# Thermodynamics and Magnetism of SmFe\({}_{12}\) Compound Doped with Co and Ni: An Ab Initio Study
## 3 Thermodynamic Properties of the Sm(FeCo-Ni)\({}_{12}\) Alloys
In order to explore the stabilizing effects of the addition of nickel metal to the Sm(Co\({}_{1-x}\)Fe\({}_{x}\))\({}_{12}\) alloys, we performed EMTO-CPA calculations of the formation energy of the quasi-binary SmNi\({}_{4}\)(Fe\({}_{1-x}\)Co\({}_{x}\))\({}_{8}\) alloys relative to the unary systems \(\alpha\)-Sm, \(\alpha\)-Fe, \(\alpha\)-Co, and \(\alpha\)-Ni, where 4 Ni atoms occupy the 8/ sublattice and occupation of the 8/ and sublattices gradually changes from pure Fe to pure Co metals. As can be seen from Figure 3, the formation energies of the quasi-binary SmNi\({}_{4}\)(Fe\({}_{1-x}\)Co\({}_{x}\))\({}_{8}\) alloys are negative (stable alloys) within the whole compositional range. Even if the calculated heat of formation of the unstable SmFe\({}_{12}\) compound (\(+7.38\) mRy/atom) is approximately seven times as large as the heat of formation of the also unstable SmCo\({}_{12}\) compound (\(+1.06\) mRy/atom), the heats of the formation of the end points, SmNi(8)\({}_{4}\)Co(8,8)\({}_{8}\) and SmNi(8)\({}_{4}\)Fe(8,8)\({}_{8}\) compounds, are almost equal at \(-1.65\) mRy/atom and \(-1.71\) mRy/atom, respectively. Consequently, the occupation of the 8/ sublattice by Ni atoms plays a critical role in defining the sign and magnitude of the heat of formation of the Sm (Fe,Co,Ni)\({}_{12}\) compound. EMTO calculations revealed that the equilibrium atomic volumes of the SmNi(8)\({}_{4}\)Fe(8,8)\({}_{8}\) and SmNi(8)\({}_{4}\)Co(8,8)\({}_{8}\) compounds are \(\Omega_{0}=12.98\) A\({}^{3}\) and \(12.40\) A\({}^{3}\), respectively. As we mentioned in the Introduction, [PERSON] et al. 2018 [145] found that the calculated formation energy of the \(RE\)Fe\({}_{12}\) compound in respect to the \(RE\)\({}_{2}\)Fe\({}_{17}\) compound and \(\alpha\)-Fe is positive, indicating that the \(RE\)Fe\({}_{12}\) compound is unstable toward the decomposition of the \(RE\)\({}_{2}\)Fe\({}_{17}\) compound and \(\alpha\)-Fe. The possible reactions of the \(RE\)Fe\({}_{12}\) compound have been suggested in [85; 92; 145; 177]. In this paper, we studied the reactions suggested by [PERSON] et al. [104], assuming the decomposition of the Sm(Fe\({}_{1-x}\)\({}_{x}\))\({}_{12}\) alloys to the Sm\({}_{2}\)Fe\({}_{17}\) compound, \(\alpha\)-V, and \(\alpha\)-Fe:  
\[\Delta H_{f}(SmFe_{12-x}V_{x})=\frac{1}{26}[2E(SmFe_{12-x}V_{x})-E(Sm_{2}Fe_{ 17})-2 xE(\alpha-V)-(7-2x)E(\alpha-Fe)] \tag{1}\]  
Figure 2: The formation energy of Sm(Fe\({}_{1-x}\)Co\({}_{x}\))\({}_{12}\) alloys. [PERSON] et al. [104] performed calculations for 26 atoms (doubled) of ThMn\({}_{12}\)-type supercells. Taking into consideration that in the EMTO method the total energy is calculated in the [Ry/atom] unit, \(x\) = 4 for the SmNi\({}_{4}\)Fe\({}_{8}\) compound, while Equation (1) transforms (in the EMTO formalism) to:  
\[\Delta H_{f}(SmFe_{12})=\frac{1}{26}[26E(SmFe_{12})-19E(Sm_{2}Fe_{17})-7~{}E( \alpha-Fe)], \tag{2}\]  
\[\Delta H_{f}(SmFe_{8}Ni_{4})=\frac{1}{26}[26E(SmFe_{8}Ni_{4})-19E(Sm_{2}Fe_{1 7})-8E(\alpha-Ni)+E(\alpha-Fe)] \tag{3}\]  
\[\Delta H_{f}(SmCo_{12})=\frac{1}{26}[26E(SmCo_{12})-19E(Sm_{2}Co_{17})-7~{}E( \alpha-Co)], \tag{4}\]  
\[\Delta H_{f}(SmCo_{8}Ni_{4})=\frac{1}{26}[26E(SmCo_{8}Ni_{4})-19E(Sm_{2}Co_{1 7})-8E(\alpha-Ni)+E(\alpha-Co)]. \tag{5}\]  
Table 2 presents the decomposition energy of the reaction \(RE\)(Fe,Co,Ni)\({}_{12}\)\(\rightarrow\)\(RE_{2}\)(Fe, Co)\({}_{17}\) + TMs. As previously mentioned, the SmFe\({}_{12}\) and SmCo\({}_{12}\) compounds are unstable (positive heat of formation reported in Figure 2). Moreover, each of theThMn\({}_{12}\)-type compounds without Ni is unstable relative to decomposition into the Th\({}_{2}\)Zn\({}_{17}\)-type compound and TMs. On the contrary, the SmNi\({}_{4}\)Fe\({}_{8}\) and SmNi\({}_{4}\)Fe\({}_{8}\) compounds have negative formation energies (see Figure 3) and are also stable toward decomposition into the Th\({}_{2}\)Zn\({}_{17}\)-type compounds and TMs. These results highlight the promising phase stability behavior for the SmNi\({}_{4}\)Fe\({}_{8}\)- and SmNi\({}_{4}\)Fe\({}_{8}\)-based magnets.