Evaluation of neutron nuclear data on antimony isotopes

Abstract

Neutron nuclear data on ^{121,123,124,125,126}Sb have been evaluated for the next version of JENDL general purpose file in the energy region from 10^{− 5} eV to 20 MeV. In the low-energy region, the latest resolved resonance parameters were taken into account for ^{121, 123}Sb. The thermal capture cross sections of ^{125, 126}Sb, for which no measurements exist, were determined by using a simplified formula. The statistical model was applied to calculate the cross sections above the resolved resonance region. In the calculations, coupled-channel optical model parameters were used for the interaction between neutrons and odd-mass targets. Preequilibrium and direct-reaction processes were considered in addition to the compound process. The present evaluation is consistent with available experimental data. The evaluated data are compiled into an ENDF-formatted data file.

Keywords:

• neutron nuclear data
• evaluation
• antimony isotopes
• cross section
• JENDL
• coupled-channel optical model
• statistical model

1. Introduction

The fourth version of the Japanese Evaluated Nuclear Data Library (JENDL-4.0 [1]) was released in May 2010. In this library, 215 nuclides are regarded as fission products in the region from Z = 30 to 68. Although the resolved resonance parameters (RRPs) were examined by reviewing experimental data for all of those nuclides, the fast neutron region above resolved resonances was re-evaluated for about 170 nuclides due to time limit. Hence, the fast-neutron cross sections for the rest of the fission-product nuclides may need improvements for the next version of JENDL general purpose file. Antimony isotopes are classified into such nuclides. Moreover, neutron-induced activation cross sections of Sb isotopes are important [2] for the decommissioning of light-water reactors. Therefore, it is required to re-examine the cross sections of Sb.

The data of ^121,123,124Sb were evaluated for JENDL-2 [3] in 1984. For JENDL-3.1 [4], the ¹²⁵Sb data were newly evaluated in 1990. After that, the ^121,123Sb data were revised by considering the evaluations for JENDL/F-99 [5] in 1994, and the revised data were adopted by JENDL-3.2 [6]. The ^{121,123,124,125}Sb data mentioned above were essentially carried over to JENDL-3.3 [7] and JENDL-4.0 [1], whereas a new evaluation of ¹²⁶Sb was performed for JENDL-4.0. In the above evaluations except for ¹²⁶Sb, the statistical-model code CASTHY [8] was used to calculate the total, elastic and inelastic scattering, and capture cross sections, while the cross sections for the competing reactions such as (n, 2n) and (n, p) were obtained from the codes PEGASUS [9] or GNASH [10]. The direct inelastic scattering was considered for ^121,123Sb by using the distorted-wave Born approximation (DWBA). However, gamma-ray production data were not compiled into the library except for ¹²⁶Sb.

The present work was undertaken to improve the evaluated data on ^{121,123,124,125,126}Sb by considering the latest knowledge on experimental and theoretical nuclear physics. Evaluated are the total, elastic and inelastic scattering, (n, γ), (n, p), (n, d), (n, t), (n, ³He), (n, α), (n, np), (n, nd), (n, nα), (n, 2n), (n, 3n) reaction cross sections, the angular distributions of emitted neutrons and charged-particles, and the energy distributions of emitted particles and γ-rays in the incident-neutron energy region from 10^{− 5} eV to 20 MeV. The Q-values of the reactions were calculated from the 2003 version of the mass table (AME2003) obtained by Audi et al. [11] and are listed in Table 1, together with isotopic abundances [12]. The RRPs of ^121,123Sb were modified by considering the latest measurements. The thermal capture cross sections of ^125,126Sb were determined by using a simplified formula. The unresolved resonance parameters (URPs) were obtained by fitting to the calculated total and (n, γ) cross sections.

Table 1. Isotopic abundances and reaction Q-values of antimony isotopes.

