The zones above the first

The cell nobody chose builds the one region of the plane that involves no convention: the points closer to a given lattice point than to any other. In the reciprocal lattice that region is the first Brillouin zone, and it is where every band structure ever drawn is drawn.

There is no reason to stop at the first, and the reason to carry on is a statement that reads as though it cannot be true.

Every zone has the same area. The second, the third, the seventh — each is a different number of disconnected fragments, sitting in a different part of reciprocal space, and each has exactly the area of one cell of the reciprocal lattice.

What the n-th zone is #

The definition is a counting statement and has no geometry in it. The n-th zone is the set of wavevectors that have exactly n − 1 reciprocal lattice points nearer to them than the origin is.

The first zone is therefore the points nearest to the origin, which is the Wigner–Seitz cell of the reciprocal lattice. The second is the points with exactly one competitor nearer; the third, two. Since “nearer to a lattice point than to the origin” is a half-plane bounded by the perpendicular bisector of that lattice vector, the boundaries of every zone are bisectors and nothing else. The whole family is decided by one arrangement of lines, and the zone index is constant on each cell of that arrangement, because the index changes only when a bisector is crossed.

That makes the construction as exact as the first zone’s. The bisector of a lattice vector v is the set where |k|² = |k − v|², an equality between two integer-valued forms in the Gram matrix, and which side of it a cell lies on is an integer comparison. The corners are the one thing that needs a square root, exactly as before.

Why the areas are equal #

The proof is not a computation. Take a fragment of the n-th zone and subtract the reciprocal lattice vector nearest to it: what is left is a set of points closer to the origin than to any other lattice point, which is inside the first zone. Do that to every fragment and the translated pieces cover the first zone exactly once.

So the n-th zone is not a region that happens to have the same area as the first. It is the first zone, cut into pieces along the lines of the arrangement and moved by lattice vectors. Every zone is the same region, dissected differently.

This is the operation a solid-state physicist calls zone folding, and it is the reason a band structure can be plotted on one small polygon rather than on the whole plane. A free electron’s energy is a smooth paraboloid over all of reciprocal space; folded into the first zone it becomes a stack of sheets, one per zone, and those sheets are the free-electron bands. The whole classical construction of a Fermi surface — draw a circle of the right area, cut it by the zone boundaries, fold each piece back — is this dissection applied to a disc.

The hexagonal case is worth putting beside the square one, because the two disagree about everything except the areas. The square lattice’s zones break into 1, 4, 8 and 12 fragments; the hexagonal lattice’s into 1, 6, 6 and 6. Same construction, same theorem, different combinatorics — and the combinatorics are a fingerprint of which bisectors happen to meet at a point, which is a question about the lattice’s symmetry rather than about the zones.

The identity as a gate on its own construction #

Building the fragments needs a sweep. The arrangement’s cells are found by sampling: a point’s pattern of sides across all the bisectors identifies the cell it is in, and each distinct pattern is then turned into an exact polygon by intersecting the half-planes it names. Sampling can miss a cell thinner than the sample spacing.

That is exactly what the equal-area statement catches. A missed fragment makes its zone come out short, and nothing else does. So the theorem is used as the gate on the construction that illustrates it: the areas are compared with the reciprocal cell’s, and where they disagree the sampling is doubled and the sweep repeated.

It fired on the first run. The square, hexagonal, rhombic and oblique lattices all came out at a ratio of one to machine precision at the first sampling; the rectangular lattice’s fourth zone came out at 0.993, because its fragments include slivers whose short dimension is smaller than the sample spacing on an elongated lattice. Doubling the sampling recovered them. The number of doublings needed is a fact about how thin a lattice’s high-zone slivers are, and it is reported rather than hidden.

The folding needed a second decomposition, and finding that out took the check #

The first version of the folding was wrong in a way that looked right, and the check refused it.

Each fragment was moved by the lattice vector nearest to the fragment’s own sample point. That is the correct rule for a point and not for a fragment: which lattice point is nearest is constant on the cells of the Voronoi decomposition — the translates of the first zone — and the zone fragments are cells of a coarser arrangement, the bisectors through the origin only. A fragment can straddle two Voronoi cells, and moving it whole lands part of it outside the first zone.

