ESR linewidth of rare-earth ions in dilute rare-earth alloys

The exchange coupling between the $4f$ magnetic electrons and the $5d$-like screening electrons is calculated for Er and Dy dilute magnetic alloys, in the limit that the strength of the spin-orbit coupling of the $5d$ virtual bound state is much greater than its half-width. Significant cancellation between the spin and orbital components of the exchange is found. This leads to a drastic reduction in the calculated linewidth from the value calculated in the absence of spin-orbit coupling.