Nonlocal Optical Potentials Research Articles

Density and energy dependent semi-microscopic optical potential of nucleon–nucleus elastic scattering

A nonlocal velocity-dependent optical potential was used to study the angular distribution of the differential cross section of nucleon–nucleus elastic scattering. The potential parameters were calculated by fitting elastic angular distributions over a wide range of energy (10-65 MeV) and mass (12-208). The real part of the local optical potential was calculated using the CDM3Y6-Paris interaction, whereas the imaginary part was a Woods-Saxon form factor and its derivative.

Microscopic bell-shape nonlocality: The case of proton scattering off 40Ca at 200 MeV

This work is part of an ongoing effort to build microscopicallydriven nonlocal optical potentials easily tracktable in scattering codes. Based on the separable ‘JνH’ structure proposed recently [1], where the potential can be cast as the product of a radial and nonlocality form factors, we investigate its angular dependence in momentum space. We find that scattering observables have a weak angular dependence between the momentum transfer q = k − k′, and K = (k + k′)/2. The study is focussed on proton elastic scattering off 40Ca at 200 MeV, where the JνH structure is found to be inadequate. We conclude that any improvement of the JνH structure of the potential can be made to the lowest order in multipole expansions in Kq−representation of the potential.

Cluster scattering in the non-local model

Scattering of weakly bound nuclei with a pronounced cluster structure is strongly affected by their breakup. Usually, this mechanism is accounted for in a three-body model with pairwise potentials. The interaction potentials between complex systems are non-local due to the existence of excitation channels and antisymmetrization. However, a common practice is to use local optical potentials in cluster scattering studies. To assess the validity of replacing non-local optical potentials by their local equivalents, we extend the local-equivalent continuum-discretized coupled-channel (LECDCC) approach proposed by us for deuteron scattering in [Phys. Rev. C98, 011601(R) (2018)] to the case of cluster scattering. We consider the case of 6Li + 120Sn at 27 and 60 MeV, and compare the angular distributions and reaction cross sections for elastic and breakup cross sections with those obtained in the standard continuum-discretized coupled-channel (CDCC) method with local equivalents of non-local potentials. We found that while elastic scattering is not significantly affected by non-locality, the breakup observables could be affected by up to 20% depending on kinematical conditions of their observation.

Nonlocal optical potential with core excitation in [formula omitted] and [formula omitted] reactions

We propose a new nonlocal form of the nucleon-nucleus optical potential and demonstrate its reliability. We extend the nonlocal potential to include the excitation of the nuclear core and develop energy-independent proton-Be10 potential reasonably reproducing the experimental data at low energies. We apply the new potential to the study of deuteron stripping and pickup reactions Be10(d,p)11Be and Be11(p,d)10Be using rigorous three-body Faddeev-type equations for transition operators that are solved in the momentum-space partial-wave framework. The achieved description of the experimental data is considerably more successful as compared to previous studies with local potentials. The values of spectroscopic factors consistent with the data are determined, exhibiting only weak energy dependence. The results possibly indicate an increased predicting power of the proposed calculational scheme.

Velocity dependent potential for alpha 40Ca-elastic scattering

Angular distribution of the differential cross section of α40Ca elastic scattering has been investigated in the anomalous large angle scattering (ALAS) region by a nonlocal velocity dependent optical potential with surface term. The geometrical potential parameters were fixed and the depths of the potential were determined by fitting elastic angular distribution over the energy range (23–53.9 MeV). The real and the imaginary parts of the optical potential used here are of Woods–Saxon form factors. An energy dependence of the potential depths has been investigated at fixed geometrical potential parameters.

On the separability of microscopic optical model potentials and emerging bell-shape Perey–Buck nonlocality

After nearly sixty years since its introduction, the phenomenological bell-shape Perey-Buck spatial nonlocality in the optical model potential for nucleon-nucleus scattering has remained unaccounted for from a microscopic standpoint. In this article we provide a quantitative account for such nonlocality considering fully nonlocal optical potentials in momentum space. The framework is based on a momentum-space in-medium folding model, where infinite nuclear matter $g$ matrices in Brueckner-Hartree-Fock approximation are folded to the target one-body mixed density. The study is based on chiral next-to-next-to-next-to-leading order (N3LO) as well as Argonne $v_{18}$ nucleon-nucleon bare interaction models. Applications focus on $^{40}$Ca($p,p$) scattering at beam energies in the range 11-200 MeV, resulting in the identification of a separable structure of the momentum-space optical potential of a form we coin as $JvH$, with a nonlocality form factor as one of its terms. The resulting nonlocaliy form factor features a bell-shape with nonlocality range $\beta$ between 0.86 and 0.89 fm, for both proton and neutron beams at energies below 65 MeV. An analytic toy model is introduced to elucidate the underlying mechanism for the nonlocality in the optical model, providing an estimate of its range based on the Fermi motion of the target nucleons and the long-range part of the $NN$ interaction.

