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

International Tables for Crystallography (2016). Vol. A, ch. 2.1, pp. 155-157

Section 2.1.3.6.4. Orthorhombic space groups and orthorhombic settings

Th. Hahna and A. Looijenga-Vosb

2.1.3.6.4. Orthorhombic space groups and orthorhombic settings

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The space-group tables contain a set of four diagrams for each orthorhombic space group. The set consists of three projections of the symmetry elements [along the c axis (upper left), the a axis (lower left) and the b axis (upper right)] in addition to the general-position diagram, which is given only in the projection along c (lower right). The projected axes, the origins and the projection directions of these diagrams are illustrated in Fig. 2.1.3.5[link]. They refer to the so-called `standard setting' of the space group, i.e. the setting described in the space-group tables and indicated by the `standard Hermann–Mauguin symbol' in the headline.

[Figure 2.1.3.5]

Figure 2.1.3.5 | top | pdf |

Orthorhombic space groups. Diagrams for the `standard setting' as described in the space-group tables ([\hbox{\sf G}] = general-position diagram).

For each orthorhombic space group, six settings exist, i.e. six different ways of assigning the labels a, b, c to the three ortho­rhombic symmetry directions; thus the shape and orientation of the cell are the same for each setting. These settings correspond to the six permutations of the labels of the axes (including the identity permutation); cf. Section 1.5.4.3[link] : [{\bf abc} \quad {\bf ba}\overline{{\bf c}} \quad {\bf cab} \quad \overline{{\bf c}}{\bf ba} \quad {\bf bca} \quad {\bf a}\overline{{\bf c}}{\bf b.}]The symbol for each setting, here called `setting symbol', is a shorthand notation for the (3 × 3) transformation matrix P of the basis vectors of the standard setting, a, b, c, into those of the setting considered (cf. Chapter 1.5[link] for a detailed discussion of coordinate transformations). For instance, the setting symbol cab stands for the cyclic permutation [{\bf a}' = {\bf c},\quad {\bf b}' = {\bf a},\quad {\bf c}' = {\bf b}]or [({\bf a}',{\bf b}',{\bf c}') = ({\bf a}, {\bf b}, {\bf c})\,{\bi P}=({\bf a},{\bf b}, {\bf c}) \pmatrix{0 &1 &0\cr 0 &0 &1\cr 1 &0 &0\cr} = ({\bf c},{\bf a},{\bf b}),]where a′, b′, c′ is the new set of basis vectors. An interchange of two axes reverses the handedness of the coordinate system; in order to keep the system right-handed, each interchange is accompanied by the reversal of the sense of one axis, i.e. by an element [\bar{1}] in the transformation matrix. Thus, [{\bf ba}\overline{{\bf c}}] denotes the transformation [({\bf a}',{\bf b}',{\bf c}') = ({\bf a},{\bf b},{\bf c}) \pmatrix{0 &1 &0\cr 1 &0 &0\cr 0 &0 &\bar{1}\cr} = ({\bf b},{\bf a},\overline{{\bf c}}).]The six orthorhombic settings correspond to six Hermann–Mauguin symbols which, however, need not all be different; cf. Table 2.1.3.4[link].1

Table 2.1.3.4| top | pdf |
Numbers of distinct projections and different Hermann–Mauguin symbols for the orthorhombic space groups

The space-group numbers are given in parentheses. The space groups are listed according to point group as indicated in the column headings.

