A twin leaves an edge that nothing else can

A twinned crystal is two orientations of one structure grown together, and a detector cannot tell which orientation sent a given photon. Each spot it records is a sum: a fraction $1 - \alpha$ of one reflection’s intensity and a fraction $\alpha$ of the intensity of the reflection the twin law carries onto it. A twin hides in the statistics found the fraction without looking at any particular spot. Adding two independent intensities narrows their distribution, and the second moment of all the intensities together, $\langle I^2 \rangle / \langle I \rangle^2$, falls from two toward one and a half as $\alpha$ rises toward a half.

That test needs no knowledge of the twin law, which is its strength, and it reads the whole distribution at once, which is its weakness. Anything else that narrows or widens the distribution moves the moment too. A translation that is nearly there is the standard case, where a pseudo-translation widens the distribution so far that the test reports a centre of symmetry the structure does not have, and a twin underneath would be invisible.

The other test takes the law in hand and compares each reflection with its own partner. It is usually presented as a second statistic with its own blind spots. It is better than that, because of one exact identity, which turns a statistical distribution into a hard geometric fact and gives the test a feature that nothing other than a twin can produce.

Twinning keeps a pair’s sum and shrinks its difference #

Take a pair of reflections related by the twin law, with untwinned intensities $I_1$ and $I_2$. Twinned at a fraction $\alpha$ they become

$$J_1 = (1 - \alpha) I_1 + \alpha I_2, \qquad J_2 = \alpha I_1 + (1 - \alpha) I_2.$$

Add them and the $\alpha$ cancels: $J_1 + J_2 = I_1 + I_2$. Subtract them and it does not: $J_1 - J_2 = (1 - 2\alpha)(I_1 - I_2)$. Twinning is a mixing of the pair that keeps their total and multiplies their difference by $1 - 2\alpha$, for every pair, exactly, whatever the structure. The two numbers are the eigenvalues of the twin’s mixing matrix, one for the sum and $1 - 2\alpha$ for the difference, and nothing statistical has entered yet.

The consequence is a wedge. If one intensity of a pair is zero and the other is not, the twinned pair is in the ratio $\alpha$ to $1 - \alpha$, and that is as unequal as a twinned pair can be. At $\alpha = 0.3$ no pair can have one member less than three sevenths of the other. The left half of the figure has pairs in every direction from the origin, some with one member almost nothing beside the other. The right half has none outside two straight lines, and the lines are not fitted to the points: they are drawn from the fraction alone, before any intensity is computed. This is not a tendency of the twinned distribution, the kind of statement the moment test makes, but a bound that every single pair obeys.

Uniform up to an edge #

The bound becomes a distribution through the contrast of a pair, $H = |J_1 - J_2| / (J_1 + J_2)$, which is zero for equal partners and one when one partner is absent. By the identity, a twinned pair’s contrast is its untwinned contrast multiplied by $1 - 2\alpha$, pair by pair. Over three hundred pairs and three fractions the identity holds to $3 \times 10^{-16}$, which is the rounding of the arithmetic.

What the untwinned contrast looks like is a fact about structures in general. For a structure without a centre of symmetry, with enough atoms that its intensities follow the acentric Wilson distribution that whether there is a centre is a statistic derives, two reflections not related by any symmetry of the structure have independent intensities, each exponentially distributed with the same mean. The ratio of one to their sum is then uniform between zero and one, and so is the contrast. The untwinned contrast is uniform on $[0, 1]$, and the twinned contrast is uniform on $[0, 1 - 2\alpha]$. Its mean is $\frac12 - \alpha$, and its largest value is $1 - 2\alpha$.

The cumulative distributions in the figure are straight lines, as uniform distributions must be, and each stops at its edge. Between the untwinned line and the twinned ones there is no change of shape, only of scale. That is the content of the identity seen through a distribution: twinning does not reshape the contrast, it shrinks it. Yeates made the mean of $H$ into the standard test in 1988, and the edge is where the same identity shows most plainly.

Three readings of one fraction #

The identity gives two ways to read $\alpha$ from the pairs. The mean contrast gives $\frac12 - \langle H \rangle$. The largest contrast gives $(1 - \max H)/2$. The second moment of all the intensities gives a third, independent reading, from the quadratic $2 - 2\alpha(1 - \alpha)$.

For exact intensities the edge is the best of the three by a long way. Three hundred contrasts drawn uniformly up to an edge include one within about a three-hundredth of it, so the largest measured contrast reads the fraction as 0.000, 0.100, 0.200, 0.300, 0.400 and 0.450, each to the third decimal. The mean reads it to within a hundredth, 0.009 to 0.451, which is the scatter of an average of three hundred numbers. The moment drifts upward, reading 0.33 for a twin of 0.30 and 0.48 for one of 0.40, and at 0.45 it refuses, because the sample’s moment has crossed below the perfect twin’s value of one and a half. The moment is a function of the whole distribution’s shape and pays for it; the pairs are a function of the twin alone.

