Three-body Coulomb Systems Research Articles

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.

Classification of Coulombic three-body systems in hyperspherical coordinates.

Hyperspherical coordinates are used to study properties of Coulombic three-body systems of arbitrary masses. Consider a system {ital ABA}, which consists of two identical particles {ital A} and a third particle {ital B}, each with one unit of charge. We examine the evolution of the approximate quantum numbers that are used for classifying bound and resonance states of the system as the mass ratio {lambda}={ital m}{sub {ital A}}/{ital m}{sub {ital B}} changes from the atomic limit ({lambda}{r arrow}0 as in H{sup {minus}}) to the diatomic molecular limit ({lambda}{much gt}1 as in H{sub 2}{sup +}). It is shown that for states which exhibit rovibrational behaviors in the atomic limit ({lambda}{r arrow}0), a single set of approximate quantum numbers can be used to describe three-body systems of any {lambda}'s. For states that display independent-particle behavior in the atomic limit, such as singly excited states, it is shown that these states display rovibrational behaviors only in the large-{lambda} limit. The evolution of the spectroscopy of the three-body systems from the shell model of atoms to the rovibrational model of molecules is thus analyzed. Calculations of potential curves in hyperspherical coordinates were carried out for Ps{sup {minus}} and {ital d}{sup +}{mu}{sup {minus}}{ital d}{sup +}more » that serve as the intermediate steps for the study of the evolution of the approximate quantum numbers from H{sup {minus}} to H{sub 2}{sup +}.« less

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.

Relativistic and QED contributions to the coulomb three-body system in muon-catalyzed fusion

Energy shifts in the shallowest level of the muonic moleculedtμ are estimated by using an accurate wave function of the point-charge three-body system which gives e110= −660.3 meV. Contributions of nuclear finite size, vacuum polarization, relativistic effect and internuclear interaction are evaluated to be +10.4 meV, +17.2 meV, +0.9 meV and ∼ +10−4 meV, respectively. These corrections to e110 amount to 28.5 meV, which is significantly different from the corresponding value 20.6 meV of the Soviet group, with serious dependence on the dtμ, formation rate.

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.

Methods of solving Coulombic three-body problems in hyperspherical coordinates.

A method for solving three-body problems in mass-weighted hyperspherical coordinates in the adiabatic approximation is presented. The adiabatic channel function is expanded in terms of analytical functions expressed in different sets of Jacobi coordinates to describe each dissociation limit naturally. Evaluation of matrix elements between functions in different Jacobi coordinates is achieved through the known transformation properties of hyperspherical harmonics in these coordinates. The method is applied to study the lowest channels for various Coulombic three-body systems such as H/sup -/, e/sup -/e/sup -/e/sup +/, p/sup +/p/sup +/..mu../sup -/, d/sup +/d/sup +/..mu../sup -/, e/sup +/p/sup +/e/sup -/, p/sup +/d/sup +/..mu../sup -/, and d/sup +/t/sup +/..mu../sup -/ using only one or two analytical basis functions, and the results are compared with some known calculations. The behavior of the potential curves with respect to the variation of the masses of the three-body system is also examined.

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.

Muonic three-body problem and muon- catalyzed fusion

Recently, the muonic three-body systemdtμ, has aroused considerable attention in relation to the realization of a useful muon-catalyzed fusion [1, 2]. This bound system must be solved up to 6 digits in energy to establish the muon catalyzed fusion process. We applied the ATMS method [3] and the coupled-channel method [4] to the Coulomb three-body system and obtained the detailed information on the energy levels and the wave functions of the bound states. Further we investigated the effect of the nuclear interaction on the fusion rate in the bound states [5] and developed a formalism to calculate the muon sticking to4He [6]. In this paper we report the results obtained in our collaborations: Figure 2 shows up our main results.

New method for treating excitonic molecules

The results for binding energies for excitonic molecules in the region 0<σ=m ∗ e/m ∗ h⩽1 are presented in this paper. The application of the so-called adiabatic method, which has widely been applied to three-body Coulomb systems, gives an increase in the binding of these simplest four-body systems with Coulomb interaction, compared to previous variational results.

Excitonic molecule in the adiabatic representation.

The binding energy for four-body systems with Coulomb interaction is investigated using the adiabatic method usually applied to three-body Coulomb systems. Our investigation gives an increase in the binding energies of these systems compared to previous theoretical results.

