## 3 Single Bragg-Dip Profile-Fitting Analysis Method  
### Crystallographic Information Obtained from a Bragg-Dip Profile  
In this study, the wavelength where Bragg-dip appears, the width (FWHM) of Bragg-dip and the integrated intensity of Bragg-dip were analyzed by fitting a Gaussian function to a single Bragg-dip. Bragg-dips in a neutron transmission spectrum are caused by diffraction phenomenon of neutrons in a single crystal. Generally, crystal orientation is evaluated from the wavelength where Bragg-dips appear, mosaic spread angle is evaluated from the width of Bragg-dips and crystal thickness is evaluated from the integrated intensity of Bragg-dips.  
These relations between the crystallographic information and the Bragg-dip profiles are explained as follows. Bragg-dips appear as the attenuations of neutron transmission  
Figure 2: Schematic of setup of pulsed neutron transmission imaging experiment.  
intensities caused by single crystal diffractions. Diffracted neutrons causing Bragg-dips satisfy [PERSON]'s law:  
\[\lambda=2d_{hkl}\sin\theta_{hkl}(0^{\circ}\leq\theta\leq 90^{\circ}). \tag{1}\]  
Here, \(\lambda\) is the neutron wavelength, \(d_{hkl}\) is the crystal lattice spacing of (_hkl_) plane and \(\theta_{hkl}\) is the Bragg angle. According to Equation (1), the wavelength \(\lambda\), where Bragg-dip appears, depends on the lattice spacing \(d_{hkl}\) and the Bragg angle \(\theta_{hkl}\). Therefore, the information relating to the macro-strain depending on \(d_{hkl}\), and the change in crystal orientation relating to \(\theta_{hkl}\), can be evaluated from the Bragg-dip wavelength \(\lambda\).  
The width of Bragg-dip, \(\Delta\lambda\), depends on the width of the crystal lattice spacing distribution \(\Delta d_{hkl}\) which relates to the micro-strain and the crystallite size, and the fluctuation of Bragg angle \(\Delta\theta_{hkl}\) which relates to the mosaic spread angle of the crystal. Therefore, the information relating to the crystal defect such as micro-strain and mosaic can be evaluated from the width of Bragg-dip. Here, the other parameters which can affect the width of Bragg-dip, \(\Delta\lambda\), are discussed. The width of Bragg-dip, \(\Delta\lambda\), is also affected by the instrumental resolution, such as the wavelength resolution and the beam angular divergence. The wavelength resolution depends on the neutron wavelength. Moreover, when the neutron supermirror guide tube is used in the neutron beamline, the beam angular divergence also depends on the wavelength. Hence, the width of Bragg-dip, \(\Delta\lambda\), can also change depending on the wavelength. However, actually, the change in the wavelength of a certain Bragg-dip is small in a certain crystal because the changes in macro-strain and crystal orientation are small in the same crystal. Therefore, the width of Bragg-dip, \(\Delta\lambda\), is not affected by the change in the instrumental resolution so much because the Bragg-dip wavelength is not changed. Thus, it is generally considered that the change in the width of Bragg-dip represents the changes in crystal defect and mosaic spread angle.  
Finally, the information evaluated from the integrated intensity of a Bragg-dip is explained. In the neutron diffraction experiment, the diffracted intensity is proportional to the crystal volume. In the Bragg-dip neutron transmission method, the integrated intensity of Bragg-dip is proportional to the diffracted intensity along the neutron transmission path. Thus, this type of diffracted intensity is proportional to the crystal thickness where neutrons are transmitted. Therefore, the information of the crystal thickness can be evaluated from the integrated intensity of Bragg-dip. However, because of the extinction effect, the integrated intensity of Bragg-dip is not simply proportional to the crystal thickness. The principle of the extinction effect is described in Section 3.3.