# The XTRA Option at the NEUTRA Facility--More Than
## 2 Data Interpretation and Data Fusion  
The transmission image obtained by neutrons and by X-rays can be described by the Beer-Lambert law  
\[I(y,\,z)=I_{0}(y,\,z)\cdot e^{-\int_{0}^{d}A\cdot dx} \tag{1}\]  
For a homogenous material of the thickness d in beam direction \(x\) this is  
\[I(y,\,z)=I_{0}\,(y,\,z)\cdot e^{-A\cdot d} \tag{2}\]  
where \(I_{0}\) is the incident intensity, which can be measured without a sample in the beam, while \(I\) is the transmitted intensity. The incident beam is attenuated exponentially depending on the linear attenuation coefficient \(A\) of the material in the beam and the thickness \(d\) of  
\begin{table}
\begin{tabular}{c c c c} \hline  & **Neutron Setup** & \multicolumn{2}{c}{**X-ray Setup**} \\ \hline Aperture D (mm) & 20 & & \\ Focal spot (mm) P & & 3 & 5.5 \\ Position 2 distance L (m) & 7.29 & 3.29 & 3.29 \\ L/D & 350 & 1096 & 598 \\ Position 3 distance L (m) & 10.55 & 6.90 & 6.90 \\ L/D & 550 & 2301 & 1255 \\ \hline \end{tabular}
\end{table}
Table 2: Beam formation conditions for the neutron and X-ray setups.  
Figure 3: Pneumatic scintillator exchange setup at position 2.  
the material layer, which enters the equation through the integral along the flight path in the x-direction.  
The difference between neutron and X-ray attenuation is solely reflected by the specific linear attenuation coefficients, which we here denote as \(\sum\) and \(\mu\), respectively. As outlined earlier, both the neutron and X-ray linear attenuation coefficients are highly energy dependent. The linear attenuation coefficient \(\sum\) is related to the tabulated total microscopic cross sections \(\sigma\) (absorption and scattering) [2; 3] through the number density \(N\) of atoms per cm\({}^{3}\) by \(\sum=N^{\ast}\sigma\) and, thus, has the unit of a reciprocal length (cm\({}^{-1}\)). For composite materials (with \(i\) as the index of the specific material) it is  
\[\Sigma=\sum_{i}N_{i}\cdot\sigma_{i} \tag{3}\]  
In the case of X-rays, there are no tabulated microscopic cross sections available. The database is directly in the unit cm\({}^{-1}\) or even as \"mass-attenuation coefficient \(\mu/\rho\)\" in cm\({}^{2}\)/g, with the density \(\rho\)[3].  
The transmission T refers to the exponential term of the Beer-Lambert law (Equation (2)) as  
\[T=\frac{I}{I_{0}}=e^{-A\cdot d} \tag{4}\]  
From this it becomes obvious that for a single transmission image, only the \(\int_{0}^{d}A\cdot dx\) can be retrieved without a priori knowledge about material phases and thickness of the sample. Through tomography, which requires a number of projection images to be taken on, e.g., a range of 180\({}^{\circ}\), the local attenuation coefficient \(A(x,y,z)\) can be retrieved with the corresponding spatial resolution of a specific measurement and the reconstructed discrete voxels according to \(A_{i,j,k}\)[9].  
Using the XTRA setup, two individual data sets, \(T_{\theta,i,j}\), where \(\theta\) refers to the projection angle, and tomographic reconstructions of \(\sum_{i,j,k}\) and \(\mu_{i,j,k}\) can be obtained. Both data sets (T\({}_{\text{n}}\) and T\({}_{\text{x}}\)) and tomograms (\(\sum_{i,j,k}\) and \(\mu_{i,j,k}\)) can be correlated in the best case pixel, voxel-wise. Different strategies to combine the data, depending on the aim of a specific investigation, can be applied.