Faddeev Formalism Research Articles

Collision Dynamics of Three Interacting Atoms: The Faddeev Equations

An approach to atom-diatom collisions is developed in terms of sums of spin-dependent atomic pair interactions. Elastic, inelastic, atom-exchange, and dissociative collisions are simultaneously considered within the Faddeev formalism, neglecting electronic excitation or ionization. These events may be described as successions of pair-encounters where atomic spins are coupled in singlet or triplet states. At high energies a justification is given for the ``spectator stripping'' model for the reaction A+BC → AB+C and the corresponding probability amplitude is obtained. It is also found that the first correction to this model involves a ``displacement'' mechanism and an effective interaction between A and C.

The D(p,2p)n reaction at 51.8 MeV

The results of the deuteron breakup by protons, measured in coplanar symmetric geometry at forward angles, were compared with calculations based on the three-body Faddeev formalism and other theories.

E-H Scattering by the Faddeev Approach

$1s\ensuremath{-}1s$ elastic and $1s\ensuremath{-}2s$ inelastic total cross sections for $e$-H collision have been calculated with and without exchange effects using the Faddeev formalism as developed by Sloan and Moore. We have considered the effect of coupling to $1s$ and $2s$ intermediate states. Our theoretical results have been compared with the existing experimental findings and other theoretical calculations. In the intermediate-energy region, our results for the inelastic process are found to be in close agreement with the experimental observations. Our results for both systems indicate that coupling to states other than the initial and final ones causes a reduction in the cross sections, and even up to 500 eV the effect of exchange is not negligible.

Binding energy of triton and scattering lengths in neutron-deuteron collision in Faddeev formalism with local potential

A separable representation for the off-shell two-body t-matrix for a local Hulthén potential is presented, in which deuteron states are chosen as the expansion bases. Using the Faddeev equations with these t-matrices as input, the ground state energy of the triton and doublet and quartet scattering lengths in neutron-deuteron scattering, have been computed. The results have been compared with the experimental findings and the theoretical results of Sitenko et al. obtained in the Sturmian function representation with the same Hulthén potential.

Calculation of trinucleon binding energies and charge form factors using the boundary condition model

Trinucleon binding energies and charge form factors are calculated using the Faddeev formalism and a two nucleon s-wave interaction given by the boundary condition model. The calculated binding energy is compared with that obtained using Brayshaw's modification of the Faddeev equations.

Ground-State Energies of the He Atom and theLi+Ion in the Faddeev Approach

The ground-state energies of the He atom and the ${\mathrm{Li}}^{+}$ ion have been calculated using the Faddeev formalism. For the off-shell two-body collision amplitude $t$ matrix), Sturmian-function (SF) as well as Coulomb-function representations have been employed. For the He atom, calculations have been performed retaining $1s$, $1s$ and $2s$, and $1s$, $2s$, and $3s$ states in the SF representation for the two-body $t$ matrix, whereas only the $1s$ term has been retained in the mixed-mode (MM) representation. For the ${\mathrm{Li}}^{+}$ ion, computations have been done after retaining terms up to $2s$ in the SF representation and only the $1s$ term in the MM representation. The results obtained by retaining only the $1s$ term are in marked disagreement with the experimental values in both representations. It is noticed that the results for the MM representation are almost as much below the experimental values as those for the SF representation are above. The results for the MM representation are slightly better than those for the SF representation. This is seen more clearly in the case of the ${\mathrm{Li}}^{+}$ ion. The results in the SF representation are found to be in good agreement with the experimental findings after inclusion of the $2s$ term.

ΛdScattering as a Probe of theΛNCore Potential

Nonlocal separable (NLS), $S$-wave, spin-dependent, central potentials and a Faddeev formalism are used to solve the $\ensuremath{\Lambda}d$ scattering problem. The $\mathrm{NN}$ interaction is represented by an attractive NLS potential. Each of the $\ensuremath{\Lambda}N$ interactions is represented by the sum of an attractive and a repulsive (core) NLS potential. Three different forms are used for this core potential, including no core at all. For a given $\ensuremath{\Lambda}N$ spin each $\ensuremath{\Lambda}N$ potential is adjusted to give the same low-energy scattering parameters, and the potentials with a core are adjusted to give the same phase shift over a wide energy range, as that given by a phenomenological local potential with a hard core. The $\ensuremath{\Lambda}N$ wave functions and off-shell transition matrices are compared to see how each reflects the presence of the $\ensuremath{\Lambda}N$ core potential. For incident $\ensuremath{\Lambda}$ laboratory momentum in the range 100-300 $\frac{\mathrm{MeV}}{c}$ the $\ensuremath{\Lambda}d$ cross sections are reduced by (13-25)%, and they do depend on the off-shell behavior of the $\ensuremath{\Lambda}N$ amplitude.

Singular Core Interactions and Three-Body Theory

It is shown that the Faddeev formalism must and can be modified when the short-range two-body behavior is specified by a hard core or boundary-condition model. A new integral equation is formulated for this purpose which, in the special case of boundary-condition model or hard core alone, reduces to an equation in a single vector variable. This feature provides an apparently unique example of a nonseparable interaction for which the three-body equations can be solved exactly.

Binding energy of helium atom by Faddeev formalism

The energy for the three-particle ground state of the (e-, He+) system, namely that of the helium atom has been calculated using Faddeev theory and following the method of Ball et al. (1968). Retaining only 1s and 2s terms in the Sturmian set of expansion for two-body collision amplitude, the result is found to be in good agreement with experimental measurement.

