1 Introduction
Careful study of nucleon-induced reactions, in general, is a crucial step for the further development of nuclear reaction theories. Complete information in this field is also strongly needed for many applications. The applications require accurate nuclear reaction data of common cross sections and, especially, data of neutron and proton induced energy-angle correlated spectra of secondary light particles (neutron, proton, deuteron, triton, heliums-3, and alpha-particles), as well as γ-ray production cross sections and γ-ray production energy spectra. Accurate nuclear data for neutron-induced reactions on 23Na are of critical importance for research and development of the Generation IV fast neutron reactors. The results of the advanced modeling tools applied for design, testing, and evaluation of new systems rely extensively on the input data. The sensitivity of the key reactor parameters to the nuclear data has been thoroughly investigated [1, 2], and the newly established requirements for data development are listed on the High Priority Request List (HPRL) for nuclear data. The inelastic neutron scattering of 23Na is of great concern for the research and development of the sodium-cooled fast reactors (SFR), which are one of the principal proposed systems supported by a significant amount of research and technical experience [3, 4]. Improved 23Na data is also required for the advanced breeder test reactor (ABR) and European fast reactor (EFR).
The neutron inelastic cross-section measurements for 23Na were performed at the GELINA time-of-flight facility with a white neutron source in resent years. The differential gamma production cross-sections were measured with eight high purity germanium detectors at 110 and 150, and transitions up to the 7th excited state were observed. The integral gamma production cross-sections, excitation functions, and total inelastic cross-sections were determined up to the threshold of the highest observed level at 3838.9 keV. The resulting cross-sections are in excellent agreement with existing data, particularly in the energy region up to 1.6 MeV, and the experimental uncertainties are improved to 2.2% in the energy region up to 2.23 MeV [5].
The evaluated data for n+23Na reactions in the ENDF/B-VII library [6] were provided over the incident neutron energy range from 1.010-5 eV to 20 MeV, based on the evaluated cross sections from the experimental data and the calculated results using theoretical model codes. In the JENDL-4.0 library [7], the total cross sections were obtained from least-squares fit to the experimental data of the n+23Na reaction. The other reaction cross sections, angular distributions, energy spectra of particles emitted in the compound, and pre-compound reactions for the incident neutron energy range from 1.010-5 eV to 20 MeV were calculated by theoretical models. In the JEFF-3.2 library [8], the evaluated data were also given at incident neutron energies from 1.010-5 eV to 20 MeV, the total cross sections are from the evaluation of the high-resolution data, the other reaction cross sections, angular distributions, and energy spectra of particles emitted are also mainly calculated by theoretical models. Recently, Ma et al. provided an investigation of neutron induced reactions on 23Na [9] by means of theoretical calculation.
Since the experimental data of neutron-induced 23Na reactions are scarce and the discrepancies exist in this experimental data from different laboratories, consistent calculations and analyses are very important and interesting when using nuclear theoretical models. Better nuclear data libraries for n+23Na reactions are also required by the applications over the incident neutron energy up to 200 MeV.
In the present work, all reaction cross sections, angle-integrated spectra, and double differential cross sections of neutron, proton, deuteron, triton, helium-3, and alpha emissions for n+23Na reactions are calculated in the incident neutron energy region of En ≤ 200 MeV. And the recent experimental data are used to guide the calculation. The calculated results are analyzed consistently and compared with the experimental data available.
2 Theoretical models and parameters
The optical model is the core model used in this theoretical calculation, which is used to describe the measured neutron-induced total, nonelastic, and elastic cross section and the elastic scattering angular distributions by means of calculating the transmission coefficients of the compound nucleus and the preequilibrium emission process. The optical model potential (OMP) considered here is in the most common form, which is made up of the real part potential, Vr, and the imaginary part potential, Wv(r), of volume absorption in the Woods-Saxon form [10], the imaginary part potential, Ws(r), of surface absorption in the derivative Woods-Saxon form, the spin-orbit potential, Vso(r), in the Thomas form, and the Coulomb potential, Vc(r). The energy dependency of each part of neutron optical potential is expressed as follows:
Where the incident neutron energy, E, is in the center of the mass system, Z, N, and A are proton, neutron, and mass numbers of the target nucleus, respectively. The detailed description of the form of the OMP can be found elsewhere [11].
