Determining the average prompt-fission-neutron multiplicity for $^{239}$Pu($n$,$f$) via a $^{240}$Pu($\alpha$,$\alpha^{\prime}f$) surrogate reaction

TL;DR

This study uses a surrogate reaction method to indirectly measure the average prompt-fission-neutron multiplicity for $^{239}$Pu($n$,$f$), providing results consistent with direct measurements across a broad energy range.

Contribution

First application of the surrogate-reaction method to determine $ar{ u}$ for $^{239}$Pu($n$,$f$) using $^{240}$Pu($ extalpha$,$ extalpha^{ extprime}f$) reactions.

Findings

Values of $ar{ u}$ agree with direct measurements.

Measured $ar{ u}$ over 0.25--26.25 MeV energy range.

Validated surrogate method for fission neutron multiplicity.

Abstract

The average prompt-fission-neutron multiplicity $\bar{\nu}$ is of significance in the areas of nuclear theory, nuclear nonproliferation, and nuclear energy. In this work, the surrogate-reaction method has been used for the first time to indirectly determine $\bar{\nu}$ for $^{239}$Pu($n$,$f$) via $^{240}$Pu($\alpha$,$\alpha^{\prime}f$) reactions. A $^{240}$Pu target was bombarded with a beam of 53.9-MeV $\alpha$ particles. Scattered $\alpha$ particles, fission products, and neutrons were measured with the NeutronSTARS detector array. Values of $\bar{\nu}$ were obtained for a continuous range of equivalent incident neutron energies between 0.25--26.25~MeV, and the results agree well with direct neutron measurements.

Tables

Table 1. Table 1: Properties of the 240 Pu target used in the experiment. In addition to 240 Pu, a small amount of 238 Pu was also present.

Property	240Pu	238Pu
Activity ($\mu$Ci)	$67.374\left(334\right)$	$0.250\left(3\right)$
Weight Percent (%)	$99.995\left(718\right)$	$0.00493\left(6\right)$
Thickness ($\mu$g/cm2)	$104.078\left(528\right)$	$0.00513\left(6\right)$

Table 2. Table 2: Equivalent incident neutron energies E n subscript 𝐸 𝑛 E_{n} and corresponding ν ¯ ¯ 𝜈 \bar{\nu} values from the present work.

$E_n$ (MeV)	$\overline{\nu }$	$E_n$ (MeV)	$\overline{\nu }$
$0.25$	2.97(8)	$13.75$	4.87(15)
$0.75$	3.08(9)	$14.25$	4.95(16)
$1.25$	3.17(9)	$14.75$	4.92(16)
$1.75$	3.17(9)	$15.25$	4.95(16)
$2.25$	3.23(9)	$15.75$	5.13(17)
$2.75$	3.35(10)	$16.25$	5.24(17)
$3.25$	3.48(10)	$16.75$	5.24(17)
$3.75$	3.48(10)	$17.25$	5.16(17)
$4.25$	3.55(11)	$17.75$	5.26(17)
$4.75$	3.62(11)	$18.25$	5.36(17)
$5.25$	3.78(12)	$18.75$	5.38(18)
$5.75$	3.79(12)	$19.25$	5.67(19)
$6.25$	3.89(12)	$19.75$	5.58(19)
$6.75$	4.00(12)	$20.25$	5.48(20)
$7.25$	4.06(12)	$20.75$	5.80(20)
$7.75$	4.16(12)	$21.25$	5.83(21)
$8.25$	4.36(13)	$21.75$	6.06(23)
$8.75$	4.33(13)	$22.25$	5.96(24)
$9.25$	4.26(13)	$22.75$	6.14(25)
$9.75$	4.36(13)	$23.25$	6.06(26)
$10.25$	4.46(14)	$23.75$	6.07(27)
$10.75$	4.49(14)	$24.25$	6.03(29)
$11.25$	4.49(14)	$24.75$	6.19(34)
$11.75$	4.67(15)	$25.25$	6.53(39)
$12.25$	4.73(15)	$25.75$	6.34(44)
$12.75$	4.93(16)	$26.25$	6.56(55)
$13.25$	4.84(15)

