Every reflection, several times over

Counting the reflections inside a sphere of resolution gave a number: how many reciprocal lattice points a measurement can reach, and therefore how many observations are available to determine a structure. That count treated every point as distinct.

Most of them are not. Symmetry relates hkl to the other members of its orbit under the crystal’s Laue class, and those are not different reflections but the same reflection reached from another direction. A measurement that visits all of them has measured one quantity several times.

Two numbers with different jobs #

Multiplicity is a property of a single reflection: the size of its orbit under the Laue group. It says how many of the reciprocal lattice’s points carry the same intensity.

Redundancy is a property of a data set: how many measurements it contains per distinct reflection. It is what an experiment is planned around, because repeated measurements of quantities symmetry says must agree are the only estimate of precision that does not come from a model.

The two are related and are not the same, and the difference is the special reflections.

Why redundancy is below the group’s order #

A first guess is that redundancy equals the order of the Laue class: a group of order eight relates each reflection to seven others, so each is measured eight times.

It does not, because some reflections are fixed by some operations. A reflection lying on a symmetry axis is carried to itself by rotations about that axis; a reflection in a mirror plane is carried to itself by that mirror. Its orbit is shorter than the group, and it is measured fewer times.

Those are the special positions of reciprocal space, and they are the same kind of object as the special positions of direct space — the Wyckoff positions where a molecule sits on a symmetry element and its multiplicity drops. The arithmetic is identical: an orbit’s size is the order of the group divided by the order of the stabiliser.

What the classes buy #

The census sorts the eleven Laue classes by what they are worth to an experiment, and the range is large.

At the bottom is 1̄, the triclinic class, whose only operation besides the identity is the inversion that diffraction supplies whether or not the crystal has it. Its redundancy is exactly two and every orbit has exactly two members: hkl and −h−k−l, always, with no exceptions and no special positions at all.

At the top is m3̄m, of order forty-eight, whose redundancy is twenty-one. That is far short of forty-eight, and the shortfall is the largest of any class, because a cubic group has a great many symmetry elements for a reflection to lie on.

In between the pattern is monotone but not proportional: doubling the order does not double the redundancy, because a larger group has more elements and also more places for a reflection to be special.

What redundancy is for #

Three things, and the third is the one that is usually left implicit.

Precision. The spread among measurements that must agree estimates the error on each. That estimate — the merging residual — is reported with every crystal structure ever published, and it has no meaning at all without redundancy: a data set measured once has nothing to merge.

Detection of systematic error. Two measurements of one reflection made at different crystal orientations pass through different amounts of the crystal, are absorbed differently, and see different parts of the detector. Their disagreement is a measurement of those effects, which is how absorption corrections are derived from the data rather than from a model of the crystal’s shape.

Completeness. A crystal cannot be turned through every orientation, so some parts of reciprocal space are inaccessible. High symmetry means the inaccessible parts are copies of accessible ones, and a complete data set can be gathered from a smaller sweep. That is the practical reason a cubic crystal is quicker to measure than a triclinic one.

The orbit is the unit, and partial orbits are refused #

A detail of the counting turns out to matter and is worth recording, because getting it wrong produces a table that looks right.

The census counts orbits inside a block of indices. A hexagonal operation carries a reflection out of a cube of indices, so a block contains partial orbits: some members inside, some outside. Counting the inside members as measurements while counting the whole orbit as one reflection inflates the measurement total and depresses the redundancy — by half a unit on the trigonal classes, which is enough to make a table look plausible and be wrong.

So an orbit is counted only when it lies wholly inside the block. That drops the trigonal and hexagonal counts to a smaller sample and leaves them correct, which is the right trade: the orbit is the unit being counted, so a partial orbit is not counted at all.

This is the same discipline as the ancillary counts elsewhere in this field, where a shell of reflections is counted with its edge handled explicitly rather than by rounding.

A worked class, index by index #

4/mmm is the clearest case to walk through, because its sixteen operations produce four different orbit sizes and each has a legible reason.

