Hyperbolique law of a ferroïc single crystal at low applied constraint amplitudes

The response of a single crystal to the application of low constraint amplitudes (electric field E or mechanic stress σ) is generally depicted by the empirical Rayleigh law. According to the model describing the motion of the domain walls and presented else where in this congress, we establish, in the limit of the Rayleigh domain, the variation of the dielectric constant ε and of the piezoelectric coefficient d with the amplitude of the applied constraint. For low amplitudes, we have to consider two different mechanisms: the vibration of the wall in a pinning centre and its jumps between sites. For the vibration mechanism, we show that ε and d are nondependent of the amplitude of E or σ, while these quantities vary linearly with the amplitude for the jump processes. So, the jumps contribution becomes important only if the amplitude is higher than a threshold value. This one is a function of the frequency and of the temperature. This results in a hyperbolic law which is a generalization of the Rayleigh law. A parallel between the electrical and mechanical responses ε and d is done and the corresponding hyperbolic laws are plotted with normalized variables. In this representation, the curves ε(E) and d(E) are identical because each response is due to the same mechanism, the motion of the domain walls. The link between the two quantities is the electrostrictive coefficient. Data from the literature confirms this theoretical result. Finally, studies at higher amplitudes shows that the response deviates from the hyperbolic law. The value of the amplitude for which a deviation from the hyperbolic law is significant strongly depends on the difference between the frequency used and the frequency of the relaxation of domain walls. This is in agreement with data of the literature.