Photons Produced in Electron-Positron Annihilation: Energy and Direction

When an electron and a positron come together and annihilate, they typically result in the emission of two photons. This process is a fundamental example in particle physics that illustrates the principles of conservation of energy and momentum. Here, we delve into the key aspects of the energy and direction of these photons.

Energy of the Photons

Conservation of Energy: The first principle at play is the conservation of energy. The total energy in the system before the annihilation is the sum of the rest mass energy and any kinetic energy that the electron and positron possess. This is expressed as:

[ E_{total} E_{rest} E_{kinetic} ]

Where the rest mass energy of an electron or positron is given by:

[ E_{rest} mc^2 ]

Here, ( m 0.511 , text{MeV}/c^2 ), the mass of an electron or positron, and ( c ) is the speed of light. For a scenario where both the electron and positron are at rest, the total energy before annihilation is:

[ E_{total} 2 times 0.511 , text{MeV} 1.022 , text{MeV} ]

Each photon produced will carry energy ( E hf ), where ( h ) is Planck's constant, and ( f ) is the frequency of the photon. When the electron and positron are at rest, the energy of each photon will be approximately 0.511 MeV. This total energy of the photons is the same as the initial rest mass energy of the electron and positron system.

Direction of the Photons

Conservation of Momentum: The second conservation law to consider is the conservation of momentum. At rest before annihilation, the total momentum of the system is zero. To preserve this conservation, the two photons must be emitted in opposite directions. This ensures that the momentum vectors of the two photons cancel each other out, maintaining the overall momentum of the system at zero.

This emission in opposite directions can be understood through the following example. If two detectors, motionless with respect to the center of mass of the electron and positron, register the photons, the detection points of the two photons will form a straight line through the point of annihilation. In other words, the photons will travel in exactly opposite directions from the point of annihilation.

Summary

Energy: Each photon has an energy of 0.511 MeV when the electron and positron are at rest, totaling 1.022 MeV. This energy is the same as the initial rest mass energy of the electron and positron system.

Direction: The two photons are emitted in opposite directions to conserve momentum, resulting in a net momentum of zero if the electron and positron were initially at rest. The detection points of the photons will form a straight line through the annihilation point.

This annihilation process epitomizes the principles of conservation of energy and momentum in particle physics, highlighting the fundamental laws that govern interactions at the subatomic level.