International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. D, ch. 1.1, p. 8
Section 1.1.3.4. Tensor nature of physical quantities^{a}Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France |
Let us first consider the dielectric constant. In the introduction, we remarked that for an isotropic medium
If the medium is anisotropic, we have, for one of the components, This relation and the equivalent ones for the other components can also be written using the Einstein convention.
The scalar product of D by an arbitrary vector x is
The right-hand member of this relation is a bilinear form that is invariant under a change of basis. The set of nine quantities constitutes therefore the set of components of a tensor of rank 2. Expression (1.1.3.3) is the contracted product of by .
A similar demonstration may be used to show the tensor nature of the various physical properties described in Section 1.1.1, whatever the rank of the tensor. Let us for instance consider the piezoelectric effect (see Section 1.1.4.4.3). The components of the electric polarization, , which appear in a medium submitted to a stress represented by the second-rank tensor are where the tensor nature of will be shown in Section 1.3.2 . If we take the contracted product of both sides of this equation by any vector of covariant components , we obtain a linear form on the left-hand side, and a trilinear form on the right-hand side, which shows that the coefficients are the components of a third-rank tensor. Let us now consider the piezo-optic (or photoelastic) effect (see Sections 1.1.4.10.5 and 1.6.7 ). The components of the variation of the dielectric impermeability due to an applied stress are
In a similar fashion, consider the contracted product of both sides of this relation by two vectors of covariant components and , respectively. We obtain a bilinear form on the left-hand side, and a quadrilinear form on the right-hand side, showing that the coefficients are the components of a fourth-rank tensor.