1 Introduction

We have demonstrated in the previous chapters the relevance of surface effects in nanoparticles and nanostructures: the Néel surface anisotropy can induce additional macroscopic anisotropies which affect the magnetic behavior of the system. For nanoparticles in chapter 2 we have investigated the influence of surface anisotropy at zero temperature, in particular highlighting the effects of spin non-collinearities. The most noticeable effect of these non-collinearities is the appearance of effective cubic anisotropy caused by large values of the surface anisotropy constant $K_s$. It is reasonable to ask the question, what happens to these effective anisotropies with the additional spin non-collinearities due to the thermal disordering. Also from the Callen-Callen theory one can see that cubic and uniaxial type anisotropies have very different temperature dependencies. Thus it is an open question whether the surface induced cubic anisotropy in nanoparticles also shows such a strong temperature dependence. This is what we aim to address in the present chapter.

In the present chapter we consider spherical and truncated octahedral nanoparticles cut from the face-centered cubic (fcc) and simple cubic (sc) internal structures. Nanoparticles have the diameter $D\approx 3 nm$ with approximately $N=1200\div1500$ atoms depending on the underlying structure and shape, and $D\approx 7 nm$ with approximately $N=6300$ atoms. To model the magnetic behavior we use an anisotropic Heisenberg model similar to that described in section 4.1.

We have taken into account the exchange interaction with nearest-neighbor exchange energy only ( we use $J=1.0\times 10^{-13} erg$ for sc and $J=5.6\times10^{-14} erg$ for fcc lattices). For core spins, we use the magneto-crystalline anisotropy in the uniaxial or cubic form, with the core anisotropy value $K_c=4.16 \times 10^6 erg/cm^3$. For surface spins (spins with not full coordination number) we use the Neél surface anisotropy model, with varying $K_{s}$.