Evaluation of neutron nuclear data for gallium

Abstract

Neutron nuclear data on ^69,71Ga have been evaluated for the next version of the Japanese Evaluated Nuclear Data Library (JENDL) general purpose file in the energy region from eV to 20 MeV. The resolved resonance parameters at negative energies were adjusted so as to reproduce measured thermal capture cross sections. 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 neutrons. Pre-equilibrium and direct-reaction processes were taken into account 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
• gallium isotopes
• resonance parameter
• 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 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 the time limit. Hence, the fast-neutron cross sections for the rest of the fission-product nuclides may need improvements for the next version of the JENDL general purpose file. Gallium (Ga) isotopes are classified into such nuclides. Moreover, Ga is an important semiconductor material. Therefore, it is required to re-examine the cross sections of Ga.

Ga data were evaluated for JENDL-3.2 [2] in 1994. The Ga data of JENDL-3.3 [3] and JENDL-4.0 [1] are essentially based on those of JENDL-3.2. In the 1994 evaluation, the statistical-model code CASTHY [4] 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 multistep Hauser–Feshbach [5] code GNASH [6]. The direct inelastic scattering was considered by using the distorted-wave Born approximation (DWBA). However, gamma-ray production data were not compiled into the library.

The present work was undertaken to improve the evaluated data on stable Ga isotopes ^69,71Ga 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, α), (n, np), (n, nd), (n, nα), (n, 2n) and (n, 3n) reaction cross sections, the angular distributions of emitted neutrons and charged particles, and the energy distributions of emitted particles and gamma rays in the incident-neutron energy region from 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. [7] and are listed in Table 1, together with isotopic abundances [8]. The resolved resonance parameters at negative energies were modified so as to reproduce measured thermal capture cross sections. 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 stable Ga isotopes

	^69Ga	^71Ga
Abundances (%)	60.108	39.892
Q values (MeV)
(n, γ)	7.654	6.520
(n, p)	−0.127	−2.031
(n, α)	2.577	1.073
(n, d)	−4.385	−5.640
(n, t)	−8.326	−8.601
(n, ^3He)	−8.869	−11.264
(n, 2n)	−10.313	−9.301
(n, np)	−6.610	−7.865
(n, nα)	−4.489	−5.246
(n, nd)	−14.583	−14.858
(n, 3n)	−18.591	−16.955

This paper presents how the evaluation was performed. Section 2 deals with the resonance parameters. In Section 3, the computational methods and procedures 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 parameters

The upper boundaries of the resolved resonance regions were set to 5.9 keV and 5.6 keV for ^69,71Ga, respectively, in JENDL-4.0. No new resolved resonance parameters are available experimentally. However, in this work, attention was paid to thermal capture cross sections. Measured thermal capture cross sections are listed in Table 2 . The JENDL-4.0 evaluation is completely based on the data of Koester et al. [10], which deviate from the other measurements for both nuclides. The capture cross sections of Koester et al. were obtained from their measured total and scattering cross sections, while the other measurements were derived by using the activation method. As for ⁷¹Ga, Uddin et al. [17] obtained a capture cross section of 2.75 ± 0.14 b at 0.0536 eV by using the activation method. The capture cross sections ordinarily obey the 1/v law, where v stands for the velocity of the incident neutron. The value of Uddin et al. would amount to 4.00±0.20 b at 0.0253 eV. In the present work, it was judged that the thermal capture cross section of JENDL-4.0 is too large for ⁶⁹Ga but too small for ⁷¹Ga. Consequently, the resonance parameters at negative energies for both isotopes were re-adjusted so as to be consistent with the capture cross sections measured by the activation method. The thermal cross sections are given in Table 3 , where the capture cross sections are given at 300 K and the elastic scattering cross sections at 0 K. It is found from the table that the presently evaluated results are in better agreement with the recommendations of Mughabghab [18] than the JENDL-4.0 evaluations except for the elastic scattering cross section of ⁷¹Ga.

Table 2 Experimental data on thermal capture cross sections of ^69,71Ga

Target nuclei	Author	Year	Cross section (b)
^69Ga	Son et al. [9]	2011	1.69 ± 0.08 b
	Koester et al. [10]	1984	2.22 ± 0.05 b
	Rives [11]	1971	1.68 ± 0.07 b
	Pomerance [12]	1952	1.99 b ± 8%
	Seren et al. [13]	1947	1.4 b ± 20%
^71Ga	Son et al. [9]	2011	4.45 ± 0.25 b
	Karadag et al. [14]	2003	4.41 ± 0.18 b
	Huang et al. [15]	2000	4.62 ± 0.09 b
	Koester et al. [10]	1984	3.67 ± 0.10 b
	Gleason [16]	1975	4.4 ± 0.2 b
	Ryves [11]	1971	4.71 ± 0.23 b
	Pomerance [12]	1952	4.9 b ± 8%

