Direct measurements of low-energy resonance strengths of the 23 Na(p, γ ) 24 Mg reaction for astrophysics

The NeNa and the MgAl cycles play a fundamental role in the nucleosynthesis of asymptotic giant branch stars undergoing hot bottom burning. The 23 Na ( p , γ ) 24 Mg reaction links these two cycles and a precise determination of its rate is required to correctly estimate the contribution of these stars to the chemical evolution of various isotopes of Na, Mg and Al. At temperatures of 50 (cid:2) T (cid:2) 110 MK, narrow resonances at E p = 140 and 251 keV are the main contributors to the reaction rate, in addition to the direct capture that dominates in the lower part of the temperature range. We present new measurements of the strengths of these resonances at the Laboratory for Underground Nuclear Astrophysics (LUNA). We have used two complementary detection approaches: high eﬃciency with a 4 π BGO detector for the 140keV resonance, and high resolution with a HPGe detector for the 251keV resonance. Thanks to the reduced cosmic ray background of LUNA, we were able to determine the resonance strength of the 251keV resonance as ωγ = 482 ( 82 ) μeV and observed new gamma ray transitions for the decay of the corresponding state in 24 Mg at E x = 11931 keV. With the highly eﬃcient BGO detector, we observed a signal for the 140keV resonance for the ﬁrst time in a direct measurement, resulting in a strength of ωγ 140 = 1 . 46 + 0 . 58 − 0 . 53 neV (68% CL). Our measurement reduces the uncertainty of the 23 Na ( p , γ ) 24 Mg reaction rate in the temperature range from 0.05 to 0.1GK to at most + 50% − 35% at 0.07GK. Accordingly, our results imply a signiﬁcant reduction of the uncertainties in the nucleosynthesis calculations. 3

The NeNa and the MgAl cycles play a fundamental role in the nucleosynthesis of asymptotic giant branch stars undergoing hot bottom burning. The 23 Na(p, γ ) 24 Mg reaction links these two cycles and a precise determination of its rate is required to correctly estimate the contribution of these stars to the chemical evolution of various isotopes of Na, Mg and Al. At temperatures of 50 T 110 MK, narrow resonances at E p = 140 and 251 keV are the main contributors to the reaction rate, in addition to the direct capture that dominates in the lower part of the temperature range. We present new measurements of the strengths of these resonances at the Laboratory for Underground Nuclear Astrophysics (LUNA).

Keywords:
Experimental nuclear astrophysics Underground nuclear physics Hydrogen burning Stellar evolution We have used two complementary detection approaches: high efficiency with a 4π BGO detector for the 140 keV resonance, and high resolution with a HPGe detector for the 251 keV resonance. Thanks to the reduced cosmic ray background of LUNA, we were able to determine the resonance strength of the 251 keV resonance as ωγ = 482(82) μeV and observed new gamma ray transitions for the decay of the corresponding state in 24 Mg at E x = 11931 keV. With the highly efficient BGO detector, we observed a signal for the 140 keV resonance for the first time in a direct measurement, resulting in a strength of ωγ 140 = 1.46 +0.58 −0.53 neV (68% CL). Our measurement reduces the uncertainty of the 23 Na(p, γ ) 24 Mg reaction rate in the temperature range from 0.05 to 0.1 GK to at most +50% −35% at 0.07 GK. Accordingly, our results imply a significant reduction of the uncertainties in the nucleosynthesis calculations.

