The star of a wavevector

The reciprocal lattice entered this collection as a bookkeeping device for diffraction: the places where a periodic structure scatters, indexed by three integers, with the whole geometry of a diffraction experiment following from where those points sit. It has a second life, and the second life is where most of the calculations anybody performs on a crystal happen.

A wave in a periodic medium is labelled by a wavevector. Two waves whose wavevectors differ by a reciprocal lattice vector are the same wave — they agree at every lattice point, which is where the crystal is — so the labels live not in the plane of all wavevectors but in that plane modulo the reciprocal lattice, which is a torus. The natural picture of it is the cell nobody chose, taken in reciprocal space, and it is called the Brillouin zone.

The action, and the thing that surprises #

An operation {M | t} sends the point x to Mx + t. Its action on a wavevector is not the same map. A wavevector pairs with a position to give a number — the phase of a plane wave — and the pairing has to be preserved, so k goes to k M⁻¹ with the translation part contributing nothing at all.

That is the decisive difference between the two actions, and the whole ladder follows from it. The translation part of an operation moves every point of the crystal and no wavevector whatsoever. A screw axis and a plain rotation act identically on reciprocal space; a glide and a mirror are indistinguishable there.

The translation does not vanish, though. It comes back as a phase factor: applying the operation to a wave multiplies it by $e^{-2\pi i\,\mathbf{k}\cdot\mathbf{t}}$, and the factor is invisible until two operations are composed, at which point it decides whether the operators multiply the way the group does. Three rungs of this ladder are about what the factor does, and the glide essay is where it stops being bookkeeping.

The star: what a wavevector becomes #

The star of k is its orbit under the point group, taken modulo the reciprocal lattice. It is the set of wavevectors the crystal cannot distinguish from k: whatever a symmetric operator does at one of them, it does at all.

Its size is a count of how much has been saved. A generic wavevector in p4m has a star of eight, so seven eighths of the zone is a copy of the remaining eighth. A wavevector at the centre has a star of one, because every operation fixes it.

The little group: what stays behind #

The little group of k is the set of operations that leave it where it is, modulo the reciprocal lattice. It is a subgroup, and it is the object every later rung is about, because a symmetric operator restricted to one wavevector is invariant under the little group and under nothing larger.

“Modulo the reciprocal lattice” is doing real work in that definition, and it is the point most often skated over. An operation sending (½, 0) to (−½, 0) has left the wavevector alone, because the two differ by a whole reciprocal lattice vector and label the same wave. That is why the edge of the zone is special: a two-fold rotation fixes no interior wavevector except the centre, and it fixes every point of the boundary that lies opposite itself.

One identity, checked everywhere #

The relation between the two is the orbit–stabiliser theorem, which in this setting says

$$|\text{star of } k| \times |\text{little group of } k| = |\text{point group}|$$

It is not a deep statement — the members of the star are in one-to-one correspondence with the cosets of the little group — and it is exactly the kind of identity worth computing on every case rather than quoting once. A modular-arithmetic error in the action on wavevectors would break it immediately and would break nothing else visibly, which is the signature of a bug worth having a test for.

It is checked here on all seventeen groups at every wavevector of a grid, which is a few hundred cases and takes no measurable time. Each is integer arithmetic: k is a pair of rationals, M is an integer matrix, and congruence modulo one is a comparison of numerators.

The zone, and why it is the cell nobody chose #

The set of labels is a torus and a torus is awkward to draw, so a representative region is wanted: one wavevector from each class modulo the reciprocal lattice. Any cell of the reciprocal lattice would do, and the one everybody uses is the Brillouin zone, which is the set of wavevectors nearer the origin than any other reciprocal lattice point.

That is precisely the Wigner–Seitz cell of the reciprocal lattice, and it is chosen for exactly the reason that essay gives: it needs no basis and no convention. A parallelogram cell would work as well arithmetically and would be a fact about somebody’s choice of axes; the zone is a fact about the lattice, and it has the lattice’s full point symmetry, which a parallelogram cell generally does not.

