## 3 Results and Discussion
### Resonant Ultrasound Spectroscopy  
In order to determine mechanical properties of a compact specimen (in our case a ceramic cylinder), the resonance structure is scanned by RUS [22; 23; 29; 33]. This analysis comes in completion and adds more information related to the structural analysis performed above. In comparison to other ultrasound methods, resonant techniques are particularly interesting because they allow for easy and inexpensive detection of both internal and surface defects with a single test and it has a suite of advantages, among them is its applicability to small volume specimens. RUS is based on the principle that the mechanical resonant response of solids depend strongly on its elastic moduli, shape, and density. Resonant (or natural) frequencies of a system can be either measured or calculated by solving equations of motion for the known shape [34]. The reverse is also true: if resonant frequencies of an object are known, its elastic properties can be determined [23; 34; 35]. \begin{table}
\begin{tabular}{c|c|c|c|c|c|c} \hline \multirow{3}{*}{**Atomic Parameters**} & \multicolumn{6}{c}{**Ce Concentration \(x\)**} \\ \cline{2-7}  & \multicolumn{2}{c|}{\(x=0.09\)} & \multicolumn{2}{c|}{\(x=0.13\)} & \multicolumn{2}{c|}{\(x=0.17\)} \\ \cline{3-7}  & & **P2\(\downarrow\)/c** & **P4\(/\)nmc** & **P2\(\downarrow\)/c** & **P4\(/\)nmc** & **P4\(/\)nmc** \\ \hline \multirow{3}{*}{Zr/Ce} & \(x\) & \(0.2953\pm 0.0002\) & 0.75000 & \(0.2885\pm 0.0001\) & 0.75000 & 0.75000 \\ \cline{2-7}  & \(y\) & \(0.0287\pm 0.0001\) & 0.25000 & \(0.0227\pm 0.0001\) & 0.25000 & 0.25000 \\ \cline{2-7}  & \(z\) & \(0.2277\pm 0.0003\) & 0.75000 & \(0.2463\pm 0.0003\) & 0.75000 & 0.25000 \\ \cline{2-7}  & \(\mathrm{B}_{\mathrm{uo}}\) (Å\({}^{2}\)) & \(0.400\pm 0.000\) & \(0.407\pm 0.000\) & \(0.412\pm 0.000\) & 0.410(0) & 0.433(0) \\ \hline \multirow{3}{*}{O1} & \(x\) & \(0.0640\pm 0.0001\) & 0.25000 & \(0.0630\pm 0.0001\) & 0.25000 & 0.25000 \\ \cline{2-7}  & \(y\) & \(0.3291\pm 0.0003\) & 0.25000 & \(0.3301\pm 0.0003\) & 0.25000 & 0.25000 \\ \cline{2-7}  & \(z\) & \(0.3291\pm 0.0001\) & \(0.04654\pm 0.0001\) & \(0.3281\pm 0.0001\) & \(0.0467\pm 0.0001\) & \(0.0203\pm 0.0001\) \\ \cline{2-7}  & \(\mathrm{B}_{\mathrm{uo}}\) (Å\({}^{2}\)) & \(0.704\,\pm 0.001\) & \(0.246\pm 0.003\) & \(0.860\pm 0.002\) & \(0.419\pm 0.001\) & \(0.519\pm 0.001\) \\ \hline \multirow{3}{*}{O2} & \(x\) & \(0.4251\pm 0.0001\) & - & \(0.4291\pm 0.0001\) & - & - \\ \cline{2-7}  & \(y\) & \(0.7561\pm 0.0002\) & - & \(0.7610\pm 0.0001\) & - & - \\ \cline{2-7}  & \(z\) & \(0.4844\pm 0.0003\) & - & \(0.4801\pm 0.0001\) & - & - \\ \cline{2-7}  & \(\mathrm{B}_{\mathrm{uo}}\) (Å\({}^{2}\)) & \(0.800\pm 0.000\) & - & \(0.920\pm 0.000\) & - & - \\ \hline \multicolumn{3}{c}{d\({}_{\mathrm{Zr-O1}}\)(Å)} & 2.1946 & - & 2.2210 & - & - \\ \hline \multicolumn{3}{c}{d\({}_{\mathrm{Zr-O2}}\)(Å)} & 2.2135 & - & 2.2387 & - & - \\ \hline \multicolumn{3}{c}{d\({}_{\mathrm{Zr-O}}\)(Å)} & - & 2.1933 & - & 2.1987 & 2.2257 \\ \hline \end{tabular}
\end{table} 
Table 4: Atomic positions, isotropic thermal parameters, and Zr-O distances of Zr\({}_{1\cdot x}\)Ce\({}_{x}\)O\({}_{2}\) (\(x=0.9\)–0.17) calculated using the Rietveld method from XRD data. (P 2\(\downarrow\)/c: Zr (4e) (\(xyz\)); O1 (4e) (\(xy\),\(z\)); O2 (4e) (\(x\),\(y\),\(z\)); P 4\({}_{2}\)/nmc: Zr (2a) (0.75, 0.25, 0.75); O (4d) (0.25, 0.25, \(z\)). Inhomogeneity in an object may be identified from a resonant frequency spectrum by resonant frequency shifts, peak splitting, increases in peak width, and changes in amplitude. The method is based on the estimation of resonant eigenfrequencies [27], based on an eigenvalue and eigenfunction method [26]. For these, we have used an equipment configuration as the one shown in Figure 1. The ceramic sample is supported by two piezoelectric ultrasound transducers, for emission and reception respectively, placed at opposite edges of the ceramic cylinders.