Crystallography & Symmetry
patterns derived from their groups
Generator

The first 4 zones of the square lattice

The first 4 zones of the square lattice
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.

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.

2 essays call zone-stack. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about. Every one of this site's 393 essays names its parameters at the call site, which the standard pass of 2026-08-09 established and param-floor holds.

Where it is called

Changing this generator changes every one of these figures.

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. Lattices

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.

 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. Lattices

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.

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