Nuclear Level Density and $\gamma$-ray Strength Function of ${}^{67}\mathrm{Ni}$ and the impact on the $i$-process

The origin and production mechanism of the heavy elements has been under investigation for a long time. The first direct observations of nucleosynthesis of heavy elements was reported by P. Merrill in 1952 [1]. Soon after, the mechanism behind the production of heavy elements through neutron capture was outlined in the famous B²FH paper in 1957 [2].

Nucleosynthesis beyond the Fe/Ni mass region is due to the slow neutron capture (s-process), rapid neutron capture (r-process), photo-disintegration processes, and more recent results call for an intermediate neutron capture process (i-process) [3, 4]. These processes are sensitive to nuclear properties such as the nuclear level densities (NLD) and $\gamma$-ray strength functions ($\gamma$SF). The availability and uncertainties in nuclear data affect the ability to reliably calculate reaction rates, in particular for the r- and i-processes as they involve neutron-rich nuclei for which data is non-existing or at the very least sparse. This can have significant impact on the neutron capture rates and on the final abundance distribution of elements and their isotopes predicted by models. Currently, the vast majority of nuclei for which NLDs and $\gamma$SFs have been measured lie at, or very near, the line of stability [5]. Experimentally measured NLD and $\gamma$SF can provide significant constraints on calculated neutron capture rates which are relevant for nucleosynthesis models [6, 7].

Away from stability, directly measured $(n,\gamma)$ cross sections are not available and NLDs and $\gamma$SFs can provide the necessary constraints. The Oslo method is particularly suitable as it extracts NLD and $\gamma$SF simultaneously [8, 9]. These quantities, together with the optical model potentials, are the main ingredients of the Hauser-Feshbach theory [10] to calculate neutron capture cross-sections and reaction rates. The Oslo Method has been extended to include total absorption spectroscopy following $\beta$-decay leading to the $\beta$-Oslo method [11] and has demonstrated the versatility of the Oslo method and its applicability even for nuclei far from stability [11, 12, 13, 14, 15]. The $\beta$-Oslo method requires specific $\beta$-Q value conditions for the mother-daughter pair placing a limit on the nuclei which can be measured with this technique.

Another approach is to use inverse kinematics with radioactive beams. Here, the beam intensity is the main limiting factor. Applying the Oslo method to experimental data from an inverse kinematics experiment was first demonstrated in Ref. [16] extracting the NLD and $\gamma$SF of ${}^{87}\mathrm{Kr}$ following a $\mathrm{d}(^{86}\mathrm{Kr},\mathrm{p})^{87}\mathrm{Kr}$ pick-up reaction. In this paper, we have applied this inverse-Oslo method to data from a radioactive ion beam experiment to extract the NLD and $\gamma$SF of ${}^{67}\mathrm{Ni}$. Based on these results, the neutron capture rate of ${}^{66}\mathrm{Ni}$ was constrained using the Hauser-Feshbach theory, which is of particular importance for our understanding of the weak i-process. In the weak i-process the neutron density is large ($\sim 10^{15}$), but exposure is low due to a quick and abrupt termination of the neutron production. In such a case only the first n-capture peak elements are produced significantly [17]. This was studied in Ref. [17], where it was found that the ${}^{66}\mathrm{Ni}(n,\gamma)$ reaction have a major impact on the overall production rate of heavier elements, essentially acting as a bottleneck for the entire weak i-process. Constraining the ${}^{66}\mathrm{Ni}(n,\gamma)$ reaction could improve our understanding of the i-process nucleosynthesis.

The experiment was performed using the HIE-ISOLDE facility at CERN [18], where 1.4 GeV proton beam from the PSBooster bombarded a uranium carbide target inducing fission. The evaporated fission products were ionized by the resonance ionization laser ion source (RILIS) [19] and the ${}^{66}\mathrm{Ni}$ ions were separated by the General Purpose Separator (GPS) to be collected and cooled by a penning trap (REXTRAP) [20]. The ions were then charge bread to a $16^{+}$ charge state by an electron-beam ion source (REXEBIS) [21, 22]. The highly ionized ${}^{66}\mathrm{Ni}$ ions were re-accelerated by the HIE-ISOLDE linear accelerator to an energy of 4.47(1) MeV/u. The $\approx 3.5\times 10^{6}$ pps intense ${}^{66}\mathrm{Ni}$ beam impinged on a $\approx 670$ $\mu$g/cm² thick secondary deuterated polyethylene target (99% enrichment in deuterium) [23] placed at the Miniball experimental station and produced the desired $\mathrm{d}(^{66}\mathrm{Ni},\mathrm{p})^{67}\mathrm{Ni}$ reactions. The total time of beam on target was approximately $140$ hours. The same reaction has previously been measured at ISOLDE by Ref. [24], but at a much lower beam energy of $2.5$ MeV/u. That data-set was not considered as it only reached excitation energies up to 3.5 MeV, much less than what is required for the Oslo Method.

