## 2 Methods  
The thermodynamic properties are calculated on the basis of [PERSON]'s semi-empirical model [23; 24], also called the macroscopic-atom model. The atoms are taken in simplified form, as building blocks with the boundaries of the Wigner-Seitz cell. The enthalpy is the main thermodynamic quantity that is discussed throughout the paper and its variation is triggered by the contact of two blocks of atoms of different elements. Such an alloying process is thermodynamically described by three quantities that differentiate each element from the periodic table: (i) molar volume (\(V\)), (ii) potential similar to work function of an electron (\(\varphi\)), and (iii) density at the boundary of the Wigner-Seitz cell (\(n_{\text{t}ws}\)). The interfacial enthalpy \(\Delta H^{inter}(A\) in \(B)\) for solving one mole of \(A\) atoms in an excess of \(B\) is defined as follows:  
\[\Delta H^{inter}(A\text{ {in} }B)=\frac{V_{A}^{2}}{\frac{1}{2}\bigg{(}\frac{1}{n_{ \text{t}ws}^{1/3}}+\frac{1}{n_{\text{t}ws}^{1/3}}\bigg{)}}\bigg{\{}-P(\Delta \varphi)^{2}+Q\Big{(}\Delta n_{ws}^{1/3}\Big{)}^{2}\bigg{\}}, \tag{1}\]  
where \(P\) and \(Q\) are empirical constants that depend on the alloying elements. To determine the possibility of the formation of various phases, one should compare their formation enthalpies. The formation enthalpy of the amorphous phase is expressed as the following:  
\[\Delta H^{am}=\Delta H^{chem}+\Delta H^{topo}, \tag{2}\]  
where \(\Delta H^{chem}\) is the chemical contribution equal to the following:  
\[\Delta H^{chem}=c_{A}c_{B}\Big{(}c_{B}^{s}\Delta H^{inter}(A\text{ {in} }B)+c_{A}^{s}\Delta H^{inter}(B\text{ {in} }A)\Big{)}, \tag{3}\]while \(c_{A}\), \(c_{B}\) and \(c^{s}_{A}\)(= \(\frac{c_{A}V_{A}^{\frac{2}{3}}}{c_{A}V_{A}^{\frac{2}{3}}+c_{B}V_{B}^{\frac{2}{3} }}\)), \(c^{s}_{B}\) are the volume and surface fractions of each atom type. \(\Delta H^{topo}\) is a topological contribution reflecting the disorder in amorphous state and strongly dependent on the melting temperature. \[\Delta H^{topo}=3.5(c_{A}T_{m,A}+c_{B}T_{m,B})\cdot 10^{-3}. \tag{4}\]  
The formation enthalpy of the solid solution is defined as follows:  
\[\Delta H^{ss}=\Delta H^{chem}+\Delta H^{elast}+\Delta H^{struct}. \tag{5}\]  
The first of the contributions unique for a solid solution is the elastic misfit enthalpy Equation (6), connected with the atom size mismatch:  
\[\Delta H^{elast}(A\ in\ B)=\frac{2K_{A}G_{B}(W_{A}-W_{B})^{2}}{4G_{B}W_{A}+3 K_{A}W_{B}}, \tag{6}\]  
where:  
\[W_{A}=V_{A}+\alpha\frac{(\varphi_{A}-\varphi_{B})}{n_{us}^{A}},\ \text{and}\ \alpha=1.5\frac{V_{A}^{\frac{2}{3}}}{\frac{1}{n_{us}^{A} \frac{1}{3}}+\frac{1}{n_{us}^{\frac{2}{3}}}}. \tag{7}\]  
\(K\) is a bulk modulus, \(G\) is a shear modulus, and \(W_{A}\) Equation (7) and \(W_{B}\) are the corrected molar volumes (because of electron transfer from \(B\) to \(A\) and from \(A\) to \(B\)). The structural contribution \(\Delta H^{struct}\) originates from the valence and the crystal structure of the solvent and the solute atoms. As a structural term can be only roughly estimated, according to [PERSON] [24], even for binary transition metal systems, we omitted it from the analysis. Moreover, it should be underlined that the structural contribution for the analyzed transition metal and rare earth element substitutions should not provide much energy difference, according to the structural enthalpies of binaries [24], especially when focussing on Fe-rich compositions. The structural enthalpy for solving one mole of Zr (four valence electrons) in Fe (eight valence electrons) is equal to 0 kJ/mol, while for solving Ce or Nd (both \(\sim\) 3 valence electrons) in Fe it is equal to 2 kJ/mol. The emphasis was put on the calculations of enthalpies of formation of amorphous phases and solid solutions, complementary to the vast number of literature data on intermetallic compounds, so as to compare the formation enthalpies of all phases that compete during the solidification process.