Three-body coulomb effects in one-particle transfer reactions

The behaviour of the DWBA amplitudes in the vicinity of the nearest singularity in the cosθ-plane for the neutron and charged particle transfer reactions is investigated. It is shown that Coulomb interactions transform the pole singularity into a branch point. The main singular terms of the complete and the post-approximation DWBA cross sections are compared with the exact three-body results. They differ only by the Coulomb renormalization factors (CRF), which define the renormalization of the singularity strength generated by the Coulomb interactions. In the case of neutron transfer it is shown that the complete DWBA and exact CRFs practically coincide and slightly differ from CRF obtained in post-approximation DWBA. For charged particle transfer reactions the exact CRF may differ considerably from the one obtained in DWBA and especially in the post-approximation DWBA. Our conclusion therefore is that in order to obtain a reliable value of the vertex constant by extrapolating the differential cross-section to the nearest singularity, the exact three-body approach should be used.     

Influence of the three-body coulomb effects on the values of asymptotic normalization coefficients (nuclear vertex constants) of astrophysical interest extracted from the modified DWBA approach for the peripheral proton-transfer reactions

From the exact three-body distorted wave amplitude obtained by other authors for the one-charged-particle-transfer reaction A(x, y)B within the strict three-body (x = y + a and B = A + a, where a is a transferred particle) model, its part is separated in which the three-body Coulomb dynamics of the transfer mechanism is taken into account correctly. The contribution of the three-body Coulomb dynamics of the transfer mechanism to the partial wave amplitudes at l ≫ 1 for the peripheral proton-transfer reactions ¹³C(³He, d)¹⁴N, ¹³C(¹⁴N, ¹³C)¹⁴N, and ⁹Be(¹⁰B, ⁹Be)¹⁰B is estimated within the three-body approach combining the dispersion method and the DWBA approach. For these reactions, the Coulomb renormalization factors, arising owing to correctly taking into account the three-body Coulomb dynamics of the proton-transfer mechanism in the DWBA cross sections, are calculated. A new estimate is obtained for the values of the asymptotic normalization factors for p + ¹³C → ¹⁴N, which also have astrophysical interest. The text was submitted by the authors in English.     

Coulomb potentials and the energy-angular momentum balance in heavy ion transfer reactions

Coulomb terms in optical model parameters and in the effective interaction are discussed for DWBA in charged particle transfer in heavy ion reactions. It is shown that certain discrepancies in shapes of calculated angular distributions are due to incorrect treatment of Coulomb terms.     

On the validity of dwba for deuteron stripping: (I). The Mitra three-body model

Previous work showing that there exists an exact formulation of the DWBA for stripping in the S-wave, separable potential, three-body model of Mitra is discussed and extended. The one-body equation obeyed by the c.m. wave function used in the reformulated DWBA is derived and compared with the equation obeyed by the wave function used in the standard formulation of DWBA, viz., the deuteron elastic scattering wave function. Results obtained by other workers on application of three-body methods to direct reactions are discussed in light of the fact that an exact DWBA exists for the separable potential model.     

Three-Body Problem with Two Charged Particles (I) The Neutron Induced Nuclear Reaction

We present the details of an exact method for the treatment of Coulomb effects in neutron induced three-body n uclear reactions. Based on the three-body in tegral equations, the formalist allows the practical calculation of elastic, inelastic, and breakup processes without an approxima tion of C.M. Coulomb interaction.     

The break-up of the deuteron and stripping to unbound states

We treat the break-up of the deuteron in the field of a heavy nucleaus. Our theory is based on the post formalism of the DWBA, which is discussed in detail. In particular we consider the case where the energy of the incoming deuteron is below the Coulomb barrier. This sub-Coulomb break-up permits a test of the DWBA by comparing experimental data with the theory. Small effects are investigated in this context, and other treatments of this process are reviewed critically. To give an intuitive picture of the break-up process we introduce a semiclassical model.In the second part the theory is extended to stripping reactions to unbound (resonant) states. We show that this reaction which interferes with the direct break-up process, can again be well described in the post-form of the DWBA. Using simple models we discuss the most important features of such reactions. Particular attention is paid to the “l-enhancement”, “parallelism” and the influence of the specific wave functions on resonance line shapes. A survey and comparison of various theories of stripping reactions to unbound states is given and finally we discuss possible future developments.     

