Cross section of He(,)Be around the Be proton separation threshold
The He(,)Be reaction is a widely studied nuclear reaction; however, it is still not understood with the required precision. It has a great importance both in Big Bang nucleosynthesis and in solar hydrogen burning. The low mass number of the reaction partners makes it also suitable for testing microscopic calculations.
Despite the high number of experimental studies, none of them addresses the He(,)Be reaction cross sections above 3.1-MeV center-of-mass energy. Recently, a previously unobserved resonance in the Li(p,)Be reaction suggested a new level in Be, which would also have an impact on the He(,)Be reaction in the energy range above 4.0 MeV. The aim of the present experiment is to measure the He(,)Be reaction cross section in the energy range of the proposed level.
For this investigation the activation technique was used. A thin window gas-cell target confining He gas was irradiated using an beam. The Be produced was implanted into the exit foil. The Be activity was determined by counting the rays following its decay by a well-shielded high-purity germanium detector.
Reaction cross sections have been determined between MeV with 0.04-MeV steps covering the energy range of the proposed nuclear level. One lower-energy cross-section point was also determined to be able to compare the results with previous studies.
A constant cross section of around 10.5 barn was observed around the Be proton separation energy. An upper limit of 45 neV for the strength of a He(,)Be resonance is derived.
Zentrum Dresden-Rossendorf (HZDR), Dresden, Germany ]firstname.lastname@example.org; Present address: Helmholtz-
The He(,)Be reaction is important both in models of Big Bang nucleosynthesis (BBN) Coc and Vangioni (2017) and in the pp chain of solar hydrogen burning Serenelli et al. (2013). It was extensively studied in recent years to gain high-precision information on the reaction rate in both scenarios. Most of the studies concentrated on the cross section at low energy to constrain the extrapolations (Adelberger et al., 2011, and references therein) to the solar energy range ( keV). A pioneering study by the LUNA collaboration Bemmerer et al. (2006); Confortola et al. (2007); Gyürky et al. (2007); Costantini et al. (2008) provided direct high-precision cross-section data in the BBN energy range, which ruled out the He(,)Be nuclear physics solution of the cosmological Li problem Fields (2011). Even though the solar energy range is not accessible using the current experimental techniques, the reaction rate can be predicted by extrapolation. One way of extrapolation is the R-matrix approach, which is based on experimental data. The R-matrix framework was applied by Descouvemont et al. Descouvemont et al. (2004) to determine BBN reaction rates; however, the He(,)Be reaction was treated only as purely external capture leading to a constant S-factor towards higher energies. This assumption was supported by the data of Parker and Kavanagh Parker and Kavanagh (1963) which was the only available experimental data set at that time extending to higher energies. Later, a precision measurement with many data points had been performed by the ERNA collaboration ranging up to MeV Di Leva et al. (2009). These new data contradict the former one suggesting a higher slope of the S factor. This discrepancy ignited a few new studies, which all support the positive slope of the S factor Carmona-Gallardo et al. (2012); Bordeanu et al. (2013); Kontos et al. (2013), but there are no experimental cross-section data above MeV. In the most recent experimental work by Kontos et al. Kontos et al. (2013), an extensive R-matrix analysis was also performed using their new results as well as the post-2000 capture data Nara Singh et al. (2004); Bemmerer et al. (2006); Confortola et al. (2007); Gyürky et al. (2007); Costantini et al. (2008); Brown et al. (2007); Di Leva et al. (2009); Carmona-Gallardo et al. (2012); Bordeanu et al. (2013). Taking into account the internal capture contribution to the cross section, this R-matrix analysis successfully reproduced simultaneously the capture and scattering cross sections. The R-matrix study has been extended by deBoer et al. deBoer et al. (2014) with a detailed Monte Carlo analysis, which predicts the solar reaction rate with a 3-4% precision; however, this is still not sufficient for, e. g., precise solar models. New experimental data are thus called for at higher energies.
Additionally, a recent study of the Li(p,)Be reaction showed a resonance like structure He et al. (2013). This has consequences on the level scheme of Be, where a positive parity level was proposed around MeV excitation energy corresponding to MeV in the He(,)Be reaction. This energy range was never investigated before, therefore there is no experimental constraint from this reaction on the proposed level.
In this paper we present a measurement of the He(,)Be reaction cross section in the energy range of MeV with 0.04-MeV energy steps, to cover the range of the proposed resonance. The present experiment answers the call for a higher-energy dataset and also provides a constraint on the strength of a possible resonance in He(,)Be corresponding to the proposed Be level.
