Derivation of Compton Effect

Compton effect is defined as the effect that is observed when x-rays or gamma rays are scattered on a material with an increase in wavelength. Arthur Compton studied this effect in the year 1922. During the study, Compton found that wavelength is not dependent on the intensity of incident radiation. It is dependent on the angle of scattering and on the wavelength of the incident beam. It is given in the following mathematical form:

$\lambda _{s}-\lambda _{0}=\frac{h}{m_{0}c}(1-cos\Theta )$

Where,

Ө: angle at which radiation scattered

m0: rest mass of an electron

$\frac{h}{m_{0}c}$ : Compton wavelength of the electron

λs and λ0: radiation spectrum peaks.

Derivation of Compton effect equation

Considering the elastic collide between a photon and an electron, following is the derivation:

$h\nu _{0}$ : energy of photon

$p_{i}=\frac{h\nu _{0}}{c}$ :momentum of the photon

$p_{i}=p_{f}cos\Theta +p_{e}cos\phi (1)$ (conservation of momentum in x direction)

$0=-p_{f}sin\Theta +p_{e}sin\phi (2)$ (conservation of momentum in y direction)

$p_{e}^{2}=p_{e}^{2}(cos^{2}\phi +sin^{2}\phi )$

$=(p_{i}-p_{f}cos\Theta )^{2}+p_{f}^{2}sin^{2}\Theta$

$=p_{i}^{2}+p_{f}^{2}-2p_{i}p_{f}cos\Theta$

$h\nu _{0}+m_{0}c^{2}=h\nu +\sqrt({m_{0}^{2}}c^{4}+p_{e}^{2}c^{2})$

$m_{0}^{2}c^{4}+p_{e}^{2}c^{2}=(h\nu _{0}-h\nu +m_{0}c^{2})^{2}$

$=(h\nu _{0}-h\nu)^{2} +m_{0}^{2}c^{4}+2m_{0}c^{2}(h\nu_{0}-h\nu )$

$p_{e}^{2}c^{2}=(h\nu _{0}-h\nu)^{2} +2m_{0}c^{2}(h\nu_{0}-h\nu )$

$p_{i}^{2}c^{2}+p_{f}^{2}c^{2}-2p_{i}p_{f}cos\Theta c^{2}=(h\nu _{0}-h\nu)^{2}+2m_{0}c^{2}(h\nu _{0}-h\nu )$

$h\nu \nu _{0}(1-cos\Theta )= m_{0}c^{2}(\nu _{0}-\nu)$

$∴ \lambda _{s}-\lambda _{0}=\frac{h}{m_{0}c}(1-cos\Theta )$

Therefore, above is the Compton effect equation and$\frac{h}{m_{0}c}\equiv \lambda _{c}$ is Compton wavelength of an electron.

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