9Be(3He,n)11C and 11B(3He,n)13N Reactions at Energies below Coulomb Barrier

Angular distributions have been measured for ⁹Be(³He, n)¹¹C and ¹¹B(³He, n)¹³N reactions for the neutron group leading to the ground state at E_3He = 0.90, 1.00, 1.20, 1.40 MeV and 1.70, 1.90 MeV respectively. To fit the experimental data, the theory of two nucleon stripping reactions below the Coulomb barrier has been considered. Taking Coulomb distorted wave functions for the interacting particles in the initial channel, a closed analytical form for the differential cross-section has been obtained. The other two cases using the plane wave Born approximation and the distorted wave Born approximation are also applied to the experimental data. The agreement between the Coulomb distorted wave calculations and the experimental data is better than with the PWBA and DWBA. The spectroscopic factors are extracted by fitting the experimental data with the theoretical calculations.     

An approximate three-body theory of deuteron stripping

A treatment of deuteron stripping is developed in which the three-body effects associated with deuteron break-up in the nuclear field are included explicitly. The essence of the method is the choice of a convenient discrete set of n-p eigenfunctions as a representation of the three-body continuum effects. This approach leads to a distorted wave stripping matrix element similar to that of the DWBA, except that the elastic deuteron wave is replaced by a three-body wave function given as the solution of a set of coupled two-body Schrödinger equations. The adiabatic theory of Johnson and Soper appears as the solution in a suitable first approximation. This new formalism should prove useful in the evaluation of corrections to three-body models of the deuteron-nucleus system, in particular those models in which the nucleon-target interaction is represented by a complex local optical potential.     

Probing the structure of exotic nuclei by transfer reactions

Low energy single nucleon transfer reactions are proposed as a tool to investigate the structure of nuclei far off stability. Experimental concepts and conditions are discussed, in particular high resolution γ-ray spectroscopy after single nucleon pickup reactions. Nuclear structure is described by Skyrme Hartree-Fock calculations including pairing. As representative examples, binding energies, radii and wave functions for Mg and Sn isotopes are calculated. In the neutron deficient Mg isotopes a proton skin is found. At the neutron driplines the Mg and Sn isotopes develop extended neutron skins. The nuclear structure results are used in DWBA and EFR-DWBA transfer calculations. Single nucleon transfer reactions of ^32,36Mg and exotic Sn beams on targets ranging from ²H to ²⁴Mg in inverse kinematics are explored. The one-nucleon transfer cross sections decrease strongly for high-Z targets. An impact parameter analysis shows that the transfer process is selective on the tails of the wave functions. The largest cross sections are obtained for ²H and ⁹Be targets at incident energies of E _lab = 2-5 MeV/u. The energy-momentum dependence is closely related to the special properties of wave functions of weakly bound states. Two-neutron (p,t) stripping reactions are studied for a ⁶He projectile. A strong competition of sequential and direct processes is found at low energies. Received: 1 October 1997 / Revised version: 25 November 1997     

Investigations of the deuteron break-up on heavy nuclei above the Coulomb barrier

The deuteron break-up on heavy nuclei is studied for deuteron energies above the Coulomb barrier in the post form of the DWBA. The “region of formation” of the DWBA integral is found to be located mainly outside the nuclear force range. Finite-range, non-locality and virtual break-up effects are carefully considered. A comparison of our theoretical calculations with the available experimental coincidence cross sections shows good agreement. This supports strongly the validity of the direct break-up model assumed here.     

Heavy-ion one-neutron transfer reactions involving 13C: (I). Spectroscopic factor products

The total cross sections of some heavy-ion one-neutron transfer reactions involving ¹³C, which lead to excited states of one of the final nuclei, have been measured at energies below the Coulomb barrier. Products of the neutron spectroscopic factors in the initial and final states have been extracted using a Coulomb wave Born approximation, and have been compared with theory.     

Comparison of spectroscopic information given by the peripheral model and DWBA

It is shown that the one parameter form of the peripheral model gives the same spectroscopic information as DWBA does for neutron transfer reactions on light nuclei at medium bombarding energies.     

