## 3 Results and Discussion  
Figure 1a-c shows the XRD patterns of the Zn-Ti, Mn-Ti, and Zn-Zr substituted SrM (\(0\leq x\leq 2.0\)) powders after second calcination at 1250 \({}^{\circ}\)C. The diffraction peaks of the samples were indexed based on the international center for diffraction data (PDF search number; SrM: 00-033-1340) and the hexagonal magnetoplumbite structure with the space group \(\mathrm{P6}_{3}/mmc\) (ICDD 0801198). As shown in Figure 1a, Zn-Ti substituted SrM (Zn-Ti:SrM) exhibits a single M-type hexaferrite. It is believed that Zn-Ti is fully soluble in the SrM phase in the substitution range of \(x\leq 2.0\). For the case of Mn-Ti:SrM shown in Figure 1b, a single M-type phase can be identified for \(x\leq 1.5\) and a small number of second phase peaks of Fe\({}_{2}\)TiO\({}_{5}\) can be observed. Unidentified secondary phase peaks are denoted by an asterisk (*). Unlike Zn-Ti and Mn-Ti:SrM, Zn-Zr:SrM exhibits large second phase peaks of ZnFe\({}_{2}\)O\({}_{4}\), SrZrO\({}_{3}\), and ZrO\({}_{2}\) for \(x\geq 1.5\) in Figure 1c. For \(x=2.0\), ZrFe\({}_{2}\)O\({}_{4}\) exhibits a primary peak. In order to reveal the solubility limit of Zn-Zr (\(x\)) in SrM, additional XRD analysis was carried out on samples with \(x\) between 0.6 and 1.1 with an interval of 0.1. A clear second phase peak of the ZrFe\({}_{2}\)O\({}_{4}\) phase starts appearing at \(x=1.0\) and its intensity grows larger with increasing \(x\). In the inset of Figure 1c, the (220) peaks of the samples around \(2\theta=65^{\circ}\) are presented. Going from \(x=0\) to \(x=0.5\), the peak shift to the left is large, and the peak position slightly moves to the left between \(x=0.5\) and \(x=1.0\), but beyond that the peak position does not change. For all samples shown in Figure 1a-c, the lattice parameters, \(a\) and \(c\), are calculated from the values of \(d_{hkl}\) corresponding to the (2011) and (220) peaks according to the following equation:  
\[d_{hkl}=\{\frac{4(h^{2}+hk+k^{2})}{3a^{2}}+\frac{l^{2}}{c^{2}}\}^{-1/2}, \tag{2}\]  
where \(d_{hkl}\) is the interplanar spacing, and \(h\), \(k\), and \(l\) are the Miller indices. The values of \(a\), \(c\), and the cell volume of the sample are listed in Table 1, and their % changes are plotted in Figure 2a-c. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Composition** & \(x\) & **a(Å)** & **c(Å)** & **c/a(Å)** & **vol(Å)** \\ \hline  & 0 & 5.883 & 23.06 & 3.919 & 691.0 \\  & 0.5 & 5.884 & 23.06 & 3.919 & 691.2 \\ SrFe\({}_{12-2x}\)Zn\({}_{x}\)Ti\({}_{x}\)O\({}_{19}\) & 1 & 5.890 & 23.06 & 3.916 & 693.0 \\  & 1.5 & 5.892 & 23.07 & 3.916 & 693.6 \\  & 2 & 5.896 & 23.09 & 3.916 & 695.1 \\ \hline  & 0 & 5.882 & 23.04 & 3.916 & 690.2 \\  & 0.5 & 5.892 & 23.07 & 3.915 & 693.4 \\ SrFe\({}_{12-2x}\)Mn\({}_{x}\)Ti\({}_{x}\)O\({}_{19}\) & 1 & 5.893 & 23.09 & 3.917 & 694.3 \\  & 1.5 & 5.898 & 23.09 & 3.915 & 695.8 \\  & 2 & 5.901 & 23.11 & 3.915 & 696.8 \\ \hline  & 0 & 5.883 & 23.06 & 3.919 & 691.0 \\  & 0.5 & 5.905 & 23.22 & 3.932 & 701.3 \\  & 0.6 & 5.909 & 23.27 & 3.938 & 703.4 \\  & 0.7 & 5.913 & 23.30 & 3.941 & 705.5 \\ SrFe\({}_{12-2x}\)Zn\({}_{x}\)Zr\({}_{x}\)O\({}_{19}\) & 0.8 & 5.917 & 23.32 & 3.942 & 707.2 \\  & 0.9 & 5.920 & 23.36 & 3.946 & 709.2 \\  & 1 & 5.921 & 23.36 & 3.945 & 709.4 \\  & 1.1 & 5.922 & 23.36 & 3.945 & 709.6 \\  & 1.5 & 5.922 & 23.37 & 3.945 & 709.7 \\  & 2 & 5.922 & 23.35 & 3.943 & 709.3 \\ \hline \hline \end{tabular}
\end{table}