A set of optimal neutron optical model potential parameters are automatically determined by APMN [12] for n+23Na reactions to fit the experimental data of the total, nonelastic, and elastic cross sections and elastic scattering angular distributions at the same time. The experimental neutron total cross sections for n+23Na reactions were given by different laboratories at incident neutron energies below 120 MeV. The experimental data [13-17] of the total cross sections are used here to obtain the best neutron optical model potential parameters. The experimental nonelastic scattering cross sections are all at incident neutron energies below 5 MeV. There is some experimental data of elastic scattering cross sections and elastic scattering angular distributions at incident neutron energies below 15.0 MeV. The experimental total cross section [18] for n+27Al reactions above an incident neutron energy 20 MeV and the experimental nonelastic scattering cross sections [19-22] for n+nat.Mg and n+27Al reactions below 150 MeV are also used here as references data. These experimental data are taken from the EXFOR library. The adjustment of the optical potential parameters is performed to minimize the χ2, which represents the deviation of the calculated results from the experimental data. Subsequently, a set of optimal neutron optical model potential parameters are obtained for n+23Na reaction at an incident neutron energy from 0.1 to 250 MeV. The parameter obtained is given in Table 1, and among them, V3, W2, and Vso are taken from the Becchetti’s results [23].
| Parameters | Values | Units |
|---|---|---|
| V0 | 43.15001 | MeV |
| V1 | -0.19732 | |
| V2 | 0.0001229 | MeV-1 |
| V3 | -24.0 | MeV |
| W0 | 13.47293 | MeV |
| W1 | -0.19162 | |
| W2 | -12.0 | MeV |
| U0 | -1.84325 | MeV |
| U1 | 0.17239 | |
| U2 | -0.0002695 | MeV-1 |
| Vso | 6.2 | MeV |
| ar | 0.61993 | fm |
| as | 0.50422 | fm |
| av | 0.52772 | fm |
| aso | 0.75 | fm |
| rr | 1.29278 | fm |
| rs | 1.16945 | fm |
| rv | 1.13485 | fm |
| rso | 1.01 | fm |
The optical potential parameters for protons are obtained by fitting the experimental data for p+23Na reactions. The optical potential parameters are taken from Han’s results [11] for deuterons and Xu’s results for helium-3 [24] for energies below 250 MeV. The Xu’s helium-3 optical potential parameters are also used as the optical potential parameters both for triton and alpha particles.
The direct inelastic scattering cross sections and angular distributions of discrete levels for n+23Na reactions are pre-calculated by the code DWUCK4 [25] in the distorted wave Born approximation (DWBA) using the optical model potential parameters obtained in this work. The discrete levels are taken into account from the ground (3/2+) state to the 15th (5.7660 MeV, 3/2+) excited state. The discrete levels are taken from Nuclear Data Sheets [26]. Meanwhile, levels above the 15th excited state are assumed to be overlapping and the level density formula is used. Due to the angular momentum and parity conservations in nuclear reactions, the direct inelastic scattering cross sections are mainly contributed by the excited states with the spins and parities 1/2+, 5/2+, 7/2+, and 9/2+. The experimental inelastic scattering cross sections and angular distributions for the discrete levels are used to guide theoretical calculations.
The nuclear reaction equilibrium and preequilibrium decay processes have been described by the unified Hauser-Feshbach and exciton models [27], with parity conservation and angular momentum conservation developed by J.S. Zhang at incident neutron energies below 20 MeV. The emissions from the compound nucleus to the discrete levels and continuum states of the residual nuclei are described by the Hauser-Feshbach model with width fluctuation correction in equilibrium processes, and by the angular momentum dependent exciton model in the preequilibrium process. The emissions of the discrete levels and the continuum states in the multi-particle emissions are included for all opened channels. At incident neutron energies below 20 MeV, the secondary particle emissions are described by the multi-step Hauser-Feshbach model. The preequilibrium and equilibrium decay processes are described by the preequilibrium statistical theory based on the exciton model, the evaporation models, and the Hauser-Feshbach theory with width fluctuation correction at incident neutron energies above 20 MeV, and also by the intranuclear cascade model above 150 MeV. The composite particle (d, t, 3He, α) emissions are described by the improved Iwamoto-Harada model [28-31] in compound nuclei. The improved Iwamoto-Harada model is included in the exciton model for light composite particle emissions. The double differential cross section can be calculated by the generalized master equation to get the angular momentum dependent lifetime with the Legendre expansion form. Instead, it is calculated with the angular dependent formula form of the Kalbach phenomenological approach [32, 33] at incident neutron energy above 20 MeV. The physical basis for many features of the widely used phenomenological systematics of Kalbach and a framework for understanding the systematical properties of continuum angular distributions by using the linear momentum dependent exciton state density model [34] were provided by Chadwick and Oblozinsky [35].
