Generation of the two fission fragments

The mass of the light fission fragment M_L was randomly sampled with a probability proportional to the yields shown in Fig. 2 [22]. The mass of the paired heavy fission fragment M_H is 252u-M_L. Then, the total kinetic energy TKE and its corresponding uncertainty σ_TKE were determined using the total kinetic energy distribution and the corresponding uncertainty distribution as functions of the light fragment mass, as shown in Fig. 3. The specific total kinetic energy of the sampled fission fragment pair E_T was specified as , where ΔTKE was randomly sampled from the normal distribution with a standard deviation of σ_TKE. The kinetic energy of the light fission fragment E_L was determined as:(1)The kinetic energy of the heavy fission fragment E_H was equal to . Finally, the direction of the light fission fragment in the laboratory system was sampled from an isotropic distribution, and the direction of the heavy fission fragment was opposite to that of the light fragment.

Fig. 2Full-screen View

The yields of the fission fragments with different masses [22]

Fig. 3Full-screen View

The distributions of the total kinetic energy (a) and the corresponding uncertainty (b) versus the fragment mass [22]

Generation of the prompt fission neutrons

The evaporation of the neutrons was simulated via a cascade process for the light fission fragments, followed by the heavy fission fragments. The number of neutrons emitted from fission fragments with different masses and the corresponding uncertainties are shown in Fig. 4 [22]. Because σ_TKE is known, σv can be calculated. The number of neutrons N evaporated from the sampled light fission fragment was , where Δv was randomly sampled from the normal distribution with a standard deviation of σv. In almost all cases, N was a non-integer. Thus, N was decomposed into an integer part IN (the largest integer smaller than N) and a decimal part DN (where ). Then, a random number e in [0, 1] was sampled and compared with DN. If e was smaller than DN, the actual number of neutrons RN emitted from the light fission fragment was ; otherwise RN equaled IN.

Fig. 4Full-screen View

The number (a) and the corresponding uncertainty (b) of the neutrons evaporated from fission fragments with different masses [22]

According to the standard nuclear evaporation theory, the fission neutron energy spectrum Φ in the center-of-mass system of the fission fragment can be described as [22]:(2)where λ is the cascade neutron emission coefficient and T denotes the temperature of the fission fragment, as shown in Fig. 5 [22]. According to the energy distribution calculated using Eq. (2), the energy of a neutron in the center-of-mass system was sampled. The direction of the neutron was sampled from an isotropic distribution. Thereafter, the energy and direction of the neutron in the laboratory system, together with those of the light fission fragments, were calculated under the constraints of energy and momentum conservation. The mass of the light fission fragment equals M_L-n, where n represents the number of neutrons evaporated from the light fission fragment until this point. Finally, the energy and direction of the neutron were stored in a queue.

Fig. 5Full-screen View

The cascade neutron emission coefficients (a) and temperatures (b) versus the fragment masses [22]

Neutrons were evaporated one by one using the same process until RN neutrons have been evaporated from the light fission fragment. After the generation of neutrons evaporated from the light fission fragment, the neutrons evaporated from the heavy fission fragment were processed in a similar manner.

Statistics of the neutrons

After the generation of prompt fission neutrons, they were tracked to obtain their energy correlations. As shown in Fig. 6, 36 surface detectors (with a radius of 5 cm) were positioned on a circle (with a radius of 60 cm), and a ²⁵²Cf source was located at the center of the circle. The angle between the adjacent surface detector normals was 10°. The fission events, stored in the queue with the energy and directional information of the emitted neutrons, were tracked individually. If a neutron reached any of the detectors, it was counted as a single event. As shown in Fig. 7, the simulated energy spectrum of the single events agrees well with the ²⁵²Cf prompt fission neutron energy spectrum recommended by IAEA with the relative deviation smaller than 0.6% [23]. After all the neutrons were tracked, if multiple neutrons emitted in the same fission event reached the detectors, each two neutrons were paired as a group. In this process, the neutron reaching the detector with a smaller serial number was designated as the first one. In the present work, the correlation events at 0° to 180° with the interval of 30° (the angle between the two neutron directions) were counted.

