Nonlocal Optical Potentials Research Articles

Optical potentials in algebraic scattering theory

Using the theory of induced representations new realizations for the Lie algebras of the groups , , are found. The eigenvalue problem of the Casimir operators yield Schrodinger equations with non-Hermitian interaction terms (i.e. optical potentials). For the group we have a two-parameter family of (matrix-valued) potentials containing terms of Poschl-Teller and Gendenshtein type. We calculate the S-matrices for special values of this two-parameter family. In particular we also include a derivation of the S-matrix for the two-dimensional scattering problem on a complex Gendenshtein potential. The canonically transformed realization results in a non-local optical potential.

Microscopic model analyses of the elastic scattering of 65 MeV protons from targets of diverse mass

Nonlocal coordinate space optical potentials for the scattering of 65 MeV protons from nuclei ranging in mass from ${}^{6}\mathrm{Li}$ to ${}^{238}\mathrm{U}$ have been defined by folding a complex, medium-dependent effective interaction with the density matrix elements of each target. The effective interaction is based upon solutions of the Lippmann-Schwinger and Brueckner-Bethe-Goldstone equations having the Paris potential as input. The nuclear structure information required in our folding model is the one-body density matrix elements for the target and the single- nucleon bound state wave functions that they weight. For light mass nuclei, very large basis shell model calculations have been made to obtain the one-body density matrix elements. For medium and heavy mass nuclei, a very simple shell model prescription has been used. The bound-state single-particle wave functions that complete the nuclear density matrices are either Woods-Saxon or harmonic oscillator functions. The former are employed in most cases when large basis structure is available. For light nuclei $(Al~16)$ Woods-Saxon potential parameters and harmonic oscillator lengths are determined from fits to electron scattering form factors. For all other nuclei, we use harmonic oscillator functions with the oscillator lengths set from an ${A}^{1/6}$ mass law. Using this microscopic model, optical potentials result from which differential cross sections, analyzing powers, and spin rotations are found. In general the calculated results compare very well with data when the effective interactions are determined from a mapping of nucleon-nucleon g matrices. This is not the case when effective interactions built from a mapping of (free) t matrices are used.

Microscopic model analyses of select scattering of 135 MeV protons from12C

An effective interaction based upon two nucleon $g$ matrices has been used in a fully microscopic calculation of the nonlocal proton-${}^{12}$C optical potential at 135 MeV. Excellent predictions of the differential cross section from elastic scattering result. The same effective interaction, combined with one body density matrix elements specified by a large basis shell model for ${}^{12}$C ${(}^{12}$N), and with the relative motion wave functions from the nonlocal optical potentials define the distorted wave approximation that is employed to analyze cross sections from inelastic proton scattering. With that approximation, inelastic scattering cross sections from various states in ${}^{12}$C and from the charge exchange reactions leading to the low excitation states of ${}^{12}$N have been evaluated. Our results indicate a problem with the structure model when 3$\ensuremath{\Elzxh}\ensuremath{\omega}$ components are considered.

Fully microscopic model analyses of the elastic scattering of 200 MeV protons from targets of diverse mass

Non-local optical potentials for the scattering of 200 MeV protons from 22 nuclei, ranging in mass from to , have been defined by folding a complex, medium-dependent effective interaction with density matrix elements of each target. The effective interaction is based upon solutions of the Lippmann - Schwinger and Brueckner - Bethe - Goldstone equations having the Paris potential as input. The bound-state single-particle wavefunctions that specify the nuclear density matrices are Woods - Saxon functions for and harmonic oscillator functions thereafter. The resulting differential cross sections, analysing powers and spin rotations all compare well with the known data.

Multi-

Elastic and inelastic scattering data from the scattering of 160 MeV protons from $^{14}\mathrm{N}$, and of 200 MeV protons from $^{16}\mathrm{O}$ have been analyzed using a fully microscopic distorted wave approximation. The analyses involve large space (multi-\ensuremath{\Elzxh}\ensuremath{\omega}) shell model wave functions, an effective nucleon-nucleon interaction that is energy and medium dependent, and fully microscopic (nonlocal) optical potentials built with that same effective interaction. The results for $^{14}\mathrm{N}$ and $^{16}\mathrm{O}$ correlate with analyses of elastic and inelastic electron scattering form factors indicating that improvements are needed in the shell model interactions used to obtain the nuclear wave functions. \textcopyright{} 1996 The American Physical Society.

Fully microscopic model analysis of the elastic and inelastic scattering of protons from 12C and for energies in the range 200 to 800 MeV.

