The cell nobody chose

The cell is a choice is this site’s standing warning: a lattice has infinitely many unit cells, crystallography picks one by rule, and a symbol read against the wrong rule means something else. The warning is unavoidable for a conventional cell, and there is exactly one construction that escapes it.

Take a lattice point and keep every point of the plane that is closer to it than to any other lattice point. The result is a region that tiles the plane by translation, has exactly the area of a unit cell, and was built without choosing a basis, an origin beyond the lattice point itself, or an axis. Two people who agree about the lattice cannot disagree about this cell.

It has several names, and which one is used says what it is being used for. Mathematicians call it the Voronoi cell, after the construction Georgy Voronoi published in 1908 for arbitrary point sets. Solid-state physicists call it the Wigner–Seitz cell, after the 1933 paper in which Eugene Wigner and Frederick Seitz used it to compute the cohesive energy of sodium. Crystallographers call it the Dirichlet domain, after the earlier and more general construction. It is one object.

The whole decision is an inequality between two integers #

A Voronoi cell looks like a geometry problem — perpendicular bisectors, circumcentres, square roots — and the decisions in it are not geometric at all.

A lattice vector v contributes a face when the midpoint v/2 is strictly closer to the origin and to v than to any other lattice point. Written out and cleared of halves, that is

$$|v - 2u|^2 > |v|^2 \quad\text{for every lattice vector } u \ne 0, v,$$

which is a comparison of two lengths squared. And a squared length in a lattice is an integer, provided lengths are measured with the Gram matrix — the table of dot products of a basis, G[i][j] = aᵢ · aⱼ, whose entries are integers for every lattice this site draws.

So the combinatorics of the cell, its symmetry, and its area are exact, and the coordinates of its corners are the one thing that needs a square root. That is a sharper statement of where the exactness stops than this site usually gets to make: not the claims are exact and the drawing is approximate, but this specific list of decisions is integral and this specific list of numbers is not.

The vectors that survive the inequality have a name — the relevant vectors — and for the hexagonal lattice there are six of them, the six shortest. That is not a general rule. The relevant vectors of a very oblique lattice include one pair that is not among the shortest, and a construction that took the six shortest vectors and bisected them would draw the wrong hexagon.

Why it carries the whole symmetry, and a conventional cell does not #

Every symmetry of the lattice permutes the lattice points, so it permutes the distances from the origin to all of them, so it maps the set of points nearer the origin than to anything else onto itself. The cell therefore has the lattice’s whole point group, always, by construction.

A conventional cell does not. The primitive cell of the centred rectangular lattice is a rhombus, and drawing it displays two of that lattice’s four operations: the rhombus is invariant under the identity and the half-turn, and the two mirrors are symmetries of the lattice that the rhombus does not show. Choosing the centred rectangle instead displays the mirrors and hides the fact that the cell has twice the area it needs, which is the trade why the bigger cell wins is about.

The symmetries themselves are derived rather than assumed. An integer matrix M is a symmetry of the metric exactly when MᵀGM = G — every length and every angle preserved, and the lattice preserved because the matrix is integral — and searching a small box for both returns 2, 4, 4, 8 and 12 for the five types. That is five lattices and no others arriving from the metric side, and it is a check on this construction rather than the point of it: the cell is then required to be invariant under every one of those matrices, and it is.

Four faces or six, and never anything else #

Read across the five: the two right-angled lattices have four-sided cells and the other three have six-sided ones. Nothing has five sides, or seven, or eight.

That is a theorem rather than an observation about these five. A Voronoi cell of a lattice in d dimensions has at most 2(2^d − 1) faces — six in the plane, fourteen in space — because each face is bisected by a relevant vector, relevant vectors come in ± pairs, and each pair is determined by a distinct non-zero class of the lattice modulo 2Λ, of which there are 2^d − 1.

The bound is worth stating with its consequence in three dimensions attached: fourteen is why the Wigner–Seitz cell of the body-centred cubic lattice is a truncated octahedron — eight hexagonal faces and six square ones, fourteen in all — which is the shape a physicist meets as the Brillouin zone of every bcc metal.

In the plane the bound is reached by the generic case. Sweeping every reduced integer Gram matrix in a box gives six faces for two thirds of them and four for the rest, and the four-faced ones are exactly the metrics with a right angle. A hexagon is what a lattice cell looks like by default, and the rectangle is the special case — which is the reverse of the impression a page of conventional cells gives.

That sweep is a negative test rather than an illustration, and the distinction is the one this collection keeps insisting on. A figure of the five plane lattices showing four faces twice and six three times is a picture of the answer; sixty-five metrics chosen by an enumeration rather than by hand, every one of them computed, is a search that had the opportunity to produce a five or a seven and did not. The enumeration is complete over the box because the reduction conditions — twice the off-diagonal entry no larger than the first diagonal one, and the diagonal entries in order — pick exactly one metric out of each lattice’s infinitely many, so no lattice in that range is tried twice and none is missed.

