## 3 Single Bragg-Dip Profile-Fitting Analysis Method
### Profile-Fitting Function  
_Tr(\(\lambda\))_, called the neutron transmission spectrum, is expressed as the following equation:  
\[\textit{Tr}(\lambda)=\exp[-N\sigma_{\text{tot}}(\lambda)]. \tag{2}\]  
Here, \(\lambda\) is the neutron wavelength, \(N\) is the projected atomic number density of the crystal and \(\sigma_{\text{tot}}(\lambda)\) is the microscopic neutron total cross-section that represents the probability of the interaction between neutron and nucleus. The negative value of the natural logarithm of transmission is calculated as the following equation:  
\[-\ln(\textit{Tr}(\lambda))=N\sigma_{\text{tot}}(\lambda), \tag{3}\]  
and this value is proportional to the neutron total cross-section. \(-\ln(\textit{Tr}(\lambda))\) is usually used in this study for the following reasons. The neutron total cross-section includes the neutron coherent elastic scattering cross-section. The neutron coherent elastic scattering cross-section, as described later, is proportional to the diffracted intensity. The diffracted intensity is proportional to the crystal volume. Therefore, by evaluating the neutron transmission spectrum in the dimension of a cross-section like \(-\ln(\textit{Tr}(\lambda))\), the crystallographic information can be evaluated more quantitatively. The neutron total cross-section \(\sigma_{\text{tot}}(\lambda)\) is expressed as the following equations:  
\[\sigma_{\text{tot}}(\lambda)=\sigma_{\text{Bragg}}(\lambda)+\sigma_{\text{non Bragg}}(\lambda), \tag{4}\]  
\[\sigma_{\text{tot}}(\lambda)=\sigma_{\text{ela}}^{\text{coh}}(\lambda)+\sigma_{ \text{ela}}^{\text{incoh}}(\lambda)+\sigma_{\text{inela}}(\lambda)+\sigma_{ \text{abs}}(\lambda). \tag{5}\]  
\(\sigma_{\text{Bragg}}(\lambda)\) and \(\sigma_{\text{nonBragg}}(\lambda)\) satisfy the following equations:  
\[\sigma_{\text{Bragg}}(\lambda)=\sigma_{\text{ela}}^{\text{coh}}(\lambda), \tag{6}\]  
\[\sigma_{\text{nonBragg}}(\lambda)=\sigma_{\text{ela}}^{\text{incoh}}(\lambda)+ \sigma_{\text{inela}}(\lambda)+\sigma_{\text{abs}}(\lambda). \tag{7}\]  
Here, \(\sigma_{\text{Bragg}}(\lambda)\) is the Bragg scattering cross-section, \(\sigma_{\text{nonBragg}}(\lambda)\) is the non-Bragg scattering cross-section, \(\sigma_{\text{ela}}^{\text{coh}}(\lambda)\) is the coherent elastic scattering cross-section, \(\sigma_{\text{ela}}^{\text{incoh}}(\lambda)\) is the incoherent elastic scattering cross-section, \(\sigma_{\text{inela}}(\lambda)\) is the inelastic scattering cross-section and \(\sigma_{\text{abs}}(\lambda)\) is the absorption cross-section. Figure 3a shows the neutron transmission spectrum of Grain 3, one of the crystalline grains in the sample measured in the experiment. Figure 3b shows the negative value of the natural logarithm of transmission, \(-\ln(Tr(\lambda))\). Figure 3c shows the procedure of the profile fitting and the Bragg-dip profile information obtained through the profile fitting. In Figure 3b, the peak indicated by the Bragg scattering component is caused by the Bragg scattering cross-section, \(\sigma_{\text{Bragg}}(\lambda)\), and the baseline indicated by the non-Bragg scattering component is caused by the non-Bragg scattering cross-section, \(\sigma_{\text{nonBragg}}(\lambda)\). The profile of the non-Bragg scattering cross-section, \(\sigma_{\text{nonBragg}}(\lambda)\), can be approximated by the following linear function (see Figure 3c left):  
\[\sigma_{\text{nonBragg}}(\lambda)=a\lambda+b. \tag{8}\]  
Here, \(a\) and \(b\) are the parameters refined by the fitting. Furthermore, the Bragg scattering cross-section, \(\sigma_{\text{Bragg}}(\lambda)\), is added to the non-Bragg scattering cross-section. In this study, the Gaussian function is fitted to the Bragg scattering cross-section, \(\sigma_{\text{Bragg}}(\lambda)\), to analyze the wavelength where Bragg-dip appears, the width and the integrated intensity of Bragg-dip.