Equations

$$n{+}^{239}\text{Pu}{\to }^{240}{\text{Pu}}^{\ast }\to \text{LF}+\text{HF}+\nu n$$

$$\alpha {+}^{240}\text{Pu}\to {\alpha }^{\prime }{+}^{240}{\text{Pu}}^{\ast }\to {\alpha }^{\prime }+\text{LF}+\text{HF}+\nu n,$$

$${\sigma }_{n,f}(E_n)=\sum _{J,\pi }{\sigma }_n^{CN}(E_{\text{ex}},J,\pi )G_f^{CN}(E_{\text{ex}},J,\pi ),$$

$${\sigma }_{n,f}(E_n)={\sigma }_n^{CN}(E_{\text{ex}})G_f^{CN}(E_{\text{ex}}).$$

$${\sigma }_{\alpha ,{\alpha }^{\prime }f}(E_{\alpha })={\sigma }_{\alpha ,{\alpha }^{\prime }}^{CN}(E_{\text{ex}})G_f^{CN}(E_{\text{ex}}).$$

$$\bar{\nu }(E_n)=\frac{N_n(E_n)}{N_{\alpha -f}(E_n){ϵ}_n},$$

$$E_{\text{lin}}=[(E_{\Delta E}+E_E{)}^{1.75}-E_E^{1.75}{]}^{1/1.75}.$$

$$E_{\text{ex}}=E_{\alpha }-E_{{\alpha }^{\prime }}-E_r.$$

$$E_n=\frac{m_t+m_n}{m_t}(E_{\text{ex}}-S_n),$$

Full text

I Introduction

The average prompt-fission-neutron multiplicity $\bar{\nu}$ following ($n$,$f$) reactions is important to both basic and applied physics. In nuclear theory, measurements of $\bar{\nu}$ can be used to validate fission models and provide constraints on the fission process itself Maslov et al. (2004). In the area of international safeguards and verification, nuclear materials are assayed with passive neutron-multiplicity counting, and here, $\bar{\nu}$ is needed to determine the amount of neutron-induced fission (or self-multiplication) in the sample Ensslin et al. (1998); Shin et al. (2017). For proposed nuclear reactor concepts, such as accelerator-driven systems (ADS) and those based on the thorium-uranium cycle, there is interest in the $\bar{\nu}$ values for short-lived actinides, as the dependence of $\bar{\nu}$ on the incident neutron energy is important for determining the criticality, safety, and lifetime of these reactors Ethvignot et al. (2005); Kerdraon et al. (2003); Nifenecker et al. (2001). In addition, $\bar{\nu}$ for short-lived actinides is also relevant to transmutation of radioactive waste with ADS Ethvignot et al. (2005); Kerdraon et al. (2003); Nifenecker et al. (2001).

Directly measuring $\bar{\nu}$ presents a number of experimental challenges, including producing high-flux neutron beams and addressing beam-related backgrounds. For short-lived actinides, $\bar{\nu}$ data are particularly sparse due to the fact that target fabrication and high target activity are also issues. These challenges can be bypassed with the surrogate-reaction method Escher et al. (2012), an indirect measurement technique that has typically been used to determine the cross sections of reactions that proceed through a highly excited, statistically equilibrated compound nuclear state. In a surrogate experiment, the desired compound nucleus (CN) is produced using an alternative (“surrogate”) reaction with a more experimentally accessible or preferable combination of projectile and target nucleus. The surrogate method has been demonstrated to work well for determining ($n$,$f$) reaction cross sections of various actinides Petit et al. (2004); Plettner et al. (2005); Burke et al. (2006); Escher and Dietrich (2006); Lesher et al. (2009); Ressler et al. (2011); the values obtained are within $\sim$5–20% of direct neutron measurements. The present work extends the applicability of this technique to determining $\bar{\nu}$. Benchmarking has been performed by using the surrogate reactions 240Pu($\alpha$,$\alpha^{\prime}f$) and 242Pu($\alpha$,$\alpha^{\prime}f$) to obtain $\bar{\nu}$ as a function of incident neutron energy for the reactions 239Pu($n$,$f$) and 241Pu($n$,$f$), respectively, for which direct-measurement data are available. The results for 239Pu($n$,$f$) are discussed in this paper, while those for 241Pu($n$,$f$) can be found in Ref. Akindele et al. (2019).

