Three-body Coulomb Research Articles

Saddle structure of the three-body Coulomb problem; symmetries of doubly-excited states and propensity rules for transitions

The separable two-centre Coulomb problem is investigated with emphasis on the saddle structure of its potential. A new quasi-degeneracy between molecular orbitals (MO) is identified. The authors demonstrate that the results have threefold relevance for the formation and character of symmetric doubly-excited states: (i) it is shown that all features of Herrick's various multiplet classifications can be explained in terms of MO saddle dynamics; (ii) for the first time quantitative results of resonance energies from the adiabatic MO approach are reported and compared to other theoretical and experimental data; (iii) the propensity rules for radiative and nonradiative transitions are summarized and details of their derivation within the framework of motion in the vicinity of the potential saddle are given.

Solution of three-body Coulomb problems for J=0.

The Schr\"odinger equation for the S state of the three-body system with arbitrary charge and mass has been solved directly using finite-element analysis. In this analysis, the wave function is approximated piecewise using polynomial interpolation functions. The energy and wave function converge to their exact values as the number of elements is increased. In contrast to standard variational calculations, the error in the expectation value of physical observables is comparable to the error in the energy. Results are reported here for the helium atom and the muonic molecular ion dd${\mathrm{\ensuremath{\mu}}}^{+}$.

Classification scheme for adiabatic hyperspherical potential curves fordt?-like systems

A quantum three-body Coulomb system of two heavy and one light particles (m1≫m3,m2≫m3) is considered in the framework of the adiabatic hyperspherical approach. Two sets of approximate quantum numbers are found that specify the adiabatic hyperspherical potential curves at small and large hyperradii. A complete classification scheme is obtained for the potential curves of the mesic moleculedtμ.

Asymptotic separability of three-body continuum wave functions for Coulomb systems

We consider a general three-body Coulomb system above the threshold for total break-up. For large particle separations the Schrodinger equation for this system is shown to be separable in terms of new suitably defined 6-dimensional parabolic coordinates. Although these coordinates are not orthogonal, mixed second derivative terms in the expression for the kinetic energy do not contribute at large particle separations to the Coulomb modification of free particle states. Rejection of mixed second derivatives at all particle configurations yields a three-body continuum wave function that satisfies exact asymptotic boundary conditions for Coulomb systems and treats all two-body interactions fully symmetric. This analysis splits the wave equation into an ‘unperturbed’ part plus a perturbation where the unperturbed part takes into account all long-range correlations.

Three-body Coulomb problem in the dipole approximation.

The doubly excited states of the two-electron atom (or ion) are considered in the case when 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 can be approximated by the dipole term in its multipole expansion. The inner electron velocity is assumed to be much larger than that of the outer electron, which permits assigning a fixed principal quantum number for the inner electron. The same approach is extended to the general Coulomb three-body problem with the particles of arbitrary mass and charge. For the three-body states with symmetry S and ${\mathit{P}}^{\mathit{e}}$, the quantum problem reduces to the three-term recursion relations, which means that the system is effectively one dimensional. The slow evolution of the classical particle trajectories under the influence of interaction is described. It is shown that, depending on the parameters of the system, two types of librations or rotation can be realized. The semiclassical quantization rules (analogous to the well-known Bohr-Sommerfeld rules) are deduced. For S and ${\mathit{P}}^{\mathit{e}}$ states they differ only by the substitution of a half-integer quantum number by an integer. Particularly simple results are obtained in the harmonic approximation, which is valid in the vicinity of the effective-potential curve extreme.

Three-body coulomb effects in one-particle transfer reactions

The behaviour of the DWBA amplitudes in the vicinity of the nearest singularity in the cosθ-plane for the neutron and charged particle transfer reactions is investigated. It is shown that Coulomb interactions transform the pole singularity into a branch point. The main singular terms of the complete and the post-approximation DWBA cross sections are compared with the exact three-body results. They differ only by the Coulomb renormalization factors (CRF), which define the renormalization of the singularity strength generated by the Coulomb interactions. In the case of neutron transfer it is shown that the complete DWBA and exact CRFs practically coincide and slightly differ from CRF obtained in post-approximation DWBA. For charged particle transfer reactions the exact CRF may differ considerably from the one obtained in DWBA and especially in the post-approximation DWBA. Our conclusion therefore is that in order to obtain a reliable value of the vertex constant by extrapolating the differential cross-section to the nearest singularity, the exact three-body approach should be used.

