Proton 0.01 MeV resonance width and low-energy $S$ factor of $p+{}^{10}\mathrm{B}$ fusion

Background: The ${}^{10}\mathrm{B}(p,\alpha ){}^7\mathrm{Be}$ reaction is of interest for nuclear reaction theory, nuclear astrophysics, and is important for neutronless (aneutronic) fusion ${}^{11}\mathrm{B}(p,2\alpha ){}^4\mathrm{He}$. At low-energies the $S$ factor of the ${}^{10}\mathrm{B}(p,\alpha ){}^7\mathrm{Be}$ reaction is contributed by the near-threshold resonance $(E_x=8.70\mathrm{MeV};{\frac{5}{2}}^{+})$ with the resonance energy $E=0.01\mathrm{MeV}$ and by higher resonances. Contrary to the $\alpha$ width, the proton resonance width of this resonance is unknown.

Purpose: In this paper, the proton resonance width of the near-threshold resonance is calculated using two different approaches. The values of the proton width are used to calculate the low-energy $S$ factor.

Method: First, the proton resonance width is estimated using the mirror symmetry of the resonance ${}^{11}\mathrm{C}[E_x=8.70\mathrm{MeV};{\frac{5}{2}}^{+}]$ and the mirror bound state ${}^{11}\mathrm{B}[E_x=9.272\mathrm{MeV};{\frac{5}{2}}^{+}]$. In the second approach, this width is estimated using the $R$-matrix definition of the observable resonance width, which is expressed in terms of the $p\text{−}{}^{10}\mathrm{B}$ resonant wave function calculated in the potential approach utilizing the spectroscopic factor.

Results: Depending on the method chosen, the calculated proton resonance width varies from $1.17\times {10}^{-19}\mathrm{MeV}$ to $3.00\times {10}^{-19}\mathrm{MeV}$. The role of the near-threshold resonance is determined using fitting of two low-energy $S$ factors from direct measurement [C. Angulo et al., Z. Phys. A 345, 333 (1993)] and from the indirect Trojan horse method (THM) [A. Cvetinović et al., Phys. Rev. C 97, 065801 (2018)]. Within the framework of the $R$-matrix method using the determined proton resonance widths and the modified THM parameters for six low-lying resonances, the low-energy $S$ factors were calculated and compared with the corresponding experimental $S$ factors. The closest agreement with the data is achieved with the proton resonance widths $1.0\times {10}^{-19}$ MeV when fitting the $S$ factor from the THM indirect measurements, and $1.68\times {10}^{-19}$ MeV and $2.5\times {10}^{-19}$ MeV when fitting the $S$ factor from [Angulo et al., Z. Phys. A 345, 333 (1993)].

Conclusion: Calculated proton resonance widths using the mirror symmetry and the $R$-matrix method are an order of magnitude larger than the phenomenological $S$ factor determined from the $R$-matrix fit of the latest measurements [Van de Kolk et al., Phys. Rev. C 105, 055802 (2022)]. Using the theoretically determined proton resonance widths I achieved excellent fits of the low-energy $S$ factors determined from the direct and indirect measurements.

Figure 1

The energy levels of ${}^{11}\mathrm{C}$ contributing to low-energy ${}^{10}\mathrm{B}(p,\alpha ){}^7\mathrm{Be}$ reaction cross section. $E_x$ is the excitation energy expressed in MeV. Only levels slightly above the proton threshold and one subthreshold bound state in the channel ${}^{10}\mathrm{B}+p$ are shown. The black numbers are data taken from [1]. The blue data are from the recent $R$-matrix fit to the $S$ factor and cross section of the ${}^{10}\mathrm{B}(p,\alpha ){}^7\mathrm{Be}$ reaction reported in [2]. The green numbers are taken from [3]. The red lines are the proton and $\alpha$-channel thresholds.

Figure 2

Internal single-particle resonance (solid red line) and bound-state (black dash-dotted line) wave functions of ${}^{11}\mathrm{C}[E_x=8.70\mathrm{MeV};{\frac{5}{2}}^{+}]$ and ${}^{11}\mathrm{B}[E_x=9.272\mathrm{MeV};{\frac{5}{2}}^{+}]$ states, respectively. Both wave functions are normalized to unity in the internal region $0\le r\le 4\mathrm{fm}$.

Figure 3

The same as in Fig. 2 but for $0\le r\le 5\mathrm{fm}$.

Figure 4

Ratio $\frac{{\mathrm{\Gamma }}_{\left(p\right)01/2}}{{\left(C_{\left(n\right)0\frac{1}{2}}\right)}^2}$ determined from the mirror symmetry of ${}^{11}\mathrm{C}[E_x=8.70\mathrm{MeV};{\frac{5}{2}}^{+}]$ and ${}^{11}\mathrm{B}[E_x=9.27\mathrm{MeV};{\frac{5}{2}}^{+}]$ states. The solid red line is determined using Eq. (5), and the black dash-dot-dotted line is determined using Eq. (8). Note that a discontinuity of the solid red line is caused by the node of the bound-state wave function at $r\approx 2$ fm, see Figs. 2 and 3.

Figure 5

Low-energy $S$ factors for the ${}^{10}\mathrm{B}(p,\alpha ){}^7\mathrm{Be}$ reaction. The green short dots are the $S$ factor from [6]. The black dash-dotted line with errors is the THM $S$ factor [3]. The solid red (magenta dashed) line is the present work best fit to data from [6] for the proton resonance width ${\mathrm{\Gamma }}_{\left(p\right)0\frac{1}{2}}=1.68\times {10}^{-19}\mathrm{MeV}({\mathrm{\Gamma }}_{\left(p\right)0\frac{1}{2}}=2.5\times {10}^{-19}$ MeV) of the resonance $E_R=0.01$ MeV. The blue dash-dot-dotted line is obtained with the proton resonance width of $2.2\times {10}^{-17}\mathrm{MeV}$ [4].

Figure 6

The low-energy $S$ factors for the ${}^{10}\mathrm{B}(p,\alpha ){}^7\mathrm{Be}$ reaction. All the notations are the same as in Fig. 5 except for the solid red line, which is now the $R$-matrix fit to the THM data from [3].