Calculations and analysis of n+⁵²Cr reactions below 20 MeV

ABSTRACT

Chromium is one of the important structural materials in the process of nuclear reaction, and the precise nuclear data of n + ⁵²Cr reaction is very important. On the basis of the spherical optical model, the unified Hauser–Feshbach theory and exciton model, all reaction cross sections, angular distributions, energy spectra, and double-differential cross sections were calculated and analysed for n + ⁵²Cr reactions below 20 MeV. The theoretical calculations were compared with the available experimental data and other evaluation data from ENDF/B-VIII.0 and JENDL-5.

KEYWORDS:

• Neutron-induced reaction
• double-differential cross section
• nuclear models theory

1. Introduction

The development of nuclear energy is important to solve the global energy crisis and environmental problems. Safety is the prerequisite to ensure the use of nuclear energy. Chromium is one of the most important structural materials during the nuclear reaction process. The study shows that Cr has the characteristics of thermostability, good corrosion resistance and abrasion resistance, and suitable for engineering application. It is used as a corrosive coating of zirconium alloy in the reactor [1]. The study of coating on the surface of zirconium alloy cladding pipe becomes a vital goal in the development of accident resistant fuel (ATF) cladding. Chromium is currently the most researched material to improve accident resistant coating on zirconium alloy surfaces [2], and it is also considered to be the most promisingly commercialized [3–6]. Thus, the development of high-quality nuclear data for chromium is particularly important. Natural Cr consists of four isotopes, and they are ⁵⁰Cr (4.345%), ⁵²Cr (83.789%), ⁵³Cr (9.501%), and ⁵⁴Cr (2.365%). In this study, we obtained a set of optical model potential parameters for the n + ^natCr reaction below 20 MeV. We systematically calculated various cross sections, angular distributions, energy spectra, double-differential cross sections and γ-ray production cross sections for the reaction of the n+ ⁵²Cr reactions. The calculated results were compared with the existing experimental data from EXFOR Library [7], together with the evaluated data from the ENDF/B-VIII.0 [8] and JENDL-5 [9].

2. Theoretical models and model parameters

2.1 Optical model and the optical potential parameters

The optical model is one of the most important theories in nuclear data calculations and evaluations, which describes the nuclear reaction process using the optical model potential. The optical potentials considered here are Woods-Saxon [10] form for the real part; Woods-Saxon and derivative Woods-Saxon form for the imaginary parts corresponding to the volume and surface absorption, respectively; and the Thomas form for the spin-orbit part.

The potential depths for the real part, the imaginary part of surface absorption, and the volume absorption are dependent on incident energies and are expressed as follows:

(1) $\mathit{V}=\mathit{\hspace{0.17em}}{V}_0+V_1\mathit{E}+V_2{E}^2+V_3\left(\mathit{N}-\mathit{Z}\right)/\mathit{A},$(1)

(2) $W_S=\mathit{\hspace{0.17em}}{W}_{\mathit{S}0}+W_{\mathit{S}1}\mathit{E}+W_{\mathit{S}2}\left(\mathit{N}-\mathit{Z}\right)/\mathit{A},$(2)

(3) $W_V={W_V}_0+{W_V}_1\mathit{E}+{W_V}_2{E}^2,$(3)

(4) $R_i=\mathit{\hspace{0.17em}}{r}_i{\mathit{A}}^{1/3},\mathit{i}=\mathit{r},\mathit{\hspace{0.17em}}\mathit{S},\mathit{\hspace{0.17em}}\mathit{V},\mathit{\hspace{0.17em}}\mathit{SO},\mathit{\hspace{0.17em}}\mathit{C},$(4)

where V is the real part potential, W_S and W_V are the imaginary part potential of surface absorption and volume absorption. Where V₀, W_S0, W_V0 are Constant factors, V₁, W_S1 and W_V1 are energy-linear term factors, V₂ and W_V2 are energy-square term factors, V₃ is charge-symmetry term factors in real potentials, W_S2 is charge-symmetry term factors of surface absorption imaginary potentials. Where r_r, r_S, r_V, r_SO, and r_C are the radius of the real part, the surface absorption, the volume absorption, the spin–orbit couple and the Coulomb potential, respectively.The spin–orbit couple potential is V_SO, and the diffusive width of the real part, the surface absorption, the volume absorption and the spin–orbit couple potential are a_r, a_s, a_v, and a_so; Where Z, N and A are charge, neutron, and mass numbers of target, respectively, E is incident neutron energy in the center of mass system. The units of the potential V, W_S, W_V, and V_SO are in MeV, the lengths r_r, r_S, r_V, r_SO, r_C, a_r, a_s, a_v, and a_so are in fm, and the energy E is in MeV.