II Surrogate-reaction technique

In the present work, the compound nucleus 240Pu in the desired reaction

[TABLE]

is produced via the surrogate reaction

[TABLE]

where LF and HF are the light and heavy fission fragments, respectively, and $\nu$ is the prompt-fission-neutron multiplicity. Assuming a statistically equilibrated CN, where the decay is independent of the method of formation Bohr (1936), the ($n$,$f$) cross section for an incident neutron energy $E_{n}$ is given by the following Hauser-Feshbach Hauser and Feshbach (1952); Fröbrich and Lipperheide (1996); Thompson and Nunes (2009) formula:

[TABLE]

where $\sigma_{n}^{CN}(E_{\text{ex}},J,\pi)$ is the cross section for forming a CN with excitation energy $E_{\text{ex}}$, angular momentum $J$, and parity $\pi$, and $G_{f}^{CN}(E_{\text{ex}},J,\pi)$ is the probability that the CN will fission. In the Weisskopf-Ewing limit of Hauser-Feshbach theory, where the decay of the CN is independent of $J$ and $\pi$, Eq. 3 reduces to

[TABLE]

Analogously, the ($\alpha$,$\alpha^{\prime}f$) cross section for an incident $\alpha$-particle energy $E_{\alpha}$ is given by

[TABLE]

In Eq. 4 and 5, $\sigma_{n}^{CN}(E_{\text{ex}})$ and $\sigma_{\alpha,\alpha^{\prime}}^{CN}(E_{\text{ex}})$ are the $J\pi$-independent CN-formation cross sections and $G_{f}^{CN}(E_{\text{ex}})$ is the $J\pi$-independent fission probability of the CN. If the Weisskopf-Ewing approximation applies, then ($n$,$f$) and ($\alpha$,$\alpha^{\prime}f$) reactions that generate the same CN with excitation energy $E_{\text{ex}}$ will have identical values of $G_{f}^{CN}(E_{\text{ex}})$ and yield the same $\bar{\nu}$. The validity of this assumption is tested by comparing the $\bar{\nu}$ values obtained with the surrogate reaction 240Pu($\alpha$,$\alpha^{\prime}f$) to those determined from direct 239Pu($n$,$f$) measurements.

III Experiment

The experiment was performed in Cave 4 of the Texas A&M University Cyclotron Institute Tabacaru et al. (2019). A 240Pu target was loaded onto a target wheel Lesher et al. (2010) located at the center of the NeutronSTARS array Akindele et al. (2017) and bombarded with a 100-pA beam of 53.9-MeV alpha particles from the K150 Cyclotron; 4.75 days’ worth of data was collected.

III.1 Targets

The 240Pu target was 99.995%-pure; it was fabricated by first epoxying a 100-$\mu$g/cm2-thick natural-carbon foil to an aluminum frame, and then electroplating plutonium onto the foil surface, covering a circular area 1.90 cm in diameter. Properties of the target are given in Table 1.

The following calibration targets were included in the experiment: a 208Pb foil to determine the beam energy; a natural-carbon foil, Mylar ((C10H8O4)${}_{\text{n}}$) foil, and empty aluminum frame to assess backgrounds due to $\alpha$ interactions with carbon, oxygen and aluminum in the 240Pu target. Two phosphor targets were also used for beam alignment and observing the beam-spot size.

III.2 Apparatus

The NeutronSTARS array is shown in Fig. 1. Charged particles, including inelastically scattered $\alpha$ particles from 240Pu($\alpha$,$\alpha^{\prime}f$) reactions, were detected with a silicon telescope located 19 mm downstream from the target and consisting of two Micron S2-type annular silicon detectors (a 152-$\mu$m-thick $\Delta E$ detector and a 994-$\mu$m-thick $E$ detector) that were separated by 4 mm. The energy loss in the two detectors was used for particle identification. A 4.44-mg/cm2-thick aluminum-foil shield was placed between the target and the telescope to prevent fission fragments and $\delta$ electrons produced in the target from damaging the $\Delta E$ detector and degrading detector performance. Fission fragments were detected with a third 146-$\mu$m-thick Micron S2 silicon detector located 19 mm upstream from the target. The silicon detectors are segmented into 48 0.5-mm-wide rings on one side and 16 22.5∘-wide sectors on the other. For this experiment, pairs of adjacent rings and sectors were bussed together to form 24 1-mm-wide rings and 8 45∘-wide sectors. The silicon detectors are also coated with 27-$\mu$g/cm2 aluminum contacts on the ring side and 500-$\mu$g/cm2 gold contacts on the sector side. The gold can significantly straggle the fission fragments, making energy separation between scattered $\alpha$ particles and fission fragments difficult. To minimize straggling, the fission detector was installed with the ring side facing downstream and the 240Pu target was mounted with the electroplated surface facing upstream.

