Crystallography & Symmetry
patterns derived from their groups
Generator

A symmetric rank-2 property in 4/mmm

A symmetric rank-2 property in 4/mmm
A symmetric rank-2 property in 4/mmm. Permittivity, thermal expansion and conductivity are all symmetric rank-2 tensors, so all three have this shape in a crystal of class 4/mmm: 2 independent components of the six a symmetric tensor starts with. The shape is obtained by averaging each component over the group, and the count agrees with a character sum computed in the lattice basis that never chooses axes at all.

Permittivity, thermal expansion and conductivity are all symmetric rank-2 tensors, so all three have this shape in a crystal of class 4/mmm: 2 independent components of the six a symmetric tensor starts with. The shape is obtained by averaging each component over the group, and the count agrees with a character sum computed in the lattice basis that never chooses axes at all.

5 essays call tensor-shape. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about. Every one of this site's 393 essays names its parameters at the call site, which the standard pass of 2026-08-09 established and param-floor holds.

Where it is called

Changing this generator changes every one of these figures.

Neumann's principle for elastic constants in 6/mmm. Each bar is one operation's contribution to the character of the representation the elastic constants live in — the stiffness of the crystal in every direction and shear. The identity contributes the unconstrained count of 21; every other operation of 6/mmm subtracts from it, and the average over all 24 is 5, which is how many independent components the property may have. The sum is exact in integers and is taken in the lattice basis, so no Cartesian frame is chosen anywhere in it. What symmetry decides

A character does not know its basis

The number of independent elastic constants a hexagonal crystal has is computed here from integer matrices in a lattice basis, having never chosen a Cartesian frame. That looks wrong the first time: a physical tensor lives in an orthonormal frame and these matrices are not orthogonal.

 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. What symmetry decides

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.

 A spontaneous vector in 222. Averaging each of the three axes over the 4 operations of 222 leaves 0 independent components. No direction survives, so the class permits no spontaneous polarisation at all — which is a statement about what is forbidden, not about any measurement. What symmetry decides

The ten with a direction of their own

A crystal has a spontaneous electric polarisation only if some direction is left completely alone by every one of its symmetry operations. Ten of the thirty-two classes have such a direction, and the same ten are computed here twice by routes that share nothing but the group.

 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. What symmetry decides

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.

 Five strains, and which of them move the atoms. A honeycomb of harmonic bonds, strained five ways, with the internal coordinate minimised at fixed cell each time. The shuffle is how far the second atom moves away from where the strain alone would have put it. Which strains produce one is decided before any energy is computed: a strain that leaves the site's three-fold axis intact forces the shuffle to vanish, because the only vector a three-fold rotation of the plane fixes is the zero vector. The prediction and the measurement are in adjacent columns. What symmetry decides

The strain the atoms do not follow

The rule that makes an elastic constant computable — deform the cell and move every atom by the same map — is exact for a lattice with one atom in it and wrong for every other, and the reason is a site symmetry rather than a mechanical one. Which strains move the atoms inside the cell is decided by what survives of the site's own group, and a wrong answer here is a constant that is too stiff by a third.

The whole library · All essays