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

Background: The ${}^8\mathrm{Li}(n,\gamma ){}^9\mathrm{Li}$ reaction plays a critical role in several reaction chains leading to the nucleosynthesis of $A>12$ nuclei. Due to unstable nature of ${}^8\mathrm{Li}$ and the unavailability of neutron targets, direct measurements of this reaction are exceedingly difficult. Only upper limits of this cross section, provided by the indirect experiments, have been obtained so far.

Purpose: In this work, we use the Gamow shell model (GSM) in the coupled-channel representation (GSM-CC) to study the properties of ${}^9\mathrm{Li}$ and the radiative capture reaction ${}^8\mathrm{Li}(n,\gamma ){}^9\mathrm{Li}$.

Method: GSM-CC is a theoretical framework allowing for the description of both nuclear structure and reaction cross sections. In GSM-CC calculations, a translationally invariant Hamiltonian is used with a finite-range two-body interaction tuned to reproduce the low-energy spectra of ${}^{8,9}\mathrm{Li}$. The reaction channels are built by coupling wave functions of the ground state $2_1^{+}$, the first-excited state $1_1^{+}$, and the first resonance state $3_1^{+}$ in ${}^8\mathrm{Li}$ with the neutron wave function of the projectile in different partial waves. In the calculation of ${}^8\mathrm{Li}(n,\gamma ){}^9\mathrm{Li}$ cross section, all relevant $E1$, $M1$, and $E2$ transitions from the initial continuum states to the final bound states ${3/2}_1^{-}$, ${1/2}_1^{-}$ and the resonance ${5/2}_1^{-}$ of ${}^9\mathrm{Li}$ are included.

Results: The GSM-CC approach reproduces the experimental low-energy spectrum, neutron emission threshold, and spectroscopic factors in ${}^9\mathrm{Li}$. The calculated reaction rate is consistent with the experimental upper limit of the reaction rate obtained in the indirect measurements at stellar energies.

Conclusion: The GSM-CC calculations suggest that the ${}^8\mathrm{Li}(n,\gamma ){}^9\mathrm{Li}$ reaction does not reduce significantly heavy-element production via the main chain ${}^7\mathrm{Li}(n,\gamma ){}^8\mathrm{Li}(\alpha ,n){}^{11}\mathrm{B}(n,\gamma ){}^{12}\mathrm{B}\left({\beta }^{+}\right){}^{12}\mathrm{C}$. Major contribution to the calculated cross section is given by the direct $E1$ transition to the ground state of ${}^8\mathrm{Li}$. The contribution of excited states to the reaction rate does not exceed $\approx 20$% of the total reaction rate.

Figure 1

GSM-CC energy levels of ${}^9\mathrm{Li}$ are compared with experimental data [37]. Energies are given relatively to the binding energy of ${}^4\mathrm{He}$.

Figure 2

The GSM-CC neutron radiative capture cross section of the reaction ${}^8\mathrm{Li}(n,\gamma ){}^9\mathrm{Li}$ is plotted as a function of the neutron projectile energy in the $n+{}^8\mathrm{Li}$ center-of-mass frame. The solid line shows the direct capture to the ground state $J^{\pi }={3/2}_1^{-}$ of ${}^9\mathrm{Li}$ and the red dashed line exhibits the total neutron radiative capture cross section which is a sum of contributions from the capture to $J^{\pi }=3/2_1^{-},1/2_1^{-}$, and $5/2_1^{-}$ final states. All lines represent the fully antisymmetrized GSM-CC results in the long-wavelength approximation [28, 29, 32]. The red points and black squares are the upper limits obtained in the Coulomb-dissociation experiment with Pb and U targets, respectively [11]. Experimental cross sections at ${\tilde{E}}_n=0.25$ and 0.75 MeV correspond to average cross sections in the two decay energy bins: $E_n\in [0.0,0.5]$ MeV and $E_n\in [0.5,1.0]$ MeV.

Figure 3

The $E1$ neutron capture cross section for the ${}^8\mathrm{Li}{(n,\gamma )}^9\text{Li}$ reaction is plotted as a function of the neutron projectile energy in the $n+{}^8\mathrm{Li}$ center-of-mass frame. The solid line represents the fully antisymmetrized GSM-CC calculation for the radiative neutron capture to both ground state $J^{\pi }={3/2}_1^{-}$ and excited states $J^{\pi }={1/2}_1^{-},{5/2}_1^{-}$ of ${}^9\mathrm{Li}$. The dashed, dotted, and the dashed-dotted-dotted lines show the separate contributions to the $E1$ capture cross section of various final states.

Figure 4

The $E1$ neutron capture cross section to different final states (a) $J^{\pi }={3/2}_1^{-}$, (b) ${1/2}_1^{-}$, and (c) ${5/2}_1^{-}$ in the reaction ${}^8\mathrm{Li}{(n,\gamma )}^9\text{Li}$. (d) Sum of the $E1$ capture cross section in panels (a)–(c). For each final state separately, the $E1$ neutron capture cross section is plotted for $s$-wave neutrons (dashed line), $d$-wave neutrons (dotted line), and a sum of $s$- and $d$-wave contributions (solid line). For more information, see the caption of Fig. 3.

Figure 5

The $M1$ neutron capture cross section for the ${}^8\mathrm{Li}{(n,\gamma )}^9\text{Li}$ reaction. The peak corresponds to the ${5/2}_1^{-}$ resonance of ${}^9\mathrm{Li}$. For more information, see the caption of Fig. 3.

Figure 6

The $E2$ neutron capture cross section for the ${}^8\mathrm{Li}{(n,\gamma )}^9\text{Li}$ reaction. (a) Separate contributions to the different final states of ${}^9\mathrm{Li}$ and their sum. (b) Sum of contributions to $J^{\pi }=3/2_1^{-},1/2_1^{-},5/2_1^{-}$ states calculated for two sets of $E2$ effective charges: $(e_{\text{eff}}^p,e_{\text{eff}}^n)=(1.26\mathrm{e},0.47\mathrm{e})$ [52], and $(e_{\text{eff}}^p,e_{\text{eff}}^n)=(e(1-\frac{2}{A}+\frac{Z}{A^2}),eZ/A^2)$ [46, 47] resulting from the recoil correction. The peak corresponds to the ${5/2}_1^{-}$ resonance.

Figure 7

The rate of the ${}^8\mathrm{Li}(n,\gamma ){}^9\mathrm{Li}$ reaction calculated in GSM-CC is shown as a function of temperature ${\mathrm{T}}_9$. The total reaction rate is depicted by the solid line. The separate contributions from the ground state $3/2_1^{-}$ and excited states $1/2_1^{-}$, $5/2_1^{-}$ are shown by dashed, dotted, and dashed-dotted lines, respectively.

Figure 8

The comparison of experimental [11, 12, 13, 14] and theoretical [3, 15, 17, 18, 19, 20, 21, 22, 23, 56] reaction rates for the direct radiative neutron capture ${}^8\mathrm{Li}(n,\gamma ){}^9\mathrm{Li}$ at $T_9=1$. The experimental upper limits in the Coulomb dissociation experiment with Pb and U targets [11] are shown separately. The GSM-CC results are labeled “Present.” The star and square symbols stand for the rate of the direct neutron capture to the ground state $3/2_1^{-}$ and total reaction rate obtained, respectively.