	^121Sb	^123Sb	^124Sb	^125Sb	^126Sb
Abundances(%)	57.21	42.79	60.2 d^*	2.75856 y^*	12.35 d^*
Q-values (MeV)
(n, γ)	6.806	6.467	8.707	6.214	8.373
(n, p)	0.391	−0.621	1.399	−1.575	0.404
(n, α)	3.282	2.157	3.867	0.968	2.674
(n, d)	−3.554	−4.343	−4.864	−5.083	−5.564
(n, t)	−6.405	−6.899	−4.553	−7.314	−5.040
(n,^3He)	−8.751	−10.243	−10.903	−11.689	−12.382
(n, 2n)	−9.242	−8.965	−6.467	−8.707	−6.214
(n, np)	−5.779	−6.567	−7.089	−7.308	−7.789
(n, nα)	−3.075	−3.945	−4.310	−4.839	−5.246
(n, nd)	−12.662	−13.156	−10.810	−13.571	−11.297
(n, 3n)	−16.260	−15.772	−15.433	−15.174	−14.921

^*Half-lives are given for ¹²⁴Sb, ¹²⁵Sb, and ¹²⁶Sb which are unstable.

This paper presents how the evaluation was performed. Section 2 deals with the resonance region. In Section 3, the computational methods and procedures above the resonance region are described. Comparisons of the evaluated results with available experimental data and evaluated data are made in Section 4. Finally, Section 5 summarizes the conclusion.

2. Resonance region

The resolved and unresolved resonance regions are given in Table 2 for each target nucleus. In the resolved resonance regions of ^121,123Sb, we took account of the parameters of the p-wave resonances obtained by Matsuda et al. [13] from the transmission measurements. In their analysis, the total spin J and the radiation width Γ_γ are unknown. The spin value was estimated from the probability distribution of J using the JCONV code [14]. The Γ_γ values were taken from the average values adopted in JENDL-4.0, i.e., 89 meV and 98 meV for ¹²¹Sb and ¹²³Sb, respectively.

Table 2. Resolved and unresolved resonance regions.

Target nuclei	Resolved resonance	Unresolved resonance
^121Sb	10^−5 eV–2.0 keV	2.0 keV–100 keV
^123Sb	10^−5 eV–2.5 keV	2.5 keV–300 keV
^124Sb	–	10 eV–100 keV
^125Sb	–	40 eV–400 keV
^126Sb	–	30 eV–300 keV

Concerning thermal cross sections, if no measurements are available, the scattering cross sections can be given by the potential scattering, i.e., (1) where R is the scattering radius, which can be estimated from nuclear model calculations in the higher energy region. However, there is not a useful method to determine the radiative capture cross section, although the 1/v behavior is normally assumed in the low energy region. Blatt and Weisskopf [15] made rough estimates of unknown capture cross sections. The present work followed the same method. Namely, the capture cross section at a thermal energy is expressed by (2) where k is the wave number of the relative motion in the incident channel, K the wave number of the incident neutron in a compound nucleus, ⟨Γ_γ0⟩ the average total radiative width for s-wave neutrons, and D₀ the average level spacing for s-wave neutrons. The derivation of Equation (2(2) ) is given in the Appendix. The unknown thermal capture cross sections of ^125,126Sb are calculated theoretically, and they are listed in Table 3 together with the recommendations of Mughabghab [16]. Although the estimates are rough, the calculated values of ^121,123Sb are in fair agreement with those of Mughabghab based on available measurements. The experimental value of ¹²⁴Sb, which is recommended by Mughabghab, is a maximum value, as mentioned by the authors [17]. Therefore, the calculated thermal capture cross sections of ^125,126Sb were adopted in the present work.

Table 3. Calculated thermal capture cross sections.

Target nuclei	Calculations	Mughabghab [16]
^121Sb	6.319 b	5.77 ± 0.11 b
^123Sb	2.946 b	3.94 ± 0.12 b
^124Sb	8.556 b	17.4 ± 2.8 b^*
^125Sb	1.333 b	–
^126Sb	2.673 b	–

^*Considered as a maximum value [17].

