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Physical Review A

Three-body Coulomb bound states

A. K. Bhatia, Richard J. Drachman

Published May 1, 1987• 44 citations
Publisher

Abstract

We reexamine the binding energies of three-particle systems containing two electrons and one positive particle of mass M, in an attempt to understand the approximate proportionality of the $^{1}\mathrm{S}^{\mathrm{e}}$ ground-state binding energies to the reduced masses, as pointed out by Botero and Greene [Phys. Rev. 56, 1366 (1986)]. Our investigation involves a careful evaluation of the contribution to the energy of the mass-polarization term ${H}_{\mathrm{mp}=\mathrm{\ensuremath{-}}(2/\mathrm{M}){\ensuremath{\nabla}}_{1}\mathrm{\ensuremath{\cdot}}{\ensuremath{\nabla}}_{2}}$ which is the only part of the Hamiltonian not scaling like the reduced mass. It does not appear that any fundamental principle is involved, since we find merely that the mass polarization decreases somewhat as the mass of the positive particle is reduced below the proton mass. In the case of the excited $^{3}\mathrm{P}^{\mathrm{e}}$ state, this reduction is not sufficient to allow binding when M approaches the electron mass. As a by-product, we compute accurately some properties of the recently observed negative muonium ion (${e}^{\mathrm{\ensuremath{-}}}$${\ensuremath{\mu}}^{+}$${e}^{\mathrm{\ensuremath{-}}}$).