A new set of neutron optical potential parameters for ^natCr were obtained to fit the experimental data of total cross sections, nonelastic scattering cross sections, elastic scattering cross sections and elastic scattering angular distributions by the code APMN [11]. The experimental data of n+ ^natCr were used in the fitting, because the measured data of n+ ⁵²Cr are scarce. The optical model potential parameter set obtained for n+ ^natCr is given in Table 1. The global OMP parameters for proton, deuteron, ³He, triton, and alpha-particle projectiles [12–16] were employed in the calculations.

Table 1. The optical model potential parameters.

V0	57.21	rr	1.171
V1	−0.4745	rS	1.315
V2	−0.00008644	rV	1.523
V3	−24.00	ar	0.5970
WS0	6.871	aS	0.6860
WS1	−0.1751	aV	0.4500
WS2	−12.00	rC	1.250
WV0	−1.115	VSO	6.200
WV1	0.1659	rSO	1.010
WV2	0.002139	aSO	0.7500

2.2 Direct inelastic scattering for discrete level

The direct inelastic scattering cross sections and angular distributions to low-lying states are important in nuclear data theoretical calculations. The code DWUCK [17] of the distorted wave Born approximation (DWBA) theory was used to pre-calculate the direct reaction cross sections and angular distributions in n+ ⁵²Cr. The optical model potential parameters obtained in this work were included as input into the DWUCK code. Based on experimental data of the direct inelastic scattering cross sections and angular distributions of discrete levels, we adjusted the deformation parameters in DWUCK. The contribution to the direct inelastic scattering comes mainly from eight discrete levels, the deformation parameters of which are listed in Table 2.

Table 2. The excited energies (ε), spins (J), parities (π), and deformation parameters (β) of eight discrete levels which make the main contribution to the direct inelastic scattering.

ε (MeV)	J	π	β
1.4341	2.0	1	0.21
2.3696	4.0	1	0.08
2.9648	2.0	1	0.10
3.1617	2.0	1	0.10
4.5630	3.0	−1	0.15
4.7020	2.0	1	0.10
4.8001	2.0	1	0.10
4.8413	2.0	1	0.10

2.3 The pre-equilibrium and equilibrium processes

The unified Hauser–Feshbach and exciton model [18] is used to describe the equilibrium and pre-equilibrium emission processes for the incident energy below 20 MeV. The UNF [19] code of the spherical optical model, the unified Hauser–Feshbach, and exciton model were developed for calculating fast neutron reaction data of structure materials with incident energies from about 1 keV up to 20 MeV. The angular momentum dependent exciton model was established to describe the emissions from compound nucleus to the discrete levels of the residual nuclei in pre-equilibrium processes, while the equilibrium processes were described by the Hauser–Feshbach model with width fluctuation correction. The emissions to the discrete level in the multi-particle emissions for all opened channels were included [18,19]. The double-differential cross sections of neutron and proton were calculated by the linear momentum dependent exciton state density [20]. Since the improved pickup mechanism has been employed based on the Iwamoto-Harada model [21–23], the double-differential cross sections of alpha-particle,³He, deuteron, and triton can be calculated by using a new method based on the Fermi gas model. The recoil effects in multi-particle emissions from continuum state to discrete level as well as from continuum to continuum state were taken into account strictly, so the energy balance was held accurately in every reaction channels [19]. If the calculated direct inelastic scattering data and the calculated direct reaction data of the outgoing charged particles are available from other codes, one can input them, so that the calculated results will included the effects of the direct reaction processes. In this work, the calculated results of DUWCK and APMN were included as input into the UNF. To keep the energy balance, the recoil effects were taken into account for all of the reaction processes. The gamma-production data were also calculated, and the gamma-ray E1 strength function [24,25] is expressed as follows:

(5) $f_{\mathit{E}1}({\epsilon }_{\gamma })=K_{\mathit{E}1}\sum _i\frac{{\sigma }_i{\epsilon }_{\gamma }{\mathrm{\Gamma }}_i^2}{{({\epsilon }_{\gamma }^2-E_i^2)}^2+{\epsilon }_{\gamma }^2{\mathrm{\Gamma }}_i^2}$(5)

where σ_i, Γ_i, E_i are photon absorption cross section, resonance width, and resonance energy of electric dipole resonance. K_E1 = 8.68 × 10⁻⁸(mb)⁻¹(MeV)⁻².

