Neutron capture nucleosynthesis is the main responsible for the production of elements heavier than Fe. Such a process can be rapid (r-process) or slow (s-process), according with the comparison between the neutron capture reaction (n,γ) rates of the involved nuclei and their half-lives [1], but anyway they are a series of subsequent neutron captures, separated by β − decays. Therefore, the rapidity of the process is determined by the intensity flux of neutron available in the stellar environment in which occurs.

In recent years, nuclear astrophysicists focused their attention to the s-process relevant cross sections, also considering that r-process abundances can be obtained by the s-process ones [2]. The s-process nucleosynthesis follows a path that wanders along the valley of β-stability close to the strongly bound isotopes of a given atomic weight (or mass number) A. The main site of the slow neutron captures is the final Asymptotic Giant Branch (hereafter AGB) phase of low- and intermediate-mass stars. Stars with mass up to 3 M_⊙ are actually responsible for the production of the main component of the s-process (i.e., nuclei from Sr to Bi) and the main neutron source is known to be the ¹³C(α,n)¹⁶O reaction [3]. It operates at typical energies of 8 keV in radiative conditions and provides a neutron density of about 10⁶–10⁷ n/cm³ [4,5]. The ²²Ne(α,n)²⁵Mg reaction is the second neutron source in AGB stars and it is activated only marginally during the thermal pulses (periodic episodes of He-convective burning) in low mass stars. Only red giants more massive than 3 M_⊙ (the intermediate mass stars) burn ²²Ne efficiently, at maximum temperatures of 3–3.2 ⋅ 10⁸ K. Even stars with larger masses, terminating their evolution as core-collapse Supernovae, are sites of s-process nucleosynthesis, and in particular they are responsible of the so-called weak component, namely that devoted to the production of nuclei with A ≤ 85 [6].

The s-process is considered as being a rather well-known nucleosynthesis mechanism because 1) its main component occurs in common astrophysical objects and 2) neutron capture reactions are not hampered by the Coulomb barrier. This fact makes the most of (n,γ) reactions relatively easy to be studied experimentally. However, there are still theoretical problems to be understood ([7]), as like as the ¹⁹F abundance in AGB stars [8]. Thus, some nuclear reactions cross sections need to be measured with high precision in the energy range of AGB star nucleosynthesis, where objective experimental difficulties are present. Among them, we appoint:

1. Reactions delivering the neutrons for the s process, in particular ¹³C(α,n)¹⁶O (Section 3) and ²²Ne(α,n)²⁵Mg (Section 4), whose cross section determination is affected by the well-known problems in measuring the reactions between charged particles at low energy [9,10];

Thanks to the indirect Trojan Horse Method (THM), described in the following section, one can extract the cross sections at the ultra-low energies of astrophysical interest for charged particle induced reactions, where this is made almost impossible by the Coulomb barrier presence. In more recent years, the method has been largely tested and applied with success to shed light on different problems, such as the different levels contributions to the cross sections [13,14], for neutron induced reactions [15–21] and for reactions induced by unstable beams [22,23], thus helping in solving the experimental difficulties of points 1 to 3.

The idea at the base of THM is to obtain the binary reaction cross section of interest for astrophysics a + x → c + C, by measuring the reaction cross section of a given A+ a → c + C + s quasi-free (QF) break-up reaction.

This is possible if we use a suitable nucleus A (called TH nucleus) that has a high probability to be found in a x-s cluster configuration, so that A = x ⊕ s. The QF break-up of A occurs if, in the interaction between A and a, in the exit channel s keeps the same momentum distribution before its break-up. In this case s is the spectator to the virtual process a + x → c + C [24]. It has been experimentally demonstrated that changes in the Trojan Horse nucleus are not impacting the applicability of the THM, despite the change in the spectator cluster for example from a charged to a neutral particle [25,26]. Figure 1 represents a sketch of the TH process, where it is possible to apply the polar approximation [27] only for small spectator momenta p _s respecting the relation in [28].

The possibility to reach ultra-low energies is given by the binding energy E _B of x inside A, which compensates for the beam energy [29–31]. One of the advantages of the THM is that, using only one beam energy, the variation of the spectator momentum values inside the limit in [28] makes it possible to scan almost the entire energy range of interest.

