The points a group treats differently

Put a point almost anywhere inside the cell of p4m and its orbit has eight members, because the group has eight operations and every one of them moves it somewhere new. Put it at the centre of the cell and the orbit has one member, because all eight operations leave it exactly where it is.

Between those two extremes there is nothing continuous. A point is fixed by some number of operations and that number divides eight; the orbit’s length is eight divided by it; and there is no arrangement in which a point is nearly on a mirror.

These are the Wyckoff positions, and they are the part of a group that a working crystallographer uses most often — because atoms sit at them, and because a structure’s chemical formula is read off their multiplicities.

The theorem, checked at every point #

The relation between the two numbers is the orbit–stabiliser theorem: the length of a point’s orbit, multiplied by the number of operations that fix it, is the order of the group.

The proof is three lines and it is not what this page is for. What the figure does instead is compute both numbers separately at every one of a hundred and forty-four sample points — the orbit by applying every operation and counting the distinct images, the stabiliser by applying every operation and counting the ones that come back to where they started — and assert that the product is the group’s order at each one.

That is not a proof and it is a good deal more than a restatement. A hundred and forty-four independent checks of a claim about two computed quantities would catch an error in either computation immediately, and the assertion runs every time the figure is drawn.

The sampling is exact. Special positions in a plane group have coordinates whose denominators divide twelve — halves from the mirrors and half turns, thirds from the threefold centres, quarters and sixths from the rest — so a grid of twelfths in exact rational arithmetic lands on every special position rather than near one. This is the decidability the whole site rests on doing something concrete: there is no question of a point being close enough to a mirror to count.

What the sizes on the figure mean #

Reading the p4m figure from small dots to large ones gives the group’s whole Wyckoff table.

Eighty of the hundred and forty-four are general. Nothing but the identity fixes them, so each has an orbit of eight. In a real structure an atom at a general position contributes eight atoms to the cell.

Sixty lie on a mirror. One reflection fixes each of them, so the orbit is four. The mirrors are lines, so this is much the largest special class — special positions on a line are common and special positions at a point are rare.

Two are fixed by everything. The corner of the cell and its centre are on four mirrors and a fourfold rotation at once, so all eight operations fix them and the orbit is one point.

The groups with no special positions #

Two of the seventeen have none at all, and asking which brings out something the classification does not otherwise say.

p1 and pg. Their operations, apart from the identity, are translations and glides — and neither has a fixed point. A translation moves every point of the plane. A glide reflects and then slides, so a point on the glide axis lands further along the axis rather than staying put. There is no point of the plane that any operation of these two groups leaves alone.

This is worth naming because it is exactly the property that makes them safe to illustrate. Every other group has positions where a motif placed carelessly acquires symmetry it was not meant to have — that is the comma rule, and its mechanism is the stabiliser being non-trivial. In p1 and pg there is nowhere to make that mistake.

The property has a name outside crystallography. A group acting on a space with no fixed points is said to act freely, and the plane groups that act freely are exactly the two whose quotients are surfaces rather than orbifolds — the torus and the Klein bottle. That is the same fact orbifold notation records as those two groups having no cone points and no mirror boundary.

The figure does not take this on trust either. Which groups have special positions is decided twice: once by the sampling, and once from the operations themselves — a rotation fixes its centre, a mirror fixes its axis, a glide and a translation fix nothing — and the two answers are required to agree. An earlier version of that check named p1, pg and pgg as the fixed-point-free groups. pgg has four half-turn centres per cell, the sampling found all four, and the assertion failed the moment it was run. A hardcoded list is exactly the thing this site’s figures are meant not to contain.

Counting the kinds, group by group #

How many kinds of special position a group has is a small invariant of it, and one worth comparing across the seventeen because it does not track anything else about them.

p2 has one kind — four half-turn centres, each fixed by two operations. p3 has one kind as well, three threefold centres, but each of those is fixed by three operations rather than two, and the orbit lengths differ accordingly. cmm has two kinds; p4m has three; p6m has four. The count rises with the number of different orders of element the group contains rather than with its size, which is why the sixfold groups are the richest: they hold rotations of order two, three and six at once, and a point can be fixed by any of them.

