Charge-changing-cross-section measurements of ${}^{12-16}$C at around $45A$ MeV and development of a Glauber model for incident energies $10A-2100A$ MeV

Abstract

We have measured for the first time the charge-changing cross sections ($\ddot{I}{ƒ}_{\text{CC}}$) of ${}^{12-16}$C on a ${}^{12}$C target at energies below $100A$ MeV. To analyze these low-energy data, we have developed a finite-range Glauber model with a global parameter set within the optical-limit approximation which is applicable to reaction cross section ($\ddot{I}{ƒ}_{\text{R}}$) and $\ddot{I}{ƒ}_{\text{CC}}$ measurements at incident energies from 10$A$ to $2100A$ MeV. Adopting the proton-density distribution of ${}^{12}$C known from the electron-scattering data, as well as the bare total nucleon-nucleon cross sections, and the real-to-imaginary-part ratios of the forward proton-proton elastic scattering amplitude available in the literatures, we determine the energy-dependent slope parameter ${\hat{I}}_{\mathrm{pn}}^2$ of the proton-neutron elastic differential cross section so as to reproduce the existing $\ddot{I}{ƒ}_{\text{R}}$ and interaction-cross-section data for ${}^{12}$C+${}^{12}$C over a wide range of incident energies. The Glauber model thus formulated is applied to calculate the $\ddot{I}{ƒ}_{\text{R}}$'s of ${}^{12}$C on a ${}^9$Be and ${}^{27}$Al targets at various incident energies. Our calculations show excellent agreement with the experimental data. Applying our model to the $\ddot{I}{ƒ}_{\text{R}}$ and $\ddot{I}{ƒ}_{\text{CC}}$ for the "neutron-skin" ${}^{16}$C nucleus, we reconfirm the importance of measurements at incident energies below $100A$ MeV. The proton root-mean-square radii of ${}^{12-16}$C are extracted using the measured $\ddot{I}{ƒ}_{\text{CC}}$'s and the existing $\ddot{I}{ƒ}_{\text{R}}$ data. The results for ${}^{12-14}$C are consistent with the values from the electron scatterings, demonstrating the feasibility, usefulness of the $\ddot{I}{ƒ}_{\text{CC}}$ measurement and the present Glauber model.

8 pages, 4 figures