Transfer Reactions As a Tool in Nuclear Astrophysics
- IJCLab, Université Paris-Saclay, CNRS/IN2P3, Orsay, France
Nuclear reaction rates are one of the most important ingredients in describing how stars evolve. The study of the nuclear reactions involved in different astrophysical sites is thus mandatory to address most questions in nuclear astrophysics. Direct measurements of the cross-sections at stellar energies are very challenging–if at all possible. This is essentially due to the very low cross-sections of the reactions of interest (especially when it involves charged particles), and/or to the radioactive nature of many key nuclei. In order to overcome these difficulties, various indirect methods such as the transfer reaction method at energies above or near the Coulomb barrier are used to measure the spectroscopic properties of the involved compound nucleus that are needed to calculate cross-sections or reaction rates of astrophysical interest. In this review, the basic features of the transfer reaction method and the theoretical concept behind are first discussed, then the method is illustrated with recent performed experimental studies of key reactions in nuclear astrophysics.
Our understanding of stellar evolution in the Universe has been largely improved thanks to the interaction between three fields: observation, stellar modeling and nuclear physics. All these fields are in constant development: new telescopes and satellites open more and more windows on the cosmos, stellar modeling relies on ever-increasing computing and nuclear physics takes advantage of new facilities (radioactive beams, high-intensity beams, underground laboratories) and sophisticated detection systems.
Nuclear reaction rates are one of the most important ingredients in describing how stars evolve. The study of the nuclear reactions involved in different astrophysical sites is thus essential to address most questions in nuclear astrophysics.
Experimental techniques for determining cross sections fall into two main categories: direct measurements, in which the reaction of interest is reproduced, even though the energy range may be different from that of the stellar site and indirect measurements, in which a different reaction is coupled with theoretical modeling to obtain the cross-section of interest or to access the spectroscopic properties (excitation energies, spins and parities, decay widths, … ) of the nuclei involved.
Direct measurements at stellar energies are very challenging - if at all possible. This is mainly due to the very small cross-sections (sub nanobarns) of the reactions of interest (in particular when charged particles are involved), and/or to the radioactive nature of many key nuclei.
Although direct measurements of the charged-particle cross-sections are possible at the energies of interest in some cases, they are often carried out at higher energies and then extrapolated down to the energies of astrophysical interest using
The other issue concerning direct measurements is due to the radioactive nature of the nuclei involved in the reactions occurring in explosive sites (classic novae, supernovae, X-ray bursts, … ) or in the radiative captures (n,γ) in the r-process [2–4] and sometimes in the s-process . Here, the cross sections at stellar energies are often substantial but their study requires either the production of radioactive beams (which intensity is often weak, rarely exceeding 105 or 106 pps) or, for nuclei with relatively long half life, the production of radioactive targets with a sufficiently large areal density, which is often very difficult. Therefore, direct measurements of such reactions are very challenging, and in the case of r-process reactions, they are impossible.
To overcome these problems (sub–threshold resonances, radioactive nuclei, … ) indirect techniques such as transfer reaction method , Coulomb dissociation method [7–9], Asymptotic Normalization Coefficient (ANC) method [10–12], surrogate reactions  and Trojan Horse Method (THM) [14–16] are good alternatives. In these various methods, the experiments are usually carried out at higher energies than the Coulomb barrier which implies higher cross-sections than in direct measurements. Moreover these methods allow also the use of stable beams to study reactions involving radioactive nuclei not far from the valley of stability. However these methods lead to results which depend on the choice of the model and its parameters in addition to the experimental errors. This is why, to reduce the overall uncertainty on the cross sections of the reactions, it is important to combine various experimental approaches.
We would like to emphasize that ANC and Trojan Horse methods as well as surrogate method are also based on transfer reactions. However, the THM and ANC method require particular kinematics conditions. For instance the transfer reactions used in ANC method need to be performed at energies where the reaction process is very peripheral in order to deduce an ANC value weakly sensitive on the potential parameters. Concerning the Trojan Horse Method, it consists in obtaining information on the two-body reaction of astrophysical interest at low relative energies by studying a three body reaction at energies above the Coulomb barrier. The basic idea of this method relies on the assumption that the three body reaction can occur via a quasi free reaction mechanism that is dominant at particular energies and angles. For the surrogate method, the transfer reaction is used to populate the resonant states of interest and then measure their decay probability to deduce the cross section of the reaction of interest from the product of the measured quantity and the calculated compound nucleus formation cross-section. All these particular transfer reactions will not be discussed in this manuscript except the ANC method which will be described a little bit more in section 3.2.4.
In this review we focus on the transfer reaction method where a composite nucleus is produced in a two body reaction by transferring one or several nucleons from a projectile to a target nucleus. Transfer reactions are a unique tool to access key spectroscopic information concerning the structure of the composite nucleus. In particular the spectroscopic factor
In the next section we will present the type of reactions (resonant and direct capture) which can be studied with transfer reactions. In Section 3, the description of the method and basic theoretical concepts behind such reactions are recalled. In Section 4, the experimental needs and challenges for transfer reaction studies are presented. Some examples of recently performed experimental studies using stable and radioactive beams together with a variety of detection systems are presented in Section 5. We then conclude with some perspectives in a last section.
