One- and two-neutron removal from the neutron-rich carbon isotopes

Reactions that involve the fast removal (knockout) of one and two neutrons from the neutron-rich carbon isotopes, ${}^{15-19}\mathrm{C}$, by light nuclear targets are studied within an eikonal reaction model. Shell model calculations are used to describe the excitation energy spectra and the structures of the carbon isotopes. The calculated one-neutron knockout cross sections from the ${}^A\mathrm{C}$ isotopes, to particle-bound configurations of the ${}^{A-1}\mathrm{C}$ residues, are in agreement with the available experimental data. The two-neutron removal cross sections, producing ${}^{A-2}\mathrm{C}$ residues, receive contributions from both the direct, single-step two-neutron knockout and the indirect mechanism, involving single-neutron removal strength to neutron-unbound excited states in the ${}^{A-1}\mathrm{C}$ system followed by neutron emission. The latter two-step reaction mechanism is shown to be dominant. The empirical odd-even staggering of the single-neutron separation energies along the carbon isotopic chain is reflected in the two-neutron removal data. This staggering and the magnitudes of the two-neutron removal cross sections are reproduced qualitatively by the theoretical calculations.

Figure 1

Schematic energy diagram showing the relevant one- and two-neutron thresholds for knockout reactions from neutron-rich ${}^{18}\mathrm{C}$ (upper part) and from ${}^{19}\mathrm{C}$ (lower part). The energy intervals for transitions applicable to the inclusive cross sections for the indirect two-neutron removal (one-neutron removal followed by neutron emission), ${\sigma }_{-1n(e)}$, and direct two-neutron removal, ${\sigma }_{-2n(d)}$, reaction mechanisms are shown. These are seen to be larger for reactions from the even-$A$ isotopes (upper part).

Figure 2

Ground-state to ground-state one- and two-neutron separation energies for the neutron-rich carbon isotopes (main plot), from Ref. [17]. The inset shows the energy windows for one- and two-neutron knockout transitions, defined as $W_{1n}=S_{1n}(A-1)$ and $W_{2n}=S_{2n}(A-1)-S_{1n}(A-1)=S_{1n}(A-2)$. Population of bound states in ${}^{17}\mathrm{C}$, in two-neutron removal from ${}^{19}\mathrm{C}$, requires either (a) direct population of ${}^{17}\mathrm{C}$ bound states with excitation energy $E^{*}<S_{1n}(17)$ or (b) population of excited particle-unbound states in ${}^{18}\mathrm{C}$ with $S_{1n}(18)<E^{*}<S_{2n}(18)$. Both of these windows are narrow for the odd-$A$ carbon isotopes, with the expectation of suppressed two-neutron removal cross sections.

Figure 3

Calculated and measured inclusive one-neutron removal cross sections for the carbon isotopes. The solid line joins the theoretical calculations of ${\sigma }_{-1n}$ for each isotope. The calculations are for a ${}^{12}\mathrm{C}$ target for the ${}^{16}\mathrm{C}$, ${}^{17}\mathrm{C}$, and ${}^{18}\mathrm{C}$ projectiles at 83, 79, and 80 MeV/nucleon, respectively, and for a ${}^9\mathrm{Be}$ target for ${}^{15}\mathrm{C}$ and ${}^{19}\mathrm{C}$ at 103 and 64 MeV/nucleon. The measured cross sections are for ${}^{15}\mathrm{C}$ [12, 34] (solid squares, solid circles), ${}^{16}\mathrm{C}$ [9, 10, 11] (open circles, solid diamonds, solid triangles), ${}^{17}\mathrm{C}$ [9, 11, 14] (open circles, solid triangles, open squares), ${}^{18}\mathrm{C}$ [11, 15] (solid triangles, open inverted triangles) and ${}^{19}\mathrm{C}$ [9, 13] (open circles, open triangles).

Figure 4

Calculated and measured inclusive two-neutron removal cross sections for the carbon isotopes. The lines join the results from the direct (dotted line and Table ) and indirect one-neutron knockout to unbound intermediate states (dashed line and Table ), two-neutron removal mechanisms. The solid line shows the summed cross sections. The open inverted triangles show the total cross sections when including the additional contributions from shell model states predicted to lie above but within 1.5 MeV of the two-neutron threshold in the mass $A-1$ system. The calculations are for a ${}^{12}\mathrm{C}$ target for the ${}^{15}\mathrm{C}$, ${}^{16}\mathrm{C}$, ${}^{17}\mathrm{C}$, and ${}^{18}\mathrm{C}$ projectiles, at 83, 83, 79, and 80 MeV/nucleon, respectively, and for a ${}^9\mathrm{Be}$ target for ${}^{19}\mathrm{C}$ at 64 MeV/nucleon. The measured cross sections are for ${}^{15}\mathrm{C}$ [34], ${}^{16}\mathrm{C}$ [10], ${}^{17}\mathrm{C}$ [14], ${}^{18}\mathrm{C}$ [15], and ${}^{19}\mathrm{C}$ [13].

Figure 5

Theoretical parallel momentum distributions for one- and two-neutron removal from ${}^{16}\mathrm{C}$, compared with the data of Ref. [10]. The one-neutron distributions (upper panel) are calculated based on the stripping mechanism. Shown are the distributions for the ground state (dotted line, $\ell =0)$ and 0.740 MeV excited state (dashed line, $\ell =2)$ transitions. The results shown include the deduced shell model suppression factor $R_s$ from Table . The two-neutron (lower panel) removal distribution (solid line) is the sum of indirect (dotted line) and direct (dashed line) two-neutron knockout contributions. The calculated parallel momentum distributions have not been folded with the experimental resolutions, though the expected broadening is rather small.

Figure 6

Theoretical parallel momentum distributions for one- and two-neutron removal from ${}^{19}\mathrm{C}$, compared with the data of Ref. [13]. The one-neutron distribution (upper panel) is calculated based on the stripping mechanism and is scaled to the measured integrated inclusive cross section. The calculation had to be offset by −10 MeV/$c$ to overlay the data. The two-neutron (lower panel) removal cross section (solid line) includes contributions from the indirect (dotted line) and direct (dashed line) mechanisms. The dotted-dashed curve results when the total theoretical distribution is scaled to the data. The two-neutron removal distributions had to be offset by −16 MeV/$c$.