Double Folding Model Calculations Research Articles

Alpha decay around shell closures and correlation of half-life times with the neutron energy levels of the emitting nuclei

The α-decay half-lives (T1/2) for 16 isotopic chains of even-even and even-odd nuclei, from 52Te to 82Pb were investigated by employing the density dependent cluster model. The α-nucleus potential was calculated by the double-folding model (DFM) using the M3Y-Reid NN interaction with the zero-range exchange part assuming spherical and deformed density distributions of the daughter nuclei. In the framework of the WKB approximation, the half-lives of the α-emitters were calculated and compared using six analytic formulas (UDL, Horoi, BKAG, UNIV, NRDX and TM formulas). We found a clear similarity in the behavior of log10⁡T1/2 versus the neutron daughter number (Nd) when the calculations performed using both DFM and these mentioned analytical formulas for each isotopic chain. In addition, the theoretical results of log10⁡T1/2 obtained with the UDL and UNIV formulas slightly better agree with the DFM calculations than those calculated from the other analytical methods. Moreover, the present work pointed out some magic or semi-magic neutron numbers of the daughter nuclei corresponding to dips in the behavior of log10⁡T1/2 with variation in Nd. The interplay of these magic (semi-magic) numbers is inspected as the total number of neutrons filling the upper neutron level in the parent nucleus. We test the predicted results of the above mentioned models by comparing them with the available experimental α-decay half-lives. Starting from the number of neutrons needed for level closure and the small available values of the confirmed spin of parent and daughter nuclei, we predicted the level schemes around the known magicities N=50, 82 and 126. We found changes in the neutron levels for isotopes of the same element which become too large for heavy nuclei. Example for these changes is the repetition of the spin value 1/2− for the three isotopes 195−199Hg and their daughters, we need five different neutron level arrangements with 3p1/2 level at the top.

Consistent mean-field description of theC12 + C12optical potential at low energies and the astrophysicalSfactor

The nuclear mean-field potential built up by the $^{12}\mathrm{C}+^{12}\mathrm{C}$ interaction at energies relevant for the carbon burning process is calculated in the double-folding model (DFM) using the realistic ground-state density of $^{12}\mathrm{C}$ and the CDM3Y3 density dependent nucleon-nucleon (NN) interaction, with the rearrangement term properly included. To validate the use of a density dependent NN interaction in the DFM calculation in the low-energy regime, an adiabatic approximation is suggested for the nucleus-nucleus overlap density. The reliability of the nuclear mean-field potential predicted by this low-energy version of the DFM is tested in a detailed optical model analysis of the elastic $^{12}\mathrm{C}+^{12}\mathrm{C}$ scattering data at energies below 10 MeV/nucleon. The folded mean-field potential is then used to study the astrophysical $S$ factor of $^{12}\mathrm{C}+^{12}\mathrm{C}$ fusion in the barrier penetration model. Without any adjustment of the potential strength, our results reproduce very well the nonresonant behavior of the $S$ factor of $^{12}\mathrm{C}+^{12}\mathrm{C}$ fusion over a wide range of energies.

Mean-field description of heavy-ion scattering at low energies and fusion

The nuclear mean-field potential built up during the $${}^{12}\hbox {C}+{}^{12}\hbox {C}$$ and $${}^{16}\hbox {O}+{}^{16}\hbox {O}$$ collisions at low energies relevant for the carbon- and oxygen-burning processes is constructed within the double-folding model (DFM) using the realistic ground-state densities of $$^{12}\hbox {C}$$ and $$^{16}$$ O, and CDM3Yn density-dependent nucleon–nucleon (NN) interaction. The rearrangement term, indicated by the Hugenholtz–van Hove theorem for the single-particle energy in nuclear matter, is properly considered in the DFM calculation. To validate the use of the density-dependent NN interaction at low energies, an adiabatic approximation was suggested for the dinuclear overlap density. The reliability of the nucleus–nucleus potential predicted through this low-energy version of the DFM was tested in the optical model (OM) analysis of the elastic $${}^{12}\hbox {C}+{}^{12}\hbox {C}$$ and $${}^{16}\hbox {O}+{}^{16}\hbox {O}$$ scattering data at energies below 10 MeV/nucleon. These OM results provide a consistently good description of the elastic angular distributions and 90 $$^\circ$$ excitation function. The dinuclear mean-field potential predicted by the DFM is further used to determine the astrophysical S factor of the $${}^{12}\hbox {C}+{}^{12}\hbox {C}$$ and $${}^{16}\hbox {O}+{}^{16}\hbox {O}$$ fusions in the barrier penetration model. Without any adjustment of the potential strength, our results reproduce the non-resonant behavior of the S factor of the $${}^{12}\hbox {C}+{}^{12}\hbox {C}$$ and $${}^{16}\hbox {O}+{}^{16}\hbox {O}$$ fusions very well over a wide range of energies.

