International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2015 
International Tables for Crystallography (2015). Vol. A, ch. 1.4, pp. 5153

Apart from the notation for the geometric interpretation of the matrix–column representation of symmetry operations discussed in detail in the previous section, there is another notation which has been adopted and is widely used by solidstate physicists and chemists. This is the socalled Seitz notation introduced by Seitz in a series of papers on the matrixalgebraic development of crystallographic groups (Seitz, 1935).
Seitz symbols reflect the fact that spacegroup operations are affine mappings and are essentially shorthand descriptions of the matrix–column representations of the symmetry operations of the space groups. They consist of two parts: a rotation (or linear) part and a translation part . The Seitz symbol is specified between braces and the rotational and the translational parts are separated by a vertical line. The translation parts correspond exactly to the columns of the coordinate triplets of the generalposition blocks of the spacegroup tables. The rotation parts consist of symbols that specify (i) the type and the order of the symmetry operation, and (ii) the orientation of the corresponding symmetry element with respect to the basis. The orientation is denoted by the direction of the axis for rotations or rotoinversions, or the direction of the normal to reflection planes. (Note that in the latter case this is different from the way the orientation of reflection planes is given in the symmetryoperations block.)
The linear parts of Seitz symbols are denoted in many different ways in the literature (Litvin & Kopsky, 2011). According to the conventions approved by the Commission of Crystallographic Nomenclature of the International Union of Crystallography (Glazer et al., 2014) the symbol is 1 and for the identity and the inversion, m for reflections, the symbols 2, 3, 4 and 6 are used for rotations and , and for rotoinversions. For rotations and rotoinversions of order higher than 2, a superscript + or − is used to indicate the sense of the rotation. Subscripts of the symbols denote the characteristic direction of the operation: for example, the subscripts 100, 010 and refer to the directions [100], [010] and , respectively.
Examples
The linear parts R of the Seitz symbols of the spacegroup symmetry operations are shown in Tables 1.4.2.1–1.4.2.3. Each symbol R is specified by the shorthand notation of its (3 × 3) matrix representation (also known as the Jones' faithful representation symbol, cf. Bradley & Cracknell, 1972), the type of symmetry operation and its orientation as described in the corresponding symmetryoperations block of the spacegroup tables of this volume. The sequence of R symbols in Table 1.4.2.1 corresponds to the numbering scheme of the generalposition coordinate triplets of the space groups of the crystal class, while those of Table 1.4.2.2 and Table 1.4.2.3 correspond to the generalposition sequences of the space groups of 6/mmm and (rhombohedral axes) crystal classes, respectively.



The same symbols R can be used for the construction of Seitz symbols for the symmetry operations of subperiodic layer and rod groups (Litvin & Kopsky, 2014), and magnetic groups, or for the designation of the symmetry operations of the point groups of space groups. [One should note that the Seitz symbols applied in the first and second editions of IT E and in the IUCr ebook on magnetic groups (Litvin, 2012) differ from the standard symbols adopted by the Commission of Crystallographic Nomenclature.]
The Seitz symbols for plane groups are constructed following similar rules to those for space groups. The rotation part R is 1 for the identity, m for reflections, and 2, 3, 4 and 6 are used for rotations. The orientation of a reflection line is specified by a subscript indicating the direction of its `normal'. Obviously, the direction indicators are of no relevance for the rotation points. The linear parts R of the Seitz symbols of the planegroup symmetry operations are shown in Tables 1.4.2.4 and 1.4.2.5. Each symbol R is specified by the shorthand notation of its (2 × 2) matrix representation, the type of symmetry operation and, if applicable, its orientation as described in the corresponding symmetryoperations block of the planegroup tables of this volume. The sequence of R symbols in Table 1.4.2.4 corresponds to the numbering scheme of the generalposition coordinate doublets of the plane group p4mm, while those of Table 1.4.2.5 correspond to the generalposition sequence of the plane group p6mm. The same symbols R can be used for the construction of Seitz symbols for the symmetry operations of subperiodic frieze groups (Litvin & Kopsky, 2014).


As illustrated in the examples above, zero translations are normally specified by a single zero in the Seitz symbols, but in cases where it is unclear whether the symbol refers to a space or a planegroup symmetry operation, an explicit indication of the components of the translation vector is recommended.
From the description given above, it is clear that Seitz symbols can be considered as shorthand modifications of the matrix–column presentation of symmetry operations discussed in detail in Chapter 1.2 : the translation parts of and coincide, while the different (3 × 3) matrices W are represented by the symbols R shown in Tables 1.4.2.1–1.4.2.3. As a result, the expressions for the product and the inverse of symmetry operations in Seitz notation are rather similar to those of the matrix–column pairs discussed in detail in Chapter 1.2 :
Similarly, the action of a symmetry operation on the column of point coordinates x is given by [cf. Chapter 1.2, equation (1.2.2.4) ].
The rotation parts of the Seitz symbols partly resemble the geometricdescription symbols of symmetry operations described in Section 1.4.2.1 and listed in the symmetryoperation blocks of the spacegroup tables of this volume: R contains the information on the type and order of the symmetry operation, and its characteristic direction. The Seitz symbols do not directly indicate the location of the symmetry operation, nor its glide or screw component, if any.
References
Bradley, C. J. & Cracknell, A. P. (1972). The Mathematical Theory of Symmetry in Solids. Oxford University Press.Glazer, A. M., Aroyo, M. I. & Authier, A. (2014). Seitz symbols for crystallographic symmetry operations. Acta Cryst. A70, 300–302.
Litvin, D. B. (2012). Magnetic Group Tables. IUCr ebook. http://www.iucr.org/publ/978–09553602–20 .
Litvin, D. B. & Kopský, V. (2011). Seitz notation for symmetry operations of space groups. Acta Cryst. A67, 415–418.
Litvin, D. B. & Kopsky, V. (2014). Seitz symbols for symmetry operations of subperiodic groups. Acta Cryst. A70, 677–678.
Seitz, F. (1935). A matrixalgebraic development of the crystallographic groups. III. Z. Kristallogr. 91, 336–366.