Larmor-Langevin susceptibility ($\chi =-{\mu }_{0}n{e}^2⟨r^2⟩/(6m)$)

The dimensionless SI volume magnetic susceptibility of number density $n$ of independent spherical bound electrons with mass $m$ and electron-weighted mean-square orbital radius. In a uniform magnetic field, the quadratic minimal-coupling Hamiltonian gives $\Delta E=e^2{B}^2⟨x^2+y^2⟩/(8m)$. Spherical symmetry gives $⟨x^2+y^2⟩=2⟨r^2⟩/3$; differentiating twice with respect to $B$ gives the displayed negative response. Atomic orbitals use the bare electron mass, whereas a shallow donor envelope within an effective mass approximation uses that envelope's effective mass.