Crystallography & Symmetry
patterns derived from their groups
Concept

Wyckoff positions — where it appears

A class of positions sharing a site symmetry, listed with its multiplicity in the International Tables. The multiplicities are what turn a structure's occupied positions into a chemical formula.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

Special positions in p4m. Every point of a 12×12 grid inside the cell of p4m, drawn at a size set by how many operations fix it. 80 of the 144 are general — nothing but the identity leaves them alone, so their orbit is the full 8 points. The other 64 are special, and fall into 3 kinds: 60 points fixed by 2 operations, with orbits of 4; 2 points fixed by 4 operations, with orbits of 2; 2 points fixed by 8 operations, with orbits of 1.

The points a group treats differently

Almost every point of a cell has an orbit as long as the group. The exceptions are the points some operation leaves alone, and they are where atoms sit, where a structure's formula comes from, and where a careless motif destroys the group it was meant to illustrate.

operations · Fundamental domain The general positions of P2₁/c. Space group P2₁/c, number 14, projected down c on a primitive monoclinic cell. 12 general positions, the orbit of a three-point asymmetric motif, each labelled with its height along c and marked with a comma where the operation that produced it reversed handedness.

One part in however many, and why it is never quite that

A crystal's contents are the asymmetric unit repeated by the group. The unit's volume is the cell's divided by the order of the group — except that it is always a little more, and the excess is exactly the special positions counted whole.

operations · Fundamental domain orders 5 and 7 reach a site of symmetry 1 and no more. A molecule whose only symmetry is one n-fold axis, and the highest site symmetry it may occupy in any of the 45 space groups this site builds. The site's symmetry has to be a subgroup of the molecule's, so the site's order must divide n and the site group must be cyclic. Orders 1, 2, 3, 4 and 6 reach a site of their own order. Orders 5 and 7 reach one, because no site symmetry in any space group contains an operation of order five or seven — the orders available are 1, 2, 3, 4, 6, computed by asking every operation of every group whether it moves a point. A five-fold molecule keeps its axis; the crystal simply has no use for it.

What a molecule gives up to sit in a crystal

A molecule brings its own symmetry. A crystal offers sites with symmetries of their own, and the two have to be compatible — the site's symmetry must be a subgroup of the molecule's. So a molecule may always keep more than its site offers, and a molecule with a five-fold axis may sit only where the crystal offers nothing at all.

restriction · Local symmetry 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 Ten chains, two phases. The Zak phase of the lower band of a two-site chain, as the ratio of the two hoppings is swept. Every value is exactly zero or exactly π and nothing lies between them, because the chain has an inversion centre and inversion maps the zone loop to itself reversed — which forces the phase to equal its own negative modulo a full turn. The switch happens where the two hoppings are equal, which is the one place the band gap closes and the phase belongs to no band.

The phase a symmetry turns into a number

Carry a band's state once across the Brillouin zone and it returns with a phase. In a chain with an inversion centre that phase is exactly zero or exactly π and never anything else — and the two values turn out to be the two positions in the cell that an inversion centre fixes. Remove the centre and the phase moves continuously, which is what a quantisation claim has to be able to lose.

space-groups · K symmetry Seventy-two positions and the sets they fall into. Every plane group with the number of its Wyckoff positions, and the number of sets those positions fall into when the positions a normaliser exchanges are counted once: on the cell the group requires, on the most symmetric cell it may sit on, and under every change of basis carrying the group onto itself. 72 positions become 53 sets on the required cells and 51 on the best ones, and the last column never goes lower. The two groups a special cell changes are pmm and cmm.

The same site under two names

A structure report puts each atom on a Wyckoff position, and two correct reports of one crystal can name different positions. The positions that can trade places are exactly the ones the normaliser exchanges — and which those are depends on the cell as measured, not only on the group.

operations · Normalisers p2's symmetries, sorted into classes by the group itself. A pattern with the symmetry of p2 over 2 by 2 cells, with its rotation centres and mirror lines marked in the International Tables' shapes and coloured by conjugacy class in the infinite group: two marks share a colour exactly when some operation of the group carries one element onto the other. Where rotations of several orders share a centre, the mark is the highest order's and so is its colour. Glides are not drawn. Classes counted: half-turns: 1 in the quotient, 4 in the group.

Two mirrors a coset cannot tell apart

Taken modulo its lattice a wallpaper group is finite, and its conjugacy classes are easy to list. But a coset holds every mirror of one direction at once, and the group itself keeps apart mirrors the list merges: pm has two classes of mirror, p2 four classes of half-turn, p3 six classes of rotation. Deciding which is which is Dehn's conjugacy problem, and for these groups it comes down to whether one vector lies in one lattice.

operations · What symmetry is

Named alongside it

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

Site symmetryStabiliserMultiplicityOrbitSpecial positionConjugationOrbit-stabiliserAsymmetric unitBand structureBrillouin zoneCharacterConjugacy class

All concepts