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. 53-55

Section 1.4.2.3. Symmetry operations and the general position

M. I. Aroyo,a G. Chapuis,b B. Souvignierd and A. M. Glazerc

1.4.2.3. Symmetry operations and the general position

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The classifications of space groups introduced in Chapter 1.3[link] allow one to reduce the practically unlimited number of possible space groups to a finite number of space-group types. However, each individual space-group type still consists of an infinite number of symmetry operations generated by the set of all translations of the space group. A practical way to represent the symmetry operations of space groups is based on the coset decomposition of a space group with respect to its translation subgroup, which was introduced and discussed in Section 1.3.3.2[link] . For our further considerations, it is important to note that the listings of the general position in the space-group tables can be interpreted in two ways:

  • (i) Each of the numbered entries lists the coordinate triplets of an image point of a starting point with coordinates x, y, z under a symmetry operation of the space group. This feature of the general position will be discussed in detail in Section 1.4.4[link].

  • (ii) Each of the numbered entries of the general position lists a symmetry operation of the space group by the shorthand notation of its matrix–column pair [({\bi W}, {\bi w})] (cf. Section 1.2.2.1[link] ). This fact is not as obvious as the more `crystallographic' aspect described under (i), but its importance becomes evident from the following discussion, where it is shown how to extract the full analytical symmetry information of space groups from the general-position data in the space-group tables of Chapter 2.3[link] .

With reference to a conventional coordinate system, the set of symmetry operations [\ispecialfonts\{{\sfi W}\}] of a space group [{\cal G}] is described by the set of matrix–column pairs [\{({\bi W,w})\}]. The set [{\cal T_G} = \{({{\bi I},{\bi t}})\}] of all translations forms the translation subgroup [{\cal T_G}\,\triangleleft\,{\cal G}], which is a normal subgroup of [{\cal G}] of finite index [i]. If [({\bi W},{\bi w})] is a fixed symmetry operation, then all the products [{\cal T_G}({\bi W},{\bi w})] = [\{({\bi I},{\bi t})({\bi W},{\bi w})\}] = [\{({\bi W,w+t})\}] of translations with [({\bi W,w})] have the same rotation part W. Conversely, every symmetry operation [\ispecialfonts{\sfi W}] of [{\cal G}] with the same matrix part W is represented in the set [{\cal T_G}({\bi W},{\bi w})]. The infinite set of symmetry operations [{\cal T_G}({\bi W},{\bi w})] is called a coset of the right coset decomposition of [{\cal G}] with respect to [{\cal T_G}], and [({\bi W},{\bi w})] its coset representative. In this way, the symmetry operations of [{\cal G}] can be distributed into a finite set of infinite cosets, the elements of which are obtained by the combination of a coset representative [({\bi W}_{j},{\bi w}_{j})] and the infinite set [{\cal T_G} = \{({\bi I},{\bi t})\}] of translations (cf. Section 1.3.3.2[link] ):[{\cal G}={\cal T_G}\cup{\cal T_G}({\bi W}_2,{\bi w}_2)\cup\cdots\cup{\cal T_G}({\bi W}_m,{\bi w}_m)\cup\cdots\cup{\cal T_G}({\bi W}_i,{\bi w}_i),\eqno(1.4.2.1)]where [({\bi W}_{1},{\bi w}_{1})=({\bi I},{\bi o})] is omitted. Obviously, the coset representatives [({\bi W}_{j},{\bi w}_{j})] of the decomposition [({\cal G}:{\cal T_G})] represent in a clear and compact way the infinite number of symmetry operations of the space group [{\cal G}]. Each coset in the decomposition [({\cal G}:{\cal T_G})] is characterized by its linear part [{\bi W}_j] and its entries differ only by lattice translations. The translations [({\bi I},{\bi t}) \in {\cal T_G}] form the first coset with the identity [({\bi I},{\bi o})] as a coset representative. The symmetry operations with rotation part [{\bi W}_2] form the second coset etc. The number of cosets equals the number of different matrices [{\bi W}_j] of the symmetry operations of the space group. This number [i] is always finite and is equal to the order of the point group [{\cal P_G}] of the space group (cf. Section 1.3.3.2[link] ).