The extreme cases are the two with none, and the reason those two are also the two easiest to draw badly by accident is a coincidence worth noticing. Freedom to place a motif anywhere and the absence of any point worth marking are the same property, and a group with no special positions offers a pattern-maker no landmarks at all.

Where a structure’s formula comes from #

The reason crystallographers know these tables by heart is arithmetic about atoms.

A structure is described by listing which Wyckoff position each kind of atom occupies. Each occupied position contributes its own multiplicity — its orbit length — to the cell’s contents, so the formula per cell is a sum of multiplicities, and the formula of the compound follows by dividing through.

An atom on a special position therefore contributes fewer atoms than an atom at a general position, and the difference is not a small correction. In p4m an atom at a general position brings eight, an atom on a mirror brings four, and an atom at the fourfold centre brings one. Placing an atom in the wrong Wyckoff position gives a structure with the wrong composition, and the error shows up as a chemically impossible formula rather than as a bad fit.

There is a second consequence that is easy to miss. An atom at a special position has the symmetry of that position forced on it. A molecule sitting on a mirror must itself be symmetric about that mirror; a molecule on a threefold centre must have threefold symmetry of its own. So the Wyckoff position an atom occupies is a constraint on what can be there, and finding a molecule with no threefold symmetry apparently sitting on a threefold centre is a sign of disorder — the structure is an average over orientations rather than a single arrangement.

What “site symmetry” means when nothing is there #

The vocabulary here is worth separating carefully, because three words are used for closely related things and they are not interchangeable.

The stabiliser is the set of group operations fixing a given point. It is a subgroup of the group, and its order is what the figures above measure.

The site symmetry is the same subgroup regarded as a point group — the abstract type it is, without reference to where it sits. Two different threefold centres in p3 have different stabilisers as sets of operations and the same site symmetry.

The Wyckoff position is the whole orbit of such a point, taken together with its site symmetry, and named by a letter in the tables. It is a set of points rather than one point, which is why a structure “occupies” a Wyckoff position rather than sitting at one.

The distinction matters most in the case where nothing is there. A Wyckoff position exists whether or not any atom is at it: it is a property of the group and of the cell, computed before any structure is proposed. A crystallographer choosing where to put an atom is choosing from a list that was fixed by the symmetry alone, and that is the sense in which the space group constrains the structure rather than describing it.

What a site symmetry constrains, besides the count #

An atom’s Wyckoff position decides how many copies of it there are. It also decides what that atom is allowed to be, and the second consequence is used as often as the first.

Neumann’s principle applies to a site exactly as it applies to a whole crystal, with the site symmetry in place of the point group: any tensor describing the atom must be invariant under every operation fixing the atom’s position. The tensor a crystallographer meets daily is the atom’s displacement ellipsoid — the anisotropic parameters describing how far it moves about its mean position — and it is a symmetric rank-two tensor, so the counts this collection computes apply to it unchanged.

The consequences are immediate and are enforced by every refinement program. An atom at a general position has all six parameters free. An atom on a mirror must have one principal axis of its ellipsoid perpendicular to that mirror, which removes two. An atom at a site with a three-fold or four-fold axis has its ellipsoid a spheroid about that axis, leaving two parameters. And an atom at a site of cubic symmetry must be isotropic: one parameter, whatever the material.

So a special position buys a shorter formula and a shorter parameter list at once, and a structure reported with an unconstrained ellipsoid on a special position has been refined against a model its own symmetry forbids.

Special positions and the fundamental domain #

The two ideas are the same one seen from opposite sides, and putting them together explains a discrepancy the domain essay has to apologise for.

A fundamental domain contains one point from every orbit. General orbits have the full |G| points, so they contribute one sample in |G| to the domain. Special orbits are shorter, so their points are represented more often per point of the plane — and a special position, being its own whole orbit, is entirely inside the domain.

