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

International Tables for Crystallography (2015). Vol. A, ch. 1.4, pp. 59-61

Section 1.4.3. Generation of space groups

H. Wondratscheke

1.4.3. Generation of space groups

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In group theory, a set of generators of a group is a set of group elements such that each group element may be obtained as a finite ordered product of the generators. For space groups of one, two and three dimensions, generators may always be chosen and ordered in such a way that each symmetry operation [\ispecialfonts{\sfi W}] can be written as the product of powers of h generators [\ispecialfonts{\sfi g}_j] (j = [1, 2, \ldots, h]). Thus,[\ispecialfonts{\sfi W}={\sfi g}_{h}^{k_h}\cdot{\sfi g}_{h-1}^{k_{h-1}}\cdot\ldots\cdot{\sfi g}_{p}^{k_{p}}\cdot\ldots\cdot{\sfi g}_{3}^{k_{3}}\cdot{\sfi g}_{2}^{k_{2}}\cdot{\sfi g}_{1},]where the powers [k_j] are positive or negative integers (including zero). The description of a group by means of generators has the advantage of compactness. For instance, the 48 symmetry operations in point group [m\bar{3}m] can be described by two generators. Different choices of generators are possible. For the space-group tables, generators and generating procedures have been chosen such as to make the entries in the blocks `General position' (cf. Section 2.1.3.11[link] ) and `Symmetry operations' (cf. Section 2.1.3.9[link] ) as transparent as possible. Space groups of the same crystal class are generated in the same way (see Table 1.4.3.1[link] for the sequences that have been chosen), and the aim has been to accentuate important subgroups of space groups as much as possible. Accordingly, a process of generation in the form of a composition series has been adopted, see Ledermann (1976[link]). The generator [\ispecialfonts{\sfi g}_{1}] is defined as the identity operation, represented by (1) [x, y, z]. The generators [\ispecialfonts{\sfi g}_{2}], [\ispecialfonts{\sfi g}_{3}], and [\ispecialfonts{\sfi g}_{4}] are the translations with translation vectors a, b and c, respectively. Thus, the coefficients [k_2], [k_3] and [k_4] may have any integral value. If centring translations exist, they are generated by translations [\ispecialfonts{\sfi g}_{5}] (and [\ispecialfonts{\sfi g}_{6}] in the case of an F lattice) with translation vectors d (and e). For a C lattice, for example, d is given by [{\bf d}=\textstyle{{1 \over 2}}({\bf a}+{\bf b})]. The exponents [k_5] (and [k_6]) are restricted to the following values:

  • Lattice letter A, B, C, I: [k_5=0] or 1.

    Table 1.4.3.1| top | pdf |
    Sequence of generators for the crystal classes

    The space-group generators differ from those listed here by their glide or screw components. The generator 1 is omitted, except for crystal class 1. The generators are represented by the corresponding Seitz symbols (cf. Tables 1.4.2.1[link] –1.4.2.3[link] ). Following the conventions, the subscript of a symbol denotes the characteristic direction of that operation, where necessary. For example, the subscripts 001, 010, 110 etc. refer to the directions [001], [010], [110] etc. For mirror reflections m, the `direction of m' refers to the normal of the mirror plane.

    Hermann–Mauguin symbol of crystal classGenerators [\ispecialfonts{\sfi g}_{i}] (sequence left to right)
    1 1
    [\bar{1}] [\bar{1}]
    2 2
    m m
    [2/m] [2,\bar{1}]
    222 [2_{001},2_{010}]
    mm2 [2_{001},m_{010}]
    mmm [2_{001},2_{010},\bar{1}]
    4 [2_{001},4^+_{001}]
    [\bar{4}] [2_{001},\bar{4}^+_{001}]
    [4/m] [2_{001},4^+_{001},\bar{1}]
    422 [2_{001},4^+_{001},2_{010}]
    4mm [2_{001},4^+_{001},m_{010}]
    [\bar{4}2m] [2_{001},\bar{4}^+_{001},2_{010}]
    [\bar{4}m2] [2_{001},\bar{4}^+_{001},m_{010}]
    [4/mmm] [2_{001},4^+_{001},2_{010},\bar{1}]
    3 [3^+_{001}]
    (rhombohedral coordinates [3^+_{111}])
    [\bar{3}] [3^+_{001},\bar{1}]
    (rhombohedral coordinates [3^+_{111},\bar{1}]
    321 [3^+_{001},2_{110}]
    (rhombohedral coordinates [3^+_{111},2_{\bar{1}01}])
    312 [3^+_{001},2_{1\bar{1}0}]
    3m1 [3^+_{001},m_{110}]
    (rhombohedral coordinates [3^+_{111},m_{\bar{1}01}])
    31m [3^+_{001},m_{1\bar{1}0}]
    [\bar{3}m1] [3^+_{001},2_{110},\bar{1}]
    (rhombohedral coordinates [3^+_{111},2_{\bar{1}01},\bar{1}])
    [\bar{3}1m] [3^+_{001},2_{1\bar{1}0},\bar{1}]
    6 [3^+_{001},2_{001}]
    [\bar{6}] [3^+_{001},m_{001}]
    [6/m] [3^+_{001},2_{001},\bar{1}]
    622 [3^+_{001},2_{001},2_{110}]
    6mm [3^+_{001},2_{001},m_{110}]
    [\bar{6}m2] [3^+_{001},m_{001},m_{110}]
    [\bar{6}2m] [3^+_{001},m_{001},2_{110}]
    [6/mmm] [3^+_{001},2_{001},2_{110},\bar{1}]
    23 [2_{001},2_{010},3^+_{111}]
    [m\bar{3}] [2_{001},2_{010},3^+_{111},\bar{1}]
    432 [2_{001},2_{010},3^+_{111},2_{110}]
    [\bar{4}3m] [2_{001},2_{010},3^+_{111},m_{1\bar{1}0}]
    [m\bar{3}m] [2_{001},2_{010},3^+_{111},2_{110},\bar{1}]
  • Lattice letter R (hexagonal axes): [k_5=0], 1 or 2.

