The cell a zone-boundary mode doubles

Every order parameter so far has had the same value in every cell of the crystal. That is what makes the frozen structure keep the parent’s lattice: the distortion repeats with the same period, so every translation of the parent survives.

Let it alternate instead — one sign in one cell, the other in the next — and the translations no longer all survive. Half of them carry the distortion to its negative rather than to itself, and the structure that results has a cell twice the parent’s. The transition has changed the lattice as well as the point symmetry, and a diffraction experiment sees the difference immediately, as reflections that were not there before.

The wavevector is the alternation #

A displacement pattern that repeats with a period longer than the cell is written, as everything periodic is, with a wavevector: the displacement in the cell at lattice vector R carries the factor $e^{2\pi i\,\mathbf{k}\cdot\mathbf{R}}$.

At k = 0 that factor is 1 in every cell, which is the zone-centre case of the previous essays. At k on the zone boundary — half a reciprocal lattice vector — the factor is $e^{i\pi n}$, which is +1 in even cells and −1 in odd ones. Nothing between: the phase is a sign.

That is what keeps this computation exact. A general wavevector needs complex phases and a structure that is not periodic at all with any small cell; a zone-boundary one needs only an alternating sign, so the displaced positions stay rational and the structure is an exact point set like every other on this site. The detector can be handed it directly.

What survives, and what does not #

Two things change relative to the zone-centre case, and only one of them is obvious.

Half the translations go. A translation by a lattice vector R with k·R = ½ carries the distortion to its negative, so it is no longer a symmetry. The surviving translations are those with k·R a whole number, which form a sublattice of index two — the doubled cell.

Some point operations survive only in combination. An operation that reverses the sign of the order parameter is not a symmetry on its own, and neither is a lost translation; but their product can be. That is how a glide appears in the frozen structure of a parent that had no glide: the mirror alone reverses the mode, the half-cell translation alone reverses it, and the two together leave it alone.

So a zone-boundary mode does not merely thin out the parent’s operations. It manufactures operations of a kind the parent did not contain, and the frozen structure can be non-symmorphic where the parent was symmorphic.

The count of available transitions #

At the zone centre each plane group offers between one and four order parameters. At the zone boundary the count rises sharply, for two reasons that are worth separating.

There are several zone-boundary wavevectors. In the plane there are three inequivalent ones for a general lattice — doubling along one axis, along the other, or along both — and each carries its own set of order parameters.

Even the trivial representation breaks something. At k = 0 the trivial representation changes nothing and is not an order parameter at all. At a zone-boundary wavevector it still alternates from cell to cell, so it destroys translations even while attaching +1 to every point operation. Every representation of the little group is an order parameter there, including the identity.

Together those take the enumeration from forty-five modes at the zone centre to a hundred and thirty-five at the boundary, across the seventeen groups, and every one of them is built, frozen and checked. The predicted operations and the detected ones agree on all of them.

The descent is of the other kind #

Crystallography has two words for the two ways a group can shrink, and this is where the second one arrives.

A translationengleiche descent keeps every translation and loses point operations: that is the zone-centre case, where the cell is unchanged and the point group drops. A klassengleiche descent keeps the point group and loses translations: the cell grows and the class stays. The zone-boundary modes here produce descents that are klassengleiche, or a mixture of the two when the mode also breaks point operations.

This collection has met the distinction before, as the two ways down, and as the same group in a bigger cell — where a group has a subgroup isomorphic to itself sitting inside it at an index. What is added here is the mechanism: an order parameter at a wavevector, whose alternation is what removes the translations, and which a physical crystal can actually carry.

A worked case: p4m doubled along one axis #

Take p4m, double the cell along x, and follow what happens to each kind of operation.

The fourfold rotation is the first casualty, and not because the mode reverses it: a fourfold axis is incompatible with a rectangular cell at all, since it would carry the long axis onto the short one. So the moment the cell doubles in one direction only, the class drops from 4mm to 2mm whatever the mode does.

The mirrors divide. The mirror perpendicular to the doubling direction relates cells that carry the same sign of the order parameter, and survives. The mirror parallel to it relates a cell to its neighbour, which carries the opposite sign, so it survives only when composed with the half-cell translation — as a glide.

The half-turn survives or not depending on the mode, and which happens depends on the sign the representation attaches to it.

The result for the mode drawn in the hero figure is pmg: one mirror, one glide, one half-turn. Every step of that reasoning is a statement about signs, and the detector’s answer confirms it without having been told any of it.

Why even the identity representation breaks something #

The claim that the trivial representation of the little group is an order parameter at a zone-boundary wavevector deserves its own paragraph, because it sounds wrong.

At the zone centre, the trivial representation attaches +1 to every operation, so the displacement field is invariant under the whole parent group and nothing changes. At a zone-boundary wavevector the same representation still attaches +1 to every operation of the little group — but the field alternates from cell to cell, and the alternation is not the representation’s doing. It is the wavevector’s.