Table 3 Thermal cross sections of ^69,71Ga

Reaction	Present	JENDL-4.0	Mughabghab [18]
^69Ga capture	1.718 b	2.201 b	1.75 ± 0.07 b
^69Ga elastic	7.626 b	8.162 b	7.75 ± 0.08 b
^71Ga capture	4.209 b	3.710 b	4.61 ± 0.15 b
^71Ga elastic	5.288 b	5.236 b	5.16 ± 0.06 b

The ASREP code [19] was used to determine the URPs by fitting to the total and capture cross sections evaluated by the nuclear model calculations described in the following section. The unresolved resonance regions are 5.9 keV–1 MeV for ⁶⁹Ga and 5.6 keV–1 MeV for ⁷¹Ga. It should be noted that the URPs obtained are used only for self-shielding calculations, since the point-wise cross sections are given in the evaluated data files. The cross sections of ⁶⁹Ga and ⁷¹Ga reconstructed from the URPs presently obtained are compared with the total and capture cross sections calculated from the nuclear models in Figures 1 and 2, respectively. In the parameter fitting, 10% errors of the nuclear model calculations were assumed. It is found from the figures that the reconstructed cross sections reproduce the nuclear model calculations very well.

Figure 1 Unresolved resonance region for n + ⁶⁹Ga

Figure 2 Unresolved resonance region for n + ⁷¹Ga

3. Computational methods and procedures above the resonance region

3.1. Nuclear models

The POD code [20] was used for calculating the neutron-induced reaction cross sections of Ga isotopes. The code is based on the spherical optical model, the one-component exciton pre-equilibrium model, the DWBA, and the multistep statistical model. In order to simulate the direct and semidirect effects on the capture reaction, the pre-equilibrium capture was considered by using the gamma-ray emission rate derived by Akkermans and Gruppelaar [21]. The original POD code was modified to allow for options of gamma-ray strength functions, i.e. generalized, extended generalized, and modified Lorentzians [22, 23] (hereafter GLO, EGLO, and MLO, respectively) for E1 radiation, in addition to the standard Lorentzian.

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 [24] 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. [25] using the coupled-channel method based on the rigid-rotator model [26]. The coupling schemes and deformation parameters are listed in Table 4. The calculated total cross section of elemental Ga is illustrated in Figure 3, together with experimental data and the spherical optical model calculations using the parameters of Koning and Delaroche [27]. It is found from the figure that the present calculations are in good agreement with the experimental data in the entire energy region. The spherical optical model parameters of Koning and Delaroche yield large cross sections in the energy region below 1 MeV. Built-in values in POD were used for charged-particle optical-model parameters, i.e. the values of Koning and Delaroche [27] for protons, those of Lohr and Haeberli [28] for deuterons, those of Becchetti and Greenlees [29] for tritons and ³He, and those of Lemos [30] potentials modified by Arthur and Young [31] for α particles. The values of Lohr–Haeberli and Becchetti–Greenlees potentials were actually taken from the compilation of Perey and Perey [32].

Figure 3 Total cross section of elemental Ga

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

Target nuclei	Coupled levels ^a	Deformation parameters
^69Ga	3/2^−(g.s.), 5/2^−(1.107), 7/2^−(1.337)	β2 = −0.205
^71Ga	3/2^−(g.s.), 5/2^−(0.965), 7/2^−(1.395)	β2 = −0.228

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 16 nuclei, i.e. ^67–72Ga, ^67–71Zn and ^65–69Cu. The discrete levels were taken from the reference input parameter library (RIPL-3) [33].

Concerning the level density, the composite formula of Gilbert and Cameron [34] was used in the present work. In the region of low excitation energy E, the level density is described by the constant temperature formula

On the other hand, the a parameter, which characterizes the Fermi-gas part of the level density ρ _F , is defined as

where E _sh is the shell correction energy and γ is 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(*), E _sh, γ, and Δ were taken from the work of Mengoni and Nakajima [35, 36]. The two parameters T and E ₀ 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 E ^max _level.

