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 7 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 Which sites a smooth error can mislead. For the eighteen pairs of disorder models at three site symmetries, under a smooth scale error of degree two (five molecule sizes) and of degree four (three sizes): how many verdicts of perfect counting go the wrong way, how many pairs no data can decide, the largest correlation between the error and the two models' difference, and how many harmonics of the error's degree the site's group leaves unchanged. At the cubic site the degree-two error correlates with nothing, to rounding, and decides no pair wrongly; the degree-four error, which the cubic site keeps one harmonic of, reaches it.

A cubic site is blind to a degree-two error

A smooth scale error left uncorrected made perfect counting choose the wrong disorder model for eight of ninety pairs and sizes, and the obvious way to predict which is to correlate the error with the two models' difference before refining anything. That correlation predicts none of them. What it does predict, exactly, is which pairs can never go wrong: at a cubic site a scale error of degree two correlates with nothing and is absorbed by nothing, because a cubic group keeps no harmonic of degree two. A degree-four error, which it keeps one of, reaches the cubic site at once.

restriction · Local symmetry The first-order prediction against the actual verdict, for every pair and size. The 85 decidable pairs of disorder models at five molecule sizes, each placed by the first-order prediction of the wrong model's misfit minus the right model's, made from the two error-free refinements and their Jacobians, and by the actual difference after refining against data with a 3% smooth error. Shaded quadrants are agreement. 77 agree. Of the 8 wrong verdicts, ringed, 4 are predicted; 4 right verdicts are predicted wrong.

The wrong model moves too far for first order

Which disorder models a smooth error will turn the wrong way was left as a calculation to make before any data exist: project the error onto the directions each refinement's parameters can move. Made, it gets the sign of the verdict right in seventy-seven of eighty-five cases, catches four of the eight wrong verdicts and raises four false alarms. Every case it gets wrong is either a near-tie or a refinement that absorbed more than twice, or less than half, of what its first-order directions allow.

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 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

Named alongside it

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

Structure factorAverage structureDisorderSite symmetryAnomalous scatteringFriedel lawPhase problemResolutionBijvoet differenceDisplacement parameterLaue classOccupancy

All concepts