Every ambiguity in a diffraction pattern, sorted by its cause

Two groups no single operation reveals found the only two places in space where a space group’s translation parts are invisible to each of its operations taken alone. The body-centred groups I2₁2₁2₁ and I2₁3 are non-symmorphic, but every one of their operations is, on its own, an ordinary rotation about some line. So they extinguish no reflection their symmorphic partners I222 and I23 do not. That essay closed on a warning about reading too much into it. A hidden class produces no absence, but two groups with identical absences need not differ by a hidden class, since a centre of symmetry changes no absence either. The finer statement would take every pair of space groups a diffraction pattern cannot tell apart and sort it by why, and it would say which of the ambiguities in determining a space group are about statistics and which are about structure.

This essay makes that sort, over all 230 groups. The count of what a pattern shows comes out at a number the literature has quoted for decades, 122, which is a check on the whole construction. The sort then finds five causes of ambiguity, of very different sizes, and the one that the hidden classes represent turns out to be the rarest of them.

What a pattern shows #

A diffraction pattern records three things about the space group of the crystal that made it. The lattice, including the reflections its centring removes, since a centred lattice is a primitive one with some reflections missing. The symmetry of the intensities, which is the point group with a centre of symmetry added whether or not the crystal has one. That is Friedel’s law, and the enlarged group is the Laue group. It shows in its orientation on the lattice, so that a three-fold axis with its mirrors across the cell edges is distinguishable from one with its mirrors along them. And the systematic absences, the reflections that every structure in the group extinguishes, which the reflections that are not there derived from the structure-factor sum. Together the three make a group’s diffraction symbol.

Two groups share a symbol when there is a choice of axes for each that makes all three agree. The choice matters, because the International Tables give many groups in several settings, and Pmc2₁ written on relabelled axes has its glide somewhere else and its absences in a different zone. A pattern does not come with the Tables’ axes attached. The experimenter picks them.

The census, and the number it reproduces #

The groups are the 219 affine types of seventy-three Smith forms, one for each orbit of the normaliser on the cohomology of an arithmetic class, each with its operations written on its lattice’s conventional cell. The 230 are the orbits under the part of the normaliser with determinant one, and an affine type that splits into two of those is a pair of mirror images. There are eleven such pairs, and $219 + 11 = 230$.

A reflection is extinguished for every structure in a group exactly when some operation of the group fixes the reflection’s index vector and shifts its phase: an operation $(g, t)$ with $h^{\mathsf T} g = h^{\mathsf T}$ and $h \cdot t$ not a whole number. The operations fixing $h$ form a group, and the structure factor sums $e^{2\pi i\, h\cdot t}$ over it, which is either zero or the group’s order. Every index vector in a box thirteen wide is tested this way, exactly, with no floating-point sum over atomic positions.

Two groups on one Bravais lattice are then compared under every change of conventional axes with entries $-1$, $0$ and $1$ that keeps the lattice’s centring: 6,960 matrices on a primitive lattice, 912 on a C-centred one, 624 on body-centred and face-centred ones, 588 on the rhombohedral one. A change qualifies when it carries one group’s Laue group onto the other’s and one group’s absences onto the other’s. That set of changes contains the six relabellings of orthorhombic axes and the monoclinic cell choices, which is what the Tables’ settings are.

The 230 groups show 122 diffraction symbols, and 50 of them name a single group. The picture at the head of this essay is the whole census: one column per symbol and one mark per group. Forty-four symbols are shared by two groups, 23 by three, 2 by four and 3 by five.

The number 122 is the one where the experiment runs out quoted from the literature, with “about fifty” uniquely determined. It was quoted there because that essay could compute the absences of only forty-five groups. It is now derived from the operations of all 230, and it comes out exactly. Nothing in the construction was tuned to reach it. A mistake in the cohomology census, in the conventional operations, in the absence test or in the set of axis changes would each move it, and some would move it a great deal. Without the monoclinic cell changes, for instance, any two monoclinic groups the census happened to write with glides sliding in different directions would count as two symbols where the Tables have one.

The lattices differ a great deal. On the primitive hexagonal lattice, which carries the trigonal and hexagonal groups together, 45 groups show 19 symbols and only one of those names a single group. On the rhombohedral lattice, 7 groups show 3 symbols and none is unique. On the primitive tetragonal lattice, 49 groups show 27 symbols, 13 of them unique. The monoclinic primitive lattice has 8 groups on 4 symbols and one unique, which is P2₁/c. The older essay found that the commonest space group in the literature is also one of the best determined, and the census confirms it: no other group on its lattice shares its symbol.

