Question

The magnetic moment of lanthanide ions is determined from which one of the following relation?

• A. $\mu \, \, = \, \sqrt{n(n+2)} $
• B. $\mu \, = g \sqrt{J(J+1)}$
• C. $\mu=g\sqrt{n(n+1)}$
• D. $\mu = 2 \sqrt{n(n+1)}$

Answer 1

The Correct Option is B

In lanthanoids, the \(4f\) orbitals are deeply buried and shielded by outer orbitals. As a result, the orbital motion of electrons is not quenched, and both spin (\(S\)) and orbital (\(L\)) angular momenta contribute to the magnetic moment. These combine to give the total angular momentum quantum number \(J\): \[ J = |L - S| \quad \text{(less than half-filled shell)}, \qquad J = L + S \quad \text{(more than half-filled shell)} \] The magnetic moment is given by: \[ \mu = g \sqrt{J(J+1)} \] where the Landé \(g\)-factor is: \[ g = 1 + \frac{J(J+1) + S(S+1) - L(L+1)}{2J(J+1)} \]