DOI
10.1103/physreva.35.4051

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We reexamine the binding energies of three-particle systems containing two electrons and one positive particle of mass M, in an attempt to understand the approximate proportionality of the 1Se^{1}\mathrm{S}^{\mathrm{e}}1Se ground-state binding energies to the reduced masses, as pointed out by Botero and Greene [Phys. Rev. 56, 1366 (1986)]. Our investigation involves a careful evaluation of the contribution to the energy of the mass-polarization term Hmp=\ensuremath−(2/M)\ensuremath∇1\ensuremath⋅\ensuremath∇2{H}_{\mathrm{mp}=\mathrm{\ensuremath{-}}(2/\mathrm{M}){\ensuremath{\nabla}}_{1}\mathrm{\ensuremath{\cdot}}{\ensuremath{\nabla}}_{2}}Hmp=\ensuremath−(2/M)\ensuremath∇1​\ensuremath⋅\ensuremath∇2​​ which is the only part of the Hamiltonian not scaling like the reduced mass. It does not appear that any fundamental principle is involved, since we find merely that the mass polarization decreases somewhat as the mass of the positive particle is reduced below the proton mass. In the case of the excited 3Pe^{3}\mathrm{P}^{\mathrm{e}}3Pe state, this reduction is not sufficient to allow binding when M approaches the electron mass. As a by-product, we compute accurately some properties of the recently observed negative muonium ion ({e}^{\mathrm{\ensuremath{-}}}$${\ensuremath{\mu}}^{+}$${e}^{\mathrm{\ensuremath{-}}}).
Various energetic, geometric, and other properties for the ground and first vibrationally excited S states (L=0) of Coulomb three-body systems having unit charges are obtained by means of accurate exponential wave functions. The mass dependence of the properties for these systems has a relatively smooth form for the ground states. The results presented should be useful as a basis for further investigation of these three-body systems. The prethreshold (weakly bound) exotic nonsymmetric systems (d+{\mathit{d}}^{+}d+t+{\mathit{t}}^{+}t+${\mathit{p}}^{\mathrm{\ensuremath{-}}}$, ${\mathrm{\ensuremath{\mu}}}^{+}$${\mathrm{\ensuremath{\pi}}}^{+}$${\mathrm{\ensuremath{\mu}}}^{\mathrm{\ensuremath{-}}}$, and ${\mathrm{\ensuremath{\mu}}}^{+}$${\mathrm{\ensuremath{\pi}}}^{+}$${\mathrm{\ensuremath{\pi}}}^{\mathrm{\ensuremath{-}}}$) considered have sharp cluster structures, i.e., the lighter positive ion moves in the field of the neutral pair (e.g., d+{\mathit{d}}^{+}d+ in d+{\mathit{d}}^{+}d+t+{\mathit{t}}^{+}t+${\mathit{p}}^{\mathrm{\ensuremath{-}}}$ moves in the field of t+{\mathit{t}}^{+}t+${\mathit{p}}^{\mathrm{\ensuremath{-}}}$ ).
Various geometrical and energetical (bound-state) properties of the positronium negative ion $({\mathrm{Ps}}^{\ensuremath{-}}$ or ${e}^{\ensuremath{-}}{e}^{+}{e}^{\ensuremath{-}})$ are determined by using highly accurate variational wave functions. These wave functions have been constructed by applying the advanced two-stage strategy proposed by Frolov [Phys. Rev. A 57, 2436 (1998)]. The determined total energy of the ground state $E=\ensuremath{-}0.2620050702329757\mathrm{a}.\mathrm{u}.$ is the lowest and most accurate value obtained for this system to date (the corresponding binding energy equals $\ensuremath{-}0.326674721317821\mathrm{eV}).$ The computation of the second-order relativistic corrections $(\ensuremath{\simeq}{\ensuremath{\alpha}}^{2})$ to the total energy is discussed also. The general form of the Breit-Pauli Hamiltonian in the relative coordinates for the ${\mathrm{Ps}}^{\ensuremath{-}}$ ion is presented.
In an effort to unravel the dependence of the dynamics of three-bound-particle systems on the relative masses of the particles, an adiabatic treatment in hyperspherical coordinates is used to study the ground state and some doubly excited states of Ps('-) (positronium negative ion) and other three-particle systems, including H('-) and M('-) (muonium negative ion). Two different methods for the solution of the adiabatic eigenvalue equation are presented; one uses a "prediagonalized" basis of hyperspherical harmonics, and the other uses an asymptotic basis which consists of hydrogenic functions for the positronium. Potential curves for Ps('-) and M('-) in the ('1)S('e), ('1)P('o), ('3)P('e), ('3)P('o) and ('1)D('e) states and for H('-) in the ('1)P('o) states are presented. A method for calculating the probability density as a function of the hyperspherical coordinates (R,(alpha),(theta)(,12)) is introduced and used for the P states of Ps('-). This helps to understand the different symmetries and to distinguish among them. A remarkable linear dependence of the binding energy of the ground state and of the doubly excited states on the reduced mass of the atom, discovered in this study, is presented and discussed.
Lower bounds to the ground-state energy of a system composed of N identical particles interacting by two-body forces (and possibly also with an external potential) are derived. They are expressed in terms of the ground-state energies of (simpler) systems composed of a smaller number of particles having masses and/or interactions different from the original ones. This reduction process may be continued all the way down to systems involving one and two particles only. The generalization to systems containing some identical and some distinguishable particles is also discussed.
Properties of some few-body systems which include one positively charged muon μ+μ^{+}μ+ and two electrons e−e^{-}e− are discussed. In particular, we consider the negatively charged muonium ion Mu−^{-}− (or μ+e2−μ^{+} e^{-}_{2}μ+e2−​) and four-body MuPs (or μ+e2−e+μ^{+} e^{-}_{2} e^{+}μ+e2−​e+) systems each of which has only one stable bound (ground) state. The problem of annihilation of the electron-positron pair(s) in the MuPs system is investigated. The hyperfine structure splitting of the ground state in the MuPs system evaluated with our expectation value of the muon-positron delta-function is Δ≈Δ\approxΔ≈ 23.05758 MHzMHzMHz. Another group of interesting four-body neutral systems investigated in this study includes the p+μ+e2−,d+μ+e2−p^{+} μ^{+} e^{-}_2, d^{+} μ^{+} e^{-}_2p+μ+e2−​,d+μ+e2−​ and t+μ+e2−t^{+} μ^{+} e^{-}_2t+μ+e2−​ `quasi-molecules'. These quasi-molecules are formed in large numbers when positively charged muons slow down in liquid hydrogen, or in liquid deuterium and/or tritium. The properties of these systems are unique, since they occupy an intermediate position between actual two-center molecules and one-center atoms.
Abstract Properties of some few-body systems which include one positively charged muon μ + and two electrons e − are discussed. In particular, we consider the negatively charged muonium ion Mu − (or μ + e 2 − ) and four-body MuPs (or μ + e 2 − e + ) systems, each of which has only one stable bound (ground) state. The problem of annihilation of the electron–positron pair(s) in the MuPs system is investigated. The hyperfine structure splitting of the ground state in the MuPs system evaluated with our expectation value of the muon-positron delta-function is Δ ≈ 23.05758 MHz. Another group of interesting four-body neutral systems investigated in this study includes the p + μ + e 2 − , d + μ + e 2 − and t + μ + e 2 − ‘quasi-molecules’. These quasi-molecules are formed in large numbers when positively charged muons slow down in liquid hydrogen, or in liquid deuterium and/or tritium. The properties of these systems are unique, since they occupy an intermediate position between actual two-center molecules and one-center