# Dielectric Spectroscopy Studies and Modelling of Piezoelectric Properties of Multiferroic Ceramics
## 3 Results and Discussion
### Calculation of Piezoelectric Parameters of 67 Bfto Ceramics  
The entry parameters used to calculate the piezoelectric parameters of BFBT ceramics, namely resonant frequencies \(\
u_{\rm r}\), anti-resonant frequencies \(\
u_{\rm a}\), impedance, and free capacitance \(C^{\rm T}\) on 1 kHz were measured by using the aforementioned impedance analyser. As shown in Figure 8, a thorough analysis of the experimental data (i.e., measured impedance spectra) was also performed. The procedure for the calculation of single coefficients is described in detail in classical textbooks (e.g., see [21, 24, 25]), scientific papers (e.g., see [22, 23]), or standards (e.g., see [30]). The results of the calculations are given in Figures 12 and 13. Figure 12: **(a)** Dependence of the planar coupling factor \(k_{\rm p}\) for the vibration along the radial direction in a circle-shaped disk of 67 BFTO ceramics on temperature; **(b)** dependence of mechanical quality factor on temperature; calculations were performed based on experimental impedance data measured under the influence of radial magnetic field \(H_{\rm ac}\) = 0.05 A/m (\(U\) = 0.1 V\({}_{\rm rms}\)) and \(H_{\rm ac}\) = 0.49 A/m (\(U\) = 2.0 V\({}_{\rm rms}\)). Figure 13: **(a)** Dependence of piezoelectric charge coefficient \(d_{31}\) on temperature; **(b)** dependence of the relative change in \(d_{31}\) on temperature; the inset in Figure 13b shows the formula for calculation of the relative change in piezoelectric modulus \(d_{31}\). Figure 12a shows dependence of the planar coupling factor \(k_{\rm p}\) for the vibration along the radial direction in a circle-shaped disk of 67 BFTO ceramics on temperature. The results of the calculations of the mechanical quality factor \(Q_{\rm m}\) are given in Figure 12b. One can see from Figure 12a that the coupling factor for both magnetic circumferential fields has a value of about \(k_{\rm p}=31\%\). Local extremes visible on the plots at \(-10\)\({}^{\circ}\)C and \(-30\)\({}^{\circ}\)C differ from the average value by about 1%. Therefore, it is reasonable to use linear regression fit for data analysis. The difference between the linear fit plots is rather small but can easily be discerned. Moreover, the linear regression fit for a higher magnetic field (\(H_{\rm ac}=0.49\) A/m) shows better stability with temperature within the measured temperature range. One can see from Figure 12b that the mechanical quality factor \(Q_{\rm m}\) exhibits non-monotonic behaviour with increasing temperature for both of the used values of magnetic circumferential field intensity. The local extremes are clearly visible. Within the temperature range \(\Delta T=(-30\)\(\rightarrow\)\(20)\({}^{\circ}\)C, the courses of the \(Q_{\rm m}\) curves are almost identical. The influence of magnetic field intensity on \(Q_{\rm m}\) becomes noticeable at about room temperature (20 \({}^{\circ}\)C). The difference between the linear fit plots can easily be discerned in Figure 12b. One can see that the linear regression fit for higher magnetic field intensity (\(H_{\rm ac}=0.49\) A/m; red line in Figure 12b) shows a negative slope with temperature within the measured temperature range, whereas the linear regression fit for \(Q_{\rm m}\) behaviour at a smaller magnetic circumferential field exhibits a positive one (\(H_{\rm ac}=0.05\) A/m; blue squares; blue line in Figure 12b). Figure 13a shows the dependence of piezoelectric charge coefficient \(d_{31}\) on the radial vibration mode of a thin disk (excited through the piezoelectric effect across the thickness of the disk).