## 3 Single Bragg-Dip Profile-Fitting Analysis Method
### Profile-Fitting Function
The probability density function which is used for the profile fitting follows a Gaussian distribution, and it is expressed as the following equation:  
\[\sigma_{\text{Bragg}}(\lambda)=S\times\exp(-(\lambda-\lambda_{hkl})^{2}/2 \sigma^{2})/\sqrt{2\pi\sigma^{2}}. \tag{9}\]  
Here, \(S\) is a scale factor of the peak height which includes the effect of projected atomic number density \(N\) during the analysis. In addition, \(\sigma\) is the standard deviation of the Gaussian function. The standard deviation, \(\sigma\), is related to the full width at half maximum (FWHM) of the Gaussian distribution, \(w\), by the following equation:  
\[ \begin{array}{ll}w&=2\sqrt{2\ln 2}\times\sigma\\ &\approx 2.35\sigma.\end{array}  \tag{10}\]  
The wavelength where Bragg-dip appears, \(\lambda_{hkl}\), is determined from the wavelength where the Gaussian function is the maximum. The FWHM of Bragg-dip, \(2.35\sigma\), is determined from the standard deviation, \(\sigma\). The height of Bragg-dip is determined as \(\sigma_{\text{Bragg}}(\lambda_{hkl})=S/\sqrt{2\pi\sigma^{2}}\). This profile information is shown in Figure 3c, right. Next, the analysis method of the integrated intensity, which is related to the crystal thickness, is explained. The value which is proportional to the integrated intensity of Bragg-dip is calculated by multiplying the FWHM and the height of Bragg-dip. The reason why the product of the FWHM and the height of Bragg-dip can be treated as the integrated intensity of Bragg-dip is described as follows. The integrated intensity of Bragg-dip is proportional to the crystal thickness. Hence, the integrated intensity should not change depending on the width and the height of Bragg-dip. On the other hand, the probability density function (Equation (9)), which follows a Gaussian distribution, has a negative correlation between the width and the height. In other words, under the condition that the integrated intensity is constant, the height decreases when the width increases, and the height increases when the width decreases. The constant value which is proportional to the integrated intensity is calculated by multiplying the height and the width of the probability density function (Equation (9)) as the following equation:  
\[ \begin{array}{ll}\sigma_{\text{Bragg}}(\lambda_{hkl})\times 2.35\sigma&= \left(S/\sqrt{2\pi\sigma^{2}}\right)\times 2.35\sigma\\ &=2.355/\sqrt{2\pi}\;(\text{constant}).\end{array}  \tag{11}\]  
It turns out that the product of the height and the width is constant. Here, \(S\), the scale factor of the height, is settled on one value under the same instrumental condition and the same crystal condition. Because the total integration of the probability density function which follows a Gaussian distribution is also constant, the value which is calculated by multiplying the height and the width of Bragg-dip can be considered as the integrated intensity of Bragg-dip. Figure 3: (**a**) The neutron transmission spectrum of Grain 3. (**b**) The negative value of the natural logarithm of transmission. (**c**) The procedure of the profile fitting and the Bragg-dip profile information obtained by the profile fitting.