Microscopic Coupled-channel Calculation Research Articles

Systematic study of the relationship between nuclear structure and reactions for the 10Be nucleus

Abstract We systematically investigate the relationship between the nuclear structure and reaction in the 10Be nucleus using a theoretical framework. The structure of the 10Be nucleus is constructed with a cluster model based on a microscopic viewpoint. In this paper, the 10Be nucleus is prepared with different structures by manipulating the parameters of an effective nucleon–nucleon interaction. The nuclear structure and expectation values of physical quantities are drastically changed by the modification. We summarize such changes and show the effects on the elastic and inelastic scatterings for proton and 12C targets in the microscopic coupled-channel calculation. Of especial interest, we recently reported the visualization of the dineutron correlation in 10Be on proton inelastic scattering in [Phys. Rev. C 104, 034613 (2021)]. In this preceding work, we found that changing the degree of dineutron correlation in 10Be leads to drastic changes of the inelastic cross section for the 2$_2^+$ state. The development (or breaking) of the dineutron correlation is governed by the strength of the spin–orbit interaction of the structure calculation. However, in the previous work, some of the realistic physical points were missing, for example, the binding energy (BE). Therefore, we reconstruct the 10Be nucleus by adjusting the effective nucleon–nucleon interaction to obtain a reasonable BE of the ground state. With this improvement, we again discuss the dineutron correlation in the 10Be nucleus. We reconfirm the way to measure the degree of development (or breaking) of the dineutron cluster structure: its sensitivity to the inelastic cross section of the 2$_2^+$ state of 10Be. Subject Index cluster model, folding model, dineutron correlation, effective nucleon-nucleon interaction.

Relation between transition density and proton inelastic scattering by C12 target at Ep=65 and 200 MeV

We calculate proton elastic and inelastic scatterings with a microscopic coupled channel (MCC) calculation. The localized diagonal and coupling potentials including the spin-orbit part are obtained by folding a complex $G$-matrix effective nucleon-nucleon interaction with a transition density. This is the first time that the present folding prescription for the spin-orbit part is applied to the proton inelastic scattering, while for the monopole transition only. We apply the MCC calculation to the proton elastic and inelastic $({0}_{2}^{+})$ scatterings by $^{12}\mathrm{C}$ target at ${E}_{p}=65 \mathrm{and} 200 \mathrm{MeV}$. The role of diagonal and coupling potentials for the central and spin-orbit parts is checked. In addition, the relation between the transition density and the proton inelastic scattering is investigated with the modified wave function and the modified transition density. Namely, we perform the investigation with the artificial drastic change rather than fine structural change. The inelastic cross section is sensitive to the strength and shape of the transition density, but the inelastic analyzing power is sensitive only to the shape of that. Finally, we make clear the property of the inelastic analyzing power derived from the transition density without an ambiguity.

Microscopic coupled-channel calculation of proton and alpha inelastic scattering to the $4^+_1$ and $4^+_2$ states of $^{24}\textrm{Mg}$

Abstract The triaxial and hexadecapole deformations of the $K^\pi=0^+$ and $K^\pi=2^+$ bands of $^{24}$Mg have been investigated by the inelastic scatterings of various probes, including electrons, protons, and alpha($\alpha$) particles, for a prolonged time. However, it has been challenging to explain the unique properties of the scatterings observed for the $4^+_1$ state through reaction calculations. This paper investigates the structure and transition properties of the $K^\pi=0^+$ and $K^\pi=2^+$ bands of $^{24}$Mg employing the microscopic structure and reaction calculations via inelastic proton and $\alpha$ scattering. In particular, the $E4$ transitions to the $4^+_1$ and $4^+_2$ states are reexamined. The structure of $^{24}$Mg was calculated employing the variation after the parity and total angular momentum projections in the framework of the antisymmetrized molecular dynamics (AMD). The inelastic proton and $\alpha$ reactions were calculated by the microscopic coupled-channel (MCC) approach by folding the Melbourne $g$-matrix $NN$ interaction with the AMD densities of $^{24}$Mg. Reasonable results were obtained on the properties of the structure, including the energy spectra and $E2$ and $E4$ transitions of the $K^\pi=0^+$ and $K^\pi=2^+$ bands owing to the enhanced collectivity of triaxial deformation. The MCC+AMD calculation successfully reproduced the angular distributions of the $4^+_1$ and $4^+_2$ cross sections of proton scattering at incident energies of $E_p=40$–100 MeV and $\alpha$ scattering at $E_\alpha=100$–400 MeV. This is the first microscopic calculation to describe the unique properties of the $0^+_1\to 4^+_1$ transition. In the inelastic scattering to the $4^+_1$ state, the dominant two-step process of the $0^+_1\to 2^+_1\to 4^+_1$ transitions and the deconstructive interference in the weak one-step process were essential.