Introduction
Asymptotic Giant Branch (AGB) stars provide a major contribution to the synthesis of the elements in the cosmos and specifically to the chemical evolution of stellar clusters and galaxies. The more massive AGB stars (M 4 M ) undergo proton captures at the base of the convective envelope (hot bottom burning, HBB) [1,2]. When the convective envelope extends down to the H-burning shell, the ashes of the CNO, NeNa and MgAl cycles appear at the stellar surface. The coupling between convection and nuclear burning is responsible for the synthesis of various isotopes of Na, Mg and Al.
Combined, the uncertainties of the 23 Na(p, γ ) 24 Mg and the 22 Ne(p, γ ) 23 Na reaction rates account for the major nuclear contribution to the uncertainty of predicted 23 Na and 24 Mg yields from these massive AGB stars [3]. The 23 Na(p, γ ) 24 Mg reaction, in particular, is the bridge between the NeNa and MgAl cycles. The 22 Ne(p, γ ) 23 Na reaction has been subject of recent experimental studies [4][5][6][7][8]; prior studies of 23 Na(p, γ ) 24 Mg are described below.
During HBB, the temperatures at the bottom of the convective envelopes range between 50 T 110 MK, increasing with the stellar mass and decreasing with the metallicity. The 23 Na(p, γ ) 24 Mg reaction rate in this temperature range is dominated by two resonances at E p = 140 and 251 keV, where E p is the proton beam energy in the laboratory system throughout this paper. At lower energies, the non-resonant capture contribution dominates the reaction rate, whereas another narrow resonance at E p = 309 keV is the principal component of the reaction rate at higher temperatures, up to T = 1 GK. The relative importance of each contribution for the reaction rate at different temperatures is illustrated in Fig. 1. The reaction rate calculations shown in this Table 1 Nuclear resonance parameters for the 23 Na(p, γ ) 24 Mg reaction rate calculation as used in [9] and shown in Fig. 1 (10) 11830.7 (15) E p (keV) [18,19], [17,9] [ 16] [ 16] 140.2(42), 250.9(2) 308.75 (6) 144.0 (16) tot (keV) [16,20]  figure use the current best upper limit for the strength of the 140 keV resonance quoted in the literature [9] and treated as in [11]. Strengths of the 251 keV and 309 keV resonances have been measured by [12][13][14][15]; and the calculations presented in Fig. 1 use the values as compiled in [16]. Information on the direct capture cross section was used as obtained indirectly in [17]. The nuclear properties of the resonances are summarized in Table 1.
The upper limit on the strength of the 140 keV resonance [9] does not exclude this resonance from being the dominant contribution to the reaction rate at T ∼ 80 − 90 MK. In fact, the uncertainty of the 23 Na(p, γ ) 24 Mg reaction rate in the discussed temperature range, associated with the upper limit of the 140 keV resonance strength and the 33% relative uncertainty of the 251 keV resonance strength, provide a strong motivation for an experimental study of these resonances.

Setup
The measurements were performed at the Laboratory for Underground Nuclear Astrophysics (LUNA), located at the Gran Sasso National Laboratory. Proton beams with energies of 130 to 400 keV were provided by the LUNA400 accelerator [22] and used to bombard water-cooled solid targets. Typical beam intensities of 100-250 μA on target were obtained.
Two complementary setups were used to study the cross section: one employed a High-Purity Germanium (HPGe) detector, the other a segmented Bismuth Germanium Oxide (BGO) summing detector. The use of two setups is advisable, as the efficiency of the HPGe detector is too low to study the weak 140 keV resonance. The energy resolution of the HPGe detector, however, significantly facilitates the study the gamma-ray branchings of the stronger resonances at 251 keV and 309 keV.
Both setups benefit from the underground location of LUNA at the Gran Sasso National Laboratory, and target and detectors were surrounded by a massive lead shielding to reduce background from environmental radiation [23][24][25].

