Equivalent Local Potential Research Articles

An exact treatment of localization of electromagnetic plus separable potential

Equivalent local potential with energy-momentum dependence is developed for the combined interaction of Hulthén modified Yamaguchi potential by developing its exact Jost solution. The generated local potentials are applied to compute scattering phase parameters for (p-p) and (p-d) systems through the phase function method (PFM). The same are also calculated for the nonlocal potential from the expression of the Fredholm determinant. Our obtained data for both the energy-momentum dependent local and the pure nonlocal interactions are in reasonable agreement with the standard data. Reasonable correspondence between the results for the phase equivalent and the nonlocal potentials indicate that our equivalent local analysis is in proper order.

The use of Laplace transform in the inverse problem from bound states

"In a previous work dedicated to non-relativistic quantum mechanics in the three-dimensional space with spherical symmetry, we have shown the possibility of reconstructing the ground state density from its moments in the case of bound states. The method is based on relationships connecting these moments to the lower energy of each state of angular momentum `. The ground state density yields the ground state wave function, which allows us to calculate the equivalent local potential by inverting the Schrodinger equation. This calculus uses the Fourier transform of the ground state density ρ(r). The aim of this work is to check how to improve the results by replacing the Fourier transform by the Laplace transform of the quantity r2 ρ(r). The method is first presented. Then examples and results are compared to those obtained previously. "

Radial overlap correction to superallowed 0+→0+ nuclear β decays using the shell model with Hartree-Fock radial wave functions

The radial overlap correction ${\ensuremath{\delta}}_{\mathrm{RO}}$ is reexamined for 30 superallowed ${0}^{+}\ensuremath{\rightarrow}{0}^{+}$ nuclear $\ensuremath{\beta}$ decays using the shell model with Hartree-Fock (HF) radial wave functions. Our mean-field calculation is based on the effective Skyrme interaction including Coulomb, charge-symmetry-breaking (CSB), and charge-independence-breaking (CIB) terms. In addition, the electromagnetic corrections such as those due to the gradient of charge density, vacuum polarization, Coulomb spin orbit, and finite size of nucleons are also considered. In order to avoid the spurious isospin mixing, the local equivalent potential is constructed from the solution of a HF calculation for the $Z=N$ nucleus with charge-dependent forces neglected. Then the obtained mean field is solved noniteratively for the parent and daughter nuclei. It turns out that the CIB term has no significant impact on ${\ensuremath{\delta}}_{\mathrm{RO}}$ throughout the mass range between 10 and 74. On the other hand, the CSB term makes ${\ensuremath{\delta}}_{\mathrm{RO}}$ increase systematically between 10% and 30%. Similarly, the gradient density leads to a further increase of ${\ensuremath{\delta}}_{\mathrm{RO}}$ between 2% and 14%, while other estimated electromagnetic corrections are negligible. The effect of the suppression of isospin spuriosity is somewhat complicated. In general, it produces a significantly larger value of ${\ensuremath{\delta}}_{\mathrm{RO}}$; however, there are a few cases for which ${\ensuremath{\delta}}_{\mathrm{RO}}$ is mostly unaffected or even reduced, especially the even-even emitters in the light mass region. All these improvements partly explain the long-standing discrepancy between the correction values obtained with Woods-Saxon (WS) and HF radial wave functions. Nevertheless, the remaining discrepancy is still significant, except for the emitters with $A\ensuremath{\le}26$. In addition, the local odd-even staggering present in the results obtained with the phenomenological potential [J. C. Hardy and I. S. Towner, Phys. Rev. C 102, 045501 (2020); L. Xayavong and N. A. Smirnova, Phys. Rev. C 97, 024324 (2018)] is not well reproduced by our calculations. This subsequently leads to a large difference in the predicted mirror $ft$ ratios. These problems might relate to the errors of the calculated charge radii as well as the deformation and other correlation effects on the data of separation energies used for constraining the asymptotic radial wave functions.

Nucleon-nucleus velocity-dependent optical model: revisited

In this work we study nucleon-nucleus elastic scattering using a nonlocal, velocity-dependent optical potential. The potential parameters are determined by fitting elastic angular distributions and polarization data for nucleon scattering off a wide range of nuclei falling in the mass range 12 ⩽ A ⩽ 208 and over the energy range 10–60 MeV. Our potential parameters lead to smoothly varying local equivalent potentials and, unlike previous works, the potential depths corresponding to the real volume, imaginary surface, and imaginary volume terms show systematic linear dependences on energy. In addition, for each nuclear target, we determined constant sets of geometric parameters. Including the polarization data in the fitting procedure helped in reducing the large variations in the depths of the spin–orbit term. Our best-fit angular distributions and polarization data are in very good agreement with measured data, and are either as good as the cross sections obtained with widely-used systematics or better.