A theoretical analysis of the 6He + 9Be reaction: phenomenological and double folding model approximations

We investigate the elastic scattering data of the 6He + 9Be reaction by using the phenomenological and double folding model potentials within the framework of the optical model. In the phenomenological calculations, we use the optical model parameters existed in literature to explain 6He + 9Be reaction. The real potentials in double folding calculations are obtained by using two different density distributions of 6He nucleus. For both the phenomenological and double folding model calculations, the imaginary potentials have the shape of the Woods-Saxon volume. We compare the phenomenological model results with the double folding model ones as well as the experimental data. We present that the folding model results are in better agreement with the experimental data than the phenomenological results for the 6He + 9Be reaction.

Reanalysis of Elastic Scattering of 6Li + 209Bi Reaction Using a New Density Distribution of 6Li Nucleus

In the present paper, the elastic scattering of 6Li + 209Bi system is reanalyzed by using the double folding model (DFM) at energies near the Coulomb barrier (ELab = 29.9 and 32.8 MeV). With this goal, a new density distribution of 6Li nucleus, the no-core full configuration (NCFC) density distribution (DD), is used to obtain the real potentials in DFM calculations. The NCFC DD results are compared with the results of both gaussian shape (GS) DD and an earlier study as well as the experimental data. This comparison provides information about the similarities and differences of the models used in calculations.

Simplification of Double Folding Model Calculations in Study of Interaction Between Two Deformed Nuclei

The Double Folding (DF) model calculation of the internuclear potential in heavy-ion interactions when the participant nuclei are deformed in their ground states involves a six-dimensional integral. Using the multipole expansion in these calculations, the DF six-dimensional integral reduce to the sum of the products of three single-dimensional integrals. In this paper we have presented a procedure for the calculation of the radius dependent functions in the multipole expansion of the nuclear density and their Fourier transforms. We have also reduced the DF model integrals to the sum of the single dimensional integrals using the obtained relations for the radius dependent functions in the multipole expansion and their Fourier transforms.

Global optical model potential forA=3projectiles

A global optical model potential (GDP08) for $^{3}\mathrm{He}$ projectiles has been obtained by simultaneously fitting the elastic scattering data of $^{3}\mathrm{He}$ from targets of $40\ensuremath{\leqslant}{A}_{T}\ensuremath{\leqslant}209$ at incident energies of $30\ensuremath{\leqslant}{E}_{\mathrm{inc}}\ensuremath{\leqslant}217$ MeV. Uncertainties and correlation coefficients between the global potential parameters were obtained by using the bootstrap statistical method. GDP08 was found to satisfactorily account for the elastic scattering of $^{3}\mathrm{H}$ as well, which makes it a global optical potential for the $A=3$ nuclei. Optical model calculations using the GDP08 global potential are compared with the experimental angular distributions of differential cross sections for $^{3}\mathrm{He}$-nucleus and $^{3}\mathrm{H}$-nucleus scattering from different targets of $6\ensuremath{\leqslant}{A}_{T}\ensuremath{\leqslant}232$ at incident energies of $4\ensuremath{\leqslant}{E}_{\mathrm{inc}}\ensuremath{\leqslant}450$ MeV. The optical potential for the doubly-magic nucleus $^{40}\mathrm{Ca}$, the low-energy correction to the real potential for nuclei with $58\ensuremath{\lesssim}{A}_{T}\ensuremath{\lesssim}120$ at ${E}_{\mathrm{inc}}<30$ MeV, the comparison with double-folding model calculations and the CH89 potential, and the spin-orbit potential parameters are discussed.

Double folding model calculation applied to fusion reactions**Supported by National Natural Science Foundation of China (60572177)

The interaction potential between a spherical and a deformed nucleus is calculated within the double-folding model for deformed nuclei. We solve the double folding potential numerically by using the truncated multipole expansion method. The shape, separation and orientation dependence of the interaction potential, fusion cross section and barrier distribution of the system 16O+154Sm are investigated by considering the quadrupole and hexadecapole deformations of 154Sm. It is shown that the height and the position of the barrier depend strongly on the deformation and the orientation angles of the deformed nucleus. These are quite important quantities for heavy-ion fusion reactions, and hence produce great effects on the fusion cross section and barrier distribution.

