## 4 Discussion
Because of the high conduction, the dielectric losses increased, similar to joule losses in conductors. For ceramics with tetragonal zirconia phase dominance, no pronounced frequency dispersion in the \(\varepsilon^{*}\)(f) spectra was observed. The dielectric loss frequency spectra showed that there was a small dispersion in the low-frequency region. As can be seen from Figure 8, the low frequency region of the tan \(\delta\) (f) curves for samples with x = 0.2-0.5 showed large measurement errors, but in the frequency range of 400-50 000 Hz, the values of the losses became 0.001 or lower. This fact demonstrates the high dielectric quality factor of high-content tetragonal zirconia ceramics. With the increase in CeO\({}_{2}\) concentration in ZrO\({}_{2}\) (samples with x = 0.00-0.20), the values of the dielectric permittivity increased significantly because of the low porosity and tetragonal phase formation (Figure 10a). This last feature contributes to the increase in permittivity value because of the smaller lattice volume. According to the Clausius-Mossotti equation, a higher bond iconicity and lower crystal unit cell volume result in increased dielectric permittivity in oxide ceramics [32]. Also, it is known that substitution of Ce\({}^{4+}\) with Zr\({}^{4+}\) can increase permittivity values [33; 34; 35; 36]. The fluctuations in permittivity values can be associated with inhomogeneity of the phase composition which occurred because of complete solid-phase reactions. With further increases in x, the value of the dielectric permittivity became lower due to the high porosity and formation of cubic zirconia and ceria phases, which have a higher crystal unit cell volume. It can be concluded that ceria ceramics doped with Zr4+ exhibited high losses because of the Maxwell-Wagner relaxation mechanism and electron hopping. Low losses in the pure ceria ceramics can be explained by a stable structure with no Ce3+ cation formation. The mechanical characteristics of the obtained ceramics were analyzed by performing microhardness measurements. The ceramics with a predominance of the cubic phase of fluorite showed lower microhardness values (1.5-2.0 times less than those of ceramics with a tetragonal structure), which are associated with the internal properties of the material. However, an increase in the microhardness value in the concentration range of x = 0.8-1.0 was associated with an increase in density, shrinkage, and a decrease in porosity (Figures 3a and 11a). It is known that the formation of pores in ceramics significantly reduces its hardness due to the fact that pores are concentrators of stress propagation, in which, the probability of destruction increases significantly [37; 38]. A sharp increase in the microhardness value is associated with the formation of the tetragonal ZrO2+ phase in the ceramic composition. The high hardness of the tetragonal phase is associated with phase transformation hardening, in which, the formation of cracks is accompanied by the t \(\rightarrow\) m transition [9]. With this strengthening mechanism, it is believed that the compressive stress created by a volumetric expansion of 4-5% in the crack formation zone reduces the intensity of its stress at the tip. In this case, the propagation of the crack stops, which generally increases the impact strength and hardness of the material. From the graph in Figure 10a, it can be observed that at values of x = 0.3-0.6, the value of microhardness HV0.2 was in the range of 1200-1450 and increased to a maximum of 1500 at x = 0.25, when a completely tetragonal phase was formed in the ceramic. A further decrease in microhardness was associated with an increase in porosity due to the low CeO2 content in the initial charge. This demonstrates that stabilization of the t-phase during cooling is impossible and macrocracks form in the samples due to the t \(\rightarrow\) m transition. Also, it is worth noting that the synthesized tetragonal ZrO2-CeO2 ceramics did not form cracks after indentation.