Constraining the destruction rate of ${}^{40}\mathrm{K}$ in stellar nucleosynthesis through the study of the ${}^{40}\mathrm{Ar}(p,n){}^{40}\mathrm{K}$ reaction

Background: ${}^{40}\mathrm{K}$ plays a significant role in the radiogenic heating of Earth-like exoplanets, which can affect the development of a habitable environment on their surfaces. The initial amount of ${}^{40}\mathrm{K}$ in the interior of these planets depends on the composition of the interstellar clouds from which they formed. Within this context, nuclear reactions that regulate the production of ${}^{40}\mathrm{K}$ during stellar evolution can play a critical role.

Purpose: In this study, we constrain for the first time the astrophysical reaction rate of ${}^{40}\mathrm{K}(n,p){}^{40}\mathrm{Ar}$, which is responsible for the destruction of ${}^{40}\mathrm{K}$ during stellar nucleosynthesis. We provide to the nuclear physics community high-resolution data on the cross section and angular distribution of the ${}^{40}\mathrm{Ar}(p,n){}^{40}\mathrm{K}$ reaction. These are important to various applications involving ${}^{40}\mathrm{Ar}$. The associated reaction rate of the ${}^{40}\mathrm{Ar}(p,n){}^{40}\mathrm{K}$ process addresses a reaction rate gap in the Joint Institute for Nuclear Astrophysics REACLIB database in the region of intermediate-mass isotopes.

Methods: We performed differential cross-section measurements on the ${}^{40}\mathrm{Ar}(p,n){}^{40}\mathrm{K}$ reaction, for six energies in the center-of-mass system between 3.2 and 4.0 MeV and various angles between $0^{\circ }$ and ${135}^{\circ }$. The experiment took place at the Edwards Accelerator Laboratory at Ohio University using the beam swinger target location and a standard neutron time-of-flight technique. We extracted total and partial cross sections by integrating the double differential cross sections we measured.

Results: The total and partial cross sections varied with energy due to the contribution from isobaric analog states and Ericson type fluctuations. The energy-averaged neutron angular distributions were symmetrical relative to ${90}^{\circ }$. Based on the experimental data, local transmission coefficients were extracted and were used to calculate the astrophysical reaction rates of ${}^{40}\mathrm{Ar}(p,n){}^{40}\mathrm{K}$ and ${}^{40}\mathrm{K}(n,p){}^{40}\mathrm{Ar}$ reactions. The new rates were found to vary significantly from the theoretical rates in the REACLIB library. We implemented the new rates in network calculations to study nucleosynthesis via the slow neutron capture process, and we found that the produced abundance of ${}^{40}\mathrm{K}$ is reduced by up to 10% compared to calculations with the library rates. At the same time, the above result removes a significant portion of the previous theoretical uncertainty on the ${}^{40}\mathrm{K}$ yields from stellar evolution calculations.

Conclusions: Our results support a destruction rate of ${}^{40}\mathrm{K}$ in massive stars via the ${}^{40}\mathrm{K}(n,p){}^{40}\mathrm{Ar}$ reaction that is larger compared to previous estimates. The rate of ${}^{40}\mathrm{K}$ destruction via the ${}^{40}\mathrm{K}(n,p){}^{40}\mathrm{Ar}$ reaction now has a dramatically reduced uncertainty based on our measurement. This result directly affects the predicted stellar yields of ${}^{40}\mathrm{K}$ from nucleosynthesis, which is a critical input parameter for the galactic chemical evolution models that are currently employed for the study of significant properties of exoplanets.

Figure 1

Flow diagram of nuclear reactions around ${}^{40}\mathrm{K}$ during $s$-process nucleosynthesis. The main avenues for destruction of ${}^{40}\mathrm{K}$ are through the $(n,p)$ and $(n,\alpha )$ reaction rates (solid black arrows). In this work we infer the ${}^{40}\mathrm{K}(n,p){}^{40}\mathrm{Ar}$ rate from a study of the ${}^{40}\mathrm{Ar}(p,n){}^{40}\mathrm{K}$ reaction (dashed black arrow).

