Scaling laws for total reaction cross sections

We analyze the neutron-nucleus total reaction cross sections for all the available experimental data, in the energy range from about 10 MeV to 400 GeV. We obtain simple scaling laws proved to be valid over all the energy ranges studied and for stable nuclei of the whole mass number region. Moreover, the proton-nucleus and nucleus-nucleus total reaction cross sections are studied for all the available experimental data in the energy range from 1 GeV to 400 GeV and from $1A$ GeV to $20A$ GeV, respectively. We show that the scaling laws found for proton(nucleus)-nucleus total reaction cross sections, for energies less than 1 GeV, in our previous works [B. Abu-Ibrahim and A. Kohama, Phys. Rev. C 81, 057601 (2010); B. Abu-Ibrahim, Phys. Rev. C 83, 044615 (2011)], are still valid in this energy region. The uncertainty of the prediction of the scaling laws is about $10\%$. Our findings, which are consistent with previously obtained results, are useful to predict total reaction cross sections for reactions involving nuclei with mass numbers larger than 8, and over a large energy scale.

Figure 1

Values of $\pi C(E,A)$ as a function of target mass number, $A$, for each energy, $E$, indicated by squares. The solid lines indicate the average values. The dashed lines indicate standard deviations around each average value. The experimental data are taken from Refs. [7, 8, 9, 10].

Figure 2

Comparison of the numerical results obtained using Eq. (1) with the experimental data of total reaction cross sections for neutron-nucleus reactions as a function of mass number, $A$, for $E$ $=25.5$ MeV, $1.069$ GeV, and 348 GeV. The solid curves are the numerical results. The values of $\pi C\left(E\right)$ are listed in Table 1. The experimental data are shown by solid circles with error bars, and they are taken from Refs. [9, 10, 12].

Figure 3

Comparison of the numerical results obtained using Eqs. (1) and (2) with the experimental data of the total reaction cross sections for neutron-nucleus reactions as a function of energy, $E$. The solid curves are the numerical results of Eqs. (1) and (2). The experimental data (solid circles with error bars) are taken from Refs. [7, 8, 9, 10, 11, 12, 13, 14, 15, 16].

Figure 4

Comparison of the numerical results of Eq. (3) with the experimental data of the total reaction cross sections for neutron-nucleus reactions as a function of target mass number, $A$. The solid curves represent the numerical results obtained using Eq. (3). The experimental data are taken from Refs. [9, 10].

Figure 5

Comparison of the numerical results with the experimental data of the total reaction cross sections for proton-nucleus reactions as a function of target mass number, $A$. The solid curves represent the numerical results obtained using Eq. (1) of Ref. [1]. The experimental data are taken from Refs. [17, 20].

Figure 6

Comparison of the numerical results with the experimental data of the total reaction cross sections for nucleus-nucleus reactions as a function of projectile mass number, $A$. The target is ${}^{12}\mathrm{C}$. The solid curves represent the numerical results obtained using Eq. (1) of Ref. [3]. The experimental data are taken from Ref. [22].

Figure 7

Values of coefficient, $\pi C\left(E\right)$, as a function of the energy per nucleon, $E$. The top panel represent the values in the energy range from $30A$ MeV to about $1A$ GeV, while the bottom panel represent the values in the energy range from $1A$ GeV to $400A$ GeV. Solid circles with error bars represent the values for proton-nucleus reactions, solid squares with error bars represent the values for neutron-nucleus reactions, and solid triangles with error bars represent the values for nucleus-nucleus reactions. The dashed curve is the prediction of Eq. (2), while the solid curve is the prediction of Eq. (3) of Ref. [1].

Figure 8

The uncertainties in numerical results of the total reaction cross sections as a function of the incident energy per nucleon, $E$. Solid circles represent the values for proton-nucleus reactions, solid squares represent the values for neutron-nucleus reactions, and solid triangles represent the values for nucleus-nucleus reactions. The dashed curves are guide for the eye. Solid line represents uncertainty of $10\%$.

Figure 9

Total cross sections for proton-proton (top) and neutron-proton (bottom). Solid curves are the fittings of Eq. (A1). The experimental data are taken from Ref. [24].