## 3 Results and Discussion  
### XRD Analysis  
Figure 1 illustrates the X-ray diffraction patterns of the prepared ZnO NCs. According to the card JCPDS 01-089-7102, all the diffracted peaks are typical to the hexagonal wurtzite structure of ZnO [15; 16]. The sharp and intense peaks indicate the good crystalline quality of the prepared ZnO samples. We notice the absence of new peaks for the doped ZnO NCs which confirms the absence of new phases and indicates the insertion of both the Eu\({}^{3+}\) and Na\({}^{+}\) ions into the ZnO lattice without any amorphous component and other additional crystalline phases. Therefore, the hexagonal phase of the oxide obtained is not affected by the doping process. Similar results were obtained for other ZnO doping with Mg [17], Sb [18] and La-Sm-Er [19]. Moreover, the reduced intensity of the peaks and their broadening with further doping are indications of the important stress and strain applied to the ZnO NCs related to the higher mismatch between the ionic radius of the dopants Eu\({}^{3+}\) (0.095 nm) and Na\({}^{+}\) (0.102 nm) with respect to Zn (0.074 nm) [20]. Figure 1: XRD patterns of undoped and doped ZnO NCs. The lattice parameters \(a\) and \(c\), as well as the volume V of the unit cell were calculated using the following expression [21]:  
\[\frac{1}{d^{2}}=\frac{4}{3}\left(\frac{h^{2}+hk+k^{2}}{a^{2}}\right)+\frac{l^{2}} {c^{2}}\]  
\[V=\frac{\sqrt{3}}{2}\;a^{2}c=0.866\;a^{2}c\]  
According to Table 1, there is a slight variation in the values of \(a\) and \(c\) which can be attributed to the successful insertions of Eu\({}^{3+}\) and Na\({}^{+}\) ions in the ZnO lattice [22]. In addition, the \(c/a\) ratio remains constant indicating that there is no change in the crystal structure following the incorporation of doping ions (Eu\({}^{3+}\) and Na\({}^{+}\)). This value is in good agreement with the standard one (1.6) obtained for ZnO [23]. Figure 2 shows a shift of the peaks toward high diffraction angle, which confirms the effective insertion of the Eu\({}^{3+}\) and Na\({}^{+}\) ions into the ZnO crystal lattice. On the other hand, since the ionic radii of doping ions are greater with respect to that of Zn\({}^{2+}\), some of the Eu\({}^{3+}\) and Na\({}^{+}\) ions can occupy interstitial positions in the ZnO lattice [24]. The crystallite size \(D\) is calculated using the Debye Scherrer equation:  
\[D=\frac{k\;\lambda}{\beta\text{cos}\theta}\]  
where \(k\) is the shape factor, \(\lambda\) is the X-ray wavelength, \(\beta\) is the peak FWHM, and \(\theta\) is the diffracting angle. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Samples** & **ZnO** & **ZnEu** & **ZnEuNa1** & **ZnEuNa2** & **ZnEuNa3** & **ZnEuNa4** & **ZnEuNa5** \\ \hline \(d_{100}\) (Å) & 2.824 & 2.815 & 2.822 & 2.793 & 2.792 & 2.797 & 2.800 \\ \hline \(d_{002}\) (Å) & 2.612 & 2.606 & 2.609 & 2.585 & 2.585 & 2.594 & 2.591 \\ \hline \(a\) (Å) & 3.261 & 3.251 & 3.258 & 3.225 & 3.224 & 3.230 & 3.233 \\ \hline \(c\) (Å) & 5.225 & 5.212 & 5.219 & 5.171 & 5.171 & 5.188 & 5.182 \\ \hline \(c/a\) & 1.602 & 1.603 & 1.601 & 1.603 & 1.603 & 1.606 & 1.602 \\ \hline \(V\) (Å\({}^{3}\)) & 48.142 & 47.712 & 47.994 & 46.598 & 46.558 & 46.884 & 46.912 \\ \hline \end{tabular}
\end{table} 
Table 1: Lattice parameters and volume V of undoped and doped ZnO NCs. Figure 2: Evolution of the preferred orientation peak (1 0 1) with the doping concentration. An estimate of the average crystallite size was made for the first three most intense peaks (Table 2). The reduction in the crystallite size following Eu\({}^{3+}\) and Na\({}^{+}\) incorporation indicates the insertion of the rare earth ions may alter the nucleation of the ZnO nanocrystallites and as the mismatch is higher only a small portion of the dopant is tolerated by ZnO. Thus, the excess of Eu\({}^{3+}\) dopant forms Eu-oxide clusters embedded in ZnO, consisting of structures with four or eight Eu atoms, as revealed recently from DFT calculations [25].