Considering nonlocality in the optical potentials within eikonal models

Background: For its simplicity, the eikonal method is the tool of choice to analyze nuclear reactions at high energies ($E>100$ MeV/nucleon), including knockout reactions. However, so far, the effective interactions used in this method are assumed to be fully local.Purpose: Given the recent studies on nonlocal optical potentials, in this work we assess whether nonlocality in the optical potentials is expected to impact reactions at high energies and then explore different avenues for extending the eikonal method to include nonlocal interactions.Method: We compare angular distributions obtained for nonlocal interactions (using the exact R-matrix approach for elastic scattering and the adiabatic distorted wave approximation for transfer) with those obtained using their local-equivalent interactions.Results: Our results show that transfer observables are significantly impacted by nonlocality in the high-energy regime. Because knockout reactions are dominated by stripping (transfer to inelastic channels), nonlocality is expected to have a large effect on knockout observables too. Three approaches are explored for extending the eikonal method to nonlocal interactions, including an iterative method and a perturbation theory.Conclusions: None of the derived extensions of the eikonal model provide a good description of elastic scattering. This paper suggests that nonlocality removes the formal simplicity associated with the eikonal model.

Three-nucleon force contribution to the deuteron channel in (d,p) reactions

The contribution of a three-nucleon (3N) force, acting between the neutron and proton in the incoming deuteron with a target nucleon, to the deuteron-target potential in the entrance channel of the $A(d,p)B$ reaction has been calculated within the adiabatic distorted wave approximation (ADWA). Four different 3N interaction sets from local chiral effective field theory ($\chi$EFT) at next-to-next-to-leading order (N2LO) were used. Strong sensitivity of the adiabatic deuteron-target potential to the choice of the 3N force format has been found, which originates from the enhanced sensitivity to the short-range physics of nucleon-nucleon (NN) and 3N interactions in the ADWA. Such a sensitivity is reduced when a Watanabe folding model is used to generate $d$-$A$ potential instead of ADWA. The impact of the 3N force contribution on $(d,p)$ cross sections depends on assumptions made about the $p$-$A$ and $n$-$A$ optical potentials used to calculate the distorted $d$-$A$ potential in the entrance channel. It is different for local and nonlocal optical potentials and depends on whether the induced three-body force arising due to neglect of target excitations is included or not.

SWANLOP: Scattering waves off nonlocal optical potentials in the presence of Coulomb interactions

We introduce the package SWANLOP to calculate scattering waves and corresponding observables for nucleon elastic collisions off spin-zero nuclei. The code is capable of handling local and nonlocal optical potentials superposed to long-range Coulomb interaction. Solutions to the implied Schrödinger integro-differential equation are obtained by solving an integral equation of Lippmann–Schwinger type for the scattering wavefunctions, ψ=ϕC+GCUSψ, providing and exact treatment to the Coulomb force Arellano and Blanchon (2019). The package has been developed to handle potentials either in momentum or coordinate representations, providing flexible options under each of them. The code is fully self-contained, being dimensioned to handle any A≥4 target for nucleon beam energies of up to 1.1 GeV. Accuracy and benchmark applications are presented and discussed. Program summaryProgram title: SWANLOPCPC Library link to program files:https://doi.org/10.17632/89gw9jdfv4.1Licensing provisions: GNU General Public License, Version 2Programming language: FORTRAN-90Nature of problem: Optical model potentials constitute a valuable tool to investigate the physics involved in nuclear collisions and reactions with nucleonic probes. As such, it becomes essential to obtain accurate results for its associated scattering observables and corresponding scattering waves. An important feature of optical potentials is their nonlocal nature, arising from the fermionic nature of the (A+1)–nucleon problem together with the fact that effective nucleon–nucleon (NN) interactions are nonlocal as well. The superposition of Coulomb interaction to these nonlocal potentials poses non-trivial difficulties to obtain scattering waves and observables in collision processes.Solution method: The code performs the calculation of scattering waves associated to nonlocal potentials in the presence of the long-range Coulomb interactions, solving a Lippmann–Schwinger type integral equation for the scattering wavefunction. The potential can be given either in coordinate or momentum space. Phase-shifts and associated elastic scattering observables are extracted from the asymptotic behavior of the solution.

