The neutron-deuteron scattering problem in the framework of the Faddeev formalism

The three-body Faddeev equations in the configuration space are solved numerically for study of the neutron-deuteron scattering process above the breakup threshold. The amplitudes for the binary scattering process and breakup are obtained by the developed asymptotic approach.     

Neutron–Deuteron Scattering Observables at E lab = 14.1 MeV

Inelastic neutron–deuteron scattering is studied on the basis of configuration-space Faddeev equations. Calculated are neutron–deuteron breakup amplitudes using AV14 nucleon–nucleon potential at incident neutron energy of 14.1 MeV. The results of calculations are presented for the differential cross sections under quasi free scattering and space–star configurations, and compared with those of the previous calculations and experimental data. The choice of the cutoff radius R _cutoff for asymptotic conditions is discussed.     

Operation of the Faddeev Kernel in Configuration Space

We present a practical method to solve Faddeev three-body equations at energies above the three-body breakup threshold as integral equations in coordinate space. This is an extension of a previously used method for bound states and scattering states below three-body breakup threshold energy. We show that breakup components in three-body reactions produce long-range effects on Faddeev integral kernels in coordinate space, and propose numerical procedures to treat these effects. Using these techniques, we solve Faddeev equations for neutron-deuteron scattering to compare with benchmark solutions.     

Three-body scattering in configuration space

The asymptotic forms of the wavefunction and Faddeev components in configuration space are shown to determine uniquely the solutions of the Schrödinger or Faddeev differential equations for 2 → (2, 3) and 3 → (2, 3) processes. An antisymmetrized form of the Faddeev differential equation for three equivalent fermions is given and its angular analysis is performed in the general case of local potentials with tensor interaction for neutron-deuteron scattering. We describe a numerical method for solving the corresponding boundary value problem and apply it to scattering and break-up at E_n^1ab = 14.4 MeV in the doublet S state for the four local potentials of Malfliet and Tjon, Reid, de Tourreil and Sprung, and de Tourreil, Rouben and Sprung. For the three realistic potentials, elastic scattering amplitudes differ by 5%, and amplitudes for break-up in the two-neutron state ¹S₀ differ by less than 4%.     

New treatment of breakup in few-body systems

A new approach to determining breakup amplitudes in few-body systems in the context of a Faddeev formalism based on lattice discretization of a continuum is described. Due to such discretization and use of finite-dimensional representations for all operators in the kernels of integral equations, breakup in few-body systems is interpreted as a partial case of multi-channel scattering and corresponds to transitions between the states of the discretized continuum of an asymptotic channel Hamiltonian. The case study is based on amplitudes of three-nucleon breakup n + d → n + n + p with semi-realistic NN interaction potentials.     

Asymptotics of the binary amplitude for a model Faddeev equation

A model equation obtained from the s-wave Faddeev equation in configuration space for three identical bosons by replacing the inhomogeneous integral term with a known function is studied. This function simulates the asymptotic decrease of the inhomogeneous term in the original Faddeev equation when y → ∞ as ～y^–3/2. The asymptotes of the amplitude functions that approach the scattering amplitudes when y → ∞ are obtained analytically for the model equation using the Green function. It is shown that the asymptotics of the binary channel amplitude function oscillates. The similar oscillating behavior of the binary amplitude function is numerically demonstrated for the original s-wave Faddeev equation describing the neutron–deuteron scattering process.     

A novel method for solving the three-body scattering problem

A numerical scheme for solving a three-body scattering problem within the framework of the configuration space Faddeev equations in three-dimension, i.e., without resort to explicit partial wave expansion, is presented. The method is applied to calculate the low-energy n-d observables.     

Calculation of the two-body scattering T-matrix in configuration space

In order to carry out a solution of the three-body Faddeev integral equations in configuration space, the calculation of the two-body scattering T-matrices and related integrals are required as an input. The formulation of the three-body Faddeev solution, as well as the computational steps used for the calculation of the T-matrices are presented, and results for the latter are illustrated for the case of the scattering of two helium atoms.     

A multi-cluster,n-particle generalization of the three-particle faddeev wavefunction equations

Recent work applying certain forms of many-body scattering theory to problems such as molecular potential energy surfaces and equations for nonequilibrium statistical mechanics indicates that a formulation of the theory based directly on multi-cluster, n-particle, wave function components could be of some utility. Such a formulation is derived in this paper using techniques from the Baer-Kouri-Levin-Tobocman and Bencze-Redish-Sloan-Polyzou theories of multi-particle scattering. It is based on components corresponding to the various multi-cluster partitions of an n-particle scattering system and is a generalization of the three-body Faddeev wave function formalism, to which it reduces when n = 3. Except for the full breakup partition, which does not enter the equations, the new components are defined for all possible m-cluster partitions of the n-particles, 2 ≤ m ≤ n ? 1. The sum of all the components yields the solution to the Schrödinger equation for scattering and either the Schrödinger equation solution or an easily identified spurious solution in the case of bound states. Both the two-cluster components and two-cluster transition operators are shown to be solutions of equations involving quantities carrying only two-cluster partition labels. Discussions of the Born term and a multiple scattering representation for the non-rearrangement transition operator and the inclusion of distortion operators in the formalism are also included.     

Continuum-discretized coupled-channels calculations for three-body models of deuteron-nucleus reactions

The formalism and results of truncated coupled channels evaluations of three-body models of deutron-induced nuclear reactions are reviewed. Emphasis is placed on breakup, elastic scattering and stripping. The relations of the coupled channels method to the Faddeev method, the adiabatic approximation and the distorted wave Born approximation are discussed extensively. Although the adiabatic approximation is seen to be excellent for the wavefunction in the elastic channel, it significantly underestimates the contributions of breakup states in stripping. Significant effects are associated with coupling to relative l = 2 breakup states.     

