The third wavelength is there to measure the edge

The two most anomalous wavelengths are the wrong pair found that a two-wavelength MAD experiment should be chosen for how much $f'$ changes between its wavelengths, not for how large $f''$ is at either. The sine of the phase difference lives in the Bijvoet difference at one wavelength, and the cosine lives in the change of $f'$ between two. The peak and the inflection, the two energies where the anomalous effects are largest, sit close together in $f'$ and recover the phase worst of all the pairs that use the edge. The same essay found that adding the high remote to the peak and the inflection brought the error lower than any pair. It asked where each of three wavelengths should go, and in what order, once radiation damage makes the last dataset the worst.

This essay runs that search, and its answer is not the one the question expected. The expected answer was a placement: a third energy somewhere on the edge that fills in what the first two left out. With the scattering factors known, the search does find a best place, and it is almost on top of the first wavelength. That is a sign that the third dataset is not doing what a third wavelength was supposed to do. It is simply more exposure. The third wavelength earns its place only when the edge itself has to be measured, and then the reason is a count of equations rather than a gain in signal.

Two ways to count a third dataset #

The equations are the ones one experiment gives the cosine, the other gives the sine set up and the previous essay solved. For each reflection and its Friedel mate at a wavelength, the intensity is a sum of three terms in the unknowns: the whole structure’s ordinary amplitude $|F_T|$, the anomalous atoms’ amplitude $|F_A|$, and the phase difference $\Delta\varphi$ between them. The coefficients of those terms are built from $f'$ and $f''$ at that wavelength. The model edge is the earlier essay’s: a selenium K edge with a white line, $f''$ as a broadened step with a peak on it, and $f'$ from $f''$ by a numerical Kramers–Kronig transform.

Solving the equations with noise for every triple of energies would be slow. The search needs a figure of merit it can evaluate twenty thousand times. The one used here is the Cramér–Rao bound: the smallest variance any unbiased estimate of $\Delta\varphi$ can have, given the intensities and their errors. It comes from the Fisher information of the six intensities with respect to the three unknowns. It depends on the experiment’s design, not on any particular solver, and the earlier essay’s refusal figure checked its solver against the same bound. The number reported is the median, over two hundred reflections with amplitudes drawn from Wilson statistics, of the bound expressed as a probable error in degrees. The intensity error is 1 per cent and the anomalous atoms carry 2 per cent of the scattering power, as before.

These are not the same numbers as the earlier essay’s measured errors. It reported 5.7 degrees for the peak, the inflection and the high remote together, from its own reflections and its own noise. The bound on this set of reflections is 6.42 degrees for the same three energies. Every comparison below is between designs on one set of reflections, which is what a ranking needs.

The figure holds two wavelengths fixed at the inflection and the high remote and moves a third across the grid, from 150 eV below the edge to 150 eV above it. It does so twice, once for each of the two ways a third dataset can be counted.

The thin curve assumes $f'$ and $f''$ are known exactly at every energy. On it the best third wavelength sits 1 eV above the edge, at 6.26 degrees, a single electronvolt from the inflection already being measured. A third measurement exactly at the inflection gives 6.50 degrees, and a third measurement exactly at the high remote gives 6.28. Repeating the remote is worth almost exactly as much as the best new energy on the whole edge. The peak, 3 eV above the edge, gives 6.42, and the low remote gives 6.99. Nothing on this curve says that a third energy adds information the first two lacked. It says that more data near the edge helps and more data at the remote helps nearly as much.

The dotted curve refines $f'$ at the energies within 20 eV of the edge instead of assuming it. It is almost the thin curve, 0.02 or 0.03 degrees above it for most of its length, except at two places. At the two energies already in the experiment it does not exist. There the bound is infinite and the phase is not determined at all. And near those gaps it rises steeply: a third wavelength 1 eV from the inflection now gives 6.41 degrees rather than 6.26, and the best third moves to 2 eV above the edge, at 6.30. The two ways of counting agree everywhere except where the third dataset is not a third energy, and there they disagree completely.

