Question

The value of Lande \( g \)-factor for \( ^2P_{3/2} \) state is:

• A. \( 2 \)
• B. \( \frac{2}{3} \)
• C. \( \frac{3}{2} \)
• D. \( \frac{4}{3} \)

Hint:

The Lande \( g \)-factor determines the energy shift in a magnetic field for fine-structure states.

Answer 1

The Correct Option is C

The Lande \( g \)-factor is given by the formula:
\[ g = 1 + \frac{J(J+1) + S(S+1) - L(L+1)}{2J(J+1)} \] For \( ^2P_{3/2} \), we have \( S = \frac{1}{2}, L = 1, J = \frac{3}{2} \). Substituting these values:
\[ g = 1 + \frac{\frac{3}{2}(\frac{3}{2} +1) + \frac{1}{2}(\frac{1}{2} +1) - 1(1+1)}{2 \times \frac{3}{2} (\frac{3}{2} +1)} \] \[ = 1 + \frac{\frac{15}{4} + \frac{3}{4} -2}{\frac{15}{4}} = 1 + \frac{16/4 - 8/4}{\frac{15}{4}} = 1 + \frac{8}{15} = \frac{3}{2} \]