Single neutron transfer on 23Ne and its relevance forthepathway ofnucleosynthesis in astrophysical X-ray bursts

We present new experimental measurements of resonance strengths in the astrophysical 23Al(p, {\gamma})24Si reaction, constraining the pathway of nucleosynthesis beyond 22Mg in X-ray burster scenarios. Specifically, we have performed the first measurement of the (d, p) reaction using a radioactive beam of 23Ne to explore levels in 24Ne, the mirror analog of 24Si. Four strong single-particle states were observed and corresponding neutron spectroscopic factors were extracted with a precision of {\sim}20{\%}. Using these spectroscopic factors, together with mirror state identifications, we have reduced uncertainties in the strength of the key {\ell} = 0 resonance at Er= 157 keV, in the astrophysical 23Al(p, {\gamma}) reaction, by a factor of 4. Our results show that the 22Mg(p, {\gamma})23Al(p, {\gamma}) pathway dominates over the competing 22Mg({\alpha}, p) reaction in all but the most energetic X-ray burster events (T>0.85GK), significantly affecting energy production and the preservation of hydrogen fuel.

Type-I X-ray bursts represent thermonuclear explosions on the surfaces of accreting neutron stars in close binary systems [1][2][3]. They exhibit dramatic, recurrent increases in luminosity and constitute the most frequent stellar eruptions to occur in our Galaxy. In between bursts, energy is generated at a constant rate by the β-limited hot CNO cycles [4,5]. However, as the temperature of the accreted material increases, the triple-α reaction becomes favourable, igniting the burst, and nucleosynthesis proceeds along the proton-rich side of stability via the αp process [6] [a series of (p, γ) and (α, p) reactions], and the rp process [3] [a series of (p, γ) reactions and β + decays], ending in the Sn-Te mass region.
Recently, advances in computing power have allowed for detailed models of X-ray burst nucleosynthesis to be constructed [6][7][8][9], incorporating complex reaction networks and hundreds of nuclear species ranging from stable isotopes up to the proton drip line. Strikingly, despite the vast number of reactions included, only a handful of nuclear processes have been highlighted as having a noticeable effect on the observational properties of X-ray bursts [10][11][12]. In particular, the 23 Al(p, γ) 24 Si reaction, which permits flow beyond masses of A = 22 in the early phases of the rp process, is postulated to have a strong influence on the inferred surface gravitational redshift (1 + z) [12]. The redshift is directly related to the neutron star compactness [13] and thus, any experimental constraints placed on the 23 Al(p, γ) reaction rate will help to reveal new facets of the underlying compact objects involved. Furthermore, at the 22 Mg, rp-process waiting point, the 22 Mg(p, γ) 23 Al(p, γ) reaction sequence is expected to compete significantly with the 22 Mg(α, p) reaction [14], affecting the overall energy generation in X-ray bursters. Specifically, a prevailing 22 Mg(p, γ) 23 Al(p, γ) pathway results in less energetic burning during the burst rise, preserving hydrogen for later burning and extending the burst tail. The exceptional measurements now available for the structure of burst light curves [15,16] are amenable to confront simulations of the burst explosions.
Previous studies of the 23 Al(p, γ) reaction [17][18][19][20] indicate that the rate is dominated by resonant capture on the 5/2 + ground state of 23 Al to excited states above the proton-emission threshold energy of 3292 (19) keV in 24 Si [21]. However, the strengths of these resonances remain uncertain, due to the scarcity of experimental data. Most recently, Wolf et al. utilised the 23 Al(d, n) reaction to investigate the properties of excited states in 24 Si [20]. In that study [20], γ decays were observed from three excited states, including the key = 0, proton-unbound resonant level at 3449(5) keV, which is expected to have the most significant influence on the 23 Al(p, γ) reaction over the temperature range of X-ray bursts. Moreover, by measuring angle-integrated cross sections of excited levels in 24 Si, Wolf et al. were able to place the first constraints on proton spectroscopic factors, reducing uncertainties in both the direct and resonant capture components of the 23 Al(p, γ) reaction [20]. However, the absolute values of spectroscopic factors reported in Ref. [20] carry large uncertainties of order 60% because their extraction relied on the use of shell-model calculations to determine the relative contributions of multiple -transfers. For example, where states are populated by a mixture of = 0 and = 2 transfer, the work of Ref. [20] was forced to use the ratio of strengths predicted by the shell model, but it has been pointed out [22] that the shell model consistently fails to predict this ratio correctly. Consequently, the rate of the 23 Al(p, γ) reaction is still weakly constrained over the temperature range of Type-I X-ray bursts and a more robust experimental measurement is demanded.
