Modified Faddeev Equations Research Articles

The Faddeev-Merkuriev Differential Equations (MFE) and Multichannel 3-Body Scattering Systems

Numerical implementation of the modified Faddeev Equation (MFE) is presented in some detail. The Faddeev channel wave function displays unique properties of each and every open channel, respectively. In particular, near resonant energies, the structures of the resonances are beautifully displayed, from which, the life-time of the resonances can be determined by simply using the uncertainty principle. The phase shift matrix, or the K-matrix, provides unique information for each and every resonance. This information enables the identification of the physical formation mechanism of the Gailitis resonances. A few of these resonances, previously known as the mysterious shape resonances, have occurred in a number of different collision systems. The Gailitis resonances are actually produced by a quantized Stark-effect within the various collision systems. Since the Stark-effect is a universal phenomenon, the Gailitis resonances are expected to occur in much broader classes of collision systems. We will present the results of a precision calculation using the MFE method in sufficient detail for interested students who wish to explore the mysteries of nature with a powerful theoretical tool.

A Possible Role of the Gailitis Resonance in Muon Catalyzed Fusion

In a previous JMP article published May 2013, a comprehensive calculation was presented for all properties of a number of long-life s-state Gailitis resonances lying just above the PS(n = 2) formation threshold in a positron-Hydrogen scattering system. The six open-channel calculation was carried out by solving a set of four hundred thousand coupled linear equations. The modified Faddeev equation was used to obtain the wave-amplitude for each of the six open channels. Details can be found in reference [1]. This note presents some qualitative properties of Gailitis resonances in the scattering systems of d + tu just above the thresholds of the opening of a new channel of the muonic atoms tu(n) or du(n), n > 2 is the principal quantum number. u is a negatively charged muon, d and t are the nuclei of the two isotopes of the Hydrogen atom with one and two neutrons in the nucleus respectively. We study the possible decay channels of some of the long-life Gailitis resonances. Of particular interest is a transition directly from a Gailitis (3-body) resonance to the bound states dtu molecular ions via a radiative emission of a photon or an external auger ejection of a nearby electron. Possible experimental evidence will be presented.

Long-Range Correlation in Positron-Hydrogen Scattering System near the Threshold of <i>Ps</i>(<i>n</i> = 2) Formation

This is a short report of a recently uncovered resonant phenomenon. The modified Faddeev equation that correctly includes all six open channels is used. The calculation is carried out in s-partial wave. We report a number of resonant peaks in the elastic cross sections as well as the wave amplitudes involved. This is the energy region where the Stark-effect induced electric dipole energy split in the target dominates the physics and the Long-Range behavior of the 3-body scattering system. It is found that when the center of mass collision energy in the new channels is in integer proportion to the corresponding electric dipole energy split, Bremsstrahlung photon mediated resonant scattering occurs. The corresponding wave amplitudes deform into wave-packets hundreds to thousands of Bohr radii in width. The physical implication of this phenomenon will be discussed.

Three-body breakup within the fully discretized Faddeev equations

A novel approach is developed to find the three-body breakup amplitudes and cross sections within the modified Faddeev equation framework. The method is based on the lattice-like discretization of the three-body continuum with a three-body stationary wave-packet basis in momentum space. The approach makes it possible to simplify drastically all the three- and few-body breakup calculations due to discrete wave-packet representations for the few-body continuum and simultaneous lattice representation for all the scattering operators entering the integral equation kernels. As a result, the few-body breakup can be treated as a particular case of multi-channel scattering in which part of the channels represents the true few-body continuum states. As an illustration for the novel approach, an accurate calculations for the three-body breakup process $n+d\to n+n+p$ with non-local and local $NN$ interactions are calculated. The results obtained reproduce nicely the benchmark calculation results using the traditional Faddeev scheme which requires much more tedious and time-consuming calculations.

Positron annihilation above the positronium formation threshold in e +–H scattering

A multichannel theory for the positron annihilation above the positronium formation threshold based on the modified Faddeev equation is presented. Its relation with the one channel effective charge formula is discussed. Application to the positron–hydrogen scattering has been carried out. A number of positron–electron direct annihilation cross sections in the Ore gap between Ps(1s) and H ( n = 2) thresholds are reported. A sharp increase in the calculated annihilation cross section near a Feshbach resonance and followed by an even more drastic decrease of the annihilation cross section across the resonant energy is found for all first S, P and D wave Feshbach resonances.

