First direct measurement of 12 C ( 12 C , n ) 23 Mg at stellar energies

Neutrons produced by the carbon fusion reaction 12C(12C,n)23Mg play an important role in stellar nucleosynthesis. Past studies have shown large discrepancies between experimental data and theory, leading to an uncertain cross section extrapolation at astrophysical energies. We present the first direct measurement which extends deep into the astrophysical energy range along with a new and improved extrapolation technique based on experimental data from the mirror reaction 12C(12C,p)23Na. The new reaction rate has been determined with a well-defined uncertainty which exceeds the precision required by astrophysics models. Using our constrained rate, we find that 12C(12C,n)23Mg is crucial to the production of Na and Al in Pop-III Pair Instability Supernovae.


Introduction
The first stars in the early Universe formed about 400 million years after the big bang.Verification of the existence of these stars is important for our understanding of the evolution of the Universe [1].It has been predicted that for Population-III stellar production yields, the abundances of odd-Z elements are remarkably deficient compared to their adjacent even-Z elements [2].Astronomers are searching for long-lived, low mass stars with the unique nucleosynthetic pattern matching the predicted yields [3].The relevance of 12 C( 12 C,n) 23 Mg in the first stars has been discussed by Woosley, Heger, and Weaver [4].By the end of helium burning in Pop-III stars, the neutron to proton ratio in the ash is almost exactly 1.In the subsequent carbon burning phase,however, frequent β + decay of produced 23 Mg converts protons into neutrons, thus increasing the neutron to proton ratio.A slight excess of neutrons would significantly affect the abundances of the odd-Z isotopes with neutron to proton ratios higher than 1, e.g. 23Na and 27 Al.
12 C( 12 C,n) 23 Mg is also a potentially important neutron source for the weak s-process occurring in Pop-I and II stars.Pignatari et al. [5] recently performed a study of the weak s-process during carbon shell burning for a 25 M stellar model using different 12 C( 12 C,n) 23 Mg rates.They found that a factor of 2 precision or better would be desirable to limit its impact on the s-process predictions to within 10%.
Stellar carbon burning has three main reaction channels:  12 C fusion reaction, their calculation could only be renormalized to the average trend of the data whereas the resonant behavior of the 12 C+ 12 C fusion reaction was ignored.The maximum deviation between the experimental result and the renormalized statistical model prediction is more than a factor of 4. Nevertheless, based on the statistical model extrapolation, this work recommended a neutron branching ratio of β n = 0.011%, 0.11%, 0.40%, and 5.4% at T 9 = 0.8, 1.0, 1.2, and 5, respectively, though no attempt was made to quantify the uncertainties in these predictions [8].

Mg at energies above E cm =3 MeV
The experimental work was performed at the University of Notre Dame's Nuclear Science Laboratory using the 11 MV FN tandem Pelletron accelerator.Carbon beams were produced at energies ranging from 5.1 to 8.7 MeV (lab frame) with typical currents on target between 0.5 and 1.5 pμA.A 1-mm  23 Mg S*-factor results from the present measurement (black squares) compared with previous data sets from Dayras 1977[8] (purple circles) and Bucher 2013 [14,16] (magenta triangles).Also shown is the neutron branching ratio calculated by Dayras applied to the total 12 C+ 12 C fusion S*-factor recommended by CF88 [9] (red solid line) and the new extrapolation from this work (blue circles).Only statistical errors are shown for the experimental data, while the extrapolation includes both statistical and a 40% systematic error.Lower: The Gamow yield for T 9 =1.1 is plotted.thick hydrogen-free Highly Ordered Pyrolytic Graphite (HOPG) target made from natural carbon was used to control the hydrogen-induced background [13].The target was cooled by circulating deionized water through the supporting flange, which was centered in a block of polyethylene containing 20 3 He proportional counters arranged around the beam axis in two concentric rings [14,15].
The main sources of beam-induced neutron background were from the reaction 13 C( 12 C,n) 24 Mg [14].With a large positive Q-value (8.99 MeV) and the relatively high natural abundance of 13 C in the target (1.1%),neutrons from 13 C( 12 C,n) 24 Mg dominate the total yield at very low beam energies near the 12 C( 12 C,n) 23 Mg reaction threshold.To estimate its contribution, the 13 C( 12 C,n) 24 Mg reaction was studied with the same setup using a 13 C beam with energies ranging between 9.5 and 5.4 MeV.Since the cross section for this reaction is much higher, relatively low beam intensities ( 50 pnA) with shorter run times were sufficient.The normalized 12 C( 13 C,n) 24 Mg yield was then subtracted from the measured total neutron yield recorded with the 12 C beam [16].
The room background rate was measured to be 9.015(92) evts/min, which dominated the yield at energies below E cm = 3.0 MeV.The background contribution from 2 H( 12 C, 13 N)n was studied using a thin TiD 2 target with thick Cu backing.After removing the room background, this contribution was found to be less than 5% of the total yield at E cm = 3.3 MeV increasing to 19% at 3.1 MeV.
The cross section for the 12 C( 12 C,n) 23 Mg reaction was determined by differentiating the thick target yield [17].In Fig. 1, it has been converted to a modified S-factor (S*) [6,17] for comparison with previous results.It is seen that the new results display good agreement with previous measurements in the overlapping energy region while extending much deeper into the astrophysical energy range.The 15% systematic uncertainty primarily results from the uncertainties in the beam current (10%), detector efficiency (6%) [15], angular distribution (5%), and stopping power (7%) [16,18].The Dayras results also have an additional systematic uncertainty of 16% [8] not shown in Fig. 1.

