Superspace groups in the plane

A modulated crystal has no space group. Its lattice is perfectly good and there is a wave running through it whose period is not a rational multiple of the lattice’s, so the structure repeats along no vector at all — and its diffraction pattern needs two integers per direction rather than one.

The extra dimension is what makes it periodic again. Lift the structure into a space of one more dimension, with the phase of the modulation as the extra coordinate; the lifted structure is periodic, and the symmetry it has up there is a superspace group. A two-dimensional crystal with one modulation lives in a superspace of three dimensions.

The question this essay answers is how many of them there are, and the answer comes out of machinery this collection already has. The seventeen were counted with no picture anywhere — as extensions of a finite group of integer matrices by a lattice, classified by a cohomology — and a superspace group is the same object with a third coordinate.

The sign, which is the whole of the new input #

An operation of the point group carries a wavevector q to qM. For the operation to survive in the modulated structure, qM has to be the same wave — so either q itself, or −q, which is the same wave with its phase reversed.

It has to be one of those exactly, and that is where an incommensurate modulation differs from everything else on this site. An ordinary reciprocal lattice vector may be carried to itself modulo the lattice, and the slack is where half the subject lives; an incommensurate q has no lattice to be reduced by, so qM = ±q is an equation and not a congruence.

The sign is a homomorphism from the point group to {±1}, because signs multiply along products. So a superspace group is an extension of the point group P by the superspace lattice ℤ³, with P acting on ℤ³ by its ordinary action on the first two coordinates and by that sign on the third.

Seven classes are gone before anything is counted #

qM = ±q is an eigenvector condition, and most rotations of the plane have no real eigenvector at all.

A three-fold rotation turns the plane by a hundred and twenty degrees. There is no direction it leaves alone and none it reverses, so no q satisfies the condition and no crystal with a three-fold axis can carry a one-dimensional incommensurate modulation. The same for four-fold and six-fold.

Six of the thirteen arithmetic classes survive: the identity, the half-turn, the mirror on a rectangular lattice and on a centred one, and 2mm on each of those two lattices. Every operation in them is the identity, a half-turn or a mirror — the operations with a real eigenvector.

That is a restriction with the same flavour as the crystallographic restriction and a different cause. The crystallographic restriction is about a lattice being carried to itself and comes out of a trace being an integer; this is about a single vector being carried to plus or minus itself and comes out of an eigenvalue being real. Both delete the high-symmetry classes and neither is the other.

It is also a real prediction about materials. A modulated structure with a single wavevector cannot have a three-fold, four-fold or six-fold axis; if a substance with such an axis modulates, it must do so with a star of several wavevectors — which is a different and larger classification, and is not this one.

Ten families, each with a wavevector to show for it #

Each surviving class has one or two sign assignments that some wavevector realises, and the table exhibits one for each. That is a deliberate choice about what counts as an answer here: realisability is a statement about the kernel of a matrix, and a dimension count with no vector shown for it is a claim rather than a computation.

Reading the table is worth a moment. On the rectangular class mp the mirror either preserves the wave or reverses it, depending on whether q lies along the mirror or across it — two genuinely different physical situations from one point group. On the oblique class 2p there is only one assignment, because the half-turn reverses every vector there is and no wave can be carried onto itself by it.

The cohomology factors, and the external factor is the seventeen #

The point group acts on the three superspace coordinates in two blocks, so the cocycle condition splits.

t(gh) = (M_g ⊕ ε_g) t(h) + t(g) is two conditions: the first two components are exactly the plane-group cocycle condition, and the third is a one-dimensional condition with the sign in it. Coboundaries — the origin shifts — split the same way. So the cohomology is a product of an external factor and an internal one, and the external factor is the computation that already gives the seventeen.

The internal factor is the whole of what is new, and it is a small computation with a large consequence. It is H¹ of the point group with the sign action: one rational per operation, modulo one, with t(gh) = ε_g t(h) + t(g) and coboundaries t(g) = (ε_g − 1)s for a shift s of the internal origin.

