GATE 2025 PH – Question 32
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) _____
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Explanation
**Hund's rules for the ground state of $4f^6$.** The $f$ subshell has 7 orbitals ($m_l=3,2,1,0,-1,-2,-3$). Six electrons are placed with parallel spins:
- **Rule 1 (maximum $S$):** six electrons with parallel spins fill six orbitals, so $S=6\times\tfrac12=3$.
- **Rule 2 (maximum $L$):** the six orbitals have $m_l=3,2,1,0,-1,-2$, giving $L=\sum m_l=3$.
- **Rule 3:** for a shell that is **less than half full** ($6<7$), $J=|L-S|=|3-3|=0$.
The ground term is $^7F_0$ with $J=0$.
**Effective magnetic moment:**
$$\mu_{eff}=g_J\sqrt{J(J+1)}\,\mu_B=g_J\sqrt{0}\,\mu_B=\mathbf{0}.$$