One matrix, four rules

Six ways to name one group establishes that Pnma and Pbnm are one group with the axes labelled differently, and ends on the operation that follows from it: given a structure written in one setting, produce the same structure in another. Every structure database performs it on every entry, and it is one of the standard places for a paper to be wrong.

The operation is a single change of basis P together with an origin shift p. What makes it error-prone is not the matrix. It is that the four things a structure report contains — the cell, the coordinates, the indices and the symmetry operations — obey four different rules under it, and two of the four look like each other’s inverses.

Why the rules differ #

The basis is the primitive thing, and everything else is defined against it. If the new basis vectors are combinations of the old ones — that is what a change of setting is — then in matrix form $(\mathbf{a}'\,\mathbf{b}'\,\mathbf{c}') = (\mathbf{a}\,\mathbf{b}\,\mathbf{c})\,P$, with the columns of P holding the new vectors’ components.

A point in the crystal does not move when the axes are relabelled, so its coordinates must change in the opposite direction: the same vector written in a bigger basis has smaller coordinates. That gives $x' = P^{-1}(x - p)$, with the origin shift subtracted first because a coordinate is measured from the origin and the origin has moved.

An index is not a coordinate. A reflection is labelled by which planes it comes from, and those planes are defined by how they cut the axes, so hkl follows the axes rather than opposing them: $(h'\,k'\,l') = (h\,k\,l)\,P$, exactly like the basis. Coordinates and indices transform by inverse rules, which is the mistake waiting to happen — the two sit in adjacent columns of the same table, both look like triples of numbers, and only one of them is a vector in the basis.

An operation has to be transformed so that it does to the new coordinates what it used to do to the old. That gives conjugation for the matrix part, $W' = P^{-1} W P$, which is the same test of sameness any change of description uses — and for the translation part it gives a term with no analogue anywhere else.

The term an origin shift adds to everything #

Write the operation as x ↦ Wx + w. In the new coordinates its translation part is

$$w' = P^{-1}\bigl(w + (W - I)\,p\bigr).$$

The second term is zero when $W$ is the identity, and not zero otherwise. So moving the origin leaves every translation alone and changes the translation part of every rotation, screw, mirror and glide in the list.

This is why two origins for one group is a practical problem rather than a bookkeeping one. A group tabulated at a centre of symmetry and the same group tabulated at the point of highest site symmetry differ by a shift, and the shift rewrites every screw and glide component in the table. A reader who copies operations from one origin and coordinates from the other gets a structure in which nothing sits where the symmetry says it should — and the operations still form a group, so nothing about the list looks wrong.

The same term explains a fact the tables state without explaining: the intrinsic part of a screw or glide is what survives every change of origin. Splitting w into the part along the axis and the part across it, the $(W - I)p$ term can only change the part across, because $(W - I)p$ is always perpendicular to the axis of $W$. A screw’s pitch is a property of the operation; where its axis sits is a property of the description.

What a measurement does under a change of setting #

A setting is a way of writing a crystal down, and a measurement is not. So every measurable quantity has to be unchanged, and the arithmetic has to show it.

The structure factor is a sum of $\exp(2\pi i\,h\cdot x)$ over the atoms, and the two rules conspire exactly: $h' \cdot x' = (hP)\cdot(P^{-1}(x-p)) = h\cdot x - h\cdot p$. The dependence on P cancels — which is the sense in which the pair of rules is right — and what is left is the origin shift, appearing as a phase and not as an amplitude. |F| is invariant and the phase is not, which is the same asymmetry the phase problem is named after, arriving here as a triviality rather than as an obstacle.

Three of the six sample reflections are absences: |F| is zero, and the phase of zero is not a number. Comparing phases there is comparing two pieces of rounding, and an early version of this computation did exactly that and reported a phase error of 1.87 radians on a reflection that does not exist. A quantity that is undefined has to be excluded from a check rather than tested loosely, which is a rule about checks and not about crystallography.

