Crystallography & Symmetry
patterns derived from their groups
How it is known

A translation that is nearly there

Half a structure copied onto the other half by a half-cell shift, with nothing exact about it. No reflection vanishes, so no extinction rule fires — and the test for a centre of symmetry answers yes about a structure that has none.

Assumes Whether there is a centre is a statistic, The reflections a superlattice adds and The average that knows the atoms and not where they are.

14 min read 7 figures From the diffraction backExactly this manySymmetry is decidable

The reflections a superlattice adds computes what happens when a structure has a translation the lattice does not: order two kinds of atom onto a sublattice, and the reflections split into two populations, one of which was there before and one of which was not. The arithmetic is exact and the effect is a rule about which reflections exist.

This is the same arithmetic with the relation only approximately obeyed — not by every atom, or not exactly, or both — and it is a different kind of statement altogether. Nothing vanishes. No extinction rule fires. What is left is a fact about the distribution of intensities, and it is the standard way of being misled by the test for a centre of symmetry.

Two populations, and neither of them empty. The reflections of a structure in which three quarters of the atoms are paired by a half-cell shift, sorted by the parity of h + k and each class scaled by its own mean. The two histograms have the same shape, which is the point: each class on its own is an ordinary acentric distribution. What differs is the scale — the odd class is a sixth of the even one on average — and no odd reflection is absent, so no extinction rule fires and nothing about the space group is affected.
Fig. 1 The reflections of a structure in which three quarters of the atoms are paired by a half-cell shift, sorted by the parity of h + k and each class scaled by its own mean. The two histograms have the same shape — each class on its own is an ordinary distribution — and differ only in scale. No odd reflection is absent.

What “nearly” costs

Take a structure and pair its atoms: for each atom at (x, y) put another at (x + ½, y + ½). The structure factor of the pair is the atom’s contribution multiplied by 1 + exp(iπ(h + k)), which is two when h + k is even and exactly zero when it is odd. Every odd reflection vanishes, and this is not a statistical statement — it is centring, and a centred cell is what the arrangement should be described in.

Now pair only three atoms in four and leave the fourth unpaired. The unpaired atoms contribute to every reflection, so the odd class is no longer empty: it carries the scattering of the quarter of the structure that does not obey the relation, and nothing else. Its mean intensity is a sixth of the even class’s here, and its weakest member is not zero but a few parts in ten thousand of the even mean.

That difference between zero and small is the whole essay. An absence is a fact about the space group and it is read off a list; a class of weak reflections is a fact about the arrangement inside the cell, it obeys no rule, and the only trace of it is statistical.

The test that gets the wrong answer

The standard test for a centre of symmetry is a statement about a distribution. A centrosymmetric structure’s amplitudes are real, so their distribution has more weight near zero and a longer tail than an acentric structure’s; the fourth moment of the normalised amplitude is three in the first case and two in the second, and whether there is a centre is a statistic is the essay that makes that precise.

The normalisation is where a pseudo-translation gets in.

The test that answers the wrong question. The fourth moment of the normalised amplitude — the statistic that decides whether a structure has a centre of symmetry — computed two ways at each degree of pairing. The pale points normalise every reflection against one overall mean; the dark ones normalise each parity class against its own. None of these structures has a centre. The pale points nevertheless climb towards the centric value as the pairing strengthens, because a mixture of two distributions with different means has a longer tail than either. Bars are one standard error of the mean.
Fig. 2 The fourth moment computed two ways at each degree of pairing. The pale points normalise every reflection against one overall mean; the dark ones normalise each parity class against its own. None of these structures has a centre. The pale points climb towards the centric value anyway.

Normalising over all the reflections at once divides both classes by a single mean, so the weak class comes out systematically below one and the strong class systematically above. What was two distributions of one shape has become one distribution with a heavy left end and a long right tail — which is precisely the shape a centre of symmetry produces, and the moment says so.

