Slope parameters determined from CREX and PREX2

[Background] Very lately, the CREX group presents a skin value $\Delta R_{\rm skin}^{48}({\rm CREX}) =0.121 \pm 0.026\ {\rm (exp)} \pm 0.024\ {\rm (model)}=0.071\sim 0.171$~fm. Meanwhile, the PREX group reported a skin value $\Delta R_{\rm skin}^{208}({\rm PREX2}) = 0.283\pm 0.071=0.212 \sim 0.354$~fm. In our previous paper, we determined both the $L$--$\Delta R_{\rm skin}^{48}$ relation and the $L$--$\Delta R_{\rm skin}^{208}$ one, using 206 EoSs, where $L$ is a slope parameter. [Purpose] We determine $L$ from $\Delta R_{\rm skin}^{48}({\rm CREX})$ and $\Delta R_{\rm skin}^{208}({\rm PREX2})$, using 207 EoSs. [Results] The $\Delta R_{\rm skin}^{48}({\rm CREX})$ yields $L({\rm CREX})=0 \sim 51$~MeV and the $\Delta R_{\rm skin}^{208}({\rm PREX2})$ does $L({\rm PREX2})=76 \sim 165$~MeV. [Conclusion] There is no overlap between $L({\rm CREX})$ and $L({\rm PREX2})$. This is a big problem to be solved

12^{12}C+12^{12}C scattering as the reference system for reaction cross section

In our previous paper, we tested the chiral (Kyushu) folding model for $^{12}$C+$^{12}$C scattering, since the profile function in the Glauber mode is constructed for the system. We found that the folding model is reliable for reaction cross sections $\sigma_{\rm R}$ in 30 \lsim E_{\rm lab} \lsim 100 ~MeV and 250 \lsim E_{\rm lab} \lsim 400 ~MeV. Accurate data are available for $^{12}$C scattering on $^{9}$Be, $^{12}$C, $^{27}$Al targets in 30 \lsim E_{\rm lab} \lsim 400 ~MeV. We determine matter radius $r_{m}({\rm exp})$ of $^{12}$C from the accurate $\sigma_{\rm R}({\rm exp})$, using the Kyushu $g$-matrix folding model. Our result is $r_{\rm m}^{12}({\rm exp}) =2.352 \pm 0.013$~fm for $^{12}$C. The model is applied for the accurate data on $^{12}$C+$^{27}$Al scattering, and yields $r_{\rm m}({\rm exp}) =2.936 \pm 0.012$~fm for $^{27}$Al. Our conclusion is that $r_{\rm m}({\rm exp}) =2.352 \pm 0.013$~fm agrees with $r_{\rm m}({\rm exp}) =2.35 \pm 0.02$~fm determined from interaction cross sections by Tanihata {\it et. al.}.Comment: arXiv admin note: text overlap with arXiv:2306.0960

Neutron skin thickness of 116,118,120,122,124^{116,118,120,122,124}Sn determined from reaction cross sections of proton scattering

