Direct measurements of the 12 C+ 12 C reactions cross-sections towards astrophysical energies

Carbon fusion reactions 12 C( 12 C,p) 23 Na and 12 C( 12 C, α ) 20 Ne play a key role in the evolution of massive stars and in explosive scenarios such as type-Ia supernovae and super-bursts in binary stars. A direct determination of their cross sections is extremely challenging and discrepancies exist between different data sets in the literature. Here we report the results of a direct measurement performed at the CIRCE Tandem Accelerator Laboratory in Caserta (Italy), using Δ E − E detectors for unambiguous charge identiﬁcation. Cross sections were measured in the energy range E c . m . = 2 . 51 − 4 . 36 MeV with energy steps between 10 and 25 keV in the centre of mass. To our knowledge these represent the ﬁnest energy steps to date. Results are presented in the form of partial and summed astrophysical ˜ S -factors for individual proton- and α -particle channels. Branching ratios of individual proton- and α -particle groups were found to vary signiﬁcantly with energy. Angular distributions, albeit limited to three angles, were also found to be non-isotropic, which could be a potential explanation for the discrepancies observed among different data sets. Further efforts are ongo-ing to extend measurements to lower energies


Introduction
Carbon fusion reactions are among the most important in nuclear astrophysics because of their far-reaching impact on stellar evolution and nucleosynthesis. Recent stellar model calculations [1][2][3] indicate that the existence of low-energy resonances in the carbon fusion reactions, such as the one predicted in Cooper et al. [4], would have a profound impact on the structure of massive stars. For example, an enhanced carbon-burning rate by a factor of 10 would affect the convection zone structure of a 25 M star and thus the nucleosynthesis processes occurring in its interior [5]. This in turn would alter the final properties of the star just before its supernova explosion [2,6], leading to smaller supernovae remnants and new values for the limiting initial stellar mass between a neutron star and a black hole [1].
An accurate knowledge of the 12 C+ 12 C fusion reaction cross sections is also essential to model X-ray bursts in binary systems [4,7], to understand explosions on the surface of neutron stars [8], and to predict the fate of white-dwarf mergers [9,10]. As carbon burning controls the ratio between thermonuclear-and core-collapse supernovae, the luminosity function of certain stars, and the ratio between different types of white dwarfs [1,2,6], it also affects the chemical enrichment of the interstellar medium arising from different types of supernova explosions.
Carbon burning takes place at typical densities of 2−5 × 10 6 g/cm 3 [11,12] and temperatures of 5 × 10 8 K, corresponding to an energy range of interest, the Gamow window,

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E c.m. = 1.5 ± 0.3 MeV [13]. At these energies, the 12 C+ 12 C reactions proceed through the 23 Na+p (Q = 2.24 MeV) and 20 Ne+α channels (Q = 4.62 MeV), while the 23 Mg+n (Q = −2.62 MeV) channel is closed. 1 At higher energies the neutron channel opens but its rate is about two orders of magnitude lower than that of the proton and alpha channels [15], thus reducing the relevant study of the 12 C+ 12 C reaction to the α and p channels alone. Since the Gamow window for the 12 C+ 12 C reactions is much lower than the height of the Coulomb barrier (6.3 MeV), carbon-fusion cross sections are very small ( 10 −9 b) and extremely difficult to measure in the laboratory [16,17]. The extrapolation of high energy data to the relevant astrophysical energy is further made uncertain by the presence of low energy resonances of still debated origin [18][19][20][21][22]. In addition, direct measurements are hindered by the intense beam induced background on 1,2 H impurities in the target [23].
Recently, low energy cross sections for the 12 C+ 12 C reactions [38] obtained using an indirect approach based on the Trojan Horse Method, sparked an intense debate [39,40], and further direct experimental investigations are therefore required.
Here we report the results of a direct thick-target measurement in the energy range E c.m. = 2.51−4.36 MeV and under controlled target-temperature conditions to minimise deuterium contaminants as a first step towards measurements in the region of astrophysical energies.
The paper is organised as follows: after a brief description of the experimental setup and procedures (Sects. 2 and 3), we discuss the data analysis to arrive at partial and total cross sections and S factors for individual proton-and alpha-channels (Sects. 3 and 4). Discussions and conclusions are presented in Sects. 5 and 6.

