A merohedral twin moves no spot at all

A twinned crystal is two crystals. The natural expectation is that its diffraction pattern is two diffraction patterns, superimposed at an angle, with spots in obviously wrong places.

For a merohedral twin, that expectation is exactly wrong. The twin law is a symmetry of the lattice; a symmetry of the lattice is a symmetry of the reciprocal lattice; so the second individual’s reflections land precisely on the first individual’s. Every spot is where a single crystal would put it. Nothing is split, nothing is doubled, nothing is anywhere unexpected.

What has changed is the intensities. The reflection at (hkl) now records a mixture of two reflections — the one a single crystal would have put there, and the one the twin law maps it from — and the mixture is weighted by how much of the crystal is in each orientation.

The arithmetic of the mixture #

Let x be the fraction of the crystal in the first orientation. The observed intensity at a reflection h is

$$I_{\text{obs}}(\mathbf{h}) = x\,I(\mathbf{h}) + (1-x)\,I(W\mathbf{h})$$

where W is the twin law acting on indices. And the reflection at Wh obeys the companion equation with the roles swapped, because applying W twice returns to the start for the laws of order two that most twinning uses.

Two equations, two unknowns. Given x, the true intensities can be recovered from the observed ones by inverting a two-by-two matrix — the procedure known as detwinning.

The determinant of that matrix is x² − (1 − x)², which is 2x − 1. At x = 0.5 it is zero. A perfectly twinned crystal cannot be detwinned at all, not for want of a good algorithm but because the information is not in the data: two orientations in exactly equal amounts contribute identically to every reflection, and no arithmetic separates two numbers from their sum alone.

Near x = 0.5 the inversion is ill-conditioned rather than impossible, and the noise in the recovered intensities blows up like 1/(2x − 1). A twin fraction of 0.45 is workable and unpleasant; 0.49 is not.

Why this is expensive #

A structure is solved by assuming a space group, phasing the reflections, and refining. The space group is chosen from the observed symmetry of the intensities and the systematic absences, which between them are supposed to name it.

An evenly twinned crystal presents intensities with the symmetry of the holohedry — the ceiling — rather than of its own class. So the natural reading of the data is that the crystal belongs to a higher-symmetry group — and that reading is consistent, self-supporting and completely wrong.

What follows is worse than a failure. The structure refines. It refines to a residual that is high but not obviously fatal, and the extra symmetry averages pairs of atoms that were genuinely in different places into single atoms at their midpoints — the same collapse a Laue class performs on a point group, applied to coordinates instead of to symbols. The result is a plausible structure of a compound that does not exist, with bond lengths that are averages of two real ones and a symmetry the substance does not have.

The literature contains a substantial number of these. The standard corrections are published as such, and the pattern in them is consistent: a structure originally reported in a holohedral group is re-determined in a subgroup with a twin fraction near a half, and the chemistry makes more sense afterwards.

How it is caught #

Three tests, and none of them looks at a picture of the pattern.

Intensity statistics. The distribution of intensities from a single crystal is known — the Wilson distribution, with a characteristic spread. Averaging pairs of intensities narrows that spread: extremes get pulled towards the middle, and very weak reflections in particular become less weak, because they are averaged with reflections that are not. The moments of the normalised intensities are therefore lower than they should be, and the deficit is a measurement of the twin fraction that needs no knowledge of the structure at all.

Merging in the higher class. If the crystal really had the higher symmetry, merging the data in that Laue class would give a good agreement factor. For an evenly twinned crystal it also gives a good one — that is the trap. But for a partial twin it gives an agreement that is too good for a coincidence and too poor for a real symmetry, and the value of that factor plotted against candidate twin fractions is the standard diagnostic.

Absences that do not fit. The twin law permutes reflections, so it can move intensity into positions that should be systematically absent. A pattern with a few weak violations of an otherwise clean extinction condition is a pattern to look at twice.

The law that changes nothing, exactly #

The Class I case deserves its own statement because it is not a matter of degree.

For a twin whose law is the inversion — modulo the crystal’s own symmetry — the two individuals give identical intensities at every reflection. Not similar, not within error: identical, as an exact consequence of Friedel’s law. The twin fraction is not merely hard to determine; it does not appear in the data at all.

