Three-body Coulomb Problem Research Articles

Quantum control of quantum triple collisions in a maximally symmetric three-body Coulomb problem

In Coulomb three-body problems, configurations of close proximity of the particles are classically unstable. In confined systems they might however exist as excited quantum states. By studying a maximally symmetric subspace of the three-body problem one obtains strong evidence for the existence of excited states for which the wavefunction is nonzero for triple collision configurations. Quantum control of such states by time-changing electromagnetic fields is discussed, with particular emphasis on the nature of the required controls.

Adiabatic representation in the Coulomb three-body problem in the united-atom limit: Nuclear widths of the energy levels of the muonic molecule $tt\mu$

Исследуется асимптотика волновой функции системы трех кулоновских частиц в пределе объединенного атома в адиабатическом представлении задачи трех тел. Этот результат используется при расчете ядерных ширин уровней энергии мезомолекул. Обсуждаются особенности подхода применительно к возбужденным состояниям мезомолекулы $tt\mu$ с ненулевым орбитальным моментом.

Adiabatic representation in the Coulomb three-body problem in the united-atom limit: Nuclear widths of the energy levels of the muonic molecule ttµ

We study the asymptotic behavior of the wave function of the system of three Coulomb particles in the united-atom limit in the adiabatic representation of the three-body problem. This result is used to calculate the nuclear widths of muonic-molecule energy levels. We discuss features of the approach with regard to excited states of the muonic molecule ttµ with a nonzero orbital angular momentum.

Dynamical adiabatic theory of atomic collisions: The structure of hidden, avoided, andL3crossings

Inelastic transitions in slow ion-atom collisions are described using an adiabatic representation of the collision system, where the transition probabilities are determined by the branch points of the potential energy curves in the complex plane of internuclear separation $R$. As an example, the ${\mathrm{HeH}}^{2+}$ system is treated, in which the evolution of the branch points related to hidden, avoided, and ${L}_{3}$ crossings in the dynamical adiabatic basis of three-body Coulomb problem as functions of parameter $\ensuremath{\omega}=\ensuremath{\rho}v$ ($\ensuremath{\rho}$ is the impact parameter and $v$ the impact velocity) is studied. Rearrangement of hidden couplings due to the passage of ${L}_{3}$ branch points with the increase of $\ensuremath{\omega}$ is discussed.

A Generalized Sturmian Treatment of (e, 3e) Processes Described as a Three-Body Coulomb Problem

Fil: Ambrosio, Marcelo Jose. Universidad Nacional del Sur. Departamento de Fisica; Argentina

Parabolic sturmians approach to the three-body continuum Coulomb problem

The three-body continuum Coulomb problem is treated in terms of the generalized parabolic coordinates. Approximate solutions are expressed in the form of a Lippmann-Schwinger-type equation, where the Green’s function includes the leading term of the kinetic energy and the total potential energy, whereas the potential contains the non-orthogonal part of the kinetic energy operator. As a test of this approach, the integral equation for the (e −, e −, He++) system has been solved numerically by using the parabolic Sturmian basis representation of the (approximate) potential. Convergence of the expansion coefficients of the solution has been obtained as the basis set used to describe the potential is enlarged.

Hyperspherical Coulomb spheroidal basis in the Coulomb three-body problem

A hyperspherical Coulomb spheroidal (HSCS) representation is proposed for the Coulomb three-body problem. This is a new expansion in the set of well-known Coulomb spheroidal functions. The orthogonality of Coulomb spheroidal functions on a constant-hyperradius surface ρ = const rather than on a constant-internuclear-distance surface R = const, as in the traditional Born-Oppenheimer approach, is a distinguishing feature of the proposed approach. Owing to this, the HSCS representation proves to be consistent with the asymptotic conditions for the scattering problem at energies below the threshold for three-body breakup: only a finite number of radial functions do not vanish in the limit of ρ→∞, with the result that the formulation of the scattering problem becomes substantially simpler. In the proposed approach, the HSCS basis functions are considerably simpler than those in the well-known adiabatic hyperspherical representation, which is also consistent with the asymptotic conditions. Specifically, the HSCS basis functions are completely factorized. Therefore, there arise no problems associated with avoided crossings of adiabatic hyperspherical terms.

Solving a Three-Body Continuum Coulomb Problem with Quasi-Sturmian Functions

The scattering problem of three particles interacting via Coulomb poten- tials is studied using generalized parabolic coordinates. The scattering solutions are obtained by solving a driven equation. The 'perturbation' operator appearing in the driven term is the non-orthogonal part of the kinetic energy operator. The approxi- mated solution appearing in the driven term is the product of two two-body Coulomb wave functions. As a test for our proposal, a simple two-dimensional model problem has been solved numerically by using so called parabolic quasi-Sturmian basis repre- sentation. Convergence of the solution has been obtained as the basis set is enlarged.

