Question

A metal target with atomic number \( Z = 46 \) is bombarded with a high-energy electron beam. The emission of X-rays from the target is analyzed. The ratio \( r \) of the wavelengths of the \( K_\alpha \)-line and the cutoff is found to be \( r = 2 \). If the same electron beam bombards another metal target with \( Z = 41 \), the value of \( r \) will be:

• A. \( 2.53 \)
• B. \( 1.27 \)
• C. \( 2.24 \)
• D. \( 1.58 \)

Hint:

Use the Rydberg formula to calculate wavelength ratios in X-ray spectra.

Answer 1

The Correct Option is A

For \( K_\alpha \)-series: \[ \frac{1}{\lambda} = R(Z-1)^2\left(1 - \frac{1}{4}\right) \]\quad \[\Rightarrow \quad \frac{1}{\lambda} = \frac{3}{4} R(Z-1)^2. \] For \( Z = 46 \): \[ r = 2 \quad \Rightarrow \quad r = \frac{45^2}{40^2} \times 2 \approx 2.53. \]