_applied_  
_sciences_  
_Article_  
Understanding the Formation of Complex Phases: The Case of FeSi\({}_{2}\)  
**[PERSON] 1,*, [PERSON] 2, [PERSON] 2, [PERSON] 3,4, [PERSON] 3(c) and [PERSON] 1(c)**  
Footnote 1: The \(\beta\)-value of the  
In this work, we ask the following question: Starting from simple, structure-based pure elements that make up an alloy at a given composition, what is the pathway to a structural transformation that leads to the formation of a complex structure at thermodynamic equilibrium for some specific alloy compositions? To better justify this question, here are some facts: Very often, the mixture between a transition metal (TM) and an _sp_-type element leads to the formation of simple (ordered) phase(s) for TM-rich alloys, whereas complex phases form for _sp_-like element-rich alloys. Note that Zn is not a TM, despite being part of the \(d\) block. Examples include Al-Ni, Al-Cu, Cu-Zn, Fe-Zn (not shown here), and Fe-Si, whose assessed phase diagrams are displayed in Figure 1 (Refs. [9; 10; 11; 12; 13; 14], respectively). In the case of **Al-Ni**, where both Al and Ni are face-centered cubic (fcc) elements, complex phases are stabilized on the Al-rich side of the diagram beyond a body-centered, cubic (bcc)-based, ordered-phase B2 (CsCl-type, cP2, Pm-3m) around an equi-atomic composition with two compounds, Al\({}_{3}\)Ni (D0\({}_{11}\), Fe\({}_{3}\)C-type, oP16, Pmma), and Al\({}_{3}\)Ni\({}_{2}\) (D5\({}_{13}\)-type, Al\({}_{3}\)Ni\({}_{2}\)-type, hP5, P-3m1), whereas in the Ni-rich region of the diagram, an fcc-based, ordered-phase L1\({}_{2}\) (L1\({}_{2}\), AuCu\({}_{3}\)-type, cP4, Pm-3m) forms. For **Al-Cu**, once again, both of the pure elements exhibit an fcc structure, whereas at least nine complex phases form at various alloy compositions, particularly a tetragonal \(\theta\)-Al\({}_{2}\)Cu-phase (CuAl\({}_{2}\)-type, tI2, I4/mcm, C16) also found in FeGe\({}_{2}\). For **Cu-Zn**, Cu and Zn exhibit fcc and hexagonal close-packed (hcp) structures, respectively, whereas an ordered bcc-based B2 phase forms around the equi-atomic composition,  
Figure 1: CALPHAD-assessed phase diagrams of Al-Ni, Al-Cu, Cu-Zn, and Fe-Si. and three complex phases form in the Zn-rich part of the phase diagram: D8\({}_{2}\)-\(\gamma\) (Cu\({}_{5}\)Zn\({}_{8}\)-type, c152, I-43m), \(\delta\) (CuZn\({}_{3}\) (at HT), hP3, P-6), and A3-\(\varepsilon\) (Mg-type, hP2, P6\({}_{3}\)/mmc), which is, in fact, based on a distorted hcp structure. In the case of **Fe-Zn**, the pure elements are based on bcc (low-temperature phase) and hcp structures, respectively, whereas, as a function of alloy composition, four complex phases form: D8\({}_{2}\)-\(\Gamma\) (Cu\({}_{5}\)Zn\({}_{8}\)-type, cl52, I-43m), D8\({}_{1}\)-\(\Gamma_{1}\) (Fe\({}_{3}\)Zn\({}_{10}\)-type, cF408, F-43m), \(\delta\) (hP555, P6\({}_{3}\)/mc), and, finally, \(\zeta\) (mC28, C2/m). Finally, in the case of **Fe-Si**, Fe stabilizes with a low-temperature bcc structure, whereas Si has a cubic diamond structure (C-type, cF8, Fd-3m). As a function of alloy composition, at least four complex phases form in this system: on the Si-rich side of the phase diagram, a semiconducting orthorhombic \(\beta\)-FeSi\({}_{2}\) (or Fe\({}_{3}\)Si\({}_{7}\); LT, FeSi\({}_{2}\), o548, Cmca, #64) and a metallic tetragonal \(\alpha\)-FeSi\({}_{2}\) (actually a slightly off-stoechiometric-phase \(\alpha\)-Fe\({}_{3}\)Si\({}_{2}\), y \(<\) 1, with vacancies on the Fe sublattice; HT, Fe\({}_{0.92}\)Si\({}_{2}\), tP3, P4/mmm, #123) are present in addition to a nearby equiatomic-phase \(\varepsilon\)-FeSi (B20, cP8, space group P2\({}_{1}\)3, #198) that is derived from the rock salt structure (Fm-3m, P2\({}_{1}\)3, #225). On the Fe-rich side of the diagram, Fe\({}_{5}\)Si\({}_{3}\) (Mn\({}_{5}\)Si\({}_{3}\)-type, hP16, P6\({}_{3}\)/mmc) and Fe\({}_{2}\)Si (Ni\({}_{2}\)Al-type, hP6, P-3m1), and at higher Fe composition (not shown in Figure 1), ordered B2 and D0\({}_{3}\) (BiF\({}_{3}\)-type, cF16, Fm-3m) phases based on the bcc structure are present.