Module 5 : MODERN PHYSICS
Lecture 25 : Reason for the unshifted peak in the spectrum
Reason for the unshifted peak in the spectrum
A free electron cannot absorb a photon and increase its energy as doing so would violate energy-momentum conservation.
  Consider a free electron at rest which absorbs a photon of energy $ h\nu$ (and momentum $ h\nu/c$). The final energy of the electron would be $ h\nu + m_0c^2$. According to relativistic principle, if the momentum of the electron is $ p$, the total energy is given by $ \sqrt{p^2c^2+m_0^2c^4}$. When the electron absorbs the incident photon, the momentum of the photon would be transferred to the electron. Since the electron was initially at rest (i.e. with zero momentum), its final momentum is $ p=h\nu/c$. Thus we have
 
$\displaystyle pc+ m_0c^2 = \sqrt{p^2c^2+m_0^2c^4}$
  which simplifies to $ 2m_0pc^3 =0$, which is not possible.
  The reason why an electron bound to an atom can absorb a photon ( as in Compton effect) is that the electron can share some of the resulting momentum with the ion which has a much larger mass
   
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