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I'm an engineer and have been reading about PET scanners and how they rely on the fact that a positron-electron annihilation will cause two photons to be emitted at 180 degrees from each other. After a bit of research I am left wondering on a couple of points.

From Electron–positron annihilation - wiki:

Conservation of energy and linear momentum forbid the creation of only one photon.

Is this simply because emitting one photon would produce a force in one direction without an equal and opposite force?

In the most common case, two photons are created, each with energy equal to the rest energy of the electron or positron.

Does it follow then that there is a (lesser) probability that 3 photons can be emitted at equal energy and equal angle of 120 degrees?

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Yes, the requirement for at least two photons is because a single photon would violate conservation of momentum. See my answer to Particle anti-particle annihilation and photon production for a (very simple!) proof of this.

Annihilation can produce more than two photons. In fact the decay of ortho-positronium to two photons is forbidden, and it (mostly) decays into three photons. This is a bit of a special case though as it's due to conservation of angular momentum in a bound state.

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The LEP accelerator and the experiments running on it were e+e- annihilations, the energies on the Z resonance and also up to 210GeV . These annihilations had a large variation in the particles detected and a great number of decay channels published in numerous papers.

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