Deuteron-induced nucleon transfer reactions within an ab initio framework: First application to $p$-shell nuclei

Background: Low-energy transfer reactions in which a proton is stripped from a deuteron projectile and dropped into a target play a crucial role in the formation of nuclei in both primordial and stellar nucleosynthesis, as well as in the study of exotic nuclei using radioactive beam facilities and inverse kinematics. Ab initio approaches have been successfully applied to describe the ${}^3\mathrm{H}(d,n){}^4\mathrm{He}$ and ${}^3\mathrm{He}(d,p){}^4\mathrm{He}$ fusion processes.

Purpose: An ab initio treatment of transfer reactions would also be desirable for heavier targets. In this work, we extend the ab initio description of $(d,p)$ reactions to processes with light $p$-shell nuclei. As a first application, we study the elastic scattering of deuterium on ${}^7\mathrm{Li}$ and the ${}^7\mathrm{Li}(d,p){}^8\mathrm{Li}$ transfer reaction based on a two-body Hamiltonian.

Methods: We use the no-core shell model to compute the wave functions of the nuclei involved in the reaction, and describe the dynamics between targets and projectiles with the help of microscopic-cluster states in the spirit of the resonating group method.

Results: The shapes of the excitation functions for deuterons impinging on ${}^7\mathrm{Li}$ are qualitatively reproduced up to the deuteron breakup energy. The interplay between $d-{}^7\mathrm{Li}$ and $p-{}^8\mathrm{Li}$ particle-decay channels determines some features of the ${}^9\mathrm{Be}$ spectrum above the $d+{}^7\mathrm{Li}$ threshold. Our prediction for the parity of the 17.298 MeV resonance is at odds with the experimental assignment.

Conclusions: Deuteron stripping reactions with $p$-shell targets can now be computed ab initio, but calculations are very demanding. A quantitative description of the ${}^7\mathrm{Li}(d,p){}^8\mathrm{Li}$ reaction will require further work to include the effect of three-nucleon forces and additional decay channels and to improve the convergence rate of our calculations.

Figure 1

(a) ${}^7\mathrm{Li}(d,p){}^8\mathrm{Li}$ reaction as described in the present work, with the mass partitions in the entrance and exit channels modeled with cluster wave functions. (b) The reaction as measured in experiment, where the detection apparatus can be based on (1) the counting of protons, (2) the $\beta$ decay of ${}^8\mathrm{Li}$, or (3) the yield of the ${}^8\mathrm{Be}$ (not shown in the figure) delayed $\alpha$'s following the ${}^8\mathrm{Li}$ beta decay.

Figure 2

Calculated (a) negative- and (b) positive-parity eigenphase shifts within the coupled $(d,{}^7\mathrm{Li})+(p,{}^8\mathrm{Li})$ NCSM-RGM basis, as a function of the relative kinetic energy in the c.m. frame with respect to the $p+{}^8\mathrm{Li}$ threshold. The $\text{SRG-N}\mathrm{LO} NN$ potential with $\mathrm{\Lambda }=2.02 {\mathrm{fm}}^{-1}$ and the HO frequency of $\hslash \mathrm{\Omega }=20$ MeV were used.

Figure 3

Eigenphase shifts for the $J^{\pi }T={\frac{5}{2}}^{+}\frac{1}{2}$ partial wave (solid blue line) compared to the $d-{}^7\mathrm{Li}$ (solid red line) and $p-{}^8\mathrm{Li}$ (solid black line) elastic phase shifts contributing to the same partial wave through a $P$ and $S$ wave respectively. All results were obtained within the coupled $(d,{}^7\mathrm{Li})+(p,{}^8\mathrm{Li})$ NCSM-RGM basis, and are plotted as a function of the relative kinetic energy in the c.m. frame with respect to the $p+{}^8\mathrm{Li}$ threshold.

Figure 4

Calculated $p-{}^8\mathrm{Li}$ elastic phase shifts within the (a) $(p,{}^8\mathrm{Li})$ NCSM-RGM and (b) $(p,{}^8\mathrm{Li})+{}^9\mathrm{Be}$ NCSMC bases as a function of the relative kinetic energy in the c.m. frame. The $\text{SRG-N}{}^3\mathrm{LO} NN$ potential with $\mathrm{\Lambda }=2.02{\mathrm{fm}}^{-1}$, the $N_{\max }=8$ basis size, and the HO frequency of $\hslash \mathrm{\Omega }=20$ MeV were used. The ${}^9\mathrm{Be}$ NCSM eigenstates used in the NCSMC calculation are specified in Table 3.

Figure 5

Calculated $d-{}^7\mathrm{Li}$ eigenphase shifts in the $J^{\pi }T={\frac{5}{2}}^{+}\frac{1}{2}$ and ${\frac{7}{2}}^{-}\frac{1}{2}$ partial waves as function of the kinetic energy of the deuteron projectile in the laboratory system, resulting from $(d,{}^7\mathrm{Li})$ NCSM-RGM (dashed lines) and $(d,{}^7\mathrm{Li})+{}^9\mathrm{Be}$ NCSMC (solid line) calculations. The $\text{SRG-N}{}^3\mathrm{LO} NN$ potential with $\mathrm{\Lambda }=2.02{\mathrm{fm}}^{-1}$, the $N_{\max }=8$ basis size, and the HO frequency of $\hslash \mathrm{\Omega }=20$ MeV were used. For the ${\frac{7}{2}}^{-}$ channel the NCSMC and NCSM-RGM eigenphase shifts are identical because the adopted set of ${}^9\mathrm{Be}$ eigenstates of Table 3 did not include states for this partial wave.

Figure 6

Resonant $d-{}^7\mathrm{Li} {}^{6}D_{{\frac{7}{2}}^{+}}$ and ${}^{6}P_{{\frac{5}{2}}^{+}}$ phase shifts as functions of the kinetic energy of the deuteron projectile in the laboratory frame, calculated within the $(d,{}^7\mathrm{Li})$ NCSM-RGM basis. The $\text{SRG-N}{}^3\mathrm{LO} NN$ potential with $\mathrm{\Lambda }=2.02 {\mathrm{fm}}^{-1}$, the $N_{\max }=8$ basis size, and the HO frequency of $\hslash \mathrm{\Omega }=20$ MeV were used.

Figure 7

Computed ${}^7\mathrm{Li}(d,d){}^7\mathrm{Li}$ differential cross sections in the c.m. frame at the deuteron scattering angle of ${90}^{\circ }$ as function of the kinetic energy of deuterons in the laboratory system, compared to the experimental data of Ref. [39]. The three sets of theoretical curves correspond to calculations within the $(d,{}^7\mathrm{Li})$ NCSM-RGM (green dashed line), $(d,{}^7\mathrm{Li})+{}^9\mathrm{Be}$ NCSMC (blue dash-dotted line), and $(d,{}^7\mathrm{Li})+(p,{}^8\mathrm{Li})$ NCSM-RGM (red solid line) model spaces.

Figure 8

${}^7\mathrm{Li}(d,p){}^8\mathrm{Li}$ integrated cross section for deuteron laboratory energies up to 2.25 MeV computed within the NCSM-RGM approach at $N_{\max }=6$ (thin-dashed line) and 8 (solid line) compared to the experimental data from Refs. [23, 25, 26, 45] (symbols). Also shown as a thick dashed magenta line is the result of the NCSM-RGM phenomenology approach (see text for details).

Figure 9

Contribution of dominant $J^{\pi } (T=\frac{1}{2})$ partial waves to the NCSM-RGM phenomenology total cross section (solid line) shown in Fig. 8.