Accurate approximants of inverse Brillouin functions

Highlights

• The polynomial part of the analytical approximation of inverse Brillouin functions was improved.

• The polynomial fit is tabulated for the inverse Brillouin functions of experimental interest.

• The investigation might interest the experimentalists due to the accessible implementation of a sextic polynomial in the inverse Brillouin functions.

Abstract

We present an investigation to improve the polynomial part of an analytical approximation of inverse Brillouin functions, recently proposed in the literature. For the analytical approximation, we employ a sextic polynomial model. To determine the best fit coefficients of the model, we apply a simple statistical analysis. We obtained the same accuracy as the only case presented in the literature, but with three orders of magnitude fewer points involved in the statistical analysis, which translates into a faster convergence of the coefficients. We extend the investigation to the most interesting cases for physical applications. The method can be applied to any other desired inverse Brillouin functions or polynomial approximations of special functions.

Introduction

The inverse Brillouin functions play a central role in ferromagnetism [1], for instance in order to obtain the Curie temperature in a positive feedback model [2], in the theory of magnetic anisotropy and magnetostriction [3], in the magnetization and ferromagnetic resonance in multilayers [4], [5], or in exact renormalization group calculations for some quantum spin systems as Ising or Kondo models [6], [7], [8]. For a recent review, see [9] and the references contained therein. As the Brillouin functions $B_{\sigma }$ are defined by

$$B_{\sigma }\left(x\right)=\frac{2\sigma +1}{2\sigma }coth\frac{2\sigma +1}{2\sigma }x-\frac{1}{2\sigma }coth\frac{1}{2\sigma }x,$$

they are rational functions of $exp\left(x/2\sigma \right)$. So, finding the inverse function $B_{\sigma }^{-1}$ is equivalent to solving an algebraic equation, namely a $2\sigma -$order one. The exact expressions of $B_{1/2}^{-1}$ and $B_1^{-1}$ are known analytically [9], [10]. Although the evaluation of $B_{3/2}^{-1}$ and $B_2^{-1}$ is an elementary, although tedious exercise, their form is too complicated to be used in practical calculations. This is why analytic calculations involving inverse Brillouin functions with $\sigma >1$ can be carried out using only analytic approximations for such functions.

In recent years, significant progress was made in obtaining exact formulas for the inverse of $B_{1/2}\left(x\right)/x$, or the Langevin function [11], [12], but even in these cases, the analytic approximations remain important due to their simplicity. For some recent papers on the approximations on inverse Brillouin and Langevin functions, see [13] and [14], [15], respectively.

In a seminal study of analytical approximations of inverse Langevin and Brillouin functions, Kröger [16] obtains an accurate formula for the approximants of $B_{\sigma }^{-1}$ (for keeping the notations simple, we shall use the same symbol for the exact inverse Brillouin function, and for its approximants)

$$B_{\sigma }^{-1}\left(y\right)=\frac{1+a_2{y}^2}{a_1+b_2{y}^2}ln\frac{1+y}{1-y},$$

with $a_1$ - linear, and $a_2,b_2$ - quadratic polynomials in $\sigma$.

Later on, Bârsan [17] proposed an approximant with the following functional form

$$B_{\sigma }^{-1}\left(y\right)=\frac{3\sigma }{1+\sigma }{P}_{\sigma }\left(y\right)ln\frac{1}{1-y},$$

where $P_{\sigma }\left(y\right)$ is a polynomial. Its specific form results from the compromise between the simplicity and accuracy of the approximant: in principle, we could expect that a higher-order polynomial would provide higher accuracy but a more complicated expression, and vice-versa.