## 3 Results and Discussion
### Characteristic Observables of CoFe\({}_{2}\)O\({}_{4}\) Nanoparticles and Study Method
The obtained results show four very important aspects for the subsequent and efficient application of superparamagnetic hyperthermia:  
(i) The specific loss power P\({}_{\rm s}\) has a narrow maximum (P\({}_{\rm sM}\)) for a very small value of the diameter (D) of the CoFe\({}_{2}\)O\({}_{4}\) nanoparticle of approx. 6-6.5 nm. This result is in good agreement with the one obtained in [1; 67]. However, the result is very different from that obtained for the magnetite nanoparticles (Fe\({}_{3}\)O\({}_{4}\)), where the maximum specific loss power was obtained at a nanoparticle diameter of \(\sim\)16-17 nm [28; 53]. This high difference in the diameters of magnetic nanoparticles is due to the very high magnetic anisotropy of cobalt ferrite, which has a much higher magnetocrystalline anisotropy constant (200 kJ/m\({}^{3}\)) [55] than that of magnetite (11 kJ/m\({}^{3}\)) [54], which makes the maximum loss power to be obtained at significantly lower values of the CoFe\({}_{2}\)O\({}_{4}\) nanoparticles diameter. The fact that CoFe\({}_{2}\)O\({}_{4}\) nanoparticles are very small is a major advantage in terms of magnetic hyperthermia, which can be achieved intracellularly, leading to a more efficient destruction of tumor cells from within. The nanoparticles being so small can easily penetrate through the phospholipids membrane inside the cells (cytoplasm or even the nucleus), thus destroying them more efficiently from inside. 2. There is a dependence of the maximum specific loss power P\({}_{\text{sM}}\) on the diameter of the magnetic nanoparticles D as a function of the frequency of the magnetic field f  
Figure 1: The specific loss power (P\({}_{\text{s}}\)) as a function of CoFe\({}_{2}\)O\({}_{4}\) nanoparticles diameter (D) and magnetic field frequency (f) for amplitude of the magnetic field of (**a**) 15 kA/m and (**b**) 30 kA/m. (Figure 2), namely: the maximum power shifts from higher values to lower values of the nanoparticle diameter when the magnetic field frequency increases from 100 kHz to 1000 kHz. Table 2 shows some values for the diameters of CoFe\({}_{2}\)O\({}_{4}\) nanoparticles for which the maximum loss power is obtained for the frequency range limits (100 kHz and 1000 kHz) as well as at the mid-value of those limits (500 kHz). Therefore, given the size dependencies of the 3 rd power of the nanoparticle diameter in the initial magnetic susceptibility (Equation (5)), the Langevin parameter (Equation (3)) and the magnetic relaxation time (Equation (6)), for a precise calculation we must take into account the diameter values corresponding to the frequencies identified (as in Table 2) when calculating the maximum loss power P\({}_{\text{sM}}\) (Equation (2)), and then when calculating the heating temperature of the nanoparticles \(\Delta\text{T}_{\text{h}}\) (Equation (8)). A small deviation of the nanoparticle diameter from those values identified to be corresponding to the maximum will lead to lower values of the specific loss power, and implicitly of the heating temperature. (iii) The maximum loss power increases both with the amplitude and the frequency of magnetic field as shown in Figure 3. However, the increase is more accentuated with the increase of the magnetic field amplitude (see Figure 1a,b). However, for the magnetic fields higher than 60-70 kA/m there is a limiting effect (saturation) of increasing the specific loss power at frequencies above 600-700 kHz. \begin{table}
\begin{tabular}{c c c} \hline \hline \(\text{P}_{\text{sM}}\) (W/g) & \(\text{f}\) (kHz) & \(\text{D}_{\text{M}}\) (nm) \\ \hline
3.46 & 100 & 6.62 \\
13.64 & 500 & 6.12 \\
24.13 & 1000 & 5.88 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The values of maximum specific loss power (P\({}_{\text{sM}}\)) and the corresponding diameter of CoFe\({}_{2}\)O\({}_{4}\) nanoparticles (D\({}_{\text{M}}\)) for three frequencies in the range of 100-1000 kHz, and H = 15 kA/m and \(\varepsilon\) = 0.15.