## 3 Results and Discussion  
### Structural Analysis  
Figure 1 illustrates an X-ray diffractogram (XRD) of the spinel MnCe\({}_{x}\)Fe\({}_{2-x}\)O\({}_{4}\) (x = 0.0, 0.1, 0.3 and 0.5) NCs in the 20 range of 20-80\({}^{\circ}\) at RT. With the presence of narrow and high diffraction peaks, the patterns show good crystallinity. The existence of high diffraction peaks corresponding to crystal planes (111), (220), (311), (222), (400), (422), (511), and (442) indicate that the cubic spinel phase is present. The XRD analysis shows the formation of a single-phase spinel structure having Fd-3m space group [39]. The lattice parameter increases with dopant concentration, reaching the limit of 8.388 A as shown in Table 1. [PERSON]'s formula was used to measure the average crystallite size of the as synthesized samples between 22 and 24 nm in size. The values of lattice parameter and crystallite size are shown in Table 1.  
To deduce the lattice parameter of the MnFe\({}_{2}\)O\({}_{4}\) and Ce:MnFe\({}_{2}\)O\({}_{4}\) NCs, Equation (1) was used.  
\[a=d_{hkl}\sqrt{(h^{2}+k^{2}+l^{2})} \tag{1}\]  
where \(d_{hkl}\) is the inter-atomic space consistent to \(h\), \(k\), and \(l\) planes (Miller indices), respectively, and \(a\) is the lattice constant. Lattice parameter '\(a\)' resembles the spinel cubic structure (Figure 2). The lattice parameter '\(a\)' value was calculated for MnFe\({}_{2}\)O\({}_{4}\) and Ce doped MnFe\({}_{2}\)O\({}_{4}\) NCs and it was found to be 8.348 A and 8.388 A, respectively [40]. The increase of '\(a\)' is due to doping of larger ionic size Ce ions (0.92 A) in place of the lower ionic size Fe cations (0.67 A), which obeys [PERSON]'s law [41; 42; 43].  
\begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{**Sample Code**} & \multirow{2}{*}{
\begin{tabular}{c} \(L\) **(nm) by [PERSON]** \\ **[PERSON]** \\ \end{tabular} } & \multicolumn{2}{c}{_D_ **(nm) by**} & \multirow{2}{*}{_a_ **(Å)**} \\  & & **Williamson–Hall Plot** & \\ \hline MnFe\({}_{2}\)O\({}_{4}\) & 26.48 & 27.13 & 8.359 \\ MnCe\({}_{0.1}\)Fe\({}_{1.9}\)O\({}_{4}\) & 25.76 & 26.47 & 8.362 \\ MnCe\({}_{0.3}\)Fe\({}_{1.7}\)O\({}_{4}\) & 23.98 & 24.56 & 8.375 \\ MnCe\({}_{0.5}\)Fe\({}_{1.5}\)O\({}_{4}\) & 24.33 & 25.84 & 8.386 \\ \hline \end{tabular}
\end{table}
Table 1: The lattice parameter (a, Å) and crystallite size (L, nm) of spinel MnCe\({}_{x}\)Fe\({}_{2-x}\)O\({}_{4}\) (x = 0.0, 0.1, 0.3 and 0.5) NCs.  
Figure 1: Powder XRD patterns of spinel MnCe\({}_{x}\)Fe\({}_{2-x}\)O\({}_{4}\) (x = 0.0, 0.1, 0.3, and 0.5) NCs.  
The particle size (D) was obtained by employing the Williamson-Hall (W-H) plot, as shown in Equation (2):  
\[\frac{\beta\cos\theta}{\lambda}=\frac{k}{D}+\frac{4\epsilon\sin\theta}{\lambda} \tag{2}\]  
where \(k\) is the constant and strain associated with samples signified by \(\epsilon\). The W-H plot is 4 sin\(\theta/\lambda\) vs. \(\beta\)cos\(\theta/\lambda\), wherever the interrupt (k/D) is used to control the actual D. The W-H plot for MnFe\({}_{2}\)O\({}_{4}\) and Ce doped MnFe\({}_{2}\)O\({}_{4}\) NCs is represented in Figure 3 and the D size attained by the W-H plot method is greater when compared with the L size obtained  
Figure 2: Crystallite size (**a**) and lattice constant (**b**) values of spinel MnCe\({}_{\rm x}\)Fe\({}_{2-\rm x}\)O\({}_{4}\) (x = 0.0, 0.1, 0.3 and 0.5) NCs.  
by the Scherer method. The variance is, chiefly, due to participation of the strain module in the plot of the W-H route. It can be perceived that, with the increase in Ce ions (x), the L size shrinks, whereas an escalation in the '_a_' can also be detected.