The unknowns against the observations

The reciprocal lattice is infinite and a measurement is not: a wavelength cuts a sphere out of it, the number of points inside is the sphere’s volume times the cell’s, and symmetry collapses the list to the reflections that carry independent information.

That is one of two counts. The other is how many numbers a structure determination is trying to find, and it is just as easy to write down. Each atom needs three coordinates and at least one displacement parameter; a scale factor covers the whole crystal. Both counts exist before any data do, and their ratio decides whether what follows is a deduction or a fit.

What counts as an observation #

Before the two numbers can be divided, both need saying carefully, and the first one has three subtractions in it that are easy to forget.

Intensities, not amplitudes. A detector records how much arrives and not when, so half of every measurement is thrown away before it is written down. That does not change the count — one reflection is one number either way — but it changes what the number constrains, which is why the ratio is a necessary condition rather than a sufficient one.

Friedel pairs are one observation, not two. With real scattering factors the intensity of a reflection and its opposite are equal exactly, so measuring both measures one thing twice. That halving is already in the count above, and forgetting it doubles the apparent data.

And reflections related by symmetry are one observation. Which is the collapse the enumeration performs, with the excess described below.

Each of those three is a subtraction, and it is worth being clear that only the last two change the count. Halving for Friedel pairs and collapsing symmetry equivalents both remove numbers from the list of observations. Discarding the phase removes half of what each surviving number would have carried and removes nothing from the list, so the ratio computed below is a ratio of counts and not of information — which is the first of several reasons it is a necessary condition rather than a sufficient one.

The volume cancels #

The obvious expectation is that a big structure is harder than a small one, and the arithmetic says otherwise.

The reflection count is proportional to the cell’s volume — a bigger cell means a denser reciprocal lattice and more points inside the same sphere. The parameter count is proportional to the cell’s volume too, because a bigger cell holds proportionally more atoms. Divide one by the other and the volume is gone.

The symmetry cancels as well, and by the same argument run twice. Symmetry divides the reflection count, because reflections related by the Laue group carry the same intensity and only one of each set is independent. It divides the parameter count by very nearly the same factor, because the asymmetric unit is the cell divided by the group’s order and only its atoms are refined. A cubic crystal and a triclinic one are in the same position.

What is left is a formula with neither volume nor symmetry in it:

$$\frac{\text{observations}}{\text{parameters}} \;\approx\; \frac{4\pi}{3\, d^{3} \rho\, p},$$

where d is the resolution, ρ the number of atoms per unit volume and p the parameters each atom needs.

The cancellation is not exact and the residue is instructive. Symmetry divides the reflection count by the order of the Laue group and the parameter count by the order of the space group, and those two differ by a factor of two whenever the crystal is not centrosymmetric. A non-centrosymmetric structure therefore has twice as many parameters per unique reflection as a centrosymmetric one of the same size at the same resolution — which is a real penalty, paid by every chiral molecule, and the only part of the symmetry that does not cancel.

Which leaves the resolution, cubed #

The curve at the top of this page is that formula measured: the ratio against resolution, on a logarithmic scale because the fall is a cube. At 0.8 Å there are sixteen observations for every unknown; at 3 Å there are three tenths of one.

Everything difficult about structure determination is in that d³.

Going from 0.8 Å to 1.6 Å is a factor of two in the resolution and a factor of eight in the ratio. Going from 0.8 to 2.4 is a factor of twenty-seven. There is no compensating for it by growing a bigger crystal, choosing a more symmetric one, or measuring more carefully — the sphere’s volume is the sphere’s volume, and the number of independent numbers inside it is fixed.

The crossing is at about two ångströms, for a molecular crystal refined with anisotropic displacement parameters. Better than that and the measurements outnumber the unknowns; worse and they do not, and the least-squares problem being solved has more variables than equations.

Which is why the rule of thumb is stated in resolution and not in anything else. A crystallographer asked whether a data set will support an anisotropic refinement answers by quoting a d-spacing, and the reason is not convention: the resolution is genuinely the only variable. Everything else that might have mattered — the size of the molecule, the symmetry of the crystal, how many molecules are in the cell — divides out of both counts.

That last figure is the only lever available. There is no way to get more reflections at a given resolution, so the only response to a poor ratio is to ask for fewer numbers: fix the displacement parameters, constrain a rigid group to move as one, or refine occupancies at a handful of sites instead of everywhere. Each is a decision about the model rather than about the crystal.

