# 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, specifically an onium. The system is unstable: the two particles annihilate each other to predominantly produce two or three gamma-rays, depending on the relative spin states. The orbit and energy levels of the two particles are similar to that of the hydrogen atom (which is a bound state of a proton and an electron). However, because of the reduced mass, the frequencies of the spectral lines are less than half of the corresponding hydrogen lines.

## States

The mass of positronium is 1.022 MeV, which is twice the electron mass minus the binding energy of a few eV. 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, 1
S
0
, with antiparallel spins (S = 0, Ms = 0) is known as para-positronium (p-Ps). It has a mean lifetime of 125 picoseconds and decays preferentially into two gamma rays with energy of 511 keV each (in the center-of-mass frame). By detecting these photons the location of the decay can be found. This process is used in positron-emission tomography. Para-positronium can decay into any even number of photons (2, 4, 6, ...), but the probability quickly decreases with the number: the branching ratio for decay into 4 photons is 1.439(2)×10−6.[1]

Para-positronium lifetime in vacuum is approximately[1]

${\displaystyle t_{0}={\frac {2\hbar }{m_{\mathrm {e} }c^{2}\alpha ^{5}}}=1.244\times 10^{-10}~\mathrm {s} .}$

The triplet state, 3S1, with parallel spins (S = 1, Ms = −1, 0, 1) is known as ortho-positronium (o-Ps). It has a mean lifetime of 142.05±0.02 ns,[2] and the leading decay is three gammas. Other modes of decay are negligible; for instance, the five-photons mode has branching ratio of ~1.0×10−6.[3]

Ortho-positronium lifetime in vacuum can be calculated approximately as:[1]

${\displaystyle t_{1}={\frac {{\frac {1}{2}}9h}{2m_{\mathrm {e} }c^{2}\alpha ^{6}(\pi ^{2}-9)}}=1.386\times 10^{-7}~\mathrm {s} .}$

However more accurate calculations with corrections to order O(α²) yield a value of 7.04 μs⁻¹ for the decay rate, corresponding to a lifetime of 1.42×10−7 s.[4][5]

Positronium in the 2S state is metastable having a lifetime of 1.1 µs against annihilation.[citation needed] The positronium created in such an excited state will quickly cascade down to the ground state, where annihilation will occur more quickly.

### Measurements

Measurements of these lifetimes and energy levels have been used in precision tests of quantum electrodynamics, confirming quantum electrodynamics (QED) predictions to high precision.[1][6][7] Annihilation can proceed via a number of channels, each producing gamma rays with total energy of 1022 keV (sum of the electron and positron mass-energy), usually 2 or 3, with up to 5 recorded.

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 other flavour)[3] in predictions based on the Standard Model, but it can be increased by non-standard neutrino properties, like 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 for p-Ps and <4.2×10−7 for o-Ps.[2]

## Energy levels

While precise calculation of positronium energy levels uses the Bethe–Salpeter equation or the Breit equation, the similarity between positronium and hydrogen allows a rough estimate. In this approximation, the energy levels are different because of a different effective mass, m*, in the energy equation (see electron energy levels for a derivation):

${\displaystyle E_{n}=-{\frac {\mu q_{\mathrm {e} }^{4}}{8h^{2}\varepsilon _{0}^{2}}}{\frac {1}{n^{2}}},}$

where:

qe is the charge magnitude of the electron (same as the positron),
h is Planck's constant,
ε0 is the electric constant (otherwise known as the permittivity of free space),
μ is the reduced mass:
${\displaystyle \mu ={\frac {m_{\mathrm {e} }m_{\mathrm {p} }}{m_{\mathrm {e} }+m_{\mathrm {p} }}}={\frac {m_{\mathrm {e} }^{2}}{2m_{\mathrm {e} }}}={\frac {m_{\mathrm {e} }}{2}},}$
where me and mp are, respectively, the mass of the electron and the positron (which are the same by definition as antiparticles).

