The same measurements, and a hole that stays

Every reflection, several times over ended on the question an experimentalist actually asks: given a target redundancy and a Laue class, how much of the sphere has to be swept? What one turn of the crystal reaches had the tools for it, since it found the two angles at which a reflection reaches the Ewald sphere about a spindle, and the region around the spindle that never does. It drew completeness against rotation range for four classes, on one axis. It did not vary the axis.

Varying the axis splits the question in two, and the two parts behave in opposite ways. Redundancy cares about the sweep and does not care about the axis. Mount a crystal any way at all, and the number of measurements collected per degree is the same to within two per cent. Completeness cares entirely about the axis. One crystal, one sweep and one count of measurements can give 76 per cent of the unique reflections on one axis and 96 on another.

And in one case, a case that symmetry defines exactly, completeness has a ceiling. Some mountings have reflections that no sweep will ever record. They are not the blind reflections of the earlier essay, which are the same share on any axis. They are the reflections that are blind along with all their symmetry equivalents, and that happens exactly when the crystal is mounted with its Laue group keeping the spindle’s line. The mounting that is fastest for the first forty degrees can be the one that never finishes.

What is being counted #

The setting is the same idealisation as the earlier essay. A reflection either is or is not in diffracting position at a given angle, with no width. The beam is perpendicular to the spindle, and the crystal turns through a range $\Delta$ from a fixed start. A reflection at reciprocal vector $p$, at distance $r$ from the axis, reaches the Ewald sphere at two angles per turn if $r \geq \lambda |p|^2/2$, and at none if not. Every calculation here uses wavelength $1$ ångström and resolution $1.4$ ångströms, with the same cells as the earlier essays, and the reciprocal lattice supplies every point being counted.

Two quantities are computed from those angles. A unique reflection is an orbit of the Laue group on the reflections, the count how many reflections there are arrived at, with Friedel’s law built in, since the Laue group includes the inversion. Completeness is the share of unique reflections having at least one observation in the sweep. Mean redundancy is the number of observations in the sweep, divided by the number of unique reflections, and it counts every observation of every member of every orbit.

The count does not see the axis #

Take a tetragonal crystal, class $4/mmm$, and try four spindles: along the four-fold axis, along a two-fold axis, along another kind of two-fold axis, and along a direction that means nothing.

The four lines coincide. At forty-five degrees the means are 2.41, 2.42, 2.44 and 2.44. They rise in proportion to the sweep, and the reason is a matter of bookkeeping with no group theory in it. Each reflection that can be reached is reached twice per full turn. A sweep of $\Delta$ degrees catches each of those two crossings with probability $\Delta/360$, averaged over reflections whose starting angles are spread round the circle, and so the expected number of observations in total is

$$2\,\frac{\Delta}{360}\,N(1 - b),$$

with $N$ the number of reflections in the limiting sphere and $b$ the blind share. Divide by the unique count to get the mean. For the tetragonal cell the formula gives 2.41 at forty-five degrees, and the count on the four-fold axis is 2.41. The axis enters only through $b$, and $b$ is about the same on every spindle, three to five per cent here.

That settles the first half of the experimentalist’s question in one line. To reach a mean redundancy $R$, sweep $\Delta \approx 180 R\,U / (N(1 - b))$ degrees, whatever the axis. For this crystal, with $N = 1{,}360$ and $U = 135$ unique reflections, a mean of three takes 55 to 57 degrees on all four spindles, and a mean of six takes 111 to 112. For the orthorhombic cell the same two targets take about 96 and 190 degrees, and for a cubic one 22 to 24 and 45 to 46. The Laue group decides the number through $U$, and that is the arithmetic of every reflection, several times over. No choice of mounting helps.

The coverage sees nothing else #

With the same crystal and the same count of measurements at every sweep, the coverage is now completely different. At forty-five degrees the four-fold spindle is at 88.9 per cent, the two-fold spindles at 76, the general one at 95.6. At ninety degrees the general spindle and both two-folds are at 100 and the four-fold is still at 88.9. It stays there at 360 degrees.

