Nucleus-nucleus Optical Potential Research Articles

Excitation of isobaric analog states from (p,n) and (He3,t) charge-exchange reactions within the G -matrix folding method

Differential cross sections of ($p,n$) and ($^3$He,$t$) charge-exchange reactions leading to the excitation of the isobaric analog state (IAS) of the target nucleus are calculated with the distorted wave Born approximation. The $G$-matrix double-folding method is employed to determine the nucleus-nucleus optical potential within the framework of the Lane model. $G$-matrices are obtained from a Brueckner-Hartree-Fock calculation using the Argonne Av18 nucleon-nucleon potential. Target densities have been taken from Skyrme-Hartree-Fock calculations which predict values for the neutron skin thickness of heavy nuclei compatible with current existing data. Calculations are compared with experimental data of the reactions ($p,n$)IAS on $^{14}$C at $E_{lab}=135$ MeV and $^{48}$Ca at $E_{lab}=134$ MeV and $E_{lab}=160$ MeV, and ($^3$He,$t$)IAS on $^{58}$Ni, $^{90}$Zr and $^{208}$Pb at $E_{lab}=420$ MeV. Experimental results are well described without the necessity of any rescaling of the strength of the optical potential. A clear improvement in the description of the differential cross sections for the ($^3$He,$t$)IAS reactions on $^{58}$Ni and $^{90}$Zr targets is found when the neutron excess density is used to determine the transition densities. Our results show that the density and isospin dependences of the $G$-matrices play a non-negligible role in the description of the experimental data.

A study of 12c +12c nuclear reaction using a new M3Y-type effective interaction

This paper is a study of nuclear reactions involving 12C + 12C nuclei carried out with a heavy-ion nucleus-nucleus optical potential derived from a new M3Y-type effective interaction, called B3Y-Fetal, within the framework of optical model at the incident energies of 112, 126.7, 240, 300, 1016 MeV. Folding analyses of the differential cross sections associated with the elastic scattering of the nuclear system, determined at these incident energies with four B3Y-Fetal-based folded potentials constructed from double folding model, have shown the DDB3Y1- and BDB3Y1-Fetal potentials to be the best in excellent agreement with previous work done with the M3Y-Reid. The agreement of the B3Y-Fetal with the famous M3Y-Reid effective interaction, which is also used for folding analysis in this work, is further buttressed and well-established by the findings of this study Herein, the values of the renormalization factor, NR ranging from 1.1117 to 0.8121, obtained with the B3Y-Fetal have been found to be slightly higher, with lower reaction cross sections, aR = 1418 - 1047 millibarns, than NR = 0.9971 - 0.8108 obtained with the M3Y-Reid effective interaction whose accompanying reaction cross sections, being higher, range from 1431 to 1050 millibarns. This depicts the B3Y-Fetal as having a better performance. Additionally, results of folding analyses have shown the best-fit folded potentials, DDB3Y1- and BDB3Y1-Fetal potentials to be in agreement at all incident energies, implying that the cold nuclear matter has an underlying soft equation of state.

Double-Folding Nucleus-Nucleus Optical Potential: Parallel MPI and OpenMP Implementations

The computation of the real part of the nucleus-nucleus optical potential based on the microscopic double-folding model was implemented within both the MPI and OpenMP parallelising techniques. Test calculations of the total cross section of the 6He + 28Si scattering at the energy 50 A MeV show that both techniques provide significant comparable speedup of the calculations.

Elastic transfer and parity dependence of the nucleus-nucleus optical potential

Background: A recent coupled reaction channel (CRC) study shows that the enhanced oscillation of the elastic $^{16}$O+$^{12}$C cross section at backward angles is due mainly to the elastic $\alpha$ transfer or the core exchange. Such a process gives rise to a parity-dependent term in the total elastic $S$-matrix, an indication of the parity dependence of the $^{16}$O+$^{12}$C optical potential (OP). Purpose: To explicitly determine the core exchange potential (CEP) induced by the symmetric exchange of the two $^{12}$C cores in the elastic $^{16}$O+$^{12}$C scattering at $E_{\rm lab}= 132$ and 300 MeV, and explore its parity dependence. Method: $S$-matrix generated by CRC description of the elastic $^{16}$O+$^{12}$C scattering is used as the input for the inversion calculation to obtain the effective local OP that contains both the Wigner and Majorana terms. Results: The high-precision inversion results show a strong contribution by the complex Majorana term in the total OP of the $^{16}$O+$^{12}$C system, and thus provide for the first time a direct estimation of the parity-dependent CEP. Conclusions: The elastic $\alpha$ transfer or exchange of the two $^{12}$C cores in the $^{16}$O+$^{12}$C system gives rise to a complex parity dependence of the total OP. This should be a general feature of the OP for the light heavy-ion systems that contain two identical cores.