The nuclear reaction model code, MEND [36], is used to complete the theoretical model calculation in the present work. The code integrates the optical model, intra-nuclear cascade model, compound reactions, and preequilibrium nuclear reactions theory and treats the direct reaction as input, and can perform the calculations in the energy range up to 250 MeV. The detailed description of the functions of MEND can be found in its author’s paper [36].
The level density parameters and pair correction parameters of the Back-Shifted Fermi gas level density [37] are used for incident neutron energies below 20 MeV. The Ignatyuk nuclear level density parameters [38] are used for incident neutron energies above 20 MeV, which include the washing-out of shell effects with increasing excitation energy, collective excitations, as well as single particle excitations, and match continuously onto a low-lying experimental discrete level. The Ignatyuk model is particularly appropriate for describing the statistical level density properties of excited nuclei at a relatively high energy. Some of the level density parameters, a, the giant dipole resonance parameters, and the pair energy correction parameters, Δ, are adjusted slightly to obtain the best results.
3 Theoretical results and analysis
The calculated results are compared to all available experimental data in the EXFOR library. Unless otherwise indicated, these calculated results are in agreement with their corresponding experimental data.
The calculated total, elastic, and nonelastic scattering cross sections for n+23Na reaction are compared to the experimental data, as shown in Figs. 1, 2 and 3, respectively. The experimental data of the total cross sections show a lot of fluctuation structures below 10.0 MeV. The calculated results reasonably pass through the experimental data, though it is impossible for the optical model to describe those structures. The evaluated total cross sections in the ENDF/B-VII, JENDL-4.0, and JEFF-3.2 libraries are given up to 20 MeV. They are all mainly based on the experimental data [17] and similar to each other, except for theoretical calculations above 2 MeV in the JEFF-3.2 library. In order to fit better to the resonance structure of the experimental data, the experimental data are adopted here for En<10.0 MeV, and the evaluated data in the JENDL-4.0 library is based on the experimental data [17] by tracking the fine structures that are used. The calculated elastic scattering cross sections also pass through the existing experimental data at an incident energy below 200 MeV. The calculated nonelastic cross sections are in reasonable agreement with the experimental data above 10 MeV, and lower than the experimental data of n+nat.Mg and n+27Al reactions above 20 MeV. The surface absorption potential decreases more rapidly than the volume absorption potential increases above 20 MeV. When the decreasing surface absorption potential becomes much smaller than the increasing volume absorption potential, at about 68 MeV, a sharp turn takes shape on the curve of the calculated nonelastic cross sections. Our present results of the total, elastic, and nonelastic cross sections are in agreement with the evaluated data in the ENDF/B-VII, JENDL-4.0, and JEFF-3.2 libraries below 20 MeV.
The calculated neutron elastic scattering angular distributions at incident neutron energy from 1.5 to 8.52 MeV are given in Fig. 4 with the experimental data [39-44]. The calculated results agree well with the experimental data and the evaluated data in the ENDF/B-VII and JENDL-4.0 libraries. The experimental neutron elastic scattering angular distributions at an incident neutron energy from 0.55 to 2.0 MeV were given [45]. The calculated results are also in agreement with the experimental data.
