Neutron skin thickness for $^{nat}$Pb, $^{159}$Tb and effects of deformation on matter radii for Mg, Na, Ar isotopes

Abstract: Data on reaction cross section $\sigma_{\rm R}$ of proton scattering are available for $^{nat}$Pb (natural lead) and $^{159}$Tb. In addition, data on $\sigma_{\rm R}$ and/or interaction ones $\sigma_{\rm I}$ are available for Mg, Na, Ar isotopes. Our aim is to determine neutron skin thickness $r_{\rm skin}$ for $^{nat}$Pb, $^{159}$Tb and to investigate effects of deformation for Mg, Na, Ar isotopes. We use the chiral (Kyushu) $g$-matrix folding model for lower energies and the folding model based on the Love-Franey (LF) $t$-matrix for higher energies. For $^{159}$Tb, our skin value is $r_{\rm skin}=0.273 \pm 0.073$~fm. As for $^{24}$Mg, our matter radius $r_{\rm m}(\sigma_{\rm R})=3.03 \pm 0.08$~fm determined from $\sigma_{\rm R}$ includes effects of deformation, whereas the corresponding $r_{\rm m}(\sigma_{\rm I})=2.79 \pm 0.15$~fm does not. Effects of deformation are seen for Mg isotopes, $^{21, 23}$Na, $^{40}$Ar. Our result $r_{\rm skin}=0.271 \pm 0.008$~fm for $^{nat}$Pb agrees with $r_{\rm skin}^{208}({\rm PREX\text{-}II}) = 0.283\pm 0.071$~fm for $^{208}$Pb, where we assume $A=208$ for $^{nat}$Pb.