Nonlocal Optical Potentials Research Articles

Calculation of three-body nuclear reactions with angular-momentum and parity-dependent optical potentials

Angular-momentum or parity-dependent nonlocal optical potentials for nucleon-${}^{16}\mathrm{O}$ scattering able to fit differential cross section data over the whole angular regime are developed and applied to the description of deuteron-${}^{16}\mathrm{O}$ scattering in the framework of three-body Faddeev-type equations for transition operators. Differential cross sections and deuteron analyzing powers for elastic scattering and ${}^{16}\mathrm{O}(d,p){}^{17}\mathrm{O}$ transfer reactions are calculated using a number of local and nonlocal optical potentials and compared with experimental data. Angular-momentum or parity-dependence of the optical potential turns out to be quite irrelevant in the considered three-body reactions while nonlocality is essential for a successful description of the differential cross section data, especially in transfer reactions.

Explicit inclusion of nonlocality in(d,p)transfer reactions

Background: Traditionally, nucleon-nucleus optical potentials are made local for convenience. In recent work we studied the effects of including nonlocal interactions explicitly in the final state for (d,p) reactions, within the distorted wave Born approximation. Purpose: Our goal in this work is to develop an improved formalism for nonlocal interactions that includes deuteron breakup and to use it to study the effects of including nonlocal interactions in transfer (d,p) reactions, in both the deuteron and the proton channel. Method: We extend the finite-range adiabatic distorted wave approximation to include nonlocal nucleon optical potentials. We apply our method to (d,p) reactions on 16O, 40Ca, 48Ca, 126Sn, 132Sn, and 208Pb at 10, 20 and 50 MeV. Results: We find that nonlocality in the deuteron scattering state reduces the amplitude of the wave function in the nuclear interior, and shifts the wave function outward. In many cases, this has the effect of increasing the transfer cross section at the first peak of the angular distributions. This increase was most significant for heavy targets and for reactions at high energies. Conclusions: Our systematic study shows that, if only local optical potentials are used in the analysis of experimental (d,p) transfer cross sections, the extracted spectroscopic factors may be incorrect by up to 40% due to the local approximation.

Forging the link between nuclear reactions and nuclear structure

A review of the recent applications of the dispersive optical model (DOM) is presented. Emphasis is on the nonlocal implementation of the DOM that is capable of describing ground-state properties accurately when data like the nuclear charge density are available. The DOM, conceived by Claude Mahaux, provides a unified description of both elastic nucleon scattering and structure information related to single-particle properties below the Fermi energy. We have recently introduced a nonlocal dispersive optical potential for both the real and imaginary part. Nonlocal absorptive potentials yield equivalent elastic differential cross sections for ${}^{40}$Ca as compared to local ones but change the $\ell$-dependent absorption profile suggesting important consequences for the analysis of nuclear reactions. Below the Fermi energy, nonlocality is essential for an accurate representation of particle number and the nuclear charge density. Spectral properties implied by $(e,e'p)$ and $(p,2p)$ reactions are correctly described, including the energy distribution of about 10\% high-momentum protons obtained at Jefferson Lab. The nonlocal DOM allows a complete description of experimental data both above (up to 200 MeV) and below the Fermi energy in $^{40}$Ca. It is further demonstrated that elastic nucleon-nucleus scattering data constrain the spectral strength in the continuum of orbits that are nominally bound in the independent-particle model. Extension of this analysis to $^{48}$Ca allows a prediction of the neutron skin of this nucleus that is larger than most predictions made so far.