Once the QF break-up data have been selected from all other reaction mechanisms with the same final state, in the Plane Wave Impulse Approximation (PWIA) the three-body cross section can be factorized as: d 3 σ d Ω c d Ω C d E C ∝ K F | ϕ p s | 2 d σ d Ω C c HOES (1) where the phase space is taken into account by the kinematical factor KF and |ϕ(p _s )|² is the squared Fourier transform of the radial wave function describing the intercluster motion inside the TH nucleus. The factor KF ⋅|ϕ(p _s )|² is calculated by means of a Montecarlo simulation where the momentum distribution has a width fixed to the value measured experimentally, to account for the distortion effects arising at low transferred momenta [32,33].

The extracted (dσ/dΩ) ^HOES obtained inverting Eq. (1) is denoted half-off-energy-shell (HOES), being x virtual and then the entrance channel off-shell, while the exit channel in on-shell (OES). This cross section has to be corrected for the penetration factor, that in case of n-induced reaction is only relative to the centrifugal barrier, and then it has to be normalized to direct data at energies higher than the Coulomb barrier, where the HOES and OES cross sections are proportional.

Details and reviews of the THM can be found in [34,35].

The main s-process neutron source has been singled out in AGB stars where, following the injection of protons from the outer layers into the carbon- and helium-rich shell, they are captured by carbon nuclei, leading to the formation of a ¹³C pocket [36]. Then, at typical temperature range between 0.8 × 10⁸ K and 1 × 10⁸ K [3], ¹³C is burnt through the ¹³C(α, n)¹⁶O reaction producing neutrons. Such temperatures correspond to ¹³C − α center-of-mass energies of about E _13C−α ∼ 140−230 keV. A recent review can be found in Xu et al. [37] and besides, theoretical evaluations are available, using microscopic cluster approach, leading to results in good agreement with the experimental data in [38] and in [39]. It is worth noting that the crucial role of the 1 2 + subthreshold state was already foreseen in [38] despite of the absence of very low energy data at the time.

Owing to its astrophysical importance, the ¹³C(α, n)¹⁶O reaction has been the subject of many direct and indirect studies. Indirect methods are mandatory since direct measurements could not span so far the energy window of astrophysical interest, which lies between 150 and 230 keV. For instance, the most recent direct measurement [40] could only reach about 230 keV in the center of mass system. Extrapolations to even lower energies are then necessary to establish the trend of the ¹³C(α, n)¹⁶O astrophysical factor inside the Gamow window. However, the understanding of the astrophysical region of the S-factor is complicated by the interplay between the rise due to the ¹⁷O 6.356 MeV 1/2⁺ level and the enhancement due to the electron screening effect (see e.g. La Cognata et al. [41] and references therein). The latter introduces a correction of about 20% at the lowest-energy data point of Drotleff et al. [42] at 279 keV, growing up exponentially approaching the energy region of astrophysical interest. Yet, larger uncertainties are introduced by the occurrence of the 6.356 MeV broad resonance in ¹⁷O. Indeed, until 2015 [43], it was commonly assumed that the aforementioned 1/2⁺¹⁷O state was lying −3 keV below the ¹⁷O →¹³C + α dissociation threshold [44]. Precision spectroscopy studies [43] have shown that the location of such near-threshold resonance should be 4.7 keV above the ¹⁷O →¹³C + α dissociation energy.

An additional challenge for direct measurements is the determination of the absolute value of the ¹³C(α, n)¹⁶O astrophysical factor, since neutron detectors are often used to measure the cross section. Already in Heil et al. [45], the scatter of the absolute normalizations of existing data was evident and as large as a factor two. This spread was realistically attributed to systematic errors affecting neutron detection efficiency as a function of its energy. This is the reason that lead Heil et al. [45] to measure the experimental detection efficiency using a setup identical to the one used for the study of the ¹³C(α, n)¹⁶O reaction and to cross check the results using a GEANT4 simulation. Then, the obtained astrophysical factor was used to scale existing ¹³C(α, n)¹⁶O data sets. Earlier THM studies of this reaction used the Heil et al. [45] astrophysical factor for normalization.

In the THM framework, the ¹³C(α, n)¹⁶O S-factor was extracted from the ¹³C(⁶Li, n ¹⁶O)²H QF cross section, measured at 8 MeV beam energy. With this method, neither Coulomb nor centrifugal barriers quell the cross section and electron screening does not shield the ¹³C −⁴He interaction. However, astrophysical energies could be reached thanks to the energy spent to break ⁶Li into to the α − d system as well as to their relative motion. Details on the THM measurement can be found in La Cognata et al. [46,47]. A drawback of the THM is the need of normalization to existing direct data. In those works, the E 13 C − α energy region between ∼ 0.6 − 1.2  MeV was selected, namely a region where the contribution of the threshold state is negligible.