Low-energy anti-hydrogen formation differential cross sections from Ps(n = 2) viathe modified Faddeev equations

Assuming CPT symmetry, we carried out calculations for the system p̄+Ps(n = 2)in the eight open channels energy gap between the Ps(n = 2) and H̄(n = 3)energy thresholds where we found anti-hydrogen formation cross sections as high as 1670πa02.We also report well converged anti-hydrogen formation differential cross sections from Ps(n = 2)targets using less than 10 partial waves. Very near the Ps(n = 2)threshold the differential cross sections are dominated by the first threepartial waves where most of the interference minimal structures seen inother energy regions are absent. Of particular interest are the large crosssection and the absence of interference minima in the differential crosssections when the target is in the metastable 2 3S1 state. The possibility ofutilizing these processes for low-energy anti-hydrogen synthesis is discussed.

Induced long-range dipole-field-enhanced antihydrogen formation in the p + Ps(n = 2) --> e(-) + H(n < or = 2) reaction.

We report high-precision calculations that correctly include the rearrangement channels by solving the modified Faddeev equations for energies between the Ps(n = 2) and H(n = 3) thresholds, which involve six and eight open channels. We find that 99% of the antihydrogen is formed in H(n = 2). Just above the Ps(n = 2) threshold the S, P, and D partial waves contribute nearly 1400pi(a(2)0) near the maximum. We find evidence that the induced long-range dipole potential is responsible for such a large H formation cross section. The possibility of utilizing this resonance to synthesize low-energy H is discussed.

Low-energy antihydrogen formation cross sectionsand differential cross sections via the modified Faddeev equations

This paper includes cross sections and differential cross sectionsfor the processes + ps(1s)→e + (n⩽2), assuming CPT symmetry and using tenpartial waves with totalangular momentum L ⩽ 9. Comparisons with previous calculationsare made wheneverfeasible and good agreement is observed. It is noted that the forward peaks ofthe differential crosssections of the formation of (2p) atoms are muchbroader than those of the formation of S-state atoms. A strong secondary maximum of one of the (2p) formation processes iscentred around 45°, in contrast to S-state processes atcomparable energies where thedifferential cross sections are strongly peaked in the forward directionwith θ < 40°.The combined differential cross sections of the two processes that leadto the formation of (2p)are also given for a number of energies.

Multichannele++Hcalculations via the modified Faddeev equations

The modified Faddeev equations are used to calculate the complete multichannel K matrices within the energy gap between the $\mathrm{Ps}(n=1)$ and $\mathrm{Ps}(n=2)$ thresholds. We report here a number of partial cross sections in this energy range including hydrogen production cross sections in the Ps entrance channel. The angular momentum states calculated are S, P, D, and F states. The number of open channels ranges from two in the Ore gap to four and five above the $\mathrm{H}(n=2)$ threshold. Comparison with previous calculations is made whenever feasible. Agreement within a few percentages is observed for most cases.

The modified Faddeev equation and multichannel positron-hydrogen scattering calculation

A bipolar spherical harmonics expansion of the modified Faddeev equation is used to calculate the K-matrices of the e++H scattering for up to five open channels and for total angular momentum L4. Partial cross sections and differential cross sections including hydrogen production channels are reported. Comparison with previous calculations are made whenever feasible. Good agreement is observed.

Applications of the modified Faddeev equations: t + dμ -

We present calculations of the cross-sections for elastic as well as for muon transfer in t + dμ- collisions below the tμ-(n = 2) threshold. The appropriate mass-scaled Jacobi coordinates are used in the direct and rearrangement channels. The K-matrix elements are obtained from the solution of multichannel Faddeev equations in the configuration space formulation. A bipolar expansion and spline representation are used in the numerical solution of the modified Faddeev equations.

Scattering of electrons and positrons on hydrogen using the Faddeev equations

Scattering of electrons and positrons on a hydrogen atom below the H(n=2) threshold is studied by means of directly solving the modified Faddeev equations in the total angular-momentum representation. Elastic phaseshifts of the e-+H scattering and those of the e++H scattering below the Ps(n=1) threshold are obtained. K-matrix elements for the e++H and p+Ps scattering between the Ps(n=1) and H(n=2) thresholds are calculated. The results are in a good agreement with previous variational calculations.

Zero-energy scattering in symmetric Coulomb systems via Faddeev approach.

A method based on the modified Faddeev equations in the total-angular-momentum representation is developed to treat the zero-energy scattering in symmetric three-body Coulomb systems. The method is applied to calculate the s-wave scattering lengths of the systems ${\mathit{e}}^{\mathrm{\ensuremath{-}}}$+(${\mathit{e}}^{\mathrm{\ensuremath{-}}}$${\mathit{e}}^{+}$), p+(p\ensuremath{\mu}), and d+(d\ensuremath{\mu}). A large negative value of the even-parity p+(p\ensuremath{\mu}) scattering length indicates the presence of a virtual state near threshold.

Faddeev calculation of e--Ps scattering lengths.

