As sharp as the sphere is wide

The previous rung counted what a wavelength can reach. This one is about the reflections it cannot, and what their absence does to a picture.

The intuition worth dislodging first is that a low-resolution map is a blurred version of a high-resolution one — the same thing, out of focus, with detail smoothed away. It is not. A Fourier sum truncated at a sphere is the true density convolved with the transform of that sphere, and the transform of a sphere is not a smooth blob. It rings.

Two effects are visible and they behave completely differently as the data improve. One of them is the reason “resolution” is a meaningful word. The other is the reason a difference map has holes in it.

The peak width is the resolution, near enough #

A peak’s width is proportional to the cut-off, at 0.6 d across a range of more than a factor of six in resolution. That is the property that makes the word resolution mean anything: a map cut at 1 Å has features about 0.6 Å wide, whatever the atoms in it are actually like, and an atom of any sharpness whatever comes out at that width.

Which is also the warning. The width of a peak in a map is a property of the sphere, not of the atom. A crystallographer reading peak shapes for information about atomic motion is reading a quantity dominated by the truncation unless the resolution is very high indeed, which is why displacement parameters are refined against the data rather than measured off the map.

The ripple is fixed, and improving the data does not help #

The second effect is stranger and it is the one this essay exists for.

The negative trough beside every peak is 22% of the peak height, at every resolution. Not approximately at coarse resolution and better at fine: the same number, from a 3 Å cut-off to a 0.5 Å one, measured on the same atom.

The reason is that the transform of a sphere is a scaled copy of itself. Widening the sphere by a factor of two narrows its transform by a factor of two and leaves the shape — including the depth of the first negative lobe relative to the central peak — completely unchanged. So the ripple gets narrower and never shallower, and no amount of better data reduces it.

This is Gibbs’s phenomenon, which a reader may have met as the overshoot at the edge of a square wave reconstructed from a Fourier series. It is the same mathematics, and it has the same resolution: the overshoot does not diminish as terms are added, it merely moves closer to the discontinuity.

The practical consequences are well known to anyone who has looked at a map, and they follow from the fixed ripple rather than from anything about the crystal:

• A heavy atom sits in a crater. The negative region round it is a truncation artefact, and reading it as an absence of density puts a hole in a structure that has none.
• A difference map near a heavy atom is untrustworthy at the scale of the ripple, and features there need a different argument.
• Adding data does not clean it up. It shrinks the region affected and leaves the depth alone.

Two atoms, and when they stop being two #

The third measurement is the one a reader most often wants: how close can two atoms be before the map stops showing two of them?

0.65 times the resolution, on perfect data. That is a better figure than the rules of thumb, and the reason is that every rule of thumb is stated for real data. Phases are approximate, amplitudes have errors, and the noise sits at exactly the scale of the feature being looked for. What the measurement here isolates is the truncation alone, and truncation alone is not the limiting factor at the separations crystallographers actually argue about.

The shape of the failure is the part to take away. Two peaks do not fade as they approach; they merge, and the merged peak is a single perfectly convincing atom sitting between the two real ones. There is nothing in the map to indicate that it should be two. The failure of a truncated Fourier sum is not blurring but misinformation, and it is the same species of error as a wrong pattern being beautiful — the picture is well formed and describes something that is not there.

The number to compare it against is not a rule of thumb but a definition. Optics has the Rayleigh criterion — two point sources are resolved when the maximum of one falls on the first zero of the other — and it is a convention rather than a theorem, chosen because it happens to correspond to a visible dip. The measurement here uses a different and blunter test, and one a map actually has to pass: are there two local maxima, or one? That is the question a crystallographer asks of a difference map, it needs no threshold to be argued about, and it is why the limit comes out below the nominal resolution rather than at it. The two criteria are answering different questions about the same curve, and quoting one against the other is how “the resolution” acquires a spurious authority it never had.

Two losses, and which one is worse #

An experiment loses the phases and it loses the outside of the sphere. Both are irreversible, and the two are worth setting beside each other because their effects on a map are not remotely comparable.

Truncation with correct phases leaves the atoms exactly where they are. Every map in this essay has its peaks at the right positions to within the grid; what changes is their width and the ringing beside them. A structure can be read out of a badly truncated map.

