We present a computational study of the compound Y(Co$_{1-x-y}$Fe$_x$Cu$_y$)$_5$ for 0 $leq x,y leq 0.2$. This compound was chosen as a prototype for investigating the cell boundary phase believed to play a key role in establishing the high coercivity of commercial Sm-Co 2:17 magnets. Using density-functional theory, we have calculated the magnetization and magnetocrystalline anisotropy at zero temperature for a range of compositions, modeling the doped compounds within the coherent potential approximation. We have also performed finite temperature calculations for YCo$_5$, Y(Co$_{0.838}$Cu$_{0.162}$)$_5$ and Y(Co$_{0.838}$Fe$_{0.081}$Cu$_{0.081}$)$_5$ within the disordered local moment picture. Our calculations find that substituting Co with small amounts of either Fe or Cu boosts the magnetocrystalline anisotropy $K$, but the change in $K$ depends strongly on the location of the dopants. Furthermore, the calculations do not show a particularly large difference between the magnetic properties of Cu-rich Y(Co$_{0.838}$Cu$_{0.162}$)$_5$ and equal Fe-Cu Y(Co$_{0.838}$Fe$_{0.081}$Cu$_{0.081}$)$_5$, despite these two compositions showing different coercivity behavior when found in the cell boundary phase of 2:17 magnets. Our study lays the groundwork for studying the rare earth contribution to the anisotropy of Sm(Co$_{1-x-y}$Fe$_x$Cu$_y$)$_5$, and also shows how a small amount of transition metal substitution can boost the anisotropy field of YCo$_5$.
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