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Register a new userNeutron skin $r_{rm skin}^{48}$ determined from reaction cross section of proton+$^{48}$Ca scattering
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{bf Background:} Using the chiral (Kyushu) $g$-matrix folding model with the densities calculated with GHFB+AMP, we determined $r_{rm skin}^{208}=0.25$fm from the central values of $sigma_{rm R}$ of p+$^{208}$Pb scattering in $E_{rm in}=40-81$MeV. The high-resolution $E1$ polarizability experiment ($E1$pE) yields $r_{rm skin}^{48}(E1{rm pE}) =0.14-0.20$fm. The data on $sigma_{rm R}$ are available as a function of $E_{rm in}$ for $p$+$^{48}$Ca scattering. {bf Aim:} Our aim is to determine $r_{rm skin}^{48}$ from the central values of $sigma_{rm R}$ for $p$+$^{48}$Ca scattering by using the folding model. {bf Results:} As for $^{48}$Ca, we determine $r_n(E1{rm pE})=3.56$fm from the central value 0.17fm of $r_{rm skin}^{48}(E1{rm pE})$ and $r_p({rm EXP})=3.385$fm of electron scattering, and evaluate $r_m(E1{rm pE})=3.485$fm from the $r_n(E1{rm pE})$ and the $r_p({rm EXP})$ of electron scattering. The folding model with GHFB+AMP densities reproduces $sigma_{rm R}$ in $23 leq E_{rm in} leq 25.3$ MeV in one-$sigma$ level, but slightly overestimates the central values of $sigma_{rm R}$ there. In $23 leq E_{rm in} leq 25.3$MeV, the small deviation allows us to scale the GHFB+AMP densities to the central values of $r_p({rm EXP})$ and $r_n(E1{rm pE})$. The $sigma_{rm R}(E1{rm pE})$ obtained with the scaled densities almost reproduce the central values of $sigma_{rm R}$ when $E_{rm in}=23-25.3$MeV, so that the $sigma_{rm R}({rm GHFB+AMP})$ and the $sigma_{rm R}(E1{rm pE})$ are in 1-$sigma$ of $sigma_{rm R}$ there. In $E_{rm in}=23-25.3$MeV, we determine the $r_{m}({rm EXP})$ from the central values of $sigma_{rm R}$ and take the average for the $r_{m}({rm EXP})$. The averaged value is $r_{m}({rm EXP})=3.471$fm. Eventually, we obtain $r_{rm skin}^{48}({rm EXP})=0.146$fm from $r_{m}({rm EXP})=3.471$fm and $r_p({rm EXP})=3.385$fm.
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For $^{48}$Ca, we determined $r_{m}$fm and $r_{rm skin}$fm from the central values of $sigma_{rm R}({rm EXP})$ of p+$^{48}$Ca scattering, using the chiral (Kyushu) $g$-matrix folding model with the GHFB+AMP densities. For $^{40}$Ca, Zenihiro {it et al.} determined $r_n({rm RCNP})=3.375$~fm and $r_{rm skin}({rm RCNP})=-0.01 pm 0.023$fm from the differential cross section and the analyzing powers for p+$^{40}$Ca scattering. For $^{40}$Ca, $sigma_{rm R}({rm EXP})$ are available with high accuracy. Our aim is to determine matter radius $r_{m}^{40}$ and skin $r_{rm skin}^{40}$ from $sigma_{rm R}({rm EXP})$ by using the Kyushu $g$-matrix folding model with the GHFB+AMP densities. We first determine $r_m({rm RCNP})=3.380$fm from the central value -0.01~fm of $r_{rm skin}({rm RCNP})$ and $r_p({rm RCNP})=3.385$fm. The folding model with the GHFB+AMP densities reproduces $sigma_{rm R}({rm EXP})$ in $30 leq E_{rm in} leq 180$MeV, in 2-$sigma$ level. We scale the GHFB+AMP densities so as to $r_p({rm AMP})=r_p({rm RCNP})$ and $r_n({rm AMP})=r_n({rm RCNP})$. The $sigma_{rm R}({rm RCNP})$ thus obtained agrees with the original one $sigma_{rm R}({rm AMP})$ for each $E_{rm in}$. For $E_{rm in}=180$MeV, we define $F$ as $F=sigma_{rm R}({rm EXP})/sigma_{rm R}({rm AMP})=0.929$. The $Fsigma_{rm R}({rm AMP})$ be much the same as the center values of $sigma_{rm R}({rm EXP})$ in $30 leq E_{rm in} leq 180$MeV. We then determine $r_{rm m}^{40}({rm EXP})$ from the center values of $sigma_{rm R}({rm EXP})$, using $sigma_{rm R}({rm EXP})=C r_{m}^{2}({rm EXP})$ with $C=r_{m}^{2}({rm AMP})/(Fsigma_{rm R}({rm AMP}))$. The $r_{m}({rm EXP})$ are averaged over $E_{rm in}$. The averaged value is $r_{m}({rm EXP})=3.380$fm. Eventually, we obtain $r_{rm skin}({rm EXP})=-0.01$fm from the averaged $r_{rm m}({rm EXP})$~fm and $r_p({rm PCNP})=3.385$fm.