That ranking is for intensities without measurement error, and it reverses for the edge once error enters. A largest value is the statistic most sensitive to a single bad measurement, and a weak reflection measured at the wrong intensity can put one pair anywhere. Practical pair tests therefore use the mean and the cumulative curve, and fit the edge rather than read it off one pair. The edge’s value lies elsewhere. It is a shape that nothing but a twin produces, and a shape survives noise in a way a single maximum does not.

What a near-symmetry does #

The pair test’s known weakness is a structure that nearly obeys the twin law. A protein crystal often holds two copies of its molecule related by a rotation that is almost, but not quite, a symmetry of the lattice, and when that rotation lies along a possible twin law, the partners under the law scatter alike without any twinning at all. Their contrasts are small because the structure resembles its own image, and the mean contrast drops as if the crystal were twinned.

The structures in the next figure are built that way: twelve atoms placed freely and each given a partner at its image under the law, displaced by a small random amount. With partners straying by up to three hundredths of a cell, the mean contrast reads a twin fraction of 0.16 from an untwinned crystal. The mean alone cannot tell this from a real twin.

The edge can. A near-symmetry makes the partners correlated, and correlated is not the same as confined. Most pairs have small contrasts, but nothing stops an occasional pair from differing completely, and the largest contrast of these structures stays between 0.97 and one. So the distribution is squeezed toward zero without being cut off, and its shape is no longer uniform. The ratio of the mean contrast to the largest measures exactly that. For a twin it cannot change: every contrast is multiplied by the same factor, which cancels from the ratio, so the ratio is the untwinned value, 0.491 for this structure, at every fraction from zero to 0.45. For the near-symmetric structures it falls to between 0.22 and 0.46, the stronger the resemblance the lower.

The resemblance also has a resolution. Two atoms a few hundredths of a cell apart scatter alike at low resolution and differently at high, because a reflection of high index resolves the displacement and a low one does not. A twin knows nothing about resolution, since it acts on intensities after they are scattered.

The figure separates the two cases at a glance. The twin’s mean contrast sits near 0.2 in every shell and its edge at 0.40 in every shell. The near-symmetric structure’s mean starts at 0.23, as low as the twin’s, and climbs to 0.44 by the fourth shell, while its largest contrast is 0.85 or more in every shell. A crystallographer who sees a low mean contrast only at low resolution, with no edge, is looking at a molecule’s resemblance to itself rather than at a twin. Near-symmetry and the tolerance found the same issue in coordinates. “Nearly symmetric” acquires a meaning only once a scale is chosen, and here the reflections choose the scale shell by shell.

When the law is exact #

Push the resemblance to the limit, with every partner exactly at its image, and the law becomes a symmetry of the structure. Every pair is then equal, every contrast is zero, and the pair test reports a perfect twin, $\alpha = \frac12$, from both its mean and its edge. The moment test, which never looks at partners, sees an ordinary acentric distribution and reports a fraction of 0.07, within the scatter its reading of the plain untwinned structure also carries.

The disagreement is correct, and it is diagnostic. A crystal that obeys its supposed twin law exactly has been assigned too low a symmetry. Its true point group contains the law, and as a twin is a symmetry the crystal lacks puts it, a twin law can only be an operation the lattice has and the structure does not. A perfect twin and a missed symmetry give the pair test the same zeros, and only the moment, at one and a half for the twin and two for the symmetry, tells them apart. The pair test cannot distinguish a perfect twin from an unrecognised symmetry, and the moment test can.

A translation that hides a twin #

The reverse case is the one the moment test cannot survive. A pseudo-translation, half the structure copied onto the other half by a shift of half a cell, sorts the reflections into a strong class and a weak class by the parity of $h + k$. Normalising them together, as the moment test does, makes the distribution bimodal and wide, and pushes the moment up toward the centric value of three. Twin that crystal at 0.30 and the moment comes out at 2.37, above the untwinned acentric value of two, so the moment test does not merely misjudge the fraction. It refuses to report one, because a moment above two is the signature of a centre, not of a twin.

The pairs read 0.317 from the mean and 0.300 from the edge. The reason is that this law, $(h, k) \to (k, h)$, keeps $h + k$, so both partners always come from the same class. Within a class the intensities are again exponential with one mean, and the contrast, being a ratio, does not care what that mean is. It is the same reason the average that knows the atoms normalises shell by shell: a ratio taken inside a population is immune to whatever scales the population as a whole. The pseudo-translation scales the two classes differently and the contrast divides the scale out. A law that exchanged the classes would mix strong reflections with weak ones and give large contrasts from the pseudo-translation alone, and the test would need to be run class by class. Whether a law keeps or swaps the classes is a fact about the law and the translation, decidable before any intensity is measured.

