Point groups

Janssen, T.

doi:10.1107/97809553602060000637

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

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. D. ch. 1.10, pp. 246-247

Section 1.10.2.2. Point groups

T. Janssen^a ^*

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

1.10.2.2. Point groups

| top | pdf |

The orthogonal transformations that leave the diffraction pattern invariant form a point group K, a finite subgroup of O(d), where d is the dimension of the physical space. All elements act on the basis of the Fourier module as in (1.10.2.1) and the matrices $[\Gamma^{*}(K)]$ form a representation of the group K, an integral representation because the matrices have all integer entries, and reducible because the physical space is an invariant subspace for $[\Gamma^{*}(K)]$ . Because K is finite, this representation is equivalent with a representation in terms of orthogonal matrices. Moreover, by construction $[\Gamma^{*}(K)]$ leaves the n-dimensional reciprocal lattice $[\Sigma^{*}]$ invariant. It is an n-dimensional crystallographic point group. The components R of $[R_{s}]$ form a d-dimensional point group $[K_{E}]$ , which is not necessarily crystallographic, and the components $[R_{I}]$ form an ( [n-d] )-dimensional point group $[K_{I}]$ .

Consider as an example an IC phase with orthorhombic basic structure and one independent modulation wavevector $[\gamma {\bf c}^{*}]$ along the c axis. Suppose that the Fourier module, which is of rank four, is invariant under the point group mmm. Then one has for the three generators $[\matrix{{m}_x{\bf a}_1^* = -{\bf a}_1^*, \hfill&{m}_x{\bf a}_2^* = {\bf a}_2^*, \hfill&{m}_x{\bf a}_3^* = {\bf a}_3^*, \hfill&{m}_x{\bf a}_4^* = {\bf a}_4^*\hfill\cr {m}_y{\bf a}_1^* = {\bf a}_1^*, \hfill&{m}_y{\bf a}_2^* = -{\bf a}_2^*, \hfill&{m}_y{\bf a}_3^* = {\bf a}_3^*, \hfill&{m}_y{\bf a}_4^* = {\bf a}_4^*\hfill\cr{m}_z{\bf a}_1^* = {\bf a}_1^*, \hfill&{m}_z{\bf a}_2^* = {\bf a}_2^*, \hfill&{m}_z{\bf a}_3^* = -{\bf a}_3^*, \hfill&{m}_z{\bf a}_4^* = -{\bf a}_4^*.\hfill\cr}]$ Therefore, the corresponding matrices $[\Gamma^{*}(R)]$ are $[\displaylines{\pmatrix{ -1&0&0&0\cr 0&1&0&0\cr 0&0&1&0\cr 0&0&0&1}, \quad\pmatrix{ 1&0&0&0\cr 0&-1&0&0\cr 0&0&1&0\cr 0&0&0&1}, \cr\pmatrix{1&0&0&0\cr 0&1&0&0\cr 0&0&-1&0\cr 0&0&0&-1}, \cr\hfill(1.10.2.6)}]$ which implies that the three generators of the four-dimensional point group are ( $[m_{x},1]$ ), ( $[m_{y},1]$ ) and ( $[m_{z},\bar{1}]$ ).

The diffraction pattern of the standard octagonal tiling has rank four, basis vectors of the Fourier module are $[(1,0),\quad(\sqrt{1/2},\sqrt{1/2}),\quad(0,1),\quad(-\sqrt{1/2},\sqrt{1/2})]$ and the pattern is invariant under a rotation of $[\pi /4]$ and a mirror symmetry. The action of these elements on the given basis of the Fourier module is $[\Gamma^{*}(R_{1}) = \pmatrix{0&0&0&-1\cr 1&0&0&0\cr 0&1&0&0\cr 0&0&1&0 \cr},\quad \Gamma^{*}(R_{2}) = \pmatrix{0&0&0&1\cr 0&0&1&0\cr 0&1&0&0\cr 1&0&0&0\cr}.]$ By a basis transformation, one may bring these transformations into the form $[\displaylines{\pmatrix{\cos(\pi{/4})&-\sin(\pi{/4})&0&0\cr \sin(\pi{/4})&\cos(\pi{/4})&0&0\cr 0&0&\cos(3\pi{/4})&-\sin(3\pi{/4})\cr 0&0&\sin(3\pi{/4}) &\cos(3\pi{/4})\cr},\cr \pmatrix{-1&0&0&0\cr 0&1&0&0\cr 0&0&-1&0\cr 0&0&0&1\cr}. }]$ Therefore, the $[\pi /4]$ rotation in physical space is combined with a $[3\pi/4]$ rotation in internal space in order to get a transformation that leaves a lattice invariant.

