Measurements of ¹⁶⁰Dy (p, γ) at Energies Relevant for the Astrophysical γ Process

Natural dysprosium is composed of seven stable isotopes with mass numbers 156, 158, 160, 161, 162, 163, and 164. ¹⁵⁶Dy and ¹⁵⁸Dy have only tiny natural abundances and reactions on them are out of reach in the current experiment conditions. Of all the (p, γ) products, ¹⁶⁵Ho is stable and ¹⁶³Ho has a half-life of 4570(20) yr and thus is too long to be studied. At 1.65 MeV proton bombarding energy the ¹⁶¹Dy(p, n)¹⁶¹Ho reaction channel opens, which results in the same final nucleus as ¹⁶⁰Dy(p, γ)¹⁶¹Ho. The threshold energies for the ¹⁶²Dy(p, n)¹⁶²Ho, ¹⁶³Dy(p, n)¹⁶³Ho, and ¹⁶⁴Dy(p, n)¹⁶⁴Ho are 2.94, 0.79, and 1.78 MeV, respectively. Hence using natural or enriched but not 100% abundance isotope Dy targets above the (p, n) reaction thresholds, (p, n) and (p, γ) reaction channels cannot be distinguished, and the resulting cross section is the weighted sum of the two cross sections.

One way to extract the (p, γ) cross sections is to employ two kinds of targets with different isotopic abundances. By coupling the (p, γ) and (p, n) channels to the same radioactive nuclide, the individual cross section can be deduced in the case of good statistics. The information of the two targets used is summarized in Table 1, one with natural Dy and the other with ¹⁶⁰Dy-enriched abundance. With the abundance distributions in two targets, we can determine the cross sections for the two reactions ¹⁶⁰Dy(p, γ)¹⁶¹Ho and ¹⁶¹Dy(p, n)¹⁶¹Ho.

The thicknesses of natural targets are around 1–2 mg cm⁻² and those of the enriched targets are 2 mg cm⁻² as displayed in Table 2. These targets are ellipses with an axis length of 9 mm × 12 mm, and the sizes of target frames are 25 mm × 12.5 mm. The self-supported natural dysprosium targets were made by the rolling method. Enriched ¹⁶⁰Dy (51.82%) targets were sputtered onto a ¹⁹⁷Au backing. The mass abundances of contamination elements are determined typically to be less than 0.1% and 1% for the natural targets and enriched targets by the energy-dispersive X-ray spectroscopy, respectively.

The investigated energies in this work ranging from 3.4 to 7 MeV are above the relevant (p, n) reaction thresholds. These energies partially cover the Gamow window for the γ process in this mass region for T₉ = [2, 2.9].

The proton beams were delivered by the 2 × 1.7 MeV and H1-13 tandem accelerators at the China Institute of Atomic Energy, Beijing, respectively. The maximum energy provided by the 2 × 1.7 MeV tandem accelerator and the minimum energy that can be steadily and intensely provided by the H1-13 tandem accelerator are 3.4 MeV and 5 MeV, respectively. The proton current throughout the irradiations was between 150 nA and 700 nA for different beam energies.

A schematic drawing of the target chamber is shown in Figure 1. The target chamber also servers as a Faraday cup with a Φ5 mm-hole collimator and insulated from the beam tube. The distance from the hole collimator to the target surface is 20 cm. A suppression voltage of −300 volts was applied to suppress secondary electrons emitted from the target. A water-cooling system for the target chamber was used to maintain the temperature and minimize surface deformation of the target. The beam current was digitized with a frequency of 1 Hz by the Keithley 6517B electrometer.

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Figure 2 shows the beam current during the beam time. The variance in beam currents I(t) can bring an up to 35% difference in the cross section when considering the maximum and minimum beam intensity. This indicates the necessity of recording the precision current with time.

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The effective center-of-mass energy (E_c.m.) for each run is calculated by considering the proton energy loss ΔE in the target material and follows

$$\begin{eqnarray}&&{E}_{{\rm{c}}.{\rm{m}}.}=\displaystyle \frac{{M}_{A}}{{M}_{p}+{M}_{A}}\left({E}_{p}-\displaystyle \frac{{\rm{\Delta }}E}{2}\right),\end{eqnarray} \tag{ 1 }$$

where M_A and M_p are the mass of the target nuclide and the proton, E_p is the proton energy. Energy losses are calculated using ATIMA, a built-in Physical Calculator in LISE++ (Tarasov & Bazin 2008). The energies used in the experiments are summarized in Table 2. The relevant uncertainties include the energy straggling and the difference caused by using different thickness of targets.