A direct measurement of the 23 Al(p, γ) reaction is not presently feasible. As such, any further experimental constraints must rely on indirect techniques. In this regard, several studies have shown that precise evaluations of proton capture reactions may be achieved via the concept of isospin [23][24][25]. Specifically, neutron spectroscopic factors of excited states in mirror nuclei, that correspond to analogs of (p, γ) resonances, can be used to accurately determine the strengths of resonances governing the rate of stellar reactions in explosive astrophysical environments [23][24][25]. In this Letter, we present a first experimental measurement of the 23 Ne(d, p) transfer reaction to study excited states in 24 Ne. These levels correspond to T = 2, mirror analogs of key resonant states in the 23 Al(p, γ) 24 Si reaction. By coupling the TIGRESS γ-array [26] to the SHARC charged-particle detection system [27], neutron spectroscopic factors were extracted to a precision of ∼20%. This reduces uncertainties in 23 Al + p resonance strengths by a factor ∼4 and, hence, defines the relative importance of the 22 Mg(p, γ) 23 Al(p, γ) and 22 Mg(α, p) reaction sequences over the temperature range of X-ray bursts.
A beam of radioactive 23 Ne 2+ ions was accelerated to 8.0 MeV/nucleon and an intensity of ∼2 × 10 4 pps, by the ISAC-II facility at TRIUMF and bombarded a 1 mg/cm 2 (CD 2 ) n foil for 93 hrs. Prompt γ rays were recorded using the TIGRESS array of 12 Compton-suppressed HPGe detectors [26], while charged particles including protons from the 23 Ne(d, p) reaction were measured in the SHARC silicon array [27]. Beyond the target, 40 cm downstream, the TRIFOIL detector [28,29] was placed (a 20 µm foil of BC400 plastic scintillator viewed by three photomultiplier tubes and mounted behind a passive stopper foil of 110 µm Al). The TRIFOIL setup (a) stopped the 23 Ne beam and 24 Ne reaction products in the scintillator and counted them, (b) stopped the 23 Na beam contaminant (∼40% of the beam) and 24 Na reaction products in the Al foil so that they had no TRIFOIL tag and (c) also in the Al, stopped fusion-evaporation products from reactions on carbon in the CD 2 target. The TRIFOIL also gave a direct measurement of the average counting rate of the beam over the entire 93 hours of data acquisition to a precision of < 1%. The rejection of 23 Na-induced events was verified by the complete removal of 24 Na γ-ray peaks when imposing the TRIFOIL requirement. The beam composition was also measured at regular intervals using a Bragg ionization detector [30] and background from other contaminant isobars was found to be negligible. Energy and efficiency calibrations were performed using standard γ-ray ( 152 Eu and 60 Co) and charged-particle (triple alpha) sources. The absolute normalisation was determined using the measured number of incident 23 Ne ions, the target thickness and the H:D ratio, as determined from elastic scattering around θ cm = 50 • measured simultaneously throughout the acquisition.     Table 1: Properties of excited states in the T = 2, A = 24 system, as determined in the present work and reported in earlier literature. Excitation energies are given in keV and shell-model spectroscopic factors were determined using the USDA interaction [31]. In Ref. [32], no uncertainties for Ex ( 24 Ne) are given, but we expect ≤ 2 keV based on HPGe calibration. The present C 2 S (d,p) is the summed = 2 strength, extracted assuming transfer to 0d 5/2 (errors and limits, see text). For comparison, C 2 S SM is the sum of USDA shell-model values for 0d 5/2 and 0d 3/2 .  Figure 1 illustrates the excitation energy of states in 24 Ne populated via the (d, p) reaction. As can be seen, four strongly populated states are observed at 0, 1981, 3871 and 4886 keV, in good agreement with previously reported 0 + 1 , 2 + 1 , 2 + 2 and 3 + 1 levels in 24 Ne [32] and the theoretical calculations of Ref. [33]. That being said, additional 4 + 1 and 0 + 2 excited states are also expected in this energy region in 24 Ne at 3962 and 4765 keV [32], respectively, which would not be resolvable using proton detection alone, due to the ∼300 keV (FWHM) excitation energy resolution of SHARC. In this regard, the simultaneous detection of γ rays is of crucial importance. In