E. O. Alt

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.

Charge-exchange reactions in a three-body eikonal approach

The three-body eikonal approach to ion-atom collisions based on the Alt-Grassberger-Sandhas equations is developed. The amplitudes for scattering and charge exchange, written in the impact parameter representation and using the eikonal approximation, are represented as a sum of two terms. One of them describes the scattering via the Coulomb potential acting between the colliding heavy particles. The second one could be factored into a product of two terms. One is an explicitly known factor which contains all the information about the interaction between the heavy particles. The remaining part does not contain any reference to this interaction any longer. For these residual amplitudes effective-two body equations are written down. Explicit expressions for the effective potentials occurring therein, for both the scattering and the transfer channels, are derived in the lowest order approximation. This approach is then used to calculate total and partial cross sections for the electron capture processes in collisions of the ions H+, He2+ and Li3+ with atomic hydrogen. The latter is taken to be in its ground state, except for the case of incident H+ when the target hydrogen atom is considered also in some low-lying excited states.

Gaussian quadrature rule for arbitrary weight function and interval

Abstract A program for calculating abscissas and weights of Gaussian quadrature rules for arbitrary weight functions and intervals is reported. The program is written in Mathematica. The only requirement is that the moments of the weight function can be evaluated analytically in Mathematica. The result is a FORTRAN subroutine ready to be utilized for quadrature. Program summary Title of program: AWGQ Catalogue identifier:ADVB Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADVB Program obtained from: CPC Program Library, Queens University, Belfast, N. Ireland Computer for which the program is designed and others on which it has been tested: Computers: Pentium IV 1.7 GHz processor Installations: 512 MB RAM Operating systems or monitors under which the program has been tested: Windows XP Programming language used: Mathematica 4.0 No. of processors used: 1 Has the code been vectorized or parallelized?: No No. of lines in distributed program, including test data, etc.:1076 No. of bytes in distributed program, including test data, etc.: 32 681 Operating systems under which program has been tested: FORTRAN Distribution format: tar.gz Nature of physical problem: Integration of functions. Method of solution: The recurrence relations defining the orthogonal polynomials for arbitrary weight function and integration interval are written in matrix form. The abscissas and weights for the corresponding Gaussian quadrature are found from the solution of the eigenvalue equation for the tridiagonal symmetric Jacobi matrix. Restrictions on the complexity of the problem: The program is applicable if the moments of the weight function can be evaluated analytically in Mathematica. For our test example the degree of the Gaussian quadrature cannot not be larger than 96. Typical running time: The running time of the test run is about 1 [s] with a Pentium IV 1.7 GHz processor.

Rotational Three-Body Resonances: A New Adiabatic Approach

In the standard adiabatic approach the motion of the fast, light particle (electron) is treated so as to produce an effective potential that governs the motion of the heavy particles (nuclei). The rotational degrees of freedom are then taken into account by adding the centrifugal J(J + 1)-term to the channel potentials and introducing rotational (Coriolis) couplings into conventional close-coupling calculations. Of course, a perturbative treatment of the rotational motion is justified only provided the rotational energy is sufficiently small. If, however, the rotation is as energetic as the motion of the fast particle, both motions should be treated on the same footing in order to produce symmetry-adapted effective potentials for the nuclear motion. Here, we present for the first time a set of adiabatic potentials of this type for two classical adiabatic systems, namely for H+ 2, for states with total angular momentum J = 35 and total spatial parity p = − 1, and for (pdμ)+-ion for states with J = 1 and p = − 1. Comparison with standard adiabatic approaches is very instructive.

Triangle diagram with off-shell Coulomb T-matrix for (in-)elastic atomic and nuclear three-body processes

The driving terms in three-body theories of elastic and inelastic scattering of a charged particle off a bound state of two other charged particles contain the fully off-shell two-body Coulomb T-matrix describing the intermediate-state Coulomb scattering of the projectile with each of the charged target particles. Up to now the latter has usually been replaced by the Coulomb potential, either when using the multiple-scattering approach or when solving three body integral equations. General properties of the exact and the approximate on-shell driving terms are discussed, and the accuracy of this approximation is investigated numerically, both for atomic and nuclear processes including bound-state excitation, for energies below and above the corresponding three-body dissociation threshold, over the whole range of scattering angles.

Definition of free hyperradial dynamics for the three-body pproblem

Following the analysis of [1,2], we define appropriate hyperradius-distorted free incoming and outgoing waves (HDFW) that incorporate unphysical long-range effects of the hyperradial-adiabatic (HA) treatment of the three-body scattering problem.

Low Energy p̄ + H Collisions in Hyperspheroidal Coordinates

Recently, Esry and Sadeghpour (2003), and Hesse, Le and Lin (2004), have reported calculations of protonium formation in p + H collisions at low energies, using hyperspherical coordinates in a hyperradial adiabatic approach. In order to make the problem tractable both groups were forced to introduce an artificial proton mass (mp′ = 17.824 a.u. and mp′ = 100 a.u., respectively) which raises doubts as to the physical relevance of their results and conclusions. Here we make use of the hyperspheroidal coordinates in order to attack the same problem in basically the same approach but without need for changing the physical particle masses.

Coulomb Fourier Transformation: Application to a Three-Body Hamiltonian with One Attractive Coulomb Interaction

Consider a three-body system consisting of one neutral particle 1 and two charged particles characterized by the indices 2 and 3 with charges of opposite sign, i.e., e2e3 < 0. We use the following notation: (x ν , y ν ), v = 1, 2, 3, denotes the (mass-renormalized) coordinate vector within the pair ν, and between the center of mass of the pair ν and particle ν, respectively. The corresponding canonically coniugate momenta are (k ν , p ν ).

Protons in collision with hydrogen atoms: Influence of unitarity and multiple scattering

Three-body integral equations when applied to collisions of protons with hydrogen atoms yield the amplitudes for direct scattering and electron exchange which automatically satisfy two-body and (if solved exactly also) three-body unitarity. As a consequence, differential cross sections (DCS) for both reactions are calculated on equal footing, and the total exchange cross section (TCS) results from the corresponding DCS. This is to be contrasted with standard models used in atomic physics which either are of the (Distorted-Wave) Born type (to be applied at higher energies), or solve the Schrodinger equation by expanding the wave function in a basis (close-coupling method which has problems especially for rearrangement processes). Hence, at higher energies in general separate models have to be developed to describe the DCS for either direct scattering or electron exchange, and frequently also for the TCS. For instance, the most sophisticated traditional models, which provide a very good reproduction of the TCS data, are the continuum distorted wave [1] and the boundary corrected first Born model [2]. Both, however, fail to describe differential cross sections which represent a much more stringent test. On the other hand, three-body integral equations suffer from the principal defect that their kernels are not compact when particles with charges of different sign are involved, as it happens in applications to atomic reactions (references can be traced from [3]). The consequence is that naive application of standard solution methods of integral equations theory would not be justified. Additional, practical difficulties arise from the complicated singularity structure of the off-shell twoparticle Coulomb T-matrix which is the basic dynamical ingredient. As is well known, the latter develops nasty singularities in the on-shell limit and, in case of attraction, has in addition an infinite number of poles.

Proton-Hydrogen Charge Exchange and Elastic Scattering in the Faddeev Approach

Results of the application of Faddeev-type integral equations to proton-hydrogen collisions are reported. The approach, realized in the impact parameter representation, incorporates the exact two-particle off-shell Coulomb T-matrices in all ‘triangle’ contributions to the effective potentials. Calculatedtotal and differential electron-transfer as well as differential elastic scattering cross sections show very good agreement with experiment, over a wide range of incident energies.