## 3 Results
### On Elemental Homogeneity  
Figure 5 shows the images of the analyzed region of the surface of the ceramic sample with different magnifications (\(\times 10\) and \(\times 100\)). With reference to the investigated objects, the measurements of X-ray fluorescence intensities were obtained in this work and the calculations of the concentrations of chemical elements located on the surface of the samples were performed. Based on a large volume of experimental data on the concentrations of basic chemical elements, the methods for the statistical determination of the homogeneity degree of the surface of the studied ceramic samples were developed. The calculated metrological characteristics of the ceramic materials, which were obtained during the statistical processing of MICRO-XRF data, allowed for the evaluation and comparison of the investigated ferroelectric ceramics produced under various technological conditions. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Sample** & \(T_{sint.}\)**, K** & \(\frac{\tau}{\tau}\)**, h.** & \(a\)**, Å & \(c\)**, Å & \(V\)**, Å\({}^{3}\)** & \(\{\it(ela-1)\times 10^{3}\)} & \(\delta\times 10^{3}\) & \(\rho_{exp}\), & \(\rho_{x-ray}\), & \(\rho_{\text{hel}}\)**, \%** \\  & & & & & & & & & & \\ CCT2 & 1500 (with & 2 & 4.005 & 4.112 & 65.95 & 26.7 & 17.8 & 7.18 & 8.16 & 87.94 \\ CCT3 & backfilling) & 5 & 3.997 & 4.120 & 65.83 & 30.7 & 20.4 & 7.87 & 8.18 & 96.21 \\ CCT1 & backfilling) & 2 & 4.018 & 4.118 & 66.47 & 25.0 & 16.6 & 7.07 & 8.10 & 87.29 \\ CCT4 & 1520 (without & 2 & 4.012 & 4.112 & 66.19 & 25.0 & 16.6 & 6.71 & 8.13 & 82.48 \\ CCT5 & backfilling) & 5 & 3.998 & 4.112 & 65.73 & 28.4 & 18.9 & 7.09 & 8.19 & 86.56 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters \(=a\), \(c\) and \(c/a\); volume \(=V\); ceramic densities: experimental \(=\rho_{exp}\); X-ray \(=\rho_{x-ray}\); and relative \(=\rho_{rel}\) of PCR-13 material under different sintering conditions. Figure 4: Fragments of microstructures of SS ceramics obtained in backfill (\(T_{sint. }~{}=~{}1500\) K (**a**); \(T_{sint. }~{}=~{}1520\) K (**b**)) and without it (\(T_{sint. }~{}=~{}1520\) K (**c**)). In particular, a statistical analysis of the degree of the homogeneity/heterogeneity of the surface was performed according to the chemical composition of the main components, which were namely Ti, Zr, Nb and Pb. The total sample size for a single sample for each chemical element was \(N~{}=~{}1000\). This corresponds to ten local areas (\(1~{}\times~{}1.5\) mm\({}^{2}\)), inside which there were 100 measurements taken of the intensities of the analytical lines of the spectrum at different points. For the ceramic samples and for those in the stage preceding the ceramic annealing, the values of the confidence intervals (\(\overline{x}\pm~{}\triangle\overline{x}\)) of the concentrations of these chemical elements were obtained, for which the condition \((\overline{x}-\Delta\overline{x})~{}\leq~{}\mu~{}\leq~{}(\overline{x}+\Delta \overline{x})\) is satisfied with a given confidence probability. The calculation of the boundary values of the confidence intervals was performed according to the Student's criterion, assuming that the concentration values included in the sample are normally distributed:  
\[(\overline{x}\pm\triangle\overline{x})=\overline{x}\pm(t(P,f)\bullet S)/n^{1/2} \tag{1}\]  
where \(t(P,f)\) are the table values of the Student's criterion. At the confidence level \(P~{}=~{}95\%\), a statistical comparison of the dispersion equation (according to [PERSON]'s F-criterion) was performed, while we obtained the reliability of the differences in mathematical expectations based on Student's \(t\)-criterion. Figure 6 shows the concentration distributions for two samples (Rand-1 and Rand-2) obtained by the fluorescence excitation of the characteristic radiation by the surface of the analyzed ceramic sample (CCT3). The statistical analysis has shown that the sample variances and calculated average values at a given confidence probability level have no significant differences.