The inelastic scattering cross sections and inelastic scattering angular distributions are from contributions of the direct reaction and compound reaction. The direct reaction cross sections are calculated by the DWBA theory, and the compound reaction cross sections are calculated by the unified Hauser-Feshbach theory and the exciton model. The calculated inelastic scattering angular distributions are compared to the corresponding experimental data, respectively. Some comparisons are given here in Figs. 5, 6, 7 and 8, and others are omitted for brevity. The experimental inelastic scattering angular distributions of some excited states are given by different laboratories [39-42, 44]. In Fig. 5, the calculated inelastic scattering angular distributions for the 1st excited state are compared to the experimental data at incident neutron energies from 1.5 to 8.52 MeV. The calculated inelastic scattering angular distributions for the 2nd and 3rd excited states are compared to the experimental data [39, 42, 44] at incident neutron energies from 4.04 to 8.52 MeV, and that for the 6th, 7th, 10th, and 11th excited states are compared to the experimental data [39, 40, 44] at incident neutron energies from 3.97 to 8.52 MeV. Only the comparisons of the 10th excited state are given in Fig. 6. The sum of the experimental data of the inelastic scattering angular distributions of some excited states are also given by different laboratories [39, 40, 44, 46]. The calculated results for the 7th to 9th excited states are compared with the experimental data [44], as shown in Fig. 7. The calculated neutron inelastic scattering angular distribution of the 2nd excited state at an incident neutron energy of 8.52 MeV is compared to the experimental data and the evaluated results in the ENDF/B-VII, JENDL-4.0, and JEFF-3.2 libraries, as shown in Fig. 8. The calculated results improve the evaluated results in the ENDF/B-VII, JENDL-4.0, and JEFF-3.2 libraries. The sum of the experimental elastic and inelastic scattering angular distributions of the 1st excited state are given by different laboratories [46-48], and the calculated results also compared. As the comparison shows, with few exceptions, the calculated results of the inelastic scattering angular distributions are in good agreement with the experimental data in most cases. Therefore, the set of optimum neutron optical model potential parameters obtained in present work gives a good description of the measured elastic and inelastic scattering angular distributions.
The results for the (n, γ) reaction cross sections and the gamma production cross section for (n, n′γ) reactions are compared with experimental data and the evaluated data in the ENDF/B-VII and JENDL-4.0 libraries, as shown in Figs. 9, 10, 11, 12, 13, and 14. In Fig. 9, the results for (n, γ) reaction cross sections are compared with the experimental data [49-54]. The calculated results decrease slowly above 20 MeV, since the contribution of the preequilibrium gamma ray emission is taken into account. The gamma production cross section for the transitions of the excited state to the ground state and different excited states of the (n, n′γ) reaction were measured in different laboratories [5, 55-57]. The measured data shows a lot of fluctuation structures below 4.0 MeV, where the calculated results reasonably pass through the experimental data [5, 56, 57]. The comparisons of the calculated results to the experimental data only for the transitions of the 1st excited state to the ground state are given in Figs. 10. The experimental data of inelastic scattering cross sections for the excited states are given [5, 39, 43, 44, 58, 59]. The measured data shows some fluctuation structures for the 1st to 6th excited states below 4.0 MeV, the calculated results are reasonably in agreement with the experimental data [5, 58, 59]. The present evaluated inelastic cross sections are from the experimental data [5] below 4.0 MeV, and from the theoretical model calculation above 4.0 MeV. The comparisons of the present results with the experimental data for the 1st and 2rd excited states are given in Figs. 11 and 12, respectively. The calculated inelastic scattering cross sections for the 7th to 15th excited states are compared to the experimental data [44]. The results for the 10th excited state are given in Fig. 13. The present results of inelastic scattering cross sections are in agreement with the experimental data [5, 60, 61] and inconsistent with the experimental data [43, 62], as shown in Fig.14. The above calculated results also compared with those in the ENDF/B-VII, JENDL-4.0 and JEFF-3.2 libraries and some improvements are given.
