## 3 Results  
Figure 3a-c shows the X-ray diffraction patterns of Ni\({}_{1-x}\)Zn\({}_{x}\)Fe\({}_{2}\)O\({}_{4}\) (x = 0.5, 0.6, 0.7) nanoparticles, which indicate a pure spinel structure for all three ferrites without any second phases regardless of the composition. Figure 4 shows the dependence of the lattice parameters of the ferrite nanoparticles on the Ni concentration, which were calculated with the following equation [28]. \[a=\frac{\lambda\sqrt{h^{2}+k^{2}+l^{2}}}{2 sin\theta_{hkl}}\]  
The lattice parameter increases monotonically from 8.340 A to 8.358 A with increasing Zn concentration from x = 0.5 to 0.7, which is due to the fact that the ionic radius of the Zn\({}^{2+}\) ion (0.74 A) is larger than that of the Ni\({}^{2+}\) ion (0.69 A). This result means that some Zn ions appear to be incorporated into Ni site, considering that the larger Zn\({}^{2+}\) ions prefer tetrahedral A-sites, but the smaller Ni\({}^{2+}\) ions prefer octahedral B-sites in the crystalline spinel structure. The lattice volume of Ni\({}_{1-x}\)Zn\({}_{x}\)Fe\({}_{2}\)O\({}_{4}\) was also increased with increasing Zn concentration, which is reasonable considering the difference in their ionic radii. Figure 5 shows the dependence of the crystallite size of ferrite nanoparticles on the Zn concentration, in which the average size (D) of crystallites was calculated from (311) peak using [PERSON]'s equation as shown below. \[\mathrm{D}=\frac{k\lambda}{\beta cos\theta}\]  
where the constant \(k\) = 0.89, \(\lambda\) = 1.5406 A, and \(\beta\) is the FWHM of the (311) diffraction peak. Regardless of the Zn concentration, the crystallite size was in the range of 46 to 51 nm and there was no significant difference in it. Figure 6 shows the hysteresis curve of NiZn ferrite nanoparticles at room temperature. As the Zn ion concentration increases from x = 0.5 to 0.7 in Ni\({}_{1-x}\)Zn\({}_{x}\)Fe\({}_{2}\)O\({}_{4}\), the saturation magnetization and the coercivity decreases from 83 to 71 emu/g and from 18.54 to 16.48 O\({}_{e}\), respectively, which is considered high compared to some other results. The magnetic properties of the Ni\({}_{1-x}\)Zn\({}_{x}\)Fe\({}_{2}\)O\({}_{4}\) depends on its chemical composition, grain size, preparation method, and the arrangement of cation between two interstitial sites [29]. The ferrite under an applied field has two magnetization processes: domain wall motion and magnetization rotation within domains. In general, domain wall motion is sensitive to extrinsic material features such as grain size and grain-boundary structure, the presence of inclusions or pores within the grains, impurity levels and stresses. Figure 7 shows the Ni\({}_{1-x}\)Zn\({}_{x}\)Fe\({}_{2}\)O\({}_{4}\) particles are considered single domain particles, taking into account the fact that the crystallite size gotten from the XRD is almost the same as the particle size obtained from the FE-SEM image. Therefore, it appears that the high saturation magnetization is mainly due to an arrangement of cations on the sub-lattices A and B. The decrease in the saturation magnetization with Zn ion content is well complied with the report that the Curie temperature, which means the transition temperature between paramagnetic and ferromagnetic phase, decreases rapidly with increasing nonmagnetic Zn ions in NiZn ferrite [29]. Figure 4: The dependence of the lattice parameters of the Ni\({}_{1-x}\)Zn\({}_{x}\)Fe\({}_{2}\)O\({}_{4}\) (x = 0.5, 0.6, 0.7) ferrite nanoparticles on the Zn concentration. Figure 5: The dependence of the crystallite size of Ni\({}_{1-x}\)Zn\({}_{x}\)Fe\({}_{2}\)O\({}_{4}\) (x = 0.5, 0.6, 0.7) ferrite nanoparticles on the Zn concentration. Figure 3: (**a–c**) X-ray diffraction patterns of Ni\({}_{1-x}\)Zn\({}_{x}\)Fe\({}_{2}\)O\({}_{4}\) (x = 0.3, 0.4, 0.5) nanoparticles of the synthesized samples: (**a**) x = 0.3, (**b**) x = 0.4, and (**c**) x = 0.5. The saturated magnetization of ferrite is proportional to the difference in sub-lattice magnetization associated with octahedral and tetrahedral sites.