Correct amplitudes with wrong phases leave nothing. The phase problem shows the same synthesis with scrambled phases and there is no structure in it at all — not a blurred structure, not a distorted one, nothing. The information sits overwhelmingly in the phases, and this site’s figure for it is one of the more startling pictures in the collection.

So the ordering is clear, and it explains the shape of the discipline. Solving a structure means recovering phases, which is why isomorphous replacement, anomalous scattering, direct methods and the Patterson function exist and why an essay on any of them is an essay about phases. Refining a structure means dealing with truncation, which is a matter of knowing which features of a map to disbelieve.

The two also fail differently under scrutiny. A wrong phase set produces a map that looks like nothing and is obviously wrong; a truncated map produces a map that looks convincing and is quietly misleading in specific, predictable places. Of the two, the second is the one that needs a rule rather than a glance.

What a coarse map is honestly good for #

The measurements above give a way of saying what a map at a stated resolution can and cannot support, and it is worth writing down because it is the practical content of the whole anchor.

At 3 Å, peaks are about 1.8 Å wide and two atoms are separated only at about 2 Å. Since a carbon–carbon bond is 1.5 Å, individual atoms are not resolved at all: what a map shows is the shape of a chain, and the atoms in it are placed by a model rather than seen.

At 1 Å, peaks are 0.6 Å wide and separations down to 0.65 Å are visible. Atoms are individually resolved, hydrogen atoms begin to appear in difference maps, and the map is genuinely a picture of the structure.

Between them the change is continuous and the language is not. The word “see” is used for both, and the previous rung’s count says why the two experiences are so different: a 1 Å data set of that cell has twenty-seven times the reflections of a 3 Å one, and the extra reflections are precisely the ones carrying the fine detail.

The site’s own figures have an edge too #

Every diffraction figure on this site draws a finite range of indices, and the range is a parameter at the call site rather than a constant — which is the fleet’s rule about parameterisation, arrived at for a different reason and useful here.

A window is not a truncation in the sense of this essay, and the distinction is worth being exact about. The figure above is not a Fourier synthesis: it plots which reflections vanish, and the reflections outside the window are not summed into anything. Widening the range adds rows to the picture and changes no statement in it.

A synthesis is different, and this site has three of them: the phase problem, the Patterson map and the superposition method. Each sums a finite set of reflections into a density, and each therefore has ripples of exactly the kind measured here. Their peaks are at the right places, which is what those essays assert; their shapes are the sphere’s, which none of them claims otherwise.

That is the reason this rung is worth having in a collection whose subject is symmetry rather than measurement. A truncated sum is the one way a figure on this site can be well formed, correctly computed, and still misleading about something it never mentioned.

What this is not about #

The phases are exact here, deliberately. The phase problem is the other and larger thing an experiment loses, and mixing the two would make it impossible to say which effect is which. Every map in this essay is synthesised from exact amplitudes and exact phases; the only information withheld is the reflections beyond the edge. So every artefact shown is attributable to truncation and to nothing else.

Nothing here is about noise. Real amplitudes have errors that grow at high angle, so the outermost data — which is most of the data — is also the worst measured. A real map is truncated and noisy, and the two interact: including weak, badly measured high-angle data can make a map worse rather than better, which is a judgement call no formula settles.

The synthesis is one-dimensional. Every claim here holds in three dimensions with the same reasoning and different constants — the transform of a three-dimensional sphere is a different function from that of an interval, and its first negative lobe is a different fraction. What the site computes is the one-dimensional case, in full, and the numbers quoted are that case’s.

Series termination has a standard remedy and it is a fudge. Multiplying the amplitudes by a function that goes smoothly to zero at the edge — a sharpening or smoothing function — removes the ringing by making the cut-off gradual, at the cost of broadening the peaks. It is a trade rather than a repair: the information outside the sphere is still absent, and what changes is which artefact it is expressed as.

Where the exactness stops #

The sums are exact and the measurements are of a sampled function. Each map is computed on a grid of several hundred points and the peak width is measured at half height on that grid, so the widths carry the grid’s resolution. Doubling the sampling changes them in the third decimal.

