Folding Model Analysis Research Articles

PROBING THE ISOSCALAR EXCITATIONS OF 12C WITH INELASTIC ALPHA SCATTERING

The robust (spin and isospin zero) α-particle remains one of the best projectiles to probe the nuclear isoscalar excitations. In the present work, a microscopic folding model analysis of the α+12 C inelastic scattering to the 2+ (4.44 MeV), 0+ (7.65 MeV), 3- (9.64 MeV), 0+ (10.3 MeV) and 1- (10.84 MeV) states in 12 C has been performed using the 3-α resonating group method wave functions. The isoscalar transition strengths of these states were carefully studied based on the coupled-channel analysis using the microscopic folded form factors. A correlation between the weak binding and/or short lifetime of the excited state and absorption in the exit channel of inelastic scattering has been established.

Missing monopole strength of the Hoyle state in the inelastic α+12C scattering

Analyses of the inelastic α+12C scattering at medium energies have indicated that the strength of the Hoyle state (the isoscalar 02+ excitation at 7.65 MeV in 12C) seems to exhaust only 7–9% of the monopole energy weighted sum rule (EWSR), compared to about 15% of the EWSR extracted from inelastic electron scattering data. The full monopole transition strength predicted by realistic microscopic α-cluster models of the Hoyle state can be shown to exhaust up to 22% of the EWSR. To explore the missing monopole strength in the inelastic α+12C scattering, we have performed a fully microscopic folding model analysis of the inelastic α+12C scattering at Elab=104–240 MeV using the 3-α resonating group wave function of the Hoyle state obtained by Kamimura, and a complex density-dependent M3Y interaction newly parametrized based on the Brueckner–Hartree–Fock results for nuclear matter. Our folding model analysis has shown consistently that the missing monopole strength of the Hoyle state is not associated with the uncertainties in the analysis of the α+12C scattering, but is most likely due to the short lifetime and weakly bound structure of this state which significantly enhances absorption in the exit α+C∗12(02+) channel.

Isovector deformation and its link to the neutron shell closure

DWBA analysis of the inelastic 30-40S(p,p') and 18-22O(p,p') scattering data measured in the inverse kinematics has been performed to determine the isoscalar (δ0) and isovector (δ1) deformation lengths of the 2+1 excitations in the Sulfur and Oxygen isotopes using a compact folding approach. A systematic N-dependence of δ0 and δ1 has been established which shows a link between δ1 and the neutron-shell closure. Strong isovector deformations were found in several cases, e.g., the 2+1 state in 20O where δ1 is nearly three times larger than δ0. These results confirm the relation δ1>δ0 anticipated from the core polarization by the valence neutrons in the open-shell (neutron rich) nuclei. The effect of neutron shell closure at N=14 or 16 has been discussed based on the folding model analysis of the inelastic 22O+p scattering data at 46.6 MeV/u measured recently at GANIL.

Folding model study of the isobaric analog excitation: Isovector density dependence, Lane potential, and nuclear symmetry energy

A consistent folding model analysis of the ($\ensuremath{\Delta}S=0,\ensuremath{\Delta}T=1$) charge exchange $(p,n)$ reaction measured with $^{48}\mathrm{Ca}$, $^{90}\mathrm{Zr}$, $^{120}\mathrm{Sn}$, and $^{208}\mathrm{Pb}$ targets at the proton energies of 35 and 45 MeV is done within a two-channel coupling formalism. The nuclear ground state densities given by the Hartree-Fock-Bogoliubov formalism and the density-dependent CDM3Y6 interaction were used as inputs for the folding calculation of the nucleon optical potential and $(p,n)$ form factor. To have an accurate isospin dependence of the interaction, a complex isovector density dependence of the CDM3Y6 interaction has been carefully calibrated against the microscopic Brueckner-Hartree-Fock calculation by Jeukenne, Lejeune, and Mahaux before being used as folding input. Since the isovector coupling was used to explicitly link the isovector part of the nucleon optical potential to the cross section of the $(p,n)$ reaction exciting the 0${}^{+}$ isobaric analog states in $^{48}\mathrm{Sc}$, $^{90}\mathrm{Nb}$, $^{120}\mathrm{Sb}$, and $^{208}\mathrm{Bi}$, the newly parametrized isovector density dependence could be well tested in the folding model analysis of the $(p,n)$ reaction. The isospin- and density-dependent CDM3Y6 interaction was further used in the Hartree-Fock calculation of asymmetric nuclear matter, and a realistic estimation of the nuclear symmetry energy was made.

