Crystallography & Symmetry
patterns derived from their groups
Concept

Least squares — where it appears

The fit that minimises the sum of squared differences between observed and calculated amplitudes. Two models' residual sums, divided by the error variance, give the likelihood ratio that chooses between them — valid only if that variance is the error the data have.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

What refinement takes back from a wrong disorder model. Every directed pair of disorder models sharing an occupancy, 462 in all: data made from one model's averaged structure, the other model refined against them with positions on its own loci, two-component displacement parameters and fixed occupancies. Across is the residual between the two ideal structures; up is the residual after refinement. Points on the dashed line lost nothing to refinement; the median point lost 58 per cent of its residual. Every point on the floor, where the wrong model fits exactly, is one whose loci contain the true model's.

What refinement takes back from a wrong model

Two disorder models sharing an occupancy leave averaged structures typically a sixth apart. Refine the wrong one against the right one's data and it takes back more than half of that — all of it wherever its loci contain the true ones. Yet the data still choose, even for a molecule of sixty-four atoms, because a likelihood summed over every reflection sees what no single reflection can, and the wrong model's partial atoms give it away through the size of their displacement parameters, not their shape.

restriction · Local symmetry Better counting makes an uncorrected smooth error choose the wrong model. Eighteen pairs of disorder models, each decided by the likelihood ratio at the 0.1 per cent level, at sixty-four and a hundred and twenty-eight atoms, as the random error falls from 3% to 1% to 0.3% of the mean amplitude. In each group, the first bar has random error only, the second adds a smooth scale error of 3% across reciprocal space, and the third adds the same error with a degree-two scale surface refined alongside the structure. With random error only, better counting decides more pairs correctly. With the uncorrected smooth error it stops helping and the number decided for the wrong model grows, to three at a hundred and twenty-eight atoms. With the surface refined, the result is the random-only one in every group.

Better counting, and the wrong model

A smooth error across reciprocal space, as large as the random one, hardly changes which disorder model the data choose: it enters both models' residuals and cancels from their difference. What it changes is where better data lead. With random error alone, counting harder decides more pairs correctly. With the smooth error left in, counting harder stops helping and starts deciding pairs for the model that did not make the data — at the 0.1% level, and more of them the better the counting.

restriction · Local symmetry 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

Named alongside it

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

ResolutionStructure factorAverage structureDisorderDisplacement parameterOccupancySite symmetryAnomalous scatteringFriedel lawPhase problemSpecial position

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