## 3 Results and Discussion
### Characteristics of Randomly Oriented Polycrystalline Samples
#### 3.1.1 Lattice Parameters and Microstructures
However, since the sintering additives of SiO\({}_{2}\) and CaCO\({}_{3}\) contribute to the increase in the lattice parameter \(c\) up to \(x=0.25\), we argue that, when the sintering additives are used, their effects should be taken into account to understand the variation in the lattice parameters as a function of \(x\) if they substitute for the cation sites. Figure 4 shows the SEM micrographs of the Sr\({}_{1-x}\)La\({}_{x}\)Fe\({}_{12-x}\)Co\({}_{x}\)O\({}_{19}\) samples (\(0.0\leq x\leq 0.3\)). The average grain size (\(d_{avg}\)) and its standard deviation (SD) are listed in Table 1. The values of both \(d_{avg}\) and SD increase from 1.20 \(\pm\) 0.15 \(\upmu\)m to 1.96 \(\pm\) 0.87 \(\upmu\)m as \(x\) increases from 0.0 to 0.3, while the average thickness of the hexagonal plate (\(t_{avg}\)) of 0.95 \(\pm\) 0.34 \(\upmu\)m insignificantly changed. To determine the \(d_{avg}\) and \(t_{avg}\) values from the SEM micrographs composed of randomly oriented grains, we selected about 150 grains with a hexagonal plate shape and also about 150 grains with a rectangular shape, respectively. For this purpose, many different SEM micrographs were used for each composition \(x\). The \(t_{avg}\) values from the rectangular grains are surely somewhat overestimated since the cross-sectional cut along the c-axis of each grain is less probable. However, although we take this overestimation into account, this \(t_{avg}\) evaluation of randomly oriented grains is very meaningful to compare with that of MFP-processed samples later. For this purpose, many SEM micrographs were used. As shown in Figure 4a, the sample of \(x=0.0\) exhibits a relatively smaller grain size and more uniform size distribution. However, the grain grows, and its size distribution becomes less uniform with the increasing \(x\), as shown in Figure 4b-f. During the sintering process, a solid-state reaction leads to grain growth and densification, while a liquid-phase sintering can cause a rapid growth and densification even at a relatively lower sintering temperature [39]. The solid-state reaction of SrCO\({}_{3}\) and Fe\({}_{2}\)O\({}_{3}\) to form SrFe\({}_{12}\)O\({}_{19}\) is typically completed at the sintering temperature of 1200 \({}^{\circ}\)C [23]. Sintering additives such as SiO\({}_{2}\) and CaCO\({}_{3}\) are utilized to create a liquid phase during sintering, which might accelerate the formation of the M-type phase. [PERSON] et al. [40] demonstrated that the grain growth behavior in polycrystalline ceramics can be explained by the relative values between the critical driving force and the maximum driving force for growth. The maximum and critical driving forces are influenced by factors such as the grain size, dopants, sintering atmosphere, and temperature. According to [PERSON] et al. [41], the addition of Co\({}_{3}\)O\({}_{4}\) to SrM as a sintering additive promotes the grain growth of M-type hexaferrite [41]. Therefore, the increase in the number of relatively larger grains shown in Figure 4b-f can be attributed to the compositional effect of La-Co substitution on these driving forces. curves shown in Figure 5a, the \(M_{S}\) and \(H_{a}\) values of the samples were calculated using the law of approach to saturation (LAS) from the following equation [42]:  
\[M\ =\ M_{s}\bigg{(}1-\frac{A}{H}-\frac{B}{H^{2}}\bigg{)}+\chi_{p}H \tag{2}\]  
where \(M\) represents magnetization induced by an applied magnetic field, \(H\); \(A\) is the inhomogeneity parameter associated with the material's inhomogeneity; \(\chi_{p}\) is the high field susceptibility; and \(B\) is the anisotropy parameter linked to magnetocrystalline anisotropy. The reported value for \(A\) is approximately 250 Oe in BaM [42], and the parameter \(\chi_{p}\) becomes negligible in high magnetic fields.