Crystallography & Symmetry
patterns derived from their groups
Concept

Supercell — where it appears

A cell that is a multiple of the true one, so that a structure described in it appears to have more atoms and fewer translations than it has. It is detected rather than assumed: a description on a supercell has extra translations among its own symmetries, and every count made against it is wrong by that factor.

Named by 10 essays across 3 fields — each of them below, with the objects they name alongside it.

A wavevector of thirds, and the boxes that cannot see it. Which sizes of box can carry the wavevector at the corner of a hexagonal zone. The characters of the box's translation group are its wavevectors, and there are exactly N² of them — the fractions with denominator dividing N. A wavevector of thirds is therefore present in a box of three, six, nine or twelve cells and absent from one of two, four or five: not approximated badly, not resolved coarsely, absent. A mechanism or a level living there is invisible to such a calculation, and that is the practical content of a mechanism count depending on the cell it was looked for in.

Crystallography in a box

A calculation over a crystal is not performed on a crystal. It is performed on a finite block with its edges glued, and the block has a symmetry group of its own — finite, complete in one direction and missing something decisive in the other.

classification · Flat space two sites in a hexagonal cell: 3 cutoffs, degree 3 to 12. Two atoms per hexagonal cell, at the positions graphite's carbons occupy, read as a net at a ladder of bonding cutoffs. Each row takes the cutoff just past a shell of neighbours and reports the net that results: how many edges it has, the degree of its vertices, whether its cycles generate the whole translation lattice, and the group of its own barycentric placement. The net is not in the coordinates. There is no bond in a list of positions; there is a cutoff, and moving it past a shell gives a different net from the same atoms. A row marked as a supercell is a net whose own translations turn out finer than the cell it was described in — the description was on too large a cell and the machinery says so.

A net is a choice of what counts as a bond

A list of atomic positions does not contain a net. It contains distances, and somebody has to decide which of them are bonds — so the net is a fact about the cutoff as much as about the crystal, and moving the cutoff past a shell of neighbours changes the answer.

applied · Nets the kagome net: one of 1 mechanism. An infinitesimal mechanism of the kagome net, drawn as a velocity at every joint. The vector is an exact solution of the rigidity matrix — a set of joint velocities and a rate of change of the cell's metric under which no bar's length changes to first order — with the two rigid translations projected out so that what is left is a motion rather than a shift. Whether it continues into a finite motion is a separate question that a first-order calculation cannot answer, and this collection answers it for one framework by constructing the motion explicitly.

A fold that keeps its symmetry

The kagome framework has exactly one mechanism, and it does not stop at first order. Every triangle turns, alternate ones the other way, the cell shrinks to half its size, and not one bar changes length — and the count that found the mechanism cannot see how many there really are.

applied · Rigidity the kagome net: 34 of 144 wavevectors carry a mechanism. The zone of the kagome net, with a mark at every wavevector whose rigidity matrix drops rank — which is to say at every wavevector that carries a motion of the bars. There are few of them and they are isolated, so enlarging the cell adds mechanisms slowly. The ranks at the half-integer wavevectors are exact; the others are computed with a stated tolerance, because the matrix there has genuinely complex entries.

A mechanism that is a wave

The framework essays found the kagome net's mechanism count growing with the cell it was looked for in, and recorded it as a finding without an explanation. Here is the explanation: the motions lie along lines in reciprocal space, and a larger cell samples a line at more places.

applied · Rigidity The figure of merit a supercell always beats. One line list, indexed on the true cell and on five multiples of it. The mean discrepancy is not merely similar down the column, it is identical to every digit: the supercell's grid of allowed Q values contains the true cell's grid exactly, so each line lands on precisely the same place and misses by precisely the same amount. Any figure of merit built on agreement alone therefore returns one number for the whole family, and cannot prefer the true cell. What falls is the last column, and the only thing in it that the fit does not already contain is the count of lines the cell says should have been seen.

The figure of merit a supercell always beats

Indexing a powder pattern returns a ranked list rather than an answer, and the ranking needs a number. The obvious number — how well the cell accounts for the lines — is exactly the number a supercell cannot lose on, because the supercell's grid contains the true cell's grid and the discrepancies are identical to every digit. What has to be paid for is the lines nobody saw.

applied · Indexing The alias accounts for every line and predicts more. The observed lines above, and below them the grid of a supercell that explains all of them. The full ticks are the observed lines, which the alias reproduces exactly; the faint ones are lines the alias predicts and nobody saw. That second set is the only thing that separates the two cells, and it is why an indexing criterion has to charge for unobserved lines rather than measure agreement.

