## 3 Results and Discussion
### Magnetic Property Analysis  
The initial magnetization curves of all samples are shown in Figure 6. From these initial magnetization curves, the \(M_{S}\) and \(H_{a}\) values of samples could be calculated using the law of approach to saturation (LAS) [38] given by the following equation,  
\[M=M_{s}\bigg{(}1-\frac{A}{H}-\frac{B}{H^{2}}\bigg{)}+\chi_{p}H \tag{2}\]  
where \(M\) is magnetization induced by an applied magnetic field \(H\), the parameter \(A\) is related to the inhomogeneity of a material, \(B\) is the anisotropy parameter related to the magnetocrystalline anisotropy, and \(\chi_{p}\) is the high field susceptibility [39]. The inhomogeneity parameter \(A\) is a factor that relies on the influence of vacancies, lattice distortion, and local concentration fluctuation that hinders the domain wall motions. The calculation of \(A/H\) term is valid in a relatively low magnetic field range, where magnetic domain wall motion contributes to \(M\). The domain wall motion, however, is non-negligible even in a relatively high field. [PERSON] [38] reported that \(A\) had about 250 Oe in the case of BaM hexaferrite. For a hexagonal crystal structure exhibiting uniaxial anisotropy, the \(B\) parameter can be expressed as follows:  
\[B=\frac{H_{a}^{2}}{15} \tag{3}\]  
Since the \(\chi_{p}\) is negligibly small compared to \(B\) in the \(H\)-field region of the consideration, Equation (2) can be approximated by the following equation:  
\[M=M_{s}\Bigg{\{}1-\frac{A}{H}-\frac{1}{15}\bigg{(}\frac{H_{a}}{H}\bigg{)}^{2} \Bigg{\}} \tag{4}\]  
Then, the \(M_{S}\) value can be obtained from the linear part on \(M\) vs. \(1/H\) curves. The \(A\) and \(H_{a}\) values can be calculated by fitting Equation (4) to the measured \(M(H)\) curves. The procedure for fitting Equation (4) to the measured \(M(H)\) data is as follows. For example, Figure 7a presents the fitting results for the sample with \(x\) = 0.15, which is representative of all samples. The black (\(A\)), blue (R\({}^{2}\)), and red (\(H_{a}\)) lines represent the fitted  
Figure 6: The initial magnetization curves for the Sr\({}_{1-x}\)La\({}_{x}\)Fe\({}_{12-x}\)Co\({}_{x}\)O\({}_{19}\) polycrystalline samples with \(x\) = 0, 0.1, 0.15, 0.2, 0.25, and 0.3.  
curves calculated in the field region of 12-26 kOe, respectively. R\({}^{2}\) is the output statistical parameter that represents the goodness of the curve fitting, and it is highly reliable when R\({}^{2}\) is higher than 0.990. Figure 7 shows the results of fixing the maximum field value and widening it to the lower field region. The values of \(A\), \(H_{a}\), and R\({}^{2}\) are calculated and plotted. \(A\) gradually increases and becomes positive below 15 kOe, while both \(H_{a}\) and R\({}^{2}\) gradually decrease. In the field region where \(A\) is about 200-300, R\({}^{2}\) shows a value of 0.990 or higher, and \(H_{a}\) shows a value of 21.7 kOe. Without \(A\), the LAS fitting to the experimental \(M(H)\) values in the same field region resulted in a relatively high \(H_{a}\) of 23.1 kOe and a low R\({}^{2}\) value of 0.9858, indicating a worse LAS fitting. Therefore, the field region of 12-26 kOe was selected for fitting with Equation (4). As shown in Figure 7b, the red LAS fitting curve fits well with the measured data in the selected field region. However, the red curve fitted to the data deviates significantly from the measured values at the lower field region of \(H\leq\) 12 kOe. Table 2 lists the calculated values for \(M_{S}\), \(H_{a}\), \(A\), and R\({}^{2}\). Figure 7: Fitting of the law of approach to saturation (LAS) to experimental data for Sr\({}_{1-x}\)La\({}_{x}\)Fe\({}_{12-x}\)Co\({}_{x}\)O\({}_{19}\) (\(x=0.15\)) samples: linear fitting by Equation (2) for (a) \(M_{s}\), A, \(H_{a}\), and R\({}^{2}\), and (b) non-linear fitting by Equation (1). The magnetic hysteresis loops of samples are shown in Figure 8. From these loops, the maximum \(M\) values measured at \(H\) = 26 kOe (\(M_{\oplus 26 kOe}\)) and intrinsic coercivity (\(H_{ci}\)) were obtainable.