Wave-packet discretization of a continuum: Path toward practically solving few-body scattering problems

A new method for discretizing a three-body continuum with the aid of the L ₂ basis of stationary wave packets is considered within the problem of three-body scattering. Substantial advantages of employing this basis in solving problems of few-body scattering are demonstrated. Specific applications of this approach are exemplified by exploring the problem of scattering of a composite particle on a heavy nucleus with allowance for the excitation of this particle to continuum states. This is done within two alternative approaches: a direct wave-packet discretization of a three-body continuum and a method that is based on the Feshbach projection formalism. It is shown explicitly that the resulting scattering amplitudes are convergent as the number of wave-packet states that are taken into account is increased. The results obtained here are compared with the results of other authors whose treatment was based on alternative methods for discretizing a continuum.     

Fusion of weakly bound projectiles around the Coulomb barrier

The effect of breakup of weakly bound projectiles on fusion is studied using two approaches. The first approach based on the coupled discretised continuum channel (CDCC) formalism is used for ¹¹Be+²⁰⁸Pb, while the second one based on the two-centre shell model (TCSM) is used for ⁹Be+²⁰⁹Bi, ⁶⁴Zn. The CDCC approach describes the breakup in terms of inelastic excitations of the projectile to the continuum, while the breakup is described by molecular single-particle effects in the TCSM approach.     

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.     

Structure of light unstable nuclei in continuum energies

The Coulomb breakup reaction of ⁶He into a ⁴He+n+n 3-body system is analyzed by applying the complex scaling method (CSM). We can derive the contributions to El and E2 transitions, not only from resonances, but also 2- and 3-body continuum states, which show the characteristic structure of the many-body unbound states in ⁶He.     

Coupling effects of resonant and discretized non-resonant continuum states in 4He + 6Li scattering at 10 MeV/A

Alpha-particle scattering from the resonant (3 ₁ ⁺ ) and non-resonant continuum states of ⁶Li is studied at incident energy 10 MeV/A. The α + d breakup continuum part within the excitation energy E _ex = 1.475–2.475 MeV is discretized in two energy bins. Unlike the results at higher incident energies, here the coupled-channel calculations show significant breakup continuum coupling effects on the elastic and inelastic scattering. It is shown that even when the continuum-continuum coupling effects are strong, the experimental data of the ground state and the resonant as well as discretized non-resonant continuum states impose stringent constraint on the coupling strengths of the non-resonant continuum states.     

Discretizing the three-body continuum: Interpretation of non-breakup results

This paper is concerned with understanding the physics responsible for the major result of the preceding article, viz., that the coupled reaction channel (CRC) ansatz responds better than any of the connected-kernel (exact) collision theories to an L² (pseudostate) discretization approximation. The analysis is carried out within the framework of the wave function equations equivalent to the transition operator formalisms employed in the preceding article. In an alternate approach comparing the CRC method with an approximate version of a multichannel collision theory, it was argued that since only the CRC ansatz is based directly on the Schrödinger equation and on orderly restrictions of the Hilbert space, then it must be the superior approximation. While this conclusion is correct, the argument underlying it is in fact inadequate. The different argument used in the present analysis is based on the fact that connected-kernel theories require contributions from the full three-body breakup continua in order to determine the two-body transition amplitudes accurately, while the effect of an L² discretization approximation on them is to eliminate much of these contributions unless very high-lying pseudostates are employed. In contrast, high-lying pseudostates are much less important for the CRC case. This is demonstrated and its ramifications are explored in some detail herein. In particular, the analyses based on this argument are seen to yield an understanding and interpretation of almost all of the trends exhibited in the results tabulated in the preceding article.     

Broad states beyond the neutron drip line

Theoretical studies of broad states in the few-body systems beyond the neutron drip line have been performed. We introduce a theoretical model which allows to incorporate the initial structure of colliding nuclei, reaction mechanism, few-body effects and final-state interactions in studies of broad unbound states. The model is directly related to the sudden-removal approximation for high-energy knock-out or break-up reactions. We apply this model to qualitative studies of some general properties of broad few-body states including correlations for emitted fragments. The theoretical ideas are illustrated mainly using the example of ⁵H. The prospect for observation of broad continuum structures corresponding to the tetraneutron ⁴n is also discussed.Received: 9 April 2003, Revised: 21 August 2003, Published online: 26 January 2004PACS: 21.45. + v Few-body systems - 21.60.Gx Cluster models - 25.10. + s Nuclear reactions involving few-nucleon systems     

Projectile breakup dynamics for 6Li + 59Co : Kinematical analysis of \alpha - d coincidences

A study of the kinematics of the $ \alpha$ -d coincidences in the ⁶Li + ⁵⁹Co system at a bombarding energy of E _lab = 29.6 MeV is presented. With exclusive measurements performed over different angular intervals it is possible to identify the respective contributions of the sequential and direct projectile breakup components. The angular distributions of both breakup components are fairly well described by the Continuum-Discretized Coupled-Channels framework (CDCC). Furthermore, a careful analysis of these processes using a semiclassical approach provides information on both their lifetime and their distance of occurrence with respect to the target. Breakup to the low-lying (near-threshold) continuum is delayed, and happens at large internuclear distances. This suggests that the influence of the projectile breakup on the complete fusion process can be related essentially to the direct breakup to the ⁶Li high-lying continuum spectrum.     

