Transformation properties of tensors

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. 51-52

Section 1.2.4.1. Transformation properties 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

1.2.4.1. Transformation properties of tensors

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A vector is an element of an N-dimensional vector space that transforms under an orthogonal transformation, an element of [O(n] ), as $[x = \textstyle\sum\limits_{i=1}^{n}\xi_{i}{\bf a}_{i} \rightarrow x' = \textstyle\sum\limits_{i=1}^{n}\xi_{i}'{\bf a}_{i} = \textstyle\sum\limits_{ij}R_{ij}\xi_{j}{\bf a}_{i},\quad \{R_{ij}\}\in O(n).]$ A tensor of rank r under [O(n)] is an object with components $[T_{i_{1}\ldots i_{r}}]$ ( $[i_{j}=1,2,\ldots, n]$ ) that transforms as (see Section 1.1.3.2 ) $[T_{i_{1}\ldots i_{r}} \rightarrow T_{i_{1}\ldots i_{r}}' = \textstyle\sum\limits_{j_{1}=1}^{n}\ldots \textstyle\sum\limits_{j_{r}=1}^{n} R_{i_{1}j_{1}}\ldots R_{i_{r}j_{r}} T_{j_{1}\ldots j_{r}}.]$ A rank-zero tensor is a scalar, which is invariant under [O(n)] . A pseudovector (or axial vector) has components $[x_{i}]$ and transforms according to $[x_{i} \rightarrow x_{i}' = {\rm Det}(R)\textstyle\sum\limits_{j}R_{ij}\xi_{j}]$ and analogously for pseudotensors (or axial tensors – see Section 1.1.4.5.3 ).

A vector field is a vector-valued function in n-dimensional space. Under an orthogonal transformation it transforms according to $[F_{i}({\bf r})' = \textstyle\sum\limits_{j=1}^{n}R_{ij} F_{j}(R^{-1}{\bf r}). \eqno (1.2.4.1)]$ Under a Euclidean transformation, the function transforms according to $[F_{i}({\bf r})' = \textstyle\sum\limits_{j=1}^{n}R_{ij} F_{j}(R^{-1}({\bf r}-{\bf a})),\quad \{R|{\bf a}\}\in E(n). \eqno (1.2.4.2)]$ In a similar way, one has (pseudo)tensor functions under the orthogonal group or the Euclidean group. So it is important to specify under what group an object is a tensor, unless no confusion is possible.

The n-dimensional vectors form a vector space that carries a representation of the group O(n). Moreover, it is an irreducible representation space. To stress this fact, one could speak of irreducible tensors and vectors. Vectors are here just rank-one tensors. The three-dimensional Euclidean vector space carries in this way an irreducible representation of O(3). Such representations are characterized by an integer l and are [(2l+1)] -dimensional. The usual three-dimensional space is therefore an irreducible [l=1] space for O(3).

Since point groups are subgroups of the orthogonal group and space groups are subgroups of the Euclidean group, tensors inherit their transformation properties from their supergroups. As we have seen in Sections 1.2.2.3 and 1.2.2.7, one can also define tensors in a quite abstract way. Irreducible tensors under a group are then elements of a vector space that carries an irreducible representation of that group. Generally, tensors are elements of a vector space that carries a tensor product representation and (anti)symmetric tensors belong to a space with an (anti)symmetrized tensor product representation.

Because the point groups one usually considers in physics are subgroups of O(2) or O(3), it is useful to consider the irreducible representations of these groups. They are not finite, but they are compact, and for compact groups most of the theorems for finite groups are still valid if one replaces sums over group elements by integration over the group.

The group O(3) is the direct product $[SO(3)\times C_{2}]$ . Therefore, there are even and odd representations. They have the property $[D^{\pm}(R)=\Delta (R),\quad D^{\pm}(-R)=\pm \Delta (R),\quad R\in SO(3).]$ The irreducible representations are labelled by non-negative integers $[\ell]$ and have character $[\chi_{\ell}(R) = {{\sin (\ell + {{1}\over{2}})\varphi}\over{\sin {{1}\over{2}} \varphi}}\eqno (1.2.4.3)]$ if R is a rotation with rotation angle $[\varphi]$ . From the character it follows that the dimension of the representation $[D_{\ell}]$ is equal to $[(2\ell +1)]$ .

The tensor product of two irreducible representations of SO(3) is generally reducible: $[D_{\ell}\otimes D_{m} = {\bigoplus\limits_{j=|\ell -m|}^{\ell +m}}D_{j} \eqno (1.2.4.4)]$ and the symmetrized and antisymmetrized tensor products are $[\eqalignno{(D_{m}\otimes D_{m})_{s} &= {\bigoplus\limits_{j=0}^{m}}D_{2j}, &(1.2.4.5)\cr (D_{m}\otimes D_{m})_{a} &= {\bigoplus\limits_{j=1}^{m}}D_{2j-1}. & (1.2.4.6)}%fd1.2.4.6]$

If the components of the tensor $[T_{i_{1}\ldots i_{r}}]$ are taken with respect to an orthonormal basis, the tensor is called a Cartesian tensor. The orthogonal transformation R then is represented by an orthogonal matrix $[R_{ij}]$ . Cartesian tensors of higher rank than one are generally no longer irreducible for the group O(n). For example, the rank-two tensors in three dimensions have nine components $[T_{ij}]$ . Under SO(3), they transform according to the tensor product of two $[\ell =1]$ representations. Because $[D_{1} \otimes D_{1} = D_{0} \oplus D_{1} \oplus D_{2},]$ the space of rank 2 Cartesian tensors is the direct sum of three invariant subspaces. This corresponds to the fact that a general rank 2 tensor can be written as the sum of a diagonal tensor, an antisymmetric tensor and a symmetric tensor with trace zero. These three tensors are irreducible tensors, in this case also called spherical tensors, i.e. irreducible tensors for the orthogonal group.

An irreducible tensor with respect to the group [O(3)] transforms, in general, according to some reducible representation of a point group $[K\in O(3)]$ . If the group K is a symmetry of the physical system, the tensor should be invariant under K, i.e. it should transform according to the identity representation of K.

Consider, for example, a symmetric second-rank tensor under [O(3)] . This means that it belongs to the space that transforms according to the representation $[D_{0} \oplus D_{2}]$ [see (1.2.4.6)]. If the symmetry group of the system is the point group [K = 432] , the representation $[D_{0}(K) \oplus D_{2}(K)]$ has character $[\matrix{R\colon\phantom{=C_{3}}\quad\varepsilon\phantom{=C_{3}}\quad\beta =C_{3}\quad\alpha^{2}=C_{4z}^{2}\quad \alpha =C_{4z}\quad\alpha \beta =C_{2}\cr \hrulefill\cr\chi(R)\colon\phantom{C_{3}}\quad6\phantom{1=C_{3}}\quad 0\phantom{1=C_{3}}\quad 2\phantom{1=C_{3}}\quad 0\phantom{1=C_{3}}\quad 2\phantom{1=C_{3}}}]$ and is equivalent to the direct sum $[\Gamma_{1} \oplus \Gamma_{3} \oplus \Gamma_{5}.]$ The multiplicity of $[\Gamma_{1}]$ is one. Therefore, the space of tensors invariant under K is one-dimensional. Consequently, there is only one parameter left to describe such a symmetric second-rank tensor invariant under the cubic group [K=432] . Noninvariant symmetric second-rank tensors are sums of tensors which transform according to the $[\Gamma_3]$ and $[\Gamma_5]$ representations. Here we are especially interested in invariant tensors.

References

International Tables for Crystallography (2006). Vol. D. ch. 1.2, pp. 51-52