Activation cross section and isomeric cross section ratio for the 76Ge(n,2n)75m,gGe process

We measured neutron-induced reaction cross sections for the 76Ge(n,2n)75m,gGe reactions and their isomeric cross section 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}$\sigma_{m}/\sigma_{g}$\end{document} at three neutron energies between 13 and 15MeV by an activation and off-line γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma$\end{document}-ray spectrometric technique using the K-400 Neutron Generator at the Chinese Academy of Engineering Physics (CAEP). Ge samples and Nb monitor foils were activated together to determine the reaction cross section and the incident neutron flux. The monoenergetic neutron beams were formed via the 3H(d,n)4He reaction. The pure cross section of the ground state was derived from the absolute cross section of the metastable state and the residual nuclear decay analysis. The cross sections were also calculated using the nuclear model code TALYS-1.8 with different level density options at neutron energies varying from the reaction threshold to 20MeV. Results are discussed and compared with the corresponding literature data.


Introduction
Activation cross sections of neutron threshold reactions on medium mass nuclei are of considerable interest for testing nuclear models. Furthermore, the data for potential first wall constituents of a fusion reactor are of practical importance, especially for estimating nuclear heating, nuclear transmutation, and radiation damage effects [1]. A lot of experimental data on neutron induced cross sections for fusion reactor technology applications have been reported and great efforts have been devoted to compilations and evaluations [2,3]. We chose to study the neutron-induced reaction cross sections of germanium-76 mainly for four reasons. First, the germanium is an important semi-conducting material for the nuclear technology and integrated circuits; second, the 76 Ge nucleus lies between the magic numbers of 28 and 50; shape coexistence plays a prominent role in its structure [4], and 76 Ge may be a rare example of a nucleus exhibiting rigid triaxial deformation in its low-lying states [5,6]; third, the germanium-75 isomeric pair is an example of the isomeric pair type in which the half-life of the metastable state is shorter than that of the ground state and decays almost entirely by isomeric transition (see fig. 1); fourth, although there are enough data for metastable cross sections for the 76 Ge(n, 2n) 75m Ge reactions in the energy a e-mail: luojh71@163.com range from 13 to 15 MeV [7][8][9][10][11][12][13][14], only three direct measurements for the ground state cross section σ g have been performed separately [7,9,14]. The experimental and theoretical data for the 76 Ge(n, 2n) 75g Ge reaction cross section are inconsistent. In the energy region around 14 MeV, the experimental cross sections [7,9,14] are clustered around 550 mb, while the results of TALYS are centered around 300 mb. Therefore, we aimed to measure the pure ground state cross section σ g directly by means of the analysis methods of residual nuclear decay [15][16][17] and to compare the experimental results to those obtained by the statistical model calculation.

Samples and irradiations
Two disks, about 0.1 and 0.19 cm in thickness and 20 mm in diameter, were formed by pressing approximately 1.7 and 3.2 g of Ge (99.99% pure) powder (natural isotopic composition) at 980 MPa to form a pellet. The samples were irradiated near the target and sandwiched between two Nb foils (99.99% pure, 0.12 mm thick) with the same diameter which were used to monitor the neutron fluence via the 93 Nb(n, 2n) 92m Nb reaction.  Irradiation of the samples was carried out at the K-400 Neutron Generator at the Chinese Academy of Engineering Physics (CAEP) and lasted approximately 3 minutes with a neutron yield (3-4)×10 10 n/s in 4π solid angle. Neutrons were produced by the T(d,n) 4 He reaction with an effective deuteron beam energy of 135 keV, beam current of 240 μA, and the diameter of the deuteron beam spot was under 0.6 cm. The groups of samples were placed at 0 • , 90 • and 135 • relative to the beam direction and centered a 0.566 mg/cm 2 thick tritium-molybdenum (T-Mo) target at a distance of ∼ 50 mm. In order to avoid the deposition of deuterium in the target, the new T-Mo target is used. The diameter of the active zone of the T-Mo target is 1.2 cm. The sample positions in the experiment are shown in fig. 2. In order to avoid the effect of low energy neutrons, samples were wrapped in cadmium foil. During irradiation, the neutron flux was monitored by accompanying α-particles so that corrections could be made for small variations of the yield. Cross sections for the 93 Nb(n, 2n) 92m Nb monitor reaction were taken from [19].