  • Lattice letter [F]: [k_5=0] or 1; [k_6=0] or 1.

As a consequence, any translation [\ispecialfonts{\sfi t}] of [{\cal G}] with translation vector[{\bf t}=k_2{\bf a}+k_3{\bf b}+k_4{\bf c}\,(+\,k_5{\bf d}+k_6{\bf e})]can be obtained as a product[\ispecialfonts{\sfi t}=({\sfi g}_{6})^{k_6}.({\sfi g}_{5})^{k_{5}}.{\sfi g}_{4}^{k_{4}}.{\sfi g}_{3}^{k_{3}}.{\sfi g}_{2}^{k_{2}}.{\sfi g}_{1},]where [k_2, \ldots, k_6] are integers determined by [\ispecialfonts{\sfi t}]. The generators [\ispecialfonts{\sfi g}_6] and [\ispecialfonts{\sfi g}_5] are enclosed between parentheses because they are effective only in centred lattices.

The remaining generators generate those symmetry operations that are not translations. They are chosen in such a way that only terms [\ispecialfonts{\sfi g}_j] or [\ispecialfonts{\sfi g}_j^2] occur. For further specific rules, see below.

The process of generating the entries of the space-group tables may be demonstrated by the example in Table 1.4.3.2[link], where [{\cal G}_j] denotes the group generated by [\ispecialfonts{\sfi g}_1, {\sfi g}_2, \ldots, {\sfi g}_j]. For [j \ge 5], the next generator [\ispecialfonts{\sfi g}_{j+1}] is introduced when [\ispecialfonts{\sfi g}_j^{k_j} \in {\cal G}_{j-1} ], because in this case no new symmetry operation would be generated by [\ispecialfonts{\sfi g}_j^{k_j}]. The generating process is terminated when there is no further generator. In the present example, [\ispecialfonts{\sfi g}_7] completes the generation: [{\cal G}_7\equiv P6_122] (178).

Table 1.4.3.2| top | pdf |
Generation of the space group [P6_122\equiv D_6^2 ] (178)

The entries in the second column designated by the numbers (1)–(12) correspond to the coordinate triplets of the general position of [P6_122].

