## 3 Results and Discussion
### Magnetic Property Analysis  
Figure 3a-d display the demagnetization curves (4\(\pi\)M-H) of samples for each composition in the four series. Additionally, to facilitate an easy assessment of the changes in the residual magnetization (4\(\pi\)M\({}_{\rm r}\)) and coercivity (H\({}_{\rm C}\)) of the samples obtained here, we have plotted them as a function of the substitution level \(x\), \(y\), and \(z\) in Figure 4a-e. The precise data values are provided in Table 1. In Figure 3a, it can be observed that the demagnetization curves vary significantly with respect to the Mn-Al-Si substitution level, \(x\). As \(x\) increases, 4\(\pi\)M\({}_{\rm r}\) decreases, while H\({}_{\rm C}\) shows an increasing trend. When there is no secondary phase, 4\(\pi\)M\({}_{\rm r}\) can be expressed in terms of the M\({}_{\rm S}\), sintered density (\(\rho\)), and squareness (\(S\)), as follows [7; 23]:  
\[4\pi\rm M_{r}=M_{S}\cdot\rho\cdot S \tag{3}\]  
here, 4\(\pi\)M\({}_{\rm r}\) is the remanent magnetization in Gauss units, M\({}_{\rm S}\) is the saturation magnetization in emu/g unit, \(\rho\) is the sintered density in g/cm\({}^{3}\), and S represents the squareness, which is a measure of the degree of alignment along the c axis in the crystallites. It can be expressed as the ratio of the remanent magnetization to saturation magnetization (\(S=\rm M_{r}/M_{S}\)). The saturation magnetization in Gauss units (4\(\pi\)M\({}_{\rm S}\)) can be estimated as twice the value of 4\(\pi\)M\({}_{\rm r}\) in the case of an isotropic magnet, where the crystalline grains are randomly oriented. Since the M\({}_{\rm S}\) of Sr M-hexaferrite in the units of emu/g is known to be 72 emu/g [2; 3], the theoretical 4\(\pi\)M\({}_{\rm S}\) can be approximately 4600 G. In this case, for a perfectly isotropic magnet with a theoretical density of 5.1 g/cm\({}^{3}\), the 4\(\pi\)M\({}_{\rm r}\) would be around 2300 G.  
In Figure 3a, the \(x=0\) sample exhibits a significantly higher \(4\pi\)M\({}_{\rm r}\) of 2570 G than the expected \(4\pi\)M\({}_{\rm r}\) for a perfectly isotropic magnet. This can be attributed to some degree to the c axis alignment in the crystallites. Based on the size and shape of the crystalline grains observed in the SEM images in Figure 2a, it can be inferred that the \(x=0\) sample maintains hexagonal plate-like crystallites even after secondary ball milling and exhibitssome c axis alignment after the molding process into disk shapes. The \(x=0.05\) sample shown in Figure 2b is also expected to show a weaker degree of c axis alignment for the same reasons. For samples with \(x\geq 0.1\), as shown in Figure 3a, considering the size and shape of the crystallites from Figure 2c-e, they are expected to be nearly perfectly isotropic. Therefore, the 4\(\pi\)M\({}_{\rm S}\) for these samples can be roughly estimated as twice the value of the 4\(\pi\)M\({}_{\rm r}\). In these samples (\(x\geq 0.1\)), the crystalline grain size and distribution are uniform. In Figure 4a, it is noticeable that the H\({}_{\rm C}\) sharply increases at \(x=0.1\). Among the samples in the series presented in Figure 3a for Sr\({}_{0.8}\)Ca\({}_{0.2}\)Fe\({}_{11-3x}\)Mn\({}_{x}\)Al\({}_{x}\)Si\({}_{x}\)O\({}_{19-\rm d}\), when considering both the 4\(\pi\)M\({}_{\rm r}\) and H\({}_{\rm C}\) values, it is concluded that the composition at \(x=0.1\), Sr\({}_{0.8}\)Ca\({}_{0.2}\)Fe\({}_{10.7}\)Mn\({}_{0.1}\)Al\({}_{0.1}\)Si\({}_{0.1}\)O\({}_{19-\rm d}\), exhibits the most optimized results for permanent magnet applications. Additionally, there is still room for improvement in 4\(\pi\)M\({}_{\rm r}\) through improving the sintering density and removing the second phase Fe\({}_{2}\)O\({}_{3}\).