Approximations for Brillouin and its reverse function

1. Introduction

The scientific study and the search for the mathematical description of magnetization as a transformation of the linear H domain into non-linear M magnetization go back in history (Brillouin, 1927; Langevin, 1905). Its mathematical formulation challenged scientists for a considerable time. In 1908, Langevin put forward a theory for the magnetization of paramagnetic materials (Langevin, 1905). The application of this theory worked well for ferromagnetic substances and has provided one of the best mathematical approaches for the description of the ferromagnetic magnetization process so far, formulated in the Langevin function L(x), in the following way:(1) $$L\left(x\right)=coth\left(x\right)-1/x$$

Later, by taking into consideration the quantum mechanical effects, Brillouin (1927) improved on Langevin’s theory by formulating the process in the mathematical equation below, which includes the dependence of the process on J, the angular momentum quantum number:(2) $$B_J\left(x\right)=\frac{2J+1}{2J}\coth \left(\frac{2J+1}{2J}x\right)-\frac{1}{2J}\coth \left(\frac{x}{2J}\right)$$

here, B_J(x) is the Brillouin function and J the angular momentum quantum number of the magnetic material used. This function is going to approach the Langevin function as J→∞ as a special case of the B_J(x) function. The function in (2) becomes a tanh function when J=1/2, leading to the proposal presented here. When J≠1/2, a very close approximation can be made with an appropriately chosen...