Introduction
The international community enforces safeguards through agreements such as the Treaty on the Non-Proliferation of Nuclear Weapons (NPT) to prevent the spread of nuclear weapons, promote the peaceful use of nuclear energy, and ensure compliance [1–3]. One key element of these safeguards is nuclear material accountancy (NMA), defined as activities carried out to establish the quantities of nuclear material present within defined areas and the changes in those quantities over specified periods. NMA primarily focuses on the verification of nuclear materials, with non-destructive assay methods being actively researched, particularly for the analysis and characterization of spent nuclear fuel [4–6]. Traditionally, techniques for characterizing spent nuclear fuel involve detecting the emission levels of neutrons and gamma rays or employing gamma spectrometry [7, 8]. Compared to these conventional methods, neutron multiplicity counting (NMC) offers the following advantages. It relies on neutrons originating solely from nuclear material rather than cladding; neutrons undergo less attenuation compared to gamma rays, they can be reliably detected even in intense gamma fields generated by short-lived isotopes, and NMC can distinguish fission neutrons from both random and (α, n) neutrons [9].
NMC detects neutrons emitted from nuclear materials. And these measurements are classified into NMC and fast neutron multiplicity counting (FNMC) [10–12]. Experimental studies on neutron multiplicity have explored various areas, including the investigation of neutron multiplicity and neutron energy-angle correlations in the induced fission of 235U, as well as the correlation between neutrons and photons during the fission of special nuclear materials (SNM) [13–15].
In NMC, a shorter die-away time is preferred, as it can reduce accidental coincidence events and prevent the degradation of measurement precision [16, 17]. Capture-based thermal NMC systems using 3He require neutron moderation and typically exhibit die-away times in the microsecond range [18, 19]. Due to the scarcity of 3He and the long die-away time of thermal NMC systems, FNMC systems employing organic scintillators, such as stilbene, anthracene, and plastic scintillators, are being developed. Scatter-based FNMC systems utilizing organic scintillators do not require neutron moderation, exhibit die-away times on the order of tens of nanoseconds, and detect neutrons through elastic scattering. Furthermore, they enable pulse shape discrimination (PSD) to distinguish between neutron and gamma-ray events [20–22]. The 3He-based active well coincidence counter developed at Los Alamos National Laboratory demonstrates a die-away time of 51 μs, whereas the fast neutron multiplicity system using 32 EJ-309 liquid scintillators (Eljen Technology) has a die-away time of 5.9 ns [23, 24]. The FNMC system, with its shorter die-away time compared to the NMC system, benefits from a reduced gate width, minimized uncertainty in fixed acquisition time, and shorter measurement duration. Trans-stilbene scintillators are widely used in organic scintillator-based FNMC systems due to their superior PSD capability for distinguishing neutron and gamma-ray signals, as well as their fast time resolution [25, 26].
The 252Cf source is considered a point source and is encapsulated to produce the characteristic neutron fission spectrum [27]. The leakage multiplication factor (ML) of 252Cf is 1, and the fraction of neutrons emitted from (α, n) reactions (α) is 0 [28]. Additionally, the progeny moments (νi) of 252Cf are well-documented and are commonly used for calibrating NMC or FNMC systems [28]. The neutron emission characteristics of 252Cf vary over time due to isotopic decay to 250Cf and subsequent α-decay products, 248Cm and 246Cm, which undergo spontaneous fission. These factors must be considered during measurements [29]. Previous studies utilizing an FNMC system composed of organic scintillators and silicon photomultipliers (SiPMs) distinguished neutrons and gamma rays using a high-energy threshold instead of PSD. Moreover, the validity of the separated data was assessed by comparing the measured singles rate (S), doubles rate (D), and triples rate (T) with GEANT4 simulations. However, the effective mass of 252Cf was not directly estimated [30].