Thus, for positronium, its reduced mass only differs from 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

${\displaystyle E_{n}=-{\frac {1}{2}}{\frac {m_{\mathrm {e} }q_{\mathrm {e} }^{4}}{8h^{2}\varepsilon _{0}^{2}}}{\frac {1}{n^{2}}}={\frac {-6.8~\mathrm {eV} }{n^{2}}}.}$

The lowest energy level of positronium (n = 1) is −6.8 electronvolts (eV). The next level is −1.7 eV. The negative sign implies a bound state. Positronium can also be considered by a particular form of the two-body Dirac equation; Two point particles with a Coulomb interaction can be exactly separated in the (relativistic) center-of-momentum frame and the resulting ground-state energy has been obtained very accurately using finite element methods of J. Shertzer.[8] The Dirac equation whose Hamiltonian comprises two Dirac particles and a static Coulomb potential is not relativistically invariant. But if one adds the 1/c2n (or α2n, where α is the fine-structure constant) terms, where n = 1,2…, then the result is relativistically invariant. Only the leading term is included. The α2 contribution is the Breit term; workers rarely go to α4 because at α3 one has the Lamb shift, which requires quantum electrodynamics.[8]

## History

The Positronium Beam at University College London, a lab used to study the properties of positronium

Croatian scientist Stjepan Mohorovičić predicted the existence of positronium in a 1934 article published in Astronomische Nachrichten, in which he called it the "electrum".[9] Other sources credit Carl Anderson as having predicted its existence in 1932 while at Caltech.[10] It was experimentally discovered by Martin Deutsch at MIT in 1951 and became known as positronium.[10] Many subsequent experiments have precisely measured its properties and verified predictions of QED. There was a discrepancy known as the ortho-positronium lifetime puzzle that persisted for some time, but was eventually resolved with further calculations and measurements.[11] Measurements were in error because of the lifetime measurement of unthermalised positronium, which was only produced at a small rate. This had yielded lifetimes that were too long. Also calculations using relativistic QED are difficult to perform, so they had been done to only the first order. Corrections that involved higher orders were then calculated in a non-relativistic QED.[4]

## Exotic compounds

Molecular bonding was predicted for positronium.[12] Molecules of positronium hydride (PsH) can be made.[13] Positronium can also form a cyanide and can form bonds with halogens or lithium.[14]

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, Riverside.[15][16]

## 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.[17]

## 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 b 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). Levchenko, B.B.; Savrin, V.I., eds. Decays of Positronium. Proceedings of the International Workshop on High Energy Physics and Quantum Field Theory (QFTHEP). 14. MSU-Press. pp. 538–544. arXiv:hep-ph/9911410. Bibcode:1999hep.ph...11410C.
4. ^ a b Kataoka, Y.; Asai, S.; Kobayashi, t. (9 September 2008). "First Test of O(α2) Correction of the Orthopositronium Decay Rate" (PDF). International Center for Elementary Particle Physics.
5. ^ Adkins, G. S.; Fell, R. N.; Sapirstein, J. (29 May 2000). "Order α2 Corrections to the Decay Rate of Orthopositronium". Physical Review Letters. 84 (22): 5086–5089. arXiv:hep-ph/0003028. Bibcode:2000PhRvL..84.5086A. doi:10.1103/PhysRevLett.84.5086.
6. ^ 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.
7. ^ 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.
8. ^ a b 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.
9. ^ Mohorovičić, S. (1934). "Möglichkeit neuer Elemente und ihre Bedeutung für die Astrophysik". Astronomische Nachrichten. 253 (4): 94. Bibcode:1934AN....253...93M. doi:10.1002/asna.19342530402.
10. ^ a b "Martin Deutsch, MIT physicist who discovered positronium, dies at 85" (Press release). MIT. 2002.
11. ^ Dumé, Belle (May 23, 2003). "Positronium puzzle is solved". Physics World.
12. ^ Usukura, J.; Varga, K.; Suzuki, Y. (1998). "Signature of the existence of the positronium molecule". arXiv:physics/9804023v1 [physics.atom-ph].
13. ^ ""Out of This World" Chemical Compound Observed" (PDF). p. 9.
14. ^ Saito, Shiro L. (2000). "Is Positronium Hydride Atom or Molecule?". Nuclear Instruments and Methods in Physics Research B. 171: 60–66. Bibcode:2000NIMPB.171...60S. doi:10.1016/s0168-583x(00)00005-7.
15. ^ 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.
16. ^ "Molecules of positronium observed in the lab for the first time". Physorg.com. Retrieved 2007-09-07.
17. ^ 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.