## 3 Results
### Catalyst Characterization
#### 3.1.3 Optical and Dielectric Proprieties
The Surface and Volume energy loss functions (SELF and VELF) were calculated using the dielectric constants as the relations [52]:  
\[\mathrm{SELF}=\frac{\varepsilon\mathrm{i}\left(\lambda\right)}{\left(\left( \varepsilon\mathrm{r}\left(\lambda\right)+1\right)^{2}+\varepsilon\mathrm{i} \left(\lambda\right)^{2}\right)};\;\mathrm{VELF}=\frac{\varepsilon\mathrm{i} \left(\lambda\right)}{\left(\varepsilon\mathrm{r}\left(\lambda\right)^{2}- \varepsilon\mathrm{i}\left(\lambda\right)^{2}\right)} \tag{7}\]  
The variations of the volume and the surface energy loss functions as functions of the energy (hv) are shown in Figure S5 in the Supplementary Data. The figure shows that both SELF and VELF are very small, especially at lower photon energies, almost equal to zero; this indicates that ZBO has a low dielectric loss. The dissipation factor \(\tan\)\(\delta\) can measure the loss rate of electrical and mechanical power mode in all-dielectric materials, usually in the form of heat [51]. for that, the dissipation factor \(\tan\)\(\delta\) can characterize the energy lost as heat in the spinel oxides, which is calculated from the relation [53]:  
\[\tan\;\delta=\frac{\varepsilon\mathrm{i}\left(\lambda\right)}{\varepsilon \mathrm{r}\left(\lambda\right)} \tag{8}\]  
Figure 5a shows the change of \(\tan\)\(\delta\) as an energy function (hv). A significant increase in the range (2.4-2.9 eV) is shown in the dependency, clearly indicating that the activation level is located within this region. The band-gap energy Eg of ZBO was determined 2.78 eV, which is in agreement with the dielectric index values; this indicated that ZBO has a significant optical characteristic in the UV region because its band-gap is smaller than ZnO, which is widely recognized by its efficacy in the UV region [54]. Figure 4: Dependence of the real and imaginary parts of the dielectric constant as a function of the energy. The dielectric relaxation time \(\tau\) allows to confirm the value of the optical bandgap Eg and also determine the limit of the recombination of the charge carrier, the relaxation time \(\tau\) is calculated from the relation [55]:  
\[|\tau|\,=\,\frac{\varepsilon_{\infty}-\varepsilon_{\tau}}{\omega\varepsilon_{ \mathrm{i}}} \tag{9}\]  
\(\varepsilon_{\infty}\) is calculated using Equation (10) [51] and by drawing \(\varepsilon_{\mathrm{r}}\) as a function of \(\lambda^{2}\), as showing in Figure S6 in the Supplementary Data:  
\[\mathrm{K}(\lambda)=\frac{\alpha(\
u)\lambda}{4\pi};\ \alpha(\
u)=\alpha_{0} \mathrm{e}^{\mathrm{h}\
u/\mathrm{E}_{\mathrm{u}}} \tag{4}\]0  
Figure 5b illustrated the dielectric relaxation time \(\tau\) as a function of the energy (h\(\
u\)). The cross-point in the curve of dielectric relaxation time at h\(\
u\) = 2.76 eV may be attributed to the band-to-band absorption because this maximum occurs at h\(\
u\) = Eg which corresponds to the energy gap for ZBO. This is in agreement with the dielectric index and the dissipation factor gap values. The optical bandgap value Eg can also be estimated and confirmed using the optical \(\mathrm{\ddot{o}opt}\) as well as electrical \(\mathrm{\sigma e}\) conductivities of ZBO, which are calculated using the absorption coefficient \(\alpha\) from the relations (11) [48]:  
\[\mathrm{K}(\lambda)=\frac{\alpha(\
u)\lambda}{4\pi};\ \alpha(\
u)=\alpha_{0} \mathrm{e}^{\mathrm{h}\
u/\mathrm{E}_{\mathrm{u}}} \tag{4}\]1  
where C is light velocity and \(\alpha\) the absorption coefficient. Figure S7 in the Supplementary Data shows the variation of the optical \(\mathrm{\ddot{o}opt}\) and electrical \(\mathrm{\sigma e}\) conductivities as a function of energy (h\(\
u\)). The cross-point between the curves in this figure confirmed once again the optical band gap Eg (\(\sim\)2.72 eV). This indicated that ZBO has good optical properties in the UV region, where the photons' conduction is the dominant mechanism.