What is the minimum number of collisions for photon required to lose all of its energy in an electron cloud? The electron cloud I refer to is a free electron cloud. Please help in this question. Is the answer infinite? If it is then it would take infinite time,  which it doesn't.
 A: As you know that electron loses its energy in scattering with free electron via Compton scattering and the change in wavelength of the photon can be written as 
$\lambda'-\lambda=\frac{h}{m_ec}(1-\cos\theta)$
after some manipulation the above equation can be written as
$\Delta E_{photon}=\frac{2E_{photon}^2}{m_ec^2}(1-\cos\theta)$
(this equation is valid for small change in the energy of photon during the scattering event) you may find that with the decrease in photon energy the change in the phton energy reduces and you will never reach to a situation that the photon can loose all of its energy via scattering from free electrons.
Physically we can say that the momentum transfer from photon to electron reduces as the photon loses its energy (similar to the negligible momentum transfer during collision of two particles with largely different mass). This is the reason that while working with optical photons the Compton scattering changed into elastic Thomson scattering.
Hope this will clear your doubts