The paper is organized as follows. In Sec. II details about the experiment are given. In Sec. III the data analysis is described, in Sec. IV the experimental results are presented, and the impact of the new data on the properties of the proposed resonance is discussed in Sec. V. Finally a summary is given in Sec. VI.
Ii Experimental details
For the cross-section determination, the activation technique was used. The number of radioactive Be reaction products was determined via spectroscopy by measuring the 477.6-keV rays following 10.44% of the Be decays Tilley et al. (2002).
For the irradiations, a thin window gas cell was used, similar to the one in previous investigations Bordeanu et al. (2013); Halász et al. (2016). Both ends of the cell were closed by aluminum foils with 10-m nominal thickness to confine the target gas (see Fig. 1). The entrance foil was pressed against an o-ring with a steel disk that had an opening of 12-mm diameter. The exit foil was glued on a steel disk with a hole of 14-mm diameter. A special heat-curing polymer was used for the glue111Vacseal® II Vacuum Leak Sealant.
A specific entrance foil was typically used for a few subsequent irradiations. The closing foils served also as the catcher of the produced Be, therefore they were replaced after every irradiation. From the reaction kinematics the energy of the Be was calculated, and its energy loss was estimated using the SRIM code Ziegler et al. (2010). The implantation energy varied between MeV, therefore the backscattering loss was negligible. The average range of Be in aluminum at these energies is between m, thus the catcher was thick enough to stop the activated particles. The particles going through the cell were stopped in a water cooled tantalum sheet. The double window configuration was used to reduce the beam power deposition in surfaces with direct contact with the gas volume.
For the irradiations, the MGC-20 cyclotron of Atomki was used. The gas cell was fixed onto an activation chamber acting as a Faraday cup. The chamber had a beam defining aperture with 5-mm opening and a secondary electron suppressing aperture supplied with V. The beam entered the chamber and gas cell through the apertures, which were isolated from the beam line allowing us to measure the beam current (see Fig. 1). The irradiation parameters are summarized in Table 1, where the measured average electric current of the beam is also shown.
ii.3 -ray countings
After the irradiation the catcher foil implanted with Be was transported to a high-purity germanium detector with 100% relative efficiency in commercial 10-cm-thick lead shielding. A single sample was counted for a few days, during which the area of the 477.6-keV gamma peak was detected with less than 5% uncertainty. A typical spectrum is shown in Fig. 2. The counting was repeated multiple times, to follow the decay of the samples. No parasitic activity populating the 477.6-keV peak was observed; the decay curve shown in Fig. 3 followed the half-life of Be (53.22 days). The final cross-section result was derived from the sum of the spectra to reduce the statistical uncertainty.
|Al thickness||Initial He pressure||Final pressure||t||I||BH correction||t|
|/ MeV||/ m||/ mbars||/ mbars||/ h||/ A||/ %||/ days|
Iii Data analysis
iii.1 Target thickness
The areal number density of the target was determined from the measured initial pressure and temperature of the gas, and the known length of the cell. The target thickness was chosen to optimize reaction yield, while limiting the energy uncertainty.
About 100 mbars 99.999% isotopically pure He gas were filled in the cell before each irradiation (see Table 1). The pressure in the cell was continuously monitored with an MKS Baratron (modell 722B) capacitive manometer gauge with a precision of about 0.1 torr at this pressure range. Without beam the pressure was measured to be constant for several days. During the irradiations an increase of not more than about 20 mbars was observed with a saturating trend caused by air desorption of the stainless steel wall and Al foils during the irradiation. The increased pressure was lower when a previous irradiation was performed with the same entrance foil, and the cell was not exposed to air for a longer time. After switching off the beam the pressure stayed constant at the increased level. Since the cell was closed after the filling, the number of He atoms was considered to be constant. The additional pressure is considered to be due to air, and was taken into account in the effective energy calculation (see Sec. III.2). The local-density reduction of the gas, the so-called beam heating effect, is at maximum about % calculated with the equation given in Marta et al. (2006). In Table 1, the beam-heating correction factors for each run are given, as a conservative estimate the correction factor is also considered as statistical uncertainty in the target thickness determination (see Table 3).
The temperature of the cell was constant at 23 2 C determined by the cooling water flowing through the body of the cell.
