Crystallography & Symmetry
patterns derived from their groups
Concept

Anomalous scattering — where it appears

Scattering at a wavelength near an absorption edge, where an atom's factor acquires an imaginary part and Friedel's law fails. That failure is what hands back the handedness of a structure, which a measurement of intensities alone cannot reach.

Named by 12 essays across 2 fields — each of them below, with the objects they name alongside it.

What each group extinguishes. The extinction conditions of 9 space groups, each derived by summing the structure factor over that group's own operations and reading the surviving rule off the result: P1 — nothing; P1̅ — nothing; P2 — nothing; Pm — nothing; P2/m — nothing; P222 — nothing; Pmm2 — nothing; P4 — nothing; P23 — nothing. 9 of the 9 extinguish nothing at all, and diffraction alone cannot distinguish those from each other.

Where the experiment runs out

Absences narrow the space group down and often not to one. Two groups can extinguish exactly the same reflections and scatter with exactly the same symmetry, and telling them apart needs something the diffraction pattern does not contain.

diffraction · Diffraction Friedel's law, as an equality rather than a resemblance. Each pair of bars is a reflection and its opposite for a structure of three atoms in no particular arrangement. They are the same height, and not approximately: with real scattering factors, negating the indices conjugates the structure factor, and conjugation does not change a modulus. The test behind this figure requires the largest difference over 40 pairs to be below 10⁻⁹ and it comes back exactly zero. This is why a diffraction pattern is centrosymmetric whatever the crystal is, and why the thirty-two classes collapse to eleven before a structure is even proposed.

The law that hides handedness

With real scattering factors, negating the indices conjugates the structure factor and leaves the intensity exactly alone — so every diffraction pattern is centrosymmetric whatever the crystal is. The escape is an imaginary component that the negation does not touch, and it is how the handedness of a molecule is measured.

diffraction · Friedel's law Two candidates, and the sign that chooses. The phase of reflection (2, 3), recovered from three measurements and no model. The circle is every complex number of the measured amplitude; the isomorphous difference fixes the cosine of the angle between the unknown phase and the heavy atom's, leaving the two candidates marked; the anomalous difference fixes the sine, which picks one. The recovered phase agrees with the true one to fifteen decimal places, and the true phase was never used in the calculation.

One experiment gives the cosine, the other gives the sine

Friedel's law holding exactly is what makes the phase unreachable. Its breaking is what hands it back: an isomorphous difference fixes the cosine of the phase and leaves two candidates, and the anomalous difference fixes the sine, which chooses.

diffraction · Friedel's law One number, and it is the fraction. Data simulated from crystals that are nought, a quarter, a half, three quarters and wholly inverted, each fitted for the single parameter. The fitted values sit on the diagonal to better than five parts in a hundred, which is what makes the parameter a measurement of composition rather than a test of a hypothesis: a crystal is allowed to be part one hand and part the other, and a value near a half is a real answer about the specimen rather than a failure of the determination.

How much of it is the other hand

A crystal of one enantiomer is a hypothesis, not an observation. What the diffraction actually measures is a fraction — how much of the specimen is the inverted structure — and the useful part of that measurement is the uncertainty on it.

diffraction · Friedel's law The same structure, mapped from intensities and from differences. Left, the ordinary Patterson map of the structure: 14762 interatomic vectors, a continuous field of overlapping peaks, and the two vectors between the anomalous scatterers — circled — nowhere among its strongest. Right, the map from squared Bijvoet differences over the same reflections: two peaks after the origin, and they are those two vectors. The difference map is 7381 times smaller a problem to read.

A map of the atoms that break the law

Feed a Patterson synthesis the differences between the two halves of each Friedel pair instead of the intensities, and the map that comes back holds the vectors between the anomalous scatterers and nothing else. Two atoms among a hundred and twenty-two: 14,762 vectors become two.

diffraction · The Patterson function The worst edge pair is the two most anomalous wavelengths. For each pair of the four wavelengths, and for the peak, inflection and high remote together, the median error of the phase difference between the whole structure and its anomalous atoms, recovered from both Friedel mates at each wavelength with a 1% error on every intensity. Beside each, the spread of f′ across the chosen wavelengths. The pairs that include a remote wavelength and one on the edge phase to seven or eight degrees; the peak and the inflection together, the two with the largest anomalous effect, to eighteen; the two remotes to fifty-five.

The two most anomalous wavelengths are the wrong pair

A MAD experiment measures both Friedel mates at two wavelengths near an absorption edge and solves for a phase. The obvious choice is the two wavelengths where the anomalous effects are largest, the peak of f″ and the dip of f′. Of all the pairs that use the edge, it is the worst. The sine of the phase is carried by f″, but the cosine is carried by how much f′ changes between the two wavelengths, and two points on the same edge barely change it.

diffraction · Friedel's law Three difference maps of one substructure, and the map they approximate. Patterson syntheses over 1,681 reflections of a structure with 120 light atoms and two anomalous scatterers, from squared Bijvoet differences at one wavelength, squared dispersive differences between two wavelengths, and the two normalised and summed; the fourth panel is the Patterson of the anomalous atoms' own intensities. Circles mark the two vectors between the anomalous atoms. Each map's correlation with the exact one is printed beneath it: Bijvoet 0.775, dispersive 0.838, combined 0.999.

