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Register a new user$g$-matrix folding-model approach to reaction cross sections for scattering of Ca isotopes on a C target
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We first predict the ground-state properties of Ca isotopes, using the Gogny-D1S Hartree-Fock-Bogoliubov (GHFB) with and without the angular momentum projection (AMP). We find that $^{64}$Ca is an even-dripline nucleus and $^{59}$Ca is an odd-dripline nucleus, using $A$ dependence of the one-neutron separation energy $S_{1}$ and the two-neutron separation energy, $S_{2}$. As for $S_{1}$, $S_{2}$ and the binding energies $E_{rm B}$, our results agree with the experimental data in $^{40-58}$Ca. As other ground-state properties of $^{40-60,62,64}$Ca, we predict charge, proton, neutron, matter radii, neutron skin and deformation. As for charge radii, our results are consistent with the experimental data in $^{40-52}$Ca. For $^{48}$Ca, our results on proton, neutron, matter radii agree with the experimental data. Very lately, Tanaka et. al. measured interaction cross sections for $^{42-51}$Ca scattering on a $^{12}$C target at an incident energy per nucleon of $E_{rm lab}=280$MeV. Secondly, we predict reaction cross sections $sigma_{rm R}$ for $^{40-60,62,64}$Ca, using a chiral $g$-matrix double-folding model (DFM). To show the reliability of the present DFM for $sigma_{rm R}$, we apply the DFM for the data on $^{12}$C scattering on $^{9}$Be, $^{12}$C, $^{27}$Al targets in $30 < E_{rm lab} < 400 $MeV, and show that the present DFM is good in $30 < E_{rm lab} < 100 $MeV and $250 < E_{rm lab} < 400 $MeV. For $110 < E_{rm lab} < 240 $MeV, our results have small errors. To improve the present DFM for $sigma_{rm R}$, we propose two prescriptions.
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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.
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 (GHFB) with and without the angular momentum projection (AMP). Using the chiral $g$-matrix folding model, we predicted $sigma_{rm R}$ for Ca scattering on a $^{12}$C target at 280 MeV/nucleon, since Tanaka {it el al.} measured interaction cross sections $sigma_{rm I} (approx sigma_{rm R})$ for $^{42-51}$Ca in RIKEN. After our prediction, they determine $r_{rm m}({rm RIKEN})$, $r_{rm skin}({rm RIKEN})$, $r_{rm n}({rm RIKEN})$. In this paper, we reanalyses the $sigma_{rm I}$, since they assumed the Wood-Saxon densities for $^{42-51}$Ca. The $sigma_{rm R}$ calculated with the folding model with GHFB and GHFB+AMP densities almost reproduce the $sigma_{rm I}$. We then scale proton and neutron densities so that $r_{rm p}$ and $r_{rm n}$ may agree with the central values of $r_{rm p}(rm exp)$ and $r_{rm n}({rm RIKEN})$, respectively. The $sigma_{rm R}$ calculated with the scaled densities do not reproduce the central values of $sigma_{rm I}$ perfectly. We then determine the $r_{rm m}$ that agree with the central values of $sigma_{rm I}$, using the chiral $g$-matrix folding model. The fitted $r_{rm m}$ do not reproduce the central values of $r_{rm m}({rm RIKEN})$ perfectly, but are in one $sigma$ level. Finally, we show the $r_{rm skin}$, $r_{rm n}$ determined from the fitted $r_{rm m}$ are close to the original ones except for $r_{rm skin}^{48}$. The fitted $r_{rm skin}^{48}$ is 0.105 fm, while the central value of $r_{rm m}^{48}({rm RIKEN})$ is 0.146 fm. When we fit $r_{rm m}$ to the upper bound of $sigma_{rm I}$, the fitted $r_{rm skin}^{48}$ is 0.164~fm and near the central vale 0.17 fm of the high-resolution $E1$ polarizability experiment.
We systematically study total reaction cross sections of carbon isotopes with N=6-16 on a proton target for wide range of incident energies, putting an emphasis on the difference from the case of a carbon target. The analysis includes the reaction cross sections of ^{19,20,22}C at 40 AMeV, the data of which have recently been measured at RIKEN. The Glauber theory is used to calculate the reaction cross sections. To describe the intrinsic structure of the carbon isotopes, we use a Slater determinant generated from a phenomenological mean-field potential, and construct the density distributions. To go beyond the simple mean-field model, we adopt two types of dynamical models: One is a core+n model for odd-neutron nuclei, and the other is a core+n+n model for 16C and 22C. We propose empirical formulas which are useful in predicting unknown cross sections.
Background: Using the chiral (Kyushu) $g$-matrix folding model with the densities calculated with Gogny-HFB (GHFB) with the angular momentum projection (AMP), we determined the central values of matter radius and neutron skin from the central values of reaction cross sections $sigma_{rm R}({rm EXP})$ of p+$^{40,48}$Ca and p+$^{208}$Pb scattering. As for p+$^{58}$Ni scattering, $sigma_{rm R}({rm EXP})$ are available as a function of incident energy $E_{rm in}$. Aim: Our aim is to determine matter radius $r_{m}$ and skin $r_{rm skin}$ for $^{58}$Ni from the $sigma_{rm R}({rm EXP})$ of p+$^{58}$Ni scattering by using the Kyushu $g$-matrix folding model with the GHFB+AMP densities. Results: For p+$^{58}$Ni scattering, the Kyushu $g$-matrix folding model with the GHFB+AMP densities reproduces $sigma_{rm R}({rm EXP})$ in $8.8 leq E_{rm in} leq 81$MeV. For $E_{rm in}=81$MeV, we define the factor $F$ as $F=sigma_{rm R}({rm EXP})/sigma_{rm R}({rm AMP})=0.9775$. The $Fsigma_{rm R}({rm AMP})$ be much the same as the center values of $sigma_{rm R}({rm EXP})$ in $8.8 leq E_{rm in} leq 81$MeV. We then determine $r_{rm m}({rm EXP})$ 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 $r_{m}({rm EXP})$ thus obtained are averaged over $E_{rm in}$. The averaged value is $r_{m}({rm EXP})=3.697$fm. Eventually, we obtain $r_{rm skin}({rm EXP})=0.023$fm from $r_{rm m}=3.697$fm and $r_p({rm EXP})=3.685$fm of electron scattering.
We discuss the role of pairing anti-halo effect in the observed odd-even staggering in reaction cross sections for $^{30,31,32}$Ne and $^{36,37,38}$Mg isotopes by taking into account the ground state deformation of these nuclei. To this end, we construct the ground state density for the $^{30,31}$Ne and $^{36,37}$Mg nuclei based on a deformed Woods-Saxon potential, while for the $^{32}$Ne and $^{38}$Mg nuclei we also take into account the pairing correlation using the Hartree-Fock-Bogoliubov method. We demonstrate that, when the one-neutron separation energy is small for the odd-mass nuclei, a significant odd-even staggering still appears even with finite deformation, although the degree of staggering is somewhat reduced compared to the spherical case. This implies that the pairing anti-halo effect in general plays an important role in generating the odd-even staggering in reaction cross sections for weakly bound nuclei.
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