How much of it is the other hand

The law that hides handedness establishes the obstacle and its escape. With real scattering factors, negating the indices conjugates the structure factor and leaves the intensity exactly alone, so every diffraction pattern is centrosymmetric whatever the crystal is; near an absorption edge the scattering acquires an imaginary part, the conjugation argument fails, and the two halves of a Friedel pair differ by a percent or so. Bijvoet used that in 1951 to settle which hand a tartrate is, and with it the sign convention the whole of organic chemistry had been guessing at.

One experiment gives the cosine, the other gives the sine then names two rungs above itself, and this is the first of them. It is a change of question rather than of method: not what are the phases, but which hand is on the diffractometer — and, it turns out, how much of it.

A crystal need not be all one hand #

The tidy version of absolute configuration has two possibilities: the structure as modelled, or its mirror image. Real specimens are not obliged to choose.

A crystal can be an inversion twin — two orientations of one structure, related by a centre of symmetry, grown together in one lump. That is an ordinary kind of twin and this collection has the machinery for it: a twin is a symmetry the lattice has and the crystal does not, and an inversion is available to every lattice, so the operation is always a candidate. What the detector then records is a sum of the two individuals’ intensities in whatever proportion the crystal happens to have:

I(h) = (1 − x)·|F(h)|² + x·|F(−h)|²

with x the fraction of the inverted component. That is the Flack parameter, introduced by Howard Flack in 1983, and it is the number a structure report means when it says the absolute configuration was determined. Zero says the model is right; one says the model should be inverted; a half says the specimen is a racemic twin and the question has no answer for that crystal.

The model is linear in x once both calculated intensities are in hand, so fitting it is a division rather than a search. There is no possibility of a refinement that failed to converge being mistaken for a result, which matters for what follows.

The value is not the measurement #

The fit above recovers the fraction to within a few parts in a hundred at every value between nought and one, on data with three per cent noise. That is the easy half.

The two intensities differ only through the imaginary part of the scattering factor. Set every f″ to zero and they are equal, exactly, which is Friedel’s law — so the model’s derivative with respect to x vanishes identically and the fit has no information in it at all. It is not that the answer is wrong; there is no answer, and the arithmetic says so by returning an infinite uncertainty.

Between those two situations lies the whole practical difficulty of the technique. A structure with a sulphur in it, measured with copper radiation, has f″ of about half an electron and Bijvoet differences of a few percent: decisive. A structure containing nothing heavier than oxygen has f″ of a hundredth of an electron, differences smaller than the noise, and an uncertainty on x of a third or more. The value is still unbiased. It is simply useless, and this is the reason the crystallographic literature carries absolute configurations that were asserted rather than measured — the number was quoted and the number beside it was not.

So the uncertainty is the measurement. Every figure here reports it, and the ladder above is a picture of a quantity that stays right while becoming meaningless.

Where the difference actually is #

Nothing in the paragraph above is visible on a diffraction pattern as an image. The Bijvoet differences are small, they are not systematic in any pattern the eye can catch, and they occur where the structure happens to put them.

The exactness claim underneath the whole method is worth restating in its strong form, because it is what makes the small differences trustworthy. Invert a structure and every Bijvoet difference changes sign, exactly. Not approximately, not on average: pair by pair, to machine precision, because inverting the structure exchanges the two halves of every pair. So a fit that gets x wrong is a fit that got the sign of a systematic quantity wrong, and that is a much easier failure to detect than a small bias would be.

What symmetry has to say first #

A determination that cannot be made at all is worth ruling out before any data are collected, and this collection has the arithmetic for it.

A crystal with a centre of symmetry contains both hands already, so it has no absolute configuration to determine and x is meaningless — the two models are the same model. Beyond that, the eleven Laue classes are what a diffraction experiment reports before anything anomalous is considered, and a structure in one of the twenty-one non-centrosymmetric classes is the only case where the question arises.

There is a second, subtler filter. The parameter measures the fraction of the inverted component, and inversion has to be a distinguishable operation: in a class where the inverted structure is related to the original by an operation the crystal already has, there is nothing to measure. That is the same arithmetic as twinning by merohedry, where the number of twin laws is the index of the class in its lattice’s point group, and it is why an inversion twin is a possibility for the sixty-five Sohncke groups and a non-question elsewhere.

