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

International Tables for Crystallography (2013). Vol. D, ch. 1.1, p. 14

Section Tensors of higher rank

A. Authiera*

aInstitut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France
Correspondence e-mail: Tensors of higher rank

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If the rank of the tensor is higher than 2, the tensor may be antisymmetric with respect to the indices of one or several couples of indices.

  • (i) Tensors of rank 3 antisymmetric with respect to every couple of indices. A trilinear form [T({\bf x},{\bf y},{\bf z}) = t_{ijk}x^{i}y\hskip1pt^{j}z^{k}] is said to be antisymmetric if it satifies the relations [\left.\matrix{T({\bf x},{\bf y},{\bf z}) & = - T({\bf y},{\bf x},{\bf z})\hfill\cr & = - T({\bf x},{\bf z},{\bf y})\hfill\cr & = - T({\bf z},{\bf y},{\bf x}).\cr}\right\}]

    Tensor [t_{ijk}] has 27 components. It is found that all of them are equal to zero, except [t_{123}= t_{231}= t_{312}= - t_{213}= - t_{132}= - t_{321}.]

    The three-times contracted product with the permutations tensor (Section[link]), [(1/6)\varepsilon_{ijk}t_{ijk}], is a pseudoscalar or axial scalar. It is not a usual scalar: the sign of this product changes when one changes the hand of the reference axes, change of basis represented by the matrix[\pmatrix {\bar 1 & 0 & 0\cr 0 & \bar 1 & 0\cr 0 & 0 & \bar 1}.]

    Form [T({\bf x},{\bf y},{\bf z})] can also be written [T({\bf x},{\bf y},{\bf z}) = Pt_{123},]where[P=\varepsilon_{ijk}x^iy^jz^k=\left|\matrix{x^1&x^2&x^3\cr y^1&y^2&y^3\cr z^1&z^2&z^3\cr}\right|]is the triple scalar product of the three vectors x, y, z:[P=({\bf x}, {\bf y}, {\bf z})=({\bf x}\wedge{\bf y}\cdot{\bf z}).]It is also a pseudoscalar. The permutation tensor is not a real tensor of rank 3: if the hand of the axes is changed, the sign of P also changes; P is therefore not a trilinear form.

    Another example of a pseudoscalar is given by the rotatory power of an optically active medium, which is expressed through the relation (see Section[link] ) [\theta = \rho d,]where θ is the rotation angle of the light wave, d the distance traversed in the material and ρ is a pseudoscalar: if one takes the mirror image of this medium, the sign of the rotation of the light wave also changes.

  • (ii) Tensor of rank 3 antisymmetric with respect to one couple of indices. Let us consider a trilinear form such that [T({\bf x},{\bf y},{\bf z}) = - T({\bf y},{\bf x},{\bf z}).]Its components satisfy the relation [t\hskip1pt^{iil} = 0; \quad t\hskip1pt^{ijl} = -t\hskip1pt^{jil}.]

    The twice contracted product [t\hskip1pt_{k}^{l} = \textstyle{1 \over 2}\varepsilon_{ijk}t\hskip1pt^{ijl}]is an axial tensor of rank 2 whose components are the independent components of the antisymmetric tensor of rank 3, [t\hskip1pt^{ijl}].


  • (1) Hall constant. The Hall effect is observed in semiconductors. If one takes a semiconductor crystal and applies a magnetic induction B and at the same time imposes a current density j at right angles to it, one observes an electric field E at right angles to the other two fields (see Section[link] ). The expression for the field can be written [E_{i} = R_{H \ ikl}\hskip1pt j_{k}B_{l},]where [R_{H \ ikl}] is the Hall constant, which is a tensor of rank 3. However, because the direction of the current density is imposed by the physical law (the set of vectors B, j, E constitutes a right-handed frame), one has [R_{H \ ikl} = -R_{H \ kil},]which shows that [R_{H \ ikl}] is an antisymmetric (axial) tensor of rank 3. As can be seen from its physical properties, only the components such that [i \neq k \neq l] are different from zero. These are [R_{H \ 123} = - R_{H \ 213}; \quad R_{H \ 132} = - R_{H \ 312}; \quad R_{H \ 312}; \quad R_{H \ 321}.]

  • (2) Optical rotation. The gyration tensor used to describe the property of optical rotation presented by gyrotropic materials (see Section[link] ) is an axial tensor of rank 2, which is actually an antisymmetric tensor of rank 3.

  • (3) Acoustic activity. The acoustic gyrotropic tensor describes the rotation of the polarization plane of a transverse acoustic wave propagating along the acoustic axis (see for instance Kumaraswamy & Krishnamurthy, 1980[link]). The elastic constants may be expanded as [c_{ijkl}(\omega, {\bf k}) = c_{ijkl}(\omega) + \hbox{i} d_{ijklm}(\omega)k_{m} + \ldots,]where [d_{ijklm}] is a fifth-rank tensor. Time-reversal invariance requires that [d_{ijklm} =- d_{jiklm}], which shows that it is an antisymmetric (axial) tensor.


Kumaraswamy, K. & Krishnamurthy, N. (1980). The acoustic gyrotropic tensor in crystals. Acta Cryst. A36, 760–762.

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