## 3 Results and Discussions
At the same time, the conductivity values for the acceptor-doped BaLa\({}_{2-x}\)Sr\({}_{x}\)In\({}_{2}\)O\({}_{7-0.5x}\) compositions are close to the values for the co-doped BaLa\({}_{1.9-x}\)Sr\({}_{x}\)Gd\({}_{0.1}\)In\({}_{2}\)O\({}_{7-0.5x}\) compositions, except for the compositions with \(x\) = 0.1. The conductivity values for the BaLa\({}_{1.9}\)Sr\({}_{0.1}\)In\({}_{2}\)O\({}_{6.95}\) composition are higher than for the BaLa\({}_{1.8}\)Sr\({}_{0.1}\)Gd\({}_{0.1}\)In\({}_{2}\)O\({}_{6.95}\) composition at the same oxygen vacancy concentration. It is clear that the conductivity values depend on both geometric and concentration factors. Because the lattice parameters and interlayer space are larger for the acceptor-doped composition than for the co-doped composition, the mobility of oxygen ions may be higher for the BaLa\({}_{1.9}\)Sr\({}_{0.1}\)In\({}_{2}\)O\({}_{6.95}\) composition than for the BaLa\({}_{1.8}\)Sr\({}_{0.1}\)Gd\({}_{0.1}\)In\({}_{2}\)O\({}_{6.95}\) composition. The proton conductivity values were obtained as the difference between the ionic conductivity under wet conditions (wet Ar) and dry conditions (dry Ar). The concentration dependence of the protonic conductivity for the compositions BaLa\({}_{1.9-x}\)Sr\({}_{x}\)Gd\({}_{0.1}\)In\({}_{2}\)O\({}_{7-0.5x}\) is shown in Figure 6a. As can be seen, the protonic conductivity increases with increasing dopant concentration. For the correct analysis of these dependencies, the values of the proton concentrations are required. Figure 6b shows the results of thermogravimetry (TG), mass spectrometry (MS) and differential scanning calorimetry (DSC) analyses for BaLa\({}_{1.8}\)Sr\({}_{0.1}\)Gd\({}_{0.1}\)In\({}_{2}\)O\({}_{6.95}\) as an example. Mass loss occurs at the temperatures below 400 \({}^{\circ}\)C (TG curve) and is solely due to the release of water (MS(H\({}_{2}\)O) curve). The values of water uptake were close for all doped BaLa\({}_{1.9-x}\)Sr\({}_{x}\)Gd\({}_{0.1}\)In\({}_{2}\)O\({}_{7-0.5x}\) and undoped BaLa\({}_{2}\)In\({}_{2}\)O\({}_{7}\) compositions and are about \(\sim\)0.17 mol of water per formula unit (Table 1). In other words, the proton concentration (\(c_{H^{+}}\)) was close for all compositions. Consequently, the increase in proton conductivity (\(\sigma_{H^{+}}\)) with increasing \(x\) in the formula BaLa\({}_{1.9-x}\)Sr\({}_{x}\)Gd\({}_{0.1}\)In\({}_{2}\)O\({}_{7-0.5x}\) is due to the increase in proton mobility (\(\mu_{H^{+}}\)):  
\[\sigma_{H^{+}}=z\times e\cdot\mu_{H^{+}}\times c_{H^{+}} \tag{2}\]  
Figure 6: The concentration dependencies of protonic conductivity for the solid solution BaLa\({}_{1.9-x}\)Sr\({}_{x}\)Gd\({}_{0.1}\)In\({}_{2}\)O\({}_{7-0.5x}\) at 300 \({}^{\circ}\)C and 400 \({}^{\circ}\)C (**a**); the results of TG (blue line), DSC (orange line) and MS(H\({}_{2}\)O) (violet line) investigations for the composition BaLa\({}_{1.8}\)Sr\({}_{0.1}\)Gd\({}_{0.1}\)In\({}_{2}\)O\({}_{6.95}\) (**b**). The proton conductivity value for the most conductive co-doped composition BaLa\({}_{1.75}\)Sr\({}_{0.15}\)Gd\({}_{0.1}\)In\({}_{2}\)O\({}_{6.925}\) is 8\(\cdot\)10\({}^{-6}\) S/cm at 400 \({}^{\circ}\)C. The conductivity values for the undoped BaLa\({}_{2}\)In\({}_{2}\)O\({}_{7}\), composition, the isovalent-doped BaLa\({}_{1.9}\)Gd\({}_{0.1}\)In\({}_{2}\)O\({}_{7}\) composition and the acceptor-doped BaLa\({}_{1.75}\)Sr\({}_{0.15}\)In\({}_{2}\)O\({}_{6.925}\) composition were 0.3\(\cdot\)10\({}^{-6}\), S/cm, 2\(\cdot\)10\({}^{-6}\) S/cm and 15\(\cdot\)10\({}^{-6}\) S/cm, respectively. Thus, the acceptor doped composition was more conductive than the co-doped one. However, co-doping increases the proton conductivity values by up to 1.5 orders of magnitude.