The ripple’s 22% is measured, not derived. The value for a sharp one-dimensional cut-off follows from the Dirichlet kernel and is about 21.7%; the measurements land between 21.7 and 22.2 per cent, and the variation across resolutions is sampling rather than physics — which is precisely why the claim is that the ripple does not change, rather than a claim about its third digit.

The separation limit depends on the peak criterion. A peak here is a local maximum above three-tenths of the largest value in the map. A more generous criterion would report two peaks slightly closer together, and a stricter one slightly further apart; the figure of 0.65 d carries that choice with it. What does not depend on the criterion is the scaling — halving the resolution halves the limit — which is the claim the second figure is for.

Who found it, and when #

Series termination was a known problem before there were many structures to have it. The early Fourier syntheses of the 1920s and 1930s — W. L. Bragg’s on diopside, Robertson’s on phthalocyanine — were made from a few dozen reflections, at resolutions where the ripples are as large as the features, and the literature of the period is full of arguments about which peaks in a map are real.

A. D. Booth worked out the corrections in the 1940s, and “Booth’s corrections” for series termination were standard practice for bond lengths measured off maps until refinement against the structure factors replaced map-reading altogether. That replacement is the real fix: modern practice does not measure anything off the density, it fits a model to the observed amplitudes, and the truncation then affects the picture without affecting the numbers.

Gibbs’s phenomenon is older than any of it, described by Henry Wilbraham in 1848, forgotten, and rediscovered by J. Willard Gibbs in 1898 in a letter to Nature about the Fourier series of a sawtooth. The crystallographic case is the same phenomenon in three dimensions with a spherical rather than a rectangular cut-off, and the constant differs while the argument does not.

The map where the ripple cancels #

The ripple is described above as something better data narrow and never remove. There is one map in which it very largely cancels, it is the map a crystallographer actually reads, and the cancellation is a consequence of the same linearity that produced the problem.

A difference synthesis is computed not from the observed amplitudes but from the differences between what was observed and what the current model predicts, with the model’s phases. Both terms are truncated at the same resolution and both acquire the same ripple, so subtracting them subtracts the ripple as well — to the extent that the model is right.

What is left is the part of the structure the model does not contain, with the truncation artefacts of everything the model does contain removed. That is why a missing atom shows in a difference map as a clean peak even at a resolution where the ordinary map is a landscape of overlapping ripples, and why the difference map is where atoms are found rather than the map itself.

The cancellation is exact only where the model is exact. An atom placed slightly wrong leaves a difference feature with a characteristic shape — positive on one side and negative on the other — which is the model’s ripple failing to cancel the observation’s, and is read as move it that way. The failure of the cancellation is itself the signal, which is the property that makes the map useful for refinement rather than only for finding what is absent.

The error the ripple puts into a bond length #

There is a second consequence of the ripple that is quantitative and was for a long time a systematic error in published structures.

The ripples of a heavy atom are large, because the ripple’s height is a fixed fraction of the peak’s. A light atom sitting a short distance away sits on top of one of them, and its own peak — much smaller — is shifted towards whichever side of it the ripple slopes. The apparent position of the light atom moves, and the bond length measured from the map is wrong by an amount depending on the resolution, the heavy atom’s strength and the separation.

That is a termination error, it is systematic rather than random, and it does not average away over many reflections. In the era when structures were read off maps rather than refined, it produced bond lengths that disagreed between structures at different resolutions and pointed in a consistent direction.

The repair is again the difference synthesis. Refining against the observations, rather than reading positions off a map, removes the error entirely — because the calculated amplitudes carry the same truncation as the observed ones, and the comparison is between two equally truncated quantities. The error was one of reading a map as though it were a structure, and it disappeared when the method stopped doing that.

Where this anchor ends #

Two rungs, and between them a statement of what a diffraction experiment can produce.

A finite set, whose size is a volume: thousands of reflections for a small cell, hundreds of thousands for a large one, shrunk by the order of the Laue group and not quite by that factor.

A map as sharp as that set is wide, with peaks at six-tenths of the resolution and a ripple at twenty-two per cent of the peak height that better data narrow and never remove.

Both are consequences of the edge alone, with everything else assumed perfect. What an experiment actually loses is more than this — the phases, the orientation in a powder, the handedness under Friedel’s law — and each of those is a separate essay on this site with its own measurement. The edge is the one that is pure geometry, and the one nobody can buy their way out of.