Analysis ofD(p,n)2pReaction at 45.5 MeV by the Faddeev Formalism with Coulomb Force

Analysis of<i>D</i>(<i>p</i>,<i>n</i>)2<i>p</i>Reaction at 45.5 MeV by the Faddeev Formalism with Coulomb Force

Dynamical three-body calculations with complex potentials

Abstract A three-body model of the deuteron-nucleus collision with absorptive s-wave interactions is investigated by the Faddeev formalism. The absorption is found to produce important effects on the reaction cross sections.

Three-Body Problem with Local Hard-Core Potential

The Faddeev formalism is used to compute the ground-state energy of the triton for a simple model in which the nucleon-nucleon force is given by a spin-independent $s$-wave Herzfeld potential (hard core with outside square well). The effective nucleon-nucleon potential used corresponds to the usual weighted average of triplet and singlet potential strengths. The two-variable Faddeev equation is solved using the complete off-shell $t$ matrix given by: (1) an exact analytic expression, and (2) an integral-equation formalism recently proposed. Three-body calculations based on both evaluations of the two-body $t$ matrix lead to a triton binding energy of 8.13 MeV with about 0.5% accuracy.

Analysis ofD(p,n)2pReaction at 45.5 MeV with the Faddeev Formalism

Analysis of<i>D</i>(<i>p</i>,<i>n</i>)2<i>p</i>Reaction at 45.5 MeV with the Faddeev Formalism

Coulomb Scattering in the Three-Body Problem

Using the Faddeev formalism for the three-body problem with two charged particles, with separable nuclear interactions and an approximate form for the Coulomb wave function and Coulomb Green's function, a simplified model is set up for $p\ensuremath{-}d$ scattering. The differential scattering cross section is calculated numerically for different incident proton energies from 2.08 to 14.0 MeV. The calculations are repeated with the Coulomb potential turned off, and the results are compared with the Coulomb scattering.

A Generalized Lovelace Equation in the Relativistic Faddeev Formalism. II

By using the relativistic Faddeev formalism for three-particle scattering based on the techniques of Blankenbecler and Sugar to multiladder diagram, a generalized relativistic Lovelace equation is proposed for composite-particle scattering in order to treat bound states and unstable particles in a unified way. In order to obtain our equation from the rela­ tivistic Faddeev formalism, we introduce relativistic off-shell form factors and propagators corresponding to composite particles by applying the Fredholm theory to the Blankenbecler­ Sugar kernel of the two-body scattering equation. These functions, the off-shell form factors and propagators, are defined by an analytic extension from the energy of bound state or resonance state and the two-body scattering matrix can be separated into a part including a composite-particle pole and a remaining part. By use of these functions, we can derive a generalized Lovelace equation for composite particle scattering as an an~lytically extended equation of the so-called Lovelace equation given by Freedman, Lovelace and Namyslowski. From the analytically extended properties of these functions, it is emphasized that our equation is more useful in applying to unstable particle scattering and also to production process en­ hanced by final interaction in the relativistic three-body scattering problems.

Variational Approach to the Faddeev Equations

A variational principle is formulated as a means for obtaining approximate solutions to the Faddeev integral equations for the nonrelativistic three-body scattering problem. Bound states are described by the homogeneous version of the Faddeev equations. In this case a minimum principle for the binding energies is obtained in close analogy with the familiar Rayleigh-Ritz method. An effective potential is defined as the solution of a set of modified Faddeev equations. The variational principle therefore applies to the effective potential. This is shown to be a minimum principle for the eigenphase shifts for energies below the three-particle breakup threshold. Some formal applications of the variational principle are described. These include derivations, within the Faddeev formalism, of a normalization condition for bound-state wave functions, a distorted-wave theory, and a perturbation theory for discrete states.

P-Wave Interactions inτDecay

We have calculated the effect of final-state $s$- and $p$-wave interactions on the spectra in $K\ensuremath{\rightarrow}3\ensuremath{\pi}$ decay using the Faddeev formalism derived in an earlier paper. In our model, $p$ waves are not produced directly by the weak interaction but only by the recoupling of pion pairs in the multiple scattering series. For large $s$-wave scattering lengths ($\ensuremath{\sim}1.0{{m}_{\ensuremath{\pi}}}^{\ensuremath{-}1}$), the $p$ state has very little influence on the spectrum, and a fit to the experimental spectra requires ${{a}_{2}}^{2}&gt;{{a}_{0}}^{2}$. However, for small $s$-wave scattering lengths ($\ensuremath{\sim}0.2{{m}_{\ensuremath{\pi}}}^{\ensuremath{-}1}$) the $p$ wave can become dominant, and fits to the data can be achieved with ${{a}_{2}}^{2}&lt;{{a}_{0}}^{2}$. These $p$-wave-dominant solutions are cutoff-sensitive, so that there is no unique fit. The final-state interaction phase is also calculated.

Three-Body Correlations in Reaction-Matrix Calculations

An improved method of treating three-body cluster correlations in reaction-matrix calculations is presented and applied to nuclear matter. A three-particle reaction matrix is defined which allows three particles to interact with each other in all possible ways after specifying the particles involved in the initial and terminal interactions. The method is similar to that proposed by Bethe but, since all off-energy-shell terms can be treated separately, full advantage can be taken of the Faddeev formalism as applied to the three-nucleon system. The Pauli exclusion operator is included in calculating on-energy-shell reaction-matrix elements but is neglected in the off-energy-shell terms. Numerical calculations are presented for two spin-independent, $s$-state, separable potentials, one of which contains a hard-shell repulsive term. Higher-order terms are found to be important for the hard-shell potential but not for the simple attractive potential. In both cases the total three-body correlation energy is found to be small.

An Extension of the Faddeev Formalism to Composite-Particle Scattering

An Extension of the Faddeev Formalism to Composite-Particle Scattering

An Extension of the Faddeev Formalism to Composite-Particle Scattering

An Extension of the Faddeev Formalism to Composite-Particle Scattering