Magnetic polarization clouds, percolation, and the critical concentration for ferromagnetism

The magnetic properties of many alloys are described in terms of magnetic polarization clouds. This is true in particular of dilute alloys of magnetic atoms in strongly exchange-enhanced hosts and of concentrated but magnetically dilute alloys of Ni in nonmagnetic hosts. In such alloys the locally averaged magnetization density is observed by diffuse neutron scattering to fall off as a Yukawa function away from the center of a polarization cloud. We present numerical calculations which confirm this Yukawa form and present an approximate theoretical derivation which gives a formula for the inverse range κ of a polarization cloud in an alloy in terms of the uniform exchange-enhancement factor α and the band susceptibility Γ(q) for the alloy. Assuming ferromagnetism in such alloys to arise from percolation amongst the polarization clouds, we also derive a relationship between the critical concentration c*cr, of polarization clouds for the onset of ferromagnetism and the average magnetization μ*cr and inverse range κcr of those clouds. This relationship is found to be in good agreement with experiment. Approximate calculated values of c*cr, μ*, and κ are presented for two cases. The effects of chemical ordering and clustering are considered briefly.