## 4 Results and Discussion
### Temperature-Induced Phase Transformation in \(\gamma\)-FeSi\({}_{2}\)
[47; 48; 50] for more details), is very sensitive to temperature and associated with different combinations of phonon modes activated at different temperatures. Here, we note that the formation of the \(\gamma\)1 and \(\gamma\)2 phases could only exist because of the monoclinic  
Figure 11: Total energy (E\({}_{\rm{T}}\)) and lattice parameters (a, b, c, \(\alpha\), \(\beta\), \(\gamma\)) of \(\gamma\)-FeSi\({}_{2}\) as functions of simulation time at 200 K. For comparison, the E\({}_{\rm{T}}\)(t) dependences obtained after FPMD simulations at T = 600 K and 1400 K are presented. (For references to colors in this figure, the reader is referred to the web version of this article). strains \(\varepsilon_{xy}\
eq\varepsilon_{yx}\
eq\varepsilon_{xz}\) (\(\gamma 1\)) and \(\varepsilon_{xy}\
eq\varepsilon_{yz}=\varepsilon_{xz}\) (\(\gamma 2\)), since the monoclinic strains have \(\Gamma_{5+}\) symmetry [47]. Indeed, Figure 11 shows that the intermediate-phase \(\gamma 2\) forms due to a change in cell angles \(\alpha\approx\beta\
eq\gamma\) associated with \(\varepsilon_{xy}\
eq\varepsilon_{yz}=\varepsilon_{xz}\) strains. \begin{table}
\begin{tabular}{c c c c c} \hline
**Phase,** & **a, b, c** & **\(\alpha\)**\(\beta\), \(\gamma\)** & **Wyckoff Position** \\
**Temperature (K)** & **(Å)** & **(Degree)** & **Wyckoff Position** \\ \hline \(\gamma\) & 5.400 5.400 5.400 & 90.00 90.00 90.00 & Fe 1a 0.000 0.000 0.000 \\ T \(<\) 100 & & & Si 2c 0.250 0.250 0.250 \\ \hline \(\gamma 1\) & 3.319 3.523 3.552 & 94.56 116.79 116.60 & Fe 1h 0.500 0.500 0.500 \\ \(100\leq T<200\) & & & Si 1i 0.003 0.776 0.230 \\ \hline \(\gamma 2\) & 4.731 5.268 3.234 & 90.00 129.854 90.00 & Fe 2d 0.000 0.500 0.500 \\ \(200\leq T<400\) & & & Si 4g 0.000 0.226 0.000 \\ \hline \(\gamma 3\) & & & Fe 4i 0.186 0.000 0.726 \\ \(400\leq T\leq 800\) & 6.333 3.742 5.974 & 90.00 118.44 90.00 & Si 4i 0.598 0.000 0.875 \\  & & & Si 4i 0.827 0.000 0.643 \\ \hline \(\gamma 4\), \(\gamma 5\) & & & Fe 4i 0.243 0.000 0.881 \\ T \(>\) 800 K & 7.125 9.342 3.756 & 90.00 100.44 90.00 & Fe 4h 0.000 0.783 0.500 \\  & & & Si 8j 0.278 0.126 0.414 \\  & & & Si 4g 0.000 0.119 0.000 \\  & & & Si 4g 0.000 0.634 0.000 \\ \hline \end{tabular}
\end{table} 
Table 6: Structures of the intermediate and final structures identified during FPMD simulations of \(\gamma\)-FeSi\({}_{2}\) at different temperatures (T). The space groups of the identified structures were Fm-3m (\(\gamma\)), P-1 (\(\gamma 1\)), and C2/m (\(\gamma 2\)\(\gamma 5\)). \begin{table}
\begin{tabular}{c c c c c} \hline
**Phonon Mode** & \(\Gamma_{5+}\) & \(\Gamma_{3-}\) & \\ \hline Phase & \(\gamma 1\) & \(\gamma 2\) & \(\gamma 3\) & \(\gamma 4\) & \(\gamma 5\) \\  & P-1, \#2 & C2/m, \#12 & C2/m, \#12 & C2/m, \#12 & C2/m, \#12 \\ \hline N\({}_{\rm a}\) & 1 & 1 & 6 & 12 & 12 \\ \hline P & SI(a,b,c) & C2(a,a,b) & P2(a,a,000000) & C12(a,b,a,b,0,0,0,0) & C18(a,b,b,a,0,0,0,0) \\ \hline Strain & \(\varepsilon_{0}\), \(\varepsilon_{1}\), \(\varepsilon_{2}\), & \(\varepsilon_{0}\), \(\varepsilon_{1}\), & \(\varepsilon_{0}\), \(\varepsilon_{1}\), \(\varepsilon_{2}\), & \(\varepsilon_{0}\), \(\varepsilon_{1}\), \(\varepsilon_{2}\), \(\varepsilon_{xy}\) & \(\varepsilon_{0}\), \(\varepsilon_{1}\), \(-\varepsilon_{xz}\)\(+\varepsilon_{yz}\), \(-\varepsilon_{xy}\) \\  & \(\varepsilon_{xy}\), \(\varepsilon_{yz}\), \(\varepsilon_{xz}\) & \(\varepsilon_{xz}\)\(+\varepsilon_{yz}\), \(\varepsilon_{xy}\) & \(\varepsilon_{xz}\)\(+\varepsilon_{yz}\), \(\varepsilon_{xy}\) \\ \hline \end{tabular}
\end{table}