## 3 Results and Discussions  
The homogeneity range of the solid solution BaLa\({}_{1.9-x}\)Sr\({}_{x}\)Gd\({}_{0.1}\)In\({}_{2}\)O\({}_{7-0.5x}\) was defined by XRD analysis. The full profile Le Bail refinement of the obtained data showed that compositions with \(x\)\(\leq\) 0.15 are single phase and have orthorhombic symmetry, Pbca space group (Figure 1). All compositions were isostructural to the matrix composition BaLa\({}_{2}\)In\({}_{2}\)O\({}_{7}\), which has the structure of bilayer perovskite [51]. The higher dopant content \(x\)\(>\) 0.15 led to the appearance of impurities. Table 1 shows the lattice parameters of the compositions studied. As shown, doping led to an increase in lattice parameters and unit cell volume. The comparison of these data with the results for the related compositions obtained by doping the La\({}^{3+}\) sublattice with Sr\({}^{2+}\) ions BaLa\({}_{2-x}\)Sr\({}_{x}\)In\({}_{2}\)O\({}_{7-0.5x}\)[47] and Gd\({}^{3+}\) ions BaLa\({}_{2-x}\)Gd\({}_{x}\)In\({}_{2}\)O\({}_{7}\)[48] (Figure 1e) shows that the largest increase in parameters during doping is achieved for the acceptor-doped solid solution BaLa\({}_{2-x}\)Sr\({}_{x}\)In\({}_{2}\)O\({}_{7-0.5x}\), where the ionic radii of the dopants are larger (\(r_{\rm La^{3+}}\) = 1.216 A; \(r_{\rm Sr^{2+}}\) = 1.31 A; \(r_{\rm Gd^{3+}}\) = 1.107 A [52]). However, there is no direct correlation between the dopant radius and the change in the lattice parameters. The introduction of Gd\({}^{3+}\) ions into the La\({}^{3+}\) sublattice led to an increase in the interlayer space (lattice parameter \(c\)) due to the appearance of local distortions caused by the effects of mutual repulsion of ions of different electronegativity in the same sublattice [37]. It is obvious that the presence of three different cations also leads to the local distortion of the crystal lattice, and this is the reason for the significant decrease in the lattice parameter \(c\) for the solid solution BaLa\({}_{1.9-x}\)Sr\({}_{x}\)Gd\({}_{0.1}\)In\({}_{2}\)O\({}_{7-0.5x}\) compared to BaLa\({}_{2-x}\)Sr\({}_{x}\)In\({}_{2}\)O\({}_{7-0.5x}\). Therefore, the increasing in the lattice parameter c is observed for all three solid solutions, but this increasing is different. Figure 1: The results of XRD investigations (red dots are experimental data, black line is calculated pattern, blue line is the difference curve, green symbols are calculated peak positions) for the compositions BaLa\({}_{2}\)In\({}_{2}\)O\({}_{7}\) (**a**), BaLa\({}_{1.85}\)Sr\({}_{0.05}\)Gd\({}_{0.1}\)In\({}_{2}\)O\({}_{6.975}\) (**b**), BaLa\({}_{1.8}\)Sr\({}_{0.1}\)Gd\({}_{0.1}\)In\({}_{2}\)O\({}_{6.95}\) (**c**), BaLa\({}_{1.75}\)Sr\({}_{0.15}\)Gd\({}_{0.1}\)In\({}_{2}\)O\({}_{6.925}\) (**d**), and the concentration dependencies of the lattice parameter \(c\) for the solid solutions BaLa\({}_{1.9-x}\)Sr\({}_{x}\)Gd\({}_{0.1}\)In\({}_{2}\)O\({}_{7-0.5x}\) (blue line), BaLa\({}_{2-x}\)Sr\({}_{x}\)In\({}_{2}\)O\({}_{7-0.5x}\) (**b**) (orange line) and BaLa\({}_{2-x}\)Gd\({}_{x}\)In\({}_{2}\)O\({}_{7}\) (**c**). Morphological analysis of the compositions was carried out using scanning electron microscopy (SEM). The diameter of the grains was about \(\sim\)2-5 \(\upmu\)m (Figure 2). Energy dispersive X-ray spectroscopy (EDS) analysis confirmed the presence of all cations in the samples and the agreement of their ratio with the theoretical values (Table 2). The electrical conductivities were obtained using the impedance spectroscopy method. Figure 3 shows the typical EIS plots for the sample BaLa\({}_{1.75}\)Sr\({}_{0.15}\)Gd\({}_{0.1}\)In\({}_{2}\)O\({}_{6.925}\). The semicircle starting from zero coordinates (high frequency semicircle) corresponds to the resistivity of the grain volume of polycrystalline sample. This is proved by the small capacitance value \(\sim 10^{-12}\) F/cm. The conductivities were calculated from the resistivity values taken at the intersection of the high-frequency semicircle with the abscissa axis.