Relativistic Optical Model Research Articles

Core Potential Correction from a Relativistic Optical Model

Core Potential Correction from a Relativistic Optical Model

Theoretical study of elastic electron scattering by zinc atoms in the framework of relativistic optical potential model

A theoretical study of integral and differential cross sections as well as of spin-polarization effects is reported for elastic electron scattering by zinc atoms at collision energies up to 3 keV. It has been shown that a P-wave shape resonance appears in the low-energy range of the integral cross sections. Its energy is about 0.19 and 0.20 eV, while its width is about 0.309 and 0.356 eV for the j = 3/2 and j = 1/2 total angular momentum of the electron, respectively. The differential cross sections of scattering and the Sherman-functions S(E,θ) are computed by the parameter-free complex optical potential method. The calculated data are in a good overall agreement with the available experimental and theoretical data in the literature. The energy and angular positions have been located for five critical minima in the differential cross sections. The low-energy minimum is located at [6.63 eV; 102.34°], while the high-energy minimum is at [347.53 eV; 124.11°]. Ten points for the scattered electrons' total spin-polarization (S=±1) have been found in the vicinity of the critical minima along with the energy and the angular widths of the spin-polarization peaks (where |S|≥0.9).

Positron Scattering from the Group IIB Metals Zinc and Cadmium: Recommended Cross Sections and Transport Simulations

Results from the application of our optical potential and relativistic optical potential models to positron scattering from gas-phase zinc (Zn) and cadmium (Cd) are presented. In particular, integral cross sections (ICSs) for elastic scattering, positronium formation, summed discrete electronic-state excitation, and ionization scattering processes are reported for both species and over an extended incident positron energy range. From those ICSs, the total cross section is subsequently constructed by taking their sum. We note that there are currently no experimental data available for any of these scattering processes for either species, with earlier computational results being limited to the elastic channel and restricted to relatively narrow incident positron energy regimes. Nonetheless, we construct recommended positron cross section datasets for both zinc and cadmium over the incident positron energy range of 0–10 000 eV. The recommended positron cross section data are subsequently employed in a multi-term Boltzmann equation analysis to simulate the transport of positrons, under the influence of an applied (external) electric field, through the background Zn and Cd gases. Qualitatively similar behavior in the calculated transport coefficients was observed between both species. Finally, for the case of zinc, the present positron transport coefficients are compared against corresponding results from electron transport with some significant differences now being observed.

Positron Scattering from Gas-Phase Beryllium and Magnesium: Theory, Recommended Cross Sections, and Transport Simulations

Results from the application of our optical potential and relativistic optical potential models to positron scattering from gas-phase beryllium (Be) and magnesium (Mg) are presented. Specifically, total cross sections and integral cross sections for the elastic, positronium formation, summed discrete electronic-state excitation, and ionization scattering processes are reported for both species and over an extended incident positron energy range. Where possible, these results are compared against the existing theoretical and experimental data, although it must be noted here that no current measurements are yet available for Be and those that are available for Mg are largely restricted to the total cross section. Nonetheless, on the basis of that comparison, recommended cross section datasets, for all the aforementioned cross sections, are formed. Those recommended cross section data are subsequently employed in a Boltzmann equation analysis to simulate the transport of positrons, under the influence of an applied (external) electric field, through the background Be and Mg gases. Note that relativistic optical potential results for the elastic momentum transfer cross section are also reported, to allow us to account for anisotropy effects in our transport simulations. Finally, our positron simulation results for quantities such as the ionization rate coefficients and flux and bulk drift velocities are compared with the corresponding electron transport results with significant differences being observed.

Microscopic optical model potential based on a Dirac Brueckner Hartree Fock approach and the relevant uncertainty analysis

A relativistic microscopic optical model potential, named CTOM , for nucleon-nucleus scattering is investigated in the framework of Dirac-Brueckner-Hartree-Fock approach. The microscopic feature of CTOM is guaranteed through rigorously adopting the isospin dependent DBHF calculation within the subtracted T matrix scheme. In order to verify its prediction power, a global study n , p + A scattering are carried out. The predicted scattering observables coincide with experimental data within a good accuracy over a broad range of targets and a large region of energies only with two free items, namely the free-range factor t in the applied improved local density approximation and minor adjustments of the scalar and vector potentials in the low-density region. In addition, to estimate the uncertainty of the theoretical results, the deterministic simple least square approach is preliminarily employed to derive the covariance of predicted angular distributions, which is also briefly contained in this paper.

