Density-dependent Effective Interaction Research Articles

Structure of 9Be from proton scattering at 180 MeV.

Cross section and analyzing power measurements for the scattering of 180 MeV protons are used to investigate the structure of $^{9}\mathrm{Be}$. Data were collected for 24 states below 21 MeV of excitation. Detailed line-shape analysis was used to isolate several broad states. Most notably, the dependence of apparent peak position upon momentum transfer was used to separate the strong resonance listed as 6.76 MeV in standard compilations into two contributions identified as the 7/${2}^{\mathrm{\ensuremath{-}}}$ member of the ground-state rotational band, located at 6.38 MeV, and the 9/${2}^{+}$ weak-coupling state, located at 6.76 MeV. Calculations of proton scattering were made using a density-dependent effective interaction in the local density approximation. The quadrupole contribution to elastic scattering was included in distorted wave Born approximation and found to have an important effect upon the analyzing power. For states dominated by a single multipolarity, neutron transition densities were fitted to (p,p') data and compared with the corresponding proton transition densities fitted to (e,e') data. We find that excitation of the rotational states and the 9/${2}^{+}$ state are essentially isoscalar. Shell model calculations were performed in the full 0\ensuremath{\Elzxh}\ensuremath{\omega} and 1\ensuremath{\Elzxh}\ensuremath{\omega} model spaces. Suggested assignments for most states are made by comparisons of shell model calculations with data for both (p,p') and (e,e'). Similar calculations for analog states observed with the $^{9}\mathrm{Be}$(p,n${)}^{9}$B reaction at 135 MeV support the proposed assignments. Finally, several relatively narrow states are observed between 18 and 21 MeV that are candidates for positive-parity states with T=3/2.

A simple calculation of the heavy-ion optical potential using a finite-range density-dependent Brink-Boeker effective interaction

Analytical results of the energy density functionals are obtained by using a finite-range density-dependent Brink-Boeker effective interaction to calculate the energy dependent heavy-ion optical potential in the energy density model (EDM) and folding model (FM) with similar computational benefit as achieved by short-range interaction. The resulting potentials and elastic scattering cross sections for the system 40Ca+40Ca (ECM=120 MeV) are compared with those obtained previously from the short-range Skyrme interaction.

The 16O( [formula omitted]) reaction at 35 MeV

The 16O( p, p′ ) reaction was studied with a 35 MeV polarized proton beam. The cross sections and analyzing powers for excited states in 16O up to and including the 2 −, T = 1 state at 12.97 MeV were measured and compared with DWBA calculations. Both density-independent and densitydependent effective interactions were used in the calculations. Transition densities calculated using shell-model amplitudes and Woods-Saxon single-particle wave functions give poor accounts of the data. The use of macroscopic transition densities obtained from electron scattering data give a much better description of the data, especially when used in combination with the densitydependent interaction.

Thermostatic properties of finite and infinite nuclear systems

From a constructed finite-range momentum- and density-dependent effective interaction, we arrive at the equation of state for infinite nuclear matter. This interaction reproduces ground-state properties of finite nuclear systems; in addition it gives a proper energy dependence of the single-particle potential. For symmetric and asymmetric hot nuclear matter, we find the critical and phase-separation temperature, the specific heats, incompressibility and variations of effective mass with temperature and density. For finite nuclei, we also find the limiting temperature, i.e. the maximum temperature that such nuclear systems can sustain.

12C+12C elastic scattering analysis above E/A=6 MeV using deep real potentials.

We present phenomenological and folding optical-model descriptions of {sup 12}C+{sup 12}C elastic scattering between 6 and 120 MeV/nucleon. The potentials have a deep real part, a weakly absorbing imaginary part, and are energy dependent. The folding-model real potential is calculated from the energy- and density-dependent effective interaction DDM3Y. It requires renormalizations of about 1.1 below 10 MeV/nucleon, and close to unity above. The phenomenological real potential is about 300 MeV at the center for the lower energies, slowly reducing to 120 MeV for the highest energies. The discrete ambiguities for the real part found in the analysis at energies below 10 MeV/nucleon are resolved by demanding consistency with the higher energy results. The volume integrals for the folding-model and phenomenological potentials agree and display a smooth and slow energy dependence that is consistent with simple dispersion-relation predictions. The values for the total reaction cross sections show the transparency effect expected from microscopic calculations. A possible extension of this description towards lower energies is discussed.

Momentum distribution of nuclei in a one-dimensional nuclear model

The momentum distributions of finite nuclei are calculated using Monte Carlo techniques for a one-dimensional model having a two-body potential with a repulsive core and a long-range attraction. The results obtained in mean-field using a density-dependent effective interaction are compared to the exact solution.

Medium effects in deuteron elastic scattering

Nuclear medium modifications to the deuteron-nucleus folded potential are discussed using a density-dependent effective interaction derived from Brueckner-Hartree-Fock calculations of the optical model potential in nuclear matter. The medium effects generate central and TR type tensor potentials which are found to improve the agreement with differential cross section data for the 208Pb(d, d)208Pb reaction at 56 MeV.

