Performance of real-time neutron/gamma discrimination methods

Performance of real-time neutron/gamma discrimination methods 1 corresp first-author liusx@hfut.edu.cn liusx@hfut.edu.cn liusx@hfut.edu.cn https://orcid.org/0000-0002-9906-1780 2 corresp * zhangweid@cnpe.cc, * zhangweid@cnpe.cc, zhangweid@cnpe.cc 1 2 3 corresp caohr@ipp.ac.cn, caohr@ipp.ac.cn, caohr@ipp.ac.cn https://orcid.org/0000-0001-8301-4085 2 3 3 https://orcid.org/0000-0002-6235-933X School of Electronics, Hefei University of Technology, Hefei 230009, China School of Electronics, Hefei University of Technology, Hefei 230009, China China Nuclear Power Engineering Co., Ltd., Beijing 100840, China China Nuclear Power Engineering Co., Ltd., Beijing 100840, China Institute of Plasma Physics, Hefei Institutes of Physical Science, Chinese Academy of Sciences, Hefei 230031, China Institute of Plasma Physics, Hefei Institutes of Physical Science, Chinese Academy of Sciences, Hefei 230031, China Nuclear security usually requires the simultaneous detection of neutrons and gamma rays. With the development of crystalline materials in recent years, Cs 2 LiLaBr 6 (CLLB) dual-readout detectors have attracted extensive attention from researchers, where real-time neutron/gamma pulse discrimination is the critical factor among detector performance parameters. This study investigated the discrimination performance of the charge comparison, amplitude comparison, time comparison, and pulse gradient methods and the effects of a Sallen–Key filter on their performance. Experimental results show that the figure of merit (FOM) of all four methods is improved by proper filtering. Among them, the charge comparison method exhibits excellent noise resistance; moreover, it is the most suitable method of real-time discrimination for CLLB detectors. However, its discrimination performance depends on the parameters t s , t m , and t e . When t s corresponds to the moment at which the pulse is at 10% of its peak value, t e requires a delay of only 640 to 740 ns compared to t s , at which time the potentially optimal FOM of the charge comparison method at 3.1–3.3 MeV is greater than 1.46. The FOM obtained using the t m value calculated by a proposed maximized discrimination difference model(MDDM) and the potentially optimal FOM differ by less than 3.9%, indicating that the model can provide good guidance for parameter selection in the charge comparison method. Charge comparison Maximized discrimination difference model Pulse filtering Real-time n-γ discrimination Introduction 1 Neutron and gamma detection is an essential technique for many nuclear security applications, such as isotope identification [ 1 1 , 2 2 ] , radiation monitoring [ 3 3 - 5 5 3-5 ] , and the search for fissile material [ 6 6 , 7 7 ] . The detection of neutrons and gamma rays usually requires a combination of both types of radiation detectors, which must be portable and compact in many detection scenarios. In recent years, a new Cs 2 LiLaBr 6 (CLLB) detector in the elpasolite scintillator family has been developed [ 8 8 ] . It has the advantages of small size, high light output, and excellent energy resolution; it enables the detection of two types of radiation using pulse shape discrimination technology, making handheld instruments possible [ 9 9 - 12 12 9-12 ] . Since the 1960s, researchers have found that scintillators produce different pulse shapes when they interact with different particles [ 13 13 ] , and many methods of discriminating the waveform characteristics have been developed. These methods are divided into three categories: time-domain, frequency-domain, and intelligent methods. Time-domain methods include charge comparison [ 14 14 ] (CC), time comparison (TC), amplitude comparison (AC), and pulse gradient (PG) methods [ 15 15 ] . Frequency-domain methods include frequency gradient [ 16 16 ] , wavelet transform [ 17 17 ] , and fractal spectrum [ 18 18 ] methods. Intelligent methods include multilayer perceptron [ 19 19 ] , support vector machine [ 20 20 , 21 21 ] , and deep learning network [ 22 22 ] approaches. Time-domain methods are simple to calculate but usually have a low figure of merit (FOM). Frequency-domain methods involve many Fourier calculations that are complex and difficult to run in real-time; however, they are less sensitive to noise than time-domain methods. Intelligent methods usually have the highest FOM but require many floating-point matrix operations and thus a complicated computational process. In addition, the discrimination performance of intelligent methods usually depends on the training set size and number of neural network parameters. It is almost impossible to complete the computations required by frequency-domain and intelligent methods using