BGO detector phase: 140 keV resonance
The segmented BGO summing detector was used to search for the 140 keV resonance with a large detection efficiency, utilizing a digital data acquisition system to acquire independent spectra of the six individual crystals and generating a sum energy spectrum offline [25,26, and references therein]. The response of the BGO detector was estimated through a Monte Carlo simulation using Geant4 [27,26], which was validated with calibration measurements. The efficiency for the detection of the full gamma-ray energy depends on the gamma-ray branching ratios and the lowenergy threshold of the individual crystal segment. In the case of the 309 keV resonance, for example, the detection efficiency above 10 MeV is about 50% [25]; the efficiency of this detector is further discussed in subsection 3.2. In the search for the E p = 140 keV resonance, measurements were taken at beam energies around 130, 140, 145 and 147 keV, in view of the uncertainty of the resonance energy (see Table 1). Due to a narrow resonance in 18 O(p, γ ) 19 F ( Q = 7994 keV) at E p = 151 keV, the background from random coincidence summing (pile-up) of gamma rays from this reaction prohibited the 23 Na(p, γ ) 24 Mg resonance search for beam energies above 151 keV. At energies of 140 keV and below we observed yields compatible with zero within the statistical uncertainty. For the runs at 147 keV (147 keV E p 149 keV) we observe a statistically significant excess of counts over the beam-induced background. This beam energy is compatible with the resonance energy in literature, and we analyzed this data set considering these runs as on-resonance, taking into account the target thickness (described in sec. 2.4).

HPGe detector phase: 251 keV and 309 keV resonances
The HPGe detector was employed in close geometry at an angle of 55 • to study the resonance strengths and gamma-ray branchings of the resonances at 251 keV and 309 keV (cf. [28, and references therein]). The efficiency calibration was based on radioactive sources ( 137 Cs, 60 Co) and gamma rays from narrow resonances at E p = 326 keV in 27 Al(p, γ ) 28 Si and at E p = 278 keV in 14 N(p, γ ) 15 O. An empirical calibration curve was fitted to the total and full energy peak efficiency, using the parametrization in [29]. Summing effects that arise from the coincident detection of multiple gamma rays of the same cascade (enhanced by the close detector geometry) were taken into account, based on the parametrized efficiency curve; angular correlations were neglected for the calculation of the summing effects. The parameters of the efficiency model were determined in a global maximum likelihood fit. We considered an uncertainty of 8% for the efficiency calibration curve, taking into account the scatter of the calibration data points, the repeatability of the (p, γ ) calibration measurements, and the few available sources to calibrate the efficiency at gamma energies above 8 MeV.

Targets
All targets were produced by evaporation of sodium compounds onto a high purity tantalum backing at the Institute for Nuclear Research, Hungarian Academy of Sciences (MTA Atomki). The backing was chemically cleaned and heated in vacuum before the evaporation. We tested several sodium compounds (NaCl, Na 2 SiO 3 , Na 2 WO 4 ), and obtained the best results (i.e., reproducible target properties and stability under beam) with Na 2 WO 4 (as in [9]). In a test evaporation of Na 2 WO 4 onto a carbon backing we observed no major deviations in the stoichiometry of the sodium compound with respect to the nominal composition in a Rutherford backscattering measurement. Materials supplied by different vendors were tested in an effort to reduce beam-induced backgrounds from light contaminants, and the cleanest material was chosen for target production.
Typical target thicknesses for the Na 2 WO 4 targets were 50 μg/cm 2 (beam energy loss of about 14 keV at 150 keV proton energy). Between experimental runs we regularly performed measurements of the target thickness by means of nuclear resonance analysis, scanning over the 309 keV resonance in 23 Na(p, γ ) 24 Mg.
Targets were replaced when the target thickness (width of the plateau in a 309 keV resonance scan) had decreased strongly, or a drop in on-resonance yield was observed. An example for the development of the target scan under bombardment is shown in Fig. 2. Typical accumulated charges on a single target ranged between 10 and 30 C.
Elastic Recoil Detection Analysis (ERDA) was performed on selected targets at the Ion Beam Center at the Helmholtz-Center Dresden-Rossendorf, using a 43 MeV Cl 7+ beam and detecting the recoils at an angle of 31 • (cf. [30] for a description of the setup). Analysis of the data with the program NDF [31] resulted in depth profile information and allowed for the study of the chemical composition of various targets during the initial target development phase. Furthermore, two targets were produced under identical conditions: one was used to measure the 251 keV and 309 keV resonances with the HPGe detector setup at LUNA, the other was not bombarded. Both of these targets were also analyzed with ERDA to study the chemical composition for the analysis of the HPGe measurements. Depth profiles were obtained and the stoichiometry of Na, W, O and H in the main target layer resulted in effective stopping powers that were on the order of 10% larger compared to the nominal composition of pure Na 2 WO 4 . Traces of carbon were detected in the bulk of the samples, but the amount was irrelevant in the context of our measurement.

