## 2 Structures, Synthesis, and Magnetic Properties of Fe\({}_{3}\)O\({}_{4}\) Nanoparticles  
Fe\({}_{3}\)O\({}_{4}\) NPs exhibit either superparamagnetic (SPM) or ferrimagnetic (FM) behavior. Figure 3a shows examples of magnetization curves of the SPM curve (green) and FM hysteresis loop (orange color) of magnetic NPs represented by the magnetization value (emu/g) versus the applied magnetic field (Oe). In the presence of an external magnetic field, the magnetic material is magnetized and reaches a saturated value of magnetization called \"saturation magnetization\" (M\({}_{\text{S}}\)), which is the highest value of magnetization that the material can achieve. In addition to M\({}_{\text{S}}\), the hysteresis curve of FM materials also exhibits coercivity (H\({}_{\text{C}}\)) and remnant magnetization (M\({}_{\text{R}}\)) values, which indicate how difficult the materials are to demagnetize and how much magnetization is retained in the absence of an applied magnetic field, respectively. For SPM NPs, H\({}_{\text{C}}\) and M\({}_{\text{R}}\) are equal to zero. Consequently, SPM materials do not exhibit magnetic properties without an applied magnetic field; however, they respond magnetically in the presence of an external magnetic field. SPM NPs have several advantages, such as preventing the agglomeration of NPs (caused by magnetic attraction) and a sensitive response to a remote-controlled magnetic field. In contrast, FM materials exhibit a certain magnetization value in the absence of an external magnetic field. Therefore, FM NPs always retain strong magnetic properties, which are potentially useful for applications that always require the existence of strong magnetic properties/signals. Figure 2: Fe\({}_{3}\)O\({}_{4}\) NPs with various nanostructures and sizes used in emerging biomedical, biosensing, environmental, and energy applications. At the nanoscale, Fe\({}_{3}\)O\({}_{4}\) nanoparticles have different magnetic properties that are influenced by their structures, including size, morphology, crystallinity, and surface properties [2, 42, 45, 46]. These parameters are strongly affected by the synthetic methods and chemicals used in their syntheses, such as iron precursors, surfactants, reducing agents, and solvents. Thus, the appropriate selection of a method to synthesize Fe\({}_{3}\)O\({}_{4}\) NPs is very important to ensure the success of their applications. In the scope of this review, we focus on analyzing the magnetic features of Fe\({}_{3}\)O\({}_{4}\) NPs at room temperature (RT) due to the special interest in their use in various technologies and applications. The superparamagnetic (SPM) and ferrimagnetic (FM) behaviors of Fe\({}_{3}\)O\({}_{4}\) NPs depend on size, shape, crystallinity, and surface properties and are even affected by synthetic methods [46, 47, 48, 49, 50]. Figure 3b illustrates the relationship of the Fe\({}_{3}\)O\({}_{4}\) NP size (diameter) with its magnetic behavior and coercivity. Two important size-magnetic property transitions of Fe\({}_{3}\)O\({}_{4}\) NPs are superparamagnetic size (r\({}_{\rm SP}\)) and single-domain size (r\({}_{\rm SD}\)), which usually exist in NPs with diameters of \(\sim\)25 nm and \(\sim\)80 nm, respectively [45, 51]. Superparamagnetic size (r\({}_{\rm SP}\)) is the point at which the magnetic property of NPs transitions from the superparamagnetic to the ferromagnetic state, as NPs with a size larger than r\({}_{\rm SP}\) have a coercivity larger than 0 (H\({}_{\rm C}>0\)). The features of the superparamagnetic behavior of NPs are coercivity and remnant magnetization equal to zero (H\({}_{\rm C}=0\) and M\({}_{\rm R}=0\)) at temperatures above their blocking temperature (T\({}_{\rm B}\)). In particular, NPs exhibit superparamagnetic behaviors at temperatures above T\({}_{\rm B}\) (T \(>\) T\({}_{\rm B}\)) and ferromagnetic behavior at temperatures below T\({}_{\rm B}\) (T \(<\) T\({}_{\rm B}\)).