The data of discrete level, and the parameters of giant dipole resonance, were taken from Reference Input Parameter Library for Calculation of Nuclear Reactions and Nuclear Data Evaluations (RIPL) [26], the initial parameters of the level densities and pair corrections were taken from RIPL, and were adjusted in order to fit the experimental data of reaction cross sections. The final parameters values are listed in Table 3. The calculated neutron reaction data can be output in the ENDF-6 format.

Table 3. The level density and pair correction parameters.

	a	Δ
(n, γ)	6.080	−3.20
(n, n’)	5.416	0.100
(n, p)	5.7	−1.90
(n, α)	6.235	0.05
(n, ^3He)	5.868	0.130
(n, d)	6.255	−0.8
(n, t)	7.034	−0.460
(n,2n)	6.232	−2.80
(n, nα)	6.140	2.890
(n,2p)	6.524	1.530
(n,3n)	5.736	2.560

3. Theoretical results and analysis

The calculated results of the total and elastic scattering cross sections for n+⁵²Cr reaction were compared with experimental data [27–37] and evaluated data from ENDF/B-VIII.0 [8] and JENDL-5 [9] in Figure 1.

Figure 1. Calculated (n, tot) and (n, el) cross sections (solid line) compared with experimental data (symbols) for n+⁵²Cr reaction.

For the total cross sections, the measured data demonstrate a lot of fluctuations structure below 5.0 MeV for total cross sections of n+ ⁵²Cr and n+ ^natCr reactions and it was impossible for the optical model to describe those structures, though calculated results pass reasonably through the experimental data. In order to fit the resonance structure of the experimental data, R-matrix theory must be used for energy En <5 MeV. For the elastic scattering cross-section, compared with the existing experimental data below 15 MeV, the calculated results passed through the data.

D. Schmidt et al. [35] and A. B. Smith et al. [38] measured the elastic scattering angular distributions at incident neutron energies 7.95~14.76 MeV and 4.5~9.9 MeV. The comparisons of the calculated results of the elastic scattering angular distributions with the experimental data are shown in Figure 2.

Figure 2. Calculated neutron elastic scattering angular distributions (solid curves) compared with the experimental data (symbols) for incident energy in the range 7.95 to 14.76 MeV and 4.5 to 9.99 MeV. From 7.95 to 14.76 MeV and 4.5 to 9.99 MeV, the results were offset by factors of 10. The data at the top of the figure have not been adjusted.

The theoretical calculation of the total cross section, elastic scattering cross section, and elastic scattering angular distributions were in good agreement with the experimental data. Therefore, the optical potential parameters determined by APMN code are reasonable, which are the basis of our theoretical calculations.

Moreover, D. Schmidt et al. [35] and A. B. Smith et al. [38] measured the inelastic scattering angular distributions for the first excited state. Figure 3 shows the comparisons of calculated results for inelastic scattering angular distributions of the first excited state with experimental data at incident neutron energies 7.95~14.76 MeV and 4.5~14.8 MeV.

Figure 3. Calculated neutron inelastic scattering angular distributions (solid curves) for the first excited state (1.434 2+) compared with the experimental data (symbols). From 7.95 to 14.76 MeV and 4.5 to 14.8 MeV, the results were offset by factors of 10. The data at the top of the figure have not been adjusted.

Figure 4 shows the comparisons of calculated results for the total inelastic scattering cross section with the experimental data. The calculated results below 5 MeV agreed well with the experimental data, and in the high-energy region were higher than the experimental data [39–43] but were consistent with the evaluated data. The calculation of total inelastic scattering cross section considered of the contribution of discrete levels and continuous levels, so it was reasonable that the theoretical results were higher than the experimental data. Meanwhile, our calculation results have good consistency with ENDF/B-VIII.0 and JENDL-5.