The targets were irradiated for 30–60 minutes (irradiation time) and then followed by 30–1000 minute measurements (measurement time), while in between there were 10 to 20 minutes (waiting time) to release the vacuum, dismount the target, and place the target in the position for off-line measurement. The details of measurement are displayed in Table 2.

The γ radiations following the electron-capture decays of the produced Ho isotopes were measured with a CLOVER detector in a low background shielding system as shown in Figure 3. The CLOVER consists of four coaxial N-type high-purity Germanium detectors, each with a diameter of 60 mm and a length of 60 mm. The energy resolution for the CLOVER is 2.2 keV (FHWM) for the 1.332 MeV γ rays of ⁶⁰Co. The relative efficiency is 38% for each Germanium crystal. The shielding system is a cylinder with a radius of 64 cm and a height of 66.1 cm. It consists of layers of iron, lead, copper, and plexiglass from the outside to the inside. The lead layer is used to shield most of the low energy environment background, and the copper layer aims to absorb the characteristic X-rays of lead. This system has been used to investigate the intrinsic radiation background of a B380 LaBr₃(Ce) detector (Cheng et al. 2020).

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The irradiated targets were placed 3 cm away from the front surface of the detector. The absolute efficiency was calibrated with one ¹⁵²Eu and six monoenergetic sources (¹³⁷Cs, ²⁴¹Am, ⁵⁴Mn, ⁸⁸Y, ¹⁰⁴Cd, and ⁶⁵Zn). The sources were placed at the same position as the targets. The absolute full-energy peak efficiency curve was described using the EFFIT program in the Radware (Radford 1995) package. The efficiency curve was further corrected for the summing coincidence effect by a dedicated GEANT4 simulation (He et al. 2018). The impact of geometric acceptance due to the size of irradiation target, is estimated to be less than 1%. This effect is much smaller than the statistic error.

Signals of the CLOVER detector were recorded by the VME data acquisition system (DAQ). The timestamp of each event was digitized in a precision of 10 ms. This information is useful to determine the decay curve of a specified γ activity and can be used as an independent check of the radioactive product. Dead time correction was also added in DAQ for the absolute counting.

There is a low-lying 1/2⁺ isomeric state with a half-life of 6.67 (7) s in ¹⁶¹Ho. Nevertheless, this isomer does not affect the determination of cross sections because its half-life is much shorter than the irradiation time of typically more than half an hour. All the isomeric states would have decayed to the ground state by isomeric transition (IT) in the off-line activity measurement.

Figure 4 shows the typical activation γ-ray spectra below 1500 keV taken from the natural Dy and enriched Dy targets irradiated by 6.5 MeV protons. We identified the characteristic γ-rays at 103 keV from ¹⁶¹Ho, the γ-rays at 197, 728, 879, etc., from ¹⁶⁰Ho, the γ-rays at 185, 283, 937, 1220, 1320, and 1373 keV from ¹⁶²Ho, and the γ-ray at 91 keV from ¹⁶⁴Ho. The 103-keV γ transition with a relative intensity of 3.9% per decay of ¹⁶¹Ho was used for data analysis. The purity of the γ ray is verified by examining its time evolution. The deduced half-life of 148.5(24) minutes is in good agreement with 148.8(30) minutes reported in Reich (2011)

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The count of the 103 keV γ peak was analyzed using the Radware package (Radford 1995) and corrected with the dead time of DAQ. Decay parameters of Ho isotopes, the decay constant (λ) and the relative intensity (η_γ ) of the characteristic γ ray, are from the Nuclear Data Sheets (NDS; Singh & Chen 2014). By decoupling the total counts of characteristic γ rays measured in both types of targets at the same proton energies, we can deduce the cross sections for ¹⁶⁰Dy(p, γ)¹⁶¹Ho and ¹⁶¹Dy(p, n)¹⁶¹Ho. The derivation of cross section from the activation γ rays is described in detail in Section appendix.

The key ingredients in the Hauser–Feshbach calculations (Rauscher et al. 2013) include γ strength functions (GSF), optical model potential (OMP), the nuclear level densities (NLD) and mass models. TALYS offers a variety of options for the description of these inputs and is used hereafter to make the sensitivity studies (Rapp et al. 2006; Mei et al. 2015).

