Gamow shell model description of the radiative capture reaction ${}^8\mathrm{B}(p,\gamma ){}^9\mathrm{C}$

Background: In low metallicity supermassive stars, the hot $pp$ chain can serve as an alternative to produce the CNO nuclei. In the astrophysical environment of high temperature, the proton capture of ${}^8\mathrm{B}$ can be faster than its $\beta$ decay, so that the ${}^8\mathrm{B}(p,\gamma ){}^9\mathrm{C}$ reaction plays an important role in the hot $pp$ chain. Due to the unstable nature of ${}^8\mathrm{B}$ and the unavailability of ${}^8\mathrm{B}$ beams, the study of the ${}^8\mathrm{B}(p,\gamma ){}^9\mathrm{C}$ reaction can only be achieved by indirect methods, so that large uncertainties exist.

Purpose: The Gamow shell model in the coupled-channel representation (GSM-CC) is applied to study the proton radiative capture reaction ${}^8\mathrm{B}(p,\gamma ){}^9\mathrm{C}$.

Method: The GSM-CC is a unified microscopic theory for the description of nuclear structure and nuclear reaction properties. A translationally invariant Hamiltonian is considered for that matter, making use of a finite-range two-body interaction, whose parameters are adjusted to reproduce the low-energy spectra of ${}^8\mathrm{B}$ and ${}^9\mathrm{C}$. The reaction channels are then built through the coupling of the wave functions of the ground state $2_1^{+}$, the first excited state $1_1^{+}$, and the second excited state $3_1^{+}$ in ${}^8\mathrm{B}$ to a projectile proton wave function in different partial waves. For the calculation of the ${}^8\mathrm{B}(p,\gamma ){}^9\mathrm{C}$ astrophysical factor, all $E1, M1$, and $E2$ transitions from the initial continuum states to the final bound states ${3/2}_1^{-}\mathrm{of}{}^9\mathrm{C}$ are considered. The resonant capture to the first resonant state ${1/2}_1^{-}$ of ${}^9\mathrm{C}$ is also calculated.

Results: The experimental low-energy levels and the proton emission threshold in ${}^9\mathrm{C}$ are reproduced by the GSM-CC. The calculated astrophysical factor agrees with experimental data obtained from indirect measurements. The reaction rates from the direct capture and resonant capture are calculated for the temperature range of astrophysical interest.

Conclusion: The calculated total astrophysical $S$ factor is dominated by the $E1$ transition to the ground state of ${}^9\mathrm{C}$. The GSM-CC calculations suggest that $S$ first increases with the energy of the center of mass $E_{\mathrm{c}.\mathrm{m}.}$, and then decrease with the energy. This agrees with existing data, which has smaller values around zero energy and larger values in the energy range of 0.2 MeV $\le E_{\mathrm{c}.\mathrm{m}.}\le$ 0.6 MeV.

Figure 1

The GSM energy levels of ${}^8\mathrm{B}$ and GSM-CC energy levels of ${}^9\mathrm{C}$ are compared with the experimental data [30, 31] in (a) and (b), respectively. Energies are given relatively to that of the ${}^4\mathrm{He}$ core. The measured ${5/2}_1^{+}$ state has a very broad width, which is not shown in the figure. For more details see Table 4 and discussion in the text.

Figure 2

The calculated GSM one-body density of valence protons in the ground state $2_1^{+}$ of ${}^8\mathrm{B}$.

Figure 3

The $E1$ astrophysical factor of the ${}^8\mathrm{B}(p,\gamma ){}^9\mathrm{C}$ reaction is plotted as a function of proton projectile energy in the $p+{}^8\mathrm{B}$ center of mass frame. The solid line represents the fully antisymmetrized GSM-CC calculation for the radiative proton capture to the ground state of ${}^9\mathrm{C}$. The dashed, dashed-dotted, and dotted lines show the contributions from the radiative proton capture to the ${}^9\mathrm{C}$ ground state from the initial $p+{}^8\mathrm{B}$ composite scattering states with $J_i^{\pi }=3/2^{+}, 5/2^{+}$, and $1/2^{+}$, respectively.

