## 3 Results
The calculation of the chemical formulas from the mole fractions of oxides and from the EDX composition data were significantly different. This can be explained by the following reasons: (1) the roughness of the cross-sectional surfaces of the ceramics obstructs the precise detection of all elements; (2) the EDX method of oxygen concentration detection has a high error value; and (3) the analyzed square of the sample can be inhomogeneous (oxygen deficiency and unfinished solid phase reaction). The elemental mapping of the surfaces showed that for ceramics with a pure cubic, tetragonal, or monoclinic phase, the distribution of elements was uniform. In mixed-phase samples with an equal ratio of cubic and tetragonal phases, or with a tetragonal domination, the distribution of elements appeared to not be uniform. For example, the elemental mapping of the surface of the sample with x = 0.55 is shown in Figure 7.  
From the obtained SEM images, the average grain size d was calculated for each sample by using the intercept line method and ImageJ 1.54g software [24]. The results of the calculation are shown in Figure 8. As can be seen from the graph for the ceramics with high CeO\({}_{2}\) concentrations, the values of d were between 10 and 28 \(\upmu\)m. When x decreased, a sharp drop to \(\sim\)2 \(\upmu\)m at a value of x = 0.55 was observed. The frequency dependences of dielectric permittivity and loss tangent of some samples are shown in Figure 9. It was observed that for ceramics with a high content of CeO\({}_{2}\) in the initial mixture, their electrical properties (permittivity, tangent loss values) varied greatly for samples with identical compositions. We believe that during the sintering process, some oxygen vacancies can unpredictably occur, leading to a decrease in the valence of the Ce cation from 4+ to 3+. This feature can change the electrical properties because of the increasing conductivity of the electron hopping mechanism (the formation of small polarons) [25]. For further research, we chose samples with the highest measured AC resistivity for each composition. It can be seen from the graph in Figure 9a that for high-content CeO\({}_{2}\) ceramics, a low-frequency tail occurred in the \(\epsilon^{\prime}\)(f) spectra. To understand how the composition of sintered ceramics influences their dielectric properties, the dependencies of \(\epsilon^{\prime}\)(x) and tan \(\delta\)(x) in the plateau region, where changes in the dielectric parameter with frequency are small, were plotted. In the case of conductive samples, the tan \(\delta\) value used for comparison was taken from the middle of the plateau region, which is shown in Figure 9b. As can be seen from the graph in Figure 10a, the dielectric permittivity of zirconia was low because of cracks and its high porosity. From the tan \(\delta\) (x) dependence (Figure 10b), it can be seen that a pronounced rise in the loss value in the plateau region occurred when x = 0.6. When the CeO\({}_{2}\) content decreased to close to this concentration, the formation of high-content tetragonal ceramic was observed. Figure 11a shows the results of the microhardness measurements using the Vickers method at a force of 0.2 kgf for the synthesized ceramics. The microhardness measurements were performed on unpolished surfaces to avoid the influence of tooling on the mechanical properties of the zirconia-ceria ceramics (Figure 11b). Figure 5: SEM micrographs of the cross-section of samples with characteristic features of the microstructure of the obtained ceramics. (**a**) x = 0.0; (**b**) x = 0.2; (**c**) x = 0.3; (**d**) x = 0.45; (**e**) x = 0.8; (**f**) x = 1.0. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**x** & **O Concentration, at. \%** & **Ce Concentration, at. \%** & **Zr Concentration, at. \%** \\ \hline
1 & 61.6 & 38.4 & - \\  
& 72.2 & - & 27.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Elemental composition of the sintered samples calculated from EDX spectra. Figure 6: The EDX spectra of the samples with x = 0; 0.25; 0.5; 0.75; 1.