1 Introduction

In order to improve the public acceptance of nuclear energy and facilitate the sustainable use of uranium resources, an increasing number of studies on accelerator-driven sub-critical systems (ADSs) are being conducted globally. The ADS is a promising approach for burning the uranium, plutonium, and minor actinides in the spent fuel with the fast neutrons to generate energy while transmuting the long-lived radioactive fission products with the high-flux neutrons. The new nuclear energy system places new requirements of high-precision nuclear data of neutron-induced reactions on many related nuclides at incident energies below 200 MeV [1, 2]. Plutonium is an important component of the spent fuel and depletion chain, and thus, accurate nuclear data of neutron-induced reactions on plutonium isotopes are essential for the design of an ADS. In our previous study [3], the theoretical analysis and evaluation of an n+²³⁹Pu reaction were introduced. In this study, we focus on neutron-induced reactions on ²⁴⁰Pu, ²⁴²Pu, and ²⁴⁴Pu. In addition, the theoretical study of n+^240,242,244Pu reactions is important for understanding the reaction mechanism of fertile nuclei and their structure.

The evaluation data of n+^240,242,244Pu reactions for an incident energy below 20 MeV are provided in the JENDL-4.0 database [4]. The nuclear data of an n+²⁴⁰Pu reaction for an incident energy below 30 MeV are given in ENDF/B-VII.1 [5], and the data for the n+^242,244Pu reactions from JENDL-4.0 are adopted in the ENDF/B-VII.1 with a few modifications to the resonance cross sections and (n,γ) cross section for the n+²⁴²Pu reaction. The data regarding light-charged-particle (proton, deuteron, triton, helium-3, and alpha) emission are not available in ENDF/B-VII.1 and JENDL-4.0. For meeting the ADS design requirements, the energy maximum of the nuclear data is required to be extended to 200 MeV. Subsequent to the evaluations for ENDF/B-VII.1 and JENDL-4.0, some new experimental data have been provided; in particular, some fission and capture cross sections were measured at the Los Alamos Neutron Science Center (LANSCE) using world-class advanced experimental facilities in recent years. These new experimental data are also required to be considered in the new evaluation.

Experimental data for n+^240,242,244Pu reactions are lacking, and there exist some discrepancies among the experimental data from various groups. Thus, self-consistent calculation of each reaction using theoretical models is important. In this work, all cross sections, angular distributions, energy-angle distributions of neutron and light-charged-particle emissions, number of neutrons per fission, and prompt fission neutron spectra are calculated for an incident energy below 200 MeV. The optical model, distorted-wave Born approximation (DWBA), preequilibrium and equilibrium emission theories, fission model, intranuclear cascade model, and recent experimental data are used. The calculated results are analyzed and compared with the experimental data and the evaluation data obtained from ENDF/B-VII.1 and JENDL-4.0.

In Sect. 2, the theoretical models are briefly outlined, and in Sect. 3, the present results are compared with the experimental data and analyzed. Finally, a summary is presented in Sect. 4.

2 Theoretical model

The optical model is the basis of nuclear reaction analyses. Here, the phenomenological global optical model potential (GOP) from Ref. [6], which has a good prediction power for nucleon-actinide reactions, is adopted to describe the interactions between a neutron and ^240,242,244Pu. The total, nonelastic, elastic cross sections, and elastic scattering angular distributions are calculated using the GOP. The DWBA is applied to obtain the direct inelastic scatterings to the discrete levels of ^240,242,244Pu. The experimental data of the inelastic cross sections and angular distributions are used to guide the theoretical calculation. The GOP obtained from Ref. [6] is also applied in the DWBA calculation. The corresponding calculation code applied is DWUCK4 [7].

The equilibrium and preequilibrium emission processes at incident energies below 20 MeV are analyzed using the unified Hauser–Feshbach and exciton model [8]. The Hauser–Feshbach theory with width fluctuation correction describes the compound-nucleus decay. The angular-momentum-dependent exciton model, which takes into consideration the parity and angular momentum conservations, describes the preequilibrium emission process. The summation of the occupation probabilities of preequilibrium and equilibrium states is 1. The multi-step Hauser–Feshbach model is used to calculate secondary particle emissions. The back-shifted Fermi gas level densities [9] are applied to describe the continuum states of compound nuclei. The recoil effect is considered in all reaction processes. The energy-angle distributions of emitted particles are calculated using the angular-momentum-dependent exciton state lifetimes, which are obtained by solving the generalized master equation. The improved Iwamoto–Harada pickup mechanism [10, 11] is used in the description of the complex particle (d, t, ³He, α) emissions.

