Transformation of tensors

Janssen, T.

doi:10.1107/97809553602060000629

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. 1.2, p. 54

Section 1.2.5.3. Transformation of tensors

T. Janssen^a ^*

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

1.2.5.3. Transformation of tensors

| top | pdf |

Vectors and tensors transforming in the same way under Euclidean transformations may behave differently when time reversal is taken into account. As an example, both the electric field $[{\bf E}]$ and magnetic field $[{\bf B}]$ transform under a rotation as a position vector. Under time reversal, the former is invariant, but the latter changes sign. Therefore, the magnetic field is called a pseudovector field under time reversal. Under spatial inversion, the field $[{\bf E}]$ changes sign, as does a position vector, but the field $[{\bf B}]$ does not. Therefore, the magnetic field is also a pseudovector under central inversion. The electric polarization induced by an electric field is given by the electric susceptibility, a magnetic moment induced by a magnetic field is given by the magnetic susceptibility and in some crystals a magnetic moment is induced by an electric field via the magneto-electric susceptibility. Under the four elements of the group generated by [T=1'] and $[I=\bar 1]$ , the fields and susceptibility tensors transform according to $[\matrix{\phantom{\chi_{mm}}& \hfill E & \hskip1pt\phantom{-}\bar{1} & \phantom{-}1' & \hskip -4pt\phantom{-}\bar{1}'\cr}\over\matrix{{\bf E} & \hfill1 &\hfill-1 &\hfill1 &\hfill-1 \cr {\bf B} & \hfill1 & \hfill1 & \hfill-1 & \hfill-1 \cr \chi_{ee} & \hfill1 & \hfill1 & \hfill1 & \hfill1 \cr \chi_{mm} & \hfill1 & \hfill1 & \hfill1 & \hfill1 \cr \chi_{me} & \hfill1 & \hfill-1 & \hfill-1 & \hfill1 }]$ Here $[{\bar 1}'={\bar 1}1']$ .

In general, a vector transforms as the position vector $[{\bf r}]$ under rotations and changes sign under $[\bar{1}]$ , but not under [1'] . A pseudovector under $[\bar{1}]$ or (respectively and) [1'] gets an additional minus sign. The generalization to tensors is straightforward. $[gT_{i_{1}\ldots i_{n}} = \varepsilon_{P}\varepsilon_{T} \textstyle\sum\limits_{j_{1}\ldots j_{n}} \left(\textstyle\prod\limits_{k=1}^{n} R_{i_{k}j_{k}}\right) T_{j_{1}\ldots j_{n}}, \eqno (1.2.5.1)]$ where $[\varepsilon_{P}]$ and $[\varepsilon_{T}]$ are $[\pm 1]$ , depending on the pseudotensor character with respect to space and time reversal, respectively.

Under a rotation [ $[R\in SO(d)]$ ], a vector transforms according to a representation characterized by the character $[\chi (R)]$ of the representation. In two dimensions $[\chi =2\cos \varphi]$ and in three dimensions $[\chi =1+2\cos \varphi]$ , if $[\varphi]$ is the rotation angle. Under [IR] the character gets an additional minus sign, under [RT] it is the same, and under [RIT] there is again an additional minus sign. For pseudovectors , either under I or T or both, there are the extra factors $[\varepsilon_{P}]$ , $[\varepsilon_{T}]$ and $[\varepsilon_{P} \varepsilon_{T}]$ , respectively. As an example, the character of the representations corresponding to the electric and magnetic fields in two orthorhombic point groups ( [222] , [2'2'2] and [2'mm'] ) are given in Table 1.2.5.1.

Table 1.2.5.1 | top | pdf |
Character of the representations corresponding to the electric and magnetic fields in point groups [222] , [2'2'2] and [2'mm']

$[n_{i}]$ is the number of invariants.

$[{\bf E}]$				$[n_{i}]$	$[{\bf B}]$				$[n_{i}]$
3	−1	−1	−1	0	3	−1	−1	−1	0
3	−1	−1	−1	0	3	1	1	−1	1
3	−1	1	1	1	3	1	−1	1	1

The number of invariant components is the multiplicity of the trivial representation in the representation to which the tensor belongs. The nonzero invariant field components are $[B_{z}]$ for [2'2'2] , $[E_{x}]$ and $[B_{y}]$ for [2'mm'] . These components can be constructed by means of projection-operator techniques, or more simply by solving the linear equations representing the invariance of the tensor under the generators of the point group. For example, the magnetic field vector B transforms to ( $[-B_{x},B_{y},-B_{z}]$ ) under $[m_{y}]$ and to ( $[B_{x},B_{y},-B_{z}]$ ) under $[m_{z}]$ , and this gives the result that all components are zero except $[B_{y}]$ .

References

International Tables for Crystallography (2006). Vol. D. ch. 1.2, p. 54