limited computational and storage resources. In addition, as the sample points of the waveform increase, the computational complexity of frequency-domain and intelligent methods can rarely maintain linear growth. Therefore, time-domain methods are still preferred for real-time analysis. Consequently, it is necessary to study suitable discrimination methods and preprocessing techniques to identify methods with a good FOM and some noise immunity that are suitable for real-time analysis using modest computational resources. Zuo et al. [ 23 23 ] explored the discrimination effect of several time-domain methods on plastic scintillator detectors and found that filtering could enhance the discrimination performance of these methods. However, the applicability of this finding to CLLB detectors is unknown. In addition, time-domain methods are all sensitive to their parameters. To optimize the discrimination performance of these methods, one must continuously search for the appropriate parameters by trial and error, which is very laborious. To solve these problems, we examine the discrimination effects of the CC, TC, AC, and PG time-domain methods on CLLB detectors and the effects of preprocessing by a Sallen–Key (SK) filter on the methods in real time. A new SK recursive expression based on the second-order Runge–Kutta method is proposed to reduce the error and applied in an experiment. The experimental results show that appropriate filtering can enhance the discrimination performance of these time-domain methods, where the CC method is the most suitable real-time analysis method for CLLB detectors. However, the performance of the CC method is affected by the parameter settings. We discuss the effects of the CC parameters on its discrimination performance and propose a maximized discrimination difference model (MDDM) to guide parameter selection in the CC method. This tool can be used to model the parameter settings of other time-domain methods and guide parameter selection. The remainder of the paper is organized as follows. The experimental setup for the work is described in Sect. 2. The basic principles of the numerical methods used are described in Sect. 3. The discrimination effects of the methods and the effects of filtering and the MDDM on their performance are discussed in Sect. 4. Conclusions are presented in Sect. 5. Neutron/gamma pulse discrimination experiment 1 Detector and radioactive source 2 A CLLB crystal detector from Saint-Gobain [ 24 24 ] was used; the scintillator crystal has a diameter of 2 in. and is sealed inside an aluminum housing. It was coupled directly to a photomultiplier tube (PMT) (Hamamatsu R6231-100), as shown in Fig. 1 Fig. 1 . An ORTEC 556 high-voltage power supply was used during the experiment to supply a voltage of 800 V to the detector. A 137 Cs gamma source was used as the energy calibration source for the detector, and a 252 Cf source was selected to provide the neutron/gamma hybrid radiation field. Data acquisition system 2 A data acquisition system developed by our research group was used for the experiment; it includes a PXIe chassis with an embedded controller and an electronics card. The signal input of the main electronics card was directly connected to the CLLB detector. The main electronics card consists mainly of analog signal conditioning components, high-speed analog-to-digital converters (ADCs) (14 bits and 500 Msps), a clock jitter cleaner, and a field-programmable gate array (FPGA) (type xc7k325tffg900). As shown in Fig. 4 Fig. 4 , the rise time of the electrical signal of the CLLB detector is <math id="M1"><mrow><msub><mi>t</mi><mi>r</mi></msub><mo>≥</mo><mn>30</mn><mtext> ns</mtext></mrow></math> . If the signal is processed directly by a 500 Msps ADC, there are no fewer than 15 sampling points on the rising edge. A signal with a rising edge <math id="M2"><mrow><msub><mi>t</mi><mi>r</mi></msub><mo>≥</mo><mn>30</mn><mtext> ns</mtext></mrow></math> corresponds to the frequency spectrum of 16.7 MHz, which is much lower than the sampling rate of the waveform digitization module for 500 Msps ADCs. The hardware of the main electronics card is almost identical to that of the main electronics card for neutron flux monitoring in the Experimental Advanced Superconducting Tokamak [ 25 25 ] , except that the gain of the analog signal conditioning component is 8 V/V to accommodate the signal amplitude of the CLLB detector and PMT output ( < 100 mV), and the FPGA code is different. Data processed by the FPGA algorithm are transmitted to the chassis-embedded controller through the PXIe bus in direct memory access mode for display or storage [ 26 26 ] . The pulse acquisition mode of the FPGA is shown in Fig. 2 Fig. 2 . <math id="M3"><mrow><mo> </mo><msub><mi>L</mi><mi>r</mi></msub></mrow></math> is