Positronium (Ps) is a system consisting of an
electron and its
anti-particle, a
positron, bound together into an
exotic atom, specifically an
onium. Unlike hydrogen, the system has no
protons. 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
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 those for 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 lowest energy orbital state of positronium is 1S, and like with hydrogen, it has a
hyperfine structure arising from the relative orientations of the spins of the electron and the positron.
The
singlet state, 1 S 0, with
antiparallelspins (
S = 0, Ms = 0) is known as para-positronium (p-Ps). It has a mean lifetime of 0.12
ns and decays preferentially into two gamma rays with energy of 511
keV each (in the
center-of-mass frame). 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]
The
triplet states, 3S1, with
parallel spins (S = 1, Ms = −1, 0, 1) are known as ortho-positronium (o-Ps), and have an energy that is approximately 0.001 eV higher than the singlet.[1] These states have 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 ≈10−6.[3]
Ortho-positronium lifetime in vacuum can be calculated approximately as:[1]
However more accurate calculations with corrections to
O(α2) yield a value of 7.040
μs−1 for the decay rate, corresponding to a lifetime of 142 ns.[4][5]
Positronium in the 2S state is
metastable having a lifetime of 1100 ns against
annihilation.[6] The positronium created in such an excited state will quickly cascade down to the ground state, where annihilation will occur more quickly.
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 gamma ray photons recorded from a single annihilation.
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]
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, μ, in the energy equation (see
electron energy levels for a derivation):
where:
qe is the
charge magnitude of the electron (same as the positron),
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
The lowest energy level of positronium (n = 1) is −6.8 eV. The next level is −1.7 eV. The negative sign is a convention that implies a
bound state. Positronium can also be considered by a particular form of the
two-body Dirac equation; Two 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
Janine Shertzer.[9] Their results lead to the discovery of anomalous states.[10][11]
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.[9]
Formation and decay in materials
After a radioactive atom in a material undergoes a
β+ decay (positron emission), the resulting high-energy positron slows down by colliding with atoms, and eventually annihilates with one of the many electrons in the material. It may however first form positronium before the annihilation event. The understanding of this process is of some importance in
positron emission tomography. Approximately:[12][13]
~60% of positrons will directly annihilate with an electron without forming positronium. The annihilation usually results in two gamma rays. In most cases this direct annihilation occurs only after the positron has lost its excess kinetic energy and has thermalized with the material.
~10% of positrons form para-positronium, which then promptly (in ~0.12 ns) decays, usually into two gamma rays.
~30% of positrons form ortho-positronium but then annihilate within a few nanoseconds by 'picking off' another nearby electron with opposing spin. This usually produces two gamma rays. During this time, the very lightweight positronium atom exhibits a strong zero-point motion, that exerts a pressure and is able to push out a tiny nanometer-sized bubble in the medium.
Only ~0.5% of positrons form ortho-positronium that self-decays (usually into three gamma rays). This natural decay rate of ortho-positronium is relatively slow (~140 ns decay lifetime), compared to the aforementioned pick-off process, which is why the three-gamma decay rarely occurs.
History
The Croatian physicist
Stjepan Mohorovičić predicted the existence of positronium in a 1934 article published in Astronomische Nachrichten, in which he called it the "electrum".[15] Other sources incorrectly credit
Carl Anderson as having predicted its existence in 1932 while at
Caltech.[16] It was experimentally discovered by
Martin Deutsch at
MIT in 1951 and became known as positronium.[16] Many subsequent experiments have precisely measured its properties and verified predictions of quantum electrodynamics.
A discrepancy known as the ortho-positronium lifetime puzzle persisted for some time, but was resolved with further calculations and measurements.[17] Measurements were in error because of the lifetime measurement of unthermalised positronium, which was produced at only a small rate. This had yielded lifetimes that were too long. Also calculations using relativistic quantum electrodynamics are difficult, so they had been done to only the first order. Corrections that involved higher orders were then calculated in a non-relativistic quantum electrodynamics.[4]
In 2024, the
AEgIS collaboration at
CERN was the first to cool positronium by laser light, leaving it available for experimental use. The substance was brought to −100 °C (−148 °F) using
laser cooling.[18][19]
Exotic compounds
Molecular bonding was predicted for positronium.[20] Molecules of
positronium hydride (PsH) can be made.[21] Positronium can also form a cyanide and can form bonds with halogens or lithium.[22]
Unlike
muonium, positronium does not have a nucleus analogue, because the electron and the positron have equal masses.[26] Consequently, while muonium tends to behave like a light isotope of hydrogen,[27] positronium shows large differences in size, polarisability, and binding energy from hydrogen.[26]
Natural occurrence
The
events in the early universe leading to
baryon asymmetry predate
the formation of atoms (including exotic varieties such as positronium) by around a third of a million years, so no positronium atoms occurred then.
Likewise, the naturally occurring positrons in the present day result from high-energy interactions such as in
cosmic ray–atmosphere interactions, and so are too hot (thermally energetic) to form electrical bonds before
annihilation.
^
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Cooke, D. A.; Crivelli, P.; Alnis, J.; Antognini, A.; Brown, B.; Friedreich, S.; Gabard, A.; Haensch, T. W.; Kirch, K.; Rubbia, A.; Vrankovic, V. (2015). "Observation of positronium annihilation in the 2S state: towards a new measurement of the 1S-2S transition frequency". Hyperfine Interact. 233 (1–3): 67–73.
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