For each space group, a set of coset representatives [\{({\bi W}_{j},{\bi w}_{j}), 1\leq j \leq [i]\}] of the decomposition [({\cal G}:{\cal T_G})] is listed under the general-position block of the space-group tables. In general, any element of a coset may be chosen as a coset representative. For convenience, the representatives listed in the space-group tables are always chosen such that the components [w_{j,k}, k=1,2,3], of the translation parts [{\bi w}_j] fulfil [0 \leq w_{j,k} \,\lt\, 1] (by subtracting integers). To save space, each matrix–column pair [({\bi W}_j,{\bi w}_j)] is represented by the corresponding coordinate triplet (cf. Section 1.2.2.3[link] for the shorthand notation of matrix–column pairs).

Example

The right coset decomposition of [P2_1/c], No. 14 (unique axis b, cell choice 1) with respect to its translation subgroup is shown in Table 1.4.2.6[link]. All possible symmetry operations of [P2_1/c] are distributed into four cosets:

  • The first column represents the infinitely many translations [\ispecialfonts{\sfi t}] = [({\bi I}, {\bi t})] = [x+u_1, y+u_2, z+u_3] = [\{1|u_1, u_2, u_3\}] of the translation subgroup [{\cal T}] of [P2_1/c]. The numbers [u_1], [u_2] and [u_3] are positive or negative integers. The identity operation [({\bi I}, {\bi o})] is usually chosen as a coset representative.

    Table 1.4.2.6| top | pdf |
    Right coset decomposition of space group [P2_1/c], No. 14 (unique axis b, cell choice 1) with respect to the normal subgroup of translations [{\cal T}]

    The numbers [u_1], [u_2] and [u_3] are positive or negative integers.