Effect of the nucleon-nucleon interaction on the fusion cross-section within the relativistic mean field formalism

We establish a relationship between the nucleon-nucleon interaction potential and the nuclear fusion reaction cross-sections at low energies. The axially deformed self-consistent relativistic mean field is used with the non-linear NL3* interaction parameter set. The Wong formula is used to estimate the fusion cross-section for 58Ni + 58Ni and 48Ca + 238U systems, which are known to display fusion hindrance phenomena. The results of the application of the nucleus-nucleus optical potential for the fusion cross-section from the recently developed effective relativistic NN-interaction (R3Y) potential is compared with the well-known, phenomenological M3Y potential. The results obtained from our present calculation for the R3Y interaction are reasonable good as compared to the M3Y potential concerning the available experimental at barrier energies. The present analysis pursues a full microscopic studies of fusion process at low energies by taking the R3Y potential along with the relativistic mean field density instead of taking the M3Y interaction within the double folding approach.

Investigation of the 6Li + 40Ca elastic scattering using different folding models

In the present study we investigate the 6Li + 40Ca elastic scattering through the energy range 26-240MeV in the framework of the optical model and $ \alpha$ -cluster structure of the colliding nuclei. The double folding (DF) calculations for the real central part of the nuclear optical potential are performed by folding the $ \alpha$ -$ \alpha$ and $ \alpha$ -n effective interactions over the density distributions of $ \alpha$ -clusters in the target ( 40Ca nucleus and considering the $ \alpha$ -d structure of the projectile ( 6Li nucleus. The imaginary part of the optical potential is expressed in the phenomenological Woods-Saxon form. The measured angular distributions of the elastic scattering differential cross section have been successfully reproduced using the derived semi-microscopic potentials for nine sets of data all over the measured angular range. However, it is found that introducing a real renormalization factor smaller than unity is essential in order to obtain a successful description of the data at bombarding energies larger than 10MeV/nucleon. The obtained results confirm the validity of the $ \alpha$ -cluster structure to generate a realistic representation of nucleus-nucleus optical potentials. For the sake of comparison, the same considered sets of data are reanalyzed using microscopic DF optical potentials based upon the Sao Paulo potential. The energy dependence of the corresponding reaction cross sections and real and imaginary volume integrals of the considered reaction are also investigated.

Fusion cross section for Ni-based reactions within the relativistic mean-field formalism

In this theoretical study, we establish an interrelationship between the nucleon-nucleon interaction potential and the nuclear fusion reaction cross-sections at low energies. The axially deformed self-consistent relativistic mean field with non-linear NL3$^*$ force is used to calculate the density distribution of the projectile and target nuclei for fusion. The Wong formula is used to estimate the fusion cross-section and barrier distribution from the nucleus-nucleus optical potential for Ni-based systems, which are known for fusion hindrance phenomena. The results of the application of the so obtained nucleus-nucleus optical potential for the fusion cross-section from the recently developed relativistic $NN-$interaction (R3Y) are compared with the well-known, phenomenological M3Y effective $NN$ potential. We found a relatively good results from R3Y interactions below the barrier energies as compare to the M3Y potential concerning the experimental data. We also observe the density dependence on the nuclear interaction potential in terms of nucleon-nucleon optical potentials.

From ab initio structure predictions to reaction calculations via EFT

We study how EFT can improve the description of reactions used to study the structure of exotic nuclei. Using Halo EFT helps constraining the potential that simulates the interaction between the projectile constituents. It enables us to include ab initio results within the reaction model and to clearly identify the nuclear properties probed by the reaction. New local chiral EFT nucleon-nucleon interactions can also be used to derive nucleus-nucleus optical potentials from first principles, which can provide reliable inputs in reaction modelling.

