The eleven a diffraction pattern reports

Point a beam at a crystal and measure where and how strongly it scatters. The pattern has a symmetry, and that symmetry is the first thing a structure determination reads off it.

It is not the crystal’s point group. It is the crystal’s point group with an inversion centre added, whether the crystal has one or not — and there are only eleven such groups.

Friedel’s law #

The intensity of a reflection is the squared modulus of the structure factor,

$$F(hkl) = \sum_j f_j \, e^{2\pi i (hx_j + ky_j + lz_j)}$$

summed over the atoms of the cell. Replace (hkl) by (h̅k̅l̅) and every exponent changes sign, so F(h̅k̅l̅) is the complex conjugate of F(hkl) — and conjugates have equal modulus.

$$|F(hkl)|^2 = |F(\bar h\bar k\bar l)|^2$$

That is Friedel’s law, stated in 1913, and it is one line of algebra with a large consequence: the measured pattern always has a centre of symmetry, because opposite reflections always have equal intensity. A crystal with no inversion centre produces a diffraction pattern that has one.

The experiment adds symmetry the crystal does not have, and cannot be persuaded to stop. What diffraction cannot tell apart is the site’s general treatment of that loss; this essay is the arithmetic of exactly how much of it there is.

Adjoining the inversion, thirty-two times #

The computation is mechanical. Take each class, throw in the matrix −I, close the group, and see what comes out.

Eleven distinct groups come out, and this site derives their symbols the same way it derives everything else — off the symmetry directions — rather than looking them up. They are 1̅, 2/m, mmm, 4/m, 4/mmm, 3̅, 3̅m, 6/m, 6/mmm, m3̅ and m3̅m.

Two observations, and neither is obvious in advance.

The eleven Laue classes are exactly the eleven centrosymmetric crystal classes. That has to be true — adjoining the inversion to a group that already has it changes nothing, and the result of adjoining it to any group is centrosymmetric — but it means the Laue classes are not a new list. They are a sublist, and a class is its own Laue class precisely when it contains the inversion.

The collapse is uneven. 1 and 1̅ both go to 1̅, so that row has two members. But 422, 4mm, 4̅2m and 4/mmm all go to 4/mmm, so that row has four, and so does 6/mmm’s. A crystallographer who reads 4/mmm off a pattern has narrowed thirty-two possibilities to four; one who reads 1̅ has narrowed them to two.

What the eleven buy, which is more than it sounds #

Reading eleven answers instead of thirty-two looks like a poor bargain, and in one sense it is: twenty-one of the thirty-two classes cannot be identified by the symmetry of a diffraction pattern at all, and the eleven-way answer is where symmetry-from-intensities stops.

But the Laue class is not nothing, and what it settles is the part of the problem that matters first.

It settles the crystal system, and therefore which cell parameters are independent and which are constrained — a cubic Laue class means one lattice parameter to refine rather than six. It settles the asymmetric unit, since the size of the unique portion of the cell follows from the order of the group. And combined with the systematic absences, which report the translations rather than the point symmetry, it usually narrows the space group to one or two candidates: the absences give the screws and glides and the centring, the Laue class gives the point symmetry, and between them there is often nothing left to choose.

That is the actual workflow, and it is why the eleven-way answer is the entry point rather than an obstacle. Reading a space group from its absences is the other half of the same procedure.

Why eleven and not some other number #

The collapse is a map from thirty-two things to eleven, and the sizes of its fibres are worth reading, because they say where diffraction is informative and where it is not.

Adjoining the inversion to a group G gives G ∪ (−1)·G. If G already contains the inversion the result is G, and the fibre over that Laue class contains G itself. If G does not, the result has twice as many operations — and the classes landing on a given Laue class L are exactly the subgroups of L of index one or two that contain no inversion, together with L.

So the fibre sizes are decided by how many index-two subgroups each Laue class has that miss the centre. 1̅ has one such subgroup and a fibre of two. 4/mmm has three and a fibre of four. The largest fibres are the two of size four, at 4/mmm and 6/mmm, and the tetragonal and hexagonal systems are therefore where a diffraction pattern’s symmetry is least decisive.

That is the same arithmetic as the index-two subgroups the two-colour essay counts, asked of a different family of groups. An index-two subgroup is a way of splitting a group in half, and here the half that is kept is the half without the centre.

Where the collapse actually hurts #

Three places, and they are the three where the missing information is exactly what somebody wants.

