## 1 Introduction  
Spinel ferrites (AFe\({}_{2}\)O\({}_{4}\), where A = Ni, Cu, Mg and Zn, etc.) are a well-known family of scientific materials and have been well studied owing to their unique magnetic properties, as they are endowed with high electrical resistivity [1; 2; 3; 4; 5]. The excellent properties of spinel ferrites are well utilized in modern technological applications. Ferrites often have a spinel structure with the formula AB\({}_{2}\)O\({}_{4}\), where A and B are various metal cations, with iron being the most common (Fe). Spinel ferrites typically adopt FCC oxides (O\({}^{2-}\)) with A cations reside in one-eighth of the tetrahedral holes and B cations reside in half of the octahedral holes, i.e., instead of the normal spinel structure, ferrite crystals have the inverse spinel structure: B cations occupy (one-eighth of the tetrahedral holes, A cations occupy one-fourth of the octahedral sites, and B cations occupy the remaining one-fourth octahedral sites. Spinel ferrites with a mixed structure can also be possible with the formula [M\({}_{1-}\)\({}_{8}\)\({}^{2+}\) Fe\({}_{5}\)\({}^{3+}\)][M\(\delta\)\({}_{2}\)\({}_{4}\)Fe\({}_{2-}\)\({}_{8}\)\({}^{3+}\)]O\({}_{4}\), where \(\delta\) represents the degree of inversion [6; 7; 8; 9; 10]. Additionally, ferrites are hard and brittle, contain iron, are generally grey or black, and are polycrystalline materials. They can be magnetized or drawn to a magnet since they are ferrimagnetic. Conflicting with other ferromagnetic materials, most of the ferrites are insulating materials, making them ideal for transformers to defeat eddy currents for applications such as magnetic cores [11; 12; 13; 14; 15]. Spinel ferrites have vast applications in the present technological world. The main applications are precursors for ferro-fluids, radio, magnetic guided drug-delivery agents, telecommunication, biomedicine, magnetic storage, gas sensor, television, microwave, military devices, microwave absorbers, satellite communication, humidity sensors and magnetic storage [16; 17]. Among them, manganese ferrite (MnFe\({}_{2}\)O\({}_{4}\)) is an important material with good chemical stability, moderate magnetization, high sensitivity, band gap of \(\sim\)2.0 eV, and high electrical resistivity, and it is widely used in the transformer core, gas sensor, anode materials, photocatalysis, microwave absorber, and magnetic hyperthermia applications [18]. MnFe\({}_{2}\)O\({}_{4}\) nanoparticles exhibit a spinel cubic structure. Manganese ferrite (MnFe\({}_{2}\)O\({}_{4}\)) is a kind of lenient magnetic material among different ferrites with little saturation magnetization (Ms) and coercivity (Hc). They are used for sensing gases and temperature, and for catalytic applications. MnFe\({}_{2}\)O\({}_{4}\) nanoparticles have been shown to be superparamagnetic or ferrimagnetic depending on its size. MnFe\({}_{2}\)O\({}_{4}\) nanoparticles with a size smaller than 15 nm behave as a superparamagnetic material and those above 15 nm behave as ferrimagnetic materials. The oxygen atoms in ferrites surround the metal ions, resulting in a super exchange interaction between the A- and B-sublattices, which disturbs the ferrites' magnetic properties [19]. These interesting properties motivated us to choose MnFe\({}_{2}\)O\({}_{4}\) magnetic nanoparticles for the present study. The bulk MnFe\({}_{2}\)O\({}_{4}\) crystallizes mostly in a normal spinel structure with interstitial sites, such as A- and B-sites. Among the two sites, the manganese ions occupy the A-sites, whereas the iron ions occupy the B-sites [20]. Compared with the bulk form, nanocrystalline ferrites exhibit unusual behaviors, such as metastable cation distribution, super paramagnetism, enhanced coercivity and improved electrical resistivity [21]. Hence, the preparation and performance evolution of nanocrystalline ferrites attract increasing scientific attention. Additionally, the surface area and crystallite size of the ferrites are leads to the properties of the materials.