Energy-dependent Potential Research Articles

Static versus energy-dependent nucleus-nucleus potential for description of sub-barrier fusion dynamics of reactions**Supported by Dr. D. S. Kothari Post-Doctoral Fellowship Scheme sponsored by University Grants Commission (UGC), New Delhi, India

The static and energy-dependent nucleus-nucleus potentials are simultaneously used along with the Wong formula for exploration of fusion dynamics of reactions. The role of internal structure degrees of freedom of colliding pairs, such as inelastic surface vibrations, are examined within the context of coupled channel calculations performed using the code CCFULL. Theoretical calculations based on the static Woods-Saxon potential along with the one-dimensional Wong formula fail to address the fusion data of reactions. Such discrepancies can be removed if one uses couplings to internal structure degrees of freedom of colliding nuclei. However, the energy-dependent Woods-Saxon potential model (EDWSP model) accurately describes the sub-barrier fusion enhancement of reactions. Therefore, in sub-barrier fusion dynamics, energy dependence in the nucleus-nucleus potential governs barrier modification effects in a closely similar way to that of the coupled channel approach.

Non-relativistic continuous states in arbitrary dimension for a ring-shaped pseudo-Coulomb and energy-dependent potentials

In this research, the non-relativistic particle scattering has been investigated for an alternative pseudo-Coulomb potential plus ring-shaped and an energy-dependent potentials in D-dimensional space. The normalized wave functions of continuous states on the k/2π scale are expressed in terms of the hyper-geometric series, and formula of phase shifts is presented. Analytical properties of the scattering amplitude and thermodynamics properties are discussed. Some of the numerical results of energy levels have been calculated too. Copyright © 2015 John Wiley & Sons, Ltd.

On integral and differential representations of Jordan chains and the confluent supersymmetry algorithm

We construct a relationship between integral and differential representation of second-order Jordan chains. Conditions to obtain regular potentials through the confluent supersymmetry algorithm when working with the differential representation are obtained using this relationship. Furthermore, it is used to find normalization constants of wave functions of quantum systems that feature energy-dependent potentials. Additionally, this relationship is used to express certain integrals involving functions that are solution of Schrödinger equations through derivatives.

Few-body calculations of η-nuclear quasibound states

We report on precise hyperspherical-basis calculations of ηNN and ηNNN quasibound states, using energy dependent ηN interaction potentials derived from coupled-channel models of the S11N⁎(1535) nucleon resonance. The ηN attraction generated in these models is too weak to generate a two-body bound state. No ηNN bound-state solution was found in our calculations in models where ReaηN≲1 fm, with aηN the ηN scattering length, covering thereby the majority of N⁎(1535) resonance models. A near-threshold ηNNN bound-state solution, with η separation energy of less than 1 MeV and width of about 15 MeV, was obtained in the 2005 Green–Wycech model where Re aηN≈1 fm. The role of handling self consistently the subthreshold ηN interaction is carefully studied.

Systematic failure of static nucleus–nucleus potential to explore sub-barrier fusion dynamics

This paper addresses the validity of the static Woods–Saxon potential and the energy dependent Woods-Saxon potential (EDWSP) for description of sub-barrier fusion dynamics. The low lying surface vibrations of colliding nuclei and neutron transfer channels are found to be major factors responsible for fusion enhancement at sub-barrier energies. Theoretical calculations based upon the static Woods–Saxon potential obtained using the one-dimensional Wong formula fail to explain the energy dependence of the sub-barrier fusion cross-section of systems. The role of inelastic surface vibrations is properly entertained within the context of coupled channel calculations performed using the CCFULL code. However, the EDWSP model, in conjunction with the one-dimensional Wong formula, accurately explains the sub-barrier fusion enhancement of systems and simulates the influence of nuclear structure degrees of freedom, such as the inelastic surface vibrational states of colliding pairs. In EDWSP model calculations, a wide range of diffuseness parameters, ranging from to which is much larger than a value extracted from the elastic scattering data, is required to bring the observed fusion enhancement.

