The law that hides handedness

The symmetry diffraction adds establishes the fact and measures it: a diffraction pattern is more symmetric than the crystal that made it, and the thirty-two classes collapse to eleven before a single reflection is indexed. This essay is about the one line of algebra underneath that, and about the physics that voids it — because the law has an exception, the exception is a few percent of an intensity, and everything that is known about the handedness of anything comes out of it.

One line, and the reason it is exact #

The structure factor is a sum over the atoms of a cell,

$$F(h) = \sum_j f_j \exp(2\pi i\, h\cdot r_j),$$

and negating the index vector negates every phase. If each fⱼ is a real number, then F(−h) is the complex conjugate of F(h), term by term. Conjugation does not change a modulus, so

$$|F(-h)| = |F(h)|,$$

exactly, for every structure, with no symmetry assumed anywhere. That is Friedel’s law, published by Georges Friedel in 1913, and it says that the measured intensities always have a centre of inversion whether or not the crystal does.

The exactness is worth dwelling on because this site’s usual claims are of the form the residual is below a tolerance. Here there is nothing to be below a tolerance. The two sums are conjugates of one another as real arithmetic expressions, so their moduli are equal in the same sense in which 2 + 2 is 4, and a computation that reported a difference of 10⁻¹² would be reporting a rounding error rather than a physical effect.

What it costs: the eleven Laue classes #

The consequence is the collapse the eleven that contain inversion is about. A diffraction experiment measures the intensity’s symmetry rather than the crystal’s, the intensity’s symmetry contains the inversion for free, so what an experiment reports is the crystal’s point group with an inversion added: the Laue class. There are eleven of them, and each is the answer for two or three of the thirty-two classes at once.

A structure determination therefore begins by answering a coarser question than the one it wants answered — and what diffraction cannot tell apart is where the rest of that shortfall is measured. The absences narrow the space group; the Laue class narrows the point group to one of eleven; and what remains — is there a centre, and if not, which of the two hands is this — is not decidable from the intensities at all. That is the situation Friedel’s law creates, and for forty years it was believed to be permanent.

The escape: a scattering factor that is not real #

The premise of the one-line proof is that each fⱼ is real, and near an absorption edge it is not.

An atom scatters X-rays by having its electrons driven by the incident field. Far from any resonance the electrons respond in phase and the scattering factor is a real number f₀, falling off with angle. Close to an absorption edge — where the photon energy is near that of an inner-shell transition — the response acquires a phase lag, and the scattering factor becomes

$$f = f_0 + f' + i f'',$$

with f′ a real correction and f″ the absorptive part, which is positive for every atom.

An imaginary component is not negated when the index is. The conjugation argument therefore fails: F(−h) is no longer the conjugate of F(h), the two moduli differ, and the difference between them is the Bijvoet difference.

The effect is small in a real measurement: a percent or two of an intensity for a light-atom structure with sulfur in it, more for a heavy atom near its edge. Everything about handedness is fought for in that margin, which is why anomalous work needs high-quality data and careful absorption corrections, and why it was not routine until diffractometers were.

What the difference measures #

The measurement is the sharpest thing in this essay, and it is worth stating as an identity rather than as a correlation.

Invert the structure — replace every position r by −r — and the two halves of every Friedel pair exchange. So every Bijvoet difference changes sign, and nothing else changes at all: the same magnitudes, the same pattern, the same everything, with one column of signs reversed.

So the experiment is: measure the differences, compute what they would be for each of the two hands, and compare. One prediction matches and the other is its negative. There is no ambiguity left, and no continuous parameter to fit — the two hypotheses are as far apart as two hypotheses can be.

Johannes Bijvoet did this in 1951, on sodium rubidium tartrate, using zirconium radiation near the rubidium edge. The result settled a convention that organic chemistry had been carrying on faith since Emil Fischer arbitrarily assigned a configuration to glyceraldehyde in 1891 — and it settled it in Fischer’s favour, which was a one-in-two piece of luck that saved a great deal of relabelling.

Why the effect needs a heavy atom, and how heavy #

The size of a Bijvoet difference is worth putting a number on, because it explains the whole practice around the measurement.

Write the structure factor as the sum of a normal part and an anomalous one. The intensity difference between h and −h is proportional to the product of the anomalous scatterer’s f″ with the rest of the structure’s contribution, and to the sine of the phase angle between them. Two things follow immediately.

