Energy of Photons in Electron-Positron Annihilation and Its Dependence on Kinetic Energy

When an electron and a positron annihilate each other, they convert their rest mass energy into a pair of high-energy photons. This process is governed by the principles of special relativity and quantum mechanics, leading to fascinating insights into the nature of energy and particle interactions.

Rest Mass Energy of Electron and Positron

The energy of a photon produced by the annihilation of an electron and a positron is primarily determined by the rest mass energy of the particles involved. The rest mass energy of an electron (or positron) is given by Einstein's famous equation:

E mc2

where:

m mass of the electron or positron (approximately 9.11 x 10-31 kg) c speed of light in a vacuum (approximately 3.00 x 108 m/s)

Calculating the rest mass energy of an electron or positron:

Erest 9.11 x 10-31 kg x (3.00 x 108 m/s)2 approx; 8.19 x 10-14 J

Approximately, this is equivalent to 0.511 MeV ( megaelectronvolts).

Total Energy of Annihilation

During a typical electron-positron annihilation event, two photons are produced, each carrying energy equal to the rest mass energy of the electron and positron combined:

E_{total} 2 x 0.511 MeV 1.022 MeV

Dependence on Kinetic Energy

If the electron and positron have additional kinetic energy due to their motion before annihilation, this kinetic energy contributes to the total energy of the system. The total energy of the annihilation can be expressed as:

E_{total} 2 x mc2 K.E.

Where K.E. is the total kinetic energy of the electron and positron. Therefore, the energy of the photons produced will also include this kinetic energy, meaning that the total energy of the two photons will be:

E_{photons} 1.022 MeV K.E.

Doppler Broadening of Emission Photons

When the positronium (a bound state of an electron and positron) has kinetic energy, the emitted photons experience a phenomenon known as Doppler broadening. This effect modifies the energy distribution of the emitted photons depending on the motion of the positronium. In the case of 2-photon emission, the energy distribution will be constant within the kinematic limits:

frac{gamma M}{1 - beta / 2}

Here, M is the positronium rest mass, which is very close to the sum of the masses of the electron and positron. (gamma) is the Lorentz factor, and (beta) is the velocity of the system divided by the speed of light c.

Conclusion

The energy of the photons produced by the annihilation of an electron and a positron is at least 1.022 MeV due to their rest mass. If the particles have kinetic energy, the energy of the photons will increase accordingly, depending on the total kinetic energy of the approaching particles. Thus, the energy of the photons does depend on the kinetic energy of the approaching particles.