In the case of spherical nanoparticles the same relation has been suggested , leading to the formula .
We would like to emphasize that this formula has been introduced in an "ad hoc" manner, and it is far from evident that the surface should contribute into an effective uniaxial anisotropy in a simple additive manner. Actually one cannot expect to coincide with the atomistic single-site surface anisotropy, especially when strong deviations from non-collinearities leading to "hedgehog-like" structures appear. The effective anisotropy appears in the literature in relation to the measurements of energy barriers of nanoparticles, extracted from the magnetic viscosity or dynamic susceptibility measurements. Generally speaking, the surface anisotropy should affect both the minima and the saddle points of the energy landscape in this case. It is clear that while the measurement of viscosity are related to the saddle point, the magnetic resonance measurements depend on the stiffness of the energy minima modified by the surface effects. Thus the meaning of the is different for different measurement techniques.
Despite its rather "ad hoc" character, this formula has become the basis of many experimental studies with the aim to extract the surface anisotropy (see, e.g., Refs. [8,57,32]) because of its mere simplicity.
In Fig. 1.6 we present some experimental results (from Ref. ) where the effective anisotropy is plotted as a function of the inverse of the diameter of Co nanoparticle (which is supposed to have a spherical shape) and is fitted to the expression of similar to Eq.(1.17).