Crystallography & Symmetry
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Friedel's law, as an equality rather than a resemblance

Friedel's law, as an equality rather than a resemblance
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.

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.

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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. How it is known

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.

 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 it is known

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.

 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. How it is known

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.

 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°. How it is known

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.

 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. How it is known

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.

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