Crystallography in a box

Every group in this collection is infinite. A plane group contains translations by every vector of a lattice, and the arguments that classify the seventeen depend on the translation subgroup being ℤ²: the crystallographic restriction, the split into point group and translations, the whole of the extension theory.

A calculation is not infinite. It takes a block of N × N cells and glues its edges — the top row’s neighbour is the bottom row, the right column’s is the left — which makes a torus. Everything computed about a crystal by anybody, on any machine, is computed on such a block, and the block is a different object from the crystal.

What survives the gluing #

More than a first look suggests, and it is worth being clear about it before the loss.

Every operation of the point group still acts. An integer matrix acts perfectly well modulo N, so a rotation or a mirror of the plane group is a permutation of the box’s cells. Nothing is lost there.

The translations become a finite group. They are (ℤ/N)², of order N², rather than ℤ². Still transitive on the cells, still abelian, still normal in the whole group.

So a pattern on the box has a symmetry group of order N²|P|, finite, containing everything the infinite group had except that translations now repeat. p4m on a four-by-four box has a group of order a hundred and twenty-eight; p1 on the same box has a group of order sixteen.

One thing does have to be checked rather than assumed: a group whose translations are halves — a glide, a screw — needs an even N, because half a cell has to land on a cell. On an odd box those operations have nowhere to go, and the machinery refuses them rather than rounding.

Counting the patterns a box holds #

With a finite group acting on a finite set, Burnside’s lemma applies directly, and this collection has had the machinery since the counting essays: the number of distinct patterns is the average, over the group, of the number of colourings each element leaves alone.

A three-by-three box has nine cells, so five hundred and twelve two-colourings before any identification. Under p1’s translations alone, sixty-four remain. Under p4 twenty-eight. Under p4m twenty-six.

The averages come out whole every time, which is the property that makes Burnside’s lemma a check as well as a count: the terms are wildly different in size — the identity fixes every colouring, a general rotation fixes very few — and a mistake in the fixed-point counting almost never leaves the average an integer.

One entry in the top half of that table is smaller than the arithmetic predicts, and it is the two-by-two box. A half turn sends a cell at x to one at −x, and modulo two −0 is 0 and −1 is 1 — so on a box of two cells the half turn sends every cell to itself and is indistinguishable from the identity. p2’s group on that box has order four rather than eight, and p4m’s has eight rather than thirty-two. The box has not lost the operations; it has merged them, because it is too small to tell them apart. That is the first appearance of this essay’s whole subject in the direct-space half: a box answers questions about itself, and only sometimes about the crystal.

The average, term by term #

Burnside’s average deserves one look at its terms, because the way it comes out whole is more interesting than the number it comes out as.

The identity fixes every colouring: 2⁹ = 512 of them on a three-by-three box. A translation by one cell in one direction fixes only the colourings that are constant along that direction, which on a three-by-three box is 2³ = 8. A four-fold rotation fixes fewer still.

So the terms range over two orders of magnitude and the average of them is a small integer. That is not a coincidence — the average counts orbits, and orbits are things — but nothing in the individual terms hints at it, which is why a computation that gets a fractional average has found a bug rather than a subtlety.

The machinery here raises an error in that case rather than rounding, which is the same discipline as everywhere else in this collection: a count that comes out fractional is a count that has failed.

Where the loss is #

The gluing costs nothing in direct space and something decisive in reciprocal space, and the accounting is exact.

The wavevectors a box can carry are the characters of its translation group — the irreducible representations of (ℤ/N)², which are N² in number and are indexed by pairs of integers modulo N. In the language of the star of a wavevector, they are the fractions with denominator dividing N.

There is no room for any other. A wavevector with denominator five is not represented coarsely in a box of four cells, and not approximated by the nearest available one: it is not present, because the space the calculation works in has no vector with that periodicity.

That is a stronger statement than a discretisation error and it is the one worth carrying. A calculation on a box does not compute an approximation to the crystal’s behaviour at every wavevector; it computes the exact behaviour at N² of them and says nothing whatever about the rest.

The consequence the previous essays met #

Two rungs of this ladder ran into that fact before naming it.

A framework’s mechanism count grows with the cell it is looked for in — one, four, seven for the kagome net at cells of one, four and nine — and the reason is that its motions lie along lines of the zone, which a bigger box samples at more points. The count is not a property of the framework; it is a property of the framework and the box.

