# Positronium

An electron and positron orbiting around their common centre of mass. This is a bound quantum state known as positronium.

Positronium (Ps) is a system consisting of an electron and its anti-particle, a positron, bound together into an "exotic atom". The system is unstable: the two particles annihilate each other to produce two gamma ray photons after an average lifetime of 125 picoseconds or three gamma ray photons after 142 nanoseconds in vacuum, depending on the relative spin states of the positron and electron. The orbit of the two particles and the set of energy levels is similar to that of the hydrogen atom (electron and proton). However, because of the reduced mass, the frequencies associated with the spectral lines are less than half of those of the corresponding hydrogen lines.

## States

The ground state of positronium, like that of hydrogen, has two possible configurations depending on the relative orientations of the spins of the electron and the positron.

The singlet state with antiparallel spins (S = 0, Ms = 0) is known as para-positronium (p-Ps) and denoted 1S
0
. It has a mean lifetime of 125 picoseconds and decays preferentially into two gamma quanta with energy of 511 keV each (in the center of mass frame). Detection of these photons allows for the reconstruction of the vertex of the decay and is used in the positron emission tomography. Para-positronium can decay into any even number of photons (2, 4, 6, ...), but the probability quickly decreases as the number increases: the branching ratio for decay into 4 photons is 1.439(2)×10−6.1

$t_{0} = \frac{2 \hbar}{m_e c^2 \alpha^5} = 1.244 \times 10^{-10} \; \text{s}$

The triplet state with parallel spins (S = 1, Ms = −1, 0, 1) is known as ortho-positronium (o-Ps) and denoted 3S1. The triplet state in vacuum has a mean lifetime of 142.05±0.02 ns2 and the leading mode of decay is three gamma quanta. Other modes of decay are negligible; for instance, the five photons mode has branching ratio of ~1.0×10−6.3

$t_{1} = \frac{\frac{1}{2} 9 h}{2 m_e c^2 \alpha^6 (\pi^2 - 9)} = 1.386 \times 10^{-7} \; \text{s}$

Positronium in the 2S state is metastable having a lifetime of 1.1 µs against annihilation.citation needed If the positronium is created in such an excited state then it will quickly cascade down to the ground state where annihilation will occur more quickly. Measurements of these lifetimes, as well as of the positronium energy levels, have been used in precision tests of quantum electrodynamics.14

Annihilation can proceed via a number of channels each producing one or more gamma rays. The gamma rays are produced with a total energy of 1,022 keV (since each of the annihilating particles have mass of 511 keV/c2), the most probable annihilation channels produce two or three photons, depending on the relative spin configuration of the electron and positron. A single photon decay is only possible if another body (e.g. an electron) is in the vicinity of the annihilating positronium to which some of the energy from the annihilation event may be transferred. Up to five annihilation gamma rays have been observed in laboratory experiments,5 confirming the predictions of quantum electrodynamics to very high order.

The annihilation into a neutrino–antineutrino pair is also possible, but the probability is predicted to be negligible. The branching ratio for o-Ps decay for this channel is 6.2×10−18 (electron neutrino–antineutrino pair) and 9.5×10−21 (for each non-electron flavour)3 in predictions based on the Standard Model, but it can be increased by non-standard neutrino properties, like mass or relatively high magnetic moment. The experimental upper limits on branching ratio for this decay (as well as for a decay into any "invisible" particles) are: <4.3×10−7 (p-Ps) and <4.2×10−7 (o-Ps).6

## Energy levels

While precise calculation of positronium energy levels uses the Bethe–Salpeter equation, the similarity between positronium and hydrogen allows for a rough estimate. In this approximation, the energy levels are different between the two because of a different value for the mass, m*, used in the energy equation

$E_n = - \frac{\mu q_e^4}{8 h^2 \epsilon_{0}^2} \frac{1}{n^2} \,.$
See Electron energy levels for a derivation.
$q_e$ is the charge magnitude of the electron (same as the positron)
$h$ is Planck's constant
$\epsilon_{0}$ is the electric constant (otherwise known as the permittivity of free space) and finally
$\mu$ is the reduced mass

The reduced mass in this case is

$\mu = {{m_e m_p} \over {m_e + m_p}} = \frac{m_e^2}{2m_e} = \frac{m_e}{2},$
where
$m_e$ and $m_p$ are, respectively, the mass of the electron and the positron—which are the same by definition of particles and antiparticles.

Thus, for positronium, its reduced mass only differs from the rest mass of the electron by a factor of 2. This causes the energy levels to also roughly be half of what they are for the hydrogen atom.