Na
An extrapolation is required to estimate the reaction cross section at the lower energies beyond experimental reach.As mentioned earlier, Dayras et al. provided a renormalized statistical model calculation for this purpose.However, the large discrepancy between the experimental data and their theory calls into question the reliability of the extrapolation.To provide a better prediction including the effect of the molecular resonances in the entrance channel, a novel extrapolation method has been developed based on experimental information from the mirror reaction 12 C( 12 C,p) 23 Na.The predicted neutron cross section, σ n(pred) , is obtained using the formula where N is the highest available decay channel in the residual 23 Mg, which depends on the reaction energy.For E cm ≤ 4.6 MeV, only the n 0 and n 1 channels are open.The theoretical ratio, σ p i (th) , is calculated using TALYS [19] combined with entrance channel spin populations supplied from a coupledchannels calculation by Esbensen [20].The resonances in 12 C( 12 C,n i ) 23 Mg and 12 C( 12 C,p i ) 23 Na originate from both the molecular resonances in the entrance channel and the characteristic resonances in the final decay channels.The traditional statistical model calculation employed by Dayras uses the optical model and assumes a high level density to describe the entrance and exit channels and therefore could only reproduce the average trend of the experimental data.In our approach, the complicated molecular resonance associated with the entrance channel is embodied in the experimental cross sections (σ p i (exp) ) of 12 C( 12 C,p i ) 23 Na, the mirror system of 12 C( 12 C,n i ) 23 Mg, whereas the statistical model is only used to predict the decay width ratio between the n i and p i channels.Since the proton energy resolution in the Zickefoose experiment from Ref. [21] was insufficient to resolve p 0 from p 1 , only the sum, σ p 0 + σ p 1 , is available for E cm < 4 MeV.Eq. 1 has been modified to accommodate the combination of p 0 and p 1 .Additionally, the measurements of 12 C( 12 C,p i ) 23 Na performed by Fang et al. in the energy range 3 < E cm < 6 MeV [22] have also been used to predict the 12 C( 12 C,n) 23 Mg cross section [14].In this case, up to N=6 possible decay channels are required for the prediction calculated in Eq. 1.
Figure 2 shows the ratios between our measured 12 C( 12 C,n) 23 Mg cross section σ n(exp) and the two σ n(pred) based on the Zickefoose and Fang proton data sets plotted as a function of E cm .The average ratios (standard deviations) for the Zickefoose and Fang predictions are 0.9(4) and 0.9(3), respectively.The ratios to the Dayras calculation are also shown for comparison.The large deviation at E cm 4.8 MeV has been eliminated by our approach.The fluctuations, which are larger than the quoted statistical uncertainties, reflect the systematic errors associated with our extrapolation.They consist of the systematic errors in the proton measurements, the assumed entrance channel spin populations, and the TALYS calculation used in the prediction of σ p i (th) .To provide better consistency with the experimental 12 C( 12 C,n) 23 Mg data, our extrapolation has been renormalized by the factor 0.9.We have adopted 0.4 as the systematic error in accordance with the Zickefoose-based prediction since that data set was used for the extrapolation, being the only one to reach sufficiently low energies.

Na and its astrophysical impact
The new cross section defined by our extrapolation and experimental data has been used to calculate the 12 C( 12 C,n) 23 Mg reaction rate.To highlight the important stellar energy range for a typical carbon shell burning temperature of T 9 = 1.1, the Gamow yield is computed and shown in Fig. 1.Our

Figure 1 .
Figure 1.(Color online) Upper: The 12 C( 12 C,n)23 Mg S*-factor results from the present measurement (black squares) compared with previous data sets fromDayras 1977[8]  (purple circles) and Bucher 2013[14,16] (magenta triangles).Also shown is the neutron branching ratio calculated by Dayras applied to the total 12 C+ 12 C fusion S*-factor recommended by CF88[9] (red solid line) and the new extrapolation from this work (blue circles).Only statistical errors are shown for the experimental data, while the extrapolation includes both statistical and a 40% systematic error.Lower: The Gamow yield for T 9 =1.1 is plotted.