Hyperfine Splitting in a Metal of a Localized Moment

Abstract

Phenomenological equations describing the hyperfine splitting of a localized electron resonance in a metal are studied. The transverse susceptibility χ+(ω) is obtained for the case of nuclear spin I=½, and its poles and residues are displayed. The corresponding eigenvectors are determined, enabling a physical generalization to arbitrary nuclear spin. The validity of the generalization procedure is verified by an examination of the generalized susceptibility. The conditions for the observation of the hyperfine splitting in a dilute magnetic alloy are detailed. Three distinct regions are found to be important, depending on the strength of the exchange coupling. Region (1), corresponding to a very weak exchange coupling, exhibits hyperfine splittings of the same character as in an insulator. The linewidth of each hyperfine component equals Δse+ΔsL, where Δse is the second-order localized-conduction electron exchange relaxation rate, and ΔsL the localized spin lattice relaxation rate. Region (2) corresponds to sufficiently larger values of the exchange coupling for a magnetic resonance bottle-neck to obtain, but for which ωhf/Δse≫1, where ωhf is the electronic hyperfine splitting frequency. In region (2) the hyperfine splitting is the same as in region (1), but the width of each component equals [2I/(2I+1)]· [Δse+ΔsL]. Region (3) corresponds to yet larger values of the exchange coupling, such that ωhf/Δse≪1. In this more strongly bottlenecked region the hyperfine split resonance is narrowed into a single line, and the resonance properties are those one would derive in the complete absence of a hyperfine interaction. Recent successful observations of the hyperfine splitting of a localized moment resonance for rare-earth alloys belong to category (1). Unsuccessful measurements in transition metal alloys indicate their character is appropriate to region (3). It is suggested that lower temperature measurements may shift some of these materials into region (2) (because Δse ∝T), allowing for a direct determination of the magnitude of the exchange coupling constant, even though the resonance system is still in the bottlenecked regime.