2 Nuclear Reactions of Astrophysical Interest
Thermonuclear reaction rates are key physical inputs to computational stellar models, and they are defined per particle pair (in cm3 s-1) as :
where µ is the reduced mass of the interacting nuclei, k is the Boltzmann constant, T is the temperature at which the reaction rate is evaluated, and
FIGURE 1. (Color online) Schematic view of a resonant (left panel) and direct (middle panel) capture process for a radiative capture reaction. The contribution of these two processes is shown in the case of the 17O(p,γ)18F reaction (right panel) where the energy dependence of the astrophysical S-factor is presented (adapted from ref. 18).
2.1 Resonant Capture
In a resonant capture reaction
where λ is the de Broglie wavelength,
Transfer reactions are powerful tools to derive several quantities needed to calculate the cross-section given by Equation 2. They can be used to determine the transferred orbital angular momentum
The single-particle width is the decay probability of the compound nucleus state
2.2 Direct Capture
The direct capture is an electromagnetic process which can not be neglected at low energies, and which may even be dominant for radiative captures where the level density is low  and the compound states lie at higher energies than the energies of interest .
where E is the energy of the incident projectile and the sum runs over all available final bound states i of the residual nucleus,
In case of a dipole transition,
3 Transfer Reaction Method
Transfer reactions in which one nucleon or a cluster of nucleons are exchanged between the target and the projectile are often used in nuclear structure studies to determine the energy position and the orbital occupation of the excited states of many nuclei. Likewise it is widely used in nuclear astrophysics to determine the partial decay widths of nuclear states involved in resonant reactions, and to evaluate the direct capture cross-section.
3.1 General Concepts
Let’s consider the simple case of a radiative capture
FIGURE 2. Sketch of a transfer reaction where the particle x is transferred from the projectile a to the target nucleus A forming the final state
Once the particle x is transferred to the target A, the projectile component c will continue its movement and should be detected. By measuring its emission angle and energy, the energy of the populated states in nucleus C can be obtained using two-body kinematic properties if the masses of the interacting nuclei are known. A precise measurement of the energies of the excited states of interest is very important to calculate accurately the resonance energies involved in the evaluation of the thermonuclear reaction rates.
From a comparison of the shape of the measured angular distributions to those predicted by theory, it is possible to deduce the transferred angular momentum
When the direct transfer mechanism is dominant the measured transfer angular distributions are often analyzed using the Distorted Wave Born Approximation (DWBA) formalism (see Section 3.2.1). However other reaction mechanisms such as the compound nucleus mechanism, the multi-step transfer reaction mechanism, the projectile breakup and the transfer to continuum  can occur. The contribution of these mechanisms can be evaluated by using Hauser Feshbach calculations , coupled reaction channel calculations (CRC), adiabatic distorted wave approximation (ADWA) (see Section 3.2.5) and continuum discretized coupled channel (CDCC) calculations , respectively.
The angular distributions of direct reactions display a characteristic shape which often shows a forward protuberant peak and smaller peaks at larger center-of-mass angles (see Figure 5 in Section 5.2). This is in contrast to the compound nucleus mechanism where the angular distribution shows an almost flat and symmetric shape with respect to 90° in the center-of-mass. Hence to be more sensitive to the direct reaction mechanism, transfer measurements need to be performed at relatively small detection angles (typically
3.2 Elements of Theory
3.2.1 Distorted Wave Born Approximation
The most commonly used theoretical model to describe direct transfer reaction cross-sections is the Distorted Wave Born Approximation (DWBA) which relies on the following assumptions:
• the entrance and exit channels processes are dominated by the elastic scattering
• the transfer process is weak enough to be treated as a first order perturbation
• the nucleon(s) transfer occurs directly between the two active channels
• the transferred nucleon(s) is directly deposited on the final state with no rearrangement of the core configuration
The spectroscopic factor
The product of the spectroscopic factors corresponding to the configuration of the
3.2.2 Finite-Range and Zero-Range Calculations
The calculation of the DWBA transfer reaction cross-section involves the evaluation of the transition amplitude (see Equation 7) which is of the form of a six-dimensional integral over the two relative coordinate variables
The zero-range (ZR) approximation is usually a good assumption when calculating the cross-section of a direct transfer reaction induced by light projectile. In the case of the typical
3.2.3 Reduced and Partial Decay Widths
where µ is the reduced mass of the
In case of unbound states the partial decay width
A common procedure to determine the partial decay width for an unbound state is to use the weakly-bound approximation. In this approach the radial form factor is calculated for a very weakly bound state (typical binding energies between 5 and 50 keV) and is further used to calculate the reduced decay width using Equation 11. The partial width is then obtained with Equation 13 evaluated at the energy of the resonance. It has been shown that for proton or neutron resonances having single-particle widths small compared to their resonance energy, the weakly-bound approximation gives spectroscopic information within 15% with results obtained using an unbound form factor .