the length of the entire recorded pulse; <math id="M4"><mrow><msub><mi>L</mi><mi>b</mi></msub></mrow></math> is the length of the sliding average of the baseline calculation; <math id="M5"><mrow><msub><mi>L</mi><mi>o</mi></msub></mrow></math> is the offset of the pulse trigger, which determines the starting point of the baseline calculation; and <math id="M6"><mrow><msub><mi>T</mi><mi>g</mi></msub></mrow></math> is the moment at which the pulse is triggered. In this experiment, the average value of 48–64 ns sampling points before pulse triggering was defined as the baseline of the pulse, and the total recording time of the pulse was 1008 ns. Pulse preprocessing 2 First, the detector was energy-calibrated using a 137 Cs gamma source. Then, a pulse waveform with an equivalent gamma energy of 3.1–3.3 MeV (the energy band containing the thermal neutron peak) was extracted from the 252 Cf source; therefore, we investigated the neutron/gamma discrimination performance in this energy band. A total of 33,782 waveforms were measured. Pulse width and amplitude analysis was used to remove severely overlapped or chopped waveforms; 33,246 pulse waveforms remained after this process. Finally, each pulse baseline value was subtracted from the sampled pulse sample data. Principle of numerical method 1 SK filter 2 In 1955, the SK filter based on discrete components was first proposed by R.P. Sallen and E.L. Key, and the Gaussian shaping of pulse signals was successfully realized. SK filters are widely used for signal filtering and pulse shaping, where they outperform other methods in terms of energy resolution and computational workload [ 27 27 ] . The SK filter has a second-order filter circuit with a simple structure and is widely used in nuclear signal pulse shaping [ 28 28 , 29 29 ] . Fig. 3 Fig. 3 shows the SK low-pass filter. When R 1 = R 2 = R = 3.99 kΩ and C 1 = C 2 = C = 1.5 pF, the corresponding bandwidth is 16.7 MHz. Real-time discrimination method 2 The CC method uses the different integration values of the neutron and gamma pulse tails as the basis for discrimination. The principle is illustrated in Fig. 4a Fig. 4a ; the discrimination is calculated using Eq. (1): <math display="block" id="M7"><mrow><msub><mi>P</mi><mrow><mtext>CI</mtext></mrow></msub><mo>=</mo><mfrac><mrow><mi>Q</mi><mo>−</mo><msub><mi>Q</mi><mtext>f</mtext></msub></mrow><mi>Q</mi></mfrac><mo>=</mo><mfrac><mrow><msub><mi>Q</mi><mtext>s</mtext></msub></mrow><mrow><msub><mi>Q</mi><mtext>f</mtext></msub><mo>+</mo><msub><mi>Q</mi><mtext>s</mtext></msub></mrow></mfrac></mrow></math> where <math id="M8"><mi>Q</mi></math> is the total pulse integral, and Q f and Q s represent the pulse front and tail integrals, respectively. The range of the integration intervals of Q f is determined by t s and t m , and the integration interval of <math id="M9"><mi>Q</mi></math> is determined by t s and t e . If the interval between t s and t m is set too large, it will cause a loss of the difference between the falling portions of the neutron and gamma pulse signals in Eq. (1), where P CI thus tends to 0. By contrast, if the interval between t s and t m is set too small, the information on useless pulses in Eq. (1) will increase, causing P CI to tend to 1. Therefore, the integration interval needs to be selected according to the pulse waveform of the detector response in practical applications. In the CLLB detector, the neutron pulse will have smaller P CI values than the gamma pulse because neutrons decay faster than gammas, unlike the typical case of liquid scintillators. The principle of the AC method is shown in Fig. 4b Fig. 4b , where t c after the peak is selected as the moment of amplitude comparison. The different amplitudes of neutrons and gammas at this moment are used as the basis for discrimination, where <math id="M10"><mrow><msub><mi>A</mi><mtext>γ</mtext></msub></mrow></math> and <math id="M11"><mrow><msub><mi>A</mi><mtext>n</mtext></msub></mrow></math> represent the amplitude of the gamma and neutron pulses, respectively, at the <math id="M12"><mrow><msub><mi>t</mi><mtext>c</mtext></msub></mrow></math> moment. This method requires the calculation of only one sampling point but is very susceptible to the effects of noise. Therefore, the amplitudes were normalized before comparison with the other methods. The principle of the TC method is shown in Fig. 4c Fig. 4c . It can be considered as the inverse function of the AC method. A fixed A c threshold line is chosen, and the difference in the time it takes for neutron and gamma pulses to decay from their peaks to the threshold line is used as the basis for discrimination. <math id="M13"><mrow><msub><mi>T</mi><mtext>γ</mtext></msub></mrow></math> and <math