## 4 Conclusions  
The substitution of the Zr with Ce in Zr\({}_{1\cdot x}\)Ce\({}_{x}\)O\({}_{2}\) leads to a change of the phase composition, a gradual transition from the monoclinic to tetragonal structure. Concerning the substitution of Zr with Ce we observed a difference of the phase composition between the surface layer of the sample and the phase composition of the bulk samples. We attributed this difference to the various oxygen concentration in the surface layer and in the bulk sample. The investigations performed by means of neutron diffractometry, which \"viewed\" a large volume of the samples, indicated a transition from the monoclinic to tetragonal phase for \(x=0.09\). On other hand, XRD investigations, which \"viewed\" only a thin layer have shown that the phase composition of the samples corresponding to \(x=0.09\) and \(0.13\) represents a mixture between a monoclinic phase and a tetragonal phase, the tetragonal phase concentration increasing with the increase of Ce concentration. The unit cell volumes of the tetragonal \begin{table}
\begin{tabular}{c|c|c|c|c|c|c} \hline \hline \multicolumn{1}{c|}{**Results Corresponding to the Natural Face**} & \multicolumn{3}{c}{**Results Corresponding to the Face after Cutting with Diamond Wire**} \\ \hline \(x\) & \(S\) (\(\mu\)m\({}^{2}\)) & \(H\) (\(\mu\)m) & \(HV\) (GPa) & \(x\) & \(S\) (\(\mu\)m\({}^{2}\)) & \(H\) (\(\mu\)m) & \(HV\) (GPa) \\ \hline
0.09 & 54.47 & 1.4 & \(9.0\pm 0.7\) & 0.09 & 53.39 & 1.4 & \(9.2\pm 0.7\) \\ \hline
0.13 & 37.60 & 1.2 & \(13.0\pm 0.1\) & 0.13 & 34.51 & 1.1 & \(14.2\pm 0.7\) \\ \hline
0.17 & 22.43 & 0.9 & \(21.9\pm 0.6\) & 0.17 & 27.18 & 1.0 & \(18.0\pm 0.7\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Results of microhard ness measurements (\(x\)—nominal Ce concentration in the samples; \(S\)—the surface area of the resulting indentation; \(h\)—indentation depth; \(HV\)—microhardness of the sample). \begin{table}
\begin{tabular}{c|c|c} \hline \hline \multicolumn{1}{c|}{**Nominal**} & \multicolumn{1}{c|}{**Chemical Composition at the Surface of**} & \multicolumn{1}{c}{**Chemical Composition at the Surface Obtained**} \\ \multicolumn{1}{c|}{**Composition**} & \multicolumn{1}{c|}{**the Samples**} & \multicolumn{1}{c}{**after Fracture**} \\ \hline Zr\({}_{0.91}\)Ce\({}_{0.09}\)O\({}_{2}\) & Zr\({}_{0.88\pm 0.03}\)Ce\({}_{0.12\pm 0.09}\)O\({}_{1.82\pm 0.03}\) & Zr\({}_{0.84\pm 0.03}\)Ce\({}_{0.16\pm 0.03}\)O\({}_{2.00\pm 0.03}\) \\ \hline Zr\({}_{0.85}\)Ce\({}_  
and monoclinic phases and Zr-O distances increase with the increase of Ce concentration, for the inner part of the samples. The calculated and measured densities increase also with the increase of Ce concentration. A maximum of the average size of coherent blocks and of the microstrain appear in the tetragonal phase for the sample with \(x\) = 0.13. The increase of Ce concentration leads also to an increase of the volumes of both monoclinic and tetragonal unit cells. From the variation of unit cell volume of the surface layer, compared with those corresponding to the unit cell volume of the sample core, we conclude that a small but systematic decrease of the large radii cation concentration takes place at the surface layer. An increase of the relative density and of the mechanical parameters (elasticity and shear moduli) was obtained with the increase of Ce concentration. RUS is a reliable technique, which emphasizes the eigenfrequency intervals which exhibit a slight change as a function of samples composition, for monitoring structural modifications related to sample density. Slight material anisotropy leads to splitting of the higher modes but not of the fundamental torsion mode. In the case of the studied samples, whose ratio is around the unit, the interpretation is favorable because the torsional mode is the lowest one, well separated from the others for \(\
u>0\), allowing immediate extraction of the shear modulus and its damping.