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. 154-158

Section 2.1.3.6. Space-group diagrams

Th. Hahna and A. Looijenga-Vosb

2.1.3.6. Space-group diagrams

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The space-group diagrams serve two purposes: (i) to show the relative locations and orientations of the symmetry elements and (ii) to illustrate the arrangement of a set of symmetry-equivalent points of the general position.

With the exception of general-position diagrams in perspective projection for some space groups (cf. Section 2.1.3.6.8[link]), all of the diagrams are orthogonal projections, i.e. the projection direction is perpendicular to the plane of the figure. Apart from the descriptions of the rhombohedral space groups with `rhombohedral axes' (cf. Section 2.1.3.6.6[link]), the projection direction is always a cell axis. If other axes are not parallel to the plane of the figure, they are indicated by the subscript p, as [a_{p},\ b_{p}] or [c_{p}] in the case of one or two axes for monoclinic and triclinic space groups, respectively (cf. Figs. 2.1.3.1[link][link] to 2.1.3.3[link]), or by the subscript rh for the three rhombohedral axes in Fig. 2.1.3.9[link].

The graphical symbols for symmetry elements, as used in the drawings, are displayed in Tables 2.1.2.2[link] to 2.1.2.7[link].

In the diagrams, `heights' h above the projection plane are indicated for symmetry planes and symmetry axes parallel to the projection plane, as well as for centres of symmetry. The heights are given as fractions of the shortest lattice translation normal to the projection plane and, if different from 0, are printed next to the graphical symbols. Each symmetry element at height h is accompanied by another symmetry element of the same type at height [h + {1 \over 2}] (this does not apply to the horizontal fourfold axes in the diagrams for the cubic space groups). In the space-group diagrams, only the symmetry element at height h is indicated (cf. Section 2.1.2[link]).

Schematic representations of the diagrams, displaying the origin, the labels of the axes, and the projection direction [uvw], are given in Figs. 2.1.3.1[link][link][link][link][link][link][link][link] to 2.1.3.10[link] (except Fig. 2.1.3.6). The general-position diagrams are indicated by the letter [\hbox{\sf G}].

2.1.3.6.1. Plane groups

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Each description of a plane group contains two diagrams, one for the symmetry elements (left) and one for the general position (right). The two axes are labelled a and b, with a pointing downwards and b running from left to right.

2.1.3.6.2. Triclinic space groups

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For each of the two triclinic space groups, three elevations (along a, b and c) are given, in addition to the general-position diagram [{\sf G}] (projected along c) at the lower right of the set, as illustrated in Fig. 2.1.3.1[link].

[Figure 2.1.3.1]

Figure 2.1.3.1 | top | pdf |

Triclinic space groups ([\hbox{\sf G}] = general-position diagram).

The diagrams represent a reduced cell of type II for which the three interaxial angles are non-acute, i.e. [\alpha,\beta,\gamma \geq\! 90^{\circ}]. For a cell of type I, all angles are acute, i.e. [\alpha,\beta,\gamma \,\lt\,90^{\circ}]. For a discussion of the two types of reduced cells, see Section 3.1.3[link] .

2.1.3.6.3. Monoclinic space groups (cf. Sections 2.1.3.2[link] and 2.1.3.15[link])

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The `complete treatment' of each of the two settings contains four diagrams (Figs. 2.1.3.2[link] and 2.1.3.3[link]). Three of them are projections of the symmetry elements, taken along the unique axis (upper left) and along the other two axes (lower left and upper right). For the general position, only the projection along the unique axis is given (lower right).

[Figure 2.1.3.2]

Figure 2.1.3.2 | top | pdf |

Monoclinic space groups, setting with unique axis b ([\hbox{\sf G}] = general-position diagram).

[Figure 2.1.3.3]

Figure 2.1.3.3 | top | pdf |

Monoclinic space groups, setting with unique axis c ([\hbox{\sf G}] = general-position diagram).

The `synoptic descriptions' of the three cell choices (for each setting) are headed by a pair of diagrams, as illustrated in Fig. 2.1.3.4[link]. The drawings on the left display the symmetry elements and the ones on the right the general position (labelled [{\sf G}]). Each diagram is a projection of four neighbouring unit cells along the unique axis. It contains the outlines of the three cell choices drawn as heavy lines. For the labelling of the axes, see Fig. 2.1.3.4[link]. The headline of the description of each cell choice contains a small-scale drawing, indicating the basis vectors and the cell that apply to that description.

[Figure 2.1.3.4]

Figure 2.1.3.4 | top | pdf |

Monoclinic space groups, cell choices 1, 2, 3. Upper pair of diagrams: setting with unique axis b. Lower pair of diagrams: setting with unique axis c. The numbers 1, 2, 3 within the cells and the subscripts of the labels of the axes indicate the cell choice (cf. Section 2.1.3.15[link]). The unique axis points upwards from the page. [\hbox{\sf G}] = general-position diagram.

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].

