Ab initio folding potentials for nucleon-nucleus scattering based on no-core shell-model one-body densities

Background: Calculating microscopic optical potentials for elastic nucleon-nucleus scattering has already led to large body of work in the past. For folding first-order calculations the nucleon-nucleon ($NN$) interaction and the one-body density of the nucleus were taken as input to rigorous calculations in a spectator expansion of the multiple scattering series.

Purpose: Based on the Watson expansion of the multiple scattering series we employ a nonlocal translationally invariant nuclear density derived from a chiral next-to-next-to-leading order (NNLO) and the very same interaction for consistent full-folding calculation of the effective (optical) potential for nucleon-nucleus scattering for light nuclei.

Methods: The first order effective (optical) folding potential is computed by integrating over the nonlocal, translationally invariant NCSM one-body density and the off-shell Wolfenstein amplitudes A and C. The resulting nonlocal potential serves as input for a momentum-space Lippmann-Schwinger equation, whose solutions are summed to obtain the nucleon-nucleus scattering observables.

Results: We calculate scattering observables, such as total, reaction, and differential cross sections as well as the analyzing power and the spin-rotation parameter, for elastic scattering of protons and neutrons from ${}^4\mathrm{He}$, ${}^6\mathrm{He}$, ${}^{12}\mathrm{C}$, and ${}^{16}\mathrm{O}$, in the energy regime between 100 and 200 MeV projectile kinetic energy, and compare to available data.

Conclusions: Our calculations show that the effective nucleon-nucleus potential obtained from the first-order term in the spectator expansion of the multiple scattering expansion describes experiments very well to about 60 degrees in the center-of-mass frame, which coincides roughly with the validity of the NNLO chiral interaction used to calculate both the $NN$ amplitudes and the one-body nuclear density.

Figure 1

Diagram for the matrix element of the effective (optical) potential for the single scattering term.

Figure 2

Wolfenstein amplitudes A and C as function of the scatting angle and momentum transfer for $np$ scattering at 100 MeV laboratory kinetic energy. The solid (red) line stands for the CD-Bonn potential [50] and the dash-dotted (green) line for the $\mathrm{NNLO}{}_{\mathrm{opt}}$ chiral interaction [44]. The solid diamonds represent the extraction from the GW-INS analysis [49].

Figure 3

Same as Fig. 3 for $np$ scattering at 200 MeV laboratory kinetic energy.

Figure 4

The angular distribution of the differential cross section divided by the Rutherford cross section for elastic proton scattering from ${}^{16}\mathrm{O}$ at 200 MeV laboratory kinetic energy as function of the c.m. angle calculated with the $\mathrm{NNLO}{}_{\mathrm{opt}}$ chiral interaction [44]. The different values of $N_{\max }$ are indicated in the legend. From top to bottom, the three sets of lines correspond to $\hslash \omega =24,20$, and 16 MeV, respectively.

Figure 5

The angular distribution of the differential cross section divided by the Rutherford cross section for elastic proton scattering from ${}^4\mathrm{He}$ at 100, 150, and 200 MeV laboratory kinetic energy as function of the momentum transfer and the c.m. angle calculated with the $\mathrm{NNLO}{}_{\mathrm{opt}}$ chiral interaction [44]. The dashed line represents the calculation based on nonlocal densities using $\hslash \omega =16\mathrm{MeV}$, the solid line with 20 MeV, and the dash-dotted line with 24 MeV. For all calculations $N_{\max }=18$ is employed. The data for 100 MeV are taken from Ref. [74], for 156 MeV from Ref. [75], and for 200 MeV from Ref. [76]. The dashed vertical line in each figure indicates the momentum transfer $q=2.45{\mathrm{fm}}^{-1}$ corresponding to the laboratory kinetic energy 125 MeV of the $np$ system.

Figure 6

The angular distribution of the analyzing power for elastic proton scattering from ${}^4\mathrm{He}$ at 100, 150, and 200 MeV laboratory kinetic energy as function of the momentum transfer and the c.m. angle calculated with the $\mathrm{NNLO}{}_{\mathrm{opt}}$ chiral interaction [44]. The lines follow the same notation as in Fig. 5. The data for 150 MeV are taken from Ref. [77], and for 200 MeV from Ref. [76].

Figure 7

The angular distribution of the differential cross section divided by the Rutherford cross section for elastic proton scattering from ${}^{16}\mathrm{O}$ at 100, 135, and 200 MeV laboratory kinetic energy as function of the momentum transfer and the c.m. angle calculated with the $\mathrm{NNLO}{}_{\mathrm{opt}}$ chiral interaction [44]. The dashed line represents the calculation based on nonlocal densities using $\hslash \omega =16\mathrm{MeV}$, the solid line with 20 MeV, and the dash-dotted line with 24 MeV. For all calculations $N_{\max }=10$ is employed. The data for 100 MeV are taken from Ref. [78], for 135 MeV from Ref. [79], and for 200 MeV from Ref. [80]. The dashed vertical line in each figure indicates the momentum transfer $q=2.45{\mathrm{fm}}^{-1}$ corresponding to the laboratory kinetic energy 125 MeV of the $np$ system.

Figure 8

The angular distribution of the analyzing power for elastic proton scattering from ${}^{16}\mathrm{O}$ at 100, 135, and 200 MeV laboratory kinetic energy as function of the momentum transfer and the c.m. angle calculated with the $\mathrm{NNLO}{}_{\mathrm{opt}}$ chiral interaction [44]. Also included is the angular distribution of the spin rotation parameter for elastic proton scattering from ${}^{16}\mathrm{O}$ at 200 MeV. The lines follow the same notation as in Fig. 7. The data for 100 MeV are taken from Ref. [78], for 135 MeV from Ref. [79], and for 200 MeV from Ref. [80].

