## 3 Results and Discussion
### Optical Properties
#### 3.5.1 UV-Vis-NIR Analysis with Tauc's Plots  
Figures 6 and 7 display absorption spectra in the 400-2200 nm range, with transitions numbered. B0H without holmium does not show peaks in Figure 6, but the strength of the peaks grew with the increased concentration of Ho\({}^{3+}\)-ions in the B0H set. The ground condition (\({}^{5}\)I\({}_{8}\)) of Ho\({}^{3+}\)-ions with a 4f transition was detected in seven transitions in the UV-Vis-NIR area, as can be seen in the standard spectra (Figure 7) of the B05H sample. Peak transitions of 417,450, 486, 538, 642, 1152 and 1948 nm are assigned to (\({}^{5}\)G, \({}^{3}\)G)\({}_{5}\), \({}^{5}\)G\({}_{6}\), \({}^{5}\)F\({}_{3}\), \({}^{5}\)S\({}_{2}\), \({}^{5}\)F\({}_{5}\), \({}^{5}\)I\({}_{6}\) and \({}^{5}\)I\({}_{7}\), respectively [13; 18; 29; 30]. Among all transitions, \({}^{5}\)G\({}_{6}\) is a hypersensitive transition that is specific to the surrounding atmosphere of the Ho\({}^{3+}\)-ions; it follows the selection rules \(\mid\Delta S\mid=0\), \(\mid\Delta L\mid\leq 2\) and \(\mid\Delta J\mid\leq 2\). Electric-dipole interactions cause transitions in the 400-700 nm range, while magnetic-dipole interactions cause transitions around 1950 nm [28; 29; 30; 31]. The absorption co-efficient was estimated using,  
\[\alpha(h\
u)=\frac{2.303\times A}{t} \tag{7}\]  
where, \(A\) is absorbance recorded for the B0H set and \(t\) is the thickness of the B0H set. Figure 6: Absorption spectra in UV-Vis region for B0H set. Details on electronic states may be revealed by the higher energy spectral region, referring to interband electronic transitions. Photon absorption is initially utilized to drive electrons from a full band into an empty band, and the coefficient of absorption increases afterwards. This abrupt change in \(\alpha(h\
u)\), and the difference in energy is the resultant energy, is the fundamental absorption edge [32; 33]. [PERSON] and [PERSON] suggested a Tauc's [34; 35] relation between \(\alpha(h\
u)\) and the incident radiations energy (photon energy), which can be expressed as,  
\[\alpha(h\
u)=\frac{\left[t_{p}\left(h\
u-E_{g^{opt}}\right)^{n}\right]}{h\
u} \tag{8}\]  
where \(t_{p}\) is the band tailing parameter (constant), \(E_{g}^{opt}\) is the optical band gap energy, and n is depending on the form of transition (direct or indirect). The transitions with \(n=2\), \(3\), \(1/2\), and \(1/3\) will be indirect allowed, indirect forbidden, direct allowed, and direct forbidden, respectively. This n is often affected by the material's composition, such as crystalline or amorphous. The indirect transition curve is extrapolated in the linear component using photon energy \(\
u\)/s \((\alpha h\
u)^{1/2}\) to achieve optical energy gap (\(E_{g^{opt}}\)) and the estimated values are tabulated in Table 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Samples**} & **Indirect Band Gap (eV)** & **Urbach Energy (eV)** & **Steepness Parameter** \\  & **(\(\pm\)0.01)** & **(\(\pm\)0.01)** & **(\(\pm\)0.01)** \\ \hline B0H & 2.50 & 0.34 & 0.07 \\ \hline B01H & 2.89 & 0.23 & 0.11 \\ \hline B03H & 2.88 & 0.22 & 0.11 \\ \hline B05H & 2.78 & 0.26 & 0.09 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of Optical parameter such as Indirect band gap, Urbach energy, Steepness parameter for B0H set of samples. Figure 7: Absorption spectra in UV-Vis-NIR region with assigned transitions for B0H sample. Localized states with expanded optical band gaps called exponential tails, also known as Urbach tails, occur in amorphous materials (disordered structure). In an absorption coefficient curve, this tail is above the optical band gap. The Urbach analytical rule [33] expresses the relationship between the absorption co-efficient and photon energy as follows:  
\[\alpha(h\
u)=\alpha(h\
u)_{o}exp\frac{h\
u}{\Delta E} \tag{9}\]  
Here \(\alpha(\
u)_{o}\) is a constant and \(\Delta E\) is known as Urbach energy. The straight-line equation can be obtained by adding logarithm to both sides of the above equation.