particular, by placing gates across the observed energy peaks at 3871 and 4886 keV, and viewing coincident γ rays within the TIGRESS array, it was possible to rule out any significant population of the 4 + 1 , 3962-keV and 0 + 2 , 4765-keV excited states via the 23 Ne(d, p) reaction. For example, when a gate was placed across the 3871-keV proton peak, we observe 1890and 1981-keV γ-ray peaks of equal intensity from the cascade decay from 3871 keV (inset of Fig. 1). The numbers of counts are 76 ± 10 and 84 ± 10, respectively. The surplus for the 1981-keV peak is 8 ± 14 which is consistent with zero and gives a 2σ upper limit (allowing for the double counting) of 12% of the combined population of the two states. This is the basis of the limit on the spectroscopic factor for the 3962-keV state in Table 1. Consequently, we conclude that the 3962-keV excited state in 24 Ne was not appreciably populated and, based on an upper limit analysis of 1981-keV transitions originating from the 4 + 1 state, we set a stringent upper limit on its spectroscopic factor, C 2 S ( =2) ≤ 0.012. This is in agreement with our shell model calculations using NuShellX [34] with the USD-A interaction [31], which predict C 2 S ( =2) = 0.01 for the 4 + 1 level in 24 Ne. In contrast, a similar procedure to the above was not possible for the expected 4765/4886-keV doublet due to a considerable level of background in the γ-ray energy region of interest. Whilst the observed = 0 angular distribution for this doublet, shown in Fig. 2, may be ascribed entirely to the known 3 + , 4886-keV level in 24 Ne [32], we adopt an upper limit of 0.19 for the = 2 component of the spectroscopic factor for both the 4765-and 4886-keV excited states in 24 Ne. An angular distribution analysis of the 0-, 1981-, 3871and 4886-keV excited states in 24 Ne, shown in Fig. 2, confirms the spin-parity assignments of Ref. [32]. However, with the exception of the ground state, which necessarily exhibits a pure = 2 character, the measured distributions indicate strong mixing between = 0 and = 2 Table 2: Properties of resonant states in the 23 Al(p, γ) reaction used in the present analysis, together with a comparison to resonance strengths based on spectroscopic factors reported in Ref. [20]. Excitation energies in 24 Si and present C 2 S values are as in Table 1. These were used to calculate Γp and (together with the Γγ from USDA shell-model calculations, see text) ωγ. For states shown in Table 1 with = 0 and = 2 contributions, only the = 0 is included here since it overwhelmingly dominates the resonance strength. Upper limits have been determined to a 68% confidence level. (5) 157 (20) 2 + 0 0.44(9) 8.2(17) × 10 −5 1.8 × 10 −2 3.4(7) × 10 −5 5.4(31) × 10 −5 3471 (6) 179(20) (11) 3.0 × 10 4 8.9 × 10 −3 5.2 × 10 −3 d a Previous resonance strengths have been estimated based on spectroscopic factors reported in Ref. [20] b For the 179-keV resonance, we currently favour a 4 + assignment based on mirror energy difference arguments. However, for completeness, we provide resonance strength determinations for both 4 + 1 and 0 + 2 assignments. c Adopted from Ref. [33] d Resonance strength determination dominated by theoretically calculated γ-ray partial width transfer for all levels. These observed distributions were then compared with reaction calculations in the Adiabatic Distorted Wave Approximation (ADWA) performed, using the code TWOFNR [35]. Here, the Johnson-Soper adiabatic model [36] was employed with standard parameters [37] using zero range and the Koning-Delaroche [38] global nucleon-nucleus optical potential. We estimate an uncertainty in the overall cross section normalization of ∼20%, with the dominant contribution coming overwhelmingly from the modelling of the (d, p) reaction itself [37]. The solid angle was calculated accurately from the known geometry, the fitted position of the beam spot and omitting the detector strips excluded from the analysis. The systematic uncertainty in the normalisation of the data arises principally from the uncertainty in the target thickness (taken as 10%) since the total number of incident particles was precisely given by a direct measurement included continuously in the data stream. A summary of the properties of excited states in 24 Ne determined