The areas still summed correctly, because translation preserves area. The tiling check did not: for the hexagonal lattice’s third zone it reported half the sampled points of the first zone uncovered and none covered twice, which is the signature of pieces landing on top of each other somewhere else. Intersecting each fragment with each translate of the first zone before moving it fixes it, and the counts then come out exact for every lattice and every zone tried.

Two checks and only one of them could see the error. The area identity is necessary and not sufficient; the tiling test — sample the first zone, count how many moved fragments contain each point, require one — is the one that carries the content. A construction verified only by the identity it illustrates would have shipped.

It is worth restating what the zones are made of, because the word reciprocal does a lot of quiet work. Each point of the reciprocal lattice is one family of rows of the direct lattice, at a distance from the origin equal to the inverse of that family’s spacing. So the first zone’s boundaries are the bisectors of the closest-spaced families of rows, and the higher zones are the bisectors of progressively wider-spaced ones. A zone diagram is a statement about which planes in a crystal are close together, drawn in a space where close becomes far.

That is also why the same picture is what a diffraction pattern is indexed on. Each reciprocal lattice point is a possible reflection, and the bisector between the origin and that point is where the Bragg condition is met for it — which is the geometrical content of the Ewald construction and the reason the zone boundaries are exactly the places a wave in the crystal is diffracted.

How far the sweep has to go #

The first zone and the higher ones differ in a way that matters as soon as either is computed, and it is a statement about which lattice vectors can possibly contribute.

For the first zone the answer is famously small. A bisector contributes a face to the Wigner–Seitz cell only if it is a relevant vector, and Voronoi’s criterion says a vector is relevant exactly when no other lattice point lies inside the region it would cut off — which restricts the candidates to the non-zero classes of the lattice modulo twice itself. In the plane that is three pairs, so the cell nobody chose has at most six sides. In space it is seven pairs, so a three-dimensional Wigner–Seitz cell has at most fourteen faces, and the truncated octahedron of the body-centred cubic lattice is the case that attains it. Both bounds are exact and neither depends on the lattice being special: they hold for every one of the five plane lattices and for all fourteen in space.

None of that survives to the higher zones. The n-th zone is defined by counting how many bisectors a point lies beyond, and a bisector that contributes no face to the first zone still contributes a cut to the second and above. So the economy that makes the first zone cheap is unavailable, and the sweep must include every lattice vector out to a radius large enough that no further bisector can reach the region under construction.

That radius is what has to be argued rather than guessed. A point of the n-th zone lies beyond exactly n − 1 bisectors, so it is within the n-th nearest-neighbour shell of the origin in the reciprocal lattice, and a vector more than twice that far away has its bisector entirely outside. Taking the sweep to that bound is safe; taking it to a fixed multiple of the shortest vector is the sort of shortcut that works on a square lattice and fails on an elongated rectangular one — which is exactly the lattice the area identity was run on, and exactly why it was run there.

The general shape of this is worth keeping. A bound that is tight for the first case is not a bound for the general one, and a construction inheriting the first case’s bound will produce something plausible on the symmetric lattices and wrong on the flat ones. The area identity is what turns that from a hazard into a caught error, since a missing cut leaves a fragment too large and the sum too big.

One thing the construction inherits is worth naming. Everything on this page is a statement about the reciprocal lattice rather than about the crystal, and the reciprocal lattice of each of the five plane lattices is again one of the five — so the zones of a square lattice are the zones of a square lattice, and nothing about which of the fourteen a crystal has changes the kind of construction, only its shape.

Where the exactness stops #

The zone indices and the combinatorics are exact. Which cell has which index is a count of integer comparisons in the Gram matrix.

The equality of areas is exact as a theorem and measured as a number. The fragments’ corners are intersections of bisectors and carry square roots, so the ratio is compared to one to within 10⁻⁸. Every lattice tried reports agreement at the 10⁻¹⁶ level once the sampling is fine enough, which is the floating-point floor rather than a residual.

The tiling check is a sample. It tests a finite set of interior points rather than the whole region, and points landing exactly on a cut — which are in two closed fragments at once — are reported separately rather than counted as overlaps. A statement that two closed regions share a boundary is not a statement that they overlap.

What the fragment counts say about a lattice #

The areas are all equal and the fragment counts are not, and the counts are the part that varies between lattices.