Fig. 6Full-screen View

(Color online) The layout of the detectors in the simulation

Fig. 7Full-screen View

The ratio of the simulated ²⁵²Cf prompt fission neutron energy spectrum to that recommended by IAEA [23]

As presented in Fig. 8, the distributions of the correlated events at 90° and 180° are noticeably different from each other. The neutron energies corresponding to the highest counts at 90° and 180° are 2.6 MeV and 3.0 MeV, respectively. To present the energy correlation more clearly, the counts at each angle were normalized in two steps, as shown in Fig. 9. First, the total counts in each column with neutron energies ranging from 1–5 MeV were normalized. Second, the count distributions in each column were normalized to the simulated ²⁵²Cf prompt fission neutron energy spectrum. As Fig. 9 shows, the neutron energy correlations vary with angle. The second neutron energy trends toward lower values for the correlated events at near 90°, whereas it trends toward higher values for the correlated events at near 0° or 180°. These trends increased with the first neutron energy. Because the crosstalk effect in the measurement is non-negligible for the correlated events at angles smaller than 75° [14], the energy correlations at 90° and 180° are measured in the present work as representative cases.

Fig. 8Full-screen View

(Color online) The simulated correlation events at 90° (a) and 180° (b)

Fig. 9Full-screen View

(Color online) The simulated neutron energy correlations at 0° (a), 30° (b), 60° (c), 90° (d), 120° (e), 150° (f) and 180° (g)

In the measurement, a triple coincident measurement is required, resulting in a very low detection efficiency. According to the test, the time resolution of the measurement was approximately 1.5 ns. Therefore, the length-of-flight of neutrons was set to 62.5 cm to balance both neutron detection efficiency and energy resolution measured using the TOF (time-of-flight) technique.

Experimental setup

The experimental setup is illustrated in Fig. 10. A ²⁵²Cf source with a fission rate of approximately 1.3×10⁵ s^-1 was used for the experiment. The source was sealed in a stainless steel capsule (~2 mm in diameter and ~5 mm in height) and suspended vertically at a height of 1.5 m above the floor using a fine nylon thread to minimize neutron scattering effects on the detection by the EJ309 liquid scintillation detectors. All detectors were mounted on aluminum supports. Every four EJ309 liquid scintillation detectors were combined into an array. The angle between the normals of the adjacent arrays was fixed at 90°. The distance from the ²⁵²Cf source to the front face of the eight CLYC detectors was 6±0.1 cm, and that to the front face of the sixteen EJ309 liquid scintillation detectors was 60±0.1 cm. Since the thickness of the EJ309 liquid scintillation detector is 5.08 cm, the uncertainty of the neutron flight path length σ_L is 2.15 cm calculated as:(3)where H is the thickness of the EJ309 liquid scintillation detector, and σ_p represents the uncertainty in the distance from the ²⁵²Cf source to the front faces of the EJ309 liquid scintillation detectors.

Fig. 10Full-screen View

(Color online) The experimental setup photograph

The detector signals were acquired using two CAEN VX1730B digitizers. Each digitizer has 16 input channels with a sampling frequency of 500 MS/s. One digitizer was connected to the eight CLYC detectors, and the other was connected to the sixteen EJ309 liquid scintillation detectors. Both digitizers were operated in list mode with the time-stamp (T), long and short gate integrated charges (Q_L and Q_S) of every event being saved for off-line analysis. The pulse height (PH) and pulse shape discrimination (PSD) parameter can be calculated as [24, 25]:(4)(5)where C_A and C_P are calibration factors of PH and PSD, respectively.

Calibration of the detectors

Before the measurement, the 8 CLYC detectors and the 16 EJ309 liquid scintillation detectors were calibrated with ⁶⁰Co (1.33 and 1.17 MeV) and ²²Na (1.28 and 0.55 MeV) γ sources as shown in Fig. 11. The signal gains were almost equivalent for the eight CLYC detectors by adjusting their working high voltages. Similar adjustments to the working high voltage were also applied to the sixteen EJ309 liquid scintillation detectors. In the calibration of the eight CLYC detectors, both the photo peaks and Compton edges were used, and in that of the sixteen EJ309 liquid scintillation detectors, only the Compton edges were utilized. The method for determining the Compton edges was described in Ref. [26, 27]. The corresponding Compton electron energy E_e is(6)where Eγ denotes the γ-ray energy and represents the electron rest mass energy (0.511 MeV).

Fig. 11Full-screen View

(Color online) The pulse height spectra induced by the ⁶⁰Co and ²²Na γ sources for a CLYC detector (a) and a EJ309 liquid scintillation detector (b)

In Fig. 11(a), the Compton edge corresponding to the Compton electron energy of 1.12 MeV was obscured because it overlapped with the tail of the 1.17 MeV Photo peak. The position of the Compton edge corresponding to the Compton electron energy of 0.38 MeV could not be determined accurately because it was significantly affected by the measurement threshold. Consequently, these two Compton edges were not used in the calibration of the eight CLYC detectors.