In Figure 6(a), the (p, γ) experimental results are compared with the predictions by the TALYS-1.9 code utilizing different NLD models, while all the other inputs are kept the same as the default ones. Using different NLDs gives similar trends, so we present the selective calculations including the one with the default parameters as well as those of the lowest and highest cross sections, as limits of these calculations. The default NLD in TALYS is the Constant temperature + Fermi gas (CTFG) level density (Gilbert & Cameron 1965). It sets the lower limit of cross sections, while the upper limit is obtained by using the microscopic NLD from Hilaires table (HT) (Hilaire & Goriely 2008). As shown in Figure 6(a), the (p, γ) cross section is sensitive to the NLDs from about 4 MeV on, and has a less significant impact in the Gamow window, i.e., the low energy data. Variation of the level density modes could result in a factor of 4.5 difference in cross section at 7 MeV. Our data are close to the predicted lowest limit, and thus provide an direct constraint for the NLD and rule out many NLDs in this mass range.

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Similarly, we present TALYS predictions using three GSFs in Figure 6(b). The default GSF in the TALYS is the traditional standard Lorentzian model of Kopecky-Uhl generalized Lorentzian (KUGL; Kopecky & Uhl 1990). The lowest and highest cross sections are obtained by using the KUGL and the Gogny D1M HFB+QRPA models (Goriely et al. 2018), respectively. Generally, the (p,γ) cross section depends sensitively on both GSF and NLD modeling.

We also examined the (p, γ) cross sections by using various compound models, pre-equilibrium models and optical models in TALYS. They have only a minor effect on the computed cross sections. The optical potential is expected to be important at the astrophysically relevant low energies for charged particle captures and photodisintegrations. The one and only data point at 3.34 MeV can already help to exclude several models.

Further, it is worth noting that nuclear reaction calculations involve several (sub)models to compute the key input quantities like level density, γ strength function, optical potential, nuclear masses, pre-equilibrium, and the compound process. Their interplay makes the model evaluation almost impossible. Instead of developing a consistent reaction model, we attempt to find the best model parameters of TALYS, which can describe the known experimental data. We are aware that the parameters might be mass range dependent or only hold to a certain precision in applying to nuclei nearby; therefore, we concentrate on the mass range of 160. This strategy is practically useful to reduce the nuclear physics inputs in γ process simulations (Rauscher et al. 2016; Nishimura et al. 2018). We thus made a Monte Carlo calculation by randomly varying the key physical input parameters (Koning et al. 2019), and employed the least-square method to find the calculation that matches best both the ¹⁶⁰Dy(p, γ)¹⁶¹Ho and ¹⁶²Er(p, γ)¹⁶³Tm data. The results are shown in Figure 6.

The TALYS results constrained by the present experimental data, as presented in the previous section, represent a solid basis for the reaction rate calculation. We thus employ the optimized parameters to compute the astrophysical rate of ¹⁶¹Ho(γ, p). It is obtained by integrating the cross section over a Maxwell–Boltzmann distribution of energies E at the given temperature T. In addition, the thermal excitation effect in hot astrophysical plasma is taken into account.

The calculated stellar reaction rates sharply depend on the temperature and span over 250 orders of magnitude in the temperature range of T₉ = [0.1, 1.0]. The rates are summarized in Table 4. Figure 7 shows the ratios of the deduced reaction rates to the NON-SMOKER predictions in the temperature of T₉ = [0.1,10]. The theoretical uncertainty (Koning et al. 2019) is shown by the shadowed band. The ratio is up to one order of magnitude, and NON-SMOKER underproduced the rates at T₉ = [1, 2]. The difference would affect the yield of ¹⁶⁰Dy in the γ process and will be discussed in the next subsection.

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As a possibly important region for the γ process of the A ∼ 160 region, we tested trajectories with peak temperatures T_9p = [2, 2.9] and peak density ρ_p = 10⁶ g cm⁻³ assuming the γ process layer in supernovae (SNe) Ia. They are assumed to decrease exponentially as ${T}_{9}={T}_{9p}\exp [-t/(3\tau )]$ and $\rho ={\rho }_{p}\exp (-t/\tau )$ with τ = 1 s. The initial nuclear composition is adopted from Case A1 of Kusakabe et al. (2011), which is based on an efficient s-process during thermal pulses in the presupernova stage. The nucleosynthesis calculation is then performed with a variable-order network code (Kusakabe et al. 2019), which is updated with new theoretical estimates on the rates of ¹⁶O(n,γ)¹⁷O (He et al. 2020) and ¹⁷O(n,γ)¹⁸O (Zhang et al. 2021).