Protons from the reaction were measured with the C-REX particle detector array [25, 26], while coincident $\gamma$-rays were measured using the Miniball detector array [27, 28]. Only the downstream particle array was considered due to low yields in the upstream array. The particle detectors considered covered angles between $25^{\circ}$ and $49^{\circ}$ in the laboratory frame (CM angles $40^{\circ}$ to $76^{\circ}$). Two of the eight Miniball high-purity germanium (HPGe) clusters [28] were replaced by six large-volume ($3.5\times 8$ inch) LaBr₃:Ce detectors to increase the high $\gamma$-ray energy efficiency of the array. The placement of each detector is given in table 1. Signals from the C-REX detectors were processed by an analog data acquisition system (DAQ) which fed four 32-channel analogue-to-digital converter (ADC) modules from Mesytec. The signals from the $\gamma$-ray detectors were processed by a digital DAQ consisting of XIA Digital Gamma Finders (DGF-4C). The two systems were synchronized, read out simultaneously, and the data written to disk.

Prompt particle-$\gamma$ coincidences were selected from the data by applying time gates on the time difference between the detected $\gamma$-rays and particles, while the $(\mathrm{d},\mathrm{p})$ reaction was selected by applying the appropriate energy cut on the $\Delta\text{E}-\text{E}$ matrix. Background events were selected in a similar fashion by placing time gates of half the size as the prompt gate on either sides of the prompt peak. This resulted in a total of $3.2\times 10^{5}$ and $1.1\times 10^{6}$ background subtracted proton-$\gamma$ coincidences with the LaBr₃:Ce detectors and the Miniball clusters, respectively. The low number of proton-$\gamma$ coincidences with the LaBr₃:Ce compared to those with the Miniball detectors is due to the lower geometric efficiency of the LaBr₃:Ce detectors which were located considerably further from the target. Despite having only a quarter of the Miniball coincidences, the more efficient LaBr₃:Ce detectors measured the majority of high-energy $\gamma$-ray transitions. For this reason, the results presented here are based on the analysis of proton-$\gamma$ coincident events from the LaBr₃:Ce detectors only.

Due to the considerable residual kinetic energy of ${}^{67}\mathrm{Ni}$ the $\gamma$-ray energies were Doppler corrected. The maximum deflection of the residual ${}^{67}\mathrm{Ni}$ was less than $1.5^{\circ}$ and the maximum spread in velocity was less than $4.2$%. The correction was performed using a constant factor based only on the $\gamma$-detector angle relative to the beam axis.

For each proton-$\gamma$ event the excitation energy was calculated from the kinematic reconstruction of the two-body reaction. The proton-$\gamma$ coincidences were collected in an excitation-$\gamma$ energy matrix shown in Fig. 1(a). Due to the Lorentz boost associated with inverse-kinematics there were no well-resolved levels in the particle spectra suitable for an in-beam calibration and the FWHM of the excitation energy was $\approx 1.1$ MeV.

The first step to obtain NLDs and $\gamma$SFs from excitation-$\gamma$ matrices is to correct for the detector response using the unfolding method [29] by using a detector response deduced from a Geant4 [30] simulation of the experimental setup [31]. This resulted in the unfolded matrix shown in Fig. 1(b).

Next, an iterative subtraction method [32] is applied to the unfolded matrix in order to deduce the distribution of primary $\gamma$-rays emitted from each excitation bin, resulting in the first generation matrix shown in Fig. 1(c).

