The second spindle must lean past one Bragg angle

The same measurements, and a hole that stays separated two things a diffraction experiment is usually asked to deliver at once. The mean number of times each reflection is measured depends on how far the crystal is turned and on almost nothing else. Completeness, the fraction of distinct reflections measured at least once, depends on the axis. And on one kind of axis it has a ceiling that no amount of turning lifts: an axis the Laue group keeps. A tetragonal crystal turned about its four-fold axis, at a resolution of 1.4 ångström, never measures one reflection class in nine, however many full turns are made.

The essay closed by saying how the hole must be filled. Any second axis that the group does not keep will see the missing reflections eventually, so the second sweep does not have to repeat the first. It only has to reach a small set of reflections lying close to the first axis. How much it has to turn, as a function of the Laue class and of the angle between the two axes, was left as a clean calculation with an answer in degrees. This essay makes it, and the answer turns out to have a threshold with a precise physical meaning.

Where the hole is #

A crystal turned about an axis brings each reciprocal lattice point through the Ewald sphere twice per turn, except for points close enough to the axis. Those never touch the sphere. The condition is simple to state in angles. A reflection with Bragg angle $\theta$, so that $\sin\theta = \lambda / 2d$ for spacing $d$ and wavelength $\lambda$, is blind about an axis exactly when its own direction makes an angle with the axis smaller than $\theta$. The blind region is a cone about the axis, widening with resolution, and at 1.4 ångström with a wavelength of one ångström its widest part is twenty-one degrees.

A blind reflection is not lost if a symmetry mate of it is measured, because the two have the same intensity and only one needs to be seen. That is what what one turn of the crystal reaches relied on: for a general axis the blind cone holds reflections whose mates lie elsewhere, and completeness reaches one hundred per cent. When the group keeps the axis, every operation carries the axis to itself and each reflection to another reflection at the same angle from it. So a blind reflection’s mates are all blind, and the whole orbit is missing.

The figure shows the hole for the tetragonal case, on a cell eight by eight by fourteen ångström. The reflections lie in two groups. Most sit exactly on the axis: the $00l$ reflections from $l = 1$ to $10$, each with a Bragg angle that grows with $l$. A few sit slightly off it, at eleven to seventeen degrees, but with Bragg angles larger still. Every blind reflection is in the hole, as the argument above requires, and the census checks it: no blind reflection has a mate that is seen.

The threshold #

Now add a second axis, tilted from the first by an angle $\chi$. A reflection in the hole is reachable from the second axis if its angle from that axis exceeds its Bragg angle. For a reflection lying on the first axis, its angle from the second axis is exactly $\chi$. So such a reflection becomes reachable as soon as $\chi$ exceeds its Bragg angle, and not before.

The reflections on the first axis are therefore the hardest to reach, and among them the outermost, with the largest Bragg angle, sets the bar. For reflections off the axis the second axis can do better or worse depending on which way it leans, but in every case computed they become reachable earlier. So the least tilt that fills the hole is the Bragg angle of the outermost reflection on the first axis. It is a property of one reflection, and so of the cell length along the axis and of the resolution.

The figure tests the rule on three crystals. For the tetragonal one, the outermost axial reflection is $(0, 0, 10)$, at exactly 1.4 ångström, with a Bragg angle of 20.92 degrees; the count of unreachable orbits reaches zero at 21 degrees on a quarter-degree grid. For an orthorhombic cell seven by eleven by thirteen, turned about $c$, the outermost axial reflection is $(0, 0, 9)$ at 1.44 ångström, with a Bragg angle of 20.25 degrees; the count reaches zero at 20.5. For a monoclinic crystal turned about its two-fold axis $b$, twelve ångström long, the outermost axial reflection is $(0, 8, 0)$ at 1.5 ångström, because $(0, 9, 0)$ falls outside the resolution limit. Its Bragg angle is 19.47 degrees, and the count reaches zero at 19.5.

That last case is the instructive one. The threshold is not the widest point of the blind cone, which is 20.9 degrees at this resolution for every crystal. It is set by which reflection actually lies on the axis, and the monoclinic cell’s length along $b$ happens to leave a gap between its last axial reflection and the resolution limit. A crystal measured to the same resolution with a slightly different cell would need a slightly different tilt.

The steps on the way down are the other axial reflections. As $\chi$ increases, the $00l$ reflections become reachable one at a time, lowest $l$ first. The off-axis members of the hole drop out somewhere in the middle. The last reflection to drop out is always the one that defines the threshold.

