Derivation of Compton Effect

What is 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.

Difference Between Compton Effect and Photoelectric Effect

Compton effect Photoelectric effect
This is the effect caused by the inelastic scattering of high-energy photons that are bound to free electrons.  This is the effect caused by the weakly bound electrons that are ejected from the surface of the material when electromagnetic radiation interacts with the electrons.
Arthur Compton explained the effect. Albert Einstein explained the effect.
The energy associated with the free electrons is mid-energy. The energy associated with the electrons is low-energy.
The wavelength of the scattered photon is higher than that of the incident photon. The wavelength is not observed as the photon disappears after interacting with the electrons.

This is a step by step explanation on the derivation of the Compton effect equation derivation. Stay tuned with BYJU’S to know more other related Physics topics.

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