How many reflections there are

The reciprocal lattice is infinite in every direction, and every figure on this site that draws one has quietly drawn whatever part of it fitted on the page. An experiment has the same problem and cannot solve it by choosing a page size.

Bragg’s law puts a hard edge on what is reachable. A reflection with spacing d is observed at an angle satisfying λ = 2d sin θ, and the sine cannot exceed one, so a wavelength λ reaches only the reflections with d ≥ λ/2. In reciprocal space, where a reflection sits at distance 1/d from the origin, that is a sphere of radius 2/λ, and everything outside it is not unmeasured but unreachable — no exposure time, no detector and no patience gets at it.

So the measurable set is finite, and the first question worth asking about it is how large it is.

The count is a volume #

The number of lattice points inside a large region is the region’s volume divided by the volume per point, up to a surface correction. In reciprocal space the volume per point is 1/V, where V is the cell volume, so the number of reflections out to a resolution d is

N ≈ (4π/3) d⁻³ V

and that is worth reading twice, because both of its consequences are ones a crystallographer lives with daily. A larger cell has proportionally more reflections — a protein cell of 60 Å has 216 times the reflections of a 10 Å cell at the same resolution. And improving the resolution costs the cube — going from 2 Å to 1 Å multiplies the number of measurements by eight.

The estimate and the enumeration are separate calculations, and the site’s habit is to run both. The enumeration loops over a box large enough to contain the sphere — the bound on each index comes from the reciprocal cell edge, not from a guess — and counts what falls inside. That the two agree to a couple of per cent, and agree better as the sphere grows, is a check on the enumeration; that they do not agree exactly is a fact about lattice points rather than an error, and it is the reason the count is enumerated rather than computed.

The check that the enumeration is not silently truncating is arithmetic: halving the resolution must multiply the count by exactly eight, since the count is a volume. Measured on the cubic cell it comes to 8.05, and a box that was too small in any direction would show up here as a factor well under eight.

The surface term has a sign, and it is worth knowing which. A sphere of lattice points contains more points than its volume suggests rather than fewer, because the origin sits on a lattice point and the shells of points nearest the boundary are counted whole. The excess is proportional to the area of the boundary, so it falls as the sphere grows — it is a few per cent at the counts in the table here and a fraction of one per cent for a protein cell, where the sphere holds hundreds of thousands of points. That is the general shape of every lattice-point count in this collection: an exact statement about a volume, plus a term about a surface that no volume argument can supply and that shrinks out of the way as soon as the numbers are large.

What symmetry removes #

Reflections related by the crystal’s point group have equal intensities, so an experiment measures one of each set and treats the rest as redundancy. The group doing the relating is the Laue class — the point group with a centre of symmetry added, because Friedel’s law puts one there whether the crystal has it or not.

The obvious calculation is to divide: 4,168 reflections in a cubic cell, a Laue group of order 48, so 87 independent ones. That answer is wrong, and it is wrong in a direction worth understanding.

A reflection on a symmetry axis has a short orbit. The reflection 400 in a cubic crystal lies on a four-fold axis, and the operations of that axis carry it to itself; its orbit has six members rather than forty-eight. The reflection 440 lies on a mirror; 444 lies on a three-fold. Each of these is the reciprocal-space form of exactly the same phenomenon as a special position in the cell — a point whose stabiliser is bigger than the identity, with an orbit smaller than the group in the same proportion.

So the unique count is always larger than the total divided by the order, and by exactly the amount the short orbits contribute. Every orbit size divides the group order, as orbit-stabiliser requires, and the figure’s distribution shows the sizes actually occurring: 48, 24, 12, 8 and 6 in the cubic case.

The practical form of this is a ratio worth carrying. Per thousand cubic ångström of cell, a cubic crystal at 1 Å needs about 140 independent measurements and a triclinic one needs about 2,080 — a factor of fifteen for the same volume of matter at the same resolution. The symmetry is doing the work, and it is why a highly symmetric crystal is a cheaper experiment in a way that has nothing to do with how hard it is to grow.

The redundancy is the other half of the same number, and it is what makes the data accurate. An experiment does not throw away the reflections that share an orbit; it measures them and averages them, and the spread among measurements that symmetry says must agree is the only estimate of precision available that does not come from a model. So the order of the Laue group is simultaneously the factor by which the list of independent numbers shrinks and the number of times the average one is measured — and because the short orbits are short, the redundancy a data set actually achieves is always a little below the group’s order rather than equal to it. A cubic crystal turned through the same range as a triclinic one comes back with a fifteenth as many unknowns, each measured many times over; the triclinic one comes back with everything measured twice.

