Question

When an electron and positron annihilate, both their masses are destroyed, creating two equal energy photons to preserve momentum.

(a) Confirm that the annihilation equation \({e^ + } + {e^ - } \to \gamma + \gamma \) conserves charge, electron family number, and total number of nucleons. To do this, identify the values of each before and after the annihilation.

(b) Find the energy of eachγ ray, assuming the electron and positron are initially nearly at rest.

(c) Explain why the twoγ rays travel in exactly opposite directions if the centre of mass of the electron-positron system is initially at rest.

Short answer

(a) It is confirmed that the annihilation equation \[{{\rm{e}}^{\rm{ + }}}{\rm{ + }}{{\rm{e}}^{\rm{ - }}} \to \gamma {\rm{ + }}\gamma \]conserves charge, electron family number, and total number of nucleons as before and after reaction the value of each quantity is zero.

(b) The energy of each γ ray, assuming the electron and positron are initially nearly at rest is 0.511 MeV.

(c) As the centre of mass of the electron-positron system is initially at rest, so the no external force is there and momentum is also conserved. This happens when the γ rays travel in exactly opposite directions.

Step-by-step solution

Concept Introduction

The product of a particle's mass and velocity is called momentum. Momentum is a vector quantity in the sense that it has both a magnitude and a direction.

Nuclear energy is a type of energy that is emitted from the nucleus, which is made up of protons and neutrons and is the core of all atoms. This type of energy can be generated in one of two ways: fission (when atom nuclei split into several pieces) or fusion (when nuclei fuse together).

Annihilation Equation

(a)

The initial charge of electron in- 1 and positron is + 1 hence, total charge before reaction is 0. Charge of γ is zero, hence, total charge after reaction is zero. Hence, total charge is conserved.

The electron number of electron is + 1 and positron is - 1 hence, total number is zero. The electronfamily number of γ is zero, hence, after reaction total electron family number is zero. Hence, electron family number is conserved before and after reaction.

Total number of nucleons before reaction is zero and after reaction is zero. Hence, total number ofnucleons is conserved.

Therefore, the annihilation equation is conserved.

Energy of the γ  Ray

(b)

Since the electrons are nearly at rest before annihilation, the rest mass energy of electron and positron will be converted to the energy of γ ray.

The rest mass energy of both electron and positron is 0.511 MeV. Hence, using the conservation of mass and energy, we can say that the energy of each γ photon is 0.511 MeV.

Therefore, the value for energy is obtained as 0.511 MeV.

Direction of γ Rays

(c)

Since centre of mass of the electron was at rest before annihilation, the total momentum of the electron - positron system was zero. Now, since, there is no external force, the momentum remains conserved after annihilation. This can only happen if the γ photons travel exactly in opposite direction, then total momentum will be zero and hence momentum will be conserved before and after annihilation.

Therefore, the γ rays travel in opposite directions.