1 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 \(\simeq 25~M_{\odot }\) 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 \(2{-}5\times 10^6\) g/cm\(^3\) [11, 12] and temperatures of \(5\times 10^8\) K, corresponding to an energy range of interest, the Gamow window, \(E_{\mathrm{c.m.}}=1.5\pm 0.3~{\mathrm{MeV}}\) [13]. At these energies, the \(^{12}\)C+\(^{12}\)C reactions proceed through the \(^{23}\)Na+p (\(Q=2.24\) MeV) and \(^{20}\)Ne+\(\alpha \) channels (\(Q = 4.62\) MeV), while the \(^{23}\)Mg+n (\(Q = -2.62\) MeV) channel is closed.Footnote 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 \(\alpha \) 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 (\(\ll 10^{-9}\) 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].

Considerable efforts have been made over the past four decades to determine the \(^{12}\)C+\(^{12}\)C reactions cross-sections with the precision required for astrophysics. Experiments were performed using either charged particle spectroscopy [14, 24,25,26], gamma-ray spectroscopy [16, 27,28,29,30,31,32,33,34] or a combination of both [35,36,37]. Data at \(E_{\mathrm{c.m.}}< 3.0\) MeV still carry large uncertainties and show significant discrepancies between different data sets. Furthermore, no direct measurement has been possible at energies below \(E_{\mathrm{c.m.}}=2.14\) MeV. As a result, the extrapolation of the astrophysical S factor into the Gamow window remains highly uncertain [18,19,20,21,22].

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 \(E_{\mathrm{c.m.}} =2.51{-}4.36\) 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.

2 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 \(^{12}\)C-beam currents of up to 20 p\(\upmu \)A on target. The accelerator beam-transport system comprises a series of magnetic and electrostatic filters to deliver beams of sufficient mass (\(\varDelta M/M = 1.38 \times 10^{-3}\)) and energy (\(\varDelta E/E = 1.43 \times 10^{-3}\)) 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 (\(25\times 25\) 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 beam-current reading directly on target. Typical beam currents on target were of few tens of \(\upmu \)A resulting in target temperatures of up to \(\sim 1200\,^{\circ }\)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 \(\upmu \)m).

Fig. 1
figure 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])

The pressure in the scattering chamber was maintained below \(5\times 10^{-7}\) 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 \(\varDelta E-E\) modules, each comprising an ionisation chamber (IC, \(\varDelta E\) stage) and a large area silicon strip detector (SSD, \(E_{\mathrm{rest}}\) stage). Further details on the full detector array and its commissioning are reported in [43]. Briefly, each module consists of an aluminium truncated pyramidal structure with square base, hosting the IC and the SSD detectors. The IC is equipped with an entrance Havar window (\(23\times 23\) mm\(^2\) large and 2.6 ± 0.2 \(\upmu \)m thick) acting as a cathode; an aluminised Mylar foil (\(52.5 \times 52.5\) mm\(^2\), 1.6 ± 0.2 \(\upmu \)m thick, metallised with 50 \(\upmu \)m/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 \(\upmu \)m diameter). The IC was operated with CF\(_4\) maintained at a constant pressure (within \(\pm ~0.05\) mbar) by means of an automatic system [43]. The second stage of each GASTLY module consists of a large (\(58.0\times 58.0\) mm\(^2\) active area, \(300 \pm 15\) \(\upmu \)m 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].

At the time of the \(^{12}\)C+\(^{12}\)C measurement, only four \(\varDelta E-E\) modules were available: three were mounted on a vertical plane at \(121^{\circ }\) and \(156^{\circ }\) (above and below the beam axis), and one was mounted on the horizontal plane at an angle of \(143^{\circ }\) to the beam axis (see Fig. 1). Detectors at \(121^{\circ }\) and \(143^{\circ }\) were mounted closest to the target and covered an angular range of 19\(^{\circ }\), subtending a solid angle of 80 msr. Detectors at \(156^{\circ }\) had to be placed farther away from the target to accommodate for the cold finger. They covered an angular range of 13\(^{\circ }\) 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).

3 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 \(400^{\circ }\)C (using intense beams) to reduce deuterium contaminationFootnote 2 on target by up to 90% its original value, as found in our previous study [42].