The reaction cross section $sigma_R$ is useful to determine the neutron radius $R_n$ as well as the matter radius $R_m$. The chiral (Kyushu) $g$-matrix folding model for $^{12}$C scattering on $^{9}$Be, $^{12}$C, $^{27}$Al targets was tested in the incident energy range of $30 lsim E_{rm in} lsim 400 $ MeV, and it is found that the model reliably reproduces the $sigma_R$ in $30 lsim E_{rm in} lsim 100 $ MeV and $250 lsim E_{rm in} lsim 400$ MeV. item[Aim] We determine $R_n$ and the neutron skin thickness $R_{rm skin}$ of ${}^{208}{rm Pb}$ by using high-quality $sigma_R$ data for the $p+{}^{208}{rm Pb}$ scattering in $30 leq E_{rm in} leq 100$ MeV. The theoretical model is the Kyushu $g$-matrix folding model with the densities calculated with Gongny-D1S HFB (GHFB) with the angular momentum projection (AMP). item[Results] The Kyushu $g$-matrix folding model with the GHFB+AMP densities underestimates $sigma_{rm R}$ in $30 leq E_{rm in} leq 100$~MeV only by a factor of 0.97. Since the proton radius $R_p$ calculated with GHFB+AMP agrees with the precise experimental data of 5.444 fm, the small deviation of the theoretical result from the data on $sigma_R$ allows us to scale the GHFB+AMP neutron density so as to reproduce the $sigma_R$ data. In $E_{rm in}$ = 30--100 MeV, the experimental $sigma_R$ data can be reproduced by assuming the neutron radius of ${}^{208}{rm Pb}$ as $R_n$ = $5.722 pm 0.035$ fm. item[Conclusion] The present result $R_{rm skin}$ = $0.278 pm 0.035$ fm is in good agreement with the recent PREX-II result of $r_{rm skin}$ = $0.283pm 0.071$ fm.
