Two origins for one group

One group, three symbols shows that a change of axes renames a group: P2₁/c, P2₁/a and P2₁/n are one group written on three cell choices, and deriving the symbol from the operations shows exactly why. Underneath that convention sits a second one, invisible in the symbol, and it has caused more published errors than the first.

Where is the origin?

A space group’s operations are written with translation parts, and those parts depend on where the origin was put. Moving it by s sends (M, t) to (M, t + (I − M)s), which changes every number in the table and no fact about the crystal. So a convention is needed, the Tables have one, and for some groups the convention gives two answers.

The two rules, and where they disagree #

The International Tables place the origin by two rules that usually agree.

Origin choice 1 is at the point of highest site symmetry — the position that the most operations leave alone, which is the natural centre of the arrangement.

Origin choice 2 is at a centre of inversion, when the group has one. That is the choice that makes the arithmetic of a structure factor simplest, since a centrosymmetric structure written about a centre has real structure factors and phases of 0 or π rather than arbitrary values.

For most centrosymmetric groups the point of highest site symmetry is an inversion centre and there is nothing to choose. For twenty-four of the two hundred and thirty it is not, and both settings are printed — with a note saying so, which everyone who works with those groups learns to read and everyone else does not.

Deciding which case a group is in is a computation on its operations and nothing else. The site symmetry of every point of a grid is the number of operations that fix it; the inversion centres are the exact solutions of 2x ≡ t for each inversion (M = −I, t); and the question is whether the two sets meet.

Fddd, and the eighth that separates the settings #

The example this site has to hand is Fddd, and it is the standard one for good reason.

Its highest site symmetry is of order four — a point where three twofold axes meet, the site symmetry 222 — and it has no inversion at that point. Its inversion centres sit an eighth of a cell away in each coordinate. So the two settings are related by a shift of (⅛, ⅛, ⅛), every coordinate in the second table differs from the first by that vector, and the operations’ translation parts differ accordingly.

A grid that cannot represent an eighth reports that the two origins coincide. That is worth saying because twelfths are the natural grid for this subject — halves, thirds, quarters and sixths all live on them — and a survey done at twelfths would find Fddd unremarkable. The grid here is twenty-fourths, chosen after the failure rather than before it.

The eighths are not an accident of this group: they come from the d glide — the fifth of the glide planes, the one that exists only where centring has already made a half-diagonal a lattice vector — whose intrinsic translation is a quarter along each of two axes. Halving a quarter gives an eighth, and halving is what locating an inversion centre does. Any group with d glides has the same problem, which is why the F-centred orthorhombic and cubic groups are heavily represented among the twenty-four.

The site-symmetry map, which is where the answer comes from #

The computation behind the two origins is a sweep, and it is worth describing because it is the same object under two names.

For every point of a grid of twenty-fourths, count the operations that leave it exactly where it is. That count is the point’s site symmetry, the sweep produces a map of it, and the map’s maxima are the candidate origins. This is Wyckoff positions in three dimensions: the same orbit–stabiliser arithmetic, the same special positions, the same fact that a point’s stabiliser order times its orbit length is the group’s order.

So the two rules the Tables use are two different readings of that one map. Choice 1 takes its global maximum. Choice 2 takes a point of a particular kind — an inversion centre — whether or not it is the maximum. When the maximum happens to be of that kind, the readings agree.

There is a pleasing consequence in the arithmetic. The number of distinct candidate origins of each kind is the number of orbits of that Wyckoff position, and those orbits are permuted by the equivalent origins below — so the whole business is one orbit computation asked three ways: which points are special, which of them are inversion centres, and which shifts between them leave the table unchanged.

What goes wrong, and why nothing catches it #

A structure published on origin choice 1 and read as though it were on choice 2 has every atom in the wrong place by (⅛, ⅛, ⅛). The failure has a characteristic and unpleasant shape.

Every internal check passes. The distances between the atoms are unchanged, because the whole structure moved rigidly; the bond lengths and angles are the ones the authors reported; the formula is right; the density is right. Nothing about the coordinate list, examined on its own, is wrong.

Everything that uses the cell’s frame is wrong. The positions relative to the symmetry elements are wrong, so the Wyckoff assignments are wrong, the site symmetries are wrong, and any calculation that places the structure against the group’s own elements — a Madelung sum, a symmetry-mode analysis, a comparison with a related structure — is wrong by that shift.

