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. 142-144

Section 2.1.1. Conventional descriptions of plane and space groups

Th. Hahna and A. Looijenga-Vosb

2.1.1. Conventional descriptions of plane and space groups

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2.1.1.1. Classification of space groups

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In this volume, the plane groups and space groups are classified according to three criteria:

  • (i) According to geometric crystal classes, i.e. according to the crystallographic point group to which a particular space group belongs. There are 10 crystal classes in two dimensions and 32 in three dimensions. They are described and listed in Chapter 3.2[link] and in column 4 of Table 2.1.1.1[link]. [For arithmetic crystal classes, see Chapter 1.3[link] and Table 2.1.3.3[link] in this volume, and Chapter 1.4[link] of International Tables for Crystallography, Vol. C (2004)[link].]

    Table 2.1.1.1| top | pdf |
    Crystal families, crystal systems, conventional coordinate systems and Bravais lattices in one, two and three dimensions

    Crystal familySymbolCrystal systemCrystallographic point groupsNo. of space groupsConventional coordinate systemBravais lattices
    Restrictions on cell parametersParameters to be determined
    One dimension
    [Scheme scheme1] 2 None a [{\scr p}]
    Two dimensions
    Oblique m Oblique [Scheme scheme2] 2 None a, b mp
    (monoclinic)         γ§  
    Rectangular o Rectangular [Scheme scheme3] 7 [\gamma = 90^{\circ}] a, b op
    (orthorhombic)           oc
    Square t Square [Scheme scheme4] 3 [a = b] a tp
    (tetragonal)       [\gamma = 90^{\circ}]    
    Hexagonal h Hexagonal [Scheme scheme5] 5 [a = b] a hp
            [\gamma = 120^{\circ}]    
    Three dimensions
    Triclinic a Triclinic [Scheme scheme6] 2 None [a,b,c] aP
    (anorthic)         [\alpha, \beta, \gamma]  
    Monoclinic m Monoclinic [Scheme scheme7] 13 b-unique setting [a,b,c] mP
            [\alpha = \gamma = 90^{\circ}] β § mS (mC, mA, mI)
            c-unique setting [a, b, c] mP
            [\alpha = \beta = 90^{\circ}] γ § mS (mA, mB, mI)
    Orthorhombic o Orthorhombic [Scheme scheme8] 59 [\alpha = \beta = \gamma = 90^{\circ}] a, b, c oP
    oS (oC, oA, oB)
    oI
    oF
    Tetragonal t Tetragonal [Scheme scheme9] 68 [a = b] a, c tP
            [\alpha = \beta = \gamma = 90^{\circ}]   tI
    Hexagonal h Trigonal [Scheme scheme10] 18 [a = b] a, c hP
            [\alpha = \beta = 90^{\circ},\ \gamma = 120^{\circ}]    
          7 [a = b = c] [\alpha = \beta = \gamma] (rhombohedral axes, primitive cell) [a = b] [\alpha = \beta = 90^{\circ}, \gamma = 120^{\circ}] (hexagonal axes, triple obverse cell) a, α hR
        Hexagonal [Scheme scheme11] 27 [a = b] a, c hP
            [\alpha = \beta = 90^{\circ}, \gamma = 120^{\circ}]    
    Cubic c Cubic [Scheme scheme12] 36 [a = b = c] a cP
            [\alpha = \beta = \gamma = 90^{\circ}]   cI
                cF
    The symbols for crystal families (column 2) and Bravais lattices (column 8) were adopted by the International Union of Crystallography in 1985; cf. de Wolff et al. (1985)[link].
    Symbols surrounded by dashed or full lines indicate Laue groups; full lines indicate Laue groups which are also lattice point symmetries (holohedries).
    §These angles are conventionally taken to be non-acute, i.e. [\geq 90^{\circ}].
    For the use of the letter S as a new general, setting-independent `centring symbol' for monoclinic and orthorhombic Bravais lattices, see de Wolff et al. (1985)[link].
  • (ii) According to crystal families. The term crystal family designates the classification of the 17 plane groups into four categories and of the 230 space groups into six categories, as displayed in column 1 of Table 2.1.1.1[link]. Here all `hexagonal', `trigonal' and `rhombohedral' space groups are contained in one family, the hexagonal crystal family. The `crystal family' thus corresponds to the term `crystal system', as used frequently in the American and Russian literature.

    The crystal families are symbolized by the lower-case letters a, m, o, t, h, c, as listed in column 2 of Table 2.1.1.1[link]. If these letters are combined with the appropriate capital letters for the lattice-centring types (cf. Table 2.1.1.2[link]), symbols for the 14 Bravais lattices result. These symbols and their occurrence in the crystal families are shown in column 8 of Table 2.1.1.1[link]; mS and oS are the standard setting-independent symbols for the centred monoclinic and the one-face-centred orthorhombic Bravais lattices, cf. de Wolff et al. (1985)[link]; symbols between parentheses represent alternative settings of these Bravais lattices.

