Crystallography & Symmetry
patterns derived from their groups
Concept

Pseudo-translation — where it appears

A shift that carries most of a structure nearly onto itself without being a lattice vector. It splits the reflections into a strong and a weak class, so no extinction rule fires and every intensity statistic computed over the mixture is pulled toward the centric answer.

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

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 A twin confines every pair of partner reflections to a wedge. Each of 300 pairs of reflections related by the twin law (h, k) → (k, h), for a structure of 24 atoms in a square cell with no symmetry of its own, plotted as one intensity against its partner's, in units of the mean (the few pairs beyond four times the mean are off the scale). Left, untwinned: the pairs fill the quadrant. Right, the same structure twinned at a fraction of 0.3: every pair lies between the two lines of slope 0.429 and 2.333, because twinning keeps each pair's sum and multiplies its difference by exactly 1 − 2α. Nothing statistical confines them; the wedge is arithmetic.

A twin leaves an edge that nothing else can

Compare each reflection with its partner under the twin law and twinning turns out to keep their sum and multiply their difference by exactly 1 − 2α. So no pair of a crystal twinned at thirty per cent can differ by more than forty per cent of its sum, and the distribution of contrasts stops dead at that edge. A structure that merely resembles its own twin image lowers the average contrast as a twin does, and leaves no edge. A pseudo-translation that hides a twin from the moment test leaves the pairs reading the fraction to three places.

diffraction · Statistics

Named alongside it

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

Intensity statisticsCentrosymmetryFriedel lawNormalised structure factorPseudosymmetrySecond momentSuperstructureSystematic absenceTwin fractionTwin lawTwinningWilson statistics

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