Five reasons to share a symbol #

Take any two groups with one diffraction symbol. They have the same lattice, the same Laue group in the same orientation, and the same absences. They can still differ in one of five ways, and each needs different evidence to settle.

A centre of symmetry. One group has one and the other does not: P2/m and P2, Pmmm and Pmm2, P4/mmm and P4mm. Friedel’s law puts a centre into every pattern, so the absences and the Laue symmetry cannot see whether the crystal has one. The distribution of intensities can. A structure with a centre has real structure factors and a different spread of intensities, which is what whether there is a centre is a statistic measures.

Two point groups without a centre. P2 and Pm; P222 and Pmm2; P422, P4mm and P4̄2m. Neither group has a centre, both have the same Laue group, and intensity statistics sees them as the same, since both are acentric. What separates them is anomalous scattering, which breaks Friedel’s law and lets the intensities show the point group itself rather than the Laue group.

One point group set on the lattice two ways. Two groups of the same geometric class whose point groups sit differently on the lattice, in a way the Laue group cannot see. P4̄2m and P4̄m2 have the same point group turned by an eighth of a turn about the four-fold axis, and the Laue group 4/mmm is symmetric under that turn. Anomalous scattering separates them too, since the point group’s own orientation shows once Friedel’s law is broken.

Mirror images. P4₁ and P4₃, P3₁21 and P3₂21, and the rest of the eleven pairs. Their absences agree because a screw extinguishes the same reflections whichever way it turns. Anomalous scattering separates them, which is how the absolute configuration of a crystal is determined at all, for the reason the law that hides handedness sets out: Friedel’s law is exactly the statement that a pattern cannot see a mirror.

A translation part no absence shows. Two groups with the same point group in the same orientation on the same lattice, with the same absences, differing only in their translation parts. No measurement of symmetry separates them, since everything symmetry can show about them is the same. Only a solved structure can.

The census counts every pair, treating mirror images as separate groups. Of the 155 pairs that share a symbol, 95 differ by a centre, 41 are two acentric point groups, 6 are one point group set two ways, 11 are mirror images, and 2 differ only in their translation parts. Over nine pairs in ten are Friedel’s law doing what it always does, adding a centre the crystal may not have. Almost all the rest yield to anomalous scattering. Two do not.

The sort is made by asking, for each pair, a sharper version of the question that put them in one family. The same change of axes that carries one Laue group and set of absences onto the other must also carry one point group onto the other. If one does, the pair differs only in translations. If none does but the point groups belong to one geometric class, the pair is one point group oriented two ways. Otherwise the point groups are different.

One point group, two polar axes #

Five of the six pairs in the third class are the familiar ones: P4̄2m and P4̄m2, I4̄2m and I4̄m2, P6̄2m and P6̄m2, and Cmm2 and Amm2, whose two-fold axis stands either perpendicular to the centred face of the cell or in it. All eight of those groups are symmorphic, and none has an absence beyond its lattice’s centring.

The sixth pair is less familiar, and it is non-symmorphic.

Pmc2₁ has a two-fold screw axis along c and a glide plane that slides by half of c. Its absences are the reflections of the h0l zone with l odd. Pma2 has an ordinary two-fold axis along c and a glide that slides by half of a. Relabel Pma2’s axes so that its glide slides along c instead, which the Tables call the P2cm setting, and its absences become exactly Pmc2₁’s: the h0l zone with l odd, and nothing else. What differs is the polar axis. In those axes Pma2’s two-fold lies along a, and Pmc2₁’s along c. The Laue group mmm has a mirror across every axis, so the intensities cannot say which axis is polar, and a pattern with that absence pattern is consistent with both. A third group, the centrosymmetric Pmma in the matching setting, completes the family.

The same thing happens once more on the C-centred lattice, between Cmc2₁ and Ama2. In both cases what the absences come from, a glide plane sliding along one axis, is shared, and the two-fold axis that distinguishes them lies along one of the other two, with nothing in the Laue group to say which.

The largest families #

The five largest families are the symmorphic groups of a large Laue class on one lattice, with nothing extinguished beyond the centring. P4/mmm, P422, P4̄2m, P4̄m2 and P4mm share one symbol; so do the five body-centred tetragonal groups of that class, and the five primitive hexagonal groups of 6/mmm. Two families of four sit on the C-centred and body-centred orthorhombic lattices. A symmorphic group has no absences past its lattice’s, so every one of them lands in a family determined by its lattice and Laue class alone. The 73 symmorphic groups fall into 24 such families, which hold 75 groups between them. The two extra are the non-symmorphic halves of the hidden pairs.

