## 4 Results and Discussion  
### Ground State Analysis of the Ising Model  
Based on the description of the three approximants given in Section 2, ordering processes can be treated as a lattice problem, i.e., in terms of an Ising model on the partially occupied TM-\(\mathcal{V}\) sublattice, as was successfully carried out in the case of carbide and nitride compounds [51]. From this point of view, the Si sublattice is ignored in the atomic description of the model. In this Ising model, the configurational (C) part of the energy [52,53] can be written, up to a constant, as:  
\[\Delta E^{C}\sim\frac{1}{2}{\sum_{R}q_{R}}J_{R} \tag{1}\]  
where \(J_{R}\) is the pair interaction between TM atoms separated by a distance R on the simple cubic lattice, and \(q_{R}\) corresponds to the number of TM atomic pairs. The search for the most stable ordered structures at T = 0 K for a finite set of interaction parameters and a given composition reduces to the problem of finding the ground states of this Ising model [52]. This problem has been solved exactly in the case of the simple cubic lattice in the whole range of interaction parameters and composition with first- and second- [54] and up to third-neighbor pair interactions [55]. The results of the analysis are shown in Figure 4, where the ground-state (top) structures are shown together with an ordering map (bottom) that indicates the zones of existence of the various ground states in the space spanned by \(J_{1}\) and \(J_{2}\). Most relevant to this study where the focus is on TMSi\({}_{2}\), i.e., for which the composition is set to the generic composition (TM\({}_{1-C}\)\(\mathcal{V}\)\({}_{c}\))Si\({}_{8}\) with \(c=1/2\), only three ordered  
the small reshuffle of the atoms inside the \(\gamma\)-unit cell (C1, CaF\({}_{2}\)-type) to obtain the atomic configuration of the \(\beta\)-phase, it is worth noting that the creation of antiphase boundaries every two planes along the (100) direction leads to transformation from the approximant \(\alpha\)-FeSi\({}_{2}\) to \(\beta\)-FeSi\({}_{2}\), and similarly, the creation of antiphase boundaries along the (011) direction every two planes transforms the approximant \(\beta\)-FeSi\({}_{2}\) in the one of the \(\Theta\) phase (C16, Al\({}_{2}\)Cu-type), as shown in Figure 6. Strong similarities between these three structures are observed despite their original complexity. Below, we will show that \(\gamma\leftrightarrow\alpha\) and \(\beta\leftrightarrow\alpha\) transformations can also occur through monoclinic-type deformation of the unit cells.  
Figure 5: Supercells for the \(\beta\)- (**a**), \(\gamma\)- (i.e., approximant of \(\gamma\)-FeSi\({}_{2}\)) (**b**), and \(\alpha\)- (**c**) phases. The supercells (**a,b**) were constructed via the (\(2\times 2\times 2\)) translation of the corresponding 6-atom tetragonal cells. The large red circles are Fe atoms, and small blue circles are Si atoms. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article).  
Figure 6: Transformations from \(\alpha\)-FeSi\({}_{2}\) to \(\beta\)-FeSi\({}_{2}\) to \(\Theta\)-C16. Big (small) black circles refer to TM (Si), and open circles to vacancies, V.