HPGe detector phase: 251 keV and 309 keV resonances
With the effective stopping power of the material at the resonance energy, ε eff , the absolute strength ωγ of the narrow resonances can be determined through the well-known relation Y = λ 2 2 ωγ ε eff [32], where Y is the reaction yield (number of reactions per incident projectile), and λ the de Broglie wavelength of the projectile. Stopping power tables were generated using SRIM 2013.00 [33].
Branching ratios and resonance strengths were determined solving the system of equations: the vector elements N i are the number of counts in the primary gamma-ray peaks, and the matrix elements B ij correspond to the probability that a decay through the primary branch i is registered in the primary peak j (including detection efficiency and summing effects). The vector elements P i are the primary branching ratios that fulfill the normalization condition P i = 1 and Q and q p are the accumulated charge on target and the charge of a single projectile (proton), respectively. When calculating the summing effects, gamma-ray branching ratios for secondary transitions were taken from literature [18]. The uncertainties of Y and the P i were determined by a Monte Carlo approach, repeating the analysis after varying the N i within the respective experimental uncertainties and the secondary branching ratios within their uncertainties given in literature. The uncertainty of ε eff , derived from the uncertainty of the chemical composition of the target as determined with ERDA, typically amounted to 15% and was the dominant contribution to the resonance strength uncertainty. A systematic uncertainty of 3% was assumed for the measurement of the accumulated charge on target.
Among the 23 Na(p, γ ) 24 Mg resonances in Table 1, the strength of that at E p = 309 keV is the best known in literature [16]. To verify our results, we redetermined the resonance strength and gamma-ray branching ratios as described above. The branching ratios of the 12 primary transitions in literature [18] were found to be in agreement with our measurement within the experimental uncertainties. The height of the plateau of two resonance scans on target and the composition of the not bombarded target as determined by ERDA resulted in a resonance strength of ωγ 309 = 108 (19) meV, in good agreement with the literature value of 105 (19) meV [16].
The strength of the resonance at E p = 251 keV is known with a larger uncertainty of about 33% (Table 1). Additionally, the gammaray branchings of the corresponding level in 24 Mg are not given in current compilations (such as [19]), and the only publication of these branching ratios known to the authors indicates only two strong transitions from this level [34]. A scan of the 251 keV resonance, followed by a long run ( Q ≈ 11 C) at an energy on the resonance, was used to determine the strength of this resonance and the gamma-ray branching ratios of the corresponding level in 24 Mg. The most intense transitions are marked in the spectrum in Fig. 3 and the resulting branching ratios are given in Table 2. In addition to the two major transitions described in [34], we identified multiple minor transitions and determined their branching ratios. The intensities of several minor peaks in the spectrum, most prominently the transition to the ground state of 24 Mg, were consistent with the expectation from true coincidence summing (summing-in) alone and are thus not included in Table 2.
With the yield of the long run on the resonance plateau and using the same target stoichiometry as in the analysis of the 309 keV resonance, we determined a resonance strength of ωγ 251 = 482(82) μeV, in agreement with the literature value but with an improved relative uncertainty of 17% (previously 33% [16]) that is now comparable to that of the 309 keV resonance.