Probing negative-parity states of Mg24 probed with proton and α inelastic scattering

[Background:] The band structure of the negative-parity states of $^{24}$Mg has not yet been clarified. The $K^\pi=0^-$, $K^\pi=1^-$, and $K^\pi=3^-$ bands have been suggested, but the assignments have been inconsistent between experiments and theories. [Purpose:] Negative-parity states of $^{24}$Mg are investigated by microscopic structure and reaction calculations via proton and alpha inelastic scattering to clarify the band assignment for the observed negative-parity spectra. [Method:] The structure of $^{24}$Mg was calculated using the antisymmetrized molecular dynamics~(AMD). Proton and alpha inelastic reactions were calculated using microscopic coupled-channel (MCC) calculations by folding the Melbourne $g$-matrix $NN$ interaction with the AMD densities of $^{24}$Mg. [Results:] The member states of the $K^\pi=0^+$, $K^\pi=2^+$, $K^\pi=0^-$, $K^\pi=1^-$, and $K^\pi=3^-$ bands of $^{24}$Mg were obtained through the AMD result. In the MCC+AMD results for proton and alpha elastic and inelastic cross sections, reasonable agreements were obtained with existing data, except in the case of the $4^+_1$ state. [Conclusions:] The $3^-$ state of the $K^\pi=3^-$ band and the $1^-$ and $3^-$ states of the $K^\pi=0^-$ bands were assigned to the $3^-_1$(7.62 MeV), $1^-_1$(7.56 MeV), and $3^-_2$(8.36 MeV) states, respectively. The present AMD calculation is the first microscopic structure calculation to reproduce the energy ordering of the $K^\pi=0^-$, $K^\pi=1^-$, and $K^\pi=3^-$ bands of $^{24}$Mg.

Neutron dominance in excited states of Mg26 and Be10 probed by proton and α inelastic scattering

Isospin characters of nuclear excitations in $^{26}$Mg and $^{10}$Be are investigated via proton($p$) and alpha($\alpha$) inelastic scattering. A structure model of antisymmetrized molecular dynamics (AMD) is applied to calculate the ground and excited states of $^{26}$Mg and $^{10}$Be. The calculation describes the isoscalar feature of the ground-band $2^+_1$($K^\pi=0^+_1$) excitation and predicts the neutron dominance of the side-band $2^+_2$($K^\pi=2^+$) excitation in $^{26}$Mg and $^{10}$Be. The $p$ and $\alpha$ inelastic scattering off $^{26}$Mg and $^{10}$Be is calculated by microscopic coupled-channel (MCC) calculations with a $g$-matrix folding approach by using the matter and transition densities of the target nuclei calculated with AMD. The calculation reasonably reproduces the observed $0^+_1$, $2^+_1$, and $2^+_2$ cross sections of $^{26}$Mg+$p$ scattering at incident energies $E_p=$24 and 40 MeV and of $^{26}$Mg+$\alpha$ scattering at $E_\alpha=$104 and 120 MeV. For $^{10}$Be+$p$ and $^{10}$Be+$\alpha$ scattering, inelastic cross sections to the excited states in the $K^\pi=0^+_1$ ground-, $K^\pi=2^+$ side-, $K^\pi=0^+_2$ cluster-, and $K^\pi=1^-$ cluster-bands are investigated. The isospin characters of excitations are investigated via inelastic scattering processes by comparison of the production rates in the $^{10}$Be+$p$, $^{10}$Be+$\alpha$, and $^{10}$C+$p$ reactions. The result predicts that the $2^+_2$ state is selectively produced by the $^{10}$Be+$p$ reaction because of the neutron dominance in the $2^+_2$ excitation as in the case of the $^{26}$Mg+$p$ scattering to the $2^+_2$ state, whereas its production is significantly suppressed in the $^{10}$C+$p$ reaction.

