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

International Tables for Crystallography (2006). Vol. D, ch. 1.1, p. 8

Section Tensor nature of physical quantities

A. Authiera*

aInstitut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France
Correspondence e-mail: Tensor nature of physical quantities

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Let us first consider the dielectric constant. In the introduction, we remarked that for an isotropic medium [{\bf D} = \varepsilon {\bf E}.]

If the medium is anisotropic, we have, for one of the components, [D^{1} = \varepsilon^{1}_{1} E^{1} + \varepsilon^{1}_{2} E^{2}+ \varepsilon^{1}_{3} E^{3}.]This relation and the equivalent ones for the other components can also be written [D^{i} = \varepsilon^i_{j}E\hskip1pt^{j} \eqno(]using the Einstein convention.

The scalar product of D by an arbitrary vector x is [D^{i}x_{i} = \varepsilon^{i}_{j} E\hskip1pt^{j} x_{i}.]

The right-hand member of this relation is a bilinear form that is invariant under a change of basis. The set of nine quantities [\varepsilon^{i}_{j}] constitutes therefore the set of components of a tensor of rank 2. Expression ([link] is the contracted product of [\varepsilon^{i}_{j}] by [E\hskip1pt^{j}].

A similar demonstration may be used to show the tensor nature of the various physical properties described in Section 1.1.1[link], whatever the rank of the tensor. Let us for instance consider the piezoelectric effect (see Section[link]). The components of the electric polarization, [P^{i}], which appear in a medium submitted to a stress represented by the second-rank tensor [T_{jk}] are [P\hskip1pt^{i} = d\hskip1pt^{ijk}T_{jk},]where the tensor nature of [T_{jk}] will be shown in Section 1.3.2[link] . If we take the contracted product of both sides of this equation by any vector of covariant components [x_{i}], we obtain a linear form on the left-hand side, and a trilinear form on the right-hand side, which shows that the coefficients [d\hskip1pt^{ijk}] are the components of a third-rank tensor. Let us now consider the piezo-optic (or photoelastic) effect (see Sections[link] and 1.6.7[link] ). The components of the variation [\Delta \eta^{ij}] of the dielectric impermeability due to an applied stress are [\Delta \eta^{ij} = \pi^{ijkl}T_{jl}.]

In a similar fashion, consider the contracted product of both sides of this relation by two vectors of covariant components [x_{i}] and [y_{j}], respectively. We obtain a bilinear form on the left-hand side, and a quadrilinear form on the right-hand side, showing that the coefficients [\pi^{ijkl}] are the components of a fourth-rank tensor.

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