Tables of superspace groups

Janssen, T.; Janner, A.; Looijenga-Vos, A.; Wolff, P. M. de

doi:10.1107/97809553602060000624

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 9.8, pp. 916-935

Section 9.8.3.3. Tables of superspace groups

T. Janssen,^a A. Janner,^a A. Looijenga-Vos^b and P. M. de Wolff^c ^‡

^a Institute for Theoretical Physics, University of Nijmegen, Toernooiveld, NL-6525 ED Nijmegen, The Netherlands,^bRoland Holstlaan 908, NL-2624 JK Delft, The Netherlands, and ^cMeermanstraat 126, 2614 AM, Delft, The Netherlands

9.8.3.3. Tables of superspace groups

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9.8.3.3.1. Symmetry elements

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The transformations [g_s] belonging to a (3 + 1)-dimensional superspace group consist of a point-group transformation [R_s] given by the integral matrix Γ(R) and of the associated translation. So the superspace group is determined by the arithmetic crystal class of its point group and the corresponding translational components. The symbol for the arithmetic crystal class has been discussed in Subsection 9.8.3.2. Given a point-group transformation [R_s] , the associated translation is determined up to a lattice translation. As in three dimensions, the translational part generally depends on the choice of origin. To avoid this arbitrariness, one decomposes that translation into a component (called intrinsic) independent of the origin, and a remainder. The (3 + 1)-dimensional translation $[\upsilon_s]$ associated with the point-group transformation [R_s] is given by $[\upsilon_s=\textstyle\sum\limits^{3+1}_{i=1}\upsilon_i\,a_{is}. \eqno (9.8.3.4)]$ Its origin-invariant part $[\upsilon^o_s]$ is given by $[\upsilon^o_s=\textstyle\sum\limits^4_{j=1}\upsilon^o_j\,a_{sj}\quad\hbox{with}\quad \upsilon^o_j= \displaystyle{1\over n}\,\sum^n_{m=1}\,\sum^4_{k=1}\Gamma(R^m)_{jk}\upsilon_k, \eqno (9.8.3.5)]$ where n is now the order of the point-group transformation R so that [R^n] is the identity. As customary also in three-dimensional crystallography, one indicates in the space-group symbol the invariant components $[\upsilon^o_j]$ . Notice that this means that there is an origin for [R_s] in (3 + 1)-dimensional superspace such that the translation associated with [R_s] has these components. This origin, however, may not be the same for different transformations [R_s] , as is known in three-dimensional crystallography.

Written in components, the non-primitive translation $[\upsilon_s]$ associated with the point-group element [(R,R_I)] is $[({\bf v}, \upsilon_I)]$ , where $[\upsilon_I]$ can be written as $[\delta-{\bf q}\cdot {\bf v}]$ . In accordance with (9.8.1.12), δ is defined as $[\upsilon_4]$ . The origin-invariant part $[\upsilon^o_s]$ of $[\upsilon_s]$ is $[\upsilon^o_s=({\bf v}^o,\upsilon^o_I) = {1\over n}\, \sum^n_{m=1}\, (R^m{\bf v}, R^m_I\upsilon_I) = ({\bf v}^o, \tau-{\bf q}\cdot{\bf v}^o), \eqno (9.8.3.6)]$ where $[\tau=\upsilon^o_4=\upsilon^o_I+{\bf q}\cdot{\bf v}^o.]$ The internal transformation [R_I(R)] = ɛ(R) = ɛ is either +1 or −1. When ɛ = −1 it follows from (9.8.3.6) that $[\upsilon^o_I=0]$ . For $[\varepsilon=+1]$ , one has $[\upsilon^o_I=\upsilon_I]$ . Because in that case $[{\bf q}\cdot {\bf v}^o={1\over n}\,\sum^n_{m=1}\,{\bf q}\cdot R^m{\bf v}={\bf q}^i\cdot {\bf v}, \eqno (9.8.3.7)]$ it follows that $[\tau=\upsilon_I+{\bf q}\cdot{\bf v}^o=\delta-{\bf q}\cdot{\bf v}+{\bf q}\cdot{\bf v}^o=\delta-{\bf q}^r\cdot {\bf v}. \eqno (9.8.3.8)]$

For [R_s] of order n, [R^n_s] is the identity and the associated translation is a lattice translation. The ensuing values for τ are $[0,{1\over2}, \pm{1\over3}, \pm{1\over4}]$ or $[\pm{1\over6}]$ (modulo integers). This remains true also in the case of a centred basis. The symbol of the (3 + 1)-dimensional space-group element is determined by the invariant part of its three-dimensional translation and τ. Again, that information can be given in terms of either a one-line or a two-line symbol .