Measurement of radioactivity
High-resolution gamma-ray spectroscopy was applied to the activated disks. The measurements were carried out using low-background high-purity germanium (HPGe) detector (ORTEC, model GEM 60P, crystal diameter 70.1 mm, crystal length 72.3 mm) with a relative efficiency of ∼ 68% and an energy resolution of 1.69 keV at 1.332 MeV for 60 Co. The distance from sample to detector was 2.0 cm. To avoid excessive death time, the sample was cooled for 2 minutes after irradiation. Figure 3 shows the typical spectra acquired from the Ge samples during the measurement of the isomeric and ground state, where the γ-rays of interest have been marked. The γ-ray intensities and half-lives used in the analysis are summarized in table 1 [18]. The detector was pre-calibrated for energy and efficiency by using the standard gamma ray sources 54 Mn, 57 Co, 60 Co, 109 Cd, 133 Ba, 137 Cs, 152 Eu, 241 Am and 226 Ra.

Calculation of cross sections and their uncertainties
The cross sections were calculated by the following formula [16,17]: where the subscript 0 represents the term corresponding to the monitor reaction and subscript x corresponds to the measured reaction; ε is the full-energy peak efficiency of the measured characteristic gamma-ray; Iγ is the gammaray intensity; η is the abundance of the target nuclide; M is the mass of the sample; D = e −λt1 − e −λ(t1+t2) is the counting collection factor; S = 1− e −λT is the growth factor of the product nuclide, T is the total irradiation time; t 1 is the total cool time and t 2 is the total measurement time; A is the atomic weight; C is the measured full energy peak area; λ is the decay constant; K is the neutron where L is the number of time intervals into which the irradiation time is divided; Δt i is the duration of the i-th time interval; T i is the time interval from the end of the i-th interval to the end of irradiation; Φ i is the neutron flux averaged over the sample during Δt i ; Φ is the neutron flux averaged over the sample during the total irradiation time T . F is the total correction factor of the activity: where f s and f g are correction factors for the selfabsorption of the sample at a given gamma-ray energy and the counting geometry, respectively. The gamma ray attenuation correction factor in the Ge pellet, f s and the geometry correction, f g were calculated by eqs. (3) and (4), where μ (in cm −1 ) is the linear attenuation coefficient in Ge for gamma rays at each of the photon energies, E (see table 2), h (in cm) is the thickness of the sample and D (in cm) is the distance from the measured sample to the surface of the Ge crystal. The mass attenuation coefficients, μ/ρ for the germanium, which are 0.3122 and 0.1319 cm 2 /g at gamma-ray energies of 139.68 and 264.6 keV respectively, were obtained by interpolating values from the literature [20]. The linear attenuation coefficients in Ge were then calculated according to ρ = 5.323 g/cm 3 . The correction factors at 139.68 and 264.6 keV gamma-rays are given in table 2.
Irradiating Cooling Measuring While calculating the cross sections of the 76 Ge(n, 2n) 75g Ge reaction, C x in (1) should be the result of the measured full-energy peak area (at 264.6 keV γ-ray) minus the contribution from 75m Ge via 75m Ge According to the regulation of growth and decay of artificial radioactive nuclide we can deduce a formula to calculate the number of the daughter nucleus 75m Ge at any moment t during the irradiation (see fig. 4) as follows: where σ m is the cross sections for formation of the metastable, λ m is the decay constant of this state, φ 0 is the mean neutron flux in neutrons/cm 2 /sec, and N is the number of target nuclei. At any moment t during the irradiation, the number of 75g Ge from the 75m Ge → 75g Ge procedure meets the following equation: where P mg is the fraction of disintegrations of the metastable state that produces ground state nuclides (branching ratio), λ g is the decay constant of 75g Ge.
Using eqs. (5) and (6) and the initial condition: t = 0, N g (0) = 0, and working out N g (t), At the moment of the end of the irradiation (t = T ), the numbers of 75m Ge and 75g Ge from 75m Ge → 75g Ge are N m (T ) and N g (T ), respectively, which can be obtained by using eqs. (5) and (7).
At any moment t after the irradiation, the number of 75m Ge is (8) At any moment after the irradiation t , the number of 75g Ge from 75m Ge → 75g Ge meets eq. (6). Using eqs. (6) and (8) and the initial condition t = 0, N g (0) = N g (T ) (the number of 75g Ge from 75m Ge → 75g Ge is equal at the end of the irradiation and the start of cooling) and working out N g (t ), Let t in eqs. (8) and (9) equal t +t 1 . t 1 is the time interval from the end of the irradiation to the start of counting. We can obtain the number of 75m Ge at any moment t after beginning to detect the characteristic γ ray of 75m Ge and the number of 75g Ge from the 75m Ge → 75g Ge procedure at any moment t after beginning to detect the characteristic γ ray of 75g Ge During the period t 2 of detecting the characteristic γ ray, the full-energy peak (FEP) counts C m of the characteristic γ ray of 75m Ge and C mg of the characteristic γ ray of 75g Ge from the 75m Ge → 75g Ge procedure are Using eqs. (12) and (13), C mg can be written as where S m = 1− e −λmT and S g = 1− e −λgT ; I m and I g are the gamma ray intensities of the measured metastable and ground state, respectively; ε m and ε g are the full-energy peak efficiencies of the characteristic gamma-rays of the measured metastable and ground state, respectively; K m is neutron fluence fluctuation factor of the metastable state; D m and D g can be written as