Faddeev equations and the method of hyperspherical harmonics in the problem of three-nucleon continuum

A method for solving the Faddeev equations in configuration space is developed for a three-nucleon system in the continuum by using the decomposition over a hyperspherical basis. The wave functions of Nd-system, phase shifts, and cross sections of Nd-scattering at subthreshold energies are calculated. Also, within the framework of this method, one-dimensional integral equations are formulated for the problem of infinite motion of all three strongly interacting particles, and the Faddeev equations for a system of three hadrons with Coulomb interaction in the continuum are modified. Similar methods of investigation of three-particle systems are reviewed.     

Scattering from a bound target in the Faddeev approach: (I). Neutron scattering from protons in heavy hydrocarbons in the eV region

We consider the Faddeev approach to the scattering of a projectile from a target bound to a residual core under the assumptions that the projectile-target and target-core forces are separable, that the projectile and core do not interact, and that the core is infinitely heavy. As a first application of our formalism we calculate the scattering of neutrons from the protons in hydrocarbon molecules, the so-called chemical binding problem. Upon solving the Faddeev equations by inversion for this situation, we find that the impulse approximation, the driving term in the Faddeev equations, describes the exact scattering to within ≦ 0.45%. Further we examine the energy region where molecular dissociation is possible and find that the bound-final-state and molecular breakup cross sections balance to give the experimentally observed asymptotically constant cross section σ_free^np.     

R-matrix approach to the 3-body problem

The asymptotic form of the Faddeev amplitude in coordinate space is derived in various orders. This form and the structure of the Faddeev equations allow by aR-matrix method to establish a set of equations directly for the 3-body on-shellT-matrix elements. The procedure is equally well suited for local and nonlocal interactions.     

Solving Faddeev equations for a bound state and a continuous spectrum of a three-nucleon system by the method of <Emphasis Type="Italic">K</Emphasis>-harmonic expansions

A method for solving Faddeev equations in configuration space for a bound state and a continuous spectrum of the system of three nucleons was developed on the basis of expansions in K harmonics. Coulomb interaction and particle spins were not taken into account in this study. The method in question was used to describe the triton bound state and differential cross sections for neutron-deuteron scattering at subthreshold incident-neutron energies. The Volkov, Malfliet-Tjon, and Eikemeier-Hackenbroich local nucleon-nucleon potentials were employed in the present calculations.     

Calculations of elastic n-d scattering for the Reid potential

A perturbational scheme is used to study neutron-deuteron elastic scattering with the Faddeev equations in momentum space. The pure s-wave parts of the two-nucleon T-matrix are treated in an exact way, while the higher partial-wave components are retained in first order. This is done for the full Reid soft-core potential. The lab energies considered in this study are between 5 and 50 MeV. It is found that the perturbation method can give insight in the sensitivity to details of the potential, and at higher energies can give quantitatively reliable results, for most of the observables.     

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 g \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.     

Faddeev approach to the three-body problem in total-angular-momentum representation

For a system of three charged particles the Faddeev equations are derived in the total-angular-momentum representation. They have the form of coupled sets of partial differential equations in three-dimensional space and can be used to develop new efficient numerical procedures to tackle the three-body Coulomb problem. The asymptotic conditions at large distances corresponding both to binary scattering and bound-state problems are presented. The behaviour of the Faddeev components near the triple and double collision points is studied.     

Simple functional-differential equations for the bound-state wave-function components

We present a new method of a direct derivation of differential equations for the wave-function components of identical-pariticles systems. The method generates in a simple manner all the possible variants of these equations. In some cases they are the differential equations of Faddeev of Yakubovskii. It is shown that the case of the bound states allows to formulate very simple equations for the components which are equivalent to the Schrödinger equation for the complete wave function. The components with a minimal antisymmetry are defined and the corresponding equations are derived.     

Two-body scattering without angular-momentum decomposition: Fully off-shell T-matrices

Two-body scattering is studied by solving the Lippmann-Schwinger equation in momentum space without angular-momentum decomposition for a local short-range interaction plus Coulomb. The screening and renormalization approach is employed to treat the Coulomb interaction. Benchmark calculations are performed by comparing our procedure with a configuration space calculation, using the standard partial-wave decomposition, for ¹²C - ¹⁰Be elastic scattering. The fully off-shell T -matrices are also calculated for the final goal of studying the three-body scattering by solving Faddeev/AGS equations.     

The three-body problem with short-range interactions

The quantum mechanical three-body problem is studied for general short-range interactions. We work in coordinate space to facilitate accurate computations of weakly bound and spatially extended systems. Hyperspherical coordinates are used in both the interpretation and as an integral part of the numerical method. Universal properties and model independence are discussed throughout the report. We present an overview of the hyperspherical adiabatic Faddeev equations. The wave function is expanded on hyperspherical angular eigenfunctions which in turn are found numerically using the Faddeev equations. We generalize the formalism to any dimension of space d greater or equal to two. We present two numerical techniques for solving the Faddeev equations on the hypersphere. These techniques are effective for short and intermediate/large distances including use for hard core repulsive potentials. We study the asymptotic limit of large hyperradius and derive the analytic behaviour of the angular eigenvalues and eigenfunctions. We discuss four applications of the general method. We first analyze the Efimov and Thomas effects for arbitrary angular momenta and for arbitrary dimensions d. Second we apply the method to extract the general behaviour of weakly bound three-body systems in two dimensions. Third we illustrate the method in three dimensions by structure computations of Borromean halo nuclei, the hypertriton and helium molecules. Fourth we investigate in three dimensions three-body continuum properties of Borromean halo nuclei and recombination reactions of helium atoms as an example of direct relevance for the stability of Bose–Einstein condensates.