# Neutron and Gamma Pulse Shape Discrimination by Robust Determination of the Decay Shape
## 3 Results and Discussion  
The determination of \(\tau\) (characteristic decay time) and \(\delta A\) (residual amplitude) for each pulse, in accordance with Equation (2), will be used for discrimination between neutron and gamma pulses. As a reference, the FOM value that is obtained with the charge comparison method for the same dataset (raw values) is taken. In [11], it is said that the FOM value for the application of the charge comparison method to this dataset is 1.3285, but a close inspection of the calculation of this value has revealed that a factor of ln(2) was missing in the denominator of their formula for FOM computation (QCSCM.m file); that is, 1.667 was used instead of 2.35 in Equation (15), so that the correct FOM value for the charge comparison method applied to this dataset is 0.9405. First, the results obtained in each pulse for \(\tau\) (in ns) and \(\delta A\) (unitless) in Equation (2) after their determination by standard least squares adjustments are presented. As can be seen in Figure 4, the values of \(\tau\) alone obtained by least squares do not permit the separation of the neutron from gamma pulses: the corresponding FOM value is 0.4064. Figure 4: Discrimination by tau (\(\tau\)) values obtained by least squares: the resulting FOM of 0.4064, associated by the visual fact that there is no clear distinction between gamma and neutron lobes, indicate a poor discrimination capability of the method. \(\tau\) values in ns. Figure 3: Scheme of FOM definition: the distance S between gamma and neutron contributions is evaluated, as the full widths at half maximum for each gamma and neutron lobes, respectively. The \(\tau\) values obtained by robust estimation allow for a better discrimination, Figure 5, but are still insufficient for a good separation; the FOM value is 0.6128. If the values of \(\delta A\) alone obtained by least squares and robust estimation are analyzed, Figures 6 and 7 are obtained, respectively, with corresponding FOM values of 0.9521 and 0.9525, which are very similar, and only slightly better than the value of 0.9405 obtained by the charge comparison method. The fact that higher values for \(\tau\) are obtained in the case of neutron pulses (values around 13 ns) than for gamma pulses (values around 11 ns) and, at the same time, higher values for \(\delta A\) are obtained in the case of neutron pulses (values around 0.044) than for gamma pulses (values around 0.024)--which can be explained by the above-mentioned contribution of the delayed emission--may suggest that the estimator obtained by multiplying the two, that is \(\tau\delta A\), will provide a much better discrimination. In any case, having a look at the variances and covariances in the covariance matrix, it was observed that the correlation coefficient between \(\tau\) and \(\delta A\) has an average (very stable) value of 0.4688. Thus, the variables are moderately correlated and it can be hoped that the discrimination can indeed be slightly enhanced by the combination of the two variables. The quantity \(\tau\delta A\) obtained for each pulse after the estimation of \(\tau\) and \(\delta A\) by least squares (Figure 8) and robust estimation (Figure 9) provides in effect, a much better discrimination between gamma  
Figure 5: Discrimination by tau (\(\tau\)) values obtained by robust estimation: a slightly better discrimination ability (FOM value 0.6128) than in the previous case is achieved. \(\tau\) values in ns. Figure 6: Discrimination by deltaA (\(\delta A\)) values obtained by least squares. There is an evident separation between gamma and neutron contributions (the FOM value is 0.9521). \(\delta A\) values are unitless. (lower values) and neutron (higher values) pulses. Their corresponding FOM values are 0.9265 and 1.0356, for the cases of least squares and robust estimation, respectively.