Three-body optical potentials in $(d,p)$ reactions and their influence on indirect study of stellar nucleosynthesis

Model uncertainties arising due to suppression of target excitations in the description of deuteron scattering and resulting in a modification of the two-body interactions in a three-body system are investigated for several $(d,p)$ reactions serving as indirect tools for studying the astrophysical $(p,\gamma )$ reactions relevant to $rp$ process. The three-body nature of the deuteron-target potential is treated within the adiabatic distorted-wave approximation (ADWA) which relies on a dominant contribution from the components of the three-body deuteron-target wave function with small $n\text{−}p$ separations. This results in a simple prescription for treating the explicit energy dependence of two-body optical potentials in a three-body system requiring nucleon optical potentials to be evaluated at a shifted energy with respect to the standard value of half the deuteron incident energy. In addition, the ADWA allows for leading-order multiple-scattering effects to be estimated, which leads to a simple renormalization of the adiabatic potential's imaginary part by a factor of two. These effects are assessed using both nonlocal and local optical potential systematics for ${}^{26}\mathrm{Al}$, ${}^{30}\mathrm{P}$, ${}^{34}\mathrm{Cl}$, and ${}^{56}\mathrm{Ni}$ targets at a deuteron incident energy of 12 MeV, which is typical for experiments with radioactive beams in inverse kinematics. The model uncertainties induced by the three-body nature of deuteron-target scattering are found to be within $40\%$ both in the main peak of angular distributions and in total $(d,p)$ cross sections. At higher deuteron energies, around 60 MeV, model uncertainties can reach 100% in the total cross sections. A few examples of application to astrophysically interesting proton resonances in ${}^{27}\mathrm{Si}$ and ${}^{57}\mathrm{Cu}$ obtained using $(d,p)$ reactions and mirror symmetry are given.

Figure 1

Comparison of (a), (b), (d), (e) ${}^{26}\mathrm{Al}(d,p){}^{27}\mathrm{Al}$ and (c), (f) ${}^{56}\mathrm{Ni}(d,p){}^{57}\mathrm{Ni}$ cross sections calculated at $E_d=12$ MeV with GRZ potentials with (dashed lines) and without (solid lines) I3B terms for $NN$ models from Table 2. The results with Hulthén, AV18, CD-Bonn, and $\chi \mathrm{EFT}$ $NN$ model choices are shown by green, black, red, and blue lines, respectively.

Figure 2

The ADWA angular distributions for (a), (b) ${}^{26}\mathrm{Al}(d,p)$; (c), (d) ${}^{34}\mathrm{Cl}(d,p)$; and (e), (f) ${}^{56}\mathrm{Ni}(d,p)$ reactions and populating (c) $l=0$; (b), (f) $l=1$; (a), (d) $l=2$; and (e) $l=3$ states calculated at $E_d=12$ MeV using the nonlocal GRZ potential with (dashed) and without (solid) I3B terms in comparison with those obtained using standard Johnson-Tandy model with the local KD03 potential (dot-dashed).

Figure 3

The ADWA angular distributions for (a), (b) ${}^{26}\mathrm{Al}(d,p)$; (c) ${}^{30}\mathrm{P}(d,p)$; (d) ${}^{34}\mathrm{Cl}(d,p)$; and (e), (f) ${}^{56}\mathrm{Ni}(d,p)$ reactions and populating (a), (d) $l=0$; (f) $l=1$; (b) $l=2$; and (c), (e) $l=3$ states calculated at $E_d=12$ MeV using the KD03 potential evaluated at a shifted by 57 MeV energy with (dashed) and without (solid) I3B terms in comparison with those obtained using standard Johnson-Tandy model with the same KD03 potential (dot-dashed).

Figure 4

The ADWA angular distributions for (a), (b) ${}^{26}\mathrm{Al}(d,p)$; (c) ${}^{30}\mathrm{P}(d,p)$; (d) ${}^{34}\mathrm{Cl}(d,p)$; and (e), (f) ${}^{56}\mathrm{Ni}(d,p)$ reactions and populating (a), (d) $l=0$; (f) $l=1$; (b) $l=2$; and (c), (e) $l=3$ states calculated at $E_d=12$ MeV using the nonlocal equivalent of the KD03 potential with (dashed) and without (solid) I3B terms in comparison with those obtained using standard Johnson-Tandy model with the same KD03 potential (dot-dashed).