    Table 2.1.1.2| top | pdf |
    Symbols for the conventional centring types of one-, two- and three-dimensional cells

    SymbolCentring type of cellNumber of lattice points per cellCoordinates of lattice points within cell
    One dimension
    [{\scr p}] Primitive 1 0
    Two dimensions
    p Primitive 1 0, 0
    c Centred 2 0, 0; [{1 \over 2}], [{1 \over 2}]
    h Hexagonally centred 3 0, 0; [{2 \over 3}], [{1 \over 3}]; [{1 \over 3}], [{2 \over 3}]
    Three dimensions
    P Primitive 1 0, 0, 0
    C C-face centred 2 0, 0, 0; [{1 \over 2}], [{1 \over 2}], 0
    A A-face centred 2 0, 0, 0; 0, [{1 \over 2}], [{1 \over 2}]
    B B-face centred 2 0, 0, 0; [{1 \over 2}], 0, [{1 \over 2}]
    I Body centred 2 0, 0, 0; [{1 \over 2}], [{1 \over 2}], [{1 \over 2}]
    F All-face centred 4 0, 0, 0; [{1 \over 2}], [{1 \over 2}], 0; 0, [{1 \over 2}], [{1 \over 2}]; [{1 \over 2}], 0, [{1 \over 2}]
    R [\cases{\hbox{Rhombohedrally centred}\cr \hbox{(description with `hexagonal axes')}\cr \hbox{Primitive}\cr \hbox{(description with `rhombohedral axes')}\cr}] 3 [\!\openup1pt{\cases {0,{\hbox to 1pt{}} 0,{\hbox to 1pt{}} 0{\hbox{; }}{\hbox to 2pt{}}{2 \over 3},{\hbox to -1.5pt{}} {1 \over 3},{\hbox to -1.5pt{}} {1 \over 3}{\hbox{; }}{\hbox to 1pt{}}{1 \over 3},{\hbox to -1pt{}} {2 \over 3},{\hbox to -1pt{}} {2 \over 3} \hbox{ (`obverse setting')}\cr 0,{\hbox to 1pt{}} 0,{\hbox to 1pt{}} 0{\hbox{; }}{\hbox to 1.5pt{}}{1 \over 3},{\hbox to -1pt{}} {2 \over 3},{\hbox to -1.5pt{}} {1 \over 3}{\hbox{; }}{\hbox to 1pt{}}{2 \over 3}, {1 \over 3},{\hbox to -1.5pt{}} {2 \over 3} \hbox{ (`reverse setting')}\cr}}]
    1 0, 0, 0
    H§ Hexagonally centred 3 0, 0, 0; [{2 \over 3}], [{1 \over 3}], 0; [{1 \over 3}], [{2 \over 3}], 0
    The two-dimensional triple hexagonal cell h is an alternative description of the hexagonal plane net, as illustrated in Fig. 1.5.1.8[link] . It is not used for systematic plane-group description in this volume; it is introduced, however, in the sub- and supergroup entries of the plane-group tables of International Tables for Crystallography, Vol. A1 (2010)[link], abbreviated as IT A1. Plane-group symbols for the h cell are listed in Section 1.5.4[link] . Transformation matrices are contained in Table 1.5.1.1[link] .
    In the space-group tables (Chapter 2.3[link] ), as well as in IT (1935)[link] and IT (1952)[link], the seven rhombohedral R space groups are presented with two descriptions, one based on hexagonal axes (triple cell), one on rhombohedral axes (primitive cell). In the present volume, as well as in IT (1952)[link] and IT A (2002)[link], the obverse setting of the triple hexagonal cell R is used. Note that in IT (1935)[link] the reverse setting was employed. The two settings are related by a rotation of the hexagonal cell with respect to the rhombohedral lattice around a threefold axis, involving a rotation angle of 60, 180 or 300° (cf. Fig. 1.5.1.6[link] ). Further details may be found in Section 1.5.4[link] and Chapter 3.1[link] . Transformation matrices are contained in Table 1.5.1.1[link] .
    §The triple hexagonal cell H is an alternative description of the hexagonal Bravais lattice, as illustrated in Fig. 1.5.1.8[link] . It was used for systematic space-group description in IT (1935)[link], but replaced by P in IT (1952)[link]. It is used in the tables of maximal subgroups and minimal supergroups of the space groups in IT A1 (2010)[link]. Space-group symbols for the H cell are listed in Section 1.5.4[link] . Transformation matrices are contained in Table 1.5.1.1[link] .
  • (iii) According to crystal systems. This classification collects the plane groups into four categories and the space groups into seven categories. The classifications according to crystal families and crystal systems are the same for two dimensions.

