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I understand that gamma-rays can produce positron-electron pairs and positrons and electrons can annihilate to two gamma-rays.

My question is: If a gamma-ray photon creates a positron-electron pair, will the two particles "instantly" annihilate again because of their positive-negative-attraction and create two photons (each photon with half the energy of the original gamma-ray photon)? Or do they travel apart from each other (and then do stuff on their own)?

For the first case: Is this occuring until such low energy photons are created which then can't produce matter-antimatter pairs anymore?

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Electrons, positrons and gammas are in the realm of quantum mechanics.

pair creation

The particles are described by energy-momentum four vectors, their mass being the "length" of the four vector.

A photon by itself, because of its zero mass, no matter how much energy it has cannot decay into an electron positron pair because of energy and momentum conservation laws. In the center of mass of the e+e- the minimum energy they have is twice the mass of the electron, whereas a photon does not have a center of mass. So it is necessary that the photon interacts with a field of another particle so that energy and momentum are conserved. The minimum energy of the photon should be a bit above the sum of the the two masses, to include the momentum and energy carried away by the nucleus Z .

Generally, once created, the electron positron in their center of mass move in opposite directions and cannot recombine as the attraction of the opposite charges is not strong enough. Only if they are produced at rest this phenomenon can happen.

An electron hitting a positron may annihilate in two gamma rays, of equal and opposite fourvectors in the center of mass of the system. For high energies, as happened in the LEP collider a number of other alternative particle creations dominate.

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