A nonlocal optical potential with a Gaussian nonlocality for proton elastic scattering off light 1p-shell nuclei

In a recent work we used the nonlocal optical model of Perey and Buck (Nucl Phys A 32:353, 1962) to fit the angular distribution data corresponding to neutron elastic scattering off light nuclei from $$^6\hbox {Li}$$ to $$^{18}\hbox {O}$$ over the 9–35 MeV energy range. Although such nuclei posses diffuse edges that complicate the scattering process, the model resulted in improved angular distribution fits compared to those obtained using local optical models. In this work we extend the treatment to the case of proton elastic scattering off light 1p-shell nuclei, but consider the wider 10–70 MeV energy range. No single global set of nonlocal parameters adequately described the measured data across the entire energy range. Instead, we obtained two global sets of fixed nonlocal parameters; one corresponds to the 10–39 MeV range, the other is for the higher 40–70 MeV incident energies. Each set resulted in angular distribution fits which are in very good agreement with experimental data even in the large-angle scattering region, which is usually endowed by nonlocal effects. The fits were improved even further by expressing the imaginary volume and imaginary surface depths as linear functions of energy and the $$(N-Z)/A$$ asymmetry term. This indicates that there is still need for energy-dependent parameters even when nonlocality is explicitly included in the potential. We also used the model to predict elastic angular distributions for proton scattering off nuclei not included in the fitting procedure. The theoretical predictions showed very good agreement with experiment. As a further test of the nonlocal model, we calculated total reaction cross sections for proton scattering off light 1p-shell nuclei. Our predictions are in fair overall agreements with the experimental data.

SIDES: Nucleon–nucleus elastic scattering code for nonlocal potential

We introduce the package SIDES (Schrödinger Integro-Differential Equation Solver) that solves the integro-differential Schrödinger equation for elastic scattering of a nonlocal optical potential in coordinate space. The code is capable of treating the Coulomb interaction without restrictions. The method is based on previous developments by Jacques Raynal in the DWBA07 code. Elastic scattering observables such as differential and integral cross sections, as well as analyzing power and spin rotation functions for both neutron and proton projectiles are evaluated, with no restriction on the type of nonlocality of the potential nor on the beam energy. The corresponding distorted wavefunctions are calculated as well. The SIDES package includes a Perey–Buck potential generator with two parametrizations. It includes as well local potential parametrizations and allows for mixing local and nonlocal contributions. Benchmarks are performed and discussed. Program summaryProgram Title: SIDESProgram Files doi:http://dx.doi.org/10.17632/cmpjgyrngr.1Licensing provisions: GNU General Public License, Version 2Programming language: FORTRAN-90Nature of problem: The description of nucleon elastic scattering off a target nucleus involves solving the Schrödinger’s wave equation for positive incident energy. The determination of scattering observables calls for accurate treatments of the continuum. The effective coupling between the projectile and the target is accounted for by an optical potential, an operator which is by nature complex, energy-dependent and nonlocal. The coupling becomes long-range in the case of charged projectiles. In a general scenario under nonlocal potentials, Schrödinger’s equation becomes an integro-differential equation.Solution method: SIDES solves the Schrödinger integro-differential equation numerically by matrix inversion using Gibbs, Numerov or a modified Numerov method with a uniform radial mesh in a box. The solution is refined by an iterative procedure until a specified precision is achieved. To obtain elastic scattering observables, the associated phase-shifts are calculated via matching of the numerical solution with its analytic asymptotic behavior.

Effects of an induced three-body force in the incident channel of ( d,p ) reactions

A widely accepted practice for treating deuteron breakup in $A(d,p)B$ reactions relies on solving a three-body $A+n+p$ Schr\"odinger equation with pairwise $A$-$n$, $A$-$p$ and $n$-$p$ interactions. However, it was shown in [Phys. Rev. C \textbf{89}, 024605 (2014)] that projection of the many-body $A+2$ wave function into the three-body $A+n+p$ channel results in a complicated three-body operator that cannot be reduced to a sum of pairwise potentials. It contains explicit contributions from terms that include interactions between the neutron and proton via excitation of the target $A$. Such terms are normally neglected. We estimate the first order contribution of these induced three-body terms and show that applying the adiabatic approximation to solving the $A+n+p$ model results in a simple modification of the two-body nucleon optical potentials. We illustrate the role of these terms for the case of $^{40}$Ca($d,p$)$^{41}$Ca transfer reactions at incident deuteron energies of 11.8, 20 and 56 MeV, using several parameterisations of nonlocal optical potentials.

