Class 12 > Unit # 25: Quantum Physics > Compton Effect

 

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Compton Effect and Derivation of Compton Shift

Video Lecture: Compton Effect

Watch the complete video lecture explaining the Compton effect and the mathematical derivation of Compton shift:

Compton's Effect

“When a photon strikes a stationary electron, the wavelength of the photon is increased after collision. This phenomenon is referred to as Compton’s effect.”

Alternatively: “It is the phenomenon in which a photon of frequency $f$ is scattered by an electron, and the scattered photon has a frequency less than that of the incident photon.”

Mathematical Representation & Setup

Consider a photon of frequency $f$ and wavelength $\lambda$ striking a stationary electron of rest mass $m_0$. After collision, the photon is scattered at an angle $\theta$ with its original line of action, while the recoil electron moves forward at an angle $\phi$ with its original direction.

Fig: Scattering of a photon by a stationary electron (Compton scattering).

Energy Conservation

• Energy of photon before collision = $E_1 = hf$
• Energy of electron before collision = $E_2 = m_0 c^2$ (Rest mass energy)
• Energy of photon after collision = $E_1' = hf'$
• Energy of electron after collision = $E_2' = mc^2$ (Relativistic total energy)

Since the collision between the photon and electron is elastic, total energy is conserved:

$\text{Total Energy Before Collision} = \text{Total Energy After Collision}$

$hf + m_0 c^2 = hf' + mc^2$     --- (i)

Momentum Conservation

• Momentum of photon before collision = $\frac{hf}{c}$
• Momentum of electron before collision = $0$
• Momentum of photon after collision = $\frac{hf'}{c}$
• Momentum of electron after collision = $mv$

Momentum Equation Along X-Axis:

$\frac{hf}{c} = \frac{hf'}{c}\cos\theta + mv\cos\phi$     --- (ii)

Momentum Equation Along Y-Axis:

$0 = \frac{hf'}{c}\sin\theta - mv\sin\phi$
$\frac{hf'}{c}\sin\theta = mv\sin\phi$     --- (iii)

Derivation of Compton Shift

Solving equations (i), (ii), and (iii) simultaneously by eliminating the electron's velocity and angle $\phi$, we obtain the relationship for the change in wavelength (Compton Shift):

$\Delta\lambda = \lambda' - \lambda = \frac{h}{m_0 c}(1 - \cos\theta)$

Where $\frac{h}{m_0 c}$ is called the Compton’s wavelength, and its value is $2.426 \times 10^{-12}\text{ m}$ ($0.00242\text{ nm}$).

Conclusion

Thus, the frequency $f'$ of the scattered photon after the collision will be less than the original frequency $f$, resulting in an increase in wavelength ($\lambda' > \lambda$).

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