The cross sections of SDR in the Sb isotopes have been measured. Within the model used, the neutron-skin thicknesses $r_{\rm skin}({\rm exp})$ deduced $0.12 \pm 0.06$fm for $^{116}$Sn, $0.13 \pm 0.06$fm for $^{118}$Sn, $0.18 \pm 0.07$fm for $^{120}$Sn, $0.22 \pm 0.07$fm for $^{122}$Sn, $0.19 \pm 0.07$fm for $^{124}$Sn. We tested the chiral (Kyushu) $g$-matrix folding model for $^{12}$C+$^{12}$C scattering, and found that the Kyushu $g$-matrix folding model is reliable for reaction cross sections $\sigma_{\rm R}$ in $30 < E_{\rm in} < 100$MeV and $250 < E_{\rm in} < 400$MeV. We determine neutron skin thickness $r_{\rm skin}({\rm exp})$, using measured $\sigma_{\rm R}$ of $^{4}$He+$^{116,120,224}$Sn scattering. The results are $r_{\rm skin}({\rm exp})=0.242 \pm 0.140$fm for $^{116}$Sn, $r_{\rm skin}({\rm exp})=0.377 \pm 0.140$fm for $^{120}$Sn, $r_{\rm skin}({\rm exp})=0.180 \pm 0.142$fm for $^{124}$Sn. The $\sigma_{\rm R}$ are available for proton scattering on $^{116,118,120,122,124}$Sn with high accuracy. Our aim is to determine $r_{\rm skin}({\rm exp})$ for $^{116,118,120,122,124}$Sn with small errors by using the Kyushu $g$-matrix folding model. Our model is the folding model with the densities scaled from the D1S-GHFB+AMP neutron density. The proton radii of D1S-GHFB+AMP agree with those calculated with the isotope shift method based on the electron scattering. We then scale the neutron densities so as to reproduce the $\sigma_{\rm R}({\rm exp})$. In $30 < E_{\rm in} < 65$MeV, we determine $r_{\rm skin}({\rm exp})$ from measured $\sigma_{\rm R}$. The values are $r_{\rm skin}({\rm exp})=0.118 \pm 0.021$~fm for $^{116}$Sn, $0.112 \pm 0.021$fm for $^{118}$Sn, $0.124 \pm 0.021$fm for $^{120}$Sn, $0.156 \pm 0.022$fm for $^{124}$Sn. As for $^{122}$Sn, the skin value in $30 < E_{\rm in} < 50$MeV is $0.122 \pm 0.024$fm. Our results are consistent with the previous values.Comment: arXiv admin note: text overlap with arXiv:2211.1668

Determination of matter radius and neutron-skin thickness of 60,62,64^{60,62,64}Ni from reaction cross section of proton scattering on 60,62,64^{60,62,64}Ni targets

In our previous work, we determined matter radii $r_{\rm m}({\rm exp})$ and neutron-skin thickness $r_{\rm skin}({\rm exp})$ from reaction cross sections $\sigma_{\rm R}({\rm exp})$ of proton scattering on $^{208}$Pb, $^{58}$Ni, $^{40,48}$Ca, $^{12}$C targets, using the chiral (Kyushu) $g$-matrix folding model with the densities calculated with Gogny-D1S-HFB (D1S-GHFB) with angular momentum projection (AMP). The resultant $r_{\rm skin}({\rm exp})$ agree with the PREX2 and CREX values. As for $^{58}$Ni, our value is consistent with one determined from the differential cross section for $^{58}$Ni+$^{4}$He scattering. As for p+$^{60,62,64}$N scattering, $\sigma_{\rm R}({\rm exp})$ are available as a function of incident energies $E_{\rm in}$, where $E_{\rm in}=22.8 \sim 65.5$~MeV for $^{60}$Ni, $E_{\rm in}=40,60.8$~MeV for $^{62}$Ni, $E_{\rm in}=40, 60.8$~MeV for $^{64}$Ni. Our aim is to determine matter radii $r_{\rm m}({\rm exp})$ for $^{60,62,64}$Ni from the $\sigma_{\rm R}({\rm exp})$. Our method is the Kyushu $g$-matrix folding model with the densities scaled from D1S-GHFB+AMP densities, Our skin values are $r_{\rm skin}({\rm exp})=0.076 \pm 0.019,~0.106 \pm 0.192,~0.162 \pm 0.176$~fm, and $r_{\rm m}({\rm exp})=3.759 \pm 0.011,~3.811 \pm 0.107,~3.864 \pm 0.101$~fm for $^{60,62,64}$Ni, respectively

Folding-model approach to reaction cross section of 4,6,8^{4,6,8}He+12^{12}C scattering at 790 MeV

Tanihata {\it et al.} determined matter radii $r_{m}(\sigma_{\rm I})$ for $^{4,6,8}$He from interaction cross sections $\sigma_{\rm I}$ of $^{4,6,8}$He+$^{12}$C scattering at 790 MeV per nucleon, using the optical limit of the Glauber model. Lu {\it et al.} determined proton radii $r_{p}({\rm AIS})$ for $^{4,6,8}$He with the atomic isotope shifts (AIS). We investigate whether the Love-Franey $t$-matrix folding model is good for $^{4,6,8}$He+$^{12}$C scattering at 790 MeV per nucleon

