## 3 Results and Discussion  
Dry ceria nanoparticles synthesized via the precipitation method can be seen in Figure 1a. Such nanoparticles were used as initial material for the thermal treatment method. These nanoparticles have a pale yellow color, which is distinctive for nanocrystalline cerium dioxide with a hydrated surface. Performance of the thermal treatment method led to both crystallite size changes and dehydration of the nanoparticles. Both of these effects had a major impact on the nanoparticles' color. The dehydration had a major impact at the beginning of the thermal treatment, which is documented in the STA measurements shown in Figure 1c. Based on the STA analysis, it was proved that nanocrystalline CeO\({}_{2}\) contained approximately 17 wt.% of water. The dehydration process ended at approximately 300 \({}^{\circ}\)C, as can be seen from both the DTA and TG curves. Due to this fact, it is reasonable to prepare the first sample at 300 \({}^{\circ}\)C and the remaining samples at higher temperatures. No color change could be observed when comparing the thermally treated samples; however, as is apparent from Figure 1, the nontreated sample was notably different from the heated ones. Due to the dehydration, the samples first turned brighter while, at higher temperatures, the effect of the crystallite growth turned the color to slightly pink. A summary of all prepared samples can be seen in Table 1.  
The diffraction pattern of the precipitated ceria is shown in Figure 2a. The diffraction pattern corresponded with ICDD 04-013-9853 (cubic structure, space group Fm-3m) and the ceria was in the form of a fluorite structure, as expected. There was apparently no or only a low amount of the amorphous phase left after the processing of the precipitated sample (Figure 2a). According to the Scherrer equation, the crystallite size was approximately 18 A for the (111) crystallographic plane. Let us note that the approximate crystallite size was calculated as an average of the values obtained for the individual crystal planes. The diffraction on the (111) plane was the most significant and hence the most relevant. It is essential to mention that the calculated results can be affected by some error, especially for those crystallographic planes whose reflections overlapped. The diffractions belonging to the (220) and (311) planes were clearly recognizable, but no more reflections could be seen due to the very small particle sizes. For sample TT-300 (Figure 2b), there were no significant changes in comparison to the non-treated sample, TT-25. According to the estimation made by the Scherrer equation, the average crystallite size was around 22 A. The average crystallite size of sample TT-600 (Figure 2c) reached 35 A. We saw a new peak at approximately 20 = 70\({}^{\circ}\), which was attributed to the (222) plane. The diffraction pattern of sample TT-900 is shown in Figure 2d. Here, the crystallite size value increased to 81 A. The last sample, TT-1100, is shown in Figure 2e. The final size of the TT-1100 crystallites was 358 A. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Temperature Used for Treatment** & \(\emptyset\) & **300 \({}^{\circ}\)C** & **600 \({}^{\circ}\)C** & **900 \({}^{\circ}\)C** & **1100 \({}^{\circ}\)C** \\ \hline Sample name & TT-25 & TT-300 & TT-600 & TT-900 & TT-1100 \\ Average crystallite size & 20 Å & 23 Å & 35 Å & 82 Å & 357 Å \\ \hline \end{tabular}
\end{table} 
Table 1: Overview of prepared samples. Figure 1: (**a**) Photography of ceria nanoparticles without thermal treating, (**b**) TT-1100 and (**c**) results of the simultaneous thermal analysis (STA) measurement. The temperature dependence of the crystallite size is shown in Figure 3. As mentioned, the crystallite sizes were calculated from the Scherrer equation, making use of all crystal planes with the most intense signals: (220), (111) and (200). The average crystallite size was calculated and is also visualized in Figure 3. In this figure, the non-treated sample TT-25 is not included because its crystallite sizes were too small.