## 3 Results  
### Impedance Measurements as a Function of the Temperature  
Figure 1A,B shows the measured complex modulus and phase, respectively, measured in the interval from 100 kHz to 1.9 MHz for a BNBT6 thin disk (\(m\) = 0.950 g, \(t\) = 1.91 mm and \(D\) = 10.60 mm) at 35 \({}^{\circ}\)C. Figure 1C,D show also an alternative representation of the complex impedance (Z) as the peaks of the real part (Resistance (R)) of Z and the real part (Conductance (G)) of the complex admittance (Y). These peaks are used for the determination of the frequencies of interest for the iterative analysis.  
As the fundamental radial resonance is sharp and since the number of points to measure in the spectrum was determined so as to make the problem manageable, the accuracy of determination of \(|\)Z \(|\) \(\upmu\)m for this mode was not sufficient to clearly show the tendency of its thermal evolution. It is here analyzed by the evolution of the overtones (R\({}_{2}\), R\({}_{3}\), R\({}_{4}\)).  
The spectrum comprises the fundamental radial extensional mode and three overtones of this, together with the fundamental thickness extensional mode and other complex modes [6].  
Figure 2A,B shows the evolution of the frequency of resonance and the modulus of the electric impedance for all temperatures. The resonances highlighted in Figure 2 can be observed for all temperatures in the selected range.  
From the analysis of Figure 2, we can note a different behavior in the frequencies for the radial modes compared with the thickness and the complex modes. Qualitatively, the frequencies of all modes decrease linearly up to a temperature close to 85 \({}^{\circ}\)C. Above this temperature, the frequency increases rapidly. To estimate the temperature of the minimum frequency, a quadratic interpolation is performed near the minimum of each response. Table 1 shows the results of this interpolation.  
Figure 1: Experimental response at 35 \({}^{\circ}\)C (continuous blue line) and the numerical simulation using the optimized parameters (black dots). (**A**) Modulus of the electric impedance; (**B**) phase of the electric impedance; (**C**) electric conductance G; (**D**) electric resistance R. Fundamental radial resonance and overtones are indicated by a Rn (\(n=1\), 2, 3, 4), thickness resonance by a TH and the coupled modes by \(Cm\) (\(m=1\), 2).  
Figure 2: Thermal evolution of resonance modes: (**A**) Resonance frequency; (**B**) Modulus of the complex impedance. Results are expressed as a percentage of the value at room temperature.  
As for the impedance modulus, the one for the radial modes decreases monotonically until 85 \({}^{\circ}\)C and at a faster rate above this temperature. The inflection point for this decrease is at 100 \({}^{\circ}\)C for TH mode and at 90 \({}^{\circ}\)C for C\({}_{2}\) mode. However, such an inflection point is not found for C\({}_{1}\) in the measured interval.  
Therefore, we found three different tendencies for the thermal evolution of resonances of the: (i) Rn (\(n\) = 2, 3, 4) modes; (ii) the TH and C\({}_{2}\) modes; and (iii) the C\({}_{1}\) mode, revealing the electromechanical anisotropy of the thermal depolarization process.