On the Lipschitz stability of inverse nodal problem for p-Laplacian Schrödinger equation with energy dependent potential

In this study, we consider reconstruction and stability issues of an inverse nodal problem for a p-Laplacian Schrodinger equation with energy dependent potential. We solve Lipschitz stability of the inverse nodal problem for this p-Laplacian operator. Furthermore, we show that the space of all potential functions q is homeomorphic to the partition set of all asymptotically equivalent nodal sequences induced by an equivalence relation.

Microscopic positive-energy potential based on the Gogny interaction

We present a nucleon elastic scattering calculation based on Green's function formalism in the random-phase approximation. For the first time, the finite-range Gogny effective interaction is used consistently throughout the whole calculation to account for the complex, nonlocal, and energy-dependent optical potential. Effects of intermediate single-particle resonances are included and found to play a crucial role in the account for measured reaction cross sections. Double counting of the particle-hole second-order contribution is carefully addressed. The resulting integro-differential Schr\"odinger equation for the scattering process is solved without localization procedures. The method is applied to neutron and proton elastic scattering from $^{40}\mathrm{Ca}$. A successful account for differential and integral cross sections, including analyzing powers, is obtained for incident energies up to 30 MeV. Discrepancies at higher energies are related to a much-too-high volume integral of the real potential for large partial waves. This work opens the way to simultaneously assess effective interactions suitable for both nuclear structure and reactions.

Essential K̄ cluster "K−pp" studied with a coupled-channel Complex Scaling Method + Feshbach method

We have investigated an essential K̄ cluster "K−pp", which is the simplest system of kaonic nuclei, with a coupled-channel Complex Scaling Method (ccCSM). Combining the ccCSM with Feshbach method we can handle a coupled-channel problem effectively as a singlechannel problem. As a result of a study of the K−pp with the ccCSM+Feshbach method using an energy-dependent chiral-theory based potential, it is found that the K−pp is shallowly bound with the binding energy of around 20-35 MeV. The mesonic decay width is rather dependent on the interaction parameters and ansatz; the half decay width is ranging from 20 to 65 MeV.

20Ne+20Ne elastic scattering in the α +16O model of 20Ne

A folding potential for elastic 20Ne + 20Ne scattering is constructed based on the model of 20Ne nucleus. The elastic scattering angular distributions at energies of 62.1–74.5 MeV can be reasonably described. Linear expressions of energy-dependent of the renormalization factor N and the central imaginary depth W0 are obtained from the analyzes of the elastic angular distributions. The excitation function for the elastic 20Ne + 20Ne scattering at in the energy region of Ec.m. 30–38 MeV can be well reproduced by the linear energy-dependent potential.

Universal and nonuniversal effectiveN-body interactions for ultracold harmonically trapped few-atom systems

We derive the ground-state energy for a small number of ultracold atoms in an isotropic harmonic trap using effective quantum field theory (EFT). Atoms are assumed to interact through pairwise energy-independent and energy-dependent delta-function potentials with strengths proportional to the scattering length $a$ and effective range volume $V$, respectively. The calculations are performed systematically up to order $l^{-4}$, where $l$ denotes the harmonic oscillator length. The effective three-body interaction contains a logarithmic divergence in the cutoff energy, giving rise to a non-universal three-body interaction in the EFT. Our EFT results are confirmed by nonperturbative numerical calculations for a Hamiltonian with finite-range two-body Gaussian interactions. For this model Hamiltonian, we explicitly calculate the non-universal effective three-body contribution to the energy.

Faddeev calculations of theK¯NNsystem with a chirally motivatedK¯Ninteraction. II. TheK−ppquasibound state

New calculations of the quasi-bound state in the $K^- pp$ system using Faddeev-type equations in AGS form with coupled $\bar{K}NN$ and $\pi \Sigma N$ channels were performed. A chiral $\bar{K}N$ potential together with phenomenological models of $\bar{K}N$ interaction with one- and two-pole structure of the $\Lambda(1405)$ resonance were used. All three potentials reproduce experimental data on low-energy $K^- p$ scattering and kaonic hydrogen with the same level of accuracy. New method of calculating the subthreshold resonance position and width in a three-body system was proposed and used together with the direct search of the resonance pole. We obtained binding energy of the $K^-pp$ quasi-bound state $\sim 32$ MeV with the chirally motivated and $47 - 54$ MeV with the phenomenological $\bar{K}N$ potentials. The width is about $50$ MeV for the two-pole models of the interaction, while the one-pole potential gives $\sim 65$ MeV width. The question of using an energy dependent potential in few-body calculations is discussed in detail. It is shown that ``self-consistent'' variational calculations using such a potential are unable to produce a reasonable approximation to the exact result.

