Spatially Random Heisenberg Spins at Very Low Temperatures. III. Multicomponent Ferromagnetic Alloy with Substitutional Short-Range Order

Abstract

A formulation is given for calculating both the partial and the total single-site magnetic densities of states in a multicomponent Heisenberg ferromagnetic alloy in the presence of arbitrary amounts of nearest-neighbor substitutional short-range order (SRO). The number $n$ of the alloy constituents $\ensuremath{\lambda}$ and their concentrations ${m}^{\ensuremath{\lambda}}$ are assumed to be arbitrary. Only isotropic nearest-neighbor exchange integrals are treated. Moreover, except for the special case of a noncommunicating alloy (where all interspecies exchange is vanishing), the positivity of the intraspecies exchange ${J}^{\ensuremath{\lambda},\ensuremath{\lambda}}$ is assumed throughout. By computing and comparing the exact results for the frequency moments of the total density of states with those given by the present theory, we confirm that the first four frequency moments are exactly conserved (for lattices without nearest-neighbor triangles) and frequency moments of arbitrary order are given exactly for all lattices to the leading order in the ${z}^{\ensuremath{-}1}$ power expansion. For certain favorable situations (which include the cases of a dilute ferromagnet, the symmetrical dilute ternary, and the $n$-component noncommunicating alloy) our theory also enables us to calculate the system Curie temperature. We discuss these cases in detail and show how in a dilute ferromagnet the critical concentration is strongly dependent upon the substitutional SRO. Indeed, for perfect clustering of all magnetic atoms, the critical concentration is effectively zero. We also show how by manipulating (i) the relative strength of the interspecies to intraspecies exchange coupling and (ii) the substitutional SRO among the constituents, first-order magnetic phase transitions may be induced. For the noncommunicating alloy we show that under suitable conditions ferromagnetic and antiferromagnetic long-range order (LRO) may coexist. Lastly, we compare our results for the Curie temperature (for a random symmetrical binary) with those given by Murray. These results agree to the two leading orders in a ${z}^{\ensuremath{-}1}$ expansion. When $z$ is small, e.g., in a simple-cubic lattice, these results differ somewhat: Ours always lies lower than Murray's.