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arXiv:1611.03124

Quasi-atomic three- and four-body systems with muonium

Alexei M. Frolov

Published Nov 9, 2016• 0 citations
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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 $Δ\approx$ 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 atoms.

DOI
10.48550/arxiv.1611.03124

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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
The $({e}^{\ensuremath{-}},{e}^{+})$-pair annihilation is considered in the positron-helium e+[He(23S)]{e}^{+}[\mathrm{He}(2\phantom{\rule{0.2em}{0ex}}^{3}S)]e+[He(23S)] four-body ion, in which the positron forms a bound state with the helium atom in its lowest triplet state [i.e., in the 23S(L=0)2\phantom{\rule{0.2em}{0ex}}^{3}S\phantom{\rule{0.3em}{0ex}}(L=0)23S(L=0) state]. A number of bound-state properties of this system have been determined for the e+[He(23S)]{e}^{+}[\mathrm{He}(2\phantom{\rule{0.2em}{0ex}}^{3}S)]e+[He(23S)] ions, where the $^{\ensuremath{\infty}}\mathrm{He}$, 4He^{4}\mathrm{He}4He, and 3He^{3}\mathrm{He}3He helium nuclei are considered. Some applications related to the positron annihilation in the e+[He(23S)]{e}^{+}[\mathrm{He}(2\phantom{\rule{0.2em}{0ex}}^{3}S)]e+[He(23S)] ion are also discussed.
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+${\mathit{p}}^{\mathrm{\ensuremath{-}}}$ ).
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{-}}}$).
In the framework of nonrelativistic quantum mechanics we derive a necessary condition for four Coulomb charges (m1(+), m2(-), m3(+), m4(-)), where all masses are assumed finite, to form the stable system. The obtained stability condition is physical and is expressed through the required minimal ratio of Jacobi masses. In particular, this provides the rigorous proof that hydrogen-antihydrogen and muonium-antimuonium molecules and hydrogen-positron-muon systems are unstable. It also proves that replacing hydrogen in the hydrogen-antihydrogen molecule with its heavier isotopes does not make the molecule stable. These are the first rigorous results on the instability of these systems.
Bound state properties, positron annihilation and hyperfine structure splittings in the four-body positron-helium e+[3e^{+} [^{3}e+[3He(23Se2^{3}S_e23Se​)] and e+[4e^{+} [^{4}e+[4He(23Se2^{3}S_e23Se​)] ions