Other forms of the piezoelectric constants

Authier, A.

doi:10.1107/97809553602060000628

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

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

Section 1.1.5.2. Other forms of the piezoelectric constants

A. Authier^a ^*

^a Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France
Correspondence e-mail: aauthier@wanadoo.fr

1.1.5.2. 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 (1.1.5.1): $[\left.\matrix{T_{ij} = (c_{ijkl})^{E,\Theta} S_{kl}\hfill &-\,(e_{ijn})^{S,\Theta} E_{n}\hfill &-\, (\lambda_{ij})^{E,S}\,\,\delta\Theta \hfill\cr D_{n} = (e_{nij})^{E,\Theta} S_{ij}\hfill &+ \,(\varepsilon_{nm})^{S,\Theta} E_{m}\hfill &+\, (p_{n})^{S} \,\,\delta\Theta\hfill \cr \delta\sigma = (\lambda_{ij})^{E} S_{ij}\hfill &+\, (p_{n})^{S} E_{n}\hfill &+ \,\rho C^{E,S}\,\, \delta\Theta/\Theta, \hfill\cr}\right\}\eqno(1.1.5.2)]$ 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 (1.1.5.1) and (1.1.5.2), 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.

References

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