The comparison of the calculated reaction cross sections of the emitted charged particles to all available experimental data are done, and most of them are given in Figs. 15, 16, 17, 18, 19 and 20. The experimental results of the (n, p) reaction cross sections for the ground state and the first excited states are given [63, 64]. The measured data shows some fluctuation structures for the ground excited state at incident neutron energy below 7.0 MeV. The present evaluated cross sections are from the experimental data [64] below 7.0 MeV, and from the theoretical model calculation above 7.0 MeV. The comparisons of the present results with the experimental data for the ground state are given in Fig. 15. The comparisons of the calculated results of the (n, p) reaction cross sections with the experimental data and the evaluated data are given in Fig. 16. The calculated results are in agreement with the experimental data [63-80]. The only experimental data [81] from the (n, t) reaction cross sections were given at an incident neutron energy of 14.5 MeV. The calculated curves pass through the experimental data within error bars, and the comparison is omitted. The experimental (n, α) reaction cross sections for the ground state to the 5th excited states are given [64]. The calculated curves for the ground state, and the 1st, 3rd, 4th, and 5th excited states pass through the experimental data within error bars. The calculated results for the 2nd excited state are larger than the experimental data. The comparison for the third excited states are given in Fig. 17. The cross sections of the (n, α) reaction have some experimental data [63, 64, 66-68, 70, 71, 73, 75, 78, 81-86]. The present results are in good agreement with the experimental data [63, 64, 66-68, 70, 73, 75, 83, 84]. The comparisons of the calculated results with the experimental data and the evaluated results in the ENDF/B-VII, JENDL-4.0, and JEFF-3.2 libraries are given in Fig. 18. The experimental (n, 2n) reaction cross sections were given in different laboratories [87-96]. The calculated (n, 2n) reaction cross sections are in good agreement with the experimental data [87, 88, 91-95] below 20.0 MeV, and inconsistent with the experimental data above 20.0 MeV, as shown in Fig. 19. The experimental data [81, 97] of the (n, np) and (n, d+np) reaction cross sections were given at incident neutron energies of 14.5 and 14.6 MeV, respectively. The calculated curves of the (n, np) reaction cross sections pass through the experimental data [97] within error bars. The calculated (n, d+np) reaction cross sections are lower than the experimental data [81]. The calculated curves for the (n, nα) reaction cross sections pass through the experimental data [83] within error bars, as shown in Fig. 20. There are no experimental data for other reaction cross sections as of now, all reaction cross sections are predicted by theoretical models. The calculated n+23Na reaction cross sections for all channels are similar to the evaluated results in the ENDF/B-VII, JENDL-4.0, and JEFF-3.2 libraries concerning curve shapes, but fit the experimental data much better for some channels.
Based on the agreement of the calculated results with the experimental data for all reaction cross sections and angular distributions, the energy spectra, as well as double differential cross sections, for neutron, proton, deuteron, triton, helium and alpha emissions are consequently calculated and compared with the corresponding experimental data. A.Takahashi et al. measured the neutron emission double differential cross sections of 23Na(n, xn) reactions at incident neutron energies from 13.56 to 14.67 MeV and emission angles from 45 to 135 [98]. The comparisons between the calculated results and the experimental data [98] are given in Fig. 21. The structures of the double differential cross sections of neutron emissions are from the contributions of the inelastic scattering cross sections from the discrete levels and the elastic scattering cross sections. The calculated neutron emission double differential cross sections for n+23Na reactions are also compared to the experimental data for n+27Al reactions, given by Baba et al. [99] at incident neutron energies of 14.1 and 18.0 MeV. The comparison at the incident neutron energy of 18.0 MeV, as shown in Fig.22. Since there is no corresponding experimental data for n+23Na reactions, the calculated energy spectra and the double differential cross sections of emission neutrons, protons, deuterons, tritons, helium-3 and alpha for n+23Na reactions are compared to the experimental data [100-102] of the n+27Al reaction above 20.0 MeV. Some of the comparisons are given in Figs. 23-24. The shape and magnitude of the calculated results curve are all in agreement with the corresponding experimental data.
4 Conclusion
According to the experimental data of the total, nonelastic cross section and elastic scattering angular distribution of n+23Na, n+nat.Mg, n+27Al reactions, a set of optimal neutron optical potential parameters are obtained up to 200 MeV. All cross sections of neutron induced reactions, angular distributions, energy spectra, and double differential cross sections are consistently calculated using nuclear theory models at incident neutron energies from 0.1 to 200 MeV. The energies for the whole reaction process are balanced, since the recoil effects are taken into account. Good agreement is generally observed between the calculated reaction cross sections and angular distributions and the corresponding experimental data. The calculated results also are compared with the evaluated data in the ENDF/B-VII, JENDL-4.0, and JEFF-3.2 libraries and a number of improvements have been achieved. The evaluated data is given in the ENDF/B format.
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