What a threshold would cost here, and why there is none #

This site has one essay whose whole subject is what happens when a decidable claim meets measured coordinates: near-symmetry, and the tolerance that is not here. Its rule is that the moment a threshold is chosen, the answer becomes a fact about the threshold as much as about the object.

The temptation here is a rule of the form report the configuration as determined when x is within so many standard uncertainties of nought. Flack’s own recommendation avoids it, and the reason is instructive. The parameter is a physical quantity with a value between nought and one, and the useful output is the value with its uncertainty rather than a verdict; a specimen with x = 0.08 ± 0.04 is telling a story about a slightly twinned crystal, and a rule that converts it to determined or not determined throws that away.

What has replaced the temptation in practice is a better statistic rather than a better threshold. The Bijvoet differences can be fitted directly, pair by pair, against the differences the model predicts, and the slope of that line is the same parameter with a much smaller uncertainty because the quantities being compared are the ones that carry the signal. This is what a modern determination reports, and the improvement is a factor of several in σ for light-atom structures — which moves some of them from useless to decisive without any change in the data.

Why the fraction is a fraction of what it is #

There is a subtlety in the model worth drawing out, because it is where the parameter’s meaning comes from and it is easy to read past.

The two intensities being mixed are those of a structure and of its inverse, and the mixing is incoherent — intensities added, not amplitudes. That is a physical statement about the specimen: the two individuals are separate regions of crystal, each many cells across, each scattering on its own, and what arrives at the detector is the sum of two intensities because the regions are not coherent with one another over the distances involved.

A crystal in which the two hands were interleaved at the scale of a unit cell would be a different object entirely. Its structure factor would be the average of the two, amplitudes added, and what would result is not a twin but a disordered structure with a centre of symmetry on average — the same distinction the symmetry of an average is about, where a refined structure carries symmetry that no individual molecule has.

So the Flack parameter measures the composition of a mixture of macroscopic domains, and it is silent about disorder at the scale of a cell. The two situations are told apart by other means: an inversion twin has sharp reflections and correct intensities, while a disordered structure has diffuse scattering and displacement parameters that come out too large. The distinction is the same one the average scatters sharply and the rest does not puts on an exact footing.

Two hands and one lattice #

There is a lattice fact underneath the possibility of an inversion twin, and it is the same one every twin on this site rests on.

A twin is a symmetry the lattice has and the crystal does not: the operation relating the two individuals must not be a symmetry of the crystal, or there would be nothing to see, and it must be a symmetry of the lattice, or the boundary between them would be a crack rather than a plane of coincidence. Inversion satisfies both conditions for every non-centrosymmetric crystal, because every lattice is centrosymmetric — the negative of a lattice vector is a lattice vector, always, with no condition on the cell at all.

So an inversion twin is available to every crystal that could have one, and it costs almost nothing in energy: the two individuals share their whole lattice exactly, at index one, so there is no coincidence arithmetic to satisfy and no misfit at the boundary. Twinning by merohedry counts the twin laws available to each class as the index of the class in its lattice’s point group minus one, and the inversion is always among them for the twenty-one non-centrosymmetric classes.

That is why the Flack parameter is a routine part of a structure report rather than an exotic measurement. The twin it measures the fraction of is the easiest twin there is to form, it leaves no trace in the geometry of the diffraction pattern — nothing splits, nothing moves — and the only evidence for it anywhere in the experiment is the one-percent difference between the halves of a Friedel pair.

Where the exactness stops #

Computed here: the intensities of an inversion-twinned crystal at any fraction; the fit for that fraction and its standard uncertainty from the curvature; the recovery of the fraction across the full range on noisy data; the growth of the uncertainty as f″ is turned down through six values; and the exactness of Friedel’s law and of the sign reversal on inversion, both at machine precision.

Simulated rather than measured: everything. The data here are exact intensities from three point atoms with a single relative error added, and the errors that matter in practice — absorption, extinction, radiation damage, a scale that drifts — are all absent. A real determination is dominated by systematic errors that are correlated between the two halves of a pair, which is exactly the correlation the method depends on being able to ignore.

Quoted: that fitting the Bijvoet differences directly gives a smaller uncertainty than fitting the intensities. That is Parsons’ method, and it is a statement about real data rather than about this model.

The measurement in the collection’s own terms #

It is worth putting this rung beside the site’s ordinary way of settling a question, because it is a rare case of the two being genuinely different.

Almost everything here is decided by integer arithmetic on a point set: generate, forget the group, rediscover it, and compare the two answers exactly. Handedness is decidable in that sense too — a set of points either has an orientation-reversing symmetry or it does not, and the detector settles it without a tolerance anywhere.