With the edge known, the third dataset is exposure #

The first way of counting deserves to be followed to its end before the second is explained. If $f'$ and $f''$ are known, a three-wavelength experiment is a two-wavelength experiment with a third helping of photons. The fair comparison is therefore not with the pair measured once each, but with the pair given the third dataset’s exposure too.

The best pair on the grid is 1 eV above the edge with the high remote, at 7.28 degrees. It is the inflection-and-remote pair of the earlier essay moved one electronvolt up, where $f''$ has begun to rise and $f'$ has not yet climbed. The best triple of distinct energies is those two with the energy 2 eV above the edge added, at 6.14 degrees. Measuring the edge point twice instead, with the same three datasets’ worth of exposure, also gives 6.14. Measuring the remote twice gives 6.18. And measuring the best pair once each, but at one and a half times the exposure, so that the total equals three datasets, gives 5.95 degrees. That beats every triple on the grid.

That is the comparison the expected answer would have lost. With the scattering factors known, the best use of a third dataset is to spend it on the first two energies, evenly. A third energy on the edge is a slightly worse way of doing the same thing, because it puts a third of the photons where $f'$ and $f''$ are not quite as useful as at the best pair. Every reflection, several times over made the general point about redundancy: repeated measurement shrinks the random error and changes nothing else. Under this counting, a third wavelength is redundancy with a slightly worse aim.

The classic triple, peak and inflection and high remote, fits the same reading. At 6.42 degrees it is worse than the best triple and worse than the balanced pair. Its peak dataset adds $f''$ that the inflection-and-remote pair did not need. The earlier essay’s refused claim, that measuring where $f''$ is largest is enough, has a milder cousin here: that a dataset where $f''$ is largest is worth more than another look where $f'$ changes most. With the scattering factors known, it is not.

What a pair cannot measure #

The scattering factors are not known. The imaginary part $f''$ is close to directly measured, because an absorption scan across the edge records it up to a scale. The real part $f'$ is not measured at all. It follows from $f''$ only through a Kramers–Kronig transform over the whole spectrum, and just above the edge it changes by nearly an electron per electronvolt. In this model $f'$ is $-7.50$ at the edge, $-7.36$ one electronvolt above, $-6.81$ two above and $-5.85$ three above. The whole dispersive difference between the inflection and the high remote is 5.7 electrons. An energy calibration off by one electronvolt, or a chemical environment that shifts the edge, can move the edge $f'$ by a sixth of the very difference the cosine is read from. Programs that phase MAD data treat the edge values of $f'$ as parameters to refine along with everything else, and the dotted curve does the same.

That refinement is where two wavelengths run out. The reason is visible in the equations the earlier essay wrote down. At each wavelength a Friedel pair gives two numbers, a sum and a difference. The sum contains $|F_T|^2$, the $|F_A|^2$ term and the cosine term, with coefficients built from $f'$. The difference is

$$I(+h) - I(-h) = 2c\,|F_T||F_A|\sin\Delta\varphi , \qquad c = 2f''/f_0 .$$

Both Bijvoet differences are the same product, scaled by $f''$. Two wavelengths give two differences, but their ratio is fixed in advance by the two known values of $f''$. Between them they carry one number, $|F_T||F_A|\sin\Delta\varphi$, measured twice. So a pair supplies three independent numbers per reflection, the two sums and the one product, for three unknowns per reflection. There is no equation left over. An unknown $f'$ at the edge energy is a fourth unknown with nothing to determine it. It is shared by every reflection, but every reflection is already exactly determined, so adding more reflections adds as many unknowns as equations.