The ordinary reasoning accounts for the first part. Rotating about a four-fold axis repeats what the crystal’s own symmetry would do, so each $90^\circ$ of sweep contains the same reflections, and a mirror containing the axis halves it to forty-five degrees. Nothing in the first forty-five degrees is wasted, which is why it is fast. A two-fold axis perpendicular to the four-fold needs a quarter turn for a similar reason. The general axis has no such coincidence, and it collects reflections with fewer repeats.

The histograms show where the same number of observations goes. On the four-fold spindle they fall in clumps at two and at four, because a four-fold rotation lands a reflection’s whole orbit on the sphere together, in pairs. On the general spindle each member of an orbit is on its own schedule, and the counts spread out around the mean. The mean does not change, and the shape does, and the shape is what decides how many reflections have zero. Spread observations are better for completeness, since an observation that repeats another is one more measurement and no new reflection. That is the same tension as the merging residual has, in the opposite direction. Clumped equivalents, which a twin’s statistics also depend on for a merged average, measured close together in time and under the same conditions, are the better ones for calibrating errors between them, which is why a data collection is sometimes planned for that on purpose.

The hole that never closes #

Now for the ceiling. It occurs on the four-fold spindle in $4/mmm$, on both main axes of $mmm$ tested, and on $b$ and on a direction across $b$ in $2/m$. It does not occur on any general spindle. It does not occur on any spindle in the cubic system.

The explanation is exact and uses one fact about the group. The earlier essay’s blind cusp consists of reflections too close to the spindle, those with $r < \lambda |p|^2/2$. Any spindle has one. Whether it leaves a hole in the unique set depends on where a blind reflection’s symmetry equivalents are. Its length $|p|$ is the same as theirs, and all that differs is $r$. If any equivalent is farther from the axis, it may be reachable and the orbit is safe.

Now suppose the Laue group sends the spindle’s line to itself: every operation of the group either fixes that line or reverses it. An operation that does this keeps the distance $r$ from the line, for every reflection. So every member of an orbit has the same $r$, and all have the same $|p|$. If one is blind, then all are. The orbit is blind as a whole, and no rotation about that spindle can ever see it. Conversely, when some operation moves the spindle’s line, it moves a reflection’s distance from it, and every orbit in the census had a member within reach.

The test in the calculation is direct and independent of the geometry of the cusp. For each spindle it checks whether every orbit stays at a single distance from the axis, which is what “the group keeps the line” amounts to, and it then compares that with whether the table has a ceiling. They agree on all fourteen spindles. So the rule is an exact one, and it can be stated without any computation: a mounting has a permanent gap exactly when every operation of the Laue group fixes or reverses the spindle’s line. That is an axis of the group, or a direction in a mirror plane when nothing else in the group moves it.

That last clause explains the case that first looks wrong. In $2/m$ the direction called “across $b$” is not a symmetry axis at all, and it has a ceiling of 94.7 per cent. But it lies in the mirror plane, and the mirror fixes it, and the two-fold axis reverses it, so the whole group keeps its line. Any direction at all in the plane across $b$ has that property. In a monoclinic crystal, then, an entire plane of orientations has a permanent gap, whereas in $4/mmm$ the bad spindle is a single direction. And in the cubic classes there is none at all: no direction is kept by the whole group, since the three-fold axes move every line. All four cubic spindles here reach 100 per cent, and they do it in 27 to 39 degrees.

The practical lesson is familiar to anyone who has collected data, and here it has an exact statement. Crystals are commonly mounted a little away from a principal axis to avoid the cusp. The usual reasoning is that it fills in the blind region. The computation says when that is necessary and when it is not. A small offset is enough only if it makes the group move the spindle, and a mounting that is off the principal axis but still in a mirror plane of a monoclinic crystal gets no benefit at all.

The size of the hole is small but not negligible. Fifteen of 135 unique reflections on the four-fold spindle are 11 per cent, and they are not a random 11 per cent. They are the reflections with the longest component along that axis, those at $l$ large and $h, k$ small, which are the very ones that give resolution along that direction. A map computed without them has anisotropic resolution, of the kind as sharp as the sphere is wide computed for a truncated sum, for a reason that is a simple result of how the crystal was glued on. And a structure’s count of unknowns against observations is reduced by exactly those reflections.

