Standard Woods-Saxon Potential Research Articles

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.

Investigation of fusion hindrance in a soft asymmetric system deep below the barrier

A steeper fall of fusion excitation function, compared to the predictions of coupled-channel (CC) models, at energies below the lowest barrier between the reaction partners, is termed as deep sub-barrier fusion hindrance. This phenomenon has been observed in many symmetric and nearly symmetric systems. Different physical origins of the hindrance have been proposed though a complete understanding is yet to be achieved. This work reports the measurement of the fusion (evaporation residue) cross sections for the system 19F+181Ta, from above the barrier down to the energies where fusion hindrance is expected to come into play. CC calculation with standard Woods–Saxon potential gives a fair description of the fusion excitation function down to energies ≃14% below the barrier. This is in contrast with the observation of increasing fusion hindrance in asymmetric reactions induced by increasingly heavier projectiles, viz. 6,7Li, 11B, 12C and 16O. The most likely reason for non-observation of fusion hindrance in the present system is the ‘softness’ of the collision partners. The ‘threshold’ for the onset of fusion hindrance in reactions involving well-deformed nuclei is expected to be at an energy lower than that estimated from the systematics. It is also noted that the asymmetric reactions, which have not shown any signature of fusion hindrance within the measured energy range, are induced by projectiles with low α-particle break-up threshold and have positive Q-values for most light particle pick-up channels. Since CC calculation reproduces the fusion excitation functions for these systems, probable roles of projectile break-up and particle transfer in fusion deep below the barrier are not conclusively proven. Further measurements and inclusion of break-up and transfer channels within the framework of CC formalism would be of interest.

Revised analysis ofCa40+Zr96fusion reactions

Fusion data for $^{40}$Ca+$^{96}$Zr are analyzed by coupled-channels calculations that are based on a standard Woods-Saxon potential and include couplings to multiphonon excitations and transfer channels. The couplings to multiphonon excitations are the same as used in a previous work. The transfer couplings are calibrated to reproduce the measured neutron transfer data. This type of calculation gives a poor fit to the fusion data. However, by multiplying the transfer couplings with a $\sqrt{2}$ one obtains an excellent fit. The scaling of the transfer strengths is supposed to simulate the combined effect of neutron and proton transfer, and the calculated one- and two-nucleon transfer cross sections are indeed in reasonable agreement with the measured cross sections.

Fusion ofTi48+Fe58andNi58+Fe54below the Coulomb barrier

Background: No data on the fusion excitation function of $^{48}\mathrm{Ti}+^{58}\mathrm{Fe}$ in the energy region near the Coulomb barrier existed prior to the present work, while fusion of $^{58}\mathrm{Ni}+^{54}\mathrm{Fe}$ was investigated in detail some years ago, down to very low energies, and clear evidence of fusion hindrance was noticed at relatively high cross sections. $^{48}\mathrm{Ti}$ and $^{58}\mathrm{Fe}$ are soft and have a low-lying quadrupole excitation lying at $\ensuremath{\approx}800--900$ keV only. Instead, $^{58}\mathrm{Ni}$ and $^{54}\mathrm{Fe}$ have a closed shell (protons and neutrons, respectively) and are rather rigid.Purpose: We aim to investigate (1) the possible influence of the different structures of the involved nuclei on the fusion excitation functions far below the barrier and, in particular, (2) whether hindrance is observed in $^{48}\mathrm{Ti}+^{58}\mathrm{Fe}$, and to compare the results with current coupled-channels models.Methods: $^{48}\mathrm{Ti}$ beams from the XTU Tandem accelerator of INFN-Laboratori Nazionali di Legnaro were used. The experimental setup was based on an electrostatic beam separator, and fusion-evaporation residues (ERs) were detected at very forward angles. Angular distributions of ERs were measured.Results: Fusion cross sections of $^{48}\mathrm{Ti}+^{58}\mathrm{Fe}$ have been obtained in a range of nearly six orders of magnitude around the Coulomb barrier, down to $\ensuremath{\sigma}\ensuremath{\simeq}2 \ensuremath{\mu}\mathrm{b}$. The sub-barrier cross sections of $^{48}\mathrm{Ti}+^{58}\mathrm{Fe}$ are much larger than those of $^{58}\mathrm{Ni}+^{54}\mathrm{Fe}$. Significant differences are also observed in the logarithmic derivatives and astrophysical $S$ factors. No evidence of hindrance is observed, because coupled-channels calculations using a standard Woods-Saxon potential are able to reproduce the data in the whole measured energy range. Analogous calculations for $^{58}\mathrm{Ni}+^{54}\mathrm{Fe}$ predict clearly too large cross sections at low energies. The two fusion barrier distributions are wide and display a complex structure that is only qualitatively fit by calculations.Conclusions: It is pointed out that all these different trends originate from the dissimilar low-energy nuclear structures of the involved nuclei. In particular, the strong quadrupole excitations in $^{48}\mathrm{Ti}$ and $^{58}\mathrm{Fe}$ produce the relative cross section enhancement and make the barrier distribution $\ensuremath{\approx}2$ MeV wider, thus probably pushing the threshold for hindrance below the measured limit.

