Elastic differential cross sections for space radiation applications

The eikonal, partial wave (PW) Lippmann-Schwinger, and three-dimensional Lippmann-Schwinger (LS3D) methods are compared for nuclear reactions that are relevant for space radiation applications. Numerical convergence of the eikonal method is readily achieved when exact formulas of the optical potential are used for light nuclei $\left(A\le 16\right)$, and the momentum-space representation of the optical potential is used for heavier nuclei. The PW solution method is known to be numerically unstable for systems that require a large number of partial waves, and, as a result, the LS3D method is employed. The effect of relativistic kinematics is studied with the PW and LS3D methods and is compared to eikonal results. It is recommended that the LS3D method be used for high-energy nucleon-nucleus reactions and nucleus-nucleus reactions at all energies because of its rapid numerical convergence and stability.

Figure 1

Elastic differential cross sections for $p+{}^{16}\text{O}$ reactions for projectile laboratory kinetic energies of (a) 150 MeV/nucl, (b) 500 MeV/nucl, (c) 1000 MeV/nucl, and (d) 20 000 MeV/nucl. Eik. represents eikonal, LS3D represents three-dimensional Lippmann-Schwinger, and PW represents partial wave. Nonrelativistic results are denoted “NR” and relativistic results are denoted “REL.”

Figure 2

Elastic differential cross sections for $p+{}^{56}\text{Fe}$ reactions for projectile laboratory kinetic energies of (a) 150 MeV/nucl, (b) 500 MeV/nucl, (c) 1000 MeV/nucl, and (d) 20 000 MeV/nucl. Eik. represents eikonal, LS3D represents three-dimensional Lippmann-Schwinger, and PW represents partial wave. Nonrelativistic results are denoted “NR” and relativistic results are denoted “REL.”

Figure 3

Elastic differential cross sections for ${}^4\mathrm{He}$ + ${}^{16}\mathrm{O}$ reactions for projectile laboratory kinetic energies of (a) 150 MeV/nucl, (b) 500 MeV/nucl, (c) 1000 MeV/nucl, and (d) 20 000 MeV/nucl. Eik. represents eikonal, LS3D represents three-dimensional Lippmann-Schwinger, and PW represents partial wave. Nonrelativistic results are denoted “NR” and relativistic results are denoted “REL.”

Figure 4

Elastic differential cross sections for ${}^{12}\mathrm{C}$ + ${}^{56}\mathrm{Fe}$ reactions for projectile laboratory kinetic energies of (a) 150 MeV/nucl, (b) 500 MeV/nucl, (c) 1000 MeV/nucl, and (d) 20 000 MeV/nucl. Eik. represents eikonal, LS3D represents three-dimensional Lippmann-Schwinger, and PW represents partial wave. Nonrelativistic results are denoted “NR” and relativistic results are denoted “REL.”

Figure 5

Elastic differential cross sections for (a) $p+{}^{32}\text{S}$ at $T_{\mathrm{Lab}}=1$ GeV/nucl [42], (b) $p+{}^{40}\text{Ca}$ at $T_{\mathrm{Lab}}=500$ MeV/nucl [43], (c) $p+{}^{58}\text{Ni}$ at $T_{\mathrm{Lab}}=1$ GeV/nucl [44], and (d) ${}^4\mathrm{He}$ + ${}^{40}\mathrm{Ca}$ at $T_{\mathrm{Lab}}=347$ MeV/nucl [45].