id="M14"><mrow><msub><mi>T</mi><mtext>n</mtext></msub></mrow></math> represent the time from the intersection of the decaying portion of the gamma and neutron pulse, respectively, with the threshold line A c to the peak. The PG method selects peak and post-peak sampling points for gradient calculation; the principle is illustrated in Fig. 4d Fig. 4d . It can be written as Eq. (2): <math display="block" id="M15"><mrow><msub><mi>P</mi><mtext>G</mtext></msub><mo>=</mo><mfrac><mrow><msub><mi>A</mi><mtext>p</mtext></msub><mo>−</mo><msub><mi>A</mi><mtext>b</mtext></msub></mrow><mrow><msub><mi>t</mi><mtext>p</mtext></msub><mo>−</mo><msub><mi>t</mi><mtext>b</mtext></msub></mrow></mfrac></mrow></math> where <math id="M16"><mrow><msub><mi>P</mi><mtext>G</mtext></msub></mrow></math> is the pulse gradient, <math id="M17"><mrow><msub><mi>A</mi><mtext>p</mtext></msub></mrow></math> is the pulse peak amplitude, and <math id="M18"><mrow><mo> </mo><msub><mi>A</mi><mtext>b</mtext></msub></mrow></math> is the amplitude at a specific time interval after the pulse peak. In addition, <math id="M19"><mrow><msub><mi>t</mi><mtext>p</mtext></msub></mrow></math> and <math id="M20"><mrow><msub><mi>t</mi><mtext>b</mtext></msub></mrow></math> are the corresponding moments of the pulse peak and post-peak sampling points, respectively. Similarly, the gradient is susceptible to strong amplitude effects and is calculated using normalized amplitudes. Evaluation criteria 2 The FOM was introduced as an evaluation criterion to objectively assess the performance of the discrimination method, as shown in Fig. 5 Fig. 5 . The FOM is defined [ 30 30 ] as shown in Eq. (3): <math display="block" id="M21"><mrow><mtext>FOM</mtext><mo>=</mo><mfrac><mi>D</mi><mrow><msub><mrow><mtext>FWHM</mtext></mrow><mtext>n</mtext></msub><mo>+</mo><msub><mrow><mtext>FWHM</mtext></mrow><mi>γ</mi></msub></mrow></mfrac></mrow></math> <math id="M22"><mrow><msub><mrow><mtext>FWHM</mtext></mrow><mi>n</mi></msub></mrow></math> and <math id="M23"><mrow><msub><mrow><mtext>FWHM</mtext></mrow><mi>γ</mi></msub></mrow></math> are the half-height widths of the neutron and gamma peaks, respectively, and D is the distance between the neutron and gamma peaks. A larger FOM indicates that the method exhibits better neutron/gamma discrimination. MDDM 2 Among the four methods described above, only the CC method uses all the waveform information. Therefore, it has better noise immunity. The discrimination performance of the CC method depends on three parameters, t s , t m , and t e . The values of t s and t e are usually taken as the moments of the upper edge of the pulse and the end of pulse decay, respectively, and the parameter t m is the most difficult to determine. We propose a model that maximizes the difference used for discrimination to determine the parameter t m and achieve excellent performance by the CC method. When neutrons and gamma rays deposit energy in the CLLB detector, the photon pulse generated by the scintillation crystal can be written as shown in Eq. (4). <math display="block" id="M24"><mrow><mi>I</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><msub><mi>I</mi><mtext>f</mtext></msub><msup><mi>e</mi><mrow><mo>−</mo><mi>t</mi><mo>/</mo><msub><mi>τ</mi><mi>f</mi></msub></mrow></msup><mo>+</mo><msub><mi>I</mi><mtext>s</mtext></msub><msup><mi>e</mi><mrow><mo>−</mo><mi>t</mi><mo>/</mo><msub><mi>τ</mi><mi>s</mi></msub></mrow></msup></mrow></math> where <math id="M25"><mrow><msub><mi>τ</mi><mtext>f</mtext></msub></mrow></math> and <math id="M26"><mrow><msub><mi>τ</mi><mtext>s</mtext></msub></mrow></math> are the fast and slow components of the decay time constant, respectively. <math id="M27"><mrow><msub><mi>I</mi><mtext>f</mtext></msub></mrow></math> and <math id="M28"><mrow><msub><mi>I</mi><mtext>s</mtext></msub></mrow></math> represent the proportions of the fast and slow components, respectively. In the linear region, the pulse output from the CLLB detector can be represented by the convolution of the photon pulse with the response function of the PMT and readout electronics. However, the final expression of <math id="M29"><mrow><mi>V</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></math> still contains the factors <math id="M30"><mrow><msup><mi>e</mi><mrow><mo>−</mo><mi>t</mi><mo>/</mo><msub><mi>τ</mi><mtext>f</mtext></msub></mrow></msup></mrow></math> and <math id="M31"><mrow><msup><mi>e</mi><mrow><mo>−</mo><mi>t</mi><mo>/</mo><msub><mi>τ</mi><mtext>s</mtext></msub></mrow></msup></mrow></math> [ 31 31 ] . Because the pulse rises rapidly, this part of the output can be replaced by a linear function; thus, the gamma and neutron pulse response described above can be written as shown in Eqs. (5) and (6), respectively. <math