In the SNe γ process environment, a sudden heating followed by a nearly adiabatic cooling occurs. The heating is triggered either by a deflagration or detonation wave in thermonuclear explosion corresponding to SNe Ia, or a shock in gravitational collapse type explosion corresponding to SNe ii. Then, early in the heating epoch, first neutron source nuclei are burned to proceed with the neutron-capture process. As a result, neutron-rich nuclei are produced from seed s nuclei. As the temperature increases further, the (γ, n) reactions start operating, and abundant nuclei shift from neutron-rich nuclei to neutron-deficient nuclei. If the peak temperature is high enough, (γ, p) and (γ, α) reactions also work and heavy nuclei are disintegrated to lighter nuclei (e.g., Kusakabe et al. 2011). We estimate a sensitivity of nuclear yields to the reaction rate of ¹⁶¹Ho(γ, p)¹⁶⁰Dy. Any effect of changing the ¹⁶¹Ho(γ, p)¹⁶⁰Dy rate always occurs via changes of ¹⁶¹Ho and/or ¹⁶⁰Dy abundances in the nuclear reaction network in the γ process. We confirm that the current rate determination indicates that the possible uncertainty in the reaction cross section is small enough that it does not remarkably affect evolution of ¹⁶¹Ho or ¹⁶⁰Dy abundances in the γ process. Accordingly, the change of the reaction rate does not affect yields of nuclei with A < 160 produced through the pathway of ¹⁶¹Ho(γ, p)¹⁶⁰Dy.

Figure 8(a) shows time evolution of abundances on the test trajectory with T_9p = 3. The temperature is rather high, and first neutrons are released shortly. Instantaneous neutron captures make a flow of nuclear abundances to the neutron-rich region on the nuclear chart at t ∼ 10⁻⁷ s, and abundances of ^159,160 Dy and ¹⁶¹Ho decrease. After the neutron abundance decreases, successive (γ, n) reactions move back the flow toward the proton-rich region. Then, abundances of ^159,160 Dy and ^160,161Ho increase at t ∼ 10⁻⁵–10⁻⁴ s and eventually decrease due to the photodisintegration into a more proton-rich and low A region. Figure 8(b) shows fractional differences in the abundances between the standard case and the case of the JINA REACLIB rate for ¹⁶¹Ho(γ, p)¹⁶⁰Dy. Each band bounded by two lines delineates a variation caused by the uncertainty in the evaluated rate. Variations of ${ \mathcal O }(1) \%$ are seen when the photodisintegration is effectively destroying those nuclei. The smaller ¹⁶¹Ho(γ, p)¹⁶⁰Dy rate in our standard case at T₉ = 3 increases abundances of ¹⁶¹Ho, which survives that reaction and ¹⁶⁰Ho produced via ¹⁶¹Ho(γ, n)¹⁶⁰Ho. Also, the ¹⁶⁰Dy abundance slightly decreases due to the hindered production via the ¹⁶¹Ho(γ, p)¹⁶⁰Dy. The uncertainty in the current rate corresponds to at most ${ \mathcal O }(0.1) \%$ differences in abundances.

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The effect of the change in the ¹⁶¹Ho(γ, p)¹⁶⁰Dy rate is small even at t ∼ 10⁻⁵–10⁻⁴ s when the photodisintegration operates predominantly. This is because the ¹⁶¹Ho(γ, n) rate is much larger than the (γ, p) rate for T₉ ≥ 2 relevant to the γ process. At such high temperatures of T₉ ≲ 3, all heavy seed nuclei are disintegrated, and yields are zero (Figure 9). At lower temperatures, although the changes in abundances could remain in the final abundances, those are constrained to be at the ${ \mathcal O }(0.1) \%$ level.

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Figure 9(a) shows final yields of stable nuclei with mass numbers A ≲ 160 in the γ process in SN ia as a function of T_9p . Since those nuclei are located immediately downstream of the pathway ¹⁶¹Ho(γ, p)¹⁶⁰Dy, changes are expected in abundances of those nuclei if that reaction is important in determining the final abundances. As a standard case, we adopt the rate for ¹⁶¹Ho(γ, p)¹⁶⁰Dy derived from the ¹⁶⁰Dy Best parameters. Figure 9(b) shows fractional differences in the final yields between the case with the ¹⁶¹Ho(γ, p)¹⁶⁰Dy rate 100 times as large as the adopted rate and the standard case. Even when the rate is thus extremely large, the yields vary by at most ${ \mathcal O }(10) \%$. Since the uncertainty in the ¹⁶¹Ho(γ, p)¹⁶⁰Dy rate is estimated to be much smaller (Figure 7), it is found that the current determination is sufficiently precise to obtain accurate results of the γ process.