Second-order Optical Potential Research Articles

Optical potential approach toK+dscattering at low energies

We study the $K^{+}d$ scattering at low energies using the optical potential. Our optical potential consists of the first-order and second-order terms. The total, integrated elastic and elastic differential cross sections at incident kaon momenta below 800 MeV/c are calculated using our optical potential. We found that our results are consistent with the Faddeev calculation as well as the data and especially the second-order optical potential is essential to reproduce them at low energies. We also discuss the multiple scattering effects in this process.

Effects of the range of the pion-nucleon interaction on pion-nucleus scattering

The range of the pion-nucleon interaction is a model-dependent quantity. The range affects the off-shell behavior of the pion-nucleus optical potential that is calculated from it. This will affect the differential cross sections in turn. In this paper, we make an extensive investigation of the effects of varying the range of the pion-nucleon interaction on the differential cross sections of 14C with a first-order and a second-order optical potential. This is done for π + → π +, π + → π 0 and π + → π − reactions at 80, 162 and 292 MeV. It is found that varying the range of the interaction from half to twice its nominal value will produce large differences in the differential cross section including changes in the position of the first minimum. These calculations are performed in momentum space with the optical potential constructed by calculating the Fermi integral. Previous calculations that investigated the effect of the range of the pion-nucleon interaction on the differential cross section, were usually done with a factorized approximation. The results found here are contrary to previous results that agreed with the general argument that the location of the minima should be set by the location of the interaction region in the nucleus. We also investigate taking the infinite-range (zero-range in configuration space) limit. It is found that the differential cross sections change significantly in this limit for the charge-exchange reactions. At 80 and 292.5 MeV it is found that the pion-nucleon P 33 channel still dominates the differential cross section.

Microscopic coupled-channel description of pion inelastic scattering from rotational nuclei.

We establish the applicability of a microscopic coupled-channel theory to pion inelastic scattering from rotational nuclei for the purpose of studying details of neutrons and proton densities. The theory is completely specified as a result of our taking the second-order pion-nucleus optical potential from previous phenomenological studies of elastic and charge-exchange data on spherical nuclei and of our using nuclear wave functions obtained from a Hartree-Fock + BCS calculation with the Skyrme ${\mathrm{SKM}}^{\mathrm{*}}$ interaction. Making multipole expansions leads to a set of coupled equations, which is solved for the scattering cross sections in a standard manner. Numerical studies are presented for the specific nucleus $^{152}\mathrm{Sm}$. The theory is shown to be stable with respect to truncation of the expansions: we find rapid convergence of the cross sections for elastic scattering and inelastic scattering to the first ${2}^{+}$ and ${4}^{+}$ states as transition densities of successively higher multipolarity are added, and as the number of coupled rotational nuclear levels is increased. Medium modifications are included and have a small but significant effect on the cross section: for scattering to the ${2}^{+}$ state, their effect is about 10% for both ${\mathrm{\ensuremath{\pi}}}^{\mathrm{\ensuremath{-}}}$ and ${\mathrm{\ensuremath{\pi}}}^{+}$. We also quantify the extent of sensitivity of pion scattering cross sections to variations in the size and shape of the neutron and proton distribution using parametrized Woods-Saxon forms for the density.

Influence of nuclear structure on the characteristics of light pionic atoms.

An extended version of the standard pion-nucleus optical potential is presented using the momentum-space formulation of the multiple scattering theory. This potential is used for studying the influence of nonlocal effects and nuclear structure on the characteristics of light pionic atoms. A method is suggested to treat the long-range two-nucleon correlations involved in the second order optical potential. Particularly, the spin-orbital Fermi motion correction to the first-order optical potential, the center-of-mass correlations, and the spin-isospin corrections to the second-order optical potential are discussed in some detail. The results are presented in comparison with the experimental data for the 1s and 2p levels of light pionic atoms up to $^{44}\mathrm{Ca}$.

Pion inelastic scattering and the neutron density in 152Sm

We apply a microscopic coupled-channel theory of pion elastic and inelastic scattering to study the neutron and proton densities of deformed 152Sm obtained from a Hartree-Fock+BCS calculation with the Skyrme SkM ∗ interaction. The approach utilizes a second-order pion-nucleus optical potential which has been determined from a previous phenomenological study of elastic and charge-exchange data on spherical nuclei. We obtain an excellent representation of the 180 MeV π + data. The π − data are not quite as well reproduced, suggesting that the radius of the neutrons in 152Sm is smaller than the one predicted by our SkM ∗ Hartree-Fock theory. However, we find the deformation of the neutrons and protons as given by the Hartree-Fock theory ( β 2n = 0.263, β 2p = 0.287) to be consistent with the experimental data.

