Status and future directions for direct cross-section measurements of the ¹³C(a,n)¹⁶O reaction for astrophysics

Half of the chemical elements heavier than iron in the Universe are produced in stars via slow neutron captures (the s-process) through sequences of neutron capture reactions and β decays. Spectroscopic observations combined with stellar models support the ¹³C(α,n)¹⁶O reaction as the neutron source for s-process in low-mass Thermally Pulsing Asymptotic Giant Branch (TP-AGB) stars [1, 2]. A TP-AGB star is composed of a degenerate carbon–oxygen core surrounded by a thin He-rich shell and an extended, convective H-rich envelope. Periodically, these stars undergo thermonuclear instabilities caused by flashes of He-burning shell, called thermal pulses (TPs). Each He-flash generates a convective zone that mixes the carbon produced by the triple-α reaction up to the top of the He shell. During the TP, the shell H burning is extinguished, while after a TP the external convection may penetrate the He-rich shell. Because of this recurrent mixing process, so-called third dredge up, carbon enriched material is brought up to the stellar surface. As a byproduct of these recursive mixing episodes, a so-called ¹³C-pocket can be formed at the top of the He shell through the reaction sequence ¹²C(p,γ)¹³N (β⁺,ν)¹³C [1, 3]. In between the TP events (referred later as an interpulse period), the ¹³C(α,n)¹⁶O reaction is activated at temperatures of about 90 MK and provides a neutron flux with relatively low density of around 10⁷ neutrons cm⁻³ for about 10⁴ years. Starting from Fe-group nuclei, neutron capture reactions followed by β-decays along the valley of stability of nuclides [4] produce heavier chemical elements in the He-rich region reaching up to ²⁰⁹Bi, the highest mass stable nucleus. To predict chemical abundances in TP-AGB, the complex study of stellar structure, composition and mixing phenomena needs to be supported by accurate and precise reaction rates derived from reaction cross sections in the relevant astrophysical energy window.

Furthermore, the ¹³C(α,n)¹⁶O cross section around E_c.m. = 0.14−0.25 MeV is crucial to estimate the energy balance and neutron flux in the TPs for those stellar models where the ¹³C nuclei can survive the interpulse period, be ingested into the convective shell, and burn at higher temperature of around 200 MK [5, 6]. It is also significant for one of the astrophysical scenarios assumed to explain the observed surface abundances of a fraction of the Carbon Enhanced Metal Poor stars, post-AGB stars, and of Sakurai’s object [7–9], where protons are ingested directly into a TP. This mixing results in the production of ¹³C nuclei that ignite the ¹³C(α,n)¹⁶O reaction at temperatures around 200 MK, providing a neutron flux with a neutron density up to 10¹⁴ neutrons cm⁻³, and driving the intermediate neutron capture, so-called, i-process, i.e. neutron capture further away from the valley of beta stability [2]. The associated Gamow energies in this case are E_c.m. = 0.2−0.54 MeV.

The ¹³C(α,n)¹⁶O reaction can become an important source of background in ultra-low event rate research, such as underground nuclear, neutrino research and double-beta decay experiments [10–12]. In fact the α-emitters of the ²³²Th and ²³⁸U decay chains combined with an environment of carbon-rich construction material, e.g. plastic for neutron shielding or electrical insulation, and the detector materials of organic scintillators can become a significant contributor to neutron background, mimicking neutrino signal.

To constrain the phenomena described above, an accurate knowledge of the total and differential reaction cross section (σ(E) and dσ(E)/dΩ(E), respectively), over a wide energy range is crucial. These quantities can be extracted, e.g. from the experimental yield (Y), defined as the number of neutrons per charge, and expressed by the formula:

where η(E_n ) is the neutron detection efficiency at a given neutron energy, E_α is the beam energy and _eff(E) is the effective stopping power. ΔE is the energy lost by the beam in the target. Since the measured yield is proportional to the integral of σ(E)/_eff(E), the cross section needs to be extracted by an iterative approach and the σ(E) can be converted to so-called astrophysical S(E)-factor defined by:

where η represents the Sommerfeld parameter [13]. Each parameter to obtain the cross section requires careful and specific evaluation of the corresponding uncertainties.

Here, we summarize the experimental parameters of the LUNA measurement [14–17] and discuss in details the efforts made to keep under control the systematic effects due to target characterization. This is one of the main, and often overlooked, source of uncertainty for ¹³C(α,n)¹⁶O cross-section measurements. We also guide the way forward to an improved determination of the cross section. Section 2 describes the state of the art of cross-section measurements of the ¹³C(α,n)¹⁶O reaction; section 3 provides the details of the experimental parameters affecting the uncertainty budget of cross-section data. Section 4 gives details for the selection of experimental data sets to the LUNA extrapolation of the cross section to Gamow energies. Section 5 provides the work of LUNA for low-energy extrapolation and the comparison with more recent measurements of the ¹³C(α,n)¹⁶O reaction. The conclusions and prospects of upcoming experiments are discussed in section 6.