Figure 4

The solid line shows the $E1$ astrophysical factor $S^{E1}$ as a function of proton projectile energy in the $p+{}^8\mathrm{B}$ center of mass frame for the proton radiative capture to the ground state of ${}^9\mathrm{C}$. The $E1$ radiative capture of $s$- and $d$-wave protons is shown as the dashed-dotted and dotted lines, respectively.

Figure 5

The $M1$ astrophysical factor for the radiative proton capture to the ground state of ${}^9\mathrm{C}$ as a function of proton projectile energy is plotted (solid line). The peak corresponds to the ${1/2}_1^{-}$ resonance of ${}^9\mathrm{C}$. The dashed, dashed-dotted, and dotted lines show the contributions from the radiative proton capture to the ${}^9\mathrm{C}$ ground state from the initial $p+{}^8\mathrm{B}$ composite scattering states with $J_i^{\pi }=1/2^{-}, 3/2^{-}$, and $5/2^{-}$, respectively.

Figure 6

The $E2$ astrophysical factor is depicted as a function of proton projectile energy. In the upper panel (a), the solid line shows the sum of individual contributions of different initial $p+{}^8\mathrm{B}$ composite scattering states $J_i^{\pi }=3/2_1^{-},1/2_1^{-},5/2_1^{-}$, and $7/2_1^{-}$ to the astrophysical $S^{E2}$ factor of the radiative proton capture to the ground state of ${}^9\mathrm{C}$. The $E2$ effective charges $(e_{\text{eff}}^p,e_{\text{eff}}^n)=(1.26e,0.47e)$ [38] are used. The lower panel (b) compares two different sets of $E2$ effective charges. The solid line is the same as in (a). The dotted line is calculated for the $E2$ effective charges issued from the recoil of the center of mass [Eq. (20)].

Figure 7

(a) The $E1$ astrophysical factor for the radiative proton capture to the ground state of ${}^9\mathrm{C}$ is plotted as a function of proton projectile energy for different values of the scaling factor $f_{E1}$ of $E1$ effective charges [see Eq. (19)]. The total astrophysical $S$ factor is plotted in (b). The different lines correspond to GSM-CC calculations involving different $f_{E1}$ values. The same $E2$ effective charges $(1.26e,0.47e)$ [38] are utilized. Standard $g$ factors are used in $M1$ contribution $S^{M1}$. The astrophysical factor of the proton radiative capture reaction in the limit $E_{\mathrm{c}.\mathrm{m}.}\to 0$ has been extracted by using an expansion from a quadratic polynomial (dashed line), whose parameters are obtained by fitting the calculated function $S\left(E_{\mathrm{c}.\mathrm{m}.}\right)$ in the energy range $0.1\le E_{\mathrm{c}.\mathrm{m}.}\le 0.3$ MeV. The experimental data for the astrophysical factor in the limit $E_{\mathrm{c}.\mathrm{m}.}\to 0$, taken from Refs. [3, 4, 5, 6, 7, 10], are labeled as “E03”, “T02”, “B01”, “G05”, “F12”, and “F15”, respectively. The astrophysical factor extracted from the Coulomb dissociation experiment [9] in the energy range 0.2 MeV $\le E_{\mathrm{c}.\mathrm{m}.}\le$ 0.6 MeV is labeled as “M03”.

Figure 8

Calculation of the total astrophysical factor of the ${}^8\mathrm{B}(p,\gamma ){}^9\mathrm{C}$ reaction with GSM-CC. The capture reaction to the ground state of ${}^9\mathrm{C}$ is shown by the solid line, while the capture to the first excited resonance state, $1/2_1^{-}$, is provided by the dotted line.

Figure 9

Comparison between different experimental analyses [3, 4, 5, 6, 7, 10] and various theoretical calculations [1, 12, 13] of the total astrophysical $S$ factor at $E_{\mathrm{c}.\mathrm{m}.}=0$. The GSM-CC result is labeled as “Present”.

Figure 10

The reaction rate of ${}^8\mathrm{B}(p,\gamma ){}^9\mathrm{C}$ as a function of the temperature ${\mathrm{T}}_9$ calculated by GSM-CC. The solid line represents the direct capture rate to the ground state $3/2_1^{-}$ and the dotted line depicts the capture to the first excited resonance state $1/2_1^{-}$.