## 3 Results and Discussion  
XRD patterns, presented in Figure 1a, confirm the single-phase nature of all the prepared samples. No characteristic peaks for impurities can be observed, which confirms the purity of the studied compounds. All studied compounds crystallize in a cubic Fd-3m crystal structure. The lattice parameters, determined by Rietveld analysis, indicate a nearly linear increase in the lattice parameter and unit cell volume, obeying Vegard's law, as the Mn content rises, as shown in Figure 2. This outcome is consistent with expectations, considering the higher ionic radius of Mn when compared to Co ions (0.083 nm for Mn\({}^{2+}\) and 0.078 nm for Co\({}^{2+}\)). Selected detailed experimental patterns, the Rietveld calculated ones, the Bragg positions, and the differences between theoretical and experimental data for the samples with \(x=0.3\) and 0.8 are presented in Figure 1b,c. The errors in the lattice parameter values were calculated from chi2 data obtained by Rietveld refinement. The observed lattice parameter values and their dependence on Mn content closely resemble the findings from previous studies within a more limited concentration range [35; 36; 47]. The average crystallite-sizes, calculated using the Debye-Scherrer formula, after subtracting the instrumental peak broadening were found to increase for \(x=0.1\) followed by a decrease up to \(x=0.5\) and then increasing again. The composition dependence of the crystalline sizes, determined from XRD data, is presented in Figure 3. Similar behavior was reported earlier for Mn in the concentration range up to 0.5, but the nanoparticles sizes were much higher [36]. TEM investigations of the Co\({}_{1-x}\)Mn\({}_{x}\)Fe\({}_{2}\)O\({}_{4}\) samples reveal that the nanoparticles are well defined and present a small tendency to agglomerate due to their powerful magnetic interaction that competes with the much powerless electrostatic repulsion (Figure 4). The particles consist of some regular and irregular elongated spherical structures. The study of the TEM images reveals the existence of nanoparticles of sizes between 7 nm and 15 nm, having a mean diameter near to the values determined from XRD; this suggests that most of the Co\({}_{1-x}\)Mn\({}_{x}\)Fe\({}_{2}\)O\({}_{4}\) nanoparticles are single crystals and have high crystallinity. The composition dependences of the nanoparticle sizes determined from TEM measurements are presented in Figure 3. The values obtained are close to those obtained from XRD data. Elemental analysis by EDS confirmed the presence of Co, Fe, Mn, and O elements in all prepared samples. As an example, the SEM image and EDS spectra for Co\({}_{0.8}\)Mn\({}_{0.2}\)Fe\({}_{2}\)O\({}_{4}\) nanoparticles are given in Figure 5a,b. The EDS spectra confirmed that there are no impurities in the studied nanoparticles. The elemental mapping for the same sample with \(x=0.2\) is shown in Figure 5c-f. The images confirm that the Co, Mn, Fe, and O elements are distributed almost homogeneously over the selected zone of the investigated sample. We have obtained similar results for all prepared compounds. The elemental abundances show that the concentrations of Mn, Co, and Fe atoms in the prepared nanoparticles are in good agreement with the proposed stoichiometric ratio. The Raman features of spinel ferrites, having Fd-3m symmetry and face-centered cubic unit cells, were described in detail in our previous publications [42; 48]. Considering the group theory [49], Mn ferrite should exhibit a slightly more complex spectrum than cobalt doped iron oxide magnetite due to the additional content of manganese cations in addition to cobalt, iron, and oxygen. Experimentally, we recorded Raman spectra of Co\({}_{1-x}\)Mn\({}_{x}\)Fe\({}_{2}\)O\({}_{4}\) featuring five distinct modes (Figure 6).