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. 861-862

Section 3.6.3.6. Symmetry operations

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

3.6.3.6. Symmetry operations

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Listed under the heading of Symmetry operations is the geometric description of the symmetry operations of the magnetic group. A symbol denoting the geometric description of each symmetry operation is given. Details of this symbolism, except for the use of prime to denote time inversion, are given in Sections 1.4.2[link] and 2.1.3.9[link] . For glide planes and screw axes, the glide and screw part are always explicitly given in parentheses by fractional coordinates, i.e. by fractions of the basis vectors of the coordinate system of [{\cal F}] of the superfamily of the magnetic group. A coordinate triplet indicating the location and orientation of the symmetry element is given, and for rotation-inversions the location of the inversion point is also given. These symbols, with the addition of a prime to denote time inversion, follow those used in the present volume, IT E and Litvin (2005[link], 2008b[link]). In addition, each symmetry operation is also given in Seitz (1934[link], 1935a[link],b[link], 1936[link]) notation (see Section 3.6.2.2.3[link]), e.g. see Table 3.6.3.1[link] for the symmetry operations of the magnetic space group 51.14.400 P2bmma′.

Table 3.6.3.1| top | pdf |
Symmetry operations of magnetic space group 51.14.400 P2bmma

Symmetry operations
For (0, 0, 0)+ set
(1) 1 (2) 2 1/4, 0, z (3) 2′ 0, y, 0 (4) 2′ (1/2, 0, 0) x, 0, 0
{1∣0} {2001∣1/2, 0, 0} {2010∣0}′ {2100∣1/2, 0, 0}′
(5) [\overline 1'] (6) a′ (1/2, 0, 0) x, y, 0 (7) m x, 0, z (8) m 1/4, y, z
[\{\overline 1|0\}'] {m001∣1/2, 0, 0}′ {m010∣0} {m100∣1/2, 0, 0}
       
For (0, 1, 0)′+ set
(1) t′ (0, 1, 0) (2) 2′ 1/4, 1/2, z (3) 2 (0, 1, 0) 0, y, 0 (4) 2 (1/2, 0, 0) x, 1/2, 0
{1∣0, 1, 0}′ {2001∣1/2, 1, 0}′ {2010∣0, 1, 0} {2100∣1/2, 1, 0}
(5) [\overline 1] 0, 1/2, 0 (6) n (1/2, 1, 0) x, y, 0 (7) mx, 1/2, z (8) b (0, 1, 0) 1/4, y, z
[\{\overline 1|0,1,0\}] {m001∣1/2, 1, 0} {m010∣0, 1, 0}′ {m100∣1/2, 1, 0}′

The corresponding coordinate triplets of the General positions, see Section 3.6.3.9[link], may be interpreted as a second description of the symmetry operations, a description in matrix form. The numbering (1), (2), …, (p), … of the entries in the blocks Symmetry operations is the same as the numbering of the corresponding coordinate triplets of the General position, the first block below Positions. For all magnetic groups with primitive lattices, the two lists, Symmetry operations and General position, have the same number of entries.

For magnetic groups with centred cells, only one block of several (two, three or four) blocks of the general positions is explicitly given, see Table 3.6.3.2[link]. A set of two, three or four centring translations is given below the subheading Coordinates. Each of these translations is added to the given block of general positions to obtain the complete set of blocks of general positions. While one of the several blocks of general positions is explicitly given, the corresponding symmetry operations are all explicitly given. Each corresponding block of symmetry operations is listed under a subheading of `centring translation + set' for each centring translation listed below the subheading Coordinates.

Table 3.6.3.2| top | pdf |
General positions of magnetic space group 51.14.400 P2bmma

Positions Coordinates
  (0, 0, 0)+ (0, 1, 0)′+
16 l 1 (1) x, y, z [u, v, w] (2) [\overline x +1/2, \overline y ,z] [[\overline u ,\overline v ,w]] (3) [\overline x ,y, \overline z ] [[u, \overline v ,w]] (4) [x+1/2, \overline y , \overline z ] [[\overline u ,v,w]]
  (5) [\overline x ,\overline y ,\overline z ] [[\overline u ,\overline v ,\overline w ]] (6) [x+1/2,y, \overline z ] [[u,v, \overline w ]] (7) [x, \overline y ,z] [[\overline u ,v, \overline w ]] (8) [\overline x +1/2,y,z] [[u, \overline v ,\overline w ]]

References

Litvin, D. B. (2005). Tables of properties of magnetic subperiodic groups. Acta Cryst. A61, 382–385.
Litvin, D. B. (2008b). Tables of crystallographic properties of magnetic space groups. Acta Cryst. A64, 419–424.
Seitz, F. Z. (1934). A matrix-algebraic development of the crystallographic groups. Z. Kristallogr. 88, 433–459.
Seitz, F. Z. (1935a). A matrix-algebraic development of the crystallographic groups. Z. Kristallogr. 90, 289–313.
Seitz, F. Z. (1935b). A matrix-algebraic development of the crystallographic groups. Z. Kristallogr. 91, 336–366.
Seitz, F. Z. (1936). A matrix-algebraic development of the crystallographic groups. Z. Kristallogr. 94, 100–130.








































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