The preequilibrium and equilibrium particle emission in the incident energy region above 20 MeV are calculated using the intranuclear cascade model, exciton model, evaporation model, and Hauser–Feshbach theory with width fluctuation correction. The 1st to the 18th particle emissions from equilibrium states are analyzed using the Hauser–Feshbach theory with the width fluctuation correction and evaporation model. Ignatyuk’s description of nuclear level densities [12] is applied to describe the continuum states of compound nuclei. The first to the fifth particle emissions in the preequilibrium reactions are analyzed using the exciton model, and the cascade emissions of one to four nucleons are calculated using the empirical formula presented in Ref. [13]. The energy-angle distributions of emitted particles at incident energies greater than 20 MeV are obtained by using the Kalbach systematics [14, 15]. The energy-dependent improved Iwamoto–Harada pickup mechanism is also used in this energy region above 20 MeV to describe the composite particle emissions. The UNF code [16] and MEND code [17] are applied below and above 20 MeV, respectively, as in Ref. [18].

The optical model potentials for a nucleon [6], deuteron [19], triton [20], helium-3 [21], and alpha [22] are used to calculate inverse cross sections and transmission coefficients.

Fission is considered as a compound-nucleus decay channel. Uncoupled fission barriers are used to describe (n,f), (n,nf), (n,2nf),..., (n,15nf) fissions. There are a series of transition states characterized by the excited energy above the barrier, spin, and parity at each barrier. The continuum transition states are described by the Gilbert–Cameron level density [23]. The Bohr–Wheeler theory [24, 25] is used to calculate the fission rate. The height and curvature parameters of the fission barriers, level density parameters, and pair corrections at saddle points are obtained by fitting the experimental data. The fission spectra are calculated using the Madland–Nix formula [26].

3 Results and analysis

3.1 n+²⁴⁰Pu reaction

The calculated total cross section of the n+²⁴⁰Pu reaction is compared with experimental data [27-29] and evaluated data obtained from ENDF/B-VII.1 and JENDL-4.0 in Fig. 1. The obtained result reproduces the experimental data rather well and has a slight discrepancy with the ENDF/B-VII.1 and JENDL-4.0 data. It should be noted that all the resonance cross sections in this work are obtained from JENDL-4.0 and are not plotted in the figures. Each curve reduces to 0 at the upper limit of the resonance energy region.

Fig. 1.Full-screen View

(Color online) Calculated total cross section (solid line) for the n+²⁴⁰Pu reaction as compared with the experimental data (solid circles [27], solid squares [28], and crosses [29]) and the evaluated data obtained from ENDF/B-VII.1 (short-dashed line) and JENDL-4.0 (dashed line).

A set of consistent experimental data [27] of the cross section and angular distribution of neutron elastic scattering from ²⁴⁰Pu was presented by Smith et al. in 1972. The calculated elastic scattering cross section is in good agreement with the experimental data for an incident energy of less than 1 MeV but is less than that of the experimental data for an incident energy between 1 and 2 MeV, as shown in Fig. 2. Similarly, the calculated result of the elastic scattering angular distribution is in good agreement with the experimental data for an incident energy less than 0.8 MeV but is less than that of the experimental data for back angles for an incident energy of 1.2 MeV, as shown in Fig. 3. Another set of experimental data [30] regarding the scattering cross section contributed by the neutron elastic scattering and inelastic scatterings to the first, second, and third excited states of ²⁴⁰Pu was presented by Smith et al. in 1982. The calculated result reproduces the later experimental data well in the incident energy range of 1 to 4 MeV, as shown in Fig. 4. It should be noted that the two sets of experimental data contradict each other at incident energies greater than 1 MeV, where the experimental data on the elastic scattering cross section obtained from Ref. [27] are even greater than those obtained from Ref. [30], which comprise the contribution of not only the elastic scattering but also some inelastic scatterings.

Fig. 2.Full-screen View

(Color online) Same as Fig. 1, but for neutron elastic scattering cross section.

Fig. 3.Full-screen View

Elastic scattering angular distribution for n+²⁴⁰Pu reaction. The data are shifted downward by factors of 10⁰, 10^-1, 10^-2, and 10^-3.

