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Physical Review Letters

Positronium negative ion: Molecule or atom?

Jan Michael Rost, D. Wintgen

Published Oct 26, 1992• 36 citations

Abstract

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 ${\mathrm{H}}_{2}^{+}$ quantum numbers by treating the interelectronic axis of ${\mathrm{Ps}}^{\mathrm{\ensuremath{-}}}$ as an adiabatic parameter. We report and interpret the existence of $^{1}$S 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}${\mathit{Z}}_{\mathit{A}}$/${\mathit{Z}}_{\mathit{B}}$\ensuremath{\Vert}=1 and masses ${\mathit{m}}_{\mathit{A}}$/${\mathit{m}}_{\mathit{B}}$\ensuremath{\ge}1.

DOI: 10.1103/physrevlett.69.2499

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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

{\mathrm{H}}_{2}^{+}

quantum numbers by treating the interelectronic axis of

{\mathrm{Ps}}^{\mathrm{\ensuremath{-}}}

as an adiabatic parameter. We report and interpret the existence of

^{1}

S 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}

{\mathit{Z}}_{\mathit{A}}

/

{\mathit{Z}}_{\mathit{B}}

\ensuremath{\Vert}=1 and masses

{\mathit{m}}_{\mathit{A}}

/

{\mathit{m}}_{\mathit{B}}

\ensuremath{\ge}1.

An alternative approach is introduced for adiabatic and nonadiabatic calculations in the ABA Coulomb systems from

{\mathrm{H}}_{2}^{+}

to the positronium negative ion ${\mathrm{Ps}}^{\mathrm{\ensuremath{-}}}$ (${\mathit{e}}^{\mathrm{\ensuremath{-}}}$

{\mathit{e}}^{+}

${\mathit{e}}^{\mathrm{\ensuremath{-}}}$) so that they are treated as molecular systems. The adiabatic vibrational solutions point out the most relevant ``nuclear'' configurations that should appear in the construction of the nonadiabatic wave function and reveal the importance of the AA vibrations at this stage. Applications provide evidence for the separability of the ground state of ${\mathrm{Ps}}^{\mathrm{\ensuremath{-}}}$ and give energy values for six S bound states of ${\mathrm{\ensuremath{\mu}}}^{+}$${\mathit{e}}^{\mathrm{\ensuremath{-}}}$${\mathrm{\ensuremath{\mu}}}^{+}$ that seem not to be reported so far.

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.

An adiabatic treatment in hyperspherical coordinates is used to predict the existence of resonant structures in ${\mathrm{Ps}}^{\ensuremath{-}}$ (Ps denotes positronium) photodetachment. This highly correlated system displays a weak series of narrow Feshbach resonances lying just below the

\mathrm{Ps}(n=2)

threshold, while a single strong shape resonance dominates the photodetachment spectrum above this threshold as in ${\mathrm{H}}^{\ensuremath{-}}$. Use of a "prediagonalized" representation of hyperspherical harmonics dramatically improves the convergence of the adiabatic calculation.

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}

.