Crystallography & Symmetry
patterns derived from their groups
Concept

Independent components — where it appears

How many numbers a measurement of a property may return in a crystal of a given class, counted as the dimension of the space the group leaves alone. A zero is a genuine prohibition; a non-zero count is a permission and not a prediction.

Named by 3 essays across 3 fields — each of them below, with the objects they name alongside it.

How many components a property may have, at each rank and in each class. One row per plane point group, one column per rank of a fully symmetric property tensor, with the number of independent components in each cell — counted by averaging the tensor over the group index by index, and equal at every entry to the Molien coefficient of that degree. A symmetric property of rank r is a form of degree r, so Neumann's principle and the invariant ring are the same arithmetic in two notations. The last column is the elastic tensor, which is not fully symmetric — symmetric within each pair of indices and under exchanging the pairs — and its counts are not in the table to its left. A property with its own symmetries needs its own average, and that is why the elastic constants are not read off a degree.

The parts a property splits into

A symmetric property of rank r is a polynomial of degree r wearing indices, so the number of components a class permits it is a coefficient of an invariant ring's series. The elastic tensor is not a polynomial in disguise, and its counts are not in that table — which is the most useful thing about it.

point-groups · Neumann's principle Which descents change the shape of the cell, and into how many shapes. Each descent the modes produced, with the number of independent strain components the parent class permits and the number the child permits. A transition is ferroelastic exactly when the second is larger — the child leaves alone a distortion the parent moves — and the difference is a spontaneous strain the crystal acquires without being pushed. The count of distinct shapes is the orbit of that strain under the parent, which can be smaller than the number of domains: two domains may differ in something a change of shape cannot show. Every count here is a rank of an averaged set of quadratic forms, computed twice — once by averaging, once from a character.

The strain that arrives with the transition

A crystal that loses symmetry usually changes shape, and whether it does is a subtraction: how many strain components the child permits, minus how many the parent did. The difference is a distortion nobody applied, and it is what makes a domain visible in a microscope.

applied · Domains Every crystal class is a rotation group, read one of three ways. The 32 crystal classes sorted by their rotations. Each row is one of the 11 proper classes; beside it is the class obtained by adjoining the inversion, which doubles the order, and the classes obtained by negating the half of the group outside a subgroup of index two, which keeps it. The columns hold 11, 11 and 10 classes, and every class appears exactly once. 3 rows have nothing in the last column, because 1, 3, 23 have no subgroup of index two to leave alone. At most 2 classes share a row, which happens where a proper class has halves of two different kinds.

Eleven, eleven and ten

Twenty-one of the thirty-two crystal classes contain a mirror, a centre or a rotoinversion, and not one of them is a new group. Each is a group of rotations with the inversion added, or a group of rotations with half of itself negated — and which half is left alone is the whole of the choice.

restriction · Finite groups

Named alongside it

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

Property tensorAizu criterionAveraging projectorCouplingCrystal classDeterminantDomain stateElastic constantsFerroelasticityFinite groupHomomorphismImproper operation

All concepts