Crystallography & Symmetry
patterns derived from their groups
Concept

Star of k — where it appears

The orbit of a wavevector under a crystal's point group, taken modulo the reciprocal lattice. Its members are indistinguishable to any symmetric operator, so a calculation visits one of them and the rest are copies.

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

p4m: 10 of 36 wavevectors have to be visited. The Brillouin zone of the square lattice with a grid of 36 wavevectors on it, of which 10 are drawn solid: one per star, which is everything a calculation over a p4m-symmetric operator has to visit. The share is 27.8 per cent against the 12.5 per cent that the order of the point group would give, and it is larger for a reason worth naming — the wavevectors on the boundary of the wedge have short stars, so they are over-counted by any argument that only divides by the group order. The identity that is checked is that the star sizes add to the whole grid.

The domain in reciprocal space

A fundamental domain is the piece of a pattern the group repeats, and this collection has drawn several. The same idea in reciprocal space is what makes a calculation over a crystal affordable — and its share of the zone is larger than one part in the group's order, for a reason worth measuring.

operations · Fundamental domain p4: (1/2, 0) has a star of 2. The first Brillouin zone of the square lattice, with the reciprocal lattice points at its corners and centre, and the whole star of the wavevector (1/2, 0) under p4. The star has 2 members and the little group — the operations that leave the wavevector where it is, modulo the reciprocal lattice — has order 2. The two multiply to the order of the point group, which is the orbit–stabiliser theorem and is checked rather than displayed. Each member is drawn at whichever of its equivalent copies lies nearest the origin, because that is where a reader expects a wavevector to be.

The star of a wavevector

A plane group acts on the plane, and this collection has spent two hundred essays watching it. It also acts on the reciprocal lattice, where the action is different in one decisive way: the translations move no wavevector at all, and come back instead as a phase.

space-groups · K symmetry

Named alongside it

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

Brillouin zoneOrbit-stabiliserWavevectorBloch stateFundamental domainIrreducible wedgeLittle groupReciprocal latticeSpecial position

All concepts