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, , with antiparallel spins (S = 0, Ms = 0) is known as para-positronium (p-Ps). It has a mean lifetime of and decays preferentially into two gamma rays with energy of 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 .
Para-positronium lifetime in vacuum is approximately
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. These states have a mean lifetime of , and the leading decay is three gammas. Other modes of decay are negligible; for instance, the five-photons mode has branching ratio of ≈.
Ortho-positronium lifetime in vacuum can be calculated approximately as:
However more accurate calculations with corrections to O(α2) yield a value of −1 for the decay rate, corresponding to a lifetime of .
Positronium in the 2S state is metastable having a lifetime of against annihilation. 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.
Annihilation can proceed via a number of channels, each producing gamma rays with total energy of (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 (electron neutrino–antineutrino pair) and (for other flavour) 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 < for p-Ps and < for o-Ps.
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, μ, in the energy equation (see electron energy levels for a derivation):
where:
is the charge magnitude of the electron (same as the positron), is the Planck constant, is the electric constant (otherwise known as the permittivity of free space), is the reduced mass: where and 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.