A level’s degeneracy at the corner of a hexagonal zone is invisible to a box whose size is not a multiple of three, because the corner has denominator three. A calculation on a four-by-four box would report a honeycomb with a gap where the infinite crystal has a crossing, and nothing in its output would say so.

Both are the same statement. A box is a sampling of reciprocal space, the sample points are decided by its size, and a feature between them is not seen at all.

What the box does to a symmetry operation #

The direct-space side deserves one careful paragraph, because “the point group still acts” hides a small subtlety about fixed points.

A two-fold rotation of the infinite plane fixes the points of a lattice of centres — infinitely many, spread out. The same rotation on an N × N torus fixes exactly the cells satisfying 2x ≡ 0 modulo N, which is four cells when N is even and one when N is odd.

So an operation acquires a definite, finite number of fixed points, and the number depends on N in a way that has nothing to do with the crystal. Those counts are precisely what Burnside’s lemma consumes, which is why the pattern counts in the table above differ between box sizes in ways that are not simply proportional.

The general statement is that the fixed points of an operation on the torus are the solutions of a linear congruence, and the number of them is a divisor count. Crystallography on a box is therefore modular arithmetic wearing a familiar coat, and the counts it produces are number-theoretic rather than geometric.

The last line of that table is the one to carry away. The average number of cells an operation holds still is one, on every box and for every group — which is Burnside’s lemma run with a single colour, where the count of orbits is the count of orbits of cells, and the box’s translations make that one. It is a small identity and it is a strong check: every permutation in the group has to be built correctly for the average to land on one, and a single one built wrongly moves it off.

Choosing a box size is choosing what to be able to see #

The practical rule follows immediately and is worth stating as a rule.

A box of N cells sees exactly the wavevectors of denominator dividing N. So a structure expected to modulate with a period of three cells needs N a multiple of three; a transition at the zone corner of a hexagonal lattice needs N divisible by three; a doubling needs N even.

That is why supercells in the literature come in the sizes they do — 2×2, 3×3, √3 × √3, 2×1 — and why a calculation searching for an unknown modulation has to try several. A box chosen for its size in memory rather than for its arithmetic can be blind to the thing being looked for.

And an incommensurate modulation, whose wavevector is irrational, is held by no box at all. That is not a limitation to be worked around by taking N large; it is a statement that periodic boundary conditions cannot represent the object. The remedy is to describe the structure in superspace instead, where it is periodic again in a higher dimension.

An arithmetic aside: which boxes hold which points #

The rule “denominator divides N” is worth turning into the small table it deserves, because the pattern is a divisibility one and reads oddly at first.

The centre of the zone is in every box. The corner of a square zone, at halves, is in every even box. The corner of a hexagonal zone, at thirds, is in every box divisible by three. A wavevector at fifths needs a box divisible by five, and so on.

So the boxes that see the most are the ones with the most divisors — twelve sees halves, thirds, quarters and sixths; sixteen sees only halves, quarters and eighths, despite being larger. A bigger box is not automatically a more informative one, and a box of twelve cells is a better choice than one of sixteen for a structure whose modulation is unknown.

That is an unusual sort of practical advice to fall out of pure arithmetic, and it is the reason this essay exists in a collection about symmetry rather than in a manual.

Two boxes that are not the same box #

A subtlety worth naming: an N × N box and an N × M box with NM the same size are different objects, and neither is better.

A 2 × 8 box holds sixteen wavevectors, all with first coordinate in halves and second in eighths. A 4 × 4 box holds sixteen too, all in quarters. They sample completely different sets, and a feature at (¼, ¼) is in the second and not the first.

More generally the box need not be rectangular at all: any sublattice of the translation lattice will do as the box’s own translations, and the sublattices of a given index are exactly the choices available. A box is a sublattice, its wavevectors are the characters of the quotient, and choosing one is choosing which quotient to work in.

That reframing connects this essay to the sublattice counting done several sizes ago: the number of distinct boxes of a given size is the number of sublattices of that index, which is a divisor sum. Most of them are never used, because the rectangular ones are easier to think about — and the non-rectangular ones sometimes sample exactly the wavevector a rectangular one misses.

The other flat surfaces #

The flat-space anchor began with a related question: which surfaces a plane pattern can be folded onto. The answer there was the torus and the Klein bottle, being the quotients of the plane by the two groups that act freely — p1 and pg — and the three-dimensional version is Bieberbach’s ten flat manifolds.

