Crystallography & Symmetry
patterns derived from their groups
Concept

Diffraction symbol — where it appears

The record of what a diffraction pattern shows of a crystal's symmetry: the Laue class it exhibits together with the systematic absences observed. It narrows the space group to a short list and often not to one, because two groups can extinguish exactly the same reflections.

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

What each group extinguishes. The extinction conditions of 9 space groups, each derived by summing the structure factor over that group's own operations and reading the surviving rule off the result: P1 — nothing; P1̅ — nothing; P2 — nothing; Pm — nothing; P2/m — nothing; P222 — nothing; Pmm2 — nothing; P4 — nothing; P23 — nothing. 9 of the 9 extinguish nothing at all, and diffraction alone cannot distinguish those from each other.

Where the experiment runs out

Absences narrow the space group down and often not to one. Two groups can extinguish exactly the same reflections and scatter with exactly the same symmetry, and telling them apart needs something the diffraction pattern does not contain.

diffraction · Diffraction 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.

Structure factorSystematic absenceAnomalous scatteringBessel functionCentrosymmetricCrystallographic restrictionFriedel lawLaue classReciprocal latticeRod groupScrew axisSelection rule

All concepts