The second half of the finding is sharper than the bound and is not a theorem anybody quotes. Four faces happens exactly when the off-diagonal entry is zero — no exceptions in sixty-five, in either direction. A zero off-diagonal entry means the two basis vectors are perpendicular, so the statement is that a lattice’s cell is a rectangle precisely when the lattice has a right angle available to it, and a hexagon otherwise. Nothing about that is obvious from the construction, and it is the kind of clean split that a sweep can find and a proof by five examples cannot.

How it is actually built, and the way that fails #

The construction has an algorithm and the algorithm has a characteristic failure, both worth stating because the failure is silent.

Start with a large square and clip it with one half-plane per candidate vector: keep the side of the perpendicular bisector on which the origin lies, discard the rest. Do that for every lattice vector out to a few shells and what remains is the cell. It is the dullest of the available methods — a sweep over a sorted list of angles is faster — and its failure mode is a missing corner rather than a wrong one, which is the property worth paying for.

The silent failure is the shell count. Clip with too few shells and the polygon is too large; it is still convex, still symmetric, still perfectly plausible, and its area is wrong. Nothing about the picture betrays that, which is the same shape of error this site’s round trip exists to catch in patterns. So the area is asserted against √det G, computed from the metric and sharing no code with the clipping, and a cell built from too few shells fails that assertion immediately.

That figure is the essay’s whole claim in one panel, and the part worth checking is the faint outlines rather than the solid one. Any pair of vectors whose integer matrix has determinant ±1 is a basis of the lattice, and there are infinitely many of them: shear one vector by any whole number of the other and the determinant is unchanged. Every one of those bases gives a primitive cell, every one of those cells has the same area, and none of them is more correct than any other. Crystallography picks one by a rule about lengths and angles, and the rule is a convention with a history rather than a fact about the lattice.

The determinant is the whole of the check, and getting it wrong is not an abstract hazard. A matrix of determinant two draws a perfectly plausible parallelogram twice the size, and what it is a cell of is a sublattice — a different lattice that happens to sit inside this one. Nothing in the drawing distinguishes the two cases, which is why the figure computes the determinant and the area rather than trusting the vectors it was handed.

The candidate list has a bound that is not an assumption either. A vector longer than twice the covering radius cannot bisect anything, because some closer lattice point already cuts off everything on that side; so a search over a few shells is complete, and widening it is asserted to change nothing.

The same construction in the reciprocal lattice #

The dual lattice builds the reciprocal lattice as one point per family of lattice rows, at the inverse of the spacing. For this construction the reciprocal lattice is needed only as a metric, and the metric of the dual is the inverse of the metric of the original — one line of arithmetic, and everything above applies unchanged.

The Wigner–Seitz cell of the reciprocal lattice is the first Brillouin zone, and it is the region of wavevectors a periodic medium cannot tell apart from any other. A wave whose wavevector differs from another’s by a reciprocal lattice vector produces the same displacement at every lattice point, so the two are the same wave as far as the crystal is concerned, and the zone is one representative of each class. Every band structure ever drawn is drawn on it.

Two things are worth noticing. The zone of the hexagonal lattice is a hexagon turned thirty degrees from the lattice’s own cell, because the reciprocal of a hexagonal lattice is a hexagonal lattice rotated by thirty degrees. And the zone’s symmetry is the lattice’s symmetry, by the argument above applied to the dual — which is why a band structure has the point group of the crystal without anybody imposing it.

What ordering does to the zone #

A crystal that orders two species onto a sublattice has not moved an atom: the lattice of positions is untouched, and the repeat of the contents is n times as large. In reciprocal space that makes the lattice n times denser, so the zone shrinks by a factor of n and everything that lived in the large zone is folded into the small one — n points of the old zone becoming one point of the new.

That is the reciprocal-space face of the ordering arithmetic, and the reflections a superlattice adds is the diffraction face of the same fact: n − 1 new reflections per parent cell, at exactly the points the folded zone’s new boundaries pass through. One integer — the index — decides both, and how many sublattices there are of that index says how many different foldings a crystal has to choose between.

Where the construction is used, and where it misleads #

The cell earns its keep in three places, and it is worth separating them because they want different things from it.

In solid-state physics it is the natural cell because it has the crystal’s full symmetry, so a calculation done in it needs no symmetry-breaking basis and the irreducible wedge is a genuine wedge of the point group. That is what Wigner and Seitz wanted it for.

In counting neighbours it settles a question that “nearest neighbour” leaves open, and it does so on the same lattice the reciprocal lattice is built from. Two lattice points are Voronoi neighbours when their cells share a face, and that is a property of the lattice rather than of a cutoff distance — which is why coordination numbers computed from Voronoi faces are stable where ones computed from a radius are not.

In deciding the lattice type it is a fingerprint. The number of faces and the pattern of face areas classify the five plane lattices and, in three dimensions, distinguish the Bravais types by the shape of the cell alone. That is Delaunay’s method, and it is one of the two standard routes to reduction — the other being the shortest basis, which reaches the same classification through a basis rather than through a shape.

Where it misleads is in being pretty. The cell of a slightly oblique lattice is a hexagon with two very short faces, and a drawing at a coarse resolution shows a rectangle. Nothing in the picture says that the two nearly-vanished faces are there; only the integer inequality does, and it does not care how short the face is. A cell drawn by eye from a lattice plot will lose them every time.