2.1.3.6.5. Tetragonal, trigonal P and hexagonal P space groups

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The pairs of diagrams for these space groups are similar to those in the previous editions of IT. Each pair consists of a general-position diagram (right) and a diagram of the symmetry elements (left), both projected along c, as illustrated in Figs. 2.1.3.7[link] and 2.1.3.8[link].

[Figure 2.1.3.7]

Figure 2.1.3.7 | top | pdf |

Tetragonal space groups ([\hbox{\sf G}] = general-position diagram).

[Figure 2.1.3.8]

Figure 2.1.3.8 | top | pdf |

Trigonal P and hexagonal P space groups ([\hbox{\sf G}] = general-position diagram).

2.1.3.6.6. Trigonal R (rhombohedral) space groups

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The seven rhombohedral space groups are treated in two versions, the first based on `hexagonal axes' (obverse setting), the second on `rhombohedral axes' (cf. Sections 2.1.1.2[link] and 2.1.3.2[link]). The pairs of diagrams are similar to those in IT (1952)[link] and IT A (2002)[link]; the left or top one displays the symmetry elements, the right or bottom one the general position. This is illustrated in Fig. 2.1.3.9[link], which gives the axes a and b of the triple hexagonal cell and the projections of the axes of the primitive rhombohedral cell, labelled arh, brh and crh. For convenience, all `heights' in the space-group diagrams are fractions of the hexagonal c axis. For `hexagonal axes', the projection direction is [001], for `rhombohedral axes' it is [111]. In the general-position diagrams, the circles drawn in heavier lines represent atoms that lie within the primitive rhombohedral cell (provided the symbol `−' is read as [1 - z] rather than as [- z]).

[Figure 2.1.3.9]

Figure 2.1.3.9 | top | pdf |

Rhombohedral space groups. Obverse triple hexagonal cell with `hexagonal axes' a, b and primitive rhombohedral cell with projections of `rhombohedral axes' arh, brh, crh. Note: In the actual space-group diagrams the edges of the primitive rhombohedral cell (dashed lines) are only indicated in the general-position diagram of the rhombohedral-axes description ([\hbox{\sf G}] = general-position diagram).

The symmetry-element diagrams for the hexagonal and the rhombohedral descriptions of a space group are the same. The edges of the primitive rhombohedral cell (cf. Fig. 2.1.3.9[link]) are only indicated in the general-position diagram of the rhombohedral description.

2.1.3.6.7. Cubic space groups

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For each cubic space group, one projection of the symmetry elements along [001] is given, Fig. 2.1.3.10[link]; for details of the diagrams, see Section 2.1.2[link] and Buerger (1956)[link]. For face-centred lattices F, only a quarter of the unit cell is shown; this is sufficient since the projected arrangement of the symmetry elements is translation-equivalent in the four quarters of an F cell. It is important to note that symmetry axes inclined to the projection plane are indicated where they intersect the plane of projection. Symmetry planes inclined to the projection plane that occur in classes [\bar{4}3m] and [m\bar{3}m] are shown as `inserts' around the high-symmetry points, such as [0,0,0]; [\textstyle{{1}\over{2}},0,0]; etc.

[Figure 2.1.3.10]

Figure 2.1.3.10 | top | pdf |

Cubic space groups. [\hbox{\sf G}] = general-position diagram, in which the equivalent positions are shown as the vertices of polyhedra.

The cubic diagrams given in IT (1935)[link] are different from the ones used here. No drawings for cubic space groups were provided in IT (1952)[link].

2.1.3.6.8. Diagrams of the general position (by K. Momma and M. I. Aroyo)

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Non-cubic space groups. In these diagrams, the `heights' of the points are z coordinates, except for monoclinic space groups with unique axis b where they are y coordinates. For rhombohedral space groups, the heights are always fractions of the hexagonal c axis. The symbols [+] and − stand for [+z] and [-z] (or [+y] and [-y]) in which z or y can assume any value. For points with symbols [+] or − preceded by a fraction, e.g. [{1 \over 2}+] or [{1 \over 3}-], the relative z or y coordinate is [{1 \over 2}] etc. higher than that of the point with symbol [+] or −.

Where a mirror plane exists parallel to the plane of projection, the two positions superimposed in projection are indicated by the use of a ring divided through the centre. The information given on each side refers to one of the two positions related by the mirror plane, as in [-\def\circhyp{\mathop{\bigcirc\hskip-.95em{{\raise0.15em\hbox{$\scriptscriptstyle{{\hbox to 1.5pt{}}\displaystyle{,}}$}}\hskip-0.05em \lower0.2em\hbox{\vrule height 0.9em}}}\;\;}\circhyp +].

