Crystallography & Symmetry
patterns derived from their groups
Concept

Primitive cell — where it appears

A cell containing exactly one lattice point, which is the smallest a cell can be. Crystallography often prefers a larger centred cell anyway, because that cell buys operations that read as whole numbers along the axes.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

Two cells of equal area on one rhombic lattice. One lattice — the rhombic lattice that cm sits on — with two parallelograms drawn on it: the conventional cell, and a sheared cell whose edges are the integer combinations (1, 0) and (1, 1) of it. The points are identical in both outlines; only the description changes. Each cell's contents were counted by writing every lattice point in that cell's own coordinates and sharing each one out between the cells that meet at it — a quarter at a corner, a half on an edge, one inside — and the totals come to 1 and 1, which are the determinants of the two matrices. The alternative has determinant one, so its inverse is integral and it generates exactly the same lattice; that is the whole condition, and it is why a lattice has infinitely many bases and no arithmetic can prefer one.

The cell is a choice, the lattice is not

Every lattice has infinitely many unit cells and infinitely many bases, and crystallography picks one by convention. Knowing which convention is in force is the difference between a symbol that means something and a symbol that means nothing.

lattices · Lattice One lattice, two cells, and the absences the choice creates. The same set of points described on a centred rectangular cell and on its primitive rhombic cell, with what the centred description scatters. Half the reflections vanish, and they vanish because of how the cell was drawn rather than because of anything the crystal does.

Centring, and why cm is not pm

A centred cell has a lattice point in the middle and twice the area it needs, and crystallography prefers it anyway. The preference has a price, and the price is paid in reflections that vanish for reasons that have nothing to do with the crystal.

lattices · Centring Centring the five lattices. Each of the five plane lattices with the midpoint of every cell added, and the type of lattice that results — read off the reduced basis of the new point set rather than looked up. Every centring halves the cell area, so the original lattice is a sublattice of index two in the centred one, and every centred lattice is again one of the five. Two of the five come back as themselves and are therefore no richer for being centred. The rectangular and rhombic lattices exchange, which is what makes them one family under two descriptions. And the hexagonal lattice centred is rectangular — its holohedry falls from 12 to 4, so centring destroys the symmetry it was meant to display.

Centring, counted as a sublattice

Adding the centre of every cell to a lattice produces another lattice, containing the first with index two. Doing it to each of the five in turn shows why the list is five rather than ten, and why only one of the five has a centred description worth keeping.

lattices · Centring One lattice, two cells, and the absences the choice creates. The same set of points described on a centred rectangular cell and on its primitive rhombic cell, with what the centred description scatters. Half the reflections vanish, and they vanish because of how the cell was drawn rather than because of anything the crystal does.

Why the bigger cell wins

A centred cell has twice the area it needs and crystallography prefers it anyway. The preference is not conservatism — it buys operations that read as whole numbers along the axes, and the price is a set of reflections that vanish for reasons having nothing to do with the crystal.

lattices · Centring 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.

lattices · Wigner–Seitz cells 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

Named alongside it

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

CentringCentred latticeLattice typeOrigin choiceSystematic absenceUnit cellBasis reductionBonding cutoffBravais latticeBrillouin zoneConventionCoordination shell

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