Monoclinic, tetragonal, trigonal, hexagonal 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. 836-838

Section 13.1.2.1. Monoclinic, tetragonal, trigonal, hexagonal 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

13.1.2.1. Monoclinic, tetragonal, trigonal, hexagonal systems

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If W is the matrix corresponding to a rotation about the c axis, $[{\bi W}' = {\bi W}]$ holds if the positive direction is the same for c and c′.² In consequence, W must commute with S [cf. equation (13.1.1.2a)]. This condition imposes relations on the coefficients $[S_{ij}]$ of the matrix so that S and $[\det ({\bi S})]$ take the following forms:

Monoclinic system $[{\bi M}_{c} = \pmatrix{S_{11}\hfill &S_{12}\hfill &0\hfill\cr S_{21}\hfill &S_{22}\hfill &0\hfill\cr 0\hfill &0\hfill &S_{33}\hfill\cr},\quad \det ({\bi M}_{c}) = S_{33}(S_{11}S_{22} - S_{12}S_{21});]$ or if b instead of c is used $[{\bi M}_{b} = \pmatrix{S_{11}\hfill &0\hfill &S_{13}\hfill\cr 0\hfill &S_{22}\hfill &0\hfill\cr S_{31}\hfill &0\hfill &S_{33}\hfill\cr},\quad \det ({\bi M}_{b}) = S_{22}(S_{11}S_{33} - S_{13}S_{31}).]$
Tetragonal system $[{\bi T}_{1} = \pmatrix{S_{11}\hfill &-S_{21}\hfill &0\hfill\cr S_{21}\hfill &\phantom{-}S_{11}\hfill &0\hfill\cr 0\hfill &\phantom{-}0\hfill &S_{33}\hfill\cr},\quad \det ({\bi T}_{1}) = S_{33}(S_{11}^{2} + S_{21}^{2}).]$
Hexagonal and trigonal systems $[\eqalign{{\bi H}_{1} &= \pmatrix{S_{11}\hfill &-S_{21}\hfill &0\hfill\cr S_{21}\hfill &\phantom{-}S_{11} - S_{21}\hfill &0\hfill\cr 0\hfill &\phantom{-}0\hfill &S_{33}\hfill\cr},\hfill\cr \det ({\bi H}_{1}) &= S_{33}(S_{11}^{2} + S_{21}^{2} - S_{11}S_{21}).}]$ For rhombohedral space groups, the matrix $[{\bi H}_{1}]$ applies only when hexagonal axes are used. If rhombohedral axes are used, the matrix S has the form $[\eqalign{{\bi R}_{1} &= \pmatrix{S_{0} &S_{2} &S_{1}\cr S_{1} &S_{0} &S_{2}\cr S_{2} &S_{1} &S_{0}\cr},\cr \det ({\bi R}_{1}) &= S_{0}^{3} + S_{1}^{3} + S_{2}^{3} - 3S_{0}S_{1}S_{2}\cr &= (S_{0} + S_{1} + S_{2})\cr &\quad \times (S_{0}^{2} + S_{1}^{2} + S_{2}^{2} - S_{0}S_{1} - S_{1}S_{2} - S_{2}S_{0}).}]$

13.1.2.1.1. Additional restrictions

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If mirror or glide planes parallel to and/or twofold rotation or screw axes perpendicular to the principal rotation axis exist, further conditions are imposed upon the coefficients $[S_{ij}]$ and these are indicated below (cf. Bertaut & Billiet, 1979).

Monoclinic system

The matrices $[{\bi M}_{c}]$ and $[{\bi M}_{b}]$ apply without any further restrictions on the coefficients.

Tetragonal system

The matrix $[{\bi T}_{1}]$ is valid for all space groups belonging to the crystal classes 4, $[\bar{4}]$ and [4/m] .

For all other space groups, restrictions apply to the coefficients $[S_{21}]$ according to the following two rules which are consequences of equation (13.1.1.2a):

(i) If the last two letters of the Hermann–Mauguin symbol are different, $[S_{21} = 0]$ ; the corresponding matrix is called $[{\bi T}_{2}]$ .

