Crystallography & Symmetry
patterns derived from their groups
Concept

Ordering — where it appears

The arrangement of two or more kinds of atom onto distinct sites of what was one crystallographic position. It requires the symmetry to fall far enough for the position to split, produces superlattice reflections, and is measured by an order parameter that vanishes above the transition.

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

Two species on one lattice, ordered at index 2. Every position is a lattice point of the parent and none of them has moved. What has changed is which atom sits where: the larger marks are a sublattice of index 2, the smaller ones its other 1 coset, and the outlined cell is the new repeat. The lattice of positions is untouched and the repeat of the contents is 2 times as large, which is the whole of what an ordering transition does and the reason its signature is in reciprocal space rather than in the positions.

The reflections a superlattice adds

Centring a lattice makes reflections vanish. Ordering two kinds of atom onto a sublattice makes new ones appear, exactly n − 1 of them per parent cell, and their intensity is a difference rather than a sum — which is why an ordered alloy of two neighbouring elements can be invisible to X-rays and obvious to neutrons.

lattices · Sublattices One orbit of p4m, two of p4. The general position of p4m — 8 points in a cell, all equivalent under that group — with each point coloured by which orbit of p4 it belongs to. Losing half the operations does not move a single point; it changes which of them are related, and the one orbit becomes 2. An atom sitting on this position in the parent becomes 2 crystallographically distinct atoms in the child, which may then be different elements, or move independently, or order.

What a position becomes on the way down

Cool a crystal through a transition and it loses operations. Nothing moves — and one crystallographic site becomes two, which is how an ordering transition finds somewhere to put a second kind of atom.

operations · Fundamental domain

Named alongside it

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

CosetOrbit-stabiliserReciprocal latticeSite symmetrySplitting schemeStructure factorSubgroup indexSublatticeSuperlatticeSuperstructureSystematic absenceWyckoff positions

All concepts