What is not decidable in that sense is which of two mirror-image structures produced a set of measured intensities. There the answer is a fitted number with an uncertainty, and no amount of care removes the uncertainty because the signal is a physical effect of a definite size. Whether there is a centre is a statistic makes the same point about a different question, and the two together mark out the boundary of the site’s usual method: the structure is decidable and the measurement is not.

That boundary is not a defect of the experiment. It is a statement about what a diffraction pattern contains — which is the subject of where the experiment runs out, and the reason a structure determination is a chain of inferences rather than a computation.

Who found it, and when #

Bijvoet, Peerdeman and van Bommel measured sodium rubidium tartrate with zirconium radiation in 1951 and found that Fischer’s arbitrary convention of 1891 had been right by luck — a fifty per cent guess that the whole of carbohydrate nomenclature had been resting on.

Howard Flack introduced the parameter in 1983, in a paper about how to refine it rather than about what it means; the meaning followed, and the parameter is now printed in every structure report of a non-centrosymmetric crystal. Simon Parsons and Flack introduced the differences-based estimate in the 2000s. The name that stuck is Flack’s; the quantity is a composition, and the most useful thing about it is the number printed after the plus and minus.

Where the ladder goes next #

Back, to the law this rung exists to escape. The law that hides handedness is the exact statement and the exact reason the escape is small.

Sideways, to the same anomalous difference used for something else: one experiment gives the cosine, the other gives the sine, where the differences phase a structure rather than orienting one.

And outward, to the other places a crystal turns out to be a mixture. A twin hides in the statistics detects a twin fraction from the intensity distribution alone, with no model at all — the same shape of measurement as this one, and the same reason for taking the uncertainty more seriously than the value.

What decides whether the uncertainty is usable #

The measurement’s difficulty is entirely a matter of how large the imaginary part of the scattering is, and that quantity is not a property of the crystal alone — it depends on which elements are present and which radiation is used, both of which are choices.

The imaginary part rises steeply near an absorption edge. An element scatters anomalously when the incoming photon has enough energy to promote one of its inner electrons, and the effect is largest just above that threshold. Far from any edge it is very small.

So the wavelength matters more than the diffraction quality. For a compound of carbon, nitrogen and oxygen only, the imaginary part of oxygen’s scattering factor is about 0.006 electrons with molybdenum radiation and about 0.032 with copper — five times larger, for a change that costs nothing but a different tube. A determination hopeless with one is routine with the other, on the same crystal.

A single heavier atom transforms the situation. Sulfur’s imaginary part with copper radiation is around 0.56 electrons, nearly twenty times oxygen’s, and it needs to appear only once in the molecule. That is why the standard remedy for an unmeasurable absolute configuration is chemical rather than crystallographic: crystallise the compound as a salt with a counter-ion carrying a sulfur, a chlorine or a bromine, and determine the configuration of the resulting crystal instead.

The absorption that comes with it is the cost. The same edge proximity that raises the anomalous signal raises the absorption, so the crystal attenuates the beam more and the corrections become larger. The trade is almost always worth taking, and it is a trade rather than a free improvement.

Estimators better than the fit #

The linear fit above is the definition of the quantity, and it is not the best way to measure it — a distinction worth drawing, because the uncertainty is what the technique lives on and the estimator is where a factor of two is available.

The fit uses every intensity. Most of them carry no information about the fraction at all: a reflection whose Friedel mate has the same calculated intensity contributes equally to both terms of the model, so it constrains nothing and adds its noise to the total.

A better estimator uses the pairs directly. Form, for each Friedel pair, the difference between the two measured intensities and compare it with the difference the model predicts. Reflections with no predicted difference drop out instead of diluting, and the resulting estimate has a smaller uncertainty from the same data — often by a factor of two or more.

A Bayesian version goes further. Rather than returning a fraction and an uncertainty, it returns the probability that the structure is one hand rather than the other, computed from the same pair differences with an explicit prior about whether the sample is enantiopure. That is the quantity a chemist actually wants, and it is stated as a probability rather than as a number near zero with a bar attached.

None of this changes what is being measured. The fraction is defined by the model at the head of this essay and every estimator is estimating it. What changes is how much of the data’s information about it is used — and on a light-atom structure, where the whole signal is a few parts in a thousand, that is the difference between an answer and a shrug.