The first-generation matrix is proportional to the $\gamma$-ray transmission coefficients $\mathcal{T}(E_{\gamma})$ and the NLD $\rho(E_{f}=E_{x}-E_{\gamma})$ at the final excitation energy $E_{f}$ following the first $\gamma$ decay of the cascade [8]

The NLD and transmission coefficients are extracted by normalizing the first-generation $\gamma$-ray spectra for each excitation bin and by fitting the experimental matrix ($\Gamma(E_{\gamma},E_{x})$) with a theoretical matrix [8]

by minimizing

where $\mathcal{T}(E_{\gamma})$ and $\rho(E_{f}=E_{x}-E_{\gamma})$ are treated as free parameters for each $\gamma$-ray energy $E_{\gamma}$ and final energy $E_{f}$, respectively. The denominator, $\Delta\Gamma(E_{\gamma},E_{x})$, represents the experimental uncertainty in the first-generation matrix. Considering the experimental resolution and to ensure that only primary transitions, originating in the quasi-continuum, are considered a minimum $\gamma$-ray energy of $1.5$ MeV was selected. Similarly, only excitation energy bins between $3.5$ and $5.6$ MeV were included in the fit. These $E_{\gamma}$ and $E_{x}$ energy regions have been chosen based on the guidelines of Ref. [33].

The resulting $\gamma$-ray transmission coefficients are converted to $\gamma$SF by

under the assumption that the transmission coefficients are dominated by dipole transitions.

The theoretical first-generation matrix, (2), is invariant under transformation of NLD and $\gamma$SF [8]

where $A$, $B$ and $\alpha$ are transformation parameters. This means that the extracted NLD and $\gamma$SF do not represent the physical NLD and $\gamma$SF, but contain information on the functional shape. To deduce the absolute values of the NLD and $\gamma$SF the extracted NLD and $\gamma$SF will have to be normalized to auxiliary data.

Many nuclei along the island of stability have auxiliary data available with which the normalization of the NLDs and $\gamma$SFs is achieved. Typical data used for this purpose are the level density from resolved discrete states, the level density at the neutron separation energy ($S_{n}=5.8$ MeV) from s- and/or p-wave neutron resonance spacing data, and the average radiative widths of s-wave neutron resonances. When applying the Oslo method, NLDs do not extend to $S_{n}$ as the method probes the NLD at the final excitation energy $E_{f}=E_{x}^{\text{max}}-E_{\gamma}$, the maximum energy possible to be considered in the first-generation matrix is $E_{x}^{\text{max}}$, and is typically the neutron separation energy (depending on reaction kinematics) [33], but in this case slightly below at $5.6$ MeV, which corresponds to a final and highest excitation energy of 4.1 MeV. Hence, the measured NLDs are typically interpolated to the NLDs from resonance data at $S_{n}$.

The impact on the uncertainty of NLDs and $\gamma$SF on the availability of neutron resonance data has recently been demonstrated [34]. In the absence of neutron resonance data, the resonance spacing or radiative widths of neutron resonances are generally estimated from systematics of neighboring nuclei, see for instance Refs.[35, 36, 37]. Similarly, for $\beta$-Oslo measurements various combinations of solutions have been used ranging from systematic information of nearby nuclei [11, 38], model calculations [14, 39, 38], or the Shape method [39, 38, 40, 41]. Neither for ${}^{67}\mathrm{Ni}$ nor for surrounding nuclei are neutron resonance data available. The situation is further aggravated by an incomplete level scheme at low excitation energies resulting in an unreliable level density of resolved discrete states which, together with the low experimental resolution, makes the application of the Shape method not ideal. With the absence of any reliable normalization points the only practical solution for normalizing the NLD and $\gamma$SF is the reliance on model estimates.

The Oslo method relies on external nuclear data for the normalization. In the absence of those, additional uncertainties may be induced and model dependencies may become significant. This is apparent through the relatively large uncertainties toward $S_{n}$ on the measured NLD for ${}^{67}\mathrm{Ni}$.

The challenge specific to inverse kinematic reactions is the Lorentz boost which causes a strong angular dependence in the kinematic reconstruction of the residual excitation energy. This leads to an excitation-energy resolution which is limited by the angular opening of the particle telescope’s active areas. The consequences are most apparent in the NLD where very little structures are visible.

In contrast, the measured $\gamma$SF still retains noticeable features and clearly exhibits a well established enhancement for $E_{x}<4$ MeV similar to those found in other $\mathrm{Ni}$ isotopes [79, 80, 34, 44, 81, 12]. Its observation indicates that the upbend is a structure which exists also away from stability. The upbend in ${}^{67}\mathrm{Ni}$ is predicted to be due to M1 strength, based on large-scale shell model calculations [46], and shown by the black solid line in Fig. 3. The significant strength in the enhancement and the simultaneous absence of a measurable scissors resonance may be supportive of the suggested connection of the two structures [82, 61], although results on the $\gamma$SF in ${}^{142,144-151}\mathrm{Nd}$ seem to contradict this [83].

