Measurement of the ¹⁸Ne($\alpha,p$)²¹Na reaction with ANASEN at $E_{c.m.}=$ 2.5 - 4 MeV

Type-I X-ray bursts occur when hydrogen- and helium-rich matter is accreted onto a neutron star in a close-binary system Iliadis (2007); Schatz and Rehm (2006). In this environment, hydrogen is converted into helium via the hot-CNO (HCNO) cycles, in which the energy release is limited by the $\beta$-decay rates of the ¹⁴O ($t_{1/2}=$ 70.6 sec), ¹⁵O ($t_{1/2}=$ 122 sec) and ¹⁸Ne ($t_{1/2}=$ 1.67 sec) isotopes, and thereby is mostly independent of temperature. The temperature-dependent energy production, observed in explosive hydrogen burning, can only occur after the breakout from the closed HCNO cycles and is initiated through the activation of the $\mathrm{{}^{15}O(\alpha,\gamma)^{19}Ne}$ and ¹⁸Ne($\alpha,p$)²¹Na reactions Wallace and Woosley (1981); Wiescher et al. (1999). Therefore, these reactions carry a large impact on the timescale and onset conditions for the explosion. The role of individual nuclear reactions on the properties of X-ray burst events has been studied in significant detail, e.g. by Parikh et al. Parikh et al. (2008, 2013), and Cyburt et al. Cyburt et al. (2016). The ¹⁸Ne($\alpha,p$)²¹Na reaction has been identified as one of the most important reactions in its impact on the light curve and nucleosynthesis in X-ray burst events.

The available experimental information on the ¹⁸Ne($\alpha,p$)²¹Na reaction is limited and somewhat contradictory. Direct measurements have remained difficult, as they require cross section measurements significantly below 10 mbarn with beams of a T_z=-1 short-lived isotope at low energies, combined with the use of a gas-target to provide the $\alpha$ particles. The only experimental data on the $(\alpha,p)$ reaction was reported in Bradfield-Smith et al. Bradfield-Smith et al. (1999) and Groombridge et al. Groombridge et al. (2002), which presented results from experiments performed at the Louvain-la-Neuve laboratory. Both utilized a ¹⁸Ne beam at reaction energies between 1.7 and 3 MeV in the center-of-mass system and an experimental setup of a gas-filled chamber with silicon detectors. These experimental results were published as a set of resonance parameters, which were used to deduce thermal reaction rates.

Experiments on the time-inverse reaction ²¹Na($p,\alpha)^{18}$Ne by Sinha et al. Sinha (2005) and Salter et al. Salter et al. (2012) have reached a higher sensitivity towards lower reaction energies than the time-forward studies, owing to the higher-intensity ²¹Na beam and the use of solid polyethylene targets. However, their interpretation is complicated by the fact that the ¹⁸Ne($\alpha,p$)²¹Na reaction may proceed to excited ²¹Na states even at the lowest reaction energies, which cannot be examined by the time-inverse reaction with the ²¹Na ground state. It was noted in Salter et al. Salter et al. (2012) that their results point to much lower cross sections than Groombridge et al. Groombridge et al. (2002) reported.

Additional experimental information to determine the astrophysical reaction rates is available in the form of the resonance parameters in the ²²Mg compound system, which determine the astrophysical ¹⁸Ne($\alpha,p$)²¹Na reaction rate. Examples of efforts to characterize the resonance properties include experiments with the ²⁴Mg($p$,$t$)²²Mg reaction by Matic et al. Matic et al. (2009) and Chae et al. Chae et al. (2009), and more recently with the resonant proton scattering ²¹Na($p,p$)²¹Na reaction in inverse kinematics by He et al. He et al. (2013), Zhang et al. Zhang et al. (2014) and Ruiz et al. Ruiz et al. (2005).

