A Classical model for diamagnetism

Question

I'm trying to derive a classical model to describe diamagnetism, and I'd like to understand if it is possible to understand the basic properties of diamagnetic materials with it i.e.:

''A diamagnetic material placed in an external magnetic field $B_{ext}$ develops a magnetic dipole moment directed opposite $B_{ext}$. If the field is nonuniform, the diamagnetic material is repelled from a region of greater magnetic field toward a region of lesser field.''

or as much information as possible.

I started considering only the orbital motion of the electron in the atom. Consider an electron in orbit with angular velocity $\omega$. Its magnetic moment would be

\begin{align} \vec{\mu} = \frac{-e}{2m_e}\vec{L}_{orb} \end{align}

Subject to a field ${\bf H}$, with induction, $\mathbf{B} = \mu_0\mathbf{H}$, this magnetic moment will experience a torque

\begin{align} \vec{\tau} &= \frac{d\vec{L}}{dt} \\ &= \vec{\mu} \times \vec{B} \\ &= \mu_0 \frac{-e}{2m_e} \vec{L} \times \vec{H} \\ &= \mu_0 \frac{e}{2m_e} \vec{H}\times\vec{L} \\ &= \vec{\omega} \times \vec{L} \end{align}

Now, it seems from the last relation that this magnetic moment is in precession. It should be this magnetic moment to cause the effects of diamagnetism, but I'm not sure how to treat this preceding magnetic moment and show that it is antiparallel to the external magnetic field and its repulsion to it.

Answer 1

The Hamiltonian $H,$ for a system of charged particles interacting via a potential energy $U$ is: $$ H\left(\boldsymbol{r}_{k}, \boldsymbol{p}_{k}\right)=\sum_{j=1}^{N} \frac{\boldsymbol{p}_{j}^{2}}{2 m_{j}}+U\left(\boldsymbol{r}_{k}\right) $$ Particles defined with, mass $m_{j},$ position vector $r_{j},$ and momenta $p_{j}$. Under external magnetic field with vector potential $A(r)$ changes Hamiltonian to: $$ H\left(\boldsymbol{r}_{k}, \boldsymbol{p}_{k}\right)=\sum_{j=1}^{N} \frac{\left[\boldsymbol{p}_{j}-\frac{e_{j}}{c} \boldsymbol{A}\left(\boldsymbol{r}_{j}\right)\right]^{2}}{2 m_{j}}+U\left(\boldsymbol{r}_{k}\right) $$

where $e_{j}$ are the charges of masses $m_{j}$. Using the above equation, statistical mechanics predicts that the energy-the thermal average of the Hamiltonian-does not depend on the external field. So, the system exhibits neither a paramagnetic nor a diamagnetic response.

The above result is known as Bohr-van Leeuwen theorem where it states that there is no classical magnetism. Also, the basics of magnetism depend on the relativistic approach where the charge densities are evaluated by different observers respectively. Basically, classical statistical mechanics and electromagnetism cannot explain diamagnetism.

As a source: Feynman lectures10 (Vol. 2, Sec. 34-6) and Understanding dia- and para-magnetism by Andreas Wacker