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. These functions are named after French physicists Paul Langevin and Léon Brillouin who contributed to the microscopic understanding of magnetic properties of matter.

Brillouin function

edit

The Brillouin function[1][2] is a special function defined by the following equation:

 

The function is usually applied (see below) in the context where   is a real variable and   is a positive integer or half-integer. In this case, the function varies from -1 to 1, approaching +1 as   and -1 as  .

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   on the applied magnetic field   and the total angular momentum quantum number J of the microscopic magnetic moments of the material. The magnetization is given by:[1]

 

where

  •   is the number of atoms per unit volume,
  •   the g-factor,
  •   the Bohr magneton,
  •   is the ratio of the Zeeman energy of the magnetic moment in the external field to the thermal energy  :[1]
 
  •   is the Boltzmann constant and   the temperature.

Note that in the SI system of units   given in Tesla stands for the magnetic field,  , where   is the auxiliary magnetic field given in A/m and   is the permeability of vacuum.

Takacs[3] proposed the following approximation to the inverse of the Brillouin function:

 

where the constants   and   are defined to be

 
 

Langevin function

edit
 
Langevin function (blue line), compared with   (magenta line).

In the classical limit, the moments can be continuously aligned in the field and   can assume all values ( ). The Brillouin function is then simplified into the Langevin function, named after Paul Langevin:

 

For small values of x, the Langevin function can be approximated by a truncation of its Taylor series:

 

An alternative, better behaved approximation can be derived from the Lambert's continued fraction expansion of tanh(x):

 

For small enough x, both approximations are numerically better than a direct evaluation of the actual analytical expression, since the latter suffers from catastrophic cancellation for   where  .

The inverse Langevin function L−1(x) is defined on the open interval (−1, 1). For small values of x, it can be approximated by a truncation of its Taylor series[4]

 

and by the Padé approximant

 
 
Graphs of relative error for x ∈ [0, 1) for Cohen and Jedynak approximations

Since this function has no closed form, it is useful to have approximations valid for arbitrary values of x. One popular approximation, valid on the whole range (−1, 1), has been published by A. Cohen:[5]

 

This has a maximum relative error of 4.9% at the vicinity of x = ±0.8. Greater accuracy can be achieved by using the formula given by R. Jedynak:[6]

 

valid for x ≥ 0. The maximal relative error for this approximation is 1.5% at the vicinity of x = 0.85. Even greater accuracy can be achieved by using the formula given by M. Kröger:[7]

 

The maximal relative error for this approximation is less than 0.28%. More accurate approximation was reported by R. Petrosyan:[8]

 

valid for x ≥ 0. The maximal relative error for the above formula is less than 0.18%.[8]

New approximation given by R. Jedynak,[9] is the best reported approximant at complexity 11:

 

valid for x ≥ 0. Its maximum relative error is less than 0.076%.[9]

Current state-of-the-art diagram of the approximants to the inverse Langevin function presents the figure below. It is valid for the rational/Padé approximants,[7][9]

 
Current state-of-the-art diagram of the approximants to the inverse Langevin function,[7][9]

A recently published paper by R. Jedynak,[10] provides a series of the optimal approximants to the inverse Langevin function. The table below reports the results with correct asymptotic behaviors,.[7][9][10]

Comparison of relative errors for the different optimal rational approximations, which were computed with constraints (Appendix 8 Table 1)[10]

Complexity Optimal approximation Maximum relative error [%]
3   13
4   0.95
5   0.56
6   0.16
7   0.082


Also recently, an efficient near-machine precision approximant, based on spline interpolations, has been proposed by Benítez and Montáns,[11] where Matlab code is also given to generate the spline-based approximant and to compare many of the previously proposed approximants in all the function domain.

High-temperature limit

edit

When   i.e. when   is small, the expression of the magnetization can be approximated by the Curie's law:

 

where   is a constant. One can note that   is the effective number of Bohr magnetons.

High-field limit

edit

When  , the Brillouin function goes to 1. The magnetization saturates with the magnetic moments completely aligned with the applied field:

 

References

edit
  1. ^ a b c d C. Kittel, Introduction to Solid State Physics (8th ed.), pages 303-4 ISBN 978-0-471-41526-8
  2. ^ Darby, M.I. (1967). "Tables of the Brillouin function and of the related function for the spontaneous magnetization". Br. J. Appl. Phys. 18 (10): 1415–1417. Bibcode:1967BJAP...18.1415D. doi:10.1088/0508-3443/18/10/307.
  3. ^ Takacs, Jeno (2016). "Approximations for Brillouin and its reverse function". COMPEL - the International Journal for Computation and Mathematics in Electrical and Electronic Engineering. 35 (6): 2095. doi:10.1108/COMPEL-06-2016-0278.
  4. ^ Johal, A. S.; Dunstan, D. J. (2007). "Energy functions for rubber from microscopic potentials". Journal of Applied Physics. 101 (8): 084917. Bibcode:2007JAP...101h4917J. doi:10.1063/1.2723870.
  5. ^ Cohen, A. (1991). "A Padé approximant to the inverse Langevin function". Rheologica Acta. 30 (3): 270–273. doi:10.1007/BF00366640. S2CID 95818330.
  6. ^ Jedynak, R. (2015). "Approximation of the inverse Langevin function revisited". Rheologica Acta. 54 (1): 29–39. doi:10.1007/s00397-014-0802-2.
  7. ^ a b c d Kröger, M. (2015). "Simple, admissible, and accurate approximants of the inverse Langevin and Brillouin functions, relevant for strong polymer deformations and flows". J Non-Newton Fluid Mech. 223: 77–87. doi:10.1016/j.jnnfm.2015.05.007. hdl:20.500.11850/102747.
  8. ^ a b Petrosyan, R. (2016). "Improved approximations for some polymer extension models". Rheologica Acta. 56: 21–26. arXiv:1606.02519. doi:10.1007/s00397-016-0977-9. S2CID 100350117.
  9. ^ a b c d e Jedynak, R. (2017). "New facts concerning the approximation of the inverse Langevin function". Journal of Non-Newtonian Fluid Mechanics. 249: 8–25. doi:10.1016/j.jnnfm.2017.09.003.
  10. ^ a b c Jedynak, R. (2018). "A comprehensive study of the mathematical methods used to approximate the inverse Langevin function". Mathematics and Mechanics of Solids. 24 (7): 1–25. doi:10.1177/1081286518811395. S2CID 125370646.
  11. ^ Benítez, J.M.; Montáns, F.J. (2018). "A simple and efficient numerical procedure to compute the inverse Langevin function with high accuracy". Journal of Non-Newtonian Fluid Mechanics. 261: 153–163. arXiv:1806.08068. doi:10.1016/j.jnnfm.2018.08.011. S2CID 119029096.