Set of supplementary notes.

134 CHAPTER 9. ELECTRONIC INSTABILITIES
9.2 Magnetism
By magnetism, in the widest sense, we understand the capacity of materials to change the magnetic
field in their environment. It is not possible for classical systems in thermal equilibrium
to be magnetic. This remarkable result, the Bohr-van Leeuwen theorem 2 (see, e.g., Feynman
Lectures on Physics, Vol. 2) implies that all magnetic phenomena are rooted in the protection
of orbital or spin angular momentum afforded by quantum mechanics.
Phenomenologically, we distinguish between materials which are diamagnetic, paramagnetic
or magnetically ordered. A magnetically ordered material can exhibit magnetism even without
an applied magnetic field. Diamagnets and paramagnets, on the other hand, only have a nonzero
magnetisation, when a field is applied. They differ in the direction of the response to the
field. In a diamagnet, the magnetisation induced by an applied magnetic field will point in the
direction opposite to the direction of the applied magnetic field, whereas the magnetisation in
a paramagnet points along the direction of the applied field.
A diamagnetic response is a fundamental property of charged, quantum mechanical particles
in a magnetic field. Because it is a very weak effect, however, it is only usually observed in materials
with completely filled shells. When there are partially filled shells, or unpaired electrons,
then the orbital and spin angular momentum of these electrons gives rise to a paramagnetic
response, which usually far exceeds the diamagnetic moment produced by the remaining, paired
electrons.
9.2.1 Local moments, Curie law susceptibility
The paramagnetic response of an isolated magnetic moment, or ‘local moment’, provides a very
useful model system. Here, we discuss a classical calculation of the magnetic susceptibility of
local moments; the corresponding quantum mechanical calculation forms one of the problems
on the accompanying problem sheet. The fact that a classical calculation in this case does give
rise to a finite magnetisation in an applied magnetic field does not contradict the Bohr-van
Leeuwen theorem mentioned above, because the starting point of the calculation, an isolated
magnetic moment of fixed magnitude, can only arise from quantum mechanics.
We consider a local moment m of fixed magnitude µ = |m|, in a small applied magnetic
field H → 0. The dipole energy of this moment is given by E = −µ 0 m · H, and the probability
of finding the moment pointing in a particular direction, at finite temperature T , is
p(m) = e −E(m)β /Z (9.7)
where β = 1
k B T and Z =
∫
p(m)d 2 m (9.8)
|m|=µ
2 “At any finite temperature, and in all finite applied electrical or magnetical fields, the net magnetization of
a collection of electrons in thermal equilibrium vanishes identically.” (van Vleck, 1932)