Action of superspace groups

Janssen, T.

doi:10.1107/97809553602060000637

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. 1.10, pp. 247-248

Section 1.10.3.1. Action of superspace groups

T. Janssen^a ^*

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

1.10.3.1. Action of superspace groups

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The action of the symmetry group on the periodic density function $[\rho_s]$ in n dimensions is given by (1.10.2.9). The real physical structure, however, lives in physical space. One can derive from the action of the superspace group on the periodic structure its action on the quasiperiodic d-dimensional one. One knows that the density function in $[V_{E}]$ is just the restriction of that in $[V_{s}]$ . The same holds for the transformed function. $[g\rho_{s}({\bf r}_{s}) = \rho_{s}(g^{-1}{\bf r}_{s}) \rightarrow g\rho ({\bf r}) = \rho_{s}[R^{-1}({\bf r}-{\bf a}_{E}),-R_{I}^{-1} {\bf a}_{I}].\eqno(1.10.3.1)]$ This transformation property differs from that under an n-dimensional Euclidean transformation by the `phase shift' $[-R_{I}^{-1}{\bf a}_{I}]$ . Take for example the IC phase with a sinusoidal modulation. If the positions of the atoms are given by $[{\bf n} + {\bf r}_{j} + {\bf A}_{j}\cos (2\pi{\bf Q}\cdot{\bf n}+\varphi_{j}),]$ then the transformed positions are $[R({\bf n} + {\bf r}_{j}) + R{\bf A}_{j}\cos (2\pi{\bf Q}\cdot{\bf n}+\varphi_{j} -R_{I}^{-1}{\bf a}_{I}) + {\bf a}_{E}.\eqno(1.10.3.2)]$ If the transformation g is a symmetry operation, this means that the original and the transformed positions are the same. $[R({\bf n} + {\bf r}_{j}) + {\bf a}_{E} = {\bf n^\prime} + {\bf r}_{j^\prime}]$ and $[R{\bf A}_{j}\cos (2\pi{\bf Q}\cdot{\bf n}+\varphi_{j}-R_{I}^{-1}{\bf a}_{I}) = {\bf A}_{j^\prime}\cos (2\pi{\bf Q}\cdot{\bf n^\prime}+\varphi_{j^\prime}).]$ This puts, in general, restrictions on the modulation.

Another view of the same transformation property is given by Fourier transforming (1.10.2.9). The result for the Fourier transform is $[g\hat{\rho}_{s}({\bf k}_{s}) = \hat{\rho}_{s}(R_{s}^{-1}{\bf k}_{s}) \exp (-i{\bf k}_{s}\cdot{\bf a}_{s})\eqno(1.10.3.3)]$ and because there is a one-to-one correspondence between the vectors $[{\bf k}_{s}]$ in the reciprocal lattice and the vectors k in the Fourier module one can rewrite this as $[g\hat{\rho}({\bf k}) = \hat{\rho}(R^{-1}{\bf k})\exp (-i{\bf k}\cdot{\bf a}_{E} -{\bf k}_{I}\cdot{\bf a}_{I}).\eqno(1.10.3.4)]$ For a symmetry element one has $[g\hat{\rho}({\bf k})=\hat{\rho}({\bf k})]$ . Therefore, the superspace group element g is a symmetry transformation of the quasiperiodic function $[\rho]$ if $[\hat{\rho}({\bf k}) = \hat{\rho}(R^{-1}{\bf k})\exp (-i{\bf k}\cdot{\bf a}_{E} -{\bf k}_{I}\cdot{\bf a}_{I}).\eqno(1.10.3.5)]$ This relation is at the basis of the systematic extinctions. If one has an orthogonal transformation R such that this in combination with a translation ( $[{\bf a}_{E},{\bf a}_{I}]$ ) is a symmetry element and such that $[R{\bf k} = {\bf k}]$ , then $[\hat{\rho}({\bf k}) = 0\;\;{\rm if}\;\;{\bf k}\cdot{\bf a}_{E}+{\bf k}_{I}\cdot{\bf a}_{I} \neq 2\pi \times\, {\rm integer}.\eqno(1.10.3.6)]$ Because the structure factor is the Fourier transform of a density function which consists of $[\delta]$ functions on the positions of the atoms, for a quasiperiodic crystal it is the Fourier transform of a quasiperiodic function $[\rho ({\bf r})]$ . Therefore, symmetry-determined absence of Fourier components leads to zero intensity of the corresponding diffraction peaks. Therefore, although there is no lattice periodicity for aperiodic crystals, systematic extinctions follow in the same way from the symmetry as in lattice periodic systems if one considers the n-dimensional space group as the symmetry group.

References

International Tables for Crystallography (2006). Vol. D. ch. 1.10, pp. 247-248