What one turn of the crystal reaches

How many reflections there are counts the reciprocal lattice points inside the limiting sphere and calls them the measurable ones. That is the right count for an experiment with unlimited freedom: every point inside that sphere can be brought into diffracting position by some orientation of the crystal.

A real experiment does not have every orientation. A crystal is glued to a fibre and turned about one axis, and what a single axis reaches is less than everything — in two ways that are different in kind, and both of which are decided by a condition on one line.

The condition, and what rotation can supply #

The Ewald construction is the geometry of Bragg’s law: a reflection is in diffracting position when its reciprocal lattice point lies on a sphere of radius 1/λ passing through the origin, with the incident beam along the diameter. Written as a condition on the point p, that is

p · x̂ = −λ|p|²/2

where x̂ is the beam direction. The right-hand side depends only on the length of p, and it is what the experiment must supply.

Now turn the crystal about an axis â perpendicular to the beam. The point p moves on a circle: its component along the axis is fixed, and its component along the beam sweeps between −r and +r, where r is the point’s distance from the axis. So the reflection can be brought into diffracting position exactly when

r ≥ λ|p|²/2

and if it cannot, no amount of turning helps. The points that fail are those close to the axis relative to their length, and they fill a cusp-shaped region about it — a genuine hole in the data with a shape that geometry fixes and patience does not touch.

For an orthorhombic cell at 1.4 ångström with copper-like radiation, sixty-six of fifteen hundred and eighteen reflections are in it: four per cent, and always the same four per cent for a given mounting.

Why the wavelength makes it worse #

The condition has λ in it, and only on one side.

This is the same trade-off that decides how much of reciprocal space is reachable at all, arriving in a less obvious place. A long wavelength makes the limiting sphere small, which is the familiar cost; it also makes the Ewald sphere more curved, which makes the blind region larger as a fraction of what is left. Both push in the same direction, and neither is a defect of the instrument.

The practical answer is the same in both cases: shorter wavelengths. The reason a modern experiment uses radiation nearer half an ångström than one and a half is not only that more reflections fall inside the limiting sphere. It is also that the sphere is flatter, so the geometry of what a single axis can reach is kinder.

Completeness, which is a question about the Laue class #

The second limitation is patience rather than geometry, and symmetry is what pays for it.

A crystal turned through less than a full circle collects less than a full circle’s worth of data. But a reflection missed at one angle may have a symmetry equivalent that was caught, and symmetry relates a reflection to the others of its orbit with identical intensity — so the question is not how many reflections were measured but how many unique ones.

The numbers here are the ones an experiment is planned against. A cubic crystal is complete after a quarter turn; an orthorhombic one after ninety degrees as well, since its Laue class has order eight; a monoclinic one needs more; and a triclinic crystal, whose only symmetry is the centre that Friedel’s law supplies anyway, needs a full half turn and then some.

The shortfall at the right-hand end of every curve is the blind region, and it does not close. A triclinic crystal turned through a full three hundred and sixty degrees still misses the cusp; the only remedy is to take the crystal off the spindle and put it back on a different face, which is a second experiment.

Two angles per reflection, and what they are #

The condition above is a cosine set equal to a number, so it has two solutions in a full turn, and every reachable reflection is therefore in diffracting position at exactly two rotation angles.

That pair is not the Friedel pair, and the difference is worth keeping straight. The two angles are the two places on its circle where one reciprocal lattice point crosses the sphere; the Friedel mate is a different point, at minus the indices, with two angles of its own. So a full turn records four crossings for a general reflection and its mate, and a half turn records two of the four — which is the arithmetic behind the familiar statement that a hundred and eighty degrees suffices for a triclinic crystal when Friedel’s law is being relied on.

The two angles coincide exactly when r equals λ|d*|²/2, which is the boundary of the blind region. A reflection sitting on that boundary is tangent to the Ewald sphere: it touches and does not cross, its two angles have merged, and in a real experiment it is recorded partially or not at all. The edge of the cusp is therefore not a sharp edge in practice but a band of unreliable measurements, which is a second reason to keep the axis away from directions that put important reflections near it.

What the redundancy buys, and why it is collected anyway #

The completeness curves above flatten well before a full turn, which raises an obvious question: why does anybody collect more than the minimum wedge?

The answer is that a reflection measured once is a number with no error estimate attached to it.

