 Site-symmetry group

The site-symmetry group of a point $\mathbf{r}$ in a crystal structure is the subgroup of its space group $G$ that fixes that point:It is the stabilizer subgroup for the action on points. Although the space group contains infinitely many lattice translations, the site-symmetry group is finite and is isomorphic to a subgroup of the corresponding crystallographic point group.^[ref]

For an original example, take a lattice with inversion symmetry and no other point operations, the space-group type conventionally denoted $P\bar{1}$. At the origin, both the identity and inversion $\mathbf{r}\mapsto -\mathbf{r}$ fix the site, giving a group of order2. A generic site has only the identity in its site-symmetry group; inversion instead produces a distinct site at $-\mathbf{r}$.

The equality in the definition fixes the actual point. At $\mathbf{r}=\mathbf{a}/2$, inversion alone moves it to $-\mathbf{a}/2$. Inversion followed by translation by $\mathbf{a}$ fixes it. Reducing fractional coordinates modulo integers can conceal that translation, but the full space group operation retains it.

The crystallographic site multiplicity consequently satisfiesIn this primitive inversion example the general multiplicity is2, so the origin has multiplicity1 while a generic site has multiplicity2. This is why high site symmetry reduces the number of distinct sites generated in a cell.^[ref]