Complex, energy-independent, local potential reproducing an absorptive phase shift and a bound state

The triton binding energy, and the partly real and partly complex neutron-deuteron doublet channel elastic scattering phase shifts, calculated by means of the exact three-body theory, are used as input in the fixed-[ital l] inverse scattering theory of Marchenko. The local potentials obtained hereby are independent of energy, and complex. Their strong imaginary part reflects the strong absorption in the doublet channel arising from the opening of the deuteron breakup channel. For total orbital angular momentum [ital l] different from zero the potentials are unique, reproducing the input phase shift in the whole energy region. For [ital l]=0 where there exists, in addition, a bound state we …

Definition of an appropriate free dynamics and the physical S-matrix in multichannel hyperradial adiabatic scattering

In the hyperradial adiabatic (HA) treatment of the three-body problem the total wave function is i expanded as ΨHA(R, ξ, η) = R−5/2 ∑iχi(R)φi(R|ξ, η),where R denotes the hyperradius and (ξ , η) are internal hyperangles. Integration over ξ and η converts the Schrödinger equation into a system of coupled hyperradial equations. It is a well-known fact that, within the HA approach, the non-adiabatic corrections that couple channels converging to the same asymptotic configuration can show an unphysical long-range behavior ∼ 1/R. Though the latter is of purely kinematic origin and arises from the use of the hyperradius instead of the pertinent Jacobi variables, it is nevertheless the source of the…

Proton-induced deuteron breakup reaction2H(p, pp)n

The “screening and renormalization” approach allows for a mathematically correct incorporation, in three-body scattering theory, of the long-ranged Coulomb interaction between charged particles. It is based on first calculating the transition amplitudes using screened Coulomb potentials. Then, after renormalization the zero-screening limit, leading to the amplitudes pertaining to unscreened Coulomb potentials, is performed numerically. Within this formalism the proton-induced breakup of deuterons is investigated, with the Coulomb repulsion between the two protons taken into account. Kinematically complete differential cross sections in various kinematic configurations are calculated and com…

News in Charged-Composite Particle Scattering: Theory and Applications

A brief summary is given of recent developments in the theory of charged-composite particle scattering, and of practical applications.

Coulomb effects in deuteron breakup by proton impact

We present the first results of a calculation of kinematically complete differential cross sections for the proton-induced deuteron breakup reaction, obtained by using a three-body formalism based on momentum space integral equations which correctly takes into account the Coulomb repulsion between the two protons. Comparison with experimental data is made.

Proton-Deuteron Break Up Including Coulomb Effects

The first results of the calculation of proton-deuteron break-up cross sections are presented and compared with experimental data.

Spheroidal and hyperspheroidal coordinates in the adiabatic representation of scattering states for the Coulomb three-body problem

Recently, an involved approach has been used by Abramov (2008 J. Phys. B: At. Mol. Opt. Phys. 41 175201) to introduce a separable adiabatic basis into the hyperradial adiabatic (HA) approximation. The aim was to combine the separability of the Born–Oppenheimer (BO) adiabatic basis and the better asymptotic properties of the HA approach. Generalizing these results we present here three more different separable bases of the same type by making use of a previously introduced adiabatic Hamiltonian expressed in hyperspheroidal coordinates (Matveenko 1983 Phys. Lett. B 129 11). In addition, we propose a robust procedure which accounts in a stepwise procedure for the unphysical couplings that are …

The Few-Body Coulombian Problem

Recent advances in the treatment of scattering of charged composite particles are reviewed. In a first part I report on developments of the theory. Specifically I describe the recent completion of the derivation of the co-ordinate space asymptotic behaviour of the wave function for three charged particles in the continuum. This knowledge is increasingly being made use of in attempts to ‘derive’ three-Coulomb particle wave functions to be used in all of configuration space which are solutions of the Schrodinger equation, though not everywhere but at least in one or preferably all of the asymptotic regions. Their practical application in approximate calculations of ionisation and breakup proc…

Collision Theory for Two- and Three-Particle Systems Interacting via Short-Range and Coulomb Forces

In two- and three-particle reactions with light nuclei, a rich body of precise experimental data exists in which both projectile and target and/or the fragments occurring in the final state are charged. In order to make optimal use of these data for extracting physically interesting information about the nuclear interactions, the effects of the Coulomb force must be separated out in a reliable manner. For this purpose the mastering of the intricacies of charged-particle scattering theory is of vital importance.