For optimal detection of \(\alpha \) particles and protons, the ICs and SSDs detectors were calibrated using, respectively, a certified mixed \(^{239}\)Pu–\(^{241}\)Am \(\alpha \) source and proton beams at various energies in the range \(E_\text {lab}~=~1{-}5\) MeV (see [45]). At high energies, the ICs were operated with CF\(_4\) at a pressure of 35 mbar, sufficient to separate \(\alpha \) 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 \(\varDelta E-E_{\mathrm{rest}}\) matrix for detector D121 at a pressure of 35 mbar, obtained with a \(^{12}\)C\(^{+3}\) beam at \(E_\text {lab}=8.72\) MeV on the HOPG target. The two loci correspond to protons and \(\alpha \) particles from the \(^{12}\)C(\(^{12}\)C,p)\(^{23}\)Na and \(^{12}\)C(\(^{12}\)C,\(\alpha \))\(^{20}\)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 \(\mathrm \varDelta E>\) 1 MeV were treated as \(\alpha \) particles.

Fig. 2
figure 2

Typical calibrated \(\varDelta E-E_\text {rest}\) matrix showing the \(\alpha \) particles and protons loci. This matrix was obtained using D121 with 35 mbar of CF\(_4\) in the IC and with a E\(_\text {lab}\) = 8.72 MeV \(^{12}C^{+3}\) beam impinging on the HOPG target

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 \(\varDelta E-E_{\mathrm{rest}}\) matrix). Depending on energy, some protons and \(\alpha \)-particles were stopped in the IC and were therefore recorded with zero \(E_{\mathrm{rest}}\). 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 daysFootnote 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 \(\alpha \)-particle spectra at each beam energy. Figures 3 and 4 show typical raw spectra for the proton and \(\alpha \) channel, respectively and the background spectra used for analysis.

Fig. 3
figure 3

Raw spectra for the proton channel taken with D156 at \(E_\text {lab}\) = 7.71 MeV (in blue) and its corresponding time-normalised background run (in pink)

Fig. 4
figure 4

Raw spectra for the \(\alpha \) channel taken with D121 at \(E_\text {lab}\) = 5.86 MeV (in blue) and its corresponding time-normalised background run (in pink)

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 (\(\ll \) 1%) at all energies studied here [45].

3.1 Fitting procedure

After background subtraction, proton and \(\alpha \)-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.

3.1.1 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 \(E_{\mathrm{rest}}\) 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.

Fig. 5
figure 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

Distinct proton groups are labelled according to the excitation state of the recoiling nucleus.Footnote 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:

$$\begin{aligned} f(x)=A\Bigg (\frac{\exp \bigg [\frac{\sigma '^2+2\tau (x-\mu )}{2\tau ^2}\bigg ]\mathrm{Erfc}\bigg [\frac{x-\mu +\frac{\sigma '^2}{\tau }}{\sigma '\sqrt{2}}\bigg ]}{2\tau }\Bigg ), \end{aligned}$$

where A is the amplitude of the distribution, \(\mu \) and \(\sigma '\) are its centroid and standard deviation, \(\tau \) corresponds to the shape parameter and Erfc(x) is the complementary error function. Specifically, the sum of three skewed Gaussians was used to fit \({\mathrm{p}}_0\), \({\mathrm{p}}_1\) and the satellite peak (where visible) at the left of \({\mathrm{p}}_1\), while the sum of four skewed Gaussians was used to fit \({\mathrm{p}}_2\), \({\mathrm{p}}_3\), \({\mathrm{p}}_{4+5}\) (since \({\mathrm{p}}_4\) and \({\mathrm{p}}_5\) cannot be separated) and \({\mathrm{p}}_6\). 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 \(\varDelta {\mathrm{E}}\) 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, \(\delta N\), was calculated as a sum in quadrature of the statistical error in background counts, \(\sqrt{N_{\mathrm{bkg}}}\), under the relevant proton peak and the error, \(\delta A\), on the parameter A.

For runs taken at beam energies \(E_\text {lab}< 5.5\) 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 \(N_{\mathrm{ROI}}\). Uncertainties \(\delta N\) were calculated by replacing \(\delta {\mathrm{A}}\) with the statistical error \(\sqrt{N_{\mathrm{ROI}}}\).

