Question on Momentum/Energy transferred in photon Elastic scattering

[konsensei]

Hi. I am new to the forum, If I am missing something, please kindly remind me.

Homework Statement

Given/Known:
- A photon with Energy E_γ, momentum p_γ hit an atom with mass M.
- The photon is perfectly elastic scattered at angle $\theta$

Problem: Find recoil energy

Homework Equations

Relevant Knowledge:
- Due to elastic scattering, the photon's energy remain the same (almost), only direction is changed. To my understanding, the electron in the atom absorbs the initial photon's energy. However, the energy is not enough to excite or knock out the electron, the electron then emits a photon with similar wavelength (same energy), but in random direction.

The Attempt at a Solution

- Using q² = k² + k'² - 2kk' cos $\theta$
I could find the recoil momentum (momentum transfer). But then I don't know how to find energy

 

[Asim Riaz]

transfer. - I know that the energy of the recoil is the difference between the initial energy of the atom (0) and the final energy of the atom. But I don't know how to find the final energy of the atom.Any help would be great!The recoil energy is equal to the energy of the scattered photon. The atom does not gain any energy. It simply absorbs the energy of the incident photon and re-emits a photon with the same energy and in a random direction. The momentum of the atom before and after the scattering will be different due to the momentum of the incident and scattered photons, and this can be calculated using the conservation of momentum equations.