Table 5 Level density parameters for each nucleus

	a(*)	Δ	E sh	T	E 0	E m	E ^max level
Nuclei	(1/MeV)	(MeV)	(MeV)	(MeV)	(MeV)	(MeV)	(MeV)
^72Ga	9.658	0.000	3.408	0.897	−2.795	7.185	0.684
^71Ga	9.215	1.424	3.011	0.843	−0.395	7.131	2.396
^70Ga	9.327	0.000	2.337	0.973	−2.875	7.627	1.598
^69Ga	8.997	1.445	2.185	0.902	−0.433	7.476	2.668
^68Ga	9.282	0.000	1.568	1.037	−3.150	8.243	1.344
^67Ga	8.778	1.466	1.090	1.056	−1.152	9.129	2.457
^71Zn	9.825	1.424	3.275	0.785	−0.371	6.833	1.421
^70Zn	9.521	2.869	2.799	0.874	0.622	9.247	3.246
^69Zn	9.482	1.445	2.644	0.956	−1.539	9.105	1.633
^68Zn	8.534	2.910	1.955	1.171	−1.144	12.907	3.732
^67Zn	9.462	1.466	1.848	1.042	−2.076	10.192	1.875
^69Cu	8.997	1.445	2.252	0.631	1.308	3.894	2.800
^68Cu	9.282	0.000	1.714	0.576	0.091	1.826	0.722
^67Cu	8.778	1.466	1.483	0.699	1.265	4.284	2.940
^66Cu	9.511	0.000	0.918	0.842	−1.246	4.898	1.577
^65Cu	8.558	1.488	0.522	1.103	−1.078	9.278	3.036

3.2.3. Gamma-ray transition

The GLO form was used for E1 radiation in the present work, since the calculations with GLO could reproduce measured gamma-ray spectra for other nuclei [37– 39] satisfactorily. The giant resonance parameters were taken from the compilation of Kopecky [40]. The resonance energy and width parameters are given by

for E1, M1 and E2 radiations, respectively. The symbol A stands for the mass number of the nucleus. The contributions of M1 and E2 relative to E1 were estimated [6] such as

As for compound nuclei ^70,72Ga, 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 gamma-ray strength functions S _γ0 obtained are compared in Table 6 with the values recommended by Mughabghab [18]. Normalization factors for the other nuclei were obtained by using the calculated average level spacing D ₀ 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 [41] by

where S _n is the separation energy of a neutron from a compound nucleus. These two quantities lead to

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

Compound nuclei	Present work	Mughabghab [18]
^70Ga	9.549	8.4 ± 1.3
^72Ga	7.958	6.75 ± 0.87

3.2.4. Pre-equilibrium parameters

For pre-equilibrium capture, Lorentzian-shaped photon absorption cross sections are required [21] to calculate the gamma-ray emission rate. The resonance energy and width are given in Equations (3) and (4), respectively. The peak cross section is given [40] by

where Z is an atomic number and N = A − Z. The single-particle state density g, which is required to describe the pre-equilibrium state density for a residual nucleus, is set to A/13 MeV⁻¹ for all nuclei but ⁶⁶Cu. Comparing with measured ⁶⁹Ga(n, α) data, the g value was adjusted for ⁶⁶Cu, i.e. 2.6 MeV⁻¹. There remain three kinds of normalization parameters for pre-equilibrium 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, K _pickup and K _knockout. These parameters were determined so as to fit measured cross sections. The factors K _pickup were actually adjusted by considering the measured (n, α) cross sections with K _knockout = 1.0. The values obtained are listed in Table 7 .

Table 7 Pre-equilibrium parameters used in the present work

Target nuclei	K(MeV^3)	K pickup	K knockout
^69Ga	115	0.5	1.0
^71Ga	100	1.0	1.0

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

The presently evaluated data are compared with experimental and other evaluated data. The experimental data are taken from the EXFOR database [42], 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 [43], and JEFF-3.1.2. [44]. Only elemental Ga data that were taken from JEND-3.2 [2] are contained in JEFF-3.1.2. In the fast neutron energy regions, ⁶⁹Ga and ⁷¹Ga of ENDF/B-VII.1 were taken from JENDL-3.3 [3] and CENDL-3 [45], respectively. As described in Section 1, the Ga data of JENDL-4.0 are almost the same as those for JENDL-3.2 and 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 [46] and JEFF-3.1/A [47] are used for comparison with such activation data.

The radiative capture cross sections of ⁶⁹Ga and ⁷¹Ga are illustrated in Figures 4 and 5, respectively. The presently evaluated cross sections are in good agreement with measurements, but slightly larger than the JENDL-4.0 values. It is found from Figure 6 that the present evaluation reproduces measured elemental data below 200 keV. Figure 7 shows the Maxwellian-averaged capture cross sections of elemental Ga as a function of kT. The present evaluation is consistent with the data of Walter [48] at kT = 20–50 keV, while the JENDL-4.0 and ENDF/B-VII.1 evaluations are slightly lower than them.

