Microscopic approach to ${}^3\mathrm{He}$ scattering

We propose a practical folding model to describe ${}^3\mathrm{He}$ elastic scattering. In the model, ${}^3\mathrm{He}$ optical potentials are constructed by making the folding procedure twice. First the nucleon-target potential is evaluated by folding the Melbourne $g$ matrix with the target density and localizing the nonlocal folding potential with the Brieva-Rook method, and second the resulting local nucleon-target potential is folded with the ${}^3\mathrm{He}$ density. This double single-folding model well describes ${}^3\mathrm{He}$ elastic scattering from ${}^{58}\mathrm{Ni}$ and ${}^{208}\mathrm{Pb}$ targets in a wide incident-energy range from 30 to 150 MeV/nucleon with no adjustable parameter. Spin-orbit force effects on differential cross sections are found to be appreciable only at higher incident energies such as 150 MeV/nucleon. Three-nucleon breakup effects of ${}^3\mathrm{He}$ are investigated with the continuum discretized coupled-channels method and are found to be appreciable only at lower incident energies around 40 MeV/nucleon. Effects of knock-on exchange processes are also analyzed.

Figure 1

Total reaction cross section ${\sigma }_{\mathrm{R}}$ as a function of $E_{\mathrm{in}}/A_{\mathrm{P}}$ for (a) ${}^3\mathrm{He}+{}^{58}\mathrm{Ni}$ scattering and (b) ${}^3\mathrm{He}+{}^{208}\mathrm{Pb}$ scattering. The circles (squares) stand for results of the DSF (DF-FDA) model. The spin-orbit force is not included in both the models. The experimental data are taken from Refs. [48, 49].

Figure 2

Differential cross sections as a function of transfer momentum $q$ for (a) ${}^3\mathrm{He}+{}^{58}\mathrm{Ni}$ scattering and (b) ${}^3\mathrm{He}+{}^{208}\mathrm{Pb}$ scattering. The cross section at each $E_{\mathrm{in}}/A_{\mathrm{P}}$ is multiplied by the factor shown in the panel. The solid and dashed lines denote results of the DSF and DF-FDA models, respectively. The spin-orbit force is not included in both the models. The experimental data are taken from Refs. [52, 53, 54, 55, 56].

Figure 3

Projectile-breakup and spin-orbit force effects on (a) differential cross sections and (b) total reaction cross sections for ${}^3\mathrm{He}+{}^{58}\mathrm{Ni}$ scattering. In panel (a), the differential cross section at each $E_{\mathrm{in}}/A_{\mathrm{P}}$ is multiplied by the factor shown in the panel. The dot-dashed and dashed lines denote results of the DSF model with and without the spin-orbit force, respectively, while the solid lines correspond to results of CDCC in which the spin-orbit force is neglected. In panel (b), the circles stand for results of the DSF model without the spin-orbit force, while the triangles correspond to results of CDCC in which the spin-orbit force is neglected. Results of the DSF model with the spin-orbit force agree with those of the DSF model without the spin-orbit force. The experimental data are taken from Refs. [48, 49, 52, 53, 54]. For $E_{\mathrm{in}}/A_{\mathrm{P}}=72.3$ MeV in panel (a), the CDCC result (solid line) is somewhat different from the DSF one (dashed line) at $1.5<q<2.5{\mathrm{fm}}^{-1}$, although the two results are close to each other at backward angles $q>2.5{\mathrm{fm}}^{-1}$; note that the dashed line agrees with the dot-dashed line at $1.5<q<2.5{\mathrm{fm}}^{-1}$. The few percent deviation between CDCC and DSF results in panel (b) for ${\sigma }_{\mathrm{R}}$ mainly comes from the difference between CDCC and DSF results in panel (a) at $1.5<q<2.5{\mathrm{fm}}^{-1}$.

Figure 4

Projectile-breakup effects on elastic $S$-matrix elements for ${}^3\mathrm{He}+{}^{58}\mathrm{Ni}$ scattering at $E_{\mathrm{in}}/A_{\mathrm{P}}=40$ MeV. The filled (open) circles correspond the elastic $S$-matrix elements calculated with the DSF model (CDCC); here note that the spin-orbit force is not included. The elastic $S$-matrix elements tend to a point $(1,0)$ as $L$ increases.

Figure 5

Comparison of DSF, DF-TDA, and DF-FDA models for differential cross sections in (a) ${}^3\mathrm{He}+{}^{58}\mathrm{Ni}$ scattering and (b) ${}^3\mathrm{He}+{}^{208}\mathrm{Pb}$ scattering. The differential cross section at each $E_{\mathrm{in}}/A_{\mathrm{P}}$ is multiplied by the factor shown in the panel. The dashed (dot-dashed) lines denote results of the DF-TDA (DF-FDA) model, while the solid lines stand for results of the DSF model. The spin-orbit force is not included in both of the models. The experimental data are taken from Refs. [52, 53, 54, 55, 56].

Figure 6

Microscopic optical potentials for ${}^3\mathrm{He}+{}^{58}\mathrm{Ni}$ scattering at $E_{\mathrm{in}}/A_{\mathrm{P}}=40$ MeV. Here $V\left(R\right)$ [$W\left(R\right)$] is the real (imaginary) part of the potentials. The dashed (dot-dashed) line stands for the DF-TDA (DF-FDA) potential, whereas the solid line denotes the DSF potential.