    x y z [\bar{x}] [y+\textstyle{{1 \over 2}}] [\bar{z}+\textstyle{{1 \over 2}}] [\bar{x}] [\bar{y}] [\bar{z}] x [\bar{y}+\textstyle{{1 \over 2}}] [z+\textstyle{{1 \over 2}}]
    x + 1 y z [\bar{x}+1] [y+\textstyle{{1 \over 2}}] [\bar{z}+\textstyle{{1 \over 2}}] [\bar{x}+1] [\bar{y}] [\bar{z}] x + 1 [\bar{y}+\textstyle{{1 \over 2}}] [z+\textstyle{{1 \over 2}} ]
    x + 2 y z [\bar{x}+2] [y+\textstyle{{1 \over 2}}] [\bar{z}+\textstyle{{1 \over 2}}] [\bar{x}+2] [\bar{y}] [\bar{z}] x + 2 [\bar{y}+\textstyle{{1 \over 2}}] [z+\textstyle{{1 \over 2}} ]
      [\vdots]     [\vdots]     [\vdots]     [\vdots]  
    x y + 1 z [\bar{x}] [y+\textstyle{{3}\over {2}}] [\bar{z}+\textstyle{{1 \over 2}}] [\bar{x}] [\bar{y}+1] [\bar{z}] x [\bar{y}+\textstyle{{3}\over {2}}] [z+\textstyle{{1 \over 2}} ]
    x + 1 y + 1 z [\bar{x}+1] [y+\textstyle{{3}\over {2}}] [\bar{z}+\textstyle{{1 \over 2}}] [\bar{x}+1] [\bar{y}+1] [\bar{z}] x + 1 [\bar{y}+\textstyle{{3}\over {2}}] [z+\textstyle{{1 \over 2}} ]
    x + 2 y + 1 z [\bar{x}+2] [y+\textstyle{{3}\over {2}}] [\bar{z}+\textstyle{{1 \over 2}}] [\bar{x}+2] [\bar{y}+1] [\bar{z}] x + 2 [\bar{y}+\textstyle{{3}\over {2}}] [z+\textstyle{{1 \over 2}} ]
      [\vdots]     [\vdots]     [\vdots]     [\vdots]  
    x y + 2 z [\bar{x}] [y+\textstyle{{5}\over {2}}] [\bar{z}+\textstyle{{1 \over 2}}] [\bar{x}] [\bar{y}+2] [\bar{z}] x [\bar{y}+\textstyle{{5}\over {2}}] [z+\textstyle{{1 \over 2}} ]
    x + 1 y + 2 z [\bar{x}+1] [y+\textstyle{{5}\over {2}}] [\bar{z}+\textstyle{{1 \over 2}}] [\bar{x}+1] [\bar{y}+2] [\bar{z}] x + 1 [\bar{y}+\textstyle{{5}\over {2}}] [z+\textstyle{{1 \over 2}} ]
    x + 2 y + 2 z [\bar{x}+2] [y+\textstyle{{5}\over {2}}] [\bar{z}+\textstyle{{1 \over 2}}] [\bar{x}+2] [\bar{y}+2] [\bar{z}] x + 2 [\bar{y}+\textstyle{{5}\over {2}}] [z+\textstyle{{1 \over 2}} ]
      [\vdots]     [\vdots]     [\vdots]     [\vdots]  
    x y z + 1 [\bar{x}] [y+\textstyle{{1 \over 2}}] [\bar{z}+\textstyle{{3}\over {2}}] [\bar{x}] [\bar{y}] [\bar{z}+1] x [\bar{y}+\textstyle{{1 \over 2}}] [z+\textstyle{{3}\over {2}}]
    x + 1 y z + 1 [\bar{x}+1] [y+\textstyle{{1 \over 2}}] [\bar{z}+\textstyle{{3}\over {2}}] [\bar{x}+1] [\bar{y}] [\bar{z}+1] x + 1 [\bar{y}+\textstyle{{1 \over 2}}] [z+\textstyle{{3}\over {2}}]
    x + 2 y z + 1 [\bar{x}+2] [y+\textstyle{{1 \over 2}}] [\bar{z}+\textstyle{{3}\over {2}}] [\bar{x}+2] [\bar{y}] [\bar{z}+1] x + 2 [\bar{y}+\textstyle{{1 \over 2}}] [z+\textstyle{{3}\over {2}}]
      [\vdots]     [\vdots]     [\vdots]     [\vdots]  
    x y + 1 z + 1 [\bar{x}] [y+\textstyle{{3}\over {2}}] [\bar{z}+\textstyle{{3}\over {2}}] [\bar{x}] [\bar{y}+1] [\bar{z}+1] x [\bar{y}+\textstyle{{3}\over {2}}] [z+\textstyle{{3}\over {2}}]
    x + 1 y + 1 z + 1 [\bar{x}+1] [y+\textstyle{{3}\over {2}}] [\bar{z}+\textstyle{{3}\over {2}}] [\bar{x}+1] [\bar{y}+1] [\bar{z}+1] x + 1 [\bar{y}+\textstyle{{3}\over {2}}] [z+\textstyle{{3}\over {2}}]
    x + 2 y + 1 z + 1 [\bar{x}+2] [y+\textstyle{{3}\over {2}}] [\bar{z}+\textstyle{{3}\over {2}}] [\bar{x}+2] [\bar{y}+1] [\bar{z}+1] x + 2 [\bar{y}+\textstyle{{3}\over {2}}] [z+\textstyle{{3}\over {2}}]
      [\vdots]     [\vdots]     [\vdots]     [\vdots]  
    [x+u_1] [y+u_2] [z+u_3] [\bar{x}+u_1] [y+u_2+\textstyle{{1 \over 2}}] [\bar{z}+u_3+\textstyle{{1 \over 2}}] [\bar{x}+u_1] [\bar{y}+u_2] [\bar{z}+u_3] [x+u_1] [\bar{y}+u_2+\textstyle{{1 \over 2}}] [z+u_3+\textstyle{{1 \over 2}} ]
      [\vdots]     [\vdots]     [\vdots]     [\vdots]  
  • The third coset of the decomposition [({\cal G}:{\cal T_G})] represents the infinite set of inversions [(-{\bi I}, {\bi t})] = [\bar{x}+u_1, \bar{y}+u_2, \bar{z}+u_3] = [\{\bar{1}|u_1,u_2,u_3\}] of the space group [P2_1/c] with inversion centres located at [{u_1}/{2}, {u_2}/{2}, {u_3}/{2}] (cf. Section 1.2.2.4[link] for the determination of the location of the inversion centres). The inversion in the origin, i.e. [\bar{x}, \bar{y}, \bar{z}=\{\bar{1}|0\}], is taken as a coset representative.