And there is a floor the resolution cannot go below. Bragg’s law puts a hard edge on the sphere: a wavelength λ reaches only spacings of λ/2 and more, so a copper source at 1.54 Å cannot measure past 0.77 Å however long the experiment runs. The best possible ratio for a molecular crystal on a laboratory source is therefore about eighteen observations per unknown, and everything below that is a property of the crystal rather than of the equipment. Shorter wavelengths move the edge; nothing else does.

The excess, and why the counts are enumerated #

The closed form above is an approximation in two respects, and the site’s own count is not.

Reflections on a symmetry axis have short orbits. A general reflection is one of a set of |Laue| related ones; a reflection lying on an axis is one of fewer. So dividing the total by the group’s order under-counts the unique set, and the more symmetric the crystal the more it under-counts. That is the reciprocal-space form of a special position, and it is why the count of reflections here is enumerated rather than divided.

And a sphere is not a lattice. The count of lattice points inside a sphere is its volume plus a surface term, and the surface term matters when the count is small. Both effects push the same way — the enumerated ratio exceeds the closed form and never falls short of it — and both shrink as the counts grow.

For a triclinic cell at 0.8 Å the two agree to better than one per cent. For a cubic cell with two hundred and fifty unique reflections they differ by a third. The closed form is the argument; the enumeration is the number.

The powder case, where the count collapses #

Everything above is about a single crystal, where each reflection is measured in its own place on a detector. Grind the crystal up and the three-dimensional pattern collapses onto one axis, and the observation count collapses with it.

The arithmetic is worth doing on the small case in the figure, because the two kinds of loss in it are not the same kind. A square lattice out to six indices holds a hundred and sixty-eight reflections and produces twenty-six lines — a factor of six and a half. Nearly all of that is symmetry and costs nothing: the lattice’s point group has eight operations, so it relates up to eight reflections at a time, and eight measurements of one number were always one observation. The part that does cost is the rings holding reflections the point group does not relate, where two independent numbers have been added together with no way of recovering either — and those rings are marked in the figure rather than assumed, by computing the orbits and looking for a ring spanning more than one.

And the ratio between the two kinds of loss moves the wrong way as the count grows. The number of reflections inside a sphere goes as the cube of its radius; the number of distinct spacings goes as the radius, since the lengths are spread along a line. So reflections per line rises as the resolution improves, and the excess over the symmetry multiplicity — the accidental part — rises with it. A better instrument separates spacings that were merely close and does nothing at all about spacings that are exactly equal.

What is measured is no longer a reflection but a line, and two reflections at the same spacing are one line whatever their symmetry relation. In a cubic cell that is a heavy loss — the multiplicities are large and exact coincidences are common — and in a triclinic cell the loss is different in kind: the lines are distinct in principle and overlap in practice, because a diffractometer has a finite width and a low-symmetry cell has many lines close together.

So a powder refinement has the parameter count of a single-crystal one and a fraction of the observations, and the fraction is worst exactly where the structure is hardest. That is the arithmetic behind a fact every powder crystallographer knows: the method works well for small, symmetric cells and badly for large, unsymmetric ones, and the boundary is not about the equipment.

There is one honest reply to that, and it is the reason powder work exists at all: a powder pattern is available for materials that never grow a single crystal, and a poor ratio on a sample in hand beats a good ratio on a sample nobody has. The arithmetic says what the method costs and not whether to use it.

What a protein does instead #

A protein crystal diffracting to 2.5 Å is on the wrong side of the crossing, and everybody involved knows it. The response is not to refine anyway but to add observations that are not measurements.

A bond length between two bonded atoms is known to within a few thousandths of an ångström from chemistry that has nothing to do with this crystal. So is a bond angle, so is the planarity of an aromatic ring, so is the handedness of an amino acid. Each of those is entered into the refinement as an observation with a weight, and there are of the order of four of them per atom.

They shift the balance by a fixed amount per atom rather than by a factor, which is the important structural fact about them: restraints do not scale with the data and cannot rescue an arbitrarily bad resolution. Four per atom against nine parameters per atom is a ceiling of about four ninths added to the ratio, whatever the resolution — a constant, where the shortfall grows as a cube. At 2.5 Å they take a ratio of about 1.3 to about 1.8. At 3.5 Å they would not take it past one.

And the distinction they blur is the one the ratio exists to keep sharp. A refined structure with more restraints than measurements behind a given feature is a structure in which that feature was assumed. Nothing in the deposited coordinates says which is which; the ratio is the only thing that does, and it is a number available before the experiment.

Where the rule of thumb comes from #

The number crystallographers quote is not one but about ten: a small-molecule structure is expected to have of the order of ten observations per parameter before an anisotropic refinement is trusted. The arithmetic above says where that comes from and what it is a proxy for.

