Structure of space groups

Janssen, T.

doi:10.1107/97809553602060000629

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 1.2, pp. 46-47

Section 1.2.3.1. Structure of space groups

T. Janssen^a ^*

^a Institute for Theoretical Physics, University of Nijmegen, 6524 ED Nijmegen, The Netherlands
Correspondence e-mail: ted@sci.kun.nl

1.2.3.1. Structure of space groups

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The Euclidean group E(n) in n dimensions is the group of all distance-preserving inhomogeneous linear transformations. In Euclidean space, an element is denoted by $[g = \{ R|{\bf a}\}]$ where $[R\in{O}(n)]$ and $[{\bf a}]$ is an n-dimensional translation. On a point $[{\bf r}]$ in n-dimensional space, g acts according to $[\{R|{\bf a}\} {\bf r} = R{\bf r} + {\bf a}. \eqno (1.2.3.1)]$ Therefore, $[|g{\bf r}_{1}-g{\bf r}_{2}|=|{\bf r}_{1}-{\bf r}_{2}|]$ . The group multiplication law is given by $[\{R|{\bf a}\}\{R'|{\bf a}'\} = \{RR'|{\bf a}+R{\bf a}'\}. \eqno (1.2.3.2)]$ The elements $[\{E|{\bf a}\}]$ form an Abelian subgroup, the group of n-dimensional translations T(n).

An n-dimensional space group is a subgroup of E(n) such that its intersection with T(n) is generated by n linearly independent basis translations. This means that this lattice translation subgroup A is isomorphic to the group of n-tuples of integers: each translation in A can be written as $[\{E|{\bf a}\} = \textstyle\prod\limits_{i=1}^{n} \{E|{\bf e}_{i}\}^{n_{i}} = \{E|\textstyle\sum\limits_{i=1}^n n_i {\bf e}_i\}. \eqno (1.2.3.3)]$ The lattice translation subgroup A is an invariant subgroup because $[g\{E|{\bf a}\}g^{-1} = \{R|{\bf b}\}\{E|{\bf a}\}\{R|{\bf b}\}^{-1} = \{E|R{\bf a}\}\in A.]$ The factor group [G/A] , of the space group G and the lattice translation group A, is isomorphic to the group K formed by all elements R occurring in the elements $[\{R|{\bf a}\}\in G]$ . This group is the point group of the space group G. It is a subgroup of O(n).

The unit cell of the space group is a domain in n-dimensional space such that every point in space differs by a lattice translation from some point in the unit cell, and such that between any two points in the unit cell the difference is not a lattice translation. The unit cell is not unique. One choice is the n-dimensional parallelepiped spanned by the n basis vectors. The points in this unit cell have coordinates between 0 (inclusive) and 1. Another choice is not basis dependent: consider all points generated by the lattice translation group from an origin. This produces a lattice of points $[\Lambda]$ . Consider now all points that are closer to the origin than to any other lattice point. This domain is a unit cell, if one takes care which part of the boundary belongs to it and which part not, and is called the Wigner–Seitz cell. In mathematics it is called the Voronoi cell or Dirichlet domain (or region).

Because the point group leaves the lattice of points invariant, it transforms the Wigner–Seitz cell into itself. This implies that points inside the unit cell may be related by a point-group element. Similarly, space-group elements may connect points inside the unit cell, up to lattice translations. A fundamental region or asymmetric unit is a part of the unit cell such that no points of the fundamental region are connected by a space-group element, and simultaneously that any point in space can be related to a point in the fundamental region by a space-group transformation.

Because $[\{E|R{\bf a}\}]$ belongs to the lattice translation group for every $[R\in K]$ and every lattice translation $[\{E|{\bf a}\}]$ , the lattice $[\Lambda]$ generated by the vectors $[{\bf e}_{i}]$ ( $[i=1,2,\ldots, n]$ ) is invariant under the point group K. Therefore, the latter is a crystallographic point group. On a basis of the lattice $[\Lambda]$ , the point group corresponds to a group $[\Gamma (K)]$ of integer matrices. One has the following situation. The space group G has an invariant subgroup A isomorphic to $[{\bb Z}^{n}]$ , the factor group [G/A] is a crystallographic point group K which acts according to the integer representation $[\Gamma (K)]$ on A. In mathematical terms, G is an extension of K by A with homomorphism $[\Gamma]$ from K to the group of automorphisms of A.

