Methods of derivation of self-consistent equations for an Ising magnet

Bethe approximation in the Ising model with mobile impurities

“…To calculate M and p approximately, let us use the following procedure [7][8][9]. Let us determine local exchange field h i and crystalline field ϕ i of the ith site as h i = Σσ j and ϕ j = Σ (summation is performed over all sites neighboring to the ith sites).…”

“…Then we will derive from (2) and (3) (6) (7) and from (4) and (5) (8) (9) where Set of equations (6)- (9) at h e > 0 (in this case, mag netization M is always positive) can be presented as follows. Let us introduce the notation Then we will derive from (6)-(9) (10) (11) (12) where (13) In fact, these equations are parametric depen dences M(p, K, h e ) and μ(p, K, h e ) since, as we can eas ily convince, such range of values of parameter w always occurs at h e > 0, for which p calculated by for mula (11) runs the entire range from 0 to 1.…”

“…The Bethe approximation for a pure magnet can be obtained as the solution of the Ising problem for the Bethe lattice (tree) [5] or as the relation between the magnetization of the central atom and the magne tization of the atom of the first coordination sphere [6]. However, we can also consider the Bethe approxi mation as one of self consistent methods for which the general construction scheme was presented in our pre vious papers [7][8][9]. These methods are based on the averaging over local exchange fields.…”