A comprehensive analysis of the available experimental information on the ²²Mg compound system was performed by Mohr et al. Mohr and Matic (2013); Mohr et al. (2014). In the narrow, non-interacting resonance approximation, the stellar reaction rate $N_{A}<\sigma v>$ of the ¹⁸Ne($\alpha,p$)²¹Na reaction depends on the sum of the contributing resonances:

with resonance energies $E_{i}$ and resonance strengths $(\omega\gamma)_{i}$. The resonance strength for a resonance with spin $J$ for the ¹⁸Ne($\alpha,p$)²¹Na reaction is determined by

where the approximations $\Gamma_{p}\approx\Gamma$ and $\Gamma_{\alpha}\ll\Gamma_{p}$ are used. The analysis by Mohr et al. Mohr and Matic (2013); Mohr et al. (2014), employs the total widths $\Gamma$ and the excitation energies of the ²²Mg compound nucleus, which define the resonance energies $E_{i}$ in the ¹⁸Ne($\alpha,p$)²¹Na reaction, from the ²⁴Mg($p,t$)²²Mg transfer experiment Matic et al. (2009). The $J^{\pi}$ assignments were obtained initially from Ref. Matic et al. (2009); Chae et al. (2009) and confirmed from the latest experiments on resonant proton scattering He et al. (2013); Zhang et al. (2014). However, the transfer reactions cannot provide the required $\Gamma_{\alpha}$ widths, and thus the resonance strengths $\omega\gamma_{\alpha p}$. As a result, these have to be determined from theory or obtained from direct experiments. Since the partial widths $\Gamma_{\alpha}$ of ²²Mg levels in the relevant energy window have not been measured directly yet, they were estimated using the assumption of mirror symmetry in the wave functions of corresponding ²²Mg and ²²Ne levels in Ref. Mohr et al. (2014). For levels with no spectroscopic information, the partial widths $\Gamma_{\alpha}$ were calculated from a Porter-Thomas distribution following Ref. Pogrebnyak et al. (2013). Additionally, while the time-reverse ²¹Na($p,\alpha$)¹⁸Ne reaction information Salter et al. (2012); Sinha (2005) was used in the Mohr et al. studies Mohr and Matic (2013); Mohr et al. (2014), to determine a lower limit for the time-forward reaction cross section, the results of Groombridge et al. Groombridge et al. (2002) were excluded from the study due to the large disagreement with the time-reverse-reaction data. Both studies of Ref. Mohr and Matic (2013) and Mohr et al. (2014) arrived at cross sections and reaction rates significantly lower than the ones determined by Groombridge et al. Groombridge et al. (2002).

This work presents the results of a new experiment to measure the excitation function of the ¹⁸Ne($\alpha,p$)²¹Na reaction at energies relevant to X-ray burst events at peak temperatures ($T\sim 2$ GK), and to clarify the discrepancies between the direct experimental data and the information obtained from indirect methods.

The ¹⁸Ne($\alpha,p$)²¹Na reaction was studied in inverse kinematics using a beam of the short-lived isotope ¹⁸Ne, produced with the RESOLUT radioactive beam facility Wiedenhöver et al. (2014) at the John D. Fox Superconducting Accelerator Laboratory of Florida State University. The ¹⁸Ne beam was produced via the $\mathrm{{}^{16}O(^{3}He,n)^{18}Ne}$ reaction and selected at 80 MeV and at an average beam intensity of 2$\cdot$10³ pps, which constitutes 14% of the particles delivered. Offline, the events associated with the ¹⁸Ne beam particles were identified and discriminated from the ¹⁶O contaminant to 98% purity through individual time-of-flight signals relative to the accelerator RF-reference signal.

The experiment was performed with the active-target detector Array for Nuclear Astrophysics and Structure with Exotic Nuclei (ANASEN). A schematic of the detection scheme is displayed in Fig. 1, and more details on the design and operation of ANASEN are described in Ref. Koshchiy et al. (2017). ANASEN was filled with ⁴He as the target gas including a 4% admixture of CO₂ as a quenching gas, at a pressure of 379 Torr. The beam particles entered the gas-volume through an 8.9 $\mu$m Kapton window and continuously lost energy in the target gas, such that events from the energy range of interest occurred in an area of high geometric efficiency. The ANASEN detector systems were arranged in concentric layers surrounding the beam axis in a cylindrical geometry, starting from an inner set of 24 position-sensitive proportional-counter (PC) anodes at 3.75 cm radius, to a layer of 24 position-sensitive silicon-strip detectors of the Micron Semiconductor “Super-X3” design at a radius of 8.9 cm. In addition, 4 silicon-strip detectors of the Micron-Semiconductor “QQQ3” design cover forward laboratory angles in an annular geometry. The $(\alpha,p)$ protons were first detected in the proportional counter as they left the central beam volume, using the charge division on the resistive anode wire to determine the z-coordinate of interaction on the beam axis, along with the characteristic energy loss of the particles. The protons proceeded to the outer perimeter of ANASEN, where they were stopped in the silicon detectors, which allowed to determine the residual energy and position.