Every reflection, several times over is the essay about what that buys: the agreement among symmetry equivalents is the statistic that says whether the data are good, and it exists only because the same quantity was measured more than once. A data set that is exactly complete and exactly non-redundant is a data set with no way of knowing whether it is right.

So the rotation range is chosen against three things at once — completeness, redundancy, and the radiation damage that accumulates while the crystal is being turned — and the first of the three is the one this essay computes.

The two effects, told apart #

It is worth stating the difference sharply, because a completeness figure quoted without it is ambiguous.

A range shortfall is bookkeeping. Turn further and it goes away. It depends on the Laue class, and a crystallographer choosing a rotation range is choosing how much time to spend against how much redundancy is wanted.

A blind region is geometry. Turning further does nothing at all. It depends on the wavelength, the resolution, and where the axis happens to point in the crystal — and that last dependence is why it can sometimes be arranged to matter less: mounting the crystal so that the axis lies along a symmetry direction puts the cusp where its reflections have equivalents elsewhere in the sphere, and the unique set survives even though the same reflections are missing.

The reason that trick works is worth spelling out, because it is a symmetry argument doing something practical. A cusp about a two-fold axis contains reflections whose orbit under the Laue class includes points outside the cusp, so the unique reflections are recovered from their equivalents; a cusp about a general direction contains reflections whose whole orbit is inside it. So the mounting is chosen with the point group in mind, and the completeness that results is a fact about the crystal class as much as about the diffractometer.

What is exact here, and what an experiment adds #

Everything above is geometry on the reciprocal lattice: which points satisfy a condition, and at which rotation angles. It is exact and it is idealised in a specific way worth naming.

A reflection here is a point, and a real reflection has a width. A crystal is a finite object with a mosaic spread, so a reciprocal lattice point is a small volume rather than a point, and it passes through the Ewald sphere over a range of angles rather than at an instant. That is what makes a rotation exposure possible at all: an image collected over half a degree records every reflection that crossed the sphere during it. It also softens the edge of the blind region into a band where reflections are partially recorded.

And nothing here is about intensity. A reflection can be reachable and unmeasurable because it is weak, absorbed, or behind the beamstop — the last of which removes a small region at the centre of reciprocal space that is exactly the complement of the resolution limit, and is why the lowest-angle data are often missing.

Where the axis points, and why it is chosen #

Everything above depends on one thing the geometry does not fix: which direction in the crystal the rotation axis happens to be.

That direction is set when the crystal is mounted, which in practice means when a small object of no particular shape is picked up in a loop and frozen. For a routine small-molecule structure it is not chosen at all; for a difficult one it is, and the reasoning is the one the completeness curves make visible.

A cusp about a symmetry axis is cheap. Mount the crystal with a two-fold or four-fold axis along the spindle and the reflections in the blind region have symmetry equivalents outside it, so the unique set survives even though those particular reflections are never recorded. A cusp about a general direction is expensive, because a general reflection’s whole orbit can lie inside it.

The same reasoning applies to the range. A high-symmetry crystal mounted about a symmetry axis is complete after a wedge whose width is the angle between neighbouring elements of its Laue class — which is why a cubic crystal can be finished in a few minutes and a triclinic one cannot.

None of that is available before the crystal’s orientation is known, and the orientation is determined from the first few images. So a modern data collection has a shape that this arithmetic explains: a short exposure to find the cell and the orientation, a calculation of exactly the strategy above, and then the collection itself — which is planned by running the computation in this essay on the crystal actually on the instrument.

Where the exactness stops #

Computed here: for every reciprocal lattice point of a named cell within a stated resolution, its distance from the rotation axis, its height along it, whether the Ewald condition can be satisfied by any rotation, and the two angles at which it is satisfied when it can; the blind fraction at four wavelengths; and completeness against rotation range for four Laue classes, counting a unique reflection as collected when any of its equivalents was.

Idealised: point-like reflections, a perfect crystal, a beam perpendicular to the axis, and no beamstop. Each of those changes the numbers a few per cent and none of them changes the shape of any curve.

Not attempted: the choice of mounting, which is the practical use of all of this, and which needs the orientation matrix of a real crystal rather than a cell.

The same limitation, seen from the powder side #

A powder experiment has none of this trouble, and the reason is instructive about what the trouble actually is.

