A study on fast digital discrimination of neutron and gamma-ray for improvement neutron emission profile measurement

Abstract

Neutron and γ-ray (n-γ) discrimination with a digital signal processing system has been used to measure the neutron emission profile in magnetic confinement fusion devices. However, a sampling rate must be set low to extend the measurement time because the memory storage is limited. Time jitter decreases a discrimination quality due to a low sampling rate. As described in this paper, a new charge comparison method was developed. Furthermore, automatic n-γ discrimination method was examined using a probabilistic approach. Analysis results were investigated using the figure of merit. Results show that the discrimination quality was improved. Automatic discrimination was applied using the EM algorithm and k-means algorithm.

I. INTRODUCTION

Neutron measurement is an important diagnostic task for the control of burning plasmas. A neutron emission profile measurement is extremely important for fusion reactor development because it monitors time behavior and space behavior of neutron emissions, which is useful for plasma transport study of energetic ions.¹ In such measurements, neutron and γ-ray (n-γ) discrimination has been conducted for signals from an organic scintillation detector because it generally has sensitivity not only to neutrons but also to γ-rays, which are a unfavorable noise source.^2–4 Recently, a digital signal processing (DSP) system equipped with a fast flash analog-to-digital converter (Flash ADC) has been used for such measurements.^3–9 The output waveform from an anode of a photomultiplier tube (PMT) is recorded using a Flash ADC and is analyzed off-line. The DSP system is applicable for measurement under high-counting-rate conditions in which analog processing is impossible to use.^1,8,10,11 A charge comparison method was applied to pulse signals from the scintillation detector, where neutrons and γ-rays are discriminated through comparison of pulse shape attenuation.^3–8,11 In the conventional pulse comparison method, each pulse signal is integrated in two time intervals. Using the results, a two-dimensional (2D) map was drawn.¹¹ However, to increase the total measuring time, the sampling rate of the Flash ADC must be set low because the memory installed on the DSP system is limited. A decrease of the sampling rate increases the time jitter, which results in the broadening of the distribution in the 2D map. As described in this paper, a new charge comparison method was developed to improve n-γ discrimination quality, even at a low sampling rate. Furthermore, the possibility is examined of automatic n-γ discrimination method using a probabilistic approach.

II. DIGITAL SIGNAL PROCESSING SYSTEM

In this study, the desktop waveform digitizer (CAEN; DT5751) used for this study has a 10-bit vertical resolution and a sampling rate of 1G samples/s. Experiments with a ²⁵²Cf neutron source were conducted, where an NE213 liquid scintillation detector coupled to a PMT (R878; Hamamatsu Photonics KK) was used. Figure 1 presents an example of a pulse waveform from the scintillation detector measured using the DSP system. For the pulse produced by a neutron, the attenuation time is longer than that by γ-rays. In the conventional method, the 2D map was generated using the ratio of integral values designated as Q_fast, Q_slow, and Q_total as presented in Fig. 1.¹¹ Figures 2(a) and 2(b) respectively portray examples of the 2D map measured at a sampling rate of 1G samples/s and 200 M samples/s. The result of 200 M samples/s was generated using the data of 1G samples/s. The distribution in Fig. 2(a) is broader than that in Fig. 2(b). This broadening stems from the increase in the time jitter due to the lower sampling rate. This means that the degradation of the discrimination quality occurred in the case of 200 M samples/s. To resolve this problem, the rise-time region of each pulse designated as Q_rise was introduced into analyses to reduce the influence of time jitter in detecting the pulse peak. In addition, a new region was set, Q_middle, which covers the boundary of fast and slow regions to reduce the fluctuation of the boundary line between Q_fast and Q_slow regions. Q_middle region was neglected to exclude the effects of the time jitter. Figure 3 presents an example of a 2D map with the new charge comparison method. In Fig. 3, each segment corresponding to neutron and γ-ray exists in a narrow region. It has been demonstrated that the time jitter influence was reduced as compared to the conventional procedure.

FIG. 1.

Normalized pulse waveform from the NE213 scintillation detector. The sampling rate was set 200 M samples/s.

FIG. 2.

2D map generated using the conventional charge comparison method. (a) 1 Gsamples/s. (b) 200 Msamples/s.

FIG. 3.

2D map generated using the new charge comparison method.