Nuclear model calculations
The excitation functions for the reactions were studied theoretically using the numerical nuclear model code TALYS-1.8 [21]. The theoretical calculations were computed using the default parameter values and only changing the choice of the level density models. The level density parameters were calculated using the six different choices of level density models available in TALYS-1.8. The six level density models are given in table 3.

Discussions
The cross sections measured in this work are presented in table 4. The uncertainty analysis was carried out using the quadrature method [22]. The principal sources of uncertainty and their estimated values are given in table 5. The total uncertainty lies between 4.5 and 8.7%. The small contribution to the gamma ray activity of products from the 74 Ge(n, γ) reaction could be ignored because of the very small cross section of the (n, γ) reaction in the region of 14 MeV. Furthermore, samples were wrapped in a cadmium foil in order to reduce the contribution of thermal and epithermal effects. For the 76 Ge(n, 2n) 75m,g Ge reactions the cross sections slightly increase with the increasing neutron energy. The various reactions are discussed below. Goriely's tables [23].

Isomeric cross section ratio
The isomeric cross section ratio σ m /σ g for the isomeric pair 75m,g Ge produced in the (n,2n) reaction on 76 Ge was   fig. 8. It can be seen that our results agree well with the result of Bhattacharyya et al. [34], and the data from the TALYS-1.8 calculations with ldmodel 1, 2 and 3. The isomeric cross section ratio determined in this work has a slightly increasing trend with the increasing neutron energy, suggesting that at higher excitation energies the formation of the high-spin isomer (7/2 → 1/2) is more favored. This trend is similar to that for several other neutron-and charged-particle-induced reactions near thresholds [35][36][37][38][39][40][41][42][43][44]. In the range of 13-15 MeV, the calculated isomeric cross section ratio shows the same slightly increasing trend for the six ldmodels. Our data and the results of TALYS-1.8 with ldmodels 1, 2, and 3 are somewhat lower than the data of Hlavac et al. [12], Okumura [26], Birn et al. [28], while they are higher than the results of Vanska and Rieppo [9] and Mangal and Gill [11].

Conclusions
In the present paper, a methodical experimental campaign and TALYS-1.8 code calculations with different level density models have been carried out. Activation cross sections for 76 Ge(n, 2n) 75m Ge, 76 Ge(n, 2n) 75g Ge, and 76 Ge(n, 2n) 75 Ge reactions as well as isomeric cross section ratios for 76 Ge(n, 2n) 75m,g Ge reactions induced by 13.5, 14.1 MeV, and 14.8 MeV neutrons have been measured using the latest decay data, and by taking into account the contribution of the metastable state in the case of unstable ground state formation cross section. In order to avoid the effect of low energy neutrons, the near threshold 93 Nb(n, 2n) 92m Nb (E th = 8.972 MeV) monitor reaction was selected, samples were wrapped in a cadmium   foil and the new T-Mo target was used. The constant temperature and Fermi gas model (ldmodel 1) is to be preferred for 76 Ge(n, 2n) 75m,g Ge reactions. The results were compared with previous experimental results reported in the literature and theoretical nuclear model calculations computed using TALYS-1.8. A detailed comparison with previously reported cross sections reveals that the discrepancies in the historic data could be due to: 1) the decay data (half-life and ray intensity) used in the determination of the cross sections; 2) the system difference caused by different measuring methods (radiation detector and neutron monitoring method) and experimental conditions (neutron field characteristics); and 3) interfering reactions. The experimental results presented here may be used to more accurately describe the reaction processes and verify statistical model parameters used in their theoretical representation.