The domains a lost translation makes, which nothing optical can see

An alloy of copper and gold in the proportion three to one is, at high temperature, a face-centred cubic solid solution: every site is equally likely to hold a copper atom or a gold one, and the structure’s translations are the ordinary fcc ones.

Cool it slowly below about 390 °C and the atoms sort themselves out. This is a klassengleiche descent in the sense the previous rung set out: the point group survives and the translations do not, so the transition appears nowhere on a diagram of crystal classes. The gold takes the cube corners and the copper takes the face centres. Nothing has moved far, no atom is in a new kind of site, and the crystal class has not changed at all — it was m3̅m and it still is.

What has changed is the translations. In the disordered alloy a translation by half a face diagonal maps the structure onto itself, because the site it lands on is statistically the same. In the ordered alloy that translation takes a gold site onto a copper site, and it is no longer a symmetry. The translation lattice has shrunk to one quarter of what it was.

A crystal cooling through that transition has four choices of where to put the gold, and different regions choose differently.

The count is again an index, of a different group #

The argument is exactly the one for orientation states, with the translation subgroup in place of the point group.

Let T be the translation lattice of the disordered phase and T′ the smaller lattice of the ordered one. A domain state is a choice of where the ordering starts, and two choices differing by an element of T′ are the same state, because T′ is a symmetry of the ordered structure.

So the states are the cosets of T′ in T, and there are [T : T′] of them — which is the ratio of the two cell volumes.

For copper-gold that ratio is four. The ordered cell has the same size as the fcc conventional cell and is primitive rather than face-centred, so the three centring vectors have stopped being translations — centring read as a sublattice, and then taken away — and the four cosets are the identity together with those three. Four antiphase domain states, and they are the four possible positions of the gold sublattice.

For β-brass — copper and zinc in equal parts, ordering from body-centred cubic to the caesium chloride arrangement — the ratio is two, so there are two states.

The two cases are the same drawing at two settings of one number, and it is worth seeing them that way rather than as two pictures. An ordering is a choice of translation sublattice; the index of that sublattice is the number of states; and the sublattice is what a drawing has to be built from if the picture is to be a consequence of the arithmetic rather than an illustration of it. Widen the ordering block from two cells to two-by-two and the minority species goes from one site in two to one site in four, which is the three-to-one composition the copper-gold alloy actually has.

What makes them invisible #

An orientation domain and an antiphase domain differ in a way that decides every practical question about finding them.

Two orientation states are related by a rotation or reflection that was lost. Their cells point in different directions, so their strains differ, their refractive indices differ, and a polarising microscope sees them at once. That is why ferroelastic domains have been visible since the nineteenth century, and why a twin — which is the same thing under another name — was described long before anybody had a group to describe it with.

Two antiphase states are related by a translation that was lost. Their cells point the same way. They have the same strain, the same optical properties, the same everything that is not sensitive to which atom is at which site. No optical method distinguishes them, and none ever will, because there is nothing directional to distinguish.

What separates them is the ordering itself, and the only probes that see the ordering are the ones that see the difference between a copper atom and a gold one at a particular site.

Superlattice reflections, and where the domains show #

Ordering makes new reflections appear.

In the disordered alloy the face-centring means that reflections with mixed-parity indices are systematically absent — a standard extinction condition, and one this site computes rather than quotes. Ordering destroys the centring, so those reflections are no longer forbidden, and they appear. They are weak, because the difference between a copper scatterer and a gold one is much smaller than their sum, but they are there.

Those are the superlattice reflections, and they are the direct evidence of the ordering. Their intensity is proportional to the square of the difference in scattering power, and to the square of a long-range order parameter measuring how completely the sorting has happened. Both factors are read straight off the structure-factor sum, and the first of them is why a superlattice reflection between two neighbouring elements in the periodic table is very weak indeed.

Now the domains. Each antiphase state contributes to the superlattice reflections with a phase set by which coset it is in, and different states contribute with different phases — while contributing identically to the fundamental reflections, which do not care about the ordering. So a crystal full of antiphase domains has fundamental reflections that are sharp and superlattice reflections that are broadened, by an amount inversely proportional to the average domain size.

A width in reciprocal space becomes a length in real space, and that length is the antiphase domain size. It is measurable in a powder pattern.