Elastic scattering ofF17,O17, andF19on a heavy target in a microscopic continuum discretized coupled-channels method

Background: In the traditional continuum discretized coupled-channels (CDCC) method, the clusters of the projectile are structureless. Exotic nuclei exhibit unusual properties and often show significant couplings to the continuum. Therefore, reaction models that consider a more accurate structure of the projectile are often preferable. A microscopic description of the projectile, based on an effective nucleon-nucleon (NN) interaction, in a microscopic CDCC (MCDCC) model [Descouvemont and Hussein, Phys. Rev. Lett. 111, 082701 (2013)] has been successfully applied to $^{7}\mathrm{Li}+^{208}\mathrm{Pb}$ scattering.Purpose: The MCDCC method is applied to low-energy elastic scattering of $^{17}\mathrm{F},^{17}\mathrm{O}$, and $^{19}\mathrm{F}$ on $^{58}\mathrm{Ni}$ and $^{208}\mathrm{Pb}$ targets. The goal of the calculations is twofold: to test the adequacy and the accuracy of the MCDCC model for heavier projectiles and to study the contribution of various channels to the elastic scattering cross sections.Methods: The elastic scattering cross sections are calculated by using the MCDCC method. The nucleon-target optical potential is folded with the projectile densities resulting from an effective NN interaction, which includes central nuclear, spin-orbit, and Coulomb terms. Discretization of the continuum is achieved via the pseudostate method. Coupled equations are solved by using the $R$-matrix method on a Lagrange mesh.Results: For the test case of $^{17}\mathrm{F}$ at 10 MeV/nucleon, the cross sections are weakly sensitive to the choice of the effective NN interaction. Three different energy-dependent optical nucleon-target potentials provide a similar reasonable agreement with data. Just below the Coulomb barrier, the MCDCC significantly underestimates the cross sections at larger angles. The coupling to the continuum is not significant in most of the assessed cases.Conclusions: The MCDCC is very satisfactory in the sense that it includes the microscopic properties of the projectile in a reaction model. Well above the Coulomb barrier, the cross sections are in a good agreement with the data. The reasons for the discrepancy between the data and the calculated cross sections at the lower energies, which is also observed in a traditional CDCC, are unclear.

Role of neutron transfer in the enhancement of sub-barrier fusion excitation functions of various systems using an energy-dependent Woods-Saxon potential

The focus of the present article is to address the effects of the role of a neutron transfer channel in the enhancement of sub-barrier fusion excitation function data of various heavy-ion systems by using an energy-dependent Woods-Saxon potential (EDWSP) model in conjunction with the one-dimensional Wong's formula and the coupled-channel model by using the code ccfull. The effects of coupling to neutron transfer channels are found to have dominance over the coupling to low-lying surface vibrational states, such as the $2{}^{+}$ and $3{}^{\ensuremath{-}}$ vibrational states. Furthermore, the influence of inelastic surface excitations and the effects of six neutron transfer channels with positive ground state $Q$ values are mocked up by the EDWSP model.

Net electron-phonon scattering rates in InN/GaN multiple quantum wells: The effects of an energy dependent acoustic deformation potential

The rates of charge carrier relaxation by phonon emission are of substantial importance in the field of hot carrier solar cell, primarily in investigation of mechanisms to slow down hot carrier cooling. In this work, energy and momentum resolved deformation potentials relevant to electron-phonon scattering are computed for wurtzite InN and GaN as well as an InN/GaN multiple quantum well (MQW) superlattice using ab-initio methods. These deformation potentials reveal important features such as discontinuities across the electronic bandgap of the materials and variations over tens of eV. The energy dependence of the deformation potential is found to be very similar for wurtzite nitrides despite differences between the In and Ga pseudopotentials and their corresponding electronic band structures. Charge carrier relaxation by this mechanism is expected to be minimal for electrons within a few eV of the conduction band edge. However, hole scattering at energies more accessible to excitation by solar radiation is possible between heavy and light hole states. Moderate reductions in overall scattering rates are observed in MQW relative to the bulk nitride materials.