3.2.4 Asymptotic Normalization Coefficients
As mentioned in the introduction, the ANC method is a particular case of transfer reactions. It relies on the peripheral nature of the reaction process that makes the calculations free from the geometrical parameters (radius, diffuseness) of the binding potential of the nucleus of interest and less sensitive to the entrance and exit channel potentials. The ANC method was extensively used for direct proton-capture reactions of astrophysical interest where the binding energy of the captured charged particle is low  and also for reactions where the capture occurs through loosely sub-threshold resonance states [36, 37]. These very peripheral transfer reactions performed at sub-Coulomb energies are good tools to determine asymptotic normalization coefficients (ANCs) which are weakly sensitive to the calculations and which may be linked to the partial width of a resonance . Nevertheless, ANC’s can also be determined from transfer reactions performed at energies above the Coulomb barrier.
The asymptotic normalization coefficient C describes the amplitude of the tail of the radial overlap function at radii beyond the nuclear interaction radius and in case of a bound state it can be related to the spectroscopic factor using the following expression :
3.2.5 Adiabatic Distorted Wave Approximation
When one of the participants of the transfer reaction is a loosely bound system, the DWBA may not be suited to analyze the data since the breakup of this system becomes an important additional reaction channel to consider. This is the case when deuterons are involved in the transfer reaction since they can break up easily into their constituents due to their small binding energy (
An interesting feature of the ADWA method is that its implementation is very similar to the DWBA calculations, thus any pre-existing inputs for the DWBA calculations can be easily adapted to perform ADWA calculations. The only difference is in the optical model potential parameters (see Section 3.3.1) describing the interaction of the deuteron with the target. While in the DWBA this potential is adjusted to reproduce the elastic scattering differential cross-section, it is no longer the case for the ADWA since it includes the treatment of the deuteron breakup. Therefore the optical potential used in ADWA will not be adapted to provide a good description of the deuteron elastic scattering, but it will give instead a better description of the transfer differential cross-section.
3.3 Ingredients for a Distorted Wave Born Approximation Calculation
To calculate the transfer DWBA differential cross sections, a number of computer codes are available such as FRESCO , DWUCK , TWOFNR  and PTOLEMY , to cite a few of them. They all require the same ingredients which are the distorted waves in the entrance and exit channel, and the two overlap functions which describe the relative motion of the transferred nucleon in the projectile and in the final state (see Sec. 3.2.1). These ingredients are calculated using optical model and interaction potentials whose parameters are the main inputs for any DWBA code.
3.3.1 Distorted Waves
The distorted waves are the solution of the Schrödinger equation for elastic scattering by an appropriate optical-model potential. This potential has usually a central (both real and imaginary parts), a spin-orbit and a Coulomb component; and its most common shape is a Woods-Saxon well. The best way to determine the potential parameters is to analyze the differential cross-section of the elastic scattering in the entrance and exit reaction channel at the same energy as the reaction under study. When elastic measurements are not available, one should use potential parameters deduced from measurements performed in the mass region close to the nuclei of interest at close incident energies. Another alternative is to use global potential parametrisations obtained by fitting a large number of elastic scattering data. The radius, diffuseness and depth of the different components of the potential usually have an energy and Z, A dependence allowing to derive a potential parameter set adapted to the reaction under study. The most commonly used global parametrisations for protons, neutrons and deuterons are those of Perey and Perey , Daehnick et al.  and Koning et al. .
The radial part of the overlap function is usually approximated by the radial part of the wave-function describing the relative motion of the transferred nucleon x to the core A to form the bound state C (see Equation 8). It is obtained by solving the Schrödinger equation for an interaction potential usually having a Woods-Saxon form. In this procedure the depth of the real part of the volume component of the potential is adjusted to reproduce the binding energy of the bound state.
The shape of the bound state wave-function is dictated by the orbitals to which the nucleon or group of nucleons are transferred. In the case of a single nucleon transfer reaction, the nucleon is transferred to an orbital characterized by the usual quantum numbers
The radial form factor strongly depends on the radius and the diffuseness of the potential. Different realistic
The way the projectile is treated depends on the type of DWBA calculation. If the zero-range approximation is used, then it is enough to know the value of
3.4 Uncertainties on Spectroscopic Factors, ANCs and Reduced Widths
The uncertainty associated to the extracted spectroscopic factors depends on the accuracy of the measured differential cross-sections, and mainly on the uncertainties related to the different parameters used in the DWBA calculation. This includes the optical potential parameters used to describe the wave functions of the relative motion in the entrance and exit channels, and the geometry parameters of the potential well describing the interaction of the transferred particle with the core in the final nucleus. In case of a one-nucleon transfer reaction these uncertainties give rise to a typical uncertainty on the spectroscopic factor of about 25–35% , which is increased to 30%–40% in case of an α-particle transfer reaction [52, 53]. However, the DWBA model remains very useful and even essential for reactions that cannot be studied directly and whose uncertainty on cross sections is more than a factor two.
Concerning the reduced widths and ANCs deduced from transfer reactions (see Equations 11, 12 and Equation 14, respectively), their uncertainties depend, not only on the spectroscopic factors uncertainty but also on the potential parameters used to calculate the radial wave function of the relative motion between the transferred nucleon(s) and the core nucleus. Since the spectroscopic factor determination also depends on the aforementioned potential parameters its uncertainty is then correlated to the determination of the radial wave-function. It is then mandatory that the same optical potential parameters must be used in deriving the spectroscopic factor and the radial wave function to determine the reduced width and the ANC. Their associated relative uncertainties may therefore be different from the one of the spectroscopic factors.
