## 3 Results and Discussion
### Dielectric Analysis  
Dielectric permittivity (\(\varepsilon_{\mathrm{r}}\)) and losses (\(tan\)\(\delta\)) at (1 kHz), as a function of the Ca\({}^{2+}\) content and temperature, are shown in Figure 3. Two phase transitions corresponding to rhombohedral-tetragonal (\(R\)-\(T\)) and tetragonal-cubic (\(T\)-\(C\)) transitions were observed. The increase of the Ca\({}^{2+}\) substitution at the Ba\({}^{2+}\) sites resulted in a displacement, to lower temperatures, of the \(R\)-\(T\) phase transition temperature (\(T_{R\cdot T}\)), which was not observable for \(x\) = 0.18. The Ca\({}^{2+}\) addition increases the stability of the temperature range for the tetragonal phase [16]. Furthermore, the dielectric permittivity reached a maximum at \(x\) = 0.15, promoted by the MPB. This effect led to a higher number of grains that were favorably oriented towards an electric field, and hence, the optimum piezoelectric and ferroelectric performance could be obtained [1]. The associated physical properties of BCTZ ceramics after \(T_{C}\) are related to relaxation and conductivity processes [17; 18; 19; 20; 21]. For ferroelectric materials, it is important to perform a complete analysis of their electrical properties to separate different electroactive contributions (grain/bulk and grain boundary) and their frequency/temperature dependence. Figure 4b shows a dielectric anomaly (around 400 \({}^{\circ}\)C) for \(T>T_{C}\). Temperatures higher than \(T_{C}\) avoid the influence of ferroelectric domains on the ceramic resistance (conduction) because it is well known that defect dipoles and charged species are located at the domain walls [22], which aid the bulk (grain) conduction [11; 22]. Despite Ca\({}^{2+}\) content increases, the dielectric permittivity and losses remained constant for all compositions. Furthermore, Ca\({}^{2+}\) content shifted the permittivity maxima to lower dielectric values at higher temperatures (around 600 \({}^{\circ}\)C). More conveniently seen from the dielectric loss (\(tan\)\(\delta\)),  
Figure 3: (**a**) Dielectric permittivity and (**b**) losses measured at 1 kHz in BCTZ (\(x\) = 0.10, 0.15, 0.18) sintered ceramics (1400 \({}^{\circ}\)C; 2 h). Figure 2: (**a**) SEM micrographs for BCTZ (\(x\) = 0.10, 0.15, 0.18) sintered ceramics at 1400 \({}^{\circ}\)C for 2 h in air atmosphere; (**b**) EDS spectra of BCTZ and CaTiO\({}_{3}\) grains for \(x\) = 0.18 sintered ceramic. the dielectric anomaly was associated with a process of relaxation [17]: loss peaks shifted toward higher temperatures with an increase in the frequency (Figure 4c). For the analysis of BCTZ \(x\) = 0.15, wide relaxation peaks were overlapped in a large frequency-dependent range and shifted to higher frequency as the temperature increased. This phenomenon indicates that the dielectric loss peak is related to a thermally activated relaxation process [17]. Moreover, as the Ca\({}^{2+}\) content increased, the amplitude of the dielectric loss peak decreased, indicating a low activity in the same range of frequencies (Figure 5a-c). Figure 4: (**a**) Dielectric permittivity and (**b**) losses of BCTZ (\(x\) = 0.10, 0.15, 0.18) sintered ceramics (above \(T_{\mathrm{C}}\)) at 1 kHz; (**c**) Relaxation process obtained for BCTZ \(x\) = 0.15. Figure 5: _Cont._  
The thermally activated relaxation process follows the Arrhenius law:  
\[\tau=\tau_{0}\,\exp\,(Ea/K_{B}\cdot T), \tag{1}\]  
where \(\tau_{0}\) is the relaxation time, \(Ea\) is the activation energy of the relaxation process, \(T\) is the absolute temperature, and \(K_{B}\) is the Boltzmann constant. The resonant condition is \(\omega_{p}\tau_{p}=1\), where \(\omega\) is the angular frequency (\(\omega=2\pi f\)) and the subscript \(p\) represents the values at the peak maximum position. Figure 6 shows the Arrhenius plots of the dielectric loss peak and the linear fitting for the studied BCTZ compositions. The relaxation parameters obtained from the linear relation, according to Equation (1), are shown in Table 1.