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 show that the minimum energy the photon must have to create an electron-positron pair in the presence of a stationary nucleus is 2mc^2(1+m/M), we can use conservation of energy and momentum.
Here's how we can approach the problem:
1. Let's denote the mass of the nucleus as M, the mass of the electron as m, and the energy of the photon as E.
2. For the creation of an electron-positron pair, we need to conserve both energy and momentum in the process.
3. The total energy before the interaction is the energy of the photon E. After the interaction, the energy is divided between the electron and the positron.
4. The electron and positron will have equal energies due to symmetry, so each will have energy E_e = E/2.
5. By conservation of energy, the total energy after the interaction is E_electron + E_positron = E/2 + E/2 = E.
6. The energy of the electron and positron can be expressed in terms of their rest mass energy using Einstein's mass-energy equivalence, E_electron = mc^2 and E_positron = mc^2.
7. Therefore, we have E = mc^2 + mc^2 = 2mc^2.
8. Now, let's consider momentum conservation. Since the nucleus is stationary, the total momentum before the interaction is simply zero.
9. After the interaction, the electron and positron will move in opposite directions with equal but opposite momenta.
10. By conservation of momentum, the magnitude of the momentum of the electron and positron will be equal, let's denote it as p.
11. Therefore, the total momentum after the interaction is p - p = 0.
12. We can express the momentum in terms of the energy and rest mass using the relativistic momentum formula p = sqrt(E^2 - (mc^2)^2).
13. Substituting E = 2mc^2, we have p = sqrt((2mc^2)^2 - (mc^2)^2) = sqrt(4m^2c^4 - m^2c^4) = sqrt(3m^2c^4).
14. Now, let's use momentum conservation to relate the photon energy E to the momentum p.
15. The momentum of the photon is given by p_photon = E_photon / c, where c is the speed of light.
16. By conservation of momentum, p_photon = p = sqrt(3m^2c^4).
17. Rearranging the equation, we have E_photon = pc = sqrt(3m^2c^4) * c = sqrt(3)m^2c^3.
18. Finally, substituting the value of E_photon into the expression for the total energy E, we have E = 2mc^2(1 + m/M).
Therefore, we have shown that the minimum energy the photon must have to create an electron-positron pair in the presence of a stationary nucleus is 2mc^2(1 + m/M).