Why the reflections off the axis are easier #

The rule leaves something to explain. A reflection in the hole that sits off the first axis, at some angle $\alpha$ from it, can be closer to the second axis than $\chi$ is. If the second axis leans towards it, the angle between them is only $\chi - \alpha$, and that reflection would need a tilt of $\theta + \alpha$ to be reached, more than the axial threshold. On its own, an off-axis reflection could set a higher bar than the axial ones.

It does not, because it is never on its own. The same symmetry that put it in the hole put its mates there too, at the same angle from the first axis but spread around it. In the tetragonal crystal a reflection off the four-fold axis has mates at four or eight azimuths around it, and whichever way the second axis leans, some of those mates lie on the far side, at angles near $\chi + \alpha$ from it. Only one mate needs to be reached. So the orbit is reachable as soon as its best member is, which happens at a tilt of $\theta - \alpha$ or less, below the axial threshold. The symmetry that makes the hole is the same symmetry that makes most of it easy to fill.

The reflections on the axis get no such help. Their mates, $(0, 0, l)$ and $(0, 0, -l)$, both lie on the axis, at exactly the angle $\chi$ from any second axis whatever its azimuth. That is why the axial reflections, and only they, set the threshold, and why the threshold does not depend on which way the second axis leans. The census checks that too. For the orthorhombic crystal, leaning the second axis towards $a$, towards $b$ or towards the diagonal between them gives the same least tilt, 20.5 degrees, and arcs at a right angle of 19.2, 19.6 and 18.3 degrees. For the tetragonal crystal, leaning towards $a$ or towards the diagonal gives identical arcs to a tenth of a degree, because the four-fold axis makes those directions nearly equivalent for the hole.

There is a general lesson here about kept axes. An axis the group keeps is special for two opposite reasons at once. It is the only kind of axis that leaves a hole, because only a kept axis maps the blind cone onto itself. And the hole it leaves is mostly made of orbits that are spread around it, so any departure from it recovers them quickly. What is left is the axis itself, a single row of reflections that no rotation of the crystal about its own symmetry can move.

The table at three resolutions #

The rule holds in all nine combinations of class and resolution, to within the quarter-degree step of the search. At two ångström the hole is small, seven orbits in each class, and the threshold is about fourteen degrees. At one ångström the hole is large, from 42 orbits in the tetragonal case to 127 in the monoclinic, and the threshold is thirty degrees for all three, because at that resolution each cell has an axial reflection exactly at the limit.

Half a degree short of the threshold, some orbit in the hole is still out of reach, in every case. That is checked separately rather than inferred, because a rule found on a grid can be an artefact of the grid. It is not. A second axis leaning ten degrees from the four-fold axis of the tetragonal crystal still leaves six of its fifteen missing orbits blind, however far it turns.

The arc, once past the threshold #

Reaching the hole is necessary; reaching it cheaply is the practical question. For each tilt the calculation finds the shortest arc about the second axis, from the best starting angle, that brings every missing orbit into diffracting position at least once. Just past the threshold the arc is long, nearly eighty degrees for the tetragonal crystal, because the outermost axial reflection is barely outside the second axis’s blind cone and only touches the Ewald sphere at a narrow range of angles. As the tilt increases the arc shortens quickly: forty-two degrees at thirty, twenty-eight at forty-five, and 18.9 at a right angle.

The three curves nearly coincide. The orthorhombic and monoclinic cases need 19.2 and 19.6 degrees at a right angle. At a right angle, the missing reflections lie close to the second axis’s equator, and each one diffracts after a rotation of about its own Bragg angle from the starting position. So the arc at a right angle is close to the largest Bragg angle in the hole, a little less when the best start splits the missing reflections between the two sides. At two ångström it is about twelve degrees and at one ångström about thirty.

So the minimum for full completeness is a full turn about the first axis plus about one Bragg angle about a second axis at a right angle. The second sweep is short because the hole is small and compact. It is not the sum of two sweeps’ fractions, and the second sweep adds only one twentieth to the rotation of the first.

A repair, not a plan #

That arithmetic invites a comparison it would be easy to misread. The two-axis plan needs a full turn plus about twenty degrees. One sweep about a general axis, which the group does not keep, has no hole at all.

For the tetragonal crystal a general axis sees every orbit in 87 degrees, against 379 for the two-axis plan. For the orthorhombic and monoclinic crystals it needs 167 and 173. So if the only goal were to see every reflection once, the kept axis would be the wrong place to start. The second axis is cheap as a repair: the experiment that has already turned a crystal fully about a kept axis, perhaps because the crystal grew as a needle along it and was mounted that way, needs only one more short sweep. It is not a reason to choose a kept axis in the first place.

