Two-body Lippmann-Schwinger Equation Research Articles

Analytical solutions for three-dimensional Coulomb transition matrix at negative energy and integer values of interaction parameter

With the use of a stereographic projection of momentum space into a four-dimensional sphere of unit radius, the possibility of analytical solution of the three-dimensional two-body Lippmann–Schwinger equation with the Coulomb interaction at negative energy has been studied. Simple analytical expressions for the three-dimensional Coulomb transition matrix in the case of the repulsive Coulomb interaction and positive integer values of the Coulomb parameter have been obtained. The worked out method has also been applied to the generalized three-dimensional Coulomb transition matrix in the case of the attractive Coulomb interaction and negative integer values of the Coulomb parameter.

High-rank separable atom-atom interaction potential used for solving two-body Lippmann-Schwinger and three-body Faddeev equations.

We derive a high-rank separable potential formula of the atom-atom interaction by using the two-body wave function in the coordinate space as inputs. This high-rank separable potential can be utilized to numerically solve the two-body Lippmann-Schwinger equation and three-body Faddeev equation. By analyzing the convenience and stability of numerical calculations for different kinds of the matrix forms of the Lippmann-Schwinger and Faddeev equations, we can find the optimal forms of the kernal matrices in the two- and three-body scattering equations. We calculate the dimer bound energy, two-body scattering phase shift and off-shell t-matrix, the trimer bound energy, atom-dimer scattering length, and three-body recombination rate using the high-rank separable potentials, taking the identical 4He atoms as an application example. All the calculations converge quickly for the rank number . The high-rank separable potential is valid for two-body scattering calculation of 4He atoms, but not accurate enough for reproducing the three-body scattering results by using only two-body s-wave interaction and describing the contributions of two-body high partial-waves to the three-body scattering for the 4He3 system.

Positron scattering from hydrogen atom embedded in dense quantum plasma

Scattering of positrons from the ground state of hydrogen atoms embedded in dense quantum plasma has been investigated by applying a formulation of the three-body collision problem in the form of coupled multi-channel two-body Lippmann-Schwinger equations. The interactions among the charged particles in dense quantum plasma have been represented by exponential cosine-screened Coulomb potentials. Variationally determined hydrogenic wave function has been employed to calculate the partial-wave scattering amplitude. Plasma screening effects on various possible mode of fragmentation of the system e++H(1s) during the collision, such as 1s→1s and 2s→2s elastic collisions, 1s→2s excitation, positronium formation, elastic proton-positronium collisions, have been reported in the energy range 13.6-350 eV. Furthermore, a comparison has been made on the plasma screening effect of a dense quantum plasma with that of a weakly coupled plasma for which the plasma screening effect has been represented by the Debye model. Our results for the unscreened case are in fair agreement with some of the most accurate results available in the literature.

Positron scattering from hydrogen atom embedded in weakly coupled plasma

The positron-hydrogen collision problem in weakly coupled plasma environment has been investigated by applying a formulation of the three-body collision problem in the form of coupled multi-channel two-body Lippmann-Schwinger equations. The interactions among the charged particles in the plasma have been represented by Debye-Huckel potentials. A simple variational hydrogenic wave function has been employed to calculate the partial-wave scattering amplitude. Plasma screening effects on various possible mode of fragmentation of the system e++H(1s) during the collision, such as 1s→1s and 2s→2s elastic collisions, 1s→2s excitation, positronium formation, elastic proton-positronium collisions, have been reported. Furthermore, a detailed study has been made on differential and total cross sections of the above processes in the energy range 13.6-350 eV of the incident positron.

A new form of three-body Faddeev equations in the continuum

We propose a novel approach to solve the three-nucleon (3N) Faddeev equation which avoids the complicated singularity pattern going with the moving logarithmic singularities of the standard approach. In this new approach the treatment of the 3N Faddeev equation becomes essentially as simple as the treatment of the two-body Lippmann-Schwinger equation. Very good agreement of the new and old approaches in the application to nucleon-deuteron elastic scattering and the breakup reaction is found.

