  Using exact diagonalization, Monte-Carlo, and mean-field techniques,
characteristic temperature scales for ferromagnetic order are discussed for the
Ising and the classical anisotropic Heisenberg model on finite lattices in one
and two dimensions. The interplay between nearest-neighbor exchange, anisotropy
and the presence of surfaces leads, as a function of temperature, to a complex
behavior of the distance-dependent spin-spin correlation function, which is
very different from what is commonly expected. A finite experimental
observation time is considered in addition, which is simulated within the
Monte-Carlo approach by an incomplete statistical average. We find strong
surface effects for small nanoparticles, which cannot be explained within a
simple Landau or mean-field concept and which give rise to characteristic
trends of the spin-correlation function in different temperature regimes.
Unambiguous definitions of crossover temperatures for finite systems and an
effective method to estimate the critical temperature of corresponding infinite
systems are given.
