The wavelengths that phase best do not find the edge

The third wavelength is there to measure the edge settled what a third wavelength is for in a MAD experiment. With the anomalous atom’s scattering factors known, a third energy on the edge is only extra exposure. But they are never known exactly at the edge, where they change fastest. Once $f'$ there has to be found from the data, two wavelengths carry one equation too few, and any third energy that is not a repeat supplies it, for two hundredths of a degree of phase error.

That essay treated the edge as a nuisance, a set of numbers refined only so that the phases could be found. A crystallographer studying a metalloprotein often wants them for their own sake. An absorption edge moves to higher energy when the absorbing atom is more oxidised, by an electronvolt or more for each unit of charge, and the white line above the edge grows or shrinks with the atom’s coordination. A MAD experiment measures intensities that depend on both, and it measures them in order to solve the phase problem, not to answer the chemist. So the same datasets that phase a structure also report, in principle, the chemistry of the atom that phased it. The earlier essay asked which triple of wavelengths would measure the edge best rather than the phases, and where the two answers differ.

They differ in where the triple’s anchor goes. The triple that finds the phase best puts two wavelengths on the rising edge and the third far above it. The triple that locates the edge best puts two on either side of the step and the third far below it. No triple is best at both. Between the two lies a front of nine triples, each the best available phase for its precision on the edge, and the textbook choice of peak, inflection and high remote is one of them, at the point where the trade begins to cost more phase than it buys edge.

The edge as two numbers #

The model edge is the one the earlier essays used, a selenium K edge with $f''$ drawn as a broadened step with a white line on it and $f'$ computed from $f''$ by a numerical Kramers–Kronig transform. To ask what the experiment says about the chemistry, the edge needs parameters that the chemistry would change, and two are enough to show the conflict. The first is a shift $s$ of the whole edge along the energy axis, the quantity an oxidation state moves. The second is the amplitude $w$ of the white line, the quantity coordination changes.

Both act on both scattering factors. A shift moves $f''$ and $f'$ together, since the transform carries a translation in energy to the same translation. A taller white line raises $f''$ on the line itself and, because the transform is linear, adds its own transform to $f'$: a dip below the line and a rise above it. So the derivatives of the intensities with respect to $s$ and $w$ can be written down at every energy, and those derivatives are what a measurement can see.

The picture shows where each kind of change lives. A shift changes $f''$ only on the rising step, within a few electronvolts of the edge, where the step is steep. It changes $f'$ there too, with a narrow positive and negative excursion around the inflection, because $f'$ has its sharp minimum at the edge and moving the minimum moves both of its flanks. Twenty electronvolts away neither factor changes by a measurable amount. The white line acts a little more widely. It raises $f''$ over the few electronvolts of the line and changes $f'$ on both sides of it, with a long, weak tail above the edge, because a Kramers–Kronig transform spreads a narrow feature into a broad one.

So a shift is seen only by wavelengths on the step, and a white line by wavelengths on or just past it. Wavelengths far from the edge see neither, and they serve as the reference that fixes the ordinary scattering of each reflection.

How the bound is built #

The figure of merit is the earlier essays’ Cramér–Rao bound, the smallest variance any unbiased estimate can have given the intensities and their errors. Two hundred reflections are drawn from Wilson statistics, with the anomalous atoms carrying two per cent of the scattering, and each intensity carries a one per cent error. Each reflection has three numbers of its own: the whole structure’s amplitude, the anomalous atoms’ amplitude and the phase difference between them. The edge’s $s$ and $w$ are shared by all of them.

For one reflection the Fisher information of its six intensities, three wavelengths and both members of each Friedel pair, is a five-by-five matrix over its own three numbers and the shared two. Eliminating the reflection’s own numbers leaves what it says about the edge. Summing that over the two hundred reflections and inverting gives the edge’s covariance, and each phase’s variance is its own part plus what the uncertainty in the edge passes into it. With $s$ and $w$ fixed, the same computation gives the earlier essay’s phase bound with the edge known. Every number below is relative, and the reflections are the same two hundred the earlier essays used, so the bounds can be compared with theirs directly: a real experiment has more reflections and more systematic error, and the bounds would scale, but the ranking of designs on one set of reflections is what the comparison needs.

The search runs over every triple of the 51 energies the earlier essay used, dense through the edge and sparse on the remotes: 20,825 triples, each scored three ways, for the phase, for the shift and for the white line.

No triple is best at both #

The picture at the head of this essay is the search. Each grey point is a triple, placed across by its median phase bound and up by its bound on the edge’s shift. A triple is on the front, drawn as the line along the lower left, when no other triple is better at both. Nine are.