Three-body problem with velocity-dependent optical potentials: a case of (d, p) reactions

The change in mass of a nucleon, arising from its interactions with other nucleons inside the target, results in velocity-dependent terms in the Schrödinger equation that describes nucleon scattering. It has recently been suggested in a number of publications that introducing and fitting velocity-dependent terms improves the quality of the description of nucleon scattering data for various nuclei. The present paper discusses velocity-dependent optical potentials in the context of a three-body problem used to account for deuteron breakup in the entrance channel of (d, p) reactions. Such potentials form a particular class of nonlocal optical potentials which are a popular object of modern studies. It is shown here that because of a particular structure of the velocity-dependent terms the three-body problem can be formulated in two different ways. Solving this problem within an adiabatic approximation results in a significant difference between the two approaches caused by contributions from the high n–p momenta in deuterons in one of them. Solving the three-body problem beyond the adiabatic approximation may remove such contributions, which is indirectly confirmed by replacing the adiabatic approximation by the folding Watanabe model where such contributions are suppressed. Discussion of numerical results is carried out for the reaction where experimental data both on elastic scattering in entrance and exit channels and on nucleon transfer are available.

R-matrix method and the nonlocal nucleon optical potential

The calculable R-matrix method is applied to solve the Schr ̈odinger equationin the optical model (OM) analysis of the elastic nucleon-nucleus scatter-ing using a nonlocal nucleon optical potential (OP). The phenomenologicalnonlocal nucleon OP proposed by Perey and Buck (PB), and the two recentversions of the PB parametrization were used in the present OM study of theelastic nucleon scattering on 208 Pb and 27 Al targets at different energies. Thecomparison of the OM results given by the calculable R-matrix method withthose given by other methods confirms that the calculable R-matrix methodis an efficient tool for the OM study of the elastic nucleon-nucleus scatteringusing a nonlocal nucleon OP.

Validity of the distorted-wave impulse-approximation description of Ca40(e,e′p)K39 data using only ingredients from a nonlocal dispersive optical model

The nonlocal implementation of the dispersive optical model (DOM) provides all the ingredients for distorted-wave impulse-approximation (DWIA) calculations of the $(e,e'p)$ reaction. It provides both the overlap function, including its normalization, and the outgoing proton distorted wave. This framework is applied to describe the knockout of a proton from the $0\textrm{d}\frac{3}{2}$ and $1\textrm{s}\frac{1}{2}$ orbitals in ${}^{40}$Ca with fixed normalizations of 0.71 and 0.60, respectively. Data were obtained in parallel kinematics for three outgoing proton energies: 70, 100, and 135 MeV. Agreement with the data is as good as, or better than, previous descriptions employing local optical potentials and overlap functions from Woods-Saxon potentials - both with standard nonlocality corrections - whose normalization (spectroscopic factor) and radius were fitted to the data. The present analysis suggests that slightly larger spectroscopic factors are obtained when nonlocal optical potentials are employed than those generated with local potentials. The results further suggest that the chosen kinematical window around 100 MeV proton energy provides the best and cleanest method to employ the DWIA for the analysis of this reaction. The conclusion that substantial ground-state correlations cannot be ignored when describing a closed-shell atomic nucleus is therefore confirmed in detail. To reach these conclusions, it is essential to have a complete description of the nucleon single-particle propagator that accounts for all elastic nucleon-scattering observables in a wide energy domain up to 200 MeV. The current nonlocal implementation of the DOM fulfills this requirement.

Weak sensitivity of three-body ( d,p ) reactions to np force models

Adiabatic distorted-wave approximation (ADWA) study of three-body $(d,p)$ transfer reactions [G.W. Bailey, N.K. Timofeyuk, and J.A. Tostevin, Phys. Rev. Lett. 117, 162502 (2016)] reported strong sensitivity of cross sections to the neutron-proton $(np)$ interaction model when the nucleon-nucleus optical potential is nonlocal. The verification of this unusual finding using more reliable methods is aimed for in the present work. A rigorous Faddeev-type three-body scattering theory is applied to the study of $(d,p)$ transfer reactions. The equations for transition operators are solved in the momentum-space partial-wave framework. Differential cross sections for $^{26}$Al$(d,p)^{27}$Al reactions are calculated using nonlocal nuclear optical potentials and a number of realistic $np$ potentials. Only a weak dependence on the $np$ force model is observed, typically one order of magnitude lower than in the previous ADWA study. The shape of the angular distribution of the experimental data is well reproduced. Cross sections of $(d,p)$ transfer reactions calculated using a rigorous three-body method show little sensitivity to the $np$ interaction model. This indicates a failure of the ADWA in the context of nonlocal potentials. Some evident shortcomings of the ADWA are pointed out.

