Question

A photon of energy E is scattered by a particle of rest energy Eo. Find the maximum kinetic energy of the recoiling particle in terms of E and Eo

Answer 1

For a given incident energy $E=hc/\lambda$, the outgoing final-state photon energy, $E_{sc}$, is given by

$$E_{sc}=\frac{E}{1+\frac{E}{E_0}(1-\cos \theta)}$$

where $\theta$ is the scattering angle.

$\min E_{sc}(\theta)=E_{sc}(\theta=\pi)=\frac{E}{1+2E/E_0}$

Part of the photon energy is transferred to the recoil particle (electron). Thus the photon loses energy in the process. Kinetic energy of the recoiling particle is given by

$E_{re}=E_0-E_{sc} \rightsquigarrow \max E_{re}=E_0-\min E_{sc}=E_0-\frac{E}{1+2E/E_0}$

$\max E_{re}==\frac{E_0+E}{1+2E/E_0}$