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

International Tables for Crystallography (2006). Vol. D, ch. 1.1, p. 24

Section Introduction

A. Authiera*

aInstitut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France
Correspondence e-mail: Introduction

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Many tensors representing physical properties or physical quantities appear in relations involving symmetric tensors. Consider, for instance, the strain [S_{ij}] resulting from the application of an electric field E (the piezoelectric effect): [S_{ij} = d_{ijk}E_{k} + Q_{ijkl}E_{k}E_{l}, \eqno(]where the first-order terms [d_{ijk}] represent the components of the third-rank converse piezoelectric tensor and the second-order terms [Q_{ijkl}] represent the components of the fourth-rank electrostriction tensor. In a similar way, the direct piezoelectric effect corresponds to the appearance of an electric polarization P when a stress [T_{jk}] is applied to a crystal: [P_{i} = d_{ijk}T_{jk}. \eqno(]

Owing to the symmetry properties of the strain and stress tensors (see Sections 1.3.1[link] and 1.3.2[link] ) and of the tensor product [E_{k}E_{l}], there occurs a further reduction of the number of independent components of the tensors which are engaged in a contracted product with them, as is shown in Section[link] for third-rank tensors and in Section[link] for fourth-rank tensors.

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