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

International Tables for Crystallography (2006). Vol. D, ch. 1.1, p. 32

Section Other forms of the piezoelectric constants

A. Authiera*

aInstitut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France
Correspondence e-mail: Other forms of the piezoelectric constants

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We use here another Gibbs function, the electric Gibbs function, [{\cal G}_2], defined by[{\cal G}_{2} = {\cal U} - E_{n}D_{n} - \Theta \sigma.]

Differentiation of [\cal G] gives[d{\cal G}_2=-D_n\,{\rm d}E_n+T_{ij}\,{\rm d}S_{ij}-\sigma\,{\rm d}\Theta.]It follows that[T_{ij}={{\partial{\cal G}_2}\over{\partial S_{ij}}}\semi\quad D_n=-{{\partial{\cal G}_2}\over{\partial E_n}}\semi\quad\sigma=-{{\partial{\cal G}_2}\over{\partial\Theta}}] and a set of relations analogous to ([link]:[\left.\matrix{T_{ij} = (c_{ijkl})^{E,\Theta} S_{kl}\hfill &-\,(e_{ijn})^{S,\Theta} E_{n}\hfill &-\, (\lambda_{ij})^{E,S}\,\,\delta\Theta \hfill\cr D_{n} = (e_{nij})^{E,\Theta} S_{ij}\hfill &+ \,(\varepsilon_{nm})^{S,\Theta} E_{m}\hfill &+\, (p_{n})^{S} \,\,\delta\Theta\hfill \cr \delta\sigma = (\lambda_{ij})^{E} S_{ij}\hfill &+\, (p_{n})^{S} E_{n}\hfill &+ \,\rho C^{E,S}\,\, \delta\Theta/\Theta, \hfill\cr}\right\}\eqno(]where the components [\left(c_{ijkl}\right)^{E,\Theta}] are the isothermal elastic stiffnesses at constant field and constant temperature, [e_{ijn}=-{\partial T_{ij}\over\partial E_n}=-{\partial^2{\cal G}_2\over\partial E_n\partial S_{ij}}=-{\partial^2{\cal G}_2\over\partial S_{ij}\partial E_n}={\partial D_n\over \partial S_{ij}}=e_{nij}]are the piezoelectric stress coefficients at constant strain and constant temperature,[\lambda_{ij}=-{\partial T_{ij}\over \partial \Theta}=-{\partial^2{\cal G}_2\over \partial\Theta\partial S_{ij}}=-{\partial^2{\cal G}_2\over\partial S_{ij}\partial\Theta}={\partial\delta\sigma\over\partial S_{ij}}] are the temperature-stress constants and[p_n={\partial D_n\over\partial\Theta}=-{\partial^2{\cal G}_2\over\partial\Theta\partial E_n}=-{\partial^2{\cal G}_2\over \partial E_n\partial\Theta}={\partial\delta\sigma\over\partial D_n}]are the components of the pyroelectric effect at constant strain.

The relations between these coefficients and the usual coefficients [d_{kln}] are easily obtained:

  • (i) At constant temperature and strain: if one puts [\delta\Theta = 0] and [S_{kl} = 0] in the first equation of ([link] and ([link], one obtains, respectively, [\eqalignno{0 &= s_{klij} T_{ij} + d_{kln} E_{n}\cr T_{ij} &= - e_{ijn} E_{n},\cr}]from which it follows that [d_{kln} = s_{klij}e_{ijn}]at constant temperature and strain.

  • (ii) At constant temperature and stress: if one puts [\delta\Theta = 0] and [T_{ij} = 0], one obtains in a similar way [\eqalignno{S_{kl} &= d_{kln} E_{n}\cr 0 &= c_{ijkl} S_{kl} - e_{ijn} E_{n},\cr}]from which it follows that [e_{ijn} = c_{ijkl}d_{kln}]at constant temperature and stress.

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