Measurement

The experiment was conducted for a total measurement duration of approximately 1200 h. Because two VX1730B digitizers were used to acquire the detector signals in the experiment, they must be operated in a time-synchronized mode to obtain the accurate TOF data. To achieve this, the two digitizers shared the internal clock of the one connected to the eight CLYC detectors, and the synchronization settings were configured via the data acquisition software COMPASS, as described in the COMPASS User Manual [28]. According to our synchronization test with a pulse generator DT5810, the synchronization accuracy was better than 0.1 ns. Although this synchronization accuracy contributes to the experimental time resolution, its influence can be ignored because it is much smaller than the experimental time resolution of 1.5 ns.

Data analysis

With the experimental data, the PH-PSD two dimensional spectra of the eight CLYC detectors and the sixteen EJ309 liquid scintillation detectors were obtained, as illustrated in Fig. 12. By applying optimal discrimination thresholds, event types (γ-rays or neutrons) can be identified.

Fig. 12Full-screen View

(Color online) The PH-PSD two dimensional spectra of a CLYC detector (a) and an EJ309 liquid scintillation detector (b)

In the present measurement, the measurement thresholds of the CLYC detectors were approximately 250 channels (0.3 MeV for γ-ray), and those of the EJ309 liquid scintillation detectors were approximately 50 channels (0.5 MeV for protons). Although the low measurement threshold for the EJ309 liquid scintillation detector could lead to the misidentification of event types using PSD for low pulse height events, the probability of classifying a γ-ray event as a neutron event was low because of the significant TOF difference between γ-ray and neutron events, as shown in Fig. 13.

Fig. 13Full-screen View

(Color online) The TOF distribution for the “neutron events” decided by PSD

Figure 13 illustrates the TOF distribution of single neutron event detected by the 16 EJ309 liquid scintillation detectors. To suppress the interference of accidental coincidences, triple time correlation coincidence was adopted in the data analysis; that is, a neutron event was required to correlate with at least two γ-rays detected by either the CLYC or EJ309 liquid scintillation detectors. The start moment of TOF T₀ is(7)where T₀ denotes the time-stamp, and the subscript k represents the serial number of the corresponding detected γ-rays. The coincident time window was -200 to 200 ns. The average count S_b in the time window from -200 to -50 ns was taken as the accidental coincident background. The net single neutron events S_net can be derived by subtracting S_b from the TOF distribution shown in Fig. 13. Theoretically, S_net was proportional to the detection efficiency ε and neutron flux ϕ as(8)where the subscript i represents the serial number of the divided TOF bin.

The neutron coincident events at 90° and 180° are shown in Fig. 14. Triple time correlation coincidence was employed in the data analysis, that is, two neutrons and at least one γ-ray were required to define one coincident event. There are five types of interferential events. The first is the γ-ray event shown in the purple box. These events do not noticeably affect the measurement of the neutron event shown in the white box because the TOF difference between γ-rays and neutrons is significant. The second one, shown in the green box, indicates that the first neutron is an accidental coincident neutron. The related interference can be deducted using the counts of each column in the white box to subtract the average counts of each column in the corresponding region with TOF ranging from -150 ns to -50 ns for the first neutron. The third, shown in the red box, is that the second neutron is an accidental coincident neutron, and the related interference can be subtracted using a data analysis approach similar to that for the second. The fourth one shown in the black frame is that the γ-ray is an accidental coincident γ-ray. The corresponding interference can be deducted using the following two steps. In the first step, the region with TOF ranging from -150 ns to -50 ns for the first neutron was scanned along the diagonal line using a scanning box identical to the white box shown in Fig. 14. During the scanning process, the average count of every bin in the scanning box was calculated. In the second step, the corresponding interference was deducted using the counts in the white box to subtract the average counts of the corresponding bins in the scanning box. The fifth is the events induced by crosstalk, that is, the neutron detected by the second detector is the same one that is scattered and detected by the first detector. Although the accurate subtraction of this interference is difficult, it can be significantly mitigated by restricting the TOF to the range of 19–46 ns. This conclusion was drawn based on the Monte Carlo simulation performed by JMCT, as shown in Fig. 15 [29].

Fig. 14Full-screen View

(Color online) The neutron coincident events at 90° (a) and 180° (b)

Fig. 15Full-screen View

(Color online) The simulated cross talk event at 90° (a) and 180° (b)