Halo effects in the Li11(p,t)Li9 reaction

I investigate the $^{11}\mathrm{Li}(p,t)^{9}\mathrm{Li}$ two-neutron transfer reaction at ${E}_{\mathrm{lab}}=33$ MeV within the distorted wave Born approximation (DWBA). The $^{11}\mathrm{Li}$ and $^{3}\mathrm{H}$ nuclei are described in the hyperspherical formalism, as $^{9}\mathrm{Li}\phantom{\rule{0.16em}{0ex}}+\phantom{\rule{0.16em}{0ex}}n+n$ and $p+n+n$ configurations, respectively. The calculation of the cross section is shown to be much more complicated than in one-neutron transfer reactions, owing to the three-body structure of $^{11}\mathrm{Li}$. The $^{11}\mathrm{Li}\phantom{\rule{0.16em}{0ex}}+\phantom{\rule{0.16em}{0ex}}p$ and $^{9}\mathrm{Li}\phantom{\rule{0.16em}{0ex}}+\phantom{\rule{0.16em}{0ex}}t$ scattering wave functions are determined with the continuum discretized coupled channel (CDCC) method. Approximations derived from equivalent local potentials are used to compute the transfer cross section at the DWBA. The model reproduces experimental data reasonably well, considering that there is no adjustable parameter. I show that the cross section is fairly sensitive to the long-range part of the $^{11}\mathrm{Li}$ wave function, and therefore to its halo structure.

Systematic calculations of reactions with exotic and stable nuclei to establish a unified theoretical approach

We report on systematical optical model (OM) and continuum discretized coupled channel (CDCC) calculations applied to describe the elastic scattering angular distributions of exotic and stable nuclei projectiles on heavy targets. Our optical potential (OP) is composed of the nuclear microscopic double folding S\~ao Paulo potential (SPP), derived from the nonlocal nature of the interaction, and the Coulomb dipole polarization (CDP) potential, derived from the semiclassical theory of Coulomb excitation. The OP is compared to the trivial equivalent local potential (TELP), extracted from CDCC calculations. The OM and CDCC predictions corroborate each other and account for important differences in the nuclei reaction mechanisms, which are directly related to their structural properties. Thus, OM and CDCC establish a common basis for analyzing or even predicting exotic and stable nuclei reactions.

Equivalent Local Potentials for the coupled channel system and the optical potential

The explicit formulae of the equivalent local potentials for a coupled channel problem are calculated. We prove that the equivalent local potential of the coupled channel system coincides with the equivalent local potential of the Feshbach optical potential.

Effective Interactions: Resonances and Off-Shell Characteristics

The influence of resonances on the analytical properties and off-shell characteristics of effective interactions has been investigated. This requires, among others, the knowledge of the Jost function in regions of physical interest on the complex kplane when the potentials are given in a tabular form. The latter are encountered in inverse scattering and supersymmetric transformations. To investigate the effects of resonances on the analytical properties of the potential, we employed the Marchenko inverse scattering method to construct, phase and bound state equivalent local potentials but with different resonance spectra. It is shown that the inclusion of resonances changes the shape, strength, and range of the potential which in turn would modify the bound and scattering wave functions in the interior region. This could have important consequences in calculations of transition amplitudes in nuclear reactions, which strongly depend on the behaviour of the wave functions at short distances. Finally, an exact method to obtain the Jost solutions and the Jost functions for a repulsive singular potential is presented. The effectiveness of the method is demonstrated using the Lennard-Jones (12,6) potential.

Assessment of nonlocal nuclear potentials in α decay

Different models for the nonlocal description of the nuclear interaction are compared through a study of their effects on the half-lives of radioactive nuclei decaying by the emission of alpha particles. The half-lives are evaluated by considering a preformed $\ensuremath{\alpha}$ particle ($^{4}\mathrm{He}$ nucleus), which tunnels through the Coulomb barrier generated by its interaction with the daughter nucleus. An effective potential obtained from a density-dependent double-folding strong potential between the $\ensuremath{\alpha}$ and the daughter nucleus within the nonlocal framework is found to decrease the half-lives as compared to those in the absence of nonlocalities. Whereas the percentage decrease within the older Perey-Buck and S\~ao Paulo models ranges between 20 to 40% for medium to heavy nuclei, a recently proposed effective potential leads to a decrease of only 2 to 4%. In view of these results, we provide a closer examination of the approximations used in deriving the local equivalent potentials and propose that, apart from the scattering data, the $\ensuremath{\alpha}$ decay half-lives can be used as a complementary tool for constraining the nonlocality models.

