How many domains a transition makes is an index

Barium titanate above 120 °C is cubic, in class m3̅m — the full cubic holohedry — with forty-eight symmetry operations. Below 120 °C it is tetragonal, in class 4mm, with eight. The titanium atom has moved a few hundredths of a nanometre off the centre of its oxygen cage, and the crystal has acquired a spontaneous electric polarisation along the direction it moved.

Which direction? There is nothing to choose between the six faces of the cube. The cubic phase treats them identically, so the titanium has six equally good options, and different regions of the crystal take different ones.

Six. Not approximately six, and not “several”: forty-eight divided by eight, the index of 4mm in m3̅m, and the count is fixed before anybody grows a crystal or measures a temperature.

The argument, in four lines #

Let G be the class of the high-symmetry phase and H the class of the low-symmetry one, with H a subgroup of G.

A domain state is a way of embedding the low-symmetry structure in the frame of the high-symmetry one. Two embeddings differing by an operation of H are the same state, because H is a symmetry of the low-symmetry structure and moves it onto itself.

So the states are the cosets gH, and there are [G : H] of them. Nothing else can be a state: a coset is a complete description of an embedding up to the low-symmetry group’s own operations, so the list is exhaustive as well as correct.

Every operation of G permutes the states, and it does so transitively — for any two states there is an operation carrying the first to the second. That is why they are equally likely, why they occur in equal proportion in a crystal that has cooled without a bias, and why nothing distinguishes one as the “true” one.

That is the whole of it, and it is the identical argument the twin laws came from with the containing group changed. There the big group was the point group of the lattice; here it is the class of the parent phase. Cosets either way.

What each state is like on the inside #

A single domain of barium titanate is an ordinary tetragonal crystal in class 4mm. It has a polar axis, it is piezoelectric, it has the properties the ten polar classes permit — and it carries no memory whatever of having been cubic.

That is worth stating explicitly because it is the part that surprises. A measurement inside one domain cannot tell that the crystal ever had more symmetry. The parent phase is visible only in the relationship between domains, which is why the subject is about walls and orientations rather than about anything inside a region.

Ferroelectric, ferroelastic, ferroic #

The domains differ in something, and which something they differ in has a name.

If the low-symmetry class is polar and the parent’s is not, the states differ in the direction of a spontaneous polarisation. The material is ferroelectric, an electric field favours the states aligned with it, and the walls move. Barium titanate is the standard example and the six states are the six ⟨100⟩ directions.

If the low-symmetry class is in a different crystal system from the parent’s, the states differ in the direction of a spontaneous strain — the cell is stretched along different axes in different domains. The material is ferroelastic, a stress favours some states, and again the walls move.

Barium titanate is both. Its cubic-to-tetragonal transition changes the system, so the states are strained differently, and it also produces a polar axis. Rochelle salt is ferroelectric and, for the same transition, ferroelastic. Some transitions are one and not the other: a descent that stays inside one system but loses a mirror is ferroelectric and not ferroelastic.

The umbrella term is ferroic, coined by Aizu in the 1960s for exactly this: a material with two or more orientation states that can be switched by a field of some kind. Enumerating the parent-and-child pairs that produce one is this anchor’s last rung. What kind of field depends on which property the states differ in, and that is decided by comparing the two classes rather than by any measurement — which is Neumann’s principle doing the deciding, one class at a time.

Counting states for the same parent, three ways #

The index depends on the child, and the same parent produces very different domain structures depending on how far the symmetry falls.

m3̅m → 4/mmm. Index three. A cubic crystal that becomes tetragonal without becoming polar has three states, one for each of the cube’s three four-fold axes. The states differ only in which axis is the unique one, so they differ in strain and not in polarisation: purely ferroelastic.

m3̅m → 4mm. Index six. The three axes, each with two senses, because the polar direction has a head and a tail. Ferroelectric and ferroelastic together.

m3̅m → 3m. Index eight. The polarisation now lies along a body diagonal, of which there are four, and each has two senses. This is the low-temperature rhombohedral phase of several perovskites, and the count is the reason a rhombohedral ferroelectric has eight states where a tetragonal one has six.

That last comparison is the useful one. Both are descents from the same cubic parent, both are ferroelectric, and they have different numbers of domains because the polar axes point along different families of directions — three axes with two senses against four diagonals with two senses. The count is reading off which family, which is a fact about the child class rather than about the material.

The non-polar version of the same descent makes the point sharply. m3̅m → 3̅m has index four, not eight, because 3̅m keeps the inversion and so cannot tell one sense of a body diagonal from the other. Adding a polar axis to the child doubles the number of domains, and it does so for a reason a reader can state in one line.

