Question

The doublet splitting of the first excited state of H-atom due to spin orbit interaction (\(^2 P_{3/2} \rightarrow ^2 P_{1/2}\)) is \(0.365 \text{ cm}^{-1}\). What is the corresponding separation for \(Li^+\) atom?

• A. \( 1.095 \text{ cm}^{-1} \)
• B. \( 3.285 \text{ cm}^{-1} \)
• C. \( 9.855 \text{ cm}^{-1} \)
• D. \( 29.565 \text{ cm}^{-1} \)

Hint:

Spin-orbit interaction energy increases significantly for heavier elements due to its \(Z^4\) dependence.

Answer 1

The Correct Option is D

Spin-orbit splitting scales as \( Z^4 \). For hydrogen (\(Z = 1\)), the given splitting is \(0.365 \text{ cm}^{-1}\).
For lithium (\(Z = 3\)), the corresponding splitting is:
\[ \Delta E_{Li^+} = \Delta E_H \times \left(\frac{Z_{Li^+}}{Z_H}\right)^4 = 0.365 \times 3^4 = 29.565 \text{ cm}^{-1} \]