Coulomb T-matrix Research Articles

Triangular exchange diagram within the full off-shell Coulomb T-matrix in deuteron induced reactions

<span>The amplitude of the triangular diagrams in the three-body models of deuteron induced reactions <span>with charged particles contains the off-shell two-body <span><em>T</em><span>-matrix describing the intermediate-state <span>Coulomb scattering of charged subsystems. Up to now the latter has usually been replaced by the <span>Coulomb potential due to the computational reason. In this paper, we first investigate theoretically <span>and numerically the validity of the mentioned Coulomb-Born approximation. The results show the <span>important contribution of the higher order of the latter approximation for the (d,p) reactions off <span>heavy target.</span></span></span></span></span></span><br /><br class="Apple-interchange-newline" /></span></span></span>

The proton–deuteron system in pionless EFT revisited

We provide a detailed discussion of the low-energy proton–deuteron system in pionless effective field theory, considering both the spin-quartet and doublet S-wave channels. Extending and amending our previous work on the subject, we calculate the 3He–3H binding energy difference both perturbatively (using properly normalized trinucleon wave functions) and non-perturbatively by resumming all Coulomb diagrams in the doublet channel. Our nonperturbative result agrees well with a calculation that involves the full off-shell Coulomb T-matrix. Carefully examining the cutoff-dependence in the doublet channel, we present numerical evidence for a new three-nucleon counterterm being necessary at next-to-leading order if Coulomb effects are included. Indeed, such a term has recently been identified analytically. We furthermore make a case for a simplified Coulomb power counting that is consistent throughout the bound-state and scattering regimes. Finally, using a ‘partially screened’ full off-shell Coulomb T-matrix, we investigate the importance of higher-order Coulomb corrections in low-energy quartet-channel scattering.

Momentum space 3N Faddeev calculations of hadronic and electromagnetic reactions with proton-proton Coulomb and three-nucleon forces included

We extend our approach to incorporate the proton-proton (pp) Coulomb force into the three-nucleon (3N) momentum-space Faddeev calculations of elastic proton-deuteron (pd) scattering and breakup to the case when also a three-nucleon force (3NF) is acting. In addition we formulate that approach in the application to electron- and gamma-induced reactions on 3He. The main new ingredient is a 3-dimensional screened pp Coulomb t-matrix obtained by a numerical solution of a 3-dimensional Lippmann-Schwinger equation (LSE). The resulting equations have the same structure as the Faddeev equations which describe pd scattering without 3NF acting. That shows the practical feasibility of both presented formulations.

Coulomb off-shell T-matrix and its Haar scalet projection

Several new analytical forms for the off-the-energy-shell transition matrix are presented which, in some cases, have hypergeometric functions appearing in the expressions for them. The scale of deviation of the Coulomb T-matrix from the Coulomb potential is determined. The Coulomb T-matrix derived by Shastry et al (1970 Phys. Rev. A 1 1137) is corrected by an additional term. Further, the study derives the energy shell limit for the hypergeometric forms using the Padé approximation, in which these forms are compared to the limit of exact off-shell forms in the vicinity of the energy shell. Finally, wavelet theory and multiresolution analysis are employed to arrive at a new approximate form for the Coulomb off-shell T-matrix in the Haar scalet basis.

A novel treatment of the proton-proton Coulomb force in proton-deuteron Faddeev calculations

We present resently introduced novel approach to include the proton-proton (pp) Coulomb force into the momentum space three-nucleon (3N) Faddeev calculations. It is based on a standard formulation for short range forces and relies on a screening of the long-range Coulomb interaction. In order to avoid all uncertainties connected with an application of the partial wave expansion, unsuitable when working with long-range forces, we apply directly the 3-dimensional pp screened Coulomb t-matrix. That main new ingredient, the 3-dimensional screened pp Coulomb t-matrix, is obtained by a numerical solution of the 3-dimensional Lippmann-Schwinger (LS) equation. Using a simple dynamical model for the nuclear part of the interaction we demonstrate the feasi- bility of that approach. The physical elastic pd scattering amplitude has a well defined screening limit and does not require renormalisation. Well converged elastic pd cross sections are obtained at finite screening radii. Also the proton-deuteron (pd) breakup observables can be determined from the resulting on-shell 3N amplitudes in- creasing the screening radius. However, contrary to the pd elastic scattering, the screening limit exists only after renormalisation of the pp t-matrices.