The calculated ${}^{66}\mathrm{Ni}(n,\gamma)$ capture cross section in Fig. 5 features an uncertainty of $\approx 40\%$, constraining the cross section considerably. It is interesting to note that our cross section, lies consistently higher than the recommended values as provided in TALYS and TENDL but is smaller than those of JEFF 3.3 for $E_{n}>100$ keV. These differences highlight the necessity for measurements of NLDs and $\gamma$SFs, especially for nuclei away from stability.

Ref. [17] allows the capture rate to vary within the red band as indicated in Fig. 6, with the nominal value taken from the JINA REACLIB v1.1 database [76]. Our results constrain the capture rate significantly and show that this rate is found at the high end of the range used by Ref. [17]. This suggests a short exposure time for the weak i-process and could help pinpoint details in the stellar environment responsible to produce neutrons.

The ${}^{66}\mathrm{Ni}(n,\gamma)$ capture rate was suggested to be of key importance for the overall production of heavy elements during the i-process nucleosynthesis taking place in Sakurai’s object or rapidly accreting white dwarfs [17]. One possible astrophysical site with similar i-process conditions are low-metallicity low-mass stars during the early thermally pulsating Asymptotic Giant Branch (AGB) phase [84, 85, 86, 87, 88]. During this stage, protons can be engulfed by the convective thermal pulse. As they are transported downward by convection on a timescale of about 1 hr, they are burnt rapidly by ${}^{12}\mathrm{C}(p,\gamma)^{13}\mathrm{N}$. After the decay of ${}^{13}\mathrm{N}$ to ${}^{13}\mathrm{C}$ in about 10 min, the reaction ${}^{13}\mathrm{C}(\alpha,n)$ is activated at the bottom of the thermal pulse and leads to neutron densities of up to $\sim 10^{15}$ cm^-3. Stellar modeling was performed with the stellar evolution code STAREVOL [89, 90, 91]. A 1 $M_{\odot}$ AGB model at a metallicity of [Fe/H] $=-2.5$ is considered. This model is discussed in detail in Ref. [88].

We estimated the impact of the new ${}^{66}\mathrm{Ni}(n,\gamma)$ rate on the i-process nucleosynthesis in these objects. The effects are evaluated on both multi-zone stellar evolution models and in one-zone nucleosynthesis calculations. We used a reaction network including 1160 nuclei, linked through 2123 nuclear reactions ($n$-, $p$-, $\alpha$-captures and $\alpha$-decays) and weak interactions (electron captures, $\beta$-decays). Nuclear reaction rates were taken from BRUSLIB, the Nuclear Astrophysics Library of the Université Libre de Bruxelles²²2Available at http://www.astro.ulb.ac.be/bruslib/ [77] and the updated experimental and theoretical rates from the NETGEN interface [92]. Additional details can be found in Refs. [88, 93, 94]. In our sensitivity analysis, eight cases were investigated. We consider, for the ${}^{66}\mathrm{Ni}(n,\gamma)$ reaction, the recommended Maxwellian-averaged cross section (MACS) from this work ($10.7\pm 5$ mb at 30 keV) as well as the minimum (1 mb at 30 keV) and maximum (19 mb at 30 keV) rates predicted by a global unconstrained TALYS calculation (i.e. maximum and minimum reaction rates as prescribed in Ref. [17]). Additionally, the rate of ${}^{65}\mathrm{Ni}(n,\gamma)$ was also varied because ${}^{65}\mathrm{Ni}$ has a half-life of $\sim 2.5$ hr (55 hr for ${}^{66}\mathrm{Ni}$), which is fast enough to partially bypass the neutron capture during the i-process. For ${}^{65}\mathrm{Ni}(n,\gamma)$, we considered the minimum (10 mb at 30 keV) and maximum (70 mb at 30 keV) MACS predicted by TALYS for different NLD and $\gamma$SF models. We also considered the uncertainties related to the nominal ${}^{66}\mathrm{Ni}(n,\gamma)$ MACS evaluated in this study by varying it by $\pm 5$ mb at 30 keV (cases 1B, 1C, 2B and 2C). The different cases considered are reported in Table 5.

