Cubic and orthorhombic systems

Billiet, Y.; Bertaut, E. F.

doi:10.1107/97809553602060000528

International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

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International Tables for Crystallography (2006). Vol. A. ch. 13.1, pp. 838-839

Section 13.1.2.2. Cubic and orthorhombic systems

Y. Billiet^a ^‡ and E. F. Bertaut^b ^§

^a Département de Chimie, Faculté des Sciences et Techniques, Université de Bretagne Occidentale, Brest, France, and ^bLaboratoire de Cristallographie, CNRS, Grenoble, France

13.1.2.2. Cubic and orthorhombic systems

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Cubic system

For cubic space groups, equation (13.1.1.2a) leads to the matrix C: $[{\bi C} = \pmatrix{S\hfill &0\hfill &0\hfill\cr 0\hfill &S\hfill &0\hfill\cr 0\hfill &0\hfill &S\hfill\cr},\quad \det({\bi C}) = S^{3}.]$

Orthorhombic system

There are six choices of matrices $[{\bi O}_{i}\ (i = 1,2,3,4,5,6)]$ corresponding to the identical orientation $[({\bi O}_{1})]$ , to cyclic permutations of the three axes ( $[{\bi O}_{2}]$ and $[{\bi O}_{3}]$ ) and to the interchange of two axes ( $[{\bi O}_{4}, {\bi O}_{5}]$ and $[{\bi O}_{6}]$ ), i.e. to the six orthorhombic `settings'. $[\eqalignno{{\bi O}_{1}\hfill &= \pmatrix{S_{11}\hfill &0\hfill &0 \hfill\cr 0\hfill &S_{22}\hfill &0\hfill\cr 0\hfill &0\hfill &S_{33}\cr};\quad {\bi O}_{2} = \pmatrix{0\hfill &S_{12}\hfill &0\hfill\cr 0\hfill &0\hfill &S_{23}\cr S_{31}\hfill &0\hfill &0\hfill\cr};\cr {\bi O}_{3} &= \pmatrix{0\hfill &0\hfill &S_{13}\cr S_{21}\hfill &0\hfill &0\hfill\cr 0\hfill &S_{32}\hfill &0\hfill\cr};\quad {\bi O}_{4} = \pmatrix{S_{11}\hfill &0\hfill &0\hfill\cr 0\hfill &0\hfill &S_{23}\cr 0\hfill &S_{32}\hfill &0\hfill\cr};\cr {\bi O}_{5}\hfill &= \pmatrix{0\hfill &0\hfill &S_{13}\cr 0\hfill &S_{22}\hfill &0\hfill\cr S_{31}\hfill &0\hfill &0\hfill\cr};\quad {\bi O}_{6} = \pmatrix{0\hfill &S_{12}\hfill &0\hfill\cr S_{21}\hfill &0\hfill &0\hfill\cr 0\hfill &0\hfill &S_{33}\cr}.}]$

The determinant is always equal to the product of the three non-zero coefficients, $[\det({\bi O}_{i}) = \pm S_{1j}S_{2k}S_{3l}]$ .

The following general rule exists: only those matrices $[{\bi O}_{i}]$ are permissible for which, if the non-zero coefficients are replaced by 1, the corresponding transformation of the axes conserves the Hermann–Mauguin symbol.

Examples

(1) When the three letters of the Hermann–Mauguin symbol are the same, as in P222, Pmmm, Pnnn etc., the Hermann–Mauguin symbol does not change and all six matrices are valid.
(2) When the z axis plays a privileged role and when the x and y axes are equivalent, only $[{\bi O}_{1}]$ and $[{\bi O}_{6}]$ apply. Examples are $[P222_{1}]$ , Pbam and Ccca. In Pmma, the x and y axes are not equivalent because the interchange leads to Pmmb (the non-equivalence of the x and y axes can also be recognized by inspection of the full symbol $[P2_{1}/m\ 2/m\ 2/a]$ ).
(3) Matrix $[{\bi O}_{1}]$ always applies.

References

International Tables for Crystallography (2006). Vol. A. ch. 13.1, pp. 838-839