an approximation: it is the honest ceiling on what integer arithmetic in a lattice basis can decide. The literature’s finer merging is correct and it needs a larger group — the Euclidean normaliser, the set of all isometries of space carrying the parent onto itself, which for several classes is continuous and contains rotations of every order.

Which is the right count #

Neither, and the question is malformed. What each number counts is different, and both are worth having.

247 is the number of distinct descents relative to a fixed crystal setting. Two of them that differ by which axis was chosen are counted separately, and that is the right thing to do if the question is about a particular crystal in the laboratory, where the axes are physical directions and a domain along ⟨100⟩ is not a domain along ⟨110⟩.

212 is the number of distinct descents up to a change of frame. Two that differ only by which axis was chosen are counted once, and that is the right thing to do if the question is about kinds of ferroic behaviour, where a species is a description of what a material does rather than of how it was mounted.

Aizu, who introduced the species concept in the 1960s, was answering the second question, and the ferroics literature has used his enumeration since. It is the standard.

What this site can say is the first. It says it exactly, from a search rather than from a table, and it says which equivalence it used — which is the whole of the discipline this site’s counting essays are written under. The alternative would have been to quote 212, and quoting a number whose equivalence the machinery cannot express is exactly what the two hundred and thirty essay refused to do about the space groups.

The census, read #

The distribution across parents is steep and it is not surprising once stated.

m3̅m, 32 descents. The cubic holohedry has more subgroups than anything else, and the count follows.

6/mmm, 31. Almost as many, on half the order, because a hexagonal holohedry has subgroups of orders 24, 12, 8, 6, 4, 3, 2 and 1 — more divisors to work with than a cubic group’s.

4/mmm, 26 and mmm, 15.

Class 1, zero. A group with one element has no proper subgroup, so nothing descends from it, and this is asserted rather than merely observed — an enumeration reporting a descent from the trivial class would have a bug in it that nothing else would catch. The next entries up are almost as sparse: 1̅, 2 and m contribute one each, being the descent to the trivial class, and 4̅ and 4 contribute two. A material whose high-temperature phase is already of low symmetry has very little room to lose any.

The same reading applies to the top of the list, and the split there is milder than the tetragonal one. m3̅m has twenty-four distinct classes below it and contributes thirty-two descents, so seven of its twenty-four names split — against seven of fourteen for 4/mmm, and the splits themselves are smaller, mm2 into three placements and the rest into two. The cubic group has more operations to identify its own subgroups with, so more of the placements that a tetragonal group keeps apart are carried onto one another and merged. That is the mechanism behind the census being flatter at the top than the raw subgroup counts suggest: m3̅m has ninety-eight subgroups against 4/mmm’s thirty-five, nearly three times as many for twice the order, and yet contributes only six more descents.

There is an arithmetic reason for the shape of the distribution that is worth separating from the physical one. The number of subgroups of a finite group grows faster than the order does, because a subgroup is chosen from the subsets and closure is a weak constraint at small sizes; and the number of conjugacy classes of subgroups grows more slowly, because a group with many operations has many ways to identify its own subgroups with one another. The census is the difference of those two effects, and it is why 6/mmm at order twenty-four contributes almost as much as m3̅m at order forty-eight: the cubic group has more subgroups and merges more of them.

The steepness has a physical reading. The materials with rich domain structures are the ones whose high-temperature phases are highly symmetric, and the perovskites — cubic aristotype, m3̅m — are at the very top of the list. That is why a single structure type supplies so much of the ferroelectric and ferroelastic literature.

Ferroelastic against the rest #

The census can be split by whether the descent changes the crystal system, and the split is informative.

A descent that changes the system changes the shape of the cell — its metric, in the sense a goniometer measures — so the domain states differ in strain and the material is ferroelastic: a stress favours some states over others, and the walls can be moved mechanically.

Of the 247 descents, 202 change the system and 45 do not. So the great majority of the transitions symmetry permits are ferroelastic, which is a fact about how the classes are distributed among the systems rather than about materials — most subgroups of a high-symmetry class lie in a lower system, because the systems are nested and there are more low-symmetry classes than high-symmetry ones.

The 45 that do not change the system are the purely ferroelectric and ferrobielectric cases: a descent from 4/mmm to 4mm, say, which stays tetragonal and gains a polar axis. Those are the transitions whose domains are invisible in polarised light and detectable electrically, and they are the near neighbours of the antiphase case, where the domains are invisible to everything directional at all.

The normaliser, and why it is not a small correction #

It is tempting to read the gap between 247 and 212 as a technicality — thirty-five pairs, a fifth of the total, arising from a bookkeeping choice. It is worth resisting, because the object doing the merging is a substantial one and its behaviour is not uniform across the classes.

The Euclidean normaliser of a group G is the set of all isometries of space that carry G onto itself. It always contains G. For a class with a unique axis and nothing constraining rotations about it — class 4, say — it contains every rotation about that axis, so it is infinite and continuous. For a class with axes in several directions, such as a cubic one, it is finite and not much bigger than the class.

So the merging is severe for some parents and does nothing at all for others. It is largest exactly where the parent has a free direction, and the classes with a free direction are the polar ones, which are the ones ferroelectrics live in. Aizu’s equivalence therefore compresses precisely the region of the census that materials science cares most about, which is a reasonable thing for a classification of materials to do and a strange thing for a classification of lattices to do.