Faddeev-type calculation of(d,n)transfer reactions in three-body nuclear systems

Exact Faddeev-type three-body equations are applied to the study of the proton transfer reactions $(d,n)$ in the system consisting of a nuclear core and two nucleons. The integral equations for the three-body transition operators are solved in the momentum-space framework including the Coulomb interaction via the screening and renormalization method. For a weakly bound final nucleus the calculation of the $(d,n)$ reaction is more demanding in terms of the screening radius as compared to the $(d,p)$ reaction. Well-converged differential cross section results are obtained for $^{7}\text{Be}(d,n)^{8}\text{B}$, $^{12}\text{C}(d,n)^{13}\text{N}$, and $^{16}\text{O}(d,n)^{17}\text{F}$ reactions. A comparison with the corresponding $(d,p)$ reactions is made. The calculations fail to reproduce the shape of the angular distribution for reactions on $^{12}\text{C}$ but provide quite successful description for reactions on $^{16}\text{O}$, especially for the transfer to the $^{17}\text{F}$ excited state $1/{2}^{+}$ when using a nonlocal optical potential.

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.

Systematic nonlocal optical model potential for nucleons

Based on the Perey–Buck nonlocal optical model potential, F. Perey and B. Buck, Nucl. Phys. 32 (1962) 353, we obtain a new set of nonlocal optical model potential (NLOMP) parameters for proton and neutron scattering off nuclei. The experimental angular distributions of nucleon scattering off nuclei ranging from 27 Al to 208 Pb with incident energies around 10 MeV to 30 MeV are adopted in the fitting procedures. This NLOMP is energy independent. The chi-squares χ2 obtained in the fittings are comparable to those from the KD03 phenomenological local optical model potentials (OMP) A. Koning and J. Delaroche, Nucl. Phys. A 713 (2003) 231. Good agreement is found in comparisons between optical model calculations using this NLOMP and KD03 in their reproduction to the experimental angular distributions of elastic scattering cross-sections and analyzing powers.

Adiabatic model of(d,p)reactions with explicitly energy-dependent nonlocal potentials

We have developed an approximate way of dealing with explicit energy-dependence of non-local nucleon optical potentials as used to predict the $(d,p)$ cross sections within the adiabatic theory. Within this approximation, the non-local optical potentials have to be evaluated at an energy shifted from half the incident deuteron energy by the $n-p$ kinetic energy averaged over the range of the $n-p$ interaction and then treated as an energy-independent non-local potential. Thus the evaluation of the distorting potential in the incident channel is reduced to a problem solved in our previous work in [{\it Phys. Rev. Lett. 110, 112501(2013) and Phys. Rev. C 87, 064610 (2013)}]. We have demonstrated how our new model works for the case of $^{16}$O$(d,p)^{17}$O, $^{36}$Ar($d,p)^{37}$Ar and $^{40}$Ca$(d,p)^{41}$Ca reactions and highlighted the need for a detailed understanding of energy-dependence of non-local potentials. We have also suggested a simple way of correcting the $d-A$ effective potentials for non-locality when the underlying energy-dependent non-local nucleon potentials are unknown but energy-dependent local phenomenological nucleon potentials are available.

Analyzing power in elastic scattering of6He from a polarized proton target at 71 MeV/nucleon

The vector analyzing power has been measured for the elastic scattering of neutron-rich 6He from polarized protons at 71 MeV/nucleon making use of a newly constructed solid polarized proton target operated in a low magnetic field and at high temperature. Two approaches based on local one-body potentials were applied to investigate the spin-orbit interaction between a proton and a 6He nucleus. An optical model analysis revealed that the spin-orbit potential for 6He is characterized by a shallow and long-ranged shape compared with the global systematics of stable nuclei. A semimicroscopic analysis with a alpha+n+n cluster folding model suggests that the interaction between a proton and the alpha core is essentially important in describing the p+6He elastic scattering. The data are also compared with fully microscopic analyses using non-local optical potentials based on nucleon-nucleon g-matrices.

Effects of coupling of forward-backward modal components on continental slope ocean environment.

