The Trojan horse method in nuclear astrophysics: recent results

In this book a collection of the lecture notes given during the Eighth European Summer School on Experimental Nuclear Astrophysics is given. The school, whose first edition was first held in 2003, took place from 13 to 20 of September 2015 in Santa Tecla, a small village about 15 km north of Catania, characterized by its position on the volcanic shores of the Ionian Sea, surrounded by the spectacular "Timpa" area, a green protected park specific for its mediterranean vegetation. 80 young students and researchers from more than 20 countries attended the lectures and were also encouraged to present their work and results.

The study of the energy production in stars and related nucleosyntesis processes requires increasingly precise knowledge of the nuclear reaction cross section and reaction rates at interaction energy. In order to overcome the experimental difficulties, arising from small cross-sections involved in charge particle induced reactions at astrophysical energies, and from the presence of electron screening, it was necessary to introduce indirect methods. Trough these methods it is possible to measure cross sections at very small energies and retrieve information on electron screening effect when ultra-low energy direct measurements are available. The Trojan Horse Method (THM) represents the indirect technique to determine the bare nucleus astrophysical S-factor for reactions between charged particles at astrophysical energies. The basic theory of the THM is discussed in the case of non-resonant.

The Trojan Horse method is a powerful indirect technique to determine the astrophysical factor for binary rearrangement processes A+x→b+B at astrophysical energies by measuring the cross section for the Trojan Horse (TH) reaction A+a→B+b+s in quasi free kinematics. The Trojan Horse Method has been successfully applied to many reactions of astrophysical interest, both direct and resonant. In this paper, we will focus on direct sub-processes. The theory of the THM for direct binary reactions will be shortly presented based on a few-body approach that takes into account the off-energy-shell effects and initial and final state interactions. Examples of recent results will be presented to demonstrate how THM works experimentally.

Because of the Coulomb barrier, reaction cross sections in astrophysics cannot be accessed directly at the relevant Gamow energies, unless very favourable conditions are met (e.g. LUNA—underground experiments). Theoretical extrapolations of available data are then needed to derive the astrophysical S(0)-factor. Various indirect processes have been used in order to obtain additional information on the parameters entering these extrapolations. The Trojan Horse Method is an indirect method which might help to bypass some of the problems typically encountered in direct measurements, namely the presence of the Coulomb barrier and the effect of the electron screening. However, a comparison with direct data in an appropriate energy region (e.g. around the Coulomb barrier) is crucial before extending the method to the relevant Gamow energy. Additionally, experimental and theoretical tests are needed to validate the assumptions underlying the method. The application of the Trojan Horse Method to some cases of interest is discussed.

We discuss recent developments in indirect methods used in nuclear astrophysics to determine the capture cross sections and subsequent rates of various stellar burning processes, when it is difficult to perform the corresponding direct measurements. We discuss in brief, the basic concepts of Asymptotic Normalization Coefficients, the Trojan Horse Method, the Coulomb Dissociation Method, (d,p), and charge-exchange reactions.

We discuss indirect methods that make use of transfer reactions to determine cross sections of reactions in stellar burning processes. We focus on two of them that have been extensively used in the past decades: the asymptotic normalization coefficients method and the Trojan horse method. We provide a comprehensive description of their theoretical as well as basic experimental features.

Nuclear reactions within stars typically occur at energies significantly below 1 MeV. Consequently, the Coulomb barrier exponentially suppresses the cross section, reducing it to values as small as a few nanobarns for charged particles. This challenge in obtaining accurate input data for astrophysics has led to the introduction of indirect methods. Specifically, techniques such as ANC and THM have been employed to derive cross sections for reactions involving photons and charged particles in the exit channel, respectively, eliminating the need for extrapolation. The discussion delves into recent results from the application of these methods. For instance, the 6Li(3He,d)7Be measurement is utilized to deduce the ANC’s of the 3He+4He→ 7Be and p+6Li→ 7Be channels, along with their corresponding radiative-capture cross sections. Additionally, the THM measurement of the 27Al(p, α)24Mg cross section via the 2H(27Al,α 24Mg)n reaction is highlighted. In both cases, the cross section at astrophysical energies has been established with unprecedented accuracy.

Our knowledge on how stars evolve depends on understanding two key factors: energy production and chemical evolution. Both factors are determined by thermonuclear reactions. Unfortunately, directly measuring them in a laboratory can be challenging: often, the reaction cross section are too small or require radioactive targets that are impractical to work with. To overcome these challenges, indirect techniques have been developed. One such technique, the Trojan Horse Method, makes use of transfer reactions to determine cross sections of reactions relevant in stellar burning processes. Here we will explore the Trojan Horse Method in the relevant details, explaining how it is used in nuclear astrophysics and we will delve into examples of THM measurements.

Nuclear astrophysics aims to measure nuclear-reaction cross sections of astrophysical interest to be included into models to study stellar evolution and nucleosynthesis. Low energies, < 1 MeV or even < 10 keV, are requested for this is the window where these processes are more effective. Two effects have prevented to achieve a satisfactory knowledge of the relevant nuclear processes, namely, the Coulomb barrier exponentially suppressing the cross section and the presence of atomic electrons. These difficulties have triggered theoretical and experimental investigations to extend our knowledge down to astrophysical energies. For instance, indirect techniques such as the Trojan Horse Method have been devised yielding new cutting-edge results. In particular, I will focus on the application of this indirect method to resonance reactions. Resonances might dramatically enhance the astrophysical S(E)-factor so, when they occur right at astrophysical energies, their measurement is crucial to pin down the astrophysical scenario. Unknown or unpredicted resonances might introduce large systematic errors in nucleosynthesis models. These considerations apply to low-energy resonances and to sub-threshold resonances as well, as they may produce sizable modifications of the S-factor due to, for instance, destructive interference with another resonance.

