Question

A gamma ray of exactly 1.022 MeV produces an e⁺ and e^-. The e⁺ and e^- have zero momentum.

a) What is the momentum of the gamma ray?

b) Where does the momentum go to keep conservation of momentum.

Answer 1

This question is very famous concept of quantum mechanics named Pair Production.if you are not aware of this topic i will recommend you to use any book on modern physics in which you are comfortable ,my personal choice is Modern physics by Arthur Beiser.

(a) The momentum of Gamma ray(photon) given by the following formula

. where E=energy of Photon ,C=speed of light = 3*10⁸ m/s

since E=1.022 Mev = 1.022 * 10⁶ * 1.6 *10^^-19 joule {converting Mev in Joule because m/s is SI unit and Joule is also SI unit of energy)

so E=1.6352 * 10^-13 joule.

so put value of E & c in above equation you will get

p=E/c =(1.6352 *10^-13/3*10⁸) = 0.545067 * 10^-21 kg m/s which is momentum of Gamma Ray

(b) For pair production threshold energy of photon is 1.022 Mev which is rest mass energy of electron + positron (antiparticle of electron). basic phenomena behind pair production is energy converted into mass which is supported by the Einstein's famous equation E=mc². As in our case energy of gamma ray is threshold energy so we can get electron and positron with zero momentum. the momentum(energy) of gamma ray is used in production of electron and positron with their rest mass energy. if photon has energy grater than 1.022 mev then electron and positron will produced with rest mass energy + kinetic energy. and momentum of photon distributed among them selves in such way that total momentum of system will conserved.