## 1 Introduction  
In recent years, the exploration of nanomaterials has revolutionized numerous scientific and industrial domains, opening new frontiers in research and application. Possession of interesting and compelling properties by metal oxides has been known for a while. In the form of magnetic nanoparticles, these materials have garnered substantial interest fortheir potential use in various applications. Among the oxides, ferrites, such as CoFe\({}_{2}\)O\({}_{4}\), ZnFe\({}_{2}\)O\({}_{4}\), NiFe\({}_{2}\)O\({}_{4}\), and MnFe\({}_{2}\)O\({}_{4}\), exhibit a good number of these sought-out properties, including high coercivity, moderate magnetization, superparamagnetic behavior, high anisotropy, and large magnetostrictive coefficient [1; 2], making them useful for a range of applications from magnetic storage devices to biomedical uses such as drug delivery and magnetic imaging (MRI) [2; 3; 4; 5]  
Among the various magnetic nanoparticles, cobalt ferrite (CoFe\({}_{2}\)O\({}_{4}\)) and its derivatives are noteworthy due to their high coercivity, mechanical stiffness, chemical stability, and low toxicity [7; 8; 9; 10; 11; 12]. Due to their multifunctional properties, these materials are promising candidates for applications as actuators, transducers, and sensors, among others [13; 14]. Lowering their size to the nano-scale level allows their implementation in such devices as permanent magnets, storage devices, magnetoplasmonics-based devices, and, as mentioned above, drug delivery processes, particularly for cancer fighting drugs [15; 16; 17; 18]. From this group, (CoMn)Fe\({}_{2}\)O\({}_{4}\) nanoparticles have emerged as a focal point of interest [19; 20]. Previous investigations have revealed that the compounds have a spinel crystal structure, where oxygen ions form a face-centered cubic close-packed lattice, and metal ions occupy part of the interstitial sites. Understanding the significance of (CoMn)Fe\({}_{2}\)O\({}_{4}\) nanoparticles involves unraveling their distinct characteristics and the potential impact they can have on advancing technology, medical treatments, and sustainable practices [21; 22; 23; 24; 25; 26]. Adding to its distinctive features, the compound possesses a significant magnetostrictive coefficient, reflecting its ability to change shape or dimensions in response to an applied magnetic field. Its magnetic properties, particularly its high coercivity, make it suitable for magnetic storage devices, while its magnetostrictive characteristics open avenues for sensors and actuators [27]. The versatility of (CoMn)Fe\({}_{2}\)O\({}_{4}\) extends across various fields, with applications in electronics, catalysis, and medical technologies. In the realm of biomedicine, these nanoparticles have been considered for their potential applications in magnetic resonance imaging (MRI) and drug delivery systems, as well as in hyperthermia, leveraging their magnetic properties [28; 29; 30]. The structure and properties of Co\({}_{1-x}\)Mn\({}_{x}\)Fe\({}_{2}\)O\({}_{4}\) compounds have attracted the attention of many researchers. [PERSON] et al. substituted Co with Mn in cobalt ferrites and reported that the Curie temperature decreased and the magnetostriction coefficient increased, when manganese concentration increases in the samples that were sintered at low temperatures [31]. [PERSON] and [PERSON] [32] successfully prepared a highly magnetostrictive ferrite, CoFe\({}_{2-x}\)Mn\({}_{x}\)O\({}_{4}\) (\(0\leq x\leq 0.3\)), using magnetic field annealing. According to them, the magnetostriction coefficient increased after annealing at 573 K in the presence of a magnetic field of 400 kA /m for 30 min. [PERSON] et al. studied the influence of Mn\({}^{3+}\) ions on the structural, magnetic, electrical, and catalytic properties of a cobalt ferrites photocatalyst [33]. The increase of the lattice parameter with substitution of Mn\({}^{3+}\) ions into the cobalt ferrite was explained by the differences between the ionic radius of Mn\({}^{3+}\) ion (0.645) compared to that of Fe\({}^{3+}\) ion (0.64).