## 4 Results and Discussion
### Structural Transformations in \(\beta\)-FeSi\({}_{2}\) and \(\alpha\)-FeSi\({}_{2}\)  
Now let us return to the analysis of the \(\beta\)-to-\(\alpha\) transition. The goal is to understand which of the two scenarios is the most likely--(1) \(\beta\)-FeSi\({}_{2}\)\(\rightarrow\)\(\alpha\)-FeSi\({}_{2}\), and then, \(\alpha\)-FeSi\({}_{2}\)\(\rightarrow\)\(\alpha\)-Fe\({}_{y}\)Si\({}_{2}\)\(+\)\(\varepsilon\)-FeSi (direct transition), or (2) \(\beta\)-FeSi\({}_{2}\)\(\rightarrow\)\(\alpha\)-Fe\({}_{y}\)Si\({}_{2}\)\(+\)\(\varepsilon\)-FeSi (indirect transition)--since experiments never definitely answered this question [57; 58; 62]. Hence, the following investigations were carried out. First, the stability of both the \(\beta\)- and \(\alpha\)-phases was estimated by comparing their Gibbs free energies; then, the optimum composition of \(\alpha\)-Fe\({}_{y}\)Si\({}_{2}\) in the reaction \(\alpha\)-FeSi\({}_{2}\)\(\rightarrow\)\(\alpha\)-Fe\({}_{y}\)Si\({}_{2}\)\(+\)\(\varepsilon\)-FeSi was found, and the \(\gamma\)-phase was shown to possibly transform into the \(\alpha\)-phase under strain. Second, FPMD simulations of the \(\beta\) structure at different temperatures up to 2400 K were performed. The difference in the Gibbs free energies of \(\beta\)-FeSi\({}_{2}\) and \(\alpha\)-FeSi\({}_{2}\) (\(\Delta\)G) as a function of temperature is shown in Figure 13. The calculated transition temperature T\({}_{\rm C}\) = 1825 K was found to be higher compared to that obtained in the experiment (\(\sim\)1240 K) [57; 58; 62] and in the calculations based on the Debye model (1140 K) [24]. This finding indicates the possibility of a direct \(\beta\)-FeSi\({}_{2}\)\(\rightarrow\)\(\alpha\)-FeSi\({}_{2}\) transition. Once it is assumed that the stoichiometric \(\alpha\)-FeSi\({}_{2}\) phase could be stabilized at high temperature, let us consider the possibility of its decomposition into \(\varepsilon\)-FeSi and \(\alpha\)-Fe\({}_{y}\)Si\({}_{2}\)[57; 58]. For this purpose, one writes the formation energy of the \(\varepsilon\)-FeSi + \(\alpha\)-Fe\({}_{y}\)Si\({}_{2}\) system, \(\Delta\)G, as:  
\[\Delta\rm G(y,T)=1/3\cdot\left[x\cdot E_{T}\big{(}\alpha-Fe_{y}\mathcal{V}_{1- y}Si_{2}\big{)}-x\cdot T\cdot\Delta\rm S\big{(}\alpha-Fe_{y}\mathcal{V}_{1- y}Si_{2}\big{)}+z\cdot E_{T}(\varepsilon-FeSi)-ET(\alpha-FeSi_{2})\right] \tag{2}\]  
where x = (2 - y)\({}^{-1}\), z = 2\(\cdot\)(1 - x), \(\mathcal{V}\) denotes a vacancy on the iron sublattice, E\({}_{T}\) is the total energy, and \(\Delta\)S is the configurational entropy. The \(\alpha\)-Fe\({}_{y}\mathcal{V}_{1-y}\)Si\({}_{2}\) phases were described based on a 45-atom supercell, in which the vacancies in the Fe sublattice were distributed randomly. The lattice parameter of the cubic eight-atom e-FeSi structure (space group P21 #198) is found to be 4.450 A, which is very close to the experimental and other theoretical values of 4.489 A [56] and 4.41-4.47 A [21], respectively. Expression (2) was derived under the condition that a number of atoms in the system are preserved before and after the structural transformation, and this condition implies that the \(\alpha\)-Fe\({}_{y}\)Si\({}_{2}\) phase after decomposition should have vacancies on the iron sublattice. A negative value of \(\Delta\)G means that \(\alpha\)-FeSi\({}_{2}\) will decompose into \(\alpha\)-Fe\({}_{y}\mathcal{V}_{1-y}\)Si\({}_{2}\) and \(\varepsilon\)-FeSi with the fractions of x and z, respectively. In Figure 14, \(\Delta\)G is shown as a function of the iron composition in the a-phase at different temperatures. One can see that in the range 800-1000 K, the stoichiometric \(\alpha\)-phase should decompose into \(\varepsilon\)-FeSi and \(\alpha\)-Fe\({}_{y}\mathcal{V}_{1-y}\)Si\({}_{2}\), for y\(\sim\)0.8. This finding is in good agreement with the conclusion that, at temperatures higher than 1240 K, two phases, \(\varepsilon\)-FeSi and \(\alpha\)-Fe\({}_{2}\)Si\({}_{5}\) (\(\alpha\)-Fe\({}_{0.8}\)Si\({}_{2}\), as in our case), form.