Crystallography & Symmetry
patterns derived from their groups
Concept

Wilson statistics — where it appears

The expected intensity of a reflection, computed from the contents of the cell and the resolution alone, with the arrangement averaged out. It supplies the scale and the temperature factor that normalise a data set, and every distribution-based test is stated in terms of the normalised values.

Named by 2 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 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

Named alongside it

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

Normalised structure factorCentral limit theoremCentrosymmetricCentrosymmetryFriedels lawIntensity distributionIntensity statisticsPseudo-translationSpace group determinationStandard errorSuperstructureSystematic absence

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