Neutron skin of ${}^{48}$Ca consistent with experimental data on skins

Abstract

[Background]: In our previous paper, we predicted $r_{\mathrm{skin}}$, $r_{\mathrm{p}}$, $r_{\mathrm{n}}$, $r_{\mathrm{m}}$ for ${}^{40-60,62,64}$Ca after determining the neutron dripline, using the Gogny-D1S HFB with and without the angular momentum projection (AMP). We found that effects of the AMP are small. Very lately, Tanaka {\it et al.} measured interaction cross sections $\ddot{I}{ƒ}_{\mathrm{I}}$ for ${}^{42-51}$Ca, determined $r_{\mathrm{m}}$ from the $\ddot{I}{ƒ}_{\mathrm{I}}$, and deduced skin $r_{\mathrm{skin}}$ and $r_{\mathrm{n}}$ from the $r_{\mathrm{m}}$ and the $r_{\mathrm{p}}(\exp )$ evaluated from the electron scattering. Comparing our results with the data, we find for ${}^{42-48}$Ca that GHFB and GHFB+AMP reproduce the data on $r_{\mathrm{skin}}$, $r_{\mathrm{n}}$, $r_{\mathrm{m}}$, but not for $r_{\mathrm{p}}(\exp )$. [Aim]: Our purpose is to determine a value of $r_{\mathrm{skin}}^{48}$ by using GHFB+AMP and the constrained GHFB (cGHFB) in which the calculated value is fitted to $r_{\mathrm{p}}(\exp )$. [Results]: For ${}^{42,44,46,48}$Ca, cGHFB hardly changes $r_{\mathrm{skin}}$, $r_{\mathrm{m}}$, $r_{\mathrm{n}}$ calculated with GHFB+AMP, except for $r_{\mathrm{skin}}^{48}$. For $r_{\mathrm{skin}}^{48}$, the cGHFB result is $r_{\mathrm{skin}}^{48}=0.190$fm, while $r_{\mathrm{skin}}^{48}=0.159$fm for GHFB+AMP. We should take the upper and the lower bound of GHFB+AMP and cGHFB. The result $r_{\mathrm{skin}}^{48}=0.159-0.190$fm consists with the $r_{\mathrm{skin}}^{48}(\ddot{I}{ƒ}_{\mathrm{I}})$ and the data $r_{\mathrm{skin}}^{48}($E1$pE)$ obtained from high-resolution $E1$ polarizability experiment ($E1$pE). Using the $r_{\mathrm{skin}}^{48}$-$r_{\mathrm{skin}}^{208}$ relation with strong correlation of Ref.[3], we transform the data $r_{\mathrm{skin}}^{208}$ determined by PREX and $E1$pE to the corresponding values, $r_{\mathrm{skin}}^{48}(\mathrm{tPREX})$ and $r_{\mathrm{skin}}^{48}(\mathrm{t}$E1$pE)$. Our result is consistent also for $r_{\mathrm{skin}}^{48}(\mathrm{tPREX})$ and $r_{\mathrm{skin}}^{48}(\mathrm{t}$E1$pE)$.