Diagrams for cubic space groups (Fig. 2.1.3.10)[link]. Following the approach of IT (1935), for each cubic space group a diagram showing the points of the general position as the vertices of polyhedra is given. In these diagrams, the polyhedra are transparent, but the spheres at the vertices are opaque. For most of the space groups, `starting points' with the same coordinate values, x = 0.048, y = 0.12, z = 0.089, have been used. The origins of the polyhedra are chosen at special points of highest site symmetry, which for most space groups coincide with the origin (and its equivalent points in the unit cell). Polyhedra with origins at sites [({\textstyle{1\over 8}}, {\textstyle{1\over 8}}, {\textstyle{1\over 8}})] have been chosen for the space groups [P4_332] (212) and [I4_132] (214), and [({\textstyle{3\over 8}}, {\textstyle{3\over 8}}, {\textstyle{3\over 8}})] for [P4_132] (213). The two diagrams shown for the space groups [I\bar{4}3d] (220) and [Ia\bar{3}d] (230) correspond to polyhedra with origins chosen at two different special sites with site-symmetry groups of equal (32 versus [\bar{3}] in [Ia\bar{3}d]) or nearly equal order (3 versus [\bar{4}] in [I\bar{4}3d]). The height h of the centre of each polyhedron is given on the diagram, if different from zero. For space-group Nos. 198, 199 and 220, h refers to the special point to which the polyhedron (triangle) is connected. Polyhedra with height 1 are omitted in all the diagrams. A grid of four squares is drawn to represent the four quarters of the basal plane of the cell. For space groups [F\bar{4}3c] (219), [Fm\bar{3}c] (226) and [Fd\bar{3}c] (228), where the number of points is too large for one diagram, two diagrams are provided, one for the upper half and one for the lower half of the cell.

Notes:

  • (i) For space group [P4_132] (213), the coordinates [\bar{x}, \bar{y}, \bar{z}] have been chosen for the `starting point' to show the enantiomorphism with [P4_332] (212).

  • (ii) For the description of a space group with `origin choice 2', the coordinates x, y, z of all points have been shifted with the origin to retain the same polyhedra for both origin choices.

An additional general-position diagram is shown on the fourth page for each of the ten space groups of the [m\bar{3}m] crystal class. To provide a clearer three-dimensional-style overview of the arrangements of the polyhedra, these general-position diagrams are shown in tilted projection (in contrast to the orthogonal-projection diagrams described above).

The general-position diagrams of the cubic groups in both orthogonal and tilted projections were generated using the program VESTA (Momma & Izumi, 2011[link]).

Readers who wish to compare other approaches to space-group diagrams and their history are referred to IT (1935), IT (1952), the fifth edition of IT A (2002)[link] (where general-position stereodiagrams of the cubic space groups are shown) and the following publications: Astbury & Yardley (1924)[link], Belov et al. (1980)[link], Buerger (1956)[link], Fedorov (1895[link]; English translation, 1971), Friedel (1926)[link], Hilton (1903)[link], Niggli (1919)[link] and Schiebold (1929)[link].

References

International Tables for Crystallography (2002). Vol. A, 5th ed., edited by Th. Hahn. Dordrecht: Kluwer Academic Publishers. [Abbreviated as IT A (2002).]
International Tables for X-ray Crystallography (1952). Vol. I, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Revised editions: 1965, 1969 and 1977. Abbreviated as IT (1952).]
Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935). 1. Band, edited by C. Hermann. Berlin: Borntraeger. [Revised edition: Ann Arbor: Edwards (1944). Abbreviated as IT (1935).]
Momma, K. & Izumi, F. (2011). VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data. J. Appl. Cryst. 44, 1272–1276.
Astbury, W. T. & Yardley, K. (1924). Tabulated data for the examination of the 230 space groups by homogeneous X-rays. Philos. Trans. R. Soc. London Ser. A, 224, 221–257.
Belov, N. V., Zagal'skaja, Ju. G., Litvinskaja, G. P. & Egorov-Tismenko, Ju. K. (1980). Atlas of the Space Groups of the Cubic System. Moscow: Nauka. (In Russian.)
Buerger, M. J. (1956). Elementary Crystallography. New York: Wiley.
Fedorov, E. S. (1895). Theorie der Kristallstruktur. Einleitung. Regelmässige Punktsysteme (mit übersichtlicher graphischer Darstellung). Z. Kristallogr. 24, 209–252, Tafel V, VI. [English translation by D. & K. Harker (1971). Symmetry of Crystals, esp. pp. 206–213. Am. Crystallogr. Assoc., ACA Monograph No. 7.]
Friedel, G. (1926). Leçons de Cristallographie. Nancy/Paris/Strasbourg: Berger-Levrault. [Reprinted: Paris: Blanchard (1964).]
Hilton, H. (1903). Mathematical Crystallography. Oxford: Clarendon Press. [Reprint: New York: Dover (1963).]
Niggli, P. (1919). Geometrische Kristallographie des Diskontinuums. Leipzig: Borntraeger. [Reprint: Wiesbaden: Sändig (1973).]
Schiebold, E. (1929). Über eine neue Herleitung und Nomenklatur der 230 kristallographischen Raumgruppen mit Atlas der 230 Raumgruppen-Projektionen. Text, Atlas. In Abhandlungen der Mathematisch-Physikalischen Klasse der Sächsischen Akademie der Wissenschaften, Band 40, Heft 5. Leipzig: Hirzel.








































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