Crystallography & Symmetry
patterns derived from their groups
Concept

Orbit-stabiliser — where it appears

The identity that an orbit's size is the group's order divided by the order of the stabiliser of any of its points. It counts crystal forms, Wyckoff multiplicities, reflection multiplicities and stars of wavevectors, in four fields under four names.

Named by 5 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 Dropping one invariant of 3m makes two orbits agree. Every lattice point within four cells of the origin, coloured by the values a proper subset of 3m's invariants takes on it — the 2 generators with the first one removed, over a window of 4 cells. With the full set, the 25 orbits of the group take 25 distinct sets of values, one each, so the invariants are a complete set of coordinates on the quotient. With one removed, the two circled points — in different orbits, so no operation of the group carries one to the other — take the same values and become indistinguishable. That is the whole content of the statement that a complete set of invariants separates orbits: the completeness is what is doing the work.

An orbit is what the invariants cannot tell apart

Two points of the plane lie in the same orbit of a group exactly when every invariant polynomial takes the same value on both. One direction of that is a definition; the other is a theorem, and it is checked here by comparing every pair of points in a window both ways.

operations · Counting p4m: which distortions the atoms of each position can make. Every Wyckoff position of p4m, with the number of independent displacement patterns of each symmetry its atoms supply. A zero is a distortion those atoms cannot make however their amplitudes are chosen — an atom pinned at a rotation centre cannot move in a way that keeps less symmetry than the centre has. The row sums, weighted by the dimensions along the top, come to twice the number of atoms in the cell, which is the check that nothing has been lost. The last column is the number of free coordinates the position has, and it equals the multiplicity of the identity representation in that row: a displacement that keeps every symmetry is exactly a move of the position within its own Wyckoff set.

Which modes a site can carry

An atom on a mirror cannot move in a way that breaks the mirror while its images move with it: the displacements of a Wyckoff orbit carry a representation, and some of its pieces have multiplicity zero. The count of those pieces is a character, and the one that breaks nothing is the position's own freedom.

operations · Fundamental domain 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 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 zoneSite symmetrySpecial positionStar of kWavevectorWyckoff positionsBloch stateCharacterDisplacement modeFree parameterFundamental domainInvariant polynomial

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