Ten observations per parameter, at nine parameters per atom and one atom per eighteen cubic ångströms, is a resolution of about 0.9 Å — which is very nearly the limit of what a copper source reaches, and exactly the resolution small-molecule crystallography has organised itself around. The rule of thumb and the wavelength are the same fact seen from two sides.

The other half of the subject has organised itself around a different number, and for the same reason: a protein at 2.5 Å has about one observation per unknown, so the practice built around it takes restraints for granted, reports a free residual computed on data left out of the refinement, and treats a model as a hypothesis to be tested rather than a measurement to be quoted. Neither culture is being careless. They are answering the same arithmetic on opposite sides of a crossing.

The measurement that catches over-fitting #

The ratio is a count made before any data exist, and it says how much room a determination has. There is a measurement made after the fit that says how much of the room was used, and it is the standard defence against the failure this essay is about.

The trouble with a residual is that it is computed against the very data the parameters were adjusted to fit. Add parameters and it falls, whatever they are for — a model with as many parameters as observations fits perfectly and means nothing.

The repair is to hold some data back. Set aside a random few per cent of the reflections at the start, refine against the rest, and compute a second residual on the held-back set. Parameters that describe the structure improve both; parameters that are fitting noise improve the first and leave the second alone or make it worse.

That second number is the free residual, and it is reported with every protein structure and increasingly with small-molecule ones. Its gap from the ordinary residual is the direct measure of over-fitting: a small gap says the model is supported, a large one says parameters are absorbing noise, and the gap grows as the ratio on this page falls.

It also settles arguments the count cannot. Whether a particular data set supports anisotropic displacement parameters is a question the ratio answers in general and the free residual answers for this crystal: refine both ways and see which lowers the held-back residual. A count is a prediction and the free residual is a measurement, and where they disagree the measurement wins.

Restraints are prior information, counted the wrong way #

The essay describes restraints as shifting the balance by a fixed amount per atom, and there is a framing that makes what they are doing precise rather than metaphorical.

A restraint is a statement that a quantity has a value with an uncertainty, arrived at independently of this crystal. A bond length known to within thousandths of an ångström from a thousand previous structures is exactly that, and combining it with the diffraction data is combining two measurements of the same thing.

Counted honestly, a restraint is an observation and not a subtracted parameter. It goes on the top of the ratio with a weight set by its own uncertainty, rather than on the bottom as a parameter removed — and the two bookkeepings give different numbers whenever the restraint is loose.

That matters for what the ratio is claiming. Written with restraints as observations, the ratio says how much total evidence supports the model, from both sources. Written with them as removed parameters, it says how much diffraction evidence there is per free parameter, which is a smaller and more conservative statement.

Both are quoted and they are not the same number, and a determination reporting a comfortable ratio should say which convention produced it — since the difference between the two is precisely the amount of the structure that came from chemistry rather than from the crystal.

What the ratio does not measure #

It counts atoms and not everything a model contains. A real refinement has parameters the count above ignores: occupancies where a site is partly filled, several positions for a disordered group, a twin fraction, a solvent model with parameters of its own, and an extinction correction. Each is a number the data have to support, and none of them scales with the atom count — so they matter most for exactly the small structures that had plenty of data to spare.

It is a count, not a measure of information. Two reflections can be nearly the same measurement — a weak one and a weak one — and the ratio counts them alike. What actually constrains a parameter is the derivative of the calculated intensities with respect to it, and that varies enormously across a data set. The ratio is a necessary condition and a poor sufficient one.

The atom density is measured, not derived. One non-hydrogen atom per eighteen cubic ångströms is a regularity of molecular crystals that Kitaigorodskii established by measuring, and it is the same observation his packing argument rests on. Every number here that depends on it inherits its status: this is arithmetic on top of an empirical constant rather than arithmetic all the way down. An inorganic structure packs more densely and does worse at the same resolution; a protein crystal is half solvent and does better.

Nor does it know where the parameters are. The count treats an atom in the middle of a well-ordered molecule and an atom on a disordered side chain as one parameter each. In practice the first is determined many times over by the data and the second by almost nothing, and a global ratio of ten to one can hide a local ratio of nothing to one. The proper version of this arithmetic is per-parameter and is the diagonal of a matrix rather than a division; what the ratio supplies is the number available before that matrix exists.

And it says nothing about whether the answer is right. A structure can be over-determined and wrong — sitting in a false minimum, refined against data with an unrecognised twin, averaged over domains that no single cell contains. Counting observations against unknowns tells a crystallographer what kind of problem is being solved and not whether it was solved correctly, which is the same division of labour every permission on this site observes.