An exact solution to the two-dimensional (2-D) Helmholtz equation in an inhomogeneous environment may be expressed as a superposition of forward and backward traveling modes. The modal amplitudes of both components are generally coupled in the same manner that positron and electron amplitudes are coupled in the presence of an electric field, resulting in Zitterbewegung for the electron. In ocean acoustics, an accurate treatment of such coupling can become important over long propagation distances across continental slope environments with inhomogenities in either the surface or bottom boundary conditions, the volume of the water column, or both. The coupled integral equation for the range-dependent modal amplitudes is decomposed into the forward and backward modal components using the asymptotic boundary conditions of Green’s function operator. In addition to an exact treatment to the coupling, a perturbation approach is introduced to include the forward-backward coupling effects within a one-way integral equation. The perturbations are introduced via a channel elimination approach that results in a nonlocal optical potential or effective interaction. The nature of this potential and the perturbation convergence criteria are examined for the case of bottom sand dunes and internal waves. [Work supported by the ONR.]

Coupled channels optical theorem and non-elastic cross section sum rule

Within a coupled channels framework a theorem, similar to the optical theorem, is proved which provides a new formula for total non-elastic (inelastic, rearrangement and absorption) cross sections. An approximate cross section energy sum rule is also developed in terms of the trace of the imaginary part of an energy independent, but non-local optical potential. Corrections to sum rule are found small indicating that the result is a good approximation. The theorem and sum rule are applied to the Lane model describing charge exchange reactions.

Few-body nuclear reactions

Three- and four-body scattering is described in the framework of exact Faddeev-type integral equations. They are solved numerically using momentum-space partial-wave basis. The Coulomb interaction between charged particles is included using a novel implementation of the screening and renormalization method. The technique is applied to three- and four-nucleon scattering. Furthermore, the method has been extended successfully to elastic, transfer, and breakup reactions in three-body-like nuclear systems. Examples are deuteron scattering on stable nuclei ranging from 4He to 58Ni and proton scattering on weakly bound two-body system such as 11Be, 15C, and 17O. These calculations allow to evaluate the accuracy of traditional approximate nuclear reaction approaches like the continuum-discretized coupled-channels (CDCC) method but also to test novel dynamical models such as nonlocal optical potentials.

Three-body direct nuclear reactions: Nonlocal optical potential

The calculations of three-body direct nuclear reactions with nonlocal optical potentials are performed for the first time using the framework of Faddeev-type scattering equations. An important nonlocality effect is found for transfer reactions such as $d+^{16}\mathrm{O}\ensuremath{\rightarrow}p+^{17}\mathrm{O}$, often improving the description of the experimental data.

In-medium full-folding model approach to quasielastic (p,n) charge-exchange reactions

A microscopic description of the quasielastic ($p,n$) charge-exchange reaction (here, charge-exchange scattering between analog states) is presented and discussed. Emphasis is focused on the spin-isospin structure of the projectile-target coupling. The model is a coupled-channel extension of the full-folding optical model approach (OMP) developed for nucleon elastic scattering, where emphasis is placed on retaining the genuine off-shell behavior of realistic effective interactions in the nuclear medium. The resulting non-local optical potentials are applied to the calculation of ($p,n$) differential cross sections, with particular emphasis on small-angle Fermi ($\ensuremath{\Delta}S=0$) cross sections to isobaric analog states. These parameter-free results provide a reasonable description of the $^{14}\mathrm{C}$($p,n$) data at proton energies above \ensuremath{\sim}100 MeV, but deteriorate for heavier targets. These shortcomings are analyzed and possible ways to correct them are discussed.

Self-consistent non-local optical potential for XANES

We have studied the performance of the non local optical potential as developed by Fujikawa, Hatada and Hedin [Phys. Rev. B 62, 5387 (2000)] along the lines of the GW approximation for the photoelectron self-energy, in the case of the absorption spectrum of nickel hydrate (Ni(OH2)6). Comparison with the local Hedin-Lundqvist potential [J. Phys. C 4, 2064 (1971)] shows a substantial improvement especially in the low energy region of the spectrum.