display="block" id="M32"><mrow><msub><mi>V</mi><mrow><mtext>CLLB</mtext><mi>γ</mi></mrow></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>{</mo><mrow><mi>A</mi><mo stretchy="false">(</mo><msub><mi>I</mi><mrow><mtext>f</mtext><mi>γ</mi></mrow></msub><msup><mi>e</mi><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mi>t</mi><mo>−</mo><msub><mi>t</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow><mo>/</mo><msub><mi>τ</mi><mi>f</mi></msub></mrow></msup></mrow></mrow><mover><mrow><mo>+</mo><msub><mi>I</mi><mrow><mtext>s</mtext><mi>γ</mi></mrow></msub><msup><mi>e</mi><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mi>t</mi><mo>−</mo><msub><mi>t</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow><mo>/</mo><msub><mi>τ</mi><mtext>s</mtext></msub></mrow></msup><mo stretchy="false">)</mo></mrow><mrow><msub><mi>K</mi><mi>γ</mi></msub><mi>t</mi><mo>,</mo><mo> </mo><mo> </mo><mi>t</mi><mo><</mo><msub><mi>t</mi><mn>1</mn></msub></mrow></mover><mo>,</mo><mtext> </mtext><mi>t</mi><mo>≥</mo><msub><mi>t</mi><mn>1</mn></msub></mrow></math> <math display="block" id="M33"><mrow><msub><mi>V</mi><mrow><mtext>CLLBn</mtext></mrow></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>{</mo><mrow><mi>A</mi><mo stretchy="false">(</mo><msub><mi>I</mi><mrow><mtext>fn</mtext></mrow></msub><msup><mi>e</mi><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mi>t</mi><mo>−</mo><msub><mi>t</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow><mo>/</mo><msub><mi>τ</mi><mtext>f</mtext></msub></mrow></msup><mover><mrow><mo>+</mo><msub><mi>I</mi><mrow><mtext>sn</mtext></mrow></msub><msup><mi>e</mi><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mi>t</mi><mo>−</mo><msub><mi>t</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow><mo>/</mo><msub><mi>τ</mi><mtext>s</mtext></msub></mrow></msup><mo stretchy="false">)</mo></mrow><mrow><msub><mi>K</mi><mtext>n</mtext></msub><mi>t</mi><mo>,</mo><mtext>​</mtext><mtext> </mtext><mi>t</mi><mo><</mo><msub><mi>t</mi><mn>1</mn></msub></mrow></mover></mrow></mrow><mo>,</mo><mo> </mo><mo> </mo><mi>t</mi><mo>≥</mo><msub><mi>t</mi><mn>1</mn></msub></mrow></math> <math display="block" id="M34"><mrow><msub><mi>K</mi><mi>γ</mi></msub><mo>=</mo><msub><mi>K</mi><mtext>n</mtext></msub><mo>=</mo><mi>A</mi><mo>/</mo><msub><mi>t</mi><mn>1</mn></msub></mrow></math> <math id="M35"><mrow><msub><mi>I</mi><mrow><mtext>f</mtext><mi>γ</mi></mrow></msub></mrow></math> and <math id="M36"><mrow><msub><mi>I</mi><mrow><mtext>s</mtext><mi>γ</mi></mrow></msub></mrow></math> represent the fast and slow components, respectively, of the pulse response to gamma rays. <math id="M37"><mrow><msub><mi>I</mi><mrow><mtext>fn</mtext></mrow></msub></mrow></math> and <math id="M38"><mrow><msub><mi>I</mi><mrow><mtext>sn</mtext></mrow></msub></mrow></math> represent the fast and slow components, respectively, of the response to neutrons. <math id="M39"><mi>A</mi></math> is the response amplitude, which is proportional to the ray energy. In the actual detector impulse response, the pulse first rises and then decays, as shown in Fig. 6 Fig. 6 . The rise time of the CLLB response pulse to the peak is the same for rays of different energies. <math id="M40"><mrow><msub><mi>t</mi><mn>0</mn></msub></mrow></math> , <math id="M41"><mrow><msub><mi>t</mi><mn>1</mn></msub></mrow></math> . and <math id="M42"><mrow><msub><mi>t</mi><mn>2</mn></msub></mrow></math> represent the moments at which the pulse starts to rise, reaches its peak, and is completely decayed, respectively. S a represents the pulse integration area of the neutron and gamma pulses from <math id="M43"><mrow><mo> </mo><msub><mi>t</mi><mn>0</mn></msub></mrow></math> to <math id="M44"><mrow><msub><mi>t</mi><mn>1</mn></msub></mrow></math> . Neutrons and gammas of the same energy in the rising phase of the pulse can be considered to have the same integration area. S b represents the integration area of the neutron pulse from <math id="M45"><mrow><msub><mi>t</mi><mn>1</mn></msub></mrow></math> to <math id="M46"><mrow><msub><mi>t</mi><mn>2</mn></msub></mrow></math> . Because the slow component of the gamma response pulse is more significant in the decaying part of the pulse, S c represents the excess integration area of the gamma pulse from <math id="M47"><mrow><msub><mi>t</mi><mn>1</mn></msub></mrow></math> to <math id="M48"><mrow><msub><mi>t</mi><mn>2</mn></msub></mrow></math> compared to theeutron pulse. The <math id="M49"><mrow><msub><mi>P</mi><mrow><mtext>CI</mtext></mrow></msub></mrow></math> values of gammas and neutrons in the CC method are calculated as shown in Eqs. (8) and (9), respectively: <math