In this study, an FNMC-based nuclear material accounting system was developed using pixelated trans-stilbene arrays and SiPM arrays. Neutron data were extracted using a low-energy threshold and PSD to distinguish neutrons from gamma rays. To compare the variability of the data and the errors in mass estimation based on detector efficiency and PSD performance, experiments were conducted for two cases with different energy thresholds of 50 keVee and 100 keVee. Additionally, the contribution of spontaneous fission from 252Cf was evaluated by considering the effects of 250Cf, 248Cm, and 246Cm. For effective mass estimation of 252Cf, one- and two-parameter assay-based calibration curves were constructed to back-calculate the effective mass of 252Cf.
Materials and Methods
1. Point Model
The conventional point model was originally developed based on capture-based thermal NMC systems [28]. However, in scatter-based FNMC systems using organic scintillators, the influence of crosstalk must be incorporated into the point model [31–33]. In the context of FNMC systems, crosstalk refers to the phenomenon where a neutron generates a signal in one detector, scatters, and subsequently produces a signal in another detector. Given that the 252Cf source has values of ML=1 and α=0, the point model can be simplified. The modified point model equations, accounting for crosstalk, are given by Equations (1)–(3), while the absolute efficiency of the detector is expressed in Equation (4) [34].
where F denotes the spontaneous fission rate, ɛ denotes the neutron counting efficiency, fd denotes the double-gate ratio, ft denotes the triple-gate ratio, and k represents crosstalk rate. In vsi, s denotes spontaneous fission nuclide and i denotes the ith factorial moments.
To validate the effectiveness of the FNMC system, measurements were conducted using a capsule-shaped 252Cf source, while Monte Carlo code Monte Carlo N-Particle Extended (MCNPX)-PoliMi simulations modeled a spherical 252Cf source. For masses below 100 g, the relative error due to shape differences can be considered negligible [35]. Additionally, determining the system’s gate width requires analyzing the Rossi-α distribution, which can be calculated using Equation (5). This analysis allows for the specification of the system’s die-away time (τ) [28], as:
2. MCNPX-PoliMi Geometric Design for FNMC System
The FNMC system was modeled using MCNPX-PoliMi, as shown in Fig. 1. The system consists of five 12×12 arrays of trans-stilbene, with each array comprising 144 pixels, each measuring 4 mm×4 mm×20 mm. The FNMC system was simulated in a pentagonal configuration, with the 252Cf source positioned at the center of the system. The distance between the 252Cf source and the center of each detector is 9 cm. The interior of the system (shown in green) was set as air (density=0.001225 g·cm−3), while the exterior was modeled as a void.
3. Development of a Pentagon-Shaped FNMC System
In this study, the organic scintillator trans-stilbene was used, with a chemical composition ratio of H/C=0.857. Fig. 2A shows a SiPM array (ArrayC-30035-144P; Onsemi) with a photosensitive area of 3 mm×3 mm, alongside a pixelated trans-stilbene array (Inrad-Optics) with pixel dimensions of 4 mm×4 mm×20 mm. The SiPM array and trans-stilbene array were combined through mechanical contact to create a single detector, as shown in Fig. 2B, and coupled with a front-end electronics board to form a single sensor module [36]. The system uses analog-to-digital converters to condense the 144 signals from the 12×12 SiPM array into 12 signals for each row and column, respectively. This facilitates the rapid identification of rows and columns with the highest signal intensities, enabling precise determination of event locations. Moreover, by processing signals from each row and column independently, the system effectively eliminates multi-pixel events [37]. The developed sensor module was then connected to a data acquisition (DAQ) board to complete the detection system, as shown in Fig. 2C.
As shown in Fig. 3, a pentagon-shaped FNMC system was developed using five detection systems. Support structures for the five detector modules and a single source holder were fabricated using a three-dimensional printer. The distance from the 252Cf source to the detectors was set to 9 cm, consistent with the MCNPX-PoliMi simulation. A mainboard for acquiring and processing signals from the five detector modules is positioned at the bottom of the assembled system.
4. Time-Dependent Multiplicity Calculation of 252Cf
The multiplicity characteristics of 252Cf vary over time. When measuring 252Cf, the effects of its isotope 250Cf and the α-decay products 248Cm and 246Cm must be considered [29]. The half-lives and spontaneous fission probabilities of each nuclide are presented in Table 1.