The target wheel and silicon detectors were mounted inside a vacuum chamber, which was surrounded by a neutron detector (referred to as “NeutronBall”) consisting of a tank filled with 3.5 tons of liquid scintillator. The tank is segmented into six regions: four identical quadrants that make up the central cylinder and two endcaps. Twenty photomultiplier tubes (PMTs), three on each quadrant and four on each endcap, are used to measure scintillation light. At the time of the measurement, the central cylinder was filled with fresh EJ-335 liquid scintillator doped with 0.25-wt% of natural gadolinium Elj ; however the two endcaps contained degraded liquid scintillator with poor optical transmission. Therefore, in the present work, only events detected by the twelve PMTs on the central cylinder were included in the data analysis.

III.3 Detector calibrations

For the $\Delta E$ and $E$ detectors, the response of each ring and sector was calibrated with a 226Ra $\alpha$ point source that provided the following $\alpha$ lines: 4784, 5304, 5489, 6002, and 7687 keV Kinsey et al. (1996). At 7687 keV, the resulting 1$\sigma$ energy resolutions for the $\Delta E$ detector and $E$ detector were approximately 40 keV and 24 keV, respectively. The fission detector was calibrated with a 252Cf spontaneous fission source. The light and heavy fission-product mass peaks were used to gain match the response of the rings. For NeutronBall, 60Co and 228Th $\gamma$-ray point sources provided calibration points at 1253 keV (the average energy of the 1173-keV and 1332-keV $\gamma$ rays from 60Co) and 2615 keV (from 208Tl in the 228Th decay chain) Kinsey et al. (1996). Another calibration point was provided by the 4440-keV $\gamma$ rays Kinsey et al. (1996) that were emitted following inelastic $\alpha$ scattering with the natural-carbon target that promoted 12C to its first excited state. The energy resolution of the liquid scintillator at energy $E$ (in MeV) was $\sigma(E)/E=25\%/\sqrt{E}$ Akindele et al. (2017).

The efficiency for detecting a single neutron with the central cylinder of NeutronBall was determined to be 0.504(5) and was measured by placing a 252Cf fission source at the target position. More details will be given in Sec. IV.3.

III.4 $\alpha$-particle beam

The $\alpha$-particle beam-spot size was approximately 3 mm in diameter and was observed with an in-vacuum camera that imaged the phosphor targets. The exact beam energy provided by the K150 Cyclotron was determined from data collected for the 208Pb target. Scattering of $\alpha$ particles to discrete states in 208Pb was used as an in situ calibration. The beam energy was determined to be 53.9(1) MeV. This value allowed the excitation energy of the 208Pb nucleus to be properly reconstructed after taking into account the energy deposition in the $\Delta E$-$E$ telescope, the energy loss in dead layers (i.e., the target, the aluminum-foil shield, and the gold and aluminum contacts on the surfaces of the silicon detectors), and the recoil energy of the 208Pb nucleus. The uncertainty in the beam energy was taken to be the 1$\sigma$ width of the $\alpha$ peak corresponding to elastic scattering.

IV Analysis and results

A 240Pu($\alpha$,$\alpha^{\prime}f$) interaction was indicated by a coincidence between an $\alpha$ particle hitting the silicon telescope and a fission fragment hitting the fission detector. For a 240Pu CN with excitation energy $E_{\text{ex}}$, corresponding to an equivalent incident neutron energy $E_{n}$, the average prompt-fission-neutron multiplicity was determined from

[TABLE]

where $N_{\alpha-f}(E_{n})$ is the number of measured 240Pu($\alpha$,$\alpha^{\prime}f$) $\alpha$-fission coincidences at $E_{n}$, $N_{n}(E_{n})$ is the number of detected prompt fission neutrons associated with these coincidences, and $\epsilon_{n}$ is the single-neutron detection efficiency for the central cylinder of NeutronBall. The analysis performed to obtain the quantities in Eq. 6 is discussed in this section, and the resulting $\bar{\nu}(E_{n})$ distribution is given.