Repeating a wavelength does not help, because a repeat supplies the same sum and the same difference with the same coefficients. It lowers the error on numbers already in hand and adds no new equation. The budget figure shows the consequence in every row that has a single edge energy. The best pair, the best pair at one and a half times the exposure, and the pair with either dataset repeated are all finite with $f'$ known and undetermined with it refined. A pair measured forever does not find the edge.

A third distinct energy supplies a third sum with a different coefficient. That is four numbers per reflection for three unknowns, and two hundred spare equations across the reflections for the one or two edge values of $f'$. The refinement then costs almost nothing. The best triple goes from 6.14 degrees to 6.16, the classic triple stays at 6.42 to two decimals, and the triple with the low remote stays at 6.99. The cost scales with the counting error along with everything else: 0.002 degrees at 0.2 per cent error, 0.05 at 4 per cent.

There is one more way to fail, and it is the other half of the same count. Three distinct energies all on the edge, with no remote, are also undetermined. Every $f'$ in them is being refined, and a shift of all of them by the same amount can be absorbed into the total structure factor. Moving $|F_T|$ and its phase together by a fixed multiple of $F_A$ leaves every intensity unchanged. The remote is where $f'$ is known well enough to hold still, and it pins the zero that the edge values are measured from.

The third energy, placed for the edge #

Read this way, the dotted curve in the first figure is a trade between two needs. The third energy must be far enough from the inflection that its sum is a genuinely new equation, and close enough to the edge that its $f'$ contrasts with the remote’s. One electronvolt from the inflection is too close: the two sums are nearly the same equation, the edge values of $f'$ are poorly separated, and the bound is 6.41. Two electronvolts above is the best balance, at 6.30. Below the edge the curve rises quickly, 6.92 at one electronvolt below, because $f''$ falls away there and the new equation carries less of the sine.

With the whole triple free rather than two energies fixed, the search over every triple on the grid with $f'$ refined lands in the same place as without: 1 and 2 eV above the edge with the high remote, at 6.16 degrees. The two edge energies straddle the point where $f'$ bottoms out and $f''$ climbs through its steepest stretch. That is the shape the rule of thumb has always had, with one change of reason. Two edge energies are not there because two looks at the edge collect more anomalous signal. They are there because one edge energy cannot be measured and two can.

Damage decides the order, not the place #

The earlier essay’s question asked for radiation damage, and the model here is stated rather than measured. The dataset collected $k$-th has its relative intensity error multiplied by $1 + \delta(k-1)$, so with $\delta = 0.5$ the second dataset is half again as noisy as the first and the third is twice as noisy. The search is now over ordered triples, six times as many.

The classic triple is the clearest place to see it. Its six orders span from 8.38 degrees to 10.86 degrees, a difference larger than anything the choice of energies produced without damage. The best order is high remote, then inflection, then peak. The worst is peak, then inflection, then high remote. The inflection-and-remote pair is the one that carries the phase, and the peak adds least. The best order measures the two essential datasets while the crystal is fresh and leaves the least useful one for last.

Over every ordered triple, the best under $\delta = 0.5$ is 1 eV above the edge first, the high remote second, and 120 eV above the edge third, at 7.98 degrees. The third energy has left the edge entirely. That follows from the counting argument. The third dataset’s job is to be a distinct energy, so that $f'$ can be refined. Measured last it is the noisiest dataset, and as a noisy measurement it is worth more near the remote, where its $f'$ is known and its sum only has to agree with the others, than on the edge, where its $f'$ is a second unknown that its own noise has to pay for. The same three energies as the damage-free optimum, +1 and +2 and +150, measured in that order under the same damage, give 10.13 degrees. Measured remote first, they give 8.19.

Damage does not change what the third wavelength is for. It changes where the spare can afford to go. Without damage the spare sits beside the inflection and helps measure the edge. With damage it sits near the remote and is simply cheap, because by the time it is measured the crystal has lost the precision the edge would need.

