## 4 Results and Discussion
### Temperature-Induced Phase Transformation in \(\gamma\)-FeSi\({}_{2}\)  
In this Section, we will consider the temperature-induced phase transformations in \(\gamma\)-FeSi\({}_{2}\), and then, we will describe the possible phase transitions in the \(\beta\)- and \(\alpha\)-phases. FPMD simulations were performed for \(\gamma\)-FeSi\({}_{2}\) at different temperatures from 100 K to 2400 K. Using group-theoretical analysis and visualization of the atomic configurations at different stages of the simulations, the possible intermediate and final structures that formed during the simulations at a certain temperature were identified. In Figure 11, the total energy and structural parameters are shown for the 48-atom supercell of \(\gamma\)-FeSi\({}_{2}\) as functions of the time of FPMD simulations (t) at 200 K. For comparison, Figure 11 shows the E\({}_{\text{T}}\)(t) dependences determined at 600 K and 1400 K. An analysis of the results of the FPMD simulations enabled us to select a time interval between 300 fs and 600 fs, during  
Figure 10: Calculated heat capacity at constant volume (C\({}_{\text{V}}\)) (red dashed line) for the \(\beta\)-phase in comparison with experimental heat capacity at constant pressure (C\({}_{\text{P}}\)) determined in Ref. [62] (blue dotted line) and results from Ref. [63] (black solid line). (For interpretation of the references to color in this figure, the reader is referred to the web version of this article). Figure 9: Phonon densities of states (PHDOS) for (**a**) the \(\alpha\)-phase calculated using DFPT (red dashed line) and the Phonopy code (black solid line), and (**b**) the \(\beta\)-phase calculated using the Phonopy code in the present work (black solid line) and the results of Ref. [24] (red dashed line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article). which the system reached local stability (\"plateau\" in the E\({}_{\rm{T}}\)(t) dependences in Figure 11). During this interval, at T \(<\) 800 K, the \(\gamma\)1-\(\gamma\)3 intermediate structures were identified. A description of the intermediate structures is given in Table 6. One can see from this table that the P-1 (\(\alpha\)-Fe\({}_{0.8}\) Si\({}_{2}\)) and C2/m intermediate phases with different numbers of atoms in the unit cell form at temperatures higher than 100 K. The \(\beta\)-phase was the final structure at temperatures up to 400 K and, at T \(\geq\) 400 K, the final 12-atom C2/m structures (\(\gamma\)4, \(\gamma\)5) formed (cf. Table 6). Note that the P-1 and C2/m space groups are sub-groups of the Fm-3m and Cmca space groups [47]. These results clearly indicate that the intermediate structures are temperature-dependent. To clarify the possible origin of the identified structures, the phonon spectrum of the \(\gamma\)-phase was investigated. The calculated phonon dispersion curves for \(\gamma\)-FeSi\({}_{2}\) are shown in Figure 12. The points \(\Gamma\), X, and L and directions \(\Delta\), \(\Sigma\), and \(\Lambda\) in the Brillouin zone are associated with imaginary frequencies of the phonon modes. The phonon modes, such as \(\Gamma_{5+}\), X\({}_{5-}\), L\({}_{3-}\), and \(\Sigma_{3}\)(1/2 1/2 0) with imaginary frequencies (cf. Figure 12) could be responsible for the phase transformations in \(\gamma\)-FeSi\({}_{2}\). Here, it should be noted that the semi-empirical phonon spectrum in Ref. [27] also has imaginary frequencies at these points, except at the L point in the present results. In Table 7, the possible structures that could arise due to the softening or condensation of the \(\Gamma_{5+}\), and L\({}_{3-}\) phonon modes are listed. The \(\Gamma_{5+}\) optic mode was selected, although it has a high real frequency since, at finite temperatures, its softening can cause spontaneous strains and structural changes [47]. The results presented in Table 7 clearly indicate that the direction vector of a phase transition, **P** (cf. Refs.