The consequence worth carrying is that the zone’s shape records the lattice type. A square lattice has a square zone, a hexagonal lattice a hexagonal one, and a centred rectangular lattice a zone with two kinds of edge. Reading a zone is reading a lattice, one Fourier transform removed.

The points that have names #

Γ, X, M, K. Every account of the subject uses those letters and they are conventional labels for specific wavevectors — the centre, an edge centre, a corner. Nothing is wrong with the convention, and this collection nevertheless arrives at them the other way round: enumerate the wavevectors of a grid, sort them by how much of the group survives, and attach the letters to whatever comes out.

The reason is the same one that governs every count here. A named point is a point somebody decided to name, and the interesting question is which points are special, which is a property of the group. Enumeration answers it; a table of conventional labels records somebody else’s answer.

What comes out is that the special wavevectors are exactly those with small denominators. A rotation of order n can fix a wavevector modulo the reciprocal lattice only if the wavevector’s coordinates have denominator dividing n, so halves and thirds are where the little groups live and everything else is generic. That is a fact about integers with no geometry in it, and it explains why the same handful of letters appears for every crystal in the subject.

What the little group is not #

Two mistakes are easy here and both have consequences.

The little group is not the point group of the crystal restricted somehow. It is the subgroup fixing a wavevector, and different wavevectors have different ones. In p6m the centre has all twelve operations, the corner has six, an edge centre has four, and a generic point has one. A statement about “the symmetry at K” is a statement about a group of order six, not about p6m.

The little group’s elements still carry their translations. Two operations with the same rotation part and different translations are different elements of the plane group, and they act on a wave with different phases. Reciprocal space cannot see the difference between a glide and a mirror in the action on k, and can see it perfectly well in the action on the wave. Forgetting the translations at this stage is what makes the sticking of levels at a zone boundary look like magic later on; keeping them makes it arithmetic.

What a wavevector labels, in a collection that has avoided waves #

This is the first essay here to put a wave in a crystal, and it is worth saying plainly what is being labelled and what is not.

A function on the crystal that is periodic is a sum of terms indexed by reciprocal lattice points, which is what the diffraction essays use. A function that is not periodic but is compatible with the periodicity in a weaker sense — one whose magnitude repeats while the wave’s phase advances steadily — is labelled by a wavevector inside the zone. Such a function is a Bloch state, and every solution of a symmetric problem on a periodic structure can be sorted into them.

Nothing about that sorting is physics. It is a statement about the representations of the translation group: the translations form an abelian group, all of whose irreducible representations are one-dimensional, and its representations are exactly the phase factors labelled by wavevectors. So sorting by wavevector is decomposing the space of functions into irreducible pieces for the translation subgroup, and everything above this rung is what the rest of the group does after that decomposition has been made.

That is why the ladder belongs on a crystallography site rather than in a physics one. The wave is an instrument; the object is a group acting on functions, and the crystal is what makes the group infinite and periodic.

Why any of this is worth building #

Three things follow from the star and the little group, and the ladder above this essay is the three of them.

A calculation shrinks by the order of the group. Everything a symmetric operator does at one member of a star it does at all of them, so a sweep over the zone can be replaced by a sweep over one representative per star. The saving is close to the order of the point group, and for a cubic crystal that is a factor of forty-eight.

Degeneracies are forced at the special points and nowhere else. A generic wavevector has a trivial little group, so its levels are unprotected; a wavevector whose little group has a two-dimensional representation carries levels that cannot be separated. That is the next rung, and it is the reason band structures are drawn along paths through the named points rather than anywhere else.

The translations decide something the point group cannot see. Where the little group’s operators fail to multiply the way the group does, the failure cannot be removed, and the consequence is a doubling of every level along a whole line of the zone. That is the glide rung, and it is the sharpest thing the phase does.

A worked case: the square zone, point by point #

It is worth walking one group all the way round, because the pattern is the same in every other.