We would like to point out that the spectroscopic factors defined here are experimental quantities subject to the uncertainties mentioned above. In theory, they can be defined properly but there is a long discussion in recent years whether they can be considered as a well-defined observable . A direct use of proper many-body wave functions for the structure of the nuclei in the calculation of the matrix elements would remove the problem of defining spectroscopic factors and allow better testing of nuclear structure. However, these many-body calculations are up to now possible only for light nuclei [55–58], and not for most of the nuclei involved in the various nucleosynthesis processes studied in nuclear astrophysics.
4 Experimental Needs and Challenges for Transfer Reaction Studies
We have seen so far that the analysis of experimental angular distributions obtained from two-body transfer reactions is a unique tool to access key spectroscopic information (energy of excited states, spectroscopic factors and transferred angular momentum) concerning the composite nucleus produced by transferring one or several nucleons from a projectile to a target. A sketch of such transfer reaction is given in Figure 2 and in the vast majority of experimental approaches the goal is to measure the energy and angle of the emitted light particle c. It is then possible to determine the excitation energy of the composite nucleus by using the two-body kinematic properties of the reaction. In addition, the number of light particles detected at different angles is the main ingredient used to extract the angular distribution.
While one is usually interested in a specific transfer reaction channel characterized by the light particle c, many other processes ((in)elastic scattering, fusion-evaporation, etc … ) produce many other kinds of particles which need to be disentangled from c. It is therefore a requirement for the experimental detection system to have a good particle identification capability. Moreover it is important that the resolution in the center of mass be the best as possible in order to separate the different excited states of the composite nucleus. Another need for the detection system is to cover the forward angles in the center of mass where the direct mechanism is dominant, thus allowing a good description of the angular distribution by the DWBA method.
While the center of mass frame is best suited for describing the reaction mechanism, the experimental study occurs in the laboratory frame. There are two experimental possibilities to perform a given two-body reaction study: either the projectile is lighter than the target (direct kinematics), or the projectile is heavier than the target (inverse kinematics). Choosing one or the other option will have profound consequences on the nature of the experimental system. To illustrate this point the kinematic lines (
FIGURE 3. (Color online) Kinematic calculations for the 15O(7Li,t)19Ne reaction showing the triton energy as a function of the laboratory angle in case of direct (left panel) and indirect (right panel) kinematics. Both calculations are performed for the same center of mass energy.
The first striking difference is that the forward angles in the center of mass correspond to forward/backward angles in the laboratory frame in case of direct/inverse kinematics6. In addition the tritons have a rather large energy (about 30 MeV) in case of direct kinematics, while the energy is much smaller (about 3 MeV) in case of inverse kinematics. These two observations will dictate very different experimental setups, and the specifics concerning direct and inverse kinematics studies are now detailed.
4.1 Direct Kinematics Studies
Historically transfer reactions were performed using stable beams in direct kinematics. The first detection systems were based on collimated silicon detectors mounted in a
The differential cross section corresponding to a populated state in the residual nucleus is calculated from the light particle yield determined at each detection angle
Transfer reaction studies with stable beams in direct kinematics are rather straightforward. While the spectrometer requires a dedicated hall the complexity of the detection system is usually low with a limited number of electronic channels. The main delicate point in such approach comes from the targets. First because they must be very thin (between tens and hundreds of μg cm−2) in order to limit their contribution to the overall energy resolution budget. They are then extremely delicate to produce and fragile to manipulate. Their purity is another point which deserves a special care because reactions on any other nuclei present in the target may produce unwanted contamination peaks hindering the states of interest. It is then of uttermost importance to have isotopically enriched material when needed, to limit the backing material thickness when the target cannot be self-supported, and to choose carefully the compound form. Concerning the last point and as an example of a transfer reaction on fluorine nuclei, lithium or calcium fluoride targets will not produce the same background, and one or the other compound may be best suited depending on the reaction studied.
4.2 Inverse Kinematics Studies
The advent of radioactive ion beams (RIBs) allows to perform transfer reaction studies involving nuclei far from the valley of stability (see example in Section 5.4). Beam intensities are much smaller than for stable beams and should be preferably at least 105 pps in order to perform a transfer reaction study. The beam properties are one of the crucial aspects in such studies, and depending on how the RIB is produced it may be contaminated with other species and have a large emittance. Therefore, detectors tracking the beam position, such as CATS , PPACs  to cite a few of them, are usually used to reconstruct the position of the incident ions at the target location event by event. Identification of the incident beam with respect to other species is also undertaken with standard time of flight and energy loss techniques. In contrast to the direct kinematics case the solid state targets are much more simpler and easier to handle with CH2, CD2, and LiF being mainly used. On the other hand, gas targets can be very complex and usually relies on supersonic gas jet  or cryogenics  technology.
