Theoretical and numerical studies of the Brillouin function and its inverse

The Brillouin function is widely used in the theoretical description of paramagnetic and ferromagnetic materials. We know that as the total angular momentum J becomes infinite the Brillouin function and its inverse become equal to the Langevin function and its inverse, respectively. To illustrate the properties of the Brillouin function and its inverse they are redefined in terms of the hyperbolic tangent (tanh) function. We detail how the critical points may be calculated and identify the location and nature of the singularities of the Brillouin function and its inverse both analytically and graphically, listing the singularities of the inverse function in the first quadrant for J=1/2,1,3/2,…,10. We subtract the logarithmic singularities from the inverse Brillouin function and so define the reduced inverse Brillouin function FJ(y) and give its series expansion. Through use of Domb–Sykes plots we document properties of this series expansion for J = 1/2, …, 3. We present a Matlab program that is capable of accurately representing this reduced inverse Brillouin function to near machine accuracy.