Properties of Kπ=01+ , Kπ=2− , and Kπ=01− bands of Ne20 probed via proton and α inelastic scattering

The $K^\pi=0^+_1$, $K^\pi=2^-$, and $K^\pi=0^-_1$ bands of $^{20}$Ne are investigated with microscopic structure and reaction calculations via proton and $\alpha$ inelastic scattering off $^{20}$Ne. Structures of $^{20}$Ne are calculated with variation after parity and total angular momentum projections in the antisymmetrized molecular dynamics(AMD). The $K^\pi=0^+_1$ and $K^\pi=0^-_1$ bands have $^{16}\textrm{O}+\alpha$ cluster structures, whereas the $K^\pi=2^-$ band shows a $^{12}\textrm{C}+2\alpha$-like structure. Microscopic coupled-channel calculations of proton and $\alpha$ scattering off $^{20}$Ne are performed by using the proton-nucleus and $\alpha$-nucleus potentials, which are derived by folding the Melbourne $g$-matrix $NN$ interaction with AMD densities of $^{20}$Ne. The calculation reasonably reproduces the observed cross sections of proton scattering at $E_p=25$--35 MeV and $\alpha$ scattering at $E_\alpha=104$--386 MeV. Transition properties from the ground to excited states are discussed by reaction analyses of proton and $\alpha$ inelastic processes. Mixing of the $K^\pi=2^-$ and $K^\pi=0^-_1$ bands is discussed by detailed analysis of the $0^+_1\to 3^-_1$ and $0^+_1\to 3^-_2$ transitions. For the $3^-_1$ state, mixing of the $K^\pi=0^-_1$ cluster component in the $K^\pi=2^-$ band plays an important role in the transition properties.

Transition properties of low-lying states in Si28 probed via inelastic proton and α scattering

${0}^{+}, {1}^{\ensuremath{-}}, {2}^{+}$, and ${3}^{\ensuremath{-}}$ excitations of $^{28}\mathrm{Si}$ are investigated via proton and $\ensuremath{\alpha}$ inelastic scattering off $^{28}\mathrm{Si}$. The structure calculation of $^{28}\mathrm{Si}$ is performed with the energy variation after total angular momentum and parity projections in the framework of antisymmetrized molecular dynamics (AMD). As a result of the AMD calculation, the oblate ground and prolate bands, ${0}^{+}$ and ${3}^{\ensuremath{-}}$ excitations, and the ${1}^{\ensuremath{-}}$ and ${3}^{\ensuremath{-}}$ states of the ${K}^{\ensuremath{\pi}}={0}^{\ensuremath{-}}$ band are obtained. Using the matter and transition densities of $^{28}\mathrm{Si}$ obtained by AMD, microscopic coupled-channels calculations of proton and $\ensuremath{\alpha}$ scattering off $^{28}\mathrm{Si}$ are performed. The proton-$^{28}\mathrm{Si}$ potentials in the reaction calculation are microscopically derived by folding the Melbourne $g$-matrix $NN$ interaction with the AMD densities of $^{28}\mathrm{Si}$. The $\ensuremath{\alpha}\ensuremath{-}^{28}\mathrm{Si}$ potentials are obtained by folding the nucleon-$^{28}\mathrm{Si}$ potentials with an $\ensuremath{\alpha}$ density. The calculation reasonably reproduces the observed elastic and inelastic cross sections of proton and $\ensuremath{\alpha}$ scattering. Transition properties are discussed by combining the reaction analysis of proton and $\ensuremath{\alpha}$ scattering and structure features such as transition strengths and form factors. The isoscalar monopole and dipole transitions are focused.

