Fixed-field per-ion Curie coefficient ($C_{\mu }=(g{\mu }_B{)}^{2}J(J+1)B/(3k_B)$)

For independent localized angular-momentum-$J$ ions at fixed field, the weak-polarization mean moment is $⟨{\mu }_z⟩=C_{\mu }/T$. Unlike the volume-susceptibility Curie constant, this coefficient includes the magnetic field and no ion number density. The expansion requires $g{\mu }_B|B|J\ll k_{B}T$; the moment saturates toward $g{\mu }_{B}J$ beyond that regime.