Medium modified effective two nucleon interactions are defined for protons incident upon $^{12}\mathrm{C}$ with energies in the range 200 to 800 MeV. Those effective interactions have been folded with the ground-state density to specify nonlocal optical potentials that were then used to analyze the elastic scattering differential cross sections and analyzing powers. A select set of isoscalar and isovector, positive and negative parity, inelastic proton scattering transitions have also been analyzed using the same (microscopic) optical models to define the distorted wave functions needed in distorted wave approximation (DWA) calculations of the associated differential cross sections and analyzing powers. All results are compared with ones found using the Love-Franey effective interactions. The nuclear structure relating to these transitions was chosen from (0+2)\ensuremath{\Elzxh}\ensuremath{\omega} and (1+3)\ensuremath{\Elzxh}\ensuremath{\omega} shell model calculations of the positive and negative parity spectra of $^{12}\mathrm{C}$, respectively. In comparison to previous DWA analyses, very good fits to most data over a range of energies are achieved with no variation being made to any of the input details, save for the choice of the effective interaction. Evidence of limitations of the spectroscopy and of specifics in the defined effective interactions at higher energies is found.

Deep pionic bound states in a nonlocal optical potential.

Pion-nucleus bound states are investigated in momentum space using a microscopic optical potential with energy dependences and nonlocalities arising from elementary potential models. We confirm the existence of deep, hybrid Coulomb-nuclear 1s, 2s, and 2p bound states, but find them to be considerably broader (\ensuremath{\sim}20--300 %) than reported previously for other potentials and slightly (\ensuremath{\sim}7%) more bound. Although the states still remain nonoverlapping, their larger widths may affect experimental searches.

Description of proton elastic scattering on 6,7,9,11Li with microscopic effective interaction.

Proton elastic scattering on Li isotopes $^{6,7,9,11}\mathrm{Li}$ at ${\mathit{E}}_{\mathit{p}}$\ensuremath{\sim}60 MeV is described by a nonlocal optical model potential which is constructed by folding a microscopic density-dependent effective interaction with Hartree-Fock single-particle wave functions of the target nuclei. Experimental data are well-described systematically from $^{6}\mathrm{Li}$ to $^{11}\mathrm{Li}$ by our model calculation, implying that the underlying microscopic theory is applicable even at the extreme circumstance of the neutron drip line limit.

Analysis of neutron+nucleus scattering data with nonlocal optical potentials based on the resonating-group formulation.

A nonlocal optical model is used to analyze the scattering of neutrons by $^{12}\mathrm{C}$, $^{16}\mathrm{O}$, $^{28}\mathrm{Si}$, $^{32}\mathrm{S}$, and $^{40}\mathrm{Ca}$ in the energy region from about 13 to 40 MeV. The real part of the internuclear interaction is obtained by using the prescription of a recently proposed nucleon + nucleus model K, which is a microscopic model based on the resonating-group formulation. A simple approximation, called the spherical approximation, is introduced to treat the cases involving non-closed-subshell target nuclei. The imaginary part used is purely phenomenological, and is obtained by simply adopting the imaginary potentials which have been found previously by other authors in their analyses with the usual local optical model. The influence of the Perey effect is taken into account by the introduction of an additional multiplicative factor. The results obtained for the differential scattering cross sections and the analyzing powers are quite satisfactory. The essential characteristics of the measured angular distributions are well reproduced. This suggests that model K is a viable model of great simplicity and can be employed to make systematic and large-scale analyses of all existing nucleon + nucleus scattering data.

Analysis of the optical potential with coupled-channel scattering equations: Energy dependence and coordinate-space behavior.

We apply a Sturmian-expansion method we have recently developed to perform an exact multichannel analysis of the optical potential. By combining this method with momentum-space integral-equation techniques we can obtain a finite-rank representation of the nonlocal optical potential in which coupling effects are exactly taken into account. In the framework of this approach we provide also detailed expressions of the various contributions arising when couplings are treated perturbatively. These iterative-perturbative approximation schemes are generally employed in phenomenological optical-potential calculations. Our results do not apply to low-order terms only, but can be extended to any desired order. We find that a proper treatment of inelastic-inelastic couplings may be necessary in order to reproduce the detailed nonlocal structure of the optical potential. We then perform an energy-dependent analysis of the resulting optical potential. To this end, the rapid fluctuations in energy due to compound or quasicompound resonances are subtracted off by means of a direct algebraic procedure. The connection of this subtraction method with Feshbach's general reaction theory is clarified. We analyze the energy dependence of the smooth optical interaction in the light of a dispersion-theoretic approach, by resorting to simple analytic parametrizations of the real and imaginary parts. Both a model multichannel problem and the realistic neutron${\mathrm{\ensuremath{-}}}^{208}$Pb case have been considered.

