Phenomenological Imaginary Potential Research Articles

TFF (v.4.1): A Mathematica Notebook for the Calculation of One- and Two-Neutron Stripping and Pick-Up Nuclear Reactions

The program TFF calculates stripping single-particle form factors for one-neutron transfer in prior representation with appropriate perturbative treatment of recoil. Coupled equations are then integrated along a semiclassical trajectory to obtain one- and two-neutron transfer amplitudes and probabilities within first- and second-order perturbation theory. Total and differential cross-sections are then calculated by folding with a transmission function (obtained from a phenomenological imaginary absorption potential). The program description, user instructions and examples are discussed.

P.3.009 - New rodent model of social exclusion in depression: role of the subgenual anterior cingulate cortex and anterior insula

We present a detailed analysis of the 6He elastic scattering on different target nuclei over a wide energy range within the framework of the double folding model using a phenomenological imaginary potential. We show the microscopic real potential generated by No-core Shell model density distributions provides good agreement in explaining the elastic scattering angular distributions. In this study we also present a new formulation depending on the incident energy and atomic charge of the target nucleus for the phenomenological imaginary potential and investigate the threshold anomaly in halo nuclei by analyzing the 6He + 208Pb system.

Threshold behavior of interaction potential for the system 7Li + 64Ni: Comparison with 6Li + 64Ni

The elastic scattering angular distributions for the system 7Li + 64Ni were measured in the bombarding energy range of 12 MeV≤Elab≤26.4 MeV. A phenomenological optical model analysis was performed for the measured data. The strengths of the fitted potential components at the surface were estimated to extract their variation with energy. Further analyses of the measured angular distributions were performed with a hybrid potential composed of a renormalized folded real and a phenomenological imaginary potential. Both the model potentials predict similar energy dependent behavior for the effective interaction potential around the barrier. Unlike the heavy targets, 7Li + 64Ni does not show a normal threshold behavior. It also does not clearly exhibit a behavior similar to 6Li + 64Ni. The real potential for 7Li + 64Ni does not exhibit any significant energy dependence and the imaginary potential strength remains almost independent of energy above the Coulomb barrier (∼14 MeV). However, at energies below the barrier, a sudden drop in the imaginary potential strength is observed.

Three-body recombination of two-component cold atomic gases into deep dimers in an optical model

We consider three-body recombination into deep dimers in a mass-imbalanced two-component atomic gas. We use an optical model where a phenomenological imaginary potential is added to the lowest adiabatic hyper-spherical potential. The consequent imaginary part of the energy eigenvalue corresponds to the decay rate or recombination probability of the three-body system. The method is formulated in details and the relevant qualitative features are discussed as functions of scattering lengths and masses. We use zero-range model in analyses of recent recombination data. The dominating scattering length is usually related to the non-equal two-body systems. We account for temperature smearing which tends to wipe out the higher-lying Efimov peaks. The range and the strength of the imaginary potential determine positions and shapes of the Efimov peaks as well as the absolute value of the recombination rate. The Efimov scaling between recombination peaks is calculated and shown to depend on both scattering lengths. Recombination is predicted to be largest for heavy–heavy–light systems. Universal properties of the optical parameters are indicated. We compare to available experiments and find in general very satisfactory agreement.

Folding model analysis of the 6He elastic scattering using No-core Shell model: Indication of the breakup threshold anomaly

Abstract We present a detailed analysis of the 6 He elastic scattering on different target nuclei over a wide energy range within the framework of the double folding model using a phenomenological imaginary potential. We show the microscopic real potential generated by No-core Shell model density distributions provides good agreement in explaining the elastic scattering angular distributions. In this study we also present a new formulation depending on the incident energy and atomic charge of the target nucleus for the phenomenological imaginary potential and investigate the threshold anomaly in halo nuclei by analyzing the 6 He + 208 Pb system.

