Neutron Skin Thickness of ${}^{48}\mathrm{Ca}$ from a Nonlocal Dispersive Optical-Model Analysis

A nonlocal dispersive optical-model analysis has been carried out for neutrons and protons in ${}^{48}\mathrm{Ca}$. Elastic-scattering angular distributions, total and reaction cross sections, single-particle energies, the neutron and proton numbers, and the charge distribution have been fitted to extract the neutron and proton self-energies both above and below the Fermi energy. From the single-particle propagator resulting from these self-energies, we have determined the charge and neutron matter distributions in ${}^{48}\mathrm{Ca}$. A best fit neutron skin of $0.249\pm 0.023\mathrm{fm}$ is deduced, but values up to 0.33 fm are still consistent. The energy dependence of the total neutron cross sections is shown to have a strong sensitivity to the skin thickness.

Figure 1

(a) Comparison of experimental $n+{}^{48}\mathrm{Ca}$ elastic-scattering angular distributions [22, 27] to the best DOM fit of all data (solid curves) and to a constrained fit with the skin thickness forced to $\mathrm{\Delta }{r}_{np}=0.132\mathrm{fm}$ (dashed curves) which is consistent with the ab initio result. The higher-energy data and calculations have been offset along the vertical axis for clarity. (b) Comparison of the experimental total neutron cross sections of ${}^{48}\mathrm{Ca}$ (diamonds [28] and circles [29]) to DOM fits with constrained values of $r_n$. The curve labeled with a triangle is for the $r_n$ value of our best fit, while the curve labeled with a square is for a value consistent with the ab initio result [see (c)]. (c) The ${\chi }^2$ from fitting all data (solid curve) and its contribution from fitting the elastic-scattering angular distributions and total neutron cross section (short-dashed and long-dashed curves, respectively). Each data point corresponds to an average of fitted values with very similar $r_n$ values.

Figure 2

Comparison of the experimental (${\rho }_{\exp }$) and fitted (${\rho }_{\mathrm{ch}}$) charge distribution for ${}^{48}\mathrm{Ca}$. The neutron matter distribution is plotted as ${\rho }_n$, while the weak charge distribution is plotted as ${\rho }_w$.

Figure 3

Comparison of the integrated imaginary potential for protons on ${}^{40}\mathrm{Ca}$ (dashed curves) and ${}^{48}\mathrm{Ca}$ (solid curves) obtained from our DOM fits. Results are given for the two indicated $\ell$ values.

Figure 4

Decomposition of the $r^4$ weighted point densities for protons and neutrons in (a) ${}^{48}\mathrm{Ca}$ and (b) ${}^{40}\mathrm{Ca}$. These are subdivided into the contribution from the lower-$\ell$ orbitals [$s_{1/2}$, $p_{3/2}$, $p_{1/2}$, $d_{5/2}$, and $d_{3/2}$] designated by “$\ell <3$” and the remaining higher-$\ell$ orbitals “$\ell \ge 3$.”