Effective paramagnetic moment (${\mu }_{\mathrm{eff}}=|g_J|{\mu }_B\sqrt{J(J+1)}$)

The magnetic moment inferred from the weak-field Curie constant of a localized angular-momentum multiplet. Uniform zero-field occupation of its $2J+1$ substates gives $⟨m_J^2⟩=J(J+1)/3$, yielding ${\mu }_{\mathrm{eff}}=|g_J|{\mu }_B\sqrt{J(J+1)}$. Here $g_J$ is the Landé g-factor and ${\mu }_B$ the Bohr magneton. The dimensionless number $m_{\mathrm{eff}}={\mu }_{\mathrm{eff}}/{\mu }_B$ differs from the saturated projection $|g_J|J$.