Figure 4. Calculated (n, inl) reaction cross section (solid line) compared with experimental data (symbols) and evaluated data for n+⁵²Cr reaction.

The inelastic scattering gamma production cross sections were measured mainly by L. C. Mihailescu et al. in 2007 [43]. Figure 5 shows the comparisons of the calculated results with the existing experimental data [43–49] for the transitions of the first excited state to the ground state, the second excited state to the first excited state, the third excited states to the second excited state. At relatively lower emission energies, the calculated results were in agreement with the experimental data.

Figure 5. Calculated gamma production cross sections (solid line) for the transition of the first excited state to the ground state, the second excited state to the first excited state and the third excited states to the second excited state of ⁵²Cr compared with experimental data (symbols).

L. C. Mihailescu et al. [43] also measured the discrete level inelastic scattering cross section. The comparisons of calculated results with experimental data [34,35,41,43,50–54] for n+⁵²Cr are given in Figure 6. Below 5 MeV, the results of the discrete level inelastic scattering cross sections were in agreement with the experimental data.

Figure 6. Calculated neutron inelastic scattering cross sections (solid line) of the first excited state, the second excited state, and the third excited state compared with experimental data (symbols) and evaluated data for n+⁵²Cr reaction.

The ⁵²Cr(n,2n) reaction cross section calculated results were in agreement with the experimental data [39,55–66] as shown in Figure 7. In addition, (n,2nγ) reaction cross sections were also calculated. Figure 8 shows the comparisons between the calculated results with experimental data [65] for the (n,2nγ) cross section of the transitions of the first excited state to the ground state and the third excited state to the ground excited state.

Figure 7. Calculated (n, 2n) reaction cross section (solid line) compared with experimental data (symbols) and evaluated data for n+⁵²Cr reaction.

Figure 8. Calculated result of (n,2nɣ) reaction cross section (solid line) compared with the experimental data (symbols) at ɣ-ray energy of 749.1 KeV and 1164.4 KeV.

Figure 9 shows the comparisons of calculated results for the (n, p), (n, α), (n, d) reaction cross sections with the experimental data. For the (n, p) reaction cross section, our theoretical calculations were in good agreement with measurement of A. Fessler et al. [66]. Above 8 MeV, our results were consistent with the available experimental data [55,56,66–76].

Figure 9. Calculated (n, p), (n, α), (n, d) reaction cross section (solid line) compared with experimental data (symbols) and evaluated data for n+⁵²Cr reaction.

For the (n, α) reaction cross section, the experimental data of n+ ^natCr were used in the fitting, because the measured data of n+⁵²Cr were scarce. The natural chromium experimental data were measured mainly by E. Wattecamps et al. and A. Paulsen et al., and the results were in good agreement with the experimental data [77–79] within the error bar. For the (n, d) reaction cross section, S. M. Grimes et al. [80] measured (n, d) cross sections of ⁵²Cr at incident neutron energies 14.8 MeV, the present results and evaluated data in JENDL-5 passed through the experimental data within error bar.

D. Schmidt et al. [81] measured emission neutron double-differential cross sections of natural Cr at incident neutron energies 7.95, 9.0, 9.8, 10.79, 11.44, 12.01, 12.7, and 14.1 MeV, and the contributions of elastic scattering channel were not included in measurements. The comparisons between the calculated results and measurements at incident neutron energies 10.79, 12.7 and 14.1 MeV for different angles as shown in Figure 10. The double-differential neutron emission cross sections include the contributions of elastic scattering processes, direct inelastic scattering processes of discrete levels, and continuous inelastic scattering process. The calculated results of emission neutron double-differential cross sections at incident neutron energies 10.79, 12.7 and 14.1 MeV were basically in agreement with experimental data.

Figure 10. Calculated (n, xn’) double-differential neutron emission cross section (solid line) compared with experimental data (symbols) for different angles at 10.79 MeV, 12.7 MeV and 14.1 MeV incident energy. From 25.0 to 160.0 deg., the results were offset by factors of 10. The data at the top of the figure have not been adjusted.