IV.1 Particle identification and event selection

IV.1.1 Charged particles

For events in the silicon telescope, the energies deposited in the $\Delta E$ and $E$ detectors ($E_{\Delta E}$ and $E_{E}$, respectively) were used for particle identification (PID). Protons, deuterons, tritons, 3He, and $\alpha$ particles were distinguished by plotting the “linearized energy” $E_{\text{lin}}$ Goulding et al. (1964) versus the total energy deposition in both the $\Delta E$ and $E$ detectors, where

[TABLE]

Alpha-particle events were isolated by generating a PID plot for each $\Delta E$-detector ring (e.g., Fig. 2) and gating on the region above 3He ($E_{\text{lin}}$ approximately between 16.5–24).

IV.1.2 Fission

Fig. 3 shows the gain-matched spectrum measured by a single ring on the fission detector for $\alpha$ particles incident on the 240Pu target. A double hump is present at higher energies due to heavy and light fission fragments hitting the detector. The large peak at lower energies is primarily due to light ions from 240Pu $\alpha$ decay and $\alpha$-particle interactions with carbon and oxygen in the 240Pu target, which was confirmed by analysis of the data collected for the natural-carbon and Mylar targets. For each ring, fission events were selected and light-ion events removed by cutting above an energy deposition of 47 (arb. units).

IV.1.3 240Pu($\alpha$,$\alpha^{\prime}f$) events

In Fig. 4, the time difference between coincident $\Delta E$-$E$ $\alpha$-particle and fission-detector events is plotted; the energy deposited in the fission detector is given along the y axis. A horizontal line is drawn at the energy cut-off used to isolate fission fragments from light ions. Coincidences above the cut-off with a time difference between $-35$ ns and 86 ns (“prompt” region) were tagged as 240Pu($\alpha$,$\alpha^{\prime}f$) events. The small bursts of events present every 121 ns in Fig. 4 coincide with the K150 cyclotron frequency and are due to random coincidences such as an $\alpha$ particle hitting the $\Delta E$-$E$ telescope and a fission fragment from a 240Pu($\alpha$,$f$) reaction in the target hitting the fission detector.

IV.1.4 Neutrons

PMT signals that arrived within a coincidence window of 200 ns were assumed to come from a single event in NeutronBall, e.g., a neutron capture on gadolinium, or an interaction of a room-background $\gamma$ ray. These signals were first gain matched, as described in Ref. Akindele et al. (2017), then summed together to acquire the total energy deposited by the event. Only events with energy greater than 2 MeV were included in the data analysis to exclude most of the contribution from backgrounds and electronic noise.

For the tagged 240Pu($\alpha$,$\alpha^{\prime}f$) events, a timing gate was opened 50 $\mu$s before and closed 500 $\mu$s after the $\alpha$-fission coincidence. The time difference between a NeutronBall event occurring within this gate and the $\alpha$-fission coincidence was plotted (Fig. 5). The sharp peak around 0 $\mu$s in Fig. 5 is from the flash of prompt $\gamma$ rays following fission and from proton recoils generated during thermalization of the neutron in the liquid scintillator. The broad peak above 0 $\mu$s is attributed to prompt fission neutrons; its width is determined by the moderation time of the neutrons in the scintillator. Both features lie on top of a flat background due to random coincidences.

IV.2 Equivalent neutron energy

The excitation energy $E_{\text{ex}}$ of 240Pu following inelastic $\alpha$-particle scattering was determined from the beam energy $E_{\alpha}$, the scattered-$\alpha$-particle energy $E_{\alpha^{\prime}}$, and the 240Pu recoil energy $E_{r}$:

[TABLE]

The value of $E_{\alpha^{\prime}}$ was the total energy deposited in the $\Delta E$-$E$ telescope corrected for energy losses in the target, the $\delta$ shield, and the inert gold and aluminum contacts on the surfaces of the silicon detectors. The equivalent incident neutron energy $E_{n}$ was then determined from

