Brillouin and Langevin functions

The Brillouin and Langevin functions are a pair of special functions that appear when studying an idealized paramagnetic material in statistical mechanics.(where g is the g-factor, μB is the Bohr magneton, and x is as defined in the text above). The relative probability of each of these is given by the Boltzmann factor: The Brillouin and Langevin functions are a pair of special functions that appear when studying an idealized paramagnetic material in statistical mechanics. The Brillouin function is a special function defined by the following equation: The function is usually applied (see below) in the context where x is a real variable and J is a positive integer or half-integer. In this case, the function varies from -1 to 1, approaching +1 as x → + ∞ {displaystyle x o +infty } and -1 as x → − ∞ {displaystyle x o -infty } . The function is best known for arising in the calculation of the magnetization of an ideal paramagnet. In particular, it describes the dependency of the magnetization M {displaystyle M} on the applied magnetic field B {displaystyle B} and the total angular momentum quantum number J of the microscopic magnetic moments of the material. The magnetization is given by: