Crystallography & Symmetry
patterns derived from their groups
Concept

Electron density — where it appears

The distribution of scattering matter in a unit cell, whose Fourier coefficients are the structure factors. It is what a structure determination produces, and the difficulty is that a measurement gives the magnitudes of those coefficients and not their phases.

Named by 7 essays across one field — each of them below, with the objects they name alongside it.

Three atoms, four resolutions. A one-dimensional Fourier synthesis of the same three atoms, cut at four different resolutions. Nothing is approximate except the edge: every amplitude and every phase used is exact, and the only information withheld is the reflections outside the sphere. The peaks broaden as the cut-off comes in, and beside every peak sits a negative ripple that the coarsest map cannot distinguish from a real absence of density. Both effects are the transform of the sphere rather than anything about the structure.

As sharp as the sphere is wide

A map made from a truncated sum is not a blurred picture of the structure. It is the structure convolved with the transform of the sphere — so peaks acquire a width proportional to the resolution, and a negative ripple of twenty-two per cent that no improvement in the data ever reduces.

diffraction · Resolution A map from amplitudes alone, 0.59 grid steps out. The density after 150 cycles of flipping, with the atoms that produced the data drawn as rings — moved into the origin and the handedness the solution chose, because a phase set does not fix either and comparing without allowing for them measures the arbitrariness of the description. Every peak of the map is an atom and every atom has a peak. Nothing about the arrangement went into the calculation: the input was a list of amplitudes and a random set of phases.

The solver that knows no symmetry

Compute a map from amplitudes and random phases, reverse the sign of everything below a small threshold, transform back and keep the phases. Repeat. The structure appears — and so does its space group, which was never supplied.

diffraction · Direct methods An equality, with a known factor in it. Sayre's identity says the structure factor is proportional to the convolution of the structure factors with themselves — an equality, not a probability — with a factor that depends on the atoms and the resolution and on nothing else. Computed for a structure of Gaussian atoms, the ratio of the two sides falls with resolution exactly as predicted: the logarithm of the ratio is a straight line in the squared index, its slope is minus half the width in the atomic factor, and the ratio is very nearly real and positive, so the identity relates the phases and not only the amplitudes.

The relation that is an equality

Every relation of this kind so far is a probability — right nine times in ten, useless applied once and decisive applied ten thousand times. One is not. For a structure of equal, resolved atoms the squared density has peaks in the same places, so every structure factor is exactly a convolution of all the others, with a factor that depends only on the atom. It is exact, and on its own it is useless.

diffraction · Direct methods Heavier atoms spoil the line, and then mend it. Above, the scatter of Sayre's ratio about its fitted line, averaged over four random cells, as a quarter of the atoms is made heavier, for cells of eight and of twelve atoms. Below, one minus the square of ρ, the correlation between the density's transform and its square's, computed from the composition alone. The spread of atomic numbers grows without limit along this axis, but ρ does not keep falling: it is lowest near a weight of three and climbs back towards one as the heavy atoms come to dominate, because a cell ruled by a few equal heavy atoms is nearly a cell of equal atoms. The scatter rises and falls with it. At equal weight what scatter remains comes from atoms that overlap, more of them in the larger cell.

Unequal atoms break the equality, and very unequal ones mend it

Sayre's identity is exact for a cell of equal atoms. Give a quarter of them more weight and the line it predicts scatters, reflection by reflection, by exactly the difference between weighting atoms by Z and by Z squared. Make them heavier still and the scatter shrinks again, because a cell ruled by a few heavy atoms is nearly a cell of equal ones. The scatter measures the cell, but only if the phases are already known.

diffraction · Direct methods Success of the flipping solver against how far off the subtracted lead is. The share of forty starts on the cell of nineteen carbons and one lead that the flipping solver solves, with the lead's density subtracted at a position displaced from the true one by up to sixteen grid steps, half the cell. The rate does not fall with the offset: the points scatter without a trend about the pooled rate of 40%, the band marking two binomial standard deviations, and the original solver, with nothing subtracted, solves none. Refining the position from the map does no better than holding it.

The lead is wherever it is taken out

Charge flipping solves a cell of nineteen carbons and one lead once the lead's own density is taken out before each flip and put back after. The obvious worry was precision: a lead subtracted a fraction of a grid step from where it really is leaves a residual larger than any carbon. The residual is there, larger than a carbon at a tenth of a step. It does not matter. Subtracted two steps away, or half the cell away, the lead still lets the solver succeed at the same rate, because the solver builds the whole structure round wherever it is put.

diffraction · Direct methods Four flipping rules on four cells. The best success rate of the flipping solver, pooled over five arrangements and eight starts, on four cells: twenty carbons, a quarter sulphur, one sulphur among twenty and one lead among twenty. Four rules for what is flipped: below a number of standard deviations of the map, below a height set by the lightest atom's weight, the same height above the map's mean, and the original rule with the known heavy atom's density removed first. On the lead cell only the last solves anything, 55% against nothing; on both sulphur cells it raises the rate to near certainty; on carbon alone it is the original rule.

The lead taken out solves what no threshold could

Charge flipping never solved a cell of nineteen carbons and one lead, and two repairs were proposed: a threshold set by the composition rather than by the map's spread, and the lead's own density removed before each flip. The threshold does nothing for the lead cell at any setting, measured from zero or from the map's mean. Removing the lead takes it from nought to fifty-five per cent, and lifts both cells with sulphur in them to near certainty. The trouble was never where the threshold sat.

diffraction · Direct methods Success of the flipping solver against the error in the vector between two subtracted leads. The share of 80 starts on a cell of eighteen carbons and two leads that the flipping solver solves, with both leads' density subtracted, the first at its true place and the second displaced from its true place by an error in a random direction. The rate holds to a quarter of a grid step (21, 19, 18, 11, 0, 0, 0 successes at errors of 0, 0.1, 0.25, 0.5, 1, 2, 4 steps), falls by half at half a step, and is zero from one step on. With one lead the place did not matter at all; with two, the vector between them does.

Two leads need their vector to half a step

With one heavy atom in a cell with no symmetry, charge flipping did not care where the atom was subtracted: it rebuilt the structure round whatever place it was given. With two heavy atoms that freedom runs out, as predicted, because no translation can put both under wrong references at once. Subtract one of them a whole grid step from where it belongs and the solver never succeeds. But the tolerance is five times looser than the estimate it was predicted from: up to a quarter of a step the rate does not move, at half a step it halves, and the residual the solver absorbs is three carbons' worth, not one.

diffraction · Direct methods

Named alongside it

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

Direct methodsPhase problemStructure factorHeavy atomHeavy atom methodNormalised structure factorOrigin choiceRefinementStatisticsCharge flippingConvolutionDifference map

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