The ASREP code [18] was used to determine the URPs by fitting to the total and capture cross sections evaluated in the energy region above 10 eV. It should be noted that the URPs obtained are used only for self-shielding calculations, since the pointwise cross sections are given in the evaluated data files. The total cross sections reconstructed from the URPs presently obtained are compared in Figure 1 with the evaluated pointwise cross sections which are described in the next section. In the parameter fitting, 10% errors of the pointwise cross sections were assumed. It is found from the figure that the reconstructed cross sections reproduce the pointwise cross sections very well.

Figure 1. Total cross sections in the unresolved resonance region.

3. Computational methods and procedures above resonance region

3.1. Nuclear models

The POD code [19] was used for calculating the neutron-induced reaction cross sections of Sb isotopes. The code is based on the spherical optical model, the one-component exciton preequilibrium model, the DWBA, and the multistep statistical model. In order to simulate the direct and semidirect effects on the capture reaction, the preequilibrium capture was considered by using the γ-ray emission rate derived by Akkermans and Gruppelaar [20]. The original POD code was modified to allow for options of γ-ray strength functions, i.e., generalized, extended generalized, and modified Lorentzians [21,22] (GLO, EGLO and MLO, respectively) for E1 radiation, in addition to the standard Lorentzian (SLO).

The POD code is not capable of calculating direct reaction cross sections and transmission coefficients by the coupled-channel method. These two kinds of data are supplied to POD by a separate coupled-channel code if the use of the coupled-channel method is preferable. In the present work, the coupled-channel optical model code OPTMAN [23] supplied those values to the POD code.

3.2. Parameter determination

3.2.1. Optical-model potentials

We employed the global neutron optical-model parameters obtained by Kunieda et al. [24] using the coupled-channel method based on the rigid-rotator model (RRM) [25] with slight modifications. The real volume depth V⁰_R, imaginary surface depth , and imaginary volume depth were changed to –37.3 MeV, 10.5 MeV, and 27.0 MeV, respectively, so that the calculations could reproduce measured total cross sections. The exact functional forms of the potentials are given in Ref. 24. The coupling schemes and deformation parameters are listed in Table 4. As for ^124,126Sb, the parameters of Kunieda et al. were used as the spherical optical model potentials. The calculated total cross section of elemental Sb is illustrated in Figure 2, together with experimental data and the spherical optical model calculations using the parameters of Koning and Delaroche [26]. It is found from the figure that the present calculations are in better agreement with the experimental data than the calculations using the parameters of Koning and Delaroche especially in the energy region below 1 MeV. Figure 3 shows a comparison of the presently calculated nonelastic cross sections of the elemental Sb with measured data and JENDL-4.0. The calculations are consistent with the measured data. In the energy region above 10 MeV, the JENDL-4.0 evaluation is larger than the present and measured values, which indicates that some reaction cross sections of ^121,123Sb in JENDL-4.0 could be overestimated in the high energy region.

Figure 2. Total cross section of elemental Sb.

Figure 3. Nonelastic cross section of elemental Sb.

Table 4. Coupling schemes used in the interaction between neutrons and targets.

	Coupled	Deformation
Target nuclei	levels^*	parameters
^121Sb	5/2^+(g.s.), 7/2^+(1.0240)	β2 = −0.1075
^123Sb	7/2^+(g.s.), 9/2^+(1.0302)	β2 = −0.0953
^124Sb	No coupling
^125Sb	7/2^+(g.s.), 9/2^+(1.0673)	β2 = −0.0953
^126Sb	No coupling

^*The number in the parentheses indicates the excitation energy in MeV.