An analytically solvable three-body break-up model problem in hyperspherical coordinates

An analytically solvable S-wave model for three particles break-up processes is presented. The scattering process is represented by a non-homogeneous Coulombic Schrodinger equation where the driven term is given by a Coulomb-like interaction multiplied by the product of a continuum wave function and a bound state in the particles coordinates. The closed form solution is derived in hyperspherical coordinates leading to an analytic expression for the associated scattering transition amplitude. The proposed scattering model contains most of the difficulties encountered in real three-body scattering problem, e.g., non-separability in the electrons’ spherical coordinates and Coulombic asymptotic behavior. Since the coordinates’ coupling is completely different, the model provides an alternative test to that given by the Temkin-Poet model. The knowledge of the analytic solution provides an interesting benchmark to test numerical methods dealing with the double continuum, in particular in the asymptotic regions. An hyperspherical Sturmian approach recently developed for three-body collisional problems is used to reproduce to high accuracy the analytical results. In addition to this, we generalized the model generating an approximate wave function possessing the correct radial asymptotic behavior corresponding to an S-wave three-body Coulomb problem. The model allows us to explore the typical structure of the solution of a three-body driven equation, to identify three regions (the driven, the Coulombic and the asymptotic), and to analyze how far one has to go to extract the transition amplitude.

J-matrix method for calculations of three-body Coulomb wave functions and cross sections of physical processes

The review is devoted to a widely known method of numerical solution to the three-body Coulomb problem, namely, the J-matrix method. Special attention is paid to ways of solving the Lippmann-Schwinger integral equation without attraction of pseudostates. Difficulties related to the formulation of the integral equation in spherical coordinates, leading to the divergence of its integral part if the wave function is calculated with two asymptotically free electrons, are demonstrated. In addition, the relation between exact and approximate solutions turns out to be unclear if the matrix of a residual potential is restricted to a finite number of basis functions, with the latter being increased. It is shown that, in principle, these problems can be avoided by reformulating a problem in parabolic coordinates.

Resonant enhancement of the formation of hydrogen–helium muonic molecules

Formation of muonic molecules ( He 4 μ p ) J = 1 + + and ( He 3 μ d ) J = 0 , 1 , 2 + + , where J is rotational quantum number, in electron conversion process is investigated at collision energies between 0.004 eV and 50 eV. Corresponding reaction rates are calculated in adiabatic approximation for the three-body Coulomb problem. Significant enhancement of the rates for ( He 4 μ p ) 1 + + and ( He 3 μ d ) 2 + + is found near 7 eV and 30 eV, respectively. It is shown that the enhancement is due to resonances present in elastic ( p μ ) 1 s + He + + 4 and ( d μ ) 1 s + He + + 3 scattering at these energies. Acceleration of dμ atoms up to the resonant energy could be realized in triple H–D– 3He mixture due to the muon transfer from protium to deuterium. Experimental investigation of d – He + + 3 nuclear synthesis from ( He 3 μ d ) 0 + + molecular state directly formed in the mixture is suggested.

Symmetric eikonal model for projectile-electron excitation and loss in relativistic ion-atom collisions

At impact energies $ \stackrel{>}{\sim}1$ GeV/u the projectile-electron excitation and loss occurring in collisions between highly charged ions and neutral atoms is already strongly influenced by the presence of atomic electrons. In order to treat these processes in collisions with heavy atoms we generalize the symmetric eikonal model, used earlier for considerations of electron transitions in ion-atom collisions within the scope of a three-body Coulomb problem. We show that at asymptotically high collision energies this model leads to an exact transition amplitude and is very well suited to describe the projectile-electron excitation and loss at energies above a few GeV/u. In particular, by considering a number of examples we demonstrate advantages of this model over the first Born approximation at impact energies $\sim 1$--30 GeV/u, which are of special interest for atomic physics experiments at the future GSI facilities.

The Helium Atom as a Three-Body Coulomb Problem

The Helium Atom as a Three-Body Coulomb Problem

Spectral representation of the three-body Coulomb problem. I. Nonautoionizing doubly excited states of high angular momentum in helium

We investigate high-lying doubly excited nonautoionizing states of helium with total angular momentum L = 1,2, ...,9 with the help of a configuration interaction approach. We provide highly precise nonrelativistic energies of these states and discuss the properties of the wave functions with respect to the particle exchange operator.

Spectral representation of the three-body Coulomb problem. II. Autoionizing doubly excited states of unnatural parity in helium

A spectral approach of configuration interaction type is used to evaluate energies and widths for a wide range of singlet and triplet ${P}^{\mathrm{e}}$ resonance states of helium up to the eighth single ionization threshold. While the present data are in excellent agreement with existing theoretical results (below the $N=3$--$5$ ionization threshold) obtained within an explicitly correlated approach, there are substantial differences with the energies, the widths, and the number of resonances obtained with the stabilization method.