Phenomenological and Semi-microscopic Analysis for the Elastic Scattering of Protons from <sup>12</sup>C Nuclei at Different Energies

Analysis of the elastic scattering of protons from 12C nuclei had been performed within the framework of both the optical model and single folding model at different proton energies; 17, 30.3, 40, 49.48 and 61.4 MeV. We have obtained the global potential parameters which could fairly reproduce the experimental data for p+12C elastic scattering at the aforementioned energies. The radial and energy dependence of the real and imaginary parts of the potential were calculated. Good agreement between experimental data and theoretical predictions in the whole angular range was obtained using both phenomenological approach (Optical Model), and semi-microscopic approach (Single Folding). In single folding calculations, the real part of the potential was calculated from a more fundamental basis by the folding method in which the NN interaction VNN(r), is folded into the density of the target nuclei and supplemented with a phenomenological imaginary potential. The obtained normalization factor Nr is in the range of 0.75 - 0.9.

A study on the Fresnel diffraction of 6He by means of different microscopic density distributions

The elastic scattering of the halo nucleus 6He from heavy targets such as 197Au and 208Pb has been investigated in order to explain the Coulomb rainbow peak due to the Fresnel-type diffraction observed in the experimental data. In order to examine the role of nuclear potential to describe 6He + 197Au and 6He + 208Pb systems, we have used the no-core shell model, few-body and Gaussian-shaped density distributions at various energies. The microscopic real parts of the complex nuclear potential have been obtained by using the double-folding model for each of the density distribution and the phenomenological imaginary potentials have been taken as the standard Woods-Saxon shape. We have observed that fewbody and Gaussian-shaped density distributions have given standard Fresnel-type diffraction results, a classical scattering pattern with Coulomb rainbow peak whereas the nuclear potential obtained by using the no-core shell-model density distribution has provided the reduction at Fresnel peak and has given more consistent results with the experimental data.

Coupled-channels analysis of K^+ scattering from ^6Li

Differential cross sections for the elastic and inelastic scattering of K^+ at 715 MeV/c from ^6Li are analysed by applying the Watanabe superposition model to the real potential with a phenomenological imaginary potential. The optical potential of ^6Li is calculated in terms of alpha-particle and deuteron or triton and helion optical potentials. The calculated cross sections are consistent with the triton-helion model.

Microscopic few-body and Gaussian-shaped density distributions for the analysis of the 6He exotic nucleus with different target nuclei

The elastic scattering angular distributions of 6He projectile on different medium and heavy mass target nuclei including 12C, 27Al, 58Ni, 64Zn, 65Cu, 197Au, 208Pb and 209Bi have been examined by using the few-body and Gaussian-shaped density distributions at various energies. The microscopic real parts of the complex nuclear optical potential have been obtained by using the double-folding model for each of the density distributions and the phenomenological imaginary potentials have been taken as the Woods–Saxon type. Comparative results of the few-body and Gaussian-shaped density distributions together with the experimental data are presented within the framework of the optical model.

Simplified Resonating-Group Method for Light-Ion Scattering

A simplified version of the resonating-group method, called model K, is discussed. In this model, the simplifications made are to approximate recoil effects and to take only the direct and the knockon-exchange terms into explicit consideration. It is a microscopic model for nucleus + nucleus scattering which works well in situations where the scattering energy is relatively high, and medium and heavy-weight target nuclei are involved. Applications of this model to s-shell-ion scattering show that, with the addition of a phenomenological imaginary potential into the formulation, good fits to experimental data can be achieved with only minor parameter adjustments. The properties of the internuclear local potential, equivalent to the model-K nonlocal interaction through the WKB procedure, have also been studied. Here it is found that the energy dependence of the empirically determined local potential can be largely accounted for by a proper consideration of exchange effects inherent in calculations employing totally antisymmetric wave functions.

