Maximal t subgroups

Bertaut, E. F.

doi:10.1107/97809553602060000509

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. 4.3, pp. 72-73

Section 4.3.4.5.2. Maximal t subgroups

E. F. Bertaut^a ^‡

^a Laboratoire de Cristallographie, CNRS, Grenoble, France

4.3.4.5.2. Maximal t subgroups

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(i) Tetragonal subgroups

The class [4/mmm] contains the classes [4/m, 422, 4mm] and $[\bar{4}2m]$ . Maximal t subgroups belonging to these classes are read directly from the standard full symbol.

Examples

(1) $[P4_{2}/m\;bc\ (135)]$ has the full symbol $[P4_{2}/m\;2_{1}/b\;2/c]$ and the tetragonal maximal t subgroups: $[P4_{2}/m]$ , $[P4_{2}2_{1}2]$ , $[P4_{2}bc]$ , $[P\bar{4}2_{1}c]$ , $[P\bar{4}b2]$ .
(2) $[I4/m\;cm\ (140)]$ has the extended full symbol $[{\matrix{\noalign{\vskip 13pt}I 4/m{\hbox to 1pt{}}2/c{\hbox to 5pt{}}2/e\hfill\cr4_{2}/n2_{1}/b 2_{1}/m\hfill\cr}}]$ and the tetragonal maximal t subgroups , I422, I4cm, $[I\bar{4}2m]$ , $[I\bar{4}c2]$ . Note that the t subgroups of class $[\bar{4}m2]$ always exist in pairs.

(ii) Orthorhombic subgroups

In the orthorhombic subgroups, the symmetry elements belonging to directions [100] and [010] are the same, except that a glide plane b perpendicular to [100] is accompanied by a glide plane a perpendicular to [010].

Examples

(1) $[P4_{2}/m bc\ (135)]$ . From the full symbol, the first maximal t subgroup is found to be $[P2_{1}/b\;2_{1}/a \;2/m]$ (Pbam). The C-cell symbol is $[C4_{2}/m\;cg_{1}]$ and gives rise to the second maximal orthorhombic t subgroup Cccm, cell $[{\bf a}',{\bf b}',{\bf c}']$ .
(2) $[I4/m\;cm\ (140)]$ . Similarly, the first orthorhombic maximal t subgroup is $[\matrix{\noalign{\vskip 13pt}I c c m\hfill\crb a n\hfill\cr}]$ (Ibam); the second maximal orthorhombic t subgroup is obtained from the F-cell symbol as $[\matrix{\noalign{\vskip 13pt}F c{\hbox to 3pt{}} c{\hbox to 3pt{}} m\hfill\crm m n\hfill\cr}]$ (Fmmm), cell a′, b′, c′.

These examples show that P- and C-cell, as well as I- and F-cell descriptions of tetragonal groups have to be considered together.

(iii) Monoclinic subgroups

Only space groups of classes 4, $[\bar{4}]$ and [4/m] have maximal monoclinic t subgroups.

Examples

(1) $[P4_{1}\;(76)]$ has the subgroup $[P112_{1}\ (P2_{1})]$ . The C-cell description does not add new features: $[C112_{1}]$ is reducible to $[P2_{1}]$ .
(2) $[I4_{1}/a\ (88)]$ has the subgroup $[I112_{1}/a]$ , equivalent to $[I11 2/a\;(C2/c)]$ . The F-cell description yields the same subgroup $[F11\;2/d]$ , again reducible to .

References

International Tables for Crystallography (2006). Vol. A. ch. 4.3, pp. 72-73