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Journal of Physics B Atomic Molecular and Optical Physics

Atomic few-body systems with muonium

Alexei M. Frolov

Published Mar 30, 2017• 5 citations
Publisher

Abstract

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

DOI
10.1088/1361-6455/aa6a40

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The European Physical Journal D•Apr 1, 2025•1 citation
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
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.
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.
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{-}}}$).
The hydrogen atom is the simplest of all atoms consisting of proton of charge e and the electron of charge -e. It is a two particle system. A two-body problem can always be reduced to a one-body problem by choosing the centre of mass coordinate and the relative coordinates. The existence of degenerate energy eigenvalues implies that linear combination of the corresponding eigenfunctions are solutions of the wave equation with the same energy. The isotopes of atomic hydrogen, deuterium and tritium also provide examples of hydrogenic systems. Since the mass of deuterium is md˜2mp and that of tritium is mt˜3mp, the reduced masses of them will be slightly different from that of hydrogen. In addition to deuterium and tritium, there exists other hydrogenic systems in which the role of proton is played by another positive particle.
Various energetic, geometric, and other properties for the ground and first vibrationally excited S states (L=0) of Coulomb three-body systems having unit charges are obtained by means of accurate exponential wave functions. The mass dependence of the properties for these systems has a relatively smooth form for the ground states. The results presented should be useful as a basis for further investigation of these three-body systems. The prethreshold (weakly bound) exotic nonsymmetric systems (d+{\mathit{d}}^{+}d+t+{\mathit{t}}^{+}t+${\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+${\mathit{p}}^{\mathrm{\ensuremath{-}}}$ ).
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