There is a second reason the normaliser cannot be computed here even in principle. This site’s machinery represents an operation as an integer matrix in a lattice basis — a representation chosen at the outset because it makes every question decidable without a tolerance. A continuous group has no such representation: its elements are not integer matrices and most of them are not rational matrices either. The limitation is not that the search would be slow. It is that the objects being searched over do not exist in the language.

That is the sharpest example this site has of a boundary between what a formalism can decide and what it cannot, and it is worth having because the boundary is usually invisible. Nothing about the number 247 announces that a finer count exists.

What a species does and does not predict #

The word species suggests more than the classification delivers, so it is worth being explicit.

A species is a pair of groups. It says how many domain states there are, what kind of field couples to them, which properties differ between them, and which wall orientations are permitted. Every one of those is a consequence of the pair alone.

A species is not a material. Two materials in the same species can differ in every quantity that matters: transition temperature, coercive field, wall mobility, domain size, whether the transition is first or second order. Barium titanate and lead titanate share a species and are not interchangeable.

A species does not say whether a material exists. Of the 247 descents, a great many have no known material. Some are physically implausible; others are simply unexplored; and there is no argument from symmetry that any of them is realised.

That is permitted against present for the sixth time in this field, and the repetition is the point rather than a failure of variety. A symmetry argument produces an exhaustive list of possibilities and a set of consequences for each, and it never produces an existence claim. The value is in the exhaustiveness: a material behaving in a way not on the list is a material that has been mis-classified, and the list is closed.

What is computed and what is asserted #

The subgroups of each class are found by closure — start with the identity, adjoin each element in turn, close, and repeat until nothing new appears. That is a search rather than a table, and it produces 98 subgroups for m3̅m and 54 for 6/mmm, the two numbers the class enumeration already asserted and this count reuses.

Conjugacy is then decided by explicit conjugation: each subgroup is turned by every operation of the parent and the resulting sets compared. There is no invariant being trusted to distinguish classes — the sets themselves are compared, which is the strongest test available and the cheapest at these sizes.

Two things are asserted. Class 1 contributes nothing, which is the trivial case a bug would violate. And every subgroup of a crystal class is itself one of the thirty-two, which is not obvious and is what makes the census a statement about the classification rather than about an arbitrary collection of matrix groups.

Which walls a ferroelastic descent permits #

A species says how many domain states there are and the essay lists that among what it delivers. There is a second thing it delivers that is more specific and more useful, and it is a set of planes.

Two ferroelastic domains have different spontaneous strains, so where they meet, the two lattices must fit together across the boundary. That is a strong condition: the interface must be a plane along which both strains produce the same displacement, or the crystal is torn. Written out, it is a quadratic equation in the plane’s normal, with the two strain tensors as its coefficients.

The solutions are the permitted wall orientations, and there are generally two of them for each pair of states — so a ferroelastic material has a small, computable list of planes its domain walls may lie on, and walls are observed on those planes and on no others.

That is Sapriel’s condition, and it is worth noticing what kind of statement it is. The strains are not symmetry: their magnitudes are properties of the material, measured. But the equation is, because which components of strain differ between two states is fixed by which operations were lost — so the wall orientations come out as directions fixed by the group, with the magnitudes cancelling in almost every case.

So the species delivers three things and they are of two kinds. How many states, exactly, from the index. Which planes the walls lie on, exactly, from the coset structure. And how much strain, how thick the wall, how it moves — none of which is symmetry.

The four kinds of ferroic, and the descent that names them #

The word ferroic has been used above without being unpacked, and unpacking it says what the census is a census of.

A ferroic crystal is one whose domain states differ in some macroscopic property that a field can couple to, so that the states can be switched. Which property depends on which operations the descent removed, and there are four standard kinds.

Ferroelectric: the states differ in spontaneous polarisation, and an electric field switches between them. That requires the child to be one of the ten polar classes and the parent not to be.

Ferroelastic: the states differ in spontaneous strain, switched by a stress. That is the system-changing case above, and it is the numerous one — two hundred and two of the two hundred and forty-seven.

Ferromagnetic: the states differ in spontaneous magnetisation, switched by a magnetic field. That needs the magnetic classification rather than this one, since an ordinary point group cannot express it.

And ferrotoroidic, the fourth and most recent, where the states differ in a quantity that is odd under both space inversion and time reversal.

A species can be more than one at once — a state differing in both polarisation and strain is ferroelectric and ferroelastic together, and switching it with either field moves the other — which is where multiferroic behaviour comes from and is read directly off the pair of groups.

Where the ladder goes next #

This anchor has now counted domain states, drawn the descents that produce them, followed the case where a translation is lost instead of a rotation, and enumerated the pairs. Everything in it has been about a crystal coming apart into regions that share a lattice.

The last anchor of this field turns to the case where the two regions do not share a lattice — where two crystals of different orientation, or of different substances entirely, meet at a surface. There the question is not which symmetry was lost but how much of the two lattices can be made to coincide, and the answer is again integer arithmetic and again a short list. It just happens to be a list that comes out of number theory rather than out of group theory.