Generalized Woods Saxon Potential Research Articles

State dependence of tunneling processes and thermonuclear fusion

We discuss the sensitivity of tunneling processes to the initial preparation of the quantum state. We compare the case of Gaussian wave packets of different positional variances using a generalized Woods-Saxon potential for which analytical expressions of the tunneling coefficients are available. Using realistic parameters for barrier potentials we find that the usual plane wave approximation underestimates fusion reactivities by an order of magnitude in a range of temperatures of practical relevance for controlled energy production.

The Fractional Schrödinger Equation with the Generalized Woods-Saxon Potential

The bound state energy eigenvalues and the corresponding eigenfunctions of the generalized Woods-Saxon potential reported in [Phys. Rev. C, 72, 027001 (2005)] is extended to the fractional forms using the generalized fractional derivative and the fractional Nikiforov-Uvarov (NU) technique. Analytical solutions of bound states of the Schrodinger equation for the present potential are obtained in the terms of fractional Jacobi polynomials. It is demonstrated that the classical results are a special case of the present results at α=β=1. Therefore, the present results play important role in molecular chemistry and nuclear physics.

Bound state solution of the Schrodinger equation for the Woods–Saxon potential plus coulomb interaction by Nikiforov–Uvarov and supersymmetric quantum mechanics methods

The generalized Woods–Saxon potential plus repulsive Coulomb interaction is considered in this work. The supersymmetry quantum mechanics method is used to get the energy spectrum of Schrodinger equation and also the Nikiforov–Uvarov approach is employed to solve analytically the Schrodinger equation in the framework of quantum mechanics. The potentials with centrifugal term include both exponential and radial terms, hence, the Pekeris approximation is considered to approximate the radial terms. By using the step-by-step Nikiforov–Uvarov method, the energy eigenvalue and wave function are obtained analytically. After that, the spectrum of energy is obtained by the supersymmetry quantum mechanics method. The energy eigenvalues obtained from each method are the same. Then in special cases, the results are compared with former result and a full agreement is observed. In the [Formula: see text]-state, the standard Woods–Saxon potential has no bound state, but with Coulomb repulsive interaction, it may have bound state for zero angular momentum.

Damping the tail of the Woods–Saxon potential

Cutting of the tail of the Woods–Saxon (WS) potential influences the distribution of the poles of the [Formula: see text]-matrix considerably. A new attempt is made here to correct the tail of the cut-off WS potential in order to bring the positions of the poles closer to the WS potential without cut. New types of strictly finite range potentials are introduced in which the tail of the WS and the generalized Woods–Saxon (GWS) potentials are modified to damp them at their ends. This damping affects the energies of the bound and narrow resonant states weakly. The resonant pole positions in the new potentials were affected more significantly and the new pole positions became closer to that of the GWS potential without cut. However, the cut-off effect cannot be diminished completely. The new potentials belong to the [Formula: see text] function class.

Bound states of the D-dimensional Schrödinger equation for the generalized Woods–Saxon potential

The formal framework for quantum mechanics is an infinite number of dimensional space. Hereby, in any analytical calculation of the quantum system, the energy eigenvalues and corresponding wave functions can be represented easily in a finite-dimensional basis set. In this work, the approximate analytical solutions of the hyper-radial Schrödinger equation are obtained for the generalized Wood–Saxon potential by implementing the Pekeris approximation to surmount the centrifugal term. The energy eigenvalues and corresponding hyper-radial wave functions are derived for any angular momentum case by means of state-of-the-art Nikiforov–Uvarov and supersymmetric quantum mechanics methods. Hence, the same expressions are obtained for the energy eigenvalues, and the expression of hyper-radial wave functions transforming each other is shown owing to these methods. Furthermore, a finite number energy spectrum depending on the depths of the potential well [Formula: see text] and [Formula: see text], the radial [Formula: see text] and [Formula: see text] orbital quantum numbers and parameters [Formula: see text], [Formula: see text], [Formula: see text] are also identified in detail. Next, the bound state energies and corresponding normalized hyper-radial wave functions for the neutron system of the [Formula: see text]Fe nucleus are calculated in [Formula: see text] and [Formula: see text] as well as the energy spectrum expressions of other higher dimensions are revealed by using the energy spectrum of [Formula: see text] and [Formula: see text].

