Subtopic: Orbital Angular Momentum & Spherical Harmonics
Question Statement:
The effective magnetic moment (in units of Bohr magneton) for the ground state of
an isolated 4f ion with 6 unpaired electrons in the 4f shell according to Hund 's
rules is (in integer) _____
Official Answer Key:
0 to 0
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2025 Q.32)**
**1. Atomic Physics & Hund's Rules for 4f Shell**:
- Electronic configuration: $4f$ subshell with orbital angular momentum $l = 3$ ($2l + 1 = 7$ orbitals).
- Number of unpaired electrons: $n = 6$.
**2. Step-by-Step Evaluation**:
- **Hund's Rule 1 (Maximum Spin $S$)**:
- Distribute 6 electrons with parallel spins ($m_s = +1/2$):
$S = 6 \times \frac{1}{2} = 3$
- **Hund's Rule 2 (Maximum Orbital Angular Momentum $L$)**:
- Fill the available $m_l \in \{+3, +2, +1, 0, -1, -2, -3\}$ from highest to lowest:
$L = (+3) + (+2) + (+1) + (0) + (-1) + (-2) = 3$
- **Hund's Rule 3 (Total Angular Momentum $J$)**:
- Full $4f$ subshell capacity = $14$ electrons.
- Since $6 < 7$, the subshell is **less than half-full**.
- For less than half-full shells, the lowest energy multiplet has:
$J = |L - S| = |3 - 3| = 0$
- **Effective Magnetic Moment**:
$\mu_{\text{eff}} = g_J \sqrt{J(J + 1)} \mu_B = g_J \sqrt{0(1)} \mu_B = 0$
**3. Official Verification**:
The verified official IIT answer is **0** (Numerical range: `0 to 0`).
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