 Coordinate tripletsSymmetry operations
[\ispecialfonts{\sfi g}_{1}] (1) [x, y, z]; Identity [\ispecialfonts\sfi I]
[\ispecialfonts\!\matrix{{\sfi g}_{2}\hfill\cr {\sfi g}_{3}\hfill\cr {\sfi g}_{4}\hfill\cr}] [\!\left.\matrix{t(100)\hfill\cr t(010)\hfill\cr t(001)\hfill\cr}\right\}] [\!\left\{\matrix{\cr \hbox{Generating translations}\hfill\cr \cr}\right.]
  The group [{\cal G}_{4} \equiv {\cal T}] of all translations of P6122 has been generated  
[\ispecialfonts{\sfi g}_{5}] [(2)\ \bar{y},x - y, z + {1 \over 3}\semi] Threefold screw rotation
[\ispecialfonts{\sfi g}_{5}^{2}] [(3)\ \bar{x} + y, \bar{x}, z + {2 \over 3}\semi] Threefold screw rotation
[\ispecialfonts{\sfi g}_{5}^{3} = t(001){:}] Now the space group [{\cal G}_{5} \equiv P3_{1}] has been generated  
[\ispecialfonts{\sfi g}_{6}] [(4)\ \bar{x},\bar{y}, z + {1 \over 2}\semi] Twofold screw rotation
[\ispecialfonts{\sfi g}_{6}* {\sfi g}_{5}] [(5)\ y,\bar{x} + y,z + {5 \over 6}\semi] Sixfold screw rotation
[\ispecialfonts{\sfi g}_{6}* {\sfi g}_{5}^{2}] [x - y,x,z + {7 \over 6} \sim (6)\ x - y,x, z + {1 \over 6}\semi] Sixfold screw rotation
[\ispecialfonts{\sfi g}_{6}^{2} = t(001){:}] Now the space group [{\cal G}_{6} \equiv P6_{1}] has been generated  
[\ispecialfonts{\sfi g}_{7}] [(7)\ y,x,\bar{z} + {1 \over 3}\semi] Twofold rotation, direction of axis [110]
[\ispecialfonts{\sfi g}_{7}*{\sfi g}_{5}] [(8)\ x - y,\bar{y},\bar{z}\semi] Twofold rotation, axis [100]
[\ispecialfonts{\sfi g}_{7}*{\sfi g}_{5}^{2}] [\bar{x},\bar{x} + y, \bar{z} - {1 \over 3} \sim (9)\ \bar{x},\bar{x} + y, \bar{z} + {2 \over 3}\semi] Twofold rotation, axis [010]
[\ispecialfonts{\sfi g}_{7}* {\sfi g}_{6}] [\bar{y},\bar{x},\bar{z} - {1 \over 6} \sim (10)\ \bar{y},\bar{x},\bar{z} + {5 \over 6}\semi] Twofold rotation, axis [[1\bar{1}0]]
[\ispecialfonts{\sfi g}_{7}* {\sfi g}_{6}*{\sfi g}_{5}] [\bar{x} + y,y, \bar{z} - {1 \over 2} \sim (11)\ \bar{x} + y, y,\bar{z} + {1 \over 2}\semi] Twofold rotation, axis [120]
[\ispecialfonts{\sfi g}_{7}* {\sfi g}_{6}* {\sfi g}_{5}^{2}] [x,x - y, \bar{z} - {5 \over 6} \sim (12)\ x,x - y,\bar{z} + {1 \over 6}\semi] Twofold rotation, axis [210]
[\ispecialfonts{\sfi g}_{7}^{2} = {\sfi I}] [{\cal G}_{7} \sim P6_{1}22]  

1.4.3.1. Selected order for non-translational generators

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For the non-translational generators, the following sequence has been adopted:

  • (a) In all centrosymmetric space groups, an inversion (if possible at the origin O) has been selected as the last generator.

  • (b) Rotations precede symmetry operations of the second kind. In crystal classes [\bar{4}2m] and [\bar{4}m2] and [\bar{6}2m] and [\bar{6}m2], as an exception, [\bar{4}] and [\bar{6}] are generated first in order to take into account the conventional choice of origin in the fixed points of [\bar{4}] and [\bar{6}].

  • (c) The non-translational generators of space groups with C, A, B, F, I or R symbols are those of the corresponding space group with a P symbol, if possible. For instance, the generators of [I2_12_12_1] (24) are those of [P2_12_12_1] (19) and the generators of Ibca (73) are those of Pbca (61), apart from the centring translations.

    Exceptions: I4cm (108) and I4/mcm (140) are generated via P4cc (103) and P4/mcc (124), because P4cm and P4/mcm do not exist. In space groups with d glides (except [I\bar{4}2d], No. 122) and also in [I4_1/a] (88), the corresponding rotation subgroup has been generated first. The generators of this subgroup are the same as those of the corresponding space group with a lattice symbol P.

    Example

    [F4_1/d\bar{3}2/m] (227):

    [P4_132\ (213)\ \longrightarrow\ F4_132\ (210)\ \longrightarrow\ F4_1/d\bar{3}2/m].

  • (d) In some cases, rule (c) could not be followed without breaking rule (a), e.g. in Cmme (67). In such cases, the generators are chosen to correspond to the Hermann–Mauguin symbol as far as possible. For instance, the generators (apart from centring) of Cmme and Imma (74) are those of Pmmb, which is a non-standard setting of Pmma (51). (A combination of the generators of Pmma with the C- or I-centring translation results in non-standard settings of Cmme and Imma.)

For the space groups with lattice symbol P, the generation procedure has given the same triplets (except for their sequence) as in IT (1952[link]). In non-P space groups, the triplets listed sometimes differ from those of IT (1952[link]) by a centring translation.

References

International Tables for X-ray Crystallography (1952). Vol. I, Symmetry Groups, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Abbreviated as IT (1952).]
Ledermann, W. (1976). Introduction to Group Theory. London: Longman.








































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