At three quarters pairing the fourth moment over the mixture is 2.71 against an acentric expectation of 2 and a centric expectation of 3, with a standard error of about a quarter. Computed class by class it is 1.79. The structure has no centre of symmetry; it was built without one, and the statistic that is supposed to detect one has been fooled by a mixture.

The excess of weak reflections that is not a centre. The fraction of reflections weaker than z, drawn twice for the same structure. Normalised over everything at once, the curve rises steeply at the left: there is a surfeit of very weak reflections, which is exactly the signature a centre of symmetry produces and which is the basis of the standard test. Normalised class by class, the surfeit is gone. The weak reflections are not weak because of a centre; they are weak because they belong to a class whose mean is a sixth of the other's.
Fig. 3 The same failure in the cumulative form the test is usually applied in: the fraction of reflections weaker than z. Normalised together, the curve rises steeply at the left — a surfeit of very weak reflections, which is the signature of a centre. Normalised class by class the surfeit is gone. The weak reflections are weak because of which class they are in.

The size of the split, predicted and measured

The ratio between the two classes is not a fitted quantity. It follows from counting, and the count is worth doing because it says what the measurement is a measurement of.

Write p for the number of atoms that are paired and q for the number that are not, so the cell holds 2p + q atoms in all. A paired atom and its partner contribute together to an even reflection and cancel exactly in an odd one, so for an even reflection there are p contributions of amplitude two and q of amplitude one, and for an odd reflection there are q of amplitude one and nothing else. Averaging the squared amplitude over reflections, with the cross terms vanishing as they do in any average over a shell:

⟨I⟩even = 4p + q, ⟨I⟩odd = q

so the ratio is 4p/q + 1. Nothing about the positions enters, which is why the number is a statement about composition.

The measurement agrees. At three quarters paired — twelve pairs and eight loose atoms — the prediction is 56 against 8, a ratio of exactly seven; the measurement over a window of reflections gives 54.3 against 8.8, a ratio of 6.18. At a half it is 3.00 predicted and 2.92 measured, and at a quarter 1.67 and 1.64. The discrepancy is a finite window rather than a defect in the count, and it shrinks as the window grows.

So the class ratio measures how much of the structure obeys the relation, before any atom has been placed, and that number is worth having: a ratio of seven says three quarters of the scattering matter sits on a lattice with half the cell, and that substructure is the thing to solve first.

Why the fix is a normalisation and not a correction

The remedy is in the figures above and it is not a subtraction or a model: normalise each class on itself.

That is not a trick, and it is worth saying why. The quantity the test is about is how a reflection’s intensity is distributed given what is expected of it, and the expectation is what the content of the cell scatters at that resolution — a mean taken over reflections that ought to have the same mean. A pseudo-translation makes two groups whose expectations differ by a factor of six. Dividing both by one number does not make them comparable; it makes them incomparable in a way that looks like a result.

Real practice does exactly this, in a slightly more general form: the expectation is computed within resolution shells and within parity classes, and a data set showing a strong class difference is flagged as having a pseudo-translation before any test of symmetry is applied. The order matters, because the second test is invalid until the first has been done.

Two species on one lattice, ordered at index 2. Every position is a lattice point of the parent and none of them has moved. What has changed is which atom sits where: the larger marks are a sublattice of index 2, the smaller ones its other 1 coset, and the outlined cell is the new repeat. The lattice of positions is untouched and the repeat of the contents is 2 times as large, which is the whole of what an ordering transition does and the reason its signature is in reciprocal space rather than in the positions.
Fig. 4 The exact version of the same relation, for comparison: atoms ordered onto a sublattice, with the reflections that appear as a result. There the weak class is empty and the effect is a rule about extinctions; here three quarters of the atoms obey the relation and the class is weak rather than absent.

What else the split does

The two populations are a nuisance for one test and information for everything else, which is the usual position in this subject.