Experimental setup
The experiment was performed at the 3 MV Pelletron Tandem Accelerator Laboratory of the Centre for Isotopic Research on Cultural and Environmental heritage (CIRCE) at the Department of Mathematics and Physics of the University of Campania "Luigi Vanvitelli" in Caserta, Italy.
The accelerator is coupled to a caesium sputter ion-source loaded with 2 mm inner diameter Al cathodes filled with commercial graphite pellets and capable of delivering 12 Cbeam currents of up to 20 pµA on target. The accelerator beam-transport system comprises a series of magnetic and electrostatic filters to deliver beams of sufficient mass (ΔM/M = 1.38×10 −3 ) and energy (ΔE/E = 1.43×10 −3 ) resolution [41]; being an AMS system specialised on 12 C beams, contamination is negligible.
Full details on the experimental setup used for the 12 C+ 12 C experiment are reported in [42] and briefly described below. A large scattering chamber was used to house the carbon targets and the detection system and was enclosed in a sealed Plexiglas structure continuously flushed with dry nitrogen at a pressure slightly higher than atmospheric. The enclosure served to minimise contamination of low-mass species (mostly hydrogen and deuterium) inside the scattering chamber through inevitable air leaks. The rest-gas composition inside the chamber was monitored with a Pfeiffer quadrupole mass spectrometer (QMS).
A Highly Ordered Pyrolytic Graphite (HOPG) target (1 mm thick, 10 mm diameter, 99.99% purity) was mounted on a water-cooled target ladder and surrounded by a plastic 3D-printed hollow sphere with several opening ports (25 × 25 mm 2 each) to allow for the passage of the beam and of the particles to be detected (as shown in Fig. 1). The sphere was coated with conductive paint and kept at −300 V for electron suppression, thus allowing for a correct beamcurrent reading directly on target. Typical beam currents on target were of few tens of µA resulting in target temperatures of up to ∼ 1200 • C, depending on beam intensity and cooling water flow [42]. The target temperature was constantly monitored by means of a thermo-camera mounted outside the scattering chamber and facing the target through a germanium window sensitive to the operational wavelengths of the thermo-camera (7.5-13 µm).
The pressure in the scattering chamber was maintained below 5 × 10 −7 mbar by means of an oil-free turbo-and scroll-pump system, in combination with a cold finger kept at LN 2 temperature and protruding from the beam line to 5 cm from the target.
The detection system consisted of a two-stage detector array GASTLY (GAs Silicon Two-Layer sYstem) specifically designed to fulfil the requirements of charged particle identification from the 12 C+ 12 C reactions. The array can accommodate up to eight ΔE − E modules, each comprising an ionisation chamber (IC, ΔE stage) and a large area silicon strip detector (SSD, E rest stage). Further details on the full detector array and its commissioning are reported in [43]. Briefly, each module consists of an aluminium truncated Fig. 1 Sketch of the GASTLY detectors arrangement.The four GASTLY detectors are shown along with the target holder and the sphere surrounding it for electron suppression purposes. Detector D143 is shown in wireframe to reveal its internal components (for further details see [43]) pyramidal structure with square base, hosting the IC and the SSD detectors. The IC is equipped with an entrance Havar window (23 × 23 mm 2 large and 2.6 ± 0.2 µm thick) acting as a cathode; an aluminised Mylar foil (52.5 × 52.5 mm 2 , 1.6 ± 0.2 µm thick, metallised with 50 µm/cm 2 Al) acting as an anode and placed 116 mm behind the Havar window; and a Frisch grid, consisting of a mesh of gold-coated tungsten wires (20 µm diameter). The IC was operated with CF 4 maintained at a constant pressure (within ± 0.05 mbar) by means of an automatic system [43]. The second stage of each GASTLY module consists of a large (58.0×58.0 mm 2 active area, 300 ± 15 µm thick) silicon strip detector [44]. For the present study, however, the silicon detector was used as a single pad. In order to minimise electronic and environmental electromagnetic noise that would affect the detection of low energy particles, the GASTLY readout electronics were mounted on printed circuit boards arranged in a stack behind the silicon detector and inside the aluminium housing. The trigger of the DAQ system was generated with a logic module as the OR of the signals from the ICs and SSDs [45].

