Ch. Elster

THREE-BODY SCATTERING AT INTERMEDIATE ENERGIES

The Faddeev equation for three-body scattering at arbitrary energies is formulated in momentum space and directly solved in terms of momentum vectors without employing a partial-wave decomposition. In its simplest form, the Faddeev equation for identical bosons, which we are using, is a three-dimensional integral equation in five variables, magnitudes of relative momenta and angles. This equation is solved through Pade summation. Based on a Malfliet-Tjon-type potential, the numerical feasibility and stability of the algorithm for solving the Faddeev equation is demonstrated. Special attention is given to the selection of independent variables and the treatment of three-body breakup singularities with a spline-based method. The elastic differential cross section, semiexclusive d(N,N{sup }) cross sections, and total cross sections of both elastic and breakup processes in the intermediate-energy range up to about 1 GeV are calculated and the convergence of the multiple-scattering series is investigated in every case. In general, a truncation in the first or second order in the two-body t matrix is quite insufficient.

Three-Body Bound-State Calculations Without Angular-Momentum Decomposition

Abstract.u2002The Faddeev equations for the three-body bound state are solved directly as three-dimensional integral equation without employing partial wave decomposition. The numerical stability of the algorithm is demonstrated. The three-body binding energy is calculated for Malfliet-Tjon-type potentials and compared with results obtained from calculations based on partial wave decomposition. The full three-body wave function is calculated as function of the vector Jacobi momenta. It is shown that it satisfies the Schrödinger equation with high accuracy. The properties of the full wave function are displayed and compared to the ones of the corresponding wave functions obtained as finite sum of partial wave components. The agreement between the two approaches is essentially perfect in all respects.

First order relativistic three-body scattering

Relativistic Faddeev equations for three-body scattering at arbitrary energies are formulated in momentum space and in first order in the two-body transition operator directly solved in terms of momentum vectors without employing a partial wave decomposition. Relativistic invariance is incorporated within the framework of Poincare invariant quantum mechanics and presented in some detail. Based on a Malfliet-Tjon-type interaction, observables for elastic and breakup scattering are calculated up to projectile energies of 1 GeV. The influence of kinematic and dynamic relativistic effects on those observables is systematically studied. Approximations to the two-body interaction embedded in the three-particle space are compared to the exact treatment.

Modern NN force predictions for the total nd cross section up to 300 MeV

For several modern nucleon-nucleon potentials state-of-the-art Faddeev calculations are carried out for the nd total cross section between 10 and 300 MeV projectile energy and compared to new high precision measurements. The agreement between theory and data is rather good, with an exception at higher energies where a 10{percent} discrepancy builds up. In addition the convergence of the multiple scattering series incorporated in the Faddeev scheme is studied numerically with the result that rescattering corrections remain important. Based on this multiple scattering series the high energy limit of the total nd cross section is also investigated analytically. In contrast to the naive expectation that the total nd cross section is the sum of the np and nn total cross sections we find additional effects resulting from the rescattering processes, which have different signs and a different behavior as a function of the energy. A shadowing effect in the high energy limit only occurs for energies higher than 300 MeV. The expressions in the high energy limit have qualitatively a similar behavior as the exactly calculated expressions, but can be expected to be valid quantitatively only at much higher energies. {copyright} {ital 1999} {ital The American Physical Society}

3N scattering in a three-dimensional operator formulation

A recently developed formulation for a direct treatment of the equations for two- and three-nucleon bound states as set of coupled equations of scalar functions depending only on vector momenta is extended to three-nucleon scattering. Starting from the spin-momentum dependence occurring as scalar products in two- and three-nucleon forces together with other scalar functions, we present the Faddeev multiple scattering series in which order by order the spin degrees can be treated analytically leading to 3D integrations over scalar functions depending on momentum vectors only. Such formulation is especially important in view of awaiting extension of 3N Faddeev calculations to projectile energies above the pion production threshold and applications of chiral perturbation theory 3N forces, which are to be most efficiently treated directly in such three-dimensional formulation without having to expand these forces into a partial-wave basis.

A New Approach to the 3D Faddeev Equation for Three-body Scattering

A novel approach to solve the Faddeev equation for three-body scattering at arbitrary energies is proposed. This approach disentangles the complicated singularity structure of the free three-nucleon propagator leading to the moving and logarithmic singularities in standard treatments. The Faddeev equation is formulated in momentum space and directly solved in terms of momentum vectors without employing a partial wave decomposition. In its simplest form the Faddeev equation for identical bosons, which we are using, is an integral equation in five variables, magnitudes of relative momenta and angles. The singularities of the free propagator and the deuteron propagator are now both simple poles in two different momentum variables, and thus can both be integrated with standard techniques.

A New Treatment of 2N and 3N Bound States in Three Dimensions

The direct treatment of the Faddeev equation for the three-boson system in 3 dimensions is generalized to nucleons. The one Faddeev equation for identical bosons is replaced by a strictly finite set of coupled equations for scalar functions which depend only on 3 variables. The spin-momentum dependence occurring as scalar products in 2N and 3N forces accompanied by scalar functions is supplemented by a corresponding expansion of the Faddeev amplitudes. After removing the spin degrees of freedom by suitable operations only scalar expressions depending on momenta remain. The corresponding steps are performed for the deuteron leading to two coupled equations.

Incoherent Eta Photoproduction from the Deuteron near Threshold

Very recent data for the reaction d→�np, namely total cross sections, angular and momentum spectra, are analyzed within a model that includes contributions from the impulse approximation and next order corrections due

Separable representation of proton-nucleus optical potentials

Recently, a new approach for solving the three-body problem for (d,p) reactions in the Coulomb-distorted basis in momentum space was proposed. Important input quantities for such calculations are the scattering matrix elements for proton- (neutron-) nucleus scattering. We present a generalization of the the Ernst-Shakin-Thaler scheme in which a momentum space separable representation of proton-nucleus scattering matrix elements in the Coulomb basis can be calculated. The success of this method is demonstrated by comparing S -matrix elements and cross sections for proton scattering from C 12 , Ca 48 , and Pb 208 with the corresponding coordinate space calculations.

Coulomb wave functions in momentum space

We present an algorithm to calculate non-relativistic partial-wave Coulomb functions in momentum space. The arguments are the Sommerfeld parameter η, the angular momentum l, the asymptotic momentum q and the running momentum p, where both momenta are real. Since the partial-wave Coulomb functions exhibit singular behavior when p → q, different representations of the Legendre functions of the 2nd kind need to be implemented in computing the functions for the values of p close to the singularity and far away from it. The code for the momentum-space Coulomb wave functions is applicable for values of vertical bar eta vertical bar in the range of 10-1 to 10, and thus is particularly suited for momentum space calculations of nuclear reactions.