21.5 Pair Annihilation

Definition

Pair annihilation is the process in which a particle and its antiparticle collide and completely convert their combined mass-energy into electromagnetic radiation (gamma-ray photons). The most common example is the annihilation of an electron ($e^{-}$) and a positron ($e^{+}$):

$e^{-}+e^{+}\to \gamma +\gamma$

Pair annihilation is the reverse process of pair production: in pair production, a photon creates a particle–antiparticle pair (energy → matter), while in annihilation, a particle–antiparticle pair converts into photons (matter → energy).

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Why Two Photons Are Produced

Two photons are always produced (rather than one) to satisfy the Law of Conservation of Momentum.

• When an electron and positron are nearly at rest, their net momentum is approximately zero.
• A single photon always carries momentum $p=h/\lambda \ne 0$, so one photon cannot conserve zero net momentum.
• Two photons travelling in exactly opposite directions have equal and opposite momenta, giving a net momentum of zero — satisfying conservation of momentum.

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Energy of the Gamma-Ray Photons

The energy released comes from the rest mass energy of both particles, as given by Einstein's mass-energy equivalence:

$E=\Delta m{c}^2$

The rest mass energy of one electron (or positron) is:

$m_0{c}^2=(9.11\times 10^{-31})(3\times 10^8{)}^2\approx 8.19\times 10^{-14}\text{ J}=0.511\text{ MeV}$

When both particles are at rest, the total energy available is:

$E_{\text{total}}=2m_0{c}^2=2\times 0.511=1.022\text{ MeV}$

This energy is shared equally between the two photons, so each photon carries:

$E_{\gamma }=0.511\text{ MeV}$

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Conservation Laws in Pair Annihilation

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Mass-Energy Equivalence

Pair annihilation is a direct demonstration of Einstein's equation:

$E=\Delta m{c}^2$

The entire rest mass of the electron-positron pair is converted into photon energy. This is one of the most complete conversions of mass into energy known in physics.

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Application: PET Scanning

Pair annihilation is the physical basis of Positron Emission Tomography (PET):

• A radioactive tracer emitting positrons is introduced into the body.
• Each positron quickly annihilates with a nearby electron, producing two $0.511\text{ MeV}$ gamma-ray photons travelling in opposite directions.
• Detectors surrounding the patient detect these coincident photons to reconstruct a 3D image of metabolic activity.

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