For this experiment a new component was added to ANASEN, a high-rate, compact ion chamber, sensitive to heavy ions traveling within 2.2 cm of the beam axis, including both beam particles and reaction residues. The ion chamber drift field is created between a negatively biased (-200 V) tungsten-wire cathode on the beam axis and 48 anode wires surrounding it at 2.2 cm cylindrical radius, as schematically represented in Panel (b) of Fig. 1. A differential read-out scheme is applied between odd- and even-numbered anode wires, which were biased at 150 V and 250 V, respectively. The drifting electrons induce differential signals only once they approach the anode radius, which creates the effect of a Frisch-grid and allows the ion chamber to operate up to 10⁵ counts/second.

Because of the geometry of the thin tungsten wire, the ion chamber is referred to as the Needle-IC. The tungsten-wire cathode is mounted on a retractable shaft inserted from the end of the detector chamber, such that the active volume of the ion chamber is adjustable. During the experiment, the needle was inserted to a depth of 18 cm into the active ANASEN volume. It should be noted that the 0.5-mm-diameter tungsten-wire is extending into the path of the beam, which had a diameter of $\approx$ 1.5 cm. While this aspect has the potential to create additional background scattering and reaction events, those were suppressed by requiring the expected recoil-energy correlation in the data analysis.

The variable insertion depth of the Needle was also used to determine the range of beam particles in the target gas independently. The result of this range measurement was compared to calculations with the program SRIM Ziegler et al. (2010) and LISE$++$ Tarasov and Basin . The energy-loss measurement was consistent with LISE$++$, but the SRIM energy-loss rates had to be corrected by a factor of $\approx$ 2. The resulting energy-loss profile was found to be consistent with the profile determined from the kinematics of two-proton coincidence events described in section III.2.

The energy calibrations of the silicon detectors were determined using $\alpha$ particles from a ²²⁸Th source in vacuum, as well as by scattering accelerator-provided $\alpha$ and proton beams off a thin tantalum foil mounted on-axis within the detector gas. The position of the scattering target was also varied along the beam axis, in order to calibrate the vertex-position reconstruction from the proportional-counter anodes in conjunction with the silicon detectors. Based on these calibration runs, the reaction vertex could be reconstructed with a resolution of about 2.5 cm FWHM, limited by the position resolution of the proportional counter and the beam dispersion.

The experiment was designed to measure the excitation function of the ¹⁸Ne($\alpha,p$)²¹Na reaction reaching to the low energies relevant for nuclear astrophysics. The energy-level scheme of the reactants, displayed in Fig. 3, shows that, owing to the low proton threshold in ²²Mg, a range of excited states in ²¹Na can be populated, including states above the proton-decay threshold of ²¹Na.

The cross sections were determined from the number of detected events for a given reaction-energy interval, the simulated efficiency, the total number of beam particles, and the effective areal target thickness for the corresponding energy interval. The total number of beam particles was determined from the number of particles detected in the Needle-IC detector, whose events were sampled independently, once for every thousand triggers. The areal target density was calculated from the target-gas pressure, the target composition, and the beam-path length associated with the individual reaction-energy interval, consistent with the energy-depth profile used in the $E_{track}$ reconstruction. The events were separated into intervals of ²¹Na excitation energy, beginning with the ground-state region between -1 and +1 MeV, over the bound-state region up to 3 MeV, to the higher excitations, which were separated into intervals of 0.4 MeV width, up to excitation energies of 7 MeV. These data-intervals were corrected for the average efficiency over the equivalent interval of the simulated events. The efficiency-corrected data was then combined to the ²¹Na excitation regions discussed below and subjected to the Gaussian unfolding procedure.

