Curie's law ($\chi =C/T$)

For independent moments of angular momentum $J$ at small ${\mu }_B|B|/(k_{B}T)$, the zero-field variance is $⟨m^2⟩=J(J+1)/3$. Differentiating the canonical mean gives the magnetic susceptibilityThus susceptibility is inversely proportional to temperature. The small-field or high-temperature condition distinguishes this law from the exponentially vanishing fixed-field susceptibility of a saturated paramagnet at low temperature.

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.