The Cosmic Ray Detector · AI Generator #GEN-652972C6

Physics · modern-physics

Problem

A cosmic ray muon detector consists of a thin aluminum foil placed above a scintillator. When a high-energy cosmic ray muon passes through the aluminum, it can knock out a K-shell electron, creating a characteristic X-ray that is detected by the scintillator below. The aluminum foil has atomic number

Z = 13

. When a K-shell electron is ejected, an electron from the L-shell (with quantum number

n = 2

) drops down to fill the vacancy, emitting an X-ray photon. This X-ray then undergoes Compton scattering with a free electron in the scintillator material, deflecting at an angle of

60°

from its original direction. (a) Using Moseley's law in the form

\nu = R(Z - \sigma)^2\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)

where

R = 3.29 \times 10^{15}\;\text{Hz}

\sigma = 1

for K

_{\alpha}

transitions,

n_1 = 1

, and

n_2 = 2

, find the frequency of the characteristic X-ray emitted by aluminum. (b) Calculate the wavelength of the Compton-scattered X-ray after it deflects by

60°

. Given: Compton wavelength of electron

\lambda_C = 2.43 \times 10^{-12}\;\text{m}

, speed of light

c = 3.00 \times 10^8\;\text{m/s}

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