Complex potentials and the inverse problem

The occurrence of complex potentials with real eigenvalues has implications concerning the inverse problem, i.e. the determination of a potential from its spectrum. First, any complex potential with real eigenvalues has at least one equivalent local potential. Secondly, a real spectrum does not necessarily corresponds to a local real potential. A basic ambiguity arises from the possibility the spectrum to be generated by a complex potential. The purpose of this work is to discuss several aspects of this problem.

Complex potentials and the inverse problem

The occurrence of complex potentials with real eigenvalues has implications concerning the inverse problem, i.e. the determination of a potential from its spectrum. First, any complex potential with real eigenvalues has at least one equivalent local potential. Secondly, a real spectrum does not necessarily corresponds to a local real potential. A basic ambiguity arises from the possibility the spectrum to be generated by a complex potential. The purpose of this work is to discuss several aspects of this problem.

Potentials and phase shifts for nucleon–light nuclei systems

Two simple models based on the Coulomb-distorted phase function and supersymmetry-inspired factorisation methods are adapted to deal with the nucleon–light nuclei elastic scattering at low energies. The first one is associated with the derivation of a closed-form expression of the scattering phase shift for motion in Coulomb-distorted separable non-local potentials. The second one deals with the development of an energy-dependent phase equivalent local potential to the non-local one for s-wave and its subsequent generation of higher partial wave interactions through the formalism of supersymmetric quantum mechanics. The usefulness of our models is demonstrated through the computation of $$\upalpha $$ –nucleon scattering phase shifts at low energies up to partial waves $$\ell = 2$$ . Certain energy-dependent correction factors are also incorporated into energy-dependent higher partial wave potentials to achieve an excellent agreement with the standard data.

Construction of a local K¯N−πΣ−πΛ potential and composition of the Λ(1405)

A Kbar N-pi Sigma-pi Lambda coupled-channel potential is constructed on the basis of chiral SU(3) dynamics. Several matching conditions are introduced to formulate an equivalent local potential that reproduces the coupled-channel scattering amplitudes resulting from chiral SU(3)_L times SU(3)_R meson-baryon effective field theory. In contrast to a previously constructed effective single-channel Kbar N potential, the explicit treatment of the pi Sigma channel yields a natural description of the low-mass pole as part of the two-pole structure of the Lambda(1405) resonance. The energy dependence of the potential can now be parametrized with a minimum of polynomial orders. To study the properties of the Lambda(1405) as a quantum-mechanical quasibound state, we derive the normalization condition of its wave function generated by the energy-dependent coupled-channel potential, using the Feshbach projection method. This framework provides an improved understanding of this system from the viewpoint of the compositeness of hadrons. With the properly normalized wave function, we demonstrate and confirm that the high-mass pole of the Lambda(1405) is dominated by the Kbar N component.

NΩ interaction: Meson exchanges, inelastic channels, and quasibound state

Based on a baryon-baryon interaction model with meson exchanges, we investigate the origin of the strong attraction in the $N \Omega ( {}^{5}S_{2} )$ interaction, which was indicated by recent lattice QCD simulations. The long range part of the potential is constructed by the conventional mechanisms, the exchanges of the $\eta$ meson and of the correlated two mesons in the scalar-isoscalar channel, denoted by "$\sigma$" in the literature. The short range part is represented by the contact interaction. We find that the meson exchanges do not provide sufficient attraction. This means that most of the attraction is attributed to the short range contact interaction. We then evaluate the effect of the coupled channels to the $N \Omega ( {}^{5}S_{2} )$ interaction. We find that, while the $D$-wave mixing of the $N \Omega$ channel is negligible, the inelastic $\Lambda \Xi$, $\Sigma \Xi$, and $\Lambda \Xi (1530)$ channels via the $K$ meson exchange give the attraction of the $N \Omega ( {}^{5}S_{2} )$ interaction to the same level with the elastic meson exchanges. Although the elimination of these channels induces the energy dependence of the single-channel $N \Omega$ interaction, this effect is not significant. With the present model parameters fitted to reproduce the scattering length of the HAL QCD result of the nearly physical quark masses, we obtain the $N \Omega ( {}^{5}S_{2} )$ quasibound state with its eigenenergy $2611.3 - 0.7 i$ MeV, which corresponds to the binding energy $0.1$ MeV and width $1.5$ MeV for the decay to the $\Lambda \Xi$ and $\Sigma \Xi$ channels. From the analysis of the spatial structure and the compositeness, the quasibound state is shown to be the molecular state of $N\Omega$. We also construct an equivalent local potential for the $N \Omega ( {}^{5}S_{2} )$ system which is useful for various applications.

