## 3 Results and Discussion
### Impedance Spectroscopy Analysis  
The Nyquist diagrams (Z\" vs. Z\") of BCTZ (\(x\) = 0.10, 0.15, 0.18) ceramics, at different temperatures, are shown in Figure 7. The thermal effect over the impedance behavior of the samples became remarkable as temperature increases. Generally used for dielectric materials or ionic conductors, IS is a convenient technique to distinguish between different regions that are responsible for conduction mechanisms [10]. The two semicircles displayed in Figure 7a-c (for sintered BCTZ ceramics at 1400 \({}^{\circ}\)C for 2 h) represent the electrical contributions response. From high to low frequency, two semicircles appeared. These phenomena are attributed to the grain/bulk ferroelectric and grain boundary effects, respectively. Semiconducting grains and insulating grain boundaries are characteristic for BCTZ compositions. Moreover, as the radius of the semicircles decreased, the electric resistance dropped. Increasing the dopant concentration enhances the electronic conductivity until the material reaches a maximum value. The electrical processes, taking place within the material, were modeled for the BCTZ (\(x\) = 0.15, 420 \({}^{\circ}\)C) system. The electrical properties may be described by an appropriate equivalent circuit, based on a combination of RC (resistor-capacitor) elements in a series and/or parallel arrangement. These RC components represent the macroscopic processes involved in the charge transport due to inhomogeneity over the microstructure. To analyze systems that include a relaxation process, discrete RC elements must be combined with, or replaced by, distributed impedance components. Figure 8 represents the fitting data and the equivalent circuit for BCTZ \(x\) = 0.15 at 420 \({}^{\circ}\)C. The fitting data (Figure 8a) was done using two R-CPE elements connected in series. The CPE element (constant phase element) was used instead of a pure capacitor to suppress the dispersion and  
Figure 7: Impedance plots between 400–500 \({}^{\circ}\)C for BCTZ (**a**) \(x\) = 0.10; (**b**) \(x\) = 0.15; (**c**) \(x\) = 0.18 compositions sintered at 1400 \({}^{\circ}\)C for 2 h.  
non-linearities in the values. Then, the values of the resistance (R) and capacitance (C) for the bulk (b) and grain boundary (gb) were calculated. The CPE element is defined as [23; 24]:  
\[Z_{\text{CPE}}=[1/Y_{0}\ (jw)^{\alpha}], \tag{2}\]  
where \(w\) is the angular frequency, \(Y_{0}\) is related to the size, thickness and material properties, and \(\alpha\) (\(0\leq\alpha\leq 1\)) relates to the degree of energy dissipation; it measures the arc depression which is frequency-independent [21]. For \(\alpha\) = 0, CPE behaves as a resistor. When \(\alpha\) = 1, CPE behaves as a capacitor. For \(\alpha\) = 0.5, the CPE element becomes a series RC circuit (Warburg diffusion). The calculated \(\alpha\) value of BCTZ \(x\) = 0.15 is \(\approx\)1. Associated capacitances led to the identification of different processes: C\({}_{g}\) (10\({}^{-9}\)-10\({}^{-10}\) F) for bulk/grain ferroelectric ceramics and \(C_{gb}\) (10\({}^{-8}\) F) for the grain boundary. The fitting measured impedances for equivalent circuits elements are: \(C_{g}\) = 3.01 \(\times\) 10\({}^{-10}\) F and \(R_{g}\) = 1.44 \(\times\) 10\({}^{5}\)\(\Omega\); \(C_{gb}\) = 8.96 \(\times\) 10\({}^{-8}\) F and \(R_{gb}\) = 2.89 \(\times\) 10\({}^{5}\)\(\Omega\), for BCTZ \(x\) = 0.15 samples measured at 420 \({}^{\circ}\)C. These results reveal a well-sintered material with narrow intergranular regions [10]. Like the relaxation time, the electrical conductivity for sintered BCTZ ceramics, measured from 300 \({}^{\circ}\)C to 700 \({}^{\circ}\)C, follows the Arrhenius law (Equation (1)). Figure 8b shows the activation energy calculated for each region: the bulk/grain, grain boundary and total.