International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. D, ch. 1.1, pp. 31-32
Section 1.1.5. Thermodynamic functions and physical property tensors^{a}Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France |
[The reader may also consult Mason (1966), Nye (1985) or Sirotin & Shaskol'skaya (1982).]
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 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 (see Section 1.3.2.8 ). Let us consider the Gibbs free-energy function defined by Differentiation of gives The extensive parameters are therefore partial derivatives of the free energy: Each of these quantities may be expanded by performing a further differentiation in terms of the intensive parameters, , and Θ. We have, to the first order, To a first approximation, the partial derivatives may be considered as constants, and the above relations may be integrated: This set of equations is the equivalent of relation (1.1.1.6) of Section 1.1.1.3, which gives the coefficients of the matrix of physical properties. These coefficients are:
In a similar way,
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 (1.1.5.1) becomes under these circumstancesRemark. Equations (1.1.5.1) 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:
We use here another Gibbs function, the electric Gibbs function, , defined by
Differentiation of givesIt follows that and a set of relations analogous to (1.1.5.1):where the components are the isothermal elastic stiffnesses at constant field and constant temperature, are the piezoelectric stress coefficients at constant strain and constant temperature, are the temperature-stress constants andare the components of the pyroelectric effect at constant strain.
The relations between these coefficients and the usual coefficients are easily obtained:
By combining relations (1.1.5.1) and (1.1.5.2), it is possible to obtain relations between the pyroelectric coefficients at constant stress, , and the pyroelectric coefficients at constant strain, , also called real pyroelectric coefficients, . Let us put and in the first equation of (1.1.5.1). For a given variation of temperature, , the observed strain is From the second equations of (1.1.5.1) and (1.1.5.2), it follows that Substituting the expression and eliminating , it follows that
This relation shows that part of the pyroelectric effect is actually due to the piezoelectric effect.
Piezoelectric resonators usually operate at a high frequency where there are no heat exchanges, and therefore in an adiabatic regime . From the third equation of (1.1.5.1), we obtain a relation between the temperature variation, the applied stress and the electric field:
If we substitute this relation in the two other relations of (1.1.5.1), we obtain two equivalent relations, but in the adiabatic regime: By comparing these expressions with (1.1.5.1), we obtain the following relations between the adiabatic and the isothermal coefficients:
References
Mason, W. P. (1966). Crystal physics of interaction processes. London: Academic Press.Nye, J. F. (1985). Physical properties of cystals, revised ed. Oxford University Press.
Sirotin, Y. I. & Shaskol'skaya, M. P. (1982). Fundamentals of crystal physics. Moscow: Mir.