Using a quadrupole deformed generalized Woods–Saxon plus spin-orbit potential to describe the polarized interaction 7Li→ +12C at 34MeV

An overall description of the cross-sections and tensor analyzing powers corresponding to the elastic interaction [Formula: see text] at [Formula: see text] has been obtained with a coupled channel analysis using the rotational model. Based on the high electric quadrupole moment of the [Formula: see text] projectile, an optical model potential equal to the sum of a quadrupole deformed generalized Woods–Saxon potential and a standard Spin-Orbit term has been used. The deformed part of the potential induces reorientations and transitions that involve the first four energy states of the [Formula: see text] nucleus. These states are supposed to belong to an inverted [Formula: see text][Formula: see text][Formula: see text] rotational band proposed in the literature. Partial reproductions of the inelastic interaction [Formula: see text] have also been achieved by using the same deformed potential with the same parameter values that describe the elastic interaction. Comparisons are made with previous results obtained from optical model potential and continuum-discretized coupled-channel calculations using the cluster-folding model. The use of a deformed scattering potential introduces a tensor term that allows the reproduction of the tensor analyzing powers.

An investigation of the bound-state solutions of the Klein-Gordon equation for the generalized Woods-Saxon potential in spin symmetry and pseudo-spin symmetry limits

Recently, scattering of a Klein-Gordon particle in the presence of mixed scalar-vector generalized symmetric Woods-Saxon potential was investigated for the spin symmetric and the pseudo-spin symmetric limits in one spatial dimension. In this manuscript, the bound state solutions of the Klein-Gordon equation with mixed scalar-vector generalized symmetric Woods-Saxon potential are examined analytically within the framework of spin and pseudo-spin symmetry limits. We prove that the occurrence of bound state energy spectrum exists only in the spin symmetric limit, while in the pseudo-spin symmetric limit, the bound state spectrum does not exist. Besides the theoretical proof, the Newton-Raphson numerical methods are used to calculate the bound state energy spectra of a neutral Kaon particle, confined in a generalized symmetric Woods-Saxon potential, energy well constituted with repulsive or attractive surface interactions, for the spin and pseudo-spin symmetric limits, respectively. Numerical results are consistent with the non-existence of the bound state energy spectrum in the pseudo-spin symmetric limit.

Analytical solutions for the radial Scarf II potential

The real Scarf II potential is discussed as a radial problem. This potential has been studied extensively as a one-dimensional problem, and now these results are used to construct its bound and resonance solutions for l=0 by setting the origin at some arbitrary value of the coordinate. The solutions with appropriate boundary conditions are composed as the linear combination of the two independent solutions of the Schrödinger equation. The asymptotic expression of these solutions is used to construct the S0(k)s-wave S-matrix, the poles of which supply the k values corresponding to the bound, resonance and anti-bound solutions. The location of the discrete energy eigenvalues is analyzed, and the relation of the solutions of the radial and one-dimensional Scarf II potentials is discussed. It is shown that the generalized Woods–Saxon potential can be generated from the Rosen–Morse II potential in the same way as the radial Scarf II potential is obtained from its one-dimensional correspondent. Based on this analogy, possible applications are also pointed out.

A study of shell model neutron states in 207,209Pb using the generalized Woods–Saxon plus spin-orbit potential

The experimental binding energies of single-particle and single-hole neutron states belonging to neutron shells that extend from [Formula: see text] = 126–184 and 82–126, respectively, have been reproduced by solving the Schrödinger equation with a potential that has two components: the generalized Woods–Saxon (GWS) potential and the spin-orbit (SO) coupling term. The GWS potential contains the traditional WS potential plus a term (SU) whose intensity reaches a maximum in the nuclear surface. Our results indicate the existence of an explicit relationship between the strength of the SU potential and the orbital angular momentum quantum number [Formula: see text] of the state. This dependence has been used to make reasonable predictions for the excitation energy centroids of states located inside and outside the neutron shells investigated. Comparisons are made with results reported in previous investigations.

Exact Analytical Solution of the Klein–Gordon Equation in the Generalized Woods–Saxon Potential

The exact analytical solution of the Klein–Gordon equation for the spin-0 particles in the generalized Woods–Saxon potential is presented. The bound state energy eigenvalues and corresponding wave functions are obtained in the closed forms. The correlations between the potential parameters and energy eigenvalues are examined for π0 particles.

Corrected analytical solution of the generalized Woods–Saxon potential for arbitrary states

The bound state solution of the radial Schrödinger equation with the generalized Woods–Saxon potential is carefully examined using the Pekeris approximation for arbitrary states. The energy eigenvalues and the corresponding eigenfunctions are analytically obtained for different n and quantum numbers. The closed forms obtained are applied to calculate the single particle energy levels of a neutron orbiting around 56Fe nucleus in order to check the consistency between the analytical and the Gamow code results. The analytical results are in good agreement with the results obtained using Gamow code for .

