Derivation of Compton Effect: Class 12 Physics Notes, Definition & Formula

Compton effect can be observed when there is an increase in the wavelength caused by the dispersion of x-rays and gamma rays on the material. It was discovered by Arthur Compton in 1922.

The mathematical equation of the Compton effect is:

$\begin{array}{l}{\lambda }_s-{\lambda }_0=\frac{h}{m_{0}c}(1-cos\mathrm{\Theta })\end{array}$

Here, Ө is the angle of radiation;

m0 is the mass of an electron;

h/m0c is the wavelength;

λs and λ0 are the peaks of the radiation spectrum.

The derivation of the Compton equation is as follows:

The energy of the photon is hv0

Photon’s momentum is pi = hν0/c

Momentum conservation in x direction = pi = pf cos Θ + pe cos ϕ,

Momentum’s conservation in y direction = 0 = −pf sin Θ + pe sin ϕ.

p2e = p2e (cos2 ϕ + sin2 ϕ) = (pi − pf cos Θ) 2 + p2f sin2 Θ  

= p2i + p2f − 2pi pfcos Θ hν0 + m0 c2 = hν + (√m20 c4 + p2 ec2) 

m20 c4 + p2 ec2 = (hν0 – hν + m0c2)2 

= (hν0 − hν)2 + m20 c4 + 2m0 c2 (hν0−hν) 

p2ec2 = (hν0 − hν)2 + 2m0 c2 (hν0−hν) 

p2ic2 + p2fc2 − 2pipf cos Θ c2= (hν0 − hν)2 + 2m0c2 (hν0 − hν) 

hνν0 (1−cosΘ) = m0c2 (ν0−ν) ∴ λs − λ0 = hm0c (1 − cosΘ)

The step-by-step representation of the Compton effect’s derivation is illustrated above, and the wavelength of an electron is h/moc = λc.

Compton effect is observed when there is an increase in the wavelength caused by the dispersion of x-rays and gamma rays on the material of the substance. The incident photon is lower than the wavelength of the photon scattered in the electron. Free electron’s energy is associated with the mid-energy. 

On the other hand, the photoelectric effect is a weak bond of electrons released from the material’s surface; it occurs when the electromagnetic radiations and electrons interact with each other. The effect of photoelectric was explained by Albert Einstein. The energy here is low-energy, and the wavelength is not observed when photons disappear with electrons’ interaction. 

The chapter ‘Rotational Motion’ holds a weightage of 5 marks, consisting of two short questions. The first short question is of 2 marks and the second short question contains 3 marks.

1. Define Compton Shift. 

The shift in the scattered wavelength defines the Compton shift.  

2. What does the Compton shift depend upon?

It depends upon the angle of scattering. 

3.Give one example of non-mechanical waves.

Electromagnetic waves don’t require any medium for propagation, so they’re known as non-mechanical waves. 

FAQs on Derivation of Compton effect

Q: What’s the mathematical equation for the Compton effect?

A: The mathematical equation of the Compton effect is:

λs−λ0=h/m0c (1−cosΘ).

Q: What’s the wavelength of an electron?

A: The wavelength of an electron is h/moc = λc.

Q: Who discovered the Compton effect and when?

A: Arthur Compton, in the year 1922, discovered the Compton effect. During his discovery of the effect, he found out that the intensity of incident radiation and wavelength are not dependent upon each other.

Q: How do you calculate a photon’s momentum in the Compton effect?

A: Photon’s momentum can be calculated using this equation 🡪 Pi=hν0/c.

Q: How do you calculate the momentum’s conservation in both the X and Y direction/axis?

A: You can calculate the momentum’s conversion in both directions in the following way – 

1. Momentum conservation in x direction = pi = pf cos Θ + pe cos ϕ
2. Momentum’s conservation in y direction = 0 = −pf sin Θ + pe sin ϕ