Activation cross-sections for the 185Re(n, 2n) reaction and the isomeric cross-section ratio of 184m,gRe in the neutron energy range of 13–15 MeV

Cross-sections and their isomeric ratios (σm/σg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \sigma_{\mathrm{m}}/\sigma_{\mathrm{g}}$\end{document}) for the 185Re(n, 2n)184mRe and 185Re(n, 2n)184gRe reactions in the 13–15 MeV range were measured. The neutron activation technique was applied using the K-400 neutron generator at the Chinese Academy of Engineering Physics (CAEP). Natural Re samples and Nb monitor foils were activated jointly to determine the reaction cross-section and the incident neutron flux. The 3H(d, n)4He reaction was used to generate the neutron beam. The pure cross-section of the ground state was derived from the absolute cross-section of the metastable state using residual nuclear decay analysis. Numerical calculations using the nuclear-model-based computer code TALYS-1.8 with six level density models were used to obtain 185Re(n, 2n)184m, gRe reaction excitation functions and their isomeric cross-section ratios. Finally, experimentally determined cross-sections were compared with corresponding literature data.


Introduction
Rhenium (Re) metal is a high temperature corrosion resistant material, and its boron-based alloys are effective neutron absorbers used in the regulation of nuclear reactors. A high output of 14 MeV neutrons can be achieved via the 3 H(d, n) 4 He reaction, with a flux rate of approximately 3 × 10 14 n/s [1]. As a result, the activation of structural materials in fusion reactors has to be considered. 14 MeV neutrons can induce a series of nuclear reactions including 185 Re(n, 2n) 184m Re, 185 Re(n, 2n) 184g Re and 185 Re(n, 2n) 184m+g Re, and the related cross-sectional data are of great importance for the evaluation of safety in fusion reactors. In particular, they can be used to determine the required treatment of radioactive waste from reactor structural materials and improve radiation protection procedures.
Discrepancies in the literature mentioned above may be due to the following two reasons. The first is related to difference in decay data (i.e., the selection of characteristic rays). As shown in fig. 1, the main characteristic rays of the produced nuclei in the 185 Re(n, 2n) 184m,g Re reactions are partly coincident. However, the procedure for eliminating these effects is not found in the published literature. The second pertains to the interference reaction processing method. In the 185 Re(n, 2n) 184g Re reaction, the effects of both IT and EC of 184m Re on 184g Re should be  Metastable and ground states in the 185 Re(n, 2n) 184m,g Re reactions [2]. All energies are in keV. Transition from both the excited and ground state is represented by the bold black line, with the intensity in brackets indicating rays originating from both states. eliminated. Nevertheless, the elimination of these effects has not been addressed in detail in previous studies.
Thus, in this study we utilize the latest decay data and the related decay laws of the produced nuclei to remove the effect of interference reactions and select multiple appropriate characteristic rays. We then calculate the weighted average of the cross-section of the 185 Re(n, 2n) 184m Re reaction and the 185 Re(n, 2n) 184g Re reaction in the pure ground state and the related crosssection ratio. The obtained cross-sections are then analyzed in comparison with theoretical and previously reported results (table 1).

Material
Disk of natural rhenium metal (purity 99.99%, about 1.0 mm in thickness, 2.0 cm in diameter, China New Metal Materials Technology Co, Ltd.) was sandwiched between disks of niobium (purity 99.99%, 0.12 mm in thickness) of the same diameter. Three such samples (Nb-Re-Nb) were prepared for irradiation.

Neutron energy and irradiation
The irradiation of the rhenium disks was performed at the Chinese Academy of Engineering Physics (CAEP) K-400 neutron generator. A beam with an effective deuteron beam energy of 136 keV and current of 260 μA was used for the production of neutrons via the 3 H(d, n) 4 He reaction. The neutron-production target was a tritium-titanium (T-Ti) target 2.20 mg cm −2 in thickness. The neutron yield was ∼ (3-4) × 10 10 n/(4πs). The sample positions in the experiment are shown in fig. 2. Groups of samples were placed at 0 • , 90 • or 135 • with respect to the beam line and centered about the T-Ti target at distances of about 40 mm. Natural Re samples and Nb monitor foils were activated jointly to determine the reaction cross-section and the incident neutron flux. Further details can be found in our previously published work [11][12][13]. Neutron energies in this measurement were obtained using the formula from ref. [14], which is determined by the distance between the sample and T-Ti target, the emergent angle and the radius of the sample.