The two halves of that claim can be checked on the sum itself rather than taken from an extinction table, and doing so is worth the trouble because both halves are easy to state loosely. Evaluate the structure factor over the sites of the ordering block, once with the two species carrying different scattering powers and once with every site carrying their average — which is exactly what the disordered alloy is. The disordered sum cancels every superlattice reflection to zero, because summing a phase factor over the cells of the block is a geometric series in a root of unity and the series closes. And it leaves every fundamental reflection identical, because a reflection whose indices are multiples of the ordering repeat sees only the total scattering power of a block, which sorting the species inside it does not change.

So the superlattice reflections are not a new lattice appearing. They are absences that have stopped being absences, and they carry the difference of the two scattering powers where a fundamental reflection carries the sum — which is the whole of why they are weak.

The boundary #

Where two antiphase states meet, the ordering is out of step: two cells of the same species sit next to each other where the pattern says they should alternate.

That is an antiphase boundary, and it costs energy — not the energy of a broken lattice, since the lattice is untouched, but the energy of a few wrong neighbours in a plane. In a system where unlike neighbours are favoured, which is what drives the ordering in the first place, a boundary is a plane of like neighbours and is correspondingly unfavourable.

The consequence is that antiphase domains coarsen. Boundaries cost energy per unit area, so the system reduces its total boundary area over time — the same arithmetic that governs any interface whose cost is proportional to how badly the two sides fit, small domains disappear into large ones, and the average size grows with annealing. The kinetics of that coarsening is a whole field, and none of it is symmetry.

What symmetry does supply is the boundary’s displacement vector — the coset relating the two states. There are only [T : T′] − 1 possible ones, so a boundary is characterised by a choice from a short list, and in an alloy with four states there are three kinds of antiphase boundary and no others.

The two-colour connection #

An index-two antiphase relationship is a two-colouring, and this site has already enumerated those.

The counterchange essay counted the subgroups of index two in each of the seventeen plane groups, and found seventy-four across them. Every one of those subgroups is a candidate ordering: colour the pattern with two species so that half the operations preserve the colours and half exchange them, and the colour-preserving half is the group of the ordered structure.

Reading it the other way is the point. A two-coloured pattern is an ordered crystal with two antiphase states, and the colour-reversing operations are exactly the operations that carry one state onto the other. The seventy-four are seventy-four possible index-two orderings, and the black-and-white patterns of the ornament literature are, structurally, superstructure diagrams.

There is one result from that essay worth carrying over. p3 admits no two-colouring at all: a homomorphism onto a group of order two must kill every element of odd order, and p3 is generated by a three-fold rotation and its translations, which leaves nothing to reverse the colours. So a structure with p3 symmetry has no index-two ordering available — not one that is hard to make, one that does not exist.

Seeing them, which needs an electron microscope #

Antiphase domains were inferred before they were seen. The inference came from the broadening of superlattice reflections and from the kinetics of ordering, and it was solid; but the pictures arrived in the 1950s with transmission electron microscopy, and they settled the subject.

The method is elegant and it uses exactly the property this essay has been about. Form the image using a superlattice reflection alone — a dark-field image, with the objective aperture placed over one weak spot. Regions in different antiphase states contribute to that reflection with different phases, so on the far side of a boundary the contribution reverses, and the boundary appears as a line of contrast. Fundamental reflections give no contrast at all, because they cannot tell the states apart.

The images show what the arithmetic predicts: a network of curved boundaries, no change of orientation across any of them, and a mean spacing that grows as the square root of the annealing time. In Cu₃Au the boundaries are curved and the domains roughly equiaxed; in alloys where one boundary orientation is much cheaper than the others, the domains are plates.

The technique is the argument. It works because the contrast comes from a reflection that exists only because of the ordering, so what is imaged is the ordering and nothing else. A method sensitive to orientation would have shown a uniform crystal, which is what every optical method does show.

Long-range and short-range order are different claims #

One more distinction, because the vocabulary is slippery and the two are measured differently.

Long-range order is the claim that the sorting is correlated across the whole crystal — that knowing the species at one site predicts the species a thousand cells away. It produces sharp superlattice reflections, and its degree is the order parameter that goes to zero at the transition.

Short-range order is the claim that neighbours are correlated and distant sites are not. It produces no superlattice reflections at all, because there is no periodicity to diffract from. What it produces is diffuse scattering — broad features in the regions where superlattice reflections would be, whose shape encodes the correlation function.

An alloy just above its ordering temperature has short-range order and no long-range order. One just below has both. The domains of this essay are a feature of the long-range ordered state, and a crystal with only short-range order has no domains because it has no states to be in.

What symmetry decides here, and what it does not #

How many states. Exactly, as an index of translation lattices. Four for Cu₃Au, two for β-brass, and no argument.