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At the time of the 12 C+ 12 C measurement, only four ΔE − E modules were available: three were mounted on a vertical plane at 121 • and 156 • (above and below the beam axis), and one was mounted on the horizontal plane at an angle of 143 • to the beam axis (see Fig. 1). Detectors at 121 • and 143 • were mounted closest to the target and covered an angular range of 19 • , subtending a solid angle of 80 msr. Detectors at 156 • had to be placed farther away from the target to accommodate for the cold finger. They covered an angular range of 13 • each, with a solid angle of 40 msr [45]. In the following, detectors are referred to as D121, D143 and D156 (after checking for their consistency, data from both detectors at this angle were merged into one data group).

Data taking and data analysis
Data were taken at beam energy intervals of 20-50 keV (in the lab. system). The target temperature was maintained above 400 • C (using intense beams) to reduce deuterium contamination 2 on target by up to 90% its original value, as found in our previous study [42].
For optimal detection of α particles and protons, the ICs and SSDs detectors were calibrated using, respectively, a certified mixed 239 Pu-241 Am α source and proton beams at various energies in the range E lab = 1−5 MeV (see [45]). At high energies, the ICs were operated with CF 4 at a pressure of 35 mbar, sufficient to separate α particles and protons. At low energies, higher pressures (50 and 70 mbar) were also used to better separate protons from electronic noise in the acquired spectra. Figure 2 shows a typical calibrated ΔE − E rest matrix for detector D121 at a pressure of 35 mbar, obtained with a 12 C +3 beam at E lab = 8.72 MeV on the HOPG target. The two loci correspond to protons and α particles from the 12 C( 12 C,p) 23 Na and 12 C( 12 C,α) 20 Ne reactions. Events at the far left of the proton locus were neglected from the analysis since they are mainly spurious coincidences between alpha particles completely stopped in the ionization chamber and low energy protons arriving to the silicon detector. The ROI (green polygon) represents the proton locus whilst all events with ΔE > 1 MeV were treated as α particles.
For the analysis of each particle channel, the events of interest were selected excluding events due to electronic noise (mostly located in the bottom left corner of the ΔE − E rest matrix). Depending on energy, some protons and αparticles were stopped in the IC and were therefore recorded with zero E rest . Conversely, some protons were so energetic that their signal falls bellow the IC's threshold and deposited all their energy in the SSD. These events do not appear in the matrix but were all included in the analysis.  Background runs of several days 3 were taken in the same experimental conditions as the 12 C+ 12 C measurements and subtracted (after time normalisation) from the corresponding proton and α-particle spectra at each beam energy. Figures 3  and 4 show typical raw spectra for the proton and α channel, respectively and the background spectra used for analysis.

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In addition to the 1,2 H contamination already addressed in [42], beam-induced background on 13 C contaminant was also investigated, and its contribution found to be negligible ( 1%) at all energies studied here [45].

Fitting procedure
After background subtraction, proton and α-particle peaks from the 12 C+ 12 C reactions were identified through kinematic reconstruction (i.e., including all relevant energy losses) and via comparison with simulations [46]. In the following, we describe separately the procedure used to analyse data for each channel. 3 10 days at 35 mbar (used for the highest energies), 15 days at 50 mbar (used for medium-low energies) and 19 days at 70 mbar (used for the lowest energies).