One dimension below, and one above #

The construction does not care about the dimension, and looking at the neighbours on either side says what is special about the plane.

In one dimension the Voronoi cell of a lattice of spacing a is the interval of length a centred on a lattice point, and the two relevant vectors are ±a. The bound 2(2^d − 1) gives two faces, which are the two endpoints, and it is tight. The reciprocal statement is the one every diffraction essay on this site uses without naming it: the first Brillouin zone of a one-dimensional lattice is the interval from −π/a to π/a, and a wavevector outside it is the same wave as one inside.

In three dimensions the bound is fourteen and the shapes are the five parallelohedra Fedorov classified in 1885 — cube, hexagonal prism, rhombic dodecahedron, elongated dodecahedron and truncated octahedron. That list is a classification of lattices by the combinatorics of their cells, and it is the three-dimensional version of the four-or-six above. The truncated octahedron is the generic case, in the same sense in which the hexagon is generic in the plane: perturb any three-dimensional lattice and its cell acquires all fourteen faces.

Two things carry over unchanged and one does not. The integrality carries over: the inequality is the same, the Gram matrix is three by three, and the arithmetic is still integral. The symmetry argument carries over word for word. What does not carry over is the ease of drawing — a plane figure of a truncated octahedron needs a projection, and a projection is a choice, so the one construction on this site that involves no convention would arrive on the page wearing one.

Where the exactness stops, precisely #

Three statements, each with its own status.

The face count, the symmetry and the area are exact. They come out of integer comparisons in the Gram matrix, and the area is asserted against √det G while the figure is drawn — a check that would fail immediately if the half-plane clipping dropped a corner.

The corner coordinates are not. They involve a square root of an integer, and the figure’s own symmetry test compares them with a tolerance of 10⁻⁶ — the only tolerance in the whole construction, named where it is used.

Nothing here is about a crystal’s contents. The Wigner–Seitz cell is a fact about a lattice, and two crystals with the same lattice and utterly different structures have the same one. It says what shape the repeat is; it says nothing about what is in it, and the site’s habit of separating the lattice underneath from the pattern on it applies here in full.

Where the ladder goes next #

This anchor has one rung and it is the construction itself. Two rungs are visible above it.

The first is Delaunay reduction, the classification of lattices by the combinatorial type of the Voronoi cell — which in three dimensions gives Fedorov’s five parallelohedra and is the alternative to the Niggli route this site already has. The second is the higher zones: the second, third and fourth Brillouin zones are the regions second-nearest, third-nearest and so on, they have the same area as the first, and cutting them up and folding them back into it is the classical construction of a free-electron Fermi surface.

Both are the same integer inequality asked more times, which is the argument for taking the construction seriously rather than treating it as a picture.

Fedorov’s five #

The plane’s answer is two shapes — a hexagon in general, a rectangle when the lattice is rectangular — and the corresponding classification in space was made before either name attached to the cell was in use.

A Wigner–Seitz cell tiles space by translation alone, since the cells around all the lattice points fill space with no gaps and no overlaps and are all translates of one another. A convex body that does this is a parallelohedron.

Fedorov classified them in 1885 and found five. The cube; the hexagonal prism; the rhombic dodecahedron, which is the cell of the face-centred cubic lattice; the truncated octahedron, which is the cell of the body-centred cubic lattice; and the elongated dodecahedron. Every lattice in space has one of these five as its Wigner–Seitz cell, deformed but combinatorially unchanged.

The list is a classification of lattices by their combinatorics rather than by their symmetry. Two lattices of quite different Bravais types can share a parallelohedron type, and the generic lattice has the truncated octahedron with fourteen faces — the same statement as the plane’s generic hexagon, one dimension up.

Whether every space-filling convex body is a parallelohedron of this kind is open. Voronoi conjectured in 1908 that every convex body tiling space by translation is affinely equivalent to the Voronoi cell of some lattice, and the conjecture is proved up to five dimensions and unproved in general — a rare instance of an open question sitting directly under a picture in a crystallography textbook.

The same construction around atoms #

Nothing in the definition required the points to form a lattice, and dropping that requirement gives a tool for structures rather than for lattices.

Take the atoms of a crystal rather than the points of its lattice. Each atom gets the region of space nearer to it than to any other atom, and the regions fill the crystal. They are no longer all the same shape, since the atoms are not all equivalent, and they are still convex polyhedra.

Each face names a neighbour. Two atoms share a face exactly when no third atom lies between them in the relevant sense, so the faces of an atom’s cell are a list of the atoms it is genuinely adjacent to.

That settles a question distance cannot. Coordination number is usually defined by a cut-off — count the atoms within some radius — and the answer depends on the radius, which is chosen by whoever is counting. A structure with a continuous spread of interatomic distances has no defensible cut-off, and reasonable people report different coordination numbers for the same atom.

The face count needs no cut-off. It is decided by the arrangement rather than by a threshold, so it is the same for everyone, and it can be weighted by face area when some contacts are more marginal than others. The cell that is not a choice for a lattice is not a choice for a structure either, and for the same reason.