Crystallography & Symmetry
patterns derived from their groups
Concept

Intensity statistics — where it appears

The distribution of diffracted intensities, taken over many reflections rather than one. It carries facts no single reflection does — whether the structure has a centre of symmetry, whether the crystal is twinned, whether a translation is nearly obeyed — and it is available before any atom is placed.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The twin fraction, recovered from a moment and nothing else (12 atoms). A structure of 12 atoms twinned at each of 6 fractions, with the second moment of its intensity distribution measured and the fraction solved back out of it. The recovery is within a few hundredths as far as thirty per cent — 4 rows here — and 4 of the 6 fractions get a number at all. Beyond thirty per cent the relation flattens: the derivative of 2α(1−α) vanishes at a half, the two roots meet, and a small error in the moment becomes a large one in the fraction. Where the sampled moment falls below 1.5 the quadratic has no real root and the estimate refuses rather than clamping, which is why a nearly perfect twin is the hard case in practice rather than the easy one.

A twin hides in the statistics

A twinned crystal scatters as two orientations at once and the detector cannot separate them. What arrives is a sum of two intensities — and adding two independent quantities narrows a distribution, which is a signature no model of the structure is needed to read.

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.

Acentric distributionCentrosymmetryMerohedral twinNormalised intensityNormalised structure factorPseudo translationSecond momentSuperstructureSystematic absenceTwin fractionTwinningWilson statistics

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