Folding model analysis of proton scattering from [formula omitted] nuclei

The elastic and inelastic proton scattering on O 18 , 20 , 22 nuclei are studied in a folding model formalism of nucleon–nucleus optical potential and inelastic form factor. The DDM3Y effective interaction is used and the ground state densities are obtained in continuum Skyrme-HFB approach. A semi-microscopic approach of collective form factors is done to extract the deformation parameters from inelastic scattering analysis while the microscopic approach uses the continuum QRPA form factors. Implications of the values of the deformation parameters, neutron and proton transition moments for the nuclei are discussed. The p-analyzing powers on O 18 , 20 , 22 nuclei are also predicted in the same framework.

Folding model analysis of proton radioactivity of spherical proton emitters

Half-lives of the decays of spherical nuclei away from the proton drip line by proton emissions are estimated theoretically. The quantum mechanical tunneling probability is calculated within the WKB approximation. Microscopic proton-nucleus interaction potentials are obtained by single folding the densities of the daughter nuclei with M3Y effective interaction supplemented by a zero-range pseudopotential for exchange along with the density dependence. Parameters of the density dependence are obtained from the nuclear matter calculations. Spherical charge distributions are used for Coulomb interaction potentials. These calculations provide reasonable estimates for the observed proton-radioactivity lifetimes of proton-rich nuclei for proton emissions from 26 ground and isomeric states of spherical proton emitters.

Folding model analysis of proton scattering from mirror nuclei 18Ne and 18O

The elastic and inelastic scattering of protons from mirror nuclei 18Ne and 18O are studied in a folding model approach. For comparison, two different effective interactions are folded with Hartree–Fock densities to obtain the nuclear interaction potentials. Both of them provide equivalent descriptions to the data and the deformation parameters extracted from inelastic scattering are reasonable. The density dependence parameters obtained from nuclear matter calculations and used for present analysis also provide a good estimate for the nuclear mean free path. The present formalism unifies radioactivity, nuclear matter and nuclear scattering.

Folding model and coupled-channels analysis of 6,7 Li elastic and inelastic scattering

The double-folding optical potentials of 6,7Li nuclei have been constructed using the S1Y effective nucleon-nucleon interaction. Angular distributions of the differential cross-section of 6,7Li elastic and inelastic scattering from 12 C, 28 Si and 58 Ni targets in the energy range 12-35 MeV/u were analyzed using the derived potentials through the coupled-channels mechanism. Successful descriptions of the data were obtained using the derived potentials.

Cluster folding model analysis of 3He elastic and inelastic scattering from 12C

Angular distributions of differential cross sections for the 12C( 3He, 3He) 12C, 12C( 3He, 3He) 12C ∗ reactions at E=72 MeV have been analyzed with a double folding cluster model DFC based on five sets of the effective N–N interaction of Gaussian form with different parameters. The transition to the (2 +; 4.44 MeV) state in 12C is studied and the deformation length δ 2 is extracted. It is found that the extracted deformation length is sensitive to the nuclear model used and it is similar to the corresponding value found in the literature.

Total reaction cross sections for the 12C + 12C system from a folding model analysis

The elastic scattering angular distributions for the 12C + 12C system at ten incident energies between 70 and 1016 MeV have been analysed by using the folding model based on an alpha particle model for the nucleus 12C. This folding model can well reproduce the strong dependence of the total reaction cross section on the incident energy.

Folding model analysis of elastic and inelastic proton scattering on sulfur isotopes

The folding formalism for the nucleon–nucleus optical potential and inelastic form factor is applied to study elastic and inelastic proton scattering on 30–40S isotopes. A recently developed realistic density dependent M3Y interaction, well tested in the folding analysis of nucleus–nucleus elastic and inelastic scattering, is used as effective NN interaction. The nuclear ground state and transition densities (for the 2 + excitations in sulfur isotopes) are obtained in the Hartree–Fock-BCS and QRPA approaches, respectively. The best fit ratios of transition moments M n 2 + / M p 2 + for the lowest 2 + states in sulfur isotopes are compared to those obtained earlier in the DWBA analysis of the same data using the same structure model and inelastic form factors obtained with the JLM effective interaction. Our folding + DWBA analysis has shown quite a strong isovector mixing in the elastic and inelastic scattering channels for the neutron rich 38,40S nuclei. In particular, the relative strength of the isovector part of the transition potential required by the inelastic p+ 38 S data is significantly stronger than that obtained with the corresponding QRPA transition density.