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 \(\chi ^2\) (reduced \(\chi ^2\) values were between 2-6 for the first fit and over 400 for the latter) between both fits was of nearly 3\(\sigma \) demonstrating that the lowest statistics peak (typically, \(p_2\)) is indeed real.

3.1.2 Alpha channel

At the beam energies used in this work, only \(\alpha _0\) and \(\alpha _1\) had enough energy to completely pass through the Havar entrance window of the IC. Low energy \(\alpha \) particles (typically \(E_\alpha < 7.10\) 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 \(\alpha \) channel was therefore performed on reconstructed total energy spectra, (\(E_{\mathrm{tot}}\) \(= \varDelta E+E_{\mathrm{rest}}\)). A sample of a background-subtracted spectrum is shown in Fig. 6.

Fig. 6
figure 6

Typical \(\alpha \) (\(E_{\mathrm{tot}}\)=\(\varDelta E+E_{\mathrm{rest}}\)) 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 \(\alpha \) particles stopped in the IC. This spectrum was taken with D156 and a \(E_{\mathrm{lab}}=8.37\) MeV \(^{12}\)C beam. The \(\alpha \) peaks coming from the \(^{12}\)C+\(^{12}\)C reaction are labelled as \(\alpha _0\) and \(\alpha _1\)

Fig. 7
figure 7

Differential \({\tilde{S}}\)-factors (in 10\(^{16}\) MeV b/sr) for individual proton groups as obtained with detector D156. Open symbols represent upper limits. Errors are statistical only

To extract the number of events under each \(\alpha \) 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 \(\alpha \) 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 \(\alpha _0\) 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 \(\alpha \) peak was calculated as for the proton channel. Systematic errors are the same as for the proton analysis.

3.2 Cross-section evaluation from differential thick-target yields

From the net number of events N measured in each proton- and alpha-particle peak, thick-target yields \(Y^\infty \) were calculated, at each beam energy \(E_{\mathrm{b}}\), as follows [50]:

$$\begin{aligned} Y^\infty =\frac{N~q~e}{\eta ~Q}, \end{aligned}$$

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 \(\eta \) is the detection efficiency.

To obtain the differential cross sections d\(\sigma \)/d\(\varOmega \), we differentiated the thick target yields \(Y^\infty \) measured at two consecutive beam energies, as:

$$\begin{aligned} \frac{\mathrm{d}\sigma }{\mathrm{d}\varOmega }=\frac{\epsilon ~[Y^\infty (E_{\mathrm{b}})-Y^\infty (E_{\mathrm{b}}-\varDelta )]}{\varDelta }, \end{aligned}$$

where \(\epsilon \) is the stopping power and \(\varDelta \) is the beam energy step (20–50 keV in the lab. system).

The differential cross section was then associated to an effective energy \(E_{\mathrm{eff}}\), defined as (see [51] for the full derivation):

$$\begin{aligned} E_{\mathrm{eff}} = E_{\mathrm{b}} - \frac{1}{2} \varDelta + \frac{[1+R_2^2 \varDelta ^2/12]^{1/2}-1}{R_2}, \end{aligned}$$

where \(R_2~\equiv ~[(\frac{d^2\sigma _{m}}{d^2E})/(\frac{d\sigma _{m}}{dE})]_{E_{\mathrm{h}}}\) is calculated for \(E_{\mathrm{h}}=E_{\mathrm{b}}-\varDelta /2\). Here, \(\sigma _{m}\) represents a prior cross section at the energy at which the area under the cross section curve between \(E_{\mathrm{b}}\) and \(E_{\mathrm{b}}-\varDelta \) 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 modified \({\tilde{S}}\)-factor [52]:

$$\begin{aligned} {\tilde{S}}(E) = E~\sigma (E) \exp \Bigg ( \frac{87.236}{\sqrt{E}[\mathrm {MeV]}}+0.46~ E[\mathrm{MeV}] \Bigg ). \end{aligned}$$

Typical uncertainties in \(E_\text {eff}\) were found to be \(\sim 12\) keV (in the lab. system) by propagation of the beam energy uncertainty (\( \sim 11\) 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.

