Definition of space groups

Nebe, G.

doi:10.1107/97809553602060000541

International
Tables for
Crystallography
Volume A1
Symmetry relations between space groups
Edited by Hans Wondratschek and Ulrich Müller

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International Tables for Crystallography (2006). Vol. A1. ch. 1.5, pp. 34-35 | 1 | 2 |

Section 1.5.4.1. Definition of space groups

Gabriele Nebe^a ^*

^a Abteilung Reine Mathematik, Universität Ulm, D-89069 Ulm, Germany
Correspondence e-mail: nebe@mathematik.uni-ulm.de

1.5.4.1. Definition of space groups

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In IT A (2005), Section 8.1.6 , space groups are introduced as symmetry groups of crystal patterns.

Definition 1.5.4.1.1

(a) Let $[{\bf V}_n]$ be the n-dimensional real vector space. A subset $[{\bf L} \subseteq {\bf V}_n]$ is called an (n-dimensional) lattice if there is a basis $[({\bf b}_1,\ldots, {\bf b}_n)]$ of $[{\bf V}_n]$ such that $[{\bf L}={\bb Z} {\bf b}_1 + \ldots + {\bb Z} {\bf b}_n = \{ \textstyle\sum\limits _{i=1}^n a_i {\bf b}_i \mid a_i \in {\bb Z} \}. ]$
(b) A crystal structure is a mapping $[f: {\bb E} _{n} \rightarrow {\bb R} ]$ of the Euclidean affine n-space into the real numbers such that $[{\rm Stab}_{\tau ({\bb A} _{n}) } (f): = \{ t \in \tau({\bb A}_{n}) \mid f(P+t) = f(P) \,{\rm for \, all \,} P\in {\bb A}_{n} \}]$ is an n-dimensional lattice in $[\tau ({\bb A}_{n})]$ .
(c) The Euclidean group $[{\cal E}_{n}]$ acts on the set of mappings $[{\bb E}_{n} \rightarrow {\bb R}]$ via $[({\sf g} \cdot f)(P): = f ({\sf g} ^{-1}P) ]$ for all $[P\in {\bb E}_{n}]$ and for all $[{\sf g} \in {\cal E}_{n}]$ and $[f:{\bb E}_{n} \rightarrow {\bb R}]$ . A space group $[{\cal R}]$ is the stabilizer of a crystal structure $[f:{\bb E}_{n} \rightarrow {\bb R}]$ ; $[{\cal R} = {\rm Stab}_{\cal E}_{n}(f)]$ .
(d) Let $[{\cal R} \leq {\cal E}_{n} ]$ be a space group. The translation subgroup $[{\cal T}({\cal R})]$ of $[{\cal R}]$ is defined as $[{\cal T}({\cal R}): ={\cal R}\cap {\cal T}_{n}.]$

The definition introduced space groups in the way they occur in crystallography: The group of symmetries of an ideal crystal stabilizes the crystal structure. This definition is not very helpful in analysing the structure of space groups. If $[{\cal R}]$ is a space group, then the translation subgroup $[{\cal T}: = {\cal T}({\cal R})]$ is a normal subgroup of $[{\cal R}]$ . It is even a characteristic subgroup of $[{\cal R}]$ , hence fixed under every automorphism of $[{\cal R}]$ . By Definition 1.5.4.1.1, its image under the inverse $[\mu ']$ of the mapping $[\mu ]$ in Example 1.5.3.4.4 defined by $[\mu': {{\cal T}} \rightarrow \tau ({\bb{E}}_{n}); \left(\matrix{ {\bi I}\,\vphantom{(^2{\big(_2}}&\vrule\, &{\bi v} \cr \noalign{\vskip-1pt\hrule} \cr{\bi o}^{\rm T}\,\vphantom{{\big(^2}(_2}&\vrule\, &1 } \right) \mapsto \bi{v}]$ in $[\tau ({\bb A}_{n})]$ is a full lattice $[{\bf L}({\cal R})]$ . Since $[\mu ']$ is an isomorphism from $[{\cal T} ]$ onto $[{\bf L}({\cal R})]$ , the translation subgroup of $[{\cal R}]$ is isomorphic to the lattice $[{\bf L}({\cal R})]$ . In particular, one has $[\mu ' (t_1 t_2) = \mu ' (t_1) + \mu ' (t_2)]$ and the subgroup $[{\cal T}^{p}]$ , formed by the pth powers of elements in $[{\cal T}]$ , is mapped onto $[p{\bf L}({\cal R})]$ . Lattices are well understood. Although they are infinite, they have a simple structure, so they can be examined algorithmically. Since they lie in a vector space, one can apply linear algebra to them.

Now we want to look at how this lattice $[{\cal T}({\cal R})]$ fits into the space group $[{\cal R}]$ . The affine group $[{\cal A}_{n}]$ acts on $[{\cal T}_{n}]$ by conjugation as well as on $[\tau({\bb A}_{n})]$ via its linear part. Similarly the space group $[{\cal R}]$ acts on $[{\cal T}({\cal R})]$ by conjugation: For $[{\sf g} \!\in \!{\cal R}]$ and $[{\sf t} \!\in \!{\cal T}]$ , one gets $[\mu' ({\sf g} {\sf t} {\sf g} ^{-1}) = \overline{{\sf g} } \mu ' ({\sf t}) ]$ , where $[\overline{{\sf g} }]$ is the linear part of $[{\sf g}]$ . Therefore the kernel of this action is on the one hand the centralizer of $[{\cal T}({\cal R})]$ in $[{\cal R}]$ , on the other hand, since $[{\bf L}({\cal R})]$ contains a basis of $[\tau ({\bb E}_{n})]$ , it is equal to the kernel of the mapping $[\overline{} ]$ , which is $[{\cal R} \cap {\cal T}_n = {\cal T}({\cal R})]$ , hence $[{\cal C}_{\cal R}({\cal T}({\cal R})) = {\cal T}({\cal R}).]$ Hence only the linear part $[\overline{{\cal R}} \cong {\cal R}/{\cal T}({\cal R}) ]$ of $[{\cal R}]$ acts faithfully on $[{\cal T}({\cal R})]$ by conjugation and linearly on $[{\bf L}(\cal {R})]$ . This factor group $[{\cal R} /{\cal T}({\cal R})]$ is a finite group. Let us summarize this:

Theorem 1.5.4.1.2. Let $[{\cal R}]$ be a space group. The translation subgroup $[{\cal T}({\cal R})\! = \!{\cal R} \cap {\cal T}_{n} ]$ is an Abelian normal subgroup of $[{\cal R}]$ which is its own centralizer, $[{\cal C}_{\cal R}({\cal T}({\cal R})) = {\cal T}({\cal R})]$ . The finite group $[{\cal R}/{\cal T}({\cal R}) ]$ acts faithfully on $[{\cal T}({\cal R})]$ by conjugation. This action is similar to the action of the linear part $[\overline{{\cal R}}]$ on the lattice $[\mu ' ({\cal T} ({\cal R})) = {\bf L}({\cal R})]$ .

References

International Tables for Crystallography (2005). Vol. A, Space-group symmetry, edited by Th. Hahn, 5th ed. Heidelberg: Springer. (Abbreviated IT A.)Google Scholar

International Tables for Crystallography (2006). Vol. A1. ch. 1.5, pp. 34-35