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

A crystal structure is published as a short list of atomic coordinates and a space-group symbol. The list is usually a fraction of the atoms in the cell, and the fraction is one over the order of the group.

That convention is so routine that it is easy to miss what it assumes. It assumes the group is exactly right — because everything not published is regenerated by applying the group’s operations, and an operation that is not really a symmetry of the crystal will put atoms where there are none.

That is the asymmetric unit: the smallest piece of the cell from which the group reconstructs everything.

The arithmetic #

Orbit–stabiliser, which this site has already used in the plane: the length of a point’s orbit times the number of operations that fix it equals the order of the group.

A point in general position is fixed by nothing but the identity, so its orbit has the full order of the group. In P2₁/c the group has four operations modulo the lattice, so a general atom brings three companions and there are four of it per cell.

A point on a special position is fixed by something, so its orbit is shorter by exactly that factor. In P2₁/c the only special positions are the inversion centres, each fixed by two operations, so an atom there has an orbit of two.

The multiplicities of a group are therefore the divisors of its order that are actually realised, and which are realised is a property of the group rather than of the number.

What it costs a molecule #

The consequence is a constraint on what can crystallise where, and it is sharp.

An atom at a general position contributes four to the cell in P2₁/c. So does every other atom of the same molecule, so a molecule at a general position contributes four molecules per cell. If a structure is to have two molecules per cell instead, each must sit on a special position — which in P2₁/c means each must sit on an inversion centre, which means the molecule itself must be centrosymmetric.

Most molecules are not. So most P2₁/c structures have four molecules per cell, or eight, or twelve, and two is a reportable observation about the molecule rather than about the crystal.

Groups that act freely like this are the useful ones for chiral molecules, and P2₁2₁2₁ is the commonest space group for them. The plane has two groups that act freely, p1 and pg, found by sampling every point of a cell and asking what fixed it; space has more, and the reason is that space has screws.

The word “multiplicity” has two jobs #

A small terminological hazard, because the word is used for two related things and the difference matters when reading a Wyckoff table.

The multiplicity of a position is the length of the orbit of a point there: four for a general position in P2₁/c, two for an inversion centre. That is the number in the left-hand column of a Wyckoff table and it is what this essay has been calling multiplicity throughout.

The multiplicity of a reflection is something else entirely — how many symmetry-equivalent reflections share a spacing — and it turns up in powder diffraction, where those equivalents overlap and their intensities add.

The two are related, in that both count orbits under a group, and they count orbits of different things in different spaces. A crystallographer switching between a structure paper and a powder pattern uses both words within a page of each other, and the context is the only thing distinguishing them.

The excess #

Here is the part that is more interesting than the division.

The asymmetric unit’s volume ought to be the cell’s divided by the group’s order. Computed honestly, it is always a little larger, and the excess is not an error.

The reason is that the special positions have to go somewhere. A point on a mirror plane has an orbit of half the group’s order, so if the asymmetric unit is to contain exactly one representative of every orbit, it must contain the whole of every special position that lies on its boundary — and those boundaries are shared between the domain and its neighbours.

Convention resolves it by assigning boundary points to one side, and the assignment is arbitrary. What is not arbitrary is that a sampled count — put a point at every position of a fine grid and count how many orbits there are — comes out slightly above cell volume over group order, and the excess is exactly the measure of the special positions counted whole rather than shared.

This site found that in the plane and it was the finding that made the fundamental-domain essay worth writing: the domain is always a little more than one part in |G|, and the excess is the special positions. The same holds in space, for the same reason, and the special positions are now surfaces rather than lines.

Which groups have which multiplicities #

The available multiplicities are a property of the group and they are worth seeing side by side, because the pattern is not the one intuition suggests.

A group of order four can in principle offer multiplicities of 4, 2 and 1. Whether it offers any below 4 depends entirely on whether anything in it has a fixed point.

P2₁/c, order four: multiplicities 4 and 2. The inversion centres give the 2; nothing gives a 1, because no point is fixed by all four operations.

P222, order four: multiplicities 4, 2 and 1. The three perpendicular rotation axes give 2s along each of them, and where all three cross, a point fixed by everything gives a 1.

