Q7(c)(ii). Effective magnetic moment of Ce³⁺

1. Ce (Z=58): [Xe]4f¹5d¹6s². Ce³⁺: [Xe]4f¹.
2. One f electron: s=12, so S=12; l=3 with mₗ=+3, so L=3 (F). The shell is less than half full, so the lowest level has J=L-S=52. Ground term ²F_(5/2)
3. The 4f electron is buried under 5s²5p⁶, so the ligand field is small and the orbital moment is not quenched. Spin–orbit coupling is large, J is a good quantum number, and the moment depends on J through the Landé factor.
4. Landé g factor: 2pt] g_J=1+(J(J+1)+S(S+1)-L(L+1))/(2J(J+1))=1+(35/4+3/4-12)/(2×35/4)=1+(-2.5)/(17.5)=67=0.857
5. Moment: μ_(eff)=g_J√(J(J+1))=67√(52×72)=0.857×2.958 μ_(eff)=2.54 B.M.
6. At room temperature only the ²F_(5/2) level is populated (the ²F_(7/2) level lies too high), so the formula applies. Observed moments of Ce³⁺ salts are close to this value.
7. Comparison: spin only, √(n(n+2))=√3=1.73 B.M., is too low; √(4S(S+1)+L(L+1))=3.87 B.M. (spin and orbital moments uncoupled, as for the first transition series in weak fields) is too high. Only the J-based value fits a lanthanide ion.

Application. the same g_J√(J(J+1)) rule reproduces the double-humped variation of moments across the Ln³⁺ series, with maxima near Dy³⁺ and Ho³⁺ (about 10.6 B.M.); Sm³⁺ and Eu³⁺ deviate because excited J levels lie close enough to be populated at room temperature.