Crystallography & Symmetry
patterns derived from their groups
Concept

Bijvoet difference — where it appears

The difference between the intensities of a Friedel pair, which is zero unless an atom scatters anomalously. It is the measurable quantity that fixes the sine of a phase, and so the one that chooses between two otherwise equally good structures.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

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 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 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

Named alongside it

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

Anomalous scatteringStructure factorFriedel lawFriedels lawPhase problemAbsolute configurationChiralityLeast squaresCentrosymmetryCyclotomic polynomialsEnumerationFlack parameter

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