Number of distinct projections222mm2[2/m \,2/m\, 2/m]
6   [Pmc2_{1}\ (26)] [P\ 2_{1}/m\; 2/m\; 2/a\ (51)]
(22 space groups)   Pma2 (28) [P\ 2/n\;2_{1}/n\;2/a\ (52)]
    [Pca2_{1}\ (29)] [P\ 2/m\;2/n\;2_{1}/a\ (53)]
    Pnc2 (30) [P\ 2_{1}/c\;2/c\;2/a\ (54)]
    [Pmn2_{1}\ (31)] [P\ 2/b\;2_{1}/c\;2_{1}/m\ (57)]
    [Pna2_{1}\ (33)] [P\ 2_{1}/b\;2/c\;2_{1}/n\ (60)]
    [Cmc2_{1}\ (36)] [P\ 2_{1}/n\;2_{1}/m\;2_{1}/a\ (62)]
    Amm2 (38) [C\ 2/m\;2/c\;2_{1}/m\ (63)]
    Aem2 (39) [C\ 2/m\;2/c\;2_{1}/e\ (64)]
    Ama2 (40) [I\ 2_{1}/m\;2_{1}/m\;2_{1}/a\ (74)]
    Aea2 (41)  
    Ima2 (46)  
3 [P222_{1}\ (17)] Pmm2 (25) [P\ 2/c\;2/c\;2/m\ (49)]
(25 space groups) [P2_{1}2_{1}2\ (18)] Pcc2 (27) [P\ 2/b\;2/a\;2/n\ (50)]
  [C222_{1}\ (20)] Pba2 (32) [P\ 2_{1}/b\;2_{1}/a\;2/m\ (55)]
  C222 (21) Pnn2 (34) [P\ 2_{1}/c\;2_{1}/c\;2/n\ (56)]
    Cmm2 (35) [P\ 2_{1}/n\;2_{1}/n\;2/m\ (58)]
    Ccc2 (37) [P\ 2_{1}/m\;2_{1}/m\;2/n\ (59)]
    Fmm2 (42) [C\ 2/m\;2/m\;2/m\ (65)]
    Fdd2 (43) [C\ 2/c\;2/c\;2/m\ (66)]
    Imm2 (44) [C\ 2/m\;2/m\;2/e\ (67)]
    Iba2 (45) [C\ 2/c\;2/c\;2/e\ (68)]
      [I\ 2/b\;2/a\;2/m\ (72)]
2     [P\ 2_{1}/b\;2_{1}/c\;2_{1}/a\ (61)]
(2 space groups)     [I\ 2_{1}/b\;2_{1}/c\;2_{1}/a\ (73)]
1 P222 (16)   [P\ 2/m\;2/m\;2/m\ (47)]
(10 space groups) [P2_{1}2_{1}2_{1}\ (19)]   [P\ 2/n\;2/n\;2/n\ (48)]
  F222 (22)   [F\ 2/m\;2/m\;2/m\ (69)]
  I222 (23)   [F\ 2/d\;2/d\;2/d\ (70)]
  [I2_{1}2_{1}2_{1}\ (24)]   [I\ 2/m\;2/m\;2/m\ (71)]
Total: 59 9 22 28

In the earlier (1935 and 1952) editions of International Tables, only one setting was illustrated, in a projection along c, so that it was usual to consider it as the `standard setting' and to accept its cell edges as crystal axes and its space-group symbol as the `standard Hermann–Mauguin symbol'. In the present edition, following IT A (2002)[link], however, all six orthorhombic settings are illustrated, as explained below.

The three projections of the symmetry elements can be interpreted in two ways. First, in the sense indicated above, that is, as different projections of a single (standard) setting of the space group, with the projected basis vectors a, b, c labelled as in Fig. 2.1.3.5[link]. Second, each one of the three diagrams can be considered as the projection along c′ of either one of two different settings: one setting in which b′ is horizontal and one in which b′ is vertical (a′, b′, c′ refer to the setting under consideration). This second interpretation is used to illustrate in the same figure the space-group symbols corresponding to these two settings. In order to view these projections in conventional orientation (b′ horizontal, a′ vertical, origin in the upper left corner, projection down the positive c′ axis), the setting with b′ horizontal can be inspected directly with the figure upright; hence, the corresponding space-group symbol is printed above the projection. The other setting with b′ vertical and a′ horizontal, however, requires turning the figure by 90°, or looking at it from the side; thus, the space-group symbol is printed at the left, and it runs upwards.

The `setting symbols' for the six settings are attached to the three diagrams of Fig. 2.1.3.6[link], which correspond to those of Fig. 2.1.3.5[link]. In the orientation of the diagram where the setting symbol is read in the usual way, a′ is vertical pointing downwards, b′ is horizontal pointing to the right, and c′ is pointing upwards from the page. Each setting symbol is printed in the position that in the space-group tables is actually occupied by the corresponding full Hermann–Mauguin symbol. The changes in the space-group symbol that are associated with a particular setting symbol can easily be deduced by comparing Fig. 2.1.3.6[link] with the diagrams for the space group under consideration.

[Figure 2.1.3.6]

Figure 2.1.3.6 | top | pdf |

Orthorhombic space groups. The three projections of the symmetry elements with the six setting symbols (see text). For setting symbols printed vertically, the page has to be turned clockwise by 90° or viewed from the side. Note that in the actual space-group tables instead of the setting symbols the corresponding full Hermann–Mauguin space-group symbols are printed.

Not all of the 59 orthorhombic space groups have all six projections distinct, i.e. have different Hermann–Mauguin symbols for the six settings. This aspect is treated in Table 2.1.3.4[link]. Only 22 space groups have six, 25 have three, 2 have two different symbols, while 10 have all symbols the same. This information can be of help in the early stages of a crystal-structure analysis.

The six setting symbols, i.e. the six permutations of the labels of the axes, form the column headings of the ortho­rhombic entries in Table 1.5.4.4[link] , which contains the extended Hermann–Mauguin symbols for the six settings of each orthorhombic space group. Note that some of these setting symbols exhibit different sign changes compared with those in Fig. 2.1.3.6[link].

References

International Tables for Crystallography (2002). Vol. A, 5th ed., edited by Th. Hahn. Dordrecht: Kluwer Academic Publishers. [Abbreviated as IT A (2002).]








































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