The symmetry diffraction adds

A pattern can have more symmetry than the recipe that produced it, and a careless motif is the usual way it happens. That failure is avoidable: choose an asymmetric motif and it goes away. This essay is about a second kind of extra symmetry which is not avoidable at all, is present in every diffraction measurement ever made, and is a property of what a detector can record rather than of anybody’s carelessness.

The mechanism is one line of algebra and its consequences run through the whole practice of the subject — into which groups can be distinguished experimentally, into why handedness is invisible, and into a Dutch experiment of 1951 that recovered the missing information by breaking the assumption the algebra rests on.

One line of algebra #

The amplitude scattered in the direction labelled by two integers is a sum over the atoms, each contributing a complex exponential in the phase the atom’s position gives it. What the detector records is not that sum but its squared modulus, because intensity is what a photographic plate or a counter responds to.

Replace both indices by their negatives. Every phase changes sign, so every term in the sum becomes its own complex conjugate, and the whole sum becomes the conjugate of what it was. A conjugate has the same modulus. So the intensity at one reflection and the intensity at its opposite are equal — always, for any structure whatever.

That is Friedel’s law, and it is worth noticing how little it assumes. Nothing about the lattice, the symmetry, the atoms or the wavelength; only that intensity is a squared modulus. The figures on this page do not take it on trust either: the intensity is computed at every reflection in a window, and the equality is asserted at each one while the figure is drawn.

In three dimensions the added operation is a centre of inversion. In the plane, inversion through a point is a half turn, so what a plane diffraction pattern acquires is twofold rotational symmetry. Every plane diffraction pattern has it; a diffraction pattern without it would be evidence of an error somewhere in the measurement rather than of an unusual crystal.

What that does to a structure with no half turn #

The clearest case is a structure with as little symmetry as possible.

A p1 structure has no rotation, no mirror, no glide. Its diffraction pattern has a twofold axis. An experimenter looking only at where the intensity is strong and where it is weak would conclude the structure has a symmetry it does not have, and would be wrong in a way no repetition of the measurement corrects.

This is the sharpest available demonstration that the symmetry of the data is not the symmetry of the object. Everywhere else on this site, symmetry is decidable: the point set either has an operation or it does not, and finding out is integer arithmetic. Here is a case where the object’s symmetry is perfectly definite and the measurement systematically reports a different one.

Which groups gain, and which do not #

Adding a half turn to a point group that already contains one changes nothing. So the seventeen split into those that gain and those that do not, and the split is computed rather than recalled.

The seven that gain are the ones whose point groups contain no half turn: the groups built on 1, 3 and a bare mirror. p1 becomes twofold. p3 becomes sixfold. pm, pg and cm — each with a single mirror and no rotation — become twofold with two mirrors. The rest were already centred and are reported faithfully.

The consequence for practice is the second column of that figure read as a partition. The seventeen collapse onto six classes, and an experiment that has only the symmetry of the diffraction pattern to go on cannot distinguish the members of a class. The largest class holds seven of the seventeen.

These classes are called Laue classes, after the diffraction experiment’s first interpreter, and their three-dimensional analogue is the standard first step in any structure determination: the symmetry of the recorded intensities narrows two hundred and thirty space groups to eleven classes, and everything after that is done by other means.

Two kinds of accidental symmetry, and how they differ #

It is worth setting this beside the hazard the motif essay measures, because the two look alike and behave in opposite ways.

The motif’s extra symmetry is a property of a choice. Somebody picked a shape, the shape had symmetry, and the pattern inherited it. Choose differently and the extra operations vanish. The round trip catches it because the two point sets genuinely differ.

Diffraction’s extra symmetry is a property of the measurement. No choice produces it and none removes it. The structure is unchanged, the intensities are what they are, and the extra half turn is in the data every time. There is nothing for a round trip to catch, because nothing is wrong: the intensity really does have that symmetry, and the structure really does not.

The practical difference follows. The first is an error and is fixed by care. The second is a limit and is worked around by bringing in information from somewhere else — which is what the rest of this essay is about.

What the other means are #

If symmetry alone gets an experimenter to a class rather than a group, the rest has to come from somewhere, and this site has already built one of the two routes.

The absences. A glide plane removes whole families of reflections exactly, and which families are missing is a fingerprint the intensity symmetry does not carry. Systematic absences are how the glide-bearing groups are separated from their glide-free partners inside a Laue class, and they are the reason a space group can usually be determined despite Friedel’s law rather than in spite of it.

