Investigating the astrophysical 22 Ne ( p , γ ) 23 Na and 22 Mg ( p , γ ) 23 Al reactions with a multi-channel scattering formalism

P. R. Fraser1,2,a, L. Canton1, K. Amos2,3, S. Karataglidis3,2, J. P. Svenne4, and D. van der Kniff2 1Istituto Nazionale di Fisica Nucleare, Sezione di Padova, Padova I-35131, Italia 2School of Physics, University of Melbourne, Victoria 3010, Australia 3Department of Physics, University of Johannesburg, P.O. Box 524 Auckland Park, 2006, South Africa 4Department of Physics and Astronomy, University of Manitoba, and Winnipeg Institute for Theoretical Physics, Winnipeg, Manitoba, Canada R3T 2N2

Direct measurement of the reaction22 Mg(p, γ) 23 Al is not yet feasible, and though indirect methods have suggested that this reaction does not account for the lack of 22 Na in novae ejecta [1,2], it is instructive to investigate the reaction via theoretical means [3][4][5].Accordingly, we have begun to determine a 22 Ne(n, n) 22 Ne interaction with a multi-channel algebraic scattering (MCAS) method [6].Then, using mirror symmetry, we define a potential for 22 Mg(p, p) 22 Mg scattering, as a first step towards modelling the radiative capture 22 Mg(p, γ) 23 Al using the formalism of [7].
In MCAS, solutions of coupled-channel Lippmann-Schwinger equations are found in momentum space using finite-rank separable representations of an input matrix of nucleon-nucleus interactions.An 'optimal' set of sturmian functions is used as the expansion basis.The MCAS method has the ability to locate all compound system resonance centroids and widths, regardless of how narrow.Also, by use of orthogonalizing pseudo-potentials (OPP) in generating sturmians, it ensures the Pauli principle is not violated [8], even with the collective model formulation of nucleon-nucleus interactions used.Otherwise, some compound nucleus wave functions possess spurious components.The same sturmians determined in the elastic scattering calculations are integral ingredients in the future generation of capture cross sections, as described fully in [7].MCAS is well suited to modelling the 22 Mg(p, p) 22 Mg reaction due to the low density of low-energy states in 23 Al and the low scattering threshold.
The low-lying spectrum of 22 Ne consists of a J π = 0 + ground state, a 2 + state at 1.274 MeV, and a 4 + state at 3.357 MeV, which together suggest a collective rotor character.Directly above this comes a 2 + state at 4.456 MeV which decays by E2 transition to the ground state [9].Thus, a scattering potential based upon the Tamura collective model of rotor character [10] is used to investigate neutron scattering from this target.Fig. 1 shows a preliminary theoretical spectrum for 23 Ne as compound states of n+ 22 Ne, compared to experimental values.There is still room for improvement, but the low-lying spectrum is reasonably well reproduced.Fig. 2 shows a theoretical spectrum for 23 Al as compound states of p+ 22 Mg, EPJ Web of Conferences 03030-p.2generated by utilising mirror symmetry and adding a Coulomb potential of the same geometry as the nuclear force.As the nuclear potential is preliminary, the V 0 depth has been reduced from that of the n+ 22 Ne potential to allow a better fit to data.
2 The 22 Ne(p, γ) 23 Na reaction Another mass-23 reaction of astrophysical interest is 22 Ne(p, γ) 23 Na, currently under experimental investigation by the LUNA collaboration at Gran Sasso.Current stellar models predict that the surface abundance of elements should not change when stars ascend the red giant branch of the Hertzsprung-Russell diagram of observed stellar luminosity versus temperature.However, anti-correlations have been observed between sodium and oxygen across this branch.An explanation proposed for this is the existence of non-convective mixing that is, possibly, driven by rotation.At high temperatures, this would facilitate leakage from the CNO cycles into the NeNa cycle, which would produce sodium while depleting oxygen.The reaction that generates sodium in the NeNa cycle is 22 Ne(p, γ) 23 Na, a reaction that is not well understood [11].Better knowledge of it will help to test this idea of mixing.
To evaluate this system using MCAS, the elastic 22 Ne(p, p) 22 Ne process will first be examined in order to establish an interaction potential, after which radiative capture of the proton will be studied.Given the high proton threshold of 23 Na (8.79 MeV), care must be taken in selecting 22 Ne states to couple with the proton, such that the deeply-bound states are credibly reproduced as well as compound system states in the Gamow window that contribute to capture (including the first 1 2 + above threshold).Details of these states have been discussed in [12] (see Table 5.9 and discussion).Fig. 3 shows preliminary low-energy MCAS spectra for the 22 Ne(p, p) 22 Ne system, and its mirror, 22 Mg(n, n) 23 Mg, using just the four lowest target states.Experimental data are shown for both.Fig. 4 shows the experimental spectrum of 23 Na (where red states are assigned positive parity, blue as negative parity, black where no parity has been suggested, and dashed where no J π has been suggested), and a partial spectrum of 22 Ne (all states up to 6.9 MeV, and only states assigned as 2 + above) [9,12,13].At present, our goal is determining which target states need be included in calculations, as well as their mixings.States under consideration are marked with arrows.These results, while still preliminary, are promising.The outcomes of this project will guide future MCAS research in other deeply bound systems, including that of the reaction 22 Na(p, γ) 23 Mg mentioned in the first section, where the compound system ground state lies 7.58 MeV below the proton scattering threshold.

Figure 3 .
Figure 3. 23 Na and 23 Mg spectra from experiment and MCAS.