Positronium¶
Positronium (Ps) is the pairing of an electron and positron into a bound state analogous to a hydrogen atom (see Wikipedia). This occurs when the relative motion between the positron and electron is small [JR76] (\(\beta \rightarrow 0\), \(\beta=v/c\)).
The mean lifetime of Ps is very short, on the order of nanoseconds. It is most likely to produce 2 or 3 gamma-ray photons depending on if it is para-Ps (2 photons of 511 keV each) or ortho-Ps (3 photons). The Ps types are dependent on the spin state of the electron and so occur in a 1/4 (para-Ps, S = 0, Ms = 0) to 3/4 (ortho-Ps, S = 1, Ms = −1, 0, 1) ratio. The triplet emission produces a continuum of photon energies described in [OrePowell49].
In supernovae, Ps can form when \(\beta\)-decay occurs in the radioactive material produced as part of the explosion. The amount of Ps formed compared to immediate 2 photon annihilation in supernovae is unknown. In the galaxy, the value is determined to be \(0.94\pm0.04\) [MilneHungerfordFryer+04]. Gamma-ray transport codes typically assume either no Ps formation (and thus all \(\beta\)-decay releases 2 511 keV photons) or 100% Ps formation (and thus photons are released in pairs or triplets in a 1/4 to 3/4 ratio).
According to [JR76] the density-dependent reciprocal lifetime \(\frac{1}{\tau_2}\) (called a “cross-section”) of para-Ps is
where \(r_0\) is the classical electron radius and \(\rho\) is the electron number density. Note that the units here appear to be inverse length rather than the expected inverse time.
For ortho-Ps, the cross-section is
where \(\alpha\) is the fine structure constant. This makes the ortho-Ps cross-section (and associated lifetime) roughly 300 times smaller than that of para-Ps.