display="block" id="M50"><mrow><msub><mi>P</mi><mrow><mtext>CI</mtext><mi>γ</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>S</mi><mi>b</mi></msub><mo>+</mo><msub><mi>S</mi><mi>c</mi></msub><mo>−</mo><msub><mi>S</mi><mrow><mi>γ</mi><mi>δ</mi></mrow></msub></mrow><mrow><msub><mi>S</mi><mi>a</mi></msub><mo>+</mo><msub><mi>S</mi><mi>b</mi></msub><mo>+</mo><msub><mi>S</mi><mi>c</mi></msub></mrow></mfrac></mrow></math> <math display="block" id="M51"><mrow><msub><mi>P</mi><mrow><mtext>CIn</mtext></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>S</mi><mi>b</mi></msub><mo>−</mo><msub><mi>S</mi><mrow><mtext>n</mtext><mi>δ</mi></mrow></msub></mrow><mrow><msub><mi>S</mi><mi>a</mi></msub><mo>+</mo><msub><mi>S</mi><mi>b</mi></msub></mrow></mfrac></mrow></math> The difference in <math id="M52"><mrow><msub><mi>P</mi><mrow><mtext>CI</mtext></mrow></msub></mrow></math> between neutrons and gammas is strongly correlated with <math id="M53"><mi>D</mi></math> in Eq. (3). Therefore, the FOM of the CC method is expected to be highest when <math id="M54"><mrow><mi>δ</mi><mo>=</mo><msub><mi>t</mi><mn>1</mn></msub><mo>+</mo><msub><mi>τ</mi><mtext>m</mtext></msub></mrow></math> reaches the moment at which the difference in <math id="M55"><mrow><msub><mi>P</mi><mrow><mtext>CI</mtext></mrow></msub></mrow></math> between neutrons and gammas is maximum. <math display="block" id="M56"><mrow><mi>D</mi><mrow><mo>(</mo><mi>δ</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><msub><mi>P</mi><mrow><mtext>CIγ</mtext></mrow></msub><mo>−</mo><msub><mi>P</mi><mrow><mtext>CIn</mtext></mrow></msub></mrow><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><msub><mi>S</mi><mrow><mtext>n</mtext><mi>δ</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>S</mi><mi>a</mi></msub><mo>+</mo><msub><mi>S</mi><mi>b</mi></msub><mo>+</mo><msub><mi>S</mi><mi>c</mi></msub><mo stretchy="false">)</mo><mo>−</mo><msub><mi>S</mi><mrow><mi>γ</mi><mi>δ</mi></mrow></msub><mrow><mo>(</mo><mrow><msub><mi>S</mi><mi>a</mi></msub><mo>+</mo><msub><mi>S</mi><mi>b</mi></msub></mrow><mo>)</mo></mrow><mo>+</mo><msub><mi>S</mi><mi>a</mi></msub><msub><mi>S</mi><mi>c</mi></msub></mrow><mrow><mo stretchy="false">(</mo><msub><mi>S</mi><mi>a</mi></msub><mo>+</mo><msub><mi>S</mi><mi>b</mi></msub><mo>+</mo><msub><mi>S</mi><mi>c</mi></msub><mo stretchy="false">)</mo><mrow><mo>(</mo><mrow><msub><mi>S</mi><mi>a</mi></msub><mo>+</mo><msub><mi>S</mi><mi>b</mi></msub></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></math> If <math id="M57"><mrow><msub><mi>I</mi><mrow><mtext>f</mtext><mi>γ</mi></mrow></msub><mo><</mo><msub><mi>I</mi><mrow><mtext>fn</mtext></mrow></msub></mrow></math> , <math id="M58"><mrow><msub><mi>I</mi><mrow><mtext>s</mtext><mi>γ</mi></mrow></msub><mo>></mo><msub><mi>I</mi><mrow><mtext>sn</mtext></mrow></msub></mrow></math> and <math id="M59"><mrow><mn>0</mn><mo><</mo><msub><mi>τ</mi><mtext>m</mtext></msub><mo><</mo><msub><mi>t</mi><mn>2</mn></msub><mo>−</mo><msub><mi>t</mi><mn>1</mn></msub></mrow></math> , <math id="M60"><mrow><msub><mi>τ</mi><mtext>m</mtext></msub></mrow></math> is the moment at which the difference between the <math id="M61"><mrow><msub><mi>P</mi><mrow><mtext>CI</mtext></mrow></msub></mrow></math> . values of gammas and neutrons is largest, as shown in Eq. (11). <math display="block" id="M62"><mrow><msub><mi>τ</mi><mtext>m</mtext></msub><mo>=</mo><mfrac><mrow><msub><mi>τ</mi><mtext>f</mtext></msub><msub><mi>τ</mi><mtext>s</mtext></msub></mrow><mrow><msub><mi>τ</mi><mtext>s</mtext></msub><mo>−</mo><msub><mi>τ</mi><mtext>f</mtext></msub></mrow></mfrac><mtext>ln</mtext><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>I</mi><mrow><mtext>fn</mtext></mrow></msub><mi>p</mi><mo>−</mo><msub><mi>I</mi><mrow><mtext>f</mtext><mi>γ</mi></mrow></msub><mi>q</mi></mrow><mrow><msub><mi>I</mi><mrow><mtext>sγ</mtext></mrow></msub><mi>q</mi><mo>−</mo><msub><mi>I</mi><mrow><mtext>sn</mtext></mrow></msub><mi>p</mi></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></math> <math display="block" id="M63"><mrow><mi>p</mi><mo>=</mo><msub><mi>S</mi><mi>a</mi></msub><mo>+</mo><msub><mi>S</mi><mi>b</mi></msub><mo>+</mo><msub><mi>S</mi><mi>c</mi></msub><mo>,</mo><mi>q</mi><mo>=</mo><msub><mi>S</mi><mi>a</mi></msub><mo>+</mo><msub><mi>S</mi><mi>b</mi></msub><mo>,</mo><mi>p</mi><mo><</mo><mi>q</mi></mrow></math> By calculating <math id="M64"><mrow><msub><mi>τ</mi><mtext>m</mtext></msub></mrow></math> , the parameters the CC method that will yield the best FOM can be obtained. Experimental results and discussion 1 Comparison of methods 2 The discrimination effects of the time-domain methods discussed above are all related to the parameter values. To objectively evaluate each method, the definition domain of the discrimination parameters of each method is first given. Then the best FOM using these parameters is obtained as the discrimination performance of the method by a trial-and-error