Microscopic optical potentials derived from ab initio translationally invariant nonlocal one-body densities

Background: The nuclear optical potential is a successful tool for the study of nucleon-nucleus elastic scattering and its use has been further extended to inelastic scattering and other nuclear reactions. The nuclear density of the target nucleus is a fundamental ingredient in the construction of the optical potential and thus plays an important role in the description of the scattering process.

Purpose: In this paper we derive a microscopic optical potential for intermediate energies using ab initio translationally invariant nonlocal one-body nuclear densities computed within the no-core shell model (NCSM) approach utilizing two- and three-nucleon chiral interactions as the only input.

Methods: The optical potential is derived at first order within the spectator expansion of the nonrelativistic multiple scattering theory by adopting the impulse approximation. Nonlocal nuclear densities are derived from the NCSM one-body densities calculated in the second quantization. The translational invariance is generated by exactly removing the spurious center-of-mass (COM) component from the NCSM eigenstates.

Results: The ground-state local and nonlocal densities of ${}^{4,6,8}\mathrm{He}$, ${}^{12}\mathrm{C}$, and ${}^{16}\mathrm{O}$ are calculated and applied to optical potential construction. The differential cross sections and the analyzing powers for the elastic proton scattering off these nuclei are then calculated for different values of the incident proton energy. The impact of nonlocality and the COM removal is discussed.

Conclusions: The use of nonlocal densities has a substantial impact on the differential cross sections and improves agreement with experiment in comparison to results generated with the local densities especially for light nuclei. For the halo nuclei ${}^6\mathrm{He}$ and ${}^8\mathrm{He}$, the results for the differential cross section are in a reasonable agreement with the data although a more sophisticated model for the optical potential is required to properly describe the analyzing powers.

Figure 1

Ground-state ${}^4\mathrm{He}$ nonlocal neutron density calculated with an $N_{\max }=14$ basis space, an oscillator frequency of $\hslash \mathrm{\Omega }=20$ MeV, and a flow parameter of ${\lambda }_{\mathrm{SRG}}=2.0{\mathrm{fm}}^{-1}$.

Figure 2

Ground-state ${}^6\mathrm{He}$ proton and neutron nonlocal densities calculated with a $N_{\max }=12$ basis space, an oscillator frequency of $\hslash \mathrm{\Omega }=20$ MeV, and a flow parameter of ${\lambda }_{\mathrm{SRG}}=2.0{\mathrm{fm}}^{-1}$. Proton densities are shown in blue and neutron densities are shown in red.

Figure 3

Ground-state ${}^8\mathrm{He}$ proton and neutron nonlocal densities calculated with a $N_{\max }=10$ basis space, an oscillator frequency of $\hslash \mathrm{\Omega }=20$ MeV, and a flow parameter of ${\lambda }_{\mathrm{SRG}}=2.0{\mathrm{fm}}^{-1}$. Proton densities are shown in blue and neutron densities are shown in red.

Figure 4

Ground-state nonlocal neutron densities calculated with an $N_{\max }=8$ importance truncated NCSM basis for ${}^{12}\mathrm{C}$ and ${}^{16}\mathrm{O}$. An oscillator frequency of $\hslash \mathrm{\Omega }=20$ MeV, and a flow parameter of ${\lambda }_{\mathrm{SRG}}=1.8$ fm⁻¹ and ${\lambda }_{\mathrm{SRG}}=2.0$ fm⁻¹ are used, respectively.

Figure 5

Comparison between COM contaminated (dashed wiCOM) and translationally invariant (solid trinv) local neutron densities for ${}^4\mathrm{He}$, ${}^6\mathrm{He}$, ${}^{12}\mathrm{C}$, and ${}^{16}\mathrm{O}$.

Figure 6

$N_{\max }$ convergence for the translationally invariant (a) and the COM contaminated (b) local neutron densities for ${}^4\mathrm{He}$.

Figure 7

(a) Standard $N_{\max }$ convergence comparison. (b) Long-range $N_{\max }$ comparison. Both plots use the translationally invariant neutron density of ${}^{16}\mathrm{O}$.

