## 3 Thermodynamics Properties of the Y(Co-Fe-Ni)\({}_{5}\) Alloys
Calculated within EMTO formalism, the heat of formation of SmNi\({}_{5}\) and YNi\({}_{5}\) compounds (in the CaCu\({}_{5}\)-type structure) is \(-18.95\) mRy/atom and \(-22.91\) mRy/atom, correspondingly, which is in accord with the experimental measurements of \(-23.08\) mRy/atom (\(-30.3\) kJ/mole, SmNi\({}_{5}\), [32]) and \(-25.98\) mRy/atom (\(-34.1\) kJ/mole, YNi\({}_{5}\), [32]). As we mentioned in the Introduction, the YFe\({}_{5}\) compound as well as the SmFe\({}_{5}\) compound (see refs. [3,4]) do not exist in the equilibrium Y-Fe and Sm-Fe phase diagrams, correspondingly, thus no experimental information about the heat of formation of these hypothetical compounds is available. However, the EMTO calculations show that that the heat of formation of the YFe\({}_{5}\) compound is positive, \(+6.46\) mRy/atom, and is half of the calculated heat of formation of the SmFe\({}_{5}\) compound, \(+12.68\) mRy/atom. As a result, the calculated heat of formation of the YFe\({}_{3}\)Co\({}_{2}\) compound, \(+0.09\) mRy/atom, appears to be smaller than the calculated heat of formation of the SmFe\({}_{3}\)Co\({}_{2}\) compound, \(+2.15\) mRy/atom [4]. The computed heats of formation of the YFe\({}_{3}\)Ni\({}_{2}\) and the SmFe\({}_{3}\)Ni\({}_{2}\) compounds are both negative (stable compounds), however, the absolute value of the calculated heat of formation of the YFe\({}_{3}\)Ni\({}_{2}\) compound, \(|-6.20|\) mRy/atom, is more than twice as high as the absolute value of the heat of formation of the SmFe\({}_{3}\)Ni\({}_{2}\) compound, \(|-2.72|\) mRy/atom [4]. As a result, the region of stability of the pseudo-binary YFe\({}_{3}\)(Ni\({}_{1-x}\)Co\({}_{x}\))\({}_{2}\) alloys appears to be almost twice as wide as the region of stability of the pseudo-binary SmFe\({}_{3}\)(Ni\({}_{1-x}\)Co\({}_{x}\))\({}_{2}\) alloys. The promising phase stability of the YFe\({}_{3}\)CoNi compound highlighted by the EMTO-predicted heat of formation of the pseudo-binary YFe\({}_{3}\)(Ni\({}_{1-x}\)Co\({}_{x}\))\({}_{2}\) alloys in Figure 2a sparks interest in studying the thermodynamics and the stability of the Y(Co,Fe,Ni)\({}_{5}\) compounds versus temperature in the broader Y-Co-Fe-Ni composition space. As mentioned in Section 2, a full CALPHAD assessment of the Co-Fe-Ni-Y database requires evaluation of the constitutive binary (i.e., Co-Fe, Co-Ni, Co-Y, Fe-Ni, Fe-Y, Ni-Y) and ternary (i.e., Co-Fe-Ni, Y-Co-Fe, Y-Co-Ni, Y-Fe-Ni) systems. This study is focused on the Y-containing binary systems to leverage and confirm the ab initio findings and serve as foundation for an upcoming CALPHAD-centric study. The first Y-Co CALPHAD assessment, in which all intermetallic compounds were evaluated as stoichiometric compounds, was reported by [PERSON] et al. [88]. This assessment was revisited in [89] to consider solubility range in YCo\({}_{5}\) and Y\({}_{2}\)Co\({}_{17}\). As both structures of Y\({}_{2}\)Co\({}_{17}\) (Th\({}_{2}\)Ni\({}_{17}\)-type structure at high temperature and Th\({}_{2}\)Zn\({}_{17}\)-type structure at low temperature) are similar to the hexagonal CaCu\({}_{5}\)-type structure (YCo\({}_{5}\)), [PERSON] and [PERSON] [89] treated both Y\({}_{2}\)Co\({}_{17}\) and YCo\({}_{5}\) compounds as one phase  
Figure 2: (**a**) The heat of formation of the pseudo-binary YFe\({}_{3}\)(Ni\({}_{1-x}\)Co\({}_{x}\))\({}_{2}\) alloys predicted via ab initio (0 K) and CALculation of PHASe Diagrams (CALPHAD) (298 K) calculations. (**b**) The heat of formation of the pseudo-binary SmFe\({}_{3}\)(Ni\({}_{1-x}\)Co\({}_{x}\))\({}_{2}\) alloys predicted via ab initio calculations [4]. with the formula (Y,Co\({}_{2}\))\({}_{1}\)(Y,Co\({}_{2}\))\({}_{2}\)Co\({}_{15}\). The reasoning behind the use of a one-phase three-sublattice model is that homogeneity regions of Y\({}_{2}\)Co\({}_{17}\) and YCo\({}_{5}\) could be described by a CaCu\({}_{5}\)-type lattice in which part of the Ca sites is occupied by pairs of Co atoms [89].