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 5 essays across 3 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 several have been drawn in the plane. 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 everything about it so far has been about that action. 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 Whole stars of the rhombohedral zone, and the boxes that hold them. For eight wavevectors of the rhombohedral zone, the number of cells in the smallest periodic box holding one arm of the star, in the smallest holding every arm, and, where it costs more, in the better of the two regular grids: the rhombohedral cell repeated n times along each edge, or the hexagonal triple cell repeated n by n by m. The axis end T is held whole in two cells against six, the face centre F in four against eight, and the in-plane third K₃ in nine against twenty-seven; L needs eight either way. Every smallest box for a whole star keeps all twelve operations of 3̅m.

A rhombohedral box saves a factor of two the hexagon never offered

A rhombohedral crystal is drawn on two cells, the small rhombohedron and the hexagonal cell three times its size, and a periodic box is usually one of them repeated. The smallest box that holds a whole star of wavevectors is the star's own lattice instead, and across every star with denominators up to eight it beats the better of the two grids by one, two or three, never more. Three was the hexagon's number. Two is new, and it is the face centre's. And the boxes that keep the three-fold axis and lose the mirrors still come in mirror pairs, now at nine cells as well.

classification · Flat space Which Laue groups have a sampling grid that touches no special point. Every Laue group on every Bravais lattice it occurs on, twenty-four pairs, with the smallest grid of wavevectors that keeps the whole group and has no point fixed by any operation but the identity. Such a grid exists for 1̅, 2/m, mmm and 4/m, at exactly as many points as the group has operations. It does not exist, as far as the search reaches, for 4/mmm or for any group with a three-fold axis: 3̅, 3̅m, 6/m, 6/mmm, m3̅ and m3̅m.

A three-fold axis puts a grid point on itself

A grid of wavevectors offset from the zone centre can keep a crystal's whole symmetry and touch no mirror, no axis and no point where time reversal folds the zone onto itself, and then its share of the zone is exactly one over the order of the group, with no boundary excess at all. Four Laue groups have such a grid, as small as the group itself. Every group with a three-fold axis has none, for a reason that fits in a line: the rotoinversion fixes one point of the plane across its axis, and above that point the three-fold rotation fixes the grid.

operations · Fundamental domain Every square-symmetric grid of wavevectors has a point on a mirror. One cell of the square reciprocal lattice, centred on the zone centre, with the mirror lines of 4mm through it and three grids the four-fold rotation keeps with an offset off the axis. The quarter turn allows that offset at one place only, the second four-fold centre of the grid's own lattice, ringed. On an aligned grid of odd side the rotation fixes a point; on an aligned grid of even side the offset lies on a diagonal mirror; on a grid turned by 45 degrees it lies on an axial mirror. No grid escapes all three.

The offset a quarter turn allows lies on a mirror

A census of sampling grids found free grids for the tetragonal group 4/m and none for 4/mmm, to a hundred and twenty-eight points, with no reason. The reason is two lines long. A quarter turn lets a symmetric grid be offset to exactly one place off its axis, the second four-fold centre of the grid's own lattice, and a square lattice that a mirror keeps can only be aligned with its mirrors or turned by half a right angle. Either way that centre lies on a mirror line.

operations · Fundamental domain

Named alongside it

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

Brillouin zoneWavevectorFundamental domainIrreducible wedgeLaue classOrbit-stabiliserTime reversalBloch stateCommensurateLittle groupMirrorReciprocal lattice

All concepts