In this section we compute the energy potential as a function of the polar angle and azimuthal angle of net magnetization of the multispin particle. First, we do this for a spherical particle with uniaxial anisotropy in the core and NSA, cut from an sc and fcc lattices, and for different values of the surface anisotropy constant, .
We apply the Lagrange multiplier technique to analyze the case of a spherical multispin particle of spins on an sc lattice with uniaxial anisotropy in the core ( ) and NSA.

With the aim to extract the value of the effective uniaxial and cubic anisotropy constants, we cut the energy landscape at and obtain the energy potential, later we fit it to formula (2.23). In Fig. 2.4 we present the energy potential () of a multispin particle with uniaxial anisotropy in the core ( ) and NSA with (left), (right). The solid lines are numerical fits to formula (2.23). From this graph we see that the energy of the multispin particle is well recovered by Eq. (2.23) when is small. Consequently, such multispin particle can be treated as an EOSP with an energy that contains uniaxial and cubic anisotropies. However, as it is started to be seen in the right panel, and as was shown in Ref. [87], when the surface anisotropy increases, this mapping of the multispin particle onto an effective onespin particle is less satisfactory.
Repeating this fitting procedure for other values of we obtain the plots of and as a function of , see Fig. 2.5. Here we first see that these effective constants are quadratic in , in accordance with Eqs. (2.15) and (2.21). In addition, the plot on the right shows an agreement between the constant obtained numerically and the analytical expression (2.21), upon subtracting the pure core contribution , see Eq. (2.6). The agreement is better in the regime of small . These results confirm those of Refs. [87,97,73] that the core anisotropy is renormalized by the surface anisotropy, though only slightly in the present case.
Spherical particles cut from the sc lattice exhibit an effective fourfold anisotropy with , as we can check from the numerical results in Fig. 2.5 and analytical expression Eq. (2.23). As such, the contribution of the latter to the effective energy is positive.

Next we will analyze the case of a spherical particle with fcc lattice structure. First, in the same way that we have done in the case of an sc particle, we calculate the energy landscape, see Fig. 2.6. Comparing the energy potential in Fig. 2.3 for the sc and Fig. 2.6 for the fcc lattice one realizes that, because of the different underlying structure and thereby different spin surface arrangements, the corresponding energy potentials exhibit different topologies. For instance, it can be seen that the point is a saddle in MSPs cut from an sc lattice and a maximum in those cut from the fcc lattice.
In Figs. 2.7, we plot the energy potential for a spherical particle with fcc structure, for two values of : in the left graph and in the right graph. The solid line represents numerical fits to formula (2.23). We can observe an agreement between the numerical results and the fittings to formula (2.23). Now, we extract the values and as a function of for a fcc spherical particle. The effective cubic constant appears to be positive in contrast to sc case, see Fig. 2.8, and as for the sc lattice, it is quadratic in . As mentioned earlier, the coefficient in Eq. (2.15) depends on the lattice structure and for fcc it may become negative. To check this, one first has to find an analytical expression for the spin density on the fcc lattice, in the same way that the sc lattice density was obtained in Ref. [74] (see Eq. (6) therein). Likewise, the coefficient in Eq. (2.21) should change on the fcc lattice, thus changing the uniaxial and cubic contributions as well.

In fact that not only the value of the effective constant but even its sign depend on the underlying structure, more exactly the surface arrangement. It is an important point for general modelling. Very often and for simplicity the nanoparticles are considered cut from sc lattice, disregarding the fact that realistic nanoparticles never have this structure.
Rocio Yanes