# 2 Energy barriers from the effective one spin problem (EOSP)

In what follows we consider a magnetic system of spins in the classical many-spin approach, i.e., taking into account its intrinsic properties such as the lattice structure, shape and size. The magnetic properties of such system are described by the anisotropic Heisenberg model of classical spins (with ), described in Eq. (2.1), where the the anisotropy constants are measured in energy units per spin.

As we have shown in the previous chapter, under certain conditions the energy potential of such muti-spin particle can be mapped onto an EOSP model, in such way that the effective energy potential has the form:

 (49)

First, we investigate the minima, maxima and saddle points of the effective potential (3.7) for different values and signs of the parameters and , and calculate analytically the energy barriers in each case. The results are presented in Tab. 3.1.

 Minima Saddle points Energy barriers, (1.1) (1.2) (2) (3) (4.1) (4.2) (5.1) (5.2) (5.3) (6.1) (6.2) (6.3) (7) (8) (9) (10.1) (10.2)

In the previous chapter we have shown that although the magnetic behavior of the multi-spin particle can be mapped onto an effective one spin problem, its energy landscape can have a very complex character if the effective potential has two competing anisotropies, this fact is reflect in the results shows in Tab. 3.1. In some cases there are multiple energy barriers, but in foregoing we will consider only the relevant energy barrier for switching, corresponding to the lowest energy path between the global minima. Nevertheless, we would like remark that for large surface anisotropy , where , all energy barriers are simple linear combinations of the two effective anisotropy constants. The energy barriers for the case are plotted in Fig. 3.1 as a function of the parameter .

Here we only present analytical expressions based on the potential (3.7) with direct relevance to the results presented in the previous chapter for spherical, octahedral an elongated nanoparticles. We have seen that in the case of a spherical particle cut from an sc lattice, and in accordance with the EOSP energy potential (3.7), and (see Fig. 2.5). For a spherical particle with an fcc lattice, , as can be seen in Fig. 2.8. Finally for ellipsoidal and truncated octahedral multi-spin particles (MSPs), the results in Figs. 2.10 and 2.12 show that may become negative at some value of , since then the contributions similar to (2.21) and (2.16) become important.

So, when and , from Eq. (3.7) we find

 (50)

In the case , the energy barriers read,

 (51)

Rocio Yanes