Electric polarization limiting behavior

Ferroelectric crystals exhibit spontaneous electric polarization and hysteresis effects in the relation between polarization and electric field, as shown in Figure 1. This behavior is usually observed in a limited temperature range, ie, usually below a transition temperature (10). 

We showed in Section 2.3 that the real and imaginary parts of the electric susceptibility are connected by the dispersion relations (2.36) and (2.37). This followed as a consequence of the linear causal relation between the electric field and polarization together with the vanishing of x(<°) in the limit of infinite frequency to. We also stated that, in general, similar relations are expected to hold for any frequency-dependent function that connects an output with an input in a linear causal way. An example is the amplitude scattering matrix (4.75) the scattered field is linearly related to the incident field. Moreover, this relation must be causal the scattered field cannot precede in time the incident field that excited it. Therefore, the matrix elements should satisfy dispersion relations. In particular, this is true for the forward direction 6 = 0°. But 5(0°, to) does not have the required asymptotic behavior it is clear from the diffraction theory approximation (4.73) that for sufficiently large frequencies, 5(0°, to) is proportional to to2. Nevertheless, only minor fiddling with S makes it behave properly the function... 

The usefulness of electrical response measurements of solutions is not limited to effects linear in applied field. Transient birefringence induced by polarizing electric fields (the transient or dynamic Kerr effect) has given valuable information about biopolymers in solution the effect must by symmetry be an even function of E(t), beginning with terms in E (t). In both cases, a response theory treatment of transient behavior meets with difficulties not encountered in linear problems, but recent progress in deriving correlation function expressions for such effects is described in III. 

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