So the mode breaks the translations while keeping every point operation the little group contains. Its frozen structure is a superstructure with the same class as the parent and a doubled cell, which is a purely klassengleiche descent, and the index is exactly two.

That case is easy to overlook precisely because the character table is uninformative about it, and a computation organised around characters alone would miss it. Organising instead around the field — build it, freeze it, detect it — makes it fall out with the rest.

What the doubling does to the reciprocal lattice #

The doubled cell is the whole of what a diffraction pattern reports, and the arithmetic is worth setting out because it runs the opposite way to the intuition.

A cell twice as long in real space gives a reciprocal lattice twice as fine in that direction: the new reciprocal vectors are half-integers in the parent’s indices, and each of them is a possible new reflection. This collection derived the general relationship in the reciprocal lattice essay — long becomes short — and here it says that the superstructure’s signature is a row of reflections between the parent’s.

Which of them actually appear is a further question with a symmetry answer. The frozen structure’s own group forbids some of them outright, by the same systematic absence argument the parent obeys — and where the frozen structure has acquired a glide, as pmg has, that glide forbids alternate reflections along a row.

So a zone-boundary transition writes its signature into a diffraction pattern twice: new reflections where the cell doubled, and absences among them where the new operations are non-symmorphic. Both are consequences of the mode, and neither needs an intensity to be predicted.

Antiphase domains, which are the new kind #

The domains a zone-centre transition produces differ in orientation: the crystal chose one of several directions, and a polarising microscope can see which. The domains a zone-boundary transition produces include a kind that no orientation-sensitive measurement can see at all.

The order parameter alternates, so a region can be out of step with its neighbour by one cell — the same alternation, shifted. The two regions have identical structures in identical orientations, differing only in where the alternation starts, and the boundary between them is an antiphase boundary, which this collection has an essay about.

The count of domains is still the index, and now it factorises: the index of the point-group part counts orientation domains, and the index of the translation part counts antiphase domains. A descent of index four with a doubled cell and an unchanged class has four antiphase domains and one orientation.

Reading it in a diffraction pattern #

The reason this matters outside the algebra is that a doubled cell is the one kind of symmetry change a diffraction experiment sees immediately.

A structure with a cell twice as large has a reciprocal lattice twice as fine, so new reflections appear at positions halfway between the parent’s. Their intensity is proportional to the square of the order parameter, so they grow from nothing as the transition proceeds — which makes them a direct measurement of it.

This collection has the counting of those reflections already, in the superlattice essay: exactly n − 1 new reflections per parent cell for an n-fold cell increase, and their intensity a difference of scattering factors rather than a sum, which is why an ordered alloy of two neighbouring elements can be invisible to X-rays and obvious to neutrons.

What the present essay adds is where they come from: not a new arrangement of atoms decided by chemistry, but a mode at a wavevector, whose symmetry decides which of the new reflections are present and which are systematically absent.

There is an approach to the same reflections from above the transition, and it is worth naming because it shows that the wavevector is a physical quantity before it is a structural one. A crystal whose alternation has not settled into a structure at all still scatters at the wavevector it is trying to order at: the correlations between cells have a length that is finite rather than infinite, so the scattering is a broad maximum rather than a sharp spot. Its position is the wavevector of the mode; its width is the reciprocal of the correlation length; and the transition is what turns the maximum into a reflection by making the correlation length infinite. So the wavevector can be measured before the superstructure exists, and a diffuse maximum sitting where a superlattice reflection would go is the standard evidence that a crystal is about to double its cell.

The star, and why one wavevector is not enough #

A zone-boundary wavevector is not usually alone. The parent group’s operations carry it to other wavevectors, and the set of them is its star.

If the star has more than one arm, an order parameter has a component at each, and the frozen structure depends on how those components are chosen — all equal, one non-zero, or something between. Each choice gives a different superstructure, and enumerating them is a larger computation than the one done here.

What is computed here is the single-arm case: a mode at one wavevector of the star, with the others zero. That is the simplest superstructure a given wavevector can produce, and it is enough for the arithmetic to be checked. The general case — several arms, several relative amplitudes — is the standard machinery of superstructure enumeration, and this collection has not built it.

Saying so is not a formality. A reader who takes the modes here as a list of the superstructures available to a plane group will be short of some, and the ones missing are exactly those where two arms of a star are non-zero at once.

The mode a structure can actually carry #

One constraint has been left implicit and is worth making explicit, because it decides whether a transition is available to a real crystal rather than merely to a group.

A mode is a pattern of displacements, and the atoms have to be able to make it. Which patterns a given set of atoms can supply is the mechanical representation of their Wyckoff positions, and it has zeros in it: an atom at a high-symmetry site cannot contribute to a mode of certain symmetries at all.