Based on the half-life of 250Cf, the recommended operational lifespan of 252Cf is estimated to be approximately 15 years [38]. The total spontaneous fission rate of 252Cf is calculated as the sum of the spontaneous fission rates of the four nuclides, as shown in Equations (6)–(10) [39]:
F represents the spontaneous fission rate, N(0) is the initial number of atoms, λ denotes the decay constant, η is the spontaneous fission probability, and t represents time. The factorial moment values for the four nuclides are provided in Table 2. The average moments of neutron multiplicity for 252Cf,
v s i ¯ v s i ¯
5. Mass Calibration Using Parameter Assay
For SNMs with identical mass and elemental composition, the multiplication factor of the sample can vary depending on the geometric configuration, environmental conditions, and energy, potentially resulting in different mass estimation outcomes. The parameter assay method derives a mass calibration curve using three parameters: S, D, and T from samples with known mass. The mass calibration curve is categorized into a one-parameter assay, which utilizes only the singles rate (S), and a two-parameter assay, which employs both S and D as parameters [40, 41]. In this study, the physical mass of the 252Cf source, determined by its specific activity, was used for the experiments.
The one-parameter assay estimates mass using the measured D, similar to conventional coincidence counting. The mass calibration curve is expressed in Equation (14), utilizing the physical mass, D, and the fit parameters a and b, as:
The two-parameter assay is used to account for differences in sample multiplication. The ratios D/S and D/mass vary depending on the sample multiplication. In the case of 252Cf, where no (α, n) reactions occur, the mass calibration curve is expected to be linear. The mass calibration curve is expressed in Equation (15), utilizing the physical mass, D/S, and the fit parameters c and d, as:
The linearity between mass and parameters involving two or three parameters was analyzed, confirming that the D2/T value exhibits a quadratic relationship with mass. From this analysis, a mass calibration curve was derived. The mass calibration curve is expressed in Equation (16) and utilizes the physical mass, D2/T, and the fit parameters e, f, and g, as:
This study was conducted to examine the differences in detector efficiency and mass estimation errors based on the energy threshold. The experiments were performed by varying the energy threshold, as shown in Table 3, with thresholds set at 50 keVee and 100 keVee. The condition with a threshold of 50 keVee was designated as case 1, while the condition with a threshold of 100 keVee was designated as case 2.
In this study, 252Cf sources A and B, each initially exhibiting a radioactivity of 3.145 MBq, as well as their combined source C, were used. For the case 1 experimental data, source A had been in use for approximately 4.39 years, and source B for 1.50 years. Similarly, for the case 2 experimental data, source A had been in use for around 4.24 years, and source B for 1.37 years.
Results and Discussion
1. MCNPX-PoliMi Simulation
In MCNPX-PoliMi, all detectors are assumed to have identical efficiencies. However, in the developed FNMC system, each detector exhibits a different efficiency. To account for this, simulations were conducted accordingly. The neutron multiplicity of 252Cf and the Rossi-α distribution of the fabricated system are shown in Fig. 4. The die-away time, estimated from the Rossi-α distribution, is 1.56 ns. While the optimal gate width is calculated as (die-away time×1.257), the gate width was set to 100 ns to ensure signal collection stability and minimize the relative errors of S, D, and T [42, 43].
In the MCNPX-PoliMi simulation, neutron events were recorded when neutrons interacted with hydrogen, underwent elastic scattering, and subsequently reacted with each sensor. From the output file of the MCNPX-PoliMi simulation, instances where a single neutron event scattered and interacted consecutively within the same detector were identified as pile-up events, while cases involving reactions in two or more detectors were categorized as crosstalk events. Both types of events were excluded from the analysis.
The MCNPX-PoliMi simulation was conducted for 240 seconds, and the output file was analyzed in list-mode format. A gate width of 100 ns was applied, and the three sources listed in Table 3 were used. The calculated values for S, D, and T for the respective masses of 252Cf are presented in Table 4.
