ANC experiments for nuclear astrophysics

Among the indirect methods to determine nuclear cross-section present in literature, the so-called Asymptotic Normalization Coefficient (ANC) has proven to be useful in retrieving the direct part of a radiative capture cross-section in reactions of interest for astrophysics. In this work, the method will be presented, and some results obtained in collaboration between NPI CAS and INFN-LNS will be presented. Introduction Experimental measurements of reactions of relevance for astrophysics are a big challenge for experimentalists due to the presence of the Coulomb barrier between charged particles: in fact, it highly decreases the cross-section in the energy range of interest, that usually is between few keV and some MeV. It is therefore necessary, when direct methods are not feasible, to use indirect ones. Among these, the Asymptotic Normalization Coefficient (ANC) method allows to retrieve the ANC of the overlap function for the bound-state wave functions in the initial and final states [1, 2]. Such a method is widely used under the condition that the reaction of interest is peripheral. This is, for example, the case of radiative captures involving p,n and α-particles in stellar environment. In the following work the method will be presented, marking the key points of the theoretical framework and the advantages of its usage. Then an experimental overview of the method will be given, using as a benchmark the ongoing data analysis for the 26Mg(d,p)27Mg reaction: this reaction is interesting due to its mirror, 26Si(p,γ)27P, considered to be important as a constraint to the production of 26Al and 26mAl. The production of both is among the most debated issues concerning novae and x-ray bursts[3, 4]. The ANC method, as stated in the introduction, is usefull to study direct capture reactions of astrophysical interest by means of the study of a direct transfer reaction. It allows to study reactions , 0 (2020) Web of Conferences https://doi.org/10.1051/epjconf /2020 0 EPJ 227 1


Introduction
Experimental measurements of reactions of relevance for astrophysics are a big challenge for experimentalists due to the presence of the Coulomb barrier between charged particles: in fact, it highly decreases the cross-section in the energy range of interest, that usually is between few keV and some MeV. It is therefore necessary, when direct methods are not feasible, to use indirect ones. Among these, the Asymptotic Normalization Coefficient (ANC) method allows to retrieve the ANC of the overlap function for the bound-state wave functions in the initial and final states [1,2]. Such a method is widely used under the condition that the reaction of interest is peripheral. This is, for example, the case of radiative captures involving p,n and α-particles in stellar environment. In the following work the method will be presented, marking the key points of the theoretical framework and the advantages of its usage. Then an experimental overview of the method will be given, using as a benchmark the ongoing data analysis for the 26 Mg(d,p) 27 Mg reaction: this reaction is interesting due to its mirror, 26 Si(p,γ) 27 P, considered to be important as a constraint to the production of 26 Al and 26m Al. The production of both is among the most debated issues concerning novae and x-ray bursts [3,4].
The ANC method, as stated in the introduction, is usefull to study direct capture reactions of astrophysical interest by means of the study of a direct transfer reaction. It allows to study reactions , 0 (2020) Web of Conferences https://doi.org/10.1051/epjconf /2020 0 EPJ 227 10 2270100 10 th European Summer School on Experimental Nuclear Astrophysics 3 3 involving protons [5], neutrons [6] or α-particles [7] and it has also proven to be useful to investigate subthreshold resonances [8][9][10].
In the general case, this kind of reactions can be considered (as sketched figure 1) as with a is the transferred particle. For that case, the Distorted Wave Born Approximation (DWBA) leads to an approximation for the transition amplitude between the states before and after the reaction. Here, the matrix element for the transition can be written as where I B Aa and I X Ya are the radial overlap integral of the two systems Y +a and A+a with the nuclei B and X respectively, χ (+) i and χ (−) f are the distorted waves for the entrance and exit channels, and ΔV is the perturbative part of the total potential, composed by a real (V) and an optical (U) part (ΔV = V − U). In general, for the nuclear process between two particles a + b → c, with c = (ab), both the overlap functions in equation 1 can be written using a quantity known as spectroscopic factor (SF) -in general S ab -that has been widely used in the last fifty years to estimate the occupation of the orbitals. The overlap funcion of the bound state wave functions of particles a, b and c, I c abl c j c , can be written as: with φ n c l c j c (r ab ) as the bound state wave function [11]. Regarding the SF, the description of the cross-section depends on the binding portential and optical model parameters (OMP) [11]. The binding potential is adjusted according to the binding energy to reproduce the bound state, while the OMP are usually extracted from elastic scattering. In general then the SF is sensitive to the selection of the optical model adopted, and different families of those can reproduce the angular distributions retrieved from experimental data [12]. By means of DWBA calculation it is in the end possible to reproduce the experimental angular distribution: where dσ dΩ and σ DW l B j b l X j X are related by the two SF for the entrance and exit channel: one of those is usually known, while the other is left to be determined. In 1990 it was pointed out by Mukhamedzhanov and Timofeyuk [13] that the direct part of the radiative cross-section at low energies usually contains the same radial overlap integral (I c ab in general) as a direct transfer reaction, and that happens because such capture occurs usually at large distances from the nucleus, where the behaviour can be called Asymptotic. The asymptotic behaviour can be used to , 0 (2020) Web of Conferences https://doi.org/10.1051/epjconf /2020 0 EPJ 227 10 2270100 10 th European Summer School on Experimental Nuclear Astrophysics 3 3 determine the cross-section of reactions of interest in astrophysics, and in this case is possible to write both the radial overlap functions (Eq. 1) and bound-state wave functions (Eq. 2) as it follows: In both equations (4) and (5) W −η,l c + 1 2 stands for the Whittaker function, well known to have its asymptotic behaviour as W −η,l c + 1 2 (−2k I r) → e −k l r+η I ln(2k I ) . In equation 4 and 5 for the first time the coefficient b and C appear: those are the so-called single-particle ANC (SPANC) and ANC coefficients, respectively. Finally, by substitution of equations (4) and (5) in (3), we obtain: In the end, using DWBA, we are able to retrieve the ANC's for the direct part of the cross-section for reactions of astrophysical interest. This approach is useful because the ANC's have a small dependence on the chosen nucleon binding potential (see Fig.11 in [11], for example), so the ratio R l B , j B ,l X , j X in equation 6 has also small dependence on b 2 .
26 Si(p,γ) 27 P reaction One of the latest experiments meant to apply the ANC method at NPI CAS has been the study of the 26 Si(p,γ) 27 P reaction, that is supposed to be of some importance in the framework of the production of 26 Al. This reaction is studied using the mirror pair of 26 Si, 26 Mg: in particular, the reaction 26 Mg(d,p) 27 Mg is used with the aim to extract ANC's for the 26 Mg(n,γ) 27 Mg, and then using the mirror pair properties [14,15] to obtain information on the ANC of the reaction of interest.
In the last years, the production of 26 Al has been matter of debate for scientists: it is, in fact clearly observable in novae and x-ray bursts [3,4]. Its production chain seems to be dominated by the 24 Mg(p,γ) 25 Al(β + ) 25 Mg(p,γ) 26 Al, but the scenario has been strongly complicated by the presence of the widely known, short-lived isomer of 26 Al(T 1/2 =6.34s). The ratio between ground and isomeric states should be known to better understand the observed wide and intense distribution of 26 Al through the galactic plane [16]. A possible bypass of the chain mentioned above is represented by the 25 Al(p,γ) 26 Si(p,γ) 27 P one, and so the 27 P production has been considered of some importance as a possible competitive channel of the 26 Al production. Furthermore, the isomeric state 26m Al can be produced via the 25 Al(p,γ) 26 Si(β + ) 26m Al. In this scenario the reaction 26 Si(p,γ) 27 P is important, because 26 Si destroyed in this way would not end up feeding the isomeric state. In novae, it has been proposed that the thermal equilibrium between 26 Al and 26m Al can possibly be reached at T 9 ≈0.4 [17]. At this range of temperature, the Gamow window lies between 0.2 and 0.5 MeV for the 26 Si(p,γ) 27 P reaction.

