Low-energy $R$-matrix fits for the ${}^6\mathrm{Li}(d,\alpha ){}^4\mathrm{He}S$ factor

Background: Information about the ${}^6\mathrm{Li}(d,\alpha ){}^4\mathrm{He}$ reaction rates of astrophysical interest can be obtained by extrapolating direct data to lower energies, or by indirect methods. The indirect Trojan horse method, as well as various $R$-matrix and polynomial fits to direct data, estimate electron screening energies much larger than the adiabatic limit. Calculations that include the subthreshold resonance estimate smaller screening energies.

Purpose: Obtain the ${}^6\mathrm{Li}{(d,\alpha )}^4\mathrm{He}$ reaction $R$-matrix parameters and the bare astrophysical $S$ factor for energies relevant to the stellar plasmas by fitting $R$-matrix formulas for the subthreshold resonances to the $S$-factor data above 60 keV.

Methods: The bare $S$ factor is calculated using the single- and the two-level $R$-matrix formulas for the closest to the threshold $0^{+}$ and $2^{+}$ subthreshold states at $22.2,20.2$, and $20.1$ MeV. The electron screening potential $U{}_{\mathrm{e}}$ is then obtained by fitting it as a single parameter to the low-energy data. The calculations are also done by fitting $U{}_{\mathrm{e}}$ simultaneously with other parameters.

Results: The low-energy $S$ factor is dominated by the $2^{+}$ subthreshold resonance at $22.2$ MeV. The influence of the other two subthreshold states is small. The resultant electron screening is smaller than the adiabatic value. The fits that neglect the electron screening above 60 keV produce a significantly smaller electron screening potential. The calculations show a large ambiguity associated with a choice of the initial channel radius.

Conclusions: The $R$-matrix fits do not show a significantly larger $U_e$ than predicted by the atomic physics models. The $R$-matrix best fit provides $U{}_{\mathrm{e}}=149.5$ eV and $S_b\left(0\right)=21.7$ MeV b.

Figure 1

The astrophysical factor $S\left(E\right)$ for the ${}^6\mathrm{Li}{(d,\alpha )}^4\mathrm{He}$ reaction, resulting from a single-level $R$-matrix best fit to the experimental data [4, 7, 8, 9, 10], as a function of the relative kinetic energy $E$ in the entrance channel. The solid line and the dotted line correspond to the observed and the bare astrophysical $S$ factor, respectively. The top panel shows the best fit to data below 1 MeV that uses four free parameters: the reduced width amplitudes ${\gamma }_{\alpha }$ and ${\gamma }_d$, the $2^{+}$ subthreshold resonance energy $E{}_{\mathrm{R}}$ near the 22.2-MeV excitation energy, and the electron screening potential $U_e$. The bottom panel shows the best fit to data in a 60 keV to 1 MeV range that uses ${\gamma }_{\alpha }$ and ${\gamma }_d$ as the only parameters. The electron screening potential $U_e$ is then fit to data [4, 7, 8, 9, 10] below 1 MeV as a single parameter. Engstler et al.'s data [4] are shown as pluses for atomic target and crosses for molecular target. The circles are the bare $S_b\left(E\right)$ factor from the Trojan horse experiment [3]. Other data [7, 8, 9, 10, 11, 21] are taken from Ref. [22]. The errors of Engstler et al.'s data [4] do not include reported uncertainties arising from number of counts, angular distributions, effective energy, and target stoichiometry and hence are reduced by $5\text{–}10\%$. The error bars of experimental data of Elwyn et al. [7] are shown as in Ref. [22] but are enlarged in the calculation.