approach. Let t s be 10% of the time required for the pulse to rise to its peak, t p be the moment of the pulse peak, and t e be 700 ns after t s . For the CC method, t m satisfies Eq. (13); for the AG method, t c satisfies Eq. (14); for the PG method, t b satisfies Eq. (15); and for the TC method, the A c range is taken from 0.1 to 1 peak, at intervals of 1% of the peak. <math display="block" id="M65"><mrow><msub><mi>t</mi><mtext>s</mtext></msub><mo><</mo><msub><mi>t</mi><mtext>m</mtext></msub><mo><</mo><msub><mi>t</mi><mtext>e</mtext></msub></mrow></math> <math display="block" id="M66"><mrow><msub><mi>t</mi><mtext>p</mtext></msub><mo><</mo><msub><mi>t</mi><mtext>c</mtext></msub><mo><</mo><msub><mi>t</mi><mtext>e</mtext></msub></mrow></math> <math display="block" id="M67"><mrow><msub><mi>t</mi><mtext>p</mtext></msub><mo><</mo><msub><mi>t</mi><mtext>b</mtext></msub><mo><</mo><msub><mi>t</mi><mtext>e</mtext></msub></mrow></math> The FOM of the methods without filtering is shown in Fig. 7 Fig. 7 . Only the CC method can distinguish between neutron and gamma pulses at the current noise level, where the parameter t m is taken as 212 ns after t s to obtain the best FOM for this method. As show in Fig. 3 Fig. 3 , the shaping effect of the SK low-pass filter is affected by the value of <math id="M68"><mrow><mi>R</mi><mi>C</mi></mrow></math> . With increasing <math id="M69"><mrow><mi>R</mi><mi>C</mi></mrow></math> , the cutoff frequency decreases, the attenuation of the stop-band increases, and the filtering effect is improved. The AC method is more strongly affected by noise, which is strongly correlated with the filtering effect. The overall FOM of the TC method is low; it is challenging to distinguish neutron and gamma signals correctly because the shaped pulse has a large oscillation with time. Therefore, the TC method is not suitable for neutron/gamma discrimination. The PG method is also more strongly affected by noise. The FOM of the PG method is strongly correlated with the filtering effect. In addition, the FOM of the PG method is similar to that of the AC method because of the amplitude difference in Eq. (3) in the calculation, which indicates that the amplitude difference is the dominant factor in discrimination by the PG method. Therefore, the CC method is the most suitable processing method for real-time neutron/gamma discrimination. Effect of M DDM 2 When the t s and t e parameters of the CC method are fixed as described in Sect. 4.1, we take <math id="M70"><mrow><msub><mi>t</mi><mtext>m</mtext></msub><mo>=</mo><mo>{</mo><mi>t</mi><mo>|</mo><mi>t</mi><mo>=</mo><mn>60</mn><mo>+</mo><mn>20</mn><mo>*</mo><mi>a</mi><mo>,</mo><mi>a</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>...</mn><mo>,</mo><mn>28</mn><mo>}</mo></mrow></math> . The FOM of the CC method for these values of t m is shown in Fig. 8 Fig. 8 . The FOM curve is convex, which indicates the best FOM can be obtained by considering the FOM as a function of the single parameter t m , and confirms the suitability of the MDDM proposed in Sect. 3.4. Let <math id="M71"><mrow><msub><mi>e</mi><mn>0</mn></msub><mo>–</mo><msub><mi>e</mi><mn>1</mn></msub></mrow></math> be the <math id="M72"><mrow><msub><mi>R</mi><mn>0</mn></msub></mrow></math> region, <math id="M73"><mrow><msub><mi>e</mi><mn>1</mn></msub><mo>–</mo><msub><mi>e</mi><mn>2</mn></msub></mrow></math> be the <math id="M74"><mrow><msub><mi>R</mi><mn>1</mn></msub></mrow></math> region, and <math id="M75"><mrow><msub><mi>e</mi><mn>2</mn></msub><mo>–</mo><msub><mtext>e</mtext><mn>3</mn></msub></mrow></math> be the <math id="M76"><mrow><msub><mi>R</mi><mn>2</mn></msub></mrow></math> region. The slope of the curves in the <math id="M77"><mrow><msub><mi>R</mi><mn>0</mn></msub></mrow></math> and <math id="M78"><mrow><msub><mi>R</mi><mn>1</mn></msub></mrow></math> regions is larger, indicating that the FOM improves more rapidly when t m is close to <math id="M79"><mrow><msub><mi>e</mi><mn>1</mn></msub></mrow></math> . By contrast, the FOM decreases rapidly when t m is close to <math id="M80"><mrow><msub><mi>e</mi><mn>3</mn></msub></mrow></math> . The slope of the curve in the <math id="M81"><mrow><msub><mi>R</mi><mn>2</mn></msub></mrow></math> region is smaller, indicating a slower decrease in the FOM in this region, although the range is large. Thus, the performance of the CC method is somewhat independent of the t m value. To solve Eq. (11) for t m , only <math id="M82"><mrow><msub><mi>I</mi><mrow><mtext>fn</mtext></mrow></msub></mrow></math> , <math id="M83"><mrow><msub><mi>I</mi><mrow><mtext>sn</mtext></mrow></msub></mrow></math> , <math id="M84"><mrow><msub><mi>I</mi><mrow><mtext>f</mtext><mi>γ</mi></mrow></msub></mrow></math> , <math id="M85"><mrow><msub><mi>I</mi><mrow><mtext>s</mtext><mi>γ</mi></mrow></msub></mrow></math> , <math id="M86"><mrow><msub><mi>τ</mi><mtext>f</mtext></msub></mrow></math> , <math id="M87"><mrow><msub><mi>τ</mi><mtext>s</mtext></msub></mrow></math> and <math id="M88"><mrow><msub><mi>t</mi><mn>1</mn></msub></mrow></math> need to be calculated. They were obtained by fitting the neutron/gamma pulse waveform by Eqs. (5) and (6), as follows: 1. The neutron/gamma pulse signals were extracted separately by the CC method (using the parameters that were used to achieve the performance shown in Fig. 7 Fig. 7 ), where 2000 pulse waveforms each were extracted for neutrons and gammas. 