BGO detector phase: 140 keV resonance
During the search for the 140 keV resonance with the BGO detector, the 309 keV resonance was used as a reference to account  . Whilst the effective stopping power can vary considerably with changes in stoichiometry, the ratio of effective stopping powers at the two resonance energies is much more robust (with < 1% variation in this case).
We found two beam-induced backgrounds that were critical for the observation of the 140 keV resonance in our measurements: 7 Li(p,γ ) 8 Be with Q = 17.25 MeV, and 11 B(p,γ ) 12 C with Q = 15.96 MeV. With the Q -values of both background reactions larger than that of 23 Na(p, γ ) 24 Mg ( Q = 11.69 MeV), these reactions can contribute to the 23 Na(p, γ ) 24 Mg region of interest (ROI) in the sum energy spectrum. The contribution of these backgrounds has been modeled as shown in Fig. 4. To illustrate the potential influence of these impurities, we compare the expected yields: for a proton beam with an energy of 147 keV impinging on a 60 μg/cm 2 Na 2 WO 4 target, the yield of a narrow 23 Na(p, γ ) 24 Mg resonance with an ωγ at the current upper limit [9] is 15 reactions/C. The same reaction yield is obtained for 11 B(p, γ ) 12 C by a contamination of about 2500 ppm of boron, or for 7 Li(p,γ ) 8 Be by about 700 ppm of lithium.
Before and after each run in search of the 140 keV resonance, we scanned the 309 keV resonance to obtain a target profile and used the known gamma rays of the 23 Na(p, γ ) 24 Mg reaction to constrain the energy calibration of the individual detector segments. Slightly non-linear energy responses from the detector segments (especially at the higher gamma ray energies produced by the beam-induced backgrounds) were allowed by introducing a quadratic term in the energy calibration function. Using the energy calibrations of the individual crystals, we created the sum spectrum.
Using the sum spectrum at energies above the total gammaray energy expected for the 23 Na(p, γ ) 24 Mg reaction, we obtained to the 23 Na(p, γ ) 24 Mg ROI. A Bayesian approach using Poissonlikelihood for the bin contents in this energy region (1 MeV wide bins between 10 MeV and 18 MeV) and uniform prior distributions for the yields of the background reactions results in a posterior probability distribution for the resonance strength, based on the excess of the number of counts in the ROI for the 23 Na(p, γ ) 24 Mg reaction over the contribution from the two beam-induced backgrounds (cf. inset of Fig. 4). The Bayesian Analysis Toolkit (BAT) [35] was used to calculate the posterior probability density function.
The relation between an excess over background in the 23 Na(p, γ ) 24 Mg reaction ROI and the 23 Na(p, γ ) 24 Mg reaction resonance strength requires knowledge of the detection efficiency, i. e., the probability of an 23 Na(p, γ ) 24 Mg event to cause a detected energy deposition in the corresponding ROI. This efficiency depends on the gamma-ray branchings, and spin and parity of the resonant level, which are unknown. We considered levels in 24 Mg with known branching ratios and E x above 9 MeV, and simulated the corresponding detection efficiencies in the ROI around the respective E x . This procedure corresponds to the approach in [9], except that all levels above 9 MeV were considered in this work, covering a wider range of possible branching ratios than the states selected in [9] and resulting in a larger detection efficiency uncertainty by comparison. Random sampling from the efficiencies corresponding to the branching ratios of different states was applied to include the effect of this uncertainty on the results for the resonance strength. Different runs were performed within each chosen beam energy group. The posterior distributions for the resonance strength of the individual runs were combined to obtain a final posterior distribution. The posterior distribution obtained in the analysis for runs with 147 keV E p 149 keV is shown in Fig. 5. The posterior distribution has a mode at 1.46 neV, and the shortest interval to contain 68 % probability ranges from 0.94 and 2.05 neV (95 % interval from 0.51 to 2.74 neV).