Proton-Hydrogen Charge Exchange and Elastic Scattering in the Faddeev Approach

Results of the application of Faddeev-type integral equations to proton-hydrogen collisions are reported. The approach, realized in the impact parameter representation, incorporates the exact two-particle off-shell Coulomb T-matrices in all ‘triangle’ contributions to the effective potentials. Calculatedtotal and differential electron-transfer as well as differential elastic scattering cross sections show very good agreement with experiment, over a wide range of incident energies.

Three-Body Coulomb Effects in the Direct Coulomb Breakup of 8B into 7Be + p in the Field of a 208Pb Ion

The amplitude for the Coulomb breakup of a light nucleus in the field of a highly charged ion is considered in the framework of the distorted wave approach, with particular emphasis being laid on correctly taking into account the three-body Coulomb interactions in the final state. Numerical calculations have been performed for the double differential cross section for the reaction 208Pb(8B, 7Be p)208Pb. They clearly demonstrate the importance of long-range three-body Coulomb correlations in the astrophysically interesting regime when the ejectiles have the extremely small relative energies.

Results of Three-Nucleon Calculations

The motivation for studying the nonrelativistic three-body problem originates in the fact that three-particle collisions occur very frequently in many areas of physics a) atomic physics: the scattering of electrons, positrons and protons off hydrogen atoms b) nuclear physics: three-nucleon problem c) statistical mechanics: 3rd virial coefficient d) low-energy elementary particle physics: final-state interactions in three-body decays of hadrons.

Calculation of Proton-Deuteron Elastic Scattering at 10 MeV with a Realistic Potential

We present the first results of a calculation of the differential cross section and of polarization observables for proton-deuteron elastic scattering at 10thinspthinspMeV proton laboratory energy, for the Paris potential. The method used is the {open_quotes}screening and renormalization approach{close_quotes} which allows one to correctly take into account the Coulomb repulsion between the two protons. Comparison is made with the precise experimental data of Sagara {ital et al.}thinspthinsp[Phys.thinspthinspRev.thinspthinspC {bold 50}, 576 (1994)] and of Sperison {ital et al.}thinspthinsp[Nucl.thinspthinspPhys.thinspthinsp{bold A422}, 81 (1984)]. {copyright} {ital 1998} {ital The American …

Scattering amplitudes for two charged fragments

Relativistic scattering theory of charged spinless particles

In the context of a relativistic quantum mechanics we discuss the scattering of two and three charged spinless particles. The corresponding transition operators are shown to satisfy four-dimensional Lippmann-Schwinger and eight-dimensional Faddeev-type equations, respectively. A simplified model of two particles with Coulomb interaction can be solved exactly. We calculate: (i) The partial waveS-matrix from which we extract the bound state spectrum. The latter agrees with a fourth-order result of Schwinger, (ii) The full scattering amplitude which in the weakfield limit coincides with the expression derived by Fried et al. from eikonalized QED.

Coulomb Effects on Few-Body Scattering States

Modifications of stationary momentum space scattering theory, necessitated by the presence of Coulomb forces, are described, both in the formalism which uses unscreened Coulomb potentials and in the screening and renormalization approach. Thereby, emphasis is laid on exposing the conceptual differences, as well as the different, presently achieved status of applicability. Some of the unresolved problems in both methods are enumerated.

Coulomb Corrections in Proton-Deuteron Scattering

We present the first calculations of differential cross sections for elastic proton-deuteron scattering using a three-body formalism which correctly takes into account the Coulomb repulsion between the two protons.

