Crystal symmetry

The spatial transformations that preserve a crystal structure.

A crystal has centrosymmetry when its crystallographic point group contains spatial inversion. Its bulk electric-dipole polarization law must then satisfy $P(-E)=-P(E)$. Substituting a power expansion in E makes every even-order coefficient vanish, including the second-order nonlinear optical susceptibility. A surface, an external symmetry-breaking field or a response beyond the electric-dipole approximation can change the conclusion.

The finite orthogonal group obtained by quotienting a crystal space group by its translation subgroup. In three dimensions there are 32 crystallographic point-group types, constrained by preservation of a lattice. It describes macroscopic rotational and reflection symmetry, not the fractional translations of the full space group.

A point ${\mathbf{r}}_0$ about which inversion maps each atomic species onto equivalent sites of that species, possibly after lattice translations. An inversion center need not coincide with an atom.

The composition of a rotation about an axis with inversion through a point on the axis. A $\bar{4}$ operation rotates by a quarter turn and reverses all coordinates through the inversion center. It can preserve a basis whose height pattern is not preserved by a pure fourfold rotation.

A symmetry combining reflection in a plane with translation parallel to that plane. An $a$ glide translates by half an $a$ period, and a $c$ glide by half a $c$ period. Specifying the reflection trace alone loses the translation and can falsely suggest an ordinary mirror.

A symmetry consisting of rotation about an axis and translation parallel to that axis. A $2_1$ operation combines a half-turn with half a lattice period; applying it twice gives a complete lattice translation. It is a spatial symmetry, distinct from a screw dislocation.