Inelastic Scattering Theory of Channeled Electrons and Positrons by Optical Potential Method

Based on the optical potential method, a formal theory of electron and positron channeling is developed by taking account of spinor factors. Starting with the Lippmann-Schwinger equation, a Schrodinger like equation for channeling states is derived which includes relativistic corrections and a non-local optical potential. From an iterative formula of the optical potential, absorption coefficients for channeled electrons and positrons and the line width in chauneling radiation can be calculated in a systematic manner. Radiation from channeled electrons and positrons is clearly interpreted in terms of the optical potential.

Non-local coupled-channels optical calculation of electron scattering by atomic hydrogen at 54.42 eV

The present study incorporates the nonlocal optical potentials for the continuum within the coupled-channels optical framework to study electron scattering from atomic hydrogen at 54.42 eV. Nine-state coupled-channels calculations with nonlocal and local continuum optical potentials were performed. The results for differential, total and ionisation cross sections as well as the 2p angular correlation parameters lambda and R are comparable with other nonperturbative calculations. There are still discrepancies between theory and experiment, particularly for lambda and R at larger angles.

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.

Effects of the phonon structure on the elastic scattering probabilities of light particles from surfaces.

The effects of the phonon structure of a Cu(100) surface on the elasticscattering probabilities are considered for D/sub 2/ and He at lowenergies and temperatures. The calculation is based on iteratively solving theSchroedinger equation for the scattering of a particle from a flat surface with the Born approximation for the self-energy as a nonlocal and energy-dependent optical potential. The phonon contribution to this self-energy has been evaluated by performing surface lattice dynamics of a semi-infinite Cu(100) crystal.

Comment on "Elimination of the linear energy dependence of nonlocal optical potentials"

A paper dealing with the unitary off-shell ambiguity of effective interactions is criticized. A so-called optimal transformation of the resonating group equation is based on error and not valid.

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.

On the role of background scattering in the resonance interaction of slow electrons with molecules

The problem of interaction of slow electrons e- with molecules XY has been formulated directly for the collision T operator. A basis of continuous spectrum states has been used in which the K matrix, related to the collision operator, is additive with respect to the direct and the resonance interaction mechanism. This makes it possible to perform an analytical study on the contribution of these mechanisms to the processes of scattering and dissociative attachment (DA) of electrons as well as to define the validity of the theoretical approaches available. It has been shown that the elastic background scattering decreases the DA rate in the low-energy case and can be taken into account by a redefinition of the effective non-local optical potential which is introduced in the configuration interaction method to describe nuclear motion in intermediate states of XY-. The inelastic background scattering distorts the form of the resonance peaks and increases the DA rate due to the appearance of new reaction channels taking place when the molecule undergoes vibrational excitation in the collision.

Elimination of the linear energy-dependence of nonlocal optical potentials.

An optical nonlocal potential depending linearly on the energy is transformed into a phase-equivalent energy-independent potential which depends on the angular momentum via l(l+1). The transformation is exact, can be performed for a large class of nonlocal potentials, and can be interpreted in terms of physics. The relation between hard and structural core for phase-equivalent potentials is discussed.

Sturmian representation of the optical model potential due to coupling to inelastic channels

Eigenfunctions for a set of coupled equations with outgoing wave boundary conditions in each channel n, called Weinberg states, are used to construct a separable representation of the matrix Green's functions G/sub n/n'(r,r'). In terms of this representation, novel forms for the nonlocal optical potential in the elastic channel can be written, which take into account interchannel coupling to all orders. Expressions for the corrections to the distorted-wave Born approximation inelastic transition potentials are also given. Two expressions for the effective potential in the elastic channel are given. One involves the Weinberg eigenfunctions in the inelastic space. The other involves the Weinberg eigenfunctions for the complete (elastic plus inelastic) space. The former is more suitable for the evaluation of corrections to the distorted-wave Born approximation, the latter occurs naturally in the solution by the Sturmian method of the full set of coupled equations. Numerical approximations to the Weinberg states can be constructed by introducing auxiliary sets of Sturmian states, one set in each channel. By truncating the dimension of each set to the size M, one obtains separable approximations of rank M to the effective potentials. The procedure involves the diagonalization of a matrix of dimension (M x N)/sup 2/,more » where N denotes the number of channels included in the inelastic space.« less

Exchange contributions to optical-model potentials

The optical potential is computed exactly for a two-channel 1s-2s model in electron-hydrogen scattering. Exchange contributions in the 2s channel are calculated explicitly and compared to other terms in the nonlocal optical potential. It is concluded that exchange effects in excited states cannot be neglected in calculating optical potentials for low-energy electron-atom scattering.