The fact that RIB intensities are much smaller than in the case of stable beams experiment has a profound impact on the design of the detection setup. Let us illustrate this point with the case of the 15O(7Li,t)19Ne reaction. For a stable beam study of this reaction in direct kinematics we could consider the following typical parameters: a 7Li3+ beam intensity of about 100 pnA, a target thickness of about 100 μg/cm2, and a spectrometer solid angle
Transfer reaction studies with radioactive ion beams are very challenging and require complex experimental setups. Tremendous progresses over the past 20 years have been made concerning the development of both highly efficient and granular charged particles and γ-rays spectrometer. Despite of these achievements RIB transfer experiments typically last between one and two weeks with a limited level of accumulated statistics. However this is a unique way to explore regions of the nuclear charts where some of the most extreme astrophysical processes occur.
5 Examples of Experimental Transfer Reaction Studies
After some general considerations on the type of transfer reactions useful in nuclear astrophysics, three examples will be presented. The first two examples concern the study of the resonant part of the 30Si(p,γ)31P and 13N(α,p)16O reactions studied by means of the one proton (3He,d) reaction, and the α-particle (7Li,t) transfer reaction on the mirror reaction, respectively. The last example concerns the study of the direct capture component of the 60Fe(n,γ)61Fe reaction through the one neutron (d,p) transfer reaction.
5.1 Transfer Reactions in Nuclear Astrophysics
Several transfer reactions can be used to extract spectroscopic factor for the same states of astrophysical interest. The one-proton (3He,d), (4He,t) and (d,n) transfer reactions can be used to extract the proton spectroscopic factor of states involved in proton captures reactions. Similarly the one-neutron (α,3He) and (d,p) transfer reactions can be used to study the resonant and direct components of neutron capture cross-sections. The choice between these transfer reactions is driven by considerations on a good linear and angular momentum matching . The transferred linear momentum depends strongly on the beam energy and on the Q-value of the transfer reaction. Since in nuclear astrophysics small transferred angular momenta are relevant in most of the cases because of the low associated centrifugal barrier, transfer reactions having a smaller Q-value are generally mostly used. As such, the (d,p) and (3He,d) transfer reaction are a very common choice for one-neutron and one-proton transfer reactions, respectively.
In the case of one-proton transfer reactions both the (3He,d) reaction [74, 75] and the (d,n) reaction  have been used extensively, though the neutron detection may bring some experimental complexity. For the one-neutron transfer case the (d,p) reaction has been mostly used [77–79]. Note that the different momentum matching of two reactions transferring the same nucleon can provide useful hints on the nature of the populated states. In that case the same state is populated in a different way according to the reaction, and a distinction between low and high spins may be established (see ref. 80 for a comparison of the (3He,d) and (4He,t) reactions).
The (p,d) and (p,t) pickup reactions are very valuable tools to study proton-rich nuclei of astrophysical interest such as in classical novae and type I X-ray bursts. The Q-values of both reactions are strongly negative, and in case of the (p,t) reaction proton beam energies larger than 30 MeV are often needed favoring the use of cyclotron instead of electrostatic accelerators. The (p,d) direct reaction mechanism can be well described by the DWBA formalism and it is then possible to extract useful spectroscopic information from the analysis of the angular distributions . This is more complicated in the case of (p,t) reactions since the two neutrons can be transferred as a pair in a single step or in the possible two steps (p,d) (d,t) path which requires to know the spectroscopic factor and energy of the intermediate states. This makes the analysis of the angular distribution more delicate  and not as reliable as a single particle transfer reaction. Despite these complications the (p,t) reaction is widely used because of its selectivity which mainly populates natural spin and parity states (if a single step is assumed) of even-even nuclei.
Alpha-particle transfer reactions are very useful to study the spectroscopy of nuclei involved in α-induced reactions such as (α,γ), (α,n) and (α,p) reactions in helium rich environments. The most generally used transfer reactions are the (6Li,d) and (7Li,t) reactions. At the time of early studies the (6Li,d) reaction was used extensively because the
5.2 Case of the 30Si(p,γ)31P Reaction
Globular clusters are vital testing grounds for models of stellar evolution and the early stages of the formation of galaxies. Abundance anomalies such as the enhancement of potassium and depletion of magnesium have been reported in the globular cluster NGC 2419 . They can be explained in terms of an earlier generation of stars polluting the presently observed stars, however, the nature and properties of the polluting sites is not clear (see refs. 88 and 89 for a review). It has been shown that the potential range of temperatures and densities of the polluting sites depends on the strength of a number of critical reaction rates including 30Si
Several resonances are known in the Gamow window
The importance of low-lying resonances above the p + 30Si threshold
FIGURE 4. (Color online) Deuteron magnetic rigidity spectrum at a spectrometer angle of 16°. Excitation energies in 31P between 7.0 and 8.1 MeV are covered. The best fit of the spectrum is shown together with individual contributions for 31P states (red) and contamination peaks (blue).
The differential cross-sections corresponding to populated 31P states were calculated from the deuteron yield determined at each spectrometer angle, and examples are shown in Figure 5 for states populated by different transferred angular momentum . In all cases the rapidly varying cross-section on a limited forward center-of-mass angular range is indicative of states which are populated through a direct mechanism. The differential cross-sections also have a very characteristic shape which depends on the transferred angular momentum, e.g., the position of the first minimum of the cross-section increases with the magnitude of the transferred angular momentum. Finite-range DWBA calculations performed with the FRESCO code  are represented in blue for a selection of bound and unbound states, and a very good agreement is obtained with the experimental data. For unbound states the weakly bound approximation is assumed, and a bound form factor corresponding to a state bound by 10 keV is considered.