Built-in values in POD were used for charged-particle optical-model parameters, i.e., the values of Koning and Delaroche [26] for protons, those of Lohr and Haeberli [27] for deuterons, those of Becchetti and Greenlees [28] for tritons and ³He, and those of Lemos [29] potentials modified by Arthur and Young [30] for α-particles. The values of Lohr-Haeberli and Becchetti-Greenlees potentials were actually taken from the compilation of Perey and Perey [31].

3.2.2. Discrete levels and level density

In the present calculation, it was necessary to input the discrete levels and level density parameters for 25 nuclei, i.e., ^119−127Sb, ^119−126Sn and ^117−124In. The discrete levels were taken from the reference input parameter library RIPL-3 [32].

Concerning the level density, the composite formula of Gilbert and Cameron [33] was used in the present work. In the region of low excitation energy E, the level density is described by the constant temperature formula ρ_T, namely, (3) On the other hand, the a parameter, which characterizes the Fermi-gas part of level density ρ_F, is defined as (4) where is the shell correction energy and γ a damping factor. The energy U is expressed by E and the pairing energy Δ, i.e., U = E − Δ. In the above equations, the values of a(^*), , γ, and Δ were taken from the work of Mengoni and Nakajima [34,35]. The two parameters T and E₀ were determined so as to connect ρ_F and ρ_T smoothly at an appropriate matching energy E_m. The parameters used in this work are listed in Table 5 for individual nuclei, together with the energies of the highest discrete levels .

Table 5. Level density parameters for each nucleus.

	a(*)	Δ		T	E0	Em
Nuclei	1/MeV	MeV	MeV	MeV	MeV	MeV	MeV
^127Sb	15.073	1.065	−1.601	0.710	−0.225	5.924	2.514
^126Sb	15.534	0.000	−0.617	0.741	−2.251	6.198	0.128
^125Sb	14.870	1.073	−0.068	0.553	0.610	4.053	2.299
^124Sb	14.872	0.000	0.763	0.688	−1.937	5.528	0.643
^123Sb	14.667	1.082	1.092	0.594	0.036	5.016	1.773
^122Sb	14.721	0.000	1.681	0.610	−1.394	4.451	0.605
^121Sb	14.463	1.091	1.847	0.621	−0.375	5.680	1.659
^120Sb	14.908	0.000	2.186	0.533	−0.900	3.433	0.842
^119Sb	14.259	1.100	2.043	0.621	−0.368	5.683	1.660
^126Sn	15.615	2.138	−2.600	0.800	0.152	8.327	2.795
^125Sn	16.177	1.073	−1.437	0.622	0.148	5.007	0.855
^124Sn	15.404	2.155	−0.998	0.704	0.534	7.380	2.879
^123Sn	15.969	1.082	−0.017	0.714	−1.324	7.311	1.301
^122Sn	15.193	2.173	0.164	0.629	0.927	6.532	3.036
^121Sn	14.630	1.091	0.971	0.629	−0.241	5.555	1.403
^120Sn	14.981	2.191	0.887	0.640	0.657	6.955	2.800
^119Sn	14.223	1.100	1.470	0.652	−0.479	5.977	1.572
^124In	15.326	0.000	−0.387	0.733	−2.180	6.059	0.243
^123In	14.667	1.082	0.204	0.520	0.800	3.642	2.186
^122In	15.117	0.000	0.977	0.559	−0.857	3.517	0.229
^121In	14.463	1.091	1.390	0.614	−0.170	5.389	1.504
^120In	14.908	0.000	1.930	0.635	−1.806	5.074	0.070
^119In	14.259	1.100	2.153	0.543	0.296	4.453	1.921
^118In	14.699	0.000	2.548	0.535	−0.957	3.507	0.200
^117In	14.055	1.109	2.519	0.563	0.097	4.820	1.713

