## 3 Results and Discussion
### Impedance Spectroscopic Studies
\begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**Element**} & \multirow{2}{*}{**Parameter**} & \multicolumn{5}{c}{**Samples**} \\ \cline{3-6}  & & \(x\) **= 1 at.\% La** & \(x\) **= 2 at.\% La** & \(x\) **= 3 at.\% La** & \(x\) **= 4 at.\% La** \\ \hline \multirow{3}{*}{\(\mathrm{R_{G}}\)} & Value [\(\Omega\)] & 8.90 \(\times\) 10\({}^{5}\) & 7.73 \(\times\) 10\({}^{5}\) & 2,487,400 & 2.14 \(\times\) 10\({}^{6}\) \\ \cline{2-6}  & Absolute error [\(\Omega\)] & 2589 & 2590 & 2.85 \(\times\) 10\({}^{5}\) & 46,909 \\ \cline{2-6}  & Relative error [\%] & 0.29 & 0.34 & 11.45 & 2.19 \\ \hline \multirow{3}{*}{\(\mathrm{CPE\text{-}T_{G}}\)} & Value [F] & 1.13 \(\times\) 10\({}^{-8}\) & 1.61 \(\times\) 10\({}^{-8}\) & 3.80 \(\times\) 10\({}^{-7}\) & 1.39 \(\times\) 10\({}^{-7}\) \\ \cline{2-6}  & Absolute error [F] & 2.69 \(\times\) 10\({}^{-10}\) & 5.05 \(\times\) 10\({}^{-10}\) & 2.68 \(\times\) 10\({}^{-8}\) & 3.86 \(\times\) 10\({}^{-9}\) \\ \cline{2-6}  & Relative error [\%] & 2.39 & 3.13 & 7.05 & 2.79 \\ \hline \multirow{3}{*}{\(\mathrm{CPE\text{-}P_{G}}\)} & Value [a.u.] & 0.54 & 0.44 & 0.24 & 0.34 \\ \cline{2-6}  & Absolute error [a.u.] & 0.003 & 0.003 & 0.008 & 0.004 \\ \cline{2-6}  & Relative error [\%] & 0.58 & 0.73 & 3.17 & 1.1876 \\ \hline \multirow{3}{*}{\(\mathrm{C_{G}}\)} & Value [F] & 5.90 \(\times\) 10\({}^{-10}\) & 4.36 \(\times\) 10\({}^{-10}\) & 8.25 \(\times\) 10\({}^{-10}\) & 7.58 \(\times\) 10\({}^{-10}\) \\ \cline{2-6}  & Absolute error [F] & 2.37 \(\times\) 10\({}^{-12}\) & 1.59 \(\times\) 10\({}^{-13}\) & 1.36 \(\times\) 10\({}^{-11}\) & 6.46 \(\times\) 10\({}^{-12}\) \\ \cline{2-6}  & Relative error [\%] & 0.40 & 0.04 & 1.65 & 0.85219 \\ \hline \multirow{3}{*}{\(\mathrm{R_{GB}}\)} & Value [\(\Omega\)] & 23,201 & 3.69 \(\times\) 10\({}^{5}\) & 163,960 & 353,410 \\ \cline{2-6}  & Absolute error [\(\Omega\)] & 954 & 3270 & 7762 & 4187 \\ \cline{2-6}  & Relative error [\%] & 4.115 & 0.88 & 4.73 & 1.19 \\ \hline \multirow{3}{*}{\(\mathrm{CPE\text{-}T_{GB}}\)} & Value [F] & 4.03 \(\times\) 10\({}^{-9}\) & 3.26 \(\times\) 10\({}^{-8}\) & 2.64 \(\times\) 10\({}^{-9}\) & 1.11 \(\times\) 10\({}^{-9}\) \\ \cline{2-6}  & Absolute error [F] & 1.08 \(\times\) 10\({}^{-10}\) & 3.27 \(\times\) 10\({}^{-10}\) & 1.43 \(\times\) 10\({}^{-10}\) & 1.44 \(\times\) 10\({}^{-11}\) \\ \cline{2-6}  & Relative error [\%] & 2.69 & 1.01 & 5.41 & 1.30 \\ \hline \multirow{3}{*}{\(\mathrm{R_{GB}}\)} & \(\chi^{2}\) & 1.53 \(\times\) 10\({}^{-5}\) & 7.45 \(\times\) 10\({}^{-6}\) & 4.8 \(\times\) 10\({}^{-5}\) & 9.89 \(\times\) 10\({}^{-6}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Parameters of the components of the applied electrical circuit for the impedance response of the lanthanum-modified PBZT 25/70/30 ceramics. \begin{table}
\begin{tabular}{c c c} \hline \multicolumn{2}{c}{**Samples**} & \multicolumn{2}{c}{**E\({}_{\mathbf{aG}}\) [eV]**} & \multicolumn{2}{c}{**E\({}_{\mathbf{aG}}\) [eV]**} \\ \hline \(x\) = 1 at.\% La & 1.23 & 0.83 \\ \hline \(x\) = 2 at.\% La & 1.59 & 0.8 \\ \hline \(x\) = 3 at.\% La & 2.05 & 0.66 \\ \hline \(x\) = 4 at.\% La & 1.82 & 0.6 \\ \hline \end{tabular}
\end{table} 
Table 3: Activation energy for lanthanum-doped PBZT ceramics. The linear character of the presented dependencies made it possible to determine the activation energy (\(\mathrm{E_{a}}\)) of the conductivity process in grains and grain boundaries for ceramics with various lanthanum content. Undoped PBZT 25/70/30 ceramics are characterized by the following activation energy values: 0.95 for grain conductivity and 0.85 for grain boundary conductivity [37]. The value of \(\mathrm{E_{a}}\) for grains increases rapidly with the increase in \(\mathrm{La^{3+}}\) ion content, whereas the \(\mathrm{E_{a}}\) for grain boundaries decreases slightly. The explanation for the observed changes is connected with space charge, which plays a crucial role in the case of undoped ceramics.