After the subtraction of the interference events mentioned above, the net neutron coincident events D can be obtained. To obtain the relative deviation between the neutron energy spectrum of ²⁵²Cf prompt fission neutrons and that of the neutrons correlated with different energy neutrons, three calculation steps were executed. First, the net neutron coincident events D were compared with the net single neutron events S_net to obtain the relative ratio R as:(9)where the subscripts i and j of R and D separately denote the TOF bin serials of the first and second detected neutrons, and the subscript j also represents the TOF bin serial of the net single events for S_net. It should be mentioned that the influence of the detection efficiencies for different energy neutrons is canceled out in this step. Second, the relative ratios R were normalized over the TOF₂ region ranging from 19 to 46 ns as follows:(10)where N is the bin number in the TOF region, ranging from 19 to 46 ns. Third, the neutron TOF was converted into neutron energy E_n as shown in Fig. 16 and Tables 1 and 2 using [30, 31]:(11)where L is the neutron flight length, c is the speed of light in vacuum, m_n denotes the neutron mass. In the present measurement, the influence of relativity can be neglected because the velocity of the detected neutron is much smaller than that of light in vacuum. The calculation of E_n can be simplified as [32]:(12)where the units of L and TOF are m and ns, respectively.

Fig. 16Full-screen View

(Color online) The relative ratios of the measured energy spectra correlated with different energy neutrons at 90° (a) and 180° (b) to the ²⁵²Cf prompt fission neutron energy spectrum

The measurement uncertainties (1.5%~3.2% at 90°; 1.6%~4.1% at 180°) include those of D and S_net. The uncertainties of D (1.5%~3.2% at 90°; 1.6%~4.1% at 180°) comprise statistical uncertainties (1.5%~3.1% at 90°; 1.6%~3.9% at 180°) and the uncertainties from the subtraction of accidental coincidences involving the first neutron (<0.1% at 90° and 180°), the second neutron (<0.1% at 90° and 180°) and the γ-ray (<0.1% at 90° and 180°). The uncertainties of S_net (0.2%~0.3% at 90°; 0.2%~0.4% at 180°) consist of statistical uncertainties (0.2%~0.3% at 90°; 0.2%~0.4% at 180°) and uncertainties from the subtraction of accidental coincident backgrounds (<0.1% at 90° and 180°). The uncertainty of E_n (9.5%~17.2%) is contributed by the experimental time resolution of 1.5 ns (6.5%~15.8%) and the uncertainty of the neutron flight path length of 2.15 cm (6.9%).

Discussions

As shown in Figs. 9 and 16 and Tables 1 and 2, the results indicate that the influence of a neutron on the energy of the correlated neutron is more pronounced in the high energy region, and the related effect at 180° appears to be more noticeable than that at 90°. In addition to the difference in effect magnitude, the correlated neutron energy shows opposite trends between the correlated events at 90° and those at 180°. The energy of the neutron that correlates with a neutron with relatively high energy tends toward a lower value at 90°, while it tends toward a higher value at 180°. With the measured relative ratios, the related neutron energy spectra can be derived by multiplying the relative ratios by the ²⁵²Cf prompt fission spectrum recommended by IAEA [23].

When compared with the measurement results obtained by Schuster et al. [14] and the simulation results introduced previously, the present results show the same tendency, i.e., the energy correlation between the coincident neutrons at ~180° is positive, while the energy correlation between the coincident neutrons at ~90° is negative. This further enhances the credibility of the measurement results. In addition, the fact that the measurements and simulations show the same energy correlation tendency indicates that the dominant reason for this tendency may be the impact of the velocities and directions of fission fragments on the neutron energy. As introduced previously, most neutrons are emitted from fast-moving fission fragments, which means that the moving direction angle between the high-energy neutron and its corresponding fission fragment tends to be small, whereas the moving direction angle between the low-energy neutron and its corresponding fission fragment tends to be large. For the correlated neutrons at 90°, if the energy of the first neutron is relatively high, the energy of the second neutron tends to be low. This is because the relatively high energy of the first neutron indicates a small moving direction angle between this neutron and its corresponding fission fragment with a high probability. Therefore, regardless of whether these two neutrons are emitted from the same fission fragment or two different fragments, considering the opposite directions of the two fission fragments, the moving direction angle between the second neutron and its corresponding fission fragment is relatively large with a high likelihood. This results in a relatively low energy of the second neutron. Similarly, for correlated neutrons at 180°, a relatively high energy of the first neutron tends to result in low energy of the second neutron if they are emitted from the same fission fragment, but in high energy if they are emitted from different ones. As the probability of detecting neutrons from the same fission fragment is noticeably smaller than that from different fission fragments, the total tendency shows that a high-energy neutron is more likely to be correlated with another high-energy neutron. For example, if each fission fragment emits two neutrons and the detection efficiencies of these neutrons are even, the probability of detecting neutrons from the same fission fragment is approximately half that from different fission fragments. Additionally, it should be noted that the energy correlation should also be contributed by some other factors besides the impact of the velocities and directions of fission fragments, such as the energy competition relationship among the neutrons emitted from the same fragment, as pointed out in Ref. [14].