Gamma spectroscopic measurements
The activities of product radionuclides were measured using high-purity germanium (HPGe) gamma-ray spectroscopy (∼ 68% relative efficiency, 1.69 keV resolution at 1332.5 keV of 60 Co). The detector was connected to a digital gamma spectrometry system (ORTEC, model GEM 60P) and Maestro data acquisition software. The separation between sample and detector was variable within a range of 0 to 8 cm. Figure 3 shows the typical spectra acquired from the irradiated Re samples during measurement of isomeric and ground states, with the γ-rays of interest marked. The γ-ray intensities and half-lives used in the analysis are summarized in table 2 [2].

Calculation of cross-sections and their uncertainties
The method for cross-section calculation is described in detail in our previous papers [15,16]. The standard acti- where F is the total correction factor of the activity, given by where f s and f c are the correction factors for the selfabsorption of the sample at a given gamma energy, and the coincidence sum effect of cascade gamma rays in the investigated nuclide, respectively. Rhenium is a heavy element with very pronounced gamma ray self-absorption, particularly at low energies. Values of the mass attenuation coefficient (μ/ρ) for rhenium were obtained by interpolating the values reported in [17]. Correction factors for the characteristic gammaray levels are given in table 3.
In the process of calculating the cross-sections of the 185 Re(n, 2n) 184g Re reactions, C x in eq. (1) should be the result of the measured full-energy peak area minus the contribution from 184m Re via 184m Re . Rules for the creation and decay of artificial radioactive nuclides (see [15,16] for details) provide that C 1 and C 2 can be written as where m, g, P mg , C m , I and ε are metastable state, ground state, fraction of metastable state decays that produce ground state nuclides (branching ratio), full energy peak area of the measured metastable state, gamma ray intensity and full-energy peak efficiency of the characteristic gamma-rays, respectively. S m , S g , D m and D g are given by In eq. (4), the subscript 1 and 2 represent two different characteristics of gamma rays.

Experimental uncertainty 3.1 Mean (arithmetic average)
The relation for experimental mean σ for n trial measurements, σ i ± Δσ i , with i = 1, . . . , n, is given by

Experimental standard deviation
The experimental standard deviation Δσ A is defined as Obtaining as much knowledge as possible from a limited number of measurements is one of the fundamental problems in experimental science. In particular, eq. (6) for the error Δσ A of the weighted mean can yield unphysical values for very small samples.
In order to prevent this, we introduce Δσ B which limits the contribution of individual errors to Δσ However, eq. (7) may also fail if two data points are very different and have relatively small error bars. The standard deviation Δσ of the weighted average σ may then be calculated for a limited number of measurements using the following equation: Uncertainty analysis was carried out using the quadrature method [18]. The uncertainties quoted for the present measurements are estimates of standard uncertainties and include contributions due to uncertainties in cross-section of the monitor reaction (1.1-1.5%), photopeak detection efficiency (2.0-3.0%), counting statistics (0.04-2.0%), relative gamma-ray intensity (0.1-10.0%), half-life (0.02-4.7%), sample weight (< 0.1%), timing (< 0.1%), selfabsorption of gamma-ray (∼ 0.5%), and isotopic abundance (0.06%). The uncertainty of the weighted average cross-section was between 3.2 and 4.4%.

Nuclear model calculations
Nuclear model-based calculations are of great importance since the existing measured data on cross-sections induced by neutrons for the evaluation of safety in fusion reactors are lacking or inconsistent. It is well known that nuclear reaction models are reliable means for calculating the energy and angle distributions of the reaction products or radionuclide production cross-sections [19,20]. The reaction model calculations include direct-interaction, equilibrium and pre-equilibrium processes. Level density as a function of energy is among the most important inputs for crosssection calculation within nuclear reaction models [21]. Nuclear level density (NLD) is the number of excited levels per energy interval (dN/dE) near the excitation energy. Excited nuclear levels are discrete at low energies; however, they approach a continuum as the excitation energy increases. Therefore, a nuclear model for calculating level density is needed for the continuum energy regime. An accurate and reliable description of the excited levels of a nucleus at both low and high excitation energy region is necessary for testing the quality of a reaction model used for the calculation of cross-sections [22]. The TALYS code (version 1.8) calculates the partial and total cross-section, the angular distribution, the energy spectrum, the differential spectrum and recoils. This code employs various microscopic and phenomenological nuclear level density models for obtaining the nuclear cross-sections [23]. The theoretical excitation function of the 185 Re(n, 2n) 184m Re and 185 Re(n, 2n) 184g Re reactions and their isomeric crosssection ratios at different neutron energies from threshold to 20 MeV were calculated using TALYS, with default values of the parameters and only the selected level density parameters adjusted. The details of the level density parameters were reported elsewhere [11,12,[24][25][26].

