Derivation of Compton Effect

Table of Contents

What is Compton Effect?

The 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:

\(\begin{array}{l}\lambda _{s}-\lambda _{0}=\frac{h}{m_{0}c}(1-cos\Theta )\end{array} \)

Where,

Ө: the angle at which radiation scattered

m0: rest mass of an electron

h/m0c : 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, the following is the derivation:

hν0 : energy of photon

\(\begin{array}{l}p_{i}=\frac{h\nu _{0}}{c}\end{array} \)
:momentum of the photon

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

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

\(\begin{array}{l}p_{e}^{2}=p_{e}^{2}(cos^{2}\phi +sin^{2}\phi )\end{array} \)
\(\begin{array}{l}=(p_{i}-p_{f}cos\Theta )^{2}+p_{f}^{2}sin^{2}\Theta\end{array} \)
\(\begin{array}{l}=p_{i}^{2}+p_{f}^{2}-2p_{i}p_{f}cos\Theta\end{array} \)
\(\begin{array}{l}h\nu _{0}+m_{0}c^{2}=h\nu +\sqrt({m_{0}^{2}}c^{4}+p_{e}^{2}c^{2})\end{array} \)
\(\begin{array}{l}m_{0}^{2}c^{4}+p_{e}^{2}c^{2}=(h\nu _{0}-h\nu +m_{0}c^{2})^{2}\end{array} \)
\(\begin{array}{l}=(h\nu _{0}-h\nu)^{2} +m_{0}^{2}c^{4}+2m_{0}c^{2}(h\nu_{0}-h\nu )\end{array} \)
\(\begin{array}{l}p_{e}^{2}c^{2}=(h\nu _{0}-h\nu)^{2} +2m_{0}c^{2}(h\nu_{0}-h\nu )\end{array} \)
\(\begin{array}{l}p_{i}^{2}c^{2}+p_{f}^{2}c^{2}-2p_{i}p_{f}cos\Theta c^{2}=(h\nu _{0}-h\nu)^{2}+2m_{0}c^{2}(h\nu _{0}-h\nu )\end{array} \)
\(\begin{array}{l}h\nu \nu _{0}(1-cos\Theta )= m_{0}c^{2}(\nu _{0}-\nu)\end{array} \)
\(\begin{array}{l}∴ \lambda _{s}-\lambda _{0}=\frac{h}{m_{0}c}(1-cos\Theta )\end{array} \)

Therefore, above is the Compton effect equation and

\(\begin{array}{l}\frac{h}{m_{0}c}\equiv \lambda _{c}\end{array} \)
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.

Related Physics articles:

Photoelectric Effect Kinetic Energy

Frequently Asked Questions – FAQs

Q1

What is an X-ray?

An X-ray is a penetrating type of electromagnetic radiation. Almost every x-ray has a wavelength from 10 picometers to 10 nanometers.
Q2

What is a gamma-ray?

A gamma-ray is another penetrating type of electromagnetic radiation generated from the radioactive decay of atomic nuclei. Gamma-ray is made of the shortest wavelength (among the important electromagnetic waves).
Q3

What is meant by radiation?

Radiation is the transmission or emission of energy in the form of particles or waves through a medium or vacuum.
Q4

What is meant by the photoelectric effect?

The photoelectric effect is a process in which electrons are emitted from a metal’s surface when light strikes it.
Q5

Who discovered the Compton effect?

Arthur Compton discovered the Compton effect.
Q6

What is meant by the Compton effect?

The Compton effect is the enlargement of wavelengths of X-rays and other EM waves that have been scattered by electrons. It is a fundamental way in which radiating energy is absorbed by matter.

Q7

What is wavelength?

A wavelength is a length between successive troughs or crests.

Q8

Which symbol is used to represent the Compton wavelength?

Compton wavelength is represented by the letter λ (Lambda).

Q9

What is a photon?

A photon is a fundamental particle that is the force carrier of electromagnetic force. It is massless and travels at the speed of light 299792458 m/s (in vacuum).

Q10

Who explained the photoelectric effect?

Albert Einstein explained the photoelectric effect.

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  1.  sandhya verma
    December 13, 2020 at 3:14 pm

    thankyou sir

    Reply