Nuclear mean field and double-folding model of the nucleus-nucleus optical potential

Realistic density dependent CDM3Yn versions of the M3Y interaction have been used in an extended Hartree-Fock (HF) calculation of nuclear matter (NM), with the nucleon single-particle potential determined from the total NM energy based on the Hugenholtz-van Hove theorem that gives rise naturally to a rearrangement term (RT). Using the RT of the single-nucleon potential obtained exactly at different NM densities, the density- and energy dependence of the CDM3Yn interactions was modified to account properly for both the RT and observed energy dependence of the nucleon optical potential. Based on a local density approximation, the double-folding model of the nucleus-nucleus optical potential has been extended to take into account consistently the rearrangement effect and energy dependence of the nuclear mean-field potential, using the modified CDM3Yn interactions. The extended double-folding model was applied to study the elastic $^{12}$C+$^{12}$C and $^{16}$O+$^{12}$C scattering at the refractive energies, where the Airy structure of the nuclear rainbow has been well established. The RT was found to affect significantly the real nucleus-nucleus optical potential at small internuclear distances, giving the potential strength close to that implied by the realistic optical model description of the Airy oscillation.

Inelastic scattering of heavy ions at intermediate energies with excitations of nuclear collective states

Excitation of low-lying nuclear collective states upon scattering of heavy ions with energies of several tens of MeV/nucleon has been studied. The interaction potential leading to excitation is chosen in the form of a derivative of the microscopic (or semimicroscopic) nucleus-nucleus double-folding optical potential. Elastic and inelastic scattering cross sections have been calculated within the high-energy approximation; the inelastic scattering amplitude was obtained in the first order in the deformation parameter. The cross sections are compared with the experimental data on scattering of 17O from a series of nuclei with excitation of the 2+ level.

Microscopic optical potentials in 6Li scattering by nuclei

A parameter-independent microscopic optical potential of nucleus-nucleus interaction is presented by a folding model with the isospin dependent complex nucleon-nuclear potential, which is calculated in the framework of the Dirac-Bruecker-Hartree-Fock approach. Investigations on 6Li scattering by 12C, 28Si, 40Ca, 58Ni, 90Zr, and 208Pb over a wide range of incident energy and scattering angle with the microscopic nucleus-nucleus optical potential is presented. To take account of the breakup effect of 6Li and the high order dynamic effect in the reaction a modification factor NR in the real part and an enhancing factor NI in the imaginary part of the microscopic optical potential are introduced. We take the imaginary part enhancing factor NI=3.0, which has been obtained in the previous study on 6He scattering by 12C. The modification factor NR is found to be almost constant with respect to the incident energy and target mass number. The calculations with NR≈0.5—0.6 and NI=3.0 well reproduce the experimental elastic scattering data for all targets and incident energies investigated. Our parameter-independent model should be of value in the description of the nucleus-nucleus scattering of many-body systems, especially unstable nucleus-nucleus systems.

Analysis of elastic scattering ofO16+Si28andC12+Mg24by a new optical potential

We construct a new phenomenological nucleus-nucleus optical potential based on the interesting potential developed by Ginocchio that has the versatility to control the surface and volume regions of the potential. Using this potential with suitable energy dependence of some of the parameters, we are able to fit remarkably well experimental results of the differential scattering cross section for the $^{16}\mathrm{O}+^{28}\mathrm{Si}$ system in the center-of-mass energy ${E}_{c.m.}$ range from 18.67 to 90.681 MeV and the excitation function in the ${E}_{c.m.}$ range from 13.0 to 52.0 MeV. We also fit equally well the exhaustive experimental differential scattering data available for the $^{12}\mathrm{C}+^{24}\mathrm{Mg}$ system in the ${E}_{c.m.}$ range from 10.67 to 16.0 MeV. The new optical potential used has significantly fewer parameters. The results are interpreted in terms of wave mechanical aspects leading to superposition of partial waves undergoing different phase shifts generated by an optical potential characterized by volume and surface parts.

