Crystallography & Symmetry
patterns derived from their groups
Concept

The Niggli cell — where it appears

The unique reduced cell an algorithm returns from any basis of a lattice, so two descriptions of one lattice give the same six numbers. It is the three-dimensional replacement for Gauss's reduction, and its termination is proved rather than drawn.

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

Reducing a basis. An awkward basis and the reduced one Gauss's algorithm returns. Both describe the same lattice — the change of basis has determinant one — and the reduced pair is the shortest vector together with the shortest independent of it, checked against an exhaustive search.

Reduction, and the shortest basis

Every lattice has infinitely many bases and no arithmetic picks a preferred one — until a rule is imposed. Reduction is that rule, it terminates in a handful of steps, and it is what lets a database decide whether two reported crystals are the same crystal.

lattices · Lattice 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

Named alongside it

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

Reduced cellBasis reductionBravais latticeGauss reductionGram matrixHolohedryLattice characterShortest vectorUnimodularUnimodular matrix

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