Figure 2

Diagram of the beam swinger and of the neutron time-of-flight tunnel at the Edwards Accelerator Laboratory (modified image from [16]). The beam swinger allows the beam axis to rotate around the target center, making possible measurements of neutron yields at various reaction angles without the need for multiple neutron flight paths. The swinger is part of the 4.5 MV Tandem accelerator of Ohio University.

Figure 3

Level scheme of excited levels in the final ${}^{41}\mathrm{K}$ residual of the ${}^{40}\mathrm{Ar}(p,n){}^{40}\mathrm{K}$ reaction ($Q=-2.2868$ MeV). The ${}^{41}\mathrm{K}$ compound nucleus excitation energy ranged between 11.2 and 11.7 MeV. The scheme shows the energetically allowed excited states in the residual nucleus along with the prominent gamma rays from the deexcitation of the residual nucleus. The corresponding neutron energies span 1.0–1.6 MeV for $n_0, n_1$ and 0.1–0.9 MeV for $n_2, n_3$.

Figure 4

Calculation of the total cross section from neutron angular distributions and corresponding uncertainties. The Monte Carlo method described in the text was used to estimate these uncertainties from the spread of results for the integrated angular distribution fits. (a) Example of a measured angular distribution of the $(p,n_{0,1})$ channel at $E_{CM}=3.88$ MeV. The red line shows the best Legendre fit on the experimental points, while each one of the 5000 blue lines corresponds to a fit of a randomly chosen configuration of points within the limits of each point's statistical uncertainty. (b) Distribution of the angle-integrated cross sections that were obtained from $N$ sampled fits. The mean value $\mu$ and standard deviation $\sigma$ of the distribution are extracted from the Gaussian fit of the full distribution. The final cross sections are given as $\mu \pm 3\sigma$ (Table 2).

Figure 5

Neutron time-of-flight spectrum obtained during the efficiency measurement. Neutron energy increases towards the higher channel numbers, while the corresponding time of flight increases in the opposite direction.

Figure 6

Neutron yields from the ${}^9\mathrm{Be}(d,n)$ reaction. The reference spectrum has been adopted from [20]. The intrinsic efficiency of the LENDA bar is reflected in the difference between the two curves. Each data point marks the center of the corresponding energy bin.

Figure 7

Efficiency curve of the LENDA bar for two different threshold levels on the output pulse heights of the detector. The intrinsic efficiency below 1 MeV increases dramatically for the lower threshold and reaches a peak of approximately 35% at 500 keV.

Figure 8

$\gamma$-ray spectrum taken with the LaBr detector at $E_{\mathrm{lab}}=3.8$ MeV. The marked photopeaks are associated with the interactions of the proton beam with the aluminum windows of the gas cell and the argon gas. Unlabeled peaks come from interactions of neutrons with the ${\mathrm{LaBr}}_3$ crystal and the surrounding material in the experimental area, and from the detector's self-activity.

Figure 9

Gamma-ray relative yields from the $(p,n_2)$ channel. The two broad structures observed in the excitation function for incident energies above 3.2 MeV are attributed to two ${}^{41}\mathrm{Ar}$ isobaric analog state resonances folded with Ericson type statistical fluctuations of the cross section. In the gamma spectra, the Ericson fluctuations are suppressed due to the 25 keV energy resolution of the measurement that makes them appear mostly as a slowly fluctuating “background” continuum underneath the two broad IAS structures. The fluctuations were much more pronounced in the high-resolution time-of-flight spectra. There, the shape of the neutron peaks revealed a significant number of overlapping resonancelike structures (see discussion and Fig. 10).

Figure 10

Neutron time-of-flight spectra for two different target gas pressures, i.e., for two different target thicknesses. Both spectra correspond to the same beam energy (before interacting with the target window) and angle. The structured patterns on the neutron peaks are the result of the convolution of the Ericson fluctuations with the resonant neutron yields of the ${}^{41}\mathrm{Ar}$ isobaric analog states in ${}^{41}\mathrm{K}$. For the blue spectrum, the reaction energy ranges between 3.4 and 3.6 MeV. For the red spectrum, the reaction energy ranges between 3.5 and 3.6 MeV. The difference in reaction energy range results in an integration of a larger portion of the fluctuating excitation function in the blue spectrum than in the red one. Consequently, more structures appear in the neutron peaks, consistent with what would be expected by a fluctuating cross section (see discussion and Fig. 9).