Bound states of the Dirac equation for the generalized Woods–Saxon potential in pseudospin and spin symmetry limits

Bound state solutions of the Dirac equation for the generalized Woods–Saxon potential are examined for arbitrary κ states by using the approximation to the Coulomb and centrifugal potentials in pseudospin symmetry (PSS) and spin symmetry (SS) limits, respectively. The energy eigenvalues and corresponding eigenfunctions are obtained in closed forms. Some numerical examples are given for proton or anti-proton in a nucleus. The correlations between the energy splitting and some parameters of the effective potential in PSS limit are examined for several pseudospin doublets.

Generalized Nuclear Woods-Saxon Potential under Relativistic Spin Symmetry Limit

By using the Pekeris approximation, we present solutions of the Dirac equation with the generalized Woods-Saxon potential with arbitrary spin-orbit coupling number under spin symmetry limit. We obtain energy eigenvalues and corresponding eigenfunctions in closed forms. Some numerical results are given too.

Modified Rosen-Morse potential-energy model for diatomic molecules

By employing the dissociation energy and the equilibrium bond length for a diatomic molecule as explicit parameters, we generate an improved expression for the generalized Woods-Saxon potential. It is exactly shown that the generalized Woods-Saxon potential and the well-known Rosen-Morse potential are the same empirical potential-energy function for diatomic molecules. Based on the measure of inner-shell radii of two atoms, we propose a modified Rosen-Morse potential-energy model. Evaluation of the average deviations from the experimental data is carried out on six molecules. The modified Rosen-Morse potential is found to be more accurate than the Morse and Rosen-Morse potentials in fitting experimental data for the six molecules examined.

Approximate analytical solutions of the generalized Woods-Saxon potentials including the spin-orbit coupling term and spin symmetry

Abstract We study the approximate analytical solutions of the Dirac equation for the generalized Woods-Saxon potential with the pseudo-centrifugal term. We apply the Nikiforov-Uvarov method (which solves a second-order linear differential equation by reducing it to a generalized hypergeometric form) to spin- and pseudospin-symmetry to obtain, in closed form, the approximately analytical bound state energy eigenvalues and the corresponding upper- and lower-spinor components of two Dirac particles. The special cases κ = ±1 (s = $$ \tilde l $$ = 0, s-wave) and the non-relativistic limit can be reached easily and directly for the generalized and standard Woods-Saxon potentials. We compare the non-relativistic results with those obtained by others.

APPROXIMATE ℓ-STATE SOLUTIONS TO THE KLEIN–GORDON EQUATION FOR MODIFIED WOODS–SAXON POTENTIAL WITH POSITION DEPENDENT MASS

The radial part of the Klein–Gordon equation for the generalized Woods–Saxon potential is solved by using the Nikiforov–Uvarov method with spatially dependent mass within the new approximation scheme to the centrifugal potential term. The energy eigenvalues and corresponding normalized eigenfunctions are computed. The solutions in the case of constant mass are also obtained to check out the consistency of our new approximation scheme.

APPROXIMATE ℓ-STATE SOLUTIONS OF A SPIN-0 PARTICLE FOR WOODS–SAXON POTENTIAL

The radial part of Klein–Gordon equation is solved for the Woods–Saxon potential within the framework of an approximation to the centrifugal barrier. The bound states and the corresponding normalized eigenfunctions of the Woods–Saxon potential are computed by using the Nikiforov–Uvarov method. The results are consistent with the ones obtained in the case of generalized Woods–Saxon potential. The solutions of the Schrödinger equation by using the same approximation are also studied as a special case, and obtained the consistent results with the ones obtained before.

Solutions for a generalized Woods–Saxon potential

The s-wave Schrödinger and, to clarify one interesting point encountered in the calculations, Klein–Gordon equations are solved exactly for a single neutron moving in a central Woods–Saxon plus an additional potential that provides the flexibility to construct the surface structure of the related nucleus. The physics behind the solutions and the reliability of the results obtained are discussed carefully with the consideration of other related works in the literature. In addition, the exhaustive analysis of the results reveals the fact that the usual Woods–Saxon potential cannot be solved analytically within the framework of non-relativistic physics, unlike its exactly solvable relativistic consideration.

Exact solution of the Klein-Gordon equation for the PT-symmetric generalized Woods-Saxon potential by the Nikiforov-Uvarov method

The exact solution of the one-dimensional Klein-Gordon equation of the 𝒫𝒯-symmetric generalized Woods-Saxon potential is obtained. The exact energy eigenvalues and wavefunctions are derived analytically by using the Nikiforov and Uvarov method. In addition, the positive and negative exact bound states of the s-states are also investigated for different types of complex generalized Woods-Saxon potentials.

Polynomial solutions of the Schrödinger equation for the generalized Woods-Saxon potential

The bound state energy eigenvalues and the corresponding eigenfunctions of the generalized Woods Saxon potential are obtained in terms of the Jacobi polynomials. Nikiforov Uvarov method is used in the calculations. It is shown that the results are in a good agreement with the ones obtained before.