Every alias is a supercell

A cell that explains every line of a powder pattern is not a near miss and not a coincidence: its reciprocal grid contains the true one, which means its own cell is a superlattice of the true cell. So the ambiguity of indexing is the arithmetic of superlattices, and it can be counted — two cells with one unknown, sixteen with two, sixty-two with three, all of them accounting for the same twenty lines exactly.

applied · Indexing Every square-lattice superstructure, by the size of its cell. The number of distinct arrangements of two species on the square lattice whose primitive cell holds exactly n sites, for n from one to twelve, logarithmic, counted over every cell shape at once: every sublattice of index n up to the lattice's own symmetry, every colouring of it up to translations and the sublattice's stabiliser, and only those whose true period is that sublattice. Odd prime sizes are low, because only a few cell shapes exist; sizes with many divisors are high.

Every superstructure counted once

The 805 arrangements of two species on a four-by-four cell were never the structures with sixteen atoms: forty-nine of them repeat in a smaller cell, and every sixteen-atom structure whose cell is not a four-by-four square is missing. Counting structures rather than arrangements means counting over every shape of cell at once, and discarding every arrangement that repeats sooner. On the square lattice that gives 2, 2, 4, 11, 16, 40, 48, 148 structures for cells of one to eight atoms, and 22,413 with sixteen.

operations · Counting One arm of a star is cheap, and the whole star costs the cube. For nine wavevectors of the primitive cubic zone, the number of cells in the smallest periodic box that holds one arm of the wavevector's star, and in the smallest that holds every arm, on a logarithmic axis found by enumerating every box of each size. The zone corner R is held whole by two cells. The face centre X needs two cells for one arm and eight for its three; the thirds along an axis or a body diagonal need three for one arm and twenty-seven for the star; the quarters need four and up to sixty-four. Every smallest box for a whole star has the full symmetry of the cube.

A whole star needs a cubic box

A periodic box for a cubic crystal holds only the wavevectors its own lattice allows, and a box chosen to hold one wavevector usually holds one arm of its star and not the others. One arm of the face centre X fits in two cells; all three need eight. One arm of a third along an axis fits in three; the whole star needs twenty-seven. The smallest box that holds a whole star always keeps the cube's full symmetry, so a calculation either pays for the cube or reports a symmetry its box imposed.

classification · Flat space Whole stars of the hexagonal zone, and the boxes that hold them. For ten wavevectors of the primitive hexagonal zone, the number of cells in the smallest periodic box holding one arm of the star, in the smallest holding every arm, and, where it differs, in the smallest n by n grid of cells holding the star. The corner K is a single arm up to sign and is held whole by three cells, against nine for a grid; K raised to the top face, H, by six against eighteen; halfway to K, T, by twelve against thirty-six. M needs two cells for one arm and four for its three, and L two and eight. Every smallest box for a whole star keeps the full hexagonal symmetry.

The corner of a hexagon fits in three cells

A periodic box for a hexagonal crystal holds a whole star of wavevectors only if it keeps the crystal's full symmetry, as a cubic box must. What the cube never offered is a box turned through thirty degrees. It holds the corner of the hexagonal zone in three cells where every grid needs nine, and the saving is always exactly a factor of three and always at the same stars. Turn a box through any other angle that the six-fold axis allows and it loses the mirrors, and comes with a mirror image.

classification · Flat space Whole stars of the rhombohedral zone, and the boxes that hold them. For eight wavevectors of the rhombohedral zone, the number of cells in the smallest periodic box holding one arm of the star, in the smallest holding every arm, and, where it costs more, in the better of the two regular grids: the rhombohedral cell repeated n times along each edge, or the hexagonal triple cell repeated n by n by m. The axis end T is held whole in two cells against six, the face centre F in four against eight, and the in-plane third K₃ in nine against twenty-seven; L needs eight either way. Every smallest box for a whole star keeps all twelve operations of 3̅m.

A rhombohedral box saves a factor of two the hexagon never offered

A rhombohedral crystal is drawn on two cells, the small rhombohedron and the hexagonal cell three times its size, and a periodic box is usually one of them repeated. The smallest box that holds a whole star of wavevectors is the star's own lattice instead, and across every star with denominators up to eight it beats the better of the two grids by one, two or three, never more. Three was the hexagon's number. Two is new, and it is the face centre's. And the boxes that keep the three-fold axis and lose the mirrors still come in mirror pairs, now at nine cells as well.

classification · Flat space

Named alongside it

The objects these essays reach for when they reach for this one.

SublatticeWavevectorCommensurateBrillouin zoneBurnside lemmaFigure of meritIndexingMechanismRigid unit modeTorusBonding cutoffBravais lattice

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