The asymmetry between the two cases is the interesting part. When the sign is trivial — every operation carries the wave onto itself — every ε_g − 1 is zero, so an internal origin shift does nothing at all and there are no coboundaries to divide by. The classes are then exactly the homomorphisms from the point group into ℚ/ℤ. When the sign is non-trivial, the internal origin can be moved and some classes are absorbed.

That is what the last character of a superspace symbol records. p2(αβ)0 and pm(α0)s differ in the internal shift attached to the operation, and the s means a half: a mirror that, in superspace, also slides the phase by half a period.

What the internal shift is, physically #

The internal cocycle is a rational per operation and it is easy to read as bookkeeping, so it is worth saying what it is a fact about.

The extra coordinate of superspace is the phase of the modulation. A point of the real structure at position r sits in superspace at (r, q·r), and moving along the internal direction slides the phase of the wave without moving anything in the plane. So an operation of the superspace group is a symmetry of the plane together with a phase shift, and the internal cocycle records how much phase each operation carries.

A mirror with an internal shift of a half is therefore a real statement about a structure: reflect the crystal in that mirror and the modulation comes back half a period out of step, so the mirror is a symmetry of the modulated structure only when the wave is shifted along with it. Nothing in the plane distinguishes that from a mirror with no shift; the two structures have the same average positions and the same lattice, and they differ in where the wave’s crests sit relative to the mirror.

The experiment that separates them is the one the whole subject rests on. The satellite reflections of a structure with an internal half-shift are systematically absent in the rows where the shift makes the phase cancel — the same argument as a glide’s extinctions, one dimension up, with the satellite index playing the part the ordinary index plays. So the internal cocycle is not a label attached to a group; it is the thing a diffraction pattern of a modulated crystal measures, and the reason the classification exists at all.

That also explains the asymmetry in the counting. When every operation carries the wave onto itself, the internal origin cannot be moved to absorb a phase — there is nothing to move it against — and every distinct phase assignment is a distinct structure. When some operation reverses the wave, sliding the internal origin changes the phases and some assignments become the same structure written twice.

Where a product of two quotients goes wrong #

Thirty-one cocycle names, and the groups are the names up to the changes of basis that are relabellings. There are two kinds and both act on the whole name.

A change of the plane’s basis normalising the point group acts on the external cocycle in the usual way and relabels the operations — so it moves the internal cocycle by permuting its argument, and it moves the sign assignment the same way. And the internal coordinate may be negated, which is the choice of q against −q; that negates the internal cocycle and leaves everything else alone.

Quotienting the two factors separately gives twenty-seven and the answer is twenty-one. The rectangular class 2mm is where it shows. Its external cohomology is four classes — pmm, pmg, pgm, pgg — of which the swap of the two axes merges two, giving the three that appear in the seventeen. In superspace that same swap also exchanges the two mirrors, which changes both the sign assignment and the internal cocycle, so the merge happens for some internal cocycles and not others. A product of two quotients assumes it happens for all of them.

The general form of that is worth keeping, because it is the commonest way a count of this kind goes wrong. A quotient of a product is not the product of the quotients unless the group acts on the factors independently, and here it does not: the same basis change moves both.

Why the count is small, and what that says #

Twenty-one is a small number beside the seventeen and very small beside the seven hundred and seventy-five superspace groups of (3+1) dimensions, and the reason is worth stating because it is the same reason twice.

Seven of the thirteen arithmetic classes are deleted outright, which removes every plane group with a three-, four- or six-fold rotation in it — ten of the seventeen. What survives is the six classes whose operations all have real eigenvectors, carrying between them seven plane groups: p1, p2, pm, pg, cm, pmm, pmg, pgg and cmm in the external factor, merged as the plane calculation merges them.

So the superspace classification is not seventeen things each acquiring a few internal variants. It is seven or so things acquiring internal variants, and ten things ceasing to exist, and the second effect is larger than the first. A modulation is a strong constraint on a crystal’s symmetry rather than a decoration on it.

In three dimensions the arithmetic is friendlier, which is why the number there is large. A three-fold axis in space fixes the direction along its own axis, so a wavevector parallel to it satisfies qM = q exactly and the class survives; the restriction bites only on wavevectors in the plane perpendicular to the axis. The plane has no third direction to put the wave in, so every rotation of order more than two kills the class outright. The smallness of this count is a fact about two dimensions, and it is the same fact that makes a planar modulated crystal a much more constrained object than a three-dimensional one.