  • The coset representative of the second coset is the twofold screw rotation [\{2_{010}|0,\textstyle{{1 \over 2}}, \textstyle{{1 \over 2}}\}] around the line [0,y,\textstyle{{1 \over 4}}], followed by its infinite combinations with all lattice translations: [\bar{x}+u_1,] [ y+\textstyle{{1 \over 2}}+u_2, \bar{z}+\textstyle{{1 \over 2}}+u_3 =\{2_{010}|u_1, \textstyle{{1 \over 2}}+u_2, \textstyle{{1 \over 2}}+u_3\}]. These are twofold screw rotations around the lines [{{u_1}/{2}},y,{{u_3}/{2}}+\textstyle{{1 \over 4}}] with screw components [\pmatrix{ 0 \cr \textstyle{{1 \over 2}}+u_2 \cr 0}].

  • The symmetry operations of the fourth column represented by [x+u_1], [\bar{y}+\textstyle{{1 \over 2}}+u_2], [ z+\textstyle{{1 \over 2}}+u_3=\{m_{010}|u_1, \textstyle{{1 \over 2}}+u_2, \textstyle{{1 \over 2}}+u_3\}] correspond to glide reflections with glide components [\pmatrix{u_1 \cr 0 \cr \textstyle{{1 \over 2}}+u_3 }] through the (infinite) set of glide planes at [x,\textstyle{{1 \over 4}},z]; [x,\textstyle{{3}\over{4}},z]; [x,\textstyle{{5}\over{4}},z]; …; [x,(2u_2+1)/{4},z]. As usual, the symmetry operation with [u_1=u_2=u_3=0], i.e. [x], [\bar{y}+\textstyle{{1 \over 2}}], [ z+\textstyle{{1 \over 2}}] = [\{m_{010}|0, \textstyle{{1 \over 2}}, \textstyle{{1 \over 2}}\}], is taken as a coset representative of the coset of glide reflections.

The coordinate triplets of the general-position block of [P2_1/c] (unique axis b, cell choice 1) (cf. Fig. 1.4.2.1[link]) correspond to the coset representatives of the decomposition of the group listed in the first line of Table 1.4.2.6[link].

When the space group is referred to a primitive basis (which is always done for `P' space groups), each coordinate triplet of the general-position block corresponds to one coset of [({\cal G}:{\cal T_G})], i.e. the multiplicity of the general position and the number of cosets is the same. If, however, the space group is referred to a centred cell, then the complete set of general-position coordinate triplets is obtained by the combinations of the listed coordinate triplets with the centring translations. In this way, the total number of coordinate triplets per conventional unit cell, i.e. the multiplicity of the general position, is given by the product [[i]\times[p]], where [i] is the index of [{\cal T_G}] in [{\cal G}] and [p] is the index of the group of integer translations in the group [{\cal T_G}] of all (integer and centring) translations.

Example

The listing of the general position for the space–group type Fmm2 (42) of the space-group tables is reproduced in Fig. 1.4.2.2[link]. The four entries, numbered (1) to (4), are to be taken as they are printed [indicated by (0, 0, 0)+]. The additional 12 more entries are obtained by adding the centring translations [(0, \textstyle{{1 \over 2}},\textstyle{{1 \over 2}}),(\textstyle{{1 \over 2}},0,\textstyle{{1 \over 2}}),(\textstyle{{1 \over 2}},\textstyle{{1 \over 2}}, 0)] to the translation parts of the printed entries [indicated by [(0, \textstyle{{1 \over 2}},\textstyle{{1 \over 2}})+], [(\textstyle{{1 \over 2}},0,\textstyle{{1 \over 2}})+] and [(\textstyle{{1 \over 2}},\textstyle{{1 \over 2}},0)+], respectively]. Altogether there are 16 entries, which is announced by the multiplicity of the general position, i.e. by the first number in the row. (The additional information specified on the left of the general-position block, namely the Wyckoff letter and the site symmetry, will be dealt with in Section 1.4.4[link].)








































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