Ground-state energy of three-body Coulomb systems through hyperspherical harmonics

The authors use the hyperspherical formalism in order to obtain the ground-state energy for the ions e-e+e-, H- and for the He atom through numerical methods. When studying the convergence trend of the successive approximations they find it behaves comparably with other known applications (although made for short-range forces) and they make use of previous experience to propose an extrapolation formula. The results thus obtained are then compared with those provided by other techniques for the aforementioned systems.

Evaluation of off-shell matrix elements of Faddeev equations

The first-order exact off-shell matrix elements for the scattering processes to a three-body Coulomb system with arbitrary masses in which three-particle states are made of two-body bound states and a third free-particle state have been evaluated. These elements are required if one is interested in applying the tractable form (Sloan and Moore) of the Lovelace-Faddeev equations. The amplitudes have been obtained for both the direct and rearrangement channels. The two-body bound states in both the initial and final states are, however, restricted to be the ground states. It may be mentioned that one can always get the amplitudes for higher excited states by appropriate parametric differentiations. Two other approximate forms of the amplitudes have also been given. In all, the forms the matrix elements in the direct channel are found to be singular in the forward scattering angle. The expressions devoid of singularities have been obtained in the present investigation.

Correlated wave functions for three-body Coulombic systems. Asymmetric mesonic molecular ions and heliumlike atoms

The previously reported general correlated wave function for three-particle Coulombic systems is used to construct compact and accurate wave functions for the asymmetric mesonic molecular ions pμd, pμt and dμt, and for helium and heliumlike positive ions Li+ and Be++. With these wave functions the following ground-state energy values have been obtained: −2.903406 a.u. for He, −7.278210 a.u. for Li+, −13.654953 a.u. for Be++, −0.510483 mesonic a.u. for pμd, −0.517284 mesonic a.u. for pμt and −0.536451 mesonic a.u. for dμt. All the wave function parameters are tabulated together with a number of useful expectation values.

Three-body Coulomb systems using generalized angular-momentumSstates

An expansion of the three-body Coulomb potential in generalized angular-momentum eigenfunctions developed earlier by one of the authors is used to compute energy eigenvalues and eigenfunctions of bound S states of three-body Coulomb systems. The results for He, H(-), e(-)e(+)e(-), and pmu(-)p are compared with the results of other computational approaches.

Multiple-Scattering Expansions for Nonrelativistic Three-Body Collision Problems. IV. Application of the Faddeev-Watson Expansion to Scattering Processes

The high-energy limit of the differential scattering cross section, according to the exact Born approximation can have either an ${E}^{\ensuremath{-}2}$ or an ${E}^{\ensuremath{-}6}$ energy dependence, depending on the scattering angle and masses of the system. When the target is made of equal-mass particles having charges which are equal but opposite in sign, the first-order Born approximantion predicts a zero for the elastic scattering amplitude. This anomalous behavior does not appear in the first-order Faddeev-Watson multiple-scattering approximation. An application of the first-order Faddeev-Watson multiple-scattering approximation to a three-body Coulomb system with arbitrary masses is considered. The three-body scattering amplitude is derived and expressed in terms of an integral consisting of contributions coming from (a) the two-body bound-state poles in the initial and final wave functions, (b) the branch-point singularities in the off-shell two-body Coulomb $T$ matrices, and (c) the intermediate two-body bound- and antibound-state poles.

Multiple scattering expansions for nonrelativistic three-body collision problems. Ii. Convergence of the faddeev-watson expansion for rearrangement collisions

The convergence of the faddeev-watson multiple scattering expansion is investigated for three-body rearrangement collisions, 1+(2,3)=(1,2)+3, for systems with coulomb interactions and arbitrary masses. The matrix elements of the expansion are evaluated to the second order in the two-body t-matrix using the nutt procedures. The result suggests that the expansion for rearrangement transition amplitudes (psi1(3)?tr?psi1(1)) between ground two-body states converges in the high energy limit for a nonrelativistic three-body coulomb system with arbitrary masses. The effect of the constituent particle masses on the rate of approaching this high-energy behaviour is discussed. It is pointed out that the controversies regarding the role of proton-proton interaction in electron-transfer collisions for the (p+,h) system arise from the improper high-energy behaviour of the born series.

Orbital-Angular-Momentum Decomposition of the Three-Body Coupling Scheme: Muonic Molecules and the Hydrogen Molecular Ion

In obtaining the wave functions of three-body Coulomb systems in which two heavy equally charged, and one light oppositely charged, particles are involved, one can use a type of Born-Oppenheimer coupling scheme, to obtain nonadiabatic effects. In this paper we decompose the coupled differential equations of such a scheme in terms of the total orbital angular momentum and parity of the states, and reduce these equations to one-dimensional ones in a form suitable for calculation.