Three-body model of deuteron breakup and stripping

For a three-body model Hamiltonian, the scattering eigenfunction that corresponds to an incident deuteron is expanded in terms of eigenfunctions of the neutron-proton relative Hamiltonian, as suggested by Johnson and Soper. In this expansion, breakup is represented by an integral over the continuum of neutron-proton scattering states. Only states of zero relative angular momentum are included; the validity and advantages of this approximation are discussed. The continuum is divided into five discrete channels, whose coupling to each other and to the deuteron channel is treated by solving coupled differential equations with appropriate boundary conditions. It is found necessary to use a simple WKB method to take account of the long-range coupling among breakup channels; this method introduces potential matrices W and S that describe local and derivative coupling of the channels. The reaction of breakup on the elastic channel is neglected.The properties of W and S and the breakup wavefunction are examined for the case of 22.9 MeV deuterons incident on a target of mass number A ≈ 40. The Coulomb interaction is ignored, and a local Gaussian shape is used for both the real and imaginary parts of the nucleon-nucleus optical potential.It is found that a rather broad spectrum of n-p continuum states is excited, especially for low center-of-mass angular momentum. This result weakens the justification for the Johnson-Soper adiabatic theory, which emphasizes breakup into states of low relative energy.The breakup part of the wavefunction at zero n-p separation is comparable with the elastic part, but is important only over a surprisingly short range in the center-of-mass coordinate, with the result that breakup cross sections are quite small. Nevertheless, breakup produces major modifications of (d, p) cross sections. These modifications can to some extent be simulated by the Johnson-Soper method. The breakup wavefunctions show several interesting effects in their dependence on angular momentum and radius.     

New approach to solving the problem of composite-particle scattering on nuclei

An essentially new approach to solving the problem of elastic and inelastic scattering of a composite particle on stable nuclei is described. Within this approach, all channels of virtual breakup and stripping in the intermediate states are included in a nonlocal complex-valued interaction operator with the aid of the projection-operator technique.The three-particle continuum spectrum of the Hamiltonian for intermediate states in Q space is calculated within the orthogonalizing-pseudopotential method by introducing a pseudo-Hamiltonian, which is diagonalized in a full space in terms of a relevant oscillator basis. As was shown by a number of authors, the use of special quadratures makes it possible to reduce integration over the continuous spectrum of intermediate states to summation over a discretized continuum. On the basis of the formalism developed in this study, a closed Schrödinger equation with a nonlocal complex potential for partial waves is derived for describing elastic scattering of a composite particle by a target, and an explicit approximate formula for the amplitude of three-particle breakup is obtained on the same basis. This method has a number of obvious advantages over currently well-known approaches of the type of the discretized-continuum coupled-channel method, where solving the problem in question reduces to solving a cumbersome set of coupled equations.     

On three particle stripping reactions

The formalism for the coupled-channel analysis of stripping reactions of complex nuclear projectiles by nuclei is presented. The general adiabatic approximation is developed. The indirect transitions that are considered are those which arise via intermediate rotational excitations of the target and product nuclei, i.e. the adiabatic approximation for rotational bands is used. At the same time the generalized DWBA procedure is considered for the intrinsic states. Also the antisymmetrization problem is solved. A new method for the calculation of the matrix elements of the (α,n) stripping reaction with finite range effective forces is developed. The method is based on an expansion of the radial functions which describe the relative motion in terms of harmonic oscillator wave functions multiplied by a Gauss one. Effective forces of Gauss type are assumed between the outgoing neutron and each captured nucleon     

Second-order breakup corrections to elastic deuteron-nickel scattering between 13 and 80 MeV

Breakup corrections to the elastic scattering matrix elements are calculated in the second-order distorted-wave Born approximation at deuteron incident energies of 45 and 85 MeV. The effects of spin are included. The size of the corrections are found to be generally as large as those obtained in a previous study at 13 and 21.6 MeV. The breakup cross section is calculated to first order in the breakup matrix elements by a distorted-wave Born treatment. Comparison with fully coupled calculations shows that the DWBA method overestimates the breakup cross section by a factor of about three.The continuum of breakup states up to a n-p relative momentum 1 fm^?1 is included in the calculations. This continuum is discretized by subdividing it first into two bins, and then into four bins. The finer discretization does not make a large difference in either the elastic cross section or the breakup cross sections. The higher bins give only a small contribution to either quantity.     

Breakup reaction models for exotic nuclei

Breakup reactions are one of the main tools for the study of exotic nuclei. In particular, Coulomb breakup is expected to provide information on spectroscopic properties of halo nuclei and on astrophysical S factors for radiative-capture reactions. The simplest studies are based on perturbation theory and especially on its first order. However the validity of the first-order approximation may be limited for extended systems such as halo nuclei and its conditions are not always satisfied in existing experiments. More elaborate reaction models are available: resolution of the semi-classical time-dependent Schr?dinger equation, eikonal and dynamical eikonal approximations, method of coupled discretized-continuum channels, adiabatic approximation. These methods are reviewed, summarized and illustrated. Their interest and limitations are discussed.     