FIGURE 5. (Color online) Selection of experimental differential cross-sections of 31P states populated with the 30Si(3He,d)31P transfer reaction. Each panel correspond to a different transferred relative angular momentum, and the blue solid lines represent finite-range DWBA calculations normalized to the data. Zero-range calculations are also reported in case of the 7.945 MeV state (see text for details).
While the optical potential parameters for the entrance channel come from an experimental study of the same transfer reaction at the same bombarding energy , the parameters for the exit channel come from set F of Daehnick et al. global deuteron potentials . The proton form factor was obtained by adjusting the depth of a standard Woods-Saxon well in order to reproduce the experimental proton separation energy of each 31P state. The geometry of the well had a radius and diffuseness of
The finite-range calculations have been performed using the
The proton spectroscopic factors obtained from the finite-range DWBA analysis have been used to derive the proton partial widths using Equation 13. The weakly-bound approximation was assumed and the validity of this assumption was explored by performing zero-range DWBA calculations to unbound states with the DWUCK4 code  which relies on the Vincent and Fortune complex integration procedure of the radial integrals . In addition, DWUCK4 calculates the proton partial widths and a maximum difference of 15% was observed with the weakly-bound approximation. This is related to the fact that the proton wave functions, even for unbound levels, are well described in the weakly-bound approximation because the high Coulomb barrier leads to a strong suppression of the wave function at large radii.
5.3 Case of the 13N(α,p)16O Reaction
It has been recently suggested that hydrogen ingestion into the helium shell of massive stars could lead to high 13C and 15N excesses when the shock of a core-collapse supernova (CCSN) passes through its helium shell . This prediction questions the origin of extremely high 13C and 15N abundances observed in rare presolar SiC grains which is usually attributed to classical novae . In this context the 13N(α,p)16O reaction plays an important role since it is in competition with 13N
The evaluation of the 13N(α,p)16O reaction rate in the temperature range of interest between 0.4 and 1 GK requires a detailed knowledge of the structure of the compound nucleus 17F within around 2.5 MeV above the 13N+α threshold. Spins and parities are known in most cases and the energy and total widths of the states are known experimentally . Given that the 13N+α threshold
The 13N(7Li,t)17F transfer reaction would be the most evident reaction to perform. However, while not impossible, such an experimental study in inverse kinematics would require the use of an intense radioactive 13N beam (
Examples of differential cross-sections for positive and negative parity 17O states populated with different transferred angular momentum L are shown in Figure 6, together with finite-range DWBA calculations performed with the FRESCO code. An excellent agreement is observed between the theory and the experiment which supports a single step direct mechanism for the population of 17O states using the 13C(7Li,t)17O reaction. However unlike the single nucleon transfer reactions the angular distributions obtained from (7Li,t) reactions are usually less pronounced with much less marked angular minima and maxima. Also, the shape of the angular distributions is not so sensitive to the transferred angular momentum L as can be observed in Figure 6 which makes its determination more delicate.
FIGURE 6. (Color online) Selection of experimental differential cross-sections of 17O states populated with the 13C(7Li,t)17O transfer reaction. Negative- and positive-parity states are in the upper and lower row, respectively. Solid lines represent finite-range DWBA calculations normalized to the data for different values for the number of quanta Q in the relative α+13C motion (see text for more details).
Details about the ingredients needed for the FR-DWBA calculations, such as the optical potential parameters, the overlap between the α+t and 7Li systems, and the geometry of the Woods-Saxon potential used to compute the wave-function describing the α+13C relative motion, can be found in ref. 99. An important ingredient is the number of nodes N (defined here as excluding the origin) of the radial part of the α+13C wave-function. Even though there is a limited sensitivity of the angular distributions to the number of nodes N, its determination should be whenever possible guided by microscopic considerations such as a cluster description of the states or from the insight of shell-model calculations. The link between these two views is not straightforward but there have been recent efforts to perform FR-DWBA calculations of the 12C(7Li,t)16O reaction with the
In the case of positive-parity states in 17O, the 5p-4h configuration is not expected to be populated in direct α-particle transfer since the shell model overlap between a 5p-4h configuration in 17O and the 13C in its ground state and an α-particle would be zero. Since the experimental angular distributions are well described by a one step direct reaction mechanism this would mean that the reaction mechanism is sensitive to the 3p-2h configuration, either because the states have a dominant 3p-2h configuration, or, if they have a dominant 5p-4h configuration, because the (7Li,t) mechanism is sensitive to the small admixture of 3p-2h configuration. When the two neutrons and protons of the transferred α-particle are positioned on the orbitals respecting a 3p-2h configuration one obtains
In a similar way for negative-parity states in 17O, the 2p-1h and 4p-3h configurations can be associated to the number of quanta
TABLE 1. Alpha-particle spectroscopic factors and widths for negative-parity 17O states obtained when considering a number of quanta in the relative α+13C motion
The 13N(α,p)16O reaction rate was calculated based on the previous spectroscopic information, and it was found to be within a factor of two of the previous evaluation done by Caughlan and Fowler . A detailed Monte-Carlo study was then used to propagate the nuclear uncertainties to the reaction rate, and a factor of uncertainty of two to three was obtained. This translates into an overall uncertainty in the 13C production of a factor of 50 when using the lower and upper reaction rates .