Quartz’s Brazil twinning is of this kind, and so is the corresponding law in every one of the twenty-one non-centrosymmetric classes. Detecting it needs anomalous scattering, which is a different experiment: tune the wavelength near an absorption edge and Friedel’s law stops holding, at which point the two individuals differ and their proportions can be fitted.

Non-merohedral twins, which are the easy case #

Everything above assumed the twin law is a symmetry of the lattice. If it is only a symmetry of a sublattice — reticular merohedry, the mechanism the spinel law and the Japan law use, and the subject of the coincidence site lattice — the two reciprocal lattices do not coincide.

They share a sublattice of reflections and disagree everywhere else. The pattern is two interpenetrating lattices of spots, most of them from one individual or the other and a fraction from both, and it looks exactly like what a naive reading expects a twin to look like.

That makes it far easier to handle. The two lattices can be indexed separately, the overlapping reflections identified, and the non-overlapping ones used directly with no detwinning at all. A non-merohedral twin is a nuisance in the data processing and it is not a threat to the answer, because nothing about it is disguised.

The uncomfortable ranking is therefore: a twin whose law is a lattice symmetry is dangerous, a twin whose law is only a sublattice symmetry is inconvenient, and a twin whose law is the inversion is invisible. Danger goes with how well the two individuals agree, which is the opposite of the intuition that agreement is benign.

Pseudo-merohedry, where the exactness runs out #

There is a case between the two, and it is where this field’s arithmetic stops being able to decide anything.

A monoclinic crystal with β very close to ninety degrees has a lattice that is nearly orthorhombic. An operation of the orthorhombic holohedry is not a symmetry of its lattice, but it is nearly one — the mismatch after applying it is a fraction of a degree — and if the mismatch is small enough compared with the width of a reflection, the two reciprocal lattices overlap to within the resolution of the experiment.

Such a crystal twins as though it were orthorhombic, and its diffraction pattern behaves as though the twin were merohedral. The quantity that decides is the obliquity, the angle by which the law fails to be a lattice symmetry, and there is no threshold in the mathematics that separates twinning from not twinning. There is only a comparison between the obliquity and the mosaic spread of the crystal, and both are measured.

That is the tolerance problem again, in the middle of an exact subject and for the same reason it always arrives: a claim decided in integers becomes a claim about a threshold the moment a real crystal is put in front of it, and the threshold belongs to the instrument.

A twin is not a disorder, and the difference is measurable #

One more distinction, because the two are routinely confused and they have different remedies.

A twinned crystal has regions, each of which is a perfect single crystal. A disordered crystal has the two alternatives mixed at the scale of individual cells. Both produce intensities that are averages, and the averages can be numerically identical.

What separates them is the coherence. Reflections from a region scatter in phase with one another; reflections from different regions do not, because the regions are far apart compared with the coherence length of the beam. So a twin adds intensities — the equation at the top of this essay — while a disordered structure adds amplitudes and then squares, which is a different function of the same two structures and produces cross terms.

The measurable difference shows up in the diffuse scattering. A twin has none: each domain is perfect, and the sharp reflections carry everything. Disorder produces diffuse streaks or sheets between the reflections, because the correlation between neighbouring cells falls off over a finite distance and a finite correlation length is a broad feature in reciprocal space.

So a crystal whose intensities look averaged and whose background is clean is twinned, and one whose background is not is disordered — and the models used to fit them are correspondingly different, one with a fraction and one with a correlation length.

What the figures here assert #

The pattern figures compute intensities from the structure-factor sum over the point set that the round trip has already verified, so the diffraction side and the pattern side share the atom positions and nothing else — which is the arrangement running through this whole site.

On top of that, two claims are checked while each figure draws.

For a Class II twin, the law must move intensity between reflections that were different. If it did not, the figure would be drawing two identical panels under a caption saying they differ, so the figure refuses to appear at all unless some intensity has moved. Handing the routine a Class I law where a Class II one is described is what found that the plane groups p1 and pm have exactly one twin law each and that for both of them it is the invisible one.

For a Class I twin, no intensity may change at all, asserted as an exact equality over every reflection in the window. That is the stronger of the two, because it is a claim that something is precisely zero and those are the claims worth making.

The reflections the law does not move #

The mixture arithmetic pairs each reflection with its image under the twin law. Some reflections are their own image, and those behave differently from every other in a way worth extracting.

