## 3 Results and Discussion
In the case of the samples studied in this work, an accurate estimation of the theoretical density is difficult because most samples are multiphase, and, as shown in the X-ray diffraction studies, the density of the phases can vary significantly (Table 2). The apparent density of the sintered tablets was determined by measuring their geometric parameters and weights. Figure 8a shows the change in the dielectric constant and apparent density as a function of the dopant concentration. Up to a concentration of \(\mathrm{C_{w}}=0.20\), the change in the dielectric permittivity generally coincides with the change in the apparent density. The dielectric constant of the undoped sample was lower than that of the ceramics studied herein (24 vs. 16.4) [44]. This can be explained by the presence of martensitic phase transitions \(\mathrm{ZrO_{2}\text{-}m}\rightarrow\mathrm{ZrO_{2}\text{-}t}\) (1170 \({}^{\circ}\)C) and \(\mathrm{t}\rightarrow\mathrm{m}\), after which mechanical stresses are formed in the undoped sample, leading to the appearance of internal microcracks that increase the porosity. This can also be observed when measuring the apparent density, which is 4.75 g/cm\({}^{3}\) versus the theoretical density of 5.8 g/cm\({}^{3}\) for the monoclinic zirconium oxide. The decrease in the apparent density of the doped ceramics is due to the increase in the number of pores in the ceramics with higher calcium carbonate concentration in the charge. The maximum apparent density was observed for ceramics with a dopant concentration of \(\mathrm{C_{w}}=0.05\), which can be explained by the small number of pores that are formed by the release of \(\mathrm{CO_{2}}\) gas as well as by the peculiarities of the microstructure, in which small grains are densely packed. With a further decrease in the apparent density of ceramics with \(\mathrm{C_{w}}=0.10\), the dielectric constant does not decrease, which may be due to the increase in the fraction of the cubic phase. In this case, as stated earlier, the cubic structure, which has a smaller crystal cell volume, is stabilized. According to the Clausius-Mossotti equation, the dielectric permittivity of the crystals is related to the polarizability and unit cell volume as follows [45]:  
\[\frac{\varepsilon^{\prime}-1}{\varepsilon^{\prime}+2}=\frac{\alpha}{3V \varepsilon_{0}} \tag{1}\]  
where \(\alpha\) is the polarizability of the unit cell, V is the volume of the unit cell, and \(\varepsilon_{0}\) is the dielectric constant. In the case of the cubic lattice of ZrO\({}_{2}\)-c, in addition to the volume reduction, an increase in the ionicity of the crystal bonding between the anions and cations can be assumed, which increases the polarizability \(\alpha\). Then, according to Equation (1), there will be an increase in the dielectric permittivity. Similar behavior of dielectric properties in ceramics is observed for Y-stabilized ZrO\({}_{2}\), in which at low concentrations of yttrium  
Figure 8: Dependences of apparent density and dielectric constant on mass concentration of dopant (**a**) and dependence of microhardness HV on dopant concentration (**b**). oxide, a sharp increase in the dielectric constant is observed; for crystals with tetragonal structures, an almost twofold increase is obtained [44; 46]. At C\({}_{\rm w}\) = 0.15 and C\({}_{\rm w}\) = 0.20, the determining factor in the dielectric permittivity is porosity, which can be seen from the curves in Figure 8. However, at the highest concentration of the dopant, the dielectric permittivity increases to 21, which is close to the maximum value for the series of the studied ceramics. This effect can be explained by the phase composition of the ceramics, which, according to Table 3, has a 64.58% cubic phase and a 34.42% CaZrO\({}_{3}\) phase. Calcium zirconate has a perovskite structure with a non-centrosymmetric crystal lattice, which gives rise to a large dipole moment in the cell [47].