Faddeev calculations for low-energy p-d scattering.

The Coulomb-modified Faddeev equations were solved in momentum space for different separable potential models at ${\mathit{E}}_{\mathrm{lab}}^{\mathit{p}}$=2.5 and 3.0 MeV. The best agreement with available experimental data of zero- and first-order spin polarization observables was achieved by employing a separable approximation of the Paris potential in the calculation. These results, together with predictions of second-order p-d polarizations, demonstrate the necessity of a correct treatment of three-body Coulomb corrections. Moreover, no indication of model sensitivity is found in our results. Three-nucleon phase shift parameters are given and the validity of a simple approximate two-body description of the three-body Coulomb corrections is discussed.

Positional correlation in laser-excited three-body Coulomb systems.

Using a novel six-laser excitation scheme we have created heliumlike planetary states (nlNd) in barium with n\ensuremath{\ge}30, N\ensuremath{\ge}60, and l\ensuremath{\ge}4, which are equivalent of a true three-body Coulomb system. Extremely strong positional correlation effects were observed and quantitatively explained by the repulsion between a ``frozen'' outer electron and the highly polarizable inner electron. The model was experimentally verified by removing the outer electron and recording the inner electron excitation in the presence of an equivalent electric field.

The ground-state energies of three-body Coulomb systems calculated by the short-time Green function Monte Carlo method

The short-time Green function Monte Carlo method is used to calculate the ground-state energies of the three-body systems e–2 e+, e–2µ+, H–(e–2 p+) and H–(∞). In each case high accuracy is attained by employing a carefully optimised trial wavefunction and by using an extrapolation technique to correct for the short-time approximation.

Possible rigid rotations in two-electron atoms.

The path-integral formulation of quantum mechanics has promoted the study of classical periodic trajectories. In this spirit, the three-body Coulomb problem with a nucleus of infinite mass is considered in the special case of a rigid rotation. Using only Newton's fundamental law for a classical atomic system, one can show that both electrons must be equidistant from the nucleus, and one obtains the solutions previously derived by Klar [Phys. Rev. Lett. 57, 66 (1986); Z. Phys. D 3, 353 (1986); J. Opt. Soc. Am. B 4, 788 (1987)]. Nevertheless, if one allows the charge at rest to be arbitrary, one finds that for Z less than unity unsymmetrical solutions may exist. The stability of each of these configurations is considered. In each case an exponentially increasing solution is present, but the unstable character is less marked in the unsymmetrical configuration. Possible consequences for atomic systems are discussed.

Triply-differential cross sections for ionisation of hydrogen atoms by electrons and positrons

A derivation is given of the exact form of the three-body Coulomb wavefunction in the asymptotic region where the separation of all particles tends to infinity. Using a modification of the method of Pluvinage (1951), an approximate three-body scattering wavefunction is derived that satisfies this boundary condition. Triply-differential cross sections (TDCS) for electron impact ionisation of atomic hydrogen calculated with this scattering wavefunction, which contains no free parameters, show excellent agreement with measurements at impact energies greater than 150 eV. The corresponding TDCS for positron impact ionisation are also presented.

Resonance formation in few (3) body Coulomb systems

The standard physical picture of resonance formation breaks down for three-body Coulomb systems close to the threshold for total fragmentation. Near this threshold the complex as a whole performs collective motions localised essentially along potential energy surface ridges. Above threshold these motions are nonperiodic and describe total fragmentation, below threshold unstable periodic motions lead to new resonances. A semiclassical theory which describes these phenomena for two-electron atoms in excellent agreement with experiments is generalised to the ppp system.

