## 3 Numerical Results and Discussion
### Influence of the SPI and EPI on the Phonon Spectrum of FMS  
As a next step, we will study the influence of the EPI and SPI on the phonon spectra. Therefore, we have calculated from Equation (10) the temperature dependence of the phonon energy of the phonon mode \(\omega\) = 445 cm\({}^{-1}\) for EuO [42]. It can be seen from Figure 9, curves 1 and 2, that the phonon energy decreases with increasing temperature in accordance with the experimental data [42; 43]. We have taken for EuO the SPI \(R<0\)[15]. At the Curie temperatures \(T_{\rm C}=\) 69 K, there is an anomaly due to the anharmonic SPI \(R\). We have studied the phonon energy changing both interactions. Increasing the SPI and EPI leads to reducing \(\omega\) (see Figures 9 and 10, curves 1 and 2). The contribution from the SPI is largest at low temperatures and vanishes near the phase transition temperature \(T_{\rm C}\) (where the magnetization \(M\) is zero), whereas the EPI remains finite for temperatures above \(T_{\rm C}\), \(T>T_{\rm C}\). We have calculated from Equation (10) also the temperature dependence of the phonon energy of the phonon mode \(\omega\) = 151 cm\({}^{-1}\) for EuSe [44] for \(R>0\)[15] and observed a different contribution of the SPI and EPI compared to EuO. The results are presented in Figures 9 and 10, curves 3 and 4. It can be seen that the phonon energy increases with increasing temperature \(T\). At the phase transition temperature \(T_{\rm C}\) = 4.6 K, there is a kink due to the anharmonic SPI. Decreasing the SPI leads to increasing the phonon energy \(\omega\), whereas increasing the EPI reduces it. This can be seen also from the analytical expression for the phonon energy (see Equation (10)). The SPI contribution is important at lower temperatures; it vanishes at the phase transition temperature \(T_{\rm C}\). The EPI is small at low temperatures but remains finite at higher temperatures as well as above \(T_{\rm C}\). Let us emphasize that our results are in very good agreement with the experimental data of [PERSON] et al. [23]. This is evidence that our model and method are appropriate for describing the properties of FMS. Figure 8: (Color online) Magnetic field dependence of the band gap energy for (1) EuO (\(I\) = 0.2 eV) and (2) UTe (\(I\) = \(-\)0.2) for \(R<0\), \(T=2\) K.  
Figure 7: (Color online) Magnetic field dependence of the band gap energy for (1) EuO (\(I\) = 0.2 eV) and (2) UTe (\(I\) = \(-\)0.2) for \(R>0\), \(T=2\) K.  
In systems with strong SPI, an anomalous temperature behavior of the phonon modes is characteristic. In the Inset in Figure 9, we present the dependence of the renormalized phonon modes for \(T=30\) K. For \(R<0\) with increasing \(\mid R\mid\) at a fixed temperature, a hardening of the phonon mode is observed, i.e., the phonon energy increases, whereas for \(R>0\) with increasing \(R\) at fixed temperature, a softening of the phonon mode is obtained, i.e., it decreases. The SPI renormalizes the phonon mode (see Equation (10)). The increase in the value of the EPI \(A\) leads to a softening of the phonon mode (see Inset in Figure 10). Thus, in the case of \(R<0\), the two mechanisms are in competition in order of their influence on the lattice oscillations. By \(R>0\), both mechanisms lead to the softening of the phonon mode. In the last Figure 11, the temperature dependence of the phonon damping \(\gamma\) for EuSe is shown, which represents the full width at half maximum (FWHM) of the Raman line. It can be seen that the damping \(\gamma\) increases with increasing temperature \(T\). Let us emphasize that the damping \(\gamma\) increases with \(T\) for both cases \(R>0\) and \(R<0\) because it is proportional to \(R^{2}\). There is again a kink at \(T_{C}\) due to the anharmonic SPI \(R\). The SPI and EPI contribute additively to the damping \(\gamma\), i.e., both contribute to the broadening of the Raman lines, the SPI more at low temperatures, vanishing at the phase transition temperature \(T_{C}\), and the EPI more at higher temperatures and above \(T_{C}\).