Examination of the ${}^{22}\mathrm{C}$ radius determination with interaction cross sections

A nuclear radius of ${}^{22}\mathrm{C}$ is investigated with the total reaction cross sections at medium- to high-incident energies in order to resolve the radius puzzle in which two recent interaction cross-section measurements using ${}^1\mathrm{H}$ and ${}^{12}\mathrm{C}$ targets show the quite different radii. The cross sections of ${}^{22}\mathrm{C}$ are calculated consistently for these target nuclei within a reliable microscopic framework, the Glauber theory. To describe appropriately such a reaction involving a spatially extended nucleus, the multiple scattering processes within the Glauber theory are fully taken into account, that is, the multidimensional integration in the Glauber amplitude is evaluated using a Monte Carlo technique without recourse to the optical-limit approximation. We discuss the sensitivity of the spatially extended halo tail to the total reaction cross sections. The root-mean-square matter radius obtained in this study is consistent with that extracted from the recent cross-section measurement on ${}^{12}\mathrm{C}$ target. We show that the simultaneous reproduction of the two recent measured cross sections is not feasible within this framework.

Figure 1

Rms radii of ${}^{22}\mathrm{C}$ as a function of potential strengths for $l=0$ with different numbers of MC configurations, $M$. Exact calculations with the center-of-mass contribution are plotted for comparison. The center-of-mass-free rms radii with $M={10}^8$ (corrected) are also plotted for comparison. See text for details.

Figure 2

Tests of total reaction cross sections of ${}^{22}\mathrm{C}$ on ${}^1\mathrm{H}$ target incident at 40 MeV as a function of the center-of-mass uncorrected rms radii with different number of the MC configurations. “Exact” values are also plotted for comparison. See text for details.

Figure 3

Total reaction cross sections of ${}^{22}\mathrm{C}$ on ${}^{12}\mathrm{C}$ target incident at 240 MeV as a function of the center-of-mass–corrected rms radii with different number of the MC configurations. The center-of-mass contribution is exactly excluded in the calculations.

Figure 4

Total reaction cross sections of ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ and ${}^{12}\mathrm{C}+{}^1\mathrm{H}$ collisions as a function of incident energies. Experimental data of the total reaction (${\sigma }_R$) and interaction (${\sigma }_I)$ cross sections are taken from Refs. [44, 45, 46, 47, 48, 49, 50, 51, 52] for ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ and Refs. [53, 54] for ${}^{12}\mathrm{C}+{}^1\mathrm{H}$.

Figure 5

Same as Fig. 4 but of ${}^{20}\mathrm{C}$. Experimental interaction cross-section data are taken from Refs. [11, 12, 42].

Figure 6

Total reaction cross sections of ${}^{22}\mathrm{C}$ on ${}^{12}\mathrm{C}$ and ${}^1\mathrm{H}$ targets incident at 240 MeV and 40 MeV, respectively, as a function of rms radii of ${}^{22}\mathrm{C}$. Thin lines denote the cross sections with the HO wave function. See text for details. Experimental data are taken from Refs. [11, 12].

Figure 7

Difference between two reaction probabilities of the ${}^{22}\mathrm{C}$ wave functions with different rms radii on ${}^{12}\mathrm{C}$ target at 240 MeV and on ${}^1\mathrm{H}$ target at 40 MeV as a function of impact parameters. The reference reaction probability ($R=3.20$) is subtracted from each probabilities with $R=3.40$, 3.60, 3.80, and 4.00 fm. See text for details. Thick lines denote the results with the halo wave functions, while thin lines denote those with the HO wave functions.

Figure 8

Total reaction cross sections of ${}^{22}\mathrm{C}$ on ${}^{12}\mathrm{C}$ and ${}^1\mathrm{H}$ targets as a function of incident energies. The rms radius of ${}^{22}\mathrm{C}$ is set to be $3.38\pm 0.10\mathrm{fm}$. The central value and the lower and upper bounds of the cross sections are indicated by dotted and solid lines, respectively. Experimental data are taken from Refs. [11, 12].

Figure 9

Total reaction cross sections of ${}^{12}\mathrm{C}$, ${}^{20}\mathrm{C}$, and ${}^{22}\mathrm{C}$ on ${}^{12}\mathrm{C}$ target as a function of incident energies calculated with the complete Glauber amplitude, the NTG approximation, and the OLA. See text for details. The rms radii of ${}^{12}\mathrm{C}$, ${}^{20}\mathrm{C}$, and ${}^{22}\mathrm{C}$ employed in the calculations are 2.33, 3.03, and 3.38 fm, respectively.