The low-energy behaviour of the ¹³C(α,n)¹⁶O reaction cross section is determined by several resonances and their interference patterns [14, 18–20]. Thus the low-energy extrapolation is affected by the uncertainties of the resonance parameters at E_α = −641 keV (J^π = 3/2⁺), E_α = −569.7 keV (J^π = 1/2⁻), E_α = −2.7 keV (J^π = 1/2⁺) and E_α = 1162.6 keV (J^π = 3/2⁺). In particular, the tail of the broad near-threshold resonance with J^π =1/2⁺ and total resonance width (equal with the neutron partial width) Γ_total = Γ_n = (136 ± 5) keV is found to cause an increase of the S(E)-factor below E_c.m. < 0.3 MeV [19]. Table 1 summarizes the parameters of the resonances in ¹³C(α,n)¹⁶O relevant for the S(E)-factor extrapolation below E_c.m. < 0.8 MeV and figure 1 shows the ¹⁷O level scheme.

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Despite their marginal effect in the S(E)-factor extrapolation, sharp resonances between E_c.m. = 0.7−1.2 MeV can be used to constrain experimental parameters, such as target thickness and neutron detection efficiency [21]. Therefore, their parameters (except the resonance strength ω γ) are also indicated in table 1. It is worth mentioning that parameters of sharp resonances show discordance in literature, e.g. the works of [22, 23] and [24, 25] suggest ≈12 and 17 eV, respectively, for the ω γ of the E_c.m. ≈ 0.8 MeV resonance. Although [25] attributes this discrepancy to the contribution of the non-resonant cross section at the resonance energy and the stopping power of α particles in carbon, a dedicated measurement of the resonance parameters is advisable to confirm that.

Because of the 2.215 MeV [26] Q-value of the ¹³C(α,n)¹⁶O reaction, at beam energies E_c.m. = 0.23−1.53 MeV, neutrons are emitted over a wide energy range E_n = 2.1−4.1 MeV, depending also on the emission angle θ. The neutron angular distribution will be discussed in section 3.2.

To properly estimate the α-induced neutron background generated on ¹³C nuclei, the reaction cross section needs to be determined up to several MeV range.

For example, ²¹⁰Po with the emission of E_c.m. = 4.0 MeV initiates the ¹³C(α,n)¹⁶O reaction, in turn, opening the α, n_0,1,2 branches and populating the E_x = 0(n₀), 6.05(n₁) and 6.13(n₂) MeV states in ¹⁶O, respectively [12, 27]. Therefore, the ¹³C(α,n)¹⁶O reaction has been extensively studied by past and recent experiments over a wide energy range between E_c.m. = 0.23−6.1 MeV.

Experimental S(E)-factor data of ¹³C(α,n)¹⁶O [14, 18, 22, 28–32], combined with theoretical calculations [18, 33] with and without the effect of the near-threshold resonance, are shown in figure 2.

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Beside direct measurements, indirect studies were performed with the goal of determining the energy of the near threshold levels by γ-spectroscopy [19]; the squared Coulomb-modified asymptotic normalization coefficient (ANC) [34, 35]; or the similar spectroscopic factor [36–40]. The latter were used to calculate the low-energy astrophysical S(E)-factor and the ¹³C(α,n)¹⁶O reaction rate. Reference [41] reported that the S(E)-factor calculation based on the Coulomb renormalized α-particle ANC () obtained in different indirect measurements bears roughly 20% uncertainty, which is also highlighted by [42], and the inconsistency of direct measurements gives another ∼40% to the uncertainty budget in the s-process temperature range. The significant effect of such uncertainties in different astrophysical scenarios was further emphasized by [43].

In the last three years new experimental data has been published, which potentially reduce the uncertainties of experimental cross-section data. Direct measurements performed underground by the LUNA collaboration [14] in the E_c.m. = 0.23–0.31 MeV region using the thin target approach and the work done by the JUNA collaboration [31] in E_c.m. = 0.24-1.90 MeV using thick and thin targets, have allowed to significantly reduce the uncertainty below E_α < 0.40 MeV providing improved constraint to calculate s-process nucleosynthesis. Together with a recent measurement above ground with high angular resolution in the E_c.m. = 0.61–4.8 MeV [32] reports that based on their R-matrix analysis using 714 angular distributions and the combination of Bayesian uncertainty estimation, the uncertainty of the extrapolation of cross section significantly reduced to the level of ∼5% over the entire Gamow energy range.

In this section, we review the experimental parameters affecting the determination of the cross section of ¹³C(α,n)¹⁶O towards its astrophysically relevant energy region with a focus on the LUNA experiment. The most important experimental parameters of the direct cross-section measurements are summarized in table 2.

3.1. Experimental apparatus and its background

3.2. Neutron detection efficiency

In contrast to γ-ray spectroscopy, the determination of the neutron detection efficiency as a function of neutron energy η(E_n ) is challenging mainly due to the limited choices of sources with accurately known energy spectra and/or angular distributions (in the case of nuclear reactions based source) and, in some cases, to the limited availability of accurately calibrated sources. A standard procedure is to employ radioactive sources (²⁵²Cf, AmBe), which emit neutrons with a continuous energy spectrum, in combination with Monte Carlo simulations [29, 30, 53]. This approach can be made more robust using nuclear reactions, e.g. the ⁵¹V(p,n)⁵¹Cr reaction [16, 21, 54–57]. To constrain the uncertainty of the efficiency determination, the design of neutron detection setup should be optimized to obtain a relation between the detection efficiency and neutron energy as flat as possible along the energy range of interest [21]. A list of the widely used neutron emitters (both radioactive sources and nuclear reactions) is presented in table 3.