Relativistic nucleon optical potentials with isospin dependence in a Dirac-Brueckner-Hartree-Fock approach

The relativistic optical model potential (OMP) for nucleon-nucleus scattering is investigated in the framework of Dirac-Brueckner-Hartree-Fock (DBHF) approach using the Bonn-B One-Boson- Exchange potential for the bare nucleon-nucleon interaction. Both real and imaginary parts of isospin-dependent nucleon self-energies in nuclear medium are derived from the DBHF approach based on the projection techniques within the subtracted T -matrix representation. The Dirac potentials as well as the corresponding Schrodinger equivalent potentials are evaluated. An improved local density approximation is employed in this analysis, where a range parameter is included to account for a finite-range correction of the nucleon-nucleon interaction. As an example the total cross sections, differential elastic scattering cross sections, analyzing powers for n, p + 27Al at incident energy 100 keV < E < 250 MeV are calculated. The results derived from this microscopic approach of the OMP are compared to the experimental data, as well as the results obtained with a phenomenological OMP. A good agreement between the theoretical results and the measurements can be achieved for all incident energies using a constant value for the range parameter.

Distorted wave emission function (DWEF) calculations of RHIC HBT and spectra

The emission of pions produced within a dense, strongly-interacting system of matter in the presence of strong radial flow and absorption is described using a relativistic optical model formalism, replacing the attenuated or unattenuated plane waves of earlier emission function approaches with "distorted wave" solutions to a relativistic wave equation including a complex optical potential. The resulting distorted-wave emission function model (DWEF) is used in numerical calculations to fit HBT correlations and the resonance-corrected pion spectrum from central-collision STAR Au+Au pion data at <FONT FACE=Symbol>Ö</FONT>s = 200 GeV. Excellent agreement with the STAR data are obtained. This allows us to predict HBT radii over a range of centralities for both Au+Au and Cu+Cu collisions.

Second-order eikonal corrections for [formula omitted

The first-order eikonal approximation is frequently adopted in interpreting the results of A(e,e′p) measurements. Glauber calculations, for example, typically adopt the first-order eikonal approximation. We present an extension of the relativistic eikonal approach to A(e,e′p) which accounts for second-order eikonal corrections. The numerical calculations are performed within the relativistic optical model eikonal approximation. The nuclear transparency results indicate that the effect of the second-order eikonal corrections is rather modest, even at Q2≈0.2 (GeV/c)2. The same applies to polarization observables, left–right asymmetries, and differential cross sections at low missing momenta. At high missing momenta, however, the second-order eikonal corrections are significant and bring the calculations in closer agreement with the data and/or the exact results from models adopting partial-wave expansions.

Relativistic aspects of few-body physics

After a brief review of alternative approaches to the relativistic treatment of few-body problems, I discuss (i) recent progress in the study of NN and NNN systems, (ii) a new method for solving the Bethe-Salpeter equation in Minkowski space, (iii) new relativistic treatment of spin 3/2 fields, and (iv) construction of conserved electromagnetic currents in a relativistic optical model.

Polishing the lens: I. Pionic final state interactions and HBT correlations: distorted wave emission-function (DWEF) formalism and examples

The emission of pions produced within a dense, strongly interacting system of matter in the presence of strong radial flow and absorption is described using a relativistic optical model formalism, replacing the attenuated or unattenuated plane waves of earlier emission-function approaches with ‘distorted wave’ solutions to a relativistic wave equation including a complex optical potential. The resulting distorted wave emission-function model (DWEF) is used in numerical calculations to fit HBT correlations and the resonance-corrected pion spectrum from central-collision STAR Au+Au pion data at . Excellent agreement with the STAR data is obtained. The applications are extended by applying linear participant scaling to the spacetime parameters of the model while keeping all other parameters fixed. This allows us to predict HBT radii over a range of centralities for both Au+Au and Cu+Cu collisions. Good agreement is found with STAR HBT data for all but the most peripheral collisions. The squares of pionic distorted wavefunctions, obtained as exact numerical solutions to the wave equation, are displayed and significant differences with the results of using the familiar eikonal approximation are found. Using the eikonal approximation leads to a qualitative accounting for the effects of the imaginary part of the optical potential, but fails entirely to include the effects of the real part of the optical potential. A simple example is used to illustrate that an attractive optical potential can have large effects on extracting radii and can also lead to oscillations in radii measured at low momenta.