3.2.3. Gamma-ray transition

The MLO form was used for E1 radiation in the present work, since it is recommended to use in RIPL-3 [32]. The giant resonance parameters for E1 were taken from RIPL-3, where the parameters were obtained by a fit of theoretical photo-absorption cross sections to experimental data. Unless the parameters are available for particular nucleus in RIPL-3, one can use the systematics given in Ref. 32. The resonance energy and width parameters for M1 and E2 are also obtained from another systematics [36] and given by (5) (6) (7) (8) The symbol A stands for the mass number of a nucleus. The contributions of M1 and E2 relative to E1 were estimated [10] such as (9) (10) As for compound nuclei ^122,124Sb, the functions are normalized so as to reproduce measured capture cross sections of the corresponding target nuclei around several tens of keV. The s-wave γ-ray strength functions S_γ0 obtained are compared in Table 6 with the values recommended by Mughabghab [16] and those used in the JENDL-4.0 evaluations. Normalization factors for the other nuclei were obtained by using the calculated average level spacing D₀ for s-wave neutrons and the thermodynamical formula for the average total radiative width <Γ_γ0> for s-wave neutrons. The latter quantity is expressed [37] by (11) where S_n is the separation energy of a neutron from a compound nucleus. It should be noted that S_n and Δ are given in the unit of MeV and a in MeV^{− 1}. The quantities D₀ and<Γ_γ0> lead to (12)

Table 6. Gamma-ray strength functions for s-wave neutrons in units of 10⁻⁴.

Compound	Present
nuclei	work	Mughabghab [16]	JENDL-4.0
^122Sb	71.62	90.2 ± 15.7	54.5
^124Sb	31.89	40.6 ± 4.0	24.2
^125Sb^*	109.66	–	197.2
^126Sb^*	16.60	–	52.5
^127Sb^*	34.10	–	34.1

^*Calculated from Equation (12(12) ).

3.2.4. Preequilibrium parameters

For preequilibrium capture, SLO-shaped photon absorption cross sections are required [20] to calculate the γ-ray emission rate. The peak cross section, resonance energy and resonance width are calculated from the systematics [32]. If the experimentally determined parameters are available in RIPL-3, the systematics are replaced with them. The single-particle state density g, which is required to describe the preequilibrium state density for a residual nucleus, is set to A/13 MeV⁻¹ for all nuclei but ^{121,123−126}Sn. Comparing with measured proton emission data, the g values were adjusted for these residual nuclei, i.e., 7.0 MeV⁻¹. There remain three kinds of normalization parameters for preequilibrium particle emission in the POD code: K, which determines the absolute square of the average matrix element for the residual two-body interaction and the scale factors for a pickup reaction and α-particle knockout, and . The values used are listed in Table 7.

Table 7. Preequilibrium parameters used in the present work.

Target nuclei
^121Sb	180	1.0	1.0
^123Sb	180	1.0	1.0
^124Sb	180	1.0	1.0
^125Sb	180	1.0	1.0
^126Sb	180	1.0	1.0

4. Comparison with experimental and other evaluated data in fast neutron energy region

The presently evaluated data are compared with exp- erimental and other evaluated data. The experimental data are taken from the EXFOR database [38], which is available in the international data centers such as the OECD/NEA Data Bank. As for the general-purpose nuclear data, there are three major libraries, i.e., JENDL-4.0 [1], ENDF/B-VII.1 [39], and JEFF-3.1.2 [40]. However, they are not independent of each other. The ^121,123Sb data of JEFF-3.1.2 were taken from JENDL-3.3, which means that the two isotopic data of JEFF-3.1.2 are identical to those of JENDL-4.0. On the other hand, the ^124,125Sb data of ENDF/B-VII.1 were taken from JENDL-3.3. Identical evaluated data are not drawn in the following figures. It should be noted that the general-purpose libraries do not contain any partial activation cross sections leading to the ground or isomeric state. The activation cross-section files JENDL/A-96 [41] and JEFF-3.1/A [42] are used for comparison with such activation data. We do not discuss most of the reactions where experimental data are unavailable, since it is difficult to judge whether the present evaluation is reasonable without measurements.

