## 3 Results
### Optimized Parameters  
After the needed changes for the best fit of the initial conditions, a minimization approach for the best fit of the model to the experimental data was carried out. Table 4 shows the set of parameters obtained after minimization of the numerical calculation of the complex impedance spectra of the thin disk as a function of the temperature. The experimental impedance curve was fitted for all temperatures and the parameters are optimized to minimize the quadratic error in the modulus of the electric impedance. The complex part of the model was adjusted, minimizing the error in G and R curves, see Figure 1C,D. Figure 5 shows the numerical calculation, optimized for 100 \({}^{\circ}\)C, here we can observe the differences in shape and magnitude of the spectrum against the data in Figure 1. Although the fitting is poorer than that obtained at room temperature, all resonances are well represented by the numerical model. Figures 6 and 7 show the evolution of the real and imaginary part, respectively, of the optimized parameters. are related to the polarization parallel to the electric field (three-axis). Thus, the observed changes in the parameters from the optimized numerical calculation of the spectra are compatible with a phase transition with strong change in the polar structure of the material from a ferro-piezoelectric phase to a highly polarizable and non-piezoelectric one. Both the real and the imaginary parts of all parameters related with the TH mode are high sensitivity ones and their thermal evolution (red lines in Figures 6 and 7) shows smooth changes. The real part of the elastic \(c_{33}^{E}\) follows the evolution of the resonance frequency (Figure 2), having a clear minimum before 100 \({}^{\circ}\)C. At the same temperature, the net decrease of both the real and imaginary parts of \(e_{33}\) is observed, as piezoelectric activity vanishes at the ferroelectric to relaxor phase transition [1]. Figure 5: Result of the minimization of the numerical calculation (black dots) and experimental spectrum (continuous blue line) at 100 \({}^{\circ}\)C. (**A**) Modulus of the complex electric impedance; (**B**) Phase of the impedance. Figure 6: Thermal evolution of the real part of the elastic, dielectric, and piezoelectric parameters. Parameters related with the TH mode (high sensitivity) are represented in red. Those related with the radial modes are represented in black and those related with the complex resonances C\({}_{1}\) and C\({}_{2}\) in blue. Due to the local maxima at the phase transitions [1], near 100 \({}^{\circ}\)C, a faster increase of real and imaginary parts of \(\varepsilon_{33}^{S}\) is observed as the temperature increases, corresponding to the dielectric anomaly at the ferroelectric to relaxor phase transition. Also, the increase in the mechanical losses (imaginary part of the elastic \(c_{33}^{E}\)) is observed. As for the elastic parameters related with the radial mode of resonance, both the high sensitivity (\(c_{11}^{E}\), \(c_{13}^{E}\)) and medium sensitivity ones (\(c_{12}^{E}\)), show small but fluctuating changes with the temperature, both in their real and imaginary parts. This is connected with the poorer determination of the characteristic of the sharper resonances. However, even being a low sensitivity parameter, a decrease of the real part of \(e_{13}\) seems to be clear from 80 \({}^{\circ}\)C, in agreement with the evolution of both the resonance frequencies and impedance modulus of the radial modes (Figure 2). Parameter values for 16 \({}^{\circ}\)C were used in the FEM modeling of the thin disk to determine the deformation at resonance (see Supplementary Materials Figures S4-S7). This confirms the purely dilatational character of the fundamental radial (R\({}_{1}\)) and thickness (TH) modes. It also shows that the deformation at C\({}_{1}\) and C\({}_{2}\) modes involves a great deal of shear movements.