Proton channel
Given that all analysed protons at the energies used in this work arrive to the SSD, only its spectra were used in the proton analysis. Sample E rest proton spectra, taken at four beam energies with D156 and at a CF 4 pressure of 35-70 mbar in the IC, are shown in Fig. 5.
Distinct proton groups are labelled according to the excitation state of the recoiling nucleus. 4 Despite choosing experimental conditions that minimised the level of deuterium contaminants in the targets (see [42]), some deuterium-induced peaks are still visible in the proton spectra, specifically the peak around 1 MeV corresponding to the 2 H( 12 C,p) 13 C reaction. Note that, because of inverse kinematics, its position remained essentially unchanged with decreasing beam energy, while the 12 C+ 12 C proton peaks moved to lower energies as expected. In most cases, it was possible to disentangle this beam-induced contribution from the peaks of interest, but where this was not the case, the affected proton peaks were discarded from further analysis.
Another peak, probably due to an electronic effect, was also observed on the lower-energy shoulder of p 1 , however its contribution cancels out when we calculate the differential yield for the cross section (as explained in Sect. 3.2).
As many proton peaks overlap (Fig. 5), the number of events within each was extracted using the maximum likelihood method from a combined fit of skewed Gaussian functions f (x), each of the form: where A is the amplitude of the distribution, μ and σ are its centroid and standard deviation, τ corresponds to the shape

Fig. 5
Fitted sample spectra taken with D156 at four different beam energies and with 35-70 mbar of gas in the IC. Note how the various proton peaks from the 12 C( 12 C,p) 23 Na reaction shift towards lower energies at lower beam energy (as expected) thus progressively merging with the proton peak from the d( 12 C,p) 13 C direct reaction parameter and Erfc(x) is the complementary error function. Specifically, the sum of three skewed Gaussians was used to fit p 0 , p 1 and the satellite peak (where visible) at the left of p 1 , while the sum of four skewed Gaussians was used to fit p 2 , p 3 , p 4+5 (since p 4 and p 5 cannot be separated) and p 6 . Note that the positions of the different peaks in the fit were constrained to the energy range expected on the basis of kinematic calculations taking into account energy losses in the ΔE stage and the detector passive layers using SRIM [47]. The amplitude A of each fitted individual distribution (Eq. 1) represents the net number of events N (background subtracted) associated to a given transition, allowing for the extraction of its corresponding yield [48]. The statistical uncertainty, δ N , was calculated as a sum in quadrature of the statistical error in background counts, N bkg , under the relevant proton peak and the error, δ A, on the parameter A.
For runs taken at beam energies E lab < 5.5 MeV, low statistics prevented a direct identification of the proton peaks. Thus, we relied on regions of interest (ROI) identified by kinematic reconstruction at each beam energy, taking into account all energy losses. We then integrated the counts in each ROI to arrive at the net number of events N ROI . Uncertainties δ N were calculated by replacing δA with the statistical error √ N ROI . In addition to statistical uncertainties, the following sources of systematic errors were considered: accumulated charge (< 1%, [49]), solid angles (0.9−1.2% [45]) and stopping power (about 8%, [50]). The covariance matrices indicated that the correlation of the estimates of the parameters is negligible.
In order to verify the validity of our peak identification procedure, two different fits were performed in various spectra in the energy regions where statistics were critically low (p 2 -p 6 ). A first fit was perform using four skewed Gaussian functions (as discussed previously). The second fit was done over the same energy region but using only three skewed Gaussian for p 3 to p 6 . The difference in χ 2 (reduced χ 2 values were between 2-6 for the first fit and over 400 for the latter) between both fits was of nearly 3σ demonstrating that the lowest statistics peak (typically, p 2 ) is indeed real.

