## 3 Results and Discussion
Figure 4a shows that as the amount of SiO\({}_{2}\) increases, \(\Delta g_{c}\) increases until it is greater than \(\Delta g_{max}\), resulting in no grain growth in the initial stage of  
Figure 4: Schematic showing the growth rate of a crystal as a function of the driving force for diffusion (dashed lines) and a mix of diffusion and interface reaction control mechanisms (solid lines). **(a)** Plot of three mixed control curves for no-Si (0 Si), 0.5 wt% SiO\({}_{2}\) (0.5 Si), and 1.0 wt% SiO\({}_{2}\) (1.0 Si) SCLFCO samples sintered at the same temperature. **(b)** Plots of three mixed control curves for 0.5 wt% SiO\({}_{2}\) SCLFCO samples sintered at 1220 \({}^{\circ}\)C, 1240 \({}^{\circ}\)C, and 1250 \({}^{\circ}\)C.  
Figure 3: SEM micrographs of SCLFCO samples sintered at 1250 \({}^{\circ}\)C with varying amounts of SiO\({}_{2}\) additions using the pre-Si method: **(a)** No-Si, **(b)** 0.5 wt%, **(c)** 1.0 wt%, and **(d)** 2.0 wt%. sintering. The changes in the microstructure of samples with the addition of SiO\({}_{2}\), as shown in Figure 3, are in line with the grain growth theory and are similar to previous studies on hexaferrites [24; 26]. To confirm the non-linearity of the grain growth rate with \(\Delta g\) and understand the microstructural changes in samples with SiO\({}_{2}\) addition, the compacted powders of pre-Si samples with 0.5 wt% SiO\({}_{2}\) were sintered at various temperatures. The SEM images of these samples are shown in Figure 5. As seen in Figure 5b, a bimodal grain size distribution is observed in the sample sintered at 1240 \({}^{\circ}\)C; aside from a few grains that grew rapidly, most grains did not grow. No grain growth occurred in the samples sintered at 1200 \({}^{\circ}\)C and 1220 \({}^{\circ}\)C, resulting in a small grain size. In the grain growth theory, the \(\Delta g_{c}\) increases and the diffusion rate decreases as the sintering temperature decreases [20; 22; 39]. The pore and grain shape are dependent on the interface (grain boundary and surface) structure [20; 21]. The anisotropy of grain and pore increases as the fraction of the atomically ordered interface increases [14; 21]. The \(\Delta g_{c}\) increases as the fraction of the of the atomically ordered interface increases [14; 18]. The spherical pore shape can explain that the anisotropy decreases and the \(\Delta g_{c}\) decreases as the sintering temperature increases. The slope of the dashed line in Figure 4b decreases as the diffusion rate and sintering temperatures decrease [22]. As the sintering temperature decreases from 1250 \({}^{\circ}\)C to 1240 \({}^{\circ}\)C, the \(\Delta g_{c}\) increases and is closer to, but still below, \(\Delta g_{max}\). This results in an abnormal growth of a few grains with a \(\Delta g\) greater than \(\Delta g_{c}\)\({}^{1240\text{C}}\), while most grains cannot grow. Figure 4b indicates that as the \(\Delta g_{c}\) increases and the sintering temperature decreases, \(\Delta g_{c}\)\({}^{1220\text{C}}\) becomes larger than \(\Delta g_{max}\). This results in no grain growth as none of the grains have a driving force for growth at the low sintering temperatures of 1200 \({}^{\circ}\)C and 1220 \({}^{\circ}\)C. The abnormal grain growth phenomena of SCLFCO can be explained by the grain growth theory and observed via the addition of SiO\({}_{2}\) and the change in sintering temperature. The grain growth mechanism of SCLFCO can be explained by the migration of the facet plane, consistent with previous studies [6; 23; 24; 25; 26; 27]. The M-H curves of sintered samples are plotted in Figure 6a-c. The \(M_{s}\) and \(H_{c}\) values of these samples are presented in Table 2, where \(M_{s}\) is the magnetization value at \(H\) = 25 kOe. Comparing samples sintered at the same temperature, the no-Si samples have the largest \(M_{s}\) and the pre-Si samples have the smallest \(M_{s}\). These results show that a Si substitution into an Fe site in the hexaferrite matrix deteriorates the magnetism.