Grinding a crystal up makes every orientation present at once, so every reflection inside the limiting sphere is in diffracting position somewhere in the sample at every instant. There is no blind region and no rotation range: the completeness is total and immediate. What is paid for it is the collapse a powder pattern suffers — the three-dimensional pattern folded onto one axis, with reflections of equal spacing arriving on top of one another and no way to separate them.

So the two experiments are trading the same commodity in opposite directions. A single crystal on a spindle keeps the directions and loses the orientations it never visits; a powder keeps every orientation and loses the directions. Both are complete in one sense and incomplete in the other, and which one is the right instrument is decided by whether the cell is small enough for the overlaps to be resolvable.

Who found it, and when #

Paul Peter Ewald introduced the sphere in 1913, within a year of von Laue’s experiment, and it is the construction that made the reciprocal lattice indispensable rather than merely convenient — a second lattice in which long has become short, with an experimental condition drawn on it as a sphere through the origin.

The blind region was a practical nuisance for as long as crystals were mounted on single-axis cameras, and the four-circle diffractometer of the 1960s exists to defeat it: with three rotations available, any reflection can be brought into diffracting position, and the cusp disappears. Modern area-detector experiments went back to a single rotation axis for speed, and got the cusp back with it — which is why a data collection strategy is planned in software before a crystal is exposed, and why the number the software optimises is the completeness computed above.

Where the ladder goes next #

Back, to the count this rung qualifies. How many reflections there are is the limiting sphere and the collapse symmetry performs on it.

Sideways, to what the redundancy is worth once collected: every reflection, several times over turns the multiplicity into a measurement of the errors, which is the reason a range longer than the minimum is usually collected anyway.

And to what a resolution limit costs in the map rather than in the count: as sharp as the sphere is wide, where the truncated sum produces a ripple that no improvement in the data ever removes.

When the blind region is not blind #

The cusp is a fact about the geometry and not always a fact about the experiment, and the difference is decided by where the crystal’s own symmetry axes are relative to the spindle.

The blind reflections have symmetry equivalents. A reflection in the cusp is one of a set related by the operations of the Laue class, and if any member of that set lies outside the cusp then the intensity is measured after all — the crystal never presents the blind reflection itself, and it presents an equal one instead.

So the question is whether the cusp is carried onto itself by the symmetry. If the spindle axis is parallel to a symmetry axis of the crystal, every operation about that axis maps the cusp to itself, and a blind reflection’s equivalents are blind too. The loss is then real and no amount of turning recovers it.

If the spindle is off the symmetry axis, most of the cusp is recovered. The operations carry the blind region to other places in reciprocal space, and a reflection missing from one is present in another. The cusp is still there geometrically; it simply no longer coincides with a set of equivalents.

Which reverses the intuition an experimenter starts with. Aligning a crystal neatly with its own axis along the spindle feels like good practice and is the worst case for completeness. Mounting it at some general angle, so that no symmetry axis lies along the spindle, is what makes the symmetry pay for the geometry.

And when alignment cannot be avoided, the remedy is a second setting. Re-mount the crystal, or use a goniometer with more than one axis so the crystal can be tilted between sweeps; the union of two blind regions in different orientations is empty, and the reflections missing from the first sweep are ordinary reflections in the second.

Completeness is a number per shell #

The overall completeness figure quoted for a data set is an average, and the average is the statistic that hides the problem this essay describes.

The cusp widens with resolution. The condition sets the distance from the axis against the square of the reflection’s length, so a longer reciprocal-lattice vector needs to sit proportionally further from the axis to be reachable. Low-order reflections almost all get there; high-order ones near the axis do not.

So the loss is concentrated in the outer shell. A data set can be 98% complete overall and 80% complete in the resolution range that decides how well the model is determined — which is exactly the range a refinement is most sensitive to, and exactly the range the average conceals.

That is why completeness is tabulated by shell. Splitting the measurable reflections into rings of equal thickness in resolution and reporting the fraction collected in each is a report on where the data are missing, and the last row is the one to read.

The same widening explains the wavelength dependence. A longer wavelength puts the same reflections at a larger $\lambda|p|^2/2$, so the cusp opens further at every length, and the shells that lose reflections start further in. The two effects compound: a longer wavelength both shrinks the limiting sphere and blinds a larger fraction of what remains inside it.

None of which changes the count of unique reflections. The number a structure needs is fixed by the cell and the resolution; this essay is about which of them a particular apparatus can deliver, and the answer is a fraction that depends on the mounting, the wavelength and the crystal’s own symmetry — three things chosen before any data are collected.