Skip to main content

Mathematics for Physics

156 Annihilation

The process in which a particle and its antiparticle collide and convert their mass into electromagnetic energy, typically gamma-ray photons.

Total mass-to-energy transmutation. When a particle meets its antiparticle, their rest masses can vanish and reappear as the energy of massless photons. An antiparticle is the counterpart of a particle with the same mass and opposite charge. This principle is used to compute the photon energy from \(E=mc^{2}\).

Kinematic forbiddenness of one-photon annihilation. Annihilation in vacuum cannot produce a single photon, because energy and momentum cannot both be conserved. A photon is a massless quantum of electromagnetic radiation that travels at \(c\) and carries \(p=\dfrac{E}{c}\). This principle is used to show that at least two photons must emerge.

Back-to-back emission in the center-of-momentum frame. If the pair annihilates at rest, the two photons leave in opposite directions with equal energy. The center-of-momentum frame is the frame in which the total spatial momentum is zero. This principle is used to detect annihilation as a pair of opposite gamma rays.

Conservation of total relativistic energy. Rest mass of the pair is not conserved, while the total relativistic energy is conserved. Total relativistic energy is rest energy plus kinetic energy. This principle is used to equate the pair’s energy to the sum of the photon energies.

156.1 References

  1. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — source for the heading explanation.
  2. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — pair annihilation into photons; two-photon kinematics; back-to-back emission.
  3. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — one-photon annihilation forbidden; energy conservation.
  4. Shankar, R. Fundamentals of Physics I. Yale University Press, 2019. — rest mass into radiation; total energy conserved.
  5. Carroll, S. M. Spacetime and Geometry. Cambridge University Press. — mass-energy transmutation in annihilation.