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Physical Review A

Probable nonexistence of a Pe3 metastable excited state of the positronium negative ion

Allen Paine Mills

Published Dec 1, 1981• 44 citations
Publisher

Abstract

The ${\mathrm{H}}^{\ensuremath{-}}$ ion is known to have a metastable $2{p}^{2}^{3}P^{e}$ triplet excited state. To see if an analog of this state is present in the positronium negative ion ${\mathrm{Ps}}^{\ensuremath{-}}$ the Coulomb binding energy ${E}_{b}$ of the lowest-energy even-parity $L=1$ configuration of two identical-charge $\ensuremath{-}e$ fermions of mass ${m}_{1}$ plus one spinless particle of mass ${m}_{2}$ and charge $+e$ is calculated. We find a value of ${E}_{b}$ below the $n=2$ level of the neutral atom for $\frac{0\ensuremath{\le}{m}_{2}}{M\ensuremath{\le}0.17}$ and $\frac{0.90<{m}_{2}}{M\ensuremath{\le}1}$, where $M=2{m}_{1}+{m}_{2}$ is the total mass of the ion. However, we cannot say that the $^{3}P^{e}$ state exists for ${\mathrm{Ps}}^{\ensuremath{-}}$, where $\frac{{m}_{2}}{M}=\frac{1}{3}$.

DOI
10.1103/physreva.24.3242

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The H\ensuremath−{\mathrm{H}}^{\ensuremath{-}}H\ensuremath− ion is known to have a metastable 2{p}^{2}^{3}P^{e} triplet excited state. To see if an analog of this state is present in the positronium negative ion Ps\ensuremath−{\mathrm{Ps}}^{\ensuremath{-}}Ps\ensuremath− the Coulomb binding energy Eb{E}_{b}Eb​ of the lowest-energy even-parity L=1L=1L=1 configuration of two identical-charge \ensuremath−e\ensuremath{-}e\ensuremath−e fermions of mass m1{m}_{1}m1​ plus one spinless particle of mass m2{m}_{2}m2​ and charge +e+e+e is calculated. We find a value of Eb{E}_{b}Eb​ below the n=2n=2n=2 level of the neutral atom for 0\ensuremath≤m2M\ensuremath≤0.17\frac{0\ensuremath{\le}{m}_{2}}{M\ensuremath{\le}0.17}M\ensuremath≤0.170\ensuremath≤m2​​ and 0.90<m2M\ensuremath≤1\frac{0.90<{m}_{2}}{M\ensuremath{\le}1}M\ensuremath≤10.90<m2​​, where M=2m1+m2M=2{m}_{1}+{m}_{2}M=2m1​+m2​ is the total mass of the ion. However, we cannot say that the 3Pe^{3}P^{e}3Pe state exists for Ps\ensuremath−{\mathrm{Ps}}^{\ensuremath{-}}Ps\ensuremath−, where m2M=13\frac{{m}_{2}}{M}=\frac{1}{3}Mm2​​=31​.
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.
A highly accurate calculation is supplemented by an adiabatic approximation to explore the resonance spectrum of positronium ${\mathrm{Ps}}^{\mathrm{\ensuremath{-}}}$. Surprisingly, the spectrum can be understood and classified with H2+{\mathrm{H}}_{2}^{+}H2+​ quantum numbers by treating the interelectronic axis of ${\mathrm{Ps}}^{\mathrm{\ensuremath{-}}}$ as an adiabatic parameter. We report and interpret the existence of 1^{1}1S shape resonances, a phenomenon so far unknown in three-body Coulomb systems. The new results on ${\mathrm{Ps}}^{\mathrm{\ensuremath{-}}}$ combined with previous results for ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ suggest the existence of a resonance spectrum and its similarity for all ABA Coulomb systems with charges \ensuremath{\Vert}ZA{\mathit{Z}}_{\mathit{A}}ZA​/ZB{\mathit{Z}}_{\mathit{B}}ZB​\ensuremath{\Vert}=1 and masses mA{\mathit{m}}_{\mathit{A}}mA​/mB{\mathit{m}}_{\mathit{B}}mB​\ensuremath{\ge}1.
The positronium molecule (Ps2)({\mathrm{Ps}}_{2})(Ps2​) has not been experimentally observed yet because its tiny (4.5 eV) binding energy cannot be detected when the molecule annihilates by emitting two photons with energy of 0.51 MeV each. It is shown in this paper that the electric dipole transition between the recently found L=1L=1L=1 excited state and the L=0L=0L=0 ground state with its characteristic photon energy of 4.94 eV is a clear signature of the existence of the positronium molecule and the possibility of its experimental observation is realistic. The probability of this transition is about 17% of the total decay rate. Another Coulomb four-body system containing positron HPs (the positronium hydride or hydrogen positride), is also included for comparison.
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.
We have continued our investigation into the elastic scattering of positronium atoms by hydrogen atoms, extending the work to include the spin-one (triplet) state of the two electrons. Once again we have extracted the SSS-wave phase shifts from normalizable trial functions by examining the spatial behavior of the center-of-mass wave functions. An effective-range expansion linear in the positronium energy fits the results very well and leads to a scattering length ${a}^{(\ensuremath{-})}\ensuremath{\simeq}2.36{a}_{0}$, compared with the static result ${a}^{(\ensuremath{-})}=2.476{a}_{0}$ of Hara and Fraser. The SSS-wave total and ortho-para conversion cross sections are calculated by making use of both 1S^{1}S1S and 3S^{3}S3S results. In the 3S^{3}S3S case, we see no indication of any resonances below the excitation threshold at 5.1 eV.
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 ${H}_{\mathrm{mp}=\mathrm{\ensuremath{-}}(2/\mathrm{M}){\ensuremath{\nabla}}_{1}\mathrm{\ensuremath{\cdot}}{\ensuremath{\nabla}}_{2}}$ 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 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{-}}}$).