They say a substructure exists. A class of strong reflections whose intensities behave like an ordinary structure with a smaller cell is a statement that some of the atoms sit on a lattice finer than the crystal’s. That is a substructure worth solving on its own — solve the strong reflections alone as a small structure, then use it as a starting point — and it is the oldest trick in the heavy-atom repertoire.

They break the assumptions of the phasing methods. The relations that direct methods rest on are statements about normalised structure factors, whose distribution is assumed to be one thing. With two populations the normalisation is wrong for both, the triplet probabilities are wrong with it, and a procedure that would have worked fails for reasons that have nothing to do with the structure being difficult. This is the classic hard case for the tangent formula, and charge flipping — which needs no normalisation, no triplets and no statistical model — is largely indifferent to it.

And they are not a twin. A twin hides in the statistics is the other way a mixture disturbs the distribution: there, two orientations are added and the distribution becomes narrower, because adding two independent quantities narrows a distribution. Here it becomes wider. The two effects are opposite in sign and both are detected before any structure is known, which is what makes intensity statistics worth doing first.

N(z): the fraction of reflections weaker than z. The cumulative distribution of normalised intensities, measured on two structures built from the same atoms — one with an inversion centre, one without — and drawn against the two closed forms, 1 − e^(−z) without a centre and erf(√(z/2)) with one. The curves are furthest apart at small z, which is the useful end: a centrosymmetric structure has far more nearly-absent reflections, because its structure factor is a single real number that can pass through zero rather than a complex one that rarely does.
Fig. 5 The two curves the test is comparing against, for reference: the fraction of reflections weaker than z for an acentric structure and for a centric one. The pseudo-translation curve above sits between them, which is the shape of the trap — an answer that looks like a partial or uncertain version of the right one rather than like an artefact.

Where the pseudo-translation comes from

A structure does not acquire a near-relation by accident, and the two usual causes are worth separating because they are told apart by exactly the measurement above.

A substructure that is more symmetric than the whole. A compound whose heavy atoms sit on a small, high-symmetry lattice with lighter atoms arranged around them in a way that breaks it is the commonest case: the heavy atoms alone would give a cell half the size, and the light ones are what make the true cell what it is. The class ratio then measures the scattering power of the heavy substructure against the rest, and it can be very large — which is why the effect is more pronounced in a structure containing a metal than in an organic one.

An ordering transition caught halfway. A superstructure formed by ordering begins with a disordered phase in which the relation holds on average and ends with an ordered one in which it fails; in between, the weak class grows continuously from nothing. Its intensity is proportional to the square of the order parameter, which makes the weak reflections a direct measurement of how far the transition has gone — the same relation an order parameter is a representation sets up from the symmetry side.

The two cases are the same arithmetic and completely different physics, and nothing in the statistics distinguishes them. What does is the temperature dependence: heat the crystal and an ordering superstructure fades while a heavy-atom substructure does not.

The window the measurement is made in

One choice in the computation deserves stating, because a reader could reasonably ask whether the effect is an artefact of it.

The intensities are collected over a square window of reflections rather than over a resolution shell, and the two classes are compared over the same window. A resolution shell would be the crystallographic choice: the mean intensity falls with angle for reasons that have nothing to do with symmetry — the content of the cell and its temperature factor — so a comparison between two populations at different average resolutions would be measuring the fall-off as well as the split.

Here the two parity classes are interleaved at every resolution, so both are sampled identically by any window, and the ratio between them is unaffected by the fall-off. That is a property of the pseudo-translation rather than of the window: the classes differ by the parity of h + k, which is uncorrelated with the length of the reflection vector.

It matters because it is what makes the class ratio a clean measurement in practice, where a fall-off is always present and always uncertain. A quantity that requires the temperature factor to be known first is a quantity that has to wait; the class ratio does not, and it is available before any scaling has been done.

