“Other” indirect methods for nuclear astrophysics

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.

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.

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

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.

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.

In this paper we address the indirect method, which can provide a powerful technique to obtain information about radiative capture reactions at astrophysically relevant energies. The idea of the indirect method is to use the indirect reaction $A(a, s\,\gamma)F$ to obtain information about the radiative capture reaction $A(x,\,\gamma)F$, where $a=(s\,x)$ and $F=(x\,A)$. The main advantage of using the indirect reactions is the absence of the penetrability factor in the channel $x+A$, which suppresses the low-energy cross sections of the $A(x,\,\gamma)F$ reactions and does not allow to measure these reactions at astrophysical energies. A general formalism to treat indirect resonant radiative capture reactions is developed when only a few intermediate states do contribute and statistical approach cannot be applied. Angular dependence of the triple differential cross section at fixed scattering angle of the spectator $s$ is the angular $\gamma-s$ correlation function. Using indirect resonant radiative capture reactions one can obtain the information about important astrophysical resonant radiative capture reactions, like $(p,\,\gamma), \,\,(\alpha,\,\gamma)$ and $(n,\,\gamma)$ on stable and unstable isotopes. The indirect technique makes accessible low-lying resonances, which are close to the threshold, and even subthreshold bound states located at negative energies. In this paper, after developing the general formalism, we demonstrated the application of the indirect reaction ${}^{12}{\rm C}({}^{6}{\rm Li},d\,\gamma){}^{16}{\rm O}$ proceeding through $1^{-}$ and $2^{+}$ subthreshold bound states and resonances to obtain the information about the ${}^{12}{\rm C}(\alpha,\,\gamma){}^{16}{\rm O}$ radiative capture at astrophysically most effective energy $0.3$ MeV what is impossible using standard direct measurements.

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.

Some general features of indirect approaches for determining cross sections of astrophysically relevant reactions at low energies are discussed from a theoretical point of view. The methods of Coulomb dissocation (CD), of the asymptotic normalization coefficient (ANC) and the Trojan-horse method (THM) are considered.

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 relativistic radioactive beams

Indirect Methods in Nuclear Astrophysics with Transfer Reactions

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.

Based upon a consistent description of the wave functions of the captured state and the final state, a unified formalism has been established to investigate the nonstatistical effects in radiative nucleon capture reactions at energies ranging from 0.1 to 20 MeV. This model includes the direct-semidirect capture and radiative capture in the compound elastic and inelastic channels. By applying this model theory and including also the statistical contributions of the compound-nucleus process, the radiative neutron capture cross sections in $^{27}\mathrm{Al}$, $^{55}\mathrm{Mn}$, $^{89}\mathrm{Y}$, and $^{208}\mathrm{Pb}$ have been calculated and the results are compared with the measured data. We have also investigated the relative importance of the operation for each nonstatistical mechanism in the total capture cross sections.

The DRAGON recoil mass separator at TRIUMF exists to study radiative proton and alpha capture reactions, which are important in a variety of astrophysical scenarios. DRAGON experiments require a data acquisition system that can be triggered on either reaction product ($\gamma$ ray or heavy ion), with the additional requirement of being able to promptly recognize coincidence events in an online environment. To this end, we have designed and implemented a new data acquisition system for DRAGON which consists of two independently triggered readouts. Events from both systems are recorded with timestamps from a $20$ MHz clock that are used to tag coincidences in the earliest possible stage of the data analysis. Here we report on the design, implementation, and commissioning of the new DRAGON data acquisition system, including the hardware, trigger logic, coincidence reconstruction algorithm, and live time considerations. We also discuss the results of an experiment commissioning the new system, which measured the strength of the $E_{\text{c}.\text{m}.} = 1113$ keV resonance in the $^{20}$Ne$\left(p, \gamma \right)^{21}$Na radiative proton capture reaction.