A general reflection hkl with h, k and l all different and all non-zero has the full sixteen. Nothing fixes it.

A reflection hhl — equal first two indices — lies in a diagonal mirror, which fixes it, so its orbit is eight.

A reflection h0l lies in an axial mirror, and again the orbit is eight.

A reflection 00l lies on the four-fold axis, which fixes it entirely along with the mirrors containing the axis; its orbit is two, being itself and its Friedel opposite.

Adding those up over a block gives the distribution in the figure above, and the redundancy of 9.8 is the weighted mean. A reader who wants to check the arithmetic can do it on a corner of an envelope, which is the reason this class rather than a cubic one is the worked example.

Multiplicity in a powder pattern, which is a different thing #

The word “multiplicity” appears in powder diffraction with a related but distinct meaning, and confusing the two is easy.

In a powder the crystallites take every orientation, so all the reflections of one orbit arrive at the same scattering angle and their intensities add. The multiplicity is then a factor in the intensity of a line rather than a count of measurements: a line from a reflection with multiplicity twenty-four is twenty-four times as strong as the same reflection alone would be.

That is why a powder pattern loses so much: a single-crystal experiment measures the members of an orbit separately and can check that they agree, while a powder measures their sum and cannot separate them again. The same number counts a redundancy in one experiment and an inseparable overlap in the other.

What a merging residual measures #

Redundancy exists to be spent, and the thing it is spent on is a single number reported with every structure: the merging residual, which compares the measurements within each orbit.

Its definition is a sum over orbits of the deviations of the members from their mean, divided by the sum of the measurements — so it is a relative spread, and it is only defined where an orbit has more than one member. A data set of redundancy exactly one has no merging residual at all.

Two things make it the most informative single number in a data set’s header. It is model-free: nothing about the structure enters, only the symmetry, so it is a measurement of the experiment rather than of the interpretation. And it is sensitive to the wrong class: a data set merged in a class larger than the crystal’s shows a residual far worse than its counting statistics allow, which is often the first sign that a space group has been assigned optimistically.

That second use is the one worth remembering. A residual that is bad in one class and good in a subgroup is a data set telling the experimenter which class it belongs to — and it does so with the same orbit arithmetic that this essay counts.

The unique set, and where its boundary is #

A data set is usually reduced to a unique set: one representative per orbit, with the measurements of each orbit merged. Choosing the representative is a convention, and the conventional choice is the one satisfying a set of inequalities on the indices — the asymmetric unit of reciprocal space.

That region has the same awkwardness as the asymmetric unit in direct space: its interior is unambiguous and its boundary needs a rule, because a reflection on the boundary is equivalent to another point of the boundary and only one of them may be kept. The International Tables specify the boundary for each class, and the specification is a convention rather than a theorem, exactly as the direct-space one is.

Nothing about the count depends on the convention. The number of orbits is a property of the group and the block, and any consistent choice of representatives finds the same number of them — which is why this collection’s habit of counting orbits rather than tabulating regions gives an answer no convention can move.

What the count assumes #

Two assumptions, both of which are ordinary and both of which can fail.

That the crystal has the symmetry it is assigned. If a structure is assigned to a Laue class larger than its own, reflections that are not related will be merged, and the merging residual will be poor. That is a diagnostic rather than a disaster: a data set merging badly in a class and well in a subgroup is a data set saying which class it belongs to.

That the intensities are the same for related reflections. Friedel’s law makes hkl and its opposite equal only in the absence of anomalous scattering, and the whole determination of absolute structure rests on measuring the difference between them. A data set merged as though the law held exactly has thrown that difference away.

Both are cases of a symmetry being assumed rather than measured, in a field whose central discipline is that diffraction adds a symmetry the crystal need not have.

Completeness, and the wedge it is measured against #

A data set’s completeness is the fraction of the unique reflections in a resolution range that were actually measured, and it is quoted alongside redundancy as a second summary of coverage.