Coulomb Fourier Transformation: Application to a Three-Body Hamiltonian with One Attractive Coulomb Interaction

Consider a three-body system consisting of one neutral particle 1 and two charged particles characterized by the indices 2 and 3 with charges of opposite sign, i.e., e2e3 < 0. We use the following notation: (x ν , y ν ), v = 1, 2, 3, denotes the (mass-renormalized) coordinate vector within the pair ν, and between the center of mass of the pair ν and particle ν, respectively. The corresponding canonically coniugate momenta are (k ν , p ν ).

Rotational States of the Helium Trimer in the Symmetry-Adapted Hyperradial-Adiabatic Approach

We have searched for bound rotationally excited states of the helium trimer using the symmetry-adapted hyperradial adiabatic approach. Since the calculated J p = 2+ and J p = 1− potential curves are both completely repulsive, we infer that there are no bound rotational states of 4He3. A recent adiabatic calculation [1] based on the direct solution of the Coriolis-coupled Schrodinger equation agrees with this conclusion.

Three-body approach to proton-hydrogen charge exchange and elastic scattering

The impact-parameter Faddeev approach to atomic three-body collisions which has been developed for, and successfully applied to, ion-atom scattering processes, has now been developed further by including, instead of the Coulomb potentials, the full two-particle off-shell Coulomb {ital T} matrices in all {open_quotes}triangle{close_quotes} contributions to the effective potentials. Results of calculations of proton-hydrogen collisions with only the ground states of the hydrogen retained in both the direct and the rearrangement channels are presented. Total and differential electron transfer, as well as differential elastic scattering cross sections, are obtained simultaneously in very good a…

Exact and approximate triangle amplitudes for (in-)elastic three-body processes with charged particles

The triangle amplitudes, which within the framework of the multiple-scattering approach represent the leading contribution to the amplitude for three-body elastic and inelastic reactions, contain the off-shell Coulomb T-matrix describing the intermediate-state scattering of the projectile off each of the target particles. We present results of the exact numerical calculation of that amplitude in which the rescattering particles have charges of opposite sign (`attractive case'), for several atomic processes. This is facilitated by a `new' representation of the Coulomb T-matrix which turns out to be very effective for numerical purposes. One interesting result is that the charge sensitivity o…

Rotational Three-Body Resonances: A New Adiabatic Approach

In the standard adiabatic approach the motion of the fast, light particle (electron) is treated so as to produce an effective potential that governs the motion of the heavy particles (nuclei). The rotational degrees of freedom are then taken into account by adding the centrifugal J(J + 1)-term to the channel potentials and introducing rotational (Coriolis) couplings into conventional close-coupling calculations. Of course, a perturbative treatment of the rotational motion is justified only provided the rotational energy is sufficiently small. If, however, the rotation is as energetic as the motion of the fast particle, both motions should be treated on the same footing in order to produce s…

Coulomb corrections to the three-body correlation function in high-energy heavy ion reactions

Starting from an asymptotically correct three-body Coulomb wave-function, we determine the effect of Coulomb final state interaction on the three-particle Bose-Einstein correlation function of similarly charged particles. We numerically estimate that the Riverside approximation is not precise enough to determine the three-body Coulomb correction factor in the correlation function, if the characteristic HBT radius parameter is 5 - 10 fm, which is the range of interest in high-energy heavy ion physics.

Three-body reactions with charged particles

Time dependent approach to the collision of two charged composite particles

Three-Body Coulomb Final-State Interaction Effects in the Coulomb Breakup of Light Nuclei

Coulomb breakup of a projectile in the Coulomb field of a fully stripped heavy nucleus is at present one of the most popular experimental methods to obtain information on reactions of interest in nuclear astrophysics. Its theoretical interpretation presents, however, considerable difficulties, due to the three-body nature and the infinite range of the Coulomb forces involved. Among the uncertainties affecting present analyses, the possible modification of the dissociation cross section by three-body Coulomb final-state interactions plays a major role. Various methods which have been proposed to deal with it are briefly reviewed. However, none of them is based on a consistent and mathematica…

Few-body problems in nuclear astrophysics

Few-body methods provide very useful tools to solve different problems important for nuclear astrophysics. Some of them are discussed below.