The initial abundances of the one-zone model correspond to the chemical composition in the thermal pulse of the AGB stellar model, just before the start of the proton ingestion event. The temperature and density were fixed to 220 MK and 2500 g cm^-3, respectively, which are typical values in the pulse of low-metallicity AGB models. The nucleosynthesis was calculated over a total time of $2\times 10^{5}$ s ($\sim 2.3$ days). To mimic the proton ingestion episode in the one-zone model, an initial abundance of protons was set to $2\times 10^{-5}$ and $3\times 10^{-4}$ leading to maximum neutron densities of $N_{\rm n,max}=1.5\times 10^{14}$ cm^-3 and $6.7\times 10^{14}$ cm^-3, respectively. For a low initial proton abundance, a weak i-process develops and elements up to the first peak ($A\lesssim 90$) are synthesized (Fig. 7, black pattern). With more protons, a larger neutron density is reached and elements of the third peak (Fig. 7, red pattern) are produced. For this particular case, the final chemical composition is found to be very similar to that of the full multi-zone AGB model where densities up to $N_{n}=2.1\times 10^{15}$ cm^-3 are reached.

As shown in Fig. 8, increasing the rate of ${}^{65}\mathrm{Ni}(n,\gamma)$ (case 2) leads to a smaller production of ${}^{65}\mathrm{Cu}$ (as a result of ${}^{65}\mathrm{Ni}$ decay) by a factor of $\sim 3$ in both weak and strong i-process cases. Lowering ${}^{66}\mathrm{Ni}(n,\gamma)$ (case 3) has the strongest impact since ${}^{66}\mathrm{Ni}$ now acts as a bottleneck. In the weak i-process case (Fig. 8, top panel), it leads to an overproduction of ${}^{66}\mathrm{Zn}$ by a factor of $\sim 10$ to the detriment of $70<A<90$ isotopes ($30<Z<40$), which are underproduced by a factor of $\sim 3$ at maximum. For a higher neutron density (Fig. 8, bottom panel), the production of $66<A<135$ isotopes ($28<Z<54$) is enhanced by a factor of typically $2-4$. The uncertainties on the nominal ${}^{66}\mathrm{Ni}(n,\gamma)$ MACS obtained in this work ($\pm 5$ mb at 30 keV) induce a variation of about a factor of $3$ at $A=66$ (dashed lines in Fig. 8, top panel).

Ref. [17] investigated the case of a weak i-process using one-zone models with an initial metallicity of [Fe/H] $=-1.6$ ([Fe/H] $=-2.5$ here) and different reaction rates (most of ($n,\gamma$) rates coming from the JINA REACLIB library [76]). Although the physical inputs are different, in the case of a weak i-process the results are qualitatively similar to those of Ref. [17]: an underproduction of the elements with $32<Z<42$ when adopting a low ${}^{66}\mathrm{Ni}(n,\gamma)$ rate.

However, when considering a full multi-zone AGB calculation, the impact of ${}^{66}\mathrm{Ni}(n,\gamma)$ is strongly dampened (Fig. 9). The overproduction of ${}^{66}\mathrm{Zn}$ in case 3 is clearly present but the enrichment is a factor of $\sim 3$ lower. In the stellar model, multiple zones with different temperatures, densities, chemical compositions and irradiations are mixed together by convection leading to a dilute production of the elements, an effect that cannot be captured by one-zone calculations.

The impact of the ${}^{66}\mathrm{Ni}(n,\gamma)$ reaction appears to be marginal in low-metallicity AGB stars experiencing i-process nucleosynthesis. It shows that one-zone calculations are inherently approximate and cannot guarantee to reproduce the accurate physics picture for a system. We should also emphasize that only one i-process site was investigated and we cannot exclude a different impact of ${}^{66}\mathrm{Ni}(n,\gamma)$ in other sites such as e.g. rapidly accreting white dwarfs [95].

In this paper we have presented experimental NLD and $\gamma$SF of the unstable ${}^{67}\mathrm{Ni}$ nucleus. It is the first time the Oslo method has been applied to an inverse kinematics experiment with a radioactive beam. Due to the lack of reliable nuclear data for ${}^{67}\mathrm{Ni}$ the NLD and $\gamma$SF are normalized to model predictions resulting in relatively large uncertainties. Hauser-Feshbach calculations were performed with the extracted NLD and $\gamma$SF to find the neutron capture cross section of ${}^{66}\mathrm{Ni}$. The result of these calculations showed a rather high capture rate compared to the theoretical range, and can help improve our understanding of the i-process nucleosynthesis. In agreement with Ref. [17], we find that the reaction ${}^{66}\mathrm{Ni}(n,\gamma)$ acts as a bottleneck when using one-zone models. Nevertheless, the impact is strongly dampened in multi-zone low-metallicity AGB stellar models experiencing i-process nucleosynthesis. This reaction may however have a different impact in other i-process sites.