Handedness. A chiral molecule crystallises in one of the eleven enantiomorphic classes, and its mirror image crystallises in the same class in the opposite hand. Friedel’s law says the two give identical patterns. Determining which hand a structure has — the absolute configuration — is therefore impossible from intensities alone, and this was a live problem from 1913 until 1951.

Polarity. A polar class has a unique direction with a head and a tail, and reversing the crystal reverses it. The diffraction pattern is unchanged, so the sense of a polar axis is not determined either.

Centrosymmetry itself. The single most consequential question about a structure — does it have an inversion centre — is the one question the pattern’s symmetry is guaranteed not to answer. It is answered instead by statistics on the intensity distribution, which are different for centrosymmetric and non-centrosymmetric structures, and by the phase relationships used in direct methods.

The pattern’s symmetry and the crystal’s are different objects #

There is a habit of speech worth interrupting, because it makes the whole business sound like a limitation of instruments rather than a fact about the quantity being measured.

A diffraction pattern is not a picture of a crystal that has lost some detail. It is a picture of a different object: the squared modulus of the Fourier transform of the electron density. That object has its own symmetry, and its symmetry is related to the crystal’s by a rule — the reciprocal lattice has the same point symmetry as the direct one, and squaring the modulus discards the phase and adds the centre.

Read that way, Friedel’s law stops being a loss and becomes a description. Two crystals related by inversion have electron densities related by inversion; their Fourier transforms are related by conjugation; their squared moduli are equal. Nothing has been thrown away by the instrument. The two crystals genuinely have the same |F|², and an experiment measuring |F|² is measuring the thing it measures.

What is lost is the phase, and the phase is where the centre lives. That is the phase problem seen from the symmetry side: the missing half of the transform is exactly the half that would distinguish a structure from its inverse. Every route past Friedel’s law — anomalous dispersion, direct methods, isomorphous replacement — is a route to phases, and none of them is a route to a more symmetric measurement.

The law is not exact, and the correction is how the problem was solved #

Friedel’s law rests on the structure factors being sums of real scattering factors. They are not, quite. Near an absorption edge an atom’s scattering acquires an imaginary component — anomalous dispersion — and the conjugate relation breaks:

$$|F(hkl)|^2 \ne |F(\bar h\bar k\bar l)|^2$$

The difference between a Friedel pair is small, often a per cent or less, and it is a real signal. Bijvoet measured it in 1951 on sodium rubidium tartrate and settled the absolute configuration of tartaric acid — which fixed the sign convention for every organic stereochemistry that had been assigned relative to it, and which turned out to be the one Fischer had guessed.

So the eleven-way collapse is a consequence of an approximation rather than a theorem about diffraction, and the size of the departure is a measurement. The symmetry argument gives the eleven; the physics gives a way past it that the symmetry argument cannot see. That distinction is worth keeping, because a reader who takes Friedel’s law as exact will conclude that absolute configuration is undeterminable, and it is determined routinely.

The name, and the experiment behind it #

The classes are called Laue classes after the 1912 experiment rather than after anybody’s classification, and the reason is that the eleven were what the first diffraction photographs could report.

Friedrich and Knipping, working on von Laue’s suggestion, put a copper sulfate crystal in a beam and got spots. Within months the technique was being used to determine structures, and the immediate practical question was what a photograph settles. The answer — the point symmetry, plus a centre — is Friedel’s, and the eleven groups became the working vocabulary of the field before anybody needed to say that they were the centrosymmetric subset of a list from 1830.

That order of discovery leaves a fossil in the notation. The Laue classes are written as point-group symbols, because they are point groups, but they are conventionally listed as eleven rather than folded into the thirty-two — so a reader meets 4/mmm twice, once as a crystal class and once as a Laue class, and has to notice that these are the same group being used to answer two different questions. It is a crystal class when it describes a crystal and a Laue class when it describes what a photograph shows.

Where the exactness stops #

Eleven is a count of Laue classes, not of answers. A diffraction pattern’s symmetry is measured, from intensities with error bars, and deciding which of the eleven it shows is a statistical judgement rather than a lookup. Weak higher-symmetry violations get lost in noise, and the standard failure is to report the higher class.

Twinning defeats it. A twinned crystal produces a superposition of patterns, and if the twin law is an operation of the holohedry that the crystal lacks — merohedral twinning — the superposition has higher symmetry than either component. The measured Laue class is then wrong in a way nothing in the pattern reveals.

