Origin of Fe substitutions in ${\mathrm{Nd}}_2{\mathrm{Fe}}_{17\mathrm{-}\mathrm{\delta }}{X}_{\delta }$

In ${\mathrm{Nd}}_2{\mathrm{Fe}}_{17\mathrm{-}\mathrm{\delta }}{X}_{\delta }$ ($X=\mathrm{Al},$ Si, Ti, V, Cr, Mn, Co, Ga, Nb, Mo, W, and $\delta =0,\hspace{0.5em}0.5$) the $X$ atoms substitute for Fe without changing the crystal structure. Neutron-diffraction measurements show that Al, Si, Co, and Ga tend to substitute for Fe with a strong preference for the $18h$ sites while Ti, V, Cr, Mn, Nb, Mo, and W tend to substitute for Fe with a strong preference for the $6c$ sites. This substitution can be understood by considering a model which includes contributions from the enthalpy of solution of the $X$ atom at each Fe site, the elastic energy due to the differences in the volume of the site and the volume of the $X$ atom, and a structural-dependent contribution reflecting the difference in the average number of valence electrons at each Fe site.