# Neutron and Gamma Pulse Shape Discrimination by Robust Determination of the Decay Shape
## 2 Materials and Methods
### Discrimination Methods  
As previously mentioned, the charge comparison method is currently the most common one in PSD. In this method, the discrimination between neutrons and gamma rays is based on the analysis of the ratio \(R\) between the integrated charges of the slow \(Q_{s}\) and the total \(Q\) components of the scintillation pulses (Figure 1), respectively. \[R=\frac{Q_{s}}{Q} \tag{1}\]  
Another simple PSD method, as acknowledged in the introduction, is based on the computation of the slopes in the falling edge region, Figure 2. The slopes of the straight lines connecting two points--one upper threshold and one lower threshold for each line--are easy to compute, but it is less evident how to properly select these threshold points for them to maximize the discrimination capabilities of the neutron or gamma process taking place. Since the decay observed is not linear but exponential, the approach proposed in the present paper is the robust determination of the exponential decay shape after the maximum amplitude for each pulse, thus encapsulating the ideas in the two former methods. That is, it is assumed that the relationship between amplitude \(A\) and time \(t\) after the maximum amplitude of the pulse (decaying part) can be modelized as  
\[\mathbf{A}(t)=\mathbf{A}_{max}e^{-\Delta t/\tau}+\delta\mathbf{A} \tag{2}\]  
Figure 1: Values of the total integrated charge \(Q\) and the slow integrated \(Q_{s}\) for the charge comparison method. The ratio \(Q_{s}/Q\) is the main criteria of the method to classify pulses as neutron or gamma interactions with the scintillator. The ranges of integration (here, from 40 to 220 ns) have been chosen to maximize discrimination capabilities. In the figure, as in the rest of the article, pulse amplitudes are unitless after having been normalized to the range [0, 1]. where \(A_{max}\) is the maximum amplitude of the pulse, which is achieved at time \(t_{0}\) (the starting time for the analysis), \(\Delta t\) is the difference between time \(t\) and the initial time for the analysis \(t_{0}\) (that is, the time increment \(t-t_{0}\)), \(\mathbf{\tau}\) is the characteristic decay time, and \(\delta A\) in the residual amplitude. The latter term, amplitude \(\delta A\), seems pertinent to introduce, first, because experimentally the decay does not seem to decay to zero, and, second, because neutron pulses have typically thicker tails that can be explained by the physical superposition to the main effect (prompt fluorescence) of a delayed emission which \"originates from the collisional interaction of pairs of molecules (or excitons) in the lowest excited \(\pi\)-triplet states\" [10]. The inclusion of this term effectively increases the power of discrimination, as will be shown in the numerical examples. \(A_{max}\) and its corresponding time \(t_{0}\) are easy to determine by inspection of the pulse data. However, to obtain the unknowns \(\tau\) and \(\delta A\), there is a non-linear relationship, Equation (2). For all the data points in every pulse, it is possible to compose a system of equations of the form  
\[\mathbf{Ax}=\mathbf{k}+\mathbf{r} \tag{3}\]  
where the \(A\) matrix (in bold, not to be confused with a particular amplitude \(A\)) and the independent term vector \(k\) have the forms  
\[\mathbf{A}= \begin{pmatrix}A_{max}\frac{\Delta t_{1}}{\tau_{approx}}e^{-\Delta t/ \tau_{approx}}&\mathbf{1}\\ A_{max}\frac{\Delta t_{2}}{\tau_{approx}}e^{-\Delta t_{2}/\tau_{approx}}&\mathbf{1 }\\ \dots&\dots\\ A_{max}\frac{\Delta t_{n}}{\tau_{approx}}e^{-\Delta t_{n}/\tau_{approx}}&\mathbf{1 }\\ \end{pmatrix}  \tag{4}\]  
\[\mathbf{k}= \begin{pmatrix}A_{1}-A_{max}e^{-\Delta t/\tau_{approx}}\\ A_{2}-A_{max}e^{-\Delta t_{2}/\tau_{approx}}\\ \dots\\ A_{n}-A_{max}e^{-\Delta t_{n}/\tau_{approx}}\\ \end{pmatrix}  \tag{5}\]  
Figure 2: Scheme of the falling edge slope method: an upper threshold is selected, as well as a lower threshold for each pulse (as indicated on the picture). The slope is defined as the straight line between the two points.