The S, D, and T values exhibit linearity with respect to the physical mass (R2>0.999), as shown in Fig. 5. The standard deviations are represented as error bars in the graph, and the derived values follow a Poisson distribution with inherent statistical uncertainties. For the case 1 simulation, the standard deviation is approximately 0.2% for S and 1.5% for D. In Fig. 5, the error bars for these values are small and barely noticeable. However, for T, the smallest source (A) shows a standard deviation of 13.8%, while the largest source (C), exhibits a standard deviation of approximately 8.5%. The detection probability of T is proportional to the cube of the neutron detection efficiency (ɛ3), as expressed in Equation (3). Since T represents rarer events compared to S and D, the smaller number of events leads to relatively higher standard deviations due to the characteristics of the Poisson distribution, where fewer events result in larger relative uncertainties.
The case 2 simulation was also conducted for 240 seconds, similar to case 1, and the output file was analyzed in list-mode format. A gate width of 100 ns was applied, and the three sources listed in Table 3 were used. The calculated values for S, D, and T for the respective masses of 252Cf are presented in Table 5.
The S, D, and T values exhibit linearity with respect to the physical mass (R2>0.999), as shown in Fig. 6. For the case 2 simulation, the standard deviation for S is approximately 0.3%, while that for D is 3.5%, making the error bars noticeable in Fig. 6. However, for T, A has a standard deviation of 42.8%, whereas C exhibits a standard deviation of approximately 21.5%. As in case 1, T, being a rarer event compared to S and D, shows relatively higher relative standard deviation values. Due to the characteristics of the Poisson distribution, where smaller datasets result in higher relative standard deviations, the relative standard deviation of T in case 2 is higher than that in case 1.
2. Pulse Shape Discrimination Assessment
FNMC system using organic scintillators employs PSD to distinguish between the pulse shapes of neutrons and gamma rays. The PSD value is calculated as shown in Equation (17), where neutron signals exhibit a slower decay compared to gamma-ray signals, resulting in a longer pulse tail and, consequently, a higher PSD value, as:
The PSD performance metric, determined by light collection efficiency, is referred to as the figure of merit (FOM), which visualizes the separation of neutrons and gamma rays across various energy regions. An FOM value greater than 1.27 is considered indicative of good discrimination [44, 45]. In case 1, with an energy threshold set to 50 keVee, the FOM was assessed within the 50–150 keVee energy range. For case 2, with the energy threshold set to 100 keVee, the FOM was evaluated within the 100–200 keVee energy range. For case 1, with an energy threshold of 50 keVee, the PSD plots for each sensor are shown in Fig. 7. When the neutron-gamma separation baseline is set as a constant function, there is a possibility of overlap between neutron and gamma data within the 50–100 keVee range. This overlap, caused by an incorrectly defined baseline, can compromise the reliability and accuracy of the data, potentially distorting the interpretation of experimental results. To address these issues, an exponential fitting method was applied to define the neutron-gamma separation baseline as an exponential function, as done in [25], and represented by the white line in Fig. 7.
The FOM graph for case 1 is shown in Fig. 8. In the 50–150 keVee energy range, the FOM value is 1.438, which exceeds the threshold value of 1.27. Additionally, the FOM value increases to 1.846 in the 150–250 keVee energy range and 2.055 in the 250–350 keVee energy range.
For case 2, with an energy threshold of 100 keVee, the PSD plots for each sensor are shown in Fig. 9. In energy ranges above 100 keVee, the signal characteristics are more distinctly separated, resulting in clearer neutron-gamma separation compared to case 1. This improved separation is attributed to the higher energy threshold in case 2, which effectively reduces overlapping signals by better distinguishing the neutron and gamma regions. Consequently, a simple linear baseline, in the form of a constant function, is sufficient to effectively differentiate between neutron and gamma events.
The FOM graph for case 2 is shown in Fig. 10. In the 100–200 keVee energy range, the FOM value is 1.642, exceeding the threshold value of 1.27. Additionally, the FOM value rises to 1.755 in the 200–300 keVee energy range and 2.211 in the 300–400 keVee energy range. It is evident that, with an energy threshold of 100 keVee, neutron-gamma separation in the low-energy region is more distinct compared to the case with a 50 keVee threshold.