While a THM S-factor in good agreement with the direct ones scaled to match the Heil et al. [45] absolute value was found, a Coulomb modified Asymptotic Normalization Coefficient (ANC) [47] for the 1/2⁺¹⁷O threshold state equal to 7.7 ± 0 . 3 stat − 1.5 n o r m + 1.6 was determined, contradicting the existing independent assessments of the Coulomb modified ANCs, whose weighted average is 3.9 ± 0.5 fm⁻¹. This may be attributed to the systematic errors affecting direct data absolute normalization. This fact triggered further studies, which ended up in the work of Trippella and La Cognata [48]. Indeed, to unequivocally size the contribution of the ¹⁷O threshold state to the reaction rate, many coherent measurements of its ANC were carried out [49–52].

The ANC method is another powerful indirect method meant to extract the direct radiative part of the cross section for reactions of importance for astrophysics: with this technique, in principle, it is in fact possible to study (p,γ), (n,γ) and (α, γ) reactions of astrophysical importance using p-, n- or α-transfer. Such transfer, at low energies, must show a mostly peripheral character, and such a condition must be verified experimentally. The difference between ANC and THM lies in which kind of reaction can be studied–direct and resonant capture, respectively–and in the reaction products: the ANC method, in fact, can be applied just for reactions that have γ-rays in the exit channel (see also [53] for a recent review on the ANC and, particularly [54], for proton [55], for neutron and [56] for alpha captures at low energies from suitable transfer reactions, also using mirror nuclei [57]).

Therefore, a change in the paradigm usually adopted in the THM application was implemented and the THM S-factor was normalized to the ANC of 6.356 MeV state. This approach lead to a concordance scenario for the ¹³C (α, n)¹⁶O S-factor, where both direct (some data sets with a more appropriate normalization) and indirect data (both ANC and THM) accurately agree.

With the normalization of Trippella and La Cognata [48], the astrophysical factor in Figure 2 (shown as an orange band upper, where middle and lower orange lines are used for recommended + 1σ, recommended and recommended − 1σ THM S-factor) was deduced. An ANC of 3.6 ± 0.7 fm⁻¹ has been determined, in perfect agreement with the weighted average of the results in the literature. The corresponding THM S-factor at E _c.m. = 140 keV was found to be 1.8 0 − 0.17 + 0.50 × 1 0 6 MeVb, to be compared with the astrophysical factor in Heil et al. [45] S ( 140 k e V ) = 2 . 2 − 0.8 + 1.1 × 1 0 6 MeVb. More details can be found in Trippella and La Cognata [48]; a recent direct measurement reported in Ciani et al. [40] has confirmed such results, as it is clearly demonstrated in Figure 2, where other direct measurements with their original normalizations are shown.

A key role in the s-process is played by the ²²Ne(α,n)²⁵Mg reaction, the major neutron producer for its weak component (60 < A < 0) in massive stars, influencing all the late evolution of stars, until their death. Despite of the importance of this reaction, in the energy region of astrophysical interest (E _cm = 600 ± 300 keV) three orders of magnitude discrepancies in the cross section of data available in literature are still present, making those data essentially not useful for astrophysical needs [62].

The trend of the excitation function measured so far shows a steep drop in the yield which makes it possible to provide only upper limits to the cross section already at center-of-mass energies of the order of 800 keV, that is at the edge of the region of interest. A clear picture of the status of the art can be found in [63]. Indeed, direct measurements at such low energies are very challenging due to the exponential Coulomb damping (Coulomb barrier is located at 3.5 MeV) of the cross section to values less than 1 μb, pushing the signal-to-noise ratio essentially to zero.

Indirect measurements, such as capture and transfer reactions [63], at low energy are thus needed to supply resonance parameters, such as spectroscopic factors, to be used in the calculation of the reaction rate. In this regard, a recent paper [10] recalculated the reaction rates of ²²Ne(α,n)²⁵Mg and its competitor ²²Ne(α,γ)²⁶Mg, updating energies and spin-parities of all the ²⁶Mg levels, particularly focusing on the results in [64], obtained with ²²Ne(⁶Li,d)²⁶Mg, and ²²Ne (⁷Li,t)²⁶Mg transfer reactions. The authors find that, excluding the results in [64], s-process nucleosynthesis will not substantially change, but including them leads to a better agreement of Ba and Zr isotopic ratios with the data from presolar SiC grains. However, the results reported in [10,64] suggest the need for further measurements, particularly focusing on the excitation function trend, to include also levels interference in the reaction rate evaluation.

