Diamagnetism ($\chi <0$)

Diamagnetism is a magnetic response in which the induced magnetization opposes the applied magnetic field intensity. In a linear isotropic description, $\mathbf{M}=\chi \mathbf{H}$ with negative magnetic susceptibility $\chi$.

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 $T_0=3m_e{\mu }_a^2/(e^2{k}_B⟨r^2⟩)$.

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.

A small weakly susceptible sphere has induced magnetic energy $-\chi V{B}^2/(2{\mu }_0)$ and vertical force $\chi VB{B}^{\prime }/{\mu }_0$. In field magnitude $B=B_0{r}_0/z$, balancing gravity gives the title height for $\chi <0$. The vertical force varies as $z^{-3}$, giving a restoring derivative at that equilibrium. Three-dimensional trapping cannot be established from an axial field magnitude alone; lateral field geometry and Maxwell constraints also matter.

Landau diamagnetism is the orbital response of mobile electrons to an applied magnetic field, arising from quantization of their transverse motion. For a noninteracting three-dimensional isotropic electron gas with quadratic dispersion, electron spin with $g=2$ and magnetic fields weak enough for the smooth-response limit, its susceptibility is ${\chi }_L=-{\chi }_P/3$, where ${\chi }_P$ is the Pauli paramagnetism susceptibility. Band effective masses, interactions, dimensionality and resolved magnetic quantum oscillations can change this simple relation.