Coulomb breakup of ${}^{22}\mathrm{C}$ in a four-body model

Breakup cross sections are determined for the Borromean nucleus ${}^{22}\mathrm{C}$ by using a four-body eikonal model, including Coulomb corrections. Bound and continuum states are constructed within a ${}^{20}\mathrm{C}+n+n$ three-body model in hyperspherical coordinates. We compute continuum states with the correct asymptotic behavior through the $R$-matrix method. For the $n+n$ potential, we use the Minnesota interaction. As there is no precise experimental information on ${}^{21}\mathrm{C}$, we define different parameter sets for the ${}^{20}\mathrm{C}+n$ potentials. These parameter sets provide different scattering lengths, and resonance energies of an expected $3/2^{+}$ excited state. Then we analyze the ${}^{22}\mathrm{C}$ ground-state energy and rms radius, as well as $E1$ strength distributions and breakup cross sections. The $E1$ strength distribution presents an enhancement at low energies. Its amplitude is associated with the low binding energy, rather than with a three-body resonance. We show that the shape of the cross section at low energies is sensitive to the ground-state properties. In addition, we suggest the existence of a low-energy $2^{+}$ resonance, which should be observable in breakup experiments.

Figure 1

${}^{22}\mathrm{C}$ energy $E_0$ (a) and rms radius (b) as a function of the $0d_{3/2}{}^{21}\mathrm{C}$ resonance energy $E_R$, and for different scattering lengths $a_0$ (in fm). The circles refer to the three potential sets.

Figure 2

Convergence of the ${}^{20}\mathrm{C}+n+n$ ground state energy $E_0$ with the maximum hypermomentum $K_{\max }$ for the potential sets 1 (circles), 2 (filled squares), and 3 (open squares).

Figure 3

Three-body ${}^{20}\mathrm{C}+n+n$ eigenphases for $J=1^{-}$ (a), $J=0^{+}$ (b), and $J=2^{+}$ (c). With set 2, a sharp $2^{+}$ resonance below 0.5 MeV is not shown (see text for details).

Figure 4

Electric dipole strengths distributions of ${}^{22}\mathrm{C}$. The colors correspond to the three potential sets for the ground state. The continuum state is defined by set 1 (solid lines), set 2 (dashed lines), or set 3 (dotted lines).

Figure 5

Total breakup cross sections of ${}^{22}\mathrm{C}$ on ${}^{208}\text{Pb}$ at 240 MeV/nucleon with different ${}^{20}\mathrm{C}+n$ potentials. The dashed lines represent the cross sections with the dipole contribution only (see text for details).