Folding model analysis of 6,7Li elastic scattering at 12.5–53 MeV/u

6,7Li-nucleus double-folding optical potentials have been constructed using the JLM and a gaussian shape of the effective nucleon–nucleon interaction. Angular distributions of the differential cross sections of 6,7Li elastic scattering from 12C, 28Si, 40Ca, 58Ni, 90Zr and 208Pb targets at 12.5–53 MeV/u were analyzed using the derived potentials. An energy-dependent reducing renormalization coefficient was required for the real component of the JLM optical potential in order to obtain successful predictions of the observed cross sections even when the projectile density-dependence was considered. A similar behavior was found for calculations based on the density-independent gaussian version of the effective nucleon–nucleon interaction. However, no renormalization was needed when a density-dependence approximation was involved. The energy- and target mass number-dependence in the renormalization factor, real volume integrals and total reaction cross sections are investigated.

Folding model analysis of the excitation of low-lying states and the high energy octupole resonance in116Snby 240 MeV α scattering

The sum rule strength of the high energy octupole resonance (HEOR) and the transition rates of low-lying ${2}^{+}$ and ${3}^{\ensuremath{-}}$ states of ${}^{116}\mathrm{Sn},$ excited by 240 MeV \ensuremath{\alpha} scattering, have been determined from deformed potential and folding model analyses. Deformed potential cross sections for both the low-lying ${3}^{\ensuremath{-}}$ state and the HEOR are greater than folding cross sections by a factor of 1.18. The high energy octupole resonance was found to exhaust $(70\ifmmode\pm\else\textpm\fi{}15)%$ and $(83\ifmmode\pm\else\textpm\fi{}15)%$ of the $E3$ energy-weighted sum rule from the two analyses, respectively. The data for the low-lying states are fit well by the calculations made with both models using electromagnetic values for the transition rates. Optical-model parameters were obtained from fits to elastic scattering data. The differential cross sections for the elastic scattering and inelastic scattering exciting the low-lying ${2}^{+}$ and ${3}^{\ensuremath{-}}$ states in ${}^{116}\mathrm{Sn}$ were measured over the angle range from ${\ensuremath{\theta}}_{\mathrm{c}.\mathrm{m}.}=1.6\ifmmode^\circ\else\textdegree\fi{}$ to 35.2\ifmmode^\circ\else\textdegree\fi{}.

Monopole resonance strengthsin58Niand208Pb

Giant monopole resonance strengths were obtained from small angle inelastic alpha scattering ${\mathrm{on}}^{58}$Ni ${\mathrm{and}}^{208}$Pb using deformed potential and folding models. Folding model analyses increase the sum rule strength in both nuclei, with 160% of the E0 sum rule required to fit the 13.7-MeV state in Pb. Significant contributions from other multipolarities could be excluded. The E0 strength identified ${\mathrm{in}}^{58}$Ni is shown to be about 42% of that identified ${\mathrm{in}}^{208}$Pb.

Folding analysis of the elastic 6Li+12C scattering: Knock-on exchange effects, energy dependence, and dynamical polarization potential.

Versions of the M3Y effective nucleon-nucleon interaction, with different prescriptions for the knock-on exchange contributions and density dependence, have been used in a study of the energy dependence of the nucleon optical potential in nuclear matter as well as in a systematic folding model analysis of the elastic $^{6}\mathrm{Li}$${+}^{12}$C scattering data at ${\mathit{E}}_{\mathrm{lab}}$=60--318 MeV to study the energy dependence of the $^{6}\mathrm{Li}$${+}^{12}$C optical potential. Contributions from the (breakup) dynamical polarization potential to the real part of the $^{6}\mathrm{Li}$${+}^{12}$C optical potential are simulated by a surface correction using splines added to the real folded potential. This is shown to be strongest at ${\mathit{E}}_{\mathrm{lab}}$=99 MeV. The correction needed to fit the data is found to be qualitatively similar to that predicted theoretically for breakup of the $^{6}\mathrm{Li}$. The optical model analysis of the refractive $^{6}\mathrm{Li}$${+}^{12}$C scattering data, using different types of real folded potential, shows that the most successful is the folded potential built upon density-dependent interactions, which have parameters chosen to reproduce the saturation properties of nuclear matter and which predict a nuclear incompressibility K around 200 MeV.