P2₁2₁2₁, order four: multiplicity 4 only. Three screws, no fixed points anywhere.

Three groups of the same order and three different answers, decided by what kind of operations they contain rather than by how many.

What “asymmetric” means, and does not #

The word is a slight misnomer and the misreading it invites is common enough to name.

The asymmetric unit is not a region with no symmetry. It is a region that contains exactly one point from each orbit — a fundamental domain for the group’s action — and its shape is chosen for convenience rather than derived. The Tables give one for each group, as a box or a wedge with stated boundaries, and any region with the same property would do.

So two crystallographic programs can produce different asymmetric units for the same group and both be right, and a structure “moved into the asymmetric unit” has had its coordinates changed by the application of a group operation, which changes nothing about the structure.

Z and Z′ #

The published notation for all of this is two numbers and it is worth decoding, because they are the arithmetic above written down.

Z is the number of formula units in the unit cell. Z′ is the number of formula units in the asymmetric unit — which is Z divided by the order of the group, and it is usually 1.

Z′ = 1 means one molecule in the asymmetric unit, at a general position, and Z equals the group’s order.

Z′ = ½ means the molecule sits on a special position that halves its orbit — on an inversion centre, or a two-fold axis, or a mirror — and the crystal contains half a molecule per asymmetric unit, which is a sentence that alarms people and is only arithmetic. What it means is that the molecule has that symmetry itself and the group is using it.

Z′ = 2 means two crystallographically independent molecules, both at general positions, which are chemically identical and geometrically different — two conformations, or two environments, coexisting in one crystal. That is a real and interesting observation about a structure and it is reported for that reason.

The plane’s version, and the number it produced #

This is the fourth rung on an anchor whose first three are in two dimensions, so the comparison is available and worth making.

The Wyckoff essay sampled an exact twelfth-grid across a plane group’s cell, computed each point’s orbit and stabiliser separately, and required their product to equal the group’s order at every one of a hundred and forty-four points. It found that in p4m, eighty of the hundred and forty-four are general, sixty lie on a mirror and two are fixed by everything — and the shape of that distribution is the useful finding: special positions on a line are common and special positions at a point are rare.

The same holds in space with the dimensions shifted up. A mirror plane is a two-dimensional sheet of special positions and there are a lot of points on it; a rotation axis is a line; a point fixed by the whole group is one point. So the multiplicity distribution in any group with mirrors is dominated by the mirror positions, and the fully-fixed positions are always a handful.

The consequence for chemistry is the one worth carrying. A molecule that is going to economise by sitting on a special position will almost always be sitting on a mirror or an inversion centre, because those are the positions that exist in quantity and because a planar molecule and a centrosymmetric molecule are both common. Sitting at a three-fold or four-fold axis requires the molecule to have that symmetry, which is much rarer.

Why the machinery cannot decide the interesting question #

Everything above is arithmetic and the arithmetic is exact. What it cannot do is say where a molecule will sit.

Whether a molecule takes a special position depends on whether it has the symmetry the position requires — that much is decidable, and it is a question about the molecule. But a molecule with an inversion centre does not have to sit on one; plenty of centrosymmetric molecules crystallise at general positions, because packing them that way is denser or because a hydrogen bond wants a particular geometry.

So the group says what is available and the chemistry decides what happens, and the availability is a constraint rather than a prediction. That is the same shape of statement as everywhere else in this field: symmetry rules things out and never rules anything in.

What a published structure actually contains #

Putting the pieces together gives the anatomy of a structure report, and every line of it is one of the quantities above.

The space-group symbol fixes the operations, so it fixes the multiplicities available.

The cell parameters fix the volume, which with the density fixes how much material is in the cell.

The coordinate list is the asymmetric unit — one representative of each orbit — and its length is the number of independent atoms.

Z and Z′ relate the three. Z′ times the group’s order is Z; Z times the formula mass over the cell volume is the density, and the calculated density against the measured one is the oldest sanity check in the subject.

Everything the reader is not given — the other three-quarters of the atoms in a P2₁/c cell — is regenerated from the symbol by applying the group. That is the whole practical point of a space group as a working tool rather than as a classification: it is a compression scheme with a lossless expansion, and it compresses by the order of the group.

