Compton Effect (Compact)

phy-1109 PHY-1109 Physics

Compton Scattering Slide Compton Effect (Compact) Compton Effect (Detailed)

COMPTON EFFECT:

The Compton Effect demonstrates the particle nature of light (photons), treating the photon-electron interaction as a relativistic collision.

I. FUNDAMENTAL CONCEPTS

Concept	Description	Details
The Event	An x-ray photon strikes an electron (initially at rest). The photon scatters ($\varphi$), and the electron recoils ($\theta$).	Energy and momentum are conserved.
Energy Change	Scattered photon (${\nu }^{\prime }$) has less energy than incident photon ($\nu$).	Loss in photon energy $=h\nu -h{\nu }^{\prime }=$ Gain in electron kinetic energy ($T$).
Momentum	Photon momentum is related to energy: $p=E/c$. For a photon, $p=h\nu /c$.	Momentum is a vector; conservation must hold in two perpendicular directions.

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II. DERIVATION OF THE COMPTON SHIFT FORMULA ($\Delta \lambda$)

The formula relates the change in wavelength ($\Delta \lambda ={\lambda }^{\prime }-\lambda$) to the photon scattering angle ($\varphi$).

A. Conservation Equations (Momentum)

1. Along the original photon direction (x): $\frac{h\nu }{c}=\frac{h{\nu }^{\prime }}{c}\cos \varphi +p\cos \theta (6)$
2. Perpendicular to the original direction (y): $0=\frac{h{\nu }^{\prime }}{c}\sin \varphi -p\sin \theta (7)$

B. Eliminating Angles and Energies

• Step 1: Eliminate $\theta$ (Recoil Angle): Rearrange (6) and (7), square, and add to eliminate $pc\sin \theta$ and $pc\cos \theta$, yielding: $p^2{c}^2=h^2\left({\nu }^2+{\nu }^{\prime 2}-2\nu {\nu }^{\prime }\cos \varphi \right)(8)$
• Step 2: Relativistic Energy Conservation: Start with $E=h\nu -h{\nu }^{\prime }+m_0{c}^2$. Use the relativistic relation $E^2=p^2{c}^2+m_0^2{c}^4$ to find an alternate expression for $p^2{c}^2$: $p^2{c}^2=2h{m}_0{c}^2\left(\nu -{\nu }^{\prime }\right)-2h^2\nu {\nu }^{\prime }(9)$
• Step 3: Final Relation: Equate (8) and (9), simplify, and substitute $\nu =c/\lambda$ and ${\nu }^{\prime }=c/{\lambda }^{\prime }$.

C. The Compton Shift Formula

The result is Equation (10): ${\lambda }^{\prime }-\lambda =\frac{h}{m_{0}c}(1-\cos \varphi )(10)$

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III. COMPTON WAVELENGTH & IMPLICATIONS

Term / Formula	Description	Key Implications
Compton Wavelength (${\lambda }_C$)	${\lambda }_C=\frac{h}{m_{0}c}(11)$	Gives the scale of the wavelength change. For an electron: ${\lambda }_C=0.02426\text{A˚}$ ($2.426\times 10^{-12}\text{m}$).
Shifted Formula	${\lambda }^{\prime }-\lambda ={\lambda }_C(1-\cos \varphi )(12)$	The shift ($\Delta \lambda$) is independent of the incident photon’s wavelength ($\lambda$).
Maximum Shift	Occurs at $\varphi =180^{\circ }$ ($\cos \varphi =-1$): $\Delta {\lambda }_{\text{max}}=2{\lambda }_C$.	For an electron, $\Delta {\lambda }_{\text{max}}=4.852\text{ pm}$. This shift is only easily observable for X-rays.
Unshifted Peak	Some scattered X-rays retain the original wavelength ($\lambda$).	This occurs when the electron is very tightly bound to the atom. The entire atom recoils, meaning $m_0$ in Eq. (10) becomes the mass of the atom, making $\Delta \lambda$ negligible (undetectable).

Related Previous Year Questions (PYQs) - Calculations & Proofs

Question	Year(s) of Appearance	Type
What is Compton shift? Briefly describe theory and obtain expression for change in wavelength of photon undergoing Compton scattering.	2024	Derivation
Show that the Compton red shift is $\Delta \lambda ={\lambda }_c(1-\cos \theta )$. (Note: This matches Equation (12) from the source, using $\varphi$ instead of $\theta$ for the scattering angle.)	2017, 2019, 2023	Derivation
Show that the Compton shift ($\Delta \lambda$) is independent on the wavelength of the incident photon.	2021	Derivation
X-ray photon of wavelength 0.3 $\text{A˚}$ undergoes a 60° Compton scattering. Find the wavelength of scattered photon and kinetic energy imparted to the recoiling electron.	2016	Numerical
X-ray scattered by free electron. At 45°, ${\lambda }^{\prime }=0.22\text{A˚}$. What is the wavelength of direct beam?	2017	Numerical

-By Nakib(2407009)