5.4 Case of the 60Fe(n,γ)61Fe Reaction
60Fe(n,γ)61Fe plays an important role in the abundance of 60Fe which characteristic gamma-ray lines at 1173.23 and 1332.44 keV coming from the decay-chain of 60Fe-60Co-60Ni have been observed by the spacecrafts missions RHESSI in 2004  and INTEGRAL in 2007 . The observation of these gamma-ray lines indicates that the nucleosynthesis of 60Fe is still active in the Galaxy since its lifetime 2.6 million years is much smaller than the galactic time evolution which is around 10 billion years. An excess of 60Fe has also been observed in deep ocean crusts and sediments as well as in lunar soils [107–109] and in galactic cosmic rays (CRIS/ACE) . All these observations have underlined the need for accurate nuclear information concerning the stellar nucleosynthesis and destruction of this nucleus. 60Fe is mainly produced in massive stars through the weak s-process component and it is released in the interstellar medium by the subsequent core-collapse supernovae explosion . Thus, all 60Fe observations give the opportunity to test stellar models that describe the evolution of massive stars. However, the important uncertainties surrounding the cross-section of the destruction reaction 60Fe(n,γ)61Fe imply large uncertainties on the predictions of 60Fe abundance by stellar models.
The direct measurement of the cross-section of this reaction is very challenging due to the radioactive nature of 60Fe. An alternative method would be to determine the cross-section through the activation method, which was performed by Uberseder et al.  or by the (d,p) transfer reaction to determine the excitation energies, orbital angular momenta and neutron spectroscopic factors of 61Fe states that are important for the calculation of the direct component (Section 2.2) of the (n,γ) reaction cross-section in the region of astrophysical interest (Ec.m
Since 60Fe is radioactive, it is very difficult to produce an 60Fe target with enough areal density to perform the (d,p) reaction measurement with a deuteron beam. Consequently, the 60Fe(d,p)61Fe measurement was performed in inverse kinematics , using the 27 A.MeV 60Fe secondary beam produced by fragmentation at LISE spectrometer line of GANIL and a deuterated polypropylene CD2 target of 2.6 mg/cm2 to induce the reaction.
The 60Fe beam intensity produced was of about 105 pps which is the usual beam intensities one can get with radioactive beams not far from the valley of stability. As discussed in Section 4 these low intensities required the use of large area and highly segmented silicon strip detector arrays placed at backward angles in the laboratory; four MUST2 telescopes  and an S1 annular DSSSD from Micron Semiconductor Ltd., in order to increase the angular coverage of the protons detection (from 2° to 23° in center of mass) and hence the statistics. The 60Fe beam being produced by fragmentation has a large emittance. Therefore, to determine precisely the location of the proton emission point on the target and its emission angle, two multi-wire proportional chambers (MWPC) called CATS were used to track the beam. To disentangle the different populated states in 61Fe that can not be discriminated with particle detection, four Germanium clovers (EXOGAM)  were used to detect the emitted γ-rays from the decay of the populated states in 61Fe. As for the residual fragments, they were identified in mass and charge using their energy loss in the ionization chamber and the time of flight between the plastic scintillator at the end of the line and one of the CATS detectors.
The reconstructed 61Fe energy spectrum using MUST2 energy and angle measurements is displayed in Figure 7 (left panel), with and without γ coincidences.
FIGURE 7. (Color online) Left: Measured 61Fe excitation energy spectrum in coincidences with gammas (red curve) and without coincidences (blue curve). The vertical dashed line corresponds to the neutron threshold. Right: Energy spectrum of the γ-rays in coincidences with protons detected in MUST2 or S1 detectors: Black, for an excitation energy gate between 0 to 1 MeV; pink, for a gate between 1 to 2 MeV.
Two peaks are observed below the neutron threshold Sn = 5.58 MeV. The first peak is around 1 MeV and the other at 3 MeV. The width of these peaks is 1.5 and 2 MeV respectively which is much larger than the expected excitation energy resolution, namely 800 keV. This is an indication that several levels are present in the peaks observed. The importance of detecting the γ-rays is obvious in this case.
One can also observe a drop of about a factor three in the counts of the main peak around 1 MeV when comparing the excitation energy spectrum with and without γ-ray coincidence while it is between 1.6 to a factor 2 everywhere else. This is a strong indication of the population of the isomeric state at 861 keV whose γ-rays can not be detected because they are emitted when 61Fe ions are stopped in the plastic which is at a far distance from the EXOGAM detectors. Indeed the lifetime of the isomeric state (τ = 239 ns) is much longer than the time of flight of 61Fe ions (
From the observation of the gamma-ray spectra corresponding to two energy gates in the first peak, from 0 to 1 MeV and from 1 to 2 MeV in Figure 7 (right panel) and from the comparison of the excitation energy spectrum with and without γ-ray coincidence in Figure 7 (left panel), three states were clearly identified: the known 207 keV, the 391 keV and the isomeric state at 861 keV.