The radiative capture cross sections are shown in Figures 4–7. The present evaluation well reproduces the measured data of ^121,123Sb, while the JENDL-4.0 evaluation is lower than the measured data below several MeV. Concerning the elemental data, the measured data of Gibbons et al. [43] are systematically smaller than those of Spitz et al. [44], as seen in Figure 6. Both experiments employed the time of flight method, although the former one normalized the data to the shell transmission measurements obtained by Schmitt and Cook [45]. The JENDL-4.0 and ENDF/B-VII.1 evaluations obviously laid weight on the measurements of Gibbons et al. On the other hand, Macklin et al. [46] measured the capture cross sections of ^121,123Sb at 24 keV by using the activation method, and the sum weighted with the isotopic abundance is 739 ± 61 mb. This value is almost consistent with the value of 677 ± 94 mb at 22 keV measured by Spitz et al. within experimental uncertainties. The present evaluation placed emphasis on the consistency between the isotopic and elemental data. As a result, the cross sections obtained are larger than the JENDL-4.0 and ENDF/B-VII.1 evaluations below 100 keV. The capture cross sections of ^124,125Sb are systemically smaller than JENDL-4.0, as seen in Figure 7, since smaller γ-ray strength function values were used. As for ¹²⁶Sb, the same value as that for JENDL-4.0 was employed, yielding almost the same cross sections as those for JENDL-4.0.

Figure 4. Radiative capture cross section of ¹²¹Sb.

Figure 5. Radiative capture cross section of ¹²³Sb.

Figure 6. Radiative capture cross section of elemental Sb.

Figure 7. Radiative capture cross sections of ^{124, 125, 126}Sb.

The (n, 2n) reaction cross sections of ^121,123Sb are illustrated in Figures 8–13. The present evaluation is in good agreement with the data measured by Ghorai et al. [47]. The total (n, 2n) reaction cross sections of JENDL-4.0 (the sum of ground and isomeric state production) are obviously larger the present results above 12 MeV, which is consistent with the fact that the JENDL-4.0 evaluation overestimates the nonelastic cross sections in the high energy region. Concerning ¹²¹Sb(n, 2n)^120mSb, the JENDL/A-96 data are smaller than the present and JEFF-3.1/A evaluations in the entire energy region. Most of the measured partial cross sections of ¹²³Sb are larger than all evaluations, although the presently evaluated total (n, 2n) cross sections of ¹²³Sb agree with the data of Ghorai et al. [47], as mentioned above.

Figure 8. ¹²¹Sb(n, 2n)¹²⁰Sb reaction cross section.

Figure 9. ¹²¹Sb(n, 2n)^120gSb reaction cross section.

Figure 10. ¹²¹Sb(n, 2n)^120mSb reaction cross section.

Figure 11. ¹²³Sb(n, 2n)¹²²Sb reaction cross section.

Figure 12. ¹²³Sb(n, 2n)^122gSb reaction cross section.

Figure 13. ¹²³Sb(n, 2n)^122mSb reaction cross section.

Figures 14–17 show the (n, p) reaction cross sections. The ¹²¹Sb(n, p)^121gSn data of JEFF-3.1/A are not drawn in Figure 14, since they are identical to those of JENDL/A-96. It is difficult to judge which evaluation is the best because of few measurements for each reaction.

Figure 14. ¹²¹Sb(n, p)^121gSn reaction cross section.

Figure 15. ¹²³Sb(n, p)¹²³Sn reaction cross section.

Figure 16. ¹²³Sb(n, p)^123gSn reaction cross section.

Figure 17. ¹²³Sb(n, p)^123mSn reaction cross section.

Figure 18. ¹²¹Sb(n, α)¹¹⁸In reaction cross section.