Flux loss distributions due to breakup in the elastic scattering of 56 MeV deuterons from51V

The loss of flux from the elastic channel due to breakup is calculated in a model case (56 MeV deuterons on ${}^{51}\mathrm{V})$ where the breakup contribution to the imaginary part of the nucleus-nucleus optical potential has been explicitly determined. We find that most of the flux is lost in a cap-shaped region near the nuclear surface while a smaller part is lost at a refractive focus.

Nuclear incompressibility and density dependentNNinteractions in the folding model for nucleus-nucleus potentials

A generalized version of density dependence has been introduced into the M3Y effective nucleon-nucleon $(\mathrm{NN})$ interaction that was based on the $G$-matrix elements of the Paris $\mathrm{NN}$ potential. The density dependent parameters have been chosen to reproduce the saturation binding energy and density of normal nuclear matter within a Hartree-Fock scheme, but with various values for the corresponding nuclear incompressibility $K$ ranging from 176 to 270 MeV. We use these new density dependent interactions in the folding model to calculate the real parts of $\ensuremath{\alpha}$-nucleus and nucleus-nucleus optical potentials for those systems where strongly refractive scattering patterns have been observed. These provide some information on the potentials at short distances, where there is a strong overlap of the projectile and target density distributions, and hence where the density dependence of the interaction plays an important role. We try to infer, from careful optical model (OM) analyses, the sensitivity of the scattering data to different $K$ values. Results obtained for elastic $\ensuremath{\alpha}$ scattering on targets ranging from ${}^{12}$C to ${}^{208}$Pb allow us to determine unambiguously that the $K$ value favored in this approach is within the range of 240 to 270 MeV. Similar OM analyses have also been done on measurements of the elastic scattering of ${}^{12}$C+${}^{12}$C, ${}^{16}$O+${}^{12}$C, and ${}^{16}$O+${}^{16}$O at incident energies up to 94 MeV/nucleon. These data were found to be much less sensitive over such a narrow range of $K$ values. This lack of sensitivity is due mainly to the smaller maximum overlap density which occurs for these systems, compared to that which is formed in an $\ensuremath{\alpha}$-nucleus collision. This makes the effects of density dependence less substantial. Another reason is that a small difference between two folded heavy ion potentials can often be compensated for, in part, by a small overall renormalization of one of them. This renormalization is often allowed in optical model analyses, and interpreted, for example, as accounting for a contribution from a higher-order dynamic polarization potential. In an attempt to avoid this ambiguity, some OM analyses of the extensive and accurate data for ${}^{16}$O+${}^{16}$O scattering were done using the unrenormalized folded potentials, together with the explicit addition of a correction term, expressed in terms of cubic splines. This correction term can be interpreted as representing a contribution to the real potential from the dynamic polarization potential. The results of such a ``folding+spline'' analysis suggest a tendency to favor the same $K$ value range that was found in the OM analyses of $\ensuremath{\alpha}$-nucleus scattering.

On the accuracy of calculating the exchange part of the real heavy ion potential

In the present work, we test the validity of replacing the non-diagonal density matrix appearing in the exchange part of the nucleus-nucleus optical potential by an approximation based on the density matrix expansion used frequently in the nuclear structure calculations. We have found that for the recently derived BDM3Y nucleon-nucleon force the DME may produce a maximum error more than 20% in the nucleus-nucleus potential.