To extract the proton angular distributions of the identified states, a deconvolution of the first peak observed in 61Fe excitation energy spectrum around 1 MeV was performed considering the ground state (gs), the three well identified populated states at 207 keV (Jπ = 5/2-), 391 (Jπ = 1/2-) and 861 keV (Jπ = 9/2+) and also a higher level centered at 1600 keV representing a mixture of the non-identified higher states between 1.2 and 2 MeV .
An example of the extracted proton angular distributions is displayed in Figure 8 for the 861 keV state. The blue and the green curves are zero-range (ZR) and finite-range (FR) Adiabatic distorded wave approximation (ADWA) calculations, respectively (see Section 3.1.4). The magenta curve does not take into account the deuteron breakup, and is a zero range calculation using Daehnick et al. global parametrization  for the optical potential describing the entrance channel. All calculations were performed with the FRESCO code. Given the large statistical uncertainties all three calculations give a similar reduced chi-square. However, the incident beam energy of 27 A.MeV would correspond to an incident deuteron energy of 54 MeV, and deuteron breakup should then be considered, therefore favoring the ADWA calculations. The main effect of taking into account the deuteron breakup is a noticeable difference in the shape of the differential cross-section with respect to the DWBA calculation, e.g., different position of the first angular minimum. Note as well that, in the present case, the ZR- and FR-ADWA calculations give similar results both in terms of the shape of the differential cross-section, and of the spectroscopic factors which differs by only 5%.
FIGURE 8. (Color online) Experimental differential cross-sections for the 861 keV excited state, together with the different calculations normalized to the data. See text for details.
A comparison between the C2S obtained in this work  with those predicted by shell-model calculations within a
TABLE 2. Comparison of the spectroscopic factors obtained in this work and those prediced in LNPS shell-model calculations.
The direct component of 60Fe(n,γ)61Fe was calculated using the experimental C2S for the first four excited states of 61Fe and its value was found to be 0.2 mb at 25 keV. This represents 2% of the total cross-section measured in ref. 112.
6 Summary and Perspectives
In this review, we have focused in the transfer reaction method which has been widely used to derive very useful spectroscopic information (spectroscopic factors, partial widths, orbital momenta and resonance energies) needed to evaluate resonant and non-resonant reaction rates of astrophysical interest. The theoretical description of the method has been recalled and a review of its use in some recent experimental studies using stable and radioactive beams with different detection systems has been given.
The current development of exotic radioactive ion beams in many facilities around the world opens new opportunities for the study of astrophysical processes involving nuclei far from the valley of stability, such as the r- and
On the theoretical side, many progresses have been made to describe one-nucleon overlap functions as well as to understand the three-body dynamics related to the deuteron breakup degrees of freedom, including the nonlocality effects  (and references therein). Prediction of nuclear properties based on a realistic description of the strong interaction is at the heart of the ab initio effort in low-energy nuclear theory. Ab initio calculations have long been limited to light nuclei , but with the ever-increasing computing power and its associated decreasing cost, ab initio calculations for many more nuclei are now in development . These approaches are now used not only for predictions of binding energies but also to calculate one nucleon overlap functions [119, 120] and nucleon optical potentials [121, 122]. Developments of optical potentials calculations using microscopic models have also been recently undertaken  and the most recent WLH9 microscopic global optical potential could be very useful for the future transfer reaction experiments involving proton and neutron-rich isotopes . However, as Timofeyuk and Johnson pointed out so well ”providing an input from ab-initio approaches to a transfer reaction amplitude based on an oversimplified distorted-wave approximation does not make the reaction description truly microscopic. To date only four truly ab-initio calculations of one-nucleon transfer have been published” , involving light nuclei not heavier than 8Li [55–58].
Despite their use since more than 50 years, transfer reactions remain a powerful method in nuclear astrophysics which is still promised to have a bright future in the forthcoming decades to provide a better insight on the reactions that govern the Cosmos.
All authors have made a substantial, direct, and intellectual contribution to the work and approved it for publication.
Conflict of Interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
NdS thanks Anne Meyer for providing angular distributions and α-particle partial widths in case of
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6This is usually the case for stripping reactions. In the case of pick-up transfer reactions (e.g., (p,d) (d,3He)…), the forward angles in the center of mass correspond to forward angles in the laboratory frame also in inverse kinematics.
7where the xp-yh notation has the usual meaning of x particles in the
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Keywords: transfer reactions, angular distributions, distorted wave born approximation, spectroscopic factors, nuclear astrophysics
Citation: Hammache F and de Séréville N (2021) Transfer Reactions As a Tool in Nuclear Astrophysics. Front. Phys. 8:602920. doi: 10.3389/fphy.2020.602920
Received: 04 September 2020; Accepted: 01 December 2020;
Published: 30 March 2021.
Edited by:Rosario Gianluca Pizzone, Laboratori Nazionali del Sud (INFN), Italy
Reviewed by:Giovanni Luca Guardo, Laboratori Nazionali del Sud (INFN), Italy
Antonio Caciolli, University of Padua, Italy
Antonio M. Moro, Sevilla University, Spain
Stefan Typel, Technische Universität Darmstadt, Germany
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