One anomalous atom in a large structure gives a small effect. The difference scales as f″ divided by the total scattering power, so a single sulfur in a protein of five thousand atoms gives Bijvoet ratios of a fraction of a percent — measurable only with data whose random errors are smaller than that, which is a demanding requirement and the reason the technique waited for good detectors.

The wavelength matters as much as the element. f″ jumps at an absorption edge, so the same selenium atom gives an effect several times larger at a wavelength just above its K edge than at a wavelength far from it. That is why anomalous experiments are done at synchrotrons: the ability to tune to an edge is worth more than any amount of intensity elsewhere.

The practical consequence is a piece of biochemistry rather than crystallography. Selenomethionine — methionine with its sulfur replaced by selenium — is grown into proteins deliberately so that there is an atom with a usable edge in the structure, and a substantial fraction of all protein structures determined since the 1990s were phased that way.

The symmetry statement, made by measurement #

There is a way of saying all of this in the language the rest of this site uses, and it makes the two halves one statement.

Hand the intensities to the same kind of test the point-set detector makes: which operations on the indices leave every intensity where it was? With real scattering factors, the inversion h → −h is always among them whatever the structure is. With f″ present it is not.

That is Friedel’s law and its failure, reported as a symmetry rather than as an inequality — and the second half is the more interesting, because it means an anomalous measurement sees the true point group rather than the Laue class. The collapse from thirty-two to eleven is a property of a particular kind of measurement, not of diffraction as such, and choosing a wavelength near an edge partially undoes it.

A neutron does it differently #

X-rays are not the only probe, and the other two say something about how contingent the law’s failure is.

Neutrons scatter from nuclei, and nuclear scattering lengths are real for most isotopes — so Friedel’s law holds for ordinary neutron diffraction exactly as it does for X-rays, and for the same algebraic reason. A few isotopes have nuclear resonances near thermal energies and do scatter with an imaginary component; cadmium-113 and samarium-149 are the standard examples, and neutron anomalous scattering with them is possible and rare.

Electrons scatter from the potential, and multiple scattering inside a crystal is strong enough that the kinematic sum this essay is built on does not describe an electron diffraction pattern at all. That failure breaks Friedel’s law too, and in a way that is a nuisance rather than a tool: intensities of h and −h differ because the electron took different paths, not because of anything about the structure’s hand.

The pattern in all three is the same. The law follows from the sum being a sum of real terms, and any physics that puts a phase into a term breaks it. Which physics does so decides whether the breaking is useful.

Where this decides a space group #

The practical consequence goes beyond molecules. Two hundred and thirty or two hundred and nineteen is about the eleven pairs of space groups that are mirror images of one another — P3₁ and P3₂, P4₁ and P4₃, and so on. Nothing in a conventional diffraction pattern distinguishes the members of such a pair, because the two structures are related by the inversion that Friedel’s law supplies for free.

An anomalous measurement distinguishes them, and it is the only thing that does. So a protein crystallographer reporting P3₁ rather than P3₂ is reporting the outcome of a Bijvoet comparison, and a structure reported in the wrong member of a pair is a structure whose entire model is inverted — every helix the wrong way round.

What is exact, what is physical, and what is measured #

The three kinds of claim in this essay are worth separating, because they have different warrants.

Exact, by algebra. With real scattering factors the two halves of every Friedel pair are equal. The computation asserts it at machine precision over sixty pairs, and the assertion is a check on the code rather than on nature: a bug in the structure-factor sum would break it immediately.

Exact, by algebra. Inverting a structure reverses every Bijvoet difference and changes nothing else. Also asserted at machine precision, and also a check on the code.

Physical, taken from elsewhere. That f = f₀ + f′ + i f″ near an edge, and that f″ is positive for every atom, is dispersion theory and is quoted rather than derived. Nothing on this site computes an atomic scattering factor; the values used in the figures are real ones (sulfur’s f″ = 0.56 electrons at the copper Kα wavelength) so that the effect’s size is not invented.

Measured, on a toy. The Bijvoet ratios in the figures are those of a three-atom structure, and they are larger than a real one’s because a real structure has hundreds of atoms whose contributions to the difference partly cancel. The figures show the mechanism at a size that can be seen; the numbers are not a prediction for any material.

The centre that is there and the centre that is not #

There is a pleasing symmetry between this essay’s subject and one of the site’s oldest, and it is worth putting side by side because the two are the same trap seen from opposite ends.

The motif must be a comma shows that a single dot, orbited under p1, produces a point set with an inversion centre nobody asked for — the midpoint between a dot and its own lattice translate is one. The picture is of a group the caption never claimed, and the round trip refuses it.

