Euclidean- and affine-equivalent sub- and supergroups

Koch, E.; Fischer, W.

doi:10.1107/97809553602060000535

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

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. A. ch. 15.3, pp. 902-903

Section 15.3.4. Euclidean- and affine-equivalent sub- and supergroups

E. Koch^a ^* and W. Fischer^a

^a Institut für Mineralogie, Petrologie und Kristallographie, Philipps-Universität, D-35032 Marburg, Germany
Correspondence e-mail: kochelke@mailer.uni-marburg.de

15.3.4. Euclidean- and affine-equivalent sub- and supergroups

| top | pdf |

The Euclidean or affine normalizer of a space group $[{\cal G}]$ maps any subgroup or supergroup of $[{\cal G}]$ either onto itself or onto another subgroup or supergroup of $[{\cal G}]$ . Accordingly, these normalizers define equivalence relationships on the sets of subgroups and supergroups of $[{\cal G}]$ (Koch, 1984b):

Two subgroups or supergroups of a space group $[{\cal G}]$ are called Euclidean- or $[{\cal N}\!_{{\cal E}}]$ - equivalent (affine- or $[{\cal N}\!_{{\cal A}}]$ - equivalent) if they are mapped onto each other by an element of the Euclidean (affine) normalizer of $[{\cal G}]$ , i.e. if they are conjugate subgroups of the Euclidean (affine) normalizer.

In the following, the term `equivalent subgroups (supergroups)' is used if a statement is true for Euclidean-equivalent and affine-equivalent subgroups (supergroups), and $[{\cal N}({\cal G})]$ is used to designate the Euclidean as well as the affine normalizer.

The knowledge of Euclidean-equivalent subgroups is necessary in connection with the possible deformations of a crystal structure due to subgroup degradation. Affine-equivalent subgroups play an important role for the derivation and classification of black-and-white groups (magnetic groups) and of colour groups (cf. for example Schwarzenberger, 1984). Information on equivalent supergroups is useful for the determination of the idealized type of a crystal structure.

For any pair of space groups $[{\cal G}]$ and $[{\cal H}]$ with $[{\cal H} \lt {\cal G}]$ , the relation between the two normalizers $[{\cal N}({\cal G})]$ and $[{\cal N}({\cal H})]$ controls the subgroups of $[{\cal G}]$ that are equivalent to $[{\cal H}]$ and the supergroups of $[{\cal H}]$ equivalent to $[{\cal G}]$ . The intersection group of both normalizers, $[{\cal M}({\cal G},{\cal H}) = {\cal N}({\cal G}) \cap {\cal N}({\cal H}) \geq {\cal H}]$ may or may not coincide with $[{\cal N}({\cal G})]$ and/or with $[{\cal N}({\cal H})]$ . The following two statements hold generally:

(i) The index $[i_{g}]$ of $[{\cal M}({\cal G}, {\cal H})]$ in $[{\cal N}({\cal G})]$ equals the number of subgroups of $[{\cal G}]$ which are equivalent to $[{\cal H}]$ . Each coset of $[{\cal M}({\cal G}, {\cal H})]$ in $[{\cal N}({\cal G})]$ maps $[{\cal H}]$ onto another equivalent subgroup of $[{\cal G}]$ .
(ii) The index $[i_{h}]$ of $[{\cal M}({\cal G},{\cal H})]$ in $[{\cal N}({\cal H})]$ equals the number of supergroups of $[{\cal H}]$ equivalent to $[{\cal G}]$ . Each coset of $[{\cal M(G},{\cal H)}]$ in $[{\cal N(H)}]$ maps $[{\cal G}]$ onto another equivalent supergroup of $[{\cal H}]$ .

Equivalent subgroups are conjugate in $[{\cal G}]$ if and only if $[{\cal G} \cap {\cal N(H)} \neq {\cal G}]$ . In this case, $[{\cal G}]$ contains elements not belonging to $[{\cal N(H)}]$ and the cosets of $[{\cal G} \cap {\cal N(H)}]$ in $[{\cal G}]$ refer to the different conjugate subgroups.

