## 3 Results and Discussion
The important features revealing its structure are the modes of overlapping tetrahedral Co-O and Mn-O vibration around 600 cm\({}^{-1}\), tetrahedral Fe-O around 677 cm\({}^{-1}\), and a T\({}_{2g}\)(2) band representing an overlap  
Figure 1: XRD patterns at room temperature of Co\({}_{1-x}\)Mn\({}_{x}\)Fe\({}_{2}\)O\({}_{4}\) nanoparticles (**a**); Rietveld refinement results for the samples with \(x=0.3\) (**b**) and 0.8 (**c**). Figure 2: The manganese dependence of the lattice parameters for Co\({}_{1-x}\)Mn\({}_{x}\)Fe\({}_{2}\)O\({}_{4}\) nanoparticles. of octahedral Mn-O and Fe-O vibrations around 466 cm\({}^{-1}\). A weak shoulder due to the octahedral Co-O Ti\({}_{2g}\)(3) mode can be observed at about 570 cm\({}^{-1}\) for the ferrites with a high Co content. Weak T\({}_{2g}\)(1) and E\({}_{g}\) bands around 175 and 320 cm\({}^{-1}\), respectively, were also recorded. Thus, Raman spectra can be used to track the changes in site occupancy rates of Mn and Fe cations among the different substitutions. Typical Raman spectra and deconvoluted main bands of Mn substituted ferrites at rates of 0.2 (a) and 0.9 (b), and plots of intensity ratio of 600 and 677 cm\({}^{-1}\) modes (c) and the wavenumber position of the T\({}_{2g}\)(2) band (d) across the samples, are shown in Figure 7. Let us first consider the preferred enthalpy of Mn and Fe cations reached by thermodynamics calculations presented by [PERSON] [49]. According to those data, Mn\({}^{2+}\) has a preferred enthalpy near zero, Mn\({}^{4+}\) has \(-\)43 kJ/mole, Fe\({}^{2+}\) has \(-\)20.9 kJ/mole, and Fe\({}^{3+}\) has \(-\)22.2 kJ/mole. It is thus more likely that Fe\({}^{3+}\) would be placed on octahedral sites while Fe\({}^{2+}\) would, in turn, be displaced to tetrahedral sites while Mn ions will be placed on both sites. In practice, it is difficult to control the content of Mn\({}^{2+}\) and Mn\({}^{4+}\) in ferrites. On the other hand, Mn cations will always exhibit a Raman signature at lower wavenumbers than Fe cations due to the larger crystalline radius of Mn compared to Fe. Two trends can be observed as the Mn substitution rate increases: (i) the tetrahedral Co-O/Mn-O band around 600 cm\({}^{-1}\) increases in intensity relative to the Fe-O band around 677 cm\({}^{-1}\) (Figure 7c), and (ii) the T\({}_{2g}\)(2) band decreases in intensity and shifts from about 466 cm\({}^{-1}\) in low-Mn samples towards 445 cm\({}^{-1}\) in highly substituted samples  
Figure 4: TEM images of Co\({}_{1-x}\)Mn\({}_{x}\)Fe\({}_{2}\)O\({}_{4}\) nanoparticles and sizes distributions for the samples with \(x=0.1\) (**left**) and 0.6 (**right**). Figure 3: The composition dependence of the crystalline sizes determined from XRD (**left**) and TEM (**right**). (Figure 7d). However, it is difficult to use the intensity ratio of the A\({}_{\rm 1g}\) bands located around 600 cm\({}^{-1}\) and 677 cm\({}^{-1}\) to quantify the relative localization of Mn to tetrahedral sites since the former is the convolution of both Co-O and Mn-O T\({}_{\rm 2g}\)(2) vibration modes. The Fe-O band at 677 cm\({}^{-1}\) is clearly present in all the samples, meaning that a population of Fe cations always occupies tetrahedral sites. Figure 5: SEM image (**a**), EDS spectra (**b**), and the elemental mapping (**c**–**f**) of Co\({}_{0.8}\)Mn\({}_{0.2}\)Fe\({}_{2}\)O\({}_{4}\) nanoparticles. Mn-O and Fe-O vibrations exhibited a significant shift from 466 cm\({}^{-1}\) at Mn substitution rate of 0.2 to 445 cm\({}^{-1}\) at Mn substitution rate of 0.9. This is indicative of increasing Mn occupancy of both tetrahedral and octahedral sites, hence both Mn\({}^{2+}\) and Mn\({}^{4+}\) are present in Co\({}_{1-x}\)Mn\({}_{x}\)Fe\({}_{2}\)O\({}_{4}\) ferrites. The magnetization isotherms measured in external magnetic fields up to 12 T are presented in Figure 8. One can see that in the case of samples  
Figure 3.0 Raman spectra of selected Co\({}_{1x}\)Mn\({}_{x}\)Fe\({}_{2}\)O\({}_{4}\) ferrites.