A Predictive Model of Nucleon-Nucleus Scattering Cross Sections

Nucleon total reaction and neutron total cross sections as well as angular variables including spin observables from 25 to 300 MeV for all stable nuclei from 6Li to 238U have been predicted that are in good agreement with measured data. Those predictions have been made using non-local, energy dependent, and complex optical potentials in coordinate space formed by full folding of effective nucleon-nucleon interactions with realistic nuclear ground state densities. By inverse kinematics the same model prescription describes exotic (radioactive) nuclei scattering from hydrogen as a target and the results reveal the extended (neutron) distributions such nuclei can have.

Discerning the neutron density distribution of208Pbfrom nucleon elastic scattering

In this paper we seek a measure of the neutron density of ${}^{208}\mathrm{Pb}$ from analyses of intermediate-energy nucleon elastic scattering. The pertinent model for such analyses is based on coordinate space nonlocal optical potentials obtained from model nuclear ground-state densities. Those potentials give predictions of integral observables and of angular distributions, which show sensitivity to the neutron density. When compared with experiment, and correlated with analyses of electron scattering data, the results suggest that ${}^{208}\mathrm{Pb}$ has a neutron skin thickness of $\ensuremath{\sim}0.17 \mathrm{fm}.$

Elastic scattering of polarized protons from3Heat 800 MeV

Cross section, analyzing power, and spin transfer observables for ${p\ensuremath{-}}^{3}\mathrm{He}$ elastic scattering have been obtained using an 800-MeV polarized proton beam at the Los Alamos Meson Physics Facility (LAMPF). The results are compared with theoretical predictions for these observables using nonlocal optical potentials defined by full folding a complex nucleon-nucleon (NN) effective interaction with a ground state ${}^{3}\mathrm{He}$ wave function given by a large space, shell model nuclear structure calculation. The effective NN interaction has been derived from complex NN potentials that fit the (complex) NN scattering phase shifts at 800 MeV.

Theoretical study of the non-local optical potential in EXAFS spectra.

We study the optical potential effects on the extended x-ray absorption fine structure (EXAFS) and x-ray photoelectron diffraction (XPD) spectra. For the valence electron optical potential we use a local density approximation because the charge density changes fairly slowly, whereas we use a non-local optical potential for the core electron part based on GW-approximation. In the Br K-edge EXAFS spectra the present optical potential gives rise to the phase difference and the amplitude reduction; the agreement with the experimental result is excellent. In the N-1s XPD spectra for N2/Ni(100), the spherical wave effects enhance the effects due to the optical potential.

Theory of spin-polarized optical potential.

We develop an approximation for the non-local spin-polarized optical potential theory for atoms in solids at intermediate and high energy. The present approximation for the optical potential builds on the GW-expression. We separate the RPA polarization propagator into a core electron and a valence electron part, and can then achieve a corresponding separation of the optical potential. For the valence electron optical potential we use a local density approximation because the charge density changes fairly slowly, whereas we use a non-local optical potential for the core electron part. Both of them depend on the spin-polarization. We apply this method to electron-Fe elastic scattering in solid, and discuss the results.

Comment on “Energy dependence of theNNtmatrix in the optical potential for elastic nucleon-nucleus scattering”

Elster and Weppner [Phys. Rev. C 57, 189 (1998)], conclude from comparisons of their results with proton elastic scattering data that an optical potential formed from a model based upon the impulse approximation to the scattering is inappropriate below 200 MeV. That is certainly the case with their model, however folded models using $\mathrm{NN} g$ matrices have been shown to be appropriate. They claim that higher order terms in the spectator expansion are the required corrections and that a Faddeev calculation is needed. We show that the less than satisfactory comparisons of their calculated results with data are due to their use of free $\mathrm{NN} t$ matrices as the transition operators. Comparison of predictions obtained from the (complex, nonlocal) optical model potentials we find by folding appropriate medium modified $\mathrm{NN}$ interactions with data from the scattering of 65 and 200 MeV protons from ${}^{40}\mathrm{Ca}$ illustrate this.