Description in four dimensions

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, p. 911

Section 9.8.1.4.2. Description in four dimensions

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

9.8.1.4.2. Description in four dimensions

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The matrices Γ*(R) form a faithful integral representation of the three-dimensional point group K. It is also possible to consider them as four-dimensional orthogonal transformations leaving a lattice with basis vectors (9.8.1.14) invariant. Indeed, one can consider the vectors (9.8.1.15) as projections of four-dimensional lattice vectors $[H_s=({\bf H},H_I)]$ , which can be written as $[H_s=\textstyle\sum\limits^4_{i=1}\,h_ia^*_{si}, \eqno (9.8.1.18)]$ where [cf. (9.8.1.14)] m has now been replaced by [h_4] and $[a^*_{si}=({\bf a}^*_i,0),\quad i=1,2,3\semi \quad a^*_{s4}=({\bf q}, 1). \eqno (9.8.1.19)]$

As will be explained in Section 9.8.4, these vectors span the four-dimensional reciprocal lattice for a periodic structure having as three-dimensional intersection (say defined by the hyperplane t = 0) the modulated crystal structure (a specific example has been given in Subsection 9.8.1.3). In direct space, the point group [K_s] in four dimensions with elements [R_s] of O(4) then acts on the corresponding dual basis vectors (9.8.1.11) of the four-dimensional direct lattice as $[R_sa_{si}=\textstyle\sum\limits^4_{j=1}\,\Gamma(R)_{ji}a_{sj} \quad (i=1,2,3,4), \eqno(9.8.1.20a)]$ where Γ(R) is the transpose of the matrix $[\Gamma^*(R^{-1})]$ appearing in (9.8.1.17) and therefore for incommensurate one-dimensionally modulated structures it has the form $[\Gamma(R)=\left(\matrix{\Gamma_E(R)&0 \cr \Gamma_M(R)&\varepsilon(R)} \right). \eqno (9.8.1.20b)]$

References

International Tables for Crystallography (2006). Vol. C. ch. 9.8, p. 911