The length of the cell between the foils with compressed o-rings is 41.5 mm. The pressure difference between the beam line vacuum and target cell causes a deformation of the entrance and exit foils changing the length of the beam path in the cell and thus altering the number of target atoms at a given pressure. The deformation of the entrance and exit foils was measured off-site with a laser distance sensor as described in Halász et al. (2016). The maximum deformation of both the entrance and exit foils was found to be less than 0.2 and 0.3 mm, respectively. This total 0.5 mm was added to the cell length for the effective gas length determination. As a conservative estimate, 1.0 mm is taken as the uncertainty of the total target length stemming from the deformation and the uncertainty of the cell length measurement.
iii.2 Effective energy
As the beam is decelerated in the entrance foil, the knowledge of the accurate window thickness is crucial in the reaction energy calculation. The thickness of the entrance foils was measured by -energy loss, where the energy of the particles from a triple-nuclide source (i. e., containing three emitters as Pu, Am, and Cm) was measured after penetrating through the foils in a SOLOIST spectrometer. As a first step, the spectrum without the foil was measured and the eight peaks from this source were fitted with Gaussian plus low energy tail functions simultaneously for energy calibration (see gray points in Fig. 4). The peak area ratio for a given isotope was fixed at its literature branching ratio value. Then the spectrum of the particles penetrating the foil was recorded and fitted as follows: The energy of the particles was calculated using SRIM Ziegler et al. (2010) by assuming a given aluminum thickness. The center of the Gaussian peaks was fixed at each given energy, and as in the calibration case the branching ratios were also fixed. Only the total intensities of the peaks and the widths were varied in the fit. This kind of fit was repeated for many assumed Al thicknesses, and the reduced chi square () of each fit was calculated. The aluminum thickness-dependent values can be fitted by a parabola (see Fig. 5). The minimum of the fitted parabola was taken as the actual foil thickness. The thickness uncertainty was also calculated from the curve by taking the values222In case of 4085 degree of freedom (4096 data point and 11 parameters) the 1 uncertainty border is at . These numbers were divided by the degrees of freedom.. The thickness of each entrance foil is measured on a few different points and consistent values are obtained. Assuming uniform thicknesses, the statistically weighted average of those measurements is shown in Table 1 for a given entrance foil. The foil thickness uncertainty includes the fit uncertainty after the averaging only and does not include the uncertainty in the stopping power. The latter is taken into account only in the beam energy calculation, thus avoiding double counting.
The effective energy and its uncertainty have been determined as follows. The beam energy loss was calculated using the SRIM tables. To avoid double counting of the uncertainties, the uncertainty in the stopping power of 3.5% according to SRIM was assigned only to the energy-loss value. The energy loss in the gas volume was also calculated and a stopping power uncertainty of 4.4% was assigned. The effective energy was taken at the middle of the target and its uncertainty is based on the cyclotron beam energy uncertainty of 0.3% and the previously mentioned foil thickness and stopping power uncertainties. As a conservative estimate, the energy loss caused by the extra gas in the cell (about 0.5 keV) was added to the uncertainty of the beam energy, even though it is negligible compared to effects on the beam energy due to the uncertain stopping power of the Al entrance foil (about 40-50 keV). In total, the energy uncertainty is about 0.5% (see Table 3).
iii.3 -detection efficiency
The detector efficiency calibration was performed with multi-line calibration sources (Ba and Eu) of precisely known activity (0.75 and 1.00%) at 10- and 27-cm sample to end-cap distance to avoid the true coincidence summing effect. The obtained efficiency points were interpolated with a log-log linear function to get the efficiency at the Be -peak energy of 477.6 keV. The attenuation in the Al catcher at this energy is negligible considering the typical implantation depths. The efficiency at the actual counting distance of 1 cm was determined with a stronger Be source, which was counted in each geometry. Taking into account the time elapsed between the countings and the known half-life of Be, efficiency scaling factors were derived. Scaling both from 10 and 27 cm gave consistent results. The uncertainty of the 1-cm efficiency finally used is the quadratic sum of the interpolation (1.5%), the calibration source activity (1.0%), and scaling factor (0.9%) uncertainties.
The reaction cross sections obtained at the given center of mass energies are summarized in Table 2. In the table the statistical uncertainty is given for each point as well as the systematic uncertainty. The components of the uncertainties are summarized in Table 3.