Multiple scattering contributions to the RIA optical potential

Multiple scattering contributions to the relativistic impulse approximation (RIA) optical potential in the low-energy region as well as in the intermediate- and high-energy regions are examined. A relativistic analog of the non-relativistic multiple scattering theory is adopted and the second-order optical potential based on the RIA is derived as multiple scattering contributions. In order to evaluate the second-order optical potential, the correlation function formalism extended to the generalized RIA is used. Numerical calculations are performed for proton elastic scattering by 40Ca at 100, 200, 500 and 800 MeV. Calculated results including multiple scattering contributions show better agreement with the experimental data for both cross section and spin observables at large angles. Although multiple scattering contributions are considered to become larger in low-energy region, they are not so large that the first-order RIA calculations are changed drastically at 100 MeV.

Self-consistent method for obtaining a local optical potential from phaseshifts

A simple approach to the construction of the phase equivalent local optical potential from phaseshifts for a given energy is proposed. The local potential is expanded in a set of given functions and the expansion coefficients are determined in a self-consistent way. In each step one has to solve a number of radial equations with a local potential as well as the set of algebraic equations. The method is first tested on simple examples and then applied to the construction of the local potential equivalent to the second-order non-local optical potential for positron-hydrogen-atom scattering.

Calculation of the first-order s-wave optical potential in pionic atoms.

A microscopic calculation of the first-order s-wave optical potential in pionic atoms which allows the inclusion of the off-shell dependence in momentum of the s-wave \ensuremath{\pi}N amplitude is carried out. Binding effects are also properly taken into account. The results obtained, together with those of the second-order optical potential, are in disagreement with phenomenological optical potentials. The results of the paper suggest the need of new and precise experiments on \ensuremath{\pi}N scattering at low energies and other experiments which can provide precise values of the \ensuremath{\pi}N scattering lengths.

Self-energy corrections in the second-order relativistic impulse approximation model

Abstract The elastic scattering of protons from 16 O, 40 Ca and 208 Pb at 500 and 800 MeV is analyzed using a simplified scalar-vector form of the second-order relativistic impulse approximation optical model. Dispersion effects are introduced into the second-order optical potential by including self-energy terms in the intermediate state projectile propagator. The effect of these corrections on the predicted elastic scattering observables is discussed.

Positron-atomic-hydrogen elastic scattering: continued-fraction approach to a second-order optical model

The recently proposed method of continued fractions is applied to the second-order non-local optical potential with closure for the incident-energy range 54.4-300 eV. The method is rapidly convergent even for partial waves with small l. The results differ from those obtained by Mukherjee et al. (1982) for the same model and agree well with the recent results of the unitarised eikonal-Born series method.

Spin asymmetry in elastic scattering of electrons by hydrogen atoms.

We have performed calculations of the spin asymmetry in elastic scattering of electrons by hydrogen atoms using an optical potential appraoch. An 18-state basis set is employed. The scattering calculations are carried out explicitly for the channels corresponding to the lowest six states. The effects of other channels are included via a second-order optical potential. The present values of the asymmetry at 90\ifmmode^\circ\else\textdegree\fi{} are in agreement with the recent experimental values of Fletcher et al. and resolve the discrepancy between theory and experiment which they reported in the intermediate-energy region.

Microscopic calculation of the imaginary optical potential forPb208(p,p) at 14 MeV

A microscopic calculation of the second-order imaginary optical potential is made for $^{208}\mathrm{Pb}$(p,p) at 14 MeV incident energy using random phase approximation transition densities for intermediate excited states. The sensitivity of the surface absorption on details of the nuclear structure wave functions is discussed.

Investigation of second-order optical potential for elastic π-4He scattering

Pion elastic scattering on 4He was calculated within the second-order optical model. The roles of pion double isospin and spin flip in intermediate states and of recoil correlation were investigated in some detail. A second-order term, which represents the effects of true pion absorption, is also included in the optical potential. A comparison was made between the results obtained using the Kerman-McManus-Thaler and Watson formulations of multiple scattering theory.

Cross section calculations for nucleon-Ca40andα-Ca40elastic scattering from microscopic nonlocal optical potentials

Microscopic calculations of the second-order imaginary optical potential are made for /sup 40/Ca(n,n), /sup 40/Ca(p,p), and /sup 40/Ca(..cap alpha..,..cap alpha..) at various incident energies using random phase approximation transition densities for intermediate excited states. Elastic scattering cross sections are calculated from the microscopic nonlocal potentials and are compared to cross sections which have been calculated from the equivalent local potentials. The validity of the local approximation to the nonlocal potentials is discussed.

Isospin dependence of second-order pion-nucleus optical potential

A universal form for the dependence of the second-order pion-nuclear optical potential on the nuclear and pionic isospin and on the densities $\ensuremath{\rho}$ and $\ensuremath{\Delta}\ensuremath{\rho}$ is derived, where $\ensuremath{\rho}$ is the total density and $\ensuremath{\Delta}\ensuremath{\rho}$ is the valence neutron density. The result is characterized by five complex parameters for each partial wave channel contributing to the optical potential. Our result applies to scattering in the vicinity of the (3,3) resonance, and the parameters may be most cleanly determined phenomenologically by applying the theory to elastic, and single- and double-charge-exchange data for medium to heavy weight $J=0$, spherical nuclei. The relationship between the parameters and a microscopic cluster decomposition of the optical potential is explored. The parameters are calculated theoretically for selected second-order processes to obtian a first orientation to the magnitude of the terms. The results show, in particular, that large contributions to double charge exchange arise from nonanalog intermediate nuclear states and that these contributions have a characteristic isospin dependence which is different from that found in the simple models previously studied.

