## 3 Results and Discussion
One can see that the dependence of the saturation magnetization with the Mn content is not linear and can be explained by the changes in the inversion degree with the composition. In the case of the sample with \(x=1.0\), the experimental magnetic moments are 0.45 \(\upmu_{\text{B}}\) higher than the calculated values with the proposed cation distribution. The difference could be explained by the presence of some extra oxygen in this sample and consequently a larger amount of Fe\({}^{3+}\) ions on the octahedral site. The magnetic properties of Co\({}_{1-x}\)Mn\({}_{x}\)Fe\({}_{2}\)O\({}_{4}\) nanoparticles measured at room temperature were also studied. The saturation magnetization at room temperature follows the same trend as at 4.2 k but the obtained values are much lower. The hysteresis loops were measured at 4.2 K and 300 K in external magnetic fields between \(-2\) and 2 T and are shown in Figure 9. Large values of the coercive field were found at 4.2 K for all samples and decreases with the Mn content for \(x>0.2\) being almost zero for the sample with \(x=1\) (see Figure 10). On the other hand, one can see that the magnetic hysteresis is almost absent in all investigated samples at room temperature. This behavior can be explained by the low sizes of the prepared nanoparticles and the values of the blocking temperatures, T\({}_{\text{B}}\). ZFC and FC curves were measured in an external magnetic field of 50 Oe and at a temperature range of 4.2-300 K. Some examples of the temperature dependences of ZFC and FC are shown in Figure 11. T\({}_{\text{B}}\) values were estimated as the minimum in the derivative of the difference between the FC and ZFC curves. The blocking temperature values are lower than room temperature and decrease when the manganese content increases. Similar behaviour was reported by [PERSON] et al. [45]. The onset of irreversibility probably occurs due to the blocking of individual superspins, or, due to their freezing in a spin-glass-like behavior, as was shown before by [PERSON] et al. [52] We can conclude that the mean size of the prepared nanoparticles plays a crucial role on T\({}_{\text{B}}\) values. [PERSON] et al. have shown that the presence or absence of the distinct types of intergrain group exchanges is decided by the amount the squareness ratio, M\({}_{\text{r}}\)/M\({}_{\text{s}}\) (M\({}_{\text{r}}\) is remanence magnetization), that varies from 0 to 1 [53]. For R \(<0.5\), the particle interacts by magnetostatic interaction while for R = 0.5 it interacts according to the Stoner-Wohlfarth theory (there is an assembly of noninteracting and randomly oriented particles), while R \(>0.5\) showed the existence of exchange coupling particles. According to the Stoner-Wohlfarth model, the value for noninteracting single domain particles, which have the easy axis  
Figure 9: Hysteresis loops for Co\({}_{1-x}\)Mn\({}_{x}\)Fe\({}_{2}\)O\({}_{4}\) nanoparticles at 4.2 K (**left**) and 300 K (**right**). randomly oriented, is 0.5 for uniaxial anisotropy and 0.832 for cubic anisotropy [54]. In case of our samples, the value \(\mathrm{M_{r}}/\mathrm{M_{s}}\) at room temperature is lower than 0.5, suggesting that the prepared ferrite nanoparticles are in a single-domain regime, interacting by magnetostatic forces [55]. Keeping in mind the high values of the magnetic moments, it is our intention to use these nanoparticles to prepare magnetoelectric core-shell nanoparticles covering \(\mathrm{Co_{1-x}Mn_{x}Fe_{2}O_{4}}\) nanoparticles with BaTiO\({}_{3}\) for applications in cancer treatment. The purpose is to locally induce electric charges in the human body, which can be controlled by relatively low external magnetic field. For biomedical applications, it will be useful for the preparation of hydrogels [56] or magnetic microemulsions [57] using the prepared \(\mathrm{Co_{1-x}Mn_{x}Fe_{2}O_{4}}\) nanoparticles. Figure 11: The temperature dependences of magnetization measured in ZFC and FC regime for samples with: (**a**) \(x=0.2\); (**b**) \(x=0.4\); (**c**) \(x=0.6\); (**d**) \(x=0.8\).