Deuteron elastic scattering and stripping processes offC12as a three-body problem

Deuteron elastic scattering and stripping processes off a target nucleus consisting of A nucleons are treated within the framework of the few-body integral equations theory. By projecting the ($A+2$)-body operators onto target states, matrix three-body integral equations are derived that allow for the incorporation of the excited states of the target nucleons. This approach is applied to deuteron scattering off $^{12}\mathrm{C}$ when the latter is in its ground state before and after the reaction. For the nucleon-$^{12}\mathrm{C}$ subsystem three sets of (quasi-separable) potentials are employed. The first such potential is based on the one derived by Cattapan et al. [Nucl. Phys. A241, 204 (1975)] for orbital angular momentum states with $L\ensuremath{\leqslant}2$, which is valid for low energies. As second set we use the potential of Miyagawa and Koike [Prog. Theor. Phys. 82, 329 (1989)] fitted to semiphenomenological higher-energy phase shifts for states up to $L=6$. The third one finally consists for $3\ensuremath{\leqslant}L\ensuremath{\leqslant}5$ of the potential set of Miyagawa and Koike, whereas the potential parameters for $L\ensuremath{\leqslant}2$ are determined by simultaneously fitting the elastic-channel T matrix obtained as solution of multichannel two-body Lippmann-Schwinger equations, to the experimental low-energy and the semiphenomenological higher-energy phase shifts. For the nucleon-nucleon interaction we take one of the separable ${}^{3}{S}_{1}\ensuremath{-}{}^{3}{D}_{1}$ potentials of Phillips [Nucl. Phys. A107, 207 (1968)]. Differential cross sections for the elastic-scattering reaction $d+$$^{12}\mathrm{C}$ $\ensuremath{\rightarrow}d+$$^{12}\mathrm{C}$ and the transfer reaction $d+$$^{12}\mathrm{C}$ $\ensuremath{\rightarrow}p+$$^{13}\mathrm{C}$($^{13}\mathrm{C}$${}^{*}$) are calculated at deuteron bombarding energies of 4.66 and 15 MeV (up to 36-channel calculation), and at 56 MeV (up to 76-channel calculation) together with some selected analyzing powers, and are compared with experimental data. At the highest energy considered, the decomposition of the differential cross section into the near-side and the far-side components shows the appearance of nuclear rainbow scattering.

Formulation of few-body equations without partial waves

A formalism is developed whereby the two-body Lippmann-Schwinger equation may be solved in momentum space without partial-wave decomposition. The integral equation derived is two-dimensional and so is amenable to direct numerical solution. Because the technique uses the well-known helicity formalism, the matrices involved can be further reduced by taking advantage of symmetries common in nuclear and atomic systems (parity conservation, particle symmetry). An example is shown for nucleon-nucleon scattering, and the results are compared to those obtained from the usual partial-wave method.

Solution of the Lippmann-Schwinger Equation with an Expansion of the Wave Function inside the Potential Range

We propose a simple but rigorous formalism to solve the Lippmann·Schwinger equation, expand· ing the wave function inside the potential range. The solution gives us a separable t-matrix. The formalism can be applied to any finite range two-body interactions and is useful to solve few-body equations. The solution of the two-body Lippmann-Schwinger equation is important not only in the two-body problem alone but also in the few-body problems. It is well known that a t-matrix solution in a separable form is very useful in the few-body equations. l ) Here we propose a new formalism to solve the two-body Lippmann-Schwinger equation. The assumption is that the two-body interaction is of finite range, i.e., vanishes beyond the range R. We expand the wave function I¢) inside the potential range. Since the range is finite, it is easy to find a set of complete base functions for the expansion. The expansion is useful in the integral equation because it keeps the product Vl¢) unchanged. Then we get a set of linear equations each for the scatter­ ing problem and for the bound state problem. It will be shown that the solution of the scattering equation leads to a separable t-matrix. Thus the present formalism is very useful in the few-body equations. When it is applied to the two-body bound state problem, it easily gives the bound state wave function in the whole range, although the original expansion is limited inside the potential range. The resulting separable t-matrix in the present formalism can also be derived from a separable potential expansion method developed by Adhikari and his collabo­ rators. 2 ) Their method has been used in a recent nod scattering calculation by the present author3) adopting modified Malftiet-Tjon potential as the two-body interac-. tion. 4 ) It has been demonstrated that the phase shift and the absorption parameters of the nod scattering are obtained very accurately with low rank separable potentials. The result has been compared with those obtained from other methods. 4 ) It was found that the agreement between them are satisfactorily good. However, it is sometimes asked why a separable potential expansion works so well. The present formalism gives a simple but solid theoretical base for the nod scattering calculation done in Ref. 3). First we discuss the two-body scattering problem. Hereafter, we use the partial wave projected one-dimensional equation, for simplicity. But, it is not essential. We could also use a three-dimensional one. The scattering wave function j¢) satisfies the Lippmann-Schwinger equation,

Kernel approximation for solving few-body integral equations

This paper investigates an approximate method for solving integral equations that arise in few-body problems. The method is to replace the kernel by a degenerate kernel defined on a finite dimensional subspace of piecewise Lagrange polynomials. Numerical accuracy of the method is tested by solving the two-body Lippmann-Schwinger equation with non-separable potentials, and the three-body Amado-Lovelace equation with separable two-body potentials.