Reaction rate and astrophysical implications
To summarize our results, the obtained strengths for the 309, 251 and 140 keV resonances are shown in Table 3. The value  Table 3 Resonance strengths for 23 Na with an excess over zero of more than 2 σ for the first time.
The new experimentally determined value for the strength of the 140 keV resonance is compatible with the result of [9] when interpreted as a positive detection, but we established it as a value different from zero with a much larger statistical significance and a relative uncertainty of 38 %.
With these results we calculated the 23 Na(p, γ ) 24 Mg reaction rate and determined its uncertainty by a Monte Carlo approach.
Resonance strengths for ωγ 309 and ωγ 251 were sampled from Gaussian distributions, ωγ 140 was sampled from the posterior distribution obtained in this measurement as discussed above. The direct capture strength and resonance energies were sampled as described in [11,10]. The resulting reaction rate, compared to the rates in [9], is shown in Fig. 6. The different treatment of the uncertainty of ωγ 140 in this work and the upper limit in [9] should be noted for this comparison. The relative uncertainty of the reaction rate is decreased in the temperature window T = 0.05-0.1 GK, thanks to the improved knowledge on the 140 keV resonance and the 251 keV resonance, with uncertainties between +50% −35% at 0.07 GK and +17% −14% at 0.1 GK. At T = 0.06-0.09 GK the reaction rate based on the results of this work is slightly higher than previously determined, because of the larger median value of ωγ 140 . Below T = 0.05 GK and above T = 0.2 GK, the reaction rate is dominated by the direct capture contribution and the 309 keV resonance (plus higher resonances), respectively, and is therefore unaffected by this work.
The main limiting factors for the determination of ωγ 140 were the beam-induced backgrounds from traces of 7 Li and 11 B, the un- Fig. 6. Reaction rate and uncertainty (68% CL) based on the results presented in this work, compared to the reaction rate in [9]. Refer to the text for details on the different treatment of the uncertainty of the 140 keV resonance. Fig. 7. Network calculations to follow 20 Ne, 23 Na and 24 Mg mass fractions, using the reaction rates of [9] (top) and this work (bottom) at T = 80 MK. See text for details.
certainty of the resonance energy, and the missing branching ratios for the resonant level.
To understand the potential impact of our results, we performed a network calculation for a constant T = 80 MK and ρ = 10 g/cm 3 . The outcome of this calculation is shown in Fig. 7. The network includes all the isotopes of the NeNa cycle plus 24 Mg.
The H and He mass fractions are kept constant, at X(H) = 0.5 and X( 4 He) = 0.499, to simulate continuous replenishment via convec-tion. The initial mass fractions of Ne and Na isotopes are scaled to solar abundances, with X(Ne) + X(Na) = 0.001, and initially X( 24 Mg) = 0. The calculations cover 3 × 10 3 yr, the typical time elapsed between two subsequent thermal pulses (instabilities of the He-burning shell) in an AGB star with initial mass M = 7 M .
Except for 22 Ne(p, γ ) 23 Na, reaction rates were taken from Star-Lib (version 6) [36], which includes the 23 Na(p, γ ) 24 Mg rates of [9]. For 22 Ne(p, γ ) 23 Na we used the most recent reaction rate from [8] (which is similar to the rate of [7]). The results of these calculations are shown in the upper panel of Fig. 7. The lower panel shows the results of the same calculation when using the 23 Na(p, γ ) 24 Mg rate based on this work.
The Ne and Na isotopes almost instantaneously reach their equilibrium abundances, and their abundance ratios, e.g. the 23 Na/ 20 Ne ratio, remain constant as they only depend on the temperature and the choice of the reaction rates. The continuous depletion of the total abundance of the Ne and Na isotopes is driven by the leakage of the 23 Na(p, γ ) 24 Mg reaction. As a consequence the 24 Mg piles up. The shaded areas in Fig. 7 represent the influence of the 23 Na(p, γ ) 24 Mg reaction rate on the three most abundant Ne, Na, and Mg isotopes during this nucleosynthesis episode. The reduction of the uncertainties when passing from the previous [9] and the new rate is evident. This result will improve future calculations of the yields of massive AGB stars, providing, in particular, a more accurate understanding of the evolution of the Na and Mg isotopes in galaxies and stellar clusters.