Approximations for Brillouin and its reverse function. Issue 6 (7th November 2016) - Takacs, Jeno
Abstract : Purpose: The mathematical complexity of the B J ( x ) Brillouin function makes it unsuitable for most calculations and its application difficult for computer programming in magnetism. Here, its approximation with the tanh function is proposed to ease the mathematical operations for most cases. The approximation works with good accuracy, acceptable in practical calculations. This approximation has already formed the foundation of the "hyperbolic model" in magnetism for the study of hysteretic phenomena. The reversal of the Brillouin function is an important but difficult mathematical problem for practical purposes. Here, a proposal has been put forward for an easy approximation using an analytical expression. This provides a good workable solution for the B J ( x ) −1 function dependent on J, the angular momentum quantum number of the material used. The proposed approximation is applicable within the working range of practical applications. The paper aims to discuss these issues. Design/methodology/approach: The multi-variant Brillouin function is closely approximated by the tanh function to ease calculations. Its mathematically unsolved reversed function is approximated by a simple analytical expression with a good working accuracy. Findings: The Brillouin function and its reversal can be approximated for practical users mostly for professionals working in Magnetism. Research limitations/implications: Most if not all practical problems in Magnetism can be solved within the limitations of the two approximations. Practical implications: Both proposed functions can ease the mathematical problems faced by researchers and other users in Magnetism. Social implications: Ease the frustration of most users working in the field of Magnetism. Originality/value: The application of the tanh function for replacing the Brillouin function led to the creation of the hyperbolic model of hysteresis. To the author's knowledge, the reverse function was mathematically only solved in 2015 with a vastly complicated mathematics, and is hardly suitable for practical calculations in Magnetism. The proposed simple expression can be very useful for theorists and experimental scientists.