Two populations, and neither of them empty. The reflections of a structure in which three quarters of the atoms are paired by a half-cell shift, sorted by the parity of h + k and each class scaled by its own mean. The two histograms have the same shape, which is the point: each class on its own is an ordinary acentric distribution. What differs is the scale — the odd class is a sixth of the even one on average — and no odd reflection is absent, so no extinction rule fires and nothing about the space group is affected.
Fig. 6 Half the atoms paired rather than three quarters. The two classes now differ by a factor of three rather than six, the split is correspondingly harder to see in the distribution, and the class ratio measures the difference directly — which is why the ratio is the statistic to report rather than the shape of the histogram.

Where the exactness stops

Computed here: intensities from structures with a stated fraction of their atoms paired by an exact half-cell shift; the mean of each parity class and the ratio between them; the weakest member of the weak class, which is what distinguishes this from an absence; the fourth moment of the normalised amplitude under both normalisations, with standard errors from the sample; and the cumulative distribution under both.

Idealised: the relation here is exact for the atoms that obey it and absent for the rest. Real pseudo-translations are usually neither — the heavy atoms obey it to within a small displacement, and the displacement is itself the interesting quantity. Modelling that would need a third parameter and would not change any statement above.

And the direction of the failure is a claim about this case. A mixture of two distributions of the same shape at different scales has a fourth moment above either, so the bias is towards the centric answer. A pseudo-translation with a very small class difference will not bias anything measurably, and the ladder above is what says where the effect begins.

What the detector says about it

This collection settles questions about symmetry by handing a point set to a detector and comparing what comes back with what went in. It is worth asking what that machinery makes of a structure like this one, because the answer is instructive and slightly deflating.

The detector finds nothing. A pseudo-translation is not a symmetry: the arrangement is not carried onto itself by the half-cell shift, because a quarter of the atoms are not where the shift would put them. The exact machinery is exact, and the honest report is that the structure has whatever group it has and no more.

What the approximate detector says is a different matter, and this site has one: near-symmetry, and the tolerance that is not here runs the same search with a distance tolerance and measures how the answer depends on it. Applied here it would report the half-cell translation as soon as the tolerance exceeded the displacement of the unpaired atoms — which is to say it would report a fact about the tolerance.

The statistics do better than either, and the reason is worth stating. They do not ask whether the operation holds; they ask how much of the structure obeys it, and answer with a number that is meaningful at every value between zero and one. A yes-or-no question with a tolerance in it has been replaced by a measurement, and that is usually the right move when the object is a real crystal rather than a pattern generated from a group.

The test that answers the wrong question. The fourth moment of the normalised amplitude — the statistic that decides whether a structure has a centre of symmetry — computed two ways at each degree of pairing. The pale points normalise every reflection against one overall mean; the dark ones normalise each parity class against its own. None of these structures has a centre. The pale points nevertheless climb towards the centric value as the pairing strengthens, because a mixture of two distributions with different means has a longer tail than either. Bars are one standard error of the mean.
Fig. 7 The same ladder on a larger structure. More atoms narrow the standard errors without changing the shape: the statistic computed over the mixture still climbs towards the centric value as the pairing strengthens, and the one computed class by class still does not.

Who found it, and when

A. J. C. Wilson set out the intensity statistics in 1949, in the papers that give the method its name, and the pseudo-translation problem was recognised almost immediately: the test’s assumption is one population and a crystallographer looking at a real data set can see two.

The systematic use of the split as information came later, with the structure solution of superstructures in the 1960s and 1970s — where the strong reflections define a small average cell that can be solved first, and the weak ones then carry the departures from it. That is the same division of labour the average scatters sharply and the rest does not puts on an exact footing for disorder, arriving at the same place from a different direction: an average structure that is easy, and a difference that is where the content is.

Where the ladder goes next

Back, to the test this rung is about defeating. Whether there is a centre is a statistic is the one measurement on this site that is not decidable, and this is the standard way it goes wrong.

