Three-body Coulomb Problem Research Articles

New methods and results in the three-body Coulomb problem

New methods and results in the three-body Coulomb problem

Exploring the separability of the three-body Coulomb problem in hyperspherical elliptic coordinates

Exploring the separability of the three-body Coulomb problem in hyperspherical elliptic coordinates

Integral Equations for Three-Body Coulomb Resonances

We propose a novel method for calculating resonances in three-body Coulombic systems. The method is based on the solution of the set of Faddeev and Lippmann-Schwinger integral equations, which are designed for solving the three-body Coulomb problem. The resonances of the three-body system are defined as the complex-energy solutions of the homogeneous Faddeev integral equations. We show how the kernels of the integral equations should be continued analytically in order that we get resonances. As a numerical illustration a toy model for the three-$\alpha$ system is solved.

Coulomb three-body bound-state problem: Variational calculations of nonrelativistic energies

It is known that variational methods are the most powerful tool for studying the Coulomb three-body bound-state problem. However, they often suffer from loss of stability when the number of basis functions increases. This problem can be cured by applying the multiprecision package designed by D. H. Bailey. We consider variational basis functions of the type $\mathrm{exp}(\ensuremath{-}{\ensuremath{\alpha}}_{n}{r}_{1}\ensuremath{-}{\ensuremath{\beta}}_{n}{r}_{2}\ensuremath{-}{\ensuremath{\gamma}}_{n}{r}_{12})$ with complex exponents. The method yields the best available energies for the ground states of the helium atom and the positive hydrogen molecular ion as well as many other known atomic and molecular systems.

Two-dimensional approximate wave function for the three-body Coulomb problem

In this work we present an approximate solution of the Schr\"odinger equation for the two-dimensional three-body Coulomb problem. We study the C3 model that treats the particle's motion as if it were uncorrelated. We also show the bidimensional ${\ensuremath{\Phi}}_{2}$ approximation, based on a recently developed model, which includes some terms of the nonorthogonal Hamiltonian associated with the three-particle correlation. We point out the main similarities and differences with their three-dimensional analogs and verify the correct asymptotic behavior of the wave functions.

Complete numerical solution of electron-hydrogen model collision problem above the ionization threshold

Benchmark results are presented for electrons colliding with hydrogen atoms in the S-wave (Temkin-Poet) model collision problem, which neglects angular momentum. Complete results (elastic, inelastic, and ionization), accurate to 1%, are obtained by numerically integrating Schrodinger's equation subject to correct asymptotic boundary conditions. This marks the first time direct matching to asymptotic boundary conditions has been shown to yield convergent ionization amplitudes for a Coulomb three-body problem. Results are presented for impact energies of 54.4 and 40.8 eV, where comparison with other theories is available.

Interference effects in the decay of resonance states in three-body Coulomb systems

The lowest ${}^{1}{S}^{e}$ resonance state in a family of symmetric three-body Coulomb systems is systematically studied as a function of the mass-ratio M for the constituting particles. The Siegert pseudostate method for calculating resonances is described and accurate results obtained by this method for the resonance position $\mathcal{E}(M)$ and width $\ensuremath{\Gamma}(M)$ in the interval $0<~M<~30$ are reported. The principal finding of these calculations is that the function $\ensuremath{\Gamma}(M)$ oscillates, almost vanishing for certain values of M, which indicates the existence of an interference mechanism in the resonance decay dynamics. To clarify this mechanism, a simplified model obtained from the three-body Coulomb problem in the limit $\stackrel{\ensuremath{\rightarrow}}{M}\ensuremath{\infty}$ is analyzed. This analysis extends the range of M up to $M=300$ and confirms that $\ensuremath{\Gamma}(M)$ continues to oscillate with an increasing period and decreasing envelope as M grows. Simultaneously it points to semiclassical theory as an appropriate framework for explaining the oscillations. On the basis of Demkov's construction, the oscillations are interpreted as a result of interference between two paths of the resonance decay on the Riemann surface of adiabatic potential energy, i.e., as a manifestation of the Stueckelberg phase. It is shown that the implications of this interpretation for the period and envelope of the oscillations of $\ensuremath{\Gamma}(M)$ agree excellently with the calculated results.

Multivariable hypergeometric functions for ion–atom collisions

In this work we present a correlated wave function for a three-body continuum Coulomb problem. This state is described by the two-variables Φ2 hypergeometric function. We examine the properties of this function and their differences with previous uncorrelated models. The Φ2 wave function can be considered as a final state of ion–atom ionizing collisions, giving rise to both undistorted (Born-Φ2) and distorted (EIS-Φ2) models. We obtain double differential cross sections with the Born-Φ2 theory for proton–helium collisions in the intermediate to high energy regime. They exhibit all the main features of the electronic emission process and agree with the experimental data.

Double differential cross section of helium ionization by proton impact

A theoretical analysis for doubly differential cross sections of the single ionization of helium by proton impact as a function of the scattering angle and projectile energy loss is presented. We have used two different theoretical methods. The first approach involved an approximate solution of Faddeev-Merkuriev equations for the three-body Coulomb problem. The second approach employed the expansion of the transition amplitude in the Born series over the projectile-target interaction up to second order. A relation between these two methods was established for the case of fast collisions. It has been demonstrated that, for small scattering angles, the post-collision interaction in the final state between the charged particles strongly influences the differential cross sections, whereas double-scattering mechanisms dominate for scattering angles mrad. The results of our calculations are in a good agreement with the measurements of Schulz et al for 50-150 keV collisions.