Some average nuclear properties with an effective interaction

Nuclear matter properties — binding energy (BE), compressibility, single particle potential, re-arrangement energy and effective mass — have been studied in first order with a density dependent effective interaction. Second order contribution to BE, single particle potential, and effective mass have been estimated. Ground state energy of4He and8Be and their r.m.s. radii have also been studied.

Quadrupole scattering of 135 MeV protons by 9Be.

We have measured cross sections and analyzing powers for $^{9}\mathrm{Be}$(p,p') at ${E}_{p}$=135 MeV, observing the ${(3/2}^{\mathrm{\ensuremath{-}}}$ and ${(5/2}^{\mathrm{\ensuremath{-}}}$ members of the ground-state rotational band. Folding model calculations were performed using a density-dependent effective interaction in the local density approximation. The quadrupole contribution to elastic scattering was evaluated in distorted wave approximation. We find the elastic quadrupole contribution to be important for q>1.5 ${\mathrm{fm}}^{\mathrm{\ensuremath{-}}1}$, in good agreement with data. The effect on the elastic analyzing power data is quite large.

Information on three-body interactions from inversion of the energy equations

The inversion of the three energy equations, i.e. for the nuclear total energy, the sum of occupied single-particle state energies and the saturation condition, using the experimental data in 16O and 40Ca, is carried out to determine whether three-body effective interactions are necessary in addition to density independent and dependent two-body interactions. In order to fit the data both in a non-relativistic and a relativistic framework, the three-body interaction energy is found to be large and repulsive. We also show that density-dependent two-body effective interactions, which are another requisite in the non-relativistic potential theory, are not necessarily needed in the relativistic mean field framework but allow to increase the effective nucleon mass.

Further studies of density-dependent interactions for the excitation of collective states

The density-dependent DDM3Y effective nucleon-nucleon interaction was used in a double-folding model with macroscopic transition densities to explore further density-dependent effects on the inelastic scattering of α-particles and protons. In particular, we investigate whether transition densities with a node near the nuclear surface, such as occurs for the giant monopole (breathing mode) resonance, are particularly sensitive to density dependence in the interaction. We also study the imposition of a consistency condition on the use of density dependence for inelastic scattering. In both situations, the effects are small (≲20%) for α-particles but large for protons. The consistency condition is discussed further in an appendix.

Intrinsic radial sensitivity of nucleon inelastic scattering

A linear expansion analysis of the folding model transition amplitude is used to study the intrinsic sensitivity of the inelastic scattering of intermediate energy nucleons to the radial form of the neutron transition density, given known proton transition densities from electron scattering. Realistic density-dependent effective interactions are used to construct pseudodata. These pseudodata are then reanalyzed and the error matrix is used to calculate an error band for the radial transition density. This approach reveals the sensitivity of the extracted transition density to absorption, medium modifications of the interaction, and the extent and quality of the data in a manner that is largely free of the residual inaccuracies in reaction theory that complicate the analysis of real data. We find that the intrinsic radial sensitivity of nucleon inelastic scattering is best for projectile energies between 200 and 500 MeV, but is adequate to resolve the radial dependence of neutron transition densities even in the interior of heavy nuclei throughout the energy regime 100--800 MeV. We have also compared our method with scale-factor analyses which assume proportionality between neutron and proton densities. For states whose transition densities are similar in the surface, we find scaling to be accurate at the 20% level. However, for light nuclei substantial deviations beyond the first peak of the differential cross section reveal sensitivity to shape differences. This sensitivity is reduced for heavy nuclei. The model dependence of radial densities is also studied. A high-q constraint is used to analyze the contribution of incompleteness error to the deduced error bands and to reduce the model dependence.

Microscopic Derivation of the Nucleus-Nucleus Potential by the Use of the Density-Dependent Effective Interaction

A method to construct the microscopic nucleu!?·nucleus potential by the use of the density· dependent effective nucleon·nucleon interaction is presented based on the resonating group method in a semiclassical manner. The method is applied to the two:alpha system, where it is shown that the potential is obtained to be very attractive due to the antisymmetrization effect between the two alpha nuclei though the direct potential is even repulsive. Through the experimental and theoretical efforts, the study of the interaction between light composite nuclei is now in the stage of highly quantitative discussion. Phenomenological analyses of the elastic scattering of alpha nucleus from some target nuclei provide us with the unambiguous optical model potentials which are deep and which become slightly shallower as the bombarding energy gets higher. I) Further­ more the elastic scattering of alpha nucleus from the closed shell nuclei has been shown to be successfully described by the resonating group method (RGM) by incor­ porating phenomenological imaginary potential. 2 ) The equivalent local potentials for these RGM non-local potentials are very much similar to the corresponding phenomenological optical model potentials. It is now possible to investigate, as the origin of the energy-dependence of the nucleus-nucleus potential, the properties of the effective nucleon-nucleon interaction in the scattering process separating the effect of the antisymmetrization between the proJectile and target nuclei. As for the effective NN interaction in the scattering process, the nucleon-nucleus optical potential has come to be explained quantitatively starting from the G-matrix theory with local density approximation. The effective NN interaction thus obtained depends on the incident energy and the Fermi momentum of surrounding nucleons, and is complex reflecting the particle-hole excitation process during the propagation through the nuclear matter. In this paper we will develop the framework to derive the nucleus-nucleus potential by the use of the realistic, density-dependent, nucleon­ nucleon interaction. It is based on the extension of the method to construct the microscopic local nucleus-nucleus potential from the RGM non-local potential. In the construction of the equivalent local potential due to Refs. 3) and 4), what is necessary is the classical Hamiltonian of the nucleus-nucleus relative motion, heR, P), which is defined by