In our previous paper, we predicted $sigma_{rm R}$ for $^{40-60,62,64}$Ca+ $^{12}$C scattering at 280 MeV/u, using the Kyushu (chiral) $g$-matrix folding model with the densities calculated with D1S-GHFB with and without the AMP. Interaction cross sections $sigma_{rm I}$ are available for $^{42-51}$Ca + $^{12}$C scattering, whereas $sigma_{rm R}$ are available for p+$^{48}$Ca scattering. As for $^{48}$Ca, the high-resolution $E1$ polarizability experiment ($E1$pE) yields $r_{rm skin}^{48}(E1{rm pE}) =0.14 sim 0.20~{rm fm}$. We determine $r_{rm skin}^{48}({rm exp})$ from the data on $sigma_{rm R}$ for p+$^{48}$Ca scattering and from the data on $sigma_{rm I}$ for $^{48}$Ca+$^{12}$C scattering. We use the chiral (Kyushu) $g$-matrix folding model with the densities calculated with the Gogny-D1M Hartree-Fock-Bogoliubov with the AMP. The D1M-GHFB+AMP proton and neutron densities are scaled so as to reproduce the data under the condition that the radius $r_{rm p}$ of the scaled proton density equals the data $r_{rm p}({rm exp})$ of the electron scattering. The neutron radius $r_{rm n}$ thus obtained is an experimental value. Our results are $r_{rm skin}^{48}({rm exp})=-0.031sim 0.183$fm for p+$^{48}$Ca and $0.100 sim 0.218$fm for $^{48}$Ca + $^{12}$C scattering. Using the $r_{rm skin}^{48}$-$r_{rm skin}^{208}$ relation with a high correlation coefficient $R=0.99$, we have transformed $r_{rm skin}^{208}({rm PREXII})$ and $r_{rm skin}^{208}(E1{rm pE})$ to the corresponding values $r_{rm skin}^{48}({rm tPREXII})$ and $r_{rm skin}^{48}({rm t}E1{rm pE})$. The transformed data $r_{rm skin}^{48}({rm tPREXII})=0.190 sim 0.268$fm is consistent with $r_{rm skin}^{48}=0.102 sim 0.218$fm for $^{48}$Ca + $^{12}$C. Our final result is $r_{rm skin}^{48}=0.102 sim 0.218$fm determined from $^{48}$Ca + $^{12}$C scattering.
[Background]: In our previous paper, we predicted $r_{rm skin}$, $r_{rm p}$, $r_{rm n}$, $r_{rm m}$ 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 $sigma_{rm I}$ for $^{42-51}$Ca, determined $r_{rm m}$ from the $sigma_{rm I}$, and deduced skin $r_{rm skin}$ and $r_{rm n}$ from the $r_{rm m}$ and the $r_{rm p}(rm {exp})$ 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 $r_{rm skin}$, $r_{rm n}$, $r_{rm m}$, but not for $r_{rm p}(rm {exp})$. [Aim]: Our purpose is to determine a value of $r_{rm skin}^{48}$ by using GHFB+AMP and the constrained GHFB (cGHFB) in which the calculated value is fitted to $r_{rm p}(rm {exp})$. [Results]: For $^{42,44,46,48}$Ca, cGHFB hardly changes $r_{rm skin}$, $r_{rm m}$, $r_{rm n}$ calculated with GHFB+AMP, except for $r_{rm skin}^{48}$. For $r_{rm skin}^{48}$, the cGHFB result is $r_{rm skin}^{48}=0.190$fm, while $r_{rm skin}^{48}=0.159$fm for GHFB+AMP. We should take the upper and the lower bound of GHFB+AMP and cGHFB. The result $r_{rm skin}^{48}=0.159-0.190$fm consists with the $r_{rm skin}^{48}(sigma_{rm I})$ and the data $r_{rm skin}^{48}(rm $E1$pE)$ obtained from high-resolution $E1$ polarizability experiment ($E1$pE). Using the $r_{rm skin}^{48}$-$r_{rm skin}^{208}$ relation with strong correlation of Ref.[3], we transform the data $r_{rm skin}^{208}$ determined by PREX and $E1$pE to the corresponding values, $r_{rm skin}^{48}(rm tPREX)$ and $r_{rm skin}^{48}(rm t$E1$pE)$. Our result is consistent also for $r_{rm skin}^{48}(rm tPREX)$ and $r_{rm skin}^{48}(rm t$E1$pE)$.
In the present work, we use a finite range effective interaction to calculate the neutron skin thickness in $^{48}$Ca and correlate these quantities with the parameters of nuclear symmetry energy. Available experimental data on the neutron skin thickness in $^{48}$Ca are used to deduce information on the density slope parameter and the curvature symmetry parameter of the nuclear symmetry energy at saturation and at subsaturation densities. We obtained the constraints such as $54.5leq L(rho_0) leq 97.5$ MeV and $47.3leq L(rho_c) leq 57.1$ MeV for the density slope parameter. The constraints on the curvature symmetry energy parameter are obtained as $-170.7leq K_{sym}(rho_0) leq -43.4$ MeV and $-80.8leq K_{sym}(rho_c) leq 23.8$ MeV. A linear relation between the neutron skin thickness in $^{48}$Ca and in $^{2088}$Pb is obtained.
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