Fusion ofSi28+Si28,30: Different trends at sub-barrier energies

Background: The fusion excitation function of the system $^{28}\mathrm{Si}\phantom{\rule{0.28em}{0ex}}+\phantom{\rule{0.28em}{0ex}}^{28}\mathrm{Si}$ at energies near and below the Coulomb barrier is known only down to $\ensuremath{\simeq}15$ mb. This precludes any information on both coupling effects on sub-barrier cross sections and the possible appearance of hindrance. For $^{28}\mathrm{Si}\phantom{\rule{0.28em}{0ex}}+\phantom{\rule{0.28em}{0ex}}^{30}\mathrm{Si}$ even if the fusion cross section is measured down to $\ensuremath{\simeq}50$ $\ensuremath{\mu}\mathrm{b}$, the evidence of hindrance is marginal. Both systems have positive fusion $Q$ values. While $^{28}\mathrm{Si}$ has a deformed oblate shape, $^{30}\mathrm{Si}$ is spherical.Purpose: We investigate 1. the possible influence of the different structure of the two Si isotopes on the fusion excitation functions in the deep sub-barrier region and 2. whether hindrance exists in the $\text{Si}\phantom{\rule{4.pt}{0ex}}+\phantom{\rule{4.pt}{0ex}}\text{Si}$ systems and whether it is strong enough to generate an $S$-factor maximum, thus allowing a comparison with lighter heavy-ion systems of astrophysical interest.Methods: $^{28}\mathrm{Si}$ beams from the XTU Tandem accelerator of the INFN Laboratori Nazionali di Legnaro were used. The setup was based on an electrostatic beam separator, and fusion evaporation residues (ER) were detected at very forward angles. Angular distributions of ER were measured.Results: Fusion cross sections of $^{28}\mathrm{Si}\phantom{\rule{0.28em}{0ex}}+\phantom{\rule{0.28em}{0ex}}^{28}\mathrm{Si}$ have been obtained down to $\ensuremath{\simeq}600$ nb. The slope of the excitation function has a clear irregularity below the barrier, but no indication of a $S$-factor maximum is found. For $^{28}\mathrm{Si}\phantom{\rule{0.28em}{0ex}}+\phantom{\rule{0.28em}{0ex}}^{30}\mathrm{Si}$ the previous data have been confirmed and two smaller cross sections have been measured down to $\ensuremath{\simeq}4$ $\ensuremath{\mu}\mathrm{b}$. The trend of the $S$-factor reinforces the previous weak evidence of hindrance.Conclusions: The sub-barrier cross sections for $^{28}\mathrm{Si}\phantom{\rule{0.28em}{0ex}}+\phantom{\rule{0.28em}{0ex}}^{28}\mathrm{Si}$ are overestimated by coupled-channels calculations based on a standard Woods-Saxon potential, except for the lowest energies. Calculations using the M3Y+repulsion potential are adjusted to fit the $^{28}\mathrm{Si}\phantom{\rule{0.28em}{0ex}}+\phantom{\rule{0.28em}{0ex}}^{28}\mathrm{Si}$ and the existing $^{30}\mathrm{Si}\phantom{\rule{0.28em}{0ex}}+\phantom{\rule{0.28em}{0ex}}^{30}\mathrm{Si}$ data. An additional weak imaginary potential (probably simulating the effect of the oblate $^{28}\mathrm{Si}$ deformation) is required to fit the low-energy trend of $^{28}\mathrm{Si}\phantom{\rule{0.28em}{0ex}}+\phantom{\rule{0.28em}{0ex}}^{28}\mathrm{Si}$. The parameters of these calculations are applied to predict the ion-ion potential for $^{28}\mathrm{Si}\phantom{\rule{0.28em}{0ex}}+\phantom{\rule{0.28em}{0ex}}^{30}\mathrm{Si}$. Its cross sections are well reproduced by also including one- and successive two-neutron transfer channels, besides the low-lying surface excitations.