And this site does not compute the intensities of a real crystal. The structure factors here are sums over the atom positions of a generated orbit, with unit scattering factors and no thermal motion, no absorption and no anomalous component. They settle questions about symmetry, which is what they are used for; they are not a diffraction simulation and no essay treats them as one.

What a crystallographer does with eleven answers #

Reading the workflow through once makes it clear why the collapse is an entry point rather than a wall.

A data collection produces reflection positions and intensities. The positions give the lattice — its metric, its centring, and hence its Bravais type. The intensities’ symmetry gives the Laue class, and the intensities’ absences give the screws, glides and centring translations.

Those two together are usually decisive. There are 230 space groups; a Laue class narrows them to the ones whose point group lies in its fibre, and the extinction conditions narrow those to a handful. For a great many structures exactly one candidate survives, and for the rest the ambiguity is between a centrosymmetric group and a non-centrosymmetric subgroup of it — which is precisely the distinction Friedel’s law is blind to, arriving as the last question rather than the first.

That last ambiguity is then settled by statistics rather than by symmetry. The distribution of normalised intensities is measurably different for centrosymmetric and non-centrosymmetric structures — the centrosymmetric one has more very weak and very strong reflections, because its structure factors are real and can cancel — and the standard tests are on that distribution.

So the eleven-way answer is not where the determination stops. It is where the symmetry argument stops and where the statistical one starts, and knowing exactly which questions each can answer is what keeps a structure from being refined in the wrong group.

It is worth ending on what the eleven are not. They are not eleven kinds of crystal, and they are not a coarser classification anybody would choose. They are an artefact of one measurement technique — the most important one the subject has — and a different technique reports something else.

Electron diffraction in convergent-beam mode sees the full point group, because dynamical scattering breaks Friedel’s law. Second-harmonic generation sees whether the class is centrosymmetric, directly. Etch figures on a grown face report the face’s own symmetry. Each of those answers a question the eleven cannot, and each is used when the question matters enough to be worth the trouble.

The Laue class is what X-ray intensities report, and the sentence needs its qualification — which is the same discipline the rest of this field applies to property counts.

How the class is actually decided #

The essay says the pattern’s symmetry is measured rather than read, and the procedure that measures it is worth setting out, because it turns a statistical judgement into a comparison of two numbers.

A candidate Laue class is a group of operations on the indices. Apply it: gather the reflections it says are equivalent, average each group, and compute the spread within the groups relative to the average — the merging residual of the redundancy essay.

A correct class merges cleanly. Its groups contain reflections that really are equal, so the spread is the measurement error and the residual is a few per cent. A class that is too large merges reflections that are not equal, so the spread includes real intensity differences and the residual jumps — often by a factor of five or more, which is not a marginal call.

So the determination is: run every candidate, and take the largest class whose residual stays at the level the data’s own errors set. The comparison is against the residual of a class known to be correct — the triclinic one, which merges only Friedel pairs and is always right — so the criterion is a ratio rather than an absolute value, and it needs no assumption about how good the data are.

What makes the method work is that the classes are nested, so the candidates form a chain and the answer is where along it the residual breaks. That is a one-dimensional search with a visible discontinuity, which is a much easier thing to do reliably than fitting a symmetry.

What a partial data set does to the answer #

There is a failure of the procedure that has nothing to do with the crystal, it is common, and it produces an answer that is wrong in a specific direction.

The residual for a candidate class can only be computed where the class’s groups have more than one member measured. A data set covering a small wedge of reciprocal space has many groups with one member, those contribute nothing, and the residual is computed over whatever is left.

A very incomplete data set therefore merges beautifully in every class, including classes far too large, because there is almost nothing for the merging to disagree about. The residual is low for the wrong reason and the symmetry appears higher than it is.

The defence is to report completeness alongside the residual, per candidate class, and to distrust a class whose verdict rests on a small fraction of its groups. That is the same discipline the redundancy essay describes for the ordinary case, applied where it bites hardest — and it is why a symmetry determination is made after the data are complete rather than during collection, when the temptation to stop early is greatest.

Where this goes #

The point groups have now been enumerated, named, and shown to be more than an experiment can resolve. What remains is the reason they are worth having at all outside crystallography’s own bookkeeping: a point group decides, in advance and exactly, which components of a physical property a crystal is permitted to have.

That is Neumann’s principle, and it turns each of these thirty-two diagrams into a statement about elasticity, piezoelectricity and optical rotation — computed, in this site’s case, as one integer sum per class.