Figure 2 shows the XRD measurement results of the Na₃OX samples (5 compositions in total, Na₃OBr, Na₃OBr_0.9I_0.1, Na₃OBr_0.8I_0.2, Na₃OBr_0.7I_0.3, and Na₃OBr_0.6I_0.4). It was confirmed that all compositions up to Br_1−xI_x (x = 0 to 0.4) had the anti-perovskite structure with Pm-3m cubic symmetry. It is also demonstrated that the impurity peaks could hardly be confirmed, and the weight of the precursor substance with respect to Na₃OX (X = Br, I) was estimated by using the crystal structure analysis software PDXL2 based on the crystallographic information framework (CIF) and data obtained from the results of the Rietveld analysis [25] , which will be described later. These results are different from the XRD pattern reported by Wang et al. [13] , in which a remarkable peak split was observed when compared with the measurement results of our Br_1−xI_x (x = 0 to 0.4) eutectics. Such a peak split was attributed to be a result of vigorous segregation of two types of halogen ions because the precursor samples were not sufficiently mixed. In our sample, only when x = 0.5 was exceeded, peak split and impurity precipitation became not negligible. It is considered that segregation occurred in this case because I (iodine) exceeded the solid solution limit, which might be in the range of x = 0.4 to 0.5.

Rietveld analysis was performed for each composition using the structural analysis software EXPO2014 [26] . Currently, the only CIF data presented in major databases are X = Cl, Br, I, and Cl_0.5Br_0.5, all of which are cubic perovskites belonging to the symmetry Pm-3m. In addition, from the waveform pattern in Figure 2, it can be inferred that the symmetry of the X = Br_1−xI_x eutectics, which is currently unknown, belongs to same cubic symmetry up to x = 0.4. Therefore, for the synthesized Na₃OBr_1−xI_x eutectics, Rietveld analysis was performed from x = 0 to 0.4, but all fittings were performed assuming cubic symmetry Pm-3m. As a result, the parameters of each composition were refined, and a CIF file containing the information was obtained. According to the fitting results of Na₃OBr and Na₃OBr_1−xI_x (x = 0 to 0.5), it was confirmed for the first time that all compositions of Br_1−xI_x (x = 0 to 0.4) had the anti-perovskite structure with Pm-3m cubic symmetry.

From the above-mentioned analysis, the crystal structure for each composition could be identified, so the counter value width was extracted for each XRD peak and Williamson-Hall (WH) plots were obtained, as shown in Figure 3. The half width component has been corrected by the correction formula obtained from the standard sample. The slope of the WH plots exhibits a lattice distortion ϵ , which represents a relative change of the lattice constant (∆d/d). It is clear from Figure 3 that the introduction of significant strain was successfully achieved in the Br_1−xI_x eutectics, which gave rise to a maximum distortion at near x = 0.2. The relative distortions comparing with Na₃OBr were ϵ = 2.3, 5.5, 4.2, and 3.4 × 10⁻³, for Na₃OBr_1−xI_x (x = 0.1, 0.2, 0.3, and 0.4) eutectics, respectively. These distortions might lower the activation energy (ΔEa) by loosening the bottleneck and increase the ion conductivity in the synthesized Na₃OBr_1−xI_x eutectics. The Na₃OBr_0.5I_0.5 sample was not the anti-perovskite crystal structure.

Figure 4 shows the impedance results, the Cole-plots, for three Na₃OBr_1−xI_x (x = 0.2, 0.3, and 0.4) samples measured at 60˚C. A relatively high ionic conductivity of σ = 1.55 × 10⁻⁷ S/cm was obtained from Na₃OBr_0.6I_0.4, as shown in Figure 4. The ionic conductivities of all samples were obtained by fitting the Cole-plots of the impedance measurements and shown in Figure 5. The measurement temperature was changed from 20˚C to 60˚C. Since the conductive type of SE follows the Arrhenius type, the slope of the straight line corresponds to the conductive activation energy. It is clear that the ionic conductivity of our anti-perovskite SE Na₃OBr_0.6I_0.4 is three orders higher than that of Na₃OBr in literature [24] . This high conductivity is considered to come from the lattice distortion caused by doping I (iodine) ions. In the impedance measurements, the resistance values of the grain boundaries and bulk had been separated using the equivalent circuit fitting software pyZwx developed by NIMS [27] . So the fitting results shown in Figure 4 are relatively credible.

Furthermore, the activity energy of ionic conductivity, ΔEa, was determined by fitting the Arrhenius plots in Figure 5. As claimed in literature [24] , ΔEa for Na₃OBr was 1.14 eV, which decreased to 0.72, 0.79, and 0.64 eV for our Na₃OBr_1−xI_x (x = 0.2, 0.3, and 0.4), respectively. Although the activation energy did not completely decrease with the distortion increase, the bromine-iodine (B-I) mixture gave rise to the high conductivity performance by lowering the activity energy, which will be discussed below.

Since the B-I mixture played an important role in increasing ionic conductivity, the ratio of B/I and their distribution should be taken into account in the synthesis process. Figure 6 shows the surface morphology and ion distribution of Na₃OX crystals measured by SEM-EDS (Ultim Max, Oxford Instruments). Compared with pure Na₃OBr, the Br-I mixture Na₃OX got a broader eutectic. However, there was still obviously visible bromide crystal in the Br-I mixture Na₃OX surface, which might due to a lack of enough reaction time.

A significant improvement in conductivity was observed with the Na₃OBr_1−xI_x eutectic compared to Na₃OBr. However, the conductivity of the Na₃OBr_0.6I_0.4 eutectic was significantly superior to that of the Na₃OBr_0.8I_0.2 eutectic, even which had the largest strain. Assuming strain introduction optimizing the bottleneck as intended and destabilizing the stronger Na-O bond, a reduction in ion transfer energy and defect formation energy should be confirmed. Since the conduction activation energy is the sum of the ion transfer energy and the defect formation energy, it is considered that the conductivity reduction can be evaluated by comparing with the strain and the conductivity activation energy. It had been demonstrated that the Br-I mixture introduced additional lattice distortion in the eutectic, which led to a large reduction of the conductive activation energy from 1.14 to 0.64 eV. But the reduction of the activation energy was not completely following with the distortion increase. The reason of a rough relationship between the activation energy and the distortion was thought to be an effect of the polarizability of the introduced X ions (X: Br = 4.16, I = 6.43 [10⁻²⁴ cm³]) [28] [29] . When the ligand ions constituting the bottleneck are highly polar in the case of Na ion diffusing, they are flexibly deformed even if the bottleneck radius is narrow, and the Na ions could push away the ligands so that they could

diffuse with low energy. From these facts, it is considered that finding a compositional balance that maximizes the contribution of both strain and polarity is an effective way to optimize the effect of improving the conductivity in the Na₃OX eutectic. It is also considered that the impedance measurement result itself changes significantly due to the influence of the surrounding environment and the pellet preparation method, which will be published next.