in this work is given in Table 1, together with a comparison with our shell-model calculations using the USDA interaction [31]. A proposed matching of analog levels in 24 Si is also shown [20]. We have adopted a number of mirror assignments from earlier work [20] and, although the spin-parity assignments of the 3449-and 3471-keV excited states in 24 Si are not uniquely defined, we propose analog matchings to the 2 + 1 , 3871-keV and 4 + 1 , 3962-keV levels in 24 Ne, respectively, based on mirror energy differences. Specifically, a pairing to the 0 + 2 state would require a very large mirror energy shift of ∼1.3 MeV (although we note that a recent study [39] suggested that such an assignment may be possible).
The present results show excellent agreement with shell model calculations, especially for = 0 transfers. Notably, we find that = 2 strengths for strongly mixed states can deviate considerably from theory, as was previously highlighted in Ref. [22]. This is particularly relevant to the extraction of astrophysical data. Specifically, the authors of Ref. [20] were forced to rely on shell model ratios of = 0 and = 2 strengths in order to analyse their angleintegrated cross sections, but their extracted = 0 values are then susceptible to inaccuracies in the shell model theory (the = 0 strength determines the important resonance parameters for astrophysics). The present work measures the = 2 and = 0 strengths independently of any prior constraints and indeed we find clear differences with the results from Ref. [20]. In particular, the values of C 2 S =0 of the 2 + 1 and 2 + 2 excited levels in the T = 2 system are found, respectively, to be 0.28 (6) [compared to 0.6(2) for Ref. [20]] and 0.44 (9) [compared to 0.7(4)]. The differences in both magnitude and uncertainty have important consequences for the role of the 23 Al(p, γ) reaction in determining the development of X-ray bursters.
For an evaluation of the astrophysical 23 Al(p, γ) reaction rate, we consider the contribution of excited states in 24 Si at E x = 3449, 3471, 4170 and 4470 keV, corresponding to resonances in the 23 Al + p system at E r = 157, 179, 878 and 1178 keV, respectively (see Table 2; the direct capture component is expected to be negligible for temperatures, T ≥ 0.1 GK, and we do not foresee any significant departure from the value previously reported in Ref. [19] based on the present results). Here, we adopt spectroscopic factors obtained in the present work for the determination of proton partial widths. The spectroscopic factors of mirror analog states are expected to be nearly identical [40,41]. In assessing the validity of this statement, we performed a comparison of proton and neutron spectroscopic factors in the mirror systems: 17 [52][53][54], up to excitation energies of ∼4 − 5 MeV. We found that spectroscopic factors agree to within ∼12%, with a standard deviation of ∼10%. This is well within the known ∼20% uncertainty associated with the extraction of spectroscopic factors from experimentally measured cross sec-   [20]. In this case, the contribution of all resonances have been included, as well as uncertainties associated with the reaction Q-value.
tions. As such, we conclude that spectroscopic factors obtained for excited states in 24 Ne may be adopted for analog levels in 24 Si to a precision consistent with experimental uncertainties. In contrast, γ-ray partial widths were calculated using transition densities from our USDA shell-model calculations, adapted to the actual transition energies between the 24 Si states shown in Table 1. It should be noted that the values of Γ γ are negligible for the determination of ωγ for the 157-and 179-keV resonances, as they are significantly larger than the corresponding proton partial widths, Γ p . However, in the case of the 878-and 1178-keV states, the opposite is true. For the 1178-keV state, the present value of Γ γ is in good agreement with the USD results of Ref. [33], whereas our current estimate for the 878-keV resonance is a factor 2 smaller. In the case of the latter, we note that while there is a discrepancy between the USDA and USD calculations, the contribution of the resonance at 878 keV to the overall 23 Al(p, γ) stellar reaction rate is negligible for temperatures, T = 0.1 − 2

GK.