Table 3. Properties of some neutron emitting radioactive sources and nuclear reactions often used for neutron detection efficiency determination.

Neutron source	Q value (MeV)	Mean energy (MeV)	Half life
natUO
239PuBe	—	4.6	24110(30) years
241AmBe	—	4.0	432.6(6) years
241AmLi	—	0.8	432.6(6) years
252Cf	—	2.13 a	2.645(8) years
88Y-Be		0.152, 0.949 (0.5%) b	106.626(21) d
124Sb-Be		0.023	60.20(3) d
2H(γ,n)p	−2.23
D(d,n)3He	3.27
7Li(p,n)7Be	−1.64
9Be(γ,n)8Be	−1.66
9Be(d,n)10B	4.362
51V(p,n)51Cr	−1.535
57Fe(p,n)57Co c	−1.6186

^a From [58]. Instead, [23] and [30] indicate 2–3 MeV as an average neutron energy. ^b From [59]. ^c From [60].

In the LUNA experiment, the η(E_n ) was experimentally determined over neutron energy range 0.1−4.0 MeV using the ⁵¹V(p,n)⁵¹Cr reaction and an AmBe neutron source, combined with a detailed simulation implemented in the GEANT4 code. The GEANT4 ²⁶ toolkit [61, 62] was used to simulate the detector response of the LUNA neutron array. This simulation was compared with the experimental data. The efficiency curve obtained from GEANT4 is shown in figure 3 together with the experimental results. The plotted experimental data were corrected for the kinematic energy distribution effect and angular distributions to obtain the nominal efficiency values. To cross-check the consistencies of the simulation, the results for the inner and outer rings of the detector array are shown separately.

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Absolute neutron detection efficiencies of (37 ± 3)% (horizontal) and (34 ± 3)% (vertical) ²⁷ of the two setups were obtained in the E_n = 2.2–2.6 MeV range, which corresponds to the energies of the emitted neutrons in the LUNA experiment. The efficiency was calculated as an average of the GEANT4 simulation and the linear interpolation of the experimental data as shown in figure 3.

A discrepancy was found between the experimental and the simulated data. Therefore, a scaling factor of 0.78 ± 0.01 was applied to the simulated efficiency curves shown in figure 3 in order to match the experimental data. The presence of such a discordance between experimental and simulated efficiencies is rather general in literature [16, 21, 55, 57]. This may indicate a model dependent uncertainty of the method, which can be related to the uncertainty of angular distribution of nuclear reactions, geometrical effect of the setup, and/or physics input of MC simulation (e.g. scattering cross sections, molecular vibrational and rotational excitation modes in the moderator materials, etc) [63–69]. These sources of uncertainties are discussed in the following paragraphs.

As mentioned in section 2, the ¹³C(α,n)¹⁶O cross section may have significant energy-dependent anisotropy. Angular distribution measurements can be found in literature over a wide E_α region [12, 32, 70–72]. Walton et al [70] used a scintillator detector, which was placed at a distance from the thin target (ΔE < 20 keV) of 11.5 cm above E_α = 2 MeV and 6.4 cm below this energy. Data was taken between 0° and 150° at angular intervals of 10° and at 85°, 95°, and 155°, acquiring enough statistics to keep the uncertainty below 5%. A long counter was also used to obtain absolute cross-section data at θ = 0°, 29° and 146° at E_α = 0.8−3.5 MeV. Moreover, Prusachenko et al [72] recently published angular distribution data in the E_α = 2–6.1 MeV region using a p-terphenyl scintillator and time-of-flight (ToF) technique. Febbraro et al [12] performed measurement of ¹³C(α,n)¹⁶O cross section in E_α = 4.2−6.4 MeV using two deuterated scintillator detectors for neutron detection and a single HPGe for γ-ray detection of ¹³C(α,n γ₂)¹⁶O. Similarly, deBoer et al [41] used deuterated scintillator array (ODeSA) [73] to provide high angular resolution differential cross-section data between θ = 0° and 157.5° in E_α = 0.8−6.5 MeV. They could not measure the excited state cross section due to the background of fluorine and the resolution of their unfolding technique.

The angular distribution can also be calculated assuming pure radiation in a two-step process as described, e.g. by [13] (Appendix D.2) near sharp resonances. Moreover, R-matrix calculation can be used to obtain the angular distribution for a given α-energy [41] based on experimental elastic scattering and nuclear reaction data. This effect is discussed in literature purely in terms of the uncertainty of cross-section data. Based on theoretical calculation [31], estimates a relative deviation between the detection efficiency lower than 2.3% assuming isotropic and anisotropic angular distribution in E_α = 0.3−0.8 MeV. This uncertainty can increase significantly near sharp resonances and at higher energies. In the case of the LUNA measurement, due to the larger solid angle and low energies, the uncertainty implied by anisotropic angular distribution is far below the quoted relative 8% assigned to η(E_n ). [50] reports ≈10% and ≈14% at α energies below and beyond 5MeV, respectively, using R-matrix method and Hauser–Feshbach calculation [74].

