Bilbao Crystallographic Server

Seitz symbols for Double Space Groups

Seitz symbols { R | t } are essentially the short-hand description of the matrix-column representations of the symmetry operations of the space groups (W, w). They consist of two parts:
  1. a rotation (or linear) part R
  2. a translation part t
The Seitz symbol is specified between braces and the rotational and the translational parts are separated by a vertical line.

The translation parts t correspond exactly to the translation parts of the coordinate triplets w of the General position blocks in the International Tables for Crystallography, Volume A. The rotation parts R consists of symbols that specify the type and the order of the symmetry operation, and the orientation of the correspondig symmetry element with respect to the basis. The orientation is denoted by the direction of the axis for rotations and rotoinversions, or the direction of the normal to reflection planes.

The nomenclature for the symbol R is:
  • 1 and 1 for the identity and the inversion, respectively
  • m for reflections
  • 2, 3, 4 and 6 are used for rotations
  • 3, 4 and 6 are used for rotoinversions
For rotations and rotoinversions of order higher than 2, a superscript + or - is used to indicate the sense of the rotation. The subscripts of the symbols R denote the characteristic direction of the operation.
To differenciate the two rotational operations which act in the same way in the space but in a different way in the spin space (they differ by a π rotation), one of them has the d superscript.