## 3 Examination of Machine Learning Analysis Using Simulation Spectra  
### Preparation of Simulation Spectra for Evaluation of Machine Learning Analysis Methods  
Prior to the solid-liquid phase fraction analysis of the measured spectra, the validity of the aforementioned analysis procedure was confirmed by simulated LBE Bragg-edge transmission spectra prepared using computer simulation shown below. The simulated total cross-section spectrum of the LBE solid phase was calculated using the RITS code [6] with the systematic variation of the texture information. The parameters considered here were the direction of the crystal orientation vector (preferred orientation <_hkl>_) and the degree of development of the orientation texture (March-Dollase coefficient, MD). However, since the ratio of the \(\beta\)-phase is higher than that of the \(\gamma\)-phase in LBE [20], only the texture of the \(\beta\)-phase was considered here. Sixteen types of texture combinations were created between four types of preferred orientation vectors (\(\beta\)<100>, \(\beta\)<002>, \(\beta\)<101>, \(\beta\)<102>) parallel to the neutron transmission direction and four types of MD coefficients (MD = 0.6, 0.7, 0.8, 0.9). A completely isotropic solid-phase texture (MD = 1.0) was added and we prepared 17 types of LBE solid-phase spectra in total. The total cross-section spectrum of the LBE liquid phase was prepared from the neutron transmission spectroscopic imaging experiment (see Section 4.2). The simulated transmission spectra of various phase fractions of the LBE mixed solid-liquid phase were calculated from these cross-sections according to the next equation:  
\[Tr(\lambda)=\exp\Big{[}-\Big{\{}\Sigma_{\text{solid}}(\lambda)at+\Sigma_{ \text{liquid}}(\lambda)(1-a)t\Big{\}}\Big{]}. \tag{3}\]  
Here, \(Tr(\lambda)\) is the neutron transmission spectrum; \(\Sigma_{\text{solid}}(\lambda)\) and \(\Sigma_{\text{liquid}}(\lambda)\) are the macroscopic total cross-section of the LBE solid and liquid phases, respectively; and \(a\) is the fraction of the solid phase (\(0\leq a\leq 1\)). \(t\) is the thickness of the entire sample, and here it was set to 10 mm according to the experiment. The solid-phase fraction \(a\) was changed from 0 to 100% in 1% steps, and one type of 100% liquid phase, which is common to the LBE system, was added to make a total of 1701 types of spectral datasets for evaluation. Figure 1 shows an example of the neutron transmission spectrum variation in an LBE solid-liquid mixture when the texture of the solid phase is isotropic. It can be seen that the Bragg-edges of the solid phase appear to be relatively clear against the 100% of the liquid phase. Thus, we prepared neutron transmission spectra of the LBE for various solid-liquid phase fractions (0 to 100%) and various textures (17 types as mentioned above) of the solid phase.  
Furthermore, in this study, to use the experimentally obtained transmission spectrum as training data, we also examined the case where the neutron transmission spectrum included statistical errors. To the simulated spectra created above, errors following a Gaussian distribution with a standard deviation \(\sigma\) of 1, 2,..., 9, and 10% were randomly added to the simulated spectra. As an example, Figure 2 shows the neutron transmission spectra with \(\sigma\) = 5% error in the case of the isotropic solid-phase texture. From this, when the statistical error included in the neutron transmission spectrum becomes large, it becomes more difficult for human eyes to distinguish between the solid and liquid phases.  
Figure 1: Neutron transmission spectra (Gaussian error \(\sigma\) = 0%) of lead-bismuth eutectic alloy (LBE) simulating the solid–liquid mixed state when the solid phase had an isotropic texture (March–Dollase coefficient (MD) = 1.0).