Recent Results on Elastic and Inelastic Scattering

In this review article, which corresponds to lectures given by one of us (N.A) at the third Euroschool on Exotic Beams held in Leuven in September 1995, we present experimental results and theoretical developments in heavy-ion elastic and inelastic scattering and Giant Resonance excitation. The paper contains a short review of the field with special emphasis on more recent results and problems. We start by recalling the theoretical situation concerning the description of nucleon-nucleus elastic scattering. We show that in the framework of the local density approximation, complex potentials derived from fundamental effective nucleon-nucleon interactions, describe successfully the data. However, the main part of the discussion on elastic scattering, is dedicated to the description of intermediate energy heavy-ion elastic scattering. We present different folding models for the calculation of the real part of the nucleus-nucleus optical potential, M3Y, DDM3Y, .... The theoretical predictions are compared to experimental data mainly obtained at GANIL (20-100 Mev per nucleon). We show that a new density dependent interaction which reproduces the equilibrium density and the binding energy of normal nuclear matter, leads also to a satisfactory description of heavy-ion elastic scattering angular distributions. This interaction reproduces also the density and energy dependence of the nucleon optical potential. We present a new simple effective interaction with a real and imaginary part for peripheral heavy-ion collisions at intermediate energies. Finally the effect of the isospin and spin terms of the effective nucleon-nucleon interaction on the nucleus-nucleus folded potentials is discussed. We introduce the deformed optical model potential which is the most frequently used model, to obtain inelastic scattering transition potentials. However the most direct approach to obtain transition potentials is from the folding of the transition densities with an effective nucleon-nucleon interaction and the ground state density of the nucleus which is not excited. We show that the predictions of the optical and folding model are very different, especially for transitions dominated by nuclear excitation. The difference between the cross sections estimated within the deformed optical model and the folding model increases with multipolarity. Following the theoretical work of R. Satchler, we recommend the use of a folding model to extract deformation lengths and multipole moments from inelastic scattering measurements. We present the state of the field concerning Electric Giant Resonances and multiphonon excitations. We introduce the different sum rules which can be found usually in the literature and we show the link between them. The excitation of Giant Resonances with intermediate energy or high energy heavy ions, measured at GANIL or GSI and the technical problems met in the analysis of these experiments are discussed. Recent results concerning the two-phonon excitation of the Giant Quadrupole and Dipole mode are presented. Concerning the breathing mode, macroscopic and microscopic prescriptions introduced to access the compressibility of the nuclear matter are discussed. We show, in the light of theoretical arguments developed recently by J.P. Blaizot and collaborators, that microscopic calculations remain the most reliable tool for the determination of the nuclear matter compression modulus from the energy of the monopole vibration of nuclei.

A systematic study of nuclear properties with Skyrme forces

The author presents a systematic study of nuclear properties with Skyrme forces. The properties studied include: (i) nuclear ground state properties with Hartree-Fock (HF) and self-consistent semiclassical approaches; (ii) properties of giant resonances with a random phase approximation (RPA) and sum rule approach; (iii) microscopic nucleon-nucleus optical potentials and related quantities with the mass operator method; and (iv) nucleus-nucleus optical potentials and fusion barriers with the energy density formalism.

Skyrme Forces and Their Applications in Low Energy Nuclear Physics11Dedicated to professor XU Gong-ou on the occasion of his 70-th birthday. The project supported by National Natural Science Foundation of China through grant No. 18905002.

This paper reviews the present status of Skyrme forces and their applications in the field of low energy nuclear physics as an effective nucleon-nucleon interaction. Their applications in the following five domains are presented: 1). Hartree-Fock (HF), selfconsistent semiclassical (SCSC) calculations and nuclear ground state properties;2). random phase approximation (RPA), sum rule approach and properties of nuclear giant resonances; 3). calculations of microscopic nucleon-nucleus optical potentials and related quantities; 4). calculations of nucleus-nucleus optical potentials and fusion barriers; 5). multifragmentation, liquid-gas phase transition and instability of hot/compressed nuclei.

Optical potential and the fusion barrier of two hot nuclei.

The finite-temperature self-consistent semiclassical calculation is carried out to determine the nucleon densities of some hot nuclei at thermal equilibrium with temperature T. These densities are applied, in the energy density formalism, to calculate the nucleus-nucleus optical potential and the corresponding fusion barrier between two hot nuclei. A moderate temperature dependence is found for the optical potential and fusion barrier.

Self-consistent Semiclassical Appoach to Fusion Barrier in Frozen-Density Appoximation11Projects supported by the National Natual Science Foundation of China

Fusion barriers of heavy-ion collisions are analysed in frozen-density approximation, with ground state density of each colliding nucleus determined by a self-consistent semiclassical (SCSC) calculation. Spin-orbit contribution to the nucleus-nucleus optical potential is included, while exchange effects on kinetic energy density and spin-orbit density are neglected. The calculated barriers are in agreement with experiments.