Antisymmetry and channel coupling contributions to the absorption for [formula omitted

To understand recently established empirical p+ α potentials, RGM calculations followed by inversion are made to study contributions of the d+ 3He reaction channels and deuteron distortion effects to the p+ α potential. An equivalent study of the d+ 3He potential is also presented. The contributions of exchange non-locality to the absorption are simulated by including an phenomenological imaginary potential in the RGM. These effects alone strongly influence the shape of the imaginary potentials for both p+ α and d+ 3He. The potentials local-equivalent to the fully antisymmetrised-coupled channels calculations have a significant parity-dependence in both real and imaginary components, which for p+ α is qualitatively similar to that found empirically. The effects on the potentials of the further inclusion of deuteron distortion are also presented. The inclusion of a spin-orbit term in the RGM, adds additional terms to the phase-equivalent potential, most notably the comparatively large imaginary spin-orbit term found empirically.

Scattering of deuterons by nuclei with breakup effects

The deuteron direct-breakup reaction is explicitly incorporated into the d + nucleus formulation of model K which is a simplified version of the microscopic resonating-group method. By employing the same nucleon-nucleon potential as that used in the corresponding nucleon + nucleus system and by adopting an imaginary potential with a single adjustable parameter to crudely represent other reactions not explicitly included in the formulation, it is shown that a good agreement between calculated and experimental differential cross sections can be obtained for deuteron scattering by a wide variety of target nuclei, ranging from 28Si to 208Pb. In addition, the results also indicate that the deuteron breakup reaction is a significant, but not dominant, reaction in the d + nucleus system. To make model K into a truly microscopic model without any phenomenological imaginary potential, one must further explicitly include the (d, p) and (d, n) one-nucleon-transfer or stripping reactions into the model-K formulation.

Scattering of deuterons by nuclei with breakup effects

The deuteron direct-breakup reaction is explicitly incorporated into the d + nucleus formulation of model K which is a simplified version of the microscopic resonating-group method. By employing the same nucleon-nucleon potential as that used in the corresponding nucleon + nucleus system and by adopting an imaginary potential with a single adjustable parameter to crudely represent other reactions not explicitly included in the formulation, it is shown that a good agreement between calculated and experimental differential cross sections can be obtained for deuteron scattering by a wide variety of target nuclei, ranging from 28Si to 208Pb. In addition, the results also indicate that the deuteron breakup reaction is a significant, but not dominant, reaction in the d + nucleus system. To make model K into a truly microscopic model without any phenomenological imaginary potential, one must further explicitly include the (d, p) and (d, n) one-nucleon-transfer or stripping reactions into the model-K formulation.

Multiconfiguration resonating-group study of the six-nucleon system with cluster-rearrangement and pseudo-inelastic configurations.

The six-nucleon system is studied with a multiconfiguration resonating-group calculation which consists of the dominant configuration {ital d}+{alpha}, the cluster-rearrangement configurations {ital p}+{sup 5}He and {ital n}+{sup 5}Li, and the pseudo-inelastic configuration {ital d}{sup *}+{alpha}. The result shows that, because the deuteron cluster is easily distortable, a {ital d}+{alpha} single-configuration study is inadequate. With the addition of the other cluster configurations, the {sup 6}Li ground-state energy is improved by a large amount equal to 1.76 MeV. From an overall viewpoint, the {ital p}+{sup 5}He and {ital n}+{sup 5}Li configurations are found to contribute more significantly than the {ital d}{sup *}+{alpha} configuration, thus confirming the findings from previous seven- and eight-nucleon investigations that single-nucleon transfer processes generally make important contributions and sequential-decay processes are more important than direct-breakup processes in the relatively low-energy region. The {ital d}+{alpha} total reaction cross sections have also been computed and found to agree rather well with empirical results. At 10 MeV, for example, the calculated total reaction cross section is equal to 85% of the empirical value, which is the highest percentage obtained in our six- to eight-nucleon microscopic resonating-group calculations containing no phenomenological imaginary potentials.

Resonating-group-method study of alpha +40Ca elastic scattering and 44Ti structure.