Coulomb-Fourier representation approach to three-body scattering with charged particles

Abstract We present a novel approach for calculating charged-composite particle scattering. It consists in eliminating by means of a suitably chosen representation that part of the interaction which is of longest range and, hence, gives rise to all the troublesome feaures which plague charged particle scattering theories. In this paper only the simplest case is considered, namely that of two charged and one neutral particles which interact via pairwise strong potentials, and a repulsive Coulomb potential between the charged particles.

Test of a separable approximation to a local soft-core potential in the three-body system

Three-nucleon observables below the break-up threshold are calculated employing the pole approximation to the soft-core Malfliet-Tjon potentials. The results are compared in detail to those obtained with the local potentials and to those calculated with the usual Yamaguchi interactions.

Triangle amplitude with off-shell CoulombTmatrix for exchange reactions in atomic and nuclear physics

The lowest-order rescattering contribution (triangle amplitude) in three-body models of exchange reactions with charged particles contains the off-shell two-body T matrix describing the intermediate-state Coulomb scattering of charged subsystems. General properties of the exact exchange triangle amplitude, when the incoming and outgoing particles are on the energy shell, are derived. This includes the analytic behavior, i.e., the positions and characters of its leading singularities, in the cos\ensuremath{\vartheta} plane, where \ensuremath{\vartheta} is the scattering angle, in the vicinity of the forward- and backward-scattering directions. Since for computational reasons the Coulomb T ma…

Gaussian quadrature rule for arbitrary weight function and interval

Abstract A program for calculating abscissas and weights of Gaussian quadrature rules for arbitrary weight functions and intervals is reported. The program is written in Mathematica. The only requirement is that the moments of the weight function can be evaluated analytically in Mathematica. The result is a FORTRAN subroutine ready to be utilized for quadrature. Program summary Title of program: AWGQ Catalogue identifier:ADVB Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADVB Program obtained from: CPC Program Library, Queens University, Belfast, N. Ireland Computer for which the program is designed and others on which it has been tested: Computers: Pentium IV 1.7 GHz processor Ins…

Coulomb effects in three-body reactions with two charged particles

We present the details of a novel approach to the treatment of Coulomb effects in atomic and nuclear reactions of the three-body type in which two of the particles are charged. Based on three-body integral equations the formalism allows the practical calculation of elastic, inelastic, rearrangement, and breakup processes with full inclusion of the Coulomb repulsion or attraction in a mathematically correct way. No restrictions need to be made concerning the form of the short-range interactions between the three pairs. A particular virtue of our method lies in the fact that it corroborates, and gives precise meaning to, the intuitively anticipated conception of how to describe such reactions.

Three-body Coulomb interaction effects in the final state of thePb208(B8,Be7p)Pb208Coulomb breakup reaction

The photodissociation reaction $^{8}\mathrm{B}+\ensuremath{\gamma}\ensuremath{\rightarrow}^{7}\mathrm{Be}+p$ is used to provide information on the astrophysical ${S}_{17}$ factor of the inverse radiative capture reaction, knowledge of which is crucial for an estimation of the high-energy neutrino flux from the sun. Since, at present, the Coulomb field of a fully stripped nucleus serves as the source of the photons, an adequate analysis requires a genuine three-body treatment of this reaction. Among the uncertainties still affecting present analyses, the possible modification of the dissociation cross section by the post-decay acceleration of the fragments $^{7}\mathrm{Be}$ and p in the targ…

About Compactness of Faddeev Integral Equations for Three Charged Particles

Momentum space three-body integral equations of the Faddeev type can not be used for Coulomb-like potentials, for energies above the breakup threshold. The reason is the occurrence of singularities in their kernels which destroy the compactness properties known to exist for purely short-range interactions. Using the rigorously equivalent formulation in terms of an effective-two- body theory, we prove that the nondiagonal kernels occurring therein possess on and off the energy shell only integrable singularities, provided all three particles have charges of the same sign (ie., only repulsive Coulomb interactions). In contrast, if some of the charges have opposite signs the nondiagonal kernel…

Final State Three-Body Coulomb Effects in thePb208(B8,Be7p)Pb208Coulomb Breakup Reaction

We present results of the first calculation of the double differential cross section for the 208Pb(8B,(7)Bep)208Pb Coulomb breakup reaction which treats the postdecay acceleration of the ejectiles within a genuine three-body approach. From this we conclude that, in order to minimize postdecay Coulomb acceleration effects, experiments should be performed at as small as possible scattering angles, not too low 7Be-p relative energies, and high incident energy.