The Trojan Horse Method is a successful indirect technique for nuclear astrophysics. It allows one to measure astrophysical rearrangement reactions down to the relevant energies, providing a successful alternative path to measure the astrophysical S(E) factor. The basic features will be discussed and some recent results will be presented.

We discuss the use of one-nucleon breakup reactions of loosely bound nuclei at intermediate energies as an indirect method in nuclear astrophysics. These are peripheral processes, therefore we can extract asymptotic normalization coefficients (ANC) from which reaction rates of astrophysical interest can be inferred. To show the usefulness of the method, three different cases are discussed. In the first, existing experimental data for the breakup of 8B at energies from 30 to 1000 MeV/u and of 9C at 285 MeV/u on light through heavy targets are analyzed. Glauber model calculations in the eikonal approximation and in the optical limit using different effective interactions give consistent, though slightly different results, showing the limits of the precision of the method. The results lead to the astrophysical factor S_17(0)=18.7+/-1.9 eVb for the key reaction for solar neutrino production 7Be(p,\gamma)8B. It is consistent with the values from other indirect methods and most direct measurements, but one. Breakup reactions can be measured with radioactive beams as weak as a few particles per second, and therefore can be used for cases where no direct measurements or other indirect methods for nuclear astrophysics can be applied. We discuss a proposed use of the breakup of the proton drip line nucleus 23Al to obtain spectroscopic information and the stellar reaction rate for 22Mg(p,\gamma)23Al.

Breakup of loosely bound nuclei at intermediate energies as indirect method in nuclear astrophysics: 8B, 9C and the S17, S18 astrophysical factors

It is well known that measuring cross-sections of thermonuclear reactions at the low energies typical of astrophysical sites is very difficult. This is due to the presence of the Coulomb barrier between the interacting nuclei. For non-explosive scenarios at astrophysical sites, the relevant energies typically span from few tens to few hundreds of keV while the Coulomb barrier is in the order of MeV. The fusion processes then proceed via tunnel effect and their cross-sections are strongly depending on the probability of penetration through the barrier. In a first approximation, this probability is given by the Gamow factor. Owing to the exponential decrease in this factor with energy, the cross-section values of thermonuclear fusion processes in stellar systems often reach values as small as micro- and nanobarn and even lower ones. Neutron-induced reactions, in spite of the absence of Coulomb barrier, are also difficult to measure. Indeed, it is the possible presence of a centrifugal barrier that can hinder the measurement of the values of the cross-sections of these processes. In either cases the cross-sections of astrophysical nuclear processes result in experimental difficulties, due to the low signal-to-noise ratio, that have been a challenge for scientists since the setting of this scientific field: Nuclear Astrophysics. In the last two to three decades, experimental improvements, including the construction of underground laboratories, allowed for the first time the measurement of cross-sections of astrophysical nuclear processes in the relevant energy region for astrophysics. Also, indirect methods were developed. As a general and common feature, using these methods it is possible to relate the features --typically the cross-section-- of a process that is experimentally simpler to measure, although not directly linked to astrophysics, to those of another process that is of interest for this latter field. This didactic paper will briefly describe some of these indirect methods with a special emphasis on Trojan Horse and on its application also to reactions that involve the use of radioactive ion beams and to neutron-induced reactions.

The knowledge of nucleosynthesis and of energy production in stars requires an increasingly precise measurement of nuclear fusion reactions at the Gamow energy. Because of the Coulomb barrier reaction cross sections in astrophysics cannot be accessed directly at ultra -low energies, unless very favorable conditions are met. Moreover, the energies characterizing nuclear processes in several astrophysical contexts are so low that the presence of atomic electrons must be taken into account. Theoretical extrapolations of available data are then needed to derive astrophysical S(E)-factors. To overcome these experimental difficulties the Trojan Horse Method (THM) has been introduced. The method provides a valid alternative path to measure unscreened low-energy cross sections of reactions between charged particles, and to retrieve information on the electron screening potential when ultra-low energy direct measurements are available. While the theory has been discussed in detail in some theoretical works, present in the scientific literature, also in relation to different types of excitation functions (e.g. non-resonant and resonant), work on detailed methodology used to extract the events to be considered for the bare nucleus cross section measurements is still on going. In this work we will present some critical points in the application of THM that deserve to be discussed in more detail.

Fluorine nucleosynthesis represents one of the most intriguing open questions in nuclear astrophysics. It has triggered new measurements which may modify the presently accepted paradigm of fluorine production and establish fluorine as an accurate probe of the inner layers of asymptotic giant branch (AGB) stars. Both direct and indirect measurements have attempted to improve the recommended extrapolation to astrophysical energies, showing no resonances. In this work, we will demonstrate that the interplay between direct and indirect techniques represents the most suitable approach to attain the required accuracy for the astrophysical factor at low energies, keV, which is of interest for fluorine nucleosynthesis in AGB stars. We will use the recently measured direct astrophysical factor in the energy interval to renormalize the existing Trojan Horse Method (THM) data spanning the astrophysical energies, accounting for all identified sources of uncertainty. This has a twofold impact on nuclear astrophysics. It shows the robustness of the THM approach even in the case of direct data of questionable quality, as normalization is extended over a broad range, minimizing systematic effects. Moreover, it allows us to obtain more accurate resonance data at astrophysical energies, thanks to the improved direct data. Finally, the present work strongly calls for more accurate direct data at low energies, so that we can obtain a better fitting of the direct reaction mechanism contributing to the astrophysical factor. Indeed, this work points out that the major source of uncertainty affecting the low-energy S(E) factor is the estimate of the non-resonant contribution, as the dominant role of the 113 keV resonance is now well established.