Density dependent effective interactions in double-folding model analyses of elastic α-particle scattering

Different forms for the density dependence of two-body effective interactions are empirically studied in double-folding model analyses of elastic α-particle scattering at E α = 104 MeV and 140 MeV. The main aspects of the calculations, which are based on the M3Y interaction, are the analytical parametrisation of the density dependence and the appropriate consideration of the target and projectile density in the overlap region. For the latter purpose different relative weights of the densities were introduced, which also simulate density gradient effects. Besides the need for a separate treatment of the target and projectile densities in particular the different relative weights are found to be important. With the new modifications suggested in this study consistent results are obtained for the various parametrisations and these results are in agreement with previous implicit folding interpretations of nucleon scattering data.

Microscopic distorted-wave approximation study of low-energy nucleon scattering from 89Y.

New differential cross section data for inelastic neutron scattering to the first three excited states of $^{89}\mathrm{Y}$ at ${E}_{\mathrm{n}=11}$ MeV are studied using a microscopic folding model and three energy- and density-dependent effective interactions. Results are also presented for the corresponding transitions in inelastic proton scattering at 14.7, 24.5, and 61.2 MeV. Transition densities were obtained from a combination of available inelastic electron-scattering data and theoretical considerations. The calculated angular distributions provide a reasonable description of the experimental data for the predominantly quadrupole (\ensuremath{\Delta}J=2), ${(1/2}^{\mathrm{\ensuremath{-}}}$\ensuremath{\rightarrow}${(3/2}^{\mathrm{\ensuremath{-}}}$ (${E}_{x}$=1.509 MeV) and ${(1/2}^{\mathrm{\ensuremath{-}}}$\ensuremath{\rightarrow}${(5/2}^{\mathrm{\ensuremath{-}}}$ (${E}_{x}$=1.745 MeV) transitions in the target. It is shown that the neutron scattering data for these two transitions are sensitive to the shape differences in the transition densities suggested by theory and electron scattering. The theoretical results for the predominantly \ensuremath{\Delta}J=5, ${(1/2}^{\mathrm{\ensuremath{-}}}$\ensuremath{\rightarrow}${(9/2}^{+}$ (${E}_{x}$=0.909 MeV) transition significantly underestimate the proton and neutron scattering cross sections at ${E}_{\mathrm{p}<25}$ MeV and ${E}_{\mathrm{n}=11}$ MeV, but provide a reasonable description of the proton scattering data for this transition at ${E}_{\mathrm{p}=61.2}$ MeV.

Microscopic analysis of antiproton-nucleus elastic scattering

Abstract Elastic antiproton-nucleus scattering is analysed with a microscopic optical model potential obtained by convoluting an effective interaction with target densities. Local, energy- and density-dependent complex effective interactions are constructed from reaction matrices generated with the N N Dover-Richard and Paris potentials. Effects of nuclear medium correction on the final results are examined. Comparison is made between the calculated results and the experimental data from LEAR which include the first polarization data as well as differential cross sections.

And Single-Particle Potentials with Density-Dependent Effective Interactions

and Single-Particle Potentials with Density-Dependent Effective Interactions

CHARGE DENSITY DIFFERENCES AND THE NUCLEAR INCOMPRESSIBILITY

The influence of the nuclear matter compression modulus K∞ on the difference in the charge density distribution of 208 Pb −206 Pb is investigated in a self-consistent mean field approach with a variety of density dependent effective interactions of the generalized Skyrme type. It is found that in addition to a dependence on the compressibility associated with a given Skyrme force the results also depend strongly on the pairing correlations which, through the level density, are influenced by the nuclear matter effective mass.

Collision terms and effective interactions

Abstract It is shown that the physics of collision terms, presently of widening application in connection with kinetic extensions of mean field descriptions, largely coincides with the physics behind the construction of density-dependent effective interactions and mean fields in the context of static nuclear structure problems. The use of collision terms to extend mean field descriptions based on dynamical extrapolations of the static density-dependent mean field becomes therefore a delicate matter.

Neutron inelastic scattering from 54,56Fe.

Differential cross sections for inelastic scattering of neutrons to numerous levels in $^{54}$,56Fe have been measured at incident neutron energies of 11 and 26 MeV. The data have been analyzed in terms of a collective model using a distorted-wave Born approximation as well as a full coupled-channels treatment. Analysis has also been performed with a microscopic folding model using an energy- and density-dependent effective interaction. Both collective and microscopic analyses have also been applied to inelastic proton scattering in the same energy region in order to extract information on the isospin nature of the measured excitations.