The other way of reading the same count #

The count has a second use, and it is the one that decides whether a structure can be solved at all.

A structure has parameters and the data are equations. Three coordinates per atom, plus a displacement parameter or six, plus an occupancy where the site is not full: a small-molecule structure with fifty atoms in the asymmetric unit has of the order of five hundred parameters. The unique reflections are the observations. The ratio between them is the data-to-parameter ratio, and refinement is a least-squares fit that needs it comfortably above one.

That ratio is a resolution question in disguise, because the parameter count is fixed by the structure and the observation count goes as d⁻³. Dropping from 1 Å to 1.5 Å multiplies the data by 0.30 and changes nothing about the parameters — which is why the practical resolution limits in different fields are where they are. A small molecule is expected at 0.8 Å and refined anisotropically; a protein at 2 Å has fewer observations than parameters and is refined with restraints supplying the difference.

Symmetry enters this ratio twice and in opposite directions, which is the part worth noticing. A higher-symmetry space group shrinks the unique data by the order of the Laue group — and it shrinks the asymmetric unit, and therefore the parameter count, by the order of the point group. The two nearly cancel. What symmetry buys is not a better-determined structure but a smaller experiment for the same one, and what it decides about the structure itself is a different question again: an atom on a special position has fewer free coordinates, and that reduction has no counterpart in the data.

A shorter wavelength buys the cube #

The sphere’s radius is 2/λ, so the wavelength chosen for an experiment decides how much of the reciprocal lattice is reachable — and it decides it as a cube.

Copper radiation at 1.54 Å reaches 9,092 reflections of the 10 Å cubic cell; molybdenum at 0.71 Å reaches 93,380 — ten times as many, from a wavelength half as long. In the unique set the factor is 8.45 rather than 10.3, because the extra reflections are concentrated at high angle where more of them lie on general positions.

That is the whole argument for short-wavelength sources, and it is also the argument against them: a shorter wavelength scatters more weakly, spreads the same number of counts over more reflections, and needs a detector covering a larger solid angle. The choice between them is a trade between how many reflections are reachable and how well any one of them is measured, and it is made differently in small-molecule work and in macromolecular work for exactly that reason.

Where the data actually are #

There is one more consequence of the count being a volume, and it decides how a data set is reported.

Half of all the reflections lie beyond about 1.26 Å in a data set nominally at 1 Å. That is the cube law again, and it is why:

• the resolution of a structure is quoted at the outer edge rather than in the middle, since that is where most of the data are;
• the argument about whether a data set is good is always an argument about the last shell;
• extending a measurement by a small amount in resolution is a large amount of extra work.

What is being counted, and what is not #

Nothing here has looked at an intensity. Every number in this essay is a count of lattice points inside a region, decided by geometry — a metric, a wavelength and a group. What each of those reflections is worth to a structure determination depends on how strongly it scatters, and the strong ones are systematically at low angle while the numerous ones are at high angle. So the count and the information are not the same quantity, and an essay that counted reflections and then spoke of information would be conflating them.

The count is of lattice points, not of observations. A reflection with a structure factor of zero is inside the sphere and contributes nothing to a measurement. Systematic absences remove whole classes of them — a glide plane extinguishes half a zone — and a real data set is therefore smaller than the count here by whatever the space group’s extinction conditions remove. That subtraction is a separate calculation, made in the essay on absences, and it is not folded in here.

The sphere is the limiting sphere, not the Ewald sphere. The Ewald sphere is the construction that decides which reflections are in diffracting position for a given crystal orientation, and it has radius 1/λ. Rotating the crystal sweeps it through reciprocal space, and the region swept is the sphere of radius 2/λ drawn here. The distinction matters because the first is about an experiment’s geometry and the second is about what is possible at all.

The Laue class is assumed rather than determined. Each cell here is counted under the class its metric would permit, which is the largest available; a cubic metric does not guarantee a cubic crystal, and a crystal can hold a lower symmetry on a higher-symmetry lattice without anything in the cell dimensions saying so. Assigning the Laue class from the data is a measurement — it is the symmetry diffraction reports — and getting it wrong in the generous direction merges reflections that are not equal, which is the failure that shows up as an implausibly bad agreement between symmetry-related measurements.

No account is taken of the detector. A real measurement misses reflections that fall in the beamstop’s shadow, between detector panels, or outside the detector’s angular range, and the fraction actually recorded is the completeness. Every number here is the ideal, and a real data set is a subset of it.

