(a) Using the Compton scattering result that the maximum change in wavelength is = 2hc/Mc 2 and the approximation E hc/ 2 ,

(a) Using the Compton scattering result that the maximum change in wavelength is Δλ = 2hc/Mc² and the approximation ΔE ≈ hcΔλ/λ², show that for a photon to lose an amount of energy E_p to a proton, the energy of the photon must be at least E = [ (1/2) Mc²E_p ]^1/2.

(b) Calculate the photon energy needed to produce a 5.7 MeV proton by Compton scattering. 

(c) Calculate the energy given a ¹⁴N nucleus in a head-on
collision with a 5.7 MeV neutron. 

(d) Calculate the photon energy needed to give a ¹⁴N nucleus this energy by Compton scattering.