Many-channel microscopic cluster model of $^{8}$Be: S-factors

TL;DR

This work applies a microscopic, many-channel three-cluster model to low-energy reactions proceeding through $^{8}$Be in the entrance channels $p+^{7}$Li, $n+^{7}$Be, and $d+^{6}$Li, connecting the high-lying $^{8}$Be spectrum to astrophysical S factors. The model explicitly includes eight binary partitions and uses the Hasegawa–Nagata $NN$ potential with a tuned Majorana parameter, incorporating cluster polarization to capture both normalization and energy dependence near thresholds. The calculations reproduce the absolute scale and low-energy trends for the mirror and charge-exchange channels, while underestimating some deuteron-induced channels due to threshold shifts and missing near-threshold $2^{+}$ strength; a detailed partial-wave analysis links observed features to specific $^{8}$Be resonances discussed in Paper I. Evaluating $S(E)$ at Gamow energies reveals a hierarchy of reaction channels that quantify the relative roles of neutron- and deuteron-induced processes in the production and destruction of $^{7}$Li and $^{7}$Be, with implications for primordial and stellar lithium evolution and guiding future experiments.

Abstract

We investigate low--energy astrophysical $S$ factors for reactions proceeding through the $^{8}$Be compound system with entrance channels $p+{}^{7}$Li, $n+{}^{7}$Be, and $d+{}^{6}$Li. Using the same microscopic many--channel three--cluster framework as in our previous study of the high--lying $^{8}$Be spectrum, we calculate $S(E)$ for $^{7}$Li($p,α)^{4}$He, $^{7}$Be($n,α)^{4}$He, $^{7}$Be($n,p)^{7}$Li, $^{6}$Li($d,α)^{4}$He, $^{6}$Li($d,p)^{7}$Li, and $^{6}$Li($d,n)^{7}$Be in the energy range relevant for primordial and stellar nucleosynthesis. For the mirror pair $^{7}$Li($p,α)^{4}$He / $^{7}$Be($n,α)^{4}$He and for $^{7}$Be($n,p)^{7}$Li the calculated $S$ factors reproduce both the absolute scale and the low--energy trends of the experimental data within their quoted uncertainties, whereas the absolute $S$ factors for the deuteron--induced channels on $^{6}$Li are underestimated at low energy, consistent with the shifted $^{6}$Li+$d$ threshold and the absence of a broad subthreshold $2^{+}$ structure in the present implementation. A partial--wave analysis identifies the dominant $J^π$ contributions in each channel and relates them to specific $^{8}$Be resonances, while demonstrating that cluster polarization, previously shown to be crucial for the $^{8}$Be spectrum, is likewise essential for the normalization and energy dependence of several $S$ factors. Evaluating $S(E)$ at appropriate Gamow energies, we obtain a hierarchy of reaction channels that quantifies the relative importance of neutron-- and deuteron--induced processes for the production and destruction of $^{7}$Li and $^{7}$Be.

Figures (16)

• Figure 1: Total and partial astrophysical $S$ factors for the reaction $^{7}$Li$(p,\alpha)^{4}$He in the center-of-mass energy range $E_{\mathrm{cm}} \lesssim 1$ MeV, calculated within the present model and compared with the experimental data listed in Table \ref{['Tab:SfactExpP7LiAA']}. The dashed and dotted curves denote the $J^{\pi}=0^{+}$ and $2^{+}$ contributions, respectively, and the solid curve shows their sum. The shaded area indicates the Gamow window.
• Figure 2: Total and partial astrophysical $S$ factors for the reaction $^{7}$Be$(n,\alpha)^{4}$He, calculated within the present model and compared with the experimental data summarized in Table \ref{['Tab:Exp7BeNAA']}. All experimental data sets except those of Kawabata et al. correspond to reactions on $^{7}$Be in its ground state; the points labeled “Kawabata2017gs” and “Kawabata2017*” represent reactions on the ground and first excited states, respectively. The dashed curves show the $J^{\pi}=0^{+}$, $2^{+}$, and $4^{+}$ partial contributions, and the solid curve is their sum.
• Figure 3: Low-energy astrophysical $S$ factors for the reaction $^{7}$Be$(n,\alpha)^{4}$He, calculated within the present model separately for initial $^{7}$Be in the ground ($3/2^{-}$) and first excited ($1/2^{-}$) states, and compared with the experimental data summarized in Table \ref{['Tab:Exp7BeNAA']} (all non-Kawabata points correspond to reactions on $^{7}$Be in its ground state, while "Kawabata2017gs" and "Kawabata2017*" denote the ground and first excited states, respectively). For each initial state, the red and green curves denote the $0^{+}$ and $2^{+}$ contributions, respectively, and the black curve gives their sum.
• Figure 4: Astrophysical $S$ factors for the reaction $^{7}$Be$(n,p)^{7}$Li, calculated within the present model and compared with experimental data summarized in Table \ref{['Tab:Exp7BeNP7Li']}. The $(n,p_{0})$ and $(n,p_{1})$ channels correspond to $^{7}$Li in the ground ($3/2^{-}$) and first excited ($1/2^{-}$) states, respectively. The labels "P" and "NP" denote calculations with and without cluster polarization. All experimental data sets, except those of Borchers and Hayakawa, correspond to the $(n,p_{0})$ channel.
• Figure 5: Contribution of states with the different values of the total angular momentum $J$ to the total astrophysical S-factor of the reaction $^{7}$Be($n$,$p$)$^{7}$Li. The $(n,p_{0})$ and $(n,p_{1})$ channels correspond to $^{7}$Li in the ground ($3/2^-$) and first excited ($1/2^-$) states, respectively.