One data point was measured at keV to compare the results with literature data and validate the setup. At this particular energy the new data point agrees with the data from previous investigations.
|/ MeV||/ keV||/ barn|
|7.30||2479 23||6.23 0.16 0.27|
|10.45||3995 20||10.47 0.24 0.45|
|10.60||4057 21||10.13 0.25 0.44|
|10.65||4097 20||10.75 0.23 0.46|
|10.75||4140 20||10.39 0.23 0.45|
|10.85||4197 20||10.49 0.25 0.45|
|10.95||4238 20||10.71 0.22 0.46|
|11.05||4278 20||10.46 0.24 0.45|
|11.15||4330 20||10.80 0.26 0.46|
|11.25||4370 20||10.56 0.24 0.45|
|11.35||4422 20||10.60 0.28 0.46|
|Initial beam energy:||0.3%|
|Al foil thickness:||0.2-1.0%|
|Energy loss in Al:||0.34-0.41%|
|Energy loss in the gas:||0.02%|
As can be seen in Fig. 6, the new dataset does not show a peak-like structure in the energy range of He et al. (2013). Our result does not support the existence of a resonance in the He(,)Be channel; however, within the statistical uncertainty of the dataset it may be present, but is not observable. Similarly no evidence of a level is visible in the scattering data covering this energy range, e. g., Tombrello and Parker (1963); Spiger and Tombrello (1967). To quantify this limit, in the following the maximum allowed resonance strength is determined by a constrained R-matrix fit using a few datasets.
The R-matrix fit was performed with the AZURE2 code Azuma et al. (2010). Besides the data from this paper, three other He(,)Be datasets from Di Leva et al. (2009); Bordeanu et al. (2013); Kontos et al. (2013) were used. From the dataset of Kontos et al. (2013) partial cross sections were used for the fit, but in the plot only the total cross sections are shown. Additionally, data from the Li(,)Be Switkowski et al. (1979); He et al. (2013) reaction channels were fitted simultaneously assuming either no level at MeV or a () level as suggested in He et al. (2013). These datasets either include partial cross sections to the ground state and to the first excited state, or explicitly include the branching ratio between those for each data point. For the fit, the partial cross sections were used to better constrain the partial asymptotic normalization coefficients (ANCs), but in the plot only the total cross sections are shown. As level parameters, those in Kontos et al. (2013) were used. The positions and the widths of the levels including the background poles were kept fixed at their values presented there. In the fit only the ANCs for both reactions and the , proton, and widths of the proposed resonance (if included) were varied. The position of the resonance was fixed at MeV.
|ANC(0)||1.7 fm||2.1 fm||2.6 fm|
|ANC(0)||-2.2 fm||-2.6,fm||-2.1 fm|
|ANC(0)||3.9 fm||3.9 fm||3.9 fm|
|ANC(429)||0.2 fm||1.8 fm||2.7 fm|
|ANC(429)||-2.9 fm||-2.7 fm||-1.8 fm|
|ANC(429)||3.0 fm||3.0 fm||3.0 fm|
|(0)||–||-4.6 meV||-2.3 meV|
|(429)||–||2.5 meV||1.2 meV|
|–||150 meV||689 meV|
|calc.||–||20 neV||45 neV|
The resonance strength of the He(,)Be reaction is calculated from the widths with the following equation.
where is the spin of the excited state in Be, and are the spins of the target and projectile; and are the and proton widths of the given level, respectively; and (0)(429) is the total width.
The resulting He(,)Be cross-section fits barely differ, and all of them are consistent with the present data, therefore the derived strengths can be treated as upper limits.
It is also important to mention, that the fitted S factor has an increasing slope in each fit, and does not become constant as the data suggest. With the given number of background poles (i. e., six at 11 MeV and one at 7 MeV) keeping their widths around their Wigner limits and widths below the Wiesskopf estimate, a constant S factor cannot be obtained. Further investigation of the R-matrix parameter space will be necessary to achieve a better fit, which is out of the scope of this paper. Scattering data in a broader energy range could constrain even better the level widths, and additional capture data on either the He(,)Be or Li(,)Be reactions would fix the widths.
The cross section of the He(,)Be reaction has been measured around the proton separation energy of Be. The results show a constant cross section of about 10.5 barn in the region of the recently proposed Be level. By performing a constrained R-matrix fit, upper limits on the capture resonance strength can be derived. If the resonance in the He(,)Be reaction exists, its strength must be lower than neV.
Acknowledgements.This work was supported by National Research, Development and Innovation Office (NKFIH) (Grants No. K120666 and No. NN128072), and New National Excellence Program of the Human Capacities of Hungary (Project No. ÚNKP-18-4-DE-449). G. G. Kiss acknowledges support from the János Bolyai research fellowship of the Hungarian Academy of Sciences. The authors thank Dr. A. J. Timothy Jull for proofreading the paper.
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