Spectral representation of the three-body Coulomb problem: Perspectives for highly doubly excited states of helium

We present a spectral method of configuration-interaction type for three-dimensional helium which combines the complex rotation method with an appropriate expansion of the atom wave function in a basis of products of Coulomb-Sturmian functions of the electron radial coordinates with independent dilation parameters for the two electrons and bipolar spherical harmonics of the angular coordinates. The matrix elements of the kinetic energy and of the electron-nucleus interaction term are calculated using Gauss-Laguerre integration techniques. A combination of Gauss-Laguerre integration techniques with the generalized Wigner-Eckart theorem and recurrence relations allows an efficient and stable calculation of the matrix elements of the electron-electron interaction. The freedom of the choice of the dilation parameters permits us to access highly excited states with rather small sizes of the basis. Highly doubly excited states up to the tenth ionization threshold of singlet and triplet S states of helium are presented.

Spheroidal and hyperspheroidal coordinates in the adiabatic representation of scattering states for the Coulomb three-body problem

Recently, an involved approach has been used by Abramov (2008 J. Phys. B: At. Mol. Opt. Phys. 41 175201) to introduce a separable adiabatic basis into the hyperradial adiabatic (HA) approximation. The aim was to combine the separability of the Born–Oppenheimer (BO) adiabatic basis and the better asymptotic properties of the HA approach. Generalizing these results we present here three more different separable bases of the same type by making use of a previously introduced adiabatic Hamiltonian expressed in hyperspheroidal coordinates (Matveenko 1983 Phys. Lett. B 129 11). In addition, we propose a robust procedure which accounts in a stepwise procedure for the unphysical couplings that are inherently present in the hyperradial adiabatic multichannel scattering approach. The advantages of the new approach are demonstrated on the example of the basic reaction in muon-catalyzed fusion physics dμ + t → tμ + d.

Comparative Numerical Analysis of Quasi-Two-Dimensional Negative and Positive Trions in High Magnetic Fields

Exact diagonalization calculations in the Haldane spherical geometry are carried out for the three-body Coulomb problem in a quasi-two-dimensional quantum well, in the presence of perpendicular magnetic field. Simultaneous inclusion of several Landau levels and quantum well subbands in the configuration-interaction basis allowed for quantitative description of the coupling between the in-plane dynamics (governed by interplay of cyclotron quantization and Coulomb interactions) and single-particle excitations in the normal direction. The energy spectra of negative and positive trions are compared in symmetric and asymmetric wells with realistic widths and carrier concentrations. Effects of nearby impurities are also studied.

Periodic orbits and binary collisions in the classical three-body Coulomb problem

In the helium case of the classical three-body Coulomb problem in two dimensions with zero angular momentum, we develop a procedure to find periodic orbits applying two symbolic dynamics for one-dimensional and planar problems. Focusing our attention on binary collisions with these tools, a sequence of periodic orbits are predicted and are actually found numerically. A family of periodic orbits found has regularity in their actions. For this family of periodic orbits, it is shown that thanks to its regularity, a partial summation of the Gutzwiller trace formula with a daring approximation gives a Rydberg series of energy levels.

Hyperspherical Coulomb spheroidal representation in the Coulomb three-body problem

A new representation of the Coulomb three-body wavefunction via the well-known solutions of the separable Coulomb two-centre problem ϕj(ξ, η) = Xj(ξ)Yj(η) is obtained, where Xj(ξ) and Yj(η) are the Coulomb spheroidal functions. Its distinguishing characteristic is the coordination with the asymptotic conditions of the scattering problem below the three-particle breakup. That is, the wavefunction of two colliding clusters in any open channel is the asymptotics of the single, corresponding to that channel, term of the suggested expansion. The effect is achieved due to a new relation between three internal coordinates of a three-body system and the parameters of ϕj(ξ, η). It ensures the orthogonality of ϕj(ξ, η) on a sphere of constant hyperradius, ρ = const, in place of the surface R = |x2 − x1| = const appearing in the traditional Born–Oppenheimer approach. The independent variables ξ and η are the orthogonal coordinates on this sphere with three poles in the coalescence points. They are connected with the elliptic coordinates on the plane by means of a stereographic projection. For the total angular momentum J ⩾ 0 the products of ϕj and the Wigner D-functions form a hyperspherical Coulomb spheroidal (HSCS) basis on a five-dimensional hypersphere, ρ being a parameter. The system of the differential equations and the boundary conditions for the radial functions fJi(ρ), the coefficients of the HSCS decomposition of the three-body wavefunction, are presented.