## 3 Results and Discussion  
### Characteristics of Randomly Oriented Polycrystalline Samples  
#### 3.1.1 Lattice Parameters and Microstructures  
The powder XRD patterns of the Sr\({}_{1-x}\)La\({}_{x}\)Fe\({}_{12-x}\)Co\({}_{x}\)O\({}_{19}\) (0.0 \(\leq\) x \(\leq\) 0.4) samples sintered at 1230 \({}^{\circ}\)C for 2 h in air are shown in Figure 1. The diffraction peaks corresponding to the M-type hexaferrite phase are indexed with reference to the ICDD cards [00-033-1340]. From this figure, one can see that the M-type single phase is obtainable from x values up to 0.35. Previous studies have demonstrated that La and Co ions exhibit solubility in SrM in the x region of 0.0-0.35 [14; 16]. However, the sample with x = 0.4 exhibits minor peaks of Fe\({}_{2}\)O\({}_{3}\) [ICDD card 04-008-7623], LaFeO\({}_{3}\) [ICDD card 00-037-1493], and CoFe\({}_{2}\)O\({}_{4}\) [ICDD card 00-022-1086], evidencing that this nominal composition is above the solubility limit of La-Co co-substitution in SrM and hence the M-type single phase composed of x = 0.4 is unobtainable at the sintering conditions of 1230 \({}^{\circ}\)C for 2 h in air. Although a secondary phase is undetectable in the XRD pattern of the sample with \(x\) = 0.35 in Figure 1, we performed a SEM-EDS analysis for this sample to further identify whether it is composed of the M-type single phase or not. Figure 2 presents SEM backscattered electron (BSE) images, along with corresponding EDS element mapping micrographs illustrating the distribution of O, Fe, La, and Sr in the Sr\({}_{0.65}\)La\({}_{0.35}\)Fe\({}_{11.65}\)Co\({}_{0.35}\)O\({}_{19}\) (\(x\) = 0.35) samples. In Figure 2, a secondary phase exhibits a brighter contrast compared to the primary phase (SrM). Within this secondary phase, the Fe content is notably lower, while the Sr content is higher compared to the main phase. The phases are presumed to be (La, Sr), (Fe,Co)O\({}_{3}\), and CoFe\({}_{2}\)O\({}_{4}\), although it was undetectable in the XRD patterns of the \(x\) = 0.35 sample shown in Figure 1. The solubility limit of La-Co in the SrM is believed to exist between \(x\) = 0.3 and 0.35 at 1230 \({}^{\circ}\)C, in air, from Figures 1 and 2. Therefore, the secondary phases observed for the sample \(x\) = 0.4 in Figure 1 are presumed not to be Fe\({}_{2}\)O\({}_{3}\), LaFeO\({}_{3}\), and CoFe\({}_{2}\)O\({}_{4}\) but to be (Fe,Co)\({}_{2}\)O\({}_{3}\), (La, Sr)(Fe,Co)O\({}_{3}\), and CoFe\({}_{2}\)O\({}_{4}\) since Sr and Co can substitute for the La and Fe sites, respectively, except the spinel phase of CoFe\({}_{2}\)O\({}_{4}\). No secondary phase was observed in the SEM-EDS analysis for samples with \(x\leq 0.3\), suggesting that the M-type single phase is achievable from the samples of \(x\leq 0.3\). The solubility limit of La-Co in the SrM is believed to exist between \(x\) = 0.3 and 0.35 at 1230 \({}^{\circ}\)C, in air, from Figures 1 and 2. Figure 1: The powder XRD patterns of Sr\({}_{1-x}\)La\({}_{x}\)Fe\({}_{12-x}\)Co\({}_{x}\)O\({}_{19}\) (\(0.0\leq x\leq 0.4\)) samples sintered at 1230 \({}^{\circ}\)C for 2 h, in air. Figure 2: SEM-BSE, and the corresponding EDS element mapping micrographs for O, Fe, La, and Sr for the Sr\({}_{0.65}\)La\({}_{0.35}\)Fe\({}_{11.65}\)Co\({}_{0.35}\)O\({}_{19}\) sample. Rietveld refinement was used to calculate the lattice parameters \(a\) and \(c\), with R\({}_{\rm wp}\) (weighted profile \(R\)-value) \(<\) 9% and \(\chi^{2}\)(reduced Chi-squared value) smaller than 1.4 accuracy. The lattice parameters \(a\) and \(c\) were calculated from the \(d_{hkl}\) values obtained from powder XRD patterns based on the following equation [14],  
\[d_{hkl}\ =\ \left\{\frac{4\left(h^{2}+hk+k^{2}\right)}{3a^{2}}+\frac{l^{2}}{c^{2 }}\right\}^{-1/2} \tag{1}\]  
where \(d_{hkl}\) is the inter-planer spacing; and \(h\), \(k\), and \(l\) are the Miller indices. The cell volume (\(V_{cell}\)) was calculated from \(a^{2}\)csin120\({}^{\circ}\).