A fourth wavelength #

The earlier essay also asked whether a fourth wavelength is ever worth its dose. The same calculation answers it for this model. Added to the best triple, with every dataset at the same exposure, the best fourth energy is 120 eV above the edge, and it brings the bound from 6.16 degrees to 5.29. That is simply more exposure. Spending the same extra exposure on the three energies already in hand gives 5.33, so the fourth energy is very slightly better than a longer look at the triple, and only very slightly.

With the triple’s total exposure shared among four datasets instead, the fourth brings the bound from 6.16 to 6.11 degrees. It is the refinement argument at the next order. A fourth distinct sum is a second spare equation per reflection, and it tightens the edge values of $f'$ a little further. But three already determine them to within two hundredths of a degree of the known case, so there is almost nothing left to buy. Under any damage at all, a fourth dataset is also the noisiest one measured. A fourth wavelength earns its dose only where the edge values are themselves the object, which is not phasing.

What the model leaves out #

The bound is a floor, not an experiment. Real MAD data carry systematic errors that the Fisher information does not see: absorption by the crystal, imperfect scaling between datasets, and the anisotropy of anomalous scattering near an edge. Several of them look like a change of $f'$ between datasets. A refinement that fits $f'$ to the data will absorb some of that, which is one reason refined values of $f'$ in real experiments can differ from tabulated ones. This model has no systematic error, so its refinement has nothing to absorb, and what it costs is only the statistical price computed above.

The damage model is a single number per dataset. Real damage is specific: disulphide bonds break, and carboxylates and selenomethionines lose order, so damage changes the structure rather than merely the noise. Specific damage between wavelengths looks like a dispersive difference, and it is the practical case for interleaving wavelengths in short passes rather than completing one before starting the next. The order found here is the answer to a simpler question, which of three noisy datasets should be the noisiest.

$f''$ is held known throughout. An absorption scan measures it up to a scale, and a phasing program would refine a scale factor alongside the $f'$ values. That scale is another global parameter shared by all the reflections. This essay does not refine it, and whether three distinct energies determine it as cheaply as they determine $f'$ is not asked here.

The phase difference is still only a phase difference. As in the earlier essay, turning it into the phase of $F_T$ needs the anomalous atoms’ positions, which a map of the atoms that break the law finds from the anomalous differences. The same Bijvoet differences also decide the hand of the structure, the question the law that hides handedness posed and how much of it is the other hand turned into a measured fraction. In a MAD experiment all of those uses share the one product that the count above found to be measured once per reflection.

A model that fits a nuisance parameter to its data can mistake something else for it, and the satellite beside the zero-order reflection is a recent case. There the tail of the zero-order reflection, leaking into the neighbouring position, read as a property of the modulation until a window was applied to the sum. The refinement of $f'$ here has a check of the same kind: it is exact without systematic error, and the counting argument says in advance which designs it cannot work for.

What the search has to refuse #

The tests that hold confirm the numbers used above: the triple beats the pair, damage makes the best ordered triple worse, the order of the classic triple matters, three distinct energies determine the edge for under a tenth of a degree, and a fourth wavelength at the same total exposure gains less than that. The refusals are the counting argument turned into tests. A single edge energy, measured once, twice or for longer, must leave $f'$ undetermined. Three edge energies with no remote must leave it undetermined too, because of the common shift. A bound with $f'$ and $f''$ both zero must say that there is no phase to find. If any of those came back finite, the bound would be reporting precision that the equations cannot contain.

Still open: the edge, measured as well as used #

The calculation treats the edge values of $f'$ as nuisance parameters, refined only so that the phase can be found. In practice they are also a result. The refined $f'$ and $f''$ of the anomalous atom report its oxidation state and its environment, and some experiments are run for them rather than for the phases. The Fisher information already computed here contains the bound on the edge values themselves. Asking which triple measures the edge best, rather than which phases best, would give a second answer. Where the two answers differ is where a single experiment must choose between the structure and the chemistry of the atom that phased it.