Approximate triangle amplitude for three-body charge exchange processes.

The single-rescattering contribution to the amplitude pertaining to three-body charge exchange reactions (triangle amplitude) contains the off-shell Coulomb {ital T}-matrix {ital T}{sup {ital C}} describing the intermediate-state Coulomb scattering of charged subsystems. For ease of computation, the latter is usually replaced by the potential {ital V}{sup {ital C}} which, however, is unsatisfactory in many cases. An alternative approximation, obtained by {open_quote}{open_quote}renormalizing{close_quote}{close_quote} the {open_quote}{open_quote}triangle{close_quote}{close_quote} contribution with {ital V}{sup {ital C}} instead of {ital T}{sup {ital C}} by a simple analytic expression, is sh…

Energetic collisions of charged projectiles with atomic bound states

Abstract Use of the multiple-scattering expansion of the three-body amplitude for atomic direct and exchange reactions requires the evaluation of multidimensional integrals involving the two-body Coulomb T-operator. We present here numerical results for the first-order terms, both for the attractive and repulsive case. Furthermore, easy-to-calculate approximations are described which in their domain of validity (i) reproduce the exact amplitudes to high accuracy, and (ii) also serve to derive interesting theoretical results.

Proton-Deuteron Elastic Scattering for E &gt; 0

We report on the first reliable numerical results for proton-deuteron elastic scattering observables for energies above the deuteron breakup thresh- old, for the Paris potential. The calculations have been performed within the screening and renormalisation approach. The theoretical results are compared with recent experimental data.

Role of Levinson’s theorem in neutron-deuteron quartetS-wave scattering

The real part of the phase shift for elastic neutron-deuteron scattering in the quartet {ital S} wave channel, as calculated with the exact three-body theory, assumes at threshold the value {pi} if normalized to zero at infinity; that is, it does not comply with the expectations raised by a naive application of Levinson's theorem since no bound state exists in this channel. A description of this situation on an equivalent two-body level via a potential, constructed by means of the Marchenko inverse scattering theory, necessitates the introduction of a fictitious bound state. This predominantly attractive, equivalent local potential can be related via supersymmetry to a strictly phase equiva…

Momentum space integral equations for three charged particles: Nondiagonal kernels

Standard solution methods are known to be applicable to Faddeev-type momentum space integral equations for three-body transition amplitudes, not only for purely short-range interactions but also, after suitable modifications, for potentials possessing Coulomb tails provided the total energy is below the three-body threshold. For energies above that threshold, however, long-range Coulomb forces have been suspected to give rise to such severe singularities in the kernels, even of the modified equations, that their compactness properties are lost. Using the rigorously equivalent formulation in terms of an effective-two-body theory we prove that, for all energies, the nondiagonal kernels occurr…

Scattering amplitudes and integral equations for the collision of two charged composite particles

Transition operators for the collision of two clusters composed of an arbitrary number of charged and neutral particles are represented as a sum of pure Coulomb and Coulomb-modified short-range operators. Sandwiching this relation between the corresponding channel states, correct two-fragment scattering amplitudes are obtained by adapting the conventional two-body screening and renormalization procedure. Furthermore, integral equations are derived for off-shell extensions of the full screened amplitudes and of the unscreened Coulomb-modified short-range amplitudes. For three particles, the final results coincide with those derived previously in a different approach. The proposed theory is v…

Three-nucleon calculations for local potentials with the quasiparticle method

The three-nucleon system for energies below the breakup threshold is investigated with the help of the quasiparticle method. Two types of local potentials are used, namely purely attractive Yukawa potentials and the soft-core potentials of Malfliet and Tjon. The results obtained are compared with those of other calculations employing different methods.