International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 3.6, pp. 859-861

## Section 3.6.3.3. Diagrams of symmetry elements and of the general positions

D. B. Litvina*

aDepartment of Physics, The Eberly College of Science, Penn State – Berks Campus, The Pennsylvania State University, PO Box 7009, Reading, PA 19610–6009, USA
Correspondence e-mail: u3c@psu.edu

#### 3.6.3.3. Diagrams of symmetry elements and of the general positions

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There are two types of diagrams: symmetry diagrams and general-position diagrams. The symmetry diagrams show (1) the relative locations and orientations of the symmetry elements and (2) the absolute locations and orientations of these symmetry elements in a given coordinate system. The general-position diagrams show, in that coordinate system, the arrangement of a set of symmetry-equivalent general points and the relative orientations of magnetic moments on this set of points. Figs. 3.6.3.3 and 3.6.3.4 show the symmetry diagram and general-position diagram, respectively, of the three-dimensional magnetic space group P412′2′.

 Figure 3.6.3.3 | top | pdf |Symmetry diagram of P412′2′
 Figure 3.6.3.4 | top | pdf |General-position diagram of P412′2′.

All diagrams of three-dimensional magnetic space groups and three-dimensional subperiodic groups are orthogonal projections. The projection direction is along a basis vector of the conventional crystallographic coordinate system, see Table 1.1 of Litvin (2013). If the other two basis vectors are not parallel to the plane of the diagram, they are indicated by a subscript p, e.g. ap, bp and cp. Schematic representations of the diagrams, showing their conventional coordinate systems, i.e. the origin O and basis vectors, are given in Table 2.1 of Litvin (2013). For two-dimensional magnetic space groups and magnetic frieze groups, the diagrams are in the plane defined by the group's conventional coordinate system.

The graphical symbols used in the symmetry diagrams are listed in Table 2.2 of Litvin (2013) and are an extension of those used in the present volume, IT E and Litvin (2008b). For symmetry planes and symmetry axes parallel to the plane of diagram, for rotation-inversions and for centres of symmetry, the `heights' h along the projection direction above the plane of the diagram are given. The heights are given as fractions of the shortest translation along the projection direction and, if different from zero, are printed next to the graphical symbol, see Fig. 3.6.3.3.

In the general-position diagrams, the general positions and corresponding magnetic moments are colour coded. Positions with a z component of +z are shown as red circles and those with a z component of −z are shown as blue circles. If the z component is either h + z or hz with h ≠ 0, then the height h is printed next to the general position, see Fig. 3.6.3.4. If two general positions have the same x component and y component, but one has a z component +z and the other −z, the positions are shown as a circle with one half coloured red, the other half blue. The magnetic moments are colour coded to the general position to which they are associated, their direction in the plane of projection is given by an arrow in the direction of the projected magnetic moment. A + or − sign near the tip of the arrow indicates that the magnetic moment is inclined, respectively, above or below the plane of projection.

For magnetic space groups of the type , the symmetry diagram is that of the group . That each symmetry element also appears coupled with time inversion is represented by a red printed between and above the general-position and symmetry diagrams. Because groups of this type contain the time-inversion sym­metry, the magnetic moments are all identically zero, and no arrows appear in the general-position diagram. An example, the diagrams of the magnetic space group P41221′ are shown in Fig. 3.6.3.5. For triclinic, monoclinic/oblique, monoclinic/rectangular and orthorhombic rod groups, the colour coding of the general positions is extended according to the positive or negative values of the x and z components of the coordinates of the general position, see Fig. 3.6.3.6.

 Figure 3.6.3.5 | top | pdf |Diagrams of the magnetic space group P41221′.
 Figure 3.6.3.6 | top | pdf |General-position diagram of rod group 2.3.29 P2/c′11. The positional colour coding is red for x > 0 and z > 0; blue for x > 0 and z < 0; green for x < 0 and z > 0; and brown for x < 0 and z < 0.

VRML (Virtual Reality Modeling Language) general-position diagrams are available for the two- and three-dimensional magnetic subperiodic groups (Cordisco & Litvin, 2004), and for the one-, two- and non-cubic three-dimensional magnetic space groups (Burke et al., 2006). These diagrams can be rotated and zoomed in on to aid in the visualization of the general-position diagrams, and include both the general positions of the atoms and the general orientations of the associated magnetic moments.

### References

Burke, J. S., Cordisco, N. R. & Litvin, D. B. (2006). VRML general position diagrams of magnetic space groups. J. Appl. Cryst. 39, 620.
Cordisco, N. R. & Litvin, D. B. (2004). VRML general position diagrams of the magnetic subperiodic groups. J. Appl. Cryst. 37, 346.
Litvin, D. B. (2008b). Tables of crystallographic properties of magnetic space groups. Acta Cryst. A64, 419–424.
Litvin, D. B. (2013). Magnetic Group Tables, 1-, 2- and 3-Dimensional Magnetic Subperiodic Groups and Space Groups. Chester: International Union of Crystallography. Freely available from http://www.iucr.org/publ/978–0–9553602–2–0 .