The superparamagnetic relation
Abstract
Analytical calculations based on finite-size spin-wave theory and Monte Carlo (MC) simulations are performed to investigate the validity of the well-known relation m(H,T)=M(H,T)B_{D}[M(H,T)\cal N H/T] between the induced magnetization m of the magnetic particle and its intrinsic magnetization M for the Ising and isotropic classical models [B_{D}(x) is the Langevin function, D is the number of spin components, \cal N is the number of atoms in the particle]. It follows from general arguments and from our analytical results for the Heisenberg model at T << T_c that this relation is not exact for any finite D and nonzero temperature. Nevertheless, corrections to this formula remain very small practically in the whole range T> 1, as confirmed by our Monte Carlo calculations. At T \lesssim T_c/4 there is a good agreement between the MC and finite-size spin-wave calculations for the field dependence of m and M for the Heisenberg model with free boundary conditions.Keywords
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