Eikonal expansion in electron scattering

We extend a previously developed eikonal expansion method to the case of high-energy electron scattering from dynamic nuclei. Nuclear degrees of freedom can be included either by a DWBA-like expansion or by neglecting the nuclear excitation energies. The latter approach is considered in detail and applied to the elastic and inelastic scattering from deformed nuclei. The impact parameterT-matrix is calculated by diagonalizing the inelastic zeroth-order and first-order eikonal phases in the subspace of the nuclear rotational levels; static eikonal phases are included up to second order. For¹⁵²Sm,¹⁵⁴Sm coupled-channel calculations including three and four rotational states show large multi-step effects for the 0⁺→4⁺ and the 0⁺→6⁺ excitations compared to DWBA results.     

EIKONAL-BORN EXPANSION AND ITS APPLICATION IN LARGE-ANGLE SCATTERING

In this paper, the scattering amplitude is expanded by two eikonal wave functions which propagate along the incoming and outgoing directions respectively to replace the distorted wave functions in DWBA. Then, we obtain a method to calculate the non-relativistic large angle scattering. This method can be easily generalized to relativistically phenomenological dynamical equation.     

On the theory of nuclear heavy-ion direct transfer reactions

We review the distorted-wave approach to direct transfer reactions and draw attention to some of the shortcomings of current theories. We show that a reformulated form of the distorted-wave Born approximation (DWBA) for transfer can lead to important simplifications of the theory, which are valid for nuclear heavy-ion induced reactions at energies ? 10 MeV/nucleon. In particular, in the semiclassical limit, it leads to a new and simple formula for the transfer t-matrix which includes all the essential physics while offering several important advantages over standard “full-recoil finite-range” DWBA. One such advantage is that the new formula is more transparent in that it is amenable to interpretation and analytical manipulation. At high-energy it is shown to reduce to one earlier deduced using eikonal-DWBA.The conditions for the validity of the new theory are discussed in detail. They are shown to be generally well satisfied for small-mass transfer between heavy-ions at energies at or above those which particularly favour transfer (? 10 MeV/nucleon for transfer of valence nucleons). The restriction to small mass is not due to any recoil approximation; in fact, it is only a necessary restriction at certain energies. The theory treats recoil exactly.Consideration of the optimum dynamical conditions for transfer leads to a set of matching conditions. The presence of hitherto neglected absorption, arising from dynamical effects of poor matching, is suggested and qualitatively discussed. Conditions under which such absorption may be neglected are derived. Results of numerical calculations are presented showing that the theory is capable of good agreement with standard full-recoil finite-range DWBA, and that it is capable of giving at least as good an account of experimental data for nucleon-transfer between heavy-ions at energies ～10 MeV/nucleon.     

Transfer reactions induced by 17, 18O on 12, 13C at c.m. energies between 12.6 TO 14.0 MeV

Angular distributions for one-neutron and many-nucleon transfer reactions were measured in the systems ^{17, 18}O and ^{12, 13}C at c.m. energies between 12.6 to 14.0 MeV. All the cross sections were analyzed in terms of the full-recoil finite-range DWBA model. For multi-nucleon transfer processes an inert cluster transfer was assumed. The sensitivity of DWBA calculations to various parameter sets is discussed, and the effect of the no-recoil approximation on the phase and magnitude of the transfer amplitude is studied. For one-neutron transfer reactions the forward part of the angular distributions was reproduced reasonably well by the DWBA model, yielding satisfactory spectroscopic information. The rise of the cross section at the backward angles was not reproduced by the DWBA model. Possible contributions of cluster-exchange and compound-nucleus reactions are discussed. For multi-nucleon transfer reactions poorer fits were obtained, particularly in the ¹⁸O+¹²C system.     

Elastic scattering of positrons by helium and lithium

We present differential and integrated cross-sections for elastic scattering of positrons by helium and lithium at intermediate and high energies, calculated using the two-potential eikonal approach. The present calculations are compared with the available theoretical results.     

