Quadrupole deformation of ${}^{16}\mathrm{C}$ studied by proton and deuteron inelastic scattering

New measurements of proton/deuteron elastic and inelastic scattering to the $2_1^{+}$ state of the neutron-rich nucleus ${}^{16}\mathrm{C}$ have been performed at Radioactive Ion Beam Line in Lanzhou (RIBLL) in inverse kinematics. The angular distributions of elastic scattering were well reproduced by the systematic optical potential with normalization factors for the depths of real and imaginary parts. The neutron and proton deformation lengths of ${\delta }_n=1.25\pm 0.30$ fm and ${\delta }_p=1.07\pm 0.26$ fm were extracted from the inelastic scattering data. The ratio of neutron and proton matrix element $M_n/M_p=1.95\pm 0.47$ was determined from the two different probes. Within error bar, this result is in fair agreement with the previous measurements, demonstrating that the combination of proton and deuteron inelastic scattering offers a useful method to extract $M_n/M_p$ for the even-even nuclei. Most of the theoretical predictions by the shell model using the MK, WBT, WBT*, and YSOX interactions in $p-sd$ model space and by the ab initio in-medium similarity renormalization group are close to these experimental results.

Figure 1

The schematic view of experimental setup.

Figure 2

With coincidence of the light-charged particles detected by the telescope T2, particle identification (PID) spectrum (a) and its linearization and projection spectrum (b) measured by the telescope T0. The data were taken from the ${\left({\mathrm{CD}}_2\right)}_n$ target.

Figure 3

Kinematics for the recoil protons (a) and deuterons (b) from the ${\left({\mathrm{CH}}_2\right)}_n$ and ${\left({\mathrm{CD}}_2\right)}_n$ targets, respectively. The solid and dot-dashed lines are the calculated kinematical curves for the elastic scattering and inelastic scattering to the $2_1^{+}$ state at $E_x=1.766$ MeV in the neutron-rich ${}^{16}\mathrm{C}$, respectively. The enlarged part, which is able to discriminate the elastic and inelastic channels, is inserted in (b).

Figure 4

Angular correlation spectra between ${}^{16}\mathrm{C}$ and all the recoil light-charged particles from the ${\left({\mathrm{CH}}_2\right)}_n$ (a) and ${\left({\mathrm{CD}}_2\right)}_n$ (b) target. The red dot-dashed and solid lines are the calculated angular correlated curves of ${}^{16}\mathrm{C}+p$ and ${}^{16}\mathrm{C}+d$, respectively. The region marked by the black dotted curves in (b) represents the cut used to count the hydrogen numbers in the ${\left({\mathrm{CD}}_2\right)}_n$ target.

Figure 5

$Q$-value spectrum obtained from the energies and angles of the recoil protons in coincidence with ${}^{16}\mathrm{C}$. The blue and green histograms represent the protons emitting to the angles of ${63}^{\circ }–{74}^{\circ }$ and ${70}^{\circ }–{72}^{\circ }$, respectively.

Figure 6

Same as Fig. 5, but deduced from the recoil deuterons.

Figure 7

Experimental and calculated elastic scattering DCSs, as a ratio to the Rutherford cross sections, for ${}^{16}\mathrm{C}+p$ (a) and ${}^{16}\mathrm{C}+d$ (b). The thin curves are from the coupled channel calculations using different global OPs with their corresponding normalization factors. The grey solid curves are the best-fit ones for the experimental angular distributions using larger diffuseness parameters in the Woods-Saxon potential. See text for more details.

Figure 8

The experimental angular distributions of inelastic proton (a) and deuteron (b) scattering of ${}^{16}\mathrm{C}$ in comparison with the coupled channel calculations using different global OPs with their corresponding optimal normalization factors and deformation lengths. See text for more details.