## 3 Results and Discussion  
The structural characteristics of undoped or RE-doped CeO\({}_{2}\) samples were determined by X-ray diffraction measurements and the corresponding diffraction patterns are shown in Figure 1. All diffraction lines are well indexed to the face-centred cubic (_fcc_) fluorite structure of CeO\({}_{2}\) (JCDD PDF 034-0394). No peaks belonging to any other phase have been observed in the XRD patterns, indicating the high structural purity of the samples. The effects of the dopant on the lattice parameters and crystallite sizes--calculated using the Scherrer method--are presented in Table 1.  
It has been demonstrated that the substitution of Ce\({}^{4+}\) ions with a larger ion radius dopant cation alters the local chemistry and induces an enlargement of the crystal lattice due to steric effects [49; 50; 51]. The substitution into the ceria lattice of octahedral Ce\({}^{4+}\) ions (Ce\({}^{4+}\) = 0.97 A) with a larger sized dopant (Gd\({}^{3+}\) = 1.053 A; Eu\({}^{3+}\) = 1.066 A and Y\({}^{3+}\) = 1.019 A [52]) induced a slight increase in the lattice parameter values. Regarding the crystallite sizes it can be noted that the small amount of dopant ions stimulates the nucleation and the growth processes of the nanocrystals. This tendency can be noticed for the crystallite sizes in the cases of Gd\({}^{3+}\) and Eu\({}^{3+}\). The introduction of Y\({}^{3+}\) into the ceria lattice led to a diminishing of the crystallite size. The morphological characterization of the samples has been achieved via TEM investigations--Figure 2. TEM images show the formation of pseudo-spherical nanoparticles with a high tendency to agglomerate. Analysis of these images disclosed that the particle sizes are below 10 nm, which is in good concurrence with the XRD results. This means that each particle is composed of only one crystallite. In the case of the Y-doped ceria sample, the presence of some aggregates with an acicular shape should be noticed. The changing of nanoparticles' shapes by introducing different types of dopants indicate that the crystallisation process can be tailored. The absorption spectrum of nanoparticles in the UV-Vis-NIR domain is affected by various factors such as optical band gap, oxygen vacancies, the presence of an impurity centre, etc. Figure 3a shows the absorbance spectra of pure and RE-doped ceria nanostructures and the spectra exhibit a strong absorbance in the UV region. In general, the absorption band below 400 nm is assigned to the transitions of charge transfer between O \(2p\) and Ce \(4f\) orbitals in cerium oxide [53]. Doping with RE caused a red shift in the absorbance band of ceria, indicating that \(4f\) or \(4d\) transition metals ions induce intermediate electronic energy levels between the O \(2p\) and Ce \(4f\) orbitals of cerium oxide [54]. To evaluate the energy band gap of the samples, Tauc's relation was applied to the absorption spectra. Figure 3b displays the Tauc plots. The energy band gaps of RE-doped CeO\({}_{2}\) nanoparticles are smaller \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Sample**} & **Crystallite Size** & **Lattice Parameter** & **Energy Band Gap** \\  & **D (nm) a** & **a (Å) a** & **Eg (eV)** \\ \hline Undoped CeO\({}_{2}\) & 8.16 & 5.4105 & 2.98 \\ CeO\({}_{2}\):Eu & 8.40 & 5.4130 & 2.93 \\ CeO\({}_{2}\):Gd & 8.40 & 5.4129 & 2.88 \\ CeO\({}_{2}\):Y & 7.79 & 5.4113 & 2.93 \\ \hline \hline \end{tabular} \({}^{\text{a}}\)Calculates from XRD.  
\end{table} 
Table 1: Average crystallite sizes (D) and lattice parameters (a) of undoped and doped ceria. Figure 1: X-ray diffraction patterns of pure ceria, CeO\({}_{2}\); Y-doped CeO\({}_{2}\), CeO\({}_{2}\):Y; Eu-doped ceria, CeO\({}_{2}\):Eu; and Gd-doped ceria, CeO\({}_{2}\):Gd. All reflections have been indexed according to the JCDD PDF 034-0394 for CeO\({}_{2}\) face-centred cubic fluorite structure.