As a conclusion, the angular distribution of the ¹³C(α,n)¹⁶O reaction is rather well known and supported by experimental data above E_α = 0.8 MeV, however the absence of experimental data in the lower energy regime still requires theoretical predictions. Dedicated measurements, e.g. with a long counter setup and/or plastic scintillators using pulsed beam and ToF technique could provide data with the required precision.

In the context of low-energy nuclear astrophysics measurements, the MCNP, GEANT4 and FLUKA simulation codes are widely used to obtain parameters for cross-section calculations. To study the precision of different simulation codes, van der Ende et al [68] performed a systematic study of boron-lined neutron detector characterisation using MCNPX and GEANT4. They confirmed the reliability of GEANT4 to accurately characterise a thermal neutron detector through the use of a special thermal elastic scattering matrix S(α,β) tables for neutron energies lower than 4 eV, as well as through the use of neutron high precision models. ²⁸

Related to the uncertainties implied by the geometry of the setup [57], suggests a solution to resolve the discrepancy by adding boron to the moderator material to mimic the neutron absorption effect. Nevertheless, the uncertainty of geometry propagates negligible effect (relative <1%) in η(E_n) in the LUNA experiment.

Finally, uncertainties of cross sections (e.g. neutron capture, thermal scattering) as inputs of the simulation and their effect over the simulated detection efficiencies are poorly discussed in literature. Especially, their energy dependent effect would require special attention. It is worth noting that this source of uncertainty also becomes significant in the case of measurement using scintillator detectors, where neutron scattering on the construction materials implies to apply even a ∼30% correction [32].

In this section, we discussed the different aspects of the determination of the neutron detection efficiency. In conclusion, dedicated studies of the scaling factor in a simple neutron detection geometry combining the use of more extended monoenergetic neutron sources in wide energy range are still advisable. A possible way to access this is the study of branching of ⁵¹V(p,n)⁵¹Cr above E_n >1 MeV through the detection of gammas from excited states in ⁵¹Cr(e.g. E_x > 2 MeV) in combination with more precise knowledge of sharp resonance parameters E_α = 1.05 and 1.59 MeV of the ¹³C(α,n)¹⁶O reaction.

3.3. Target characterisation

For ¹³C, the application of solid-state targets is dominant in literature (see table 2). Carbon targets prepared from heating chemical compounds [22, 24, 51] or evaporated onto solid backing such as Ta or Cu [14, 18, 23, 29, 30, 32, 75] have been used. In the following, we review its various aspects.

3.3.1. Methods of target characterisation

Nuclear astrophysics experiments often require long exposures of target under intense beams, which can easily cause significant target degradation. Target modification processes (such as diffusion, melting, sputtering or contamination of the target surface [76, 77]) that occur under intense beam irradiation may result in significant changes of target stoichiometry as a function of depth and accumulated charge [78]. There are different non-invasive ways to obtain information about target degradation in situ, e.g. measurements of the experimental yield variation as a function of time at a reference beam energy, Nuclear Resonant Reaction Analysis (NRRA), and the γ-shape technique. These methods are discussed in the following.

The overall effect of target degradation can be followed by the continuous monitoring of neutron or γ-ray yield (Y_γ ) of a given transition in the investigated nuclear reaction [23, 31]. However, while target thinning caused by, e.g. sputtering can be well corrected in cross-section calculation from the time dependence of such a yield, other ion beam induced processes such as diffusion and mixing of the target atoms into/with the backing are more difficult to assess. The measured yields at LUNA400 of the ¹³C(p,γ)¹⁴N reaction at E_p = 0.31 MeV as a function of the accumulated charge on a single carbon target are shown in figure 4 and represent the effect of modification through ion beam mixing. The calculated yields based on cross section taken from [79] if only target sputtering is considered are also plotted. The effect of this is estimated from a simulation using the SRIM software package [80]. It is evident that the sputtering effect alone can not explain the decreasing yield. The calculated yields considering also the level of target modification obtained in NRRA method, are also plotted and agree well with the experimentally obtained yield. The NRRA method exploits the existence of a narrow and isolated resonance in a given reaction, whose cross section can be well described by the Breit–Wigner expression, σ_BW [13]. By measuring the yield as a function of beam energies in the proximity of the resonance on targets, a characteristic resonance yield curve can be obtained, which contains information on the target thickness and composition.

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In the case of the LUNA experiment solid targets were produced by evaporating ¹³C enriched powder onto tantalum backings. Details of target preparation and characterisation are described in [15]. After the target production, a proton NRRA scan using the 1.747 MeV resonance of the ¹³C(p,γ)¹⁴N reaction was performed to measure the initial target properties, such as number of active nuclei and target homogeneity. NRRA yield profile measurements at ATOMKI were repeated after different amounts of accumulated α-beam charges (done at LUNA400). The result is shown in figure 5, where the appearance of beam induced target degradation is evident.

