## 3 Results and Discussion
### Transport Properties
The temperature dependencies of partial conductivities for Ba\({}_{5}\)In\({}_{2}\)Al\({}_{2}\)ZrO\({}_{13}\) and Ba\({}_{7}\)In\({}_{6}\)Al\({}_{2}\)O\({}_{19}\) are presented in Figure 9.  
It can be seen in Figure 9 that in dry atmospheres, the oxygen-ion conductivity for Ba\({}_{5}\)In\({}_{2}\)Al\({}_{2}\)ZrO\({}_{13}\) is higher than the hole conductivity at temperatures below \(\sim\)550 \({}^{\circ}\)C. For Ba\({}_{7}\)In\({}_{6}\)Al\({}_{2}\)O\({}_{19}\), the hole conductivity is higher for the whole investigated temperature range, and only at \(\sim\)500 \({}^{\circ}\)C are the oxygen-ion conductivity values approximately equal to hole conductivity. At \(\sim\)500 \({}^{\circ}\)C, the values of oxygen-ion and hole conductivity for Ba\({}_{5}\)In\({}_{2}\)Al\({}_{2}\)ZrO\({}_{13}\) and Ba\({}_{7}\)In\({}_{6}\)Al\({}_{2}\)O\({}_{19}\) are roughly equivalent; however, at higher temperatures, the oxygen-ion and hole conductivity of Ba\({}_{7}\)In\({}_{6}\)Al\({}_{2}\)O\({}_{19}\) are higher. It should be noted that the oxygen-ion conductivity of Ba\({}_{7}\)In\({}_{6}\)Al\({}_{2}\)O\({}_{19}\) is 0.5 orders of magnitude higher than for Ba\({}_{5}\)In\({}_{2}\)Al\({}_{2}\)ZrO\({}_{13}\). The activation energy of oxygen-ion conductivity of Ba\({}_{7}\)In\({}_{6}\)Al\({}_{2}\)O\({}_{19}\) was higher (0.85 eV) than of Ba\({}_{5}\)In\({}_{2}\)Al\({}_{2}\)ZrO\({}_{13}\) (0.64 eV). We believe that the higher oxygen-ion conductivities of this phase are due to the presence of a block with cationic vacancies. As described in the literature [63; 64], the presence of cationic vacancies in the A-sublattice of perovskites promotes facilitated oxygen-ion transport as a result of an increase in the free volume of migration. In wet atmospheres, Ba\({}_{5}\)In\({}_{2}\)Al\({}_{2}\)ZrO\({}_{13}\) demonstrates predominant proton conductivity at temperatures below 600 \({}^{\circ}\)C. Ba\({}_{7}\)In\({}_{6}\)Al\({}_{2}\)O\({}_{19}\) exhibits a lower contribution of proton conductivity in wet atmospheres; only at temperatures below 500 \({}^{\circ}\)C does proton conductivity become higher than hole conductivity, and at 600 \({}^{\circ}\)C, proton and hole conductivity are approximately equal. At temperatures above 600 \({}^{\circ}\)C, proton conductivity values for Ba\({}_{7}\)In\({}_{6}\)Al\({}_{2}\)O\({}_{19}\) are higher in comparison than for Ba\({}_{5}\)In\({}_{2}\)Al\({}_{2}\)ZrO\({}_{13}\), while at temperatures 600 \({}^{\circ}\)C, they are equivalent, and at 500 \({}^{\circ}\)C, Ba\({}_{5}\)In\({}_{2}\)Al\({}_{2}\)ZrO\({}_{13}\) shows higher proton conductivity than Ba\({}_{7}\)In\({}_{6}\)Al\({}_{2}\)O\({}_{19}\). These changes in the behavior of proton conductivity with temperature are explained by the difference in their activation energies. The activation energy of the proton conduction of phase Ba\({}_{5}\)In\({}_{2}\)Al\({}_{2}\)ZrO\({}_{13}\) (0.27 eV) is much smaller compared to phase Ba\({}_{7}\)In\({}_{6}\)Al\({}_{2}\)O\({}_{19}\) (0.45 eV). Figure 9: Temperature dependencies of oxygen-ion (**a**) and proton (**b**) conductivities for Ba\({}_{5}\)In\({}_{2}\)Al\({}_{2}\)ZrO\({}_{13}\)[56] and Ba\({}_{7}\)In\({}_{6}\)Al\({}_{2}\)O\({}_{19}\). The ionic transport numbers t\({}_{i}\) were calculated from the data on the partial conductivities according to the following equation:  
\[\text{t}_{i}=\frac{\sigma_{i}}{\sigma_{\text{tot}}}, \tag{3}\]  
where \(\sigma_{\text{tot}}\) is total conductivity in dry or wet air. Figure 10 shows the temperature dependencies of the transport numbers for Ba\({}_{5}\)In\({}_{2}\)Al\({}_{2}\)ZrO\({}_{13}\) and Ba\({}_{7}\)In\({}_{6}\)Al\({}_{2}\)O\({}_{19}\). These data confirm the above-mentioned tendencies. The oxygen-ion and proton transport for the Ba\({}_{5}\)In\({}_{2}\)Al\({}_{2}\)ZrO\({}_{13}\) dominates over a wider temperature range in comparison with Ba\({}_{7}\)In\({}_{6}\)Al\({}_{2}\)O\({}_{19}\). Nevertheless, both phases are dominant oxygen-ion conductors below 500 \({}^{\circ}\)C in dry air and dominant proton conductors below 600 \({}^{\circ}\)C in wet air.