## 3 Results
### Catalyst Characterization
#### 3.1.3 Optical and Dielectric Proprieties  
The optical characteristics of the ZnBi\({}_{2}\)O\({}_{4}\) nanostructure were investigated using UV-vis diffuse reflectance spectroscopy (DRS). The spectral distribution of T(\(\lambda\)) and R(\(\lambda\)) were studied for ZBO in the wavelength range (200-900 nm) (Figure S3 in the Supplementary Data). The complex refractive index n and the dielectric complex \(\hat{\varepsilon}\) characterize the optical properties of any solid material. They are presented in Equation (3) [47; 48]. \[\hat{\mathrm{n}}=\mathrm{n}(\lambda)+\mathrm{i}\mathrm{K}(\lambda);\ \hat{ \varepsilon}=\varepsilon\ \mathrm{r}\ (\lambda)+\mathrm{i}\mathrm{i}\mathrm{i}\ (\lambda) \tag{3}\]  
The imaginary and real parts of complex refractive n are the refractive index n(\(\lambda\)) and absorption index (extinction) k(\(\lambda\)) and for the dielectric complex \(\hat{\varepsilon}\), are the normal dielectric constant \(\varepsilon\mathrm{r}(\lambda)\) and the absorption associated with radiation by free carrier \(\varepsilon\mathrm{i}\ (\lambda)\). The extinction index k(\(\lambda\)) is calculated in terms of absorption \(\alpha\) with the following equation [49]:  
\[\mathrm{K}(\lambda)=\frac{\alpha(\
u)\lambda}{4\pi};\ \alpha(\
u)=\alpha_{0} \mathrm{e}^{\mathrm{h}\
u/\mathrm{E}_{\mathrm{u}}} \tag{4}\]  
where \(\lambda\), \(\alpha(\
u)\), \(\alpha_{0}\), h, \(\mathrm{E}_{\mathrm{u}}\) and n are wavelength, the absorption coefficient, a constant, the Plank's constant (6.63 \(\times\) 10\({}^{-34}\) J/s), the Urbach energy and the light frequency respectively. And the refractive index n(\(\lambda\)) is calculated from the relation [50]:  
\[\mathrm{n}\ (\lambda)=\frac{\left(1+\mathrm{R}\ (\lambda)+\sqrt{4\mathrm{R}\ ( \lambda)-\left(1-\mathrm{R}\ (\lambda)\right)^{2}\mathrm{K}\ (\lambda)^{2}}\right.}{1-\mathrm{R}\ (\lambda)} \tag{5}\]  
The spectral dependences of both n (\(\lambda\)) and K (\(\lambda\)) as a function of wavelength are shown in Figure S4 in the supplementary data. The refractive index n (\(\lambda\)) decreases at the lower wavelengths and increases in the visible spectrum. This favors the optical properties in the UV region. For the extinction coefficient K (\(\lambda\)), the values are very small at the visible spectrum but increase at the lower wavelengths, which indicates that ZBO has a low dielectric loss in the UV region. The extinction coefficient attains a peak at a wavelength \(\lambda\) = 367 nm that is assigned to the fundamental energy gap. For the dielectric real and imaginary parts, the normal dielectric constant \(\varepsilon\mathrm{r}(\lambda)\) and the absorption associated of radiation by free carrier \(\varepsilon\mathrm{i}\ (\lambda)\) are calculated from the refractive index n(\(\lambda\)) and absorption index K (\(\lambda\)) using the following equation [51]:  
\[\varepsilon\mathrm{r}\ (\lambda)=\mathrm{n}(\lambda)^{2}+\mathrm{K}\ (\lambda)^{2};\ \varepsilon\mathrm{i}\ (\lambda)=2\ \mathrm{n}(\lambda)\ \mathrm{K}\ (\lambda) \tag{6}\]  
Figure 3: (a) SEM (**b**) Energy dispersive X-ray spectrum of ZnBi\({}_{2}\)O\({}_{4}\). The variations of the dielectric constants \(\varepsilon\)r and \(\varepsilon\)i as functions of the energy (hv) are shown in Figure 4. The optical bandgap Eg value can be determined from this figure by extrapolating a straight line from the cross-point between the curves of the real and imaginary parts of the dielectric constant to the \(x\)-axis [51], which is equal to about 2.79 eV.