Example: $[P4_{2}/mmc]$

$[{\bi T}_{2} = \pmatrix{S_{11}\hfill &0\hfill &0\hfill\cr 0\hfill &S_{11}\hfill &0\hfill\cr 0\hfill &0\hfill &S_{33}\hfill\cr},\quad \det ({\bi T}_{2}) = S_{33}S_{11}^{2}.]$
(ii) If the last two letters are the same (except for the three cases mentioned below), two matrices have to be applied, the matrix $[{\bi T}_{2}]$ introduced above and the matrix $[{\bi T}_{1}]$ with $[S_{21} = S_{11}]$ ; the corresponding matrix is called $[{\bi T}_{3}]$ . $[{\bi T}_{3} = \pmatrix{S_{11}\hfill &-S_{11}\hfill &0\hfill\cr S_{11}\hfill &\phantom{-}S_{11}\hfill &0\hfill\cr 0\hfill &\phantom{-}0\hfill &S_{33}\hfill\cr},\quad \det ({\bi T}_{3}) = 2S_{33}S_{11}^{2}.]$

The following space groups have matrices $[{\bi T}_{2}]$ and $[{\bi T}_{3}]$ : P422, P4mm, , $[P4_{1}22]$ , $[P4_{3}22]$ , $[P4_{2}22]$ , P4cc, , I422, I4mm and . The three exceptions to the rule mentioned above are the space groups , and $[I4_{1}22]$ , which allow only $[{\bi T}_{2}]$ .

Hexagonal and trigonal systems

The matrix $[{\bi H}_{1}]$ is valid for all space groups belonging to the crystal classes 6, $[\bar{6}]$ , , 3 and $[\bar{3}]$ .

For all other space groups for which the last two letters of the Hermann–Mauguin symbol are different, $[S_{22} = S_{11}]$ , and the matrix is called $[{\bi H}_{2}]$ . Examples are $[P6_{3}/mcm]$ , P312 and $[P\bar{6}2m]$ . $[{\bi H}_{2} = \pmatrix{S_{11}\hfill &0\hfill &0\hfill\cr 0\hfill &S_{11}\hfill &0\hfill\cr 0\hfill &0\hfill &S_{33}\hfill\cr},\quad \det({\bi H}_{2}) = S_{33}S_{11}^{2}.]$

If the last two letters of the Hermann–Mauguin symbol are the same, two matrices have to be applied, the matrix $[{\bi H}_{2}]$ introduced above and the matrix $[{\bi H}_{1}]$ with $[S_{11} = 2S'_{11}]$ and $[S_{21} = S'_{11}]$ ; this matrix is called $[{\bi H}_{3}]$ , $[{\bi H}_{3} = \pmatrix{2S'_{11}\hfill &-S'_{11}\hfill &0\hfill\cr \phantom{2}S'_{11}\hfill &\phantom{-}S'_{11}\hfill &0\hfill\cr \phantom{2}0\hfill &\phantom{-}0\hfill &S_{33}\hfill\cr},\quad \det({\bi H}_{3}) = 3S_{33}{S}_{11}'^{2}.]$ Examples are P622, and P6cc.
Rhombohedral space groups

For R3 and $[R\bar{3}]$ , one has the matrix $[{\bi H}_{1}]$ for hexagonal axes and $[{\bi R}_{1}]$ for rhombohedral axes. For all other rhombohedral space groups, one has $[{\bi H}_{2}]$ (hexagonal axes) and the matrix $[{\bi R}_{1}]$ with $[S_{1} = S_{2}]$ (rhombohedral axes). This last matrix is called $[{\bi R}_{2}]$ . Example: R32. $[{\bi R}_{2} = \pmatrix{S_{0}\hfill &S_{1}\hfill &S_{1}\hfill\cr S_{1}\hfill &S_{0}\hfill &S_{1}\hfill\cr S_{1}\hfill &S_{1}\hfill &S_{0}\hfill\cr},\quad \det({\bi R}_{2}) = (S_{0} + 2S_{1})(S_{0} - S_{1})^{2}.]$