The full turn buys something else, and every reflection, several times over is about it. A full turn about the four-fold axis measures each distinct reflection about nineteen times on average, counting all its symmetry mates, because each mate crosses the sphere twice. A general sweep of 87 degrees averages under five, and six of the 135 classes are measured only once. Repeated measurements of symmetry mates are what allow scaling and absorption to be corrected, and they are what a careful experiment wants. The two-axis plan delivers high redundancy everywhere except in the hole, where each reflection is measured once or twice by the short second sweep. Whether that is good enough depends on how much the hole’s reflections matter. They lie along one direction of the crystal, so missing them distorts the electron density in a predictable direction, as the unknowns against the observations warned about systematic gaps.

What changes on a real instrument #

The calculation idealises the experiment in the same way as the essays before it. A reflection is either in diffracting position or not, the crystal is a point, and the beam has a single wavelength. A real reflection has a width in angle, set by the crystal’s mosaic spread and the beam’s divergence, and as sharp as the sphere is wide discussed what that width does to the edge of the blind cone. A reflection at the very edge of the cone, like the outermost axial one just past the threshold, is recorded only partially, over a range of angles that may be smaller than its width. So in practice the threshold is soft by roughly the mosaic spread, and a second axis leaning only a fraction of a degree past it would give poor measurements of the reflection that defines it.

The wavelength enters the rule only through the Bragg angle. With copper radiation, at 1.54 ångström, the same reflection has a larger Bragg angle and the threshold rises; at 1.4 ångström resolution the widest Bragg angle is thirty-three degrees. With a shorter wavelength from a synchrotron, the threshold falls. The rule is the same in each case: find the outermost reflection on the kept axis and compute its Bragg angle.

Diffractometers with a kappa geometry tilt the second axis by a fixed construction angle combined with a rotation, and the effective tilt between two settings can be anything from zero to about a hundred degrees. The table says what range is worth using: anything less than the threshold is wasted, and anything near a right angle is efficient. A rule of thumb sometimes quoted for remounting, to tilt by “enough”, is replaced by a number computed from the cell.

What the pictures cannot show #

The figures plot angles and counts. They cannot show the geometry of the Ewald sphere sweeping through reciprocal space, which is where the blind cone comes from, and the reciprocal lattice is the place to see it. Nor do they show intensities. The hole’s reflections are as strong or weak as any others; nothing about the hole selects weak data, and nothing in this calculation weighs reflections by their usefulness for structure solution.

The calculation depends on one convention that should be stated. Completeness here counts orbits of the Laue group, which treats a reflection and its Friedel mate as one. For an experiment that needs the anomalous differences between Friedel mates, as the law that hides handedness explains, the two must be counted separately, the orbits are of the point group rather than the Laue group, and the hole changes. That case has not been computed.

Who worked out the pieces #

The Ewald sphere is Ewald’s construction of 1913, and the blind region about a rotation axis was understood from the earliest days of oscillation photography, when crystallographers remounted crystals about a second axis precisely to fill it. Strategy programs for modern diffractometers search over goniometer settings numerically to maximise completeness and redundancy, and they find good two-axis plans without stating the threshold. The rule that the least tilt is the Bragg angle of the outermost reflection on the first axis, its check across three classes and three resolutions, and the arc as a function of tilt are what this essay adds; how many reflections there are supplies the orbit counts the completeness is measured against.

What the calculation must refuse #

Each check could fail. The rule for the least tilt is compared against a direct search in all nine cases, and the search is also run half a degree below the predicted threshold to make sure the rule is not merely an upper bound. The claim that a modest lean, ten degrees, fills the tetragonal crystal’s hole is put to the calculation and refused, with six orbits still out of reach.

Still open: the anomalous hole #

The hole here is a hole in the Laue group’s orbits. A crystal without a centre of symmetry, measured to determine its absolute structure, needs each reflection and its Friedel mate measured separately, and its orbits are those of the point group. For a kept axis of a polar point group such as $4mm$, the reflection $(0, 0, l)$ and its mate $(0, 0, -l)$ are distinct and both lie on the axis, and the Friedel mates of the off-axis hole reflections may or may not be reachable from the first axis. The threshold for the second axis might be the same Bragg angle, or it might be larger if the mates sit on the far side of the blind cone. And because anomalous differences are small, each of those reflections ideally needs several measurements, so the second sweep’s length would be set by a redundancy target rather than a completeness one. That calculation uses the same reach about two axes with smaller orbits, and it has not been made.