[TABLE]

where $m_{t}$ is the mass of 239Pu, $m_{n}$ is the neutron mass, and $S_{n}$ is the neutron separation energy for 240Pu. Fig. 6 shows the $E_{n}$ distribution for 240Pu($\alpha$,$\alpha^{\prime}f$) events. The corresponding 240Pu excitation energy is also given. Fission of 240Pu starts to occur at ${E_{n}=-1.61}$ MeV (4.9-MeV 240Pu excitation energy) Northrop et al. (1959). The feature at ${E_{n}\sim 5.5}$ MeV is due to 240Pu second-chance fission Lestone and Strother (2014); Stetcu et al. (2016), and above ${E_{n}\sim 18.5}$ MeV, the number of events tapers off quickly due to the $\alpha$-Pu Coulomb barrier.

IV.3 Average prompt-fission-neutron multiplicity

The average prompt-fission-neutron multiplicity was obtained with Eq. 6 for equivalent incident neutron energies ranging between 0.25 and 26.25 MeV. The quantity $N_{\alpha-f}(E_{n})$ in Eq. 6 is the number of $\alpha$-fission coincidences in the (121-ns-wide) prompt region of Fig. 4, corrected for the contribution from random coincidences. This contribution was determined by taking the sum of $\alpha$-fission coincidences in the region 207–1901 ns and scaling down to a 121-ns-wide time window.

The number of neutrons $N_{n}(E_{n})$ was obtained by taking the difference between the total counts in the time regions 2 to 44 $\mu$s and $-45$ to $-3$ $\mu$s in Fig. 5. The contribution from random $\alpha$-fission coincidences was determined from the time-difference spectrum for NeutronBall events associated with the 207–1901-ns region in Fig. 4 (scaled down to correspond to a 121-ns-wide $\alpha$-fission time window).

A single-neutron detection efficiency of ${\epsilon_{n}=0.504(5)}$ was obtained by first recording the time-difference between 252Cf fission events in the fission detector and events in NeutronBall. The total number of prompt neutrons measured was then determined and divided by the number of fission events and $\bar{\nu}$ for 252Cf (i.e., 3.757) Boldeman and Hines (1985); Boldeman and Dalton (1967).

The $\bar{\nu}(E_{n})$ distribution obtained is given in Fig. 7. Each $\bar{\nu}$ value and its uncertainty is also provided in Table 2; the uncertainty is dominated by the statistical uncertainties in the number of $\alpha$-fission coincidences and the number of detected neutrons. Fig. 7 also shows that the results of the present work are consistent with direct neutron measurements for 239Pu($n$,$f$) Frehaut et al. (1980); Frehaut (1980); Soleilhac et al. (1969); Condé et al. (1968); Mather et al. (1965); Hopkins and Diven (1963), providing validation that the surrogate-reaction method can be used to determine $\bar{\nu}$ for actinides.

V Summary and Conclusions

240Pu($\alpha$,$\alpha^{\prime}f$) was used as a surrogate reaction to determine the 239Pu($n$,$f$) prompt-fission-neutron multiplicity as a function of incident neutron energy from 0.25–26.25 MeV. This is the first time $\bar{\nu}$ for 239Pu($n$,$f$) has been obtained continuously over this neutron energy range in a single measurement. The results of the present work are in good agreement with those from direct neutron measurements Frehaut et al. (1980); Frehaut (1980); Soleilhac et al. (1969); Condé et al. (1968); Mather et al. (1965); Hopkins and Diven (1963). Similar conclusions were drawn in Ref. Akindele et al. (2019), where the surrogate reaction 242Pu($\alpha$,$\alpha^{\prime}f$) was used to determine $\bar{\nu}$ for 241Pu($n$,$f$). The success of these two experiments opens the door to using surrogate reactions to obtain $\bar{\nu}(E_{n})$ for a whole host of short-lived actinides that are currently inaccessible via direct methods.

VI Acknowledgments

We thank the staff of the Texas A&M Cyclotron Institute for facilitating operations and facilities needed to perform this measurement. This work was performed under the auspices of the U.S. Department of Energy National Nuclear Security Administration by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344, under Award No. DE-NA0000979, and through the Nuclear Science and Security Consortium under Award No. DE-NA-0003180.