Astrophysical motivation
For all the reasons explained in section 2.1, we decided to study the 26 Si(p,γ) 27 P using the mirror pair approach: 27 P is in fact the mirror nucleus (inverted number of protons and neutrons) of 27 Mg, and so the proton ANC for the process 27 P→ 26 Si+p can be derived [14,15] as C where R can be calculated as R = F l ik p R N k p R n j l (ik n R N ) 2 , F l being the regular Coulomb wave function, j l the Bessel function of l−th order, and R N the radius of the nuclear interior, while k p and k n are related to the neutron and proton separation energy via the relation k = 2μ 2 [14,15]. Here another relation has been found [14] between Γ p and |C n | 2 as Γ p We decided then to use the 26 Mg(d,p) 27 Mg reaction, using the 19MeV deuteron beam provided by the U120M cyclotron of the NPI CAS (Řež, Czech Republic), operated by CANAM infrastructure project. About this reaction, a similar attempt using ANC has been made [18] reanalyzing data avaliable in litterature, but the extracted ANC values are in disagreement with other results obtained studying the 26 Mg(t,d) 27 Mg reaction [19]. The discrepancies between the two analyses ended up in an S-factor that in the second case is 1.7 times smaller. Further investigations are therefore necessary. Our experimental set-up was composed by three ΔE-E telescopes placed at 10 • difference between each other on one side of the beam, and two ΔE-E at 17 • and 37 • on the other side, that were meant as monitors. All the ΔE-E detectors are composed by a 500 μm silicon detector for the thin stage and a 5000 μm one for the thick one. Angular distributions are measured by moving the detectors to higher angles from the beam axis at fixed steps. The target, provided by the target laboratory of LNS-INFN was composed by 70 μg/cm 2 26 MgO with a 32 μg/cm 2 natural carbon backing.