Fig. 4.Full-screen View

Cross sections of neutron elastic and inelastic scatterings to the first, second, and third excited levels of ²⁴⁰Pu. The dashed line indicates the calculated elastic scattering cross section, while the solid line and circles denote the calculated result and experimental data of the total contribution from the elastic channel and the three inelastic scattering channels, respectively.

Figures 5, 6, 7, and 8 show the comparisons of the calculated cross sections of the neutron inelastic scatterings to the 1st, 2nd, 5th, and 9th–11th excited states of ²⁴⁰Pu with experimental data [27], respectively. The calculated results are in reasonable agreement with the experimental data.

Fig. 5.Full-screen View

Calculated cross section (solid line) of neutron inelastic scattering to the first excited level of ²⁴⁰Pu as compared with the experimental (symbols), ENDF/B-VII.1 (short-dashed line), and JENDL-4.0 (dashed line) data.

Fig. 6.Full-screen View

Same as Fig. 5, but for neutron inelastic scattering to the second excited level of ²⁴⁰Pu.

Fig. 7.Full-screen View

Same as Fig. 5, but for neutron inelastic scattering to the fifth excited level of ²⁴⁰Pu.

Fig. 8.Full-screen View

Same as Fig. 5, but for neutron inelastic scatterings to the 9th, 10th, and 11th excited states of ²⁴⁰Pu.

The calculated (n, γ) cross section is compared with the experimental data [31] presented in Fig. 9. The calculated result reproduces the experimental data well.

Fig. 9.Full-screen View

(Color online) Same as Fig. 1, but for (n, γ) cross section.

The comparison of the calculated fission cross section with the experimental data [32-36] for an incident energy less than 200.0 MeV is shown in Fig. 10. The present result is in good agreement with the recently obtained experimental data [32-34], and it is slightly greater than that of the ENDF/B-VII.1 and JENDL-4.0 data for incident energies less than 20 MeV, where the present result fits the recent experimental data obtained from Refs. [32, 33] better, while the ENDF/B-VII.1 and JENDL-4.0 data fit the experimental data obtained from Refs. [35, 36] better.

Fig. 10.Full-screen View

(Color online) Calculated fission cross section (solid lines) for n+²⁴⁰Pu reaction as compared with the experimental data (solid circles [32], crosses [33], lower triangles [34], squares [35], upper triangles [36]), and ENDF/B-VII.1 (short-dashed lines) and JENDL-4.0 (dashed lines) data. (a) Incident energies less than 20 MeV, (b) Incident energies less than 200 MeV.

The number of neutrons per fission obtained using systematics is shown in Fig. 11. A good agreement with the experimental data [37-40] is observed.

Fig. 11.Full-screen View

(Color online) Calculated result of the number of neutrons per fission (solid line) for n+²⁴⁰Pu reaction as compared with the experimental (symbols), ENDF/B-VII.1 (short-dashed line), and JENDL-4.0 (dashed line) data.

3.2 n+²⁴²Pu reaction

The calculated total cross section is shown in Fig. 12. It can be observed that the calculated result reproduces the experimental data [41, 42] well and has some discrepancies with the data obtained from JENDL-4.0 at low energies.

Fig. 12.Full-screen View

(Color online) Calculated total cross section (solid line) for n+²⁴²Pu reaction as compared with the experimental (symbols) and JENDL-4.0 (dashed line) data.

Two sets of consistent experimental data regarding the cross sections and angular distributions of neutron elastic scattering and inelastic scatterings to the first and second excited levels of ²⁴²Pu were presented by Drake et al. [43] and Haouat et al. [44], respectively. The present results reproduce the experimental data well, as shown in Figs. 13, 14, 15, 16 and 17, and are superior to the JENDL-4.0 data in fitting the experimental data of the (n,n₂) reaction cross section.

Fig. 13.Full-screen View

(Color online) Same as Fig. 12, but for neutron elastic scattering cross section.

Fig. 14.Full-screen View

Elastic scattering angular distributions for n+²⁴²Pu reaction. (a) The experimental data (symbols) are obtained from Ref. [43]. (b) The experimental data (symbols) are obtained from Ref. [44]. The data are shifted downward by factors of 10⁰, 10^-1, 10^-2, etc.