Which displacement vectors a boundary can have. Exactly, as the non-trivial cosets.

Which orderings are possible at all. Exactly, as the subgroups of the translation lattice compatible with the composition — and, as the plane case shows, sometimes the answer is none.

Everything else. Whether the alloy orders, at what temperature, how fast, how large the domains grow, what a boundary costs, and whether the ordered phase is stable against something else entirely. All energetics, all kinetics, none of it in the group.

That balance is by now the standing shape of this field, and the antiphase case is where the useful half is at its sharpest: the count of states is not an upper bound but a fact, and it is available from the two cell volumes alone.

A note on which sublattice, which the count hides #

The index says how many states there are and not what they are, and for an ordering with several inequivalent sites the difference is worth having.

Cu₃Au’s four states are the four positions of the gold sublattice, and they are related by the three face-centring vectors of the parent — a set of translations that happens to form a group with the identity. That is not automatic. An ordering whose lost translations do not form a group with the identity would not have well-defined states at all, and the reason the cosets are the right object is precisely that the translations remaining always do form a group.

There is a second subtlety in the same place. A klassengleiche descent can change which Wyckoff positions exist, splitting one orbit of the parent into several of the child — the site that was one kind of site becomes two kinds. The four gold positions of Cu₃Au are one orbit in the child and part of a larger orbit in the parent, and it is that splitting that gives the ordering something to order. A descent that split no orbit would leave the structure with nothing to distinguish, and the transition would have no order parameter to grow.

So the full description of an ordering is a subgroup and an orbit splitting, and only the first of the two is an index.

What a dislocation does when the crystal is ordered #

The boundary is described above as something a crystal makes when two regions order out of step. There is a second way to make one, it happens continuously while the material is being deformed, and it changes how the alloy behaves.

A dislocation moving through a disordered crystal leaves the crystal behind it identical to the crystal in front of it, because the Burgers vector is a lattice translation. In an ordered crystal the lattice translations of the parent are no longer all symmetries: the ones lost in the ordering are exactly the antiphase displacements. So a dislocation whose Burgers vector is one of those leaves an antiphase boundary trailing behind it.

That costs energy proportional to the area swept, so such a dislocation is dragged back — and the material’s response is to send them in pairs. The first creates the boundary and the second, with the complementary displacement, destroys it. The two are bound together at a separation fixed by the balance between the boundary’s energy and their mutual repulsion, and the pair is called a superdislocation.

The whole of that follows from the coset arithmetic. How many antiphase displacements there are is the index; which of them a single dislocation can carry is which cosets its Burgers vector lies in; and how many dislocations must travel together is the number needed to return to the identity coset. For an index-two ordering that is two; for Cu₃Au’s index of four it can be more.

The mechanical consequences are large and are not symmetry. Ordered alloys are generally harder than their disordered forms, some of them get stronger as they are heated over a range rather than weaker, and both effects are traced to the behaviour of these bound pairs. What the classification supplies is the inventory of boundaries available; what happens to a material with them is metallurgy.

How fast the domains grow #

The observation that antiphase domains coarsen has a rate, and it is one of the cleaner results about how a pattern of domains evolves.

The driving force is the boundary’s own energy: a curved boundary can lower it by straightening, so a small domain surrounded by a larger one shrinks and disappears. Nothing is conserved in the process — unlike the coarsening of a two-phase mixture, where material has to be transported — because an atom on the wrong sublattice can simply swap with a neighbour.

That distinction sets the exponent. For a boundary driven purely by its own curvature with nothing conserved, the mean domain size grows as the square root of the time, and the whole pattern at a later time looks statistically like the earlier one scaled up. The scaling is what makes the law testable: a micrograph at four times the annealing time should be indistinguishable from an earlier one magnified by two.

And the initial size is set by the transition rather than by the arithmetic. How many domains a crystal starts with depends on how many places the ordering nucleated, which depends on the cooling rate — so a quenched sample has fine domains and a slowly cooled one has coarse ones, with the same index, the same displacement vectors and the same superlattice reflections in every case.

Where the ladder goes next #

Three rungs of this anchor have each computed an index. What has not been asked is how many such computations there are — how many parent-and-child pairs the thirty-two classes admit between them, and therefore how many distinct kinds of ferroic transition symmetry allows.

That number is enumerable and this site enumerates it. It comes out at 247 under the equivalence used here, the literature records 212 under a different one, and the difference between the two is not an error on either side. It is a disagreement about when two transitions are the same transition, and resolving it turns out to need an operation that no lattice may have.