The table is the argument of this essay in one place. Each of the three tests has a blind spot, and no two share one. The moment is defeated by a pseudo-translation. The mean contrast is defeated by a near-symmetry. The edge is defeated by an exact symmetry, which it reads as a perfect twin, and by measurement error, which blurs it. A crystal about which all three agree is either untwinned or twinned at the fraction they agree on, and a disagreement points at its own cause: a low mean with no edge at a near-symmetry, an edge with a moment out of range at a pseudo-translation, zeros with a moment of two at a missed symmetry.

That is a different use of statistics from the one the centre test makes. How many reflections a centre needs found that the centricity test can only be made more reliable by measuring more reflections, because it has one statistic and one way to be wrong. The twin tests have three statistics with different failure modes, and their disagreement carries information that no amount of extra data for any one of them would supply.

Naming the law #

The pair test asks a question the moment test cannot ask: which law? The moment sees only that something has narrowed the distribution. The pairs can be formed under each candidate law of the lattice in turn, and twinning by merohedry counts how many candidates there are, the index of the crystal’s class in its lattice’s point group, less one.

Run with the true law, the test finds the edge at 0.400. Run with the law $(h, k) \to (-k, -h)$, it finds exactly the same numbers, pair for pair, because that law is the true one followed by the inversion, and Friedel’s law makes every reflection exactly as strong as its negative when there is no anomalous scattering. The pair test therefore names a law only up to inversion, which is the same ambiguity diffraction always has. Run with the two axial mirrors, the test finds edges of 0.994, no twin.

The mirrors’ mean contrast is 0.363, which by the formula would report a twin of 0.14 under a law that is not the twin law at all. That is the twin showing through the wrong question. The intensities being compared are themselves twinned, so each is the sum of two independent intensities and less variable than an untwinned one, and two less variable numbers differ less. The mean contrast under any law picks up the narrowing that the moment test measures. Only the edge belongs to the law, and so only the edge can say which law it is.

What the account has to refuse #

The refused claim is the one the wedge forbids: a crystal twinned at 0.3 with a pair whose contrast exceeds one half. The largest of three hundred is 0.3999, below 0.4 by the amount a uniform sample of that size leaves. The other tests guard the account’s moving parts: the identity to rounding error, the edge at six fractions, the near-symmetric structure’s low mean with an edge near one, the twin’s ratio fixed to six figures. They also check the pseudo-translated twin’s refusal by the moment and its exact reading by the pairs, and the Friedel equivalence of a law with its negative.

What the figures cannot show #

Every intensity here is computed exactly from a model structure, with no counting noise, no absorption and no error in the scale. That is why the edge is sharp. A real data set has an error on every reflection, and a weak reflection’s error is comparable to its intensity. A pair whose true contrast is near the edge can then be measured beyond it, and a single badly measured pair can put the largest contrast anywhere. The edge of a real data set is a slope, not a cliff, and estimating it means fitting the cumulative distribution near its top with the errors in the model. The figures show the shape that fitting is looking for and not how well it can be found.

The conventions matter too. The structures are two-dimensional, with 24 equal atoms and no centre of symmetry. For a centrosymmetric structure the untwinned contrast is not uniform, and the straight lines in the cumulative figure would be curves, though the identity, which is algebra, would still compress them by $1 - 2\alpha$. The near-symmetric structures are one kind of resemblance, every atom with a partner displaced at random, and a real molecule’s resemblance to itself is organised by its shape. Only two-domain twins are treated. A crystal with three or more domains related by a cyclic law mixes each reflection with several partners, and its sums and differences follow the eigenvalues of a larger mixing matrix.

Who worked it out #

The second-moment test for twinning is due to Stanley in 1972, the year Britton described the test that bears his name. Yeates introduced the pair-contrast statistic $H$ and its uniform distribution for acentric data in 1988, and he and Padilla added a local test less sensitive to anisotropy and pseudo-translation in 2003. That twinning multiplies each pair’s difference by exactly $1 - 2\alpha$ is the elementary algebra under all of these. The edge as the signature that separates a twin from a near-symmetry, the scale-free ratio of the mean contrast to the largest, the resolution test and the six-case table are made here.

Still open: the cost of undoing a twin #

Knowing $\alpha$ and the law, the twin can be undone. The mixing matrix is inverted, each pair’s difference divided by $1 - 2\alpha$ and its sum kept, and the untwinned intensities come back exactly, for exact data. For measured data the division multiplies every pair’s error along with its difference. As $\alpha$ approaches a half the factor grows without bound, and an estimate of $\alpha$ that is slightly too large makes some intensities negative, which no intensity can be. How the cost grows, and what the negative intensities say about the estimate, is a separate calculation with the same identity at its centre. How many reflections a centre needs asked the corresponding question of the centricity test, and every reflection several times over is where the redundancy that pays for it comes from.