In agreement with previous studies [17,20], we find that the = 0 resonance at 157 keV makes the most significant contribution to the 23 Al(p, γ) stellar reaction rate for T = 0.1 − 2 GK. However, in contrast to previous work [20], uncertainties in the strength of the 157-keV resonance have been reduced by a factor of ∼4. Consequently, in order to fully assess the astrophysical implications of the current study, we have estimated the uncertainty in the total reaction rate based on the present resonance energies (which have an uncertainty dominated by the reaction Q-value [21]) and the resonance strengths (with uncertainties dominated by the spectroscopic factors, but now much improved). A 1.5σ confidence interval was calculated to properly account for experimental uncertainties in the 23 Al(p, γ) reaction parameters, which we note also accounts for uncertainties in the 22 Mg(α, p) reaction cross section [14]. We note that the authors of Ref. [14] utilised the TALYS code to extend their data into the Gamow energy window for Type-I X-ray bursts and, as such, we presently estimate a ∼60% uncertainty in the 22 Mg(α, p) rate − this does not include uncertainties associated with centre-of-mass energies in Ref. [14]. By using the Saha equation to determine the 22 Mg(p, γ)/ 23 Al(p, γ) equilibrium [55], and comparing the present results with those of Ref. [20], we have been able to investigate the relative competition between the 22 Mg(p, γ) 23 Al(p, γ) 24 Si reaction sequence and the 22 Mg(α, p) 25 Al process path [14] (assuming ignition conditions of Ref. [6] and total accreted mass fractions consistent with the "zM" model of Ref. [7]). In particular, in defining the temperature at which the 22 Mg(α, p) reaction governs 50% of the nucleosynthetic flow in Type-I X-ray bursts as the "tipping" point between the rp-and (α, p) processes, we find that the latter will only become significant at temperatures 0.85 GK, as shown in Fig. 3. Such temperatures are only briefly reached for standard X-ray burst model calculations [6,56] and, in ruling out the previously possible lowertemperature onset of the (α, p) process [14,20], we may now conclude that the pathway through the 22 Mg(α, p) reaction is not relevant for anything but the most energetic bursters.
In summary, we have performed the first measurement of the 23 Ne(d, p) 24 Ne transfer reaction. Several strong single-particle states in 24 Ne have been identified and their associated neutron spectroscopic factors extracted to a precision of ∼20%. Using these spectroscopic factors to deduce the properties of resonant states in the astrophysical 23 Al(p, γ) reaction, we have reduced uncertainties in the strength of the key E r = 157 keV, = 0 level, in comparison with the most recent study of Ref. [20], by a factor of ∼4, considerably constraining the rate over the temperature range of X-ray bursts. In particular, we find that the 23 Al(p, γ) 24 Si reaction is effective in bypassing the 22 Mg waiting point in the rp process (according to standard modelling conditions) for temperatures up to at least 0.85 GK, while the 22 Mg(α, p) pathway might play a more prevalent role above 1 GK, the very peak temperature region only rarely reached in X-ray bursts. Further constraints on the 23 Al(p, γ) reaction would now require a precise determination of the reaction Q-value [21] and, in this regard, we understand that a new measurement of the 24 Si mass was recently performed at the National Superconducting Cyclotron Laboratory, USA [57]. The results for resonance strengths, combined with a precise Q-value determination, are now likely to constrain the uncertainties in the nuclear physics data sufficiently tightly to allow the accurate extraction of neutron star mass-radius ratios from current experimental observations of Type-I X-ray bursts [12].