Frequency dependence high-temperature resonance calculations

In order to compare calculated and experimentally observed phase portraits it is necessary to know very exactly all the coefficients of the describing nonlinear differential Equation 14.3. Therefore, different methods of determination of the nonlinear coefficient in the Duffing equation have been compared. In the paraelectric phase the value of the nonlinear dielectric coefficient B is determined by measuring the shift of the resonance frequency in dependence on the amplitude of the excitation ( [1], [5]). In the ferroelectric phase three different methods are used in order to determine B. Firstly, the coefficient B is calculated in the framework of the Landau theory from the coefficient of the high temperature phase (e.g. [4]). This means B = const, and B has the same values above and below the phase transition. Secondly, the shift of the resonance frequency of the resonator in the ferroelectric phase as a function of the driving field is used in order to determine the coefficient B. The amplitude of the exciting field is smaller than the coercive field and does not produce polarization reversal during the measurements of the shift of the resonance frequency. In the third method the coefficient B was determined by the values of the spontaneous polarization... 

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