The statistics. Whether a structure is centrosymmetric leaves a trace in the distribution of intensities rather than in their symmetry: a centred structure has more very weak and very strong reflections than an uncentred one, because its structure factors are real and can cancel. That test is statistical rather than exact, it is a standard part of structure solution, and it is a different kind of claim from anything else on this site — a hypothesis about a distribution rather than a decision about a point set.

The seven that gain, on their own #

Pulling out the groups that gain a centre makes the pattern in them visible.

Every one of them lacks a half turn, and lacking a half turn is a strong condition in the plane: it rules out ten of the seventeen immediately. What is left is the groups whose point group is 1, 3, or m — nothing else survives.

The threefold cases are the ones with the most to lose. p3 has three operations and its diffraction shows six; p3m1 and p31m have six each and their diffraction shows twelve, which is the full symmetry of the hexagonal lattice. So the two groups whose difference this site keeps returning to are not merely hard to tell apart by eye — they scatter with identical intensity symmetry, and separating them experimentally needs the absences or the intensity statistics rather than the symmetry.

That last observation generalises: the added operation does not arrive alone. Adding a half turn to a group forces the closure of the result, so a group with one mirror acquires a second, and a group with a threefold rotation acquires the sixfold that a threefold and a half turn compose to. The classes are closures, not unions — which is why the figure computes them by closing a set of matrices rather than by adding one element to a count.

The information that is genuinely lost #

Some of what Friedel’s law hides cannot be recovered from intensities at all, and the case that matters is handedness.

A structure and its mirror image scatter identically under Friedel’s law. Every intensity is the same, every absence is in the same place, and the two are indistinguishable in the data. So an experiment measuring intensities alone cannot say which of two mirror-image structures is in front of it — and for molecules that come in two handed forms, which is most of biochemistry, that is precisely the question worth asking.

The escape was found by Bijvoet and his colleagues in 1951, and it works by breaking the assumption. Friedel’s law follows from every atom contributing a real scattering factor, so the phases are the only complex part of the sum. Near an absorption edge, an atom’s scattering acquires an imaginary component — anomalous dispersion — and then the term and its conjugate are no longer equal. The intensities at a reflection and its opposite differ slightly, the difference is measurable, and its sign settles the handedness.

Bijvoet’s group applied it to a rubidium tartrate and settled the absolute configuration of tartaric acid, which had been a convention rather than a measurement since Pasteur separated the crystals a century earlier. The general lesson is worth stating plainly: an accidental symmetry in a measurement is broken by finding an assumption in the argument that produces it, not by measuring more carefully.

What the round trip checked, and how #

Both figures compute both sides, and the point of the exercise is that the two sides are computed by machinery that shares nothing.

The structure’s point group is read off the group’s own operations — the linear parts of the wallpaper group, counted and named from the highest rotation order present and whether anything reverses handedness. The intensity’s point group is measured by taking every linear part the lattice permits, applying it to the indices, and testing whether the computed intensity is unchanged at every reflection in the window. One side is group theory; the other is a sum of cosines.

Two assertions run while the figures draw. The intensity is unchanged by reversing both indices, tested at every reflection rather than argued. And the intensity’s group is at least as large as the structure’s, which is the direction Friedel’s law predicts and the direction that would fail first if the intensity computation had a sign error in it.

The collapse figure adds a third: each point group has index one or two in the class its diffraction shows. Adding a single operation to a group can at most double it, so an index of three would mean the closure had gone wrong. It is the kind of assertion that costs nothing and catches a whole category of arithmetic slips.

Whose law it is #

Georges Friedel published the relation in 1913, within a year of the first diffraction experiments, and the speed with which it appeared says something about how immediate the mathematics is once the experiment exists.

The law is not a discovery about crystals. It is an observation about what a detector records — the squared modulus of a complex sum — and anyone who wrote down the structure factor could have noticed it. What made it worth a paper was the consequence for practice: it establishes that diffraction cannot distinguish a structure from its inverse, which was the first serious limit anybody had placed on the new technique.

Friedel’s own interest was in what could be inferred about a crystal’s symmetry from its external form, a subject he worked on for decades, and the law fits that programme exactly. It says which of the possible symmetries are observable by the new method, and therefore which conclusions about symmetry can be drawn from a photograph and which cannot.