Comparative study of the energy dependent and independent two-nucleon interactions — A supersymmetric approach

By exploiting supersymmetry inspired factorization method nucleon–nucleon (n–n) potentials, both energy dependent and independent, in the partial waves 1P1 and 3P1 are generated by judicious use of appropriate ground state wave functions and interactions. The energy independent Hulthen and energy dependent equivalent local Yamaguchi potentials and their corresponding S-wave functions are used as the starting point of our calculation. The scattering phase shifts are computed for the constructed potentials through Phase Function Method (PFM) and compared with the standard results to examine the merit of our approach to the problem.

Dirac equation in low dimensions: The factorization method

We present a general approach to solve the (1+1) and (2+1)-dimensional Dirac equations in the presence of static scalar, pseudoscalar and gauge potentials, for the case in which the potentials have the same functional form and thus the factorization method can be applied. We show that the presence of electric potentials in the Dirac equation leads to two Klein–Gordon equations including an energy-dependent potential. We then generalize the factorization method for the case of energy-dependent Hamiltonians. Additionally, the shape invariance is generalized for a specific class of energy-dependent Hamiltonians. We also present a condition for the absence of the Klein paradox (stability of the Dirac sea), showing how Dirac particles in low dimensions can be confined for a wide family of potentials.

Energy-dependent microscopic optical potential for scattering of nucleons on light nuclei

We present an energy-dependent microscopic optical model potential for elastic scattering of nucleons on light nuclei. The single-folding model is used for the real part of the optical potential (OP), while the imaginary part is derived within the high-energy approximation theory. The energy dependence of the OP is determined from the parameterization of the volume integrals those calculated from the best-fit OP that fit the experimental data of the cross sections and analyzing powers. This energy-dependent OP is successfully applied to analyze the proton elastic scattering of 4,6,i8He, 6,7Li, and 9,10Be nuclei at low and intermediate incident energies up to 200MeV/nucleon.

Spinless relativistic particle in energy-dependent potential and normalization of the wave function

Abstract The problem of normalization related to a Klein-Gordon particle subjected to vector plus scalar energy-dependent potentials is clarified in the context of the path integral approach. In addition the correction relating to the normalizing constant of wave functions is exactly determined. As examples, the energy dependent linear and Coulomb potentials are considered. The wave functions obtained via spectral decomposition, were found exactly normalized.

Optical model parallel description of elastic, fusion and breakup cross sections for systems with weakly bound projectiles

A brief description is presented of the results obtained in recent years for the simultaneous analysis of elastic and fusion cross section data of nuclear reactions for several nuclear systems with weakly bound and halo projectiles. The method used in this description, consists of simultaneously determine the parameters of fusion UF and direct reaction UDR polarization potentials of Woods-Saxon geometric shapes, that fit the elastic and fusion data. As a matter of fact, UFis an energy dependent potential, with real VF and imaginary WFcomponents, that is responsible for fusion reactions. Similarly, UDR is also energy dependent with real VDR and imaginary WDR parts, that accounts for direct reactions. A general finding for all the systems presented is that, the real and imaginary parts of the fusion potential and direct reaction potentials, are related by a dispersion relation and their energy dependence around and below the Coulomb barrier, show the so-called Breakup Threshold Anomaly. The effect of breakup reactions on fusion cross sections is studied by analyzing the separate effect of the absorption potential WDR and the fusion barrier rising produced by VDR.

Meson-baryon components in the states of the baryon decuplet

We apply an extension of the Weinberg compositeness condition on partial waves of L = 1 and resonant states to determine the weight of the meson-baryon component in the Delta(1232) resonance and the other members of the baryon decuplet. We obtain an appreciable weight of pi N in the Delta(1232) wave function, of the order of 60%, which looks more natural when one recalls that experiments on deep inelastic and Drell Yan give a fraction of pi N component of 34% for the nucleon. We also show that, as we go to higher energies in the members of the decuplet, the weights of the meson-baryon component decrease and they already show a dominant part for a genuine, non-meson-baryon, component in the wave function. We write a section to interpret the meaning of the Weinberg sum rule when it is extended to complex energies and another one for the case of an energy-dependent potential.