Explicit symbols

Shmueli, U.

doi:10.1107/97809553602060000552

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 1.4, pp. 108-112

Section A1.4.2.2. Explicit symbols

U. Shmueli^a

A1.4.2.2. Explicit symbols

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As shown elsewhere (Shmueli, 1984), the set of representative operators of a crystallographic space group [i.e. the set that is listed for each space group in the symmetry tables of IT A (1983) and automatically regenerated for the purpose of compiling the symmetry tables in the present chapter] may have one of the following forms: $[\eqalignno{& \{({\bf Q},{\bf u})\},&\cr&\{({\bf Q},{\bf u})\} \times \{({\bf R},{\bf v})\},\quad\hbox{\rm or }&({\rm A}1.4.2.1)\cr&\{({\bf P},{\bf t})\} \times [\{({\bf Q},{\bf u})\} \times \{({\bf R},{\bf v})\}],&\cr}]$ where P, Q and R are point-group operators, and t, u and v are zero vectors or translations not belonging to the lattice-translations subgroup. Each of the forms in (A1.4.2.1), enclosed in braces, is evaluated as, e.g., $[\{({\bf P},{\bf t})\}=\{({\bf I},{\bf 0}),({\bf P},{\bf t}),({\bf P},{\bf t})^{2},\ldots,({\bf P},{\bf t})^{g-1}\}, \eqno({\rm A}1.4.2.2)]$ where I is a unit operator and g is the order of the rotation operator P (i.e. P ^g = I). The representative operations of the space group are evaluated by expanding the generators into cyclic groups, as in (A1.4.2.2), and forming, as needed, ordered products of the expanded groups as indicated in (A1.4.2.1) and explained in detail in the original article (Shmueli, 1984). The rotation and translation parts of the generators (P, t), (Q, u) and (R, v) presented here were adapted to the settings and choices of origin used in the main symmetry tables of IT A (1983).

The general structure of a three-generator symbol, corresponding to the last line of (A1.4.2.1), as represented in Table A1.4.2.1, is $[{\rm LSC}\${\rm r}_1{\rm Pt}_1{\rm t}_2{\rm t}_3\${\rm r}_2{\rm Qu}_1{\rm u}_2{\rm u}_3\${\rm r}_3{\rm Rv}_1{\rm v}_2{\rm v}_3, \eqno({\rm A}1.4.2.3)]$ where

L – lattice type; can be P, A, B, C, I, F, or R. The symbol R is used only for the seven rhombohedral space groups in their representations in rhombohedral and hexagonal axes [obverse setting (IT I, 1952)].
S – crystal system; can be A (triclinic), M (monoclinic), O (orthorhombic), T (tetragonal), R (trigonal), H (hexagonal) or C (cubic).
C – status of centrosymmetry ; can be C or N according as the space group is centrosymmetric or noncentrosymmetric, respectively.
$ – this character is followed by six characters that define a generator of the space group.
r_i – indicator of the type of rotation that follows: r_i is P or I according as the rotation part of the ith generator is proper or improper , respectively.
P, Q, R – two-character symbols of matrix representations of the point-group rotation operators P, Q and R, respectively (see below).
t₁t₂t₃, u₁u₂u₃, v₁v₂v₃ – components of the translation parts of the generators, given in units of $[{{1}\over{12}}]$ ; e.g. the translation part (0 $[{{1}\over{2}}]$ $[{{3}\over{4}}]$ ) is given in Table A1.4.2.1 as 069. An exception: (0 0 $[{{5}\over{6}}]$ ) is denoted by 005 and not by 0010.

The two-character symbols for the matrices of rotation, which appear in the explicit space-group symbols in Table A1.4.2.1, are defined as follows: $[\displaylines{{\tt 1A}=\pmatrix{1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1\cr}\quad{\tt 2A}=\pmatrix{1 & 0 & 0 \cr 0 & \overline{1} & 0 \cr 0 & 0 & \overline{1}\cr}\quad{\tt 2B}=\pmatrix{\overline{1} & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & \overline{1}}\cr{\tt 2C}=\pmatrix{\overline{1} & 0 & 0 \cr 0 & \overline{1} & 0 \cr 0 & 0 & 1}\quad{\tt 2D}=\pmatrix{0 & 1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & \overline{1}}\quad{\tt 2E}=\pmatrix{0 & \overline{1} & 0 \cr \overline{1} & 0 & 0 \cr 0 & 0 & \overline{1}}\cr{ \tt 2F}=\pmatrix{1 & \overline{1} & 0 \cr 0 & \overline{1} & 0 \cr 0 & 0 & \overline{1}}\quad{\tt 2G}=\pmatrix{1 & 0 & 0 \cr 1 & \overline{1} & 0 \cr 0 & 0 & \overline{1}}\quad{\tt 3Q}=\pmatrix{0 & 0 & 1 \cr 1 & 0 & 0 \cr 0 & 1 & 0}\cr{\tt 3C}=\pmatrix{0 & \overline{1} & 0 \cr 1 & \overline{1} & 0 \cr 0 & 0 & 1}\quad{\tt 4C}=\pmatrix{0 & \overline{1} & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1}\quad{\tt 6C}=\pmatrix{1 & \overline{1} & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1},}]$ where only matrices of proper rotation are given (and required), since the corresponding matrices of improper rotation are created by the program for appropriate value of the r_i indicator. The first character of a symbol is the order of the axis of rotation and the second character specifies its orientation: in terms of direct-space lattice vectors, we have $[\displaylines{{\tt A} = [100], {\tt B} = [010], {\tt C} = [001], {\tt D} = [110],\cr{\tt E} = [1\overline{1}0], {\tt F} = [100], {\tt G} = [210]\hbox{ and }{\tt Q} = [111]}]$ for the standard orientations of the axes of rotation. Note that the axes 2F, 2G, 3C and 6C appear in trigonal and hexagonal space groups.

