## 3 Results and Discussion
According to transmission line theory [28], the reflection loss (RL), which implies the EM wave absorption performance, can be calculated using the following equations:  
\[RL\ (dB)=20 log\left|\frac{Z_{in}-1}{Z_{o}}\right|, \tag{4}\]  
\[\frac{Z_{in}}{Z_{o}}=\sqrt{\frac{\mu_{r}}{\varepsilon_{r}}}tanh[j\bigg{(}\frac {2\pi d}{c}\bigg{)}\sqrt{\mu_{r}\varepsilon_{r}}], \tag{5}\]  
where \(Z_{in}\) is the input impedance of the absorber, \(Z_{0}=\sqrt{\mu_{0}\varepsilon_{0}}\) is the characteristic impedance of free space, \(c\) is the speed of light, \(f\) is the frequency of the incident EM wave, \(d\) is the thickness of the absorber, and \(\varepsilon_{r}\) and \(\mu_{r}\) are the complex permittivity (\(\varepsilon_{r}=\varepsilon^{\prime}-\mathbf{j}\varepsilon^{\prime\prime}\)) and permeability (\(\mu_{r}=\mu^{\prime}-\mathbf{j}\mu^{\prime\prime}\)), respectively. Here, the measured \(\mathbf{u}^{\prime}\), \(\mathbf{u}^{\prime\prime}\), \(\varepsilon^{\prime}\), and \(\varepsilon^{\prime\prime}\) spectra can be used to obtain \(\frac{Z_{in}}{Z_{o}}\) for any thickness \(d\) with respect to \(f\). RL calculations were plotted in square \(f\)-\(d\) maps, as shown in Figure 7a-d. The region in which RL \(\leq-10\) dB, which implies EM absorptions exceeding 90%, is outlined by solid black lines. Inside this area, the regions in which RL \(\leq-20\), \(-30\), and \(-40\) dB are also outlined by solid lines. In the RL map shown in Figure 7a-i, the strong EM absorbing area at lower \(d\), marked with a rectangular box, starts to be observed at \(x=0.7\), and it m  
Figure 7: **(a–)** RL map as functions of sample thickness (\(d\)) and frequency (\(f\)) for SrFe\({}_{12-2x}\)Zn\({}_{x}\)Zr\({}_{x}\)O\({}_{19}\) (\(x=0.5\), \(0.6\), \(0.7\), \(0.8\), \(0.9\), \(1.0\), \(1.1\), \(1.5\), \(2.0\)) powder–epoxy (10 wt%) composite samples. increase in \(x\) up to 1.0 (Figure 7c-f). The left edge of the absorption area already appeared at the right edge of Figure 7b for the sample with \(x\) = 0.6. EM absorption in the marked area is caused by a magnetic loss mechanism, that is, FMR, which produces a peak in the \(\upmu^{\prime\prime}\) spectra. Thus, the gradual movement of the EM absorbing area with \(x\) = 0.7, 0.8, and 0.9 (Figure 6c-e) is attributed to the \(\upmu^{\prime\prime}\) peak shift shown in Figure 6d. It is also observed that the absorption intensity in the RL maps for \(x\) from 1.1 to 2.0 (Figure 6g-i) becomes weaker. A decrease in the \(\upmu^{\prime\prime}\) peak height shown in Figure 3f is the reason for the weakening EM absorption. As previously mentioned, the decrease in \(\upmu^{\prime\prime}\) peak height for the \(x\) = 1.5 and 2.0 samples is due to an increase in the non-magnetic second phase. Figure 8 shows the RL spectra of the sample (\(x\) = 0.7, 0.8, 0.9, 1.0, 1.1) at the optimal thickness, at which the minimum RL (RL\({}_{\rm min}\)) point was located. The RL\({}_{\rm min}\) values and frequency of RL\({}_{\rm min}\) (\(f_{\rm RLmin}\)) are presented in Table 2. Each sample shows an RL\({}_{\rm min}\) \(<\) -30 dB, and \(f_{\rm RLmin}\) shifts to a lower frequency with increasing \(x\). \(f_{\rm RLmin}\) shifts to a lower \(f\) in large steps as \(x\) increases from 0.7 to 0.9, but it moves only slightly at \(x\)\(\geq\) 0.9. The tendency for changes in large steps up to \(x\) = 0.9, and then for changes in very small steps for \(x\)\(>\) 0.9, is the same for H\({}_{\rm C}\), \(f_{\rm FMR}\), and \(f_{\rm RLmin}\). All these changes are related to the Zn-Zr substitution amount in the M-type phase and its magnetocrystalline anisotropy change. As shown, \(f_{\rm RLmin}\) is similar to \(f_{\rm FMR}\) for each sample, and FMR is the dominant EM absorbing mechanism in hexaferrites.