  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)NH/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, 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<T_c if N >> 1, as confirmed by our Monte Carlo
calculations. At T <~ 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.
