Heisenberg Model for Dilute Alloys in the Molecular Field Approximation

Abstract

In the 1950’s several experiments were performed in which dilute concentrations of magnetic impurities were dissolved in a nonmagnetic metal host. These experiments were done prior to the discovery of the Kondo1 effect and their purpose was to study the magnetic properties of materials in which the probability of having a near neighbor is not appreciable. Therefore any cooperative magnetic phenomena which these systems exhibit necessarily arise from the long-range magnetic interaction. That long-range magnetic phenomena existed was shown by Owen et al.2 and Schmitt and Jacobs3 for Cu—Mn and by Lutes and Schmit4 for Au—Fe. This long-range interaction can be obtained theoretically by using an s—d Hamiltonian and calculating the effective Hamiltonian in second-order perturbation theory. This calculation was done by Ruderman and Kittel,5 Kasuya,6 and Yosida,7 and will be denoted in this paper as the RKKY interaction. The effective Hamiltonian ℋ for the interaction between the magnetic impurities is7 $$ H = \left( {\frac{{3n}} {{N_0 }}} \right)^2 \frac{{2\pi }} {{E_\text{F} }}J\left( 0 \right)^2 \sum\limits_{i,j} {F\left( {2k_\text{F} r_{ij} } \right)S_i \cdot S_j \equiv \sum\limits_{i.j} {v_{ij} S_i \cdot } } S_j $$ (1) where F(x) = (x cos x — sin x)/x 4, n/N 0 is the number of conduction electrons per atom, E F is the Fermi energy of the host metal, and J(0) is the s—d exchange interaction and is assumed to be a constant. Si and Sj are the spin operators of the impurities located at sites r i and r j respectively, and k F is the Fermi wave vector. The second form of Eq. (1) defines the potential v ij .