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The observed modification of the measured target yield profile can be explained by the intense diffusion between the active target layer and the target backing, even without the inclusion of the light element implantation effect. This information should be used to correct low-energy cross-section measurements. It is also worth noting that repeated measurements of ¹³C(α,n)¹⁶O experimental yield at higher E_α can be used to estimate target degradation [18], but this propagates uncertainty to the cross-section calculation at low E_α , where the contribution of the surface layers to the experimental yield is more dominant, due to the exponential drop of the reaction cross section. Thus, an alternative, in situ technique to monitor target modification is often required.

If no resonance is accessible, an alternative approach relies on the study of the shape of the γ-ray lines emitted in a suitable radiative capture reaction, for example in ¹³C(p,γ)¹⁴N, to periodically check both the thickness and stoichiometry of the target. This so-called γ-shape analysis method was applied during the ¹³C(α,n)¹⁶O campaign at LUNA [15]. To monitor the target degradation during the ¹³C(α,n)¹⁶O measurements, data taking at LUNA400 consisted of long α-beam runs with accumulated charges of ∼1 C per run, interspersed by short proton-beam runs with typical accumulated charges of 0.2 C at most, so as to minimize possible changes in target stoichiometry caused by the proton irradiation itself. The maximum accumulated charge with α beam on each target was limited to 3 C, corresponding to at most a 30% target degradation due to the modification of stoichiometry. The lowest studied energies of E_α = 0.245 and E_α = 0.233 MeV required special attention, as the statistics collected during a single run (∼1 count/target) was insufficient to obtain a reliable estimate of the cross section. Thus, all runs at the same energy with similar target degradation level (obtained from γ-shape analysis) were summed up and the cross sections for each subset were calculated to obtain the cross section for the given energy [81].

3.3.2. Stoichiometry of carbon target

The ¹³C target stoichiometry was studied with different approaches. The NRRA technique was used to measure the isotopic abundance of enriched ¹³C material based on the comparison of the plateau height of enriched and natural carbon targets with nominal ¹³C content of 99% and 1.1%, respectively. The ¹³C abundance of the powder used for LUNA was measured to be (97.1 ± 2.3)%, compatible with the 99% reported by the manufacturer.

Another approach that can be used even at low projectile energies is based on the measured γ-ray intensities of DC → g.s. of the ^12,13C(p,γ)^13,14N reactions. A target was irradiated at E_p = 0.31 MeV at LUNA400 and the E_γ = 2.23 MeV (from ¹²C(p,γ)¹³N) and 7.84 MeV (from ¹³C(p,γ)¹⁴N) lines were detected using a HPGe detector. Based on the interpolated cross-section data from [79] and [82] and on the efficiency measurement with radioactive sources and radiative capture reactions, the isotopic abundance of the ¹³C target can be extracted. In the LUNA experiment, (98 ± 2)% was obtained, which agrees well with the data obtained from the NRRA measurement and with the value indicated by the manufacturer.

3.3.3. Light element contamination

The presence of contaminants and their time dependent accumulation in the irradiated targets requires a special attention. At low energy, deuterium, boron isotopes, carbon, oxygen and fluorine are considered the main contributors of beam-induced background in proton-induced reactions [83], while in α-induced reactions the ⁹Be, boron isotopes (especially ¹¹B through the ¹¹B(α,n)¹⁴N reaction [84]), ¹³C and ¹⁷O might contribute to the neutron background of the experimental apparatus. To prevent carbon built-up on target surface during the irradiation good vacuum conditions often supplemented with liquid nitrogen cooled cooling traps are necessary. The effect of possible carbon build-up, e.g. are shown in figure 4 of [18] in the case of a thin-target and it is also considered as a potential source of uncertainty in [31] in thick target arrangements. In the following we present the different approaches used in the LUNA experiment to quantify the presence of light contaminants and their origin.

The level of boron contamination in the target was measured by Particle Induced Gamma-ray Emission (PIGE) technique at ATOMKI at E_p = 3.23 MeV ²⁹ . Boron content of the target was calculated based on the experimental thick-target yield [85] as well as on thin target data using the well-known cross section of the ¹¹B(α,n)¹⁴N reaction [86]. An upper limit of <6 ppm corresponding to about <4 × 10¹⁴ atom cm⁻² was obtained.

To further investigate the possible contribution of beam induced background in the LUNA ¹³C(α,n)¹⁶O experiment, an evaporated ¹³C target was irradiated with proton beam at E_p = 0.31 MeV with a total 6.7 C accumulated charge. This was compared to a beam induced background study on a blank target (Ta backing) with a total 2.7 C accumulated charge at E_p = 0.38 MeV. Observation were in line with the results obtained with PIGE and no evidence of, e.g. boron contaminants was observed in the gamma-ray spectrum.

To further check the effect of possible light contaminants on _eff(E), calculations using SRIM for proton energies at E_p = 0.28−0.31 MeV, and energies at E_α = 0.3−0.4 MeV relevant to the ¹³C(α,n)¹⁶O data taking campaign were performed. Assuming the presence of only one light element contaminant in the target (H, He, C or O), _eff(E) changes by less than 3% for proton projectiles and less than 5% for α particles in the projectile energy ranges used in the simulation. This can be also concluded if more than one contaminant is present at the same time. According to this observations, the γ-shape technique can quantify the stopping power modification independently from the type of contaminant [87].

