Wigner–Seitz cells — the series · Crystallography & Symmetry

The Wigner–Seitz cell of the hexagonal lattice. Every point closer to the central lattice point than to any other. The faint lines run to the 6 neighbours whose perpendicular bisectors bound the region; every other lattice point is cut off by one of them. The cell has exactly the area of a unit cell — asserted while the figure is drawn, against √det G computed from the metric — and it carries all 12 of the lattice's symmetries, which a conventional cell need not. Nothing was chosen to build it: no basis, no axes, no convention. Two people who agree about the lattice cannot disagree about this cell.

The cell nobody chose

Every unit cell on this site is a convention, and one construction escapes the warning entirely: the region of the plane closer to one lattice point than to any other. It needs no basis, no axes and no rule — and its combinatorics are decided in integers, with the square roots confined to drawing it.

part 1 · lattices

The first 4 zones of the square lattice. Zones one to 4, each in its own shade. The n-th zone is the set of wavevectors with exactly n − 1 reciprocal lattice points nearer to them than the origin is, so the boundaries are the perpendicular bisectors and nothing else. The zones get further out and break into more pieces — 1, 4, 8, 12 fragments — and every one of them has the area of a single cell.

The zones above the first

The second Brillouin zone is a scattering of disconnected fragments in a different part of reciprocal space from the first, and it has exactly the same area. So does the third, and the seventh. The reason is that each of them is the first zone, cut up and moved.

part 2 · lattices

Five shapes, and a lattice in space has no other. The five combinatorial types a Wigner–Seitz cell can have in three dimensions — cube, hexagonal prism, rhombic dodecahedron, elongated dodecahedron, truncated octahedron — each drawn from a lattice that produces it. Fedorov proved in 1885 that there are no others, and that fourteen faces is the most any of them has, which is Minkowski's bound of 2(2ⁿ − 1) in three dimensions. Each solid here is cut out by the perpendicular bisectors of nearby lattice vectors and its volume checked against the primitive cell's, which is what catches a face that failed to appear.

Five parallelohedra, and no others

The cell that needs no basis and no convention has, in three dimensions, exactly five shapes. The fourteen Bravais lattices produce all five between them — and which one a lattice gives is not decided by which of the fourteen it is.

part 3 · lattices

Covering and packing want different lattices

A lattice has two natural radii — the largest spheres on its points that do not overlap, and the smallest that leave no gap — and both are radii of the same Voronoi cell. In the plane one lattice is best at both. In space the best packer and the best coverer are different lattices, and they are duals of one another.

part 4 · lattices

A gap of exactly 1.00. The two folded bands with the ordering switched on. The faint curves are the same bands before it, crossing at the boundary of the reduced zone; the ordering couples them there and separates them by exactly twice its own strength. The gap appears at the wavevector where the superlattice's extra reflections appear, and for the same reason: both are the Fourier component of the potential at that wavevector.

A bigger cell, a smaller zone

Ordering two kinds of atom onto a sublattice adds reflections to the diffraction pattern and opens a gap in the levels. It is one fact told twice: the same Fourier component of the potential, at the same wavevector, doing the same thing.

part 5 · lattices

Four vectors summing to zero, and six numbers on the edges. A superbasis is the three basis vectors together with their negated sum, so the four sum to nothing and their pairwise products sit on the six edges of a tetrahedron. Selling's rule is: while any edge is positive, apply one transformation. The right panel is the same lattice reduced, with the vanishing parameters marked — and a vanishing parameter is a face the Voronoi cell does not have.

A reduction with one rule

Niggli's reduction is eight numbered conditions with sub-cases, applied in order until none applies. Selling's is a single rule on four vectors that sum to zero: while any of six numbers is positive, do one thing. It terminates sooner, its termination is a quantity that visibly falls, and when it stops the six numbers are the Voronoi cell — the pattern of which ones vanish gives Fedorov's five solids and nothing else.

part 6 · lattices

Three places to put the boundary. A large site and a small one, with three candidate boundaries between them. Halfway is the ordinary Voronoi cell and it cuts through the large sphere. Splitting in the ratio of the radii is the natural repair and its surfaces are not planes, so the cells do not fit together. The power plane sits where the tangent lengths agree, which is further from the large site than halfway and is still a plane — and being a plane is the whole reason the construction works.

Where the boundary goes when the atoms differ

Assigning each point of space to the nearest atom is the right rule only when every atom is the same size. Splitting the distance in the ratio of the radii is the obvious repair and it produces curved faces that do not fit together. The repair that works measures to a sphere rather than to a point: the boundary stays a plane, the cells still tile exactly, and a small enough atom loses its cell altogether — at a radius ratio of exactly one in eight.

part 7 · lattices

The five, as generator counts. Each of the five convex bodies that tile space by translation, built as the set of combinations of a handful of vectors with coefficients between zero and one. Three generators give a cube, four give either a hexagonal prism or a rhombic dodecahedron depending on whether three of them are coplanar, five give the elongated dodecahedron and six the truncated octahedron. The last column is what the same number of generators would give in general position, and the shortfall is the number of faces lost to coplanarity.

Every parallelohedron is a shadow of a cube

Take a few vectors and form every combination of them with coefficients between zero and one. All five of the convex bodies that tile space by translation come out of that recipe, from three vectors, four, four, five and six — and since the recipe is exactly the image of a cube of that many dimensions, the truncated octahedron is a three-dimensional shadow of a six-dimensional cube. The five are not the generic answers: they are the degenerate ones, and the degeneracy is what the tiling demands.

part 8 · classification