Note on basis vectors

Hahn, Th.; Looijenga-Vos, A.

doi:10.1107/97809553602060000505

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. 2.2, pp. 37-38

Section 2.2.15.5. Note on basis vectors

Th. Hahn^a ^* and A. Looijenga-Vos^b ^‡

^a Institut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany, and ^bLaboratorium voor Chemische Fysica, Rijksuniversiteit Groningen, The Netherlands
Correspondence e-mail: hahn@xtl.rwth-aachen.de

2.2.15.5. Note on basis vectors

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In the subgroup data, a′, b′, c′ are the basis vectors of the subgroup $[{\cal H}]$ of the space group $[{\cal G}]$ . The latter has the basis vectors a, b, c. In the supergroup data, a′, b′, c′ are the basis vectors of the supergroup $[{\cal S}]$ and a, b, c are again the basis vectors of $[{\cal G}]$ . Thus, a, b, c and a′, b′, c′ exchange their roles if one considers the same group–subgroup relation in the subgroup and the supergroup tables.

Examples

(1) $[{\cal G}\!\!: Pba2\ (32)]$

Listed under subgroups IIb, one finds, among other entries, $[[2]\;Pna2_{1}\;({\bf c}' = {\bf 2c})\ (33)]$ ; thus, $[{\bf c}(Pna2_{1}) = {\bf 2c}(Pba2)]$ .

Under supergroups II of $[Pna2_{1}\ (33)]$ , the corresponding entry reads $[[2]\; Pba2 \;({\bf c}' = {1 \over 2} {\bf c})\; (32)]$ ; thus $[{\bf c}(Pba2) = {1\over 2}{\bf c}(Pna2_{1})]$ .
(2) Tetragonal k space groups with P cells. For index [2], the relations between the conventional basis vectors of the group and the subgroup read (cf. Fig. 5.1.3.5 ) $[{\bf a}' = {\bf a} + {\bf b},{\hbox to 19pt{}} {\bf b}' = - {\bf a} + {\bf b}{\hbox to 21pt{}}({\bf a}', {\bf b}' \hbox{ for the subgroup}).\hfill]$ Thus, the basis vectors of the supergroup are $[{\bf a}' = {\textstyle{1 \over 2}} ({\bf a} - {\bf b}),\quad {\bf b}' = {\textstyle{1 \over 2}} ({\bf a} + {\bf b}){\hbox to 17pt{}} ({\bf a}', {\bf b}' \hbox{ for the supergroup}).\hfill]$ An alternative description is $[\displaylines{{\bf a}' = {\bf a} - {\bf b},{\hbox to 19pt{}}{\bf b}' = {\bf a} + {\bf b}\quad \quad\quad ({\bf a}', {\bf b}' \hbox{ for the subgroup})\hfill\cr {\bf a}' = {\textstyle{1 \over 2}} ({\bf a} + {\bf b}), {\hbox to 9pt{}}{\bf b}' = {\textstyle{1 \over 2}} (-{\bf a}+{\bf b})\quad ({\bf a}', {\bf b}' \hbox{ for the supergroup}).\hfill}]$
(3) Hexagonal k space groups. For index [3], the relations between the conventional basis vectors of the sub- and supergroup read (cf. Fig 5.1.3.8 ) $[{\bf a}' = {\bf a} - {\bf b},{\hbox to 19pt{}}{\bf b}' = {\bf a} + 2{\bf b}{\hbox to 23pt{}}({\bf a}', {\bf b}' \hbox{ for the subgroup}).\hfill]$ Thus, the basis vectors of the supergroup are $[{\bf a}' = {\textstyle{1 \over 3}} (2{\bf a} + {\bf b}),{\hbox to 4pt{}}{\bf b}' = {\textstyle{1 \over 3}}(-{\bf a} + {\bf b}){\hbox to 7pt{}}({\bf a}', {\bf b}' \hbox{ for the supergroup}).\hfill]$ An alternative description is $[\displaylines{{\bf a}' = 2{\bf a} + {\bf b},{\hbox to 14pt{}}{\bf b}' = - {\bf a} + {\bf b} \quad\quad\ ({\bf a}', {\bf b}' \hbox{ for the subgroup})\hfill\cr {\bf a}' = {\textstyle{1 \over 3}} ({\bf a} - {\bf b}),{\hbox to 8pt{}}{\bf b}' = {\textstyle{1 \over 3}} ({\bf a} + 2{\bf b}){\hbox to 14pt{}}({\bf a}', {\bf b}' \hbox{ for the supergroup}).\hfill}]$