Crystallographic orbit

A crystallographic orbit is the set of points generated from one point $\mathbf{r}$ by all operations of a specified space group $G$:Because the space group includes lattice translations, the full orbit is infinite. The familiar finite list of equivalent fractional positions lists representatives within one conventional unit cell.^[ref]

For an original example in $P\bar{1}$, take $\mathbf{r}=(0.2,0.3,0.4)$. Inversion gives $(-0.2,-0.3,-0.4)$, represented in the same cell as $(0.8,0.7,0.6)$. Lattice translations repeat both sites throughout space. At the origin, inversion produces no additional site, so there is only one representative per primitive cell. The smaller crystallographic site multiplicity comes from the larger site-symmetry group.

If $\mathbf{s}=g\mathbf{r}$, their site-symmetry groups obeyIndeed, an operation fixing $\mathbf{r}$ becomes an operation fixing $g\mathbf{r}$ after conjugation. This connects spatial equivalence to conjugate subgroups.

Several distinct crystallographic orbits can belong to the same Wyckoff position. For example, replacing the generic coordinates above by $(0.1,0.2,0.3)$ usually produces a different pair, while both pairs have trivial site-symmetry groups. A Wyckoff position classifies the conjugacy type; an orbit supplies one particular symmetry-generated set of positions.^[ref]