The imaginary part of the α- 40Ca optical potential

An approximate semiclassical formula for the imaginary part of the scattering phase shift due to Feshbach's second-order optical potential has been obtained. This formula has been applied to the α- 40Ca system and the imaginary phases compared with those calculated from phenomenological optical potentials fitted to the experimental data. The target nucleus was described by a harmonic oscillator shell model and the α-nucleus interaction was obtained from a phenomenological α-nucleon interaction. This model gives an absorption which is stronger than that calculated from phenomenological potentials by a factor of two for a low α-particle incident energy ( E lab = 29 MeV). The discrepancy is less for higher energies. The results are very sensitive to the strength of the α-nucleon interaction. It was found that the giant quadrupole and monopole collective states make only a small contribution to the total absorption.

Localized second-order optical potential for electron scattering in terms of imaginary-frequency susceptibilities

A local approximation to the second-order optical potential for elastic scattering of low-energy electrons from ground-state atoms is expressed in terms of the imaginary-frequency susceptibilities of the atom due to a point charge and to modified perturbing potentials. This provides a basis for the physically appealing concept of regarding the perturbation due to the projectile as having a position-dependent effective frequency associated with it. The result is extended to higher energies with the use of the concept of a local kinetic energy. With a semiclassical approximation the result reduces to a simple general form that should be useful for model potential studies of electron-atom and electron-molecule scattering. Alternatively, variational functionals for the susceptibilities can be used to calculate the approximate optical potential most rigorously without making effective-frequency, average-kinetic-energy, or semiclassical approximations. Intermediate levels of rigor are also possible.

Nuclear structure approach to the calculation of the imaginary alpha-nucleus optical potential

A microscopic calculation of the second-order imaginary optical potential for $^{40}\mathrm{Ca}$($\ensuremath{\alpha}$,$\ensuremath{\alpha}$) is made for incident energies of 31 and 100 MeV using random phase approximation transition densities for intermediate excited states. The projectile is treated as an elementary particle, and the alpha-nucleon interaction is normalized by fitting ${3}^{\ensuremath{-}}$ inelastic cross sections with a folded M3Y potential. The use of an optical Green's function for the intermediate propagator is found to be important. Equivalent local potentials are obtained and used to calculate elastic scattering cross sections. Agreement with low-angle experimental data is fair at 31 MeV but at 100 MeV the calculated cross sections indicate much too little absorption.NUCLEAR REACTIONS ($\ensuremath{\alpha}$,$\ensuremath{\alpha}$) scattering, calculation of imaginary optical potential at $E=31 \mathrm{and} 100$ MeV.

The Schwinger variational method in electron-atom and electron-molecule scattering theory with polarisation

A method of implementing the Schwinger variational expression in electron-atom and electron-molecule scattering calculations with polarisation included via an ab initio optical potential is described. The Schwinger expression is also used to obtain approximate molecular Hartree-Fock continuum orbitals at arbitrary energies which are needed to calculate the optical potential above the first electron excitation threshold. Explicit closed-form expressions of integrals required for a Gaussian orbital basis are given. The method is applied to low-energy electron-helium scattering with a second-order optical potential.

Theory of the optical potential for strongly interacting particles

We use the insight gained from the study of analyticity and unitarity to suggest a new method for the construction of the optical potential for strongly interacting particles. The calculational procedures suggested in this work are such as to associate a distinct physical process with each term in our expansion of the optical potential. By starting with various decompositions of the A-body target wave function we can obtain theories of varying complexity. Introducing a decomposition with respect to states of the (A-1) -body system, we obtain a construction of the optical potential based upon the single-scattering approximation. If we introduce a decomposition of the target ground state with respect to states of the (A-2) -body system we obtain a more complex theory. In the extended theory, the first-order optical potential of the simpler model is readily extracted along with a second-order potential that depends on the detailed structure of the two-body density matrix of the target. The role of Pauli and short-range correlations in modifying the calculation of the second-order potential is discussed. The physical picture that emerges from this analysis is different from that obtained from multiple-scattering theory, although the two theories are in general agreement as to themore » nature of the first-order optical potential. When the fixed-scatterer approximation is used to simplify calculations based upon multiple-scattering theory, the second-order optical potential is found to depend on the two-body correlation function. In contrast, for the theory presented here, we find that the inclusion of Pauli and/or short-range correlations is a relatively minor aspect in the construction of the second-order potential. Indeed, we would obtain a second-order potential for an uncorrelated system.« less