Crystallography & Symmetry
patterns derived from their groups
Concept

Enantiomorphism — where it appears

The property of coming in two forms related by a mirror and not by any rotation, which eleven crystal classes have. It is what a diffraction experiment cannot see without anomalous scattering, since Friedel's law makes the two forms scatter alike.

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

The twin laws of class 32. The point group of the lattice of class 32 has 24 operations and the class has 6, so it splits into 4 cosets: the crystal itself, and 3 twin laws. Every operation in a block produces the identical second orientation, which is why the block and not the operation is the law.

Quartz has exactly three twin laws, and its lattice is why

Class 32 on a hexagonal lattice has index four, so three twin laws and no more. They turn out to be the three the mineralogists named — Dauphiné, Brazil and the combination of the two — and reading quartz's lattice off its class instead of measuring it would have produced one law where there are three.

applied · Twinning 15 classes may rotate light, 11 of them chiral. A crystal is chiral when its point group contains no improper operation, and there are 11 such classes. A crystal may rotate the plane of polarisation when its class permits a non-zero gyration tensor, and there are 15. The four in the difference — 4̅, m, 4̅2m, mm2 — are achiral and may still rotate light, which is why the two words are not synonyms. In each of the four, symmetry forces the tensor to be traceless, so the rotation changes sign with direction and cancels in any average over directions.

Fifteen may rotate light, and eleven are chiral

Optical rotation and handedness are treated as the same thing and are not. Eleven crystal classes are chiral; fifteen permit a crystal to rotate the plane of polarisation; and the four in between are the reason quartz and sodium chlorate are the examples everybody uses.

point-groups · Optical

Named alongside it

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

Optical activityAxial tensorBirefringenceCentrosymmetricCharacterChiralityCosetGyration tensorIndicatrixMerohedryNeumann principlePhase transition

All concepts