The fifth thing, which is not in the report #

A report gives the cell as six numbers — three lengths and three angles — rather than as three vectors, and those six are the metric: the dot products of the basis vectors with each other. The metric’s rule follows from the basis’s, with P on both sides,

$$G' = P^{\mathsf{T}} G P,$$

and the reciprocal metric, its inverse, is what turns a reflection’s indices into the spacing of the planes it comes from: $1/d^2 = h\,G^{*}h^{\mathsf{T}}$.

That gives a fourth quantity to check and it is the sharpest of them, because a d-spacing is a length a diffractometer measures directly. Computed on a cell of 7 by 9 by 11 with the indices transformed by P and the metric by $P^{\mathsf{T}}GP$, every spacing comes back exactly — a difference of zero, not of one part in 10¹⁵, because the arithmetic is done on rationals and the two routes to $1/d^2$ reduce to the same fraction.

The cell was chosen with no two edges equal on purpose. A cubic cell has a metric that is a multiple of the identity and a great many wrong transformations leave it alone; with three different edges, a rule applied with P where it wanted $P^{-1}$ changes the spacings and is caught. A test on a symmetric example is a weaker test, and this is the second place in this essay where that turns out to matter.

Which mistake is caught by which check #

There are four natural ways to get this wrong, and the useful question is not whether they are wrong but what notices.

Closure is the check everyone reaches for, and it is useless here. Transforming coordinates with P instead of its inverse does not touch the operations at all; dropping the origin term gives a group at the wrong origin; conjugating the wrong way round gives a group conjugated by something else. All three are groups of eight operations, closed, with the right census of screws and glides. A procedure that transforms a structure and then checks that the operations still form a group has checked nothing that could have failed.

What catches them is comparing two routes to the same object: take the orbit of a point and transform every member of it, then transform the point and grow its orbit under the transformed operations. Those must agree as sets, and they disagree when the coordinates are transformed by the wrong rule, and when the origin term is missing. The indices are caught by the third check — the intensities move, which is the one thing a setting may never do.

A wrong rule that gives the right answer #

The mistake that survives is conjugating the operations the other way round: $P W P^{-1}$ instead of $P^{-1} W P$. On a shear it is caught by the orbit; on the ordinary axis swap — a quarter-turn about b — nothing catches it, and the reason is worth stating because it is a lesson about testing rather than about settings.

The two conjugations differ by conjugation by $P^2$. For the axis swap, $P^2$ is a two-fold rotation about b, and a two-fold rotation about b is already one of the group’s own operations up to translation, so conjugating by it maps the group onto itself. The wrong rule and the right rule then give the same set of operations, and no check can separate them because there is nothing to separate.

So the change of setting a test uses decides which mistakes the test can find. An axis swap is the first example anyone reaches for, it is what the literature’s worked examples use, and it is blind to one of the four errors. The shear — replace c by a + c, move the origin a quarter along a — is not blind to any of them, which is why every number in this essay is computed against it as well.

One operation, carried across #

The arithmetic is worth doing once in full, on the operation where the origin term does the most visible work.

Pnma has a centre of inversion, $W = -I$ with $w = (0, 0, 0)$: the point group’s inversion sitting at the origin. Move the origin to (¼, 0, 0) and leave the axes alone, so that P is the identity and only p is not. The rule gives

$$w' = P^{-1}\bigl(w + (W - I)p\bigr) = (-I - I)\,(¼, 0, 0) = (-½, 0, 0),$$

which reduced into the cell is (½, 0, 0). The operation x ↦ −x has become x ↦ −x + (½, 0, 0), and that is right: a centre of symmetry that sat at the old origin sits at −¼ along a in the new coordinates, and an inversion through the point c is written x ↦ −x + 2c. The 2c is the ½ that appeared.

Do the same for the screw with w = (0, ½, 0) and its translation part becomes (½, ½, 0): the axis has moved across itself by a quarter of a cell, which is what a quarter-cell shift of the origin does to an axis that is not at the origin. And for a pure translation, $W - I$ is zero and nothing happens at all, which is why the lattice looks unchanged while every other line of the table has moved. An origin shift is not a small change applied uniformly; it is a change that leaves the translations alone and rewrites everything else.

