## 3 Results and Discussion
### Crystalline Structure and Microstructure Analysis
The average grain size (\(D\)) was determined using a random intercept method applied to SEM micrographs and is presented in the lower right corner of each figure. Notably, when Mn-Al-Si substitutions are absent (\(x=0\)), the average grain size is the largest, at \(D=2.23\)\(\upmu\)m. With increasing \(x\), there is a consistent trend of decreasing \(D\). Given that the substitution of Mn-Al-Si does not significantly affect the MCA of M-hexaferrite, it can be explained that H\({}_{\text{C}}\) is primarily dependent on \(D\). As discussed earlier in Equation (1), the coercivity of an individual grain (\(h_{\text{C}}\)) exhibits an inverse relationship with its size (\(d\)), which can aptly account for these variations. The abnormally high intensity of the (00l) plane peaks in the XRD pattern of the \(x=0\) sample (Figure 1a) is attributed to the growth of hexagonal plate-like crystals, which can be observed in Figure 2a. In Figure 2c-e, for samples with \(x\geq 0.1\), a uniform grain size distribution is evident, with \(D\) ranging between 1.15 and 1.33 \(\upmu\)m. In Figure 2f,g, when the Mn substitution level is increased to 0.15 and 0.2, a slight increase in \(D\) is observed. However, in Figure 2h,i, as the Al substitution level increases to 0.15 and 0.2, there is a slight tendency for \(D\) to decrease. The apparent density of the sintered samples for each composition is presented in Table 1. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \begin{tabular}{c} **Cation** \\ **Composition** \\ \end{tabular} & \begin{tabular}{c} **x, y, z** \\ **(Phase)** \\ \end{tabular} & \begin{tabular}{c} **XRD** \\ **(g/cm\({}^{3}\))** \\ \end{tabular} & \begin{tabular}{c} \(\rho\) \\ **(Å)** \\ \end{tabular} & \begin{tabular}{c} \(a\) \\ **(Å)** \\ \end{tabular} & \begin{tabular}{c} \(c\) \\ **(Å)** \\ \end{tabular} & \begin{tabular}{c} **V** \\ **(Å)** \\ \end{tabular} & \
\begin{tabular}{c} **4\(\upmu\)M\({}_{\text{r}}\)** \\ **(kG)** \\ \end{tabular} &
\begin{tabular}{c} **H\({}_{\text{C}}\) \\ **(kOe)** \\ \end{tabular} \\ \hline \multirow{5}{*}{Sr\({}_{0.8}\)Ca\({}_{0.2}\)Fe\({}_{11-3x}\)} & \(x=0.05\) & M & 4.96 & 5.881 & 23.05 & 690.3 & 2.57 & 2.80 \\  & \(x=0.05\) & M & 4.89 & 5.881 & 23.04 & 690.0 & 2.30 & 3.59 \\  & \(x=0.1\) & M+Fe\({}_{2}\)O\({}_{3}\) & 4.81 & 5.875 & 23.04 & 688.8 & 2.25 & 4.72 \\  & \(x=0.15\) & M+Fe\({}_{2}\)O\({}_{3}\) & 4.90 & 5.870 & 23.04 & 687.1 & 2.20 & 4.76 \\  & \(x=0.2\) & M+Fe\({}_{2}\)O\({}_{3}\) & 4.90 & 5.871 & 23.04 & 687.7 & 2.04 & 4.90 \\ \hline \multirow{5}{*}{Sr\({}_{0.8}\)Ca\({}_{0.2}\)Fe\({}_{10.7-x-y}\)} & \(x=0.05\) & M+Fe\({}_{2}\)O\({}_{3}\) & 4.78 & 5.869 & 23.01 & 686.3 & 2.18 & 4.65 \\  & \(x=0.1\) & M+Fe\({}_{2}\)O\({}_{3}\) & 4.74 & 5.863 & 22.98 & 684.1 & 2.19 & 4.66 \\ \cline{1-1}  & \(y=0.05\) & M+Fe\({}_{2}\)O\({}_{3}\) & 4.80 & 5.852 & 22.94 & 680.4 & 2.21 & 4.83 \\  & \(y=0.1\) & M+Fe\({}_{2}\)O\({}_{3}\) & 4.85 & 5.866 & 23.00 & 685.3 & 2.20 & 5.01 \\ \cline{1-1} \cline{2-6}  & \(z=0.05\) & M & 4.81 & 5.870 & 23.01 & 686.5 & 2.22 & 5.00 \\ \cline{1-1} \cline{2-6}  & \(z=0.1\) & M & 4.84 & 5.861 & 22.99 & 684.1 & 2.22 & 5.10 \\ \cline{1-1} \cline{2-6}  & \(z=0.15\) & M & 4.76 & 5.868 & 23.01 & 686.0 & 2.16 & 5.20 \\ \cline{1-1}  & \(z=0.2\) & M & 4.70 & 5.872 & 23.03 & 687.6 & 2.13 & 5.32 \\ \hline \multirow{5}{*}{Sr\({}_{0.7+z}\)Ca\({}_{0.3-z}\)Fe\({}_{9.7}\)} & \(z=0.05\) & M & 4.88 & 5.870 & 23.00 & 686.2 & 2.21 & 4.92 \\ \cline{1-1}  & \(z=0.1\) & M & 5.00 & 5.869 & 23.00 & 686.2 & 2.23 & 4.99 \\ \cline{1-1} \cline{2-6}  & \(z=0.15\) & M & 4.86 & 5.869 & 23.00 & 686.2 & 2.18 & 5.07 \\ \cline{1-1} \cline{2-6}  & \(z=0.2\) & M & 4.73 & 5.868 & 23.02 & 686.5 & 2.14 & 5.16 \\ \hline \hline \end{tabular}  
* The error bar for \(\rho\), 4\(\upmu\)M\({}_{\text{r}}\), H\({}_{\text{C}}\) is \(\pm\) 0.3%, \(\pm\) 0.2%, \(\pm\) 1% of the original values, respectively.  
\end{table}