= Larmor-Langevin susceptibility
{c}
{title2=$\chi=-\mu_0ne^2\langle r^2\rangle/(6m)$}
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 $\Delta E=e^2B^2\langle x^2+y^2\rangle/(8m)$. Spherical symmetry gives $\langle x^2+y^2\rangle=2\langle r^2\rangle/3$; 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>.
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