Crystallography & Symmetry
patterns derived from their groups
Concept

Normalised structure factor — where it appears

A structure factor divided by its expected magnitude at its own resolution, so that the mean of its square is one. Written E, it makes reflections at different resolutions comparable, and every probability in direct methods is stated in terms of it.

Named by 9 essays across one field — each of them below, with the objects they name alongside it.

N(z): the fraction of reflections weaker than z. The cumulative distribution of normalised intensities, measured on two structures built from the same atoms — one with an inversion centre, one without — and drawn against the two closed forms, 1 − e^(−z) without a centre and erf(√(z/2)) with one. The curves are furthest apart at small z, which is the useful end: a centrosymmetric structure has far more nearly-absent reflections, because its structure factor is a single real number that can pass through zero rather than a complex one that rarely does.

Whether there is a centre is a statistic

Everything else on this site is decidable: a pattern has a symmetry or it does not, and the detector settles it in integers. Whether a structure has an inversion centre is not like that. No single reflection carries the answer — the distribution of all of them does.

diffraction · Statistics ⟨cos Φ⟩ against κ, 379 triplets. The mean cosine of the triplet, binned by the concentration κ = 2|E₁E₂E₃|/√N, for the 379 triplets of a structure of 24 atoms whose reflections all exceed |E| = 1.2. The curve is Cochran's I₁(κ)/I₀(κ), computed from the distribution and not fitted to anything; the points are measured, with the number of triplets in each bin printed above. They agree to 0.1 root-mean-square. The measured points sit slightly above the curve throughout, which is the finite structure showing: Cochran's derivation assumes atoms placed at random and there are only 24 of them.

Three phases that do not move when the origin does

A phase is a property of the description, not of the crystal: shift the origin and every one of them changes. A sum of three phases whose indices add to zero does not change, because the shifts cancel. That sum is the smallest thing about a structure that a diffraction experiment could in principle know, and it is not distributed at random.

diffraction · Direct methods B = 3.45 against 3.4, K = 0.37 against 0.37. The mean intensity of each resolution shell of a cell of 1194 reflections, divided by Σf² computed from the cell's content alone, and logged. The points fall on a line whose slope gives B = 3.45 against the 3.4 put in, and whose intercept gives a scale of 0.37 against 0.37 — both recovered before a single atom has been placed. The atoms here are independent, so the line is straight at every resolution; the shells below the cutoff are marked in the second colour.

The average that knows the atoms and not where they are

Square a structure factor and average it over a shell of reflections at one resolution. The cross terms — every one of which carries a fact about the arrangement — cancel, and what is left is a sum over the *content* of the cell with no position in it anywhere. A scale and a temperature factor come out of that before a single atom has been placed.

diffraction · Statistics Two populations, and neither of them empty. The reflections of a structure in which three quarters of the atoms are paired by a half-cell shift, sorted by the parity of h + k and each class scaled by its own mean. The two histograms have the same shape, which is the point: each class on its own is an ordinary acentric distribution. What differs is the scale — the odd class is a sixth of the even one on average — and no odd reflection is absent, so no extinction rule fires and nothing about the space group is affected.

A translation that is nearly there

Half a structure copied onto the other half by a half-cell shift, with nothing exact about it. No reflection vanishes, so no extinction rule fires — and the test for a centre of symmetry answers yes about a structure that has none.

diffraction · Statistics How many reflections the centre test needs. The error rate of two tests for a centre of symmetry against the number of reflections used, measured on 60 centrosymmetric and 60 non-centrosymmetric structures at each point. The moment test — the one in every textbook, comparing ⟨|E|² − 1⟩ to its two theoretical values — reaches one error in twenty at 160 reflections and one in a hundred at 320. A likelihood ratio, which uses each reflection's own value instead of one average, reaches the same at 40 and 80. The gap is the price of summarising a distribution by its mean, and it is about a factor of four.

How many reflections it takes to know there is a centre

The test for a centre of symmetry compares one average of the intensities against two theoretical values a quarter apart. Whether that is a measurement depends on how many reflections went into the average, and the only honest way to find out is to run the test on structures whose answer is already known and count the mistakes.

diffraction · Statistics The sign turns when the cross terms go weak. Every quartet among the strong reflections of a small structure, sorted by the mean of its three cross terms, with the mean cosine of the phase sum in each bin. Where the cross terms are strong the quartet behaves like a triplet and the cosine is near one. Where they are weakest it is -0.698 — negative — and 93 per cent of those quartets have a cosine below nought. The four reflections of the quartet are equally strong in every bin; what changes is three reflections that are not in the sum at all.

The relation that can say no

Every phase relation before this one pushes a sum towards zero, so none of them can contradict another — a phase set satisfying all of them badly is still satisfying them in the same direction. A quartet can be estimated at π instead, and it is when its three cross terms are weak, so the information arrives from the reflections nobody would have thought worth measuring.

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 Which cells the flipping solver finds. For four cells of twenty atoms — all carbon, a quarter sulphur, one sulphur, one lead — the share of random starting phase sets that charge flipping takes to the right structure within 250 cycles, against the threshold below which density has its sign reversed. Each point pools several random arrangements of atoms with several starts each, and the atoms fall off with angle as real atoms do. The cells with sulphur are solved more often than carbon alone, the quarter-sulphur cell most of all, each at a threshold near one standard deviation; the lead cell is never solved at any threshold.

The heavy atom the flipping solver cannot see past

Sayre's identity degrades as a cell's atoms become unequal, most of all for a quarter of heavy atoms three times the weight of the rest, and the obvious prediction is that a solver built on the same sparseness should struggle there too. It does not. Charge flipping solves a cell of carbon with sulphur in it more often than carbon alone, and never solves a cell with one lead atom — because a threshold stated in standard deviations belongs to whichever atom owns the map's variance.

diffraction · Direct methods

Named alongside it

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

Structure factorDirect methodsPhase problemStatisticsTripletElectron densityHeavy atomStructure invariantWilson statisticsBiasCentral limit theoremCentrosymmetric

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