Examples

(1) $[{\cal G} = Cmmm]$ has four monoclinic subgroups of type with the same orthorhombic metric and the same basis as Cmmm: $[{\cal H}_{1} = P2/m11]$ , $[{\cal H}_{2} = P12/m1]$ , $[{\cal H}_{3} = P112/m]$ ( $[\overline{1}]$ at 000), $[{\cal H}_{4} = P112/m\ (\overline{1} \hbox{ at } {1\over 4} {1\over 4} 0)]$ . According to Table 15.2.1.3 , the Euclidean normalizer of $[{\cal G}]$ is $[Pmmm({1\over 2}{\bf a}, {1\over 2}{\bf b}, {1\over 2}{\bf c})]$ . Because of the orthorhombic metric of all four subgroups, their Euclidean normalizers $[{\cal N}\!_{\cal E}({\cal H}_{1})]$ , $[{\cal N}\!_{\cal E}({\cal H}_{2})]$ , $[{\cal N}\!_{\cal E}({\cal H}_{3})]$ and $[{\cal N}\!_{\cal E}({\cal H}_{4})]$ are enhanced in comparison with the general case and coincide with $[{\cal N}\!_{\cal E}({\cal G})]$ . Hence, no two of the four subgroups are Euclidean-equivalent.
(2) $[{\cal G} = I\overline{4}m2({\bf a}, {\bf b}, {\bf c})]$ , $[{\cal H} = P\overline{4}({\bf a}, {\bf b}, {\bf c})]$ . $[{\cal N(G)} = I4/mmm({1\over 2}{\bf a}-{1\over 2}{\bf b}, {1\over 2}{\bf a}+{1\over 2}{\bf b}, {1\over 2}{\bf c})]$ is a supergroup of index 2 of $[{\cal N(H)} = P4/mmm({1\over 2}{\bf a}-{1\over 2}{\bf b}]$ , $[{1\over 2}{\bf a} +{1\over 2}{\bf b}]$ , $[{1\over 2}{\bf c}) =]$ $[{\cal M}(I\overline{4}m2]$ , $[P\overline{4})]$ . Therefore, $[I\overline{4}m2]$ has two equivalent subgroups $[P\overline{4}]$ that are mapped onto another by a centring translation of $[{\cal N(G)}]$ , e.g. by $[t(0{1\over 2}{1\over 4})]$ . Both subgroups are not conjugate in $[I\overline{4}m2]$ because $[{\cal G} \cap {\cal N(H)}]$ equals $[{\cal G}]$ . As $[{\cal N(H)}]$ coincides with $[{\cal M(G},{\cal H)}]$ , no further supergroups of $[P\overline{4}]$ equivalent to $[I\overline{4}m2]$ exist.
(3) $[{\cal G} = Fm\overline{3}({\bf a}, {\bf b}, {\bf c})]$ , $[{\cal H} = F23({\bf a}, {\bf b}, {\bf c})]$ . $[{\cal N(H)} = Im\overline{3}m({1\over 2}{\bf a}, {1\over 2}{\bf b}, {1\over 2}{\bf c})]$ is a supergroup of index 2 of $[{\cal N(G)} = Pm\overline{3}m({1\over 2}{\bf a}, {1\over 2}{\bf b}, {1\over 2}{\bf c}) = {\cal M}(Fm\overline{3}, F23)]$ . Therefore, F23 has two equivalent supergroups $[Fm\overline{3}]$ that differ in their locations with site symmetry $[m\overline{3}]$ by a centring translation of $[Im\overline{3}m({1\over 2}{\bf a}, {1\over 2}{\bf b}, {1\over 2}{\bf c})]$ , e.g. by $[t({1\over 4} {1\over 4} {1\over 4})]$ . As $[{\cal N(G)}]$ coincides with $[{\cal M(G},{\cal H)}]$ , no further subgroups of $[Fm\overline{3}]$ equivalent to F23 exist.
(4) $[{\cal G} = Pmma({\bf a}, {\bf b}, {\bf c})]$ , $[{\cal H} = Pmmn({\bf a}, 2{\bf b}, {\bf c})]$ . The intersection of $[{\cal N}_{\cal A}(Pmma) = Pmmm({1\over 2}{\bf a}, {1\over 2}{\bf b}, {1\over 2}{\bf c})]$ and $[{\cal N}\!_{\cal A}(Pmmn) = P4/mmm({1\over 2}{\bf a}, {\bf b}, {1\over 2}{\bf c})]$ is the group $[{\cal M}(Pmma, Pmmn) = Pmmm({1\over 2}{\bf a}, {\bf b}, {1\over 2}{\bf c})]$ , which is a proper subgroup of both normalizers. As $[i_{g}]$ equals 2, Pmma has two affine-equivalent subgroups of type Pmmn that are mapped onto each other by the additional translation $[t(0 {1\over 2} 0)]$ of the normalizer of $[{\cal G}]$ . As $[i_{h}]$ also equals 2, Pmmn has two affine-equivalent supergroups, Pmma and Pmmb, that are mapped onto each other, e.g. by the affine `reflection' at a diagonal `mirror plane' of $[{\cal N}\!_{\cal A}({\cal H})]$ .