Indirect methods in nuclear astrophysics: Recent results from ANC and THM

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.

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

Nuclear reactions within stellar environments typically manifest at energies well below 1 MeV. As a consequence, the Coulomb barrier strongly suppresses the cross section, diminishing it to values as minute as a few nanobarns for charged particles. This challenge in obtaining precise input data for astrophysics has prompted the utilization of indirect methodologies. Specifically, approaches such as ANC and THM have been employed to ascertain cross sections for reactions involving photons and charged particles in the exit channel, respectively, obviating the necessity for extrapolation. The discourse explores recent findings arising from the application of these methodologies. For example, the measurement of 6Li(3He,d)7Be is employed to infer the ANC’s of the 3He+4He→ 7Be and p+6Li→ 7Be channels, along with their corresponding radiative-capture cross sections. Furthermore, the THM measurement of the 27Al(p, α)24Mg cross section via the 2H(27Al,α 24Mg)n reaction is emphasized. In both instances, the cross section at astrophysical energies has been ascertained with unparalleled precision.

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.

Nuclear reactions among charged particles in stars take place at energies generally well below the Coulomb barrier, so its penetration factor exponentially suppresses the cross section down to values as small as few nanobarns or picobarns. Reaching astrophysical energies opens new challenges and calls for new approaches. In this work, the scope of nuclear astrophysics will be introduced and how experiments are usually conducted will be discussed. In particular, we will focus on the use of indirect methods as complementary approaches to direct measurements, introducing the asymptotic normalisation coefficient (ANC) technique and the Trojan Horse Method (THM), used to deduce the cross sections of reactions with photons and charged particles in the exit channel, respectively, with no need for extrapolation. Recent results of the application of the two methods will be exposed: the \(^6\mathrm {Li}({^3\mathrm {He}},d)^7\mathrm {Be}\) measurement used to deduced the ANCs of the \(^3\mathrm {He}+{^4\mathrm {He}}\to {^7\mathrm {Be}}\) and \(p+{^6\mathrm {Li}}\to {^7\mathrm {Be}}\) channels and the corresponding radiative capture cross sections. Then, the THM measurement of the \(^{27}\mathrm {Al}(p,\alpha )^{24}\mathrm {Mg}\) cross section through the \(^2\mathrm {H}(^{27}\mathrm {Al},\alpha {^{24}\mathrm {Mg}})n\) reaction will be reviewed, as well as the \(^{12}\mathrm {C}+{^{12}\mathrm {C}}\) fusion reaction cross section using \(^{14}\)N to transfer \(^{12}\)C and induce the reaction of astrophysical importance down to astrophysical energies. The indirect measurements made it possible to assess the occurrence of several resonances that are responsible for significant changes in the reaction rate at relevant temperatures. Abstract Published by the Jagiellonian University 2025 authors

Our understanding of stellar evolution depends on knowing beta-decay rates and reaction cross sections for a wide range of nuclear capture reactions. Direct laboratory measurements of important stellar reaction rates are hindered by low cross sections and, in some cases, the need for radioactive targets. Indirect techniques have been developed to determine reaction rates for systems that are particularly difficult to measure in a direct experiment. One of the indirect techniques involves measurements of Asymptotic Normalization Coefficients (ANCs) which determine the direct-capture contribution to a capture reaction. Also ANCs can be used to understand the role of subthreshold states in stellar capture. This essay gives an introduction to ANCs and describes how they are used in nuclear astrophysics. Examples are given of measurements which have been carried out with both stable and radioactive beams.

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.

Indirect methods in nuclear astrophysics with relativistic radioactive beams

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.

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.

Reactions induced by Rare Isotope Beams (RIB) at low or intermediate energies represent a major tool to enrich our knowledge about exotic nuclei (along with decay studies and mass measurements). They allowed us, in the last decade or more, to go beyond the beta‐stability line with our structure or reaction mechanism studies, and are considered to be the way low energy nuclear physics will go. I will present here a few of the types of reactions currently used: elastic scattering, single‐nucleon transfer, breakup at intermediate energies. I will focus on the experimental aspects but, while remaining general, due to my own experience, I will concentrate on reactions that are indirect methods in nuclear astrophysics. The material will, hopefully, prepare you for some of next week’s subjects.

Indirect Methods in Nuclear Astrophysics with Transfer Reactions

We discuss the use of one‐nucleon removal reactions of loosely bound nuclei at intermediate energies as an indirect method in nuclear astrophysics. The breakup reactions are proved to be good spectroscopic tools and can be used to study a large number of loosely bound proton‐ or neutron‐rich nuclei over a wide range of beam energies. As peripheral processes, they can be used to extract asymptotic normalization coefficients (ANCs) from which non‐resonant capture reaction rates of astrophysical interest can be calculated parameter free. In this talk, we present results of a proton‐breakup experiment carried out at GANIL (France) with a cocktail beam centered around 23Al at 50 MeV/nucleon. Momentum distributions of the breakup fragments, inclusive and in coincidence with gamma rays detected by EXOGAM Germanium clover array, were measured in the focal plan of SPEG energy‐loss spectrometer. We present in particular the investigations of reaction rates for 22Mg(p,γ)23Al and 23Al(p,γ)24Si important for novae and X‐ray bursts, respectively.

We discuss the use of one‐nucleon removal reactions of loosely bound nuclei at intermediate energies as an indirect method in nuclear astrophysics. These breakup reactions are good spectroscopic tools and can be used to study a large number of loosely bound proton‐ or neutron‐rich nuclei over a wide range of beam energies. They are peripheral processes that can be used to extract asymptotic normalization coefficients (ANC) from which direct capture proton reaction rates of astrophysical interest can be calculated parameter free. We emphasize the importance of reaction model calculations and of exclusive measurements to check them. We review several cases: the breakup of 8B, 9C, 15C and 23Al. Firrst we review how we have used the data for the breakup of 8B at energies from 30 to 1000 MeV/nucleon on light and heavy targets to extract the astrophysical factor S17(0) = 18.7±1.9 eV⋅b for the key reaction for solar neutrino production. Glauber model calculations in the eikonal approximation and in the optical l...