Seitz symbols for Subperiodic Groups

Seitz symbols { R | t } are essentially the short-hand description of the matrix-column representations of the symmetry operations of the subperiodic groups (W, w). They consist of two parts:

a rotation (or linear) part R
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 E. The rotation part 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 of the coordinate system. 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.

Examples:

a. Consider the coordinate triplets of the general positions of the Layer group pb2₁m (No. 29):

(1) x,y,z          (2) -x,y+1/2,-z                (3) -x,y+1/2,z                (4) x,y,-z

The corresponding geometric interpretations of the symmetry operations as in the International Tables for Crystallography, Volume E are:

(1) 1                (2) 2 (0,1/2,0) 0,y,0      (3) b 0,y,z                      (4) m x,y,0

In Seitz notation, the symmetry operations are denoted by:

(1) { 1 | 0 }      (2) { 2₀₁₀ | 0 1/2 0 }      (3) { m₁₀₀ | 0 1/2 0 }      (4) { m₀₀₁ | 0 }

b. Consider the coordinate triplets of the general positions of the Rod group p2/c11 (No. 7):

(1) x,y,z          (2) x,-y,-z+1/2                (3) -x,-y,-z         (4) -x,y,z+1/2

The corresponding geometric interpretations of the symmetry operations as in the International Tables for Crystallography, Volume E are:

(1) 1                (2) 2 x,0,1/4                   (3) -1 0,0,0       (4) c 0,y,z

In Seitz notation, the symmetry operations are denoted by:

(1) { 1 | 0 }      (2) { 2₁₀₀ | 0 0 1/2 }      (3) { -1 | 0 }      (4) { m₁₀₀ | 0 0 1/2 }

c. Consider the coordinate triplets of the general positions of the Frieze group p2mm (No. 6):

(1) x,y              (2) -x,-y                      (3) -x,y                  (4) x,-y

The corresponding geometric interpretations of the symmetry operations as in the International Tables for Crystallography, Volume E are:

(1) 1                (2) 2 0,0                      (3) m 0,y              (4) m x,0

In Seitz notation, the symmetry operations are denoted by:

(1) { 1 | 0 }      (2) { 2₀₀ | 0 0 1/2 }      (3) { m₁₀ | 0 }      (4) { m₀₁ | 0 }