The superparamagnetic relation

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 )BD[M(H,T )NH/T ] between the induced magnetization m of the magnetic particle and its intrinsic magnetization M for the Ising and isotropic classical models [BD(x) is the Langevin function, D is the number of spin components, 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 ≪ Tc 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 < Tc if N ≫ 1, as confirmed by our Monte Carlo calculations. At T . Tc/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. PACS. 75.50.Tt Fine particle systems – 75.10.Hk Classical spin models