The probability of annihilation of positrons without emission of radiation

There are two distinct processes by means of which a positron may be annihilated and the excess energy liberated in the form of radiation. These are: (1) Annihilation by a free electron. Here the energy must reappear in the form of at least two quanta to satisfy the conservation laws. (2) Annihilation by a bound electron. In this case the energy may appear in the form of a single quantum since the nucleus of the atom in which the electron is bound is available to take up excess momentum.