2. The average waveforms of neutron and gamma pulses were calculated and fitted with Eqs. (5) and (6) for neutrons and gammas, respectively, to obtain <math id="M89"><mrow><msub><mi>τ</mi><mrow><mtext>fn</mtext></mrow></msub><mo>,</mo><mtext> </mtext><msub><mi>τ</mi><mrow><mtext>sn</mtext></mrow></msub><mo>,</mo><mtext> </mtext><msub><mi>τ</mi><mrow><mtext>f</mtext><mi>γ</mi></mrow></msub><mo>,</mo><mtext> and</mtext><mtext> </mtext><msub><mi>τ</mi><mrow><mtext>s</mtext><mi>γ</mi></mrow></msub></mrow></math> . 3. The average of the rise times of the fitted neutron and gamma waveforms is obtained as <math id="M90"><mrow><msub><mi>t</mi><mn>1</mn></msub><mo>=</mo><mn>30</mn></mrow></math> ns. 4. The average of <math id="M91"><mrow><msub><mi>τ</mi><mrow><mtext>fn</mtext></mrow></msub><mtext>  and </mtext><msub><mi>τ</mi><mrow><mtext>f</mtext><mi>γ</mi></mrow></msub></mrow></math> is taken as <math id="M92"><mrow><msub><mi>τ</mi><mtext>f</mtext></msub><mo>=</mo><mn>81.51</mn></mrow></math> ns, and the average of <math id="M93"><mrow><msub><mi>τ</mi><mrow><mtext>sn</mtext></mrow></msub></mrow></math> and <math id="M94"><mrow><msub><mi>τ</mi><mrow><mtext>s</mtext><mi>γ</mi></mrow></msub></mrow></math> is taken as <math id="M95"><mrow><msub><mi>τ</mi><mtext>s</mtext></msub><mo>=</mo><mn>451.27</mn></mrow></math> ns. 5. By substituting <math id="M96"><mrow><msub><mi>τ</mi><mtext>f</mtext></msub></mrow></math> and <math id="M97"><mrow><msub><mi>τ</mi><mtext>s</mtext></msub></mrow></math> into Eqs. (5) and (6), we obtained <math id="M98"><mrow><msub><mi>I</mi><mrow><mtext>fn</mtext></mrow></msub><mo>=</mo><mn>0.868</mn><mo>,</mo><mtext> </mtext><msub><mi>I</mi><mrow><mtext>sn</mtext></mrow></msub><mo>=</mo><mn>0.132</mn><mo>,</mo><mtext> </mtext><msub><mi>I</mi><mrow><mtext>f</mtext><mi>γ</mi></mrow></msub><mo>=</mo><mn>0.837</mn><mo>,</mo><mtext> and </mtext><msub><mi>I</mi><mrow><mtext>s</mtext><mi>γ</mi></mrow></msub><mo>=</mo><mn>0.163</mn></mrow></math> . We confirm that the above parameters satisfy Eq. (11). For a fixed value of t s , we take <math id="M99"><mrow><msub><mi>t</mi><mtext>e</mtext></msub><mo>=</mo><mo>{</mo><mi>t</mi><mo>|</mo><mi>t</mi><mo>=</mo><mn>300</mn><mo>+</mo><mn>20</mn><mo>*</mo><mi>a</mi><mo>,</mo><mi>a</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>30</mn><mo>}</mo></mrow></math> . Fig. 9 Fig. 9 compares the FOM obtained using the t m values calculated using the MDDM with the potentially optimal FOM for different t e values. The potentially optimal FOM versus t m curve (black dots) is still convex, indicating that t e does not need to cover the entire pulse to obtain the best discrimination performance of the CC method, and thus the calculation time required for integration can be reduced. Experimental tests show that t e can be set between 640 and 740 ns after t s , where the potentially optimal FOM exceeds 1.46. In addition, when t e is set in this range, the difference between the FOM obtained using the t m value calculated using the MDDM and the potentially optimal FOM is less than 3.9%, confirming that it can improve the performance of the CC method. Conclusion 1 This work studied the use of four real-time discrimination methods with CLLB detectors. The FOM of the four methods was improved when the pulses were adequately filtered. 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The first draft of the manuscript was written by Zi-Han Zhang, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript. The authors thank the ASIPP team for providing a suitable experimental environment for the radioactive source. 10.1007/s41365-022-01160-5 This work was supported by cooperation projects between an enterprise (CNPE) and a research institute (ASIPP) (Y15HX16706). 2022-07-09 2022-11-23 2022-11-24 2023-01-16 2023-02-20 © China Science Publishing & Media Ltd. 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To view a copy of this licence, visit http://creat ivecommons.org/licenses/by/4. 0/. This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-022-01160-5 http://dx.doi.org/10.1007/s41365-022-01160-5 . 2023 Nuclear Science and Techniques 01 34