Crystallography & Symmetry
patterns derived from their groups
Concept

Lattice character — where it appears

A classification of lattices finer than the fourteen, distinguishing them by the shape of the cell that needs no convention. It separates lattices the fourteen types put together, and it is what a database uses to decide whether two entries are the same lattice.

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

Six integers that do not depend on the description. The same monoclinic lattice written in 4 different bases, each obtained from the last by an integer matrix of determinant one, and each reduced by Niggli's algorithm. Every one of them gives the same six integers — the squared lengths and twice the dot products of the reduced basis. That is what makes the reduced form a fingerprint of the lattice: two cells with no number in common are the same lattice exactly when their reduced forms agree, and the comparison has no tolerance in it.

The cell that settles the argument

Two determinations of one compound can report cells that share no number and describe the same lattice. Reduction is the procedure that decides — six integers that depend on the lattice and not on anybody's choice of axes, and that agree exactly when the lattices do.

lattices · Lattice 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

Named alongside it

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

Bravais latticeBrillouin zoneThe Euler characteristicGauss reductionGram matrixHolohedryMinkowski boundThe Niggli cellParallelohedronPrimitive cellReduced cellRelevant vector

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