At a zone-boundary wavevector the same question is asked of the little group rather than of the whole point group, and the answer is generally less restrictive — a site’s symmetry in the little group is smaller, so it supplies more kinds of mode. That is one reason superstructures are common in materials whose zone-centre transitions are forbidden: the atoms have nothing to contribute at k = 0 and plenty at the zone boundary.

So the list of modes in this essay is a list of what the group permits. Whether a particular structure can carry a given one is a second question, answered by the mode table of its own occupied positions, and a structure whose atoms are all at maximally symmetric sites can carry very few.

The table is short enough to read in full, and reading it is what makes the constraint concrete rather than a caveat. Each row is a Wyckoff position of the parent, each column is a symmetry of displacement, and the entry is the number of independent patterns of that symmetry the atoms of that position supply. A zero says those atoms cannot make that distortion however their amplitudes are chosen — an atom pinned at a rotation centre cannot move in a way that keeps less symmetry than the centre has. The rows sum, weighted by the dimensions along the top, to twice the number of atoms in the cell, which is the check that nothing has been lost.

What is exact and what is chosen #

Every number in this essay is exact: the phases are ±1, the displacements are rationals, the frozen positions are rationals, and the detected group is decided by comparing fractions.

Three things are chosen, and none of them affects the answer.

The amplitude. A small rational multiplier, chosen to draw. Any smaller one gives the same group.

The representative atoms. A triple of points per orbit, in no particular arrangement, chosen so that the undistorted structure is not accidentally more symmetric than the parent group — the comma rather than the dot, which this machinery ran into and had to fix.

The doubling direction. Which axis the cell doubles along, which for a square parent gives the same answer either way and for a rectangular one does not.

What the pictures cannot show #

The parent’s cell is drawn inside the doubled one, so the left panel of each figure shows the same arrangement twice. That is deliberate — the comparison needs the same window — but it means the picture does not show what a reader might expect: a small cell becoming a large one. The cell outline in the figures is the doubled cell throughout.

Nothing here is three-dimensional. In space the zone boundary has more inequivalent wavevectors, the stars are larger, and the superstructures multiply accordingly; the two hundred and thirty groups have klassengleiche subgroups this collection has not enumerated.

And no intensity is computed for the new reflections. Their positions follow from the doubled cell, which is exact; their intensities depend on how far the atoms actually moved, which is a fact about a material and not about a group.

When the star has several arms #

The doubling described here freezes one alternation, and a wavevector whose star has more than one arm offers a choice that changes the answer.

The order parameter has one component per arm. If the star of $\mathbf{k}$ contains two inequivalent wavevectors, then a distortion is specified by two amplitudes, and the group acts on the pair by permuting them along with whatever the little group does to each.

Different directions in that space give different subgroups. Freeze the first component alone and the crystal doubles along one axis. Freeze both equally and it doubles along both, giving a cell four times the parent’s. Freeze both in some other ratio and the surviving group is smaller than either.

So one representation produces several possible low-symmetry phases, one for each essentially different direction in the order-parameter space, and each is the subgroup of operations leaving that direction fixed. Enumerating them is enumerating the stabilisers of the group’s action on the space of amplitudes — the same orbit–stabiliser arithmetic this collection uses on points in a cell, applied to a space of distortions instead.

Which direction the crystal takes is not decided by symmetry. It is decided by the fourth-order terms in the energy: the quadratic part treats every direction alike, since it is an invariant of an irreducible representation, and the first term able to distinguish them is quartic. Symmetry supplies the list of candidate phases; the coefficients pick one.

Why the boundary and not somewhere in between #

A wavevector on the zone boundary is a special choice, and it is worth saying what makes it special, because most wavevectors do not behave like this at all.

A general wavevector gives no doubling. If $\mathbf{k}\cdot\mathbf{R}$ is not a half-integer or a simple fraction for every lattice vector, the frozen distortion has no period commensurate with the parent’s — the modulation repeats after some long distance or never — and the result is not a crystal with a bigger cell. It is an incommensurately modulated structure, whose description needs additional dimensions rather than a larger cell.

The condition separating the two cases has a name. A continuous transition requires the free energy to have no term linear in the gradient of the order parameter, since such a term would push the wavevector away from the value assumed and settle it at some nearby incommensurate one. The requirement is a condition on the representation — Lifshitz’s — and it is satisfied at points of high symmetry in the zone and violated at general points along lines and planes.

Zone-boundary points of a symmetric lattice satisfy it, which is why the transitions enumerated here are the ones that occur. Points that fail it are exactly where incommensurate phases are found, and there are materials whose transition sequence runs from a high-symmetry phase through an incommensurate one to a commensurate superstructure as the temperature falls — the wavevector locking onto a rational value only when it is cold enough for the higher-order terms to hold it there.