The experimental data of natural Cr for double-differential neutron emission cross sections were given in A. Takahashi [82] at incident neutron energy from 13.35 to 14.83 MeV for different angles. Figure 11 shows the comparisons between the calculated results and measurements.

Figure 11. Calculated (n, xn’) double-differential neutron emission cross section (solid line) compared with experimental data (symbols) at incident neutron energy from 13.35 to 14.83 MeV for different angles. From 14.83 to 13.35 MeV, the results were offset by factors of 10. The data at the top of the figure have not been adjusted.

Moreover, A. Takahashi [83] also measured the double-differential neutron emission cross sections at incident neutron energy 14 MeV with 16 angles. Figure 12 shows the comparisons between the calculated results and measurements. As shown in the figure, the part of the elastic scattering and the discrete level inelastic scattering agreed well with the experimental data, and the part of the continuous level was partly higher than the experimental data.

Figure 12. Calculated double-differential neutron emission cross section (solid lines) compared with experimental data (symbols) at incident energy of 14.1 MeV. From 15.0 to 80.0 deg. and 90.0 to 160.0 deg., the results were offset by factors of 10. The data at the top of the figure have not been adjusted.

S. Matsuyama et al. [84] measured double-differential neutron emission cross sections for natural Cr at incident neutron energy 14.1 MeV. The calculated results corresponded well with the experimental data as shown in Figure 13. We also calculated the inelastic scattering neutron emission spectra for ⁵²Cr at the neutron incident energy range from 7.94 to 14.1 MeV, as shown in Figure 14. Experimental data were taken from the reference [81]. Calculated results were in agreement with experimental data.

Figure 13. Calculated double-differential neutron emission cross section (solid lines) compared with experimental data (symbols) at incident energy of 14.1 MeV. From 20.0 to 150.0 deg., the results were offset by factors of 10. The data at the top of the figure has not been adjusted.

Figure 14. Calculated (n, xn’) neutron emission spectra (solid line) compared with experimental data (symbols) at 7.95, 9.0, 9.8, 10.79, 11.44, 12.01, 12.7, and 14.1 MeV incident energy. From 14.1 to 7.95 MeV, the results were offset by factors of 10. The data at the top of the figure have not been adjusted.

The double-differential cross sections of proton were calculated by the linear momentum dependent exciton state density. Since the improved pick up mechanism has been employed based on the Iwamoto-Harada model, the double-differential cross sections of alpha-particle and deuteron can be calculated by using a new method based on the Fermi gas model.

Therefore, The energy spectrum of proton, alpha-particle and deuteron for ⁵²Cr at neutron incident energy 14.8 MeV. The experimental data and calculated results of proton, alpha-particle, and deuteron emission spectra are given in Figure 15.

Figure 15. Calculated proton, alpha-particle, and deuteron emission spectra (solid line) compared with experimental data (symbols). From (n, xp) to (n, xd) MeV, the results were offset by factors of 10 except for (n, xp) reaction.

The calculated results of energy spectra for proton and alpha-particle were in agreement with experimental data. The calculation result of the energy spectrum for deuteron was higher than the experimental data [80], but the (n, d) cross section were in agreement with the experimental data, so the results needed to be further discussed.

4. Conclusion

In this study, we obtained a set of optical model potential parameters for the n + ^natCr reaction with incident neutron energy below 20 MeV. The DWBA theory was used to calculate the direct inelastic scatterings to discrete excited states of ⁵²Cr, and we determined the distorted parameters of the eight discrete energy levels with major contributions. On the basis of the spherical optical model, the unified Hauser–Feshbach theory and exciton model, we systematically calculated various cross sections, angular distributions, energy spectrum, double-differential cross sections, and γ production cross sections for the reaction of the n+⁵²Cr reactions below 20 MeV. The theoretical calculation results were generally in agreement with the existing experimental data, together with the evaluated results in the ENDF/B-VIII.0, JENDL-5.

All of the present results have been transformed into the ENDF-6 format data files for application. In the present evaluated data file, the resonance parameters data of ENDF/B-VIII.0 were adopted. The evaluated results will be included in the new version of the Chinese Evaluated Nuclear Data Library (CENDL-4.0), which will be convenient for nuclear engineering application.

Disclosure statement

No potential conflict of interest was reported by the author(s).