Binding energy constraint on matter radius and soft dipole excitations of ${}^{22}$C

An unusually large value of the ${}^{22}$C matter radius has recently been extracted from measured reaction cross sections. The giant size can be explained by a very loose binding that is, however, not known experimentally yet. Within the three-body cluster model we have explored the sensitivity of the $s$-motion-dominated ${}^{22}$C geometry to the two-neutron separation energy. A low energy of a few tens of keV is required to reach the alleged experimental lower value of the matter radius, while the experimental mean radius requires an extremely tiny binding. The dependence of the ${}^{22}$C charge radius on the two-neutron separation energy is also presented. The soft dipole mode in ${}^{22}$C is shown to be strongly affected by the loose binding and should be studied in the process of Coulomb fragmentation.

Figure 1

Convergence of two-neutron separation energy calculations relative to the cutoff in hypermoments, $K_{\max }$. The curves correspond to calculations with neutron-core potential with depth ranging from $-4.1$ MeV (the upper one) to $-3.3$ MeV (the lowest one) with steps of 0.1 MeV, with none giving rise to bound neutron-core states.

Figure 2

The dependence of the ${}^{22}$C r.m.s. matter radius on the two-neutron separation energy (on a logarithmic scale). The lower and upper curves correspond to calculations with the radius of the neutron-core potential equal to 3.5 and 4.5 fm, respectively. The horizontal solid and dashed lines are the mean value and lower boundary of the experimental matter radius [1], respectively. Bars show theoretical estimations of the $S_{2n}$ energy at which the experimental mean value and lower boundary of the matter radius can be obtained. The insert shows the same curves, but on a linear energy scale.

Figure 3

Dependence of the ${}^{22}$C r.m.s. matter radius (left axis) on the number of hypermoments. The solid lines from lower to upper correspond to calculations with cutoff $K_{\max }$ $=$ 2, 6, 10, and 14, respectively. The dashed line shows the absolute value of the scattering length (right axis, logarithmic) of the neutron-core potential used for calculations with $K_{\max }$ $=$ 14. All results correspond to calculations with the radius of the core-neutron potential equal to 3.5 fm.

Figure 4

Dependence of the ${}^{22}$C charge radius for pointlike protons on the two-neutron separation energy (on a logarithmic scale) of the ground state. The lower, middle, and upper curves correspond to calculations with the ${}^{20}$C core charge radius equal to 2.25, 2.37, and 2.50 fm, respectively. All results are given for the core-neutron potential with a radius of 4.5 fm.

Figure 5

Calculations for ${}^{22}$C dipole strength function distributions over total continuum energy $E_T$ above the breakup threshold. Calculations for ${}^{22}$C two-neutron separation energies $S_{2n}$ $=$ 400 keV (a) and 50 keV (b) are shown. The solid lines from upper to lower correspond to calculations of the dipole continuum wave functions with $K_{f,\max }$ $=$ 13, 9, 5, and 1, respectively. The dashed lines show distributions for three-body plane wave (no FSI) continuum states.

Figure 6

Calculated ${}^{22}$C dipole strength function distributions for separation energies $S_{2n}$ $=$ 50, 100, 200, and 400 keV (upper to lower curves). The insert compares dipole strength distributions for $S_{2n}$ $=$ 10 keV (upper line) and 50 keV (lower line).