## 3 Results and Discussion
### Crystalline Structure and Microstructure Analysis
As the same amount of La\({}^{3+}\) was used to substitute for Sr\({}^{2+}\) and Co\({}^{2+}\) for Fe\({}^{3+}\), all samples were composed of the M-type single phase without any second phase, as shown in Figure 1b. Therefore, it is reasonable to assume that charge neutrality is always maintained. Considering only the influence of the size difference between the substituents (La\({}^{3+}\) and Co\({}^{2+}\)) and the original cation sites (Sr\({}^{2+}\) and Fe\({}^{3+}\)), the lattice constants \(a\) and \(c\) are expected to change consistently as a function of \(x\). This consistency of the variations in \(a\) and \(c\) with \(x\) is observed for a wide range of \(x\) values, including the monotonous decrease in \(c\) up to \(x\) = 0.25 and the gradual decrease in a from \(x\) = 0.1 to 0.2, followed by almost constant values up to 0.25. The changes are surely due to the decreasing size effect of La\({}^{3+}\) substitution for Sr\({}^{2+}\), which exceeds the increasing size effect of Co\({}^{2+}\) for Fe\({}^{3+}\). However, in a limited range of \(x\) values, the variations in \(a\) and \(c\) with \(x\), including the increase in \(c\) from \(x\) = 0.25 to 0.3 and also for the increase in a from \(x\) = 0.0 to 0.1, are unexplainable only with the size difference between the substituents (La\({}^{3+}\), Co\({}^{2+}\)) and the original cation sites (Sr\({}^{2+}\), Fe\({}^{3+}\)). To understand the above peculiar variations of \(a\) and \(c\) with \(x\), it is necessary to consider, in addition to the size difference, the effect of the valence difference on the lattice parameters, since the valence of the substituents (La\({}^{3+}\), Co\({}^{2+}\)) is different from that of the original cation sites (Sr\({}^{2+}\), Fe\({}^{3+}\)). To describe the cation sites within the unit cell of SrM hexaferrite, their locations must be specified as shown in Figure 4. The M-type hexaferrites exhibit a hexagonal structure composed of alternating layers of spinel (S, Fe\({}_{6}\)O\({}_{8}\)) and hexagonal (R, MFe\({}_{6}\)O\({}^{11}\)) blocks arranged as SRS\({}^{*}\)R\({}^{*}\), where the asterisk (*) indicates a 180\({}^{\circ}\) rotation around the \(c\)-axis. The crystallographic sublattices of Fe\({}^{3+}\) within the M-type structure are further divided into five distinct sites: 2\(a\), 2\(b\), 4\(f_{1}\), 4\(f_{2}\), and 12\(k\) sites. The magnetic moments of Fe\({}^{3+}\) ions at the 12\(k\), 2\(a\), and 2\(b\) sites are oriented parallel to the \(c\)-axis (up-spin), while those at the 4\(f_{1}\) and 4\(f_{2}\) sites are oriented antiparallel (down-spin) [31]. Using the above notations for Fe\({}^{3+}\) sites, we can describe the peculiar variations in \(a\) and \(c\) with \(x\). Initially, there is an increase in a from \(x=0.0\) to \(x=0.1\), as shown in Figure 3a. This is believed to be due to the increasing effect of larger Fe\({}^{2+}\) ions (0.78 A) transformed from smaller Fe\({}^{3+}\) ions (0.65 A) at the \(2a\) site. The substitution of smaller La\({}^{3+}\) (1.22 A) by larger Sr\({}^{2+}\) (1.31 A) is seen to overcome this. Fe\({}^{3+}\) ions at the \(2a\) sites in La-substituted SrM samples have been converted to Fe\({}^{2+}\) ions according to the Mossbauer analysis results of [PERSON]. [PERSON] et al. [15]. Similarly, other research groups [23; 33; 34; 35] have reported the presence of Fe\({}^{2+}\) ions at the \(2a\) sites, even in La-Co co-substituted SrM samples, while Co\({}^{2+}\) is reported to substitute Fe\({}^{3+}\) at the \(4f_{2}\) and \(2a\) sites. The presence of Fe\({}^{2+}\) at the \(2a\) sites in La-Co co-substituted SrM is often attributed to a non-uniform distribution of La\({}^{3+}\) and Co\({}^{2+}\) substituents within the crystal structure. This results in a transformation from Fe\({}^{3+}\) to Fe\({}^{2+}\) for local charge neutrality. However, the lattice parameters of \(a\) and \(c\) in Co-substituted SrM samples are reported to be almost unchanged [36] or slightly increased [37] with an increase in the amount of Co substituent.