Kaon-nucleon scattering states and potentials in the Skyrme model

We study the (anti)kaon nucleon interaction in the Skyrme model. The kaon field is introduced as a fluctuation around the rotating Skyrmion for the nucleon. As an extension of our previous work, we study scattering states and examine phase shifts in various kaon-nucleon channels. Then we study the interaction, where we find that it consists of central and spin-orbit components for isospin channels, $I = 0, 1$, with energy dependence and nonlocality. The interaction is then fitted to a Shr\"odinger equivalent local potential for s- and p-waves.

Optical potential approach for positron scattering by metastable 23 S state of helium☆☆Project supported by the Natural Science Foundation of Heilongjiang Province of China (Grant No. 12541160).

The momentum space coupled channels optical (CCO) method for positron scattering has been extended to study the scattering of positrons by metastable helium for impact energies in the range from the positronium threshold up to high energies. Both the positronium formation and ionization continuum channels are included in the calculations via a complex equivalent local potential. The positronium formation, ionization, elastic and excitation, and total scattering cross sections are all presented and compared with the available information.

Exact calculation of a microscopic nucleon spin-orbit potential: Reexamination of Brieva-Rook localization

Brieva-Rook approximations for calculating the equivalent local microscopic nucleon-nucleus spin-orbit potential has been examined in the energy region of $65lEl500\phantom{\rule{0.16em}{0ex}}\mathrm{MeV}$. We present an exact method for calculating both the direct and the exchange parts of the spin-orbit potential. It has been shown that the series expansion used earlier overestimates both the real and the imaginary parts by about $50%$. Our results show that the exchange contribution to the total spin-orbit potential at low energies is only $15%$ (much smaller than earlier estimates) and becomes negligible around 200-MeV nucleon incident energy. Predictions of the new spin-orbit potential with minor scaling has been found to be in fair agreement with $p\text{\ensuremath{-}}^{208}\mathrm{Pb}$ scattering observables from 65 to 500 MeV.

Comparing a Local Potential with a Non-Local Potential for the Bound States of ^(16)O and ^(40)Ca

Using the non-local nuclear potentials, we calculate the binding energies and radial wave functions of a single particle, and the proton charge density for ^(16)O and ^(40)Ca nuclei. In order to obtain them, we use a matrix equation method generated from the integral form of the Schrodinger equation and find the best parameter set, which includes the potential depth, diffuseness, radius, and non-local parameter, by fitting the experimental data. The proton charge densities for the non-local potential are repulsive in comparison with those for the local potential. It is also found that the radius of the non-local potential is greater than that of the local one due to the non-local parameter (β). In addition, using the trivially equivalent local potential (TELP) method, we compare the local potentials with the non-local potentials and then find that the non-local potentials are repulsive.

Applying supersymmetry to energy dependent potentials

We investigate the supersymmetry properties of energy dependent potentials in the D=1 dimensional space. We show the main aspects of supersymmetry to be preserved, namely the factorization of the Hamiltonian, the connections between eigenvalues and wave functions of the partner Hamiltonians. Two methods are proposed. The first one requires the extension of the usual rules via the concept of local equivalent potential. In this case, the superpotential becomes depending on the state. The second method, applicable when the potential depends linearly on the energy, is similar to what has been already achieved by means of the Darboux transform.

Dynamic polarization potentials and dipole polarizabilities of11Li,6He and6Li compared

Elastic scattering data for ${}^{11}$Li, ${}^{6}$Li, and ${}^{6}$He $+$ ${}^{208}$Pb at incident energies of 29.8 MeV, 29 MeV, and 18 MeV, respectively, were analyzed by means of coupled discretized continuum channels (CDCC) calculations. Dynamic polarization potentials (DPPs) of the trivially equivalent local potential (TELP) type were derived from these calculations and compared. The dipole polarizability factor $\ensuremath{\alpha}$ for ${}^{11}$Li obtained by fitting a Coulomb polarization potential to the long-range part of the real DPP is consistent with the theoretical value of $5.7$ fm${}^{3}$. These results point to the dineutron model providing a very good description of ${}^{11}$Li breakup coupling effects. The much larger Coulomb dipole polarizability of ${}^{11}$Li suggests the persistence of large deviations from Rutherford scattering at sub-barrier energies for medium mass targets, a prediction confirmed by CDCC calculations.