The low level of contaminant in LUNA targets were further supported by additional measurements using Elastic Recoil Detection Analysis performed on irradiated targets at the Ion Beam Center of Helmholtz-Zentrum Dresden-Rossendorf [88]. The analysis confirmed that the concentration of elements such as H, He and O after the α-beam irradiation at LUNA was at most 10%.

Similarly to the LUNA experiment, in [14, 18, 23, 29, 30, 32] isotopically enriched ¹³C targets were evaporated onto heavy elements backings (e.g. Cu, Ta, Au) and the above indicated target degradation effect was controlled either with NRRA method or with yield measurements at a reference α energy. References [22, 24, 31, 51] used thick targets in their experiments. Thick target measurements could be suited to avoid systematic uncertainty, however, in these cases light particle implantation requires a special attention, discussed in the next section.

3.3.4. Effect of high dose α implantation on the experimental yield

In the case of low energy nuclear astrophysics experiments, the number of accumulated beam particles may become comparable with the number of active nuclei of thin targets. For example assuming 100 μA beam intensity and 1 h irradiation time, even ∼6 × 10¹⁸ atom cm⁻² implantation dose is reached with a typical size of a few mm lateral beam size. Therefore, the alteration of the stoichiometry of the target due to the implantation itself and thus the modification of effective stopping power may increase drastically the uncertainty of the calculated cross section.The implantation effect should also be considered in thick target measurement, especially when a single target is used in a long irradiation campaign [30, 31].

Simulation of the implantation using E_α = 0.4 MeV ³⁰ alpha particles impinging on a thin evaporated ¹³C target is shown in figure 6 using SRIM software.

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Although the simulation reveals that the bombarding particles are implanted in a finite depth range in the Ta backings, well-separated from the evaporated thin layer, they might still affect the cross-section calculation using a thick target over a wide alpha-energy range, e.g. using repeated yield measurement at high E_α for target monitoring at low E_α run [18, 31].

To estimate the effect of light nuclei implantation into carbon on the measured neutron yield in the case of a thick carbon target, another simulation was carried out using the SRIM software package. First, the target structure was modelled assuming 3 C implantation at E_α = 0.4 MeV. Then, the integrated yield at a E_α = 0.8 MeV ³¹ was recalculated and compared with the non-irradiated case. An important parameter of the simulation is the saturation level of a given light nucleus in a heavier matrix. Although irradiation of carbon with light nuclei (¹H, ⁴He) has a significant interest in nuclear technology and material science [89–91], limited experimental information is available in literature with intense ⁺He beam in the energy region of nuclear astrophysics interest. Wide range of indicated saturation level are suggested [91, 92]. In our simulation, 40% saturation level [92] was used and the results of the calculation is shown in figure 7.

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It is assumed that helium atoms above the saturation limit diffuse towards deeper layers of the target, which agree with the experimentally observed yield reduction in proton runs. If different diffusion directions (e.g. towards the surface of the target) are assumed, the predicted yield reduction disagrees with observations. Since at 3 C of accumulated charge the saturation limit is reached, adding more charge does not change the structure of layers near the surface, which mainly contribute to the observed neutron yields.

In summary, ∼1.6% deviation of the experimental yields compared to the non-irradiated case is obtained assuming 40% saturation level, which reveals the marginal role of implantation effect in cross-section calculation.

Despite many attempts to measure the ¹³C(α,n)¹⁶O cross section [18, 22, 23, 28–30, 51], direct measurements inside the s-process Gamow window E_c.m. = 0.14−0.25 MeV have been performed only recently [14, 31] and covering only its highest energy part. Thus, the theoretical approach to extrapolate the S(E)-factor in the astrophysically relevant energy region is still mandatory. Extrapolation based on experimental cross-section data is complicated by the different systematic uncertainties of the different studies and because in many cases the full description of uncertainty budget is not available. The uncertainties of the available experimental data of the ¹³C(α,n)¹⁶O reaction are summarised in table 4.

To solve the inconsistency of the different experimental data-sets, normalization factors of experimental data can be applied. Normalization factors as high as 40% have been reported by various authors, which are well beyond the reported systematic uncertainties of the experimental data, usually less than 20%.

For example, while Drotleff et al [29] did not mention this normalization issue, some inconsistencies between data sets were already noted in previous works (see, e.g. Bair and Haas [22], and Brune [23]). Harissopulos et al [30] re-scaled their data averaging their cross section at E_α = 1.00 MeV and compared it with that of Brune et al [23], the difference was <10%, inside the uncertainty bands. Moreover, their values were found to be lower by 28%–37% with respect to Bair and Haas [22], instead Heil et al [18] presented a good agreement with Davids [51], while Bair and Haas [22], and Drotleff et al [29] have a poorer agreement (15 % discrepancy) with Kellogg [28] and Harissopulos et al [30].