Results and discussion
Cross-sections measured across the energy range of 13 to 15 MeV are given in table 4. The results obtained are prone to relatively small uncertainties due to the use of the weighted average method. All the nuclear reactions investigated within the scope of this work are discussed below. Cross-sections for the 93 Nb(n, 2n) 92m Nb monitor reaction were taken from ref. [27].

185 Re(n, 2n) 184m Re reaction
In the present work, gamma-rays with energies of 104.7 keV (I γ = 13.6%), 161.3 keV (I γ = 6.56%), 226.7 keV (I γ = 1.49%), 318.0 keV (I γ = 5.81%), 384.3 keV (I γ = 3.17%), 536.7 keV (I γ = 3.34%), and 920.9 keV (I γ = 8.2%) emitted in the decay of 184m Re were used to calculate the value of the 185 Re(n, 2n) 184m Re reaction cross-section. In a previous measurement [8], the 895 keV and 904 keV gamma-rays were used to calculate the values of the 185 Re(n, 2n) 184m Re reaction crosssection. However, these two gamma-rays not only result from the product of 185 Re(n, 2n) 184m Re reaction, but also from the product of 185 Re(n, 2n) 184g Re reaction and the decay 184m Re Re. The threshold energy of this reaction is 7.900 MeV. In order to avoid the effect of low-energy neutrons, a reaction near the threshold, i.e. the 93 Nb(n, 2n) 92m Nb (E th = 8.972 MeV) monitor reaction was selected, whereas Zhu et al. [7] used the lower threshold monitor reaction 27 Al(n, α) 24 Na (E th = 3.249 MeV) and Zhang et al. [5] and Druzhinin et al. [8] did not give information about the monitor reaction. Figure 4 shows weighted averages of our results along with other published data from refs. [3][4][5][6][7][8][9]. It can be seen that there are significant differences among the data from the literature. In the energy region between 13 and 15 MeV the values of TALYS-1.8 calculation using ldmodels 1-6 are about 100% higher than the values from previous measurements [4][5][6][7][8][9], with the exception of Karam et al. [3]. Shapes of the excitation curves of the TALYS-1.8 calculation with ldmodels 1-6 exhibit a trend similar to the present data set, but the obtained results are somewhat higher than the theoretical calculations.

Conclusion
The activation cross-sections for 185 Re(n, 2n) 184m Re, 185 Re(n, 2n) 184g Re, and 185 Re(n, 2n) 184m+g Re reactions along with isomeric cross-section ratios induced by 13.5, 14.1 and 14.8 MeV neutrons, have been obtained using the latest decay data and weighted average method. The nuclear model using the TALYS code showed that the microscopic level densities (Skyrme force) from Hilaire's combinatorial tables (ldmodel 5) [38] are appropriate for the isomeric cross-section ratios for 185 Re(n, 2n) 184m Re and 185 Re(n, 2n) 184g Re reactions, while models composed of microscopic level densities (Skyrme force) from Goriely's tables [38] (ldmodel 4) and microscopic level densities (temperature dependent HFB, Gogny force) from Hilaire's combinatorial tables [38] (ldmodel 6) are found to be appropriate for the 185 Re(n, 2n) 184g Re reaction. Our experimental results were then compared to those from the literature and to numerical calculations. The comparative analysis including cross-section data from the literature revealed that the inconsistencies in the published data can stem from: 1) the decay data (the selection of characteristic gamma-rays); 2) interfering reactions. A detailed comparison with theoretical calculations revealed that σ g and σ m+g cross-sections were easily reproduced by the calculations, while for σ m (8 + ), the theoretical results could only describe the general trend of the experimental data.
Results reveal the importance of the level scheme of the residual nuclei and indicate the possibility of incomplete documentation of high-spin levels in the level schemes of these residual nuclei. Furthermore, they highlight certain limitations in the nuclear codes, particularly regarding the embedding of discrete states in the continuum, which is not currently possible and affects the reproduction of highspin isomeric cross-sections. The results presented in this work are valuable for the improvement of nuclear data libraries, verification of nuclear reaction models and other practical applications.
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