## 3 Results  
### Characterization of Calcined Powders  
Figure 1a shows the XRD spectra of electrolyte powders where a single fluorite phase of ceria was detected. The peaks reflect the Fm3m cubic unit cell with crystalline direction 111, 200, 220, 311, 222, 400, 331 420, and 422. The ICDD card no. 01-075-0162 was employed as a reference during fit with MAUD software.  
As reported in Table 2, the improvement of copper leads to the reduction of crystallographic parameters as the unit cell, the theoretical density, and the crystallite size. The results are in agreement with [PERSON]'s law. The experimental data illustrated in Figure 1b, are aligned with [PERSON]'s theory even if the values are slightly higher than calculated from the semiempirical equation for ionic radii in eightfold coordination [26]:  
\[\mathrm{X}=0.022^{\circ}\Delta\mathrm{r}_{\mathrm{i}}+0.00015\Delta z_{i} \tag{2}\]  
where \(\Delta\mathrm{r}_{\mathrm{i}}\) is the ionic radii difference (in nm) between the dopant and Ce\({}^{4+}\) in 8-fold coordination and \(z_{i}\) is the difference in charge between the dopant and Ce\({}^{4+}\).  
Using the [PERSON]'s slope theory and the knowledge of dopants size and charge, it is possible to identify the moderately undersized acceptor dopants, most likely to induce low-temperature sintering in GDC, through either liquid phase sintering or heterogeneous doping. The dopants with a negative and moderate value of Vegard's slope (as the case of copper where Vegard's slope is -4.76 \(\times\) 10\({}^{-4}\) affect the reduction of sintering temperature with an upright effect on the densities due to the increase of the concentration of oxygen vacancies near the grain boundary. So, the results indicate an improvement of density during sintering reducing the sintering temperature.  
From XRD results, it is reasonable to think the inclusion of copper inside the crystal structure of doped ceria, as the dopant has a smaller ionic radius (0.73 A) compared to cerium (0.97 A). In fact, copper addition induces the lattice shrinkage ensuing the decrease in the lattice parameter with the  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Samples** & \begin{tabular}{c} **Lattice Parameter** \\ **(Å)** \\ \end{tabular} & \
\begin{tabular}{c} **Theoretical Density** \\ **(g/cm\({}^{-1}\))** \\ \end{tabular} &
\begin{tabular}{c} **Crystallite Size** \\ **(Å)** \\ \end{tabular} \\ \hline GDC & 5.42906 \(\pm\) 4.81\(\cdot\)10\({}^{-4}\) & 7.220 & 208.81 \\ \({}^{0.5}\)Cu & 5.42805 \(\pm\) 5.09\(\cdot\)10\({}^{-4}\) & 7.204 & 191.76 \\ \({}^{1}\)Cu & 5.42804 \(\pm\) 6.10\(\cdot\)10\({}^{-4}\) & 7.184 & 157.09 \\ \({}^{2}\)Cu & 5.42613 \(\pm\) 5.19\(\cdot\)10\({}^{-4}\) & 7.153 & 190.67 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Composition of electrolyte powders prepared.  
Figure 1: XRD spectra for **(a)** Cu-doped GDC powders and **(b)** Vegard modelincrease of Cu composition in GDC. This effect depends on the attainable molecular mixing of copper during the sol-gel synthesis which favors the dopant distribution and substitution in the lattice.  
The decrease in the crystallographic parameters along with the copper addition is predictable due to its lower molecular weight compared to cerium and gadolinium. The powders morphology is reported in the TEM micrographs of Figure 2. Hard agglomerated and quasi-spheroid crystals were detected. The shape of the powders can be easily predicted due to the synthesis route performed [26; 27]. In fact, sol-gel combustion synthesis leads to an irregular distribution even if the average particle size domain remains almost constant. Powders are arranged in spherical elements agglomerated in 10-20 nm small sub-units. However as reported in Figure 2d, the sample with 2 mol% of copper shows big spherical agglomerates, (1 to 5 \(\upmu\)m), shaped by many smaller elements of 10-20 nm in size. The small units have a particle size comparable with the calculated crystallite size reported in Table 2.