Improved parametrization of the transparency parameter in Kox and Shen models of total reaction cross sections

The total reaction cross section is an essential quantity in particle and heavy-ion transport codes when determining the mean free path of a transported particle. Many transport codes determine the distance a particle is transported before it collides with the target or is stopped in the target material, with the Monte Carlo (MC) method using semiempirical parametrization models for the total reaction cross sections. In order to improve the well-known Kox and Shen models of total reaction cross sections and allow the models to be used at energies below 30 MeV/nucleon, we propose a modified parametrization of the transparency parameter. We also report that the Kox and Shen models have a projectile-target asymmetry and should be used so that the lighter nucleus is always treated as the projectile.

Figure 1

The $C\left(E\right)$ parameter as given by Kox (black dots), and a simple straight line interpolation between the dots (dashed black line). The dotted (red) line is the parametrization by Townsend and Wilson according to Eq. (6), and the dash-dotted (blue) line is the Shen model as it is used in geant4, using Eqs. (7) and (8). The proposed modified parametrization, using a combination of Eqs. (6) and (9) (see the text for details), is given as a solid black line. The line for the Townsend and Wilson model has been extrapolated somewhat below the lower limit of its given validity in order to show its behavior for lower energies.

Figure 2

Comparison of the energy dependence of total reaction cross sections for ${}^{12}\mathrm{C}$ on C calculated with the Kox and Shen models. The dashed black line is the Kox model, and the dotted (red) line is the result of the Kox model by using the Townsend and Wilson parametrization of $C\left(E\right)$. The dash-dotted (blue) line is the Shen model with the approach from geant4, and the solid black line is the modified parametrization presented in this work. Experimental data, shown as open circles, are taken from Ref. [21].

Figure 3

The energy dependence of total reaction cross sections, predicted with the different versions of the Kox and Shen models, for ${}^{12}\mathrm{C}+{}^{27}\mathrm{Al}$. Experimental data, shown as open circles, are taken from Ref. [21].

Figure 4

The energy dependence of total reaction cross sections, predicted with the different versions of the Kox and Shen models, for ${}^{12}\mathrm{C}$+Fe. Experimental data, shown as open circles, are taken from Ref. [21].

Figure 5

Total reaction cross sections for the system $p+{}^{28}\mathrm{Si}$ predicted with the Shen model using the modified parametrization of this work. The black line shows the proton as projectile while the dashed (red) line shows the case when the proton is treated as the target. Experimental data are taken from Refs. [22, 23].