In the one-line symbol , one finds: the symbol according to International Tables for Crystallography, Volume A, for the space group generated by the elements {R|v}, in parentheses the components of the modulation vector q followed by the values of τ, one for each generator appearing in the three-dimensional space-group symbol. A letter symbolizes the value of τ according to $[\matrix{ \tau\hfill&0&{1\over2}&\pm{1\over3}&\pm{1\over4}&\pm{1\over6} \cr {\rm symbol}&0&s&t&q&h}. \eqno (9.8.3.9)]$ As an example, consider the superspace group $[P2_1/m(\alpha\beta0)0s.]$ The external components $[\{R|{\bf v}\}]$ of the elements of this group form the three-dimensional space group [P2_1/m] . The modulation wavevector is αa* + βb* with respect to a conventional basis of the monoclinic lattice with unique axis c. Therefore, the point group is $[(2/m,\bar11)]$ . The point-group element $[(2,\bar1)]$ has associated a non-primitive translation with invariant part $[(\,{1\over2}{\bf c},0)=(00{1\over2}0)]$ and the point-group generator (m, 1) one with $[({\bf 0},{1\over2}\,)=(000{1\over2}\,)]$ .

In the two-line symbol, one finds in the upper line the symbol for the three-dimensional space group , in the bottom line the value of τ for the case ɛ = +1 and the symbol ` $[\bar1]$ ' when ɛ = −1. The rational part of q is indicated by means of the appropriate prefix. In the case considered, q^r = 000. So the prefix is P and the same superspace group is denoted in a two-line symbol as $[P^{P2_1/m}_{\kern5pt{\bar1}\kern7pt s}.]$ In Table 9.8.3.5, the (3 + 1)-dimensional space groups are given by one-line symbols. These are so-called short symbols . Sometimes, a full symbol is required. Then, for the example given above one has $[P112_1/m(\alpha\beta0)000s]$ and $[P^{P112_1/m}_{\kern5pt 11\bar1\kern6pt s}]$ , respectively. Note that in the short one-line symbol for τ = 0 superspace groups (where the non-primitive translations can be transformed to zero by a choice of the origin) the zeros for the translational part are omitted. Not so, of course, in the full symbol. For example, short symbol P2₁/m(αβ0) and full symbol P112₁/m(αβ0)0000. Table 9.8.3.5 is an adapted version of the tables given by de Wolff, Janssen & Janner (1981) and corrected by Yamamoto, Janssen, Janner & de Wolff (1985).

9.8.3.3.2. Reflection conditions

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The indexing of diffraction vectors is a matter of choice of basis. When the basis chosen is not a primitive one, the indices have to satisfy certain conditions known as centring conditions. This holds for the main reflections (centring in ordinary space) as well as for satellites (centring in superspace). These centring conditions for reflections have been discussed in Subsection 9.8.2.1.

In addition to these general reflection conditions, there may be special reflection conditions related to the existence of non-primitive translations in the (3 + 1)-dimensional space group, just as is the case for glide planes and screw axes in three dimensions.

Special reflection conditions can be derived from transformation properties of the structure factor under symmetry operations. Transforming the geometric structure factor by an element $[g_s=(\{R|{\bf v}\}, \{R_I|\upsilon_I\})]$ , one obtains $[S_{\bf H}=S_{R^{-1}{\bf H}}\exp[-2\pi i ({\bf H}\cdot{\bf v}+H_I\cdot \upsilon_I)]. \eqno (9.8.3.10)]$ Therefore, if RH = H, the corresponding structure factor vanishes unless $[{\bf H}\cdot{\bf v}+H_I\cdot\upsilon_I]$ is an integer.

The form of such a reflection condition in terms of allowed or forbidden sets of indices depends on the basis chosen. When a lattice basis is chosen, one has $[H_s=({\bf H}, H_I)=\textstyle\sum\limits^4_{i=1}h_i{\bf a}^*_{si}, \eqno (9.8.3.11)]$ $[\upsilon_s=({\bf v}, \upsilon_I)+\textstyle\sum\limits^4_{i=1}\upsilon_i{\bf a}_{si}. \eqno (9.8.3.12)]$ Then the reflection condition becomes $[H_s\cdot\upsilon_s=\textstyle\sum\limits^4_{i=1}h_i\upsilon_i=\hbox{integer } \quad \hbox{ for }R{\bf H}={\bf H}. \eqno (9.8.3.13)]$