Dispersion theory of nucleon transfer reactions induced by heavy ions

A new approach, the distorted wave pole approximation (DWPA) with the three-body Coulomb effects, is developed by combining the dispersion method and DWBA to analyse the heavy ion-induced neutron transfer reactions. The influence of the three-body Coulomb dynamics on the peripheral partial wave amplitudes is investigated. Differential cross sections of the neutron transfer reactions are calculated to compare the proposed model with the conventional DWBA. The values of nuclear vertex constants for virtual separation of neutron from various nuclei are obtained. The results of the calculations show that DWPA can be applied to analyse the heavy ion-induced neutron transfer reactions and that the three-body Coulomb effects are taken into account with acceptable accuracy in DWBA.     

Effect of Gaussian Potential on the Differential Cross Sections of Alpha Transfer Reactions

The differential cross sections of different alpha transfer reactions have been investigated by using the exact finite-range DWBA calculations. The double-folding density-dependent model is used for the real and imaginary nucleus-nucleus potential in the initial and final channels. The effective nucleon-nucleon interaction is assumed to be the double-Gaussian potential. The Yukawa interaction is found reasonable and exhibits a slight better agreement at energies above the barrier region. The obtained values of the extracted normalization factors are reasonable.     

Study of （e, 2e） process on potassium at 6 eV-60 eV above threshold in a second-order Born approximation

The standard distorted wave Born approximation(DWBA) method has been extended to second-order Born amplitude in order to describe the multiple interactions between the projectile and the atomic target.Second-order DWBA calculations have been preformed to investigate the triple differential cross sections(TDCS) of coplanar doubly symmetric(e,2e) collisions for the alkali target potassium at excess energies of 6 eV-60 eV.Compared with the previous first-order DWBA calculations,the present theoretical model improves the degree of agreement with experiments,especially for the backward scattering angle region of TDCS.This indicates that the present second-order Born term is capable of giving a reasonable correction to the DWBA model in studying coplanar symmetric(e,2e) problems in low and intermediate energy ranges.     

Study of (e, 2e) process on potassium at 6 eV—60 eV above threshold in second-order Born approximation

The standard distorted wave Born approximation (DWBA) method has been extended to second-order Born amplitude in order to describe the multiple interactions between the projectile and the atomic target. Second-order DWBA calculations have been preformed to investigate the triple differential cross sections (TDCS) of coplanar doubly symmetric (e, 2e) collisions for alkali target potassium at excess energies of 6 eV-60 eV. Comparing with the first-order DWBA calculations before, the present theoretical model improves the degree of agreement with experiments, especially for backward scattering angle region of TDCS. This indicates that the present second-order Born term is capable to give a reasonable correction to DWBA model in studying coplanar symmetric (e, 2e) problems in low and intermediate energy range.     

Alpha-cluster break-up and reaction mechanism in (p, α) reactions on light nuclei

The differential cross sections for the ¹¹B(p, α)⁸Be and ²³Na(p, α)²⁰Ne reactions were measured at bombarding energies between 6 and 24 MeV. The observed angular distributions can be divided into two domains: at low energies the shapes vary rapidly with incident energy indicating a compound nucleus reaction ; at higher energies rather stable diffraction patterns are seen exhibiting a direct reaction mechanism and DWBA calculations are able to describe the shapes. The change from one region to the other is rather abrupt and this behaviour seems typical for reactions having an α-like compound nucleus. The energy at which this change occurs corresponds to an excitation energy in the compound nucleus of about 20 MeV above the α-threshold.     