Figure 8

Neutron and proton nuclear density profiles for ${}^4\mathrm{He}$, ${}^{12}\mathrm{C}$, and ${}^{16}\mathrm{O}$. The results named as “local” were obtained from Eq. (26), while the results named as “nonlocal” were obtained from Eq. (24). For ${}^{12}\mathrm{C}$ and ${}^{16}\mathrm{O}$ we also include the nuclear profile computed within a relativistic mean-field description of spherical nuclei. The densities were computed with the NN-${\mathrm{N}}^4\mathrm{LO}$(500)+3Nlnl interaction with $\hslash \mathrm{\Omega }=20$ MeV, ${\lambda }_{\mathrm{SRG}}=2.0{\mathrm{fm}}^{-1}$, and for different values of $N_{\max }$.

Figure 9

(a) Differential cross sections as functions of the laboratory scattering angle computed from trinv local and nonlocal densities. The dashed line is obtained from Eq. (25) with the density profile computed using Eq. (24). (b) Differential cross sections as functions of the laboratory scattering angle computed from wiCOM and trinv nonlocal densities. All cross sections were computed for the ${}^4\mathrm{He}$(p,p)${}^4\mathrm{He}$ reaction at 200 MeV (laboratory energy) and using the bare NN-${\mathrm{N}}^4\mathrm{LO}$(500) interaction for the calculation of the free $NNt$ matrix. All densities were computed with the NN-${\mathrm{N}}^4\mathrm{LO}$(500)+3Nlnl interaction with $N_{\max }=14$, $\hslash \mathrm{\Omega }=20$ MeV, and ${\lambda }_{\mathrm{SRG}}=2.0{\mathrm{fm}}^{-1}$. Experimental data are taken from Ref. [71].

Figure 10

Differential cross sections as functions of the laboratory scattering angle computed from trinv nonlocal densities for the ${}^4\mathrm{He}$(p,p)${}^4\mathrm{He}$ reaction at 200 MeV (laboratory energy) and using the bare NN-${\mathrm{N}}^4\mathrm{LO}$(500) interaction for the calculation of the free $NNt$ matrix. The densities were computed with the SRG-evolved NN-${\mathrm{N}}^4\mathrm{LO}$(500) interaction plus the three-body SRG-induced (3Nind) or the full three-body (3Nlnl) interaction in the first two cases, while we used the only bare NN-${\mathrm{N}}^4\mathrm{LO}$(500) interaction in the third case. In the first two cases the results were obtained with $N_{\max }=14$, $\hslash \mathrm{\Omega }=20$ MeV, and ${\lambda }_{\mathrm{SRG}}=2.0{\mathrm{fm}}^{-1}$, while in the third case we used $N_{\max }=18$ and $\hslash \mathrm{\Omega }=20$ MeV. Experimental data are taken from Ref. [71].

Figure 11

Differential cross sections in the $pA$ center-of-mass frame computed from trinv local and nonlocal densities for the ${}^4\mathrm{He}$(p,p)${}^4\mathrm{He}$ reaction at the incident proton energy in the laboratory frame of (a) 72 MeV and (b) 156 MeV, respectively. For all calculations the density was computed with the SRG-evolved NN-${\mathrm{N}}^4\mathrm{LO}$(500)+3Nlnl interaction at $N_{\max }=14$ and with $\hslash \mathrm{\Omega }=20$ MeV and ${\lambda }_{\mathrm{SRG}}=2.0{\mathrm{fm}}^{-1}$, while the free $NNt$ matrix was computed with the bare NN-${\mathrm{N}}^4\mathrm{LO}$(500) interaction. Experimental data are taken from Refs. [72, 73].

Figure 12

The same as in Fig. 11 but for the analyzing power at the incident proton energy in the laboratory frame of (a) 72 MeV and (b) 200 MeV, respectively. Experimental data are taken from Refs. [71, 72].

