Neutron skin thickness of ${}^{208}\mathrm{Pb}, {}^{116,120,124}\mathrm{Sn}$, and ${}^{40}\mathrm{Ca}$ determined from reaction cross sections of ${}^4\mathrm{He}$ scattering

Background: We constructed the Kyushu chiral $g$ matrix and confirmed its reliability at $30\lesssim E_{\mathrm{in}}\lesssim 100$ MeV and $250\lesssim E_{\mathrm{in}}\lesssim 400$ MeV for ${}^{12}\mathrm{C}$ scattering. Reaction cross-section data of ${}^4\mathrm{He}$ scattering are available for some nuclides including ${}^{208}\mathrm{Pb}$. The PREX II collaboration reported a thick neutron skin for ${}^{208}\mathrm{Pb}$.

Purpose: Our purpose is to deduce neutron skin thicknesses of ${}^{208}\mathrm{Pb}$ and some other nuclides from reaction cross sections calculated in terms of the double folding model with the $g$ matrix.

Methods: We fold the $g$ matrix and densities given by mean-field calculations. To remedy the weaker constraint of the neutron sector, we renormalize densities so as to reproduce the observed cross sections.

Results: We found that a $3.4\%$ renormalization is necessary for ${}^{208}\mathrm{Pb}$. The neutron density obtained from renormalization results in $R_{\mathrm{skin}}=0.416\pm 0.146$ fm by confronting the precision proton radius.

Conclusions: Our result is consistent with PREX II and therefore supports larger slope parameter $L$. Results for ${}^{40}\mathrm{Ca}$ and ${}^{124}\mathrm{Sn}$ are also consistent with $R_{\mathrm{skin}}$ deduced from other experiments. For ${}^{116,120}\mathrm{Sn}$ the present method gives thicker skins.

Figure 1

$r$ dependence of densities, ${\rho }_{\mathrm{p}}\left(r\right), {\rho }_{\mathrm{n}}\left(r\right), {\rho }_{\mathrm{m}}\left(r\right)$, for ${}^{208}\mathrm{Pb}$ calculated with D1S-GHFB $+$ AMP and SLy7-HFB. Dashed curves from the bottom to the top denote the ${\rho }_{\mathrm{p}}\left(r\right), {\rho }_{\mathrm{n}}\left(r\right), {\rho }_{\mathrm{m}}\left(r\right)$ of D1S-GHFB, respectively. Symbols correspond to the SLy7-HFB densities.

Figure 2

$E_{\mathrm{in}}$ dependence of reaction cross sections ${\sigma }_{\mathrm{R}}$ for ${}^4\mathrm{He}+{}^{208}\mathrm{Pb}$ scattering. Note that $E_{\mathrm{in}}$ is the incident energy per nucleon. Asterisks stand for the results of D1S-GHFB $+$ AMP and SLy7-HFB. The data are taken from Ref. [16].

Figure 3

$E_{\mathrm{in}}$ dependence of reaction cross sections ${\sigma }_{\mathrm{R}}$ for ${}^4\mathrm{He}+{}^{120}\mathrm{Sn}$ scattering. Asterisks show the results of SLy7-HFB. The data are taken from Ref. [16].

Figure 4

Three panels from the top to the bottom show $E_{\mathrm{in}}$ dependence of reaction cross sections ${\sigma }_{\mathrm{R}}$ for ${}^4\mathrm{He}+{}^{116,124}\mathrm{Sn}$, and ${}^{40}\mathrm{Ca}$ scattering, respectively. Asterisks show the results of SLy7-HFB. The data are taken from Ref. [16].

Figure 5

Skin thicknesses deduced in the present and other works are summarized as a function of the difference between the proton and neutron separation energies: $-7.31, -0.28$, 0.64, 1.58, and 3.60 MeV for ${}^{40}\mathrm{Ca}, {}^{116}\mathrm{Sn}, {}^{208}\mathrm{Pb}, {}^{120}\mathrm{Sn}$, and ${}^{124}\mathrm{Sn}$, respectively. The data are taken from Refs. [2, 23, 24, 25].