4 Results

4.1 Proton channel

Differential \(\tilde{S}\)-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) low-energy 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 astrophysical \(\tilde{S}\)-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 constant \(\tilde{S}\)-factors over the relevant energy regions. To verify this assumption, we calculated a branching ratio for each proton group i defined as:

$$\begin{aligned} {\mathrm{BR}}_{\mathrm{i}}({\mathrm{E}}) = \frac{\tilde{S}_i(E)}{\tilde{S}_{\mathrm{sum}}(E)} \end{aligned}$$

at all those energies E where data from all proton groups are available, so that the sum \(\tilde{S}_{\mathrm{sum}}\) of all partial S-factors could be calculated.

Fig. 8
figure 8

Branching ratios for p\(_0\) group (p\(_0\)/(p\(_0+\cdots +\)p\(_6\))) vs centre-of-mass 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

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 \(\tilde{S}\)-factors could only be obtained at energies where all the proton groups were observed (including cases where individual proton groups could not be separated)Footnote 5.

Summed differential \(\tilde{S}\)-factors for protons and \(\alpha \) particles are shown in the supplemental material.

4.2 Alpha channel

Similar analysis procedures were adopted for the \(\alpha \) channel. In this case, however, the \(^{12}\)C+\(^{1,2}\)H reactions do not produce \(\alpha \) particles within the region of interest, thus the extraction of cross sections for the \(^{12}\)C(\(^{12}\)C,\(\alpha \))\(^{20}\)Ne reaction was more straightforward. Differential \(\tilde{S}\)-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].

Fig. 9
figure 9

Partial differential \({\tilde{S}}\)-factors (in 10\(^{16}\) MeV b/sr), obtained with each GASTLY detectors for individual \(\alpha _0\) and \(\alpha _1\) groups, as a function of centre of mass energy. Errors are statistical only

5 Comparison with previous results and discussion

To compare our results with some of the total \({\tilde{S}}\)-factors available in the literature, the angular distribution of individual proton- and \(\alpha \)-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 the \({\tilde{S}}\)-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 \(\alpha \)-particle groups. Despite the large error bars, these plots suggest the presence of anisotropies scattered along the whole energy range studied here, such that total \({\tilde{S}}\)-factors cannot be obtained by simply integrating our summed differential \({\tilde{S}}\)-factors over the full solid angle. These anisotropies might be the reason for inconsistencies among literature data sets.

Fig. 10
figure 10

Partial differential \({\tilde{S}}\)-factors for p\(_0\) as a function of centre of mass energy and for all three angles. Data are normalised to those obtained with D121 to emphasise possible anisotropies (see text for details)

This notwithstanding, we have attempted a qualitative comparison with selectedFootnote 6 data in the literature by assuming isotropic distributions for all protons and \(\alpha \)-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].

Fig. 11
figure 11

Comparison of total proton (top) and alpha (bottom) \({\tilde{S}}\)-factors obtained in this study (filled symbols) assuming isotropic angular distributions and selected data sets (opened symbols): Mazarakis et al., Becker et al. [24, 25] (angle integrated), Barron-Palos et al. and Fruet et al. [16, 36] (assuming isotropic angular distributions)

6 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,\(\alpha \))\(^{23}\)Na reaction channels in the energy range \(E_\text {c.m.} = 2.51{-}4.36\) MeV and with energy steps \(\varDelta E_{\mathrm{c.m.}} < 25\) keV (the finest energy steps to date, to our knowledge). The experiment was performed at the CIRCE Tandem Accelerator Laboratory, using an array of four \(\varDelta E-E_{\mathrm{rest}}\) detectors at backward angles for unambiguous particle identification. Thick highly ordered pyrolytic graphite targets were maintained at temperatures \(\ge 400\,^{\circ }\)C to minimise deuterium contaminants and thus beam-induced background.

Background-subtracted proton- and \(\alpha \)-particle spectra were fit with skewed Gaussian functions to obtain the net number of events under all particle peaks (p\(_0 - \mathrm{p}_6\) and \(\alpha _0 - \alpha _1\)). Cross sections were obtained by differentiating thick-target yields measured at consecutive beam energies and converted into differential \({\tilde{S}}\)-factors for individual proton- and \(\alpha \)-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 \({\tilde{S}}\)-factors. As such, only qualitative comparisons with literature results were made for both proton- and \(\alpha \)-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 \(\alpha \)-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.