## 3 Results and Discussion  
The enthalpies of formation of various phases in (Ce, Zr, Nd)-Fe-Si systems were calculated. The emphasis in the discussion is put on the compositions with a high Fe content, due to the arguments put forward in the introduction. This computational paper is intended to provide a keystone for further experimental works on the stabilization of 1:12-type hard magnetic phase. Substituting elements were chosen on the basis of a vast number of references with the intention to improve the structural stability of the 1:12-type phase in comparison with the other competing phases, bearing in mind that the superior aim is to retain the hard magnetic properties. We also check the possibility of the formation of amorphous phase in order to assess the potential of (Ce, Zr, and Nd)-Fe-Si alloys for a more sophisticated synthesis route i.e., vitrification by rapid quenching and further nanocrystallization. It can be beneficial when designing soft/hard magnetic exchange spring magnet composites, comprising of a soft magnetic Fe matrix and hard magnetic nanocrystals [3]. The contour maps of the enthalpies of formation of amorphous phase in Zr-Fe-Si, Ce-Fe-Si, and Nd-Fe-Si (Figure 1) are qualitatively similar. The most negative values are reached for Fe-free compositions. Large negative interfacial enthalpies between (Zr, Ce, and Nd) and Si are responsible for such behavior. Bearing in mind that the large atomic radii difference of alloy constituents promotes glassy state formation, the significant difference in (Zr, Ce, and Nd) and Si atomic radii, which are equal to 1.6-1.82 A and 1.17 A [37], respectively, is also fundamental here. The Fe atomic radius with its intermediate value of 1.28 A plays a minor role. This gives us a first impression of the limited or even destructive role of Fe in the process of the formation of structural disorder in the Fe-rich alloys. The enthalpies of the formation of the amorphous phase in the Zr-Fe-Si system are more negative than those for Ce- and Nd-containing alloys. The most negative values of \(\Delta H^{nm}\), obtained for Zr-Si, Ce-Si, and Nd-Si, are equal to \(-72.1\) kJ/mol, \(-56.2\) kJ/mol, and \(-53.0\) kJ/mol for Zr\({}_{47}\)Si\({}_{53}\), Ce\({}_{43}\)Si\({}_{57}\), and Nd\({}_{43}\)Si\({}_{57}\), respectively. A similar role for Zr has been already shown for (Mn, Zr)-Co-Ge alloys [38], where the enthalpies of formation of Zr-containing amorphous phases are much more negative, mainly due to the strongly negative values of the interfacial enthalpies of Zr-Co and Zr-Ge binaries. As can be seen in Figure 1, that a simultaneous decrease in Zr and Si content leads to a gradual increase in \(\Delta H^{nm}\). Positive values for \(\Delta H^{nm}\) are reached for a small number of, mainly Fe-rich, Zr-Fe-Si compositions. For Ce-Fe-Si and Nd-Fe-Si systems, this range is extended to all Ce- and Nd-rich alloys (most compositions with Ce and Nd content extending 95 at.%). Focusing on compositions of interest, namely (Zr, Ce, Nd)Fe\({}_{12-x}\)Si\({}_{x}\) (\(0\leq x\leq 2\)) and (Zr, Ce, Nd)\({}_{2}\)Fe\({}_{17-x}\)Si\({}_{x}\) (\(0\leq x\leq 2\)), crucial differences are visible again when comparing Zr-containing and Ce- and Nd-containing alloys. For nominal compositions of the ThMn\({}_{12}\)-type phase, so ZrFe\({}_{12}\), CeFe\({}_{12}\), and NdFe\({}_{12}\), we found \(\Delta H^{am}=-2.0\) kJ/mol, 7.2 kJ/mol, and 6.5 kJ/mol, respectively. For Si-substituted compositions, which are believed to have a lower enthalpy of formation of the ThMn\({}_{12}\)-type phase, namely ZrFe\({}_{10}\)Si\({}_{2}\), CeFe\({}_{10}\)Si\({}_{2}\), and NdFe\({}_{10}\)Si\({}_{2}\), we found \(\Delta H^{am}=-17.5\) kJ/mol, \(-8.9\) kJ/mol, and \(-9.3\) kJ/mol, respectively. At first, CeFe\({}_{12}\) and NdFe\({}_{12}\) compositions will unlikely form an amorphous phase, as their enthalpies of formation are highly positive, whereas the enthalpy of formation of the amorphous ZrFe\({}_{12}\) phase is slightly negative and theoretically exhibits some driving force for vitrification.