Observation of deuteron D-state and compound nucleus effects in (d,p) reactions on 46Ti and 52Cr with a polarized deuteron beam

The cross section and the vector and tensor analyzing powers have been measured for ⁴⁶Ti(d, p)⁴⁷Ti at deuteron energies of 6 and 10 MeV and ⁵²Cr(d, p)⁵³Cr at 6 MeV. Transitions were observed to the states at E_x=0.159, 1.549 and 1.793 MeV in ⁴⁷Ti and the states at E_x=0.0, 0.564, 1.006 and 2.321 MeV in ⁵³Cr. In addition, the cross sections and vector analyzing powers for deuteron elastic scattering were measured for the same targets and deuteron energies and compared to optical model calculations. The choice of optical parameters for the DWBA analysis of the (d, p) reactions is discussed. Calculations made with the DWBA method show that the deuteron D-state must be included to reproduce even qualitatively the (d, p) tensor analyzing power measurements. The j-dependence of the tensor analyzing power T₂₂ is discussed. The validity of the local energy approximation (which was used to incorporate the deuteron D-state into the DWBA calculations) is evaluated by comparison to finite range calculations. The contribution of compound nucleus reactions to the measured cross sections and analyzing powers was investigated. In order to determine the compound cross section, the Ericson fluctuations in excitation functions of cross section and vector analyzing power were measured from 5 to 7 MeV on each target. The formulas used to calculate the polarization observables from the Hauser-Feshbach theory are presented.     

氘核削裂反应理论研究及其进展

氘核削裂反应作为单核子转移反应的一种,在原子核的单粒子结构研究中一直起着重要的作用.同时,作为典型的三体核反应,它也是直接核反应理论研究的一个重要研究对象.本文简述了氘核削裂反应的研究历史,侧重介绍其反应理论的演进,以及平面波玻恩近似(Plane Wave Born Approximation,PWBA)、扭曲波玻恩近...     

On the classical-trajectory eikonal approximation for electron capture into multicharged ions

The eikonal approach to electron capture developed in earlier publications is analyzed with respect to its physical content using the Born series as a reference expansion. In the language of perturbation theory, the eikonal approach describes a close collision between electron and projectile followed by multiple distant collisions between electron and target nucleus. The distant encounters have a large weight in phase space and therefore dominate capture at intermediate energies. The post-prior discrepancy is shown to be fairly small for nonresonant 1s-1s capture.     

Interfering transfer processes in heavy-ion reactions

Heavy-ion reactions in which two different transfer processes may interfere are analyzed. Angular distributions of the reactions ¹⁴C(¹⁶O, ¹⁷O)¹³C and ¹⁴C(¹⁶O, ¹⁸O)¹²C were measured at incident energies of 20, 25 and 30 MeV. The strong oscillations observed at the Coulomb barrier together with a backward rise at higher energies are taken as evidence for the superposition of two competing transfer reactions. DWBA calculations for the two single transfer processes were performed using the fixed-range approximation, and the two transition amplitudes were summed coherently. The experimental angular distributions are well reproduced. The DWBA also explains the disappearance of the interference structures for higher transferred angular momenta l. Data on the reaction ¹¹B(¹⁶O, ¹⁵N)¹²C measured earlier are included in the analysis in order to show the systematic dependence on l-values.     

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.     

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.     

Elastic scattering of 11B from 209Bi in the energy range 49.8 ⩽ E ⩽ 84.1 MeV

The elastic scattering of ¹¹B from ²⁰⁹Bi has been measured at laboratory energies of 49.8, 51.3, 52.2, 52.8, 54.3, 55.8, 59.8, 64.8, 69.8, 74.8 and 84.1 MeV. These data have been analyzed using a microscopic optical model and the energy dependence of the real and the imaginary parts of the optical potential at near barrier energies has been determined. The “threshold anomaly” observed in the real part of the potential is found to be consistent with the dispersion relation which connects the real and the imaginary parts of the potential. Inelastic scattering and transfer reactions have also been measured at energies of 51.3, 55.8, 59.8 and 74.8 MeV. DWBA calculations for the 3⁻ state in ²⁰⁹Bi are made. From the measured transfer probabilities, using a semiclassical approach the strength of the form factors have been obtained. The fusion cross sections have been derived at these energies from the corresponding quasi-elastic scattering angular distribution data. The fusion cross sections calculated using the energy dependent barriers extracted from the energy dependent real parts of the potential compare well with, that determined from quasi-elastic scattering data and are also in good agreement with simplified coupled channels calculation for fusion incorporating important inelastic and transfer channels.