Cluster structures and monopole transitions of C14

Cluster structures of $^{14}$C were investigated with a method of antisymmetrized molecular dynamics (AMD) combined with a $3\alpha+nn$ cluster model while focusing on the monopole excitations and linear-chain $3\alpha$ band. Variation after parity and angular momentum projections was performed in the AMD framework, and the generator coordinate method was applied to take into account various $3\alpha+nn$ cluster configurations. Energy spectra and monopole and $E2$ transition strengths of $0^+$, $2^+$, and $4^+$ states were calculated to assign band structures. The $0^+_3$ state with remarkable monopole transition was obtained as a vibrational mode of the triangle $3\alpha$ configuration. In addition, the linear-chain $3\alpha$ band from the band-head $0^+_4$ state was obtained. $^{10}$Be$+\alpha$ decay widths of $0^+$, $2^+$, and $4^+$ states were evaluated. $\alpha$ inelastic scattering off $^{14}$C was also investigated by the microscopic coupled-channel calculation with the $g$-matrix folding model to propose possible observation of the $0^+_3$ state via $\alpha$ scattering experiments.

Microscopic calculation of inelastic proton scattering off O18, Be10, Be12, and C16 to study neutron excitation in neutron-rich nuclei

The microscopic coupled-channel calculation of inelastic proton scattering is performed for the study of neutron excitations in $2^+_1$ states of $^{18}$O, $^{10}$Be, $^{12}$Be, and $^{16}$C. Proton-nucleus potentials in the coupled-channel calculation are microscopically derived by folding the Melbourne $g$-matrix $NN$ interaction with matter and transition densities of target nuclei obtained by the structure model calculation of antisymmetrized molecular dynamics. The calculated result reasonably reproduces the elastic and inelastic proton scattering cross sections, and supports the dominant contribution of neutron in the $2^+_1$ excitation of $^{12}$Be and $^{16}$C as well as $^{18}$O. Sensitivity of the inelastic scattering cross sections to the neutron transition density is discussed. The exotic feature of the neutron transition density with the amplitude in the outer region in $^{12}$Be and $^{16}$C is focused.

Large-amplitude quadrupole shape mixing probed by the $(p,p^\prime)$ reaction: A model analysis

Abstract To discuss a possible observation of large-amplitude nuclear shape mixing by nuclear reaction, we employ a simple collective model and evaluate the transition densities with which the differential cross sections are obtained through the microscopic coupled-channel calculation. Assuming the spherical-to-prolate shape transition, we focus on large-amplitude shape mixing associated with the softness of the collective potential in the $\beta$ direction. We introduce a simple model based on the five-dimensional quadrupole collective Hamiltonian, which simulates a chain of isotopes that exhibit spherical-to-prolate shape phase transition. Taking $^{154}$Sm as an example and controlling the model parameters, we study how the large-amplitude shape mixing affects the elastic and inelastic proton scatterings. The calculated results suggest that the inelastic cross section of the $2_2^+$ state shows us the important role of the quadrupole shape mixing.

First microscopic coupled-channels calculation of cross sections for inelastic α scattering off O16

The $\alpha$ inelastic scattering on $^{16}$O is investigated with the coupled-channel calculation using the $\alpha$-nucleus coupled-channel potentials, which are microscopically derived by folding the the Melbourne $g$-matrix $NN$ interaction with the $^{16}$O and $\alpha$ densities. The matter and transition densities of $^{16}$O are calculated by a microscopic structure model of the variation after the spin-parity projections combined with the generator coordinate method of $^{12}$C+$\alpha$ in the framework of the antisymmetrized molecular dynamics. The calculation reproduces the observed elastic and inelastic cross sections at incident energies of $E_\alpha$=104 MeV, 130 MeV, 146 MeV, and 386 MeV. The coupled-channel effect on the cross sections is also discussed.

α scattering cross sections on C12 with a microscopic coupled-channels calculation

$\alpha$ elastic and inelastic scattering on $^{12}$C is investigated with the coupled-channel calculation using microscopic $\alpha$-$^{12}$C potentials, which are derived by folding the Melbourne $g$-matrix $NN$ interaction with the matter and transition densities of $^{12}$C. These densities are obtained by a microscopic structure model of the antisymmetrized molecular dynamics combined with and without the $3\alpha$ generator coordinate method. The calculation reproduces satisfactorily well the observed elastic and inelastic cross sections at incident energies of $E_\alpha=130$~MeV, 172.5~MeV, 240~MeV, and 386~MeV with no adjustable parameter. Isoscalar monopole and dipole excitations to the $0^+_2$, $0^+_3$, and $1^-_1$ states in the $\alpha$ scattering are discussed.