Friedel’s law is the same phenomenon in reciprocal space: the transform of a real density has a centre nobody put there, so the measured pattern is of a symmetry the crystal never claimed. And the resolution is the same in shape though not in method — the drawing is fixed by using a motif with no accidental symmetry, and the measurement is fixed by using a wavelength at which the scattering factors are not real.

The difference is instructive. In the drawing the extra symmetry is an artefact of a careless choice and can be avoided entirely. In the measurement it is a property of what a detector records — the phase problem again — and can only be worked around at cost. One is a mistake and the other is a fact about the world, and the fact that they produce identical-looking extra symmetry is why the distinction has to be made deliberately.

What it does not settle #

Three limits, each of them a real boundary.

A centrosymmetric structure has no Bijvoet differences at all. If the structure has a centre, inverting it gives the same structure, so the differences are their own negatives and hence zero — anomalous scattering or not. The measurement determines a hand, and a structure with no hand has none to determine.

Absolute configuration is not absolute structure. The measurement determines which enantiomorph the crystal is; whether that is also the configuration of the molecule requires that the crystal contain only one hand, which is a chemical fact about the sample. A racemic crystal has both, in a centrosymmetric arrangement, and the differences vanish.

Nothing here is a refinement. Modern practice fits a parameter — the fraction of the crystal in each hand — rather than comparing two hypotheses, and the statistics of doing that reliably are a subject of their own. The exact statement is that the two predictions are negatives; how confidently a noisy data set chooses between them is not decidable by symmetry and is not attempted here.

Where the ladder goes next #

This anchor is new and has one rung: the law and the way out of it. Two rungs are visible.

Multi-wavelength methods. Measuring at several wavelengths across an edge varies f′ and f″ in a known way, and the differences between data sets give the positions of the anomalous scatterers directly — which is how most protein structures are now phased. That is a route out of the phase problem rather than merely a determination of hand, and it is the largest practical consequence of this whole business.

The Patterson of the differences. The Bijvoet differences can be fed to the same map the map that needs no phases computes, and what comes out is the vector set of the anomalous scatterers alone — a handful of atoms rather than thousands. That is the anomalous difference Patterson, and it is the point at which the two halves of this field meet.

The coin that had already been flipped #

The law was stated in 1913 and the escape from it was not demonstrated for another thirty-eight years, which left a gap with an unusual consequence: an entire branch of chemistry was built on a convention that nobody could check.

Organic chemistry needed a reference. Handedness can be transferred from one molecule to another by reactions that do not disturb the relevant centre, so the configurations of thousands of compounds could be fixed relative to a single one. Emil Fischer chose glyceraldehyde as that one and assigned it a handedness — arbitrarily, because no method existed to determine it, and explicitly so.

Every relative assignment was therefore correct and every absolute one was a coin flip. If Fischer’s guess was wrong, the entire tree of configurations built on it was uniformly inverted, and no chemical experiment could have revealed it.

Bijvoet settled it in 1951. Working on a tartrate salt with rubidium in it, he measured the differences between Friedel pairs that the imaginary part of rubidium’s scattering produces, and read the handedness directly. The answer agreed with Fischer’s assignment.

The guess had been right. Nothing was corrected and every structural formula in the literature stood, which is a strange kind of result — the experiment’s whole value lay in the fact that it could have overturned the convention, and the fifty years of chemistry that would have needed relabelling if it had.

The other way the law fails #

Anomalous scattering is not the only route past Friedel’s law, and the second one matters wherever the first is too weak.

The one-line proof assumes each scattering event is independent. A beam enters, scatters once from the electron density, and leaves. That is the kinematic approximation, and it is very good for X-rays through a small imperfect crystal.

Electrons scatter far more strongly. A beam of electrons through even a thin crystal scatters many times on the way, and the amplitude arriving in a given direction is a sum over routes rather than a single term. The conjugation argument has nothing to act on: the intensity in one direction is built from interference among paths, and there is no reason for it to equal the intensity in the opposite one.

So an electron diffraction pattern is not centrosymmetric, and the asymmetry is not small. Convergent-beam electron diffraction turns that into a method — the patterns show the crystal’s point group rather than its Laue class, and dark bands appearing where dynamical contributions cancel reveal glides and screws directly.

Which reverses the usual ordering of the two techniques. X-rays give better positions and cannot see handedness without an anomalous signal; electrons give worse positions and see the symmetry the X-rays lost. A determination that needs both is not unusual, and the reason is that the law on this page is a statement about single scattering rather than about diffraction.