## Magnetic Properties of the Y(Co-Fe-Ni)\({}_{5}\) Alloys
As has been mentioned in the Introduction, the magnetic anisotropy energy (MAE) is one of the more important properties of an efficient magnet. In the quest to increase the saturation magnetic moment or energy product, by substituting cobalt for iron, one has to review the impact of the doping on the MAE as well. In Table 4 we collect calculated MAEs for the YCo\({}_{5}\)-type magnets alloyed with iron, nickel, or both. Here, for the YFe\({}_{3}\)Co\({}_{2}\) and the YFe\({}_{3}\)CoNi magnets, we keep the iron atoms on the energetically favorably 3g sites as discussed above. In the case of YFe\({}_{3}\)CoNi, we model Co and Ni on the 2c sites as two average atoms consistent with our modeling of SmFe\({}_{3}\)CoNi [3]. In all calculations, we relax the atomic volume and the c/a axial ratio of the hexagonal phase. We find some sensitivity (not shown) of the MAE to the axial ratios, suggesting that the structural relaxation is important. According to the values calculated in Table 4, we realize that both YFe\({}_{5}\) and YNi\({}_{5}\) have relatively small magnetic anisotropy and, for that reason alone, they are not particularly good magnets. YFe\({}_{5}\) is really only included in the table to provide context to the other magnets, as it does not exist in the hexagonal phase. YCo\({}_{5}\) contrarily exists and is predicted to have significant magnetic anisotropy. We primarily rely on DFT-GGA calculations for these magnetic compounds because GGA performs better for the magnetic 3d transition metals relative to the LDA or even more modern approximations [136]. It is also a well-known fact that GGA reproduces the proper magnetic ground state of iron, as opposed to the LDA. Our GGA calculations reproduce the experimental atomic volume very well but overestimate the MAE for YCo\({}_{5}\) relative to experimental data. DFT-GGA (\(T\) = 0 K) gives the unit cell volume \(V_{cell}\) = 82.65 A\({}^{3}\) and anisotropy \(K_{1}\) = 9.89 meV/cell (19.2 MJ/m\({}^{3}\)). These numbers shall be compared to experimental data at T = 4.1 K, \(V_{cell}\) = 82.50 A\({}^{3}\)[29; 55], \(K_{1}\) = 3.80 meV/cell (7.38 MJ/m\({}^{3}\)) [29; 56] and at \(T\) = 293 K, \(V_{cell}\) = 83.99 A\({}^{3}\)[33], \(K_{1}\) = 3.04 meV/cell (5.80 MJ/m\({}^{3}\)) [29; 56]. Here, the unit cell volume at \(T\) = 4.1 K, \(V_{cell}\) = 82.50 A\({}^{3}\) is identified using the experimental value of the MAE coefficient, \(K_{1}\), presented in the units of (MJ/m\({}^{3}\)), [29; 56], and (meV/cell) [55]. [PERSON] et al. [56], on the other hand, calculated \(K_{1}\) = 4.4 meV/cell using LDA and orbital polarization in better agreement with experimental data. However, their calculation was performed at the experimental cell volume and not the LDA volume, which is drastically smaller. Their agreement with experiment is thus somewhat coincidental. We believe the difference in the calculated \(K_{1}\) numbers is mostly due to electron exchange and correlation approximation. In Table 4, we also show our LDA results, and for YCo\({}_{5}\) (6.44 meV/cell), it is closer to experimental data (3.8 meV/cell) [29; 55] than our GGA result. Unfortunately, we realize that the MAE for these magnetic systems can be very sensitive to the particular DFT approximation (GGA or LDA) that is applied. Furthermore, there does not appear to be a simple systematic explanation for the difference in the MAE when comparing GGA and LDA