Worked examples — Compton scattering — wavelength shift derivation

2.3.4 · D3 · Physics › Modern Physics › Compton scattering — wavelength shift derivation

Shuru karne se pehle, wo numbers pin down kar lo jo hum har jagah reuse karte hain:

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Scenario matrix

Har Compton problem inhi cells mein se ek hoti hai. Neeche ke worked examples ko unki cell ke saath tag kiya gaya hai.

Cell	Case class	Kya cheez alag banati hai	Example
A	$\theta =0^{\circ }$ (degenerate, koi deflection nahi)	$1-\cos \theta =0\Rightarrow \Delta \lambda =0$	Ex 1
B	$\theta =90^{\circ }$ (clean midpoint)	$\cos 90^{\circ }=0\Rightarrow \Delta \lambda ={\lambda }_C$	Ex 2
C	$\theta =180^{\circ }$ (backscatter, limiting max)	$1-\cos \theta =2\Rightarrow \Delta \lambda =2{\lambda }_C$	Ex 3
D	General angle $0<\theta <90^{\circ }$	$\cos \theta$ actually compute karna padega	Ex 4
E	General angle $90^{\circ }<\theta <180^{\circ }$	$\cos \theta <0$, shift ${\lambda }_C$ se zyada ho jaati hai	Ex 5
F	Electron ko diya gaya energy (recoil KE)	energy conservation use karo, sirf shift nahi	Ex 6
G	Fractional shift $\Delta \lambda /\lambda$ (kyun X-rays, light nahi)	same $\Delta \lambda$, wildly alag fraction	Ex 7
H	Wrong-mass trap / unshifted peak	proton ya poora atom → bahut chhoti shift	Ex 8
I	Inverse problem / exam twist ($\Delta \lambda$ se $\theta$ dhundho)	$\cos \theta$ backwards solve karo, range check karo	Ex 9

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Worked examples

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Recall Matrix ke across quick self-test

Shift at $\theta =0^{\circ }$? ::: $0$ (Cell A — koi deflection nahi, koi momentum transfer nahi). Shift at $\theta =90^{\circ }$? ::: ${\lambda }_C=2.43$ pm (Cell B). Maximum possible shift aur kahan? ::: $2{\lambda }_C=4.86$ pm at $\theta =180^{\circ }$ (Cell C). Kisi bhi $\lambda$ ke liye $\theta =120^{\circ }$ par shift? ::: $1.5{\lambda }_C=3.645$ pm (Cell E — obtuse, ${\lambda }_C$ se zyada). Agar $\Delta \lambda =1.215$ pm, toh kaun sa angle? ::: $\cos \theta =0.5\Rightarrow \theta =60^{\circ }$ (Cell D/I). Visible light mein Compton scattering kyun nahi dikhti? ::: Same 2.43 pm shift ek 500 nm wave ka negligible fraction hai (Cell G). $90^{\circ }$ par proton se photon? ::: $\approx 1.32\times 10^{-3}$ pm, yaani ~1836× chhoti (Cell H). Largest valid $\Delta \lambda /{\lambda }_C$? ::: $2$; isse upar kuch bhi unphysical hai (Cell I).

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Connections

• Parent: Compton scattering derivation
• Photon momentum and energy — har example mein use hone wale $p=h/\lambda$ aur $E=hc/\lambda$ supply karta hai.
• Conservation of energy aur Conservation of momentum — shift aur recoil-energy example ke peeche ye do laws hain.
• Relativistic energy-momentum relation — isliye Ex 6 mein electron recoil ko relativistically treat kiya gaya hai.
• Photoelectric effect — wahan ke eV energy scale ko yahan ke keV recoil se compare karo.
• X-ray production and Bremsstrahlung — incoming X-rays kahan se aati hain.
• de Broglie wavelength — ek alag context mein same $h/(mc)$-style length scale.