Panel (a) of Fig. 12 displays the ¹⁸Ne($\alpha,p$)²¹Na cross sections determined in this work for all single-proton events, separating the cross sections for population of the bound-state region. As expected, the cross section data shows a steep drop towards lower energies, reaching the sensitivity limit of the experiment at around 2.5 MeV (c.m.). At the lowest energies, the total cross section is dominated by population of the ²¹Na bound states but shifts more to unbound states above 3 MeV (c.m.). Panel (b) of Fig. 12 shows the cross section of the two-proton events which are detected in coincidence and corrected for their corresponding efficiency shown in Fig. 10. These events were analyzed independently as discussed in section III.2 representing the population of unbound excitations. While the statistical uncertainties of the coincident two-proton data are much larger, this data has a higher resolution of $\approx 500$ keV due to the $E_{kine}$ method, and is therefore not subjected to the Gaussian-unfolding procedure. This data is compared to the cross section extracted from single-proton events, associated with the ²¹Na unbound-excitation energies. The cross-section of the single-proton events is slightly higher than the cross section extracted from 2p-events, but nearly consistent within the error bars.

Experimental data on the ¹⁸Ne($\alpha,p$)²¹Na reaction were obtained by Salter et al. Salter et al. (2012) in a measurement of the time-inverse reaction ²¹Na($p,\alpha)^{18}$Ne and applying the principle of detailed balance. This interpretation, however, can only be applied to reactions leading to the ²¹Na ground state, while even at the lowest reaction energies, excited-state population is energetically allowed and may contribute substantially to the reaction rate. Our experiment allows to select events associated with the unresolved ²¹Na ground state and the 0.33 MeV first-excited state. Fig. 13 shows the thus-obtained cross sections and compares them to the corresponding cross sections extracted from Salter et al. Salter et al. (2012) and Sihna et al. Sinha (2005). Within the shared energy interval and uncertainties, both experiments agree. Due to the unresolved first-excited-state contributions, we expect that our cross section is slightly larger when compared to the time-reverse measurements. Also shown is a Hauser-Feshbach calculation of the ground-state to ground-state cross section, performed with the program SMARAGD Rauscher ; Rauscher (2011). The Hauser-Feshbach calculation shows the same energy dependence as our data and the data from the time-reverse reaction experiments Sinha (2005); Salter et al. (2012).

As discussed in the introduction, the ¹⁸Ne($\alpha,p$)²¹Na cross section has been studied before by Bradfield-Smith et al. Bradfield-Smith et al. (1999) and Groombridge et al. Groombridge et al. (2002). There, the experimental results were summarized as a table of resonance parameters and widths. Fig. 14 displays the cross section derived from the resonance parameters of Ref. Groombridge et al. (2002), in comparison to our result. Our experiment shows a much lower cross section, by close to an order of magnitude.

Also displayed in Fig. 14 is the cross section derived by Mohr et al. Mohr et al. (2014) from a survey of ²²Mg resonance parameters, obtained in indirect experiments Matic et al. (2009); Chae et al. (2009); He et al. (2013); Zhang et al. (2014), or calculated assuming mirror-symmetry in the wave functions of corresponding ²²Mg and ²²Ne levels. Where no spectroscopic information was available for the ²²Mg levels, these were calculated from a Porter-Thomas distribution following Ref. Pogrebnyak et al. (2013). While the resulting cross-section curve by Mohr et al. Mohr et al. (2014) is consistent with our data in the lower-energy range, the curve is systematically too low at higher energies. This systematic deviation can be attributed to underestimating the contributions to higher-lying ²¹Na states, which evidently dominate our experimental data at higher energies, or to contributions of unknown or mis-assigned-spin resonances in the ²²Mg spectrum. Finally, Fig. 14 shows a calculation with the Hauser-Feshbach code SMARAGD Rauscher ; Rauscher (2011), which includes the cross sections leading towards the ground state and first five excited states of ²¹Na, including the unbound 2.798 MeV state. The overall agreement with our data is excellent.