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.     

Progress in all-order breakup reaction theories

Progress in breakup reaction theories, like the distorted wave Born approximation, the continuum discretized coupled channels method and the dynamical eikonal approximation, is brought into focus. The need to calculate exclusive reaction observables and the utility of benchmark tests as arbitrators of theoretical models are discussed.     

Investigation of the inelastic deuteron scattering mechanism for 24Mg at E d = 15.3 MeV

Results from measuring the angular dependence of differential cross sections of elastic and inelastic deuteron scattering by the ²⁴Mg nucleus with the formation of the ground and first excited states 2⁺, 1.369 MeV at E _d = 15.3 MeV for deuteron angles interval from 21.5° to 161.5° (lab) are presented. Experimental results are compared with calculations in various approximations of the coupled channels model. The influence of heavy particle stripping mechanisms and consecutive particle transfer is also discussed.     

Fragmentation processes in nuclear reactions

Fragmentation processes in nuclear collisions are reviewed. The main emphasis is put on light ion breakup at nonrelativistic energies. The post- and prior-form DWBA theories are discussed. The post-form DWBA, appropriate for the “spectator breakup” describes elastic as well as inelastic breakup modes. This theory can also account for the stripping to unbound states. The theoretical models are compared to typical experimental results to illustrate the various possible mechanisms. It is discussed, how breakup reactions can be used to study high-lying single particle strength in the continuum; how they can yield information about momentum distributions of fragments in the nucleus.     

Polarization effects on heavy-ion scattering in the distorted-wave method

On the basis of an exact formalism for DWBA methods we calculate the distorting potentials to be used in a standard distorted-wave Born approximation for systems with strongly coupled channels. We examine for a practically useful model some simple approximate treatments through comparisons of the polarization potentials in the case of elastic and inelastic ¹⁶O + ¹⁶O scattering. The adiabatic approximation omitting the radial kinetic energy in the propagator is found to lead to satisfactory agreement with the exact coupled-channel cross sections.     

3He Transverse Electron Scattering Response Function with �� Degrees of Freedom

A calculation of the ³He transverse (e, e??) inclusive response function, R _T , which includes ?? degrees of freedom is performed using the Lorentz integral transform method. The resulting coupled equations are treated in impulse approximation, where the NNN and NN?? channels are solved separately. R _T is calculated for the breakup threshold region at a momentum transfer of q?=?862 MeV/c.     

Deuteron elastic scattering and (d,p) reactions on 208Pb at Ed = 22 MeV and j-dependence of T20 in (d,p) reaction

An experimental study of deuteron elastic scattering and (d,p) reactions at E_d = 22 MeV was made for ²⁰⁸Pb target. A new j-dependence of T₂₀ for (d,p) reaction at backward angles was observed. A “model independent” optical potential method was applied to analyze deuteron elastic scattering in the 10–22 MeV energy range. A continuum discretized coupled channels (CDCC) analysis was performed for deuteron elastic scattering at 22 MeV and the deuteron breakup effect was elucidated. DWBA and CDCC (d,p) analyses were made for (d,p) reactions.     

Influence of breakup on elastic and α-production channels in the ~6Li+~(116)Sn reaction

The effects of breakup reactions on elastic and α-production channels for the ~6Li+~(116)Sn system have been investigated at energies below and near the Coulomb barrier. The angular distributions of α-particle production differential cross sections have been obtained at several projectile energies between 22 and 40 MeV. The measured breakup α-particle differential cross sections and elastic scattering angular distributions have been compared with the predictions of continuum-discretized coupled channels(CDCC) calculations. The influence of breakup coupling has also been investigated by extracting dynamic polarization potentials(DPP) from the CDCC calculations. From the predictions of CDCC calculations the relative importance of the nuclear, Coulomb, and total breakup contributions have also been investigated. The nuclear breakup couplings are observed to play an important role in comparison to the Coulomb breakup for the direct breakup mechanisms associated in the reaction of ~6Li projectile with ~(116)Sn target nuclei. The influence of strong nuclear breakup coupling exhibits suppression in the Coulomb-nuclear interference peak. The direct breakup cross sections from the CDCC calculations under-predict the measured α-particle differential cross sections at all energies. This suggests that the measured α particles may also have contributions from other possible breakup reaction channels.     

Quasi-elastic scattering of the proton drip line nucleus 17F on 12C at 60 MeV

The quasi-elastic scattering angular distribution of the proton drip line nucleus ¹⁷F on a ¹²C target was measured at 60 MeV. The experimental data have been compared with the theoretical analysis based onto optical model and continuum discretized coupled channels (CDCC). The couplings between breakup and elastic scattering channels, and between inelastic and elastic scattering channels resulted very weak. In order to explore the breakup effects the total reaction cross-section was deduced from the angular distribution of the quasi-elastic scattering data, and then compared with the existing data for the other weakly and tightly bound nuclei on ¹²C target using a universal function. From this comparison, we concluded that the breakup effect is not important for weakly bound projectiles on the light target as obtained also with the CDCC analysis.     

