## 3 Numerical Results and Discussion
### Dependence of the Band Gap Energy on Temperature, SPI and EPI
It can be seen from Figure 5, curve 1, that as the SPI value increases at a fixed temperature for \(I>0\) (for example, for EuO) the band gap width decreases, i.e., a red shift is observed at the optical absorption edge. We can conclude the following:  
1. For negative values of the SPI constant \(R<0\), an increase in its absolute magnitude results in a decrease in the effective exchange interaction between the spins. This, at a constant temperature, leads to a reduction in the magnetization value \(\langle S^{z}\rangle\) as indicated by expression (12). Consequently, there is a shift towards lower energies of the valence zone (VZ) peak and an upward shift of the bottom of the conductivity zone (CZ). This leads to an increase in the width of the band gap \(E_{g}\) with increasing \(\mid R\mid\). 2. If \(R>0\), an increase in the spin-phonon interaction constant enhances the effective exchange interaction between the spins. This, at a constant temperature, causes an increase in the magnetization \(\langle S^{z}\rangle\) (see Equation (12)). Consequently, there is a downward shift of the VZ peak and a shift towards higher energies of the bottom of the CZ. This results in a reduction in the band gap \(E_{g}\). From Figure 5, for curve 2, it can be seen that with increasing the SPI value at a fixed temperature for \(I<0\) (for example UTe), the width of the band gap \(E_{g}\) increases, i.e., a blue shift is observed at the optical absorption edge. We have observed the following:  
1. For negative values of the SPI constant (\(R<0\)), as previously noted, an increase in its absolute magnitude will decrease the magnetization value \(\langle S^{z}\rangle\). According to Equation (12), this results in a shift towards lower energies of the VZ peak and an upward shift of the bottom of the CZ. This leads to a decrease in the width of the band gap \(E_{g}\) with increasing \(\mid R\mid\). 2. If \(R>0\), as the value of the SPI constant increases, the magnetization value \(\langle S^{z}\rangle\) increases, too. This leads to a shift towards higher energies of the bottom of the CZ and a downward shift of the VZ peak. This results in an increase in the width of the band gap \(E_{g}\) with enhancing \(R\). Figure 6 shows that by increasing the EPI constant \(A\) at a fixed temperature, the band gap width \(E_{g}\) decreases. From Equation (12), we can obtain the expression which determines the influence of \(A\) on the band gap width \(E_{g}\). It has the form \((2A^{2}/\omega_{0})(n_{\uparrow}-n_{\downarrow})\), where \(n_{\uparrow}\) and \(n_{\downarrow}\) are the number of conduction electrons in the spin-up and spin-down bands, respectively. \((n_{\uparrow}-n_{\downarrow})\) as a function of \(A\) decreases with increasing the EPI constant (see Inset in Figure 6), which explains the decrease in \(E_{g}\) as \(A\) increases. It is logical to expect that the thermal fluctuations in the system would lead to a decrease in the energy gap between the two subbands of the up and down spins. It can be seen from Figures 5 and 6 that in the case of \(I>0\), the SPI and EPI have the same influences on the band gap width. The increase in the values of \(R\) and \(A\) leads to a decrease in \(E_{g}\), i.e., to a red shift of the optical absorption edge. For \(I<0\), however, the two interactions are in competition, and depending on which of them prevails, we can expect both an increase and a decrease in \(E_{g}\) with increasing the \(R\) and \(A\) values, i.e., the possibility of both red and blue shift of the optical absorption edge. Let us emphasize that in a previous paper [40], we discussed the influence of the Coulomb interaction \(v\) on the band gap width and observed that increasing \(v\)\(E_{g}\) decreases. Within the framework of this article, this means the following:  
1. When \(I>0\) and \(R<0\), they will still have the same influence on the width of the band gap, i.e., an increase in \(\mid R\mid\) and \(v\) will result in a blue shift of the absorption edge.