Erratum: Microscopic coupled-channels calculations of nucleus-nucleus scattering including chiral three-nucleon-force effects [Phys. Rev. C 93 , 014607 (2016)

Erratum: Microscopic coupled-channels calculations of nucleus-nucleus scattering including chiral three-nucleon-force effects [Phys. Rev. C <b>93</b> , 014607 (2016)

Repulsive three-body force and channel-coupling effects via C12+C12 scattering at 100A MeV

The angular distributions of differential cross sections of $^{12}\mathrm{C}+^{12}\mathrm{C}$ elastic and inelastic scattering populating the ground and excited states in $^{12}\mathrm{C}$ up to 15 MeV excitation energy are precisely measured for the first time at an incident energy of $100A$ MeV to study the effect of repulsive three-body forces. Using the high-resolution spectrometer Grand Raiden at the Research Center for Nuclear Physics (RCNP), Osaka University, we have obtained the differential cross sections for the ground state (${0}_{1}^{+}$) and 4.44 MeV (${2}_{1}^{+}$) excited state, as well as the summed differential cross sections for the states between 4.44 and 15 MeV in the angular range of $1.{0}^{\ensuremath{\circ}}--7.{5}^{\ensuremath{\circ}}$. The results are compared with microscopic coupled-channel calculations. The potential between the colliding nuclei is determined by the double folding method with three different complex $G$-matrix interactions, the ESC, CEG07b, and MPa interactions. The CEG07b and MPa interactions, which include repulsive three-body forces, describe the data well, whereas the ESC interaction, which does not include repulsive three-body forces, fails to reproduce the data. The results provide evidence of repulsive three-body forces in $^{12}\mathrm{C}$ and demonstrate the possible sensitivity of elastic scattering to three-body forces.

Evidence of enhanced threeαradius inα+12C inelastic scattering

The microscopic coupled-channels calculations (MCC), which is based on precise internal wave functions and a realistic nucleon-nucleon interaction, is performed for the α + 12 C inelastic scattering in the energy range of Eα = 80 to 400 MeV. The MCC calculations nicely reproduce the observed differential cross sections for the elastic and inelastic scattering, which goes to the 21+, 31−, 02+ states. The partial wave analysis for the differential cross sections has also performed. From the partial wave analysis, the nuclear radius of three α rotational state in 12 C with a life time of 10-21 second, which has been expected to have much more extended radius than the ground 12 C nucleus, is speculated. Present analysis predicts about 1.0 fm enhancement in the matter radius of the three α rotational state in comparison to the normal radius of the ground state, which is known to be proportional to the mass number to the one third. The spatial extension of the three α rotational state is comparable to the extended radius observed in the neutron halo phenomena. Constraint on the recent ab-initio calculation for the 3α states in 12 C is also discussed.

Microscopic coupled-channels calculations of nucleus-nucleus scattering including chiral three-nucleon-force effects

We analyze $^{16}$O-$^{16}$O and $^{12}$C-$^{12}$C scattering with the microscopic coupled-channels method and investigate the coupled-channels and three-nucleon-force (3NF) effects on elastic and inelastic cross sections. In the microscopic coupled-channels calculation, the Melbourne g-matrix interaction modified according to the chiral 3NF effects is used. It is found that the coupled-channels and 3NF effects additively change both the elastic and inelastic cross sections. As a result, the coupled-channels calculation including the 3NF effects significantly improves the agreement between the theoretical results and the experimental data. The incident-energy dependence of the coupled-channels and 3NF effects is also discussed.

