## 2 Crystallographic Examination and Relationships
Fe has eight Si neighbors at a distance of about 2.3 A, and Si has four Fe neighbors [6]. Note that \(\alpha\)-FeSi\({}_{2}\) is a slightly iron-deficient structure and is formed of FeSi\({}_{8}\)-joined face-sharing cubes that are very nearly regular (whereas the empty cubes along the \(c\) direction, due to Si-Si bonds, are squeezed). On the other hand, in \(\beta\)-FeSi\({}_{2}\), the FeSi\({}_{8}\) cubes that are joined by edge-sharing, as in the fluorite type, are very severely distorted, although the fluorite CaF\({}_{2}\) structure (\(\gamma\)-phase) is still recognizable [30]. To complete the list of possible structures at this 1:2 composition, one would have expected to find a similar situation for Fe-Ge. This is not the case; instead, a \(\Theta\)-FeGe\({}_{2}\) phase (C16, Al\({}_{2}\)Cu-type, tl12, I4/mcm, #140) forms, described in Table 3 with 12 atoms in the unit cell [34; 35]. Figure 2 shows the three phases described so far. To define approximants for these three structures, small atomic displacements and local rotations applied to them are required. With the appropriate atomic displacements applied to the different inequivalent sites of the oC48 \(\beta\)-FeSi\({}_{2}\) structure depicted in Figure 2 (such that the coordinates are expressed in units of n/4, where n is an integer), one obtains a cubic \(\gamma\)-FeSi\({}_{2}\) CaF\({}_{2}\)-type structure, usually defined on an fcc lattice (space group Fm-3m, #225), given in Figure 3.  
If the tetragonality of the tP3 \(\alpha\)-FeSi\({}_{2}\) structure is ignored and the Si species are located at the center of the cubes, i.e., if the structure is described on a bcc lattice, the resulting arrangement of the three different species (Fe, Si, and vacancies, \(\mathcal{V}\)) is given in Figure 3.  
Finally, to complete the description, if one applies two so-called clockwise (z = 1/4) and anti-clockwise (z = 3/4) Hyde rotations on the Archimedean square antiprisms (indicated in Figure 2), where \(\tan(\delta)=(1-4\alpha)/(2-4\alpha)\), with an angle of \(\alpha\)-0.1586, \(\delta\)-14.987\({}^{\circ}\), and distance adjustment from d = \((2\alpha^{2}-\alpha+1/4)^{1/2}\) = 0.37644 to 0.35355 (with a unit of a), the body-centered tetragonal C16 \(\Theta\)-CuAl\({}_{2}\)-type structure can also be described on an fcc (or a bcc) lattice. Indeed, note that for all three cases, the structural rep \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Site** & \(\mathbf{x_{C}}\) & \(\mathbf{y_{C}}\) & \(\mathbf{z_{C}}\) & \(\mathbf{x_{A}}\) & \(\mathbf{y_{A}}\) & \(\mathbf{z_{A}}\) \\ \hline Fe & 0 & 0 & 0 & 0 & 0 & 0 \\ Si & 1/2 & 1/2 & 0.280 & 1/2 & 1/2 & 1/4 \\ \(\mathcal{V}\) & 1/2 & 1/2 & 0.720 & 0 & 0 & 3/4 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Description of the \(\alpha\)-FeSi\({}_{2}\) structure with the coordinates of the current (C) structure in units of the lattice parameters a and c of the 2-atom unit cell, and those of the approximant (A) structure within the unit cell. \(\mathcal{V}\) refers to vacancy. described on an fcc or a bcc lattice by considering a Bain transformation [36; 37], and the results are shown in Figure 3.  
These three types of ordering on one of the simple cubic sublattices that constitute the bcc lattice (the other simple cubic sublattice is, in the three cases, fully occupied by the  
Figure 2: Crystallographic descriptions of \(\beta\)-FeSi\({}_{2}\) (**top left**), \(\alpha\)-FeSi\({}_{2}\) (**top right**), and \(\Theta\)-C16 (**bottom**). (For interpretation of the references to colors in this figure, the reader is referred to the web version of this article).