Bring a strong magnet near most materials and nothing seems to happen, but a sensitive balance shows a tiny push. Some substances are pushed away, the diamagnets, and some are pulled in, the paramagnets, and the difference lies in their atoms. The Hamiltonian of an atom in a field has two field terms. The linear one, the Zeeman energy of the atom’s moment, lowers the energy of a moment lined up with the field, so atoms with a moment are paramagnetic — although at room temperature heat keeps them almost random, and the magnetisation grows only as $B/T$, which is Curie’s law. The quadratic one, $(e^{2}/8m)(\mathbf{B}\times\mathbf{r})^{2}$ for each electron, is positive: it raises the energy and pushes the atom out of the field. Atoms with closed shells have nothing else, so they are diamagnetic. Classical physics, oddly, allows neither: a theorem of Bohr and van Leeuwen shows that a classical system in thermal equilibrium has no magnetisation at all.

Compass needles in a shaken box point every which way; a field lines them up a little, the more the gentler the shaking. That is paramagnetism. Diamagnetism is different in kind: every orbit resists a change of the flux through it, like a coil pushing back on a magnet thrust into it.

The average moment along the field of spin-$\tfrac12$ atoms, in units of $\mu_{B}$, against $$: starts along the straight line of Curie’s law and saturates at full alignment. At room temperature and 1 T the ratio on the axis is only .

To second order in a weak field $B$ along $z$, an atom’s energy is the Zeeman energy of its moment, the second-order Zeeman shift $\Delta E^{(2)}$, and the diamagnetic term of every electron $f$:

• The moment along the field is $-\partial E/\partial B$, and the susceptibility $\chi\propto-\partial^{2}E/\partial B^{2}$: negative for a diamagnet, which is pushed out of a field, positive for a paramagnet, which is pulled in.

In a closed shell the moments cancel, so only the term quadratic in the field is left. For a field along $z$ it holds the squared distance of each electron from the $z$ axis, and in a spherically symmetric atom that averages to two thirds of the squared distance from the nucleus. The energy rises with the square of the field, so the moment it induces points against the field.

Minimal coupling with $\mathbf{A}=\tfrac12\mathbf{B}\times\mathbf{r}$ for each electron, with the spins’ moments added.

A closed shell has $L=S=0$, so the linear term vanishes at first and at second order.

For $\mathbf{B}$ along $z$.

Spherical symmetry: $\braket{x^{2}}=\braket{y^{2}}=\braket{z^{2}}=\tfrac13\braket{r^{2}}$.

The first-order shift from the quadratic term: the energy rises, and the induced moment opposes the field.

A diamagnet’s energy rises in a field, so its induced moment opposes the field and it is pushed away; a paramagnet’s moment lines up and it is pulled in.

Atoms with closed shells have no moment; only the quadratic term acts, and it raises the energy.

In a spherically symmetric state the squared distance from the $z$ axis averages to two thirds of the squared distance from the centre.

Heat keeps moments random. For a spin $\tfrac12$ the average moment along the field is $\mu_{B}\tanh(\mu_{B}B/kT)$, which grows as while that is small.

Great work! You now know how to:

• show that closed shells are diamagnetic
• derive Curie’s law for a spin $\tfrac12$
• say where each kind of magnetism comes from