## 3 Results and Discussion
### The Specific Loss Power in the Linear Approximation
#### 3.3.1 Maximum Specific Loss Power
However, in this case, there is also a decrease in the maximum temperature reached, from 42.43 \({}^{\circ}\)C to 26.74 \({}^{\circ}\)C, mainly due to the decrease in the frequency limit from 1000 kHz to 83 kHz used in this case. However, this does not  
Figure 9: The time variation of heating temperature (\(\Delta\)T\({}_{\rm h}\)) of CoFe\({}_{2}\)O\({}_{4}\) nanoparticles in optimal conditions given in Table 3 for volume fraction of 0.15 and two values of magnetic field: (**a**) 5 kA/m and (**b**) 60 kA/m. affect the hyperthermic effect, increasing the temperature by 18 \({}^{\circ}\)C above room temperature being achieved, and the optimal temperature of 43 \({}^{\circ}\)C being obtained. Using 3D representations such as those in Figure 9 for the optimal conditions in Table 3 (field values in the range 5-60 kA/m, limit frequencies in the range 83-1000 kHz and the optimal diameter of nanoparticles in the range 5.88-6.67 nm) we determined all values for the optimal time intervals (\(\Delta\)t\({}_{\rm o}\)) (marked in green) necessary to reach the optimum temperature of 43 \({}^{\circ}\)C (increase with \(\Delta\)T\({}_{\rm ho}\) = 18 \({}^{\circ}\)C above room temperature), as well as the values of the maximum temperature \(\Delta\)T\({}_{\rm hm}\) that is reached and the times \(\Delta\)t\({}_{\rm m}\) required for this. The values obtained are shown in Table 4.  
However, considering that in practice smaller packing fractions can be used, in order to check the heating efficiency in these cases we also considered another much smaller fraction, namely \(\epsilon\) = 0.017. The results obtained in this case are shown in Figure 10 for the same magnetic fields as in the previous case: 5 kA/m (Figure 10a) and 60 kA/m (Figure 10b). The results show that even in these cases the optimal heating of CoFe\({}_{2}\)O\({}_{4}\) nanoparticles can be obtained at the optimum temperature (T\({}_{\rm o}\)) of 43 \({}^{\circ}\)C, only that this will be done in a longer time interval (\(\Delta\)t\({}_{\rm o}\)). For example, in the case of a magnetic field of 5 kA/m (Figure 10a), a heating of the nanoparticles by 18 \({}^{\circ}\)C in addition to the room temperature (25 \({}^{\circ}\)C) in order to reach the optimal value of 43 \({}^{\circ}\)C, will be made in a time interval (optimal) \(\Delta\)t\({}_{\rm o}\) of \(\sim\)44 s compared to only \(\sim\)5 s in the previous case. The optimal values for the time intervals (\(\Delta\)t\({}_{\rm o}\)) determined in this case for different values of the magnetic field in the range of 5-60 kA/m, in the optimal conditions and in the biologically admissible limits, are given in Table 5. From the table it is observed that increasing the amplitude of the applied magnetic field leads to a decrease in the heating time, as we expected. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**H (kA/m)** & **f\({}_{1}\) (kHz)** & **D\({}_{0}\) (nm)** & **\(\Delta\)T\({}_{\rm hm}\) (\({}^{\circ}\)C)** & **\(\Delta\)t\({}_{\rm m}\) (s)** & **\(\Delta\)T\({}_{\rm ho}\) (\({}^{\circ}\)C)** & **\(\Delta\)t\({}_{\rm o}\) (s)** \\ \hline
5 & 1000 & 5.88 & 42.43 & 23.33 & 18 & 5.01 \\
15 & 334 & 6.25 & 33.35 & 5.18 & 18 & 1.45 \\
30 & 167 & 6.46 & 29.29 & 2.09 & 18 & 0.69 \\
45 & 111 & 6.58 & 27.48 & 1.27 & 18 & 0.46 \\
60 & 83 & 6.67 & 26.74 & 0.92 & 18 & 0.35 \\ \hline \hline \end{tabular}  
* \(\Delta\)T\({}_{\rm ho}\) = 43–25 \({}^{\circ}\)C; ** (\(\pm\)0.01).  
\end{table} 
Table 4: The values of the optimal heating times (\(\Delta\)t\({}_{\rm o}\)) of the CoFe\({}_{2}\)O\({}_{4}\) nanoparticles under the optimal parameters given in Table 3, for the volume fraction of 0.15. The table also shows the values of the maximum variation of the heating temperature (\(\Delta\)T\({}_{\rm hm}\)) of the nanoparticles and their corresponding durations (\(\Delta\)t\({}_{\rm m}\)) under the given conditions.