The (n, α) cross sections of ^121,123Sb are shown in Figures 18 and 19. There exist no measured data for these reactions. The present evaluation is obviously smaller than JENDL-4.0 and ENDF/B-VII.1, since the default values were used for the preequilibrium parameters and .

Figure 19. ¹²³Sb(n, α)¹²⁰In reaction cross section.

The neutron emission spectra of elemental antimony are illustrated in Figure 20 at an incident energy of 14.1 MeV. As in the case [48] of Br, we assumed a pseudo-resonance at an excitation energy of 2.3 MeV for ^121,123Sb, which can substitute for the missing collective enhancement phenomenologically. The cross section for the resonance was calculated by DWBA using the neutron potential parameters mentioned in Section 3 with β₃ = 0.237, and it was smeared by a normal distribution with a standard deviation of 0.6 MeV. This treatment modifies the shapes of preequilibrium spectra. It is found from the figure that the present evaluation is in good agreement with the data measured by Takahashi et al. [49]. The ENDF/B-VII.1 evaluation obviously underestimates the experiment around 12 MeV.

Figure 20. Neutron emission spectra for elemental Sb at 14.1 MeV.

5. Concluding remarks

The neutron nuclear data of ^{121,123,124,125,126}Sb were evaluated in the energy region from 10⁻⁵ eV to 20 MeV. The RRPs of ^121,123Sb were modified by taking account of the latest measurements. The thermal capture cross sections of ^125,126Sb were determined by using a simplified formula. The URPs were obtained for self-shielding calculations by fitting to total and capture cross sections mainly calculated from the nuclear models.

Above the resolved resonance region, the present evaluation is mainly based on the statistical model using the POD code. The neutron transmission coefficients were obtained by the coupled-channel method for odd-mass targets, together with the total cross sections, the shape elastic scattering cross sections and the direct-reaction components of the inelastic scattering cross sections. The model parameters used in the calculations were determined more accurately than the earlier works. On the whole, the presently evaluated cross sections are in good agreement with available experimental data, and better than the existing evaluated data. Moreover, the gamma-ray production data, which were missing in JENDL-4.0 except for ¹²⁶Sb, were newly evaluated. The evaluated data are compiled into an ENDF-formatted data file for the next release of JENDL general purpose file.

Acknowledgements

The author would like to thank the members of the Nuclear Data Center, Japan Atomic Energy Agency, for their helpful comments on this work.

Appendix. Thermal capture cross section

Considering only s-wave, the radiative capture cross section at thermal energy is written as (A1) where Γ^J_n, Γ^J_γ, Γ^J, and E_J are the neutron width, the radiative width, the total width, and the resonance energy for a total spin J, respectively. The symbol k denotes the wave number of the relative motion in the incident channel, and it is related to the following equations: (A2) (A3) where m_n, m_t, and μ stand for the masses of a neutron and a target and the reduced mass, respectively. The statistical factor g_J for a target spin I is calculated from (A4)

Using the s-wave level spacing of the compound nucleus D₀, the neutron width is approximated [15] by (A5) with the wave number of the incident neutron inside the nuclear surface given by (A6) where S_n is the separation energy of a neutron from the compound nucleus, and E_F the Fermi energy. The Fermi energy is given [50] by (A7)

It is assumed that the thermal cross section is practically determined by a single resonance near E₀. Substituting Equation (A5(A5) ) into Equation (A1(A1) ), one can obtain (A8) The quantity |E₀ − E_J| is expected to be the order of D₀. Thus, we may put |E₀ − E_J| ≡ D₀/2ζ, where ζ ≥ 1. Assuming that |E₀ − E_J| ≫ Γ^J/2, we obtain (A9)

The statistical factor g_J is 1.0 for I = 0, while g_J is approximately equal to 1/2 if I ≠ 0. Furthermore, setting ζ² = 1 and replacing Γ^J_γ with the average s-wave radiative width ⟨Γ_γ0⟩, Equation (A9(A9) ) becomes (A10) (A11)