Two-state (spin-½) particles
To simplify the calculation, we are going to work with a 2-state particle: it may either align its magnetic moment with the magnetic field or against it. So the only possible values of magnetic moment are then and . If so, then such a particle has only two possible energies, when it is aligned with the field and when it is oriented opposite to the field.
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. As temperature increases and magnetic field decreases, the argument of the hyperbolic tangent decreases. In the Curie regime,
Moreover, if , then
so the magnetization is small, and we can write , and thus
In this regime, the magnetic susceptibility given by
yields
with a Curie constant given by , in kelvins (K).[2]
In the regime of low temperatures or high fields, tends to a maximum value of , 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 is linearly dependent on the magnetic field, so that the magnetic susceptibility saturates to a constant.
General case
When the particles have an arbitrary spin (any number of spin states), the formula is a bit more complicated.
At low magnetic fields or high temperature, the spin follows Curie's law, with[3]
where is the total angular momentum quantum number, and is the g-factor (such that is the magnetic moment). For a two-level system with magnetic moment , the formula reduces to
as above, while the corresponding expressions in Gaussian units are
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.