Alpha channel
At the beam energies used in this work, only α 0 and α 1 had enough energy to completely pass through the Havar entrance window of the IC. Low energy α particles (typically E α < 7.10 MeV, depending on angle) were stopped in the gas of the ionisation chamber and did not produce any signal in the silicon detectors. Unlike for the proton channel, data analysis for the α channel was therefore performed on reconstructed total energy spectra, (E tot = ΔE + E rest ). A sample of a background-subtracted spectrum is shown in Fig. 6. Fig. 6 Typical α (E tot =ΔE + E rest ) fitted spectrum using a sum of two skewed Gaussian functions and one exponential function. The area defined by the dashed data points (E < 2.1 MeV) represents the α particles stopped in the IC. This spectrum was taken with D156 and a E lab = 8.37 MeV 12 C beam. The α peaks coming from the 12 C+ 12 C reaction are labelled as α 0 and α 1

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To extract the number of events under each α peak, we followed the same approach used for the proton groups. Here, we fit the background-subtracted spectra with the sum of two skewed Gaussian functions (for the α peaks) and a third function (for the low energy part of the spectra) as shown by the solid curve in Fig. 6. In some cases, pile up appeared close to the α 0 peak, thus another skewed Gaussian was used to fit and subtract the events under this peak from the region of interest. Also in this case, the covariance matrices indicate a negligible correlation of the estimates of the parameters. The uncertainty on the number of events in each α peak was calculated as for the proton channel. Systematic errors are the same as for the proton analysis.

Cross-section evaluation from differential thick-target yields
From the net number of events N measured in each protonand alpha-particle peak, thick-target yields Y ∞ were calculated, at each beam energy E b , as follows [50]: where q and e are the charge state of the beam and the elementary charge, respectively; Q is the total charge collected on target during beam irradiation; and η is the detection efficiency. To obtain the differential cross sections dσ /dΩ, we differentiated the thick target yields Y ∞ measured at two consecutive beam energies, as: where is the stopping power and Δ is the beam energy step (20-50 keV in the lab. system).
The differential cross section was then associated to an effective energy E eff , defined as (see [51] for the full derivation): where Here, σ m represents a prior cross section at the energy

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at which the area under the cross section curve between E b and E b − Δ is divided in two equal parts.
Effective energies were finally expressed in the centre of mass system and the cross section (Eq. 3) converted into the modifiedS-factor [52]: Typical uncertainties in E eff were found to be ∼ 12 keV (in the lab. system) by propagation of the beam energy uncertainty (∼ 11 keV). Comparisons between the cross sections obtained using Eq. 4 and that of the prior cross sections were performed. Thanks to our exceptionally small energy step, the maximum difference between cross sections is < 0.1% (for energy steps of about 100 keV in the laboratory). Thus, errors introduced by Eq. 4 are negligible.

Proton channel
DifferentialS-factors for individual proton groups were obtained for each detector as shown in Fig. 7 for D156 (figures for the other detectors and data are available in the Supplemental Material). Upper-limits are shown in the form of open symbols. Where data points are missing, this was due to either: (a) difficulties in the fitting procedure due to low statistics and/or poor kinematic discrimination between proton groups (red shaded stripped area); (b) overlap with the deuterium contaminant peak at different beam energies for different proton groups (green shaded area); or (c) lowenergy protons (high proton-group number) being stopped in the entrance window of the detector.
As a result of these data gaps, one cannot arrive at the total astrophysicalS-factor, i.e. summed over all proton groups, unless missing data can be recovered. In principle, this could be achieved by interpolating available data points under the assumption of constantS-factors over the relevant energy regions. To verify this assumption, we calculated a branching ratio for each proton group i defined as: at all those energies E where data from all proton groups are available, so that the sumS sum of all partial S-factors could be calculated.
Branching ratios revealed significant variations as a function of energy as shown in Fig. 8 for data of p 0 from detector D156 (similar results were obtained for other proton groups and detectors). Given the impossibility to recover missing data, summed (over all proton groups) differential S-factors could only be obtained at energies where all the Fig. 8 Branching ratios for p 0 group (p 0 /(p 0 + · · · +p 6 )) vs centre-ofmass energy as obtained with D156. Data points where one or more proton groups were missing are not shown. Branching ratios proved to be non-constant proton groups were observed (including cases where individual proton groups could not be separated) 5 .
Summed differentialS-factors for protons and α particles are shown in the supplemental material.

