## 3 Results and Discussion  
### Characteristic Observables of CoFe\({}_{2}\)O\({}_{4}\) Nanoparticles and Study Method  
The specific parameters of interest of the magnetic material of the CoFe\({}_{2}\)O\({}_{4}\) ferrimagnetic nanoparticles are shown in Table 1. These are: spontaneous magnetization of the material M\({}_{\rm s}\), magnetocrystalline anisotropy constant K, material density \(\rho\) and specific heat c [1; 54; 55]. For the study of the specific loss power and heating temperature of CoFe\({}_{2}\)O\({}_{4}\) nanoparticles, we used a professional 3D powerful software tool for calculation. We used the formulas in Section 1, and a spatial graphical representation in order to capture all the details through a simultaneous representation of the observables of interest as a function of two variables: the parameters of the harmonic alternating magnetic field (amplitude H and frequency f) and the basic characteristics of the magnetic nanoparticles (diameter D and volume packing fraction \(\epsilon\)) given in Table 1. Given that, in practice, magnetic hyperthermia is most often in the range of 100-500 kHz and for magnetic fields in the range of 10-40 kA/m, we have extended these ranges to study possible valid results that could be obtained outside the ranges used up to now. Additionally, for the diameter of nanoparticles D, for the same reason, the range of 1-20 nm was chosen, considering the known values for magnetite and \(\gamma\)-Fe\({}_{2}\)O\({}_{3}\) nanoparticle diameters (\(\sim\)15-20 nm). For the spontaneous magnetization and magnetic anisotropy constant in Table 1 we considered the standard values for CoFe\({}_{2}\)O\({}_{4}\) ferrite [54; 55], having in view the following reasons. In the case of magnetic nanoparticles, and even more for ferromagnetic and ferrimagnetic ones in which the exchange and superexchange interactions are strong, for small sizes of nanoparticles an important contribution of surface effects (spin canting, broken exchange bonds, etc.) can occur to magnetic saturation [71; 72; 73] and magnetic anisotropy [74; 75; 76], depending on the type and size of nanoparticles. This contribution can be manifested by a surface magnetic anisotropy [77; 78; 79; 80] and decreased magnetic saturation [71; 72; 73; 81]. Additionally, if the magnetic nanoparticles are not spherical there may be an important contribution of the shape of the nanoparticles to magnetic anisotropy if the nanoparticles deviate much from the spherical shape (e.g., magnetic nanoparticles are elongated) [82]. Thus, for magnetic anisotropy an effective magnetic anisotropy constant (K\({}_{\rm eff}\)) should be considered as a contribution of the magnetocrystalline anisotropy constant (K), the surface anisotropy constant (K\({}_{\rm s}\)) and the shape anisotropy constant (K\({}_{\rm sh}\)) (K\({}_{\rm eff}\) = K + K\({}_{\rm s}\) + K\({}_{\rm sh}\)) [78; 80]. However, in the case of our study we considered spherical nanoparticles (K\({}_{\rm sh}\) = 0), a condition fulfilled in most practical cases. Additionally, in the case of cobalt ferrite nanoparticles, the magnetocrystalline anisotropy constant (K = 2 \(\times\) 10\({}^{5}\) J/m\({}^{3}\)[54]) is much larger than the surface anisotropy constant (K\({}_{\rm s}\)), which in general can be found in the range 1-5 \(\times\) 10\({}^{4}\) J/m\({}^{3}\)[78; 80; 80], depending on the size and type of nanoparticles. Therefore, in the case of CoFe\({}_{2}\)O\({}_{4}\) ferrite nanoparticles the most important contribution to magnetic