## 3 Results and Discussion
### Transport Properties  
The impedance spectra of Ba\({}_{5}\)In\({}_{2}\)Al\({}_{2}\)ZrO\({}_{13}\) and Ba\({}_{7}\)In\({}_{6}\)Al\({}_{2}\)O\({}_{19}\) are shown in Figure 5. The evolution of the hodographs with changes in temperature is shown is Figure 5a,c for Ba\({}_{5}\)In\({}_{2}\)Al\({}_{2}\)ZrO\({}_{13}\) and Ba\({}_{7}\)In\({}_{6}\)Al\({}_{2}\)O\({}_{19}\), respectively. The evolution of the hodographs in wet air is presented in Figure 5b,d for Ba\({}_{5}\)In\({}_{2}\)Al\({}_{2}\)ZrO\({}_{13}\) and Ba\({}_{7}\)In\({}_{6}\)Al\({}_{2}\)O\({}_{19}\), respectively. It can be seen that the shape of the impedance spectra for both samples is quite similar and does not undergo any changes during temperature and water partial pressure variation. For both samples, the impedance data analysis showed the presence of at least two relaxation processes in the range of the investigated frequencies. Two overlapping semicircles can be seen in the impedance spectra, and it should be noted that the second semicircle was small. Beyond that, a small contribution of the third relaxation process was observed in the low frequencies, however, its contribution to this process is also negligible. The relaxation processes in polycrystalline materials can be represented by three contributions: bulk resistance, resistance of the grain boundaries, and the electrode processes; frequencies attributed to these processes decrease in the series: bulk\(-\)grain boundaries\(-\)electrode processes [62]. The determination of the nature of the relaxation processes can be executed by analysis of the electrical capacitance values attributed to these contributions [62]. Calculated capacitance values attributed to the third relaxation process were about \(10^{-6}\) F; these values are typical for the electrode processes. The capacitance values attributed to the first and second relaxation processes (first and second semicircles) were about \(10^{-11}\) and \(10^{-10}\) F, respectively. It can, therefore, be concluded that first semicircle is corresponded to the contribution of bulk resistance and the second semicircle to the contribution of grain boundaries. The bulk conductivity was calculated using the following equation:  
\[\sigma=\frac{l}{SR_{b}}, \tag{1}\]  
where \(l\) represents the sample thickness, \(S\) represents the sample sectional area, and R\({}_{b}\) represents the value of bulk resistance. Figure 6 shows the temperature dependencies of the conductivity for Ba\({}_{5}\)In\({}_{2}\)Al\({}_{2}\)ZrO\({}_{13}\) and Ba\(\gamma\)In\({}_{6}\)Al\({}_{2}\)O\({}_{19}\) in dry and wet air. It can be seen that for both compounds over the whole investigated temperature range, the values of conductivity in wet air were higher than in dry air. The increase in conductivity in wet atmospheres is a consequence of hydration and the appearance of proton current carriers. The difference between conductivity values obtained in dry and wet air becomes insignificant at temperatures above 850 \({}^{\circ}\)C--this is a consequence of dehydration at high temperatures. Therefore, Ba\({}_{5}\)In\({}_{2}\)Al\({}_{2}\)ZrO\({}_{13}\) and Ba\(\gamma\)In\({}_{6}\)Al\({}_{2}\)O\({}_{19}\) are able to exhibit proton transport in atmospheres with high water partial pressures. It can be seen that the temperature dependencies of conductivity for Ba\({}_{5}\)In\({}_{2}\)Al\({}_{2}\)ZrO\({}_{13}\) and Ba\(\gamma\)In\({}_{6}\)Al\({}_{2}\)O\({}_{19}\) exhibit different slopes. Conductivity dependence for Ba\(\gamma\)In\({}_{6}\)Al\({}_{2}\)O\({}_{19}\) has a steeper slope in than Ba\({}_{5}\)In\({}_{2}\)Al\({}_{2}\)ZrO\({}_{13}\). Therefore, at high temperatures, the conductivity of Ba\(\gamma\)In\({}_{6}\)Al\({}_{2}\)O\({}_{19}\) is higher in than of Ba\({}_{5}\)In\({}_{2}\)Al\({}_{2}\)ZrO\({}_{13}\), while at low temperatures (below 500 \({}^{\circ}\)C), the conductivity of Ba\({}_{5}\)In\({}_{2}\)Al\({}_{2}\)ZrO\({}_{13}\) is higher. Thus, the activation energies of the conductivity of both samples are different.