Elastic proton scattering of medium mass nuclei from coupled-cluster theory

Using coupled-cluster theory and interactions from chiral effective field theory, we compute overlap functions for transfer and scattering of low-energy protons on the target nucleus ${}^{40}$Ca. Effects of three-nucleon forces are included phenomenologically as in-medium two-nucleon interactions. Using known asymptotic forms for one-nucleon overlap functions we derive a simple and intuitive way of computing scattering observables such as elastic scattering phase shifts and cross sections. As a first application and proof of principle, we compute phase shifts and differential interaction cross sections at energies of $9.6$ and $12.44$ MeV and compare with experimental data. Our computed diffraction minima are in fair agreement with experimental results, while we tend to overestimate the cross sections at large scattering angles.

Figure 1

Scattering $s$-wave functions $\varphi \left(r\right)$ obtained from diagonalization of the one-body Hamiltonian defined in Eq. (4) in momentum space (solid lines) and their asymptotic expansion defined in Eq. (5) (dashed lines) as a function of radius, provided in femtometers. Wave functions are given in units of femtometers${}^{-3}$ and their energy is written on the figure in units of million electron volts.

Figure 2

Radial overlap function $O_A^{A+1}(lj;kr)$ between the ground state of ${}^{40}$Ca and the $J^{\pi }=7/2^{-}$ ground state of ${}^{41}$Sc (solid line), also shown is the corresponding Whittaker function $C_{lj}{W}_{-\eta ,l+1/2}\left(i\kappa r\right)/r$ for the $f_{7/2}$ proton partial wave (dashed line).

Figure 3

Radial overlap functions $O_A^{A+1}(lj;kr)$ between the ground state of ${}^{40}$Ca and two $J^{\pi }=7/2^{-}$ scattering states in ${}^{41}$Sc (solid lines). Also shown are the corresponding Coulomb scattering functions $B_{lj}\left(k\right)[F_{l,\eta }\left(kr\right)-\tan {\delta }_l\left(k\right)G_{l,\eta }\left(kr\right)]$ (dashed lines).

Figure 4

Computed phase shifts for elastic proton scattering on ${}^{40}$Ca for low-lying partial waves and energies below 14 MeV.

Figure 5

Differential cross section from coupled-cluster calculations divided by Rutherford cross section for elastic proton scattering on ${}^{40}$Ca at $E_{\mathrm{cm}}=9.6$ MeV (solid line), experimental data (dots), and optical model potential fits (dashed line), taken from Ref.[31].

Figure 6

Same caption as in Fig. 5 except that the energy is $E_{\mathrm{cm}}=12.44$ MeV.