That is why a computed domain always comes out a little larger than one part in |G| of the cell. The excess is the boundary: mirrors and rotation centres, which are exactly the special positions, counted whole rather than shared. The finer the sampling grid the smaller the excess, because the boundary is one-dimensional and the interior is two-dimensional, but it never reaches zero.

Why the letters run from the bottom #

Crystallographic tables label Wyckoff positions with letters, and the convention is worth knowing because it is the reverse of what a reader expects.

The letter a goes to the position with the smallest multiplicity — the most special one, fixed by the most operations — and the letters run upwards through the special positions in decreasing site symmetry, with the general position last. So a group with four kinds of position has a, b, c special and d general, and the general position, which is where most atoms sit, is at the end of the list rather than the beginning.

The ordering makes the tables useful in a way an alphabetical-by-anything-else ordering would not. An atom is placed at the most special position compatible with the chemistry, so a structure is usually described by a few early letters and one late one, and the multiplicities add to the cell contents in that order.

It also produces a small piece of shorthand that appears everywhere in the literature. A structure written as “4a + 8f” states two positions and their multiplicities at once, and the total — twelve atoms per cell — is read off without consulting anything. That is why the multiplicity is written into the label rather than kept in a separate column.

This site computes the orders and the multiplicities and does not assign letters, because a letter is an index into a table rather than a property of the group, and reproducing the table’s ordering would mean reproducing the table.

Counting the cell’s contents #

The arithmetic that makes these tables load-bearing is worth doing once in full, because it is how a structure is stated in practice.

Suppose a plane structure in p4m has one atom of one kind at the fourfold point, one of another kind on a mirror, and one of a third at a general position. The multiplicities are one, four and eight, so the cell contains one, four and eight atoms of the three kinds — a formula of 1 : 4 : 8, or 1 : 4 : 8 divided by their common factor if there is one.

Now move the second atom off its mirror. Its multiplicity jumps from four to eight, the formula becomes 1 : 8 : 8, and the composition of the material has changed — not the arrangement of it, the composition. A structure proposal that puts an atom in the wrong Wyckoff position is not slightly wrong about geometry; it is wrong about how much of each element is present, which is checkable against an independent chemical analysis.

That is why the Wyckoff position is stated first in any structure description, before the coordinates. The coordinates refine; the position does not. And a structure whose formula from the Wyckoff multiplicities disagrees with the measured composition is rejected before anybody looks at the fit.

Where the exactness stops #

The grid is twelfths. Every special position of a plane group is caught by it, and that is a fact about plane groups rather than a property of the number twelve. In three dimensions the denominators go further, and a grid of twelfths would miss positions with eighths in them.

These are counts over samples, not areas. The figure reports how many grid points fall into each class, which is not the same as how much of the cell each class occupies — the mirrors are lines, of zero area, and they carry sixty of the hundred and forty-four samples. The classes are what matter; the counts are a property of the grid.

The site symmetry is computed, not named. Crystallographic tables give each Wyckoff position a letter and a point-group symbol — 4b, mm2, and so on. This site computes the order of the site symmetry and does not name the group it is, because naming it would need the same positional convention the notation index sets out and would add nothing the order does not already say for a plane group.

Where the ladder goes next #

The region these positions bound is the fundamental domain, and the reason its boundary is where it is.

The hazard that special positions create for anybody drawing a pattern is the motif must be a comma, where a dot at a special position turns one group into another.

The index arithmetic that relates a group’s domain to a subgroup’s — and the reason a subgroup needs a bigger piece to build from — is domains of a subgroup.

What the pictures here cannot show. The dots are drawn at sizes that encode a count, and a size is a poor way to read a number: a reader can see that some points are marked out and cannot see that a particular one is fixed by two operations rather than three. The counts in the table beside each figure are the measurement; the drawing shows where the special positions are, which is the other half of the same fact and the half a table cannot give.