## 3 Results and Discussion  
### Crystalline Structure Analysis  
In Figure 1a-d, we show the XRD patterns of the SrFe\({}_{12-2x}\)Si\({}_{x}\)Mg\({}_{x}\)O\({}_{19}\) (\(x\) = 0, 0.05, 0.1, 0.2), Sr\({}_{1-y}\)Fe\({}_{12-y}\)Si\({}_{y}\)K\({}_{y}\)O\({}_{19}\) (\(y\) = 0, 0.05, 0.1, 0.2), SrFe\({}_{12-z}\)(Si\({}_{0.6}\)Li\({}_{0.6}\))\({}_{2}\) (\(z\) = 0, 0.05, 0.1, 0.2, 0.4, 0.6), and SrFe\({}_{12-x}\)Si\({}_{x}\)O\({}_{19}\) (\(x\) = 0, 0.1) samples sintered at 1250 \({}^{\circ}\)C. The formation of a secondary phase of Fe\({}_{2}\)O\({}_{3}\) is common when the substitution level increases to a certain extent. No other secondary phases were identified. All the diffraction peaks for SrM samples were indexed based on hexagonal magnetoplumbite crystal structures with the space group \(P6_{3}/mmc\) (International Center for Diffraction Data (ICDD), Powder Diffraction File search no. 00-33-1340). When Si-Mg was co-substituted at the Fe site of the SrM (SrFe\({}_{12-2}\)Si\({}_{x}\)Mg\({}_{x}\)O\({}_{19}\)), no secondary phase was formed at \(x\) = 0.05 and \(x\) = 0.1 (Figure 1a). When Si was substituted solely at the Fe site (SrFe\({}_{12-x}\)Si\({}_{x}\)O\({}_{19}\)) with a substitution level of \(x\) = 0.1, the Fe\({}_{2}\)O\({}_{3}\) phase was formed in the sample (Figure 1d). When Si-K was co-substituted at Fe and Sr sites (Sr\({}_{1-y}\)Fe\({}_{12-y}\)Si\({}_{y}\)K\({}_{y}\)O\({}_{19}\)), a single M-type phase could not be formed even at a small substitution level of \(y\) = 0.05 because K\({}^{+}\) could not be placed at the Sr site in SrM. Thus, the effective molar ratio, [Fe + Si]/[Sr] exceeded 12, and the Fe not included in the SrM phase formed Fe\({}_{2}\)O\({}_{3}\). Interestingly, when Li-Si was co-substituted at the Fe site (SrFe\({}_{12-x}\)(Si\({}_{0.6}\)Li\({}_{0.6}\))\({}_{2}\)O\({}_{19}\)), a single SrM phase could be formed without the secondary phase of Fe\({}_{2}\)O\({}_{3}\) up to \(z\) = 0.4, implying that 0.24 Si\({}^{4+}\) and 0.24 Li\({}^{+}\) ions are soluble in the SrM phase with a composition of SrFe\({}_{11.6}\)(Si\({}_{0.24}\)Li\({}_{0.24}\))O\({}_{19}\). Therefore, it was reasonably inferred that the extra Si\({}^{4+}\) or Li\({}^{+}\) that could not occupy the Fe site (\(\sim\)17% of total substituted ions), existed in the interstitial site of the SrM lattice. Rietveld analysis results on the samples of z = 0, 0.05, and 0.4, are presented in Figure 2a-c. A single M-type phase was confirmed even for the z = 0.4 sample. The best fitting results were obtained when it was assumed that Li\({}^{+}\) is replaced with 4_f1_, 4_f2_ sites randomly and that Si\({}^{4+}\) is replaced with 2\(a\) sites of Fe\({}^{3+}\). However, in this study, some substitution elements (\(\sim\)17%) were assumed to enter the interstitial site, which was not supported by the calculation software. For the series of the SrFe\({}_{12-2x}\)Si\({}_{x}\)Mg\({}_{x}\)O\({}_{19}\) (\(x\) = 0, 0.05, 0.1, 0.2), Sr\({}_{1-y}\)Fe\({}_{12-y}\)Si\({}_{y}\)K\({}_{y}\)O\({}_{19}\) (\(y\) = 0, 0.05, 0.1, 0.2), and SrFe\({}_{12-z}\)(Si\({}_{0.6}\)Li\({}_{0.6}\))\({}_{2}\)O\({}_{19}\) (\(z\) = 0, 0.05, 0.1, 0.2, 0.4, 0.6) samples, the calculated lattice parameters \(a\) and \(c\), their ratios _c_/_a_, and cell volumes are listed in Table 1. The \(a\) and \(c\) values were calculated from the _d\({}_{hkl}\)_ value corresponding to the (107) and (114) peaks according to the following equation:  
\[d_{hkl}=\{\frac{4(l^{2}+hk+l^{2})}{3a^{2}}+\frac{l^{2}}{c^{2}}\}^{-1/2}. \tag{2}\]  
where _d\({}_{hkl}\)_ is the interplanar spacing, and \(h\), \(k\), and \(l\) are the Miller indices. The changes in these parameters according to the substitution levels, \(x\), \(y\), and \(z\), are shown in Figure 3. For the three cases (Figure 3a-c), as the substitution amount of Si increased, the value of \(a\) decreased, and the \(c\) value and _c_/_a_ ratio increased gradually. The cell volume tended to decrease with increasing substitution levels. In the case of the Si-Mg substitution, the cell volume decreased by approximately 0.35% at \(x\) = 0.2.