A second wavelength completes the map

The Bijvoet difference carries the anomalous substructure's intensity times the sine squared of a phase; the dispersive difference between two wavelengths carries it times the cosine squared. Normalised and added, the two make the substructure's own Patterson, correlation 0.999. It needs no fewer reflections than either half, because the threshold was never the average. What the dispersive half brings is a way of failing the Bijvoet half cannot have: two datasets, and a scale between them.

diffraction · The Patterson function Where the third wavelength should go. The median Cramér–Rao bound on the phase difference for two hundred reflections, measured at the inflection and the high remote and at a third wavelength whose energy runs across the edge, twice: with f′ known everywhere, and with f′ at the edge energies refined from the data. Known, the best third sits 1 eV above the edge at 6.26°, and a repeat of either wavelength is nearly as good. Refined, both repeats determine nothing, and the best third moves to +2 eV at 6.30°.

The third wavelength is there to measure the edge

A three-wavelength MAD experiment is usually justified as more anomalous signal: a third look at the edge, placed where the first two left the most out. Counted with the scattering factors known, a third wavelength buys nothing a longer look at the best pair would not buy better. Its real job appears only when f′ at the edge has to be found from the data, which it always does. Two wavelengths then carry one equation too few, and any third energy that is not a repeat supplies it, for two hundredths of a degree.

diffraction · Friedel's law A scale error between two wavelengths has a sign. The contrast of the substructure's vectors in a dispersive difference Patterson, the weaker of its two peaks over the strongest other peak, as the second dataset is scaled wrongly, with no metal atom and with two metal atoms that do not scatter anomalously. An error that weakens the second dataset loses the map at 3.7% with no metal; one that strengthens it at 50%. The dashed line is where the substructure's peak is predicted to reach zero, a quarter of the fractional change in f′ between the wavelengths. The metals shorten the strong side and hardly move the weak one.

A scale error between two wavelengths has a sign

A dispersive difference compares two datasets, so a wrong scale factor between them adds the whole structure to it. On a structure whose own Patterson is a haze, the substructure's vectors survived a scale error of fifty per cent, and heavy atoms that do not scatter anomalously were expected to take over much sooner. They do, but only in one direction. Scale the second dataset the other way and the substructure is erased at four per cent whatever else is in the crystal, because the error's linear term cancels the signal. Scaling the datasets onto each other is itself an error of that sign.

diffraction · The Patterson function No triple of wavelengths is best for both the phase and the edge. Every triple of energies on a grid across a model selenium edge, placed by two Cramér–Rao bounds with the edge's position and white line refined from the data: the median phase error across, and the error in the edge's position up, on a logarithmic scale. The front of triples that no other beats on both runs from +1, +2, +150 eV (6.15°, ±0.066 eV) to -120, -1, +3 eV (7.54°, ±0.022 eV). The classic peak, inflection and high remote sits on it, at 6.42° and ±0.026 eV.

The wavelengths that phase best do not find the edge

A MAD experiment refines the anomalous atom's scattering factors so that it can solve for the phases, and those same factors report where the absorption edge sits and how tall its white line stands, which is chemistry. Scored for the edge instead of the phase, the best triple of wavelengths moves: two straddling the step and one far below it, where the phase-best triple put two on the step and one far above. No triple is best at both, and the textbook peak, inflection and remote turns out to be the compromise between them.

diffraction · Friedel's law Three-species arrangements whose sums are parallel at every spot, with and without a centre. Every arrangement of three species, all present, on a cyclic cell of four to nineteen sites. The third column counts those whose two phase sums are parallel at every superlattice spot, the fourth those among them with a centre of inversion, the fifth those without one, and the last how many kinds the exceptions make up to translation, inversion, relabelling the spots and renaming the species. Primes are shaded. At 5, 7 and 11 sites every parallel arrangement has a centre; at 13 sites 936 do not, and they are all one arrangement; at 17 there are 17 kinds and at 19 there are 48.

Thirteen sites obey Friedel's law without a centre

At five and seven sites the earlier census found that a three-species structure has a contrast that darkens a spot exactly when it has a centre, and asked whether that is a theorem. It is not. The two phase sums are parallel exactly when the A–B pairs are as common at every separation forwards as backwards, and exactly when no anomalous scatterer can produce a Bijvoet difference anywhere in the pattern. At 5, 7 and 11 sites only a centre achieves that. At 13 sites one arrangement does it without one.

operations · Counting Fitting the outer shells makes it worse, and correcting the fit recovers it. The substructure's weaker peak as a share of its height with no scaling, on data with no error, with the scale error each scaling builds in, for five falloffs and three scalings. Fitting everything loses 42% with no falloff and less as the falloff grows. Fitting only the outer shells loses more, and at the steepest falloffs builds in a large error of the opposite sign. The fit corrected by the ratio the composition expects recovers the whole peak at every falloff, within a few per cent.

Fit the scale, then put the lean back

Scaling two wavelengths to each other builds in an error of the cancelling sign, because their means differ for a physical reason. The proposal was to fit against the part the edge does not touch, at high resolution. It does the opposite: f₀ falls with angle and f′ does not, so the outer shells lean furthest and fitting them costs more. Taking the scale from the composition alone reproduces the lean as well. What works is fitting as usual and then putting back the difference the composition expects, and that recovers the whole peak.

diffraction · The Patterson function

Named alongside it

The objects these essays reach for when they reach for this one.

Structure factorFriedel lawPhase problemBijvoet differenceHeavy atomThe Patterson functionResolutionFriedels lawLeast squaresAbsolute configurationCentrosymmetricChirality

All concepts