The modified Faddeev equations in configuration space are applied to solve numerically the problem of the zero-energy elastic scattering of an electron by positronium at zero angular momentum. The singlet and triplet scattering lengths are calculated and zero-energy limits of elastic and ortho-para conversion cross sections are established. The results confirm and improve previous variational estimations of these parameters.

Faddeev equations with three-nucleon force in momentum space.

Modified Faddeev equations that allow the inclusion of irreducible three-body forces in addition to two-body interactions are formulated and the technical apparatus for their solution in momentum space is described. Results for the triton binding energy are obtained with realistic two-nucleon interactions and the Tucson-Melbourne two-pion exchange three-nucleon force and compared with previous calculations. Excellent agreement with the results of other groups is found confirming that the accuracy of present-day techniques for handling three-nucleon forces is very high indeed.

Three-body deuteron-nucleus scattering with extra resonance channels

By methods of dimensional reduction of 6-body problem in triangle representation, the 3-body differential dynamical equations with energy-dependent interactions describing elastic scattering and break-up in (np)4He system are constructed. The energy dependence of interactions is generated by internal structure of nucleus4He. The simple models for these energy-dependent interactions are constructed by the extended Hilbert space approach. In the frames of these models the contributions of 5-body resonances5Heg.s.,5Lig.s. and 6-body resonance6Li* (1+) into elastic scattering and deuteron break-up amplitudes for αd-reaction are investigated on the basis of modified Faddeev equations.

Extended Hilbert space approach to few-body problems

A general formulation of the quantum scattering theory for a system of few particles, which have an internal structure, is given. Due to freezing out the internal degrees of freedom in the external channels, a certain class of energy-dependent potentials is generated. By means of potential theory, a modified Faddeev equation is derived both in external and internal channels. The Fredholmity of these equations is proven and this is what provides a sound basis for solving the addressed scattering problem.

Momentum space Faddeev calculation of 3He binding energy with N-N P and D waves.

The homogeneous version of Coulomb modified Faddeev equations was used to determine the binding energy of $^{3}\mathrm{He}$. In the $^{1}\mathrm{S}_{0}$ and $^{3}\mathrm{S}_{1}$${\mathrm{\ensuremath{-}}}^{3}$${\mathrm{D}}_{1}$ subsystem states, a separable representation of the Paris potential was taken for the N-N interaction; whereas less sophisticated separable potentials were employed in the P and D partial waves with j\ensuremath{\le}2. The influence of these latter higher N-N partial waves on the $^{3}\mathrm{He}$ binding energy was calculated for the first time within the nonperturbative Faddeev equation approach. The trinucleon binding energy differences were found to be compatible with Coulomb energies obtained in the framework of perturbation theory with local N-N potentials.

Coulomb forces in the three-body problem with application to direct nuclear reactions

Faddeev equations are considered in the case of three charged particles interacting with both separable nuclear two-body interactions and also including Coulomb forces. Modified Faddeev equations with Coulomb Green's functions are introduced. The three-body amplitudes are given into pure Coulomb and distorted-Coulomb amplitudes. Introducing a decomposition in the angular momentum states, a set of three-body integral equations is obtained. The effect of pure coulomb amplitudes is studied in direct nuclear reactions and found to give a large contribution to the cross sections. The three-body integral equations obtained are applied for direct nuclear reactions. The angular distributions for12C(6Li,d)16O,16O(6Li,d)20Ne, and12C(6Li,α)14N transfer reactions are calculated as well as for the6Li elastic scattering on12C. From the good agreement between the theoretically calculated and experimental data, better spectroscopic factors are extracted. The effect of including Coulomb forces in the three-body problem is found to improve the results by about 16.26%.

Lee model field theory of three-nucleon system

The $\mathrm{VNN}$ sector of the nonstatic Lee model is investigated as a prototype of the three-nucleon $\mathrm{pnn}$ system, leading to an effective three-body Hamiltonian with a new kind of disconnected three-body force. The solution of the disconnectedness problem is presented in terms of modified Faddeev equations. The state vectors of the deuteron and triton are constructed and scattering amplitudes for reactions $n+p\ensuremath{\rightarrow}n+p$, $n+p\ensuremath{\rightarrow}n+n+{\ensuremath{\pi}}^{+}$, and $n+d\ensuremath{\rightarrow}n+d$ are derived. All effects of off-shell renormalization are included in a complete and consistent way.NUCLEAR REACTIONS Lee model, $\mathrm{VNN}$ sector, model of deuteron and triton, $N+V\ensuremath{\rightarrow}N+V$, $N+V\ensuremath{\rightarrow}N+N+\ensuremath{\Theta}$, $N+d\ensuremath{\rightarrow}N+d$ reactions, renormalization, off-shell effects, effective interactions, Faddeev equations.