## 3 Results
Saturation magnetization is considered to increase with increasing Zn content according to Equations (1) and (2):  
\[Zn_{y}{}^{2+}\ {Fe_{1-y}}^{3+}\ \  
where y is the mole fraction of Zn\({}^{2+}\) ions. The theoretical value of this magnetization was inconsistent with the magnetization data which indicates a decreasing trend with increasing Zn content. However, it has been reported that the magnetization of NiZn ferrite decreases as the Zn content increases above 0.5 mole fraction in NiZn ferrite [31]. At high levels of zinc substitution, Fe\({}^{3+}\) ions in the tetragonal sites were so diluted that super-exchange interaction between tetrahedral and octahedral sites was lost and the saturation magnetization dropped. The Curie temperature declines abruptly and eventually drops below room temperature with increasing Zn content, so that the magnetic properties disappear at this temperature.  
Figure 8a-c shows the X-ray diffraction patterns of the Ni\({}_{1-\chi}\)Zn\({}_{\chi}\)Fe\({}_{2}\)O\({}_{4}\) (x = 0.5, 0.6, 0.7) samples sintered at 1250 \({}^{\circ}\)C. There are no second phases in these X-ray diffraction patterns, which means that the sintered samples have the same pure spinel structure regardless of their composition.  
Figure 9a-c displays FE-SEM images of the Ni\({}_{1-\chi}\)Zn\({}_{\chi}\)Fe\({}_{2}\)O\({}_{4}\) (x = 0.5, 0.6, 0.7) sintered at 1250 \({}^{\circ}\)C. Note that the microstructures become denser with an increasing Zn concentration in Ni\({}_{1-\chi}\)Zn\({}_{\chi}\)Fe\({}_{2}\)O\({}_{4}\) ferrite. This behavior is consistent with the study that Zn ions not only lower the sintering temperature, but also play a crucial role in increasing the sintering density. Meanwhile, it is interesting to note that although the particle size is somewhat uniform with about 5 \(\upmu\)m, the particle shape is faceted.  
Figure 10a,b shows the real and imaginary parts of the magnetic permeability of Ni\({}_{1-\chi}\)Zn\({}_{\chi}\)Fe\({}_{2}\)O\({}_{4}\) (x = 0.5, 0.6, 0.7) toroidal samples in the frequency range from 1 MHz to 1 GHz. It is clear from the Figure 10 that the real permeability (\(\upmu\)\({}^{\prime}\)) at 5 MHz increases monotonically from \(\upmu\)\({}^{\prime}\) = 106 to 217 with increasing Zn content from x = 0.5 to 0.7 in of Ni\({}_{1-\chi}\)Zn\({}_{\chi}\)Fe\({}_{2}\)O\({}_{4}\) (x = 0.5, 0.6, 0.7). The value of \(\upmu\)\({}^{\prime}\) for all three toroid samples remains constant up to 10 MHz and increases slightly with further increase in frequency.  
The imaginary part of permeability (\(\upmu\)\({}^{\prime\prime}\)) increases with increase in frequency and a broad maximum is observed at 20 MHz, where \(\upmu\)\({}^{\prime}\) decreases rapidly. This characteristic is a consequence of domain wall resonance, which consists of relaxation and resonance type dispersions. Hence, in the low frequency domain, the complex permeability spectrum of domain wall motion can be treated as the superposition of resonance and relaxation dispersion [30]. The broad maximum in \(\upmu\)\({}^{\prime\prime}\) is the result of the overlapping of the domain wall motion (DWM) resonance and spin resonance [32; 33; 34; 35]. The contributions of domain wall motion on the complex permeability spectrum decreases gradually with increasing frequency, and only the spin rotational component becomes dominant at higher frequency [36]. At frequencies below 500 kHz, the chief magnetizing mechanism of ferrite is DMW [37], while the domain wall resonance happens in the range from 1 to 100 MHz and rotational resonance occurs above 1 GHz [32; 33].  
The frequency dependence of the permeability of ferrite samples is connected to two types of magnetization mechanisms: domain wall motion and spin rotation, as shown below.  
\[\mu=1+\chi_{dw}+\chi_{sp} \tag{3}\]  
\[\chi_{dw}=\frac{3\pi M_{\mathrm{S}}{}^{2}D}{4\gamma} \tag{4}\]  
\[\chi_{sp}=\frac{2\pi M_{\mathrm{S}}{}^{2}}{K_{u}} \tag{5}\]  
where \(\chi_{dw}\) is the domain wall susceptibility, \(\chi_{sp}\) is the intrinsic rotational susceptibility, \(K_{u}\) is the anisotropy constant, \(\gamma\) is the gyromagnetic ratio, \(M_{\mathrm{s}}\) is the saturation magnetization, and \(D\) is the grain diameter.