The cells are chosen, and the metric is exact. Six numbers give the metric tensor; its inverse gives the reciprocal metric; the spacing of any reflection is one over the length of its reciprocal vector, computed in that metric. No angle is assumed to be a right angle except in the cells where it is stated to be, and the triclinic cell is there to exercise the general case.

Who found it, and when #

Bragg’s law is 1912, from W. L. Bragg’s reinterpretation of Laue’s diffraction photographs as reflection from lattice planes. The limit λ = 2d is in the law from the first line of it and was understood immediately: it is why X-rays and not visible light, since a wavelength of five thousand ångström reaches no reflection of any crystal.

Ewald’s construction dates from 1913 and is the geometrical form of the same statement. The idea of the limiting sphere — the union of every Ewald sphere over all orientations — is the natural next step and appears in the early textbooks.

Counting lattice points in a sphere is much older and much harder than it looks. Gauss’s circle problem asks for the error term in the two-dimensional case and is unsolved: the leading term is the area, the error is known to lie between the square root of the radius and the radius to the power 131/208, and the truth is conjectured to be the first. The crystallographic use never needs the sharp answer, but the surface term visible in the table is exactly the quantity that has resisted a century and a half of attention.

The unique-set calculation is old practice and modern software. Every crystallographic program computes an asymmetric unit of reciprocal space and reduces observations into it; the short orbits are handled by tabulated multiplicity factors, which is the same computation the figure here makes by enumerating orbits directly.

How large the surface correction is #

The count is a volume “up to a surface correction”, and the correction is worth a number, because it decides whether the estimate can be used as a check on the enumeration or only as a guide.

A sphere of radius R in a reciprocal lattice of cell volume v* contains about (4/3)πR³/v* points, and the points it gets wrong are the ones near the surface — a shell of thickness about one lattice spacing, whose count goes as R². So the relative error goes as 1/R, which is to say as the cube root of the number of reflections.

That gives a rule of thumb with teeth. At a hundred reflections the estimate is good to some tens of per cent. At ten thousand, to a per cent or two. At a million, to a fraction of a per cent. So the volume formula is a genuine check on an enumeration for a large count and a poor one for a small count, and the essay’s practice of running both is not redundancy: the two disagree by a predictable amount, and a disagreement outside that amount is the error the check is for.

The exact size of the fluctuation is not known, and that is worth saying rather than hiding. Counting lattice points in a sphere is a hard problem in number theory — the three-dimensional case of the same difficulty a circle has — and the best bounds on the error are considerably better than R² and considerably worse than the R that is conjectured. A crystallographic count never approaches that regime, which is precisely why the rule of thumb is safe here and would not be a theorem.

Why resolution is quoted as a length #

A data set is described by a number in ångström rather than by a count, and the choice is not arbitrary — it is what makes two experiments comparable.

The count depends on the cell: a 10 Å cubic cell at 1 Å gives a few thousand reflections and a protein cell at the same resolution gives hundreds of thousands. So a count says nothing about how well resolved a structure is; it says how large the cell is. The resolution d is the quantity that transfers, because it is the finest spacing the data can distinguish, and two structures at 1 Å are equally well resolved whatever their cells.

What does not transfer with it is the observations-to-parameters ratio. A structure’s parameters grow with the number of atoms, which grows with the cell volume — and so does the reflection count, at the same rate. The two grow together, so the ratio is nearly a function of d alone, which is the reason a resolution in ångström is a meaningful shorthand for how determined a structure is.

Nearly, and not exactly. A protein crystal is half solvent, so its cell holds fewer ordered atoms per unit volume than a small-molecule crystal does, and its ratio at a given d is correspondingly better than the atom count suggests — while its atoms are also less well ordered, which spends the advantage. Those two effects run in opposite directions and neither is symmetry, which is why “a 2 Å structure” means something rather different in the two fields while denoting the same sphere in reciprocal space.

Where this ladder goes #

Two things have been established. The measurable set is finite, and it is large — thousands of reflections for a small molecule, hundreds of thousands for a protein. And symmetry shrinks it by nearly the order of the Laue group, which is why the classification of crystals is not merely descriptive but is the thing that makes an experiment affordable.

What has not been asked is what the edge costs. The reflections outside the sphere are not noise and are not small: they are exactly the information about the fine detail of the structure, and leaving them out of the sum is not the same as leaving them out of the crystal.

The next rung runs the synthesis with everything else exact — the amplitudes right, the phases right, no noise at all — and looks at what a truncated sum does to a picture of three atoms.