Correction factors of 1.36 and 0.7 are proposed by Giorginis et al [93] and Plompen et al [94] be applied to the data of Harissopulos et al [30] and Bair and Haas [22], respectively, which may originate from the inaccuracy of target thickness determination and neutron detection efficiency calculation. Based on the analysis of [93], [95] reported that a correction of 1.36 has to be applied to the number of active nuclei due to the incorrect extraction of target thickness from the resonance profile obtained by Harissopulos et al [30] at E_p = 1747 keV in ¹³C(p,γ)¹⁴N reaction. They mention that the comparable value of target thickness and energy resolution function requires a deconvolution (via Voight profile) analysis, however, they did not discuss the target profile obtained by Harissopulos et al [30] at E_α = 1054 and 1336 keV, which is in agreement with the target thickness at E_p = 1747 keV and therefore does not support the above proposed correction factor.

Bair and Haas [22] used age-diffusion theory calculation to obtain neutron detection efficiency. Details of that work can be found in Macklin et al [96]. In this context [94], refers to the need of different corrections due to geometrical (e.g. tube pressure) and angular distribution effects recommending the reevaluation of this data.

Recent cross-section measurements will be discussed in the context of low energy extrapolation in the following section.

In spite of the available cross-section data, low energy extrapolations are still required by stellar models. R-matrix theory has been proven to be a reliable tool for such extrapolations [97]. LUNA implemented R-matrix through the code AZURE2 [82] and included the resonance parameters proposed by [20, 35] and [98] and reported in [41]. Monte Carlo fits were performed by fixing some parameters from literature and fitting others, as described in the following. The cross-section data included in this analysis were those from Heil et al [18], Drotleff et al [29] and Harissopulos et al [30].

The LUNA analysis was limited up to E_cm = 1.2 MeV, a range which includes the whole Gamow window for temperature ≤1 GK. Two broad resonances are relevant in this range: one located at E_cm = 856 keV and the sub-threshold one at E_cm = −2.7 keV. The resonance parameter of the latter was used from indirect investigation (e.g. [20]) as no close enough direct experimental data exist. Resonance parameters were allowed to vary in the calculation within their reported uncertainties, i.e. ANC = 5.44 × 10⁹⁰ fm^−1/2 (19%) from [35] (as also confirmed by [33]) and Γ_n = 124 keV (10 %) from [20]. The 856 keV resonance energy was fixed, being its influence negligible in the reaction rate evaluation. For each spin, a background pole at E_cm = 15 MeV was included to represent the non-resonant component of the cross section (e.g. the contribution from low energy tail of higher energy resonances). Based on the trends and absolute scale of experimental data, results of [28, 30] and of [18, 22, 29, 31] form two well-separated subgroups.

The effect of the normalisation factor of the experimental data [14] was investigated via a dedicated analysis. First, the data of [18] and [29] were considered as reference and a normalization factor for [30] data was included to be fit together with the resonance parameters. Second, [30] data was considered as a reference and a normalization factor was applied to [18] and [29], again to be fitted with resonance parameters. For the LUNA data, considering the description of possible systematic uncertainties, no normalization factor was applied. Results are summarized in figure 8. In the first case, the fitted normalization factor of 1.37(16), very close to 1.36 proposed by [95] was obtained (Left side of the figure), while the value of 0.73(9) was used in the second case (Right side of the figure). Inside the Gamow peak of the s-process, the effect of the different normalisation factor is only of the order of 5% over the extrapolated S(E)-factor, increasing toward higher energies.

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To estimate the S(E)-factor uncertainty, 30’000 evaluations for each normalization group were performed with AZURE2 sampling input parameters with a Gaussian probability within their experimental uncertainty as well as sub-threshold state ANC. Poles and the partial width of 856 keV resonance were fit to data. The resulting S(E)-factors, extrapolated down to zero energy, were used to build a probability density function (PDF) from which the reaction rate was evaluated at several temperatures up to 1 GK, as reported in [14, 81]. Considering the smaller tension with the data from [18] and [29], the normalization factor of 1.37(16) for [30] was used in the evaluation of best reaction rates reported in Ciani et al [14]. The ‘low-LUNA’ rates, which are the present lower limit for the reaction rate, were obtained by extracting the 5% percentile line of the S(E)-factor PDF evaluated from the normalization of [18] and [29] data with 0.73(9) (based on [30]). Considering that LUNA data are very close to the s-process Gamow window, fits are well constrained in such energy range in spite of the normalization used. At higher energies at interest for the i-process, the ambiguity on data normalization is still relevant and deserves a deeper investigation.

Since the publication of the LUNA result, new directly measured cross-section data have been published in [31] and [32].

JUNA collaboration [31] provides a consistent data set from 0.24 to 1.9 MeV obtained in two experimental campaigns at two accelerator facilities. Even if the JUNA data of different campaigns do not overlap in energy, they seem to agree with the normalisation of [18] and [29] (data are also shown in figure 8). This makes the LUNA assumption for the recommended value more robust in the whole considered temperature range. It is worth noting that JUNA performed R-matrix fit excluding other literature data to obtain ANC of sub-threshold state and extrapolated S(E)-factor at E_c.m. = 0.24 MeV and below. Their Coulomb renormalized ANC value = 2.1(5) fm⁻¹ is in tension with = 3.6(7) fm⁻¹ of [35], which was used in the LUNA analysis. The different ANC values imply uncertainty of extrapolation mainly below E_c.m. = 0.24MeV (see figure 2 of [31]), where no direct data exist.

