Crystallography & Symmetry
patterns derived from their groups
Concept

Zone folding — where it appears

The relabelling of states when a cell is enlarged: the Brillouin zone shrinks, and wavevectors that were distinct become the same. No state is created or lost by folding, and a gap opens at the new boundary only when the enlargement is a real change to the crystal rather than a change of description.

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

The first 4 zones of the square lattice. Zones one to 4, each in its own shade. The n-th zone is the set of wavevectors with exactly n − 1 reciprocal lattice points nearer to them than the origin is, so the boundaries are the perpendicular bisectors and nothing else. The zones get further out and break into more pieces — 1, 4, 8, 12 fragments — and every one of them has the area of a single cell.

The zones above the first

The second Brillouin zone is a scattering of disconnected fragments in a different part of reciprocal space from the first, and it has exactly the same area. So does the third, and the seventh. The reason is that each of them is the first zone, cut up and moved.

lattices · Wigner–Seitz cells 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.

Brillouin zoneReciprocal latticeBand gapFermi surfaceOrder parameterPerpendicular bisectorRelevant vectorSuperlatticeUnit cellWigner seitz cell

All concepts