Collision-induced spin flip in isotopes of muonic hydrogen

We present results for spin flip processes in muonic atoms. The calculations are performed in the framework of the adiabatic representation of the three-body Coulomb problem. The contribution of back decay is taken into account and comparison with available experimental data is reported.

Faddeev approach to the three-body problem in total-angular-momentum representation

For a system of three charged particles the Faddeev equations are derived in the total-angular-momentum representation. They have the form of coupled sets of partial differential equations in three-dimensional space and can be used to develop new efficient numerical procedures to tackle the three-body Coulomb problem. The asymptotic conditions at large distances corresponding both to binary scattering and bound-state problems are presented. The behaviour of the Faddeev components near the triple and double collision points is studied.

Dispersion theory of nucleon transfer reactions induced by heavy ions

A new approach, the distorted wave pole approximation (DWPA) with the three-body Coulomb effects, is developed by combining the dispersion method and DWBA to analyse the heavy ion-induced neutron transfer reactions. The influence of the three-body Coulomb dynamics on the peripheral partial wave amplitudes is investigated. Differential cross sections of the neutron transfer reactions are calculated to compare the proposed model with the conventional DWBA. The values of nuclear vertex constants for virtual separation of neutron from various nuclei are obtained. The results of the calculations show that DWPA can be applied to analyse the heavy ion-induced neutron transfer reactions and that the three-body Coulomb effects are taken into account with acceptable accuracy in DWBA.

Adiabatic representation for the three-body problem in hyperspherical coordinates. I. Statement of the problem

For the three-body Coulomb problem a hyperspherical parametrisation of independent variables is given on a five-dimensional sphere S5with a hyperradius RH, the first linear invariant of the inertia tensor. The hyperspherical adiabatic basis is defined as a complete set of eigenfunctions and eigenvalues of the Hamiltonian on the sphere S5for every fixed value of the slow variable RH. The partial wave analysis in the total momentum J representation allows the authors to separate three Euler angles and to reduce the hyperspherical problem on S5to a system of (J+1) two-dimensional problems. Classification is given of the hyperspherical adiabatic basis for small and large values of the hyperradius RH. The logarithmic Fock singularity at the point of triple collision (RH=0) is explicitly shown. The approach is assigned to computing the cross sections of mesic atomic processes in the muon catalysis problem.

Study of three-body Coulomb systems using a hyperspherical harmonics basis. I. Calculation of the ground state of the helium atom

The helium ground-state energy has been calculated by the hyperspherical harmonics method. The system of hyperradial differential equations has been solved by the method of exclusion of the continuous spectrum which reduced significantly the size of the algebraic eigenvalue problem. The optimum basis method (OBM) yields a He ground-state energy E0=-2.903 7047 au when 120 hyperspherical harmonics are employed.

Three-body Coulomb bound states.

The binding energies of three-particle systems containing two electrons and one positive particle of mass M are reexamined in an attempt to understand the approximate proportionality of the 1Se ground-state binding energies of the reduced masses, as pointed out by Botero and Green (1986). The contribution to the energy of the mass-polarization term is evaluated. No fundamental principle is involved, since the mass polarization merely decreases somewhat as the mass of the positive particle is reduced below the proton mass. In the case of the excited 3Pe state, this reduction is not sufficient to allow binding when M approaches the electron mass. Some properties of the recently observed negative muonium ion (e/-/ mu/+/ e/-/) are also computed.

Atomic physics on a synchrotron: Crucial experiments

Various types of synchrotron experiments in atomic and molecular physics are discussed. Among other things, these experiments bear on the three-body continuum Coulomb problem, strong correlation/multiple processes, photoabsorption by unusual states, and probes of small interactions.

Comment on "Muon-alpha-particle sticking probability in muon-catalyzed fusion"

C\'eperley and Alder recently reported [Phys. Rev. A 31, 1999 (1985)] a calculation of the muon sticking probability using three-body Coulomb wave functions of the dt\ensuremath{\mu} muomolecule. We comment here that such calculations require in addition the incorporation of the interplay of the nuclear reaction dynamics with the Coulomb problem.