Quantum dynamics, the RHIC HBT puzzle, and the chiral phase transition

We use a relativistic optical model formalism to describe the emission of pions produced within a dense, strongly-interacting system of matter in the presence of strong radial flow and absorption. The attenuated or unattenuated plane waves of earlier emission function approaches are replaced with “distorted wave” solutions to a relativistic wave equation including a complex optical potential. The resulting distorted-wave emission function model (DWEF) is used in numerical calculations to describe HBT correlations and the resonance-corrected pion spectrum from central-collision STAR Au+Au pion data at s = 200 GeV . The parameters of the emission function are constrained by adopting a pion formation temperature taken from lattice gauge calculations and a chemical potential equal to the pion mass suggested by chiral symmetry restoration. The optical has strong attraction as well as absorption. Excellent agreement with the STAR data are obtained. The applications are extended by applying linear participant scaling to the space-time parameters of the model, so that we predict HBT radii over a range of centralities for both Au+Au and Cu+Cu collisions. Good agreement is found with STAR HBT data for all but the most peripheral collisions. For outgoing pion momenta less than about 500 MeV/c, our pionic distorted wave functions are very different than the corresponding functions obtained by using the familiar eikonal approximation. All of the results presented in the talk are in Refs. [J.G. Cramer, G.A. Miller, J.M.S. Wu and J.H.S. Yoon, Phys. Rev. Lett. 94 (2005) 102302 arXiv:nucl-th/0411031 ], [G.A. Miller and J.G. Cramer, Polishing the lens. I: Pionic final state interactions and HBT correlations. Distorted wave emission function (DWEF) formalism and examples, arXiv:nucl-th/0507004 , submitted to Phys. Rev. C].

Relativistic optical model on the basis of the Moscow potential and lower phase shifts for nucleon-nucleon scattering at laboratory energies of up to 3 GeV

Data of a partial-wave analysis of nucleon-nucleon scattering at energies of up to Elab = 3 GeV (lower partial waves) and the properties of the deuteron are described within the relativistic optical model based on deep attractive quasipotentials involving forbidden states (as exemplified by the Moscow potential). Partial-wave potentials are derived by the inverse-scattering-problem method based on the Marchenko equation by using present-day data from the partial-wave analysis of nucleon-nucleon scattering at energies of up to 3 GeV. Channel coupling is taken into account. The imaginary parts of the potentials are deduced from the phase equation of the variable-phase approach. The general situation around the manifestation of quark effects in nucleon-nucleon interaction is discussed.

RELATIVISTIC OPTICAL MODEL FOR PROTON-NUCLEUS ELASTIC SCATTERING

The relativistic description of the proton-nucleus elastic scattering can be considered within the framework of a relativistic optical potential model. The elastic scattering of proton with the nuclei 12 C , 16 O , 20 Ne , and 24 Mg at 800 MeV and 1.04 GeV are studied for relativistic and nonrelativistic treatments. The real optical potentials and the differential cross sections of these reactions are calculated. The obtained results are compared with the corresponding results obtained from the calculation depending on the Woods–Saxon optical potential which were adjusted to fit the experimental data. The present results are in good agreement with the experimental data.

Relativistic Hartree-Bogoliubov model with density-dependent meson-nucleon couplings

The relativistic Hartree-Bogoliubov (RHB) model is extended to include density dependent meson-nucleon couplings. The effective Lagrangian is characterized by a phenomenological density dependence for the $\sigma$, $\omega$ and $\rho$ meson-nucleon vertex functions, adjusted to properties of nuclear matter and finite nuclei. Pairing correlations are described by the pairing part of the finite range Gogny interaction. The new density-dependent effective interaction DD-ME1 is tested in the analysis of the equations of state for symmetric and asymmetric nuclear matter, and of ground-state properties of the Sn and Pb isotopic chains. Results of self-consistent RHB calculations are compared with experimental data, and with results previously obtained in the RHB model with non-linear self-interactions, as well as in the density dependent relativistic hadron field (DDRH) model. Parity-violating elastic electron scattering on Pb and Sn nuclei is calculated using a relativistic optical model with inclusion of Coulomb distortion effects, and the resulting asymmetry parameters are related to the neutron ground-state density distributions.

