Curie's law
For many paramagnetic materials, the magnetization of the material is directly proportional to an applied magnetic field, for sufficiently high temperatures and small fields.
However, if the material is heated, this proportionality is reduced.
For a fixed value of the field, the magnetic susceptibility is inversely proportional to temperature, that is where Pierre Curie discovered this relation, now known as Curie's law, by fitting data from experiment.
It only holds for high temperatures and weak magnetic fields.
As the derivations below show, the magnetization saturates in the opposite limit of low temperatures and strong fields.
If the Curie constant is null, other magnetic effects dominate, like Langevin diamagnetism or Van Vleck paramagnetism.
A simple model of a paramagnet concentrates on the particles which compose it which do not interact with each other.
is the magnetic field density, measured in teslas (T).
The extent to which the magnetic moments are aligned with the field can be calculated from the partition function.
For a single particle, this is The partition function for a set of N such particles, if they do not interact with each other, is and the free energy is therefore The magnetization is the negative derivative of the free energy with respect to the applied field, and so the magnetization per unit volume is where n is the number density of magnetic moments.
[1]: 117 The formula above is known as the Langevin paramagnetic equation.
Pierre Curie found an approximation to this law that applies to the relatively high temperatures and low magnetic fields used in his experiments.
, and thus In this regime, the magnetic susceptibility given by yields with a Curie constant given by
[2] In the regime of low temperatures or high fields,
, corresponding to all the particles being completely aligned with the field.
Since this calculation doesn't describe the electrons embedded deep within the Fermi surface, forbidden by the Pauli exclusion principle to flip their spins, it does not exemplify the quantum statistics of the problem at low temperatures.
Using the Fermi–Dirac distribution, one will find that at low temperatures
At low magnetic fields or high temperature, the spin follows Curie's law, with[3] where
is the total angular momentum quantum number, and
For a two-level system with magnetic moment
For this more general formula and its derivation (including high field, low temperature) see the article Brillouin function.
As the spin approaches infinity, the formula for the magnetization approaches the classical value derived in the following section.
An alternative treatment applies when the paramagnets are imagined to be classical, freely-rotating magnetic moments.
In this case, their position will be determined by their angles in spherical coordinates, and the energy for one of them will be: where
component of the magnetization (the other two are seen to be null (due to integration over
: (This approach can also be used for the model above, but the calculation was so simple this is not so helpful.)
In fact, its behavior for small arguments is
for large values of its argument, and the opposite limit is likewise recovered.
Pierre Curie observed in 1895 that the magnetic susceptibility of oxygen is inversely proportional to temperature.
Paul Langevin presented a classical derivation of this relationship ten years later.