Reaction cross section of proton scattering consistent with PREX-II

Background: The neutron skin thickness $R_{\rm skin}^{\rm PV}$ of PREX-II is presented in Phys. Rev. Lett. {\bf 126}, 172502 (2021). The reaction cross section $\sigma_R$ is useful to determine the matter radius $R_m$ and $R_{\rm skin}$. For proton scattering, the reaction cross section $\sigma_R$ are available for $E_{\rm in} > 400$ MeV. Method and results: We determine $R_n^{\rm exp}=5.727 \pm 0.071$ fm and $R_m^{\rm exp}=5.617 \pm 0.044$ fm from $R_p^{\rm exp}$ = 5.444 fm and $R_{\rm skin}^{\rm PV}$. The $R_p^{\rm GHFB}$ calculated with D1S-GHFB with the angular momentum projection (AMP). agrees with $R_p^{\rm exp}$. The neutron density calculated with GHFB+AMP is scaled so as to $R_n^{\rm scaling}=5.727$ fm. The Love-Franey $t$-matrix model with the scaled densities reproduces the data on $\sigma_R$. Aim: Our aim is to find the $\sigma_R$ of proton scattering consistent with $R_{\rm skin}^{\rm PV}$. Conclusion: The $\sigma_R$ of proton scattering consistent with $R_{\rm skin}^{\rm PV}$ are $\sigma_R^{\rm exp}$ at $E_{\rm in} = 534.1, 549, 806$ MeV.Comment: arXiv admin note: text overlap with arXiv:2010.0245

Neutron-skin values and matter and neutron radii determined from reaction cross sections of proton scattering on 12^{12}C, 40,48^{40,48}Ca, 58^{58}Ni, 208^{208}Pb

Very lately, the PREX and the CREX collaboration present skin values, $r_{\rm skin}^{208}({\rm newPREX2}) =0.278 \pm 0.078\ {\rm (exp)} \pm 0.012\ {\rm (theor.)}\,{\rm fm}$ and $r_{\rm skin}^{48}=0.121 \pm 0.026\ {\rm (exp)} \pm 0.024\ {\rm (model)}$, respectively. We recently determined a neutron-skin value $r_{\rm skin}^{208}=0.278 \pm 0.035$fm from measured reaction cross sections $\sigma_{\rm R}({\rm exp})$ of p+$^{208}$Pb scattering in a range of incident energies 10 \lsim E_{\rm in} \lsim 100 MeV where the chiral (Kyushu) $g$-matrix folding model is reliable for $^{12}$C+$^{12}$C scattering. The data $\sigma_{\rm R}({\rm exp})$ are available for proton scattering on $^{58}$Ni, $^{40,48}$Ca, $^{12}$C targets. Our first aim is to test the Kyushu $g$-matrix folding model for p+$^{208}$Pb scattering in 20 \lsim E_{\rm in} \lsim 180 MeV. Our second aim is to determine skin values $r_{\rm skin}$ and matter and neutron radii, $r_{\rm m}$ and $r_{\rm n}$, for $^{208}$Pb, $^{58}$Ni, $^{40,48}$Ca, $^{12}$C from the $\sigma_{\rm R}({\rm exp})$. Our method is the Kyushu $g$-matrix folding model with the densities scaled from the D1S-GHFB+AMP densities, where D1S-GHFB+AMP stands for Gogny-D1S HFB (GHFB) with the angular momentum projection (AMP). As for proton scattering, we find that our model is reliable in 20 \lsim E_{\rm in} \lsim 180 MeV. For $^{208}$Pb, the skin value deduced from $\sigma_{\rm R}({\rm exp})$ in 20 \lsim E_{\rm in} \lsim 180 MeV is $r_{\rm skin}^{208}(\sigma_{\rm R})=0.299 \pm 0.020$ fm. Our results on $r_{\rm skin}$ are compared with the previous works. Our result $r_{\rm skin}^{208}(\sigma_{\rm R}) = 0.299 \pm 0.020$ fm agrees with $r_{\rm skin}^{208}({\rm PREX2}) = 0.283\pm 0.071$ fm. In addition, our result $r_{\rm skin}^{48}=0.103 \pm 0.022$ fm is consistent with the CREX value