## 3 Results and Discussion
### Crystalline Structure and Microstructure Analysis
This strongly supports that the same amount of La and Co with the initial compositions are incorporated in the M-type single phase. Meanwhile, our results are consistent with previous studies [20,23] on La-Co co-substituted SrM powder synthesized through the sol-gel process and subsequent heat treatment at 900 \({}^{\circ}\)C in air. [PERSON] et al. [20] were able to synthesize the M-type single phase up to \(x\) = 0.4 using this processing route, possibly due to a homogeneous mixture  
Figure 1: The powder XRD patterns of samples with \(x\) = 0.3 in Sr\({}_{1-x}\)La\({}_{x}\)Fe\({}_{12-x}\)Co\({}_{x}\)O\({}_{19}\) as a representative of all samples are shown in (**a**), where bottom pattern is from the conventional process (calcination at 1280 \({}^{\circ}\)C for 2 h, sintering at 1300 \({}^{\circ}\)C for 2 h), middle pattern from our modified process (calcination at 1280 \({}^{\circ}\)C for 2 h, sintering at 1230 \({}^{\circ}\)C for 2 h), and top pattern from our modified process (calcination at 1280 \({}^{\circ}\)C for 2 h, sintering at 1300 \({}^{\circ}\)C for 2 h). Powder XRD patterns from our modified process (calcination at 1280 \({}^{\circ}\)C for 2 h, sintering at 1300 \({}^{\circ}\)C for 2 h) for the samples with \(x\) = 0, 0.1, 0.15, 0.2, 0.25, 0.3, and with \(x\) = 0.4 (**b**). of cation components that facilitated the formation of the M-type single phase. However, it should be noted that their samples consist of Sr\({}_{1.05-x}\)La\({}_{x}\)Fe\({}_{12-x}\)Co\({}_{x}\)O\({}_{19}\) (\(0.0\leq x\leq 0.4\)), which is equivalent to Sr\({}_{1-x}\)La\({}_{x}\)Fe\({}_{11.43-x}\)Co\({}_{x}\)O\({}_{19}\) (\(0.0\leq x\leq 0.38\)) in our notation. Although their samples only consist of the M-type single phase up to \(x=0.4\), their \(x\) value of 0.4 corresponds to 0.38 in our notation. Furthermore, it is difficult to directly compare their results with ours because they used 11.43 instead of 12 as the molar ratio of Fe/Sr. This suggests that about 5 mol% of Fe-deficient SrM was used for La-Co co-substitution. To calculate the lattice parameters \(a\) and \(c\), Rietveld refinement was performed with \(R_{wp}\) (weighted profile R-value) less than 9% and \(\chi^{2}\) (reduced Chi-squared value) less than 1.4 accuracy. The Rietveld analysis data are shown in Figure 2a,b for \(x=0.0\) and 0.3, respectively, as representatives of all samples. From the powder XRD patterns in Figure 1b, the lattice parameters \(a\) and \(c\) of the samples were calculated using the following equation [19],  
\[d_{hkl}=\left\{\frac{4\left(l^{2}+hk+k^{2}\right)}{3a^{2}}+\frac{l^{2}}{c^{2}} \right\}^{-1/2} \tag{1}\]  
where \(d_{hkl}\) is the inter-planer spacing and \(h\), \(k\), and \(l\) are the Miller indices. The cell volume (\(V_{cell}\)) values were calculated using \(a^{2}\)csin120\({}^{\circ}\). The calculated lattice constants \(a\) and \(c\), \(c/a\) ratios, and \(V_{cell}\) values for Sr\({}_{1-x}\)La\({}_{x}\)Fe\({}_{12-x}\)Co\({}_{x}\)O\({}_{19}\) (\(0.0\leq x\leq 0.3\)) samples are listed in Table 1. The \(a\) and \(c\) values of this table are plotted as a function of \(x\) in Figure 3a. The \(c/a\) ratios and \(V_{cell}\) values of this table are also plotted as a function of \(x\) in Figure 3b. Figure 3a shows the variation in the lattice parameters \(a\) and \(c\) with increasing \(x\). Initially, \(a\) increases from \(x\) = 0.0 to 0.1 but then decreases up to \(x\) = 0.20; it then remains almost unaltered up to \(x\) = 0.3 while the lattice parameter \(c\) monotonously decreases with increasing \(x\) from 0.0 to 0.25, but it turns to an increase from \(x\) = 0.25 to 0.3. Since La\({}^{3+}\) and Co\({}^{2+}\) ions have been reported to replace the Sr\({}^{2+}\) and Fe\({}^{3+}\) sites, respectively [22, 29], we sought to understand these behaviors by the smaller La\({}^{3+}\) (1.22 A) replacing the larger Sr\({}^{2+}\) (1.31 A) based on nine oxygen ion coordination and the larger Co\({}^{2+}\) (0.75 A) replacing the smaller Fe\({}^{3+}\) (0.65 A) based on six oxygen ion coordination [30].