Fig. 15.Full-screen View

(a) Calculated angular distributions of neutron inelastic scatterings to the first and second excited levels of ²⁴²Pu (solid lines) as compared with the experimental data (symbols) obtained from Ref. [43]; (b) Calculated angular distribution of the neutron inelastic scattering to the first excited level of ²⁴²Pu (solid lines) as compared with the experimental data (symbols) obtained from Ref. [44]; (c) Calculated angular distribution of the neutron inelastic scattering to the second excited level of ²⁴²Pu (solid lines) as compared with the experimental data (symbols) obtained from Ref. [44]. The data are shifted downward by factors of 10⁰, 10^-1, 10^-2, etc.

Fig. 16.Full-screen View

Calculated cross section of neutron inelastic scattering to the first excited level of ²⁴²Pu (solid line) as compared with the experimental (symbols) and JENDL-4.0 (dashed line) data.

Fig. 17.Full-screen View

Same as Fig. 16, but for neutron inelastic scattering to the second excited level of ²⁴²Pu.

Two sets of (n, γ) cross sections were measured by Hockenbury et al. [45] in 1975 and Buckner et al. [46] in 2016, respectively. The experimental data obtained from Ref. [45] are approximately three times greater than those obtained from Ref. [46], which were measured at LANSCE. These experimental data are compared with the present model calculation and the ENDF/B-VII.1 and JENDL-4.0 data in Fig. 18. It can be observed that the data from ENDF/B-VII.1 and JENDL-4.0 pass through the experimental data obtained from Ref. [45], while our result is consistent with the new experimental data obtained from Ref. [46].

Fig. 18.Full-screen View

(Color online) Calculated result of ²⁴²Pu(n, γ) reaction cross section (solid line) as compared with the experimental (solid circles [45], solid squares [46]), ENDF/B-VII.1 (short-dashed line), and JENDL-4.0 (dashed line) data.

The comparison of the calculated fission cross section with the experimental data [32, 35, 36, 47, 48] for incident energies less than 200 MeV is shown in Fig. 19. The obtained result is in good agreement with the recent experimental data obtained from Refs. [32, 47, 48].

Fig. 19.Full-screen View

(Color online) Calculated fission cross section (solid lines) for n+²⁴²Pu reaction as compared with the experimental data (solid squares [32], lower triangles [35], upper triangles [36], crosses [47], solid circles [48]), and JENDL-4.0 data (dashed lines). (a) Incident energies less than 20 MeV, (b) Incident energies less than 200 MeV.

The number of neutrons per fission obtained using systematics is shown in Fig. 20. A good agreement with the experimental data [40] is observed.

Fig. 20.Full-screen View

(Color online) Calculated result of the number of neutrons per fission (solid line) for n+²⁴²Pu reaction as compared with the experimental (symbols) and JENDL-4.0 (dashed line) data.

3.3 n+²⁴⁴Pu reaction

At present, experimental data [35, 36, 49-51] regarding fission cross section for an n+²⁴⁴Pu reaction are limited. The calculated result is in good agreement with those obtained from Ref. [36] for incident energies less than 140 MeV but is less than those for incident energies less than 140 MeV, as shown in Fig. 21. A trend can be observed in Figs. 10, 19, and 21 that the fission cross sections decrease as the neutron number increases.

Fig. 21.Full-screen View

(Color online) Calculated fission cross section (solid lines) for n+²⁴⁴Pu reaction as compared with the experimental data (squares [35], lowertriangles [36], solid circles [49], crosses [50], upper triangles [51]), and JENDL-4.0 data(dashed lines). (a) Incident energies less than 20 MeV, (b) Incident energies less than 200 MeV.

No experimental data are available regarding the cross sections of the total inelastic scattering, (n,2n), (n,3n) reactions, energy spectra, and energy-angle distributions of neutron and light-charged-particle emissions for n+^240,242,244Pu reactions. All these data are predicted by the theoretical models.

4 Summary

All cross sections, angular distributions, energy spectra, and energy-angle distributions for neutron-induced reactions on ²⁴⁰Pu, ²⁴²Pu, and ²⁴⁴Pu are consistently calculated using theoretical nuclear models at incident energies less than 200 MeV. The calculated results are generally in good agreement with the experimental data and reproduce them better than the evaluated data from ENDF/B-VII.1 and JENDL-4.0 for some reaction channels. The present results have been transformed into ENDF-B formatted data for the purpose of their application. The code for analyzing the covariances of the data is currently under development. The covariances will be given in the next step.