The square lattice gives 1, 4, 8 and 12 pieces for its first four zones; the hexagonal 1, 6, 6 and 6; the oblique 1, 6, 12 and 18. The hexagonal lattice’s high zones stay in six pieces because its bisector arrangement has more coincidences — three bisectors meeting at a point rather than two — and the oblique lattice’s grow fastest because it has none at all.

The two columns of that table are doing opposite things, and it is worth saying which. The area column is the theorem: it is one everywhere, it would be one for any lattice in any dimension, and it therefore carries no information about which lattice is being drawn. The fragment column carries nothing but information — five lattices, five different sequences, no two alike — and it is derived from the same construction by the same arithmetic. A quantity that is constant across a family and a quantity that separates every member of it, computed together from one object, is the shape of most classification in this collection.

That is a fingerprint of the lattice’s symmetry showing up in a place nobody puts it: the number of pieces a zone breaks into is a count of degeneracies in an arrangement of lines, and degeneracies are what symmetry produces. A lattice with more symmetry has fewer, larger fragments at every level.

Every zone carries the full point symmetry of the lattice, for the same reason the first does: the definition mentions only distances to lattice points, and a symmetry of the lattice permutes those. So a zone’s fragments come in orbits under the point group, and the counts above are sums of orbit sizes — 4 = 4 for the square lattice’s second zone, 12 = 8 + 4 for its fourth.

One dimension below, where nothing is hidden #

The construction does not care about the dimension, and running it in one is the quickest way to see why the areas agree.

A lattice of spacing a has a reciprocal lattice of spacing 2π/a. The first zone is the interval from −π/a to π/a; the second is the two intervals from ±π/a to ±2π/a; the third is the next two out. Every zone is a pair of intervals of total length 2π/a, which is the first zone’s length, and the folding is a shift by one reciprocal lattice vector — visibly a rigid motion of two half-intervals onto one whole one.

Everything the plane case does is this with more directions to be cut in. What the plane adds is that the pieces have shapes, that their number grows, and that the number depends on the lattice. What it does not add is any new content in the theorem: the folding map is the same map, and in one dimension it is a translation of two segments.

The wave statement is the same too, and is the one every band structure rests on. Two wavevectors differing by a reciprocal lattice vector describe the same displacement at every lattice point, so they are the same wave as far as the crystal is concerned. The zone is one representative of each class. In one dimension that is the statement that a wave on a line of atoms cannot be distinguished from one with a shorter wavelength that samples the same atoms identically, which is aliasing — and every zone above the first is an alias of the first.

A note on what the zones are not #

Two readings of these pictures are worth refusing, because both are common and both make the construction sound like physics rather than geometry.

A zone is not a set of energies. It is a region of reciprocal space, defined by counting bisector crossings, and nothing about it mentions electrons, bands or gaps. The free-electron bands come from folding a paraboloid into the first zone, which uses the zones and is a separate step; a crystal with no electrons in it has exactly the same zones.

And the boundaries are not where anything happens. The zone boundary is where the Bragg condition is met for a particular reciprocal lattice vector, and if there is a periodic potential a gap opens there. The gap is a consequence of the potential’s strength, which is chemistry; the boundary is a consequence of the lattice, which is arithmetic. A lattice with a vanishingly weak potential has the same zones and no gaps at all.

Keeping the two apart is what lets the construction be exact. Everything drawn here is a statement about integer comparisons in a Gram matrix, and every physical consequence of it needs something this site does not compute.

Where the ladder goes next #

The construction runs in any dimension and the count of faces grows: in three dimensions the first zone of the body-centred cubic lattice is the truncated octahedron with fourteen faces, and its higher zones are the polyhedra Brillouin drew by hand in 1930 and Harrison automated in 1959. The equal-volume statement is unchanged and the dissection is far harder to see.

Two rungs sit above this one.

The free-electron Fermi surface. Draw a circle whose area is the number of electrons per cell times the zone’s area, cut it by the zone boundaries, and fold each piece into the first zone. The result is the standard first-approximation Fermi surface of a metal, and it is this dissection applied to a disc rather than to the zones themselves. That is a calculation about electrons rather than about lattices, and it is where this site’s ground ends.

The zones of a superlattice. Ordering on a sublattice of index n makes the reciprocal lattice n times denser and the zone n times smaller, so everything that lived in the large zone folds into the small one — n points becoming one. The reflections a superlattice adds is the diffraction face of the same fact, and the arithmetic that decides both is a single integer.