A three-dimensional example is the case of a quasicrystal with icosahedral symmetry. For the diffraction pattern all spots may be labelled with six indices with respect to a basis with basis vectors $[\eqalign{{\bf a}_1^* &= (0,0,1)\cr{\bf a}_2^* &= (a,0,b)\cr{\bf a}_3^* &= \left(a\cos (2\pi /5), a\sin (2\pi /5),b\right)\cr{\bf a}_4^* &= \left(a\cos (4\pi /5), a\sin (4\pi /5),b\right)\cr{\bf a}_5^* &= \left(a\cos (4\pi /5), -a\sin (4\pi /5),b\right)\cr{\bf a}_6^* &= \left(a\cos (2\pi /5), -a\sin (2\pi /5),b\right),\cr}]$ with $[a = 2/\sqrt{5}]$ and $[b = 1/\sqrt{5}]$ . The rotation subgroup that leaves the Fourier module invariant is generated by $[\displaylines{\Gamma^{*}(A) = \pmatrix{1&0&0&0&0&0\cr 0&0&1&0&0&0\cr 0&0&0&1&0&0\cr 0&0&0&0&1&0\cr 0&0&0&0&0&1\cr 0&1&0&0&0&0 \cr},\cr\Gamma^{*}(B) = \pmatrix{0&0&0&0&0&1\cr 1&0&0&0&0&0\cr 0&0&0&0&1&0\cr 0&0&-1&0&0&0\cr 0&0&0&-1&0&0\cr 0&1&0&0&0&0\cr}. \cr\hfill(1.10.2.7)}]$ Moreover, there is the central inversion [-E] . The six-dimensional representation of the symmetry group, which is the icosahedral group $[\bar{5}\bar{3}m]$ , is reducible into the sum of two nonequivalent three-dimensional irreducible representations. A basis for this representation in the six-dimensional space is then given by $[\displaylines{({\bf a}_{1}^{*},c{\bf a}_{1}^{*}) \quad({\bf a}_{2}^{*},-c{\bf a}_{2}^{*}) \quad({\bf a}_{3}^{*},-c{\bf a}_{4}^{*})\cr ({\bf a}_{4}^{*},-c{\bf a}_{6}^{*}) \quad({\bf a}_{5}^{*},-c{\bf a}_{3}^{*})\quad ({\bf a}_{6}^{*},-c{\bf a}_{5}^{*}), \cr\hfill(1.10.2.8)}]$ which projects on the given basis in $[V_{E}]$ .

The point-group elements considered here are pairs of orthogonal transformations in physical and internal space. Orthogonal transformations that do not leave these two spaces invariant have not been considered. The reason for this is that the information about the reciprocal lattice comes from its projection on the Fourier module in physical space. By changing the length scale in internal space one does not change the projection but one would break a symmetry that mixes the two spaces. Nevertheless, quasicrystals are often described starting from an n-dimensional periodic structure with a lattice of higher symmetry. For example, the icosahedral 3D Penrose tiling can be obtained from a structure with a hypercubic six-dimensional lattice. Its reciprocal lattice is that spanned by the vectors (1.10.2.8) where one puts c = 1. The symmetry of the periodic structure, however, is lower than that of the lattice and has a point group in reducible form. Therefore, we shall consider here only reducible point groups, subgroups of the orthogonal group O(n) which have a d-dimensional invariant subspace, identified with the physical space.

The fact that the spaces [V_E] and [V_I] are usually taken as mutually perpendicular does not have any physical relevance. One could as well consider oblique projections of a reciprocal lattice $[\Sigma^*]$ on [V_E] . What is important is that the intersection of the periodic structure with the physical space should be the same in all descriptions. The metric in internal space [V_I] follows naturally from the fact that there is a finite group [K_I] .

References

International Tables for Crystallography (2006). Vol. D. ch. 1.10, pp. 246-247