# Thermodynamics and Magnetism of SmFe\({}_{12}\) Compound Doped with Co and Ni: An Ab Initio Study
## 4 Magnetic Properties of the Sm(FeCo-Ni)\({}_{12}\) Alloys
Regarding the Curie temperature (\(T_{c}\)), a mean-field treatment can be formulated as [178; 179]:  
\[T_{c}=\frac{2}{3}\times\frac{E_{tot}^{DLM}-E_{tot}^{AF}}{k_{B}}\]where \(E_{tot}^{DLM}\)_and_\(E_{tot}^{AF}\) are the ground-state total energies of the DLM and the AF (antiferromagnetic) states, respectively, and \(k_{B}\) is the Boltzmann constant. Thus, an estimation of \(T_{c}\) can be accomplished from the total energy difference between the ferromagnetic (or antiferromagnetic) and paramagnetic (DLM) states. Nevertheless, in line with [179], the diversity between the total energies can be replaced by the diversity between the effective single-particle (one atomic specie) energies, which are instantly associated with AF and DLM states (the so-called MFA treatment). In the present work, \(E_{tot}^{DLM}\)and \(E_{tot}^{AF}\) are calculated at the equilibrium volumes for DLM and AF states, respectively. Figure 6 shows the calculated (within the EMTO-CPA technique) \(T_{c}\) values of the pseudo-binary SmNi\({}_{4}\)(Fe\({}_{1-x}\)Co\({}_{x}\))\({}_{8}\) alloys, where 4 Ni atoms occupy the 8\(i\) sublattice and 8 (Fe\({}_{1-x}\)Co\({}_{x}\)) atoms are shared on the 8\(i\) and 8\(f\) sublattices. The calculated \(T_{c}\) values are equal to 853 K, 928 K, 995 K, and 1120 K for the SmNi\({}_{4}\)Fe\({}_{8}\), SmNi\({}_{4}\)(Fe\({}_{0.9}\)Co\({}_{0.1}\))\({}_{8}\), SmNi\({}_{4}\)(Fe\({}_{0.8}\)Co\({}_{0.2}\))\({}_{8}\), and SmNi\({}_{4}\)Co\({}_{8}\) alloys, respectively. These values are substantially higher than the Curie temperature of the generally used Neomax (Nd\({}_{2}\)Fe\({}_{14}\)B\({}_{1}\)) magnet (588 K) [12]. In Table 4, our FPLMTO-calculated magnetic properties for the SmNi\({}_{4}\)(Fe\({}_{1-x}\)Co\({}_{x}\))\({}_{8}\) alloys are shown. The obtained properties depend on the approximation of the problematic 4\(f\) electrons on samarium. The simplest approach is a 4\(f\)-band model that ignores spin-orbit coupling. Better models include spin-orbit (SO) and orbital polarization (OP). The best approach is to confine the 4\(f\) electrons in core states in the standard rare-earth model with OP (SRM + OP) for the \(d\) states. The largest total moments are obtained from SRM + OP because here the Sm moment (anti-parallel to the total unit-cell moment) is the smallest. We can compare our FPLMTO magnetic moments with those of the fully relativistic EMTO method (see Table 3). The reason these models give different total magnetic moments is because they predict vastly different samarium magnetic moment. In Figure 7, we show the different results for the models. Figure 6: The Curie temperature of the SmNi\({}_{4}\)(Fe\({}_{1-x}\)Co\({}_{x}\))\({}_{8}\) alloys, \(x\) = 0.0, 0.1, 0.2, 1.0. Figure 7 shows that the SRM + OP model has the smallest Sm moment and the greatest total magnetic moment (Table 4). The 4\(f\)-band-EMTO approach is second. We believe there are technical differences in the EMTO and FPLMTO codes that give rise to the very different Sm moments, because relativistic DFT values (with spin-orbit coupling but no orbital polarization), i.e., EMTO and 4\(f\)-band-SO (FPLMTO), should in principle be close but they are not. For the FPLMTO calculations, the Sm magnetic moment depends very weakly on composition parameter \(x\), while there is some dependence in the EMTO. We should point out that for the 3\(d\) (Fe, Co, and Ni) metals, the magnetic moments are almost the same for all models. Next, we present our MAE results. Here, we apply the SRM + OP model for the MAE because the other approaches (4\(f\)-band) produce unrealistically large MAE due to the inappropriate treatment of the 4\(f\) electrons (not shown). Before calculating the MAE, i.e., the energy difference between the magnetic spin moment aligned in the \(x\)-\(y\) plane and along the \(z\) axis, we carefully optimize the crystal structure. In other words, both the lattice parameters \(a\) and \(c\) are optimized to give the lowest total energy of the tetragonal crystal.