## 3 Results and Discussion
The results of the analysis are presented as histograms in Figure 4. As can be seen from the figure, the average grain size decreased at C\({}_{\rm w}\) = 0.05 from 3.42 to 1.82 \(\upmu\)m. At a dopant concentration of 0.10, the average size reached a maximum value of 7.36 \(\upmu\)m and decreased again with increasing concentrations of 0.15, 0.20, and 0.25 to values of 5.33, 4.73, and 4.68 \(\upmu\)m, respectively. In the undoped sample (Figure 4a), a broad grain size distribution was observed with a peak at 2.5-3 \(\upmu\)m, comprising no more than 22% of the grains. However, at low dopant concentrations (C\({}_{\rm w}\) = 0.05, Figure 4b), the grain size distribution became narrower, with a peak in the range of 1-1.5 \(\upmu\)m. An important observation is the presence of grains with sizes that are twice the average grain size. This indicates the initiation of grain growth, likely resulting from phase transformations during the diffusion process. Samples with dopant concentrations of C\({}_{\rm w}\) = 0.10 and 0.15 (Figure 4c,d) exhibited large ranges of grains size, spanning from 0 to 22 \(\mu\)m. This indicates the occurrence of intensive processes of grain coalescence and coarsening during annealing [35]. These samples were characterized by the highest values of the fraction of the cubic phase (Table 3), signifying the decisive contribution of phase transformations during atomic diffusion to grain growth [36]. However, with increasing concentration, a decrease in the grain size range and average grain size were observed. This can be attributed to the large concentration of CO\({}_{2}\) gas released during annealing, which forms pores. The presence of pores is a critical factor in the further densification of ceramics due to the grain growth [27]. For this reason, for concentrations of 0.20 and 0.25 (Figure 4e,f), despite the intense coalescence and expansion of grains, the average grain size and distribution range decreased. It can also be argued that the CaCO\({}_{3}\) in the charge is a pore-forming agent and significantly affects the microstructure of zirconia ceramics. Figure 5 shows the spectra of the energy-dispersive analysis of the obtained ceramics. The spectra suggest that the synthesized samples contained only Ca, Zr, and O. With increasing dopant concentration, the intensity of the peak related to Ca increased, indicating an increase in the concentration of this element in the ceramic composition. The chemical composition of the obtained samples is shown in Table 4. The analysis of the peaks using Quantax 75 software over the scan area shown in Figure 6 demonstrates that the atomic percentage of Ca increased with increasing C\({}_{\rm w}\). The discrepancy between the atomic percentages and stoichiometric formulas can be explained by the large degree of roughness of the cross-sections. This roughness makes it difficult to detect X-ray photons, which can change the atomic composition from stoichiometric. Elemental mapping of the cross-section of the ceramic sample showed that oxygen and zirconium ions were uniformly distributed over the ceramic surface. However, for the sample with a concentration C\({}_{\rm w}\) = 0.05, it can be observed that Ca ions were distributed predominantly in large grains. Considering the results of the powder diffractogram analysis, it can be concluded that the small grains were predominantly composed of monoclinic phases. In the case of ceramics with high dopant concentrations, the non-uniform distribution is due to the formation of the CaZrO\({}_{3}\) phase, which contains more Ca than the substituted ZrO\({}_{2}\)--c phase. Figure 5: Energy-dispersive analysis spectra for synthesized ceramic samples. The frequency dependences of the dielectric permittivities of the synthesized ceramics are shown in Figure 7a. As is evident from the figure, the dielectric constant remains nearly constant across the frequency range of 200-200 kHz, suggesting the absence of relaxation