Low-energy $^{3}$He($α,γ$)$^{7}$Be reaction within the Skyrme potential framework

TL;DR

This work develops a unified microscopic framework based on Skyrme Hartree-Fock theory to describe both low-energy elastic scattering and radiative capture in light-nucleus systems, focusing on $p+α$ and $^{3}$He+$α$ to study the $^{3}$He($α,γ$)$^{7}$Be reaction. The nucleus–nucleus potential is constructed by folding a continuum-extended Skyrme HF nucleon–nucleus potential with $^{3}$He densities (SC and Ngo), and is calibrated with a small central-potential scaling $ abla_0$ against $p+α$ data; the folding potentials are then used to compute bound and scattering states. The radiative-capture cross section is evaluated within a direct $E1$ mechanism, with the $E1$ matrix element factorized into a single-particle piece and angular couplings, and the astrophysical $S$ factor is obtained as $S(E) = E \, oxed{ ext{sigma}(E)} \, ext{exp}(2\, ext{pi} \, oxed{ta})$. The Ngo density yields the best agreement with elastic-scattering data, $^{7}$Be ANCs, and the extrapolated $S_{34}(0)$ value, $0.610 \pm 0.024$ keV b, closely matching prior evaluations and recent Solar Fusion III analyses, demonstrating a predictive, parameter-efficient framework for light-ion reactions without relying on spectroscopic factors.

Abstract

\textbf{Background:} The $^{3}$He($α,γ$)$^{7}$Be reaction plays a crucial role in the proton-proton chain and Big Bang nucleosynthesis, affecting solar neutrino fluxes and primordial element abundances. Experimental data at astrophysical energies remain uncertain due to the extremely low cross sections. \\ \textbf{Purpose:} This work uses a microscopic potential-model approach to construct the $^{3}$He+$α$ potential from the nucleon+$α$ interaction, aiming to describe low-energy elastic scattering and to calculate the astrophysical $S$ factor of the $^{3}$He($α,γ$)$^{7}$Be reaction. \\ \textbf{Method:} The nucleon-nucleus potential is derived from self-consistent Skyrme Hartree-Fock (HF) calculations extended to the continuum. The $^{3}$He+$α$ potential is then obtained by folding the HF potential with the $^{3}$He density. A small number of scaling parameters is constrained by elastic-scattering data.\\ \textbf{Result:} The scaled Skyrme HF potential and folded potential simultaneously reproduce the low-energy $p$+$α$ and $^{3}$He+$α$ $s$-wave phase shifts, respectively. The calculated astrophysical $S$ factor of $^{3}$He($α,γ$)$^{7}$Be shows good agreement with experimental data, yielding the recommended value $S_{34}(0) = 0.610 \pm 0.024$~keV~b. A moderate sensitivity of $S_{34}(0)$ to the choice of projectile density is also observed in the folding procedure. \\ \textbf{Conclusion:} The Skyrme HF-based potential provides a unified and predictive microscopic framework for describing both elastic scattering and radiative capture in light nuclei.

Figures (5)

• Figure 1: Calculated phase shifts for $p+\alpha$ elastic scattering below 5 MeV using the SLy4 interaction. The solid and dashed curves correspond to calculations with scaling factors $\lambda_{0}=0.88$ and $\lambda_{0}=0.59$ for the central potential, respectively. Experimental data for the $s_{1/2}$, $p_{3/2}$, and $p_{1/2}$ partial waves are taken from Ref. schwandt1971.
• Figure 2: Calculated phase shifts for (a) $s$-wave ($\ell = 0$) and (b) $p$-wave ($\ell = 1$) $^{3}\mathrm{He}+\alpha$ elastic scattering below 5 MeV, obtained using folded SLy4 potentials constructed with different $^{3}\mathrm{He}$ density distributions. The shaded band in (a) corresponds to the uncertainty range $\Delta\lambda_{0} = \pm 0.02$ around the central scaling factors $\lambda_{0}$. Experimental data are taken from Refs. boykin1972hardy1972.
• Figure 3: Calculated angular distributions of $^{3}\mathrm{He}+\alpha$ elastic scattering at $E_{\text{lab}}(^{3}\mathrm{He})=$ (a) 1.2 MeV, (b) 2.0 MeV, (c) 2.5 MeV, and (d) 3.0 MeV, normalized to the Rutherford cross section. Results including all partial waves (blue) and only the $\ell=0$ component (red) are shown. Solid curves (with shaded bands) are calculated using scaled folded potentials with $\lambda_{0}=0.85\pm 0.02$ and $\lambda_{\text{ls}} =0.14$, while dashed curves correspond to unscaled results ($\lambda_{0}=\lambda_{\text{ls}}=1$). All calculations use the Ngo density. Experimental data are from Ref. mohr1993.
• Figure 4: Calculated $E1$ transitions for direct capture (DC) to the ground state ($3/2^-$) and first excited state ($1/2^-$) of $^{7}\mathrm{Be}$ using the Ngo density in the folded potential. At low energies, the capture is dominated by transitions from $s$-wave scattering states.
• Figure 5: Calculated astrophysical $S_{34}$ factor for the $^{3}\mathrm{He}(\alpha,\gamma)^{7}\mathrm{Be}$ reaction below 2 MeV using folded potentials constructed the SC and Ngo $^{3}\mathrm{He}$ densities. The shaded bands represent the uncertainties. The results are compared with experimental data from Refs. bordeanu2013dileva2009costantini2007costantini2008brown2007hilgemeier1988osborne1984nagatani1969parker1963. The calculation based on the Ngo density provides better overall agreement with the experimental data.