New Way in Few-Body Scattering Calculations

The key features of the new general approach to solution of few-body scattering problems in hadronic, nuclear and atomic physics are presented and discussed in the paper. The approach is based on a general idea of the lattice-like discretization of few-body continuum using the stationary wave-packet basis in momentum space. The new technique includes an efficient averaging and smoothing of singular kernels of the scattering integral equations over the lattice cells. So, such an averaging procedure allows us to transform the complicated singular integral kernels into usual matrices with regular and smooth matrix elements. Such a transformation is shown to lead to an enormous simplification of the solving procedure for scattering equations.     

Dielectronic recombination rate for the Be-sequence

The dielectronic recombination (DR) rate coefficient α^DR is explicitly calculated for the Mo, Fe, Ar, and Ox target ions of the Be-sequence with four electrons, in the isolated resonance approximation. This work extends the previous study of the Mo³⁸⁺ ions at 1.4, 2.8, and 5.6 keV electron temperatures. Both Δn≠0 and Δn = 0 transitions are considered in detail. The Δn≠0 contribution still dominates, but the Δn = 0 effect becomes quite large for heavy ions. An explicit LS coupling scheme is employed throughout for the dominant transitions calculated here, and contributions from many other intermediate states and cascade transitions are included by comparing the dominant contribution with the more complete Mo³⁸⁺ case and proportionately scaling their effect. Nonrelativistic Hartree-Fock wave functions are used in the evaluation of the Auger and radiative amplitudes, and the continuum wave functions are calculated using the Hartree-Fock direct potential and explicit nonlocal exchange potential. The scaling property of α^DR and its breakdown are examined, and an improved form of the phenomenological formula is proposed.     

Spin structure of the “forward” charge-exchange reaction n + p → p + n and deuteron charge-exchange breakup d + p → (pp) + n

The spin structure of the nucleon charge-exchange process n + p → p + n is investigated. Analysis shows that the spin-dependent part of the differential cross section of the process n + p → p + n at zero angle, which is proportional to the differential cross section of the deuteron charge-exchange breakup d + p → (pp) + n in the “forward” direction, plays the predominant role in the wide range of neutron momenta.     

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.     

Coupling effects of resonant and discretized non-resonant continuum states in4He +6Li scattering at 10 MeV/A

Alpha- particle scattering from the resonant (3 ₁ ⁺ ) and non-resonant continuum states of⁶Li is studied at incident energy 10 MeV/A. Theα+d breakup continuum part within the excitation energyE _ex=1.475–2.475 MeV is discretized in two energy bins. Unlike the results at higher incident energies, here the coupled-channel calculations show significant breakup continuum coupling effects on the elastic and inelastic scattering. It is shown that even when the continuum-continuum coupling effects are strong, the experimental data of the ground state and the resonant as well as discretized non-resonant continuum states impose stringent constraint on the coupling strengths of the non-resonant continuum states.     

On the Parametrization of Low-Energy Parity-Violating Observables

The parity-violating (PV) two-nucleon interaction, unlike the parity-conserving one, is still poorly constrained. Recently such study in few-body systems has attracted quite some attention, as experiments become more feasible and theoretical calculations can be done reliably. It is found that using the so-called Danilov parameters, which are the zero-energy S–P transition amplitudes, yield a model independent way to parametrize low-energy PV observables. This provides not only a common interface through which calculations using different frameworks can communicate, but also helps to distinguish for which observables the contributions from higher partial waves can no longer be ignored.     

Clustering aspects of light exotic nuclei

The discovery of light nuclei with neutron halos has opened new perspectives for nuclear structure and a possibility to study genuine few-body aspects of matter. A number of borromean systems where the system is bound but where no binary subsystems are bound, are realized by nature in the context of nuclei,¹¹Li being the nucleus which has had most recent attention. Although a large number of calculations have addressed the detailed structure of the¹¹Li neutron halo, no single model has yet explained all the data presently available. This is in part due to uncertainties in then+⁹Li interaction. For⁶He, also a borromean system, where then+core interaction is known, the best three-body and reaction models succeed in reproducing the data. In the¹¹Li case, a major uncertainty concerns the effect of possible 1s intruder states.     