Why the plane’s answer was more surprising #

The two-dimensional version of this anchor produced a finding that has no equally striking counterpart here, and the difference is instructive.

In the plane, the Wyckoff essay asked which groups have no special position anywhere and found exactly two of the seventeen: p1 and pg. Two out of seventeen is a striking answer — almost every plane group offers somewhere for a symmetric motif to economise, and only the two that act freely do not.

In space the same question has a much less dramatic answer, because screws are common and screws fix nothing. Every group whose non-translational operations are all screws or glides acts freely, and there are a lot of them: P2₁, Pc, P2₁2₁2₁, Pca2₁, Pna2₁, P4₁, Cc, and many more.

So the plane’s “only two” becomes space’s “quite a few”, and the reason is the arrival of the screw axis — which is the same reason seventeen becomes two hundred and thirty and the same reason the classification is thirteen times larger. An operation with no fixed point changes what a group can offer, and space has eleven kinds of them where the plane has one.

The box the Tables print, and where it is not a box #

The International Tables give an asymmetric unit for every group as a set of inequalities on the fractional coordinates — usually a rectangular block, sometimes with a cut through it — and the choice deserves a paragraph, because a block is not what the arithmetic produces.

A genuine fundamental domain is what the operations leave: the region nearer to the origin than to any of its images, or any other region containing one point of each orbit. For most groups that region is not a box. A three-fold axis produces a wedge; a diagonal mirror cuts a corner; the natural domain of a hexagonal group is a triangle.

The Tables print a box anyway wherever one exists, and where it does not they print a box with extra conditions attached — a further inequality slicing a corner off. The reason is entirely practical: a program deciding whether a coordinate triple lies in the asymmetric unit does it by comparing three numbers against bounds, and a wedge costs a more elaborate test for no benefit.

And the choice is invisible in the result, which is what makes it acceptable. Any region containing one point of each orbit generates the same crystal, so two programs using different conventions produce different coordinate lists and the same structure — which is the essay’s opening point about asymmetric being a misnomer, arriving as a fact about software.

Why a listed molecule is not the asymmetric unit #

There is a practical consequence of all this that catches every reader of a structure file at least once, and it is a place where the convention and the arithmetic deliberately disagree.

An asymmetric unit is a region, and moving an atom into it means applying whichever operation carries it there. Do that atom by atom to a molecule and the result is a set of atoms scattered across the cell: one here, one a translation away, one reflected — chemically the same molecule, geometrically in pieces.

So a deposited structure does not list the asymmetric unit. It lists a connected molecule, assembled by applying whatever operations are needed to bring the fragments together, and the coordinates in it may lie well outside any region the Tables would call an asymmetric unit.

The two conventions serve different readers. A symmetry calculation wants one point per orbit and does not care whether they are near each other. A chemist wants a molecule with bonds in it and does not care where in the cell it sits. Both are one point per orbit — the second is the first with operations applied — so nothing has been added or lost, and the file’s coordinates are a choice from an orbit rather than a canonical value.

The failure this causes is a comparison. Two determinations of one structure, listed by two programs with different conventions, have coordinate lists that agree in nothing. Comparing them means reducing both to a canonical description first — and that reduction has to allow for the operations that assembled the molecule as well as for the normaliser’s own freedoms.

Where the exactness stops #

Three limits.

The multiplicities are exact and the volume is not. Orbit lengths are integers computed by applying every operation and counting distinct images, with no tolerance anywhere. The asymmetric unit’s volume is a sampled quantity, converging to cell volume over group order plus a boundary term, and the boundary term is a convention.

The count is per conventional cell. In a centred group the conventional cell holds several primitive cells, so its multiplicities are correspondingly larger — C2/c has eight operations modulo the conventional cell and four modulo the primitive one, and Z counts against the conventional. Comparing Z between a centred group and a primitive one without noticing is a routine error.

This says nothing about site symmetry as a spectroscopist means it. The local environment of an atom, the point group of its site and what that does to a spectrum are questions with a molecular counterpart, and this arithmetic touches only the periodic part: how many of the thing there are, and what fraction of the cell has to be given.