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

International Tables for Crystallography (2006). Vol. D, ch. 2.4, pp. 329-330

Section 2.4.2.2. Piezoelectric media

R. Vachera* and E. Courtensa

aLaboratoire des Verres, Université Montpellier 2, Case 069, Place Eugène Bataillon, 34095 Montpellier CEDEX, France
Correspondence e-mail:  rene.vacher@ldv.univ-montp2.fr

2.4.2.2. Piezoelectric media

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In piezoelectric crystals, a stress component is also produced by the internal electric field E, so that the constitutive equation (2.4.2.2)[link] has an additional term (see Section 1.1.5.2[link] ), [T _{ij} = c_{ijk\ell }S_{k\ell } - e_{mij}E_m, \eqno (2.4.2.8)]where e is the piezoelectric tensor at constant strain.

The electrical displacement vector D, related to E by the dielectric tensor [\boldvarepsilon], also contains a contribution from the strain, [D_m = \varepsilon _{mn}E_n + e_{mk\ell }S_{k\ell }, \eqno (2.4.2.9)]where [\boldvarepsilon] is at the frequency of the elastic wave.

In the absence of free charges, [{\rm div}\,{\bf D}= 0], and (2.4.2.9)[link] provides a relation between E and S, [\varepsilon _{mn}Q_n E_m + e_{mk\ell }Q_mS_{k\ell } = 0. \eqno (2.4.2.10)]For long waves, it can be shown that E and Q are parallel. (2.4.2.10)[link] can then be solved for E, and this value is replaced in (2.4.2.8)[link] to give [T _{ij} = \left [{c_{ijk\ell } + {{e_{mij}e_{nk\ell }\hat Q_m \hat Q_n }\over {\varepsilon _{gh}\hat Q_g \hat Q_h }}}\right]S_{k\ell } \equiv c_{ijk\ell }^{(e)}S_{k\ell }. \eqno (2.4.2.11)]Comparing (2.4.2.11)[link] and (2.4.2.2)[link], one sees that the effective elastic tensor [{\bf c}^{(e)}] now depends on the propagation direction [\hat{\bf Q}]. Otherwise, all considerations of the previous section, starting from (2.4.2.6)[link], remain, with c simply replaced by [{\bf c}^{(e)}].








































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