Three-body breakup of ${}^{11}$Li with the eikonal method

The ${}^{11}$Li breakup on a ${}^{208}$Pb target is studied in the Coulomb-corrected eikonal approximation by using a ${}^9\mathrm{Li}+n+n$ three-body description of the projectile. The ${}^{11}$Li wave functions are defined in the hyperspherical formalism for bound and scattering states and are obtained from effective ${}^9\mathrm{Li}+n$ and $n+n$ interactions. We first determine $0^{+}$, $1^{-}$, and $2^{+}$ ${}^9\mathrm{Li}+n+n$ phase shifts, which suggest the existence of a narrow $1^{-}$ resonance near 0.5 MeV above threshold. The calculated breakup cross sections show a peak at low energies, in agreement with the data. The influence of monopole and quadrupole components is analyzed and shown to be non-negligible. We discuss the derivation of dipole strengths from experimental breakup cross sections and suggest that the simple Coulomb dipole approximation, traditionally used in the literature, should be replaced by more elaborate models. We also show that ${}^{11}\mathrm{Li}+{}^{208}\mathrm{Pb}$ elastic scattering measurements would provide an indirect test of the model.

Figure 1

Dominant $0^{+}$, $1^{-}$, and $2^{+}$ eigenphases of the ${}^9\mathrm{Li}+n+n$ system.

Figure 2

$E1$ strength distribution calculated with the $R$-matrix method for $K_{\max }=25$ (solid line), $K_{\max }=23$ (dashed line), and $K_{\max }=13$ (dotted line).

Figure 3

$E1$ strength distribution calculated with the $R$-matrix (solid lines) and pseudostate (dashed lines) methods. The calculations are convoluted with a Gaussian function and with three different $\sigma$ values (labels, in MeV). Experimental data are taken from Ref. [2].

Figure 4

Total breakup cross section (solid line) of ${}^{11}$Li on ${}^{208}$Pb at 70 MeV/nucleon and its partial-wave decomposition (dashed lines).

Figure 5

Breakup cross sections of the $0^{+},1^{-},2^{+}$ partial waves. The dashed curves correspond to a modified ${}^9$Li-target potential as described in Sec. V D.

Figure 6

Total breakup cross section (solid line) and $1^{-}$ contribution (dashed line) convoluted with the detector response. Experimental data are taken from Ref. [2].

Figure 7

Double differential cross sections (solid lines) as a function of the scattering angle with their partial-wave decomposition at different excitation energies (top labels in MeV). The dashed, dotted, and dash-dotted lines correspond to the $0^{+}$, $1^{-}$, and $2^{+}$ partial waves, respectively.

Figure 8

Total angular distribution (thin solid line) for the breakup of ${}^{11}$Li on ${}^{208}$Pb at 70 MeV/nucleon in the range $0\le E\le 4$ MeV, and its decomposition in the dominant partial waves $0^{+}$ (dashed), $1^{-}$ (dotted), and $2^{+}$ (dash-dotted). The thick solid line shows the total angular distribution convoluted with the detector resolution. The experimental data are taken from Ref. [2].

Figure 9

(Top) $1^{-}$ component of the double-differential cross sections $d^2\sigma /d{E}_{21}d{E}_{c\left(12\right)}$ in b/MeV${}^2$ as a function of partial energies $E_{21}$ and $E_{c\left(12\right)}$ for ${}^{11}$Li breakup on ${}^{208}$Pb at 70 MeV/nucleon. (Bottom) Same as top panel for $d^2\sigma /d{E}_{1c}d{E}_{2\left(c1\right)}$.

Figure 10

Elastic scattering angular distributions of ${}^{11}$Li and ${}^9$Li on ${}^{208}$Pb in the present four-body eikonal model and in a two-body eikonal model at 70 MeV/nucleon, respectively. The solid lines correspond to the original ${}^9$Li-target potential, and the dashed and dotted lines are obtained with the modified ${}^9$Li-target potential (see text).

Figure 11

Dependence of the experimental $B\left(E1\right)$ strength distribution on ${\theta }_c$ (in degrees). The circles correspond to the conditions of Ref. [2], and the triangles and squares correspond to other choices of ${\theta }_c$ (the lines are to guide the eye). The solid line corresponds to the present model with the experimental energy resolution.