Sideways, to the exact version of the same arithmetic. The reflections a superlattice adds has the relation obeyed by every atom, where the weak class is empty and the effect is a rule rather than a distribution.

And to the other mixture that disturbs the same statistics in the opposite direction: a twin hides in the statistics, where two orientations are averaged and the distribution narrows.

How it is actually found

The essay diagnoses a pseudo-translation from the intensity statistics, which is where it does damage. The way a crystallographer finds one is elsewhere, it is direct, and it requires no statistics at all.

A pseudo-translation shows as a peak in the Patterson map. If a large fraction of the atoms are related by a vector u, then every one of those pairs contributes an interatomic vector equal to u — so the map has a peak at u whose height is the sum of the products of the atomic numbers over all the pairs.

That peak is enormous by ordinary standards. A structure with three quarters of its atoms paired has a non-origin peak of nearly three quarters the origin peak’s height, where an ordinary interatomic vector contributes a small fraction of one atom pair’s worth. So the diagnosis is a single glance at the map: one huge peak away from the origin, at a position with simple fractional coordinates.

And the peak’s height measures the fraction directly, by the same arithmetic as the class ratio — which gives two independent measurements of the same quantity, one in real space from the Patterson and one in reciprocal space from the intensity classes.

That is the standard order of operations. Compute the Patterson before anything else, since it needs no phases; look for a large non-origin peak; and if there is one, normalise the intensities by parity class before running any test that assumes a single distribution. A determination that runs the centre test first and looks at the map afterwards has already been misled, and the map would have told it so for nothing.

Where this goes next

N(z): the fraction of reflections weaker than z. The cumulative distribution of normalised intensities, measured on two structures built from the same atoms — one with an inversion centre, one without — and drawn against the two closed forms, 1 − e^(−z) without a centre and erf(√(z/2)) with one. The curves are furthest apart at small z, which is the useful end: a centrosymmetric structure has far more nearly-absent reflections, because its structure factor is a single real number that can pass through zero rather than a complex one that rarely does. Whether there is a centre is a statistic Everything else on this site is decidable: a pattern has a symmetry or it does not, and the detector settles it in integers. Whether a structure has an inversion centre is not like that. No single reflection carries the answer — the distribution of all of them does. Two species on one lattice, ordered at index 2. Every position is a lattice point of the parent and none of them has moved. What has changed is which atom sits where: the larger marks are a sublattice of index 2, the smaller ones its other 1 coset, and the outlined cell is the new repeat. The lattice of positions is untouched and the repeat of the contents is 2 times as large, which is the whole of what an ordering transition does and the reason its signature is in reciprocal space rather than in the positions. The reflections a superlattice adds Centring a lattice makes reflections vanish. Ordering two kinds of atom onto a sublattice makes new ones appear, exactly n − 1 of them per parent cell, and their intensity is a difference rather than a sum — which is why an ordered alloy of two neighbouring elements can be invisible to X-rays and obvious to neutrons. The twin fraction, recovered from a moment and nothing else (12 atoms). A structure of 12 atoms twinned at each of 6 fractions, with the second moment of its intensity distribution measured and the fraction solved back out of it. The recovery is within a few hundredths as far as thirty per cent — 4 rows here — and 4 of the 6 fractions get a number at all. Beyond thirty per cent the relation flattens: the derivative of 2α(1−α) vanishes at a half, the two roots meet, and a small error in the moment becomes a large one in the fraction. Where the sampled moment falls below 1.5 the quadratic has no real root and the estimate refuses rather than clamping, which is why a nearly perfect twin is the hard case in practice rather than the easy one. A twin hides in the statistics A twinned crystal scatters as two orientations at once and the detector cannot separate them. What arrives is a sum of two intensities — and adding two independent quantities narrows a distribution, which is a signature no model of the structure is needed to read.

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CentrosymmetryIntensity statisticsNormalised structure factorPseudo translationSuperstructureSystematic absenceWilson statistics