The vectors $[{\bf a}]$ occurring in the elements $[\{E|{\bf a}\}\in G]$ are called primitive translations. They have integer coefficients with respect to the basis $[{\bf e}_{1},\ldots, {\bf e}_{n}]$ . However, not all vectors $[{\bf a}]$ in the space-group elements are necessarily primitive. One can decompose the space group G according to $[G = A + g_{2}A + g_{3}A +\ldots +g_{N}A. \eqno (1.2.3.4)]$ To every element $[R\in K]$ there is a coset $[g_{i}A]$ with $[g_{i}=\{R|{\bf a}(R)\}]$ as representative. Such a representative is unique up to a lattice translation. Instead of $[{\bf a}(R)]$ , one could as well have $[{\bf a}(R)+{\bf n}]$ as representative for any lattice translation $[{\bf n}]$ . For a particular choice, the function $[{\bf a}(R)]$ from the point group to the group T(n) is called the system of nonprimitive translations or translation vector system. It is a mapping from the point group K to [T(n)] , modulo A. Such a system of nonprimitive translations satisfies the relations $[{\bf a}(R) + R{\bf a}(S) = {\bf a}(RS) \,\,{\rm mod}\,\, A \quad\forall\,\, R,S \in K. \eqno (1.2.3.5)]$ This follows immediately from the product of two representatives $[g_{i}]$ .

If the lattice translation subgroup A acts on a point $[{\bf r}]$ different from the origin, one obtains the set $[\Lambda +{\bf r}]$ . One can describe the elements of G as well as combinations of an orthogonal transformation with $[{\bf r}]$ as centre and a translation. This can be seen from $[\{R|{\bf a}\} = \{E|{\bf a}-{\bf r}+R{\bf r}\}\{R|{\bf r}-R{\bf r}\}, \eqno (1.2.3.6)]$ where now $[\{R|{\bf r}-R{\bf r}\}]$ leaves the point $[{\bf r}]$ invariant. The new system of nonprimitive translations is given by $[{\bf a}'(R) = {\bf a}(R) + (R-E){\bf r}. \eqno (1.2.3.7)]$ This is the effect of a change of origin. Therefore, for a space group, the systems of nonprimitive translations are only determined up to a primitive translation and up to a change of origin.

It is often convenient to describe a space group on another basis, the conventional lattice basis. This is the basis for a sublattice with the same, or higher, symmetry and with the same number of free parameters. Therefore, the sublattice is also invariant under K and with respect to the conventional basis, which is obtained from the original one via a basis transformation S, the point group has the form $[\Gamma_{{\rm conventional}}(R) = S\Gamma_{{\rm primitive}}(R)S^{- 1}, \eqno (1.2.3.8)]$ where S is the centring matrix. It is a matrix with determinant equal to the inverse of the number of lattice points of the primitive lattice inside the unit cell of the conventional lattice. As an example, consider the primitive and centred rectangular lattices in two dimensions. Both have symmetry [2mm] , and two parameters a and b. The transformation from a basis of the conventional lattice [( [2a,0] ) and ( [0,2b] )] to a basis of the primitive lattice [( [a,-b] ) and ( [a,b] )] is given by S, and the relations between the generators of the point groups are $[\eqalign{ \pmatrix{ -1 & 0\cr 0 & 1}&= S \pmatrix{ 0 & -1\cr -1 & 0}S^{-1}, \quad S = \pmatrix{{{1}\over{2}} & {{1}\over{2}}\cr -{{1}\over{2}} &{{1}\over{2}}}\cr \pmatrix{1&0\cr 0&-1}&=S\pmatrix{0& 1\cr 1 & 0}S^{-1}.}]$

References

International Tables for Crystallography (2006). Vol. D. ch. 1.2, pp. 46-47