Double folding nucleus-nucleus potential applied to heavy-ion fusion reactions

Double folding model calculations to obtain the bare nucleus-nucleus potential have been carried out with the Reid and Paris M3Y effective nucleon-nucleon (NN) interactions. The exchange part of the interaction was taken to be of finite range and the density dependence of the NN interaction was accounted for. The calculated fusion barrier energies are generally lower than those expected taking into account potential renormalization due to coupling to collective states at high excitation energies. Fitting the potentials at the barrier radii with a Woods-Saxon form results in effective potential diffuseness of $\ensuremath{\sim}0.65--0.70\phantom{\rule{0.3em}{0ex}}\text{fm}$, smaller than the values of $\ensuremath{\sim}1\phantom{\rule{0.3em}{0ex}}\text{fm}$ generally found from fitting fusion cross sections at above-barrier energies. These discrepancies raise questions about both the determination of the bare nucleus-nucleus potential with the folding model, and the boundary of the effect of friction on the fusion process.

Surface diffuseness of nuclear potential from heavy-ion fusion reactions

Fits to high precision fusion cross sections at above-barrier energies require values of ≈1 fm for the diffuseness parameter of the Woods-Saxon potential. This is large compared with the values deduced from elastic scattering. Double folding model calculations of the bare nucleus-nucleus potential have been carried out to predict the fusion barrier energies and nuclear potential diffusenesses. Both the calculated barrier energies, and equivalent potential diffusenesses at the barrier raddi, are generally lower than those found experimentally. These discrepancies raise questions about both the determination of the bare nucleus-nucleus potential with the folding model, and the boundary of the effects of friction on the fusion process.

Optical model and coupled-channels analysis ofLi7+Si28andLi7+Ca40scattering

Angular distributions for elastic scattering and for excitation of the 1.78 MeV state in $^{28}\mathrm{Si}$ and for elastic scattering and excitation of the 3.73 and 3.90 states in $^{40}\mathrm{Ca}$ by $^{7}\mathrm{Li}$ scattering at a bombarding energy of 45 MeV have been measured and analyzed. Double folding model calculations using a realistic effective nucleon-nucleon interaction similar to that performed for $^{9}\mathrm{Be}$ + $^{28}\mathrm{Si}$ and $^{9}\mathrm{Be}$ + $^{40}\mathrm{Ca}$ scattering have been carried out for the elastic angular distributions and the real potential must be renormalized to yield agreement with the measured cross sections. Coupled channels calculations using a Woods-Saxon potential were performed in an effort to describe the inelastic angular distributions. The extracted deformation parameters are in reasonable agreement with those obtained from light and heavier ion scattering from the same target nuclei. The effect of strong excitation of the 0.48 MeV state in $^{7}\mathrm{Li}$ and of mutual excitation of target and projectile is considered in a qualitative manner.NUCLEAR REACTIONS $^{28}\mathrm{Si}$($^{7}\mathrm{Li}$, $^{7}\mathrm{Li}$)$^{28}\mathrm{Si}$, $^{28}\mathrm{Si}$($^{7}\mathrm{Li}$, $^{7}\mathrm{Li}$*)$^{28}\mathrm{Si}$* $Q=\ensuremath{-}0.48, \ensuremath{-}1.78, \mathrm{and} \ensuremath{-}2.26$ MeV, $E(^{7}\mathrm{Li})=45$ MeV; $^{40}\mathrm{Ca}$ ($^{7}\mathrm{Li}$, $^{7}\mathrm{Li}$)$^{40}\mathrm{Ca}$, $^{40}\mathrm{Ca}$($^{7}\mathrm{Li}$, $^{7}\mathrm{Li}$*)$^{40}\mathrm{Ca}$* $Q=\ensuremath{-}0.48, \ensuremath{-}3.73, \mathrm{and} \ensuremath{-}3.90$ MeV, $E(^{7}\mathrm{Li})=45$ MeV; measured $\ensuremath{\sigma}(\ensuremath{\theta})$, ${\ensuremath{\theta}}_{\mathrm{lab}}=10\ensuremath{-}70\ifmmode^\circ\else\textdegree\fi{}$;; performed optical model calculations using Woods-Saxon potential and double folding procedure, deduced optical model parameters; performed coupled-channels calculations, deduced deformation parameters.