## 3 Results and Discussion
### Structural Parameters
It is observable that the experimental points can be fitted with linear functions and consequently, the size effect is absent (\(L>3000\) A). As it can be clearly observed, the volume of the unit cell increases as the concentration of Ce cations is increased (see Figure 4). For \(x=0.09\) a change of the unit cell symmetry, from monoclinic to tetragonal, takes place. The change of the symmetry is associated with a sudden decrease of the volume associated (see Figure 4). The slope of the linear functions decreases as the concentration of Ce ions is increasing. This behavior signifies that microstrains in the investigated samples are decreasing with the concentration of Ce ions, as shown in Figure 5.  
First sample, for which \(x=0.0\), have the baddeleyite structure [32]. The modification of the crystal structure due to the substitution of Zr ions with Ce ions is related to the variation of the observed distance between Zr-O ions (see Table 2 and Figure 6). Cerium cations have two possible valence: +3 and +4. On the same type of places Ce\({}^{3+}\)/Ce\({}^{4+}\) have a larger ionic radius as Zr\({}^{4+}\) cations (\(r_{\rm Ce3+}=1.143\) A; \(r_{\rm Ce4+}=0.97\) A; \(r_{\rm Zr4+}=0.84\) A, for \(C.N.=8\), after [PERSON] [31]). The substitution of Zr\({}^{4+}\) with Ce\({}^{4+}\)/Ce\({}^{3+}\) should lead to an increase of the lattice constants, of the unit cell volume and of the microstrains. On other hand, the increase of the lattice energy can cause a change in crystal structure and can sit at the \begin{table}
\begin{tabular}{c|c|c|c|c|c|c} \hline \multirow{3}{*}{**Atomic Parameters**} & \multicolumn{4}{c|}{**Ce Concentration**} \\ \cline{2-7}  & \(x=0\) & \(x=0.09\) & \(x=0.13\) & \(x=0.17\) \\ \cline{2-7}  & **Monoclinic** & **Monoclinic** & **Tetragonal** & **Tetragonal** & **Tetragonal** \\  & **P2/c** & **P2/c** & **P4/nmc** & **P4/nmc** & **P4/nmc** \\ \hline \multirow{6}{*}{Zr/Ce} & \(x\) & \(0.2752\pm 0.0001\) & \(0.2683\pm 0.0004\) & \(0.7500\) & \(0.75000\) & \(0.75000\) \\ \cline{2-7}  & \(y\) & \(0.0405\pm 0.0002\) & \(0.0364\pm 0.0003\) & \(0.2500\) & \(0.25000\) & \(0.25000\) \\ \cline{2-7}  & \(z\) & \(0.2084\pm 0.0001\) & \(0.2084\pm 0.0004\) & \(0.7500\) & \(0.75000\) & \(0.75000\) \\ \cline{2-7}  & Biso (Å\({}^{2}\)) & \(0.116\pm 0.000\) & \(0.287\pm 0.000\) & \(0.283\pm 0.00\) & \(0.143\pm 0.00\) & \(0.169\pm 0.00\) \\ \cline{2-7}  & Occupation & \(1.0\) & \(0.903/0.097\) & \(0.89/0.09\) & \(0.87/0.13\) & \(0.83/0.17\) \\ \hline \multirow{6}{*}{O1} & \(x\) & \(0.0710\pm 0.0001\) & \(0.0533\pm 0.0005\) & \(0.2500\) & \(0.25000\) & \(0.25000\) \\ \cline{2-7}  & \(y\) & \(0.3341\pm 0.0002\) & \(0.3146\pm 0.0005\) & \(0.2500\) & \(0.25000\) & \(0.25000\) \\ \cline{2-7}  & \(z\) & \(0.3452\pm 0.0003\) & \(0.3604\pm 0.0004\) & \(0.0476\pm 0.0002\) & \(0.0474\pm 0.0001\) & \(0.0467\pm 0.0001\) \\ \cline{2-7}  & Biso (Å\({}^{2}\)) & \(0.096\pm 0.003\) & \(0.59\pm 0.04\) & \(0.37\pm 0.03\) & \(0.419\pm 0.001\) & \(0.521\pm 0.002\) \\ \cline{2-7}  & Occupation & \(1.0\) & \(1.0\) & \(1.0\) & \(1.0\) & \(1.0\) \\ \hline \multirow{6}{*}{O2} & \(x\) & \(0.4488\) (\(1\)) & \(0.4500\pm 0.0004\) & - & - & - \\ \cline{2-7}  & \(y\) & \(0.7570\) (\(3\)) & \(0.7513\pm 0.0005\) & - & - & - \\ \cline{2-7}  & \(z\) & \(0.4799\) (\(2\)) & \(0.4735\pm 0.0004\) & - & - & - \\ \cline{2-7}  & Biso (Å\({}^{2}\)) & \(0.137\) (\(3\)) & \(0.59\pm 0.04\) & - & - & - \\ \cline{2-7}  & Occupation & \(1.00\) & \(1.00\) & - & - & - \\ \hline \end{tabular}
\end{table} 
Table 2: Atomic positions and isotropic thermal parameters of Zr\({}_{1+}\)Ce\({}_{x}\)O\({}_{2}\)(\(x=0-0.17\)) calculated using the Rietveld method from neutron diffraction data. P2\({}_{1}\)/c: Zr (4e) (\(xyz\)); O1 (4e) (\(x\),\(y\),\(z\)); O2 (4e) (\(x\),\(y\),\(z\)); P4\({}_{2}\)/nmc: Zr (2a) (0.75, 0.25, 0.75); O (4d) (0.25, 0.25, \(z\)) (ND data). XRD data were obtained from the surface of the bulk samples and correspond to the structure of a thin layer (see Table 3). The common phase for these samples has a tetragonal structure, P4/nmc.