# 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  
The SmFe\({}_{12}\) compound crystallizes in the body centered orthogonal to the ThMn\({}_{12}\)-type structure (space group \(I4/mmm\), no. 139; see Figure 1). The structure contains Sm atom in the \(2a\) Wycoff position and 12 Fe atoms that occupy three inequivalent Wyckoff positions, 8\(f\), 8\(i\), and 8\(j\), respectively. According to the Pearson symbol (\(tI26\)), the usual SmFe\({}_{12}\) supercell contains 26 atoms but can be represented by the so-called reduced supercell with 13 atoms (1 Sm and 12 Fe) used in the present calculations. EMTO calculations revealed that the SmFe\({}_{12}\) compound has an equilibrium atomic volume \(\Omega_{0}\) equal to 13.19 A\({}^{3}\), which is in fair accordance with previous calculations, 13.02 A\({}^{3}\)[141], but smaller than the experimental value for the SmFe\({}_{12}\) films, \(\Omega_{0}\) = 13.64 A\({}^{3}\)[176]. The calculated formation energy of the SmFe\({}_{12}\) compound with respect to the unary systems \(\alpha\)-Sm and \(\alpha\)-Fe is \(H\) = +6.13 mRy/atom, which confirms that the SmFe\({}_{12}\) compound is thermodynamically unstable. EMTO calculations revealed that the SmCo\({}_{12}\) compound has an equilibrium atomic volume \(\Omega_{0}\) = 12.42 A\({}^{3}\). The calculated formation energy of the SmCo\({}_{12}\) compound in respect to the unary systems, \(\alpha\)-Sm and \(\alpha\)-Co, is \(H\) = +1.06 mRy/atom, which confirms that the SmCo\({}_{12}\) compound is thermodynamically unstable. Figure 2 shows the formation energy of Sm(Fe\({}_{1-x}\)Co\({}_{x}\))\({}_{12}\) alloys calculated with respect to the unary systems \(\alpha\)-Sm, \(\alpha\)-Fe, and \(\alpha\)-Co. We assume that Sm atoms solely occupies the \(2a\) Wyckoff position and Fe and Co atoms are randomly distributed on 8\(f\), 8\(i\), and 8\(j\) Wyckoff positions (see Figure 1). We could have expected the formation energy to decrease linearly as Co atoms gradually replace Fe atoms on their sites. However, this Figure shows that 20% of the Co solute decreases the formation energy of the Sm(Fe\({}_{0.8}\)Co\({}_{0.2}\))\({}_{12}\) compound by about \(\sim\)36% relative to the undoped SmFe\({}_{12}\) compound. In the compositional range between Sm(Fe\({}_{:5}\)Co\({}_{:5}\))\({}_{12}\) and SmCo\({}_{12}\), the formation energy changes very slowly from \(\sim\)+2 mRy/atom to \(\sim\)+1 mRy/atom. Figure 1: Crystal structure of the SmFe\({}_{12}\) (ThMn\({}_{12}\)-type) compound sketched using Vesta software [175]. The larger pink spheres denote the Sm atoms at Wyckoff position \(2a\), while the smaller white, gray, and black spheres are the Fe atoms at Wyckoff positions 8\(f\), 8\(i\), and 8\(j\), respectively. Adapted with permission from Ref. [139]. Reuse and Permissions License Number: RNP/22/MAY/053350. The calculated (EMTO) value of the formation energy of three configurations (a) SmNi(8)\({}_{4}\)Co(8)\({}_{4}\)Co(8)\({}_{4}\), (b) SmNi(8)\({}_{4}\)Co(8)\({}_{4}\)Co(8)\({}_{4}\), and (c) SmNi(8)\({}_{4}\)Co(8)\({}_{4}\)Co(8)\({}_{4}\), with respect to the unary systems \(\alpha\)-Sm, \(\alpha\)-Co, and \(\alpha\)-Ni, are \(+0.047\) mRy/atom, \(-0.031\) mRy/atom, and \(-1.65\) mRy/atom, respectively. Thus, assuming that \(1/3\) of Co atoms are replaced by Ni atoms, one can obtain the energetically favorable configuration when 4 Ni atoms occupy all 8/ sites and the remaining 8 Co atoms occupy 8/ and 8/ sites, resulting in a stable SmNi\({}_{4}\)Co\({}_{8}\) compound with a formation energy \(H=-1.65\) mRy/atom. This result agrees with the conclusion of [PERSON] et al. [92], who found that the Ni dopant prefers to occupy 8/ sites when stabilizing the NdFe\({}_{11}\)Ni compound.