 Langevin diamagnetic susceptibility

For identical isotropic closed-shell atoms of number density $n$, with $Z$ electrons and mean squared radius $⟨r^2⟩$ per electron, the quadratic magnetic field energy is $Z{e}^2{B}^2⟨r^2⟩/(12m_e)$. It follows from the $e^2{\mathbf{A}}^2/(2m_e)$ term with $\mathbf{A}=\mathbf{B}\times \mathbf{r}/2$. Differentiating the energy with respect to field and using $B\simeq {\mu }_{0}H$ yields the negative SI volume magnetic susceptibility. For helium, $Z=2$; for helium-3, combining it with the independent spin-one-half nuclear spin response ${\mu }_{0}n{\mu }_a^2/(k_{B}T)$ gives a density-independent sign-change scale $T_0=3m_e{\mu }_a^2/(e^2{k}_B⟨r^2⟩)$.