Figure 11

Experimental and theoretical cross-sections for the ${}^{40}\mathrm{Ar}(p,n){}^{40}\mathrm{K}$ reaction. The theoretical cross sections have been calculated using the talys code as described in Sec. 4c. The error band shows the maximum variations of the calculations due to uncertainties in the tabulated JLM optical potential parameters include with talys (default label). The “modified” model is a calculation in which the neutron and alpha transmission coefficients using the same tabulated JLM parameters [30] were adjusted to properly reproduce the experimental data.

Figure 12

Neutron angular distributions from ${}^{40}\mathrm{Ar}(p,n_{0,1})$ and ${}^{40}\mathrm{Ar}(p,n_2)$ channels at various incident energies. All the energies and the differential cross sections are given in the center-of-mass system.

Figure 13

Experimental and theoretical energy-averaged angular distributions of the ${}^{40}\mathrm{Ar}(p,n){}^{40}\mathrm{K}$ reaction. The distributions are averaged over the energy range 3.33 to 3.92 MeV. A better agreement between the results is observed when the modified transmission coefficients are used (see also Fig. 11).

Figure 14

Top: Calculated astrophysical reaction rate of the ${}^{40}\mathrm{Ar}(p,n){}^{40}\mathrm{K}$ reaction, using the modified transmission coefficients from this work. The corresponding reaction rate of REACLIB database is also included (“rath” rate by Rausher et al. [37]). Bottom: Comparison between the reaction rates from this work and the REACLIB database. The vertical axis corresponds to the ratio ${\text{Rate}}_{\mathrm{TALYS}}/{\text{Rate}}_{\mathrm{REACLIB}}$.

Figure 15

Top: Constrained astrophysical reaction rate of the ${}^{40}\mathrm{K}(n,p){}^{40}\mathrm{Ar}$ reaction (blue line). The rate is deduced from the Hauser-Feshbach calculations on the ${}^{40}\mathrm{Ar}(p,n){}^{40}\mathrm{K}$ reaction, using the principle of detailed balance [see Eq. (9)]. For temperatures below 0.4 GK an exponential extrapolation was applied as described in the text. The grey error band marks the limits of the average experimental uncertainty. The green line represents the same reaction rate but calculated using the modified transmission coefficients from this study (Hauser-Feshbach calculation with talys). In the same graph, the reaction rate from the REACLIB library is also displayed (“rath” rate by Rausher et al. [37], yellow line). The red line corresponds to the REACLIB rate after being scaled to a reference value at 0.4 GK. Bottom: Comparison between the various reaction rates, relative to the reference rate obtained from detailed balance. The deviations are calculated using Eq. (10). The REACLIB rate is systematically above the limits of the experimental uncertainty, and deviations up to $\approx 40\%$ are observed. On the other hand, the scaled REACLIB rate and the talys rate (with modified $T_L$) are in a very good agreement with the reference curve.

Figure 16

Mass fraction of ${}^{40}\mathrm{K}$ produced by the $s$ process for different rates of the ${}^{40}\mathrm{K}(n,p){}^{40}\mathrm{Ar}$ reaction. The grey error band highlights the sensitivity of the mass fraction of ${}^{40}\mathrm{K}$ to the variation of the library rate by a factor of 2. The yellow error band represents the upper limit of the associated uncertainty on the experimental reaction rate, which is $\approx 15\%$. The lower panel displays the ratio of the two mass fractions when the numerator of the ratio corresponds to the calculations using the REACLIB rate. The new experimentally constrained rate suggests a factor of 1.39 increase in the destruction rate of ${}^{40}\mathrm{K}$ and a close to 10% reduction in the production of ${}^{40}\mathrm{K}$ after a period of ${10}^5$ yrs.