## 3 Thermodynamics Properties of the Y(Co-Fe-Ni)\({}_{5}\) Alloys  
The YCo\({}_{5}\) compound crystallizes in the hexagonal CaCu\({}_{5}\)-type structure with three non-equivalent atomic sites: Y1-(1a), Co1-(2c), and Co2-(3g) (see Figure 1) with six atoms per formula unit and per computational cell. Table 1 shows the equilibrium formula unit (f.u.) volume, bulk modulus, and the pressure derivative of the bulk modulus for the YCo\({}_{5}\) compound calculated using each of the ab initio techniques. The results of the EMTO calculations are presented both without (\"SREMTO\", \"SR\" means scalar-relativistic) and with (\"FREMTO\") inclusion of SOC. The experimental value of the equilibrium f.u. volume is taken from ref. [29; 55]--see further discussion below. Notice that the equilibrium volume, calculated by FPLMTO method, agrees very well with the experimental data and that incorporating the relativistic SOC in the calculation has only a minor effect. Earlier neutron-diffraction studies of the Th(Co\({}_{1-x}\)Fe\({}_{x}\))\({}_{5}\) alloys (also based on the CaCu\({}_{5}\)-type structure) [14] show that the larger Fe atoms preoccupies the 3g-type sites, whereas the smaller Co atoms choose to occupy the 2c-type sites. This occupational inclination has been affirmed by DFT calculations for YCo\({}_{5}\)[67] and SmCo\({}_{5}\)[68] compounds. In line with these calculations, the total energy for Fe at the 3g site (\(E_{3g}\)) is lower that than for Fe at the 2c site (\(E_{2c}\)) by 0.21 eV/f.u. and 0.10 eV/f.u. for YCo\({}_{5}\) and SmCo\({}_{5}\) magnets, correspondingly. As was mentioned in refs. [19; 28], if the YCo\({}_{5}\) magnet is doped with Fe and Ni, Fe atoms occupy preferentially 3g sites, while Ni atoms favor 2c sites. Figure 2a shows the heat of formation calculated within the EMTO-CPA technique of the pseudo-binary YFe\({}_{3}\)(Ni\({}_{1-x}\)Co\({}_{x}\))\({}_{2}\) alloys where Fe atoms occupy all 3g-type sites and the occupation of the 2c-type sites continuously changes from pure Ni (the YFe\({}_{3}\)Ni\({}_{2}\) compound) to pure Co (the YFe\({}_{3}\)Co\({}_{2}\) compound). The current calculations show that the YFe\({}_{3}\)(Ni\({}_{1-x}\)Co\({}_{x}\))\({}_{2}\) alloys could remain stable until almost all Ni atoms are replaced by Co atoms. For comparison, we also show (Figure 2b) the heat of formation calculated within the EMTO-CPA formalism [3; 4] of the pseudo-binary SmFe\({}_{3}\)(Ni\({}_{1-x}\)Co\({}_{x}\))\({}_{2}\) alloys where Fe atoms occupy all 3g-type sites and the occupation of the 2c-type sites continuously changes from pure Ni (the SmFe\({}_{3}\)Ni\({}_{2}\) compound) to pure Co (the SmFe\({}_{3}\)Co\({}_{2}\) compound). These calculations show that the pseudo-binary SmFe\({}_{3}\)(Ni\({}_{1-x}\)Co\({}_{x}\))\({}_{2}\) alloys could remain stable until almost half of Ni atoms are replaced by Co atoms. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Property** & **SREMTO** & **FREMTO** & **FPLMTO** & **Expt.** [29; 55] \\ \hline Unit cell volume (Å\({}^{3}\)) & 85.07 & 85.24 & 82.65 & 82.50 \\ \hline c/a axial ratio & & & 0.80 & \\ \hline Bulk modulus (GPa) & 134.63 & 128.65 & 164 & \\ \hline Bulk modulus pressure derivative & 3.70 & 3.81 & 2.62 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Equilibrium formula unit volume, bulk modulus, and its pressure derivative of the YCo\({}_{5}\) compound as functions of the ab initio technique. The unit cell volume is measured at \(T\) = 4.1 K.  
In order to understand the difference between Figure 2a,b, one needs to take into consideration several of the following arguments. Nickel metal forms the stable CaCu\({}_{5}\)-type compounds with both yttrium and samarium metals.