Anisotropic exchange for dilute rare-earth alloys

The exchange coupling between the $4f$ magnetic electrons and the $5d$-like screening electrons is calculated for Dy, Er, and Yb dilute magnetic alloys. The coupling is shown to be anisotropic, with significant orbital contributions for Dy and Yb. The g-shift expression is derived for arbitrary crystal-field splitting of the $5d$ virtual bound states. The linewidths are computed in the limit of no crystal-field splittings. Comparison with experiments on Dy and Er in Ag and Al hosts reveals good agreement.