I cannot find the answer to the above question. I know that para-positronium is created with a probability of $25\%$ and decays into 2 photons, while ortho-positronium is created with a probability of $75\%$ and decays into 3 photons. I also know that ortho-positronium has a way longer life time than para-positronium. This, in my understanding, should not affect the number of decays per time, but just means that the ortho-positronium will decay LATER into three photons. But in the end there should be $75\%$ 3-photon-decays and $25\%$ 2-photon-decays. But in reality 2-photon-decay happens about 300 times more often than 3-photon-decay. What information am I missing? Thank you!
Let us start with the wiki article:
The singlet state with antiparallel spins (S = 0, Ms = 0) is known as para-positronium (p-Ps) and denoted 1S0. 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
It is called conservation of angular momentum. An even number of photons allow to match the S=0 M_s=0 angular momentum. Two photons can add up to spin either 0 or 2 as each carries a spin of 1. The 0 matches the quantum numbers of para positronium.
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 ns and the leading mode of decay is three gamma quanta
Because with three gammas one can match the S=1 angular momentum quantum numbers.
One has to remember that electrons and positrons annihilate to two gammas when not in a bound state as in positronium. The ground state at S=0 has a probability of electrons and positrons to be found at the center of each other and annihilate. The closer to the ground state the shorter the lifetime of the bound system.
I'm afraid the responses so far are either misleading or do not answer the question. In fact in a dense medium a positron when it forms the longer lived ortho-positronium can "pick-up" an electron from an adjacent atom and then decay into 2 x 511 keV photons long before the 3-photon ortho-positronium state would have decayed. Look up "pick-up effect".
Jezstarski is mostly correct,
The para-positronium (p-PS) state ends up being the main mode of annihilation of positronium (PS). Positrons can annihilate in at least eight different ways but once ortho-positronium (o-PS) forms in a void/vacuum, it has additional time to undergo another mode of annihilation.
P-PS annihilates in under 125 picoseconds. O-PS annihilates at or less than 142 nanoseconds (vacuum).
A consequence of this long lived lifetime is "pick-off" (the correct term) annihilation where an opposite spin electron from some surrounding material will annihilate in para-orientation before the ortho-bound electron can collapse and annihilate with the positron.
Source: Member of the Positron Science Lab at UMKC.