To try to deal with this issue, a new measurement was performed at INFN-LNS of Catania (Italy), aimed at bypassing the challenges that prevented to obtain the result until now. To avoid problems related to the poor efficiency in neutron detection, the idea was to obtain the desired cross section applying the detailed balance principle to the inverse reaction: ²⁵Mg(n,α)²²Ne. Then, to avoid the use of neutron beam (and the consequent worsening of the energy resolution) the THM was used, exploiting the well-known p-n structure of the deuteron, used as neutron virtual source. The experiment to measure the 3-body ²H(²⁵Mg,αp)²²Ne cross section was performed in the Camera2000 scattering chamber, because of its 1 m radius that allowed to put detectors as far as it was needed to keep a good angular resolution to apply the THM. In Figure 3 is evident how the use of THM also helped to bypass complex setups. Moreover, THM allowed the use of thin CD₂ targets, to minimize the beam energy spread on the target. The first ²⁵Mg beam of LNS was accelerated with the Tandem accelerator for this experiment.

This measurement was performed with two different experimental approaches, and thus setups. At first, coincidences between α particles and the spectator proton, using Position Sensitive Detectors (PSD), were detected, reconstructing the ²²Ne kinematics offline, identifying the desired ions with ΔE-E technique using thin PSD (35 μm thickness) as ΔE stage. Dealing with the very low energy of the spectator proton below 1 MeV, forced to reduce the acquisition system threshold to its minimum. The second approach tried was to detect ²²Ne and α particles in coincidence, reconstructing the proton kinematics offline. This has implied to change beam energy to stay focused on the region where QF kinematics is most probable, as required by THM, from ⁽⁷⁺⁾²⁵Mg at 74 MeV to ⁽⁶⁺⁾²⁵Mg at 50 MeV. The ΔE stage PSDs were substituted with two Ionization Chambers (5 cm deep), filled with isobutane (p = 65 mb) to detect and separate ²²Ne. The analysis of these data is still ongoing and additional validity tests are mandatory in order to safely apply THM to this specific case.

It is worth noting that many other possibilities of indirect measurements can be considered. The possible extension of the THM to p-wave intercluster motion nuclei could open the doors to the use of ²³Ne as TH nucleus for its ²²Ne + p configuration, and another good candidate TH nucleus can be the ²⁶Mg as ²²Ne+α in s-wave. In any case, the possibility of new highly ⁴He-enriched solid targets obtained with NESTOR [65], the new ECR ion source at LNS, will be pivotal for these measurements.

The presence of certain nuclei lighter than Fe produced in the previous stages of stellar evolution, causes the capture of a large fraction of the neutrons produced by the ¹³C(α,n)¹⁶O and ²²Ne(α,n)²⁵Mg reactions. These neutron poisons reactions significantly constrain the s-process efficiency [1]. As an example, due to the large abundances of ¹²C and ¹⁶O, the ¹²C(n,γ)¹³C and ¹⁶O(n,γ)¹⁷O reactions are serious neutrons consumer, removing a non-negligible fraction of the produced neutrons [5].

The use of THM is helping in achieving neutron producers and poisons reaction cross sections at energies relevant for the s-process in stars, where they are otherwise unreachable, because of the Coulomb barrier presence or the struggling with the neutron detection.

For ¹³C(α,n)¹⁶O the results obtained with THM has already moved forward our knowledge in this sense, as it summarized in Trippella and La Cognata [48]. On the contrary, much work remains to be done for ²²Ne(α,n)²⁵Mg, where the results of an ongoing analysis of a TH measurement can boost our knowledge on its reaction rate. This has not been found out yet, despite of recent indirect measurements clarifying the role of crucial levels in ²⁶Mg. The same holds for the neutron poisons reactions, which affect the neutron flux available for the s-process nucleosynthesis, mainly ¹⁷O(n,α)¹⁴C and ¹⁴N(n,p)¹⁴C. Also in these cases, ongoing analysis on TH measurements will help in completing the picture of our understanding of the neutron involving reactions in the stellar plasma.

Thus, the overall s-process nucleosynthesis understanding can be strongly constrained by the results of these reaction rates, jointly with the measurements of half-lives of radioactive nuclei at the s-process path branching points.