3. Measurement of Neutron Multiplicity for 252Cf
When using the FNMC system, background measurements are performed to eliminate noise and interference, facilitating clearer separation and analysis of the target signal. Theoretically, neutron coincidence events can be produced by high-energy charged particles from cosmic rays interacting with the nuclei of high atomic number materials, resulting in spallation reactions [42]. However, in this system, the effects of spallation reactions are considered negligible. To verify whether the system could erroneously count D and T events in the absence of a source, background measurements were conducted for 1,800 seconds without a source. The results are presented in Table 6. If the D value is less than 0.01 cps, background correction is not required [40].
The Rossi-α distribution for the fabricated system was obtained using a 252Cf source which is commonly employed in neutron multiplicity measurements due to its well-characterized neutron emission properties. As shown in Fig. 11, the Rossi-α distribution indicates a die-away time of approximately 10.17 ns.
The fabricated system has a time resolution of 11 ns, and the pulse shaping time was set to 1 μs to ensure signal acquisition stability. Due to these characteristics, the system exhibits a longer die-away time compared to the MCNPX-PoliMi simulation, resulting in the gate width being set to 100 ns. Using this gate width and the three 252Cf sources listed in Table 3, the multiplicity of 252Cf was measured. The S, D, and T values were calculated, taking into account the crosstalk ratio, defined as (D+T)/(S+D+T), and were compared with the MCNPX-PoliMi simulation data [34].
The S, D, and T values collected during the experiment under the case 1 conditions are presented in Table 7. In the case 1 experiment, the S value exhibited a standard deviation of 0.2%, while D showed a standard deviation of 1.5%. For T, the standard deviation was 15.2% for A and 8.7% for C. The absolute neutron detection efficiency was measured at 1.2%, and the crosstalk ratio was approximately 1.4%. The experimental data were compared with the MCNPX-PoliMi simulation results, as shown in Fig. 12.
As observed in Fig. 13, the data from the fabricated system demonstrated linearity with respect to mass (R2>0.999). Although the MCNPX-PoliMi simulation data were calculated based on the efficiency of the fabricated system, the counting rates in the simulation were found to be higher than those measured by the fabricated system. This discrepancy is attributed to the influence of the PSD threshold, which can hinder the complete separation of neutron and gamma-ray data, possibly causing neutron data below the energy threshold to be undetected. This effect is particularly significant in the D and T measurements.
In the experiment conducted under case 2 conditions, the S value exhibited a standard deviation of 0.3%, while D showed a standard deviation of 3.3%. For T, the standard deviation was 45% for A and 21.8% for C. The absolute neutron detection efficiency was measured at 0.51%, and the crosstalk ratio was 0.69%.
Under case 2 conditions, the experimental data demonstrated linearity with respect to mass (R2>0.999) and showed values that were similar to those obtained from the MCNPX-PoliMi simulation data. This trend contrasts with the results observed in case 1, suggesting that effective neutron-gamma separation was achieved using PSD in case 2. Energy threshold is very important in NMC. As the energy threshold increases, the values of S, D, and T decrease. Compared to Tables 7 and 8, the reduction rates of D and T are higher than that of S, resulting in a sharp decrease in the D/S and T/S ratios. If the D/S or T/S values are measured too high or too low, it could lead to misidentification of nuclides or misestimation of the amount of nuclear material. Therefore, an appropriate energy threshold is necessary.
4. Time-Dependent Multiplicity Measurement of 252Cf
The experiment was conducted by setting the activity ratio of 250Cf generated during the production of 252Cf to 0.025 [46]. Using Equations (6)–(10), the total spontaneous fission rate of 252Cf and the contributions of each nuclide were calculated. Additionally, the factorial moments
Δ v i ¯
Table 9 shows that both sources A and B demonstrated a contribution of over 99.8% from 252Cf. Fig. 14A presents the time-dependent neutron intensity of 252Cf, with the contribution from 246Cm being negligible and therefore excluded from the analysis.
As time progresses, the factorial moments of 252Cf decrease due to changes in the neutron emission ratios contributed by each nuclide as 252Cf decays. Fig. 14B illustrates the variability of the factorial moments of 252Cf over time, and Table 10 compares the factorial moments of 252Cf at its initial state with those reduced over time. All sources (A to C) exhibited relative errors within 0.04%, and the S, D, and T values were recalculated using the updated factorial moments of the 252Cf used.