A simple effective interaction for peripheral heavy-ion collisions at intermediate energies

A very simple phenomenological nucleon-nucleon effective interaction is derived by folding-model analyses of 36 sets of heavy-ion elastic-scattering data at energies E/A between 10 and 100 MeV. It is represented by a single Yukawa term with a complex strength that varies slowly with energy. It is appropriate for peripheral collisions at intermediate energies. It is proposed for use in consistent analyses of elastic- and inelastic-scattering measurements. The possible role of spin-orbit coupling in heavy-ion elastic scattering at the higher energies is discussed.

Folding model analysis of 58Ni+ 64Ni elastic and inelastic scattering at Elab = 203.8 and 219.2 MeV

Angular distributions of elastic scattering of 58Ni by 64Ni and inelastic scattering leading to the first 2 + state in 58Ni or in 64Ni have been measured at E lab=203.8 and 219.2 MeV. The data have been analyzed in the frame of the folding model. A renormalization N of the folding potential consistently lower than unity ( N ≈ 0.65) is needed to reproduce the elastic-scattering data. Coupled-channel calculations including the main inelastic channels explain partly this discrepancy. They reproduce the elastic- and inelastic-scattering data with a renormalization of N ≈ 0.8 at both energies. Interference between Coulomb and nuclear excitation is shown to play an important role in the repulsive character of the polarization potential.

Microscopic analysis of 34S(d

A microscopic deformed folding model analysis is carried out for cross sections and vector analyzing power angular distribution of $^{34}\mathrm{S}$(d\ensuremath{\rightarrow},d'${)}^{34}$${\mathrm{S}}^{\mathrm{*}}$ for the first and second ${2}^{+}$ states, at ${\mathit{E}}_{\mathit{d}}$=11.8 MeV, using improved s-d shell transition densities of Wildenthal and Brown. While the description of the first ${2}^{+}$ state is excellent, the negative sign of ${\mathit{M}}_{\mathit{n}}$/${\mathit{M}}_{\mathit{p}}$ for the second ${2}^{+}$ state as predicted in the shell-model calculation produces too small cross sections. The (d,d') data corroborate previous findings in (\ensuremath{\alpha},\ensuremath{\alpha}') and (p,p') work that the shell-model calculations are wrong for this state.

A study of 36S by high-resolution inelastic proton and alpha scattering

In an inelastic hadron scattering study of 36S both a high resolution (p, p′) experiment at E p = 28.0 MeV and an (α, α′) experiment at E α = 31.6 MeV were performed. A highly enriched target was used. Angular distributions for levels up to 6.55 MeV have been obtained. Five previously unknown levels have been observed. A macroscopic CC analysis with a phenomenological potential was performed for the inelastic levels, yielding J π assignments for several states. The different angular distributions for the 2 1 + and 2 2 + states could be accounted for by assuming a transition potential that has a contribution in the nuclear interior. Apart from the 3 − state at E x = 4.193 MeV a 4 − state and 5 − state were identified, for which phenomenological octupole band calculations have been performed. Explicit folding-model calculations, essential for a consistent description of 0 + and 1 − states, have been performed for several levels, using the JLM nucleon-nucleon interaction both for (p, p′) and for (α, α′), A rather good description of the differential cross sections of several levels, including the 0 + and 1 − states, is obtained. A direct comparison between the results from a (p, p′) and an (α, α′) folding-model analysis is made, resulting in isoscalar B(E λ) values. Microscopic calculations have been performed for the (p, p′) results. The transition densities are obtained from shell-model (SM) and extended-RPA (E-RPA) calculations. No effective charges were introduced. The E-RPA calculations gave a fair description of the differential cross sections of several levels. In the SM calculations the neutron part of the transition density is severely underestimated.

Folding model analysis of 12,13C+12C and 16O+12C scattering at intermediate energies using a density-dependent interaction

Abstract Scattering data for 12 C+ 12 C, 13 C+ 12 C, and 16 O+ 12 C at energies E A = 9 to 120 MeV per nucleon have been analyzed using a folded potential based on the density-and energy-dependent DDM3Y interaction. The renormalization required for the real potential is about unity at all energies. A unified description of these data is obtained with a relatively weakly absorbing imaginary potential. Furthermore, only this type of potential provides a successful description of the 12 C+ 12 C data at 140 and 159 MeV which cover almost the whole angular range. Potential ambiguities are discussed.