Crystallography & Symmetry
patterns derived from their groups
Concept

Superlattice — where it appears

A lattice finer or coarser than a crystal's own, produced by ordering atoms onto a sublattice of it. It shows in diffraction as extra reflections whose intensity measures the ordering, and in the levels as a folding of the zone with a gap where the new reflections appear.

Named by 2 essays across one field — 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 A gap of exactly 1.00. The two folded bands with the ordering switched on. The faint curves are the same bands before it, crossing at the boundary of the reduced zone; the ordering couples them there and separates them by exactly twice its own strength. The gap appears at the wavevector where the superlattice's extra reflections appear, and for the same reason: both are the Fourier component of the potential at that wavevector.

A bigger cell, a smaller zone

Ordering two kinds of atom onto a sublattice adds reflections to the diffraction pattern and opens a gap in the levels. It is one fact told twice: the same Fourier component of the potential, at the same wavevector, doing the same thing.

lattices · Wigner–Seitz cells

Named alongside it

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

Reciprocal latticeBand gapBrillouin zoneCosetOrder parameterOrderingStructure factorSublatticeSuperstructureSystematic absenceZone folding

All concepts