A. V. Matveenko

Hyperradial-adiabatic approach to Coulombic three-body systems

The variational approach to the solution of the hyperradial-adiabatic three-body Hamiltonian is analysed in a new way. The hyperspheroidal coordinates, introduced earlier by Matveenko and Abe, are used to construct the channel-independent pair-collision- and triple-collision-limit solutions of the HA problem. The breakdowns of the adiabaticity in the vicinity of the avoided-crossing points and in the regions of the two-body dissociation limits are critically overlooked. In order to treat the avoided-crossing regions we define the separable HA Hamiltonian which naturally follows from the exact one, and has proper behaviour at the singular points and is simpler than the one introduced recently by Tolstikhin et al. The dissociation-limit problem is clearly demonstrated to be the asymptotic tail of the unphysical hyperradial coupling: we discuss the possibility of the general solution of the problem for a rearrangement scattering process in the hyperradial-adiabatic approach.

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.

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.

Hyper-radial adiabatic expansion for a muonic molecule dtμ

Abstract By using the hyper-radius, adiabatic potential energy curves with correct asymptotic energies are obtained for the Coulomb three-body problem. The bound state energies of the muonic molecules dtμ with total angular momentum J =0 calculated adopting the three lowest adiabatic potential energy curves are −318.72 and −34.36 eV for vibrational quantum numbers ν =0 and 1, respectively.

Asymptotically adapted adiabatic representation

The equations of the adiabatic representation for a collision problem are rederived in a new and compact way, and the kinematic nature of the long-range matrix elements which arise in this representation is shown explicitly. The matrix transformation of the adiabatic representation is introduced, which greatly simplifies its asymptotic form; the asymptotic values of matrix elements are effectively subtracted. The relation of this transformation with the translational factor operator is demonstrated. In the case of a one-electron diatomic system, the explicit form of the matrices introducing the asymptotically adapted adiabatic representation is found.

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.

Low energy structure of the H (1s) + μ+ → Mu (1s) + p total cross section

Abstract Detailed calculations of the cross section for the ground state muonium Mu(1s) formation in the center of mass energy interval 10 −3 eV ⪕ E ⪕ 5 eV are presented. The oscillatory structure of the cross section is analysed.