Figure 13

Differential cross sections in the $pA$ center-of-mass frame computed from trinv local and nonlocal densities for the ${}^{12}\mathrm{C}$(p,p)${}^{12}\mathrm{C}$ reaction at the incident proton energy in the laboratory frame of (a) 122, (b) 160, and (c) 200 MeV, respectively. For all calculations the densities were computed with the SRG-evolved NN-${\mathrm{N}}^4\mathrm{LO}$(500)+3Nlnl interaction at $N_{\max }=8$ and with $\hslash \mathrm{\Omega }=20$ MeV and ${\lambda }_{\mathrm{SRG}}=1.8{\mathrm{fm}}^{-1}$, while the free $NNt$ matrix was computed with the bare NN-${\mathrm{N}}^4\mathrm{LO}$(500) interaction. Experimental data are taken from Ref. [74].

Figure 14

The same as in Fig. 13 but for the analyzing power.

Figure 15

The same as in Fig. 13 but for the ${}^{16}\mathrm{O}$(p,p)${}^{16}\mathrm{O}$ reaction at the laboratory energy of (a) 100, (b) 135, and (c) 200 MeV, respectively. Experimental data are taken from Ref. [75].

Figure 16

The same as in Fig. 15 but for the analyzing power.

Figure 17

(a) Differential cross section and (b) analyzing power in the $pA$ center-of-mass frame computed from local RMF and trinv nonlocal densities for the ${}^{12}\mathrm{C}$(p,p)${}^{12}\mathrm{C}$ reaction at the incident proton energy in the laboratory frame of 200 MeV. In all calculations the free $NNt$ matrix was computed with the bare NN-${\mathrm{N}}^4\mathrm{LO}$(500) interaction, while the trinv nonlocal density was computed with the NN-${\mathrm{N}}^4\mathrm{LO}$(500)+3Nlnl interaction. Experimental data are taken from Ref. [74].

Figure 18

The same as in Fig. 17 but for the ${}^{16}\mathrm{O}$(p,p)${}^{16}\mathrm{O}$ reaction. Experimental data are taken from Ref. [75].

Figure 19

The same as in Fig. 8 but for ${}^{6,8}\mathrm{He}$.

Figure 20

(a) Differential cross section and (b) analyzing power in the $pA$ center-of-mass frame computed from trinv local and nonlocal densities for elastic scattering of ${}^6\mathrm{He}$ at the projectile energy of 71 MeV/nucleon. For all calculations the densities were computed with the SRG-evolved NN-${\mathrm{N}}^4\mathrm{LO}$(500)+3Nlnl interaction at $N_{\max }=12$ and with $\hslash \mathrm{\Omega }=20$ MeV and ${\lambda }_{\mathrm{SRG}}=2.0{\mathrm{fm}}^{-1}$, while the free $NNt$ matrix was computed with the bare NN-${\mathrm{N}}^4\mathrm{LO}$(500) interaction. Experimental data are taken from Refs. [78, 79].

Figure 21

The same as in Fig. 20 but for ${}^8\mathrm{He}$ and $N_{\max }=10$. Experimental data are taken from Ref. [80, 81].

Figure 22

Convergence pattern for the differential cross section in the $pA$ center-of-mass frame computed from (a) trinv local and (b) nonlocal densities for elastic scattering of ${}^8\mathrm{He}$ at the projectile energy of 71 MeV/nucleon. For all calculations the densities were computed with the SRG-evolved NN-${\mathrm{N}}^4\mathrm{LO}$(500)+3Nlnl interaction at $N_{\max }=10$ and with $\hslash \mathrm{\Omega }=20$ MeV and ${\lambda }_{\mathrm{SRG}}=2.0{\mathrm{fm}}^{-1}$, while the free $NNt$ matrix was computed with the bare NN-${\mathrm{N}}^4\mathrm{LO}$(500) interaction. Experimental data are taken from Ref. [80, 81].

Figure 23

The same as in Fig. 20 but at the projectile energy of 200 MeV/nucleon.

Figure 24

The same as in Fig. 21 but at the projectile energy of 200 MeV/nucleon.