If an atom is exposed to an external magnetic field, its energy levels will be splitted due to (at least) the Zeeman effect. Thus the hamiltonian of the system is:
We want to evaluate and . Being the energy levels split, we have:
Using (Canonical ensemble) we get:
Being , where is called Lande factor and it takes on the value for the spin operator, we get:
In a more general case, that is if we do consider both the spin and the angular momentum, we have:
Since this new hamiltonian is of the form , we have that . Hence and thus . In conclusion we get the Curie's law:
Where the quantity is called magnetic susceptibility, while is the Curie constant. Actually all these quantities have to be defined per number of particles per unit of volume, that is:
This is a quantum result, but it can be derived classicaly. In this case we'd have:
Being , this function is called Langevin function.
- ↑ We do have: . Being (if we do consider only spin), and is the Lande (or gyromagnetic) factor which takes on the value for . Concluding that: .