## 3 Results and Discussion
### The Specific Loss Power in the Linear Approximation  
#### 3.3.1 Maximum Specific Loss Power  
Using the formulas for the calculation of the specific loss power P\({}_{8}\) given by Equation (2) and Equation (7), the calculations made for the maximum specific loss power (P\({}_{\text{sM}}\)) at the frequency of 500 kHz for two relatively distant magnetic field values, one usual of (a) 15 kA/m and the other relatively high (b) 60 kA/m, lead to the results shown in Figure 4. Using Equation (7) we obtained the diagrams (a1) for the magnetic field of 15 kA/m and the diagram (b1) for the magnetic field of 60 kA/m. Using Equation (2) we obtained the diagrams (a2) for 15 kA/m and (b2) for 60 kA/m. The results show that there are no noticeable differences between the maximums of the specific loss powers P\({}_{\text{sM}}\) obtained in the case of Equation (2) and for the higher magnetic field of 60 kA/m (Figure 4b2). Figure 3: The variation of specific loss power (P\({}_{8}\)) as a function of the amplitude (H) and frequency (f) of magnetic field for the diameter (D) of the CoFe\({}_{2}\)O\({}_{4}\) nanoparticles of 6.25 nm. The explanation for this behavior which is very different from the case of magnetite is the following: Co ferrite nanoparticles have high magnetic anisotropy [55; 83] which leads to a very small size of the nanoparticle diameter D which determines the maximum specific loss power, as previously shown (Figure 1). The small size of the nanoparticles also causes a low magnetic susceptibility (\(\chi_{0}\)), which also varies very little with the increase of the magnetic field even when using very large magnetic fields of approx. 100 kA/m, as seen in Figure 5.  
Calculating the maximum loss powers (P\({}_{\rm{SM}}\)) for those two cases (as in Figure 4), using Equations (2) and (7), we find that the differences between them appear only at large values of the magnetic field, over \(\sim\)70 kA/m as shown in Figure 6. The yellow curve represents the variation of the maximum specific loss power with the increase of the magnetic field up to the value of 100 kA/m in the linear approximation given by Equation (7), considering the magnetic susceptibility constant and equal to the initial one (\(\chi_{\rm{j}}\)). The green curve represents the maximum specific loss power calculated using the formula given by Equation (2), when  
Figure 4: The specific loss power (Ps) as a function of CoFe\({}_{2}\)O\({}_{4}\) nanoparticles diameter (D) calculated at the magnetic field frequency (f) of 500 kHz, using: (**a1**) Equation (7) and the magnetic field amplitude H = 15 kA/m, (**b1**) Equation (7) and H = 60 kA/m, (**a2**) Equation (2) and H = 15 kA/m, and (**b2**) Equation (2) and H = 60 kA/m. the susceptibility depends on the magnetic field. The maximum specific loss powers were calculated at a frequency of 500 kHz and for the corresponding nanoparticle diameter of 6.12 nm. The explanation for this behavior in the case of CoFe\({}_{2}\)O\({}_{4}\) nanoparticles which is very different from that of magnetite is given in the next section. Figure 5: The variation of magnetic susceptibility of CoFe\({}_{2}\)O\({}_{4}\) nanoparticles as a function of the amplitude of magnetic field H up to 100 kA/m, for the diameters of nanoparticles of 6.6-6.7 nm. Figure 6: The dependence of the maximum specific loss power (P\({}_{\rm{SM}}\)) on the magnetic field (H) for CoFe\({}_{2}\)O\({}_{4}\) nanoparticles, calculated at the frequency (f) of 500 kHz using Equation (7) (yellow curve) and Equation (2) (green curve). 3.2 Magnetic Behavior of Small CoFe\({}_{2}\)O\({}_{4}\) Nanoparticles and Linearity of Magnetization  
When calculating the magnetization (\(M\)) in the case of CoFe\({}_{2}\)O\({}_{4}\) nanoparticles using Equation (4), where the parameter of the Langevin function is given by Equation (3), for an average nanoparticle diameter of \(\sim\)6 nm we obtained the curve \(M=f(H)\) shown in Figure 7a.