Low-energy direct nuclear reactions

The resonating group method is a successful method for treating nuclear many-body problems in an approximate way. Although the method uses the basic nucleon-nucleon interactions and treats the antisymmetrization of identical particles in an exact way, correction terms cannot be introduced without employing an overcomplete set of basis states for the wave function and the associated theoretical and numerical problems. We present an alternative derivation of the resonating group method, which incorporates bound state approximation for the resolvent operator corresponding to the two-body fragmentation channel appearing in an antisymmetrized version of the N-body Lippmann-Schwinger equation for the wave functions. Similar considerations when applied to the transition matrix Lippmann-Schwinger equation yield the transition matrix version of the resonating group method. Such derivation of the resonating group method allows us to introduce correction terms in a controlled way. The antisymmetrized N-body Lippmann-Schwinger equation for the transition matrix connecting specific two-body channels is reduced to a two-body Lippmann-Schwinger equation with an optical potential and a multiple scattering series for the optical potential, which unlike the usual multiple scattering series is expected to converge at low energy. The lowest order approximation to the optical potential is the resonating group method interaction. We show howmore » to introduce corrections to the optical potential due to excited discrete and certain continuous spectrum of the relevant resolvent operator. The present scheme is also applicable for the calculation of nuclear bound states.« less

Adiabatic limit of the Faddeev equations

The adiabatic scattering of two heavy particles, interacting by the exchange of a third light particle, is considered from the Faddeev viewpoint. The Faddeev equations are reduced to an equivalent infinite set of coupled integral equations, which like the equations of the close-coupling scheme have two-body kinematics, and effective two-body interactions, but unlike the close-coupling equations, properly include the effects of breakup states. Complete sets of eigenstates of the kernels of the two-body Lippmann-Schwinger equations (Sturmian states) are used to effect the expansions. For the special case of separable interactions the equivalent set is shown to be finite. The effective interactions of the theory are examined in the adiabatic and static limits and compared with the potentials of the close-coupling scheme. In the limiting case the interactions are determined by a reduced form of the three-particle propagator and found to be local, leading to an alternative set of coupled differential equations for the three-body elastic amplitude. The effective potentials and their limiting forms are constructed for three specific examples, the Yamaguchi interactions, a nonrelativistic model of the nucleon-nucleon interaction, and a model of heavy-ion transfer scattering using $\ensuremath{\delta}$-shell interactions.NUCLEAR REACTIONS Three-body scattering in molecular-mass limit. Sturmian reduction of Faddeev equations. Adiabatic and static limits of effective two-body interactions for Yamaguchi, $\ensuremath{\delta}$-shell potentials, model nucleon-nucleon interaction.

Upper and Lower Bounds on Phase Shifts for Three-Body Systems:n−dScattering

A variational-bound formulation of the three-body problem based on the Faddeev equations has been derived previously. This method involves a variational calculation of the exact effective potential between the incident particle and the bound two-body target. Variational upper and lower bounds on this effective potential are obtained by constructing approximate separable three-body Green's operators. The eigenphase shifts determined by using the effective potential in the two-body Lippmann-Schwinger equation are upper and lower bounds on the true eigenphases for energies below the three-body breakup threshold. The method is applied to the neutron-deuteron system, and a set of variationally converged phase shifts is obtained.

Lower Bounds on Phase Shifts for Three-Body Systems:n−dQuartet Scattering

The Faddeev equations are used to provide a variational-bound formulation of the three-body scattering problem. The present method has the distinct advantage that the Feshbach projection operators, which enter into previous formulations and which are generally difficult to construct, do not appear. The method requires the calculation of a variational approximation to the exact effective potential for the scattering of a particle by a bound two-body system. A reaction matrix is determined by using this effective potential as input to a two-body Lippmann-Schwinger equation which is easily solved numerically. The eigenphase shifts thus obtained provide lower bounds on the true eigenphases for energies below the three-body breakup threshold. To test its practicability, the method is applied to the problem of neutron-deuteron scattering in the quartet state. The results are in agreement with previous calculations and with experiment.

Integral equation approach to electron-hydrogen collisions

A recent formulation of the three-body collision problem in the form of coupled multi-channel two-body Lippmann-Schwinger equations is applied to the electron-hydrogen collision problem. The relation between these equations and the close-coupling equations is studied by casting the latter into integral form. A simple high-energy approximation, which amounts to a unitary version of the Born-Oppenheimer approximation, is derived, and used in its simplest form to calculate elastic scattering. The results indicate that exchange effects and unitarity corrections are significant at least up to 400 ev.