If Wh = h — the twin law fixes the reflection — then the two orientations contribute the same intensity, and the observed value is x I(h) + (1 − x) I(h) = I(h). The reflection is untwinned whatever the fraction is.

That has two consequences and they pull in opposite directions.

Those reflections carry no information about the twin fraction. They look exactly as they would from a single crystal, so a fraction estimated from them alone is unmeasurable — and a statistic computed over a data set dominated by them will under-report the twinning.

And they are the reflections a refinement can trust. They need no detwinning, no fraction and no inversion of an ill-conditioned matrix, so they are the ones whose intensities are known as well after the twinning as before it.

Which reflections they are is decided by the law. A two-fold twin law about c fixes every reflection with h = k = 0; a mirror law fixes the whole zone lying in the mirror. So the untwinned set is a row or a plane of reciprocal space, computable from the law alone, and knowing which one it is says where in the data the twinning is invisible and where it is worst.

Three laws at once #

A class of index four has three twin laws available, and a crystal can carry all three — which turns the two-by-two inversion of the essay into a four-by-four one and makes the conditioning question sharper.

With four orientations in fractions summing to one, each observed intensity is a weighted sum of four true intensities, and each true intensity appears in four observations. The system is four equations in four unknowns, with a matrix whose entries are the four fractions arranged by the coset structure — a group matrix, since the laws compose as the quotient group does.

Its determinant factorises, and the factors are the sums of the fractions weighted by the characters of the quotient group. So the inversion fails wherever one of those character sums vanishes, and the worst case is not one fraction but any combination making a character sum zero — equal fractions being the obvious one, and not the only one.

That is a genuinely worse situation than the two-orientation case. Two orientations fail only at exactly a half; four fail on a whole surface in the space of fractions, and a crystal can be badly conditioned without any pair of its fractions being equal. Quartz, with three laws available and all three observed, is the standard place this arises.

Friedel’s classification, which is this arithmetic #

Georges Friedel set the terms of the whole subject in his Leçons de cristallographie of 1926, and the categories he used are the ones this field has been computing.

He divided twins by what the twin law preserves. Twinning by merohedry: the law is a symmetry of the lattice, and the two lattices coincide completely. Twinning by reticular merohedry: the law is a symmetry of a sublattice of index n, and the two lattices share one point in n. Twinning by pseudo-merohedry and by reticular pseudo-merohedry: the same two cases with “is a symmetry of” weakened to “is nearly a symmetry of”, with the obliquity recording how nearly.

Four categories, and every one of them is a statement about a subgroup relationship. The first two are exact and computable; the second two are the first two with a measured tolerance attached, and Friedel introduced the obliquity precisely so that the tolerance would be a number rather than an opinion.

The same man gave his name to the law that makes one twin invisible, and to the Bravais–Friedel rule about which faces grow. Three results, all about the relationship between a structure and its lattice, all still in use.

Detwinning is not always the answer #

The practical alternative to recovering the true intensities is to refine against the observed ones with the twin fraction as a parameter — computing the mixture forward at every cycle rather than trying to undo it.

That works better, and the reason is worth stating because it generalises well beyond crystallography. Detwinning inverts a badly conditioned system before fitting, so it amplifies the noise and then fits a model to the amplified noise. Refining against the observations applies the forward operation to the model, which is well conditioned, and compares the result with data whose errors are still the errors the detector made.

Undoing a mixture is unstable; predicting one is not. So the mixing is left in the model and the data are left alone.

What that costs is one extra fitted parameter and a small loss of information, because a perfectly twinned crystal still contributes only sums. What it buys is that the answer degrades gracefully as the fraction approaches a half instead of falling apart, and that the twin fraction comes out with an error bar attached.

Where the ladder goes next #

This anchor has moved from what a twin is, through how many are possible, to what one does to the measurement. What has not been asked is where twins come from.

Two answers have already appeared in passing and they are different. A growth twin nucleates while the crystal is forming, and whether it does is a question about surface energies. A transformation twin is left behind by a phase transition, and there the number of individuals is not a matter of chance at all: it is the index of the low-symmetry group in the high-symmetry one, and every one of them must appear.

The next anchor takes up that second case, where the twin count is a group-theoretic prediction rather than an upper bound — and where the same coset arithmetic turns out to be the standard tool of a field that calls the result a domain rather than a twin.