Separable wave equation for three Coulomb interacting particles

We consider a separable approximation to the Schr\"odinger equation for the three-body Coulomb problem and found its exact solution above the ionization threshold. This wave function accounts for different possible asymptotic behaviors and reduces to the well-known product of three two-body Coulomb waves (C3) for scattering conditions. The momenta and position-dependent modifications recently proposed for the Sommerfeld parameters, as an improvement to the C3 model, are analyzed. We show how these changes can be included in our model as a suitable physically based variations in the separable approximation for the wave equation.

A potential harmonic method for the three-body coulomb problem

A potential harmonic method that is suitable for the three-body coulomb systems is presented. This method is applied to solve the three-body Schroedinger equations for He and e+e−e+ directly, and the calculations yield very good results for the energy. For example, we obtain a ground-state energy of −0.26181 hartrees for e+e−e+, and −2.90300 hartrees for He with finite nuclear mass, in good agreement with the exact values of −0.26200 hartrees and −2.90330 hartrees. Compared with the full-set calculations, the errors in the total energy for ground and excited states of e+e−e+ are very small, around −0.0001 hartrees. We conclude that the present method is one of the best PH methods for the three-body coulomb problem.

Relationship between the adiabatic hyperspherical and the Born-Oppenheimer approach to the Coulomb three-body problem

Relationship between the adiabatic hyperspherical and the Born-Oppenheimer approach to the Coulomb three-body problem

Correlated continuum wave functions for three particles with Coulomb interactions

We present an approximate solution of the Schr\"odinger equation for the three-body Coulomb problem. We write the Hamiltonian in parabolic curvilinear coordinates and study the possible separation of the wave equation as a system of coupled partial differential equations. When two of the particles are heavier than the others, we write an approximate wave equation that incorporates some terms of the Hamiltonian that before had been considered as a perturbation. Its solution can be expressed in terms of a confluent hypergeometric function of two variables. We show that the proposed wave function includes a correlation between the motion of the light particle relative to the heavy particles and verifies the correct asymptotic behavior when all particles are far from each other. Finally, we discuss the possible uses of this function in the calculation of transition matrices and differential cross sections in ionizing collisions.

Rigid-Rotator and Fixed-Shape Solutions to the Coulomb Three-Body Problem

Solutions to the general classical Coulomb three-body problem in the form of rigid-rotator and fixed-shape configurations are studied. In the collinear case, some necessary and/or sufficient conditions for the existence of the so-called charge-symmetrical, (−)(+)(−), and charge-asymmetrical, (−)(−)(+), configurations are stated. These conditions involve relations between the geometrical and dynamical parameters of the system under study. The impossibility of the existence of a planar Coulombic rigid rotator is demonstrated. In the two-dimensional case, fixed-shape solutions are studied analytically, and it is shown that, in the three-dimensional case, only fixed-shape solutions involving a triple collision and a static case are possible. Finally, some numerical experimentation, mostly based upon theoretical predictions of the work, is performed, and new bound (although unstable) rotating-oscillating orbits for systems such as the positronium negative ion and helium are found.

Three-body Coulomb problem in the dipole approximation: and states

In the asymmetric doubly excited states of the two-electron atom one electron is excited much more than the other. The characteristic separation of the strongly excited electron from the atomic nucleus significantly exceeds that for the weakly excited electron . The electron - electron interaction is approximated by the first non-trivial (dipole) term of its expansion over . The same approach is easily extended to the general Coulomb three-body problem with the particles of arbitrary mass and charge. The dynamics of three-body states with and symmetry is analysed in quantum, semiclassical and classical approximations in comparison with the S and states considered previously. Mathematically, the problem is reduced to two coupled three-term recursion relations which were not studied previously. The exact and approximate (semiclassical) degeneracies of the levels with different values of the total orbital momentum L are revealed, and the energies of the `planetary atom' states are estimated.

Muon-catalyzed fusion and fundamental physics

The essence and main features of muon-catalyzed fusion are exposed. Some important and interesting applications of muon-catalyzed fusion in fundamental physics are presented: nuclear physics, quantum electrodynamics, weak interaction physics, the Coulomb three-body problem and the physics of exotic atoms.

The three-body continuum Coulomb problem and the 3α structure of 12C

We introduce an approach, based on the coordinate space Faddeev equations, to solve the quantum mechanical three-body Coulomb problem in the continuum. We apply the approach to compute measured properties of the first two 0 + levels in 12C, including the width of the excited level, in a 3α-particle model. We use the two-body interaction which reproduces the low energy α-α scattering data and add a three-body force to account for the compound state admixture.

Reduced adiabatic hyperspherical basis in the Coulomb three-body bound state problem

A new version of the adiabatic hyperspherical approach (AHSA) is suggested which has significant advantages for the calculation of three-body states with total angular momentumJ> 0. The binding energies of all bound states of mesic molecules with normal parity are calculated by the suggested method. Comparison with results of variational calculations and the fast convergence of the method confirm its high efficiency.

Investigation of muonic hydrogen isotopes scattering from H2 molecule

Knowledge of the cross sections for scattering of µp, µd and µt on molecules of hydrogen isotopes is necessary not only for checking the algorithmic solution of the Coulomb three-body problem but also for a general and correct description of the kinetics of muonic and molecular processes in mixtures of hydrogen isotopes. We plan to measure the scattering cross-section energy dependence of the reactions µx+ H2 →>µx+H2 (x=d, t) in the energy collision range from 0.1 to 45 eV, using a multilayered target system recently developed at TRIUMF.

The three-body continuum Coulomb problem and the 3α structure of 12C

We introduce an approach, based on the coordinate space Faddeev equations, to solve the quantum mechanical three-body Coulomb problem in the continuum. We apply the approach to compute measured properties of the first two $0^+$ levels in $^{12}$C, including the width of the excited level, in a 3$\alpha$-particle model. We use the two-body interaction which reproduces the low energy $\alpha$-$\alpha$ scattering data and add a three-body force to account for the compound state admixture.