Recently, a new value for the ANC = 2.8(5) fm⁻¹ has been proposed [42], which decreases the estimate of [35] about 22%. We performed an additional analysis including this value (used [18, 29] as normalization reference) and found a difference of 15% in the reaction rate at 90 MK. If this new ANC is confirmed, a deeper investigation of its effects on reaction rates would be desirable.

Reference [32] published high resolution differential cross-section data in E_c.m. = 0.6–5.0 MeV. Their result, similarly to [31], favours the absolute cross section published in [18, 29]. Their angle integrated cross section obtained from Legedre fit of differential cross-section data are also shown in figure 8. These authors performed a detailed R-matrix analysis to obtain cross section over the Gamow energy range. First, an R-matrix analysis completed with Bayesian uncertainty analysis was done using the experimental data of [14, 18, 29, 31] and ¹⁶O(n, total) data. The obtained best fit suggests a ≈10% uncertainty over the Gamow energy range. Then, the analysis was repeated using their differential cross-section data. The two fits are consistent, but the latter results a reduced ≈5% uncertainty of S(E)-factor in Gamow energy. Experimental data were rescaled by [32] in their analysis by a larger factor in the case of [14] than the quoted systematic uncertainty. Moreover, the two data set of JUNA work [31] were handled independently to obtain their best fit. Overall, this analysis highlighted the importance of low-energy angle integrated cross-section measurement and the potential of their fit in Gamow energy range, they concluded that a full evaluation of all the literature data is still needed for the uncertainty analysis of the extrapolated S(E)-factors.

We have provided an overview of direct measurements of the ¹³C(α,n)¹⁶O reaction cross section and summarised the conclusions of the LUNA experiment, which measured this cross section covering partially for the first time its s-process Gamow window located at E_c.m. = 0.14−0.25 MeV.

In spite of the low-energy measurement, extrapolation using direct cross-section data over a wide energy range is still required. Recent and past measurements at higher energy deviate from each other by almost 30%. Based on our and earlier evaluation, the systematic uncertainty assigned to the neutron detection efficiency and target characterisation can be the main source of the observed discordance. To further constrain the S(E)-factor of the ¹³C(α,n)¹⁶O reaction below E_α = 0.3 MeV, special experimental efforts are needed to reduce systematic and statistical uncertainties over a wide E_α range.

As the systematic uncertainty of neutron detection efficiency is still the main contributor to the uncertainty budget, extension of monoenergetic neutron sources produced by nuclear reactions towards E_n ≥ 2 MeV is required, e.g. via the study of the branching of different neutron groups in neutron emitting nuclear reactions (⁵¹V(p,n)⁵¹Cr, ⁵⁷Fe(p,n)⁵⁷Co, etc). Another task is to constrain the uncertainties of the angular distribution of the ¹³C(α,n)¹⁶O reaction below E_α = 0.8 MeV. This could be done via dedicated measurements with, e.g. a long counter setup and/or plastic scintillators using a pulsed beam and Time of Flight technique, which could provide data with the required precision and improve the total cross-section calculations.

Moreover, revision of angular distribution and nuclear properties (resonance strength, energy and width) of sharp resonances in ¹³C(α,n)¹⁶O around e.g. E_α = 1.05 and 1.59 MeV can be another way to better constrain the uncertainty of neutron detection efficiency over a wide E_α region. We have found that in many cases, comparison of simulated and experimental neutron detection efficiency parameters shows discordance. This problem can also be better studied with well known, monoenergetic neutron sources and/or via the study of sharp resonances of ¹³C(α,n)¹⁶O. Proper monitoring of target properties and their modification under beam-bombardment also deserve special attention in future experiments.

Phenomenological methods for the extrapolation of experimental data, e.g. the R-matrix approach, rely on nuclear physics inputs, such as the parameters of near threshold resonances. Thus, future indirect measurements of, e.g. spectroscopic factors including comprehensive evaluations with old data are needed. As concluded by the most recent work of [32], multi-channel R-matrix analysis combined with experimental input data and their uncertainty budget is crucial to obtain a more robust low-energy S-factor extrapolation.

Financial support by INFN, the Italian Ministry of Education, University and Research (MIUR) through the ‘Dipartimenti di eccellenza’ project ‘Science of the Universe’, the European Union (ERC Consolidator Grant project STARKEY No. 615604, ERC-StG SHADES No. 852016, and ChETEC-INFRA No. 101008324), Deutsche Forschungsgemeinschaft (DFG, BE 4100-4/1), the Helmholtz Association (ERC-RA- 0016), the Hungarian National Research, Development and Innovation Office (NKFIH K134197), the European Collaboration for Science and Technology (COST Action ChETEC, CA16117) are gratefully acknowledged. UK group acknowledge STFC (Grant No. ST/V001051/1). For the purpose of open access, the author has applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising from this submission. ML was supported by the the Lendület Program LP2023-10 of the Hungarian Academy of Sciences. and NKFIH excellence Grant TKP2021-NKTA-64.

No new data were created or analysed in this study.