Variational principles and linked-cluster exp S expansions for static and dynamic many-body problems

The exp S formalism for the ground state of a many-body system is derived from a variational principle. An energy functional is constructed using certain n-body linked-cluster amplitudes with respect to which the functional is required to be stationary. By using two different sets of amplitudes one either recovers the normal exp S method or obtains a new scheme called the extended exp S method. The same functional can be used also to obtain the average values of any operators as well as the linear response to static perturbations. The theory is extended to treat dynamical phenomena by introducing time dependence to the cluster amplitudes. This allows the calculation of both nonlinear dynamical behaviour and of dynamical linear response and Green's functions. Practical approximation schemes are considered. In a SUB n approximation the m-body amplitudes are restricted to the order $\text{m ? n}$ and the energy functional is a finite-order multinomial in the amplitudes to be variationally determined. It is shown that the solution corresponds to summing well-defined subsets of Goldstone diagrams. These subsets are conveniently specificed in terms of tree structures, the normal or extended generalized time ordering g.t.o. trees. The extended exp S method is in the SUB n approximation able to sum, in addition to the normal SUB n diagrams, a set which contains m-body cluster amplitudes of arbitrarily high order (m > n) in the ordinary sense. The article also discusses how the SUB n truncation schemes must be modified to be able to treat a system with a strong repulsive core in the two-body interaction. The method is formulated for the general cases of Bose and Fermi systems which may or may not conserve total particle number. It is shown that the simplest approximation, SUB 1, in the extended exp S method agrees with the mean field theory, which is the coherent-state approximation in the boson case or the Hartree-Fock approximation in the fermion case. It is argued that the extended exp S method already in low-order approximations can realistically treat a great variety of diverse many-body problems, even including systems which may undergo ground-state phase transitions. A few applications are described in more detail. The Bose liquid is treated in the extended SUB 2 approximation. It is shown that the ground-state results in the uniform limit are exact and agree with the hypernetted-chain approximation. The modifications due to hard-core interactions and the non-linear equations of motion are also discussed in this case. For Fermi systems it is shown that the supercondictive phase transition of the BCS model Hamiltonian and the deformation phase transition of the Lipkin model are properly obtained by the extended exp S method in a low-order approximation.     

Binding corrections for reactions on the deuteron in the 3-3 resonance region

Binding corrections evaluated to all orders in a phenomenological nucleon-nucleon potential are compared with the leading term and with a double scattering correction in the Watson expansion for breakup and for coherent reactions on the deuteron at intermediate pion or photon energies. As principal result, we establish that for charged pion photoproduction, and for certain pion induced “spectator” reactions, the binding correction remains sufficiently small that truncation of the Watson expansion at terms of second order in the projectile-nucleon amplitudes is justifiable, but sufficiently large that it cannot be neglected in comparison with multiple scattering corrections. For the reactions π⁺d → π⁺p + spectator neutron and π^?d → π^?n + spectator proton, our theory predicts deviations from the simplest impulse approximation which become very large in well-defined kinematical regions, and should be easily accessible to experimental investigation.     

Multiplicity distributions at hadron colliders

We analyse an extended Nambu and Jona-Lasinio model in the context of the conventional RPA treatment and of a variational approximation which corresponds to the discretization of the continuum, allowing to describe mesonic resonances that are treated as compositeq \(\bar q\) systems. We study properties of pseudoscalar, scalar, vector and axial-vector excitations. The discretization technique simulates effects of confinement, replacing the exact continuum eigensolutions by a finite number of approximate normalizable eigensolutions.     

Study of various few-body systems using Gaussian expansion method (GEM)

We review our calculation method, Gaussian expansion method (GEM), to solve accurately the Schrödinger equations for bound, resonant and scattering states of few-body systems. Use is made of the Rayleigh-Ritz variational method for bound states, the complex-scaling method for resonant states and the Kohn-type variational principle to S-matrix for scattering states. GEM was proposed 30 years ago and has been applied to a variety of subjects in few-body (3- to 5-body) systems, such as 1) few-nucleon systems, 2) few-body structure of hypernuclei, 3) clustering structure of light nuclei and unstable nuclei, 4) exotic atoms/molecules, 5) cold atoms, 6) nuclear astrophysics and 7) structure of exotic hadrons. Showing examples in our published papers, we explain i) high accuracy of GEM calculations and its reason, ii) wide applicability of GEM to various few-body systems, iii) successful predictions by GEM calculations before measurements. The total bound-state wave function is expanded in terms of few-body Gaussian basis functions spanned over all the sets of rearrangement Jacobi coordinates. Gaussians with ranges in geometric progression work very well both for shortrange and long-range behavior of the few-body wave functions. Use of Gaussians with complex ranges gives much more accurate solution than in the case of real-range Gaussians, especially, when the wave function has many nodes (oscillations). These basis functions can well be applied to calculations using the complex-scaling method for resonances. For the few-body scattering states, the amplitude of the interaction region is expanded in terms of those few-body Gaussian basis functions.