Bound State Solutions of the Schrödinger Equation for a More General Woods—Saxon Potential with Arbitrary l-State

The energy spectra and the wave function depending on the c-factor are investigated for a more general Woods—Saxon potential (MGWSP) with an arbitrary l-state. The wave functions are expressed in terms of the Jacobi polynomials. Two potentials are obtained from this MGWSP as the special cases. These special potentials are Hulthen and the standard Woods—Saxon potentials. We also discuss the energy spectrum and wave function for the special cases.

Hindrance in the fusion ofCa48+Ca48

The coupled-channels technique is applied to analyze recent fusion data for $^{48}$Ca+$^{48}$Ca. The calculations include the excitations of the low-lying $2^+$, $3^-$ and $5^-$ states in projectile and target, and the influence of mutual excitations as well as the two-phonon quadrupole excitations is also investigated. The ion-ion potential is obtained by double-folding the nuclear densities of the reacting nuclei with the M3Y+repulsion effective interaction but a standard Woods-Saxon potential is also applied. The data exhibit a strong hindrance at low energy compared to calculations that are based on a standard Woods-Saxon potential but they can be reproduced quite well by applying the M3Y+repulsion potential with an adjusted radius of the nuclear density. The influence of the polarization of high-lying states on the extracted radius is discussed.

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.

Refractive scattering and reactions, comparison of two systems:16O+16O and20Ne+12C

Elastic, inelastic scattering as well as one-neutron transfer channels have been measured over a wide angular range for systems16O+16O at the incident energy of 350 MeV and20Ne+12C at 390 MeV, respectively, using the Q3D magnetic spectrometer. In both cases differential cross sections have been measured down to about 50 nb/sr (or dσ/dσ R≤10−4) at large angles. For the16O+16O system refractive contributions are found at the level of these cross sections, whereas in the20Ne+12C case a steeper decrease of the differential cross section with the angle is observed and the refractive contribution can not be determined. The elastic scattering data have been analyzed using standard Woods-Saxon potentials and potentials calculated in different versions of the double-folding model. Some properties of these potentials are tested in the calculations for inelastic scattering and one-neutron transfer within the DWBA. With the refractive pattern observed for the16O+16O system, the scattering and transfer data are found to be sensitive to the interaction potential at small internuclear distances down to about 2.5 fm.

Optical-model potentials from potential inversion at fixed energy

A fixed-energy potential-inversion method is employed to generate an optical-model potential to fit the elastic-scattering data for 16O + 28Si at 21.1 and 34.8 MeV. The fit yields a potential that is based on a standard Woods–Saxon potential and yields minor modifications to it.

Optical model analysis of 200 MeV p

Differential cross sections and analyzing powers are presented for $^{16}$O(p\ensuremath{\rightarrow},p${)}^{16}$O elastic scattering at ${E}_{\mathrm{p}=200}$ MeV, extending over a large range of momentum transfer (up to q\ensuremath{\sim}6 ${\mathrm{fm}}^{\mathrm{\ensuremath{-}}1}$). The data were analyzed in the framework of the nonrelativistic optical model. In addition to standard Woods-Saxon potentials, nonstandard radial shapes for the central and the spin-orbit potentials were investigated. The data are not adequately described over the entire angular range using a Woods-Saxon real central potential, but two distinct parametrizations (not simply related by the usual discrete or continuous parameter ambiguities which maintain a rough phase-shift equivalence) were found which provide a far superior representation. The similarity in the angular distributions given by these two potentials can be attributed to the fact that the non-spin-flip partial-wave scattering amplitudes ${a}_{L}$ differ by only a simple rotation in the complex ${a}_{L}$ plane. This rotation has a negligible effect on the predicted cross section and the analyzing power angular distributions, but leads to sizable differences in the predicted spin-rotation function Q(\ensuremath{\theta}).

Proton scattering fromC12between 120 and 200 MeV and the optical potential

Cross sections and analyzing powers are presented for $^{12}\mathrm{C}(\stackrel{\ensuremath{\rightarrow}}{p}, p)^{12}\mathrm{C}$ elastic scattering at 122 and 160 MeV. The data cover a large angular range (up to ${\ensuremath{\theta}}_{\mathrm{lab}}=154\ifmmode^\circ\else\textdegree\fi{}$) and extend to lower energies an earlier, similar measurement at 200 MeV. The complete data set is analyzed in the framework of the optical model. Besides the standard Woods-Saxon potential, nonstandard radial shapes for the central and the spin-orbit potentials are investigated. Although the number of free parameters for the nonstandard potential is substantially reduced from the standard potential, it provides an improved fit to the data and a description of $^{12}\mathrm{C}(p, p)^{12}\mathrm{C}$ scattering with a simple monotonic energy dependence in the region 100-200 MeV. The characteristic features of the nonstandard potential are reproduced by microscopic theories.