Figure 4 Radiative capture cross section of ⁶⁹Ga

Figure 5 Radiative capture cross section of ⁷¹Ga

Figure 6 Radiative capture cross section of elemental Ga

Figure 7 Maxwellian-averaged capture cross section of elemental Ga

The (n, 2n) reaction cross sections are shown in Figures 8 – 10. As for ⁶⁹Ga, the present evaluation goes through the recent measurements [49– 52], but is much smaller than the data of Bormann et al. [53] especially above 15 MeV. Although they provide important activation data in a wide energy range, the measured data are sometimes considerably larger than other experimental data or evaluated data, as seen in Refs. [39, 54, 55]. After changing various model parameters, it was found that fitting to the data of Bormann et al. [53] led to poor reproduction of the other reaction cross sections of ⁶⁹Ga. Therefore, the solid curve in Figure 8 was finally adopted as our evaluation. On the other hand, the measured data of ⁷¹Ga are well reproduced by the present evaluation, which is almost consistent with JENDL-4.0 and ENDF/B-VII.1. Concerning elemental data, the data of Frehaut et al. [56] are solely available. However, it is well known that their measurements are systematically smaller than others. Vonach et al. [57] suggested that a normalization factor of 1.078 be applied to all of their data. In Figure 10, the data multiplied by the normalization factor are also shown. In the energy region below 15 MeV, all the evaluated data agree with the renormalized data of Frehaut et al.

Figure 8 ⁶⁹Ga(n, 2n)⁶⁸Ga reaction cross section

Figure 9 ⁷¹Ga(n, 2n)⁷⁰Ga reaction cross section

Figure 10 (n, 2n) reaction cross section of elemental Ga

The charged-particle emission cross sections are shown in Figures 11 –17. On the whole, the present statistical-model calculations are in good agreement with experimental data except for a few cases. The ⁶⁹Ga(n, p)^69g Zn cross sections, which were measured by Nesaraja et al. [58] using an enriched sample (0.46% for ⁶⁹Ga and 99.54% for ⁷¹Ga) and β⁻ counting in the energy region from 6 to 12 MeV, are not reproduced by all the evaluations, although the evaluations are consistent with the measurement of Demichelis et al. [59] at 14 MeV. Considering the very small isotopic composition of ⁶⁹Ga, the present work did not place emphasis on the experiment of Nesaraja et al. for ⁶⁹Ga(n, p)^69g Zn. Concerning the ⁷¹Ga(n, p)^71g Zn reaction, the present evaluation laid weight on the measurements of Nesaraja et al. [58], whereas JENDL/A-96 and JEFF-3.1/A are obviously based on the data of Molla and Qaim [60]. As for the ⁷¹Ga(n, nα)⁶⁷Cu reaction, both the present and JENDL-4.0 evaluations are smaller than the experimental data of Bormann et al. [61] above 15 MeV, although those evaluations are almost consistent with the other measurements around 14 MeV. The calculation attempted to reproduce experimental data on as many reactions as possible rather than to focus on a particular experimental data set. Therefore, the inconsistency with the data of Bormann et al. was allowed in this work.

Figure 11 ⁶⁹Ga(n, p)^69g Zn reaction cross section

Figure 12 ⁶⁹Ga(n, p)^69m Zn reaction cross section

Figure 13 ⁶⁹Ga(n, α)⁶⁶Cu reaction cross section

Figure 14 ⁷¹Ga(n, p)^71g Zn reaction cross section

Figure 15 ⁷¹Ga(n, p)^71m Zn reaction cross section

Figure 16 ⁷¹Ga(n, α)^68m Cu reaction cross section

Figure 17 ⁷¹Ga(n, nα)⁶⁷Cu reaction cross section

The neutron emission spectra of elemental Ga are illustrated in Figure 18 at an incident neutron energy of 14.63 MeV. It is found from the figure that the present evaluation is in good agreement with the data measured by Hermsdorf et al. [62]. JENDL-4.0 and ENDF/B-VII.1 slightly underestimate the measurements between 10 and 12 MeV. The spectra below 4 MeV are essentially formed by the (n, 2n) reaction. As mentioned above, the ⁶⁹Ga(n, 2n) cross section was reduced by 9% at 14.5 MeV compared with the JENDL-4.0 evaluation. This reduction does not cause any deficiencies in the low-energy spectra, although the measurements are available down to 2 MeV.

Figure 18 Neutron emission spectra for elemental Ga at 14.63 MeV

5. Concluding remarks

The neutron nuclear data of ^69,71Ga were evaluated in the energy region from 10⁻⁵ eV to 20 MeV. The resolved resonance parameters at negative energies were adjusted so as to reproduce the measured thermal capture cross sections of both isotopes, while the other resolved resonance parameters remain unchanged from JENDL-4.0. The URPs were obtained for self-shielding calculations by fitting to total and capture cross sections 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, together with the total cross sections, the shape elastic scattering cross sections and the direct-reaction components of the inelastic scattering cross sections. On the whole, the presently evaluated cross sections are in good agreement with available experimental data, and better than the JENDL-4.0 data. Moreover, the gamma-ray production data, which were missing in JENDL-4.0, were newly evaluated. The evaluated data are compiled into an ENDF-formatted data file for the next release of the 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.

Notes

^aThe numbers in parentheses indicate the excitation energy in MeV.