III. STATISTICAL APPROACH FOR AUTOMATIC DISCRIMINATION

Conventionally, linear discriminant functions have been used to perform n-γ discrimination automatically on 2D maps.¹¹ However, categorization of data by a discrete line is not the optimum method when data on a 2D map comprise the overlapped region of neutron and γ-ray distributions. In this study, a statistical approach for n-γ discrimination on the 2D map was examined. Parameters of probabilistic distributions are calculated on the assumption that each datum x = (x, y) follows a Gaussian Mixture Model (GMM). GMM is defined as

$$P\left(\mathbf{X}|\mathit{\Theta }\right)=\sum _{k=1}^K{\pi }_k\mathrm{N}\left(\mathbf{X}|{\mathit{\theta }}_k\right),$$ (1)

where N(X|θ_k) is the kth normalized Gaussian function constructed by the grand data groups X ≡ {x_t|t = 1, …, N} and the model parameter of the Gaussian function θ_k = {Σ_k, μ_k, ∑_k}. π, $\mathit{\mu }$, and Σ respectively represent the mixing coefficient, average of a two-dimensional vector, and covariance matrix of a (2 × 2)-dimensional matrix. Θ ≡ {θ_k|k = 1, …K} denotes the parameters of the grand Gaussian function. K represents the number of clusters. In this case, K is set to 2 because the data on the 2D map are divided into the neutron and γ-ray groups. A stochastic model can be determined using the maximum likelihood estimation method. We adapted the expectation-maximization (EM) algorithm for that purpose. The EM algorithm repeats the following E-step and M-step.¹²

• •E-step
  Using parameter θ, the posterior probability of the kth Gaussian function calculated for each datum x is calculated

  $$P\left(k|{\mathbf{x}}_n,{\mathit{\theta }}_k\right)=\frac{P\left({\mathbf{x}}_n,k|{\mathit{\theta }}_k\right)}{\sum _{k=1}^{K}P\left({\mathbf{x}}_n,k|{\mathit{\theta }}_k\right)}.$$ (2)
• •M-step
  The log-likelihood ln P(X|Θ) is maximized for data groups on the 2D map. For maximizing the log-likelihood, ∂ln P(X|Θ) / ∂Θ = 0 is calculable as shown below

  $${\mathit{\mu }}_k=\frac{1}{N_k}\sum _{n=1}^{N}P\left(k|{\mathbf{x}}_n,{\mathit{\theta }}_k\right){\mathbf{x}}_n,$$ (3)

  $${\sum }_k=\frac{1}{N_k}\sum _{n=1}^{N}P\left(k|{\mathbf{x}}_n,{\mathit{\theta }}_k\right)\left({\mathbf{x}}_n-{\mathit{\mu }}_k\right){\left({\mathbf{x}}_n-{\mathit{\mu }}_k\right)}^T,$$ (4)

  $${\pi }_k=\frac{N_k}{N},$$ (5)

  $$N_k=\sum _{n=1}^{N}P\left(k|{\mathbf{x}}_n,{\mathit{\theta }}_k\right).$$ (6)

Therein, (·)^T denotes a transpose. In the calculation, the iteration was stopped when the difference between the old and the updated log-likelihood, Δln P(X|Θ), is less than 1 × 10⁻¹². Figure 4 presents an example of the relationship between the log-likelihood and the iteration. Although the log-likelihood value saturated with the iteration number of more than 20, the iteration was continued until it becomes less than 1 × 10⁻¹² for taking sufficient margin. The EM algorithm is a sophisticated method, but it is known to have the shortcoming that setting of the initial values strongly affects its results. In the present study, the k-means algorithm was applied to set those for automatic discrimination. Discrimination quality was evaluated through the Figure of Merit (FOM) between conventional and new charge comparison methods. Figure 5 presents the definition of the FOM, where the red solid line shows the contour line of standard deviation 1σ of GMM, and D stands for the distance between two points that are the average values of GMM. The values of σ_neutron and σ_γ-ray are the distances between the two intersection points between the contour and the straight line connecting the mean values of the two GMMs. The higher FOM presents the higher discrimination ability. Table I shows the FOMs for the two charge comparison methods at two different sampling rates. As a result, FOM was improved from 0.642 with the conventional charge comparison procedure to 0.699 with the new one. The results demonstrated the effectiveness of the new charge comparison method in low sampling rate experiments. Figure 6 presents an example of data measured in the JT-60U tokamak. Because the mapped result of the γ-ray component curves under a high counting rate condition, it can be a cause of misclassification.¹³ To evaluate the misclassification, the posterior probabilities to belong to the neutron clusters and γ-clusters were calculated for the γ-ray data in the solid circle in Fig. 6. The probability of misclassification as neutrons was reduced by 8% compared to the conventional procedure.

FIG. 4.

Relationship between the log-likelihood and iteration.

FIG. 5.

Method for FOM evaluation.

Table I.

The FOMs for the two charge comparison methods at two different sampling rates.

	Conventional method	New method
FOM (1G samples)	0.714	0.725
FOM (200M samples)	0.642	0.699

FIG. 6.

Curve of γ-ray component causing misclassification.

IV. SUMMARY

A new charge comparison method has been applied to reduce the influence of time jitter in n-γ pulse shape discrimination using a DSP system. In addition, the probability of automatic n-γ discrimination was examined using the k-means algorithm and EM algorithm in 2D map. Discrimination quality using Gaussian parameters was evaluated using the conventional charge comparison method and the new one. Results show that the discrimination quality has been improved from 0.642 to 0.699. By applying this algorithm, the probability of misclassification has been reduced by 8% even for the data at high counting rate condition. It is necessary to determine a suitable stochastic model when applying this method to actual neutron emission profile measurements.