At one end is the triple best for the phase, +1, +2 and +150 eV from the edge, the same triple the earlier essay found with only $f'$ refined. It reaches 6.15 degrees, a hundredth of a degree worse than with the edge known exactly. Refining the edge’s two numbers costs the phase almost nothing, which is the earlier essay’s conclusion carried over: three distinct energies carry enough equations for the edge as well as the phases. But this triple locates the edge to only ±0.066 eV in this model. At the other end is the triple best for the shift, −120, −1 and +3 eV, which locates the edge three times as well, to ±0.022 eV, and phases at 7.54 degrees. Moving from one end to the other trades 1.4 degrees of phase for a threefold gain on the edge, and the front says exactly what each intermediate step costs.

The steps are not even. From the phase-best triple, moving the lower edge energy from +1 to 0 eV, the inflection itself, costs 0.13 degrees and improves the edge by almost half. Moving the upper one from +2 to +3 eV, the peak, costs another 0.14 degrees and improves the edge by another third. That triple, 0, +3 and +150 eV, is the peak, the inflection and the high remote, the experiment the textbooks recommend. It phases at 6.42 degrees and locates the edge to ±0.026 eV. Past it the front flattens. Each further step costs between 0.16 and 0.29 degrees of phase and gains at most seven per cent on the edge, and two of them gain nothing visible, until the triple swaps its remote above the edge for one below.

The classic triple is the knee of the front. It was usually justified as the triple with the most anomalous signal, the peak for $f''$ and the inflection for $f'$, and the two most anomalous wavelengths are the wrong pair showed that justification to be wrong for the phase. Scored for the phase alone it is not optimal, by a quarter of a degree. Scored for the phase and the edge together it is on the front, at the point where a little phase buys a great deal of edge. That is a better reason for it than the one it was given.

Three experiments for three questions #

The three optimal triples put their wavelengths in visibly different places. The phase wants two energies just above the edge, where $f''$ has begun to rise and $f'$ is still near its minimum, and one far above, where $f'$ has recovered. Those give the largest change of $f'$ between datasets, which is what fixes the cosine of the phase, as one experiment gives the cosine, the other gives the sine explained. The shift wants one energy just below the step and one just above it, where the step’s slope is greatest and a moved edge changes $f''$ most. On those two sides of the minimum of $f'$ a shift also moves $f'$ in opposite directions, so it cannot be confused with a change of overall scale. Its third energy goes far below the edge, where nothing depends on the chemistry and the ordinary scattering is measured cleanly. The white line wants one energy on the line, at +2 eV, one past it at +7 eV, where the line has fallen but its transform still moves $f'$, and a remote above.

Each of those is a reasonable experiment on its own terms, and a crystallographer who knew which question mattered could choose one. The trouble is that a single crystal usually has to answer the structural question, and the chemistry is extracted afterwards from whatever was measured.

The table puts the named experiments beside the optimal ones. The low-remote, inflection and high-remote triple, sometimes used when the peak is hard to place, phases at 6.99 degrees and locates the edge to ±0.076 eV, worse than the classic triple on both counts. It has only one energy on the edge, and one energy cannot tell a moved edge from a taller white line: both change the scattering factors there, and a single measurement sees one combination of them. Its two edge errors are correlated at 0.84, and each is poorly known because the other is. Three remotes, two below the edge and one above, phase at 53 degrees and locate the edge to within a hundred electronvolts, which is to say not at all. And the peak with the high remote alone, the two-wavelength experiment, determines nothing once the edge is refined. Its bound is finite only when the edge is assumed known, 8.84 degrees, which is the earlier essay’s count of equations applied to two edge parameters instead of one. A raw count of unknowns against observations would not predict it, since four intensities per reflection against three numbers of its own leave two hundred to spare for two shared numbers. What the count misses is that the spare equations are all alike. With one energy on the edge, every reflection’s information about the edge arrives through the same single combination of the shift and the white line, and two hundred copies of one equation determine one number, not two.

A moved edge and a taller line look alike #

The edge’s two numbers are not determined independently, and the ellipses show how. For the phase-best triple the ellipse is long and tilted, with the two errors correlated at 0.83. Both of its edge wavelengths sit on the rising step, and there a shift to lower energy and a taller white line do nearly the same thing: each raises $f''$ at +1 and +2 eV. The data can say that $f''$ rose and not which of the two raised it. For the white-line triple the ellipse is nearly upright, correlation 0.04, because its energy at +7 eV sees the white line’s tail in $f'$ and almost nothing of a shift, and that separates them. The shift-best triple pins the shift tightly and leaves the white line loose, the tall narrow ellipse.

That correlation is the most practical result here. A crystallographer who reads an oxidation state off the refined edge of a phase-optimised experiment is reading a number that is tied to the white line at 0.83. A difference in coordination between two crystals could be reported as a difference in oxidation state. The classic triple’s correlation is 0.38, the white-line triple’s almost nothing.

What the model leaves out #

One edge shape, two numbers. A real edge has pre-edge features, a white line whose width as well as height changes with chemistry, and structure past the edge from neighbouring atoms. Each would add a parameter, and each extra parameter needs another energy to see it, so the count of wavelengths needed grows with the description of the edge. The two-number model is the smallest that shows the conflict, not a model of any particular metal.

