## 3 Results and Discussion
### Magnetic Property Analysis
Figure 3b shows the demagnetization curves of Sr\({}_{0.8}\)Ca\({}_{0.2}\)Fe\({}_{10.7-x-y}\)Mn\({}_{0.1+x}\)Al\({}_{0.1+y}\)Si\({}_{0.1}\)O\({}_{19-\rm d}\) (\(0\leq x\), \(y\leq 0.1\)), in which either of the Mn and Al substitution levels are increased from the first optimized composition in Figure 3a, Sr\({}_{0.8}\)Ca\({}_{0.2}\)Fe\({}_{10.7}\)Mn\({}_{0.1}\)Al\({}_{0.1}\)Si\({}_{0.1}\)O\({}_{19-\rm d}\) (\(x=0\)). Among the four demagnetization curves, it can be observed that there is a significant improvement in H\({}_{\rm C}\) when adding Al content with \(y=0.1\), while the 4\(\pi\)M\({}_{\rm S}\) values remain relatively consistent. In Figure 4b,c, one can easily compare the relative magnitude of the 4\(\pi\)M\({}_{\rm r}\) and H\({}_{\rm C}\) values for these samples. Therefore, based on these results, the second optimized composition can be derived as Sr\({}_{0.8}\)Ca\({}_{0.2}\)Fe\({}_{10.6}\)Mn\({}_{0.1}\)Al\({}_{0.2}\)Si\({}_{0.1}\)O\({}_{19-\rm d}\). The XRD phase analysis results presented in Figure 1 show that all four samples shown in Figure 3b contain a secondary phase of Fe\({}_{2}\)O\({}_{3}\). Therefore, the Fe content in the chemical formula was reduced from the stoichiometric composition of 10.6 to 10.2 and 9.7, respectively, while the Sr:Ca ratio was slightly adjusted. The results for these two series of compositions, Sr\({}_{0.7+z}\)Ca\({}_{0.3-z}\)Fe\({}_{10.2}\)Mn\({}_{0.1}\)Al\({}_{0.2}\)Si\({}_{0.1}\)O\({}_{19-\rm d}\) and Sr\({}_{0.7+z}\)Ca\({}_{0.3-z}\)Fe\({}_{9.7}\)Mn\({}_{0.1}\)Al\({}_{0.2}\)Si\({}_{0.1}\)O\({}_{19-\rm d}\) (\(z=0\), 0.05, 0.1, 0.15, 0.2), are presented in Figures 3c and 4d, respectively. The changes in the 4\(\pi\)M\({}_{\rm r}\) and H\({}_{\rm C}\) values with respect to \(z\) are also plotted in Figure 4d,e. In both cases, an increase in \(z\) (corresponding to a decrease in the Ca content) leads to an increase in H\({}_{\rm C}\), and a decrease in the 4\(\pi\)M\({}_{\rm S}\) values, roughly. This is related to the tendency of the Ca content to promote grain growth. Furthermore, the composition with \(z=0.1\), which corresponds to Sr:Ca = 0.8:0.2, exhibited the highest sintered density. Considering both the 4\(\pi\)M\({}_{\rm r}\) and H\({}_{\rm C}\) values, the composition with the Fe content at 10.2 and \(z=0.1\) (i.e., Sr\({}_{0.8}\)Ca\({}_{0.2}\)Fe\({}_{10.2}\)Mn\({}_{0.1}\)Al\({}_{0.2}\)Si\({}_{0.1}\)O\({}_{19-\rm d}\)) can be derived as the final optimized composition. In ferrite magnets, it is well-known that there is a trade-off relationship between 4\(\pi\)M\({}_{\rm r}\) and H\({}_{\rm C}\). To confirm this trend observed in the five series of compositions in this study, the 4\(\pi\)M\({}_{\rm r}\) and H\({}_{\rm C}\) of all the samples were plotted as points in a two-dimensional plane (\(x\), \(y\)) and presented in Figure 5. Each series of samples is represented by shapes and colors of the same type. Furthermore, a trend line with a consistent slope from the bottom-left to the top-right are drawn from the coordinate points of each series, helping distinguish the positions of the points. A specific point that trends towards the upper-right direction indicates superior permanent magnet properties. In the first series, Sr\({}_{0.8}\)Ca\({}_{0.2}\)Fe\({}_{11-3x}\)Mn\({}_{x}\)Al\({}_{x}\)Si\({}_{x}\)O\({}_{19-\rm d}\) (\(x=0\), 0.05, 0.1, 0.15, 0.2), it is confirmed that the composition with \(x=0.1\) is positioned most towards the upper-right direction. In the second series, Sr\({}_{0.8}\)Ca\({}_{0.2}\)Fe\({}_{10.7-3x}\)Mn\({}_{0.1+x}\)Al\({}_{0.1+y}\)Si\({}_{0.1}\)O\({}_{19-\rm d}\), when additional Al substitution (\(y=0.1\)) is applied, it is observed that this composition is located most towards the upper-right direction.