In the above scheme a space group is determined by one, two or at most three generators [see (A1.4.2.1)]. It should be pointed out that a convenient way of achieving a representation of the space group in any setting and relative to any origin is to start from the standard generators in Table A1.4.2.1 and let the computer program perform the appropriate transformation of the generators only, as in equations (1.4.4.4) and (1.4.4.5). The subsequent expansion of the transformed generators and the formation of the required products [see (A1.4.2.1) and (A1.4.2.2)] leads to the new representation of the space group.

In order to illustrate an explicit space-group symbol consider, for example, the symbol for the space group $[Ia\overline{3}d]$ , as given in Table A1.4.2.1: $[{\tt ICC\$I3Q000\$P4C393\$P2D933}.]$ The first three characters tell us that the Bravais lattice of this space group is of type I, that the space group is centrosymmetric and that it belongs to the cubic system. We then see that the generators are (i) an improper threefold axis along [111] (I3Q) with a zero translation part, (ii) a proper fourfold axis along [001] (P4C) with translation part (1/4, 3/4, 1/4) and (iii) a proper twofold axis along [110] (P2D) with translation part (3/4, 1/4, 1/4).

If we make use of the above-outlined interpretation of the explicit symbol (A1.4.2.3), the space-group symmetry transformations in direct space, corresponding to these three generators of the space group $[Ia\overline{3}d]$ , become $[\displaylines{\left[\pmatrix{0 & 0 & \overline{1} \cr \overline{1} & 0 & 0 \cr 0 & \overline{1} & 0}\pmatrix{x \cr y \cr z}+\pmatrix{0 \cr 0 \cr 0}\right]=\pmatrix{\overline{z} \cr \overline{x} \cr \overline{y}},\cr\left[\pmatrix{0 & \overline{1} & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1}\pmatrix{x \cr y \cr z}+\pmatrix{{{1}\over{4}} \cr {{3}\over{4}} \cr {{1}\over{4}}}\right]=\pmatrix{{{1}\over{4}}-y \cr {{3}\over{4}}+x \cr {{1}\over{4}}+z}, \cr \left[\pmatrix{0 & 1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & \overline{1}}\pmatrix{x \cr y \cr z}+\pmatrix{{{3}\over{4}} \cr {{1}\over{4}} \cr {{1}\over{4}}}\right]=\pmatrix{{{3}\over{4}}+y \cr{{1}\over{4}}+x \cr {{1}\over{4}}-z}.}]$

The corresponding symmetry transformations in reciprocal space, in the notation of Section 1.4.4, are $[\left[(hkl)\pmatrix{0 & 0 & \overline{1} \cr \overline{1} & 0 & 0 \cr 0 & \overline{1} & 0}:-(hkl)\pmatrix{0 \cr 0 \cr 0}\right]=[\overline{k} \overline{l} \overline{h}: 0]\hbox{;}]$ similarly, $[[k \overline{h} l: -131/4]]$ and $[[k h \overline{l}: -311/4]]$ are obtained from the second and third generator of $[Ia\overline{3}d]$ , respectively.

The first column of Table A1.4.2.1 lists the conventional space-group number. The second column shows the conventional short Hermann–Mauguin or international space-group symbol, and the third column, Comments, shows the full international space-group symbol only for the different settings of the monoclinic space groups that are given in the main space-group tables of IT A (1983). Other comments pertain to the choice of the space-group origin – where there are alternatives – and to axial systems. The fourth column shows the explicit space-group symbols described above for each of the settings considered in IT A (1983).

References

International Tables for Crystallography (1983). Vol. A. Space-group symmetry, edited by Th. Hahn. Dordrecht: Reidel. (Present distributor Kluwer Academic Publishers, Dordrecht.)Google Scholar

International Tables for X-ray Crystallography (1952). Vol. I. Symmetry groups, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press.Google Scholar

Shmueli, U. (1984). Space-group algorithms. I. The space group and its symmetry elements. Acta Cryst. A40, 559–567.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 1.4, pp. 108-112