A structure whose Bijvoet differences vanish. Half of what locates the edge is the Bijvoet difference, which carries $f''$. Thirteen sites obey Friedel’s law without a centre found an ordered arrangement in which every Bijvoet difference is zero for any scattering factors, and for a crystal like that the edge would have to be read from the dispersive differences alone. The bounds here assume ordinary reflections, whose Bijvoet differences are not systematically zero.

No damage, no systematic error. The earlier essay found that radiation damage, penalising the last dataset, reorders the best triple. That applies here without change and is not repeated. Absorption, scaling and a drift of the monochromator’s energy calibration are all left out. The last matters most. A calibration drift is a shift of the edge that belongs to the instrument rather than the atom, and no amount of design can separate the two without a reference foil measured alongside.

Bounds, not estimates. Every number is a Cramér–Rao bound. The earlier essays checked that a full solver reaches the phase bound within a few per cent, but no solver for the edge parameters has been run. A bound says what an experiment could deliver at best, and it ranks experiments correctly when the estimates come close to it.

The absolute numbers are the model’s. Two hundred reflections at one per cent error locate the edge to hundredths of an electronvolt here, and measuring every reflection several times over shrinks a random error without touching a systematic one. A real experiment has many thousands of reflections and a few per cent of systematic error, and the absolute precision would be different. The ratios between designs, threefold for the shift and fourfold for the white line, are the claim.

Who worked out the pieces #

The MAD method grew from Karle’s analysis of 1980 and Hendrickson’s experiments through the 1980s, which established the peak, inflection and remote as the standard choice. That the refined anomalous scattering factors carry chemical information, and that a crystal’s absorption edge can be read from its diffraction, was exploited in diffraction anomalous fine structure work by Stragier and colleagues in 1992. The Cramér–Rao bound is the statistician’s, from Rao in 1945 and Cramér in 1946.

What the search must refuse #

The last test is a check on the derivatives rather than the design. A shift’s effect on $f''$, added up over every energy across the edge, must equal the total change of $f''$ from below the edge to above it, because the derivative of a shifted function integrates back to the function. The finite differences reproduce 3.171 electrons against 3.171, which is what makes the bounds on the shift trustworthy. The two refusals are the claims a reader might expect to hold. Wavelengths that avoid the edge cannot locate it, since nothing they measure depends on where it is. And two wavelengths cannot fix two edge numbers and the phases together, however well they are chosen.

A fourth wavelength buys most of both #

The front stops at three wavelengths because the earlier essays did, and the third wavelength is there to measure the edge found that a fourth gains the phase only five hundredths of a degree. Scored for the edge as well, a fourth wavelength does much more. The search was run again over every set of four energies, all 249,900 of them, with each dataset’s error enlarged so that the four together cost the same exposure as three.

The four-wavelength front lies below and to the left of the three-wavelength front along nearly its whole length. At its phase end, the set +1, +2, +120 and +150 eV phases at 6.11 degrees, better than the best triple’s 6.15, and locates the edge to ±0.032 eV, twice as well as that triple did. Further along, the peak, the inflection and two high remotes at +120 and +150 eV phase at 6.31 degrees and locate the edge to ±0.021 eV. That beats the classic triple on both counts, 6.42 degrees and ±0.026 eV, and it locates the edge better than any triple on the grid, including the one built for nothing else.

The fourth wavelength is not where the edge question seemed to want it. It is not below the step and not on the white line. It is a second remote, thirty electronvolts below the first, where neither the shift nor the white line changes $f''$ at all. What it sees is the white line’s long tail in $f'$, the slow fall the transform spreads above a narrow feature, which differs measurably between +120 and +150 eV. That separates the white line from the shift, the two errors’ correlation falls from 0.38 in the classic triple to near zero, and with the white line pinned the shift is pinned too. The two remotes together measure the edge’s shape by its far field, while the peak and inflection measure it where it changes.

So the conflict that no triple could resolve is mostly resolved by a fourth dataset, provided the fourth goes where the textbook would least expect it. The white-line error does not follow. The four-wavelength front holds it near ±0.04 electrons, twice the best triple for the white line, so an experiment whose only interest is the white line is still better off with its own triple.

Still open: the edge the instrument moves #

Everything above treats the edge’s position as a property of the atom. In practice a shift of the absorption edge is measured against the monochromator’s energy scale, and that scale drifts by fractions of an electronvolt over an experiment, which is the size of shift that distinguishes oxidation states. A drift is a shift of the edge belonging to the instrument rather than to the atom, and nothing in a set of diffraction intensities can tell the two apart. A reference foil of the same element, measured at every wavelength alongside the crystal, supplies the missing number. How much of the precision on the front survives once the shift has to be measured against such a reference, and whether the best design changes when the reference’s own noise enters the count, is the next calculation, and it is not made here.