Crystallography & Symmetry
patterns derived from their groups
Concept

Selection rule — where it appears

A condition on the indices of a reflection deciding whether it can be present at all, imposed by the symmetry rather than by the contents. A screw axis extinguishes reflections along a row, and a helix permits only certain angular orders on each layer line — the same kind of statement with a non-crystallographic rotation in it.

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

4mm: 17 of 25 transitions forbidden. Every pair of irreducible representations of 4mm, with the number of times the identity occurs in the product of the two with the vector operator. A zero is a prohibition: the integral that would give the transition rate vanishes for every choice of functions carrying those representations, whatever the material is made of. A positive number is a permission and nothing more. Rows are final states and columns initial ones; the labels are the dimensions of the representations, so the twos are the degenerate levels. Every one of the 17 prohibitions here was checked again against explicit polynomials.

What a group forbids to happen

Two levels and a thing that might carry a crystal from one to the other. Whether it can is one sum over the group — and a zero there is a prohibition that no material, no temperature and no intensity of light gets round.

point-groups · Representations The cross, from the selection rule alone. The layer lines of a helix with the first maximum of each marked on both sides. Nothing here is a picture of a photograph: each mark is at the radius where the Bessel function of the lowest order the selection rule permits on that layer line first peaks, and that radius is proportional to the order. The order rises by one per layer line until the middle of the repeat, so the maxima lie on two straight lines through the origin — the X — and the larger marks are the layer lines that reach the axis.

What a thread scatters

A helix with ten subunits in a turn is not a screw axis a crystal may have, and nothing about its diffraction pattern is lawless. The pattern lies on layer lines, and on each one only certain angular orders may contribute — a selection rule as hard as any extinction condition. The lowest permitted order rises by one per layer line, a Bessel function of order n does nothing until its argument is about n, and the maxima therefore lie on two straight lines through the origin.

classification · Subperiodic

Named alongside it

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

Bessel functionCharacterCrystallographic restrictionDegeneracyDiffraction symbolDipole operatorInvariant polynomialIrreducible representationLaportes ruleReciprocal latticeRod groupScrew axis

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