Take p4m, whose point group has order eight. At the centre the star is one and the little group is the whole eight: everything fixes the origin. At an edge centre — (½, 0) — the star is two, since the four-fold takes it to (0, ½) and nothing takes it back; the little group has order four, being the two mirrors along and across the edge, the two-fold, and the identity. At a corner — (½, ½) — the star is one again, because the four-fold sends it to (−½, ½), which differs from it by a reciprocal lattice vector; the little group is the whole group.

At a generic point the star is eight and the little group is trivial. Between the generic points and the special ones lie the lines: a wavevector on a mirror line has a star of four and a little group of two, and it is along those lines that the levels of a symmetric operator are usually drawn, because they connect the special points where something is forced.

Every one of those statements is 2 × 4 = 8, 1 × 8 = 8, 8 × 1 = 8, and the enumeration produces them without any of them being typed.

The exactness, and where it stops #

Everything in this essay is exact. A wavevector is a pair of rationals; an operation acts by an integer matrix; the star is computed by closure and the little group by a comparison of numerators modulo one. There is no tolerance anywhere and nothing that could be nearly true.

The exactness carries as far as the structure — stars, little groups, the phases the translations contribute, the factor systems those phases produce, whether a factor system can be removed. All of that is integer arithmetic in the twelfth cyclotomic ring, for the reason the whole collection’s arithmetic is exact: the orders a lattice permits divide twelve.

It stops at the levels. An eigenvalue is a root of a polynomial and in general is irrational, so every spectrum in this ladder is computed numerically and reported as a measurement, and every multiplicity read off one is a claim about a stated gap. That division — exact structure, measured values — runs through the four rungs, and each figure says which side of it the number it shows came from.

The wedge, and what a calculation does with it #

The saving the essay names is used in practice by every calculation that sums over a zone, and the way it is used has a standard form worth knowing.

A quantity that is an integral over the zone — a density, a total energy, a count of states — is computed by evaluating an integrand on a grid of wavevectors and adding. The grid is chosen to be commensurate with the reciprocal lattice, so that it is itself a lattice of points inside the zone, and then reduced: each point is replaced by the representative of its star, with a weight equal to the star’s size.

The reduction is exactly the orbit computation on this page. A grid of N points reduces to the number of orbits, the weights are the orbit sizes, and the sum over the reduced set with weights equals the sum over the whole grid — by construction, not by approximation.

Two things make that worth doing rather than merely elegant. The saving is nearly the order of the point group, so a cubic calculation runs forty-eight times faster; and the weights are exact integers, so nothing is lost. What is approximated is the integral itself, since a finite grid samples a continuum, and that error is separate from the symmetry reduction and does not interact with it.

The standard grid offsets its points from the zone centre by half a spacing, which puts more of them at general wavevectors and fewer at the special points. That is deliberate: a special point has a small star, so it carries a large weight and contributes disproportionately, and a grid that lands on many of them converges more slowly than one that avoids them.

What the star does to a representation #

There is a structural statement behind the counting, and it says why the little group is the object to work with rather than the whole group.

A state at wavevector k is carried by an operation to a state at another member of the star. So a set of states closed under the whole group must contain states at every member of the star, and its dimension is the star’s size times whatever happens at one member.

That is the standard construction: a representation of the little group at one k determines a representation of the whole group, of dimension |star| × |little group representation|, by carrying it round the star. Every irreducible representation of a space group arises this way, from a star and an irreducible representation of the little group of one of its members, and the two labels together are the complete name of one.

The practical consequence is the essay’s opening claim made precise. Working at one wavevector with its little group is not a convenience or an approximation: it is the whole of the group’s representation theory, arranged so that only a finite subgroup ever has to be handled. The translations were dealt with by the phase, the star was dealt with by the induction, and what is left is a point group and a character table.

Where this ladder goes #

The next rung asks what a little group with a two-dimensional representation does to the levels there, and answers it twice: from the character table, exactly, and from the spectrum of an operator, numerically. The rung after is about the phase the translations left behind, and about a sign that no choice of convention removes. And the corner of the honeycomb is the whole apparatus applied to one net, where the degeneracy turns out to be an identity between three cube roots of unity.