CDCC Approaches and Roles of Closed Channels in d-Nucleus Reactions

The effect of deuteron breakup in d-nucleus reaction is treated with the continuum discretized coupled channels (CDCC) approach, and the effects on the total reaction cross sections and elastic scattering angular distributions are studied by comparing the calculations of CDCC and spherical optical model with our global deuteron optical potential [Phys. Rev. C 73 (2006) 054605] below 200 MeV, for target nuclei ranging from ¹²C to ²⁰⁸Pb. The contributions from the closed channels to the total reaction and breakup cross sections, and angular distributions of elastic scattering are also seriously discussed.     

Elastic scattering and fusion cross-sections in 7Li + 27Al reaction

With an aim to understand the effects of breakup and transfer channels on elastic scattering and fusion cross-sections in the ⁷Li + ²⁷Al reaction, simultaneous measurement of elastic scattering angular distributions and fusion cross-sections have been carried out at various energies (E _lab?=?8.0–16.0 MeV) around the Coulomb barrier. Optical model (OM) analysis of the elastic scattering data does not show any threshold anomaly or breakup threshold anomaly behaviour in the energy dependence of the real and imaginary parts of the OM potential. Fusion cross-section at each bombarding energy is extracted from the measured α-particle evaporation energy spectra at backward angles by comparing with the statistical model prediction. Results on fusion cross-sections from the present measurements along with data from the literature have been compared with the coupled-channels predictions. Detailed coupled-channels calculations have been carried out to study the effect of coupling of breakup, inelastic and transfer, channels on elastic scattering and fusion. The effect of 1n-stripping transfer coupling was found to be significant compared to that of the projectile breakup couplings in the present system.     

Adiabatic treatment of inelastic deuteron-nucleus scattering

Inelastic scattering of 80 MeV deuterons from ⁵⁸Ni and ¹²⁰Sn is analyzed in terms of an adiabatic coupled-channels Born approximation (ACCBA) method. The ACCBA method is a practical means to treat deuteron plus nucleus as a three-body system. Inclusion of the breakup modes significantly reduces the inelastic cross section. The significance of considering both the s-wave and d-wave breakup modes is pointed out.     

Asymptotic method for determining the amplitude for three-particle breakup: Neutron-deuteron scattering

The process of neutron-deuteron scattering at energies above the deuteron-breakup threshold is described within the three-body formalism of Faddeev equations. Use is made of the method of solving Faddeev equations in configuration space on the basis of expanding wave-function components in the asymptotic region in bases of eigenfunctions of specially chosen operators. Asymptotically, wave-function components are represented in the form of an expansion in an orthonormalized basis of functions depending on the hyperangle. This basis makes it possible to orthogonalize the contributions of elastic-scattering and breakup channels. The proposed method permits determining scattering and breakup parameters from the asymptotic representation of the wave function without reconstructing it over the entire configuration space. The scattering and breakup amplitudes for states of total spin S = 1/2 and 3/2 were obtained for the s-wave Faddeev equation.     

Spin waves in Heisenberg-ferromagnets above the critical point

A formulation of stripping reactions into the continuum is given taking the(d, p)-reaction as a specific example. We use the DWBA and the assumption that contributions from the nuclear interior are negligible. Under these assumptions we show that the cross section can be split into a “distortion matrix element” and a term depending on the neutron target interaction only through the phase shifts of the corresponding elastic neutron scattering. The interference with the “pure break-up” process is discussed. The connection between stripping to bound and unbound states in the limit of zero binding energy is established. A comparison with recent experimental results is given, where the “distortion matrix elements” are calculated, for the sake of simplicity, in the “Butler approximation”.     

Deuteron stripping to the single particle states of 17O

Angular distributions for deuteron-¹⁶O elastic scattering and the ¹⁶O(d, p)¹⁷O reaction leading to levels with E_x = 0.0, 0.87, and 5.08 MeV have been measured at energies of 25.4, 36.0 and 63.2 MeV. The elastic deuteron data have been fit with a standard spin one optical model potential to obtain parameters for use in a DWBA analysis of the (d, p) data. The potential found in the search is shown to be consistent with other data taken in the range from 25 to 82 MeV. In addition to this deuteron optical potential, an adiabatic deuteron potential, which includes the effects of deuteron breakup, was used in the DWBA analysis. The neutron form factor was selected independent of the width of any state. The mean square radius, a single particle property, is used to find the well parameters and it determines the width of the single particle state. The spectroscopic factors obtained for the ground and first excited states are between 0.8 and 1.0 and are consistent with a large single particle parentage for these states and lower energy data. The width extracted from the DWBA analysis of the 5.08 MeV unbound state was 20% less than that obtained from elastic neutron scattering to the same state, possibly pointing up some difficulties with DWBA procedures commonly used. The adiabatic deuteron potential yields spectroscopic factors that are energy independent to 20% and gives satisfactory calculated angular distribution shapes for angles less than 40°. The conventional deuteron potential gives less satisfactory calculated shapes with the consequent introduction of some ambiguity in the derived spectroscopic factors.