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

International Tables for Crystallography (2006). Vol. D, ch. 1.1, p. 31

Section Isothermal study

A. Authiera*

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

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The energy of a system is the sum of all the forms of energy: thermal, mechanical, electrical etc. Let us consider a system whose only variables are these three. For a small variation of the associated extensive parameters, the variation of the internal energy is [\hbox{d}{\cal U} = E_{n}\ \hbox{d}D_{n} + T_{kl}\ \hbox{d}S_{kl} + \Theta \ \hbox{d}\sigma,]where Θ is the temperature and σ is the entropy; there is summation over all dummy indices; an orthonormal frame is assumed and variance is not apparent. The mechanical energy of deformation is given by [T_{kl}\,\,{\rm d}S_{kl}] (see Section[link] ). Let us consider the Gibbs free-energy function [{\cal G}] defined by [{\cal G} = {\cal U} - E_{n}D_{n} - T_{kl}S_{kl} - \Theta \sigma.]Differentiation of [{\cal G}] gives [\hbox{d}{\cal G} = - D_{n}\ \hbox{d}E_{n} - S_{kl}\ \hbox{d}T_{kl}- \sigma \ \hbox{d}\Theta.]The extensive parameters are therefore partial derivatives of the free energy: [S_{kl} = - {\partial {\cal G} \over \partial T_{kl}};\ D_{n} = - {\partial {\cal G} \over \partial E_{n}};\ \sigma = - {\partial {\cal G} \over \partial \Theta}.]Each of these quantities may be expanded by performing a further differentiation in terms of the intensive parameters, [T_{kl}], [E_{n}] and Θ. We have, to the first order, [\eqalignno{\hbox{d}S_{kl} &= \left[{\partial S_{kl} \over \partial T_{ij}}\right]_{E,\Theta} \hbox{d}T_{ij} + \left[{\partial S_{kl} \over \partial E_{n}}\right]_{T,\Theta}\hbox{d}E_{n} + \left[{\partial S_{kl} \over \partial \Theta}\right]_{E,T} \delta\Theta\cr \hbox{d}D_{n} &= \left[{\partial D_{n} \over \partial T_{kl}}\right]_{E,\Theta}\hbox{d}T_{kl} + \left[{\partial D_{n} \over \partial E_{m}}\right]_{T,\Theta}\hbox{d}E_{m} + \left[{\partial D_{n} \over \partial \Theta}\right]_{E,T}\delta\Theta\cr \hbox{d}\sigma &= \left[{\partial \sigma \over \partial T_{ij}}\right]_{E,\Theta}\hbox{d}T_{ij} + \left[{\partial \sigma \over \partial E_{m}}\right]_{T,\Theta}\hbox{d}E_{m} + \left[{\partial \sigma \over \partial \Theta}\right]_{E,T}\delta\Theta.\cr}]To a first approximation, the partial derivatives may be considered as constants, and the above relations may be integrated: [\left. \matrix{S_{kl} &= \left(s_{klij}\right)^{E,\Theta} T_{ij} &+ \,\left(d_{kl n}\right)^{T,\Theta} E_{n} &+\, \left(\alpha_{kl}\right)^{E,T}\delta\Theta\hfill\cr D_{n} &= \left(d_{nkl}\right)^{E,\Theta} T_{kl} \hfill &+ \,\left(\varepsilon_{nm}\right)^{T,\Theta} E_{m} \hfill &+ \,\left(p_{n}\right)^{E,T}\delta\Theta \hfill \cr \delta \sigma &= \left(\alpha_{ij}\right)^{E} T_{ij} \hfill &+ \,\left(p_{m}\right)^{T} E_{m} \hfill &+ \,(\rho C^{E,T} / \Theta ) \delta\Theta. \hfill \cr}\right\} \eqno(]This set of equations is the equivalent of relation ([link] of Section[link], which gives the coefficients of the matrix of physical properties. These coefficients are:

  • (i) For the principal properties: [\left(s_{klij}\right)^{E,\Theta}]: elastic compliances at constant temperature and field; [\left(\varepsilon_{nm}\right)^{T,\Theta}]: dielectric constant at constant temperatures and stress; [\rho C^{T,E}]: heat capacity per unit volume at constant stress and field ([\rho] is the specific mass and [C^{T,E}] is the specific heat at constant stress and field).

  • (ii) For the other properties: [\left(d_{kln}\right)^{T,\Theta}] and [\left(d_{nkl}\right)^{E,\Theta}] are the components of the piezoelectric effect and of the converse effect. They are represented by [3\times 9] and [9\times 3] matrices, respectively. One may notice that [d_{kln} = {\partial S_{kl} \over \partial E_{n}} = - {\partial^{2} {\cal G} \over \partial E_{n} \partial T_{kl}} = - {\partial^{2} {\cal G} \over \partial T_{kl}\partial E_{n}} = {\partial D_{n} \over \partial T_{kl}}=d_{nkl},]which shows again that the components of two properties that are symmetric with respect to the leading diagonal of the matrix of physical properties are equal (Section[link]) and that the corresponding matrices are transpose to one another.

In a similar way,

  • (a) the matrices [\left(\alpha_{kl}\right)^{E,T}] of the thermal expansion and [(\alpha_{ij})^E] of the piezocalorific effect are transpose to one another;

  • (b) the components [(p_{n})^{T}] of the pyroelectric and of the electrocalorific effects are equal.

Remark. The piezoelectric effect, namely the existence of an electric polarization P under an applied stress, is always measured at zero applied electric field and at constant temperature. The second equation of ([link] becomes under these circumstances[P_{n} = D_{n} = (d_{nkl})^{\Theta} T_{kl}.]Remark. Equations ([link] are, as has been said, first-order approximations because we have assumed the partial derivatives to be constants. Actually, this approximation is not correct, and in many cases it is necessary to take into account the higher-order terms as, for instance, in:

  • (a) nonlinear elasticity (see Sections 1.3.6[link] and 1.3.7[link] );

  • (b) electrostriction;

  • (c) nonlinear optics (see Chapter 1.7[link] );

  • (d) electro-optic and piezo-optic effects (see Sections 1.6.6[link] and 1.6.7[link] ).

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