Crystallography & Symmetry
patterns derived from their groups
Concept

Displacement parameter — where it appears

A refined measure of how far an atom spreads about its mean position, from thermal motion or static disorder, expressed as B in square ångström. A partial atom forced onto a wrong locus shows it by refining to an implausibly large value.

Named by 2 essays across one field — 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

Named alongside it

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

Average structureDisorderLeast squaresOccupancyResolutionSite symmetryStructure factorSpecial position

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