Figure 9

The angular distribution of the differential cross section divided by the Rutherford cross section for elastic proton scattering from ${}^{12}\mathrm{C}$ at 122, 160, and 200 MeV laboratory kinetic energy as function of the momentum transfer and the c.m. angle calculated with the $\mathrm{NNLO}{}_{\mathrm{opt}}$ chiral interaction [44]. The dashed line represents the calculation based on nonlocal densities using $\hslash \omega =16\mathrm{MeV}$, the solid line with 20 MeV, and the dash-dotted line with 24 MeV. For all calculations $N_{\max }=10$ is employed. The data for 122 MeV are taken from Ref. [81], for 160 MeV from Ref. [81], and for 200 MeV from Ref. [82]. The dashed vertical line in each figure indicates the momentum transfer $q=2.45{\mathrm{fm}}^{-1}$ corresponding to the laboratory kinetic energy 125 MeV of the $np$ system.

Figure 10

The angular distribution of the analyzing power for elastic proton scattering from ${}^{12}\mathrm{C}$ at 122, 160, and 200 MeV laboratory kinetic energy as function of the momentum transfer and the c.m. angle calculated with the $\mathrm{NNLO}{}_{\mathrm{opt}}$ chiral interaction [44]. The lines follow the same notation as in Fig. 10. The data for 122 MeV are taken from Ref. [81], for 160 MeV from Ref. [81], and for 200 MeV from Ref. [82].

Figure 11

The angular distribution of the differential cross section divided by the Rutherford cross section and the analyzing power for proton scattering from ${}^6\mathrm{He}$ at 200 MeV laboratory kinetic energy as function of the momentum transfer $q$ and the c.m. angle. The dashed line represents the calculation based on nonlocal densities using $\hslash \omega =16\mathrm{MeV}$, the solid line with 20 MeV, and the dash-dotted line with 24 MeV. For all calculations $N_{\max }=18$ is employed. The data are taken from [53]. The dashed vertical line in each figure indicates the momentum transfer $q=2.45{\mathrm{fm}}^{-1}$ corresponding to the laboratory kinetic energy 125 MeV of the $np$ system.

Figure 12

The total cross section for neutron scattering from ${}^{16}\mathrm{O}$ as function of the neutron incident energy. The data are taken from Ref. [83]. The solid band corresponds to calculations using the $\mathrm{NNLO}{}_{\mathrm{opt}}$ chiral interaction [44] consistently in the nonlocal density as well as in the $NN$ $t$ matrix with the band width determined by different $\hslash \omega$ values. The downward triangles use the $\mathrm{NNLO}{}_{\mathrm{opt}}$ interaction only in the $NN$ $t$ matrix, while employing a HFB density based on the Gogny-D1S interaction [60]. The squares use this density together with the CD-Bonn [50] $NN$ $t$ matrix.

Figure 13

Calculated reaction cross sections vs. calculated point-proton rms radii $r_{\mathrm{rms}}$ for proton scattering at 200 and 230 MeV laboratory projectile kinetic energies off ${}^4\mathrm{He}$, ${}^{12}\mathrm{C}$, and ${}^{16}\mathrm{O}$: (a) Correlation plot between the two observables at 230 MeV energy and targets of ${}^4\mathrm{He}$, ${}^{12}\mathrm{C}$, and ${}^{16}\mathrm{O}$; to guide the eye, the perfect correlation is indicated by the grey dashed line (see text for details). (b) and (c) Calculated cross sections as function of point-proton $r_{\mathrm{rms}}$ radii for targets of ${}^{12}\mathrm{C}$ (b) and ${}^{16}\mathrm{O}$ (c), shown together with point-proton $r_{\mathrm{rms}}$ radii extracted from NCSM calculations (labeled as “Theory”), and compared to experimental cross sections (where data are available) and experimentally deduced point-proton rms radii extracted from Ref. [62] (labeled as “Expt.”), with the corresponding errors shown by shaded areas (see text for details). For each nucleus, calculations are performed for $N_{\max }=6,8$, and 10, and for $\hslash \omega =16,20$, and 24 MeV.

Figure 14

The angular distribution of the differential cross section divided by the Rutherford cross section and the angular distribution of the analyzing power for elastic proton scattering from ${}^4\mathrm{He}$ (a) and ${}^{16}\mathrm{O}$ (b) at 200 MeV laboratory kinetic energy as function of the momentum transfer and the c.m. angle calculated with the $\mathrm{NNLO}{}_{\mathrm{opt}}$ chiral interaction [44]. The solid line represents the calculation based on nonlocal densities without the center-of-mass (CoM) contribution while the dashed line includes it. For all ${}^4\mathrm{He}$ calculations, $N_{\max }=18$ and $\hslash \omega =20\mathrm{MeV}$ are employed, while all ${}^{16}\mathrm{O}$ calculations employ $N_{\max }=10$ and $\hslash \omega =20\mathrm{MeV}$. The data for ${}^4\mathrm{He}$ at 200 MeV are taken from Ref. [76] while the data for ${}^{16}\mathrm{O}$ at 200 MeV are taken from Ref. [80]. The dashed vertical line in each figure indicates the momentum transfer $q=2.45{\mathrm{fm}}^{-1}$ corresponding to the laboratory kinetic energy 125 MeV of the $np$ system.