The trade, in degrees #

Those facts combine into a choice, and the table puts numbers on both sides of it. For each class there is a spindle that finishes quickly and leaves a hole, and a spindle that takes longer and leaves none.

In $4/mmm$ the four-fold spindle reaches everything it will ever reach by 40 degrees and stops at 88.9 per cent. A general spindle reaches the same 88.9 per cent sooner, in fact, and needs 87 degrees to finish at 100. In $mmm$ the principal axes finish by 86 to 88 degrees with 93 to 95 per cent, and the general axis needs 167 for all of it. In $2/m$ the two-fold spindle reaches 95.4 at 164 degrees, and the general one reaches 100 at 173, which is hardly more. A direction across $b$ gets 94.7 in only 88.

These numbers hold at one resolution and one wavelength; the cusp grows with the wavelength, as the earlier essay showed, so at longer wavelengths the ceiling is lower and the hole wider. So the textbook rule, a required rotation of 45, 90 or 180 degrees according to the class, is the sweep at which a principal-axis mounting stops gaining and not the sweep at which the data are complete. That difference is the ceiling. In the monoclinic class the cost of removing the ceiling is small. In $mmm$ removing it means nearly doubling the sweep, because a generic spindle sees each reflection’s equivalents separately, and the last few must wait for their own crossing. Since the mean redundancy depends on the sweep alone, a doubled sweep also doubles the mean redundancy, so that second half is not wasted: the extra observations are extra measurements, and they would have been wanted for scaling and merging anyway.

The cubic classes show what high symmetry buys beyond a smaller $U$. Their sweeps are short, 27 to 39 degrees, and no mounting has a ceiling. So in a cubic crystal mounting need not be thought about at all as far as completeness goes, and the least sweep is on a three-fold axis, by nearly a third. For every other class the mounting is the larger decision.

What the idealisation hides #

Real reflections have width, and a real rotation camera integrates over a range of angle. So “blind” is not quite as sharp as it is here, and reflections near the edge of the cusp are recorded partly. The share would shift by a little, and a rule about whole orbits would become a rule about most of an orbit. Nothing about the group argument depends on width, but the fifteen is a count for one idealised instrument.

The sweep here starts at a fixed angle and takes the range from zero. A different start changes which reflections are in a short sweep. For the general spindle that matters: its 95.6 per cent at forty-five degrees is for the one starting angle computed, and the mean over starting angles would be a little different. For the claims here that does not change the conclusion, since the means do not depend on it and the ceiling is reached at any start, but the crossing points in the curves are specific.

The mean-redundancy formula assumes starting angles spread evenly, and it is a statement about large counts. For a small cell or a short sweep the count departs from it. And all of this leaves out everything that makes a real experiment hard: absorption, radiation damage, a detector of finite size, the bias of measuring the same reflection in the same geometry each time. An experimentalist’s actual choice weighs those too, and this page counts only what symmetry and the Ewald construction require.

What the table was checked against #

The kept-line rule is tested directly against the result it explains, and it would fail at once if a spindle had a ceiling without the group keeping its line. That belief about 45 degrees is the one a person might take away from a textbook table of required rotations, which usually gives a single number per Laue class, on the assumption that the crystal is mounted along a principal axis. It is refused because on the two-fold axis of the same class, at the same 45 degrees, completeness is 76 per cent. The mean redundancy formula is compared with the count and agrees to within one per cent, so a mistaken factor of two, an easy error in counting crossings per turn, cannot hide.

Still open: two axes #

Modern diffractometers have more than one rotation axis, and a kappa geometry allows a second sweep at a different orientation. The rule here says what a second sweep must do. Any orientation the group moves away from the first spindle’s line will fill the hole. So the minimum for full completeness is not the sum of two sweeps’ fractions, and a very small second sweep, about a generic axis, might be all that is needed: the fifteen missing reflections occupy a small cone, and they could be reached with a few degrees. How much a second sweep must cover, as a function of Laue class and of the angle between the two axes, is a clean optimisation with an answer in degrees. It belongs to the same arithmetic as above, and it would turn a rule of thumb about remounting into a number. It has not been computed here.