Microscopic investigation of the ${}^8\mathrm{Li}(n,\gamma ){}^9\mathrm{Li}$ reaction

Background: The ${}^8\mathrm{Li}(n,\gamma ){}^9\mathrm{Li}$ reaction plays an important role in several astrophysics scenarios. It cannot be measured directly and indirect experiments have so far provided only cross section limits. Theoretical predictions differ by an order of magnitude.

Purpose: In this work we study the properties of ${}^9\mathrm{Li}$ bound states and low-lying resonances and calculate the ${}^8\mathrm{Li}(n,\gamma ){}^9\mathrm{Li}$ cross section within the no-core shell model with continuum (NCSMC) with chiral nucleon-nucleon and three-nucleon interactions as input.

Methods: The NCSMC is an ab initio method applicable to light nuclei that provides a unified description of bound and scattering states well suited to calculate low-energy nuclear scattering and reactions. For the capture cross section calculation, we adjust calculated thresholds to experimental values.

Results: Our calculations reproduce the experimentally known bound states as well as the lowest $5/2^{-}$ resonance of ${}^9\mathrm{Li}$. We predict a $3/2^{-}$ spin-parity assignment for the resonance observed at 5.38 MeV. In addition to the very narrow $7/2^{-}$ resonance corresponding presumably to the experimental 6.43 MeV state, we find several other broad low-lying resonances.

Conclusions: Our calculated ${}^8\mathrm{Li}(n,\gamma ){}^9\mathrm{Li}$ cross section is within the limits derived from the 1998 National Superconducting Cyclotron Laboratory Coulomb-dissociation experiment [Phys. Rev. C 57, 959 (1998)]. However, it is higher than cross sections obtained in recent phenomenological studies. It is dominated by a direct E1 capture to the ground state with a resonant contribution at $\approx 0.2$ MeV due to E2/M1 radiation enhanced by the $5/2^{-}$ resonance.

Figure 1

${}^8\mathrm{Li}$ (circles) and ${}^9\mathrm{Li}$ (diamonds) ground-state energy dependence on the size of the NCSM and for ${}^9\mathrm{Li}$ also NCSMC (triangles) basis. Extrapolations to infinite $N_{\max }$ with their uncertainties are presented on the right. The experimental values are shown by dashed-dotted lines. The SRG-evolved $\mathrm{NN}+3\mathrm{N}$(lnl) chiral interaction [26] at the resolution scale of ${\lambda }_{\mathrm{SRG}}=2.0{\mathrm{fm}}^{-1}$ and the HO frequency $\hslash \mathrm{\Omega }=20$ MeV was used. Experimental data are from Ref. [55].

Figure 2

Comparison between the NCSM-calculated and the experimental energy spectra of ${}^8\mathrm{Li}$ (left panel) and ${}^9\mathrm{Li}$ (right panel). The SRG-evolved $\mathrm{NN}+3\mathrm{N}$(lnl) chiral interaction [26] at the resolution scale of ${\lambda }_{\mathrm{SRG}}=2.0{\mathrm{fm}}^{-1}$ was used. The HO basis frequency was $\hslash \mathrm{\Omega }=20$ MeV. Experimental data are from Ref. [55].

Figure 3

Dependence of the ${}^8\mathrm{Li}+n$ eigenphase shifts (top panel) on the NCSMC basis size characterized by $N_{\max }$ for low-lying ${\frac{5}{2}}^{-}$ (red) and ${\frac{3}{2}}^{-}$ (blue and green) resonances of ${}^9\mathrm{Li}$. The same for selected ${\frac{3}{2}}^{-}$ (middle panel) and ${\frac{5}{2}}^{-}$ (bottom panel) $P$-wave phase shifts. The $\mathrm{NN}+3\mathrm{N}$(lnl) chiral interaction was used. $E_{\mathrm{kin}}$ is the kinetic energy of ${}^8\mathrm{Li}+n$ in the center-of-mass frame.

Figure 4

Energies of ${}^9\mathrm{Li}$ bound states and low-lying resonances with respect to the ${}^8\mathrm{Li}+n$ threshold. The leftmost three panels show NCSMC calculations at $N_{\max }=4,6$, and 8. The fourth panel shows the NCSMC-pheno $N_{\max }=8$ calculation. The $\mathrm{NN}+3\mathrm{N}$(lnl) chiral interaction was used. Coloured bars represent the widths of resonances. Experimental data in the rightmost panel are from Ref. [55]. Question marks are used where data is unavailable.

Figure 5

${}^8\mathrm{Li}+n$ eigenphase shifts for selected negative-parity (top panel) and positive-parity (bottom panel) channels. Dashed lines in the bottom panel represent $S$-wave phase shifts. NCSMC calculations performed in $N_{\max }=8$ space with the $\mathrm{NN}+3\mathrm{N}$(lnl) chiral interaction.

Figure 6

${}^8\mathrm{Li}+n$ eigenphase shifts obtained from the $N_{\text{max}}=8$ NCSMC-pheno calculation for selected negative-parity (solid) and positive-parity (dashed) channels.

Figure 7

${}^9\mathrm{Li}3/2^{-}$ g.s. cluster form factors. Only $P$-wave components are shown. The full lines represent the $N_{\max }=8$ NCSMC-pheno calculations, the dashed lines (for the ${}^8\mathrm{Li}{2}^{+}$ state channels only) are NCSM results. The coupling between the ${}^8\mathrm{Li}$ and neutron in the cluster state is given in Eq. (3).

Figure 8

${}^8\mathrm{Li}(n,\gamma ){}^9\mathrm{Li}$ capture cross section obtained in the NCSMC-pheno calculations. We compare $N_{\max }=6$ (dotted lines), $N_{\max }=8$ (dashed lines), the total $N_{\max }=8$ cross-section (solid line), and experimental limits from Ref. [7] (black points). Cross-section contributions from the ground state are shown in blue, contributions from the first excited state are in green.

Figure 9

${}^8\mathrm{Li}(n,\gamma ){}^9\mathrm{Li}$ reaction rate obtained in the $N_{\max }=8$ NCSMC-pheno calculations. The upper line shows the total reaction rate, and the lower line shows the ground-state contribution.