${}^6\mathrm{Li}(p,\gamma ){}^7\mathrm{Be}$ reaction rate in the light of the new data of the Laboratory for Underground Nuclear Astrophysics

We present new calculations of the astrophysical $S$ factor and reaction rate for the ${}^6\mathrm{Li}(p,\gamma ){}^7\mathrm{Be}$ reaction at energies of 10 keV to 5 MeV in the framework of a modified potential cluster model with forbidden states, including low lying resonances. The astrophysical $S\left(E\right)$ factor is compared with the available experimental data and calculations done within different models. The results for the $S$ factor are in good agreement with the data set (for $E<0.3$ MeV) and calculations (for $E<0.6$ MeV) of the LUNA Collaboration [Phys. Rev. C 102, 052802(R) (2020)]. The recommended extrapolated zero value $S\left(0\right)$ turned out to be 101 eV b. Using the theoretical total cross sections, the ${}^6\mathrm{Li}(p,\gamma ){}^7\mathrm{Be}$ capture reaction rate is calculated at temperatures ranging from $0.01T_9$ to $10T_9$ and compared with NACRE and NACRE II. Analytical expressions for the $S$ factor and reaction rate are given, and the effect of low-lying resonances on the reaction rate is estimated. We suggest updating the NACRE and NACRE II databases in light of the new LUNA data and present calculations.

Figure 1

Doublet and quartet $S$ phase shifts of elastic $p{}^6\mathrm{Li}$ scattering at low energies. The ${}^{2}S$ and ${}^{4}S$ phase shifts are taken from Ref. [42] and shown by $•$ and $▲$, respectively. Results from [16, 17] obtained according to [42] are shown by the dashed-dotted curves. Results of the present work are shown by the solid curve.

Figure 2

$P$ and $F$ phase shifts of elastic $p+{}^6\mathrm{Li}$ scattering obtained for scattering potentials with the parameters from Tables 4 and 5.

Figure 3

Schematics of the energy spectrum of ${}^7\mathrm{Be}$. The energies are given in MeV and the figure is not drawn to scale [35]. Label “ ${}^a$ ” means the above-threshold resonance at 6.73 MeV with a width of 1.2 MeV refers to the ${}^4\mathrm{He}{}^3\mathrm{He}$ channel that is not considered in the present work.

Figure 4

Astrophysical $S$ factor of the ${}^6\mathrm{Li}(p,{\gamma }_0){}^7\mathrm{Be}$ capture to GS. Experimental data are taken from $▲$ [7], $•$ [11], $◾$ [12], $◆$ [10], $★$ [46], $\square$ [6], $▿$ [8]. Parameters of the continuum potential are given in Table 4. A band shows the sensitivity to changes in $C_{\text{w}}$.

Figure 5

Astrophysical $S$ factor of the ${}^6\mathrm{Li}(p,{\gamma }_1){}^7\mathrm{Be}$ capture to FES. Experimental data $\circ$ are taken from [9]. Parameters of the continuum potential are given in Table 5. A band shows the sensitivity to changes in $C_{\text{w}}$.

Figure 6

Astrophysical $S$ factor of the ${}^6\mathrm{Li}(p,{\gamma }_{0+1}){}^7\mathrm{Be}$ capture to GS and FES. Experimental data for capture are from $▲$ [7], $•$ [11], $◾$ [12], $◆$ [10], $\square$ [6], $▿$ [8], $★$ [46], $◦$ [9]. The parameters of GS and FES potentials are listed in the figure. A band shows the sensitivity to changes in $C_{\text{w}}$.

Figure 7

Comparison of ${}^6\mathrm{Li}(p,\gamma ){}^7\mathrm{Be}$ reaction astrophysical $S$ factors. Experimental data for capture to GS and FES are from $▲$ [7], $•$ [11], $◾$ [12], $◆$ [10], $\square$ [6], $▿$ [8], $★$ [46]. Results of calculations: red curve: present work; blue curve: Ref. [12]; black and green curves: Ref. [14].

Figure 8

Total ${}^6\mathrm{Li}(p,\gamma ){}^7\mathrm{Be}$ capture reaction rate. Curves indicate the different sums of the capture rates to the GS and FES. Curves are designated as in Fig. 6. The yellow band is taken from Ref. [18].

Figure 9

Comparison of the astrophysical reaction rates in the range $0.001T_9$ to $1T_9$ from [12, 18, 47] to present work, normalized to the NACRE rate [47]. Dotted curves represent the uncertainty of the NACRE [47] rate, while shaded areas represent the uncertainties from LUNA [12] (pale red) and NACRE II [18] (yellow).

Figure 10

Comparison of the astrophysical reaction rates in the range $0.001T_9$ to $10T_9$ from [12, 18, 47] to present work, normalized to the NACRE rate [47]. Dotted curves represent the uncertainty of the NACRE [47] rate, while shaded areas represent the uncertainties from LUNA [12] (pale red) and NACRE II [18] (yellow).