In terms of external and internal shift components, the reflection condition can be written as $[H_s\cdot\upsilon_s={\bf H}\cdot{\bf v}+H_I\cdot\upsilon_I={\bf H}\cdot{\bf v}+m\upsilon_I=\hbox{integer for }R{\bf H}={\bf H}. \eqno (9.8.3.14)]$ With $[{\bf H}={\bf K}+m{\bf q}]$ and $[\upsilon_I=\delta-\bf{q\cdot v}]$ , (9.8.3.14) gives $[{\bf K}\cdot{\bf v}+m\delta=\hbox{integer$\quad$ for }R{\bf H}={\bf H}. \eqno (9.8.3.15)]$ For $[{\bf v} =\upsilon_1{\bf a}+\upsilon_2{\bf b}+\upsilon_3{\bf c}]$ and $[{\bf K} =h{\bf a}^*+k{\bf b}^*+l{\bf c}^*]$ , (9.8.3.15) takes the form (9.8.3.13): $[h\upsilon _1+k \upsilon_2+l\upsilon_3+m\delta = \hbox{integer }\quad \hbox{ for }R{\bf H}={\bf H}. \eqno (9.8.3.16)]$ When the modulation wavevector has a rational part, one can choose another basis (Subsection 9.8.2.1) such that K′ = K + mq^r has integer coefficients: $[{\bf H}={\bf K}'+m{\bf q}^i=H{\bf a}^*_c + K{\bf b}^*_c + L{\bf c}^*_c+m{\bf q}^i.]$ Then, (9.8.3.15) with $[\tau=\delta-{\bf q}^r\cdot{\bf v}]$ becomes $[{\bf K}'\cdot{\bf v}+m\tau=\hbox{integer }\quad\hbox{ for }R{\bf H}={\bf H} \eqno (9.8.3.17)]$ and (9.8.3.16) transforms into $[H\upsilon'_1+K\upsilon'_2 +L\upsilon'_3 +m\tau=\hbox{integer }\quad\hbox{ for }R{\bf H}={\bf H}, \eqno (9.8.3.18)]$ in which $[\upsilon'_1,\upsilon'_2]$ , and $[\upsilon'_3]$ are the components of v with respect to the basis $[{\bf a}_c,{\bf b}_c]$ , and $[{\bf c}_c]$ .

As an example, consider a (3 + 1)-dimensional space-group transformation with R a mirror perpendicular to the x axis, $[\varepsilon=1,{\bf v}={1\over2}{\bf b}]$ , and $[\tau={1\over4}]$ with b orthogonal to a. The modulation wavevector is supposed to be $[(\,{1\over2}{1\over2}\gamma)]$ . Then $[\delta={1\over4}+{\bf q}^r\cdot{\bf v}={1\over2}]$ . The vectors H left invariant by R satisfy the relation 2h + m = 0. For such a vector, the reflection condition becomes $[{\bf K}\cdot{\bf v}+m\delta=\textstyle{1\over2}{\bf K}\cdot{\bf b}+{1\over2}m=\displaystyle{k+m\over2}={\rm integer, \quad or\ } k+m=2n.]$ For the basis $[{1\over2}({\bf a}^*+{\bf b}^*)]$ , $[{1\over2}({\bf a}^*-{\bf b}^*)]$ , c*, the rational part of the wavevector vanishes. The indices with respect to this basis are H = h + k + m, K = h − k, L = l and m. The condition now becomes $[{\bf K}'\cdot{\bf v}+m\tau={H-K+m\over4}= {\rm integer},]$ $[{\rm or}\quad H-K+m=4n, \quad {\rm for}\ K=-H.]$ Of course, both calculations give the same result: k + m = 2n for h, k, l, −2h and H − K + m = 4n for H, −H, L, m.

The special reflection conditions for the elements occurring in (3 + 1)-dimensional space groups are given in Table 9.8.3.5.

References

International Tables for Crystallography (2005). Vol. A, edited by Th. Hahn, fifth ed. Heidelberg: Springer.Google Scholar

Wolff, P. M. de, Janssen, T. & Janner, A. (1981). The superspace groups for incommensurate crystal structures with a one-dimensional modulation. Acta Cryst. A37, 625–636.Google Scholar

Yamamoto, A., Janssen, T., Janner, A. & de Wolff, P. M. (1985). A note on the superspace groups for one-dimensionally modulated structures. Acta Cryst. A41, 528–530.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 9.8, pp. 916-935