Successful Prediction of Total $\alpha$-Induced Reaction Cross Sections at Astrophysically Relevant Sub-Coulomb Energies Using a Novel Approach

The prediction of stellar $(\gamma ,\alpha )$ reaction rates for heavy nuclei is based on the calculation of $(\alpha ,\gamma )$ cross sections at sub-Coulomb energies. These rates are essential for modeling the nucleosynthesis of so-called $p$ nuclei. The standard calculations in the statistical model show a dramatic sensitivity to the chosen $\alpha$-nucleus potential. The present study explains the reason for this dramatic sensitivity which results from the tail of the imaginary $\alpha$-nucleus potential in the underlying optical model calculation of the total reaction cross section. As an alternative to the optical model, a simple barrier transmission model is suggested. It is shown that this simple model in combination with a well-chosen $\alpha$-nucleus potential is able to predict total $\alpha$-induced reaction cross sections for a wide range of heavy target nuclei above $A\gtrsim 150$ with uncertainties below a factor of 2. The new predictions from the simple model do not require any adjustment of parameters to experimental reaction cross sections whereas in previous statistical model calculations all predictions remained very uncertain because the parameters of the $\alpha$-nucleus potential had to be adjusted to experimental data. The new model allows us to predict the reaction rate of the astrophysically important ${}^{176}\mathrm{W}(\alpha ,\gamma ){}^{180}\mathrm{Os}$ reaction with reduced uncertainties, leading to a significantly lower reaction rate at low temperatures. The new approach could also be validated for a broad range of target nuclei from $A\approx 60$ up to $A\gtrsim 200$.

Figure 1

Total reaction cross section ${\sigma }_{\mathrm{reac}}$ for $\alpha +{}^{197}\mathrm{Au}$ for different potentials (shown as astrophysical $S$ factor), compared to experimental data which are taken from the sum of $(\alpha ,\gamma )$, $(\alpha ,n)$, and $(\alpha ,2n)$ in [11, 42]. (The comparison of calculated total reaction cross sections ${\sigma }_{\mathrm{reac}}$ to the sum of partial experimental cross sections avoids any complications from other ingredients of the SM calculations like the $\gamma$-strength function or the level density.) Modifications (i) and (ii) of the imaginary MCF WS potential reduce ${\sigma }_{\mathrm{reac}}$ at low energies dramatically (dotted red lines); the reduction of the real MCF potential (iii) has minor influence at low energies (dash-dotted red). The talys and PBTM calculations are discussed in the text. The shaded area represents the wide range of talys predictions.

Figure 2

Total reaction cross section ${\sigma }_{\mathrm{reac}}$ (given as astrophysical $S$ factor) for $\alpha$-induced reactions above $A=150$. The PBTM predicts practically all experimental data [53, 54, 55, 56, 57, 58, 59, 60] within a factor of 2 (gray shaded). The dotted lines show the results from the many-parameter AOMP of [38]. The arrows indicate the $(\alpha ,n)$ and $(\alpha ,2n)$ thresholds.

Figure 3

Reaction rate $N_A⟨\sigma v⟩$ of the ${}^{176}\mathrm{W}(\alpha ,\gamma ){}^{180}\mathrm{Os}$ reaction, normalized to the present calculation in the PBTM. The gray-shaded area indicates the uncertainty of a factor of 2 (see also Fig. 2). For further discussion, see text and Supplemental Material [12].