The non-polar descent is worth drawing beside it, because it is the same picture with half as many colours and the halving has one cause.

The walls, and what symmetry says about them #

A domain wall separates two states, and there is a piece of symmetry information about it that is genuinely predictive.

Two states are related by some operation of the parent that was lost. If that operation is a mirror, the wall between them can lie in the mirror plane and the two sides will match across it exactly — a coherent wall with no strain. If it is a rotation, the possible wall orientations are those the rotation leaves invariant.

So the permitted wall orientations are computable from the pair of states, and the observed walls in ferroelastic materials do lie on those orientations. In tetragonal barium titanate the walls between states with perpendicular polarisation lie on {110} planes at 45° to both polar axes, which is exactly the plane that makes the strain match, and the crystal is visibly striped along those directions under a microscope.

What symmetry does not say is which of the permitted orientations is chosen, how thick the wall is, how large the domains grow, or how fast they move under a field. Those are questions about energy and kinetics. The group supplies the menu of orientations and the count of states, and nothing else.

One number is being read three ways here and it is worth saying so plainly, because the three readings are usually met in three separate places. The index of the child in the parent is the number of domain states. It is also the ratio of the two fundamental domains — the region of space a group’s operations reduce everything else to — so a child of index two has a fundamental domain exactly twice the size of the parent’s, and the picture of one sitting inside the other is the picture of the two domain states sitting side by side. And it is the number of cosets, which is where both of the others come from. Index, fundamental domain size and domain count are one quantity under three names, and a reader who has met any of them has met all three.

Switching, which is what makes the count matter #

A ferroic material is useful because its domain states can be chosen. Apply a field, and the states aligned with it become lower in energy; the walls move to grow those regions at the expense of the others; remove the field and the new arrangement stays. That is a memory, and it is the basis of ferroelectric storage.

The number of states is therefore the number of things the material can remember, and it is a count the arithmetic supplies. A tetragonal ferroelectric has six; a rhombohedral one has four; a material whose transition has index two has two, which is the plain binary case.

The switching is not symmetric between the states, and the reason is again a group question. Reversing a polarisation by 180° moves the polar axis onto its own opposite and involves no change of strain, so a 180° switch is purely electrical. Turning it by 90° moves the polar axis onto a different cube direction and does change the strain, so a 90° switch is mechanical as well and can be driven by stress alone. The two kinds of wall behave differently, respond to different fields, and are visible under a microscope as different features.

Which switches exist is read off the coset structure: the states related by an operation that reverses the polar axis are the 180° pairs, and the states related by one that turns it are the 90° pairs. Symmetry supplies that partition; the energy barriers on either kind are a matter for the material.

Valasek, and a material nobody expected #

Ferroelectricity was found in 1920 by Joseph Valasek, a graduate student at Minnesota, in Rochelle salt — potassium sodium tartrate, a substance chosen because it was the best piezoelectric available and easy to grow.

What he measured was a hysteresis loop: polarisation against applied field, tracing a curve that does not retrace itself, with a remanent polarisation at zero field. That is exactly the shape ferromagnetic materials had been known to give for decades, which is where the name came from — the “ferro” in ferroelectric refers to the analogy and not to any iron, and Rochelle salt contains none.

The analogy is closer than a naming convention. A ferromagnet has domain states differing in the direction of a spontaneous magnetisation, they are the cosets of a subgroup in a parent group, and the count is an index. The differences are in which property distinguishes the states and in what field couples to it, and the group theory transfers without modification.

For twenty years Rochelle salt was the only known ferroelectric, and it was widely thought to be a peculiarity of a complicated hydrated organic salt. Barium titanate arrived during the Second World War, in a search for capacitor dielectrics, and it is structurally about as simple as a compound can be — which turned the subject from a curiosity into a field.

The condition for a continuous transition #

Not every pair of classes can be joined by a transition, and the constraint is sharper than it first appears.

Landau’s condition is that a continuous — second-order — transition requires the low-symmetry group to be a subgroup of the high-symmetry one, in the sense containment was fixed three fields ago. The reasoning is that the order parameter grows continuously from zero, so arbitrarily close to the transition the structure is arbitrarily close to the parent, and the symmetry it has must therefore be one the parent already had.

A pair of classes with neither a subgroup of the other cannot be joined continuously at all. They can still be joined by a first-order transition, where the structure jumps and nothing has to vary smoothly — which is how many real transitions between unrelated groups happen, with latent heat and hysteresis.