Both are counts against the same denominator: the number of orbits. So the arithmetic of this essay decides both, and the two answer complementary questions — how many of the possible reflections were reached, and how often each of the reached ones was measured.

An experiment can trade one for the other. Sweeping a small range of orientations many times gives high redundancy and poor completeness; sweeping widely once gives the reverse. Which trade is right depends on the class, because a high-symmetry crystal reaches completeness from a small sweep and can spend the rest of the time on redundancy.

That is the practical face of the orbit arithmetic, and it is why the eleven Laue classes are not an abstract list to a practising crystallographer: they are a table of how much work a given crystal will be.

The same orbit arithmetic, four times over #

It is worth collecting, because this collection now has the same computation in four places and each time it counts something different.

A form is the orbit of a face under the point group, and its size is how many faces the crystal must show together.

A Wyckoff position is the orbit of a point in the cell, and its size is how many atoms one independent atom accounts for.

A star is the orbit of a wavevector under the point group, and its size is how much of a calculation over the zone is a copy.

A reflection’s multiplicity is the orbit of a reciprocal lattice point, and its size is how many times an experiment measures one number.

All four are the same theorem — orbit size is the group’s order divided by the stabiliser’s — worn four different ways, and a reader who has met one has met them all.

The set that is unique twice over #

There is a second unique set, it is twice the size of the one above, and which of the two an experiment wants is decided before the crystal is mounted.

Merging hkl with −h−k−l assumes Friedel’s law, which is exact only when no atom scatters anomalously. An experiment intending to use the anomalous differences must not merge them: it keeps the two halves separate, and its unique set is the Bijvoet-unique or anomalous set rather than the ordinary one.

The arithmetic is immediate. Splitting every Friedel pair doubles the number of unique reflections and halves the redundancy of each — so a data set with a redundancy of ten merged in the ordinary way has a redundancy of five on each half. The measurements have not changed; the accounting has.

That is a real trade and it decides how an experiment is designed. The anomalous differences are small, a per cent or two, so measuring them needs high redundancy on each half — which means collecting perhaps four times as much data as a structure determination alone would require, and collecting it in a way that puts the two members of each pair on the same detector region and at the same time, so that the systematic errors cancel between them rather than between distant measurements.

And a data set merged the wrong way has thrown the signal away irrecoverably. Merging Friedel pairs averages the very difference the experiment was for, and no later processing recovers it, which is why the choice is made at the start.

What the residual measures, and what replaced it #

The merging residual is described above as the most informative number in a data set’s header, and its definition has a defect that took the field a long time to correct.

The residual sums deviations from a mean and divides by the sum of the measurements. A mean of many measurements is closer to the truth than a mean of few, so the individual deviations from it are larger when there are more of them — and the residual therefore rises with redundancy even when the data are getting better. Two data sets of identical quality, one measured four times over and one twelve, report different residuals, and the better one reports the worse number.

Two repairs were made. A redundancy-weighted version corrects the statistic so that it does not depend on how many measurements went into each mean; and a precision-indicating version reports instead the error on the mean, which is what the redundancy was bought for and which falls as the square root of the count. The three numbers answer different questions and all three are quoted, which is why a modern header has several residuals where an older one had one.

The resolution cutoff moved for the same reason. Deciding where the data end by asking where the residual passes some value inherits every defect the residual has, so the modern criterion is a correlation between two halves of the data — split the measurements of each reflection into two sets, average each, and correlate the two lists. That is a direct measure of whether the shell carries reproducible signal, it does not depend on redundancy, and it is what the orbit arithmetic on this page is ultimately being spent on.

Where this goes #

The nearest question is the one an experimentalist actually asks: given a target redundancy and a Laue class, how much of the sphere has to be swept? That is a geometry problem with the orbit arithmetic inside it, and it belongs with the resolution essays rather than here.

The nearer neighbour is the twin hiding in the statistics, which is what happens when the symmetry a data set appears to have is larger than the crystal’s — and where the merging residual, computed from exactly the redundancy this essay counts, is one of the first signs that something is wrong.