A refinement can hide it. Refining a structure that is uniformly displaced from the correct one relative to the symmetry elements does not converge to the right answer; it converges to a structure that is not the reported one, and the R factor may be perfectly respectable if the shift happens to be nearly a symmetry of the arrangement.

The one thing that catches it reliably is the comparison the same pattern, described twice is about: two independent determinations, put into a common description, disagreeing by a vector that is not in the group’s normaliser.

Which shifts change nothing at all #

The normaliser is the other half of this arithmetic and the two must be kept apart carefully.

An equivalent origin is a shift that leaves the operation set exactly as it was — every translation part identical. Those are the translations of the group’s Euclidean normaliser, and a description moved by one of them is not merely equally valid but literally the same table of numbers. There is no way to detect such a shift and no consequence of making one.

An origin choice is a shift that produces a different table of numbers describing the same group. Choice 1 and choice 2 of Fddd have different translation parts; both are correct; they are different descriptions rather than the same one.

So the count of equivalent origins says how much freedom is free, and the origin choices say where a genuine convention had to be adopted. A group with many equivalent origins and one origin choice is easy to work with; Fddd has eight equivalent origins and two choices, and the eight do not include the vector that relates the two choices.

Checked rather than asserted #

The claim that the two settings describe one group is not left as a statement. The operations of Fddd are shifted by the computed vector, and the two operation sets are compared by the routine this site uses for exactly this question: are these two groups the same up to an origin shift?

That routine searches a grid of shifts and reports the one that works, and it reports (⅛, ⅜, ⅛) for this pair — a vector equivalent to the one used, modulo the group’s own translations, which is what “the same group” means here. Had the two settings been genuinely different groups, no shift would have worked and the comparison would have come back empty.

This is the same machinery that one group, three symbols uses to show that three symbols name one group, applied to origins rather than to axes. The two conventions are independent — a group can have three symbols and one origin, one symbol and two origins, or both — and a description is pinned down only when both have been stated.

Why anybody wants the second choice at all #

The obvious question is why the Tables bother with a second setting when the first is defined for every group. The answer is arithmetic and it predates computers.

Put the origin at a centre of inversion and the structure factor of a centrosymmetric crystal becomes real. Every atom at r has a partner at −r, their contributions are complex conjugates, and the sum of a conjugate pair is twice the real part — so F is a sum of cosines, the phase of every reflection is 0 or π, and the phase problem reduces to a choice of sign.

That is an enormous simplification and it is the reason centrosymmetric structures were solvable long before non-centrosymmetric ones. It only holds in the setting where the origin is at the centre; in choice 1 of Fddd the same structure has complex structure factors with phases scattered over the circle, and none of the simplification is available.

So the two settings are optimised for two different purposes: choice 1 for describing the symmetry, where the point everything acts about is the natural origin, and choice 2 for computing with the data, where the real structure factors are worth more than the tidy description. The Tables print both because crystallography does both.

A modern program does not care, and can transform between the settings in a line — which is exactly why the error survives. Nothing forces anybody to state which was used, because nothing in the calculation needs to know.

How to tell which setting a structure used #

Three tests, in increasing order of reliability.

Read the paper. The Tables’ convention is to say origin choice 2 explicitly, and most depositions carry it. Most is not all, and older work often does not.

Look at where the symmetry elements are. If the deposited coordinates place an atom at a special position whose site symmetry does not match the one the Tables give for that Wyckoff letter, the setting is wrong. This is the test a validation program makes.

Look for the inversion centre. In choice 2 of a centrosymmetric group, an atom on a general position has a partner at exactly minus its coordinates. In choice 1 it has a partner at minus its coordinates plus the shift. Computing the pairs and looking at the sum of each pair’s coordinates settles it in one pass, and it is the test that needs no external information at all.

The same trouble one dimension down #

The plane groups have the same convention and almost none of the trouble, and comparing says what causes it.

Of the seventeen, the ones with a half-turn have their rotation centres as the natural origin, and a plane group’s rotation centres are its points of highest site symmetry — so choice 1 is unambiguous. There is no analogue of choice 2, because a plane group has no inversion: the half-turn is a rotation, it is a proper operation, and its fixed point is a rotation centre rather than a centre of inversion.