Contribution of the Δ(1232) to the (e, e'p) reaction along with Coulomb distortion effects

The Δ-resonance contribution has been included in the (e, e'p) reaction along with Coulomb distortion effects. We treat the resonance via a non-relativistic Δ current operator and use a Dirac Hartree single-particle model for the ground-state single-particle wave function and a relativistic optical model for the knocked-out proton wave function. It is assumed that the π-meson created by the virtual photon is absorbed in the target nucleus following the production of a Δ-resonance. Our DWBA calculation shows that the Δ-resonance contribution to the (e, e'p) reaction cross-section is 10-15% for an energy of 250 MeV transfered to the proton knocked out of the s-shell of 40Ca, in the parallel and perpendicular kinematics.

Pseudospin and spin symmetry in nucleon-nucleon scattering

The orthogonal transformation from the spin representation to the pseudospin representation for the nucleon-nucleon scattering matrix is derived. The phase shifts and mixing angles are calculated in the pseudospin representation using the measured phase shifts and mixing angles in the spin representation. The scattering matrix in both representations is investigated and the extent of spin and pseudospin symmetry violation is determined. Finally, a speculation is made that the pseudospin symmetry generators relevant for relativistic mean field models and optical models may need to be generalized for the two-nucleon interaction.

Effects of the form factor on exclusive (e, e′p) reactions

For the exclusive (e, e′p) reaction, we discuss the possibility of observing effects due to swollen nucleons in the nuclear medium, such as 40Ca and 208Pb, by introducing form factors of the nucleon in the nuclear medium. These form factors include effectively the change of nucleon properties in the nuclear medium. This calculation is performed by using a Dirac-Hartree single particle model for a bound state and a relativistic optical model for a continuum state with inclusion of the electron Coulomb distortion. The effect of the form factor, which increases with higher momentum transfer (q ≥ 400MeV/c) is too small to be discerned from the errors on the available experimental data. But it affects the determination of spectroscopic factors to some extent.

Parity violating elastic electron scattering and neutron density distributions in the relativistic Hartree-Bogoliubov model

Parity-violating elastic electron scattering on neutron-rich nuclei is described in the framework of relativistic mean-field theory. Self-consistent ground state density distributions of Ne, Na, Ni, and Sn isotopes are calculated with the relativistic Hartree-Bogoliubov model, and the resulting neutron radii are compared with available experimental data. For the elastic scattering of 850 MeV electrons on these nuclei, the parity-violating asymmetry parameters are calculated using a relativistic optical model with inclusion of Coulomb distortion effects. The asymmetry parameters for chains of isotopes are compared, and their relation to the Fourier transforms of neutron densities is studied. It is shown that parity-violating asymmetries are sensitive not only to the formation of the neutron skin, but also to the shell effects of the neutron density distribution.

Parity violating elastic electron scattering and Coulomb distortions

Parity violating elastic electron-nucleus scattering provides an accurate and model-independent measurement of neutron densities, because the ${Z}^{0}$ couples primarily to neutrons. Coulomb distortion corrections to the parity violating asymmetry ${A}_{l}$ are calculated exactly using a relativistic optical model. Distortions significantly reduce ${A}_{l}$ in a heavy nucleus. However, even with distortions, an experiment to measure the neutron radius is feasible. This will aid the interpretation of future atomic parity violation measurements and provide fundamental nuclear structure information. Coulomb distortions and small differences between neutron and proton radii could be important for a standard model test on ${}^{4}\mathrm{He}$, ${}^{12}\mathrm{C}$, or ${}^{16}\mathrm{O}$.

Inelastic ν and [formula omitted] scattering on nuclei and “strangeness” of the nucleon

Possibilities to extract information on the strange form factors of the nucleon from neutrino (anti-neutrino) inelastic scattering on nuclei, in an energy range from 200 MeV to 1 GeV and more, are investigated in detail. All calculations are performed within two relativistic independent particle models (Fermi gas and shell model); the final state interactions of the ejected nucleon are taken into account through relativistic optical model potentials. We have shown that the values of the cross sections significantly depend on the nuclear model (especially in the lower energy range). However, the NC/CC neutrino-anti-neutrino asymmetry in a medium-high energy range shows a rather small dependence on the model and allows to disentangle different values of the parameters that characterize the strange form factors. We have calculated also the ratio of the cross sections for inelastic NC scattering of neutrinos on nuclei, with the emission of a proton and of a neutron. Our calculations show that at high neutrino energy this ratio depends rather weakly on the nuclear model and confirm previous conclusions on the rather strong dependence of this ratio upon the axial strange form factors; however, at E ν ⩽ 200 MeV, the FSI are found to significantly affect the ratio.