What the enumeration refuses #

The first is the load-bearing one. The whole argument rests on the cohomology factoring, and the factorisation is worth nothing if the external factor is not the computation it is claimed to be. Recovering seventeen from it — by summing over the thirteen classes and quotienting by the normalisers, which is the earlier essay’s calculation run again inside this one — is the check that the split is real.

The second and third are a pair. Twenty-two of the thirty-two sign assignments across the thirteen classes are refused for want of a wavevector, and each of the ten that survive has one shown. Without the first the enumeration would be over assignments rather than over structures; without the second it would rest on a rank calculation nobody had looked at.

Who did this first, and in what order #

Superspace was de Wolff’s idea in 1974 and Janner and Janssen’s framework immediately after, and the order in which it arrived is instructive.

The satellites came first. Crystallographers had been recording extra reflections beside the main ones since the 1930s, at positions that were not rational fractions of the lattice, and the accounts of them were descriptive: a “modulation”, a “satellite”, an anomaly of the specimen. The structures were solved, where they were solved at all, by treating the satellites as a perturbation of an average structure.

The move that made a subject of it was refusing to treat the average structure as the crystal. If the satellites need a second integer to be indexed, then the diffraction pattern is a projection of a four-dimensional lattice and the structure is a section of a four-dimensional periodic one — and everything crystallography knows how to do to a periodic structure becomes available again, one dimension up.

The classification followed as a matter of course, because a periodic structure has a space group and the space groups of a given dimension can be enumerated by exactly the extension arithmetic this collection uses on the seventeen. What is enumerated here is the two-dimensional case of that programme, and it is small enough that every step of it fits in one file.

The same move was made again, harder, for quasicrystals. A five-fold diffraction pattern needs six integers rather than three, and the lattice that holds them lives in six dimensions — the same argument as this one with a larger number in it, and it is why an icosahedral quasicrystal has a symmetry group after all.

Where the exactness stops #

Computed here: the thirteen arithmetic classes; every homomorphism from each into {±1}, verified on the whole multiplication table rather than on generators; the space of wavevectors realising each, by elimination over the rationals, with a witness exhibited; the external cohomology of each class, which is the plane-group calculation; the internal cohomology with the sign action, as a one-dimensional cocycle count on a grid of twelfths with the coboundaries taken on a grid twelve times finer; and the orbits of the resulting names under the plane normalisers and the sign of q.

One modulation wavevector, incommensurate. A structure with two independent modulations lives in a superspace of four dimensions and is a different enumeration; a commensurate q is a superstructure with an ordinary space group and is not in this count at all. The condition qM = ±q exactly is what “incommensurate” is doing in every argument above, and a commensurate wave may be carried to ±q plus a reciprocal lattice vector, which admits assignments this enumeration refuses.

A count is only as sharp as the equivalence it divides by, and this one is stated rather than assumed: two names are one group when a change of the plane basis normalising the point group, or a change of the sign of the internal coordinate, carries one to the other. Both are relabellings of a description and neither changes a structure. A reader comparing this twenty-one with a number from a table should check that the table’s equivalence is the same one; the thirty-one before the merge is reported beside it for exactly that reason, as the descents were.

The grid is a grid. The cocycles are enumerated on twelfths, which reaches halves, thirds, quarters and sixths — every denominator a crystallographic operation can produce. A structure needing a fifth of the internal period would be outside the search, and no crystallographic operation produces one.

Where the ladder goes next #

Back, to the structure this classifies: the satellites that need a second integer, where the modulation first appears in a diffraction pattern, and the extra dimension that makes it periodic, where the lift into superspace is constructed.

Sideways, to the machinery reused: seventeen without a picture, which is the same extension arithmetic in two dimensions and supplies the external factor whole.

Onward, to the structures with no single wavevector: two lattices, one crystal, and no cell at all, where two subsystems each modulate the other, and the crystal that rounds τ off, where a commensurate approximation to an incommensurate wave gives back an ordinary space group and a very large cell.