The test that cannot fail #

There is one more way to check a transformation, it is the first one anybody writes, and it is worthless. Transform the structure into the new setting, transform it back, and compare the result with the structure that went in.

A round trip returns the original whatever rule was used, as long as the same rule was used both ways. Carried out on a point with the coordinates rule inverted — P where the rule wants $P^{-1}$ — the forward pass gives (0.15, 0.35, 0.55) → (−0.10, 0.35, 0.55) where the right rule gives (−0.65, 0.35, 0.55), so after one pass the two disagree plainly; and after the return pass both are back at (0.15, 0.35, 0.55) exactly. The error is undone by its own inverse.

So the round trip joins closure on the list of checks that cannot fail here, and for the same underlying reason: both ask whether the result is self-consistent, and a consistently applied wrong rule is self-consistent. What catches a wrong rule is a second route to the same answer — the orbit grown two ways, or a quantity a measurement fixes, like |F| and the d-spacing. A check needs an external witness, not an internal one.

That is the general shape of a test worth running, and it is worth naming here because the setting transformation is where it is easiest to get wrong. A figure that draws a pattern and then reads its symmetry back is a second route; a figure that draws a pattern and checks that it drew what it was told to draw is a round trip. The first can fail and has; the second cannot fail and never has. Between two procedures that both report success, the useful question is which of them was ever in a position to report anything else.

What the rules are worth to a database #

The practical form of all this is a single-line service: a structure in, a structure in the target setting out. One crystal and sixteen coordinate lists is the reason it is needed — the same crystal has many correct descriptions, and a comparison of two entries has to bring them to one description first. The same site under two names is the reason the answer is not unique even then, since the normaliser leaves a set of descriptions that are all equally standard.

The transformation is also where the International Tables put their care. Volume A gives the transformation of coordinates, of indices, of the metric and of the operations as four separate formulas with the (P, p) pair defined once — not because the reader is careless but because they are genuinely four rules, and writing them as one invites the reader to apply one of them four times.

Every group and change tried here passes all three checks: five space groups across three changes of setting, with the transformed operations closed, the orbits agreeing and |F| unchanged to one part in 10¹⁵.

The checks, and what they refuse #

The last two are the ones worth having. A check suite that only demonstrates success is a check suite that has never been tested, and the two negative results — closure catches nothing, an axis swap hides a mistake — are what say which of these tests is doing work.

Who wrote the rules down #

The (P, p) notation and the four transformation rules are Part 5 of Volume A of the International Tables, developed there by Hans Wondratschek and colleagues over several editions; the same part carries the tables of transformations between settings that a database implements. The rules themselves are older than the Tables and are nothing more than linear algebra applied consistently, which is exactly why they are misapplied: each one alone is obvious, and only their differences are hard to keep hold of.

When the cell changes size #

Everything above keeps the lattice: P has determinant ±1, so the new basis spans the same lattice and the two descriptions have the same number of operations. The rules do not require that. A change from a centred description to a primitive one, or to a supercell, uses a P whose determinant is not one, and then the four rules apply unchanged while two things about the answer differ.

The operation count changes, because centring vectors that were translations of the old description are separate operations of the new one, or the reverse — which is what a centred cell is said as a transformation rather than as a symbol. And the indices stop being whole numbers for some reflections: with a fractional P, $h' = hP$ can land between integers, and those are exactly the reflections that must be absent if the smaller cell is the right one. That is a useful test of a cell choice and it is not computed here; what is computed here is the case where the lattice is fixed and only its description moves.

Where this goes: the five hundred and thirty #

The natural next step is the full enumeration. Each space group has a definite number of settings — the axis permutations, the cell choices for monoclinic groups, the two origins where the Tables give two, the rhombohedral axes — and the total over the 230 is 530 entries in the Tables’ own list. Deriving that list from the operations, rather than reading it, would be the strongest available check on the symbol derivation, since it asks the derivation to reproduce five hundred and thirty names rather than two hundred and thirty. It needs every one of the 230 groups defined, which is work not yet done here; what it would need beyond that is only the arithmetic above, applied once per setting.