The \ensuremath{\alpha}${+}^{40}$Ca elastic scattering cross sections in the energy range from 29.0 to 61.0 MeV are shown to be well reproduced by the resonating-group method by introducing a phenomenological imaginary potential. The local potential equivalent to the nonlocal potential of the resonating-group method is found to be very close to the optical potential of Delbar et al. which fits the scattering data well in a wide energy range. Since the optical potential of Delbar et al. is the so-called unique optical potential which is free from the discrete ambiguity, this result means that the resonating-group method is powerful and reliable for the study of the internucleus interaction. By calculating the bound and quasibound level spectra by the same \ensuremath{\alpha}${+}^{40}$Ca resonating-group method, it is found that the lowest positive parity rotational band is located near the observed ground rotational band of $^{44}\mathrm{Ti}$. In addition, when we use other effective two-nucleon forces so as to locate the calculated lowest ${0}^{+}$ state near the observed excited ${0}^{+}$ state with the excitation energy 8.54 or 11.2 MeV, which has large \ensuremath{\alpha} strength, the fitting of the elastic scattering data by the resonating-group method is found to be very bad. These results force us to regard that the structure of the $^{44}\mathrm{Ti}$ ground band as containing a large amount of \ensuremath{\alpha}${+}^{40}$Ca clustering component. However our present resonating-group method predicts the appearance of the negative parity rotational band with its bandhead ${1}^{\mathrm{\ensuremath{-}}}$ state below the excitation energy 10 MeV although there have been reported no such experimental indications at all of this.

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

The nuclear response and the imaginary potential for nucleus-nucleus collisions

The Fermi-gas model is used in this paper to study the nucleus-nucleus collision. The field produced by one of the nuclei is considered to act on nucleons in the other nucleus, which is treated as a Fermi gas of radius R. The imaginary part of the (non-local) nucleus-nucleus potential is then computed by evaluating the energy-conserving second-order term in which the intermediate states are particle-hole excitations produced in the Fermi gas. The equivalent local potential, obtained by using the Perey-Saxon method, is compared with phenomenological imaginary potentials. Later it is shown that, in the limit of small range of non-locality, the imaginary potential can be related to the nuclear response function. With this, one can write the nuclear friction coefficient that is used in phenomenological analyses of heavy-ion collisions in terms of the imaginary potential.

Elastic scattering of protons and neutrons on 40Ca by the density-dependent Hartree-Fock field

We use the self-consistent density-dependent Hartree-Fock field as the real part of the optical model potential. Introducing a phenomenological imaginary potential, we describe elastic scattering of protons and neutrons on 40Ca at several incident energies. Results show that the density-dependent Hartree-Fock field, including the rearrangement potential, well reproduces differential cross sections and polarizations simultaneously. This calculation explicitly shows a unified way to understand the ground-state properties and the scattering problem. Detailed study is given the properties of the non-local Hartree-Fock field, via the WKB equivalent local potential.

Folding model analysis of α-particle elastic scattering with a semirealistic density-dependent effective interaction

Differential cross sections for the elastic scattering of α-particles from 40Ca, 46,48,50Ti, 58Ni, 90Zr and 208Pb at 140 MeV and from 58,60,62,64Ni at 172 MeV are analysed using a double-folding model with a semirealistic density-dependent effective interaction based on the M3Y interaction for the real potential and various phenomenological imaginary potentials. Very good fits with consistent parameters of the model have been obtained. The application of the folding model analyses of α-particle elastic scattering to the determination of the nuclear matter density distribution is critically examined.

On the energy dependence of the imaginary optical-model potential for proton scattering from 40Ca

The energy dependence of the imaginary potentials for proton elastic scattering from 40Ca has been derived on the basis of the Feshbach formalism by taking into account the coupling with the inelastic and pick-up channels to the elastic channel. It is found that the calculated potential describes quite well the experimental differential and absorption cross sections for incident energies between 10 MeV and 50 MeV. However, at high energies above 50 MeV, the calculated imaginary potentials seem to be inadequate. We assume a Woods-Saxon form for the real part of the potential, the parameters of which are in good agreement with phenomenological values, if finite-range calculations are made for the p-d-p process. The local potentials equivalent to the non-local potentials are also calculated and are compared with the phenomenological imaginary potentials.