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in Dual nature of matter and radiation by (20 points)
X-rays of wavelength 0.2 nm are scattered from a block of carbon. If the scattered radiation is detected at 90◦to theincident beam, find (a) the Compton shift, and (b) the kinetic energy imparted to the recoiling electron.

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Given θ = 90°

(a) Compton effect scattered wavelength incident wavelength compton shift

Δλ \(=\frac h{m_ec}\) (1 - cos θ)

Δλ \(=\frac h{m_ec}\) (1 - cos 90°)

Δλ \(=\frac h{m_ec}\) (1 - 0)

 Δλ \(=\frac h{m_ec}\) 

where h = plank constant

c = speed of light

me = mass of electron

λf - λ\(=\frac h{m_ec}\) (1 - cos θ)  (θ = 90°, cos 90° = 0)

λf - 0.2 x 10-9 \(=\frac h{m_ec}\)

λf - 0.2 x 10-9 \(=\frac{6.63\times10^{-34}}{9.1\times10^{-31}\times3\times10^8}\)

λf = 0.2 x 10-9 + 0.24 x 10-11

λf = 0.20 x 10-10 + 0.024 x 10-10

λf = 0.224 x 10-10

λf = 0.0224 x 10-9 m

Compton shift Δλ = λf - λi

Δλ = 0.0224 - 0.2

Δλ = -0.1776 nm

(b) Kinetic energy imparted to the recoiling electron.

Ek \(=\frac{h^2}{2m\lambda^2}\)

Ek \(=\frac{(6.64\times10^{-34})^2}{2\times9.1\times10^{-31}\times(0.2\times10^{-9})^2}\)

Ek \(=\frac{44.08\times10^{-68}}{0.728\times10^{-18}\times10^{-31}}\)

Ek = 60.54 x 10-19 J

K.E = 37.83 eV

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