Alpha channel
Similar analysis procedures were adopted for the α channel. In this case, however, the 12 C+ 1,2 H reactions do not produce α particles within the region of interest, thus the extraction of cross sections for the 12 C( 12 C,α) 20 Ne reaction was more straightforward. DifferentialS-factors are shown for individual detectors and particle group in Fig. 9. Our results reveal the presence of resonance-like structures across the entire energy region explored in this work, as also reported in previous studies [25,28,32,33,53].

Comparison with previous results and discussion
To compare our results with some of the totalS-factors available in the literature, the angular distribution of individual proton-and α-particle groups should be studied. Unfortunately, this was not possible with the limited angular range covered by our setup. Nevertheless, we tried to identify possible anisotropies by comparing angular distribution trends for each particle group as a function of energy. As an example, Fig. 10 shows theS-factors for p 0 at the three angles investigated as a function of centre of mass energy and normalised to the data taken with D121. Similar results were obtained for all proton-and α-particle groups. Despite the large error bars, these plots suggest the presence of anisotropies scattered along the whole energy range studied here, such that totalS-factors cannot be obtained by simply integrating our summed differentialS-factors over the full solid angle. These anisotropies might be the reason for inconsistencies among literature data sets.

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This notwithstanding, we have attempted a qualitative comparison with selected 6 data in the literature by assuming isotropic distributions for all protons and α-particle groups observed in this study. Results are displayed in Fig. 11 and show a general trend in qualitative agreement with that of other data sets. We note, however, that our data are consistently higher than those of Becker et al. [25] in the proton channel and those of Barron-Palos et al. [31] and Mazarakis et al. [24] in the alpha channel, at most energies. Such discrepancies are perhaps not entirely surprising given the challenging nature of these measurements and the difficulty of detecting rare events from heavy ion reactions at low energies. As anticipated, a possible cause for such disagreements may be found in strong anisotropies not properly taken into account. In addition, other effects may also play a role, such as for example, changes in the isotopic composition of the targets over time and/or the presence of beam-induced background difficult to account for. Despite considerable efforts to keep background contamination under control, clear deviations remain between our data set and those in the literature, whose origin is not fully understood. Future studies may therefore benefit from a careful investigation of target properties (e.g. through RBS, ERDA or NRA analyses) alongside the actual cross section measurements.
We conclude by noting that our data points have the smallest energy step to date. The results presented in this work supersede those in Ref. [45].

Summary and conclusions
The 12 C+ 12 C fusion reactions are among the most important in astrophysics as they govern the evolution and fate of massive stars. Despite impressive experimental efforts over the last few decades we are still far from the required precision in their cross sections at stellar energies.
In this work, we reported on the measurement of the 12 C( 12 C,p) 23 Ne and 12 C( 12 C,α) 23 [16,36] (assuming isotropic angular distributions) to our knowledge). The experiment was performed at the CIRCE Tandem Accelerator Laboratory, using an array of four ΔE − E rest detectors at backward angles for unambiguous particle identification. Thick highly ordered pyrolytic graphite targets were maintained at temperatures ≥ 400 • C to minimise deuterium contaminants and thus beam-induced background.
Background-subtracted proton-and α-particle spectra were fit with skewed Gaussian functions to obtain the net number of events under all particle peaks (p 0 − p 6 and α 0 − α 1 ). Cross sections were obtained by differentiating thick-target yields measured at consecutive beam energies and converted into differentialS-factors for individual proton-and α-particle groups. These were finally summed over all particle groups.
Our results suggest the presence of resonance-like structures as also revealed by previous measurements. However, non-constant branching ratios and anisotropic angular distributions were observed for all particle groups at most energies, thus preventing the calculation of the total angular-integrated S-factors. As such, only qualitative comparisons with literature results were made for both proton-and α-groups. We note that existing discrepancies among various data sets available in the literature may indeed arise from incorrect assumptions on constant branching ratios and isotropic angular distributions.
Figures for individual proton groups and summed over all proton-and α-particle groups can be found in the Supplemental Material.
Improvements to our setup are currently underway to cover a wider angular range for angular distributions and to extend cross-section measurements to lower energies.