Derive the Compton shift formula and calculate the wavelength change for a photon of initial wavelength 0.15 nm scattered at 180°. What is the scattered wavelength?

hard 4 marks Class 12 · Physics · Dual Nature of Radiation and Matter

Answer

0.155 nm

The correct answer is 0.155 nm — Calculate $D e l t a l amb d a = 0.002426 \times 2 = 0.004852 nm$ and add to the initial wavelength: $λ^{'} = 0.15 + 0.004852 = 0.154852 nm \approx 0.155 nm$ .

Step-by-step solution

Write conservation of energy: $h u + m_{e} c^{2} = h u^{'} + (m_{e} c^{2})^{2} + (p_{e}^{p r im e} c)^{2}$ .
Write conservation of momentum in vector form for photon and electron.
Eliminate the electron momentum to obtain $D e l t a l amb d a = \frac{h}{m _{e} c} (1 - cos θ)$ , the Compton shift formula.
Insert constants: $h / m_{e} c = 2.426 \times 1 0^{- 12} m = 0.002426 nm$ and $θ = 18 0^{\circ}$ so $1 - cos θ = 2$ .
Calculate $D e l t a l amb d a = 0.002426 \times 2 = 0.004852 nm$ and add to the initial wavelength: $λ^{'} = 0.15 + 0.004852 = 0.154852 nm \approx 0.155 nm$ .

Common mistake: Using

λ_{C} = h / m c

incorrectly or forgetting the factor

(1 - cos θ)

Concept tested: Compton scattering wavelength shift

Published 6 September 2026 · Last updated 16 September 2026 · How this solution was prepared and checked

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