Brillouin and Langevin Functions

The Langevin and Brillouin functions are special mathematical functions that describe how paramagnetic materials respond to an applied magnetic field, named after the French physicists Paul Langevin and Leon Brillouin. The Langevin function, derived from classical statistical mechanics, describes how magnetic dipoles tend to align with an applied field. The Brillouin function is a later, quantum mechanical treatment of the same effect, and the Langevin function turns out to be the special case of the Brillouin function reached as the quantum number J goes to infinity, so that the Langevin version is the classical limit and the Brillouin version is the more general, quantum-corrected description. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/