Spatial measure of reaction size in proton scattering

The microscopic coupled-channel (MCC) calculations for proton + 12 C inelastic scattering are performed in the energy range of E p = 29.95 MeV to 65 MeV. The nuclear interactions for the proton − 12 C system are constructed from the folding model, which employs the internal wave function of 12 C, obtained from the 3 α resonating group method (3 α RGM), and an effective nucleon-nucleon interaction of the density-dependent Michigan three-range Yukawa (DDM3Y). The MCC calculation with the 3 α RGM + DDM3Y nicely reproduces all of the differential cross sections for elastic and inelastic scattering in the angular range of θ c.m. = 30° to 120°. We introduce a scattering radius, which characterizes a spatial size of the scattering area, from partial wave decompositions of an angle-integrated cross section. The scattering radii for the elastic scattering and the various inelastic channels, which involve the rotational or vibrational excitations and the 3 α excitations in 12 C, are derived. We found that the scattering radii for the inelastic channels with a well developed 3 α structure are strongly enhanced in comparison to the scattering radii for the elastic and collective channels. This enhancement of the scattering radius for the 3 α channel strongly suggests that the scattering radius is sensitive to a size of the intrinsic structure of the finally excited state in the scattering process.

Analysis of proton+C12scattering by microscopic coupled-channels calculations

The microscopic coupled-channel (MCC) calculations for proton $+^{12}\mathrm{C}$ inelastic scattering are performed in the energy range of ${E}_{p}=29.95$ to 65 MeV. The nuclear interactions for the proton-$^{12}\mathrm{C}$ system are constructed from the folding model, which employs the internal wave function of $^{12}\mathrm{C}$, obtained from the $3\ensuremath{\alpha}$ resonating group method ($3\ensuremath{\alpha}$ RGM), and an effective nucleon-nucleon interaction of the density-dependent Michigan three-range Yukawa (DDM3Y). The MCC calculation with the $3\ensuremath{\alpha}$ RGM + DDM3Y nicely reproduces all of the differential cross sections for elastic and inelastic scattering in the angular range of ${\ensuremath{\theta}}_{\mathrm{c}.\mathrm{m}.}=\phantom{\rule{4pt}{0ex}}30$ to ${120}^{\ensuremath{\circ}}$. The channel-coupling effect is analyzed by comparing the full MCC calculation with the calculation of the distorted wave Born approximation (DWBA). The effect of the spin-orbit interaction, obtained in a simplified manner with the folding procedure, is also discussed.

Measure of the spatial size for the monopole excitation in proton scattering

We formulate a scattering radius, which will be demonstrated to be a good measure of the spatial size of a general exclusive reaction. The scattering radius is presented in a framework of the partial-wave expansion method in a general two-body scattering problem. A microscopic coupled-channel calculation is performed for proton scattering by $^{12}\mathrm{C}$ in the range of the proton's incident energy, ${E}_{p}=29.95--200$ MeV, and the scattering radii are evaluated for elastic scattering and inelastic scattering, going to the Hoyle 0${}_{2}^{+}$ state with a well-developed 3$\ensuremath{\alpha}$ structure. A prominent enhancement of the scattering radius is clearly confirmed in the 3$\ensuremath{\alpha}$ final channel in comparison to the elastic channel. The scattering radius is also calculated for excitation to the giant monopole resonance (GMR) in a microscopic coupled-channel framework. The scattering radius for the 3$\ensuremath{\alpha}$ excitation is much more enhanced than the scattering radius for the GMR excitation. The proton's incident-energy dependence of the scattering radius is also investigated, and the energy systematics strongly suggest that the scattering radius can characterize the spatial size of a reaction area, which is determined by the matter radius of a nucleus excited to a final state.

O16+O16inelastic scattering studied by a complexG-matrix interaction

We reanalyze the $^{16}\mathrm{O}+^{16}\mathrm{O}$ inelastic scattering to the single ${2}_{1}^{+}$ and ${3}_{1}^{\ensuremath{-}}$ excitation channels at an incident energy of ${E}_{\mathrm{lab}}=1120$ MeV by a microscopic coupled-channel (CC) calculation with complex $G$-matrix interaction. The imaginary part of the folded potential is renormalized so as to reproduce the elastic angular distribution. It is found that the results of full CC calculations are still inconsistent with the experimental data for both ${2}_{1}^{+}$ and ${3}_{1}^{\ensuremath{-}}$ channels. It is also found that when the imaginary coupling terms are omitted, the calculated ${2}_{1}^{+}$-channel cross sections are improved, although the ${3}_{1}^{\ensuremath{-}}$-channel cross sections are slightly overestimated.