## 3 Results and Discussion  
### The Characterization of Synthesized Nanomaterials  
The synthesized nanomaterials of magnetic Fe\({}_{3}\)O\({}_{4}\) nanoparticles, Fe\({}_{3}\)O\({}_{4}\)/CS nanocomposite, and Fe\({}_{3}\)O\({}_{4}\)/CS/PEI nanocomposite were characterized using different techniques included Fourier-transform infrared spectroscopy (FT-IR), X-ray diffraction (XRD), transmission electron microscope (TEM), and vibrating-sample magnetometer (VSM). For the characterization of the presented functional groups, FT-IR spectra were performed for Fe\({}_{3}\)O\({}_{4}\) nanoparticles, Fe\({}_{3}\)O\({}_{4}\)/CS nanocomposite, and Fe\({}_{3}\)O\({}_{4}\)/CS/PEI nanocomposite as shown in Figure 1a. For Fe\({}_{3}\)O\({}_{4}\) NPs, Fe-O bond vibration, and O-H bending vibration of absorbed H\({}_{2}\)O molecules have two significant bands at 622 cm\({}^{-1}\) and 3402 cm\({}^{-1}\), respectively [48; 49]. Additionally, the significant bands of CS were recognized at 1082 cm\({}^{-1}\), 1662 cm\({}^{-1}\), 2882 cm\({}^{-1}\), and 3402 cm\({}^{-1}\) corresponding to the axial bending vibration of C-N, C=O stretching of amide I of CS, C-H vibrations of aliphatic -CH\({}_{2}\) hydrocarbon stretch, and OH besides NH stretching vibration, respectively [50]. For the nanocomposite Fe\({}_{3}\)O\({}_{4}\)/CS/PEI, the C-N-C asymmetrical stretch and the C-N vibration in PEI were represented by the two bands at 447 cm\({}^{-1}\) and 1072 cm\({}^{-1}\), respectively [51]. Additionally, the branched polymer was indicated by two characteristic bands at 1632 cm\({}^{-1}\) and 3422 cm\({}^{-1}\) corresponding to N-H deformation and N-H stretch, respectively. Moreover, the secondary amine group of the polymer was represented by 1142 cm\({}^{-1}\) and 1312 cm\({}^{-1}\) small band. The band at 1462 cm\({}^{-1}\) was related to the CH\({}_{2}\) deformations. Finally, the clear band at 622 cm\({}^{-1}\) approved the presence of Fe-O group. So, the FT-IR results indicated the successful implantation of Fe\({}_{3}\)O\({}_{4}\)/CS skeleton by the branched polymer PEI. Figure 1: FT-IR (**a**) and XRD (**b**) of the synthesized Fe\({}_{3}\)O\({}_{4}\), Fe\({}_{3}\)O\({}_{4}\)/CS, and Fe\({}_{3}\)O\({}_{4}\)/CS/PEI magnetic nanomaterials. The crystalline structure of synthesized nanomaterials was studied using XRD as shown in Figure 1b. For Fe\({}_{3}\)O\({}_{4}\) NPs, the six characteristic crystal planes (440), (551), (422), (400), (311), and (220) were represented by the reflexes at 20 = 62, 57, 53, 43, 36, and 30\({}^{\circ}\), respectively, and indicated the spinel structure of pristine Fe\({}_{3}\)O\({}_{4}\). It can be viewed that Fe\({}_{3}\)O\({}_{4}\) NPs have inverse spinel structure without another phase, only magnetite [46,49]. XRD patterns of Fe\({}_{3}\)O\({}_{4}\)/CS and Fe\({}_{3}\)O\({}_{4}\)/CS/PEI nanocomposites were found to be similar to that of pure Fe\({}_{3}\)O\({}_{4}\) indicating the retention of Fe\({}_{3}\)O\({}_{4}\) crystal forms, and the reflex positions were unchanged (i.e., the surface modification of Fe\({}_{3}\)O\({}_{4}\) by CS or polymer did not change the phase or crystalline structure of the magnetic nanoparticles). This may be related to the preparation method of the composite that allowed the coating of chitosan on iron oxide nanoparticles [46]. The average size of synthesized nanomaterials was determined using the Debye-Scherrer Equation:  
\[\text{D}=\kappa\lambda/\beta\text{ cos}\theta, \tag{7}\]  
where \(\beta\), \(\theta\), \(\lambda\), \(\kappa\), and D denote the full-width half-maximum (FWHM), diffraction angle, wavelength of X-ray beam, Scherrer constant (0.89), and the particle size, respectively. The average size of synthesized Fe\({}_{3}\)O\({}_{4}\)/CS/PEI, Fe\({}_{3}\)O\({}_{4}\)/CS, and Fe\({}_{3}\)O\({}_{4}\) magnetic nanomaterials was 12.1 \(\pm\) 0.2 nm, 10.2 \(\pm\) 0.4 nm, and 9.2 \(\pm\) 0.3 nm, respectively.