So the group-subgroup relation is a filter on which transitions can be continuous, and the index is the domain count for those that are. Both of those are questions about the pair of groups and neither of them is about the material, which is the reason a table of possible descents is worth building. That table is the next rung.

What it does not decide, again #

The list is by now familiar and it is worth keeping.

Whether the transition happens. Symmetry says a descent from m3̅m to 4mm would give six states. It does not say that barium titanate has such a transition, or at what temperature, or whether some other descent happens first.

Which descent. A cubic parent has many subgroups. Which one a given material chooses depends on its energetics, and perovskites choose differently — barium titanate goes tetragonal, others go rhombohedral, others stay cubic to absolute zero.

How much of each state appears. Nothing at all. A crystal cooled in an electric field is largely in the states aligned with it, and a crystal cooled freely has all six in roughly equal amounts, and both are consistent with the arithmetic.

That is the same shape of limitation this field has recorded at every rung, and it is worth one more restatement because the domain case is the one where the arithmetic is strongest. Here the count is not an upper bound but an exact prediction of how many distinct states exist. What remains outside is everything about how much of each there is.

There is a fourth thing it does not decide, and it is the one that matters most in practice: how small the domains are. A crystal that cools into six states can do it as six large regions or as a fine lamellar texture with walls every few hundred nanometres, and which happens is set by the competition between wall energy and the strain energy of holding a single state across a whole crystal. Ferroelectric ceramics are engineered on exactly that competition, and none of it is in the group.

Where the ladder goes next #

The index answers “how many” once a pair of classes is given. The next question is which pairs are available, and that is a structure rather than a number: the crystal classes form a lattice under inclusion, transitions run down its edges, and a transition of index six may or may not be decomposable into two of index two and three.

That structure is a Bärnighausen tree, the next rung builds one, and it has a distinction in it this essay has quietly avoided. Losing a rotation gives domains in different orientations. Losing a translation gives domains in the same orientation, displaced — and those are invisible to every optical method there is.

The states that differ in nothing but where they start #

The argument above is run on point groups, and it therefore counts the ways the low-symmetry structure can be oriented. There is a second kind of state it cannot see, and on many transitions the second kind outnumbers the first.

A transition can lose translations rather than rotations. Order two kinds of atom onto alternate sites of what was one lattice and no direction has been singled out — the ordered structure points the same way as the parent — but half of the parent’s translations are no longer symmetries, because they exchange the two kinds of atom.

The states are then the cosets of the surviving translations. Nothing in the four-line argument required the groups to be point groups; it required only a group and a subgroup. Applied to the translation lattices it gives the number of distinct places the ordered pattern can start, and two parts of a crystal that started differently meet at a boundary where the ordering is out of step.

Those are antiphase domains, and their walls are antiphase boundaries. They have identical orientation, identical strain and identical polarisation, so no measurement of direction distinguishes them; what distinguishes them is a shift, and they are visible in an electron micrograph as a boundary with no orientation change across it.

The two counts multiply. For a space-group descent the total number of states is the index of the low-symmetry space group in the parent, and that index factors into the index of the point groups times the index of the translation lattices — orientational states times antiphase states.

Cu₃Au is the clean case. The disordered alloy is face-centred cubic; the ordered structure puts gold at the cell corners and copper at the face centres, which is the same cube with the face-centring translations gone. The point group does not change at all, so there is one orientational state and four translational ones: four antiphase domains, and a microstructure of boundaries across which nothing turns.

Barium titanate is the other extreme, with six orientational states and no lost translations. Most transitions of interest sit between the two, and quoting a domain count without saying which index was taken is how the two get run together.

An index is a count of states, not of domains #

One more distinction is worth drawing, because the number this essay computes is often read as a prediction about what a micrograph will show, and it is not quite that.

A state is a way the structure can sit; a domain is a region that sits that way. A crystal cooled through the transition contains many regions, and two regions in the same state are not two states. So the index bounds how many kinds of region there are and says nothing about how many regions there will be, which depends on cooling rate, sample size, defects and history.

Nor are all pairings of states equally likely to be seen. Two domains meeting must agree along their shared surface, and the strains of the two states have to be compatible across it — which is a condition on the wall’s orientation, and for some pairs of states it has no solution at all. A sample with six available states may therefore show only some of the fifteen possible pairings, and the ones it shows are decided by which walls a strain permits rather than by the index.

What the index does predict, exactly, is the count of distinguishable variants, which is the number a measurement on a well-prepared single crystal can hope to resolve, and the number a poling or twinning experiment has to select among. That is a real prediction and an unusually firm one for this subject: it is available before the material is made, it is an integer, and it follows from two group orders and nothing else.