    For three dimensions, this applies to the triclinic, monoclinic, orthorhombic, tetragonal and cubic systems. The only complication exists in the hexagonal crystal family, for which several subdivisions into systems have been proposed in the literature. In this volume [as well as in International Tables for X-ray Crystallography (1952)[link], hereafter IT (1952), and the subsequent editions of IT], the space groups of the hexagonal crystal family are grouped into two `crystal systems' as follows: all space groups belonging to the five crystal classes 3, [\bar{3}], 32, 3m and [\bar{3}m], i.e. having 3, [3_{1}], [3_{2}] or [\bar{3}] as principal axis, form the trigonal crystal system, irrespective of whether the Bravais lattice is hP or hR; all space groups belonging to the seven crystal classes 6, [\bar{6}, 6/m], 622, 6mm, [\bar{6}]2m and [6/mmm], i.e. having 6, [6_{1}], [6_{2}], [6_{3}], [6_{4}], [6_{5}] or [\bar{6}] as principal axis, form the hexagonal crystal system; here the lattice is always hP (cf. Chapter 1.3[link] ). The crystal systems, as defined above, are listed in column 3 of Table 2.1.1.1[link].

A different subdivision of the hexagonal crystal family is in use, mainly in the French literature. It consists of grouping all space groups based on the hexagonal Bravais lattice hP (lattice point symmetry [6/mmm]) into the `hexagonal' system and all space groups based on the rhombohedral Bravais lattice hR (lattice point symmetry [\bar{3}m]) into the `rhombohedral' system. In Chapter 1.3[link] , these systems are called `lattice systems'. They were called `Bravais systems' in earlier editions of this volume.

The theoretical background for the classification of space groups is provided in Chapter 1.3[link] .

2.1.1.2. Conventional coordinate systems and cells

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A plane group or space group usually is described by means of a crystallographic coordinate system, consisting of a crystallographic basis (basis vectors are lattice vectors) and a crystallographic origin (origin at a centre of symmetry or at a point of high site symmetry). The choice of such a coordinate system is not mandatory, since in principle a crystal structure can be referred to any coordinate system; cf. Chapters 1.3[link] and 1.5[link] .

The selection of a crystallographic coordinate system is not unique. Conventionally, a right-handed set of basis vectors is taken such that the symmetry of the plane or space group is displayed best. With this convention, which is followed in the present volume, the specific restrictions imposed on the cell parameters by each crystal family become particularly simple. They are listed in columns 6 and 7 of Table 2.1.1.1[link]. If within these restrictions the smallest cell is chosen, a conventional (crystallographic) basis results. Together with the selection of an appropriate conventional (crystallographic) origin (cf. Sections 2.1.3.2[link] and 2.1.3.7[link]), such a basis defines a conventional (crystallographic) coordinate system and a conventional cell. The conventional cell of a point lattice or a space group, obtained in this way, turns out to be either primitive or to exhibit one of the centring types listed in Table 2.1.1.2[link]. The centring type of a conventional cell is transferred to the lattice which is described by this cell; hence, we speak of primitive, face-centred, body-centred etc. lattices. Similarly, the cell parameters are often called lattice parameters; cf. Chapters 1.3[link] and 3.1[link] for further details.

In the triclinic, monoclinic and orthorhombic crystal systems, additional conventions (for instance cell reduction or metrical conventions based on the lengths of the cell edges) are needed to determine the choice and the labelling of the axes. Reduced bases are treated in Chapter 3.1[link] , orthorhombic settings in Section 2.1.3.6[link], and monoclinic settings and cell choices in Section 2.1.3.15[link] (cf. Section 1.5.4[link] for a detailed treatment of alternative settings of space groups).

In this volume, all space groups within a crystal family are referred to the same kind of conventional coordinate system, with the exception of the hexagonal crystal family in three dimensions. Here, two kinds of coordinate systems are used, the hexagonal and the rhombohedral systems. In accordance with common crystallographic practice, all space groups based on the hexagonal Bravais lattice hP (18 trigonal and 27 hexagonal space groups) are described only with a hexagonal coordinate system (primitive cell), whereas the seven space groups based on the rhombohedral Bravais lattice hR (the so-called `rhombohedral space groups', cf. Section 1.4.1[link] ) are treated in two versions, one referred to `hexagonal axes' (triple obverse cell) and one to `rhombohedral axes' (primitive cell); cf. Table 2.1.1.2[link]. In practice, hexagonal axes are preferred because they are easier to visualize.

Table 2.1.1.2[link] contains only those conventional centring symbols which occur in the Hermann–Mauguin space-group symbols. There exist, of course, further kinds of centred cells which are unconventional, see for example the synoptic tables of plane (Table 1.5.4.3[link] ) and space (Table 1.5.4.4[link] ) groups discussed in Chapter 1.5[link] . The centring type of a cell may change with a change of the basis vectors; in particular, a primitive cell may become a centred cell and vice versa. Examples of relevant transformation matrices are contained in Table 1.5.1.1[link] .

References

International Tables for Crystallography (2004). Vol. C, 3rd ed., edited by E. Prince. Dordrecht: Kluwer Academic Publishers.
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).]
Wolff, P. M. de, Belov, N. V., Bertaut, E. F., Buerger, M. J., Donnay, J. D. H., Fischer, W., Hahn, Th., Koptsik, V. A., Mackay, A. L., Wondratschek, H., Wilson, A. J. C. & Abrahams, S. C. (1985). Nomenclature for crystal families, Bravais-lattice types and arithmetic classes. Report of the International Union of Crystallography Ad-hoc Committee on the Nomenclature of Symmetry. Acta Cryst. A41, 278–280.








































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