Exploration of the energy dependence of proton nonlocal optical potentials

Background: Given the importance of including nonlocality in the effective interactions in reaction models, a recent phenomenological study focusing on neutron-target nonlocal optical potentials suggests the need for the inclusion of explicit energy dependence [Lovell et al., Phys. Rev. C 96, 051601 (2017)].Purpose: In this work, we inspect whether the same is true for proton nonlocal optical potentials.Method: Similarly to the earlier work, we perform a ${\ensuremath{\chi}}^{2}$ analysis of proton elastic scattering data on $^{40}\mathrm{Ca}$, $^{90}\mathrm{Zr}$, and $^{208}\mathrm{Pb}$ at energies of $E\ensuremath{\approx}10--45$ MeV, assuming the Tian, Pang, and Ma nonlocal form for the optical potential. We introduce energy and asymmetry dependencies in the imaginary part of the potential and refit the data to obtain a global parametrization.Results: No matter which starting point is used, or whether we include backward angles in the fitting procedure, our results show the emergence of a strong energy dependence in the potential. We also show that while our parametrization represents only a modest improvement over the original energy-independent potential for those cases included in the fit, our new energy-dependent potential extrapolates much better for nuclei not included in the fit and for energies above those included.Conclusions: As for the neutron case, we conclude that nonlocality alone cannot provide a complete description of proton elastic scattering data and that a significant energy dependence is required.

Reduced sensitivity of the (d,p) cross sections to the deuteron model beyond the adiabatic approximation

It has recently been reported [Phys. Rev. Lett. 117, 162502 (2016)] that (d, p) cross sections can be very sensitive to the n-p interactions used in the adiabatic treatment of deuteron breakup with nonlocal nucleon-target optical potentials. To understand to what extent this sensitivity could originate in the inaccuracy of the adiabatic approximation we have developed a leading-order local- equivalent continuum-discretized coupled-channel model that accounts for non-adiabatic effects in the presence of nonlocality of nucleon optical potentials. We have applied our model to the astro- physically relevant reaction $^{26m}$Al$(d, p) ^{27}$Al using two different n-p potentials associated with the lowest and the highest n-p kinetic energy in the short-range region of their interaction, respectively. Our calculations reveal a significant reduction of the sensitivity to the high n-p momenta thus confirming that it is mostly associated with theoretical uncertainties of the adiabatic approximation itself. The non-adiabatic effects in the presence of nonlocality were found to be stronger than those in the case of the local optical potentials. These results argue for extending the analysis of the $(d, p)$ reactions, measured for spectroscopic studies, beyond the adiabatic approximation.

New developments in reaction theory: preparing for the FRIB era

This is a brief report on the progress made towards an exact theory for (d,p) on heavy nuclei, which is important to determine neutron capture rates for r-process nuclei. We first discuss the role of core excitation in the framework of Faddeev equations. Following that, we provide the status of the Faddeev theory being developed in the Coulomb basis with separable interactions. We then present some recent developments on nonlocal nucleon optical potentials. Finally, the progress on the theory transfer to the continuum is summarized.

Energy dependence of nonlocal optical potentials

Recently a variety of studies have shown the importance of including non-locality in the description of reactions. The goal of this work is to revisit the phenomenological approach to determining non-local optical potentials from elastic scattering. We perform a $\chi^2$ analysis of neutron elastic scattering data off $^{40}$Ca, $^{90}$Zr and $^{208}$Pb at energies $E \approx 5-40$ MeV, assuming a Perey and Buck or Tian, Pang, and Ma non-local form for the optical potential. We introduce energy and asymmetry dependencies in the imaginary part of the potential and refit the data to obtain a global parameterization. Independently of the starting point in the minimization procedure, an energy dependence in the imaginary depth is required for a good description of the data across the included energy range. We present two parameterizations, both of which represent an improvement over the original potentials for the fitted nuclei as well as for other nuclei not included in our fit. Our results show that, even when including the standard Gaussian non-locality in optical potentials, a significant energy dependence is required to describe elastic-scattering data.