## 3 Results and Discussion  
### Structural Parameters  
The experimental results obtained from X-ray and neutron data emphasize that the Zr\({}_{1\textrm{-}x}\)Ce\({}_{x}\)O\({}_{2}\) (\(x=0\)-0.17) samples do not have a homogeneous structure (see Figure 2). The sample containing pure ZrO\({}_{2}\) has a monoclinic (P2\({}_{1\textrm{/}}\)c) crystalline structure [10]. Once the concentration of Ce cations increases, the crystal structure changes, from monoclinic to tetragonal (P4\({}_{2\textrm{/}}\)nmc) (see Figure 2) with two formula units (\(Z=2\)) per unit cell. The ND diffractograms indicated that the sintered Zr\({}_{0.91}\)Ce\({}_{0.09}\)O\({}_{2}\) (\(x=0.09\)) contains both monoclinic and tetragonal phases, whereas when \(x>0.09\), the samples only contain tetragonal phases, with a unit cell volume increasing from 69.2 A\({}^{3}\) (\(x=0.13\)) to 69.7 A\({}^{3}\) (\(x=0.17\)) (see Figure 2 and Table 1). Figure 1: RUS Experimental setup schematic block diagram. Thin lines correspond to coaxial cable for sinusoidal wave transmission; thick lines are GPIB cable for digitalized data. Data were acquired at room temperature and refined using the Fullprof program [30,31], and shown in Figure 2. From the analysis of the neutron diffraction data, we were able to define with precision the phase composition, crystal lattice, and the atomic structure of each phase. Lattice parameters and atomic fractional positions, as well as background parameters, scale factors, occupancy, and isotropic thermal parameters were refined in this case (see Table 1). \begin{table}
\begin{tabular}{c|c|c|c|c|c} \hline \multirow{2}{*}{**Parameters**} & \multicolumn{4}{c}{\(x\) \\ \cline{2-6}  & \(x=0\) & \(x=0.09\) & \(x=0.13\) & \(x=0.17\) \\ \cline{2-6}  & **Monoclinic** & **Monoclinic** & **Tetragonal** & **Tetragonal** & **Tetragonal** \\  & **P2\({}_{1}\)/c** & **P2\({}_{1}\)/c** & **P4\({}_{2}\)/nmc** & **P4\({}_{2}\)/nmc** & **P4\({}_{2}\)/nmc** \\ \hline \(a\), (Å) & \(5.1453\pm 0.0001\) & \(5.2046\pm 0.0003\) & \(3.6236\pm 0.0002\) & \(3.6358\pm 0.0001\) & \(3.6452\pm 0.0002\) \\ \hline \(b\), (Å) & \(5.2091\pm 0.0001\) & \(5.2155\pm 0.0003\) & \(3.6236\pm 0.0002\) & \(3.6358\pm 0.0001\) & \(3.6452\pm 0.0002\) \\ \hline \(c\), (Å) & \(5.3116\pm 0.0001\) & \(5.3776\pm 0.0002\) & \(5.2163\pm 0.0003\) & \(5.2378\pm 0.0001\) & \(5.2471\pm 0.0002\) \\ \hline \(\beta\), (\({}^{\circ}\)) & \(99.225\pm 0.001\) & \(98.938\pm 0.0002\) & \(90.0\) & \(90.0\) & \(90.0\) \\ \hline \(V_{\rm cell}\)/Z, (Å\({}^{3}\)) & 35.130 & 36.05 & 34.247 & 34.619 & 34.860 \\ \hline \(Volume\)\(fraction\) & 1.0 & \(0.378\pm 0.004\) & \(0.622\pm 0.006\) & 1.00 & 1.00 \\ \hline \(R_{\rm up}\), \% & 9.5 & 14.3 & 10.4 & 10.4 \\ \hline \end{tabular}
\end{table} 
Table 1: Refined structural parameters of Zr\({}_{1-x}\)Ce\({}_{x}\)O\({}_{2}\) (\(x=0.0-0.17\)) at room temperature, obtained by processing the data collected using the High Resolution Fourier Diffractometer at the Joint Institute for Nuclear Research, Dubna, Russia (\(x\) represents the nominal Ce concentration). Figure 2: Diffraction patterns for the Zr\({}_{1-x}\)Ce\({}_{x}\)O\({}_{2}\) samples measured using HRFD at room temperature, the data were treated using a Fullprof program. The experimental and calculated diffractograms and the difference between calculated and observed diffractograms are shown. The difference is weighted by the mean-squares deviation for each point. Below each graph, ticks indicate the position of the calculate peaks for each phase. 1—monoclinic structure (P2\({}_{1}\)/c (SG 14)); 2—tetragonal structure (P4\({}_{2}\)/nmc (SG 137)). The \(R_{\rm wp}\) values indicate a good fit between the calculated and the experimental data (see Table 1). The refined atomic positions are indicated in Table 2. Temperature corresponding to the ions oscillations have been taken into account in the isotropic approximation. The neutron peaks broadening (W), related to the sample of ZrO\({}_{2}\), decreases as the concentration of Ce ions is increasing (see Figure 3).