## 3 Results and Discussion
### Electrical Properties  
The polycrystalline sample-based electrical properties were studied using electrochemical impedance spectroscopy (EIS). EIS is a powerful experimental method that utilizes a small-amplitude alternating current (AC) signal to probe a material's impedance characteristics. This technique measures the impedance of a system over a range of frequencies to generate an impedance spectrum for the electrochemical material under testing. It can give information on the charge transfer mechanisms such as the frequency-dependent impedance, relaxation time, grain, grain boundary, and electrode contribution in a material. For our material, EIS measurements were accomplished in a frequency range from 0.1 Hz to 1 MHz using a frequency response analyzer. The total conductivity was determined from the resistance, which is the intercept of the Nyquist plot at a high frequency. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Elements** & **X** & **y** & **z** & **Occupancy** & **U\({}_{\text{iso}}\)** \\ \hline Ca & 0.4820 & 0.1081 & 0.0226 & 1.0000 & 0.006 (5) \\ \hline Fe1 & 0.0000 & 0.0000 & 0.0000 & 1.0000 & 0.001 (5) \\ \hline Fe2 & \(-\)0.0512 & 0.2500 & 0.0680 & 1.0000 & 0.003 (8) \\ \hline O1 & 0.2838 & \(-\)0.0141 & 0.2345 & 1.0000 & 0.019 (9) \\ \hline O2 & 0.0315 & 0.1394 & 0.0811 & 1.0000 & 0.009 (4) \\ \hline O3 & 0.5878 & 0.2500 & 0.1248 & 1.0000 & 0.007 (9) \\ \hline \end{tabular}
\end{table} 
Table 1: The structural parameters for Ca\({}_{2}\)Fe\({}_{2}\)O\({}_{5}\) obtained from the refinement of powder X-ray diffraction. Space group: _Pnma_, \(a\) = 5.4247 (5), \(b\) = 14.7756 (1), \(c\) = 5.5977 (5), _Rp_ = 0.013, _wRp_ = 0.018, and \(\chi\)2 = 1.941. Figure 2: SEM image of Ca\({}_{2}\)Fe\({}_{2}\)O\({}_{5}\). Figure 4 shows the Nyquist plot for Ca\({}_{2}\)Fe\({}_{2}\)O\({}_{5}\) over a wide frequency range at different temperatures. The EIS data of the material under investigation showed a depressed semicircle where the radius of the semicircle decreased with the temperature rising. If the width of the semicircles decreases with the temperature, the decrease in resistance (or increase in conductivity) with the temperature indicates a semiconductor behavior. The depression was observed for all the semicircles at all temperatures without having centers on the real axis. Such non-Debye-type behavior was considered due to the relaxation time distribution within the bulk material. The ideal EIS of such materials exhibits a semicircle. When such an oxygen-deficient perovskite is measured as an electrolyte, the impedance contribution from the grain, grain boundary, and the electrode-material interface can be considered. Therefore, three semicircular arcs could be expected, corresponding to the grain (bulk), grain boundary, and electrode contributions in the ceramic materials. Three RC elements in a series combination could be used for the best fitted equivalent circuit of the material (Figure 4), where R1, R2, and R3 represent the resistance offered by the grain (\(R_{g}\)), grain boundary (\(R_{gb}\)), and electrode-material interface (\(R_{el}\)), respectively, and CPE1, CPE2, and CPE3 represent the capacitance of the grain, grain boundary, and electrode-material interface, respectively. A constant phase element (CPE) is generally applied for the nonideal capacitive behavior demonstrated by an imperfect semicircle. The nonideal behavior is caused by surface roughness, leakage capacitance, and nonuniform distribution [28]. In our material (Figure 4), the heights of the semicircles were observed to be smaller than half of their diameter. Therefore, the classical parallel R//C circuit used for a perfect semicircle was not the suitable equivalent circuit to model this type of data of depressed semicircles [26; 28; 29; 30; 31; 32; 33; 24]. The inset demonstrates the equivalent circuit used, where the real (\(Z^{\prime}\)) and imaginary (\(Z^{\prime}\)) part of the impedance were derived by Equations (1) and (2) [30; 31].