The two that only a structure decides #

The fifth cause occurs exactly twice, and the two pairs are the ones the earlier essay found by a different route: I222 and I2₁2₁2₁, which share their symbol with Immm and Imm2, and I23 and I2₁3, which share theirs with Im3̄. Every coincidence of absences between two groups with the same point group in the same place is a hidden cohomology class, and there are no others. The earlier essay established that the two hidden classes have no absences to show. This census establishes the converse, which it could not: the hidden classes are the only way two groups can match in point group, orientation and absences.

In each pair the two groups have the same twelve or four point operations, and differ in where the two-fold axes sit. In I222 all three sets of axes pass through common points. In I2₁2₁2₁ no axis along one edge meets an axis along another. An experiment on intensities cannot see where axes cross, because intensities depend on the positions of atoms only through the vectors between them, and the arrangement of axes is fixed by the positions of atoms rather than by anything the vectors alone can show. A solved structure shows it immediately.

So the sort answers the question the earlier essay asked. The ambiguities in determining a space group are statistical for 95 pairs, anomalous for 58, and structural for 2.

The fifty groups a pattern names outright #

The fifty uniquely determined groups have a shared character. Thirty-three of them have a centre of symmetry, which makes sense, since a centrosymmetric group’s acentric subgroups usually share its absences, and a group can be unique only when its absences are more than its subgroups can reproduce. Forty-five have a screw axis. None of them is symmorphic. Every symmorphic group sits in a family set by its Laue class, and the only way for a group to be named outright by its pattern is to have an operation with an intrinsic translation that its relatives lack, the part of a translation that no choice of origin removes. The groups a crystallographer can assign with confidence are exactly those whose glides and screws are distinctive, which is the older essay’s observation about P2₁/c, now true of all fifty.

What the census does not decide #

The symbol assumes perfect data. A systematic absence is exact in principle, but a real reflection can be weak by accident, and the absence that fills itself in found that double diffraction can put intensity where the group puts none. The census says what the ideal pattern determines, not what a measured one does.

The settings are a finite search. Axis changes with entries $-1$, $0$ and $1$ cover the relabellings and cell choices the Tables use, and the count of 122 shows that they are enough. A pair of groups related only by a larger change of axes would be counted as two symbols, and nothing here rules that out beyond the agreement with the published number.

The cause of a coincidence is decided by the first change of axes that works. A pair counted as one point group oriented two ways is a pair for which no allowed change of axes carries one point group onto the other while also matching the absences. With a larger set of changes a pair could in principle move from that column into the translation column, and none does among changes of this size.

And separating two groups is not the same as determining one. The census says which measurement could distinguish a given pair. Whether a real crystal’s anomalous scattering is strong enough, or its intensity distribution clean enough, is a question for the crystal.

The work behind the pieces #

The diffraction symbols go back to Buerger’s work of the 1930s, and the tables of them in the International Tables for Crystallography list the groups each symbol admits. The cohomological description of space groups, which provides the 219 types used here, goes back to Zassenhaus in 1948. That I2₁2₁2₁ and I2₁3 cannot be told from I222 and I23 by their absences is standard, and the hidden-class explanation for it is in two groups no single operation reveals. That they are the only such pairs follows from the sort above.

What the census must refuse #

The refused claim is the reading that the earlier essay warned against, that a shared set of absences always means a hidden translation. The census finds that it means one only twice in 155 pairs. The test on the rhombohedral lattice guards against a mistake of a specific kind. A centring written with the wrong vectors would give the rhombohedral groups either no centring absences or the wrong ones, and the three rhombohedral symbols would then be wrong without anything else in the count looking odd.

Still open: four dimensions, where hidden classes may be common #

The plane has no hidden class and space has two, both body-centred. The census here suggests why they are rare. A hidden pair needs two groups with the same point operations in the same places and the same absences, and space offers only two places where the cohomology is rich enough and the operations poor enough to allow it. In four dimensions the body-centred lattices are more varied and the point groups larger, and the denominator a group actually needs already found the cohomology growing with dimension. Whether hidden classes become common there, and so whether a four-dimensional diffraction symbol, the kind used for modulated crystals, leaves structural ambiguities as a routine matter rather than twice, is a count the same computation could make on the four-dimensional classes. It has not been run here.