Indirect Exchange Coupling of Magnetic Moments in Rare-Earth Metals

The Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange mechanism is examined for its relevance to ferromagnetism in the rare-earth metals. More specifically, the experimental sensitivity of the exchange energy (or Curie temperature) to the interatomic spacing (applied pressure) of the rare earths is examined in terms of the RKKY interaction. The form in which the interaction is usually examined, i.e., with the coupling parameter $J\left(r\right)\mathrm{}$ given by a Dirac $\delta$ function, is unable to provide the observed sensitivity of Curie temperature to interatomic spacing. When this coupling parameter, or exchange interaction, is calculated with free-ion wave functions for the $4f$ electrons, instead of the $\delta$-function approximation, agreement between theory and experiment is still not obtained. In this article, a modification of the coupling parameter is proposed which effectively takes into account some contribution of the crystal in determining the wave functions to be associated with the isolated magnetic moments. The procedure is phenomenological. Both the correct shape of the Curie-temperature-pressure curve and the sensitivity of the Curie temperature to pressure are obtained. Also the correct interlayer turn angle of the various elements is obtained. Undetermined multiplicative parameters appear in the final result which can be adjusted at some point in order to make the agreement quantitative. The conclusion is that the present modification of the RKKY interaction appears to account correctly for the origin of ferromagnetism in the rare-earth metals.