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Register a new userNeutron skin of $^{48}$Ca deduced from reaction and interaction cross sections
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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.
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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.
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.
[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$pE)$.
A nonlocal dispersive-optical-model analysis has been carried out for neutrons and protons in $^{48}$Ca. Elastic-scattering angular distributions, total and reaction cross sections, single-particle energies, the neutron and proton numbers, and the charge distribution have been fitted to extract the neutron and proton self-energies both above and below the Fermi energy. From the single-particle propagator resulting from these self-energies we have determined the charge and neutron matter distributions in $^{48}$Ca. A neutron skin of 0.249$pm$0.023~fm is deduced. The energy dependence of the total neutron cross sections is shown to have strong sensitivity to the skin thickness.
The electric dipole strength distribution in Ca-48 between 5 and 25 MeV has been determined at RCNP, Osaka, from proton inelastic scattering experiments at forward angles. Combined with photoabsorption data at higher excitation energy, this enables for the first time the extraction of the electric dipole polarizability alpha_D(Ca-48) = 2.07(22) fm^3. Remarkably, the dipole response of Ca-48 is found to be very similar to that of Ca-40, consistent with a small neutron skin in Ca-48. The experimental results are in good agreement with ab initio calculations based on chiral effective field theory interactions and with state-of-the-art density-functional calculations, implying a neutron skin in Ca-48 of 0.14 - 0.20 fm.
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