So finally, the energy levels of positronium are given by

$E_n = - \frac{1}{2} \frac{m_e q_e^4}{8 h^2 \epsilon_{0}^2} \frac{1}{n^2} = \frac{-6.8 \ \mathrm{eV}}{n^2} \,.$

The lowest energy level of positronium (n = 1) is −6.8 electron volts (eV). The next lowest energy level (n = 2) is −1.7 eV. The negative sign implies a bound state. We also note that a two-body Dirac equation composed of a Dirac operator for each of the two point particles interacting via the Coulomb interaction can be exactly separated in the (relativistic) center of momentum frame and the resulting ground state eigenvalue has been obtained very accurately using the Finite element methods of J. Shertzer. The Dirac equation whose Hamiltonian comprises two Dirac particles and a static Coulomb potential is not relativistically invariant. But if one adds the 1/c^(2n) (or alpha^2n where alpha is the fine structure coefficient which is about 1/137) contributions where n=1,2,3, ... to the Hamiltonian then the result is relativistically invariant in the limit. So only the lead term in the Hamiltonian is included. The next 1/c^2 (or alpha^2) contribution are the Breit terms: workers rarely go to 1/c^4 (or alpha^4) because at alpha^3*log(alpha), one has the Lamb shift (which is a detailed calculation needing quantum electrodynamics).7

## Prediction and discovery

Croatian scientist Stjepan Mohorovičić predicted the existence of positronium in a 1934 paper published in Astronomische Nachrichten, in which he called the substance "electrum".8 Other sources credit Carl Anderson as having predicted its existence in 1932 while at Caltech.9 It was experimentally discovered by Martin Deutsch at MIT in 1951, and became known as positronium.9

## Observation of di-positronium molecules

The first observation of di-positronium molecules—molecules consisting of two positronium atoms—was reported on 12 September 2007 by David Cassidy and Allen Mills from University of California at Riverside.1011

## Natural occurrence

Positronium in high energy states has been predicted to be the dominant form of atomic matter in the universe in the far future, if proton decay is a reality.12

## References

1. ^ a b c d Karshenboim, Savely G. (2003). "Precision Study of Positronium: Testing Bound State QED Theory". International Journal of Modern Physics A [Particles and Fields; Gravitation; Cosmology; Nuclear Physics] 19 (23): 3879–3896. arXiv:hep-ph/0310099. Bibcode:2004IJMPA..19.3879K. doi:10.1142/S0217751X04020142.
2. ^ A. Badertscher et al. (2007). "An Improved Limit on Invisible Decays of Positronium". Physical Review D 75 (3): 032004. arXiv:hep-ex/0609059. Bibcode:2007PhRvD..75c2004B. doi:10.1103/PhysRevD.75.032004.
3. ^ a b Czarnecki, Andrzej; Karshenboim, Savely G. (2000). "Decays of Positronium". In Levchenko, B.B.; Savrin, V.I. Proceedings of the International Workshop on High Energy Physics and Quantum Field Theory (QFTHEP) (Moscow: MSU-Press) 14 (99): 538–544. arXiv:hep-ph/9911410. Bibcode:1999hep.ph...11410C.
4. ^ Rubbia, A. (2004). "Positronium as a probe for new physics beyond the standard model". International Journal of Modern Physics A [Particles and Fields; Gravitation; Cosmology; Nuclear Physics] 19 (23): 3961–3985. arXiv:hep-ph/0402151. Bibcode:2004IJMPA..19.3961R. doi:10.1142/S0217751X0402021X.
5. ^ Vetter, P.A.; Freedman, S.J. (2002). "Branching-ratio measurements of multiphoton decays of positronium". Physical Review A 66 (5): 052505. Bibcode:2002PhRvA..66e2505V. doi:10.1103/PhysRevA.66.052505.
6. ^ Badertscher, A.; et al. (2007). "Improved limit on invisible decays of positronium". Physical Review D 75 (3): 032004–1–10. arXiv:hep-ex/0609059. Bibcode:2007PhRvD..75c2004B. doi:10.1103/PhysRevD.75.032004.
7. ^ Scott, T.C.; Shertzer, J.; Moore, R.A. (1992). "Accurate finite element solutions of the two-body Dirac equation". Physical Review A 45 (7): 4393–4398. Bibcode:1992PhRvA..45.4393S. doi:10.1103/PhysRevA.45.4393. PMID 9907514.
8. ^ Mohorovičić, S. (1934). "Möglichkeit neuer Elemente und ihre Bedeutung für die Astrophysik". Astronomische Nachrichten 253 (4): 94. doi:10.1002/asna.19342530402.
9. ^ a b "Martin Deutsch, MIT physicist who discovered positronium, dies at 85" (Press release). MIT. 2002.
10. ^ Cassidy, D.B.; Mills, A.P. (Jr.) (2007). "The production of molecular positronium". Nature 449 (7159): 195–197. Bibcode:2007Natur.449..195C. doi:10.1038/nature06094. PMID 17851519. Lay summary.
11. ^ "Molecules of positronium observed in the lab for the first time". Physorg.com. Retrieved 2007-09-07.
12. ^ A dying universe: the long-term fate and evolution of astrophysical objects, Fred C. Adams and Gregory Laughlin, Reviews of Modern Physics 69, #2 (April 1997), pp. 337–372. Bibcode1997RvMP...69..337A. doi:10.1103/RevModPhys.69.337 arXiv:astro-ph/9701131.

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