Crystallography & Symmetry
patterns derived from their groups
Concept

Anisotropy — where it appears

The dependence of a property on direction, which is what makes a crystal's properties tensors rather than numbers. Symmetry decides how much of it a class may have, before anything is measured and without predicting any value.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

Every property count, in every class. For each of these 32 classes, how many independent components a property may have: elastic constants from 21 down to 3, dielectric tensor from 6 down to 1. Every number is a character averaged over the class's own operations, computed in the lattice basis with integer arithmetic. A zero means the symmetry forbids the property outright; a positive number means it does not forbid it, which is a weaker statement than it is usually read as.

Twenty-one, thirteen, nine, three

The number of independent elastic constants runs 21, 13, 9, 7, 6, 5, 3 down the crystal systems. Two of those systems carry two numbers rather than one, and which classes take which is not predicted by counting operations — a class with six of them can have more constants than a class with six of them.

point-groups · Neumann's principle Every property count, in every class. For each of these 32 classes, how many independent components a property may have: piezoelectric moduli from 18 down to 0, pyroelectric vector from 3 down to 0, gyration tensor from 6 down to 0. Every number is a character averaged over the class's own operations, computed in the lattice basis with integer arithmetic. A zero means the symmetry forbids the property outright; a positive number means it does not forbid it, which is a weaker statement than it is usually read as.

Permitted is not present

A symmetry argument says which components of a property may be non-zero. It is routinely read as saying they are — and the gap between the two is where every practical use of this table either works or quietly fails.

point-groups · Neumann's principle Every property count, in every class. For each of these 32 classes, how many independent components a property may have: elastic constants from 21 down to 3, piezoelectric moduli from 18 down to 0, dielectric tensor from 6 down to 1, pyroelectric vector from 3 down to 0, gyration tensor from 6 down to 0. Every number is a character averaged over the class's own operations, computed in the lattice basis with integer arithmetic. A zero means the symmetry forbids the property outright; a positive number means it does not forbid it, which is a weaker statement than it is usually read as.

A filter of great precision and no predictive power

The whole table in one place — thirty-two classes, six properties, 192 exact integers. What it settles, what it merely permits, and why knowing which of the two is happening at any moment is the entire skill of using it.

point-groups · Properties The seven groups a uniform field can have. Five of them have an axis and two do not. A cone has every rotation about its axis and mirrors containing it; a cylinder adds the mirror across the axis and the two-folds that go with it; turning either one destroys the mirrors that would reverse the turn. The sphere and the sphere made of something with a handedness are the two with no axis to speak of. Each drawing is the definition: the group is the set of motions leaving the picture unchanged.

The seven groups a field can have

Every group in this collection so far has been finite, because a lattice forbids the alternatives. A uniform field has no lattice: rotate it about its own axis through any angle at all and nothing has changed. There are exactly seven such groups, and they come out of the same closure argument that turns sixteen frieze candidates into seven.

point-groups · Curie

Named alongside it

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

Crystal classTensorNeumann principleMeasurementPoint groupAxial vectorCharacterClosureCurie principleElastic constantInfinite groupLimiting group

All concepts