Reaction cross sections of carbon isotopes incident on a proton

We systematically study total reaction cross sections of carbon isotopes with $N=6\text{−}16$ on a proton target for wide range of incident energies. An emphasis is put on the difference from the case of a carbon target. The calculations include the reaction cross sections of ${}^{19,20,22}\mathrm{C}$ at $40A$ MeV, the data of which have recently been measured at RIKEN. The Glauber theory is used to calculate the reaction cross sections. To describe the intrinsic structure of the carbon isotopes, we use a Slater determinant generated from a phenomenological mean-field potential, and construct the density distributions. To go beyond the simple mean-field model, we adopt two types of dynamical models: One is a core$+n$ model for odd-neutron nuclei, and the other is a core$+n+n$ model for ${}^{16}\mathrm{C}$ and ${}^{22}\mathrm{C}$. We propose empirical formulas which are useful in predicting unknown cross sections.

Figure 1

Density distributions of the carbon isotopes. The dotted curve shows the proton density, the dashed curve the neutron density, and the solid curve the matter density.

Figure 2

Comparison of the numerical results with the proton-${}^{12}\mathrm{C}$ total reaction cross section data as a function of energy. The experimental data are taken from Refs. [25, 26].

Figure 3

Reaction cross section for the carbon isotopes at 40 MeV. The experimental data are taken from Ref. [25]: The larger one is natural carbon at 42 MeV, and the smaller one is ${}^{12}\mathrm{C}$ at 40 MeV. The energy is converted to the case of a proton target. The preliminary data for ${}^{22}\mathrm{C}$ is about 1000 mb with a large uncertainty [13].

Figure 4

Comparison of the black square radii, $a=\sqrt{{\sigma }_{\mathrm{R}}/\pi }$, of the carbon isotopes as a function of the mass number, $A$. The open circle denotes the numerical result, and the solid line the fit using Eq. (11).

Figure 5

The values of $R(\mathrm{C})$ of Eq. (13) as a function of the mass number, $A$. The dashed lines denote 0.96±0.05 values.

Figure 6

The total reaction cross section of proton-${}^{22}\mathrm{C}$ as a function of energy (solid curve), and its decomposition to the neutron and proton contributions.

Figure 7

Reaction probability, Eq. (16), times $2\pi b$ for proton-${}^{12}\mathrm{C}$. The solid curve is the total probability. The dashed curve is the neutron contribution, while the dotted curve is the proton contribution. The choice of $c=d=1/2$ is made.

Figure 8

The same as in Fig. 7 but for proton-${}^{22}\mathrm{C}$ case. The choice of $c=d=1/2$ is made.

Figure 9

Comparison of the calculated reaction cross section at a given impact parameter, $b$, in ratio to the total reaction cross section between proton-${}^{12,19,22}\mathrm{C}$ and ${}^{12}\mathrm{C}$-${}^{12,19,22}\mathrm{C}$. The solid curve is the calculation at 800 MeV, while the dashed curve is at 40 MeV. The dotted line indicates $90\%$ of ${\sigma }_{\mathrm{R}}$.

Figure 10

The differential cross sections of the $\mathit{pn}$ elastic scattering. The solid curves denote the fitting using double Gaussians, as explained in the text, while the dashed curves the results with the parameters listed in Table . The experimental data are taken from Ref. [28].