The impact of our results on the reaction rate is illustrated in Fig. 15. We calculate reaction rates using the unfolded cross sections from this work shown in Figs. 12–14 for $E_{cm}>2.5$ MeV. For $E_{cm}<2.5$ MeV, we adopt cross sections based on the resonant properties given in Table I of Mohr et al. Mohr et al. (2014), where we are in agreement and our statistical precision is poor. Following the presentation in Mohr et al. (2014), we plot the ratio of our reaction rates to the reference rate from Ref. Mohr and Matic (2013). At peak X-ray burst temperatures $T=2.5$ GK, the ¹⁸Ne$(\alpha,p)^{21}$Na reaction rate from a previous Monte Carlo analysis in Ref. Mohr et al. (2014) (yellow, solid curve in Fig. 15) agrees with the recommended reaction rate from Ref. Mohr and Matic (2013), which was given as 55$\%$ of the reference rate (flat grey line in Fig. 15). We find the experimental reaction rate to be a factor of 1.5 larger at $T=2.5$ GK (black, solid curve Fig. 15). The increase comes primarily from reactions proceeding through proton unbound states in ²¹Na that subsequently decay by proton emission to ²⁰Ne, as can be seen in the comparison of the ($\alpha,p$) (blue, dashed curve Fig. 15) to the ($\alpha,2p$) (green, dashed-dotted curve) contributions to the reaction rate. The resonance properties in ²²Mg used in previous estimates of the reaction rate come almost exclusively from 2-neutron stripping reactions. It is not surprising that broad proton-unbound states may have been missed that would account for the enhanced contribution from ¹⁸Ne$(\alpha,2p)^{20}$Ne that we observe. In most astrophysical scenarios the ¹⁸Ne$(\alpha,2p)^{20}$Ne branch is likely to have a negligible effect due to the high rate of the ²⁰Ne($p$,$\gamma$)²¹Na reaction; however, it is advisable to include the ¹⁸Ne$(\alpha,2p)^{20}$Ne reaction rate as a separate channel, particularly in models with low hydrogen abundance, such as may result from accretion from a helium-rich companion.

Finally, it is worth noting that the Monte Carlo estimate of the ¹⁸Ne$(\alpha,p)^{21}$Na reaction rate in Ref. Mohr et al. (2014) (yellow curve in Fig. 15) is significantly higher for $T\approx 1-2$ GK than that obtained using the resonance parameters (Table I of Mohr et al. (2014)) or a direct integration using the recommended S-factor (Fig. 2 of Mohr et al. (2014)) (purple, dotted curve Fig. 15). Since we adopt the resonance parameterization for $E_{cm}<2.5$ MeV in calculating the reaction rate, our reaction rate may similarly underestimate the reaction rate in this intermediate temperature range.

Our experiment determined cross sections for the ¹⁸Ne($\alpha,p$)²¹Na reaction between about 2.5-4 MeV in the center of mass using the active-target detector system ANASEN, which was augmented by an ion-chamber component, allowing for a higher detection sensitivity without limiting the usable beam rate. The present experiment is the first measurement of an ($\alpha,p$) reaction on a radioactive beam with the ANASEN active-target detector, determining cross sections as low as 5 mbarn and in the center of the Gamow-window for a temperature $T=2$ GK. The partial cross sections for the population of the ²¹Na ground state are consistent with those derived from experiments on the time-reverse reaction by Sihna et al. Sinha (2005) and Salter et al. Salter et al. (2012). The total cross section is significantly lower than the one found in previous experiments by Bradfield-Smith et al. Bradfield-Smith et al. (1999) and Groombridge et al. Groombridge et al. (2002).

Compared to the predictions of Mohr et al. Mohr and Matic (2013); Mohr et al. (2014), which are based on indirectly determined resonance parameters, our cross sections are larger at higher reaction energies, demonstrating the contributions from the proton-unbound states of ²¹Na. While in most astrophysical scenarios the ¹⁸Ne$(\alpha,2p)^{20}$Ne branch is likely to have a negligible effect due to the high rate of the ²⁰Ne($p$,$\gamma$)²¹Na reaction, the ¹⁸Ne$(\alpha,2p)^{20}$Ne reaction rate may have a separate impact on models with low hydrogen abundance, such as those from accretion from a helium-rich companion. Finally, the cross sections of this work are consistent with the predictions of a Hauser-Feshbach calculation. Although at low energies the reaction proceeds mostly through individual resonances, which could not be resolved by our experiment, the overall agreement with Hauser-Feshbach calculations seems to justify their use as an approximation to determine the thermal reaction rate until more experimental data is available for the direct ¹⁸Ne($\alpha,p$)²¹Na.

This work was partially supported by the National Science Foundation under Grants No. PHY-1712953 and PHY-2012522, and partially supported by the U.S. Department of Energy, Office of Science under Grants No. DE-FG02-96ER40978 and No. DE-FG02-93ER40773. T.R. is partially supported by the “ChETEC” COST Action (CA16117). M.A. was supported for the writing of this manuscript (LLNL-JRNL-821361) by the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.