The exception was found thirty-eight years later, by Bijvoet, and it works by breaking the assumption rather than the argument.

Where the exactness stops #

The window is finite. The intensity symmetry is measured over a square window of reflections and an operation is tested only where both a reflection and its image are inside it. A symmetry that failed far out and held near the origin would be reported as present. Nothing in this subject makes that likely, and it is a limit of the measurement rather than of the mathematics.

Intensities are computed, not measured. Every number here comes from the same sum, with equal point scatterers, no thermal motion, no absorption and no experimental error. A real measurement has all four, and the practical difficulty of assigning a Laue class is dominated by them rather than by anything on this page.

This is two-dimensional. Friedel’s law itself is dimension-blind — it follows from squaring a modulus — but the counts are not. Seventeen groups, six classes and ten crystallographic point groups are the plane’s numbers. Space has two hundred and thirty, eleven and thirty-two, and none of them are derived here.

Where the ladder goes next #

The avoidable kind of extra symmetry, and the reason every motif on this site is a comma rather than a dot, is the motif must be a comma.

The kind that arrives when coordinates stop being exact — where the question is not what the pattern has but what counts as close enough — is near-symmetry and the tolerance that is not here.

The other half of what a diffraction experiment cannot record is the phase problem, and the way a glide’s signature survives even a powder average is what a powder pattern loses.

What the pictures here cannot show. The right-hand panel of each pair is a set of computed intensities drawn as discs, and a reader cannot check its symmetry by eye to the precision the claim needs: two intensities differing by a part in a thousand look identical at any disc size. The symmetry stated under the panel is the output of comparing the numbers, and the picture is an illustration of a measurement rather than the measurement itself.

Eleven classes in space, and what they have in common #

The plane’s seventeen groups collapse onto six classes of pattern, and the three-dimensional statement has a tidier description that is worth having, because it says what a Laue class is rather than only how many there are.

Adding a centre to a point group gives a point group. The centre commutes with everything, so the set of operations obtained by taking each operation of the class together with that operation composed with inversion is closed, and it is a crystal class in its own right — the smallest one containing the original and a centre.

Doing that to all thirty-two classes gives eleven distinct answers, and the eleven are exactly the eleven classes that already contain a centre. A class with a centre is unchanged by the operation; a class without one is carried to the centrosymmetric class immediately above it.

So the Laue classes are the centrosymmetric crystal classes, and there is nothing else to remember. The map from thirty-two to eleven is the map “add the inversion”, and its fibres are the sets of classes sharing one centrosymmetric parent — some of size one, most of size two or three.

The consequence for an experiment is a ceiling. A diffraction experiment on a crystal reports which of the eleven the crystal belongs to, and there its evidence stops. Which member of that fibre the crystal actually has is a further question, answered by absences, by intensity statistics, or by the anomalous signal — never by the pattern’s own symmetry.

How the class is actually measured #

The collapse is usually presented as a fact to be applied afterwards. In practice it is measured first, and the measurement is worth describing because it is the point at which the algebra above becomes an experimental number.

A modern data collection records each reflection several times. The crystal is rotated through a range large enough that reflections related by the candidate symmetry both fall on the detector, so the same quantity is measured from different directions and at different times.

A candidate class predicts which reflections must agree. Assume the Laue class is $mmm$ and the reflections $hkl$, $\bar hkl$, $h\bar kl$ and so on all have the same intensity. Assume $4/mmm$ instead and a larger set is required to agree. Each assumption is a testable claim about a set of numbers that were measured independently.

The test is a single figure. Average the intensities within each predicted group and add up how far the individual measurements fall from their own average, relative to the total — the merging residual, written $R_{\text{int}}$. A correct class gives a residual comparable to the experiment’s own noise, a few per cent. A class assumed too high forces genuinely different reflections into the same average, and the residual jumps.

So the class is chosen by trying all eleven and reading the residuals. The one that stays low with the largest set of equivalences is the answer, and the jump between it and the next class up is usually unmistakable.

Two things about that procedure are worth noticing. It is a measurement of the data’s symmetry, which is the thing this essay says is not the crystal’s — so the ceiling above is built into the method rather than encountered later. And it improves the data: once the class is fixed, the equivalent measurements are merged into one averaged intensity with a smaller uncertainty than any of them had, so the symmetry that cannot be removed is at least made to pay for itself.