So the second convention has nothing to be about in two dimensions, and the two rules cannot disagree. The trouble is specific to three dimensions and to the fact that the inversion is available there as a genuinely improper operation with a single fixed point — which is the step a flat surface has no room for showing up in the bookkeeping rather than in the classification.

The plane does keep one shadow of the problem, and it is the origin count rather than the origin choice. p2 has four equivalent origins and pm has a continuum, and a plane-group description is pinned down only up to those — which is the arithmetic the same pattern, described twice is entirely about.

Where the exactness stops #

Three limits.

Twenty-four is a count from the Tables, not from this computation. The site defines forty-five space groups and finds the double origin in the ones it has; the number for all two hundred and thirty is quoted, and named as quoted. What is computed here is which of these groups have it and what the shift is.

The grid is twenty-fourths. Every origin-related vector in the site’s groups has denominator dividing twenty-four, so the search is exact for them. A group with a translation part of a twelfth of a twelfth would need a finer grid, and none of the two hundred and thirty does — but that is a fact about the Tables rather than something this computation establishes.

The site symmetry map is a count, not a name. The computation reports how many operations fix a point; it does not name the point group they form. Naming it would let the two origins be described as 222 and 1̅ rather than as order four and order two, which is what the Tables print, and it is a small addition this site has not yet made.

Where the ladder goes next #

The settings anchor now has two rungs: the symbol that changes with the axes, and the origin that changes with the convention. The rung above is the one both of them point at — the standardisation problem: given a structure in an arbitrary description, put it into the standard one. That is a search over the normaliser’s cosets and the cell choices at once, it is what every structure database does to every entry, and it is the practical form of everything in these two essays.

There is also a sideways rung worth taking: what a cleave leaves found Fddd’s most symmetric sections at eighths, for exactly the reason its origins are eighths apart. A group’s awkward denominators show up in every question asked about it, and collecting where they come from — the d glide, the rhombohedral setting, the two-thirds of a hexagonal cell — would make a rung about denominators rather than about origins.

The twenty-four, and the one everybody has met #

The groups needing two origins are not scattered at random through the two hundred and thirty, and knowing where they concentrate is most of what a reader needs.

They are all centrosymmetric, since a group with no inversion has nothing for the second rule to place the origin at. They are concentrated in the systems with the most symmetry — the tetragonal and cubic ones — because those are the systems where a point of very high site symmetry is available and need not be a centre. And the offsets are the eighths, quarters and their relatives that a diagonal glide or a d glide produces.

The one every reader has met is the diamond structure. Its group Fd3̄m has two origins, and the two descriptions of diamond differ by exactly the eighth of a body diagonal this essay is about: carbon at (0, 0, 0) and (¼, ¼, ¼) on one setting, and at (⅛, ⅛, ⅛) and (⅜, ⅜, ⅜) on the other.

Both are correct and both are in the literature. Silicon, germanium, and every structure of the diamond or zinc-blende type inherits the same pair, and a comparison of two determinations that does not check which setting each used will find them displaced by an eighth and conclude that something is wrong.

That makes the case a good one to remember, because it is not obscure. The most-cited structure in solid-state physics is one of the twenty-four, and the ambiguity it carries has nothing to do with anything difficult about it.

What removes the ambiguity #

The essay’s tests are for reading somebody else’s file. There is a way of writing one’s own that makes them unnecessary, and it is now standard.

Deposit the symmetry operations themselves. A structure file can carry the group’s operations written out as coordinate expressions — x, y, z, −x+½, −y, z+½, and so on — rather than only the symbol. Those strings carry the origin: the translation parts in them are what they are on the setting actually used, so a reader applying them to the coordinates reproduces the structure whichever convention was intended.

A symbol carries a group and an operation list carries a description, which is the distinction the plan essay makes about diagrams and which applies here unchanged. The symbol is the compressed form and the operations are the complete one.

Modern structure files carry both, and the ambiguity survives chiefly in older depositions and in text — where a paper quotes a symbol, a cell and a coordinate list, and the reader has to reconstruct the setting from the coordinates themselves. That reconstruction is exactly the origin search this essay’s machinery performs, and it is worth knowing that it can be done rather than guessed.