The box here is the torus of that list, met from the computational side. Its group is not the group of the surface, though, and the difference is worth naming: the surface’s own symmetry group is what remains of the plane group after quotienting, while the group acting here is the whole finite group (ℤ/N)² ⋊ P, which contains the translations as an honest subgroup rather than dividing by them.

A calculation therefore has more symmetry available than the surface does, and uses it: the reduction of a sweep over N² wavevectors to a sweep over one representative per star is exactly the wedge argument applied to a finite set.

What a box is good for #

Nothing above is an argument against finite boxes, and it would be a poor conclusion to draw. Three things a box does that an infinite crystal does not.

It makes a calculation possible at all. An infinite object has no finite representation to compute with, and the box is the standard one.

It makes counting exact. Burnside’s lemma on a finite group gives an integer; the same question about an infinite pattern is a question about densities.

It exhibits a supercell directly. A structure that modulates with a period of three cells is periodic on a three-by-three box, so the box is not merely a sampling of the crystal — it is the natural home of that particular structure.

The rule to carry away is not “avoid boxes” but “say which wavevectors the box holds”, which is a line of arithmetic and is almost never stated in a paper reporting a supercell calculation.

What to state when reporting one #

The rule this essay ends on is a reporting rule rather than a computational one, and it is short.

Say which wavevectors the box holds. That is N² fractions with denominator dividing N, and naming N is enough — but naming it is not the same as naming the box’s size in cells, which is what papers usually give. The two are the same number and only one of them tells a reader what the calculation could see.

Say which features that set contains. If the structure has a modulation at a wavevector of thirds, a box of four cells could not have found it; if a level is degenerate at the zone corner, a box that misses the corner reports a gap. Both are statements a reader can check from N alone.

Neither costs anything to state, and both turn an unremarkable number into the piece of information that decides what the result means.

The box that holds any wavevector #

The essay’s central limitation — a box holds N² wavevectors and no others — has a standard repair, it costs nothing, and it is worth knowing because it explains why a great many calculations use a single cell and still resolve a general wavevector.

The limitation came from gluing the box’s edges identically: crossing the boundary returns to the same configuration, so a function on the box must repeat exactly, and the wavevectors are those with denominator dividing N. Glue with a phase instead. Require a function to come back multiplied by $e^{2\pi i\theta}$ on crossing one edge and by $e^{2\pi i\varphi}$ on crossing the other, and the permitted wavevectors shift by (θ, φ).

Those are twisted boundary conditions, and with θ and φ free the single cell reaches every wavevector in the zone. The calculation is no longer over real quantities — the phases make it complex — but nothing else changes, and the box’s size stops deciding which wavevectors are available.

That is Bloch’s theorem read as a boundary condition rather than as a wave, and it is why a band structure is computed on one cell at a hundred wavevectors rather than on a hundred cells. The supercell and the twist are two ways of buying the same thing, and the trade is between doing a small calculation many times with complex arithmetic and doing a large one once with real.

Where the supercell is still needed is where the structure itself changes: a modulation with a period of three cells is a different arrangement of atoms, not a different wavevector on the same arrangement, and no twisting of the boundary conditions produces it.

What kind of finite group a box has #

The box’s symmetry group is described above as containing the point group and the translations modulo N. How the two are joined is worth a paragraph, because it is not always the obvious way and the difference is exactly the distinction this collection draws between the symmorphic groups and the rest.

For a symmorphic group — one whose operations can all be taken to fix a common point — the box’s group is a semidirect product: the finite translations, with the point group acting on them, and a copy of the point group sitting inside as operations about the origin. Everything factorises.

For a non-symmorphic group it need not. A glide’s square is a lattice translation, so on the box the glide has an order fixed by N and by nothing else — and there may be no subgroup of the box’s group isomorphic to the point group at all. The extension does not split, which is the finite shadow of the fact that a glide cannot be written as a reflection about any point.

That has a practical consequence for the counting. Burnside’s average runs over the box’s group whatever its structure, so the counts are unaffected — but a calculation that assumed the group factorised, and built it as a point group times a translation group, would build the wrong group for four of the seventeen. It would also produce a plausible number, which is the failure mode this collection watches for, and the defence is the same one: generate the group by closing the operations rather than by asserting its shape.

Where this goes #

The unfinished business is the three-dimensional version, where the same accounting applies with N³ wavevectors and the box choices are correspondingly more consequential — and where a box commensurate with one modulation is usually incommensurate with another.

The nearer neighbour is the irreducible wedge, which is the other half of the same practical question: a box says which wavevectors exist, and the wedge says how few of them have to be visited.