Inhomogeneous tensors

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. 250-251

Section 1.10.4.3. Inhomogeneous tensors

T. Janssen^a ^*

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

1.10.4.3. Inhomogeneous tensors

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A vector field in d-dimensional space assigns a vector to each point of the space. This vector-valued function may, for a quasiperiodic system, have values in physical space or in superspace. In both cases one has the transformation property $[g{\bf f}_{i}({\bf r}) = \textstyle\sum\limits_{j}R_{ji}{\bf f}_{j}(g^{-1}{\bf r}). \eqno(1.10.4.9)]$ For a vector field in physical space, i and j run over the values 1, 2, 3. This vector field may, however, be quasiperiodic. This means that it may be embedded in superspace. Then $[g{\bf f}_{i}({\bf r}_{s}) = \textstyle\sum\limits_{j=1}^{3}R_{Eji}{\bf f}_{j} \left [R_{E}^{-1}({\bf r}-{\bf a}),R^{-1}_{I}({\bf r}_{I}-{\bf a}_{I})\right]. \eqno(1.10.4.10)]$ Here i = 1, 2, 3. If the vector field has values in superspace, as one can have for a displacement, one has $[g{\bf f}_{i}({\bf r}_{s}) = \textstyle\sum\limits_{j=1}^{n}R_{sji}{\bf f}_{j} \left[R_{E}^{-1}({\bf r}-{\bf a}),R^{-1}_I({\bf r}_{I}-{\bf a}_{I}) \right]. \eqno(1.10.4.11)]$ Here $[i = 1,\ldots, n]$ . For Cartesian coordinates with respect to a split basis, $[R_{s}]$ acts separately on physical and internal space and one has $[g{\bf f}_{i}({\bf r}_{s}) = \textstyle\sum\limits_{j = d+1}^{n}R_{Iji}{\bf f}_{j} \left[R_{E}^{-1}({\bf r}-{\bf a}),R^{-1}_I({\bf r}_{I}-{\bf a}_{I})\right]\eqno(1.10.4.12)]$ for $[i = d+1,\ldots, n]$ .

Just as for homogeneous tensors, inhomogeneous tensors may be divided into physical tensors with components in physical space only and others that have components with respect to an n-dimensional lattice. A physical tensor of rank two transforms under a space-group element $[g=\{R_{s}|{\bf a}_{s}\}]$ as $[(gT)_{ij}({\bf r}_{s}) = \textstyle\sum\limits_{k=1}^{d}\textstyle\sum\limits_{l=1}^{d}R_{Eki}R_{Elj}T_{kl} [R_{E}^{-1}({\bf r}_{E}-{\bf a}_{E}),R_{I}^{-1}({\bf r}_{I}-{\bf a}_{I})].\eqno(1.10.4.13)]$ This implies the following transformation property for the Fourier components: $[(g\hat{T})_{ij}({\bf k}) = \textstyle\sum\limits_{k=1}^{d}\textstyle\sum\limits_{l=1}^{d}R_{Eki}R_{Elj}T_{kl} (R_{E}^{-1}{\bf k})\exp (iR_{E}{\bf k}.{\bf a}_{E}+iR_{I}{\bf k}_{I}. {\bf a}_{I}).\eqno(1.10.4.14)]$ This gives relations between various Fourier components and restrictions for wavevectors $[{\bf k}]$ for which $[R_{E}{\bf k} = {\bf k}]$ : $[\hat{T}_{ij}({\bf k}) = \textstyle\sum\limits_{k=1}^{d}\textstyle\sum\limits_{l=1}^{d}R_{Eki}R_{Elj} T_{kl}({\bf k})\exp (iR_{E}{\bf k}.{\bf a}_{E}+iR_{I}{\bf k}_{I}. {\bf a}_{I}). \eqno(1.10.4.15)]$

For tensors with superspace components, the summation over the indices runs from 1 to n. An invariant tensor then satisfies $[\hat{T}_{ij}({\bf k}) = \textstyle\sum\limits_{k=1}^{n}\textstyle\sum\limits_{l=1}^{n}R_{ski}R_{slj} T_{kl}({\bf k})\exp (iR_{E}{\bf k}.{\bf a}_{E}+iR_{I}{\bf k}_{I}. {\bf a}_{I}).\eqno(1.10.4.16)]$ The generalization to higher-rank tensors is straightforward.

References

International Tables for Crystallography (2006). Vol. D. ch. 1.10, pp. 250-251