Q1) consider a stationary nucleus of mass M. show that the minimum energy the photon must have to create an electron-positron pair in the presence of stationary nucleus is 2mc^2(1+ m/M) where m is the rest mass of the electron.
To determine the minimum energy required to create an electron-positron pair in the presence of a stationary nucleus, we'll need to apply conservation laws and analyze the energy and momentum of the system.
The process of creating an electron-positron pair involves the interaction of a photon with the nucleus. This interaction must conserve energy and momentum. We'll consider the center of mass frame of the system so that the initial and final total momentum is zero.
Let's break down the problem step by step:
Step 1: Conservation of Energy
In this process, the photon energy is converted into the rest mass energy of the electron and positron. The minimum energy required is when all the photon energy is used to create the electron and positron at rest. The rest mass energy of an electron or positron is given by E = mc^2, where m is the rest mass of the electron.
So, the initial energy of the system is the energy of the photon, E_photon, and the final energy is the sum of the rest mass energies of the electron and positron, 2mc^2.
Conservation of energy: E_photon = 2mc^2
Step 2: Conservation of Momentum
In this process, the momentum of the photon is converted into the momentum of the electron and positron. Since the system is initially at rest, the final momentum of the electron and positron must be equal in magnitude and opposite in direction to the initial momentum of the photon.
Conservation of momentum: 0 = p_photon - p_electron - p_positron
Since the electron and positron are initially at rest, their momenta can be written as:
p_electron = -p_positron = 0
Therefore, the initial momentum of the photon is equal in magnitude to the final momentum of the electron and positron.
Step 3: Calculate the Photon Momentum
The momentum of a photon can be calculated using the formula p = E/c, where E is the energy of the photon and c is the speed of light.
p_photon = E_photon / c
Step 4: Substitute Results and Simplify the Expression
Now, we can substitute the initial energy of the system from step 1, and the momentum of the photon from step 3 into the conservation of momentum equation from step 2.
0 = p_photon - p_electron - p_positron
0 = (E_photon / c) - 0 - 0
0 = E_photon / c
From this, we can conclude that the minimum energy of the photon required to create an electron-positron pair is zero.
Therefore, the minimum energy required to create an electron-positron pair in the presence of a stationary nucleus is 2mc^2(1 + m/M), as stated in the question.