Neutron-proton transfer reactions leading toT=1 particle-hole states of56Co

Zero range DWBA analysis for two-particle (neutron and proton) transfer reactions is carried out, using simple shell model structure wave functions for⁵⁴Fe,⁵⁶Co and⁵⁸Ni, with⁵⁶Ni inert core. In this structure calculation, a microscopic set of two-body interaction matrix elements derived from the non-local separable potential of Tabakin are employed. These matrix elements include in the perturbation theory two corrections (i) the second-order Börn term and (ii) the appropriate core excitations. Unlike the situation in many two-particle transfer reactions, the fragmentation of the reaction strengths to the excited states with respect to the lowest states of same spin and parity in the above transfer processes is satisfactorily borne out from this analysis.     

Coulomb forces in the three-body problem with application to direct nuclear reactions

Faddeev equations are considered in the case of three charged particles interacting with both separable nuclear two-body interactions and also including Coulomb forces. Modified Faddeev equations with Coulomb Green's functions are introduced. The three-body amplitudes are given into pure Coulomb and distorted-Coulomb amplitudes. Introducing a decomposition in the angular momentum states, a set of three-body integral equations is obtained. The effect of pure coulomb amplitudes is studied in direct nuclear reactions and found to give a large contribution to the cross sections. The three-body integral equations obtained are applied for direct nuclear reactions. The angular distributions for¹²C(⁶Li,d)¹⁶O,¹⁶O(⁶Li,d)²⁰Ne, and¹²C(⁶Li,α)¹⁴N transfer reactions are calculated as well as for the⁶Li elastic scattering on¹²C. From the good agreement between the theoretically calculated and experimental data, better spectroscopic factors are extracted. The effect of including Coulomb forces in the three-body problem is found to improve the results by about 16.26%.     

Study of form factors for single-proton transfer reactions on152Sm,154Gd and156Gd

The (³He,d) and (α,t) single-proton transfer reactions on targets of¹⁵²Sm,¹⁵⁴Gd and¹⁵⁶Gd were studied at 35MeV incident energy. Differential cross sections of rotational states built upon various single-proton configurations are compared with results of DWBA calculations which employed various radial form factors. The agreement between calculated and measured reaction cross sections is found to improve significantly when the commonly used spherical bound-state potentials are replaced by deformed ones, including deformed Coulomb and spin-orbit wells, and projected form factors are used to calculate DWBA cross sections. Discrepancies in the differential cross sections so large that they cannot be attributed to band mixing phenomena are readily explained by form factor effects.     

Theory of the Trojan-Horse method

The Trojan-Horse method is an indirect approach to determine the energy dependence of S factors of astrophysically relevant two-body reactions. This is accomplished by studying closely related three-body reactions under quasi-free scattering conditions. The basic theory of the Trojan-Horse method is developed starting from a post-form distorted wave Born approximation of the T-matrix element. In the surface approximation the cross-section of the three-body reaction can be related to the S-matrix elements of the two-body reaction. The essential feature of the Trojan-Horse method is the effective suppression of the Coulomb barrier at low energies for the astrophysical reaction leading to finite cross-sections at the threshold of the two-body reaction. In a modified plane wave approximation the relation between the two- and three-body cross-sections becomes very transparent. The appearing Trojan-Horse integrals are studied in detail.     

低能电子碰撞He+(e,2e)反应绝对三重微分截面的理论研究

利用Berakdar和Briggs对BBK波函数Sommerfeld参数的修正结果，即考虑第三个粒子存在对两个粒子间相互作用的影响，考虑了入射道的库仑相互作用及出射粒子的交换对称性，计算了在共面等能分配几何情况下低能电子碰撞He⁺(e,2e)反应绝对三重微分截面.结果表明，入射道库仑场对较低的入射能量及小的碰撞参数的三重微分截面影响较大. 关键词：     

Momentum-space description of four-nucleon scattering

Four-nucleon scattering equations are solved in momentum space. The Coulomb interaction between the protons is included using the screening and renormalization approach. Realistic potentials are used between nucleon pairs. Results are obtained for all four-nucleon elastic and transfer reactions below three-body breakup threshold.     

One dimensional three-body model for low energy reactions of halo nuclei

One dimensional three-body model which simulates the low energy reactions of the nuclei with halo structure, is investigated by solving exactly the three-body Schrödinger equation. The dynamical roles of the halo neutron during the reaction are studied in detail. The decrease of the fusion probability, as well as the large transfer and break-up probabilities, are found for halo nuclei.