The values of S, D, and T for case 1 were recalculated using Equations (1)–(3). The recalculation results and relative errors are presented in Tables 11 and 12.
Under the conditions of case 1, the recalculated S, D, and T values using the modified factorial moments showed that the count rate of pure 252Cf differed from the original count rate by up to 0.5%. Case 2 was also recalculated in the same manner as case 1, with the recalculated results and their relative errors presented in Tables 13 and 14.
Under the conditions of case 2, the recalculated S, D, and T values indicated that the count rate of pure 252Cf differed from the original count rate by up to 0.5%.
5. Mass Estimation of 252Cf
Using the counts of 252Cf obtained in section 3.4 and the parameter assay described in section 2.5, a mass calibration curve was developed to estimate the mass of pure 252Cf. The mass calibration curve of the fabricated system was then compared with the mass calibration curve from the MCNPX-PoliMi simulation.
Under the conditions of case 1, Fig. 15A shows that the mass calibration curve based on a single parameter exhibits linearity with respect to the 252Cf mass. Fig. 15B and 15C represent linear and quadratic two-parameter assays, respectively, both displaying linearity with respect to the mass of 252Cf (R2>0.999).
The mass calibration curves for case 1 are expressed by Equations (18)–(20), which represent, in order, the one-parameter assay, the two-parameter assay (linear), and the two-parameter assay (quadratic), as:
The mass of 252Cf was estimated using the developed mass calibration curve, and the estimation results are shown in Table 15.
Under the conditions of case 1, the mass of 252Cf was estimated with a maximum relative error of less than 0.4%.
Fig. 16 exhibits a similar trend to the mass calibration curves of case 1 and demonstrate a linear relationship with respect to the 252Cf mass (R2>0.999). The mass calibration curves for case 2 are expressed by Equations (21)–(23), corresponding to the one-parameter assay, the linear two-parameter assay, and the quadratic two-parameter assay, respectively, as:
The mass estimation results for 252Cf, obtained using the developed mass calibration curves, are presented in Table 16. Under the conditions of case 2, the mass of 252Cf was estimated with a maximum relative error of less than 2.2%.
Conclusion
In this study, a FNMC system based on organic scintillators was developed using a pixelated trans-stilbene array and a SiPM array. The die-away time of the developed system was measured to be 10.17 ns, with a gate width set to 100 ns for stable DAQ. The system’s FOM values was 1.438 in the 50 keVee range and 1.642 in the 100 keVee range, both exceeding the threshold criterion of 1.27. It was confirmed that the contribution of 252Cf in the source exceeded 99.8%, with the factorial moment showing a relative error of approximately 0.05%. Using the recalculated factorial moments of 252Cf, the S, D, and T values were determined, with a maximum reduction rate of 0.5%. The absolute neutron detection efficiency was 1.2% for case 1 (50 keVee energy threshold) and 0.5% for case 2 (100 keVee energy threshold). The mass of pure 252Cf was estimated using both one- and two-parameter assays, yielding a maximum relative error of 0.4% for case 1 and within 2.2% for case 2. While higher energy thresholds resulted in data comparable to MCNPX-PoliMi simulation results, lower energy thresholds, offering higher efficiency, resulted in smaller relative errors in mass estimation. The FNMC system developed in this study addresses the limitations of conventional NMC systems, minimizing the crosstalk effects and pile-up errors inherent in FNMC systems. This system holds potential for applications in nuclear material accounting and is expected to enhance the efficiency of SNM verification. The study employing MCNPX-PoliMi simulations successfully estimated the effective mass of pure 240Pu [41]. However, real plutonium samples comprise multiple isotopes, such as 238Pu, 240Pu, and 241Pu, thus requiring the consideration of diverse isotopic compositions into the analysis. Furthermore, developing a new analytical model that accounts for the (α, n) reaction or calibrating the system with materials reflective of highly enriched uranium (HEU) characteristics is expected to enable the mass estimation of HEU samples as well.
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