Electromagnetic cascade Electromagnetic radiation of frequency v flows in the (-

Question

Electromagnetic cascade Electromagnetic radiation of frequency v flows in the (-x) direction. Free electrons (rest mass m, energy E) moving in the (+x) direction undergo a relativistic inverse Compton scattering e(E, p vector) + gamma (hv) rightarrow e(E', p' vector) + gamma (hv') In terms of E, pc and hv, express the energy hv' of the outgoing photon in the case of head on collisions with the hv' photon being scattered in the forward direction and hence having the maximal possible energy. Rewrite the energy hv' of the scattered photons in terms of just E and hv when E >> mc^2 and m^2c^4 >> Ehv. Considering a photon of energy hv' propagating in the (+x) direction, write the threshold condition on hv' for these photons to interact with hv photon to create a e^+/e^- pair Combining your answers to the two previous questions express the minimal energy of electrons whose inverse Compton scattering may result in photons with sufficient energy to participate in e^+/e^- pair creations.

Explanation / Answer

a) We will define the energy in form of power as we know that power =energy/time. Now in the case of inverse Compton scattering the electron losses energy and photons gain. So we consider an isotropic distribution of particles then differential photon number dn = dN/dV = [dN/dX] dx0

now dX and dV are the space volumes and dX is Lorentz invariant but time and energy in Lorentz transformation work the same way so we need a new invariant to calculate scattering/time/electron and that is dn/

now we consider total scattering rate as k' then scattering rate per electron is dn/dt = ^-1 dn'/dt' = ^-1 *c integration of []dn' and is total Compton cross section

using Lorentz invariant and initial energy the equation becomes (use of Doppler boost)

dn/dt = c* integration of [(1- Cos)]dn and c(1- Cos) defines relative velocity of photon and electron

Taking power as invariant we get the total power dE/dT = c^2 integration of (1- Cos)^2 dn

for isotopic distribution taking total photon density U we get

dE/dT = c^2 (1 +1/3^2 ) U

or energy E= integration of [c^2 (1 +1/3^2 ) U ] dT

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