The proton-proton scattering without Coulomb force renormalization

We demonstrate numerically that proton-proton (pp) scattering observables can be determined directly by standard short range methods using a screened pp Coulomb force without renormalization. We numerically investigate solutions of the 3-dimensional Lippmann-Schwinger (LS) equation for an exponentially screened Coulomb potential. For the limit of large screening radii we confirm analytically predicted properties for off-shell, half-shell and on-shell elements of the Coulomb t-matrix.

Protons in collision with hydrogen atoms: Influence of unitarity and multiple scattering

Three-body integral equations when applied to collisions of protons with hydrogen atoms yield the amplitudes for direct scattering and electron exchange which automatically satisfy two-body and (if solved exactly also) three-body unitarity. As a consequence, differential cross sections (DCS) for both reactions are calculated on equal footing, and the total exchange cross section (TCS) results from the corresponding DCS. This is to be contrasted with standard models used in atomic physics which either are of the (Distorted-Wave) Born type (to be applied at higher energies), or solve the Schrodinger equation by expanding the wave function in a basis (close-coupling method which has problems especially for rearrangement processes). Hence, at higher energies in general separate models have to be developed to describe the DCS for either direct scattering or electron exchange, and frequently also for the TCS. For instance, the most sophisticated traditional models, which provide a very good reproduction of the TCS data, are the continuum distorted wave [1] and the boundary corrected first Born model [2]. Both, however, fail to describe differential cross sections which represent a much more stringent test. On the other hand, three-body integral equations suffer from the principal defect that their kernels are not compact when particles with charges of different sign are involved, as it happens in applications to atomic reactions (references can be traced from [3]). The consequence is that naive application of standard solution methods of integral equations theory would not be justified. Additional, practical difficulties arise from the complicated singularity structure of the off-shell twoparticle Coulomb T-matrix which is the basic dynamical ingredient. As is well known, the latter develops nasty singularities in the on-shell limit and, in case of attraction, has in addition an infinite number of poles.

The half-on-shell Coulomb T-matrix in a double-scattering approximation to the ionization of helium by fast protons

We investigate the application of the Faddeev-Watson multiple-scattering series to the ionization of atoms by ion impact. Singularities in the first- and second-order terms are discussed. Different forms of the half-shell Coulomb T-matrix are used in impulse approximations to double-differential cross sections (DDCSs) for proton-impact ionization of helium, and are compared with experiment. By using the regularized half-shell T-matrix we derive a double-scattering approximation which includes the full projectile-residual-ion interaction. While the latter approximation does not improve on the impulse approximations for DDCSs, in contrast to other models, we conjecture that it might adequately describe triple-differential cross sections for large angles of deflection of the projectile.

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 of the full triangle amplitude apparently disappears at the elastic threshold, for all scattering angles. Furthermore, we propose a new approximation for the triangle amplitudes which can be viewed as a `renormalization' by a simple analytic expression, of the well known approximation which consists in replacing by the potential . While the latter is known to be generally inadequate, this new approximation is shown to yield results in excellent agreement with the numerically calculated exact amplitude, for atomic elastic reactions, over a wide range of (medium to high) projectile energies and scattering angles (including the near-forward-scattering direction). An even simpler approximate amplitude is derived which contains no quadratures at all. It yields similarly good results provided the masses of the two particles experiencing intermediate-state rescattering are of the same order of magnitude but differ from that of the spectator particle. In addition, the explicit forms of the approximate amplitudes are used to derive a variety of interesting theoretical results.

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 T-matrix ${\mathit{T}}^{\mathit{C}}$ describing the intermediate-state Coulomb scattering of charged subsystems. For ease of computation, the latter is usually replaced by the potential ${\mathit{V}}^{\mathit{C}}$ which, however, is unsatisfactory in many cases. An alternative approximation, obtained by ``renormalizing'' the ``triangle'' contribution with ${\mathit{V}}^{\mathit{C}}$ instead of ${\mathit{T}}^{\mathit{C}}$ by a simple analytic expression, is shown to yield results in excellent agreement with the numerically calculated exact amplitude, for atomic elastic exchange reactions, over a wide range of (medium to high) projectile energies and scattering angles (including the forward direction). \textcopyright{} 1996 The American Physical Society.

"Triangle" diagram with off-shell Coulomb T-matrix for (in-)elastic atomic and nuclear three-body processes

The driving terms in three-body theories of elastic and inelastic scattering of a charged particle off a bound state of two other charged particles contain the fully off-shell two-body Coulomb T-matrix describing the intermediate-state Coulomb scattering of the projectile with each of the charged target particles. Up to now the latter has usually been replaced by the Coulomb potential, either when using the multiple-scattering approach or when solving three body integral equations. General properties of the exact and the approximate on-shell driving terms are discussed, and the accuracy of this approximation is investigated numerically, both for atomic and nuclear processes including bound-state excitation, for energies below and above the corresponding three-body dissociation threshold, over the whole range of scattering angles.

Absolute triple differential cross-section measurements in energy sharing kinematics and a coplanar asymmetric geometry at 100 and 200 eV on He

Results of measurements for electron impact ionization of atomic helium are reported for energy-sharing kinematics in a coplanar asymmetric geometry. Incident energies of 100 eV and 200 eV are considered and comparison with previous experiments in fully symmetric kinematics allows the results to be put on an absolute scale. The cross-sections are compared with those obtained from two variants of the distorted wave Born approximation (DWBA) model. Good general agreement is obtained for the form of the angular distributions. The absolute magnitude of the cross sections is however not compatible with the choice of half-on-shell Coulomb T-Matrix currently used in the distorted wave impulse and related approximations.

The self-energy of the electron in the Coulomb field (momentum space method)

A new method for calculating the self-energy of an electron in a central Coulomb field is presented. It is a pure momentum space procedure. Singularities of the Coulomb t-matrix are avoided by suitable contour deformations. In this procedure renormalization is standard.

Two singular integrals containing a partial wave of the two-body coulomb T-matrix

The operatorsT C,l E+i0)[−G0(E+i0)]1−iδ andTC,l(E+ie)G0[−ieG0(E+ie)]iδ acting on spaces of Holder continuous, differentiable and analytic functions are investigated. The results of their action are expressed in terms of explicit singular factors and terms and Holder (differentiable, analytic) functions. The most singular part of these operators is shown to be determined by a simple functional.

Analytic expressions for the partial waves of the off-shell two body coulomb T-matrix

Closed form expressions for all partial waves of the pure Coulomb off-shell T-matrix 〈p¦ Tc,l.(k2) ¦p′〉 are obtained. All singularities appear through simple special functions, which makes it possible to study the analytic properties of 〈p¦ Tc,l(k2) ¦p′〉 as a function not only of one of the momentap, p′ but even of both of them. With use of the renormalization procedure found by Zorbas the transition to half-and on-shell values is performed reproducing known expressions. By the same method simple expressions for the partial waves of the Coulomb wave function in the momentum representation are found.

Orientation of the 2p state of atomic hydrogen produced by direct collisions with fast electrons

Values of the lambda and orientation parameters of the 2p state of atomic hydrogen, produced by fast electrons in a direct scattering process, are calculated using a method based upon the first-order Faddeev-Watson expansion. The method involves a classical-path approximation to the off-shell two-body Coulomb T-matrix. Comparison with several other predictions is made. The results for lambda agree closely with those of McDowell et al. (1976), who used a distorted-wave polarised-orbital calculation, but the agreement of the orientation parameter is only qualitative.

Coulomb T-matrix in the Glauber approximation in the presence of a low frequency electromagnetic field

The recently proposed method (see abstr. A26803 of 1974) of calculating the multiphoton bremsstrahlung cross section in the Glauber approximation (1959) has been applied to the Coulomb scattering problem. The low frequency limit of the cross section has been obtained by using the stationary phase approximation.

La réaction de transfert avec des ions lourds

The neutron transfer process in heavy-ion reactions is treated in accordance with Coulomb T-matrix (CTM) theory. On the basis of the Greider formalism, formulae giving differential and total cross sections in absolute magnitude are derived. Stress is laid upon appropriate normalization and nuclear structure considerations. The off-energy-shell contributions to the Coulomb T-matrix are therein treated in first approximation, resolving the originally incorrect energy dependence of the cross sections. The calculated angular distributions and excitation functions are in good agreement with the available data for energies well below the classical Coulomb barrier. Reduced widths are determined in good agreement with those derived on the assumption that a square potential well binds the transferred neutron to each of the heavy nuclei. The formalism is throughout compared with commensurate expressions derived from a DWBA treatment of the reaction. The CTM results are also